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 Permanent Link:
 http://ufdc.ufl.edu/AA00029937/00001
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 Title:
 NMR studies of molecular hydrogen confined to the pores of zeolite
 Creator:
 Rall, Markus, 1964
 Publication Date:
 1991
 Language:
 English
 Physical Description:
 ix, 176 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Cooling ( jstor )
Hydrogen ( jstor ) Lead ( jstor ) Liquids ( jstor ) Molecules ( jstor ) Signals ( jstor ) Solids ( jstor ) Supercooling ( jstor ) Transition temperature ( jstor ) Zeolites ( jstor ) Dissertations, Academic  Physics  UF Magnetic Resonance Spectroscopy ( mesh ) Nuclear magnetic resonance ( lcsh ) Physics thesis Ph. D
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 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1991.
 Bibliography:
 Includes bibliographical references (leaves 170175).
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Markus Rall.
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 University of Florida
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 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
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Full Text 
NMR STUDIES OF MOLECULAR HYDROGEN CONFINED TO THE PORES OF ZEOLITE
By
MARKUS RALL
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1991
ACKNOWLEDGEMENTS
It is my special pleasure to thank my advisor Prof. N. S. Sullivan for the opportunity to do this doctoral research with him. Without his support and encouragement I would not have undertaken this project in the United States. His comprehensive understanding of physics as well as his personal interaction with me made this a very productive time. In addition to being chairman of the Physics Department, he was involved in such projects as the Microkelvin facility and the National High Magnetic Field Laboratory. This could, however, not influence his concept of trusting my capabilities, which was a motivation for me to work independently, combined with his readiness to "talk physics" and help me out whenever there was a need for it. His attitude towards education has set an example for me.
The French postdoctoral associate J. P. Brison who spent a year at the University of Florida was also a wonderful friend to me. He worked on his own experiment and assisted me in my project. I enjoyed the long and frequent discussions with him and admired his attitude and intelligence.
I also thank M. D. Evans, an incoming student in my last year, for the new spirit he brought to the lab and D. Rubury, who was an undergraduate working for me in the summer and fall of 1990.
My thanks go to Profs. R. Andrew, C. F. Hooper, D. B. Tanner, E. D. Adams, J. Klauder and J. H. Simmons for their interest and support while serving on my supervisory committee. I would also like to acknowledge the many interesting discussions with Prof. Y. Takano.
Special thanks go to all the support groups that were important for the success of my work. In this context I want to name the machine shop under B. Fowler which produced high quality parts, the electronics shop with L. W. Phelps
ii
and J. Legg with their friendship and expertise, and the cryogenic group with A. Hingerty and G. J. Labbe for supplying helium and engineering advice. I also want to thank our staff K. Yocum and C. Knudsen.
I further express my gratitude to everybody who made my stay at the University of Florida such a productive and agreeable time. The friendly atmosphere among the students and between students and faculty always made work more pleasant.
This research has been supported by the National Science Foundation through a Low Temperature Physics Grant DMR8913999, by NATO grant 88/709 and by the Division of Sponsored Research at the University of Florida.
iii
TABLE OF CONTENTS
ACKNOWLEDGEMENTS.................................................................................II
LIST OF FIGURES............................................vi
A BSTRA C T ...................................................................... ..................... . viii
CHAPTERS
1. INTRODUCTION ........................................................................... 1
1.1. Quantum Nature of Hydrogen........................................... 1
1.1.1. Ortho and Parahydrogen ................................... 2
1.1.1.1. Quadrupolar interaction ......................... 3
1.1.2. ZeroPoint Motion................................................ 5
1.1.2.1. De Boer parameter................................ 8
1.1.3. Crystalline Potential and Quantum Rotor............ 9
1.2. Open Questions for Confinement ..................................... 11
2 T H EO R Y ......................................................................................... . 13
2.1. Surface Interactions...........................................................13
2.1.1. Theory of SurfaceMolecule Interaction .............. 14
2.1.2. Effects of Adsorption ............................................19
2.2. The LiquidSolid Transition ............................................... 20
2.2 1. Elastic Instability Theory...................................... 21
2.2.2. Dislocation Theory............................................... 21
2.2.3. TwoDimensional Melting Models ....................... 23
2.2.4. Molecular Dynamics Studies............................... 24
2.2.5. Implications for the Investigated System............. 25
2.3. Superfluid......................................................................... 26
2.4. Supercooling ..................................................................... 29
2.4.1. Nucleation Theory ............................................... 29
2.4.1.1. Implications for experiment.................... 33
2.4.2. Grain Boundaries ............................................... 35
2.5. OrthoPara Conversion in Molecular Hydrogen .................38
2.5.1. Diffusion ............................................................. 43
3. NUCLEAR MAGNETIC RESONANCE: NMR ................................ 45
3.1. Dipolar Hamiltonian........................................................... 45
3.2. Continuous Wave Lineshape ............................................ 47
3.3. Relaxation Times ............................................................. 51
3.3.1. Spectral Densities ............................................... 55
3.3.1.1. Rotational Motion................................... 55
3.3.1.2. Translational Motion ...............................58
3.4. Motional Narrowing and Second Moment......................... 59
3.5. Heterogeneous Spin System Relaxation .......................... 63
iv
3.6. Methods of Detection ......................................................... 66
3.6.1. Continuous W ave (cw) Method ........................... 66
3.6.1.1. Qm eterdetection.................................. 67
3.6.1.2. Bridge m ethod ....................................... 68
3.6.2 Transient Method ................................................. 70
3.6.2.1. Coherent Pulses.................................... 70
4. EXPERIM ENTAL SETUP ............................................................... 74
4.1. NMR Setup ...................................................................... 74
4.1.1. Pulse Apparatus .................................................. 74
4.1.2. Continuous W ave (cw) Apparatus....................... 81
4.2. Dilution Refrigerator ......................................................... 84
4.3. Substrates......................................................................... 91
4.3.1. Zeolite .................................................................. 91
4.3.2. Vycor .................................................................. 95
4.3.3. Exfoliated Graphite............................................. 96
5. EXPERIM ENTAL DATA.................................................................. 97
5.1. Sam ple Preparation ........................................................... 97
5.2. Pulse W ork....................................................................... 99
5.2.1. Transverse Relaxation Time ..................................99
5.2.1.1. Tem perature dependence ..................... 99
5.2.1.2. Orthopara dependence ............................104
5.2.1.3. Hysteresis..................................................106
5.2.2. Spin Echo Am plitude ...............................................106
5.2.3. Longitudinal Relaxation Tim e..................................112
5.3. Continuous Wave Work ........................ 116
5.3.1. OrthoPara Conversion ...........................................116
5.3.1.1. Im purities...................................................124
5.3.2. Lineshape................................................................125
5.3.3. Tim e Dependence...................................................129
5.3.3.1. Second mom ent ........................................129
5.3.4. Tem perature Dependence ......................................129
5.4. HydrogenDeuteride..............................................................138
5.5. Isochoric Pressure versus Tem perature Data ......................141
5.6. Zeolite 5A versus 13X...........................................................145
6. COM PUTER SIM ULATION ................................................................149
7. CONCLUSION....................................................................................161
BIBLIOG RAPHY ................................................................................................170
BIOGRAPHICAL SKETCH ................................................................................176
v
LIST OF FIGURES
Figure Page
1. Nucleation Theory .................................................................................. 31
2. Pake doublet with order parameter a=1 ................................................ 50
3. Spin echo formation ................................................................................ 72
4. Pulse NIM R diagram ................................................................................ 75
5. Continuous wave (cw) diagram ................................................................82
6. N M R cell................................................................................................ . 89
7. Structure of zeolite A .............................................................................. 92
8. Two measured transverse relaxation times.................................................100
9. Intrinsic transverse relaxation times ............................................................100
10. Transverse relaxation time transitions........................................................102
11. Supercooling transition as a function of paraH2 concentration .................105
12. Echo amplitude transitions .........................................................................108
13. Transverse relaxation time and echo amplitude vs temperature................113
14. Longitudinal relaxation times......................................................................115
15. Transverse relaxation time as a function of time........................................117
16. Echo amplitude as a function of pulse separation ......................................117
17. Orthopara conversion behavior.................................................................119
18. Time dependence of cw lineshape.............................................................126
19. Second moment as a function of time ........................................................130
20. Temperature dependence of cw lineshape ................................................132
21. Longitudinal relaxation time of hydrogendeuteride ...................................139
22. Echo amplitudes of hydrogendeuteride.....................................................139
23. Transverse relaxation times of hydrogendeuteride ...................................140
vi
Paae
24. Isochoric pressure versus temperature data..............................................143
25. Echo amplitudes for hydrogen in zeolite 5A ...............................................146
26. Temperature dependence of cw lineshape in zeolite 5A............................147
27. Pake doublet superposition ........................................................................158
28. Gaussian wavefunction ..............................................................................158
29. Synthesized cw lineshapes ........................................................................159
vii
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
NMR STUDIES OF MOLECULAR HYDROGEN
CONFINED TO THE PORES OF ZEOLITE By
MARKUS RALL
December 1991
Chairman: N. S. Sullivan
Major Department: Physics
Molecular hydrogen is interesting to investigate in a porous material. It is strongly quantummechanical with respect to its translational and rotational degrees of freedom as well as its orthopara conversion. Zeolite (13X) was selected as porous material for its almost monodisperse pore diameter of 13 A.
The studies mainly focussed on Nuclear Magnetic Resonance data. The hydrogenzeolite system was probed in coherent pulse and continuous wave
(cw) mode at 268 MHz.
The pulse data revealed a strong peak in the transverse relaxation time T2 at about 10 K. A similar peaking was observed for the nuclear spin echoes. This behavior was interpreted in terms of supercooling and implies that hydrogen in zeolite solidifies about 5 degrees below its bulk melting point. Two features are especially remarkable. Firstly, the T2 was found to be nonhysteretic and emphasizes a purely energetically driven process. Secondly, the supercooling transition temperatures exhibit a linear parahydrogen concentration dependence of 0.10 0.01 K/%. This implies that parahydrogen can be supercooled more easily than orthohydrogen.
viii
The continuous wave data show a glasstype lineshape of 130 kHz width at 4.2 K with c=0.45 0.3. It is a combination of broad wings with a narrow component in the center. This behavior is displayed upon warming from 4.2 K when the narrow line disappears and the broad contribution remains. No sharp transition at 10 K is observed. The line broadens by a factor of 1.5 due to orientational ordering when the sample is cooled to 0.8 K. This latter observation indicates that the pulse data peaks at 10 K do not originate from orientational ordering.
The orthopara conversion was extracted from the time dependence of the lineshape. The orthohydrogen contents was followed over 1600 hours which yielded two different conversion constants. For the first 500 hours the conversion is characterized by k, = 0.425 0.006 % / h which is only a fourth of the bulk value and is explained by the reduced number of nearest neighbors in the constrained geometry. For later times the conversion accelerates to k2= 2.21 0.075 %/h and is explained by clustering effects. The relative shape of the line changes over time as the narrow component of the line decays faster.
Isochoric pressure versus temperature data were recorded while the hydrogen sample was taken out. The nature of a twocomponent system was confirmed. The adsorption energy of hydrogen on the zeolite surface was determined to be 270 50 K.
Hydrogendeuteride was also adsorbed into zeolite. The interpretation of the pulse data peaks in terms of supercooling was supported by the fact that HD showed a similar behavior without orientational degrees of freedom.
Similar but more extreme results were obtained from a second zeolite
(5A) with even smaller pore size (9 A) in agreement with the previous data.
ix
CHAPTER 1
INTRODUCTION
The system to be discussed in this dissertation is hydrogen confined to zeolite, a porous material. Experimental investigations were carried out by means of Nuclear Magnetic Resonance at low temperatures. These four characteristics  molecular hydrogen, zeolite, low temperatures and NMR determine the scope of this thesis. Every one of these will receive special attention in the following chapters. In addition, the effects that are expected and observed will be discussed. The introduction will lead into the subject by explaining some general principles that are important for the understanding of the results and provide the background for the remainder of the thesis. The overview is also meant to motivate the performed experiment.
For the understanding of many effects it is necessary to be familiar with the special properties of hydrogen. Hydrogen is the simplest existing diatomic molecule and can therefore be used as a model molecule to understand fundamental interactions and phenomena. When it is cooled below 13.8 K it is also the simplest molecular solid. The beauty of hydrogen is that the involved interactions are understood in terms of first principles.
1.1. Quantum Nature of Hydrogen
Hydrogen is special in its properties. It is a very interesting substance as it is explicitly quantummechanical in its nature. There are three realizations of
1
2
this fact: the existence of two species, the large zeropoint motion (ZPM) and the quantum rotor property.
1.1.1. Ortho and Parahydrogen
One of the most fascinating properties of molecular hydrogen is the explicitly quantum mechanical classification into two species. Hydrogen occurs naturally as a diatomic molecule. Its nuclear constituents are spin1/2 protons with the electronic spin contributions paired in a symmetric 'Y" molecular ground state. The wave function can be written as a product of nuclear, rotational and vibrational wave functions. The vibrational groundstate is symmetric and does not influence the considerations. The quantummechanical requirement of a totally antisymmetric wavefunction for two indistinguishable fermions can therefore be realized in two distinct ways, leading to two distinguishable molecular species: ortho and parahydrogen. Orthohydrogen has a symmetric nuclear spin wavefunction (/ = 1) and an antisymmetric orbital wavefunction (J odd). The nomenclature is such that the species with the highest spin quantum number is called "ortho." Parahydrogen has an antisymmetric nuclear wavefunction (/ = 0) and a symmetric orbital wavefunction (J even). It is important to notice that both species of hydrogen are bosons.
Heteronuclear molecules such as hydrogen deuteride (HD) do not display distinct species because of the distinguishable nature of their constituents.
Because of the small momentofinertia / for hydrogen, the separation of h2
the rotational states is very large: Ej = Bj J(J +1) with Bj =  = 85.4 K. The 2/
quantum number J is therefore a meaningful entity at low densities. The intermolecular interactions which tend to mix different rotational states are very weak, and the groundstate is the J = 0 parahydrogen state. This still holds
3
when the surfacemolecule interaction of the restricted geometry is included, as discussed in Chapter 2.
1.1.1.1. Quadrupolar interaction
Parahydrogen is spherically symmetric with no electric or magnetic moments. The lack of nuclear spin degrees of freedom for parahydrogen makes it also undetectable by Nuclear Magnetic Resonance. This is in contrast to orthohydrogen with spin 1 which also has an electric quadrupole moment. The molecules tend to orient at low temperatures in order to minimize the anisotropic interaction
which occurs when the molecular axes are aligned at a 90 degree angle, a "T" configuration between quadrupoles. Here A is the lattice parameter, R the intermolecular distance, Fan angular function of the angles measured relative to the connecting line between molecules, and F = 0.8 K.
The order parameters specifying the degrees of freedom for this interaction are components of the second rank quadrupolar tensor with respect to the local reference axes (x, y, z)
Q, J2,5a, (JaJI+ JIJa))
which leads to two additional parameters besides the three local reference axes. The reason for only two additional parameters is the vanishing orbital angular momentum, which means that (J,)=0 is "quenched."1 This is the case when timereversal symmetry is not broken which can be assumed here, even though
4
a magnetic field is applied. The ratio of the magnetic interaction energy (mK) to
t,2
the involved rotational energies (B   =85.4 K) is ' the reason for this approximation: the magnetic energy is only of negligible influence. The argument for "quenching" is related to the nondegenerate nature of the rotational groundstate. A nondegenerate groundstate must be real otherwise real and imaginary parts of the wavefunction would separately be solutions, which contradicts nondegeneracy. The angular momentum operators are imaginary and the expectation value (J,) with i = x,y,z is therefore purely imaginary. On the other hand it must also be real due to hermiticity and the only expectation value satisfying both conditions is zero. At higher temperatures this requires more careful consideration because the J_ = +1 are also occupied and could contribute to the rotational Zeeman energy. This is not the case, however, because the transition frequency amongst these states is much higher than the dipolar frequency and is therefore averaged out. All offdiagonal matrix elements are also zero by the properties of angular momentum.
One additional order parameter for orthohydrogen is the alignment
a=Qz = 1(3j2 2)
For J, = 0 the alignment is = 1, which depicts a prolate ellipsoid probability for the wavefunction. For the case of an oblate ellipsoid where J, = 1 the alignment is a=112.
The other additional order parameter is the eccentricity described by
1 = (j2  j2)
5
and is zero for any rotationally symmetric body.
1.1.2. ZeroPoint Motion
Hydrogen is the molecule with the lowest mass. It possesses only a weakly polarizable electron cloud leading to weak intermolecular forces. These two facts combined with the groundstate energy EO = .hw of a threedimensional harmonic oscillator lead to a large zeropoint motion (ZPM). This implies that the molecule's oscillation amplitude is a sizeable fraction, 15% for hydrogen, of the intermolecular lattice spacing even at zero temperature. This is a fascinating feature and is the cause for interesting properties. It also renders calculations more difficult because a harmonic approximation about the intermolecular potential minimum is no more justified. Also correlations between molecules are important to consider when an understanding for the dynamics is developed.
In order to gain insight into the ZPM, a harmonic approximation is presented, with all the caution in place. The standard interaction potential used is the LennardJones potential
V(r) = 4e
where a is the hard core radius and e the potential minimum. The first part is a mathematically convenient form of a repulsive interaction and is a consequence of the Pauli exclusion principle when electronic clouds start to overlap. The second term is the attractive Van der Waals potential due to induced dipoledipole interaction. No permanent dipoles are involved but one dipole is created by thermal fluctuation fields and another dipole is induced in a neighboring molecule.
6
A more sophisticated potential was empirically found by Silvera and Goldman2
V(r) = exp(a #Pr  y2)+f(r) { +} /=6.8,10r r
with
f(r) = exp {1.28 1 for r < 1.28Rm
=1 for r >1.28RM
where the constants are a = 1.713, P = 1.5671, y = 0.00993 and C6 = 12.14, C8 = 215.2, C10 = 4813.9 and C9 = 143.1 all in atomic units and Rm = 3.41 A at the well minimum for which the ninth order term is excluded.
The simple LennardJones potential is expanded up to second order in the displacement u relative to the radius of minimum potential energy which leads to a harmonic oscillator with Hamiltonian
H=+1ku2 with k=ZV"
2m 2
where Z is the number of nearest neighbors. The groundstate energy of a harmonic oscillator is wellknown to be
3 3 ZV"
2 2
The corresponding wavefunction is of Gaussian form
7
(U ) e xp  U ) 21
(2,TW2r 2 IW
with width W. The width describes the spread of the wavefunction in space and the localization of particles on the lattice. The expectation values for the kinetic and potential energies over this groundstate wavefunction are
3 h2
(T ITI ) = mW2
4
The results imply two things: firstly, the shallow potential with small curvature reduces the potential energy and secondly, the small mass increases the kinetic energy raising its relative importance even more.
A calculation of the minimum energy with respect to the width leads to
W =
in terms of the relevant parameters. It is especially large for small m and small V", which is the case for quantum crystals, such as hydrogen.
Hydrogen has a large kinetic energy relative to its small potential energy and would be too localized for a bound state so that it reduces its kinetic energy by spreading out to a higher potential energy. This process of spreading the wave function leads to a new arrangement of the lattice parameter. A guideline is ro 2(u2)V2 where ro is the distance of minimal potential and u is the displacement about a lattice site. If this quantity is smaller than the hardcore
8
radius the particles spend appreciable time in each others' hard cores which is an unphysical situation. The strong repulsive interactions cost energy and a new minimum energy must be found. The solution to the dilemma is an expansion of the lattice from ro to A. This procedure reduces and increases , which reestablishes the situation of classical solids and is a compromise for a certain A  2(u2)
width W. Empirically it is found that arranges itself around unity.
1.1.2.1. De Boer parameter
It is interesting to define a measure for the quantum nature of a molecule. This can be approached by rewriting the Schroedinger equation in terms of reduced quantities by scaling the distance and the potential with the LennardJones parameters
r V
r  and V* =
which results in
 2 (V*)2 + V* = E*
2mec'
The prefactor of the reduced kinetic energy term expresses the relative dominance of the kinetic energy compared to the potential energy. This is the relevant quantity for quantum behavior and is defined as the de Boer parameter
A m=
2 m,C
9
which is large for small energy and mass. This is consistent with the earlier statements and calculations about the nature of quantum crystals. The parameter is 0.196 for hydrogen3 which is much larger than for classical crystals with values below 0.05.
1.1.3. Crystalline Potential and Quantum Rotor
At temperatures above the melting point (13.8 K) orthohydrogen oscillates rapidly between its orbital magnetic states M. This generates an effective spherical symmetry as it is perceived by its neighbors. In this respect it is similar to parahydrogen at high temperatures. At lower temperatures where hydrogen solidifies this is no longer true. The transition rate between M states decreases and the nonspherical character of orthohydrogen manifests itself. The average potential, the crystal field, at the site of an orthohydrogen molecule is of lower symmetry. This is a selfenforcing, cooperative process as more orthohydrogen molecules with lower symmetry reduce the free rotation of others.
The crystalline potential V(E,
V(is tr) = A ai + AA + AdAi
is therefore a quadratic function of the direction cosines
Ax = sinScos a
10
A =sin sin a>
Az = cos S
of the molecular axis. The eigenfunctions to this problem are for i = x,y,z
4 ir
At this point the quantum rotor property of hydrogen becomes imminent. By this statement is meant that orthohydrogen cannot be perceived as a static entity. Due to its large ZPM it is in rotation even at low temperatures. The J = 1 state is therefore not a rigid ellipsoid but its motion averages the shape into a sphere. As will be seen, the angle dependence of the dipolar interaction is governed by (3cos261) which is proportional to the spherical harmonic of second order. It is, however, of little significance to define an angle E between the connection line of the protons with the magnetic field. Instead it is necessary to define the angles #, (p between one of the crystal field axes, e.g. z, and the magnetic field and average the angular dependence over all space isotropically.
The angle E between the magnetic field and the proton axis can be rewritten in terms of the angles #, 9
cos6 = A, sin/cos p+ , sin/3sin 9+ A, cos# = cos ROO3+ sin Ssin/3cos((p w>)
and the average of the dipolar interaction along one crystal axis Az is taken over all space
11
(3COs2 a 1) = 3f(3Cos2 E  1) cos2 Ssin Sdwod3 = (3 cos2,_ )
The result is the famous reduction factor of 2/5. It expresses that in the case of a quantum rotor the dipolar interaction is rescaled to 40 %, leading to narrower lines in the quantum case than expected classically.
1.2. Open Questions for Confinement
This short overview about interesting properties of hydrogen gives a flavor for the type of experiment that can be performed with hydrogen in a porous material. The substrate represents a geometric confinement and influences the threedimensional properties of hydrogen. The questions concerning the changes that arise from such confinement are various. Some of them are as follows:
1) What impact does the surface potential have on hydrogen molecules being closer to the wall than others? Will different environments result in different subsystems?
2) Is the ZPM in a more localized environment dominant enough to introduce fluctuations that wash out some of the expected effects?
3) Does orientational ordering of quadrupoles occur and if, at what temperature? This may lead to new results when the surface potential is assisting in the dynamical slowing down of rotational motion.
12
4) Is the melting transition influenced by increased fluctuations due to lower dimensionality and a pore size that reduces the size of the critical solid nucleus in the formation of a solid as described by nucleation theory? What is the impact of grain boundaries forming along rough substrate walls?
5) Is a glass phase possible as a result of inhibited and disturbed solidification?
6) How is the orthopara conversion influenced by the surface potential, the dynamical restriction due to the geometry and the reduced number of nearest neighbors?
7) At last a very challenging idea is brought forward: would it be possible to reach a superfluid state in hydrogen if the solidification was suppressed to temperatures where superfluidity is conceivable and theoretically predicted?
This catalogue of questions is well worth a thorough experimental investigation that might give answers to these open problems.
Chapter 2 will expand on the concepts and ideas just mentioned. This will allow for an understanding and appreciation of the importance of the previous questions. The conclusion chapter 7 will try to answer them based on the obtained data. Chapter 3 focuses on some special topics in Nuclear Magnetic Resonance as they pertain to the experiment. Chapter 4 describes the experimental "hardware" and gives an introduction to the technical specialties of the experiment. The experimental data are presented in chapter 5 while chapter 6 contains the computer simulation of the NMR lineshape. Chapter 7 closes the dissertation with a conclusion that restructures the experimental data as they pertain to physical effects rather than a mere listing of facts and explanations.
CHAPTER 2
THEORY
2.1. Surface Interactions
The properties of a molecular liquid and solid are expected to be modified in contact with a surface. The main reason is the interaction potential between the molecules and the surface which has to be incorporated in a theoretical treatment. The impacts of the surface are various. The potential energy arising from an attractive potential such as the Van der Waals potential restricts the molecular motion, tending to localize the particles in potential wells. For particles adsorbed on a surface the number of nearest neighbors is reduced, which is a direct consequence of the geometrical constraints. This in turn reduces the dominance of the intermolecular interaction therefore increasing the relative importance of the external surface potential. The twodimensional character on the surface implies a reduced number of accessible modes compared to three dimensions.
These facts should lead to a modified phonon spectrum, reduced diffusivity, a distinct first monolayer behavior, induced dipole moments, reduced entropy4 and less intermolecular interaction.
As a consequence, the system could display altered macroscopic effects. These macroscopic manifestations of the surfacemolecule interaction could be modifications in viscosity, orientational ordering and solidification behavior, just to mention a few.
13
14
Being familiar with these facts, one must keep in mind the dominant zeropoint motion of hydrogen which is counteracting the localizing surface potential and might reduce the effective surface potential.
2.1.1. Theory of SurfaceMolecule Interaction
Many experiments and theoretical calculations have been performed for hydrogen, methane and helium on vycor, zeolite and twodimensional substrates, favorably grafoil.
A comparison between the properties of zeolite and grafoil is in place as the calculation to be discussed was performed for grafoil, but the experiment in this dissertation was carried out using zeolite. The attraction potential of grafoil is about a factor of two larger6 than the interaction strength of most other solids, including zeolite. In addition, the surface is twodimensional in comparison with a curved geometry in zeolite. An additional complication for zeolite is the tortuosity of the restricted topology. The question of dimensionality arises in this context. The main idea is nevertheless similar and reduces to investigating a system that interacts and is partially bound to the surface of a substrate.
The following calculations will lead to the ground state energy, the twodimensional phonon spectrum6 and the purity of the zero rotational state7 of hydrogen on a substrate.
The starting point for any calculation of this type must be the Hamiltonian for such a system. It consists of the translational and rotational kinetic energies, the moleculesubstrate interaction energy and the intermolecular interaction energy, neglecting weak anisotropic intermolecular interactions:
H = Hr + HR
15
Hz = V2 + v(R,,)+ YV'(R,)
12M
HR = L+LVA(RI,J
2/ 1
where M is the mass of the molecule, L the angular momentum operator and v the isotropic intermolecular interaction which depends only on the relative distance between molecular centers of mass. The anisotropic substratemolecule interaction VA depends on the distance itself and couples the rotational and translational parts of the Hamiltonian. For most calculations, however, the wavefunction is approximated by an independent product of spatial and rotational contributions. This allows for writing two independent eigenequations:
HTD = ETD
for the translational part and
HR = E'
for the effective rotational part, where the argument of the anisotropic substrate potential is now the position of the center of mass averaged over the translational motion.
The translational part is treated to calculate the phonon spectrum. It was emphasized6 that the calculation must incorporate the third dimension of the adsorbed layer. The translational wave function is therefore separated into an inplane and a vertically directed Gaussian part. The ground state energy is a superposition of three energy contributions
16
E =E,+E, +E,
The energy E, is the kinetic energy associated with the motion in zdirection plus the potential energy of the laterally averaged substrate potential in zdirection and is evaluated using the zcomponent of the wavefunction.
The E, term is correspondingly related to the kinetic energy in the xy plane plus the intermolecular interaction energy, averaged over the zcomponent of the wavefunction and evaluated with the inplane wavefunction. The form of the intermolecular interaction is the earlier described SilveraGoldman potential omitting the C9 term.
The third contribution, EXZ, also involves both components of the wavefunction. It represents the vertical potential energy due to the periodic variation of the adsorption potential averaged over inplane displacements.
The solution to this problem is determined variationally by a minimization of the total energy as a function of the parameters in the two Gaussian wavefunctions. This leads to a twodimensional selfconsistent phonon theory because each wave function contribution appears in two equations. This makes the process iterative and several iterations are necessary to obtain a selfconsistent system of equations. The process starts with the determination of the zcontribution of the wave function because it appears solely in the E, equation. It has been shown that the selfconsistent phonon approximation, including anharmonicities from the zeropoint motion, leads to a ground state energy which is by about 15% higher.8
The ground state is almost entirely determined by the E, contribution and the total energy is about 500 K for hydrogen on graphite. The phonon spectrum is characterized by a phonon gap of 46.6 K, a maximum in the transverse phonon mode of 64.9 K and a maximum in the longitudinal mode of 83.8 K. This
17
suggests that it is more advantageous for the system to rotate away from the symmetry direction. The phonon.density has a large width of 48 K because of the substantial zeropoint motion.
These calculations are backed by agreement with experimental data which are within 10%. The conclusion is that the third dimension must be incorporated in calculations dealing with moleculesurface interactions. It is intuitively also clear that the restricted geometry limits the long wavelength phonons which couple to the substrate phonon modes and are the predominant excitation at low temperatures.
Novaco6 initially performed the calculation for a commensurate phase but extended it later to the incommensurate phase. The result is that the energy for the incommensurate phase lies only a few Kelvin (3%) above the commensurate phase groundstate. This is of importance for zeolite where no commensurate phase can be expected due to the surface roughness and curvature.
Returning to the rotational part of the total Hamiltonian, another valuable piece of information can be extracted.7 This is related to the mixing of rotational states under the influence of an additional nonspherical potential. The ordinary rotational ground state is known to be J = 0 and could be subject to mixing with higher order states.
The sum in the Hamiltonian extends over single particle terms which can be solved individually with the rotational wave function expanded in a superposition of free rotor states IJM). The single particle energy contributions are then summed up to yield the total energy.
The use of spatial wave functions such as in the ground state calculation leading to the phonon spectrum (splitting into inplane and outofplane contributions) already presumes a J = 0 state. If the results turned out to be different from this assumption a new Ansatz would become necessary. This is
18
not the case, however, since hydrogen exists to 99.9% in the J = 0 state, confirming the general assumption of pure states.
Furthermore, as the calculation shows, hydrogen molecules adsorbed on a substrate behave like slightly hindered threedimensional rotors. It was also confirmed by analysis of neutron scattering data that hydrogen on surfaces, such as grafoil, behaves like a threedimensional rotor.9 Further experimental support for threedimensional behavior is superfluid helium in vycor.10 The threedimensional network displays a bulklike transition independent of fractional filling.
For rotationally excited states an interesting picture was deduced from perturbation theory" and numerical studies.12 Rotationally excited states couple with the vibrational states associated with motion perpendicular to the surface. This coupling results in a motion where the molecules line up with the normal of the substrate at the maximum amplitude of the zeropoint motion and lie parallel at the center. This implies a mixing of the M = 0 and the M= 1 states inside the J = 1 manifold.
Recent experimental data'3 suggest strong mixing of rotational states for hydrogen deuteride (HD) on vycor. Rotational transitions of as high as SJ =3 and JJ =5 were observed which violate the bulk selection rules. This may be a consequence of the mismatch of center of mass and center of rotation coordinates in HD.
The important result is that the standard classification of hydrogen in ortho and para species in their pure forms still holds. The same should also be true for zeolite with its weaker interaction potential and more threedimensional character. This is valuable information for the orthopara conversion.
19
2.1.2. Effects of Adsorption
Several interesting results from other experiments and calculations will be quoted because of their relevance to the experiment to be discussed in this dissertation.
Related to the phonon spectrum are heat capacity measurements of helium in vycor indicating a decreased Debye temperature14 in two dimensions. This is of importance for the creation of phonons carrying away the released energy in the orthopara conversion process.
The behavior of the first monolayer on the substrate has been observed to be different from subsequent layers. Superfluidity in helium15 decreases and disappears as the coverage approaches one layer.16 This seems to suggest that superfluidity only occurs on top of a more localized first layer smoothing out the surface.
Experiment17,18 and theory19 seem to agree upon the influence of the zeropoint motion on the separation between hydrogen molecules and the adsorbant. The expectation is that the large compressibility leads to a reduced nearest neighbor separation. This is, however, not the case because the ZPM, enhanced by the restriction from the pore environment, counteracts the attractive influence of the surface, hardly allowing the molecule to get close to the surface. This is illustrated by 3He being further away from the surface than 4He reflecting their relative ZPM.20 Methane,21 however, is forced closer to the wall at low temperatures due to smaller ZPM.
Induction of dipoles was observed17,22 from strongly enhanced infraredabsorption spectra of hydrogen in vycor. The conclusion of induced dipoles follows from the independence of the infraredabsorption signal over time. This means that ortho and parahydrogen give rise to the same signal which renders
20
the electric quadrupole moment of the ortho species irrelevant and shows the strength of the surface interaction.
A special effect on the diffusion rate is the purely geometric restriction.23 The parallelpore model24 was proposed to describe a monodisperse poresize distribution. In this case the diffusion Ds is related to the free bulk diffusion Do in an empirical relationship with the porosity (D and the tortuosity factor 325
DD
Ds = Do The tortuosity is a measure of the geometric complexity of the substrate. A numerical value was found25 to be 8= 5.6 for hydrogen in vycor of about 10 A radius. This is close to the dimensions of zeolite and can give an idea of the purely geometric influence on the diffusion rate. Using (D = 50 % for the porosity of zeolite the tortuosity effect reduces the diffusion by roughly a factor of 10. This implies strongly inhibited diffusion for the confined system.
This survey of the variety of influences caused by the surfacemolecule interaction illustrates the strong motivation for the intensive investigation of this type of system.
2.2. The LiquidSolid Transition
There have been many attempts to explain and describe the liquidsolid transition. The early phenomenological theories relate critical behavior of solid state properties to the onset of melting. Such theories are characterized by the increase in vibrational amplitudes,26 the disappearence of bulk moduli27 and the
21
proliferation of defects (dislocations)28,2 upon occurrence of the liquid state. These are models for the mechanical, rather than the thermodynamic, instability of the crystal. The study of a thermodynamic criterion, the minimum chemical potential, has been pursued by molecular dynamics studies.30 They predict melting which takes place in a layerwise fashion starting from the surface. Caution towards the results of such calculations is necessary, considering the limited number of particles involved.
2.2.1. Elastic Instability Theory
The original Born criterion of vanishing bulk moduli was revised,31 remedying the erroneous second order and homogeneous nature of the transition. The difference between the two theories hinges on the experimental observation that one of the bulk moduli does not completely disappear at the transition temperature but only inside the melt. This implies a twophase model where the dilatation on the liquid side is 4,',, which is different from S', on the solid side of the transition. The shear modulus vanishes or nearly vanishes when S = 8,,. At the transition the entropy 2RT is released upon entering the liquid state. This reduces the free energy and is the reason why there is another dilatation in the solid state which is associated with the same free energy. This point is the true transition where the isothermal and discontinuous transformation, involving a latent heat, occurs. The released entropy also represents the nucleation barrier which renders this a heterogeneous theory, as the solidification occurs at strained sites and surfaces where the free energy is higher.
2.2.2. Dislocation Theory
A more microscopic view of crystallization was taken by Cotterill et al. by using a model incorporating dislocations. The decisive assumption is that dislocations also exist in the liquid. The dislocation density is extremely high and
22
there is no crystalline material around the mutually contiguous dislocation cores. This makes it imprecise to describe the liquid as a crystal saturated with dislocations cores. Instead, the liquid can be referred to as a pseudodislocation saturation. Viscosity and other liquid properties are explainable by the rapid motion of pseudodislocations. Crystallization occurs upon cooling by elimination of pseudodislocations without recreation when the temperature is below melting. The elimination of pseudodislocations is driven by an increase in the free energy, as the liquid state exists below its equilibrium temperature. This phenomenon is called supercooling and indicates a metastable state. Nucleation theory (as discussed later in this chapter) applies to this system in a standard way. If the viscosity of the melt is high or the system is cooled rapidly into the solid temperature region the removal of dislocations is inhibited. The dislocation annihilation rate is then slower than the solidification process. This solid is microscopically similar to a liquid incorporating a high concentration of dislocations, which are immobile and incapable of diffusion and interaction. A polycrystal with a grain size of nearatomic dimension develops. Such a solid is called a "glass' and follows in a straightforward fashion from this model.
A few properties of glasses shall be mentioned in this context. The ground state of a regular solid is a single crystal which minimizes the dislocation energy. A glass, on the other hand, is one realization of a metastable state frozen in at some arbitrary configuration. The reason for its existence can be its frustration due to the incompatibility of the interactions with the lattice. The specific volume of a glass is larger than that of a crystal which is an immediate consequence of the abundance of frozen dislocations and the increased disorder. A glass temperature exists below the bulk melting point from where on the mobility of dislocations is zero, the elimination of defects impossible and the glass phase persistent.
23
2.2.3. TwoDimensional Melting Models
A twodimensional dislocation theory is especially interesting in the light of the twodimensional liquidsolid transition. Mermin,32 following Peierls'sm earlier arguments, showed rigorously that conventional crystalline longrange order is excluded in two dimensions for powerlaw potentials. This does not pertain to orientational ordering, however. Kosterlitz and Thouless,35 proposed another model for the existence of longrange order. This longrange order is of a topological type. It originates from the dislocation theory of melting.36 The idea is similar to the already discussed model of Cotterill29 et al. The liquid phase close to the melting point has a local structure similar to the solid but dislocations exist which are mobile and induce viscous flow. The solid state is rigid because the dislocations are immobile. They pair up in the solid state thereby reducing their energy from a logarithmically diverging to a finite contribution. The phase transition from the solid to the liquid state is characterized by the dissociation of pairs into single dislocations. The critical temperature is calculated to be
T = a2(1+ r)
', 47rkB
where p is the shear modulus (in two dimensions) and r Poisson's ratio. The connection with the revised Born model3' described earlier in this context is of interest. From this theory the shear modulus cannot be zero in the transition region, otherwise the transition to a solid would occur at zero temperature.
Starting from the dislocation model, Halperin and Nelson37 have worked out a more detailed theory of twodimensional melting. They assume the unbinding of dislocation pairs which are referred to as dipoles because they pair up with two different polarities. This yields an exponential decrease of the
24
translational order parameter. Instead of forming an isotropic liquid at this point, calculations show a sixfold anisotropy with the bondangle correlation decaying algebraically. At a higher temperature finally, the dissociation of these disclination dipole pairs drives another transition to an isotropic liquid. The intermediate liquidcrystal state is possibly bypassed or occurs simultaneously in the presence of a substrate due to its orientational bias. This would transform the two continuous phase transitions into one firstorder transition.
This idea of a virtual region, where two transitions are hidden in the experimentally observable firstorder transition, was adopted by Tallon.w In a fashion similar to the revised Born theory, an entropy term is included. This could be a volumeindependent term39A4 of AS = RIn2 as AV' +0 which is the
V
case in a porous material where the available space is constant as a geometric restriction and the molecules are trapped in the pores. This entropy term is possibly associated with the transition from a uniform distribution of the crystal cell size to a random one. Another possibility is the communal entropy arising from the fact that particles in the liquid state have access to the whole fluid instead of being localized. This contribution is significant for large diffusivities but small for a tortuous geometry such as zeolite. The transition becomes first order by going discontinuously from below the dissociation to above the disclination of dislocation dipoles for two and three dimensions.
This leads to the conclusion that melting in two and three dimensions is likely to be very similar.8 The difference would be the varying width and separation of the two virtual transitions, both being larger in two dimensions.
2.2.4. Molecular Dynamics Studies
Studies from a different starting point are performed using the molecular dynamics approach. In this case thermodynamic quantities such as the
25
chemical potential are investigated. Of special interest is Broughton3O et al.'s conclusion that melting occurs in a layerwise fashion starting from the outermost crystalline layer and progressing into the bulk region. The mechanism involves the promotion of particles from the solid to the liquid layer with an accompanied increase in mobility in the remaining solid layer. The empirical rule has been established that each layer melts when it reaches the bulk ratio of kinetic to potential energy at the melting point. Using this criterion, the outermost layer melts at about 72% of the bulk melting temperature, the second layer at 90%, while already the third layer is close to the bulk melting point.
2.2.5. Implications for the Investigated System
The previous remarks are of fundamental importance for a system such as hydrogen in zeolite where the purely geometrical surfacetovolume ratio is as much as 1/4 and about 80% of all hydrogens have contact with the substrate. It is therefore difficult to decide upon the dimensionality of the adsorbed system. The prediction of a similar melting process for two and three dimensions eliminates some of these problems.
Implications for the nature of the transition are also given by the existence of hysteresis upon heating and cooling. The absence of hysteresis would lead to the conclusion that the transition is continuous and the melting domain not a heterophase region.
Furthermore, it has been argued41 that in the case of grain boundaries the transition should always be first order. As will be seen later, grain boundaries may have substantial impact on the solidification process.
The above theories and projections are a rich testing ground for experimental investigations. A variety of work is necessary in order to understand the complicated process of melting and solidification. The outcome will certainly be severely affected by the impacts of dimensionality such as the
26
reduced number of nearest neighbors and the substrate character with an interaction potential that leads to limited diffusion.
2.3. The Superfluid
Superfluidity for helium is a wellknown phenomenon in the lowtemperature community. The important issue is that the freezing point of helium is below the transition to the superfluid. This is the case because helium does not solidify at any temperature without pressure. Therefore the superfluid state is feasible and transforms by Bose condensation from a normal liquid at T = 2.17 K.
Superfluidity is expected to occur for any system of bosons. Interest exists therefore in a potential superfluid hydrogen phase. Calculations have shown for an ideal Bose gas that the transition temperature to a superfluid42 is characterized by
h2 2/3 5/3
Tx =3.31 ) 112( M ) 3
*Mk g) MP
where M is the mass of the molecule, M, the proton mass, n the concentration, p the density and g the degeneracy for a singleparticle state which is unity for parahydrogen. This results in a predicted transition temperature of T, = 3 K for helium compared to the experimental value of T = 2.17 K. It is in good agreement and can therefore be applied to parahydrogen with a reasonable chance for a realistic prediction of the superfluid transition temperature. The calculation for hydrogen yields T. = 6.6 K. This low value for the transition temperature is the reason why superfluid hydrogen has not been observed
27
experimentally: hydrogen freezes at 13.8 K before it reaches the superfluid transition. It is expected, however, that hydrogen enters the superfluid state if the liquid is supercooled to below the superfluid transition temperature. The quality of the above estimate for the superfluid temperature is stressed by the fact that the transition temperature increases with a larger interaction potential. The energy scale of the LennardJones potential between hydrogen molecules is e= 37 K which is much larger than e= 10.2 K between helium atoms. The calculation is therefore likely to be a better estimate for hydrogen than for helium. Other authors44 are less confident about the value of T,. They scale the difference between T, and T, with the de Boer parameter which for hydrogen is only 2/3 of the value for helium. The de Boer parameter describes the degree of "quantumness" which leads the authors to believe in a reduced experimental transition temperature. On the other hand, it seems that a larger ZPM would rather be less favorable for a collective process, such as Bose condensation.
In summary, superfluid hydrogen can be expected below 6 K but the liquid state must be extended down to the superfluid transition temperature which is possible when the liquid is kept metastable.
Properties of the superfluid have been projected partly in analogy to helium. The dispersion curve should also exhibit phonon and roton excitations. The slope of the phonon curve is given by the sound velocity. The roton gap A is estimated by scaling the minimum roton energy of helium by the ratio of lattice spacings. This results in a gap A of around 3kT4. This would lead to a roton gap of 20 K. A recent experiment45 of hydrogen in vycor suggests a tentative explanation of heat capacity data by a roton gap of 23 K. This occurs in addition with supercooling at 10 K which seems a very high temperature in the light of superfluidity.
28
So far the discussion only pertained to parahydrogen. For orthohydrogen with a singleparticle degeneracy of g = 9 for spin and orbital angular momentum degeneracy the superfluid transition drops to 1.5 K. For this system further magnetic properties could exist which would generate additional diversity in the phase diagram.
The question arises if supercooling by about 8 K is a feasible operation. Several options and speculations have been proposed. It has been suggested' to investigate surface films on different smooth substrates where hydrogen may not wet and stay liquid. This could be done on top of a monolayer of substratecoating material (deuterium or neon) generating a less dense fluid. The setup would lead to a twodimensional superfluid. It has been shown by Widom46 that superfluidity, excluding intermolecular interactions, is indeed possible in two and even onedimensional systems. The condition for the existence is the presence of an external disturbance, for example gravity or rotation, which breaks translational invariance. This does therefore not contradict Hohenberg's47 argument which rules out lower dimensional Bose condensation in homogeneous systems, i.e. without symmetry breaking.
As a confirmation of this theory the possibility of surface superfluidity48 in parahydrogen has been put forward as a potential explanation of a gravityinduced flow below 3 K observed by Alekseeva and Krupski.49 The calculated flow rates are in agreement with experiment and result from vortex considerations analogous to the KosterlitzThouless theory.
Inclusion of impurities is another proposed method of achieving supercooling as suggested by Geilikman5O who extended the Lindeman criterion for melting to quantum crystals where the melting temperature T is much smaller than the Debye temperature ED. He stresses that the zeropoint
29
amplitude is inversely related to T, which is in turn proportional to (XO3)V. Impurities reduce the Debye temperature if they are heavy, weakly bound to the matrix and have a high solubility. By these means Tm, is reduced.
An interesting possibility at low temperatures is the creation of quantum excitations from light impurities, so called impuritons. Atomic hydrogen can be used which also obeys Bose statistics. If the impuritonimpuriton scattering is repulsive then the energy spectrum displays an acoustic branch which satisfies the superfluidity condition.51 The calculated superfluid transition temperature T. is raised to 12 K.
Another alternative is dynamic supercooling which relies on a rapid decrease in temperature and conserves the liquid state in a metastable form. Many experiments of this kind have been performed, being described in the next section. An explanation using nucleation theory is possible to demonstrate this effect of delayed and inhibited nucleation to the crystalline state. The theory will also be applicable to the understanding of supercooling in small pores which is performed in the experiment to be discussed in this dissertation. The procedure can substantially decrease the melting point, by as much as 5.5 degrees in the present experiment. Unfortunately, the lambda transition and the Fermi temperature are also depressed in small geometries as experimentally shown for 'HeS2 and 'He.53 This makes it in turn more difficult to reach superfluidity.
2.4. Supercooling
2.4.1 Nucleation Theory
Nucleation theory54 is one possibility for understanding the preference of the liquid versus the solid state. It is a dynamic description why the process of
30
solidification is inhibited and therefore the metastable liquid state preferred. It builds upon the formation of nucleation cores around which solidification occurs and describes the velocity of growth, once the nucleation cores have formed.
The starting point is the free energy difference SF between liquid and solid state required to form a solid sphere of radius R
SF(R) = 4 R2LS 3 Rns (fL  f5s)
where aLS is the surface energy55 between solid and liquid, ns the number density in the solid phase and fL, fs the free energies per molecule in the liquid and solid phase. The surface and volume energy contributions to the free energy are competing. There exists a maximum energy
_16,r at"
3 n2(ff)2
which occurs at a radius
RO = 2 aL,,
ns(fL fs)
Figure 1 shows a plot of the free energy difference versus the solid sphere size for different temperatures. For R > RF the solid sphere grows without limit because the cubic term in R takes over and decreases the free energy. If a thermal fluctuation can produce a solid sphere with a radius greater than RO, the supercooled liquid makes a transition into the solid state. At the normal bulk transition temperature of 13.8 K, fL and fs are equal. Therefore, RO, the critical radius, and the energy maximum for SF, which is the solidification barrier, both
31
NUCLEATION THEORY 5000
13 K
4000
F
r
e
e 3000
E
n
a 2000
r12K
9
1000
10K
0
0 5 10 15 20 25 30
Radius (A)
Figure 1. Nucleation Theory
32
diverge implying that for any lower temperature a large but finite size solid sphere suffices to trigger the transition to the solid. At this point the liquid state still prevails, however. As the temperature decreases further, the barrier and the radius also decrease. This implies that a finite, thermally induced solid sphere is enough to induce solidification because, once the nucleation core reaches the critical size and the top of the barrier, it is energetically favorable to grow without bounds and release energy. The maximum radius for the solid sphere is the above defined radius R. This is in contrast to the behavior in a porous material where the critical solid sphere is restricted to the pore size. The slightly different condition will be described in the next section. For low temperatures the critical radius and the solidification barrier, become small, indicating facilitated solidification.
The probability for solidification to occur per unit volume is described by
T = FTr exp(SF / kT)
with an attempt frequency per unit volume
FT =(nLkT / h)exp(LS /kT)
where nL is the number of molecules per unit volume in the liquid phase and DLS the activation energy for diffusion of atoms across the liquidsolid interface. The selfdiffusion activation energy is 45 K in liquid hydrogen and can be taken as an approximation for cILS.
An understanding for the shape of the solidification probability can be gained in the following fashion. The nucleation rate increases rapidly with a decrease in temperature due to the increased free energy difference between
33
solid and liquid at these supercooled temperatures. This implies a decreased energy barrier. At low temperatures, however, the energy barrier becomes constant but the lower temperature reduces the probability of overcoming this barrier by means of thermal fluctuations.
Nucleation by quantum tunneling through the energy barrier has also been considered. This could be of importance for quantum crystals like hydrogen and introduces an additional mechanism which is especially active at low temperatures, therefore increasing the nucleation rate.
A very important parameter for calculating realistic numbers in this context is aLs. The larger this value, the more efficiently a liquid can be supercooled. Unfortunately, this parameter is not known for any material with high accuracy but has strong impact on the outcome because it is contained in the exponent to the third power.
Another supercooling models which follows classical nucleation theory in three dimensions relies on a scaling argument. It suggests that supercooling of hydrogen takes place at a reduced temperature of 0.8 times the bulk melting temperature. This leads to a higher supercooling temperature of 11 K. Another statement is made from scaling. The involved degrees of freedom can be calculated as a ratio of the selfdiffusion activation energy and the bulk melting point. This ratio is three and implies participation of three translational and three vibrational degrees of freedom to the process of diffusion across the phase boundary.
2.4.1.1. Implications for experiment
Nucleation theory can be used to devise experiments which could be able to achieve supercooling.
34
The first set of experiments exploits the dynamical aspect of the theory. If hydrogen is cooled rapidly through the maximum nucleation rate into the relatively stable regime at low temperatures, substantial supercooling, maybe down to zero degrees, and superfluidity can be achieved.
Experiments of this kind have been carried out at Brown University,57,58 using liquid drops. They employed a helium pressure to levitate the injected droplets and have indeed observed supercooling by several degrees. Container walls and impurities act as nucleation cores and had to be avoided.
The second possibility is using confining geometries into which hydrogen is adsorbed. This introduces a cutoff radius R, into the system which is related to the pore size. Even when a thermal fluctuation is able to create a solid sphere of radius R0, from where on it would grow without limit in the bulk, it is now restricted to the cutoff radius and must stop growing. As long as the liquid state is energetically favored, which is the case for positive SF, the system will stay liquid. The cutoff radius Re, is therefore determined for SF= 0 to be
R = 3aLS
ns(fL  fs)
It is obvious that a smaller pore size results in stronger supercooling, or better, a more depressed melting point. This process is not dynamically motivated and relies entirely on the energetics.
Experiments following this reasoning have been performed mostly using vycor, a porous glass, and heliums2,9,60.61 or hydrogen.4.62,63
There is an effect for helium which is equivalent to supercooling. Helium already requires overpressure to achieve bulk solidification. The result for helium in restricted geometries is that an extra pressure of several atmospheres, in excess of the bulk pressure, is needed for solidification. This pressure is a
35
function of pore size and can be deduced from the critical radius of nucleation theory treated above:
R= 3aLS 3aLS VS
n,(fL  fS Ap VL VS
This determines the pressure difference as a function of the pore size. The above equation has been quoted in the literature52S8.59 with a factor of 2 instead of 3 which does not take the additional effect of the restriction by the cutoff radius into account.
The same considerations can be made for hydrogen, now using the latent heat term but neglecting the entropy contribution to the solidification process. This leads to
R, 3aLS VsTm
LL ST
where T is the bulk melting point and LLS is the latent heat of fusion per molecule which determines the amount of supercooling ST.
This model attributes the extra pressure and the depression of the melting point to the additional contribution of surface energy between solid and liquid. No account is made for the failure of the solid to wet the surface. The model in the next paragraph will attempt to explain this phenomenon.
2.4.2. Grain Boundaries
A microscopic models attempts to explain the inhibition of the formation of solid on the wetted surface and the following rapid nucleation. This model originated from the experimental observation65 that the solidwall interfacial
36
tension asw, also termed surface energy, was found to be greater than the liquidwall tension aew for helium. This result was obtained by measuring a contact angle e between solid and liquid on the surface of a copper container that was larger than 90 degrees and then compared with the equation
aLS cos 9 = ae w  asw
This was confirmed on a glass surface and disagrees wh the theory of Landau and Saam.6 Their theory states that the dense, absorbed film on a substrate in contact with the liquid is simply ordinary solid which is at an increased pressure as a result of the strong Van der Waals attraction of the substrate.
The increased solidwall tension is surprising, as virtually all uniform substrates favor the solid state, independent of their attraction potential. The reason is that the attractive forces interact more strongly with the denser solid than with the liquid. The enhancement of the solidwall tension therefore demands a microscopic explanation.
For the explanation a microscopically rough substrate with Van der Waals interaction is considered. The adsorbed material forms a monolayer solid on the substrate walls due to the attractive potential. The nature of this solid is crucial for the understanding of the reduced affinity for the solid state in the bulk pore. The solid conforms to the rough surface wall and is therefore irregular and microcrystalline. Any growth of a bulk solid is highly polycrystalline. This would give rise to a large solid contact surface area between the individual solid crystals and create many grain boundaries. They in turn have a high grain boundary energy which contributes to the effective interfacial energy between solid and liquid and disfavors the solid state. The pore environment is especially
37
prone for such a behavior with its large surfacetovolume ratio where the interference between crystallites is of the highest degree. The film on the substrate should appear as a foreign wall, not belonging to the rest of the liquid system. This is called classIl growth and describes the imperfect wetting of a film by the condensed bulk of the same material. This also reconciles the observation of a large contact angle on a substrate surface and a monolayer of solid directly adjacent to the wall.
The increased pressure for solidification of helium has been calculated using the increased chemical potential associated with the grain boundary energy. The excess average energy per unit area for largeangle boundaries is67 f = 0.05,pa, where a is the lattice spacing and p the shear modulus. This is multiplied by the volumetosurface ratio / = V/A of the porous material and equated with the pressure term of the energy which leads to
SP =0.05 apaI/
where a is a constant depending on the roughness of the surface. It is about equal to the number of particles in contact with the walls of the pore if the surface nucleates a crystal at every particle site. The roughness parameter has been given between 10 by Shimoda6O et al. and 40 by Dash.64 The equation leads to the approximately right overpressure for the solidification.
In the case of hydrogen one can again substitute the pressure energy term versus the latent heat and arrives at
ST =0.05 apavs T,
I LLS
38
using the same notation all along.
There is a discussion in place about the two different models proposed in the discussion of supercooling. Both models predict a reasonable value for the overpressure required to freeze helium. There is a clear distinction, however, between the two physical interpretations. In the grain boundary model a thin layer of adsorbate wets the surface and solidifies on it. This is not the case for the solidliquid nucleation model where a density fluctuation within the nonwetting fluid can become stable and solidify under the appropriate thermodynamic conditions.
2.5. OrthoPara Conversion in Molecular Hydrogen
One of the most fascinating properties of molecular hydrogen is the explicitly quantummechanical classification into two species. The quantummechanical requirement of a totally antisymmetric wavefunction for two fermions can be realized in two distinct ways, leading to two distinguishable molecular species: ortho and parahydrogen. Orthohydrogen has a symmetric nuclear spin wavefunction (/ = 1) and an antisymmetric orbital wavefunction (J odd). The opposite is true for parahydrogen.
The orthopara ratio is determined by the availability of states given by the partition function. In the hightemperature limit the even and odd sums over J in the partition function yield the same result so that the spin parts lead to a 3:1 ratio of ortho to parahydrogen which is the relative weight of the nuclear spin degeneracy for the ortho and para species. Because of the large separation in the rotational energy levels, the equilibrium concentration of orthohydrogen drops from 75% to minute amounts at temperatures below about 10 K. At low temperatures essentially only the parahydrogen groundstate and the first excited
39
state (ortho, J = 1) are occupied as a result of the large excitation energy in comparison to the thermal energy. The conversion process from ortho to para molecules requires, however, the simultaneous breaking of the spin and orbital symmetry, and the equilibrium ratio is therefore established only very slowly. For the isolated molecule the conversion is absolutely forbidden. In the condensed phase the rate is determined by the intermolecular magnetic dipoledipole interaction and magnetic field gradients caused by magnetic impurities. Because of this, ortho and parahydrogen can be treated as two different molecular species.
The first theoretical calculation of the orthopara conversion in bulk solid hydrogen was performed by Motizuki and Nagamiya68 and more recently revisited by Berlinsky and Hardy,69 and Berlinsky.70 The two main interactions responsible for the conversion are: 1) the nuclear spin dipoledipole interaction between two ortho molecules and 2) the interaction of the proton spin of one molecule with the rotational magnetic moment of the other. It is important to note that parahydrogen is not part of this process as it does not possess spin or orbital magnetic moments. The interaction Hamiltonian consists of two parts
Hi =HS, + H,
where "s" stands for the spin and "r" for the rotational contribution. The spinspin Hamiltonian has four identical contributions, expressing the combinatoric of the two pairs of proton spins involved. The rotational interaction involves only two contributions of the orbital momentum of one molecule with the two spins on the other molecule. The individual Hamiltonians are of the standard dipoledipole type (being described in chapter 3.1.) which can either be written in Cartesian or in spherical coordinates using spherical harmonics with the convention of
40
Rose.71 The total wave function is a product of the orbital and spin part. Orthohydrogen has 3 degenerate wavefunctions for the orbit, DJm, and the spin, TIM, reflecting the quantum numbers J = 1 and / = 1 and their magnetic states. This leads to nine degenerate states. Parahydrogen only has one state 4 TO. This implies that there are 81 initial states, i, and 9 final states, f, for a conversion of two ortho molecules into one ortho and one parahydrogen molecule. The conversion rate R can be calculated using Fermi's golden rule
R = 21 P l(fln H I i)12 ( E,  E,)
where P is the probability of being in one of the initial states i. The energy released in the process of conversion is
AE=E, E, =BJ(J +1)BJ(J +1)=2B=171 K
and must be taken up by the lattice. The bulk Debye temperature172,73 is 120 K and indicates that one phonon is not enough to absorb the released energy. Instead, two or three phonons are necessary.
Avoiding most of the tedious and involved calculation of matrix elements, the results for the twophonon process are discussed. The rate R is a combination of the dominant spinspin and the much weaker spinorbital contribution:
(=1 (1+ )
where p, =r10p. Finally, Rs is found to be
41
'Y2
R =3.8.106 '7i_ P )2
hkBe% RO 2MROg%
with a complicated geometric factor rl = 0.7331, depending on the Debye temperature, and the density p. The interesting feature is the strong dependence on the Debye temperature. Any changes in this quantity due to confinement would have great impact on the conversion rate. Threephonon processes are estimated to be only a few percent of the main contribution.
Onephonon processes are not allowed under bulk conditions, as discussed earlier. However, under conditions which change the Debye temperature, like a restricted geometry or a twodimensional system, this could become important. Berlinsky calculated the onephonon process and expressed the rate as a function of density and complicated functions that have to be extracted from experimental data. He found that highdensity hydrogen mainly converts by emission of single phonons. The rate for onephonon processes is proportional to E)2R* compared to e,R2 for two phonons.74
The calculations lead to conversion rates of 1.94%/h by Motizuki68 et al. and 1.67/1h by Berlinsky69 et al. which have to be compared with measured rates of 1.75%/h75,76 and 1.820/h.77 This is in excellent agreement although several approximations are introduced into the theory. The theory is using a Debye approximation with the energy proportional to the absolute value of the wave vector. This is a poor approximation for the high frequency part of the phonon spectrum. It also does not include anharmonicities due to the large ZPM. The interactions are averaged over a powder, assuming random orientations of crystallites. The conversion process is bimolecular or autocatalytic, described by
42
x = kx2
where x is the orthohydrogen concentration and k the conversion rate constant. The solution to this differential equation is
_xo
x(t) = *0
1+xakt
where xO is the initial ortho concentration, 75% in the case of hydrogen at STP.
If any paramagnetic impurities, for example oxygen, are present there is an additional process.78 These impurities produce inhomogeneous static magnetic fields around themselves to which hydrogen molecules couple. The calculation procedure corresponds to the autocatalytic process. The main difference is that the decay is monomolecular and described by
k=kx
with an exponential solution
x(t) = x. exp(kt)
In the case of oxygen,79 for example, the conversion rate is more efficient within a sphere of radius 2.95 R0 than in the autocatalytic case. In this sphere mostly parahydrogen is found. Measurements regarding the influence of paramagnetic impurities8O have been carried out showing fast exponential decay rates. The important result is a relationship between the rate constant and the square of the magnetic moment. This can be used, in an analogy with experimental data, to
43
estimate an upper bound on the concentration of impurities present in a mainly clean system.
2.5.1. Diffusion
So far, diffusion effects have not been taken into account. Diffusion homogenizes the distribution of orthohydrogen around impurities and in depletion zones. This establishes a perfect equilibrium distribution for temperatures above 8 K.81 As long as this is the case the conversion is temperature independent because the energy of phonons created by conversion is much larger than the thermal excitation kT. Thus the occupation numbers of thermal phonons are extremely low and have no impact on the conversion rate. Assuming a binomial distribution W, for the equilibrium distribution of ortho molecules, the mean number of orthohydrogen molecules around each ortho molecule is given by
N
M = JnW = Nx
n=O
where N is the number of nearest neighbors. The orthopara conversion leads to a deviation from equilibrium which can be balanced by selfdiffusion which tends to homogenize the distribution. If the diffusion is slower, the conversion rate is determined by the diffusion bottleneck. In addition, clustering occurs as a result of the anisotropic interaction between orthohydrogen molecules which is of electric quadrupolequadrupole character. In dilute bulk samples orthohydrogen molecules can diffuse by quantum tunneling until they are close to an ortho neighbor. The two molecules can then form a bound pair as a result of their anisotropic interactions if the temperature is low enough (kBT < Vaj, < 1.5 K). At low temperatures the overall conversion rate can therefore be accelerated
44
due to the conversion of the clusters. For high temperatures, thermally activated diffusion is clearly the dominant process, leaving the molecules well separated, while at temperatures around 1.6 K clustering becomes significant.
This was described by Schmidt82 for bulk solid hydrogen. He found that the diffusion is large enough at 4 < T < 12 K to sustain the same conversion rate (k = 1.9/o/h) while at 1.57 K the quadrupole effect overcompensates the reduced diffusion and leads to an increased conversion rate after 200 hours. The rate increases again after 900 hours in the bulk. This result is understood in terms of the enhanced conversion in clusters due to the reduction in the orthoortho separation compared to a homogeneous distribution. Oxygen impurities could be excluded by experimental control of oxygen contents which did not influence the conversion substantially. The reason is that oxygen freezes out at higher temperatures and has low solubility in hydrogen. This may be different for a substrate where oxygen can adsorb onto the surface and act as a catalyst for the conversion mechanism.
In the case of a porous material the majority of the molecules is in contact with the wall and only a few nearest neighbors, depending on the roughness of the adsorption site, while the remainder is in a bulklike surrounding in the center of the pore with 12 nearest neighbors. The value of the conversion rate is therefore expected to be different for a restricted geometry due to interactions with the walls, the reduced number of nearest neighbors and the changes in the phonon spectrum.
The orthopara conversion rate in an environment different from bulk needs to be determined accurately in order to characterize the hydrogen samples used in subsequent experiments.
CHAPTER 3
NUCLEAR MAGNETIC RESONANCE: NMR
3.1. Dipolar Hamiltonian
Hydrogen spins in a magnetic field experience the Zeeman interaction with the applied external field as well as inter and intramolecular dipoledipole interactions. The intermolecular forces are weak compared to the intramolecular forces because the interactions fall off with the cube of the distance so that they contribute only a small fraction
S R N. 3(l OH, (IHN1 .5) 2 .
s = + H 3.75 32
where the effect of the different spin quantum numbers for orthohydrogen and protons has been included. This small term can be neglected in the discussion.
The orbital nuclear Zeeman energy associated with the Hamiltonian Hi = hyHOJz and the spinorbit interaction Hso = hypl JH,, vanish, i. e. are Nquenched."1 The argument was discussed in the quadrupolar section of chapter 1 and is related to the nondegenerate nature of the rotational groundstate. The dominant part of the Hamiltonian is therefore the nuclear spin Zeeman energy with a perturbation from the dipolar Hamiltonian83
45
46
H=yAH( + /2)+ (?)  [1 .12 3(1'.n)(2 .n)]
where Ho is the magnetic field in zdirection, I is the nuclear spin vector with superscripts 1 and 2 for the two spins and n = r12 / r the unit radial vector between spins. The dipoledipole term can be rewritten in spherical coordinates in order to find a quantum mechanical expression
HDD r( 1.123 / cosO+sinE(/ coso+/ sin)] x .cosO+sin8(/ cosX +i sin4) }
The angle E is taken between the direction of the field and the intermolecular axis. The x and y components of the spins can be combined to generate raising and lowering operators
2
HDO = { 112 _ 3 /1 cosE+ sin(Ie + Pe) x [Icos+ jsinE(/2e + 2e+)I
and can be collected into six terms
H00 = (A+B+C+D+E+F) r3
which correspond to different transitions. The individual terms are
A =II (13cos20)
B=i13cos2 ( _I 1 . 12) = i3cos2 E)(/1 2+ 112)
47
C=Isinecosee*(//2 +/'/)
D = C= }sin+cos e*(/12+I)
E =  sin2 Ge2/1/2
F=E* = 2sing2 ee+2iI1
The reason for displaying the dipoledipole interaction in this fashion is that the different terms cause distinct transitions. Terms A and B do not change the overall magnetic quantum number, C raises it by one, D lowers it by one and E and F increase and decrease it by two, respectively. Choosing degenerate states 1+) and +) with the meaning that spin 1 is up and spin 2 is down, and vice versa, demonstrates the operation of each term. Term A directly connects either state and is therefore diagonal. Term B flipflops one state into the other and therefore connects the superposition of states, which is the proper state in the first place. It is therefore also diagonal. Offdiagonal elements only contribute by means of a second order correction to the wave function which results from the perturbation of the dipolar Hamiltonian. Terms C and D are offdiagonal in first order, E and F in second order and contribute therefore only negligibly to the eigenenergies of the dipolar system. They will be omitted from here on.
3.2. Continuous Wave Lineshape
The Hamiltonian is now in a form that can readily be evaluated for orthohydrogen with J = 1 and M = +1, 0, 1. The wave functions are ++), 1(1+)+1+)) and ). Bracketing the Hamiltonian with these states produces the eigenenergies
48
E., = yAHO + (13 30s2 e
( yn)2
E00 =  (1 3 c2 e)
2r
_ =  (HO + (1) 3COS2 )
E~ =~H0+ 4r (30s)
Transitions occur between adjacent energy levels due to the selection rule AM= 1 which leads to two different absorption frequencies that are no longer degenerate as they were in the pure Zeeman case
2
hv, = E_  E00= yhHO + 3 (1 3 cos2 )
hv2 = E  E., =yhHO  4 (13Cos2 e)
This calculation was carried out classically. As shown in the introduction the protonproton intramolecular axis is rapidly rotating, a consequence of the quantum rotor property. An average has to be invoked on the angular dependence. The frequencies v. and v_ are symmetrically located above and below the undisturbed Zeeman frequency v0. The frequencies are
vt =1 d(3 cos2 e _ 1)
1 hiy2where in the liquid case d = 3)  57.68 kHz. In solid hydrogen the
5 2,r(r3)
constant is slightly different with a value of 54.2 kHz. The angle E is measured between the external zdirection of the magnetic field and the intramolecular axis. It is physically more insightful to transform to another angle. This angle P defines the orientation of a surface or the crystal field in the local environment of
49
an ortho molecule relative to the magnetic field. The average over all space with respect to this direction yields the 2/5 factor for a quantum mechanical object. The result is, in addition, also dependent on the rotational state ("shape") of the molecule characterized by the normalized quadrupolar order parameter
1= MJ 2)
This leads to the final result
v = ad(3 Cos2I_1)
2
which will be discussed as it describes the continuous wave lineshape.
For a fixed order parameter a at a fixed angle # two sharp lines occur which are inhomogeneously broadened. This is the case for a single crystal.
If the angle P is distributed from 0 to 90 degrees, as is the case of a powder, the lineshape has the famous Pake doublet form (Figure 2). The intensity corresponding to the frequency can be found from a space transformation argument. Since the dipoles are isotropically distributed the fraction of pairs lying in d# is d(cos#). With the lineshape function being g(v) and the isotropic space intensity /(Q) = it turns out that
4 7
g(v)dv = I(Q)dK2
1 1
(v)~= 47 dv 18dacos#
27r d(cos#)
50
0.014 0.012 0.010 0.008 0.006
0.004
0.002 0.0001
200
50 100 150 200
Figure 2. Pake doublet with order parameter a= 1
100 50 0
FREQUENCY (kHz)
150
7
51
Solving for cos3 in terms of frequency and order parameter, using the frequency equation, results in the lineshape
g*(v) =3
1 8 d a 1+ v
3da
where v = . The complete lineshape is a superposition of the positive and negative branch. The "+N sign is valid for  vo < v, <+2 v0 and the "" sign for
2vo  < v0 .
The shape can be understood when the degree of space degeneracy is considered. The 90 degree case is much more probable because it covers the whole plane compared to one parallel alignment in the 0 degree case.
When the constraint of constant a is relaxed, the lineshape becomes a superposition of Pake doublets, leading to a bell shaped curve. This behavior is attributed to glassy8485 behavior.
A more detailed equation for the frequency shifts has been given86 incorporating the eccentricity r7 = (j2 _ j2) of the hydrogen molecule
v = d (3cos2#1) +3 71sin2 #cos2P#
In most cases the assumption of zero eccentricity holds.
3.3. Relaxation Times
The dipolar Hamiltonian is frequently written in tensorial form
52
HD = Fq)Aq)
q=2
where the F) are random functions of the relative positions of two spins in time and the Aq are spin operators. The products just rewrite the terms A to F from before. Terms A and B are equivalent to q = 0, C and D to q = 1, and E and F to q = 2. The F(' = F q' only contain the spherical coordinates and the A'q= Aq), which are nonhermitian, consist of all spin operator components.
The correlation function g measures the relationship between the random spatial functions at different times averaged over a statistical ensemble
ggq. F (=F (t)F (q')(t + )
Their Fourier transforms are the spectral densities Jq,.(o) which describe the interaction strength as a function of frequency
J. .(w) = gq,.(r)e'"dr
The solution of the equation of motion for the density matrix  in the interaction picture is
dt=f djH;D (t), [H&, (t  )(t)]]
0
where starred quantities indicate the interaction representation, transformed with the Zeeman Hamiltonian. This is just another terminology for the "rotating
53
framem. The observable q in an experiment is averaged over the macroscopic collection of subsystems acted upon by the operator Q
q'(t) = (Q) = tr{a (t)Q}
This is a relationship that describes the dynamics due to the dipolar Hamiltonian decoupled from the fast and trivial dynamics of the Zeeman Hamiltonian.
For the hydrogen system the operator is Q = + /. Taking the trace over the solution of the equation of motion multiplied by Q results in
dq (t) d(lI+ ) + /2
where
af= trfd4H D (tr),[H, (t),(/ 1+/2)] uz()
which defines a' and ao and introduces T,, the longitudinal or spinlattice relaxation time. It describes the time it takes for the spin system to relax back to equilibrium along the external magnetic field. Inserting
H' (t) = e ",'HD(t)eiHt = F A ea"
q
into the system of equations leads to
54
d(11+I/2) j,+/2 _q) q)
dt tr2q )A A ZIZ1 \GZ
by generating the spectral density J from the random spatial functions.
The computation of the commutators leads to an expression for the relaxation time in the high temperature regime by identification of terms
= = rh2(I+1)j( (co) + J(2)(2coo)}
where wo is the Larmor frequency.
A similar calculation leads to the transverse, or spinspin relaxation time. It describes the time dependence of the magnetization amplitude in the xy plane. The initial magnetization is assumed to lie along the xaxis with expectation value
{}' = tr{il. } = tr{eH IxeHza
The time evolution for the xcomponent of the magnetization to the zero value equilibrium is
dt 2qIM A XX]J'
d(1 +,) +
di~~T X JXr4)A4,A4,l+,
The result of the evaluation leads to the transverse relaxation rate
55
= 4h2/(/ + 1){ 2J(O) (0) + J (wo)+ J()(2wo)
The crucial difference between T, and T2 is that T2 has a contribution at zero frequency. For a correlation time r much shorter than the Larmor frequency all spectral densities become independent of frequency and equal to Jk)(Q). It is possible to determine the ratio between the different orders of spectral densities by keeping in mind that
J:JO:j = F(O) :F : F (2) = 6:t4
This implies that T, and T2 are equal in the case of short correlation times with "white" spectral densities for frequencies far above the Larmor frequency.
3.3.1. Spectral Densities
3.3.1.1. Rotational motion
Functional relationships for the spectral densities are needed. They are developed starting from the correlation function
gq(r) = F(q)(t)F (q(t + r)
In the case of rotational motion the average extends over space elements Q and 0O. The average takes place over a function p(Q,t;L0,tO) which defines the probability of taking on the value Q at t and DO at to. This can be alternatively expressed as
p(Q, t;20 ,to) = P(, t; no, to)p(, t)
56
where P is the conditional probability of taking on Q starting out with 00.
The rotation of a molecule can be compared to a sphere of radius b rotating in a medium of viscosity q7 and is described by a diffusion equation
dq1 D
where A, is the Laplacian on the sphere and D. is Stokes' rotational diffusion coefficient
kT
D8
The solution to this equation is the probability P(Q,t;QO,tO) that the two spins have the orientation Q at time t after having the orientation K2 initially. The initial condition of the problem is
T(92, 0) = 5(!  K2)
and the solution is sought for in the form of a spherical harmonics expansion
T(2, t) = I c Ym (0)
J.M
noting that AYm = J(J + 1)Ym. Introducing this into the diffusion equation leads to an equation in the expansion coefficients
dc' D8
=!j  J(J + 1) Cj
57
which describes an exponential relationship with time constant
b 2
DJ(J+1)
An expansion of the delta function can be given in terms of spherical harmonics which determines the initial coefficient and therefore the total time behavior of the expansion coefficient cm(t) = Y(Qo)e"'. This supplies the conditional probability
P(Q,t; )o,to) = YY (QO)Yj (Q)evI JAE
needed to compute the correlation function
gq(t) = F() (Q)F) (o) Yj'* (K)Yjm (Q)e "od%2 oD 4;r ffJM
using the isotropic probability p(20,tO) = . Since the random spatial functions with distance r between the two spins in the molecule are known to be
F(O)()(13cos2P) 1 167Cy(o)(Q r3 7V 5 2
F = sin# cos pe 1 8y (
r rs 15
r3 3 152
58
it implies that the exponential correlation functions are also established. By Fourier transform, the spectral densities are given by
J(O) 24
15r6 1+ )2?2
15r6 1+ )2r2
J(2) (O)_ 16 r
15r6 1+W2r2
3.3.1.2. Translational motion
The motion is again characterized by the diffusion equation. Its solution is the conditional probability for two particles to be a distance r away from each other. The integration of the correlation function involves spherical Bessel functions and has a more complicated closed form. The limits are
J1()) =  r for wr <<1
5 a3
4.n
JI(0)  4.5 for wr>>1 W a "r
where a is the nearest neighbor separation. The functions are proportional to the density n of particles in the material.
59
3.4. Motional Narrowing and Second Moment
As already mentioned, motion is an important ingredient in understanding the properties of hydrogen. In the case of NMR, the width and the shape of the absorption line are affected by this motion. When spins are in motion the field experienced by a neighboring spin is fluctuating on a time scale r,, the correlation time. These fluctuations are averaged out over a longer observation time. This average is much smaller than the instantaneous local field, thus the motional narrowing.
There is an intuitive derivation of the time scales involved. The line width Aw is the inverse of the time t it takes for two spins to dephase by an angle of order unity. It is assumed that the local field difference Aco between two spins stays the same with its sign changing at an average frequency of 1/ r, >> Awo. The dephasing angle after time r. is
&D = k r A W
and the mean square angle of this random walk motion is
A
where n = t / rc. After a unit change of angle the rate is given by
12
 = A o = A o2r
t
60
which means that the narrowing is more distinct for shorter correlation times. The effect on T is directly related to the correlation time as it is the time it takes for spins to dephase by one radian
T2 = = .r
1 1)
It also shows the dependence of T2 on the line width. The value of T2 is a function of motion and can be quantified as a function of the spectral densities. In the case of rapid motion car <<1 all spectral densities are identical while for slow motion wr >> 1 the dominant contribution originates from J(0)(0). This leads to a ratio of (2 x 6+ yx 1+ x 4) / (. x 6) = i which states that T2 is larger, the line width smaller, by a factor of 10/3 for total motional narrowing.
The effect of increased motion on the lineshape and the second moment is investigated in the following. For this purpose the correlation function of the magnetization
G(t) = tr{M,(t)M,}
is calculated. The timedependent magnetization is a result of its equation of motion and leads to
G(t) = E(EsMOE'Os) exp{iwo(t}exp if o(t)dt'
where
0(t') = (EosH00(t')EOs) (E'O S'jH00(t')\E'o s')
61
ao = (EO  E'O)
For random motion the correlation function is averaged over a frequency density f(w) and collecting the prefactor leads to an expression
G(t) = e'"*'f P())e' w"("t do) = e"' Kelf a(t)cr)
The use of the central limit theorem rewrites the average
ef = exp( j(f w(t')dt' )
=exp 2 (t  r)(o(t)(t  r))dr and finally leads to
G(t) = e'"0' exp W 2)f (t  r)g, ( )dzr
where g is the reduced correlation function with the property g,(0) =1. With this result the two limits of short and long correlation times can be investigated.
In the long correlation time limit (0o2),r2 >> 1 the reduced correlation function becomes unity and the integration yields
G(t) = e'" exp{j( 2)t2
which is Gaussian and the rigid lattice form. Its Fourier transform is the lineshape and also a Gaussian.
62
In the short correlation time limit ( 2 <<1 it is permissible to neglect r compared to t. This leads to an exponential decay
G(t) = e"0' exp{Kw2)trc }
whose Fourier transform indicates a Lorentzian lineshape.
The conclusion of this analysis is that the lineshape, in addition to the line width, is an indicator of the dynamic regime the system is in, whether motional narrowing takes place or not.
A discussion of the moments of the lineshape follows naturally. The nth moment is defined as
M= f(wwo)"f(o)do
where f(w) is the normalized lineshape function which leads to vanishing odd moments if it is symmetric around the Larmor frequency co. The previously introduced (a)2) is the second moment. Its form can be Gaussian or Lorentzian with different results for the moments.
For a Gaussian
f~)=exp 0)
the even moments are M2n = (2n  1)Y". The half width at half height (HWHH) is defined by
f(w0 + J) = f(0)
63
and leads to a value of 35= AV2i = 1.18A.
In the case of a Lorentzian
J 1
f(x) =~ 1
S32 +() ))2
.where 3 is the HWHH, the moments are diverging if they are taken over all frequencies. A cutoff must be introduced, limiting the frequency to an interval cO CoW01 a which, of course, must be much larger than HWHH. The moments,
3
neglecting terms , are then found to be
a
M2 =A2M4 2 a r 37r
The large ratio of M4 =ra indicates that this functional dependence should
(M2)2 63
only be tried under this condition. In the Gaussian case the ratio is much smaller with a value of 3.
3.5. Heterogeneous Spin System Relaxation
If spins experience different environments their relaxation times are also likely to be different. In the case of zeolite there are at least two distinguishable environments: the surface and the center of the cavity. A group of spins shall be called a spin system. The following discussion is restricted to two different systems a and b.
For the moment it shall be assumed that the spin systems, experiencing two different environments, are independent of each other. Each spin system
64
relaxes therefore independently and has a welldefined relaxation time or, expressed differently, inherent relaxation rate R, and R,. There is also a distinct fraction of magnetization p, and Pb associated with each system.
At this point the earlier assumption of independence between the spin systems is dropped. The consequence is that the spin systems interact with each other undergoing exchange processes. The exchange process can be a transfer of magnetization which happens by spin diffusion via spin flipflop, or particle diffusion. This interaction between spin systems is a more realistic assumption for physical systems. The outcome for this model is that the measured relaxation rates Z and A and measured magnetization fractions C+ and C do not correspond to the physical processes one is interested in. In addition to the previous relaxation rates, there are relaxation rates k, and kb between the spin systems which are not measured directly. They can only be extracted as a complex byproduct to the overall relaxation. A calculation87,8 will reveal the involved quantities.
For simplicity of the following equations, a reduced magnetization is introduced for the longitudinal relaxation (saturation recovery)
MI  MO  M,(t)
3MO
and the transverse relaxation
3M M(t)
SM, = MY
MO
The equations of motion for the magnetizations are Bloch equations extended by the additional cross relaxation between spin reservoirs
65
d(SM) R3Mk5M+kSM
dt
d(Mb)_ RM,  kSMb+ kSM.
dt
where the above definitions are employed. These coupled differential equations are solved for the magnetizations in terms of the measured or apparent fractions and relaxation rates
SM, = Ce +CeAt
with complicated expressions for
A* =(Ra +Rb+ka+kb i([RaR+kakb]2+4kk)2
C~ PaCa~+PbCb
with
C, = SM J (sM,0  SMO +
where subscript 0 indicates the value immediately after the pulse. Another important piece of information is the equation of detailed balance
Paka = Pbkb
66
where p, and Pb are the spin fractions of each group. By these means the apparent relaxation rates become dependent on the magnetization fractions of each group by substituting for one of the ks.
For known p, and Pb, which can be assumed for the case of zeolite from the fraction of molecules in contact with the surface, the real, inherent relaxation rates can be determined. This is achieved by inverting the A' and A: equations leading to
R+ Pbp
R. p,  p,
Rb, (,r  LYRa k.a kb
which can be plotted versus ka. Knowing the functional form of Ra and Rb, the functional form of the apparent rate A can also be generated. The ka for which the calculated value of A corresponds to the experimental value of Z is the exchange rate between the two spin systems. The values of Ra and R, can then be determined from the known k. value. The remaining difficulty is to reliably deconvolute two exponential decays. This is even a problem for sophisticated computer routines because of possible local minima in parameter space. As a consequence, initial guesses which are too far away from the true solution lead to a nonconvergent fit.
3.6. Methods of Detection
3.6.1. Continuous Wave (cw) Method
In continuous wave mode the system is in a steady state and the spin system therefore in equilibrium with the lattice.
67
3.6.1.1. Qmeterdetection
The Qmeter method relies on a change in impedance as the resonance is approached. It is therefore necessary to study the impedance of a LC circuit. At first, the induction must be determined. The flux cD through a coil which is induced by a circulating current I is defined as
(D = BA = (H + 4mM)A = Re{L(1+ 41ri7Z)Ie'"}
where rl is the filling factor, indicating the ratio between coil volume and sample volume. The magnetization is related to the field via the complex susceptibility M = XH. This functional form rescales the inductance to an effective value incorporating the effect of the sample magnetization to
Leff = L(1+4 7MX)
The voltage generated from a constant current source is proportional to the impedance Z of a parallel circuit consisting of a samplefilled inductor with a series resistance r and a capacitor C
1 =1 + 1
Z Lr+iL(1+4711X) (ioC)J
It is useful to introduce the quality factor 0 as the ratio of energy stored and the dissipated power in the coil
P_ W _ w LI2 _ cL
p ;r12 r
68
If the circuit is tuned to the resonance condition, w2LC = 1, the impedance Zcan
be rewritten in terms of a parallel shunt resistance R (coL)2
r
Z=R~ 1 +i4 iQi7X
_1+ 4=7 ilQ1
cL R
and the quality factor is expressed as Q   . This can be simplified by r o)L
assuming Q>> 1 which is a reasonable assumption for real circuits with Q ~100
Z = R[1  i4 nirQ ] = R[1  i4 r7QX'4 =7 QX"]
The relative change in Zis AZ = 4,rRQnz" and is proportional to the change in the detected signal. The dispersion is not measured with this method which is a drawback in itself. It does, however, have the advantage of avoiding the mixing of dispersion and absorption signals. Another problem with the arrangement is that the small signal voltage is superimposed on top of the large carrier voltage.
3.6.1.2. Bridge method
The last mentioned problem is eliminated with a bridge arrangement. The carrier voltage VO is partially compensated by a bridge or "Tee" so that the detection amplifier does not saturate
V = VO(1 i4 irQi7%) V,
It is preferable to keep
69
V4 V = aVe > SV
in order to generate a dispersion or absorption signal by having the correct phase on the compensating voltage. This leads to an equation for the voltage V
V = aVoe* 1 (x"cos D+ X'sinD)  i a(X'cos + X"sin )
This way, an angle D =0 leads to the Qmeterlike absorption signal but an angle 0 = I reverses the signal into pure dispersion. The problem with this
2
arrangement in practice is the adjustment of the phase angle and the constancy over time.
The appropriate apparatus is a hybrid Tee which has an inverting and a normal input power branch. One branch is connected to a 50 Ohm load, the other is the sample path and is matched to 50 Ohm with an impedance matching device. Scanning through the resonance with a sweeping magnetic field achieves measurable detuning proportional to the response of the sample.
Both detection schemes are supplemented with lockin amplification because of signaltonoise (S/N) problems. The lockin acts as a narrow band filter to improve the S/N. To achieve this, it modulates the swept magnetic field Ho with an audio frequency ! and an amplitude much smaller than the line width to avoid distortion. This modulation leads to a differentiated signal, oscillating at the modulation frequency. This signal is then mixed, i.e. multiplied by, a sinusoid at the same frequency and averaged over time T
T
S(T) f f[So sin(K t) + n(t)l sin(Q2t)dt
0
70
where n(t) is the noise term, taken to be the Nyquist noise at temperature e, which is only meaningful as its mean square value
1 sinK2T 1 kER S(T)='SOT 1si1+ T4kR
2 L Q T 2
With the approximation of long average times (QT >>1) the signal to noise ratio becomes
S ST _ 2
N )T4kR 4keR
implying that the band width of the lockin is Av = 2 / T which improves the S/N by longer integration times at the expense of reduced resolution.
3.6.2. Transient Method
3.6.2.1. Coherent pulses
The coherent pulse method opens up the opportunity of exploring line shapes much narrower than those limited by field inhomogeneities. Many pulse sequences are in use. Motionally narrowed line shapes are of special interest in this context.
The Hamiltonian for the system in the rotating frame, during the application of a pulse along the xdirection, is given by
H = y{IzHo  +IH
71
where the H, field is much larger than the field in zdirection. This justifies the approximation H = yH, for the short time of the pulse. Applying H, for a time t means rotating the magnetic moment by an angle 9 = yHlt described by the rotation operator
R(9) = e1Ht _ \;H,1,l = eig,
Due to the strength of the applied rf field, H, ideally turns the spins in a negligible time. The pulse sequence described in the following is the original 90,  r 180,  r sequence proposed by Hahn89 and is graphically shown in figure 3.
The steady field HO leads to an excess orientation of spins in zdirection (Figure 3a). After the steady state has been reached, the H,field is switched on for a short time in order to tilt the magnetization by 90 degrees (Figure 3b) from MZ into My:
MZ( = = e~*IxMz(0)e+if/ =M
2 yH,
The same coil producing H, picks up the voltage induced by the magnetization rotating in the xy plane. Without relaxation effects, as expressed by T and T2, the alternating induced voltage would persist forever. In reality T2 plays a role but the T, effect is negligible, considering T, >> T2 for a solid. This gives rise to dephasing which destroys the ideally persisting voltage by phase cancellation (Figure 3c). A free induction decay (FID) results from the superposition of all individual spin signals at slightly different fields. The signal decreases exponentially in the case of a Lorentzian line with a time constant T*. This T2* is
M,
y
) :0w
y
(e) t2r
w/2Pulse
Free induction signal
(a) (b)
M, y
x
(b) t0*
y
x
(d) tr
wpulse
Echo
'I'
'II
'3' I,'
(c) (d)
Figure 3. Spin echo formation'
72
0
2
(C)
73
a sum of all effects, especially the inhomogeneity of H. It causes some spins to advance, others to lag behind. But this is not what really determines the physics. The goal is to eliminate this unphysical influence by applying a 180 degree pulse
p{( r) = e'' (p,) ei'"', = p,
The 180 degree pulse produces a mirror image: the slowest spins are ahead of the faster spins (Figure 3d). The ;pulse acts like a time reversal operator. After the same time r between the ir/2 and the ;rpulse the fastest spins have caught up with the slowest. Since the inhomogeneity of the steady magnetic field has not changed during the time 2r the almost same FID occurs after this time (Figure 3e). The difference is that the voltage has the opposite sign of the rotated magnetization and a decreased amplitude. For the 90,  r 180,  r sequence the echo, called "solid" echo, has the same sign as the original FID. The reduction in signal amplitude is caused by effects that give insight into the microscopic behavior. Due to collisions during the two time intervals r only a diminished spin echo occurs, e.g. as a result of diffusion effects.
T, effects reduce the measured T2,, according to
1 1 1
For T, >> T2, as is the case of solids, this is not a crucial difference.
It is important to notice that this pulse sequence is only able to refocus spin and not dipolar components. This means that the echo amplitude, corrected by the spinspin relaxation time, is a measure of the liquidlike fraction of the sample.
CHAPTER 4
EXPERIMENTAL SETUP
4.1. NMR Setup
Many components of the NMR spectrometer are used for pulse as well as continuous wave mode. With the necessary expertise it is therefore possible to switch between the two modes within minutes. This makes the existing apparatus versatile for the different information attainable by NMR.
4.1.1. Pulse Apparatus
A diagram with an overview of the complete pulse NMR apparatus is shown in figure 4. The system will be discussed in detail, closely following the diagram.
The radio frequency (rf) path originates at the rfgenerator which supplies the ultrahigh frequency (UHF) of 268 MHz continuously. This frequency corresponds to a magnetic field HO of 6.3 T for hydrogen according to the Larmor relationship. This magnetic field is in the upper attainable region of the available magnet and is chosen for high resolution and sensitivity, i.e. good signal to noise ratio (S/N), obtainable at high frequencies. The S/N is a function of the magnetic field to the 3/2 power. It therefore pays to work in a high frequency regime as long as no other effects, such as the skin depth are counteracting. Another argument is the short deadtime which is proportional to the reciprocal of the frequency. A continuous rf signal is not desired for the pulse mode. For this reason the pulse generator produces rectangular signals of
74
75
Crossed Diodes
Signal Averager
1o oo
Mix
RFGenerator
Pulse Generator Gates
300
Phase Shifter
]0
Dup lexery
Magne
er 250
Figure 4. Pulse NMR diagram
76
variable length T and delay time r between them. Sequences can be programmed from the laboratory computer or selected manually. Four outlets connect four different gates which allow for a phase shift of 0, 90, 180 and 270 degrees. At the gates the rf frequency is blocked and passes only for the time adjusted by the pulse length T. To reduce noise from the rf generator pulse or leakage through the gates in the offstate, crossed diodes are placed after the gates. The diodes do not conduct for voltages less than 0.5 V which is above the noise level. For higher voltages they have no impact. A source of frequencies other than the generated 268 MHz is the rectangular shape of the pulse. A Fourier analysis shows that higher odd multiples of this frequency are present and useful power is lost. This contribution is also diminished by the crossed diodes.
Pulse NMR in solidstate physics requires a large amplitude of rf which is associated with a short pulse. The values in the present apparatus are 15 G for the H, field which is equivalent to a pulse duration of less than 20 A sec. In a
2
negligible time the spins have to be tipped over by the desired angle to achieve a quasiinstantaneous situation where time effects are negligible. Another reason for the short pulse is the need for wide coverage of the whole frequency spectrum. These conditions are met by the two high power amplifiers of 40 and 300 Watts.
A quarterwavelength line with attached crossed diodes is mounted in parallel. This device serves as noise reduction when the pulses are not turned on. For an understanding of this effect, transmission line theory90 is necessary. There are special characteristics for A./2 and A./4 lines. In the case of a A /2 line the impedance on one end is simply transformed to the other end, except for the additional resistance of the used cable. For a A/4 line the input and output
77
impedances Z,, Z, and the characteristic cable impedance  Z = 50 Q in the standard coaxial case  are related to each other by ZZ = Z2. These special cases are a consequence of the general formula for transmission lines
Z, Z,+iZtankl
Z Z+iZ0tankl
where k = is the wave number. For high voltages (above 0.5 V) the diodes conduct and represent a low impedance. For the low impedance at the end of the crossed diodes the resistance at the other end of the A/4 line is very high. This implies that the high voltage level of the frequency corresponding to the wavelength of the spectrometer is not affected. Other frequencies do not match and are dampened proportional to their frequency deviation. For voltage levels below 0.5 V the diodes do not conduct and the net effect is an almost zero resistance at the carrier frequency. This mechanism decreases this frequency noise component but is less efficient at other frequencies.
The centerpiece of the pulse spectrometer is the duplexer. Its task is to connect the transmitter, the sample and the receiver, but to simultaneously establish a separation of the transmitter from the receiver. This requirement becomes clear when the power of the transmitter (300 W) is compared to the 0.1 mW power of the sample signal: the highly sensitive receiver would be destroyed by the transmitter power. The duplexer consists of two impedance matching circuits. One transforms the cable impedance of 50 Q to the 3 92 of the series NMR resonance circuit. The other circuit matches the 3 0 to the 400 Q of the UHF amplifier with an amplification factor of approximately 1000.
The lowlevel amplifier is a narrowband, 3 stage solid state amplifier and is another key component of the spectrometer. A fourth stage is designed to
78
provide variable matching to the input impedance of the incoming signal. The amplifier gain must be high but also robust against selfoscillation. The narrow bandpass is achieved with a resonance circuit for each stage consisting of a fixed coil and a tunable capacitor. To reduce readiness for selfoscillation resistors were soldered across the coils, at the expense of increased band width and reduced gain. The used tunable capacitors were of high mechanical and electrical quality in order to reduce mechanical vibrations and allow for precise fine tuning. The DC power supply to the doublegate transistors (3N211, 3N213) was decoupled from the rf path by large inductors. The individual compartments were also carefully separated from each other to prevent feedback between amplification stages. The doublegate transistors were carefully biased in order to operate in a region of high forward admittance. This is the case for about 2 V between the source and gate 2 and zero voltage between the source and gate 1, which is the signal carrying input. Exact tuning of all resonance circuits is the key to high and stable amplification levels. The entire amplifier was placed into a brass box which was temperature regulated. This seemed necessary due to the temperature dependence of the transistors and the variable temperatures in the laboratory. A temperature regulator with a temperature probe was purchased. A heater was dimensioned to deliver enough power to keep the temperature at 32 centigrades. This value was chosen because it was above any reasonable room temperature.
The receiver is protected by two sets of crossed diodes inside the duplexer which shunt the high level transmitter power and limit the input voltage into the amplifier. This could also have been achieved with an additional A/4 line in the main line.91 Its function is described as follows. For the high transmission levels the diodes would turn on and the upper point of the A/4 line would represent a very high resistance from the transformation of the short
79
circuit at its end. This would decouple the receiver stage. The small sample signal would only see a negligible resistance and could be processed further on its way to detection. Matching problems prevented this method from being employed.
The NMR circuit is tuned to resonance (w2LC =1) at 268 MHz, where the series circuit shows a minimum: IZ12 = r2 + (1 2LC / wC). This is a desirable property of the series circuit because the small resistance of r = 3 Q only requires a few volts of applied power. An alternative parallel circuit would require several kV for the same power, which would lead to breakdown along the 1.5 m long coaxial cable to the coil. The sample coil must be dimensioned appropriately in conjunction with the tunable capacitor for the applied frequency. The inductance for small coils is determined by the formula
L= N2a2
9a+10b
where 2a is the diameter of the winding, N the winding number and b the length of the coil. As long as b > 0.8a this is a good estimate. The inductance is chosen relatively large to obtain a large quality factor. This is restricted by the corresponding value of the capacitor which should not be below a few pF. The present values are 2.6 pF and 150 nH. The quality factor is about 100. The used coaxial cable is important because it should not introduce excessive losses. A homemade cable was a combination of coppernickel tubing and flexible braid in the lower part of the cryostat. Along the cold finger a commercially available, high quality, rigid coaxial cable was used in order to minimize spurious strain effects. The pulse system is adjusted to the coil. The
80
inductance is matched with the input capacitance of the duplexer which can also be influenced by the coax cable length.
The probe signal is processed in a phase detector after amplification to eliminate the carrier frequency. Preamplification before transformation to low frequencies yields an S/N advantage, because 1/f noise effects are much smaller at high frequencies. The phase detector or mixer receives its reference signal from the rfgenerator and passes only signals which are modulated with this frequency and higher harmonics which are, however, not relevant. By these means the S/N ratio is increased. A phase shifter allows for changing the reference phase. The possibility of detecting absorption and dispersion signals arises.
Another amplification by 250 and a signal averager follow. Signalaveraging improves the signal to noise ratio by repeating the experimental procedure and adding up the digitized amplitudes of the signal. The signal improvement increases as the square root of the number of repetitions.
The broader the NMR line, the broader the bandwidth of the amplifiers must be in order to avoid omitting parts of the frequency spectrum. A larger bandwidth deteriorates the noise and gain characteristics.
The elaborate system at high frequency with the homogeneous magnet and the noise reducing schemes allows for detection of 1016 spins at 1 K.
Before cooldown the NMR system needs to be checked for pulse propagation to the sample coil. An auxiliary coil is placed around the NMR coil and the signal, as a response of the applied pulse, is induced in the pickup coil and detected. The connected oscilloscope displays a voltage of 40 V peak to peak via the weak coupling. This value represented a standard for good operating conditions.
81
4.1.2. Continuous Wave (cw) Apparatus
Many components of the pulse apparatus are also used for cw NMR. For this reason, the modifications and differences compared to the pulse system are described first. The setup is then explained following the complete cw block diagram displayed in figure 5.
The entire pulsegenerating equipment including the high power amplifiers, gates and pulse generator is bypassed. Instead of employing a duplexer there are two different methods of how continuous wave data are obtained. Both principles, the 0meter and the hybrid T have already been discussed in principle. The hybrid T supplies the receiver only with the sample response while the Qmeter is also subject to the input power.
The hybrid T is an analogue93 of the magic T at high frequencies. It works like a bridge and operates with a dummy load, which does, however, waste half of the transmitted power. The rf generator input is split in half and reaches the dummy load and the NMR circuit, which are both matched to 50 Q. As long as the sample is offresonance the hybrid T is matched so that the two paths cancel each other, because the sample branch is out of phase by 180 degrees. The matching is spoiled at resonance and only the NMR change is detectable by the receiver. This is the ideal case; in reality a small deliberate deviation from ideal phase or amplitude matching is adjusted as a signal carrier. The small error picks out the component of the NMR signal in the direction displaying a dispersion or absorption signal, respectively.94.95 The advantage is an amplification effect and a deliberate choice of signal type instead of a mixture.
Another major difference to the pulse apparatus is the modulated sweeping field. The steady field HO can be adjusted by a sweep coil in order to ramp the magnetic field through resonance. This is more practical than
82
Phase Shifter
H
Signal Averager
Power Splitter Hybrid T
50
Circuit Magnet
LIS
LockIn
Ramp
A1000*
LMixer25
weep Coil
Figure 5. Continuous wave (cw) diagram
RFGenerator
83
sweeping the frequency through resonance because all spectrometer components are carefully frequencytuned. The change in the static magnetic field in zdirection is not the only modification. In addition, the auxiliary sweeping field is sinusoidally modulated by 79.3 Hz. This frequency is deliberately chosen as an odd value, such that no other frequency matches it accidentally. The modulation changes the nature of the sample response. Instead of the absorption signal itself, its first derivative is obtained because the sine wave is projected onto the absorption lineshape. Thus, the absorption lineshape is probed yielding the strongest signal for the steepest slopes and therefore monitoring the first derivative of the line. Another change is the desired shift of the measurement to a higher frequency which reduces 1/f noise and allows for using a lockin amplifier. The lockin has the effect of narrowing the bandwidth and improving the signal to noise ratio sharply. Experimentally, the lockin parameters have to be chosen carefully. The time constant is a way to smoothing the recorded signal by integration. It is important, however, not to disguise narrow features with too generous time constants. The lockin is phasesensitive and a phase offset by 90 degrees totally eliminates any existing signal. It is clear that the right phase choice is important for maximizing the signal.
The complete cw system can now be understood easily. Again the rf path originates at the frequency generator. The power is split into the signal and reference branch. The signal branch feeds  via an adjustable attenuator to control the power input  the Qmeter or the hybrid T which leads to the detection of the sample response. The weak signal is amplified in the lowlevel amplifier already described in the pulse mode section. The specialty of the input stage of this amplifier in Qmeter operation is that it performs two tasks at once: it matches the impedance and acts as the resonance capacitor of the tank circuit.
84
This double task, which is performed by the appropriate tapping off the resonance coil of the first stage, is critical to the observed signal. In hybrid T operation the impedance matching is already provided by two capacitors, one to ground and the other to feed rf. This exterior impedance matching has also been successfully applied for the Qmeter alternative. The signal is mixed with the phase adjustable reference signal at the phase detector or mixer and then fed into another amplifier. The final stages are the lockin and the chart recorder.
4.2. Dilution Refrigerator
In order to cool down to temperatures below 4.2 K a dilution refrigerator is used. Its components and the cooling process are described.
A dilution refrigerator is operated with a 3He/4He mixture circulating in a closed gas handling system. The 4He essentially represents an inert background while 3He is cycling through the refrigerator. Two pumps drive the gas flow and determine the cooling power with their pumping speed. The gaseous 3He at room temperature returns from the pumping system, enters the cryostat and condenses at a small bath of liquid helium held at 1 K, the "1K pot". Thermal contact is established by passing the gas through a cupronickel tube which is wound around and then immersed in the bath.
The pressure drop of liquid 3He from the condensation condition (100 Torr) to values corresponding to the still temperature (3 Torr) makes it necessary to introduce a flow limiting impedance. The pressure drop for a circulation rate N is determined by Poiseuille's equation
Ap = rZNV3
85
where 7 is the viscosity, Z the flow limiting impedance, depending on tube parameters / and d, and V. the molar volume of 'He. Without the impedance, the 3He is in danger of reevaporating which gives rise to a subsequent heat load by recondensation and turbulences. On its way to the mixing chamber the 3He must be precooled because the dilution cooling mechanism only starts at temperatures below the tricritical point of 0.86 K. Below this temperature the 3He/4He mixture spontaneously separates into a 3He rich phase floating on top of a 4He rich phase due to the mass difference. The 3He tube is therefore led through the still and various heat exchangers for precooling purposes. Especially the heat exchangers represent an essential means of precooling and increase the efficiency of the refrigerator. Two different kinds are in use: continuous and discrete heat exchangers.9 The continuous kind consists of an outer tube which guides the inner warmer tube. Instead of truly discrete heat exchangers with single copper blocks, "quasicontinuous" heat exchangers are employed. The heat transfer is ensured by a large surface area of sintered silver. Despite the large contact area the flow impedance is not excessive. The simple continuous counterflow heat exchangers are limited by the Kapitza resistance and viscous heating. The Kapitza resistance is due to the acoustic mismatch between the liquids and the separating metal tube
A T
Q
where A is the effective surface area, AT the temperature gradient across the interface and 0 the thermal flux. As a consequence the simple heat exchangers fail at temperatures below 100 mK and the more sophisticated designs are needed for lower temperatures. More efficient heat exchangers allow for higher
86
circulation rates without increasing the temperatures7 When the 3He leaves the last exchanger it is colder than the tricritical temperature, once the refrigerator is working, and fills up the concentrated 3He phase. At this point the dilution cooling process sets in as 3He quasievaporates from the concentrated phase into the 4He phase, or dilute 3He phase. The cooling power can be quantified
Uc = ka[HD(T) Hc(TM)]
where T. is the temperature of the mixing chamber and H the enthalpy of dilute and concentrated 3He. The available cooling power Om at the mixer is equivalent to the external heat leak in equilibrium and is reduced by the heat load OL of the incoming, warmer 3He which has the heat loaded temperature T
OM = QCQL =3[H3,o(T) H3c(Tm) Hc(TL)+ Hc(TM)] = F3[H3(TM) H3C(TL)]
The calculation is simplified by the assumptions that only 3He is circulated and no frictional heating arises. It turns out to be important for the performance of the refrigerator to establish a high circulation rate. For further results the involved enthalpies are needed. The dilute phase can be interpreted as an ideal Fermi gas with the specific heat
C30 = 1xR T
2 TF
where TF = 0.38 K at xD = 0.064, the maximum 'He concentration in the 4He background due to the Pauli principle. For the concentrated phase
87
C3C =24T i
mole K
which is found experimentally. The chemical potential is constant between the concentrated and dilute phase (puc = L3D) at the coexistence curve
H3C TS3C =H3D TS3D
The entropy is calculated to be
=fC=1dT
0
and the enthalpy
H3c =H3(0)+fC3dT=H3(0)+12T mole K
Using this information leads to a numerical result for the cooling power
QM = N3(95T,2 12T2)
Two limits can be investigated. In the case of a single shot experiment or the use of ideal heat exchangers, i.e. when T = T., the cooling power is maximal and determined by
6.=3(83T,2) Watt mole K2
88
In the other limit the cooling power approaches zero if L= 2.83. This again TM
stresses the need for efficient heat exchangers.
The mixing chamber, where the cooling power is generated, is thermally linked to the sample cell (Figure 6) by a cold finger consisting of annealed copper bars. The heat contact between the mixing chamber and the cold finger is established by a cake of sintered silver. The cold sintering is performed at 200 atm using a hydraulic press and a specifically machined piston to exert even force on the silver powder. Before sintering, the copper surface must be cleaned with striking solution, applying voltage in reversed polarity, hence reducing any disturbing cations. The so prepared surface is then galvanized with a solution of silver salt as a base for the sintering.
After crossing the phase boundary inside the mixer, the 'He atoms return to the still through the heat exchangers, driven by an osmotic pressure gradient. The temperature in the still must be high enough to establish a sufficient 'He vapor pressure in order to maintain a certain circulation rate through the system. External power is supplied to the still
6S = Na[Ha(vap) H3(cond)]
This implies that the dilution refrigerator provides more cooling power when the still is moderately heated! Almost exclusively 3He gas leaves the still because the vapor pressure of 'He is only about 5% of the value for 3He at these temperatures. Impurities such as air, hydrogen and oil derivates are separated from the 3He by passing the gas through a 2stage liquid nitrogen trap and then pumping 3He back into the system. By circulating 'He, the process is maintained continuously. Without large heat leaks, temperatures of 50 mK can
89
From Mixing Chamber
Pressure Gauge
Sample Line
Membrane
N00QV
Sample
 I
RFCa il
 Plastic
Figure 6. NMR cell
Copper
Bars
X
90
be sustained over a long period of time by refilling the 4 K He bath. The 1 K bath is filled continuously by an inlet through a suitable impedance from the 4 K bath.
In a dilution process the number of 'He is independent of temperature but the cooling power decreases with T2. The lower the mixer temperature, the less cooling power is available. This is consistent with a version of the 3rd law of thermodynamics, stating that absolute zero temperature cannot be attained.
In an evaporation process the number of molecules removed from the vapor is proportional to the vapor pressure which decreases exponentially. The cooling capacity per molecule, however, remains approximately constant. For lower temperatures the dilution process is more efficient because T2 > Cexp(1/T) which makes the dilution refrigerator an efficient way of cooling to mK.
The presence of impurities poses a problem to the small diameters of the impedances in the closed cycle of the refrigerator . If the system has a leak to air the gas path for helium will start plugging up at nitrogen temperatures. In the present system several leaks occurred during the course of operation. In the external gas handling system three different leaks were discovered and fixed. Another leak at a quasicontinuos heat exchanger inside the vacuum can opened up. Such a leak can cause losing valuable 3He and prevents from obtaining low temperatures because the 3He acts as exchange gas to the surrounding 4.2 K helium bath. Also the shaft seals of the mechanical booster and the sealed oil pump had to be exchanged. Another problem are pump oil derivates which can also block the gas pipes. With the problem of contaminated mixture gas was dealt by cycling the gas through a trap immersed in liquid helium. This is the most efficient way of cleaning helium gas from impurities. Another way to prevent clogging in the pipes was to run the refrigerator with a small amount of helium gas upon initial cooldown from room temperature. As a
91
consequence, the mixture and any impurities are in motion and cycled through the nitrogen trap so that the probability of blocking is minimized.
4.3. Substrates
4.3.1. Zeolite
The name "zeolite' is Greek and means 'boiling stone'. It is a consequence of its property to boil and melt to a glass as it is heated in a flame. Before reaching high temperatures, above 500 centigrades, the zeolite releases water but does not disintegrate. This is in contrast to most other waterbearing crystals and results from the porous structure of zeolite. There are other names associated with zeolites alluding to their unique properties: Molecular Sieve, Solid Solvent or Ion Exchanger.
There exists a great variety98 of naturally occurring as well as synthetic zeolites. All of them are defined "as alum inosilicates with a frame work structure enclosing cavities occupied by large ions and water molecules, both of which have considerable freedom of movement, permitting ion exchange and reversible dehydration.99 They are grouped into Sodalites (Faujasites), Chabazites, Philipsites, Analcimes and Mordenites.100 The most commonly used zeolites are the synthetically prepared and commercially available (Union Carbide Corporation, Box 372, South Plainfield, NJ 070880) zeolites X and A of the sodalitefaujasite group. Only their structure will be discussed.
The structure01,172 of zeolite is complicated but completely regular and resembles a "junglegym.' Figure 7 provides a simplified view of the structure of zeolite A.103 The fundamental building block of any zeolite is a tetrahedron of four oxygen ions with a centered aluminium or silicon ion. Each oxygen carries two negative charges, one of them being compensated by Si. The remaining single charge on the oxygen enables it to combine with other Si ions and to

Full Text 
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NMR STUDIES OF MOLECULAR HYDROGEN CONFINED TO THE PORES OF ZEOLITE By MARKUS RALL A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1991
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ACKNOWLEDGEMENTS It is my special pleasure to thank my advisor Prof. N. S. Sullivan for the opportunity to do this doctoral research with him. Without his support and encouragement I would not have undertaken this project in the United States. His comprehensive understanding of physics as well as his personal interaction with me made this a very productive time. In addition to being chairman of the Physics Department, he was involved in such projects as the Microkelvin facility and the National High Magnetic Field Laboratory. This could, however, not influence his concept of trusting my capabilities, which was a motivation for me to work independently, combined with his readiness to "talk physics" and help me out whenever there was a need for it. His attitude towards education has set an example for me. The French postdoctoral associate J. P. Brison who spent a year at the University of Florida was also a wonderful friend to me. He worked on his own experiment and assisted me in my project. I enjoyed the long and frequent discussions with him and admired his attitude and intelligence. I also thank M. D. Evans, an incoming student in my last year, for the new spirit he brought to the lab and D. Rubury, who was an undergraduate working for me in the summer and fall of 1990. My thanks go to Profs. R. Andrew, C. F. Hooper, D. B. Tanner, E. D. Adams, J. Klauder and J. H. Simmons for their interest and support while serving on my supervisory committee. I would also like to acknowledge the many interesting discussions with Prof. Y. Takano. Special thanks go to all the support groups that were important for the success of my work. In this context I want to name the machine shop under B. Fowler which produced high quality parts, the electronics shop with L. W. Phelps ii
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and J. Legg with their friendship and expertise, and the cryogenic group with A. Hingerty and G. J. Labbe for supplying helium and engineering advice. I also want to thank our staff K. Yocum and C. Knudsen. I further express my gratitude to everybody who made my stay at the University of Florida such a productive and agreeable time. The friendly atmosphere among the students and between students and faculty always made work more pleasant. This research has been supported by the National Science Foundation through a Low Temperature Physics Grant DMR8913999, by NATO grant 88/709 and by the Division of Sponsored Research at the University of Florida. iii
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TABLE OF CONTENTS ACKNOWLEDGEMENTS Â» LIST OF FIGURES vi ABSTRACT viii CHAPTERS 1. INTRODUCTION 1 1.1. Quantum Nature of Hydrogen 1 1.1.1. Orthoand Parahydrogen 2 1.1.1.1. Quadrupolar interaction 3 1.1.2. ZeroPoint Motion 5 1.1.2.1. De Boer parameter 8 1.1.3. Crystalline Potential and Quantum Rotor 9 1 .2. Open Questions for Confinement 1 1 2 THEORY 13 2.1. Surface Interactions 13 2.1.1. Theory of SurfaceMolecule Interaction 14 2.1 .2. Effects of Adsorption 19 2.2. The LiquidSolid Transition 20 2.2 1. Elastic Instability Theory 21 2.2.2. Dislocation Theory 21 2.2.3. TwoDimensional Melting Models 23 2.2.4. Molecular Dynamics Studies 24 2.2.5. Implications for the Investigated System 25 2.3. Superfluid 26 2.4. Supercooling 29 2.4.1. Nucleation Theory 29 2.4.1.1. Implications for experiment 33 2.4.2. Grain Boundaries 35 2.5. OrthoPara Conversion in Molecular Hydrogen 38 2.5.1. Diffusion 43 3. NUCLEAR MAGNETIC RESONANCE: NMR 45 3.1. Dipolar Hamiltonian 45 3.2. Continuous Wave Lineshape 47 3.3. Relaxation Times 51 3.3.1. Spectral Densities 55 3.3.1.1. Rotational Motion 55 3.3.1.2. Translational Motion 58 3.4. Motional Narrowing and Second Moment 59 3.5. Heterogeneous Spin System Relaxation 63 iv
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3.6. Methods of Detection 66 3.6.1. Continuous Wave (cw) Method 66 3.6.1.1. Qmeterdetection 67 3.6.1.2. Bridge method 68 3.6.2 Transient Method 70 3.6.2.1. Coherent Pulses 70 4. EXPERIMENTAL SETUP 74 4.1. NMR Setup 74 4.1.1. Pulse Apparatus 74 4.1.2. Continuous Wave (cw) Apparatus 81 4.2. Dilution Refrigerator 84 4.3. Substrates 91 4.3.1. Zeolite 91 4.3.2. Vycor 95 4.3.3. Exfoliated Graphite 96 5. EXPERIMENTAL DATA 97 5.1. Sample Preparation 97 5.2. Pulse Work 99 5.2.1. Transverse Relaxation Time 99 5.2.1.1. Temperature dependence 99 5.2.1.2. Orthopara dependence 104 5.2.1.3. Hysteresis 106 5.2.2. Spin Echo Amplitude 106 5.2.3. Longitudinal Relaxation Time 112 5.3. Continuous Wave Work 116 5.3.1. OrthoPara Conversion 116 5.3.1.1. Impurities 124 5.3.2. Lineshape 125 5.3.3. Time Dependence 129 5.3.3.1 . Second moment 129 5.3.4. Temperature Dependence 129 5.4. HydrogenDeuteride 138 5.5. Isochoric Pressure versus Temperature Data 141 5.6. Zeolite 5A versus 13X 145 6. COMPUTER SIMULATION 149 7. CONCLUSION 161 BIBLIOGRAPHY 170 BIOGRAPHICAL SKETCH 176 v
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LIST OF FIGURES Figure Ease 1 . Nucleation Theory 31 2. Pake doublet with order parameter o = 1 50 3. Spin echo formation 72 4. Pulse NMR diagram 75 5. Continuous wave (cw) diagram 82 6. NMR cell 89 7. Structure of zeolite A 92 8. Two measured transverse relaxation times 100 9. Intrinsic transverse relaxation times 1 00 10. Transverse relaxation time transitions 102 1 1 . Supercooling transition as a function of paraH 2 concentration 105 1 2. Echo amplitude transitions 1 08 1 3. Transverse relaxation time and echo amplitude vs temperature 1 1 3 1 4. Longitudinal relaxation times 115 1 5. Transverse relaxation time as a function of time 1 1 7 16. Echo amplitude as a function of pulse separation 117 1 7. Orthopara conversion behavior 1 1 9 18. Time dependence of cw lineshape.. 126 19. Second moment as a function of time 130 20. Temperature dependence of cw lineshape 1 32 21 . Longitudinal relaxation time of hydrogendeuteride 139 22. Echo amplitudes of hydrogendeuteride 139 23. Transverse relaxation times of hydrogendeuteride 140 vi
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Page 24. Isochoric pressure versus temperature data 143 25. Echo amplitudes for hydrogen in zeolite 5A 146 26. Temperature dependence of cw lineshape in zeolite 5A 147 27. Pake doublet superposition 1 58 28. Gaussian wavefunction 158 29. Synthesized cw lineshapes 159 vii
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NMR STUDIES OF MOLECULAR HYDROGEN CONFINED TO THE PORES OF ZEOLITE By MARKUS RALL December 1991 Chairman: N. S. Sullivan Major Department: Physics Molecular hydrogen is interesting to investigate in a porous material. It is strongly quantummechanical with respect to its translational and rotational degrees of freedom as well as its orthopara conversion. Zeolite (13X) was selected as porous material for its almost monodisperse pore diameter of 13 A. The studies mainly focussed on Nuclear Magnetic Resonance data. The hydrogenzeolite system was probed in coherent pulse and continuous wave (cw) mode at 268 MHz. The pulse data revealed a strong peak in the transverse relaxation time T 2 at about 10 K. A similar peaking was observed for the nuclear spin echoes. This behavior was interpreted in terms of supercooling and implies that hydrogen in zeolite solidifies about 5 degrees below its bulk melting point. Two features are especially remarkable. Firstly, the T 2 was found to be nonhysteretic and emphasizes a purely energetically driven process. Secondly, the supercooling transition temperatures exhibit a linear parahydrogen concentration dependence of 0.10Â±0.01 K/%. This implies that parahydrogen can be supercooled more easily than orthohydrogen. viii
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The continuous wave data show a glasstype lineshape of 130 kHz width at 4.2 K with a = 0.45 Â±0.3. It is a combination of broad wings with a narrow component in the center. This behavior is displayed upon warming from 4.2 K when the narrow line disappears and the broad contribution remains. No sharp transition at 10 K is observed. The line broadens by a factor of 1.5 due to orientational ordering when the sample is cooled to 0.8 K. This latter observation indicates that the pulse data peaks at 10 K do not originate from orientational ordering. The orthopara conversion was extracted from the time dependence of the lineshape. The orthohydrogen contents was followed over 1600 hours which yielded two different conversion constants. For the first 500 hours the conversion is characterized by /c, =0.425 Â±0.006 %/h which is only a fourth of the bulk value and is explained by the reduced number of nearest neighbors in the constrained geometry. For later times the conversion accelerates to k 2 =2.21 it 0.075 %/h and is explained by clustering effects. The relative shape of the line changes over time as the narrow component of the line decays faster. Isochoric pressure versus temperature data were recorded while the hydrogen sample was taken out. The nature of a twocomponent system was confirmed. The adsorption energy of hydrogen on the zeolite surface was determined to be 270 Â±50 K. Hydrogendeuteride was also adsorbed into zeolite. The interpretation of the pulse data peaks in terms of supercooling was supported by the fact that HD showed a similar behavior without orientational degrees of freedom. Similar but more extreme results were obtained from a second zeolite (5A) with even smaller pore size (9 A) in agreement with the previous data. ix
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CHAPTER 1 INTRODUCTION The system to be discussed in this dissertation is hydrogen confined to zeolite, a porous material. Experimental investigations were carried out by means of Nuclear Magnetic Resonance at low temperatures. These four characteristics molecular hydrogen, zeolite, low temperatures and NMR determine the scope of this thesis. Every one of these will receive special attention in the following chapters. In addition, the effects that are expected and observed will be discussed. The introduction will lead into the subject by explaining some general principles that are important for the understanding of the results and provide the background for the remainder of the thesis. The overview is also meant to motivate the performed experiment. For the understanding of many effects it is necessary to be familiar with the special properties of hydrogen. Hydrogen is the simplest existing diatomic molecule and can therefore be used as a model molecule to understand fundamental interactions and phenomena. When it is cooled below 13.8 K it is also the simplest molecular solid. The beauty of hydrogen is that the involved interactions are understood in terms of first principles. 1.1. Quantum Nature of Hydrogen Hydrogen is special in its properties. It is a very interesting substance as it is explicitly quantummechanical in its nature. There are three realizations of 1
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2 this fact: the existence of two species, the large zeropoint motion (ZPM) and the quantum rotor property. 1.1.1. Orthoand Parahvdroaen One of the most fascinating properties of molecular hydrogen is the explicitly quantum mechanical classification into two species. Hydrogen occurs naturally as a diatomic molecule. Its nuclear constituents are spin1/2 protons with the electronic spin contributions paired in a symmetric 1 X* molecular ground state. The wave function can be written as a product of nuclear, rotational and vibrational wave functions. The vibrational groundstate is symmetric and does not influence the considerations. The quantummechanical requirement of a totally antisymmetric wavefunction for two indistinguishable fermions can therefore be realized in two distinct ways, leading to two distinguishable molecular species: orthoand parahydrogen. Orthohydrogen has a symmetric nuclear spin wavefunction (/ = 1) and an antisymmetric orbital wavefunction (J odd). The nomenclature is such that the species with the highest spin quantum number is called "ortho." Parahydrogen has an antisymmetric nuclear wavefunction (/ = 0) and a symmetric orbital wavefunction (J even). It is important to notice that both species of hydrogen are bosons. Heteronuclear molecules such as hydrogen deuteride (HD) do not display distinct species because of the distinguishable nature of their constituents. Because of the small momentofinertia / for hydrogen, the separation of *2 the rotational states is very large: Ej = Bj J(J +1) with Bj = Â— = 85.4 K. The quantum number J is therefore a meaningful entity at low densities. The intermolecular interactions which tend to mix different rotational states are very weak, and the groundstate is the J = 0 parahydrogen state. This still holds
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3 when the surfacemolecule interaction of the restricted geometry is included, as discussed in Chapter 2. 1.1.1.1. Quadrupolar interaction Parahydrogen is spherically symmetric with no electric or magnetic moments. The lack of nuclear spin degrees of freedom for parahydrogen makes it also undetectable by Nuclear Magnetic Resonance. This is in contrast to orthohydrogen with spin 1 which also has an electric quadrupole moment. The molecules tend to orient at low temperatures in order to minimize the anisotropic interaction which occurs when the molecular axes are aligned at a 90 degree angle, a T" configuration between quadrupoles. Here A is the lattice parameter, R the intermolecular distance, Fan angular function of the angles measured relative to the connecting line between molecules, and r = 0.8 K. The order parameters specifying the degrees of freedom for this interaction are components of the second rank quadrupolar tensor with respect to the local reference axes (x, y, z) which leads to two additional parameters besides the three local reference axes. The reason for only two additional parameters is the vanishing orbital angular momentum, which means that (J,) = 0 is "quenched." 1 This is the case when timereversal symmetry is not broken which can be assumed here, even though
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4 a magnetic field is applied. The ratio of the magnetic interaction energy (mK) to h 2 the involved rotational energies (Sj= Â— = 85.4K) is the reason for this approximation: the magnetic energy is only of negligible influence. The argument for "quenching" is related to the nondegenerate nature of the rotational groundstate. A nondegenerate groundstate must be real otherwise real and imaginary parts of the wavefunction would separately be solutions, which contradicts nondegeneracy. The angular momentum operators are imaginary and the expectation value (J,) with /' = x,y,z is therefore purely imaginary. On the other hand it must also be real due to hermiticity and the only expectation value satisfying both conditions is zero. At higher temperatures this requires more careful consideration because the J z = Â±1 are also occupied and could contribute to the rotational Zeeman energy. This is not the case, however, because the transition frequency amongst these states is much higher than the dipolar frequency and is therefore averaged out. All offdiagonal matrix elements are also zero by the properties of angular momentum. One additional order parameter for orthohydrogen is the alignment (7=0^ = ^(3^2) For J z = 0 the alignment is
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5 and is zero for any rotationally symmetric body. 1.1.2. ZeroPoint Motion Hydrogen is the molecule with the lowest mass. It possesses only a weakly polarizable electron cloud leading to weak intermolecular forces. These two facts combined with the groundstate energy E Q =jho) of a threedimensional harmonic oscillator lead to a large zeropoint motion (ZPM). This implies that the molecule's oscillation amplitude is a sizeable fraction, 15% for hydrogen, of the intermolecular lattice spacing even at zero temperature. This is a fascinating feature and is the cause for interesting properties. It also renders calculations more difficult because a harmonic approximation about the intermolecular potential minimum is no more justified. Also correlations between molecules are important to consider when an understanding for the dynamics is developed. In order to gain insight into the ZPM, a harmonic approximation is presented, with all the caution in place. The standard interaction potential used is the LennardJones potential where o is the hard core radius and e the potential minimum. The first part is a mathematically convenient form of a repulsive interaction and is a consequence of the Pauli exclusion principle when electronic clouds start to overlap. The second term is the attractive Van der Waals potential due to induced dipoledipole interaction. No permanent dipoles are involved but one dipole is created by thermal fluctuation fields and another dipole is induced in a neighboring molecule.
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6 A more sophisticated potential was empirically found by Silvera and Goldman 2 with V(r) = exp(afiryr 2 ) + f(r)\ Â£ [/=6.8,10 ' 'J 2 ' = exp1.28 Rm 1 r = 1 for r < 1.28/=?Â„ for r>1.28R. where the constants are a = 1.71 3, /? = 1.5671, / = 0.00993 and C 6 =12.14, C 8 =215.2, C 10 =4813.9 and C 9 =143.1 all in atomic units and R m =3.41 A at the well minimum for which the ninth order term is excluded. The simple LennardJones potential is expanded up to second order in the displacement u relative to the radius of minimum potential energy which leads to a harmonic oscillator with Hamiltonian H = +Â—ku 2 2m 2 with k = ZV" where Z is the number of nearest neighbors. The groundstate energy of a harmonic oscillator is wellknown to be 5=% = ^ 2 2 V m The corresponding wavefunction is of Gaussian form
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with width W. The width describes the spread of the wavefunction in space and the localization of particles on the lattice. The expectation values for the kinetic and potential energies over this groundstate wavefunction are The results imply two things: firstly, the shallow potential with small curvature reduces the potential energy and secondly, the small mass increases the kinetic energy raising its relative importance even more. A calculation of the minimum energy with respect to the width leads to in terms of the relevant parameters. It is especially large for small m and small V", which is the case for quantum crystals, such as hydrogen. Hydrogen has a large kinetic energy relative to its small potential energy and would be too localized for a bound state so that it reduces its kinetic energy by spreading out to a higher potential energy. This process of spreading the wave function leads to a new arrangement of the lattice parameter. A guideline is r 0 2(u 2 ) V2 where r 0 is the distance of minimal potential and u is the displacement about a lattice site. If this quantity is smaller than the hardcore 3 n 2 4mW 2
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8 radius the particles spend appreciable time in each others' hard cores which is an unphysical situation. The strong repulsive interactions cost energy and a new minimum energy must be found. The solution to the dilemma is an expansion of the lattice from r 0 to A. This procedure reduces and increases , which reestablishes the situation of classical solids and is a compromise for a certain A2(u 2 V /2 width W. Empirically it is found that Â— Â— arranges itself around unity. (7 1.1.2.1. De Boer parameter It is interesting to define a measure for the quantum nature of a molecule. This can be approached by rewriting the Schroedinger equation in terms of reduced quantities by scaling the distance and the potential with the LennardJones parameters r V r'=and V =a Â£ which results in 2nriÂ£o z (V*) 2 +V The prefactor of the reduced kinetic energy term expresses the relative dominance of the kinetic energy compared to the potential energy. This is the relevant quantity for quantum behavior and is defined as the de Boer parameter v / 2mÂ£cr 2
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9 which is large for small energy and mass. This is consistent with the earlier statements and calculations about the nature of quantum crystals. The parameter is 0.196 for hydrogen 3 which is much larger than for classical crystals with values below 0.05. 1.1.3. Crystalline Potential and Quantum Rotor At temperatures above the melting point (13.8 K) orthohydrogen oscillates rapidly between its orbital magnetic states M. This generates an effective spherical symmetry as it is perceived by its neighbors. In this respect it is similar to parahydrogen at high temperatures. At lower temperatures where hydrogen solidifies this is no longer true. The transition rate between M states decreases and the nonspherical character of orthohydrogen manifests itself. The average potential, the crystal field, at the site of an orthohydrogen molecule is of lower symmetry. This is a selfenforcing, cooperative process as more orthohydrogen molecules with lower symmetry reduce the free rotation of others. The crystalline potential V(0,
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10 A y =sin<5sintu X z = cos 8 of the molecular axis. The eigenfunctions to this problem are for /'= x,y,z At this point the quantum rotor property of hydrogen becomes imminent. By this statement is meant that orthohydrogen cannot be perceived as a static entity. Due to its large ZPM it is in rotation even at low temperatures. The J = 1 state is therefore not a rigid ellipsoid but its motion averages the shape into a sphere. As will be seen, the angle dependence of the dipolar interaction is governed by (3cos 2 01) which is proportional to the spherical harmonic of second order. It is, however, of little significance to define an angle 0 between the connection line of the protons with the magnetic field. Instead it is necessary to define the angles /?, cos 0 = X x sin/?cos + X z cos 0 = cos 5 cos p + sin <5sin/?cos(
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11 (3 cos 2 0 i) 2 = jJ (3 cos 2 01) cos 2 <5sin SdadS = (3cos 2 /?1) The result is the famous reduction factor of 2/5. It expresses that in the case of a quantum rotor the dipolar interaction is rescaled to 40 %, leading to narrower lines in the quantum case than expected classically. 1.2. Open Questions for Confinement This short overview about interesting properties of hydrogen gives a flavor for the type of experiment that can be performed with hydrogen in a porous material. The substrate represents a geometric confinement and influences the threedimensional properties of hydrogen. The questions concerning the changes that arise from such confinement are various. Some of them are as follows: 1) What impact does the surface potential have on hydrogen molecules being closer to the wall than others? Will different environments result in different subsystems? 2) Is the ZPM in a more localized environment dominant enough to introduce fluctuations that wash out some of the expected effects? 3) Does orientational ordering of quadrupoles occur and if, at what temperature? This may lead to new results when the surface potential is assisting in the dynamical slowing down of rotational motion.
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12 4) Is the melting transition influenced by increased fluctuations due to lower dimensionality and a pore size that reduces the size of the critical solid nucleus in the formation of a solid as described by nucleation theory? What is the impact of grain boundaries forming along rough substrate walls? 5) Is a glass phase possible as a result of inhibited and disturbed solidification? 6) How is the orthopara conversion influenced by the surface potential, the dynamical restriction due to the geometry and the reduced number of nearest neighbors? 7) At last a very challenging idea is brought forward: would it be possible to reach a superfluid state in hydrogen if the solidification was suppressed to temperatures where superfluidity is conceivable and theoretically predicted? This catalogue of questions is well worth a thorough experimental investigation that might give answers to these open problems. Chapter 2 will expand on the concepts and ideas just mentioned. This will allow for an understanding and appreciation of the importance of the previous questions. The conclusion chapter 7 will try to answer them based on the obtained data. Chapter 3 focuses on some special topics in Nuclear Magnetic Resonance as they pertain to the experiment. Chapter 4 describes the experimental "hardware" and gives an introduction to the technical specialties of the experiment. The experimental data are presented in chapter 5 while chapter 6 contains the computer simulation of the NMR lineshape. Chapter 7 closes the dissertation with a conclusion that restructures the experimental data as they pertain to physical effects rather than a mere listing of facts and explanations.
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CHAPTER 2 THEORY 2.1. Surface Interactions The properties of a molecular liquid and solid are expected to be modified in contact with a surface. The main reason is the interaction potential between the molecules and the surface which has to be incorporated in a theoretical treatment. The impacts of the surface are various. The potential energy arising from an attractive potential such as the Van der Waals potential restricts the molecular motion, tending to localize the particles in potential wells. For particles adsorbed on a surface the number of nearest neighbors is reduced, which is a direct consequence of the geometrical constraints. This in turn reduces the dominance of the intermolecular interaction therefore increasing the relative importance of the external surface potential. The twodimensional character on the surface implies a reduced number of accessible modes compared to three dimensions. These facts should lead to a modified phonon spectrum, reduced diffusivity, a distinct first monolayer behavior, induced dipole moments, reduced entropy 4 and less intermolecular interaction. As a consequence, the system could display altered macroscopic effects. These macroscopic manifestations of the surfacemolecule interaction could be modifications in viscosity, orientational ordering and solidification behavior, just to mention a few. 13
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14 Being familiar with these facts, one must keep in mind the dominant zeropoint motion of hydrogen which is counteracting the localizing surface potential and might reduce the effective surface potential. 2.1.1. Theory of SurfaceMolecule Interaction Many experiments and theoretical calculations have been performed for hydrogen, methane and helium on vycor, zeolite and twodimensional substrates, favorably grafoil. A comparison between the properties of zeolite and grafoil is in place as the calculation to be discussed was performed for grafoil, but the experiment in this dissertation was carried out using zeolite. The attraction potential of grafoil is about a factor of two larger 5 than the interaction strength of most other solids, including zeolite. In addition, the surface is twodimensional in comparison with a curved geometry in zeolite. An additional complication for zeolite is the tortuosity of the restricted topology. The question of dimensionality arises in this context. The main idea is nevertheless similar and reduces to investigating a system that interacts and is partially bound to the surface of a substrate. The following calculations will lead to the ground state energy, the twodimensional phonon spectrum 6 and the purity of the zero rotational state 7 of hydrogen on a substrate. The starting point for any calculation of this type must be the Hamiltonian for such a system. It consists of the translational and rotational kinetic energies, the moleculesubstrate interaction energy and the intermolecular interaction energy, neglecting weak anisotropic intermolecular interactions: H = H T +H R
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15 where M is the mass of the molecule, L the angular momentum operator and v the isotropic intermolecular interaction which depends only on the relative distance between molecular centers of mass. The anisotropic substratemolecule interaction V A depends on the distance itself and coupies the rotational and translational parts of the Hamiltonian. For most calculations, however, the wavefunction is approximated by an independent product of spatial and rotational contributions. This allows for writing two independent eigenequations: H T
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16 E E 2 + + E^ The energy E z is the kinetic energy associated with the motion in zdirection plus the potential energy of the laterally averaged substrate potential in zdirection and is evaluated using the zcomponent of the wavefunction. The E^ term is correspondingly related to the kinetic energy in the xy plane plus the intermolecular interaction energy, averaged over the zcomponent of the wavefunction and evaluated with the inplane wavefunction. The form of the intermolecular interaction is the earlier described SilveraGoldman potential omitting the C 9 term. The third contribution, E^, also involves both components of the wavefunction. It represents the vertical potential energy due to the periodic variation of the adsorption potential averaged over inplane displacements. The solution to this problem is determined variational^ by a minimization of the total energy as a function of the parameters in the two Gaussian wavefunctions. This leads to a twodimensional selfconsistent phonon theory because each wave function contribution appears in two equations. This makes the process iterative and several iterations are necessary to obtain a selfconsistent system of equations. The process starts with the determination of the zcontribution of the wave function because it appears solely in the E z equation. It has been shown that the selfconsistent phonon approximation, including anharmonicities from the zeropoint motion, leads to a ground state energy which is by about 15% higher. 8 The ground state is almost entirely determined by the E z contribution and the total energy is about 500 K for hydrogen on graphite. The phonon spectrum is characterized by a phonon gap of 46.6 K, a maximum in the transverse phonon mode of 64.9 K and a maximum in the longitudinal mode of 83.8 K. This
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17 suggests that it is more advantageous for the system to rotate away from the symmetry direction. The phonon .density has a large width of 48 K because of the substantial zeropoint motion. These calculations are backed by agreement with experimental data which are within 10%. The conclusion is that the third dimension must be incorporated in calculations dealing with moleculesurface interactions. It is intuitively also clear that the restricted geometry limits the long wavelength phonons which couple to the substrate phonon modes and are the predominant excitation at low temperatures. Novaco 6 initially performed the calculation for a commensurate phase but extended it later to the incommensurate phase. The result is that the energy for the incommensurate phase lies only a few Kelvin (3%) above the commensurate phase groundstate. This is of importance for zeolite where no commensurate phase can be expected due to the surface roughness and curvature. Returning to the rotational part of the total Hamiltonian, another valuable piece of information can be extracted. 7 This is related to the mixing of rotational states under the influence of an additional nonspherical potential. The ordinary rotational ground state is known to be J = 0 and could be subject to mixing with higher order states. The sum in the Hamiltonian extends over single particle terms which can be solved individually with the rotational wave function expanded in a superposition of free rotor states \JM). The single particle energy contributions are then summed up to yield the total energy. The use of spatial wave functions such as in the ground state calculation leading to the phonon spectrum (splitting into inplane and outofplane contributions) already presumes a J = 0 state. If the results turned out to be different from this assumption a new Ansatz would become necessary. This is
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18 not the case, however, since hydrogen exists to 99.9% in the J = 0 state, confirming the general assumption of pure states. Furthermore, as the calculation shows, hydrogen molecules adsorbed on a substrate behave like slightly hindered threedimensional rotors. It was also confirmed by analysis of neutron scattering data that hydrogen on surfaces, such as grafoil, behaves like a threedimensional rotor. 9 Further experimental support for threedimensional behavior is superfluid helium in vycor. 10 The threedimensional network displays a bulklike transition independent of fractional filling. For rotationally excited states an interesting picture was deduced from perturbation theory 11 and numerical studies. 12 Rotationally excited states couple with the vibrational states associated with motion perpendicular to the surface. This coupling results in a motion where the molecules line up with the normal of the substrate at the maximum amplitude of the zeropoint motion and lie parallel at the center. This implies a mixing of the M = 0 and the M = Â±1 states inside the J = 1 manifold. Recent experimental data 13 suggest strong mixing of rotational states for hydrogen deuteride (HD) on vycor. Rotational transitions of as high as SJ = 3 and SJ = 5 were observed which violate the bulk selection rules. This may be a consequence of the mismatch of center of mass and center of rotation coordinates in HD. The important result is that the standard classification of hydrogen in ortho and para species in their pure forms still holds. The same should also be true for zeolite with its weaker interaction potential and more threedimensional character. This is valuable information for the orthopara conversion.
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19 2.1.2. Effects of Adsorption Several interesting results from other experiments and calculations wiil be quoted because of their relevance to the experiment to be discussed in this dissertation. Related to the phonon spectrum are heat capacity measurements of helium in vycor indicating a decreased Debye temperature 14 in two dimensions. This is of importance for the creation of phonons carrying away the released energy in the orthopara conversion process. The behavior of the first monolayer on the substrate has been observed to be different from subsequent layers. Superfluidity in helium 15 decreases and disappears as the coverage approaches one layer. 16 This seems to suggest that superfluidity only occurs on top of a more localized first layer smoothing out the surface. Experiment 17 18 and theory 19 seem to agree upon the influence of the zeropoint motion on the separation between hydrogen molecules and the adsorbant. The expectation is that the large compressibility leads to a reduced nearest neighbor separation. This is, however, not the case because the ZPM, enhanced by the restriction from the pore environment, counteracts the attractive influence of the surface, hardly allowing the molecule to get close to the surface. This is illustrated by 3 He being further away from the surface than 4 He reflecting their relative ZPM. 20 Methane, 21 however, is forced closer to the wall at low temperatures due to smaller ZPM. Induction of dipoles was observed 1722 from strongly enhanced infraredabsorption spectra of hydrogen in vycor. The conclusion of induced dipoles follows from the independence of the infraredabsorption signal over time. This means that orthoand parahydrogen give rise to the same signal which renders
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20 the electric quadrupole moment of the ortho species irrelevant and shows the strength of the surface interaction. A special effect on the diffusion rate is the purely geometric restriction. 23 The parallelpore model 24 was proposed to describe a monodisperse poresize distribution. In this case the diffusion D s is related to the free bulk diffusion D Q in an empirical relationship with the porosity O and the tortuosity factor 5 2S The tortuosity is a measure of the geometric complexity of the substrate. A numerical value was found 25 to be 5= 5.6 for hydrogen in vycor of about 10 A radius. This is close to the dimensions of zeolite and can give an idea of the purely geometric influence on the diffusion rate. Using $ = 50 % for the porosity of zeolite the tortuosity effect reduces the diffusion by roughly a factor of 10. This implies strongly inhibited diffusion for the confined system. This survey of the variety of influences caused by the surfacemolecule interaction illustrates the strong motivation for the intensive investigation of this type of system. 2.2. The LiquidSolid Transition There have been many attempts to explain and describe the liquidsolid transition. The early phenomenological theories relate critical behavior of solid state properties to the onset of melting. Such theories are characterized by the increase in vibrational amplitudes, 26 the disappearence of bulk moduli 27 and the
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21 proliferation of defects (dislocations) 28 29 upon occurrence of the liquid state. These are models for the mechanical, rather than the thermodynamic, instability of the crystal. The study of a thermodynamic criterion, the minimum chemical potential, has been pursued by molecular dynamics studies. 30 They predict melting which takes place in a layerwise fashion starting from the surface. Caution towards the results of such calculations is necessary, considering the limited number of particles involved. 2.2.1. Elastic Instability Theory The original Born criterion of vanishing bulk moduli was revised, 31 remedying the erroneous second order and homogeneous nature of the transition. The difference between the two theories hinges on the experimental observation that one of the bulk moduli does not completely disappear at the transition temperature but only inside the melt. This implies a twophase model where the dilatation on the liquid side is 8' m which is different from S* m on the solid side of the transition. The shear modulus vanishes or nearly vanishes when 8 = 8' m . At the transition the entropy 2RT is released upon entering the liquid state. This reduces the free energy and is the reason why there is another dilatation in the solid state which is associated with the same free energy. This point is the true transition where the isothermal and discontinuous transformation, involving a latent heat, occurs. The released entropy also represents the nucleation barrier which renders this a heterogeneous theory, as the solidification occurs at strained sites and surfaces where the free energy is higher. 2.2.2. Dislocation Theory A more microscopic view of crystallization was taken by Cotterill 29 et al. by using a model incorporating dislocations. The decisive assumption is that dislocations also exist in the liquid. The dislocation density is extremely high and
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22 there is no crystalline material around the mutually contiguous dislocation cores. This makes it imprecise to describe the liquid as a crystal saturated with dislocations cores. Instead, the liquid can be referred to as a pseudodislocation saturation. Viscosity and other liquid properties are explainable by the rapid motion of pseudodislocations. Crystallization occurs upon cooling by elimination of pseudodislocations without recreation when the temperature is below melting. The elimination of pseudodislocations is driven by an increase in the free energy, as the liquid state exists below its equilibrium temperature. This phenomenon is called supercooling and indicates a metastable state. Nucleation theory (as discussed later in this chapter) applies to this system in a standard way. If the viscosity of the melt is high or the system is cooled rapidly into the solid temperature region the removal of dislocations is inhibited. The dislocation annihilation rate is then slower than the solidification process. This solid is microscopically similar to a liquid incorporating a high concentration of dislocations, which are immobile and incapable of diffusion and interaction. A polycrystal with a grain size of nearatomic dimension develops. Such a solid is called a "glass" and follows in a straightforward fashion from this model. A few properties of glasses shall be mentioned in this context. The ground state of a regular solid is a single crystal which minimizes the dislocation energy. A glass, on the other hand, is one realization of a metastable state frozen in at some arbitrary configuration. The reason for its existence can be its frustration due to the incompatibility of the interactions with the lattice. The specific volume of a glass is larger than that of a crystal which is an immediate consequence of the abundance of frozen dislocations and the increased disorder. A glass temperature exists below the bulk melting point from where on the mobility of dislocations is zero, the elimination of defects impossible and the glass phase persistent.
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23 2.2.3. TwoDimensional Melting Models A twodimensional dislocation theory is especially interesting in the light of the twodimensional liquidsolid transition. Mermin, 32 following Peierls's 33 earlier arguments, showed rigorously that conventional crystalline longrange order is excluded in two dimensions for powerlaw potentials. This does not pertain to orientational ordering, however. Kosterlitz and Thouless 34 35 proposed another model for the existence of longrange order. This longrange order is of a topological type. It originates from the dislocation theory of melting. 36 The idea is similar to the already discussed model of Cotterill 29 et al. The liquid phase close to the melting point has a local structure similar to the solid but dislocations exist which are mobile and induce viscous flow. The solid state is rigid because the dislocations are immobile. They pair up in the solid state thereby reducing their energy from a logarithmically diverging to a finite contribution. The phase transition from the solid to the liquid state is characterized by the dissociation of pairs into single dislocations. The critical temperature is calculated to be = //a 2 (1+T) Â° *nk B where ^ is the shear modulus (in two dimensions) and r Poisson's ratio. The connection with the revised Born model 31 described earlier in this context is of interest. From this theory the shear modulus cannot be zero in the transition region, otherwise the transition to a solid would occur at zero temperature. Starting from the dislocation model, Halperin and Nelson 37 have worked out a more detailed theory of twodimensional melting. They assume the unbinding of dislocation pairs which are referred to as dipoles because they pair up with two different polarities. This yields an exponential decrease of the
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24 translational order parameter. Instead of forming an isotropic liquid at this point, calculations show a sixfold anisotropy with the bondangle correlation decaying algebraically. At a higher temperature finally, the dissociation of these disclination dipole pairs drives another transition to an isotropic liquid. The intermediate liquidcrystal state is possibly bypassed or occurs simultaneously in the presence of a substrate due to its orientational bias. This would transform the two continuous phase transitions into one firstorder transition. This idea of a virtual region, where two transitions are hidden in the experimentally observable firstorder transition, was adopted by Tallon. 38 In a fashion similar to the revised Born theory, an entropy term is included. This AV could be a volumeindependent term 39 40 of AS = R\n2 as Â— a Â» 0 which is the case in a porous material where the available space is constant as a geometric restriction and the molecules are trapped in the pores. This entropy term is possibly associated with the transition from a uniform distribution of the crystal cell size to a random one. Another possibility is the communal entropy arising from the fact that particles in the liquid state have access to the whole fluid instead of being localized. This contribution is significant for large diffusivities but small for a tortuous geometry such as zeolite. The transition becomes first order by going discontinuously from below the dissociation to above the disclination of dislocation dipoles for two and three dimensions. This leads to the conclusion that melting in two and three dimensions is likely to be very similar. 38 The difference would be the varying width and separation of the two virtual transitions, both being larger in two dimensions. 2.2.4. Molecular Dynamics Studies Studies from a different starting point are performed using the molecular dynamics approach. In this case thermodynamic quantities such as the
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25 chemical potential are investigated. Of special interest is Broughton 30 et al.'s conclusion that melting occurs in a layerwise fashion starting from the outermost crystalline layer and progressing into the bulk region. The mechanism involves the promotion of particles from the solid to the liquid layer with an accompanied increase in mobility in the remaining solid layer. The empirical rule has been established that each layer melts when it reaches the bulk ratio of kinetic to potential energy at the melting point. Using this criterion, the outermost layer melts at about 72% of the bulk melting temperature, the second layer at 90%, while already the third layer is close to the bulk melting point. 2.2.5. Implications for the Investigated System The previous remarks are of fundamental importance for a system such as hydrogen in zeolite where the purely geometrical surfacetovolume ratio is as much as 1/4 and about 80% of all hydrogens have contact with the substrate. It is therefore difficult to decide upon the dimensionality of the adsorbed system. The prediction of a similar melting process for two and three dimensions eliminates some of these problems. Implications for the nature of the transition are also given by the existence of hysteresis upon heating and cooling. The absence of hysteresis would lead to the conclusion that the transition is continuous and the melting domain not a heterophase region. Furthermore, it has been argued 41 that in the case of grain boundaries the transition should always be first order. As will be seen later, grain boundaries may have substantial impact on the solidification process. The above theories and projections are a rich testing ground for experimental investigations. A variety of work is necessary in order to understand the complicated process of melting and solidification. The outcome will certainly be severely affected by the impacts of dimensionality such as the
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26 reduced number of nearest neighbors and the substrate character with an interaction potential that leads to limited diffusion. 2.3. The Superfluid Superfluidity for helium is a wellknown phenomenon in the lowtemperature community. The important issue is that the freezing point of helium is below the transition to the superfluid. This is the case because helium does not solidify at any temperature without pressure. Therefore the superfluid state is feasible and transforms by Bose condensation from a normal liquid at 7 A =2.17K. Superfluidity is expected to occur for any system of bosons. Interest exists therefore in a potential superfluid hydrogen phase. Calculations have shown for an ideal Bose gas that the transition temperature to a superfluid 42 is characterized by *2 / Â„\ 2/3 ( 11 V 5/3 j k =3.31*^  =112 Mk{g) {M p j where M is the mass of the molecule, M p the proton mass, n the concentration, p the density and g the degeneracy for a singleparticle state which is unity for parahydrogen. This results in a predicted transition temperature of = 3 K for helium compared to the experimental value of T l =2.17K. It is in good agreement and can therefore be applied to parahydrogen with a reasonable chance for a realistic prediction of the superfluid transition temperature. The calculation for hydrogen yields = 6.6 K. This low value for the transition temperature is the reason why superfluid hydrogen has not been observed
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27 experimentally: hydrogen freezes at 13.8 K before it reaches the superfluid transition. It is expected, however, that hydrogen enters the superfluid state if the liquid is supercooled to below the superfluid transition temperature. The quality of the above estimate for the superfluid temperature is stressed by the fact that the transition temperature increases 43 with a larger interaction potential. The energy scale of the LennardJones potential between hydrogen molecules is e= 37 K which is much larger than e= 10.2 K between helium atoms. The calculation is therefore likely to be a better estimate for hydrogen than for helium. Other authors 44 are less confident about the value of 7" v They scale the difference between T K and with the de Boer parameter which for hydrogen is only 2/3 of the value for helium. The de Boer parameter describes the degree of "quantumness" which leads the authors to believe in a reduced experimental transition temperature. On the other hand, it seems that a larger ZPM would rather be less favorable for a collective process, such as Bose condensation. In summary, superfluid hydrogen can be expected below 6 K but the liquid state must be extended down to the superfluid transition temperature which is possible when the liquid is kept metastable. Properties of the superfluid have been projected partly in analogy to helium. The dispersion curve should also exhibit phonon and roton excitations. The slope of the phonon curve is given by the sound velocity. The roton gap A is estimated by scaling the minimum roton energy of helium by the ratio of lattice spacings. This results in a gap A of around 3kT x . This would lead to a roton gap of 20 K. A recent experiment 45 of hydrogen in vycor suggests a tentative explanation of heat capacity data by a roton gap of 23 K. This occurs in addition with supercooling at 10 K which seems a very high temperature in the light of superfluidity.
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28 So far the discussion only pertained to parahydrogen. For orthohydrogen with a singleparticle degeneracy of g = 9 for spin and orbital angular momentum degeneracy the superfluid transition drops to 1.5 K. For this system further magnetic properties could exist which would generate additional diversity in the phase diagram. The question arises if supercooling by about 8 K is a feasible operation. Several options and speculations have been proposed. It has been suggested 43 to investigate surface films on different smooth substrates where hydrogen may not wet and stay liquid. This could be done on top of a monolayer of substratecoating material (deuterium or neon) generating a less dense fluid. The setup would lead to a twodimensional superfluid. It has been shown by Widom 46 that superfluidity, excluding intermolecular interactions, is indeed possible in twoand even onedimensional systems. The condition for the existence is the presence of an external disturbance, for example gravity or rotation, which breaks translational invariance. This does therefore not contradict Hohenberg's 47 argument which rules out lower dimensional Bose condensation in homogeneous systems, i.e. without symmetry breaking. As a confirmation of this theory the possibility of surface superfluidity 48 in parahydrogen has been put forward as a potential explanation of a gravityinduced flow below 3 K observed by Alekseeva and Krupski. 49 The calculated flow rates are in agreement with experiment and result from vortex considerations analogous to the KosterlitzThouless theory. Inclusion of impurities is another proposed method of achieving supercooling as suggested by Geilikman 50 who extended the Lindeman criterion for melting to quantum crystals where the melting temperature T m is much smaller than the Debye temperature 0 D . He stresses that the zeropoint
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29 amplitude is inversely related to T m which is in turn proportional to (M@ D ) . Impurities reduce the Debye temperature if they are heavy, weakly bound to the matrix and have a high solubility. By these means T m is reduced. An interesting possibility at low temperatures is the creation of quantum excitations from light impurities, so called impuritons. Atomic hydrogen can be used which also obeys Bose statistics. If the impuritonimpuriton scattering is repulsive then the energy spectrum displays an acoustic branch which satisfies the superfluidity condition. 51 The calculated superfluid transition temperature is raised to 12 K. Another alternative is dynamic supercooling which relies on a rapid decrease in temperature and conserves the liquid state in a metastable form. Many experiments of this kind have been performed, being described in the next section. An explanation using nucleation theory is possible to demonstrate this effect of delayed and inhibited nucleation to the crystalline state. The theory will also be applicable to the understanding of supercooling in small pores which is performed in the experiment to be discussed in this dissertation. The procedure can substantially decrease the melting point, by as much as 5.5 degrees in the present experiment. Unfortunately, the lambda transition and the Fermi temperature are also depressed in small geometries as experimentally shown for 4 He 52 and 3 He. M This makes it in turn more difficult to reach superfluidity. 2.4. Supercooling 2.4.1 Nucleation Theory Nucleation theory 54 is one possibility for understanding the preference of the liquid versus the solid state. It is a dynamic description why the process of
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30 solidification is inhibited and therefore the metastable liquid state preferred. It builds upon the formation of nucleation cores around which solidification occurs and describes the velocity of growth, once the nucleation cores have formed. The starting point is the free energy difference 5F between liquid and solid state required to form a solid sphere of radius R 5F{R) = 4nR 2 a LS Â±nR 2 n s {f L f s ) where a LS is the surface energy 55 between solid and liquid, n s the number density in the solid phase and f L , / s the free energies per molecule in the liquid and solid phase. The surface and volume energy contributions to the free energy are competing. There exists a maximum energy 0 3 n\{f L f L f which occurs at a radius a 2oc ls Â° n s (f L f s ) Figure 1 shows a plot of the free energy difference versus the solid sphere size for different temperatures. For R > R 0 the solid sphere grows without limit because the cubic term in R takes over and decreases the free energy. If a thermal fluctuation can produce a solid sphere with a radius greater than R 0 , the supercooled liquid makes a transition into the solid state. At the normal bulk transition temperature of 13.8 K, f L and f s are equal. Therefore, R 0 , the critical radius, and the energy maximum for SF, which is the solidification barrier, both
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31 NUCLEATION THEORY 5000 1 0 5 10 15 20 25 30 Radius (A) Figure 1. Nucleation Theory
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32 diverge implying that for any lower temperature a large but finite size solid sphere suffices to trigger the transition to the solid. At this point the liquid state still prevails, however. As the temperature decreases further, the barrier and the radius also decrease. This implies that a finite, thermally induced solid sphere is enough to induce solidification because, once the nucleation core reaches the critical size and the top of the barrier, it is energetically favorable to grow without bounds and release energy. The maximum radius for the solid sphere is the above defined radius R Q . This is in contrast to the behavior in a porous material where the critical solid sphere is restricted to the pore size. The slightly different condition will be described in the next section. For low temperatures the critical radius and the solidification barrier, become small, indicating facilitated solidification. The probability for solidification to occur per unit volume is described by T T =TÂ°exp(5F 0 /kT) with an attempt frequency per unit volume r?=(n L kT/h)exp{Â® LS /kT) where n L is the number of molecules per unit volume in the liquid phase and
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33 solid and liquid at these supercooled temperatures. This implies a decreased energy barrier. At low temperatures, however, the energy barrier becomes constant but the lower temperature reduces the probability of overcoming this barrier by means of thermal fluctuations. Nucleation by quantum tunneling through the energy barrier has also been considered. This could be of importance for quantum crystals like hydrogen and introduces an additional mechanism which is especially active at low temperatures, therefore increasing the nucleation rate. A very important parameter for calculating realistic numbers in this context is a LS . The larger this value, the more efficiently a liquid can be supercooled. Unfortunately, this parameter is not known for any material with high accuracy but has strong impact on the outcome because it is contained in the exponent to the third power. Another supercooling model 56 which follows classical nucleation theory in three dimensions relies on a scaling argument. It suggests that supercooling of hydrogen takes place at a reduced temperature of 0.8 times the bulk melting temperature. This leads to a higher supercooling temperature of 1 1 K. Another statement is made from scaling. The involved degrees of freedom can be calculated as a ratio of the selfdiffusion activation energy and the bulk melting point. This ratio is three and implies participation of three translational and three vibrational degrees of freedom to the process of diffusion across the phase boundary. 2.4.1.1. Implications for experiment Nucleation theory can be used to devise experiments which could be able to achieve supercooling.
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34 The first set of experiments exploits the dynamical aspect of the theory. If hydrogen is cooled rapidly through the maximum nucleation rate into the relatively stable regime at low temperatures, substantial supercooling, maybe down to zero degrees, and superfluidity can be achieved. Experiments of this kind have been carried out at Brown University, 57 58 using liquid drops. They employed a helium pressure to levitate the injected droplets and have indeed observed supercooling by several degrees. Container walls and impurities act as nucleation cores and had to be avoided. The second possibility is using confining geometries into which hydrogen is adsorbed. This introduces a cutoff radius R c into the system which is related to the pore size. Even when a thermal fluctuation is able to create a solid sphere of radius R 0 , from where on it would grow without limit in the bulk, it is now restricted to the cutoff radius and must stop growing. As long as the liquid state is energetically favored, which is the case for positive 5F, the system will stay liquid. The cutoff radius R c , is therefore determined for SF = 0 to be = _3a Li _ Â° n s (f L f s ) It is obvious that a smaller pore size results in stronger supercooling, or better, a more depressed melting point. This process is not dynamically motivated and relies entirely on the energetics. Experiments following this reasoning have been performed mostly using vycor, a porous glass, and helium 52 59 60 61 or hydrogen. 45 62 63 There is an effect for helium which is equivalent to supercooling. Helium already requires overpressure to achieve bulk solidification. The result for helium in restricted geometries is that an extra pressure of several atmospheres, in excess of the bulk pressure, is needed for solidification. This pressure is a
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35 function of pore size and can be deduced from the critical radius of nucleation theory treated above: _ 3a LS = 3a LS V s c n,(f L f s ) Ap V L V 5 This determines the pressure difference as a function of the pore size. The above equation has been quoted in the literature 52 58 59 with a factor of 2 instead of 3 which does not take the additional effect of the restriction by the cutoff radius into account. The same considerations can be made for hydrogen, now using the latent heat term but neglecting the entropy contribution to the solidification process. This leads to _ 3a LS v s T m where T m is the bulk melting point and L LS is the latent heat of fusion per molecule which determines the amount of supercooling ST. This model attributes the extra pressure and the depression of the melting point to the additional contribution of surface energy between solid and liquid. No account is made for the failure of the solid to wet the surface. The model in the next paragraph will attempt to explain this phenomenon. 2.4.2. Grain Boundaries A microscopic model 64 attempts to explain the inhibition of the formation of solid on the wetted surface and the following rapid nucleation. This model originated from the experimental observation 65 that the solidwall interfacial
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36 tension 55 a sw , also termed surface energy, was found to be greater than the liquidwall tension or LW for helium. This result was obtained by measuring a contact angle 0 between solid and liquid on the surface of a copper container that was larger than 90 degrees and then compared with the equation a LS cos0 = aL W a sw This was confirmed on a glass surface and disagrees wuh the theory of Landau and Saam. 66 Their theory states that the dense, absorbed film on a substrate in contact with the liquid is simply ordinary solid which is at an increased pressure as a result of the strong Van der Waals attraction of the substrate. The increased solidwall tension is surprising, as virtually all uniform substrates favor the solid state, independent of their attraction potential. The reason is that the attractive forces interact more strongly with the denser solid than with the liquid. The enhancement of the solidwall tension therefore demands a microscopic explanation. For the explanation a microscopically rough substrate with Van der Waals interaction is considered. The adsorbed material forms a monolayer solid on the substrate walls due to the attractive potential. The nature of this solid is crucial for the understanding of the reduced affinity for the solid state in the bulk pore. The solid conforms to the rough surface wall and is therefore irregular and microcrystalline. Any growth of a bulk solid is highly polycrystalline. This would give rise to a large solid contact surface area between the individual solid crystals and create many grain boundaries. They in turn have a high grain boundary energy which contributes to the effective interfacial energy between solid and liquid and disfavors the solid state. The pore environment rs especially
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37 prone for such a behavior with its large surfacetovolume ratio where the interference between crystallites is of the highest degree. The film on the substrate should appear as a foreign wall, not belonging to the rest of the liquid system. This is called classll growth and describes the imperfect wetting of a film by the condensed bulk of the same material. This also reconciles the observation of a large contact angle on a substrate surface and a monolayer of solid directly adjacent to the wall. The increased pressure for solidification of helium has been calculated using the increased chemical potential associated with the grain boundary energy. The excess average energy per unit area for largeangle boundaries is 67 E = 0.05/za, where a is the lattice spacing and n the shear modulus. This is multiplied by the volumetosurface ratio / = V/A of the porous material and equated with the pressure term of the energy which leads to where a is a constant depending on the roughness of the surface. It is about equal to the number of particles in contact with the walls of the pore if the surface nucleates a crystal at every particle site. The roughness parameter has been given between 10 by Shimoda 60 et al. and 40 by Dash. 64 The equation leads to the approximately right overpressure for the solidification. In the case of hydrogen one can again substitute the pressure energy term versus the latent heat and arrives at ' 'LS
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38 using the same notation all along. There is a discussion in place about the two different models proposed in the discussion of supercooling. Both models predict a reasonable value for the overpressure required to freeze helium. There is a clear distinction, however, between the two physical interpretations. In the grain boundary model a thin layer of adsorbate wets the surface and solidifies on it. This is not the case for the solidliquid nucleation model where a density fluctuation within the nonwetting fluid can become stable and solidify under the appropriate thermodynamic conditions. 2.5. OrthoPara Conversion in Molecular Hydrogen One of the most fascinating properties of molecular hydrogen is the explicitly quantummechanical classification into two species. The quantummechanical requirement of a totally antisymmetric wavefunction for two fermions can be realized in two distinct ways, leading to two distinguishable molecular species: orthoand parahydrogen. Orthohydrogen has a symmetric nuclear spin wavefunction (/= 1) and an antisymmetric orbital wavefunction (J odd). The opposite is true for parahydrogen. The orthopara ratio is determined by the availability of states given by the partition function. In the hightemperature limit the even and odd sums over J in the partition function yield the same result so that the spin parts lead to a 3:1 ratio of orthoto parahydrogen which is the relative weight of the nuclear spin degeneracy for the ortho and para species. Because of the large separation in the rotational energy levels, the equilibrium concentration of orthohydrogen drops from 75% to minute amounts at temperatures below about 10 K. At low temperatures essentially only the parahydrogen groundstate and the first excited
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39 state (ortho, J = 1) are occupied as a result of the large excitation energy in comparison to the thermal energy. The conversion process from ortho to para molecules requires, however, the simultaneous breaking of the spin and orbital symmetry, and the equilibrium ratio is therefore established only very slowly. For the isolated molecule the conversion is absolutely forbidden. In the condensed phase the rate is determined by the intermolecular magnetic dipoledipole interaction and magnetic field gradients caused by magnetic impurities. Because of this, orthoand parahydrogen can be treated as two different molecular species. The first theoretical calculation of the orthopara conversion in bulk solid hydrogen was performed by Motizuki and Nagamiya 68 and more recently revisited by Berlinsky and Hardy, 69 and Berlinsky. 70 The two main interactions responsible for the conversion are: 1) the nuclear spin dipoledipole interaction between two ortho molecules and 2) the interaction of the proton spin of one molecule with the rotational magnetic moment of the other. It is important to note that parahydrogen is not part of this process as it does not possess spin or orbital magnetic moments. The interaction Hamiltonian consists of two parts H int = H ss + H ra where "s" stands for the spin and V for the rotational contribution. The spinspin Hamiltonian has four identical contributions, expressing the combinatoric of the two pairs of proton spins involved. The rotational interaction involves only two contributions of the orbital momentum of one molecule with the two spins on the other molecule. The individual Hamiltonians are of the standard dipoledipole type (being described in chapter 3.1.) which can either be written in Cartesian or in spherical coordinates using spherical harmonics with the convention of
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40 Rose. 71 The total wave function is a product of the orbital and spin part. Orthohydrogen has 3 degenerate wavefunctions for the orbit, 0 Jm , and the spin, reflecting the quantum numbers J = 1 and / = 1 and their magnetic states. This leads to nine degenerate states. Parahydrogen only has one state O^^. This implies that there are 81 initial states, /, and 9 final states, f, for a conversion of two ortho molecules into one orthoand one parahydrogen molecule. The conversion rate flcan be calculated using Fermi's golden rule fl = ^XW",J/>l 2 S(E,E f ) n if where P, is the probability of being in one of the initial states /. The energy released in the process of conversion is AE = E,E f =BJ l {J l + yBJ f {J, + '\) = 2B = '\7 J \ K and must be taken up by the lattice. The bulk Debye temperature 1 1 72 73 is 120 K and indicates that one phonon is not enough to absorb the released energy. Instead, two or three phonons are necessary. Avoiding most of the tedious and involved calculation of matrix elements, the results for the twophonon process are discussed. The rate R is a combination of the dominant spinspin and the much weaker spinorbital contribution: 4/i s 2 where // 2 = 10/i*. Finally, R M is found to be
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41 ft 2 2MR 2 Q @ D ^=3.810 Â«Â«0Â©D with a complicated geometric factor 77 = 0.7331, depending on the Debye temperature, and the density p. The interesting feature is the strong dependence on the Debye temperature. Any changes in this quantity due to confinement would have great impact on the conversion rate. Threephonon processes are estimated to be only a few percent of the main contribution. Onephonon processes are not allowed under bulk conditions, as discussed earlier. However, under conditions which change the Debye temperature, like a restricted geometry or a twodimensional system, this could become important. Berlinsky calculated the onephonon process and expressed the rate as a function of density and complicated functions that have to be extracted from experimental data. He found that highdensity hydrogen mainly converts by emission of single phonons. The rate for onephonon processes is proportional to Q^R' 10 compared to G^ 3 /?" 12 for two phonons. 74 The calculations lead to conversion rates of 1.94%/h by Motizuki 68 et al. and 1.67%/h by Berlinsky 69 et al. which have to be compared with measured rates of 1.75%/h 7576 and 1.82%/h. 77 This is in excellent agreement although several approximations are introduced into the theory. The theory is using a Debye approximation with the energy proportional to the absolute value of the wave vector. This is a poor approximation for the high frequency part of the phonon spectrum. It also does not include anharmonicities due to the large ZPM. The interactions are averaged over a powder, assuming random orientations of crystallites. The conversion process is bimolecular or autocatalytic, described by
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42 x = kx 2 where x is the orthohydrogen concentration and k the conversion rate constant. The solution to this differential equation is X ( t )=Â—?Â° Â— w Ux Q kt where x 0 is the initial ortho concentration, 75% in the case of hydrogen at STP. If any paramagnetic impurities, for example oxygen, are present there is an additional process. 78 These impurities produce inhomogeneous static magnetic fields around themselves to which hydrogen molecules couple. The calculation procedure corresponds to the autocatalytic process. The main difference is that the decay is monomolecular and described by x = kx with an exponential solution x{t) = x 0 exp(kt) In the case of oxygen, 79 for example, the conversion rate is more efficient within a sphere of radius 2.95 R 0 than in the autocatalytic case. In this sphere mostly parahydrogen is found. Measurements regarding the influence of paramagnetic impurities 80 have been carried out showing fast exponential decay rates. The important result is a relationship between the rate constant and the square of the magnetic moment. This can be used, in an analogy with experimental data, to
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43 estimate an upper bound on the concentration of impurities present in a mainly clean system. 2.5.1. Diffusion So far, diffusion effects have not been taken into account. Diffusion homogenizes the distribution of orthohydrogen around impurities and in depletion zones. This establishes a perfect equilibrium distribution for temperatures above 8 K. 81 As long as this is the case the conversion is temperature independent because the energy of phonons created by conversion is much larger than the thermal excitation kT. Thus the occupation numbers of thermal phonons are extremely low and have no impact on the conversion rate. Assuming a binomial distribution W n for the equilibrium distribution of ortho molecules, the mean number of orthohydrogen molecules around each ortho molecule is given by M = f^nW n =Nx n=0 where N is the number of nearest neighbors. The orthopara conversion leads to a deviation from equilibrium which can be balanced by selfdiffusion which tends to homogenize the distribution. If the diffusion is slower, the conversion rate is determined by the diffusion bottleneck. In addition, clustering occurs as a result of the anisotropic interaction between orthohydrogen molecules which is of electric quadrupolequadrupole character. In dilute bulk samples orthohydrogen molecules can diffuse by quantum tunneling until they are close to an ortho neighbor. The two molecules can then form a bound pair as a result of their anisotropic interactions if the temperature is low enough (k g T
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44 due to the conversion of the clusters. For high temperatures, thermally activated diffusion is clearly the dominant process, leaving the molecules well separated, while at temperatures around 1.6 K clustering becomes significant. This was described by Schmidt 82 for bulk solid hydrogen. He found that the diffusion is large enough at 4 < T < 12 K to sustain the same conversion rate (k = 1.9%/h) while at 1.57 K the quadrupole effect overcompensates the reduced diffusion and leads to an increased conversion rate after 200 hours. The rate increases again after 900 hours in the bulk. This result is understood in terms of the enhanced conversion in clusters due to the reduction in the orthoortho separation compared to a homogeneous distribution. Oxygen impurities could be excluded by experimental control of oxygen contents which did not influence the conversion substantially. The reason is that oxygen freezes out at higher temperatures and has low solubility in hydrogen. This may be different for a substrate where oxygen can adsorb onto the surface and act as a catalyst for the conversion mechanism. In the case of a porous material the majority of the molecules is in contact with the wall and only a few nearest neighbors, depending on the roughness of the adsorption site, while the remainder is in a bulklike surrounding in the center of the pore with 12 nearest neighbors. The value of the conversion rate is therefore expected to be different for a restricted geometry due to interactions with the walls, the reduced number of nearest neighbors and the changes in the phonon spectrum. The orthopara conversion rate in an environment different from bulk needs to be determined accurately in order to characterize the hydrogen samples used in subsequent experiments.
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CHAPTER 3 NUCLEAR MAGNETIC RESONANCE: NMR 3.1. Dipolar Hamiltonian Hydrogen spins in a magnetic field experience the Zeeman interaction with the applied external field as well as interand intramolecular dipoledipole interactions. The intermolecular forces are weak compared to the intramolecular forces because the interactions fall off with the cube of the distance so that they contribute only a small fraction s = R intra V Inter J 3.75 (2) 3 V4 ) = 2.1% where the effect of the different spin quantum numbers for orthohydrogen and protons has been included. This small term can be neglected in the discussion. The orbital nuclear Zeeman energy associated with the Hamiltonian Hj=hy 2 H 0 J z and the spinorbit interaction H so = hy p \JH lnt vanish, i. e. are "quenched." 1 The argument was discussed in the quadrupolar section of chapter 1 and is related to the nondegenerate nature of the rotational groundstate. The dominant part of the Hamiltonian is therefore the nuclear spin Zeeman energy with a perturbation from the dipolar Hamiltonian 83 45
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46 H = yMM + 1]) + ^[l 1 Â• I 2 3(P Â• n)(l 2 Â• n)] where H 0 is the magnetic field in zdirection, I is the nuclear spin vector with superscripts 1 and 2 for the two spins and n = r 12 /r the unit radial vector between spins. The dipoledipole term can be rewritten in spherical coordinates in order to find a quantum mechanical expression H DD = ^{ l 1 l 2 3[/;cos0 + sinÂ©(/ ; ;cos + /;sincD)]x [/ 2 cos0 + sin0(/ 2 cosO + /Jsin)] } The angle 0 is taken between the direction of the field and the intermolecular axis. The x and y components of the spins can be combined to generate raising and lowering operators H DD = { V Â• I 2 3[l' z cosQ + i sin Â©(/>Â•* + /V*)] x [/ 2 cos0 + isin0(/ + 2 e1 * + /fe +i *)] } and can be collected into six terms H DD =^$(A + B + C + D + E + F) which correspond to different transitions. The individual terms are A = /;/ 2 (13cos 2 0) b = 1(1 3cos 2 e)(/]/ 2 1 1 Â• i 2 ) = 3cos 2 Â©)(/;/_ 2 + rii)
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47 C = f sinGcosee^/X 2 D = C =fsin0cos0e +i *(/]/_ 2 + /V 2 ) E = sin 2 0e2l */X 2 F = E* =sin 2 0e +2i */Vf The reason for displaying the dipoledipole interaction in this fashion is that the different terms cause distinct transitions. Terms A and B do not change the overall magnetic quantum number, C raises it by one, D lowers it by one and E and F increase and decrease it by two, respectively. Choosing degenerate states +) and +) with the meaning that spin 1 is up and spin 2 is down, and vice versa, demonstrates the operation of each term. Term A directly connects either state and is therefore diagonal. Term B flipflops one state into the other and therefore connects the superposition of states, which is the proper state in the first place. It is therefore also diagonal. Offdiagonal elements only contribute by means of a second order correction to the wave function which results from the perturbation of the dipolar Hamiltonian. Terms C and D are offdiagonal in first order, E and F in second order and contribute therefore only negligibly to the eigenenergies of the dipolar system. They will be omitted from here on. 3.2. Continuous Wave Lineshape The Hamiltonian is now in a form that can readily be evaluated for orthohydrogen with J = 1 and M +1, 0, 1. The wave functions are ++), vs(+) + +)) and  Â— )Â• Bracketing the Hamiltonian with these states produces the eigenenergies
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48 E ++ =^H o+ M(13cos 2 0) ^oo = ^(13cos 2 0) E__=yhH 0 + (1 3 cos 2 0) Ar ' Transitions occur between adjacent energy levels due to the selection rule A/W = Â±1 which leads to two different absorption frequencies that are no longer degenerate as they were in the pure Zeeman case 2 hv, = E_ E 00 = yhH 0 3 cos 2 0) Ar ' hv 2 = E M E ++ = yhH 0 ^(1 3cos 2 0) This calculation was carried out classically. As shown in the introduction the protonproton intramolecular axis is rapidly rotating, a consequence of the quantum rotor property. An average has to be invoked on the angular dependence. The frequencies v + and v. are symmetrically located above and below the undisturbed Zeeman frequency v 0 . The frequencies are v Â± =Â±Â— tf(3cos 2 0l\ 1 %y 2 where in the liquid case d = ' = 57.68 kHz. In solid hydrogen the 5 2n{r ) constant is slightly different with a value of 54.2 kHz. The angle 0 is measured between the external zdirection of the magnetic field and the intramolecular axis. It is physically more insightful to transform to another angle. This angle (5 defines the orientation of a surface or the crystal field in the local environment of
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49 an ortho molecule relative to the magnetic field. The average over all space with respect to this direction yields the 2/5 factor for a quantum mechanical object. The result is, in addition, also dependent on the rotational state ("shape") of the molecule characterized by the normalized quadrupolar order parameter This leads to the final result v Â± =Â±ad(3cos 2 /?l) which will be discussed as it describes the continuous wave lineshape. For a fixed order parameter a at a fixed angle fi two sharp lines occur which are inhomogeneously broadened. This is the case for a single crystal. If the angle /? is distributed from 0 to 90 degrees, as is the case of a powder, the lineshape has the famous Pake doublet form (Figure 2). The intensity corresponding to the frequency can be found from a space transformation argument. Since the dipoles are isotropically distributed the fraction of pairs lying in dp is d(cos/7). With the lineshape function being g(v) and the isotropic space intensity = Â— it turns out that An g(v)dv = l{Q)dQ *>n dv 18cfcrcos/7 2nd{cosp)
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50 0.014 0.012 P A K 0.010 E 0.008 0 u 0.006 B L E 0.004 T 0.002 0.000 200 150 100 50 0 50 100 150 200 FREQUENCY (kHz) Figure 2. Pake doublet with order parameter a = 1
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51 Solving for cos/? in terms of frequency and order parameter, using the frequency equation, results in the lineshape where v 0 = . The complete lineshape is a superposition of the positive and negative branch. The "+ H sign is valid for v 0 < v + <+2v 0 and the "" sign for 2v 0 < v_ < v 0 . The shape can be understood when the degree of space degeneracy is considered. The 90 degree case is much more probable because it covers the whole plane compared to one parallel alignment in the 0 degree case. When the constraint of constant cr is relaxed, the lineshape becomes a superposition of Pake doublets, leading to a bell shaped curve. This behavior is attributed to glassy 84 85 behavior. A more detailed equation for the frequency shifts has been given 86 incorporating the eccentricity r] = (j 2 x J 2 y \ of the hydrogen molecule V3 3do v t =Â±d oi3cos 2 /?1) + r7Sin 2 /?cos 2 /? In most cases the assumption of zero eccentricity holds. 3.3. Relaxation Times The dipolar Hamiltonian is frequently written in tensorial form
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52 2 H DD = Â£FÂ«M<Â« where the F W) are random functions of the relative positions of two spins in time and the A w are spin operators. The products just rewrite the terms A to F from before. Terms A and B are equivalent to q = 0, C and D to q = Â±1, and E and F to q = Â±2. The F { ~ q) = F [q) ' only contain the spherical coordinates and the A l ~ q) = A {q)+ , which are nonhermitian, consist of all spin operator components. The correlation function g measures the relationship between the random spatial functions at different times averaged over a statistical ensemble Their Fourier transforms are the spectral densities J^ico) which describe the interaction strength as a function of frequency The solution of the equation of motion for the density matrix a in the interaction picture is J qq ,(a } ) = jg q ,(T)e^dT where starred quantities indicate the interaction representation, transformed with the Zeeman Hamiltonian. This is just another terminology for the "rotating
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53 frame". The observable q in an experiment is averaged over the macroscopic collection of subsystems acted upon by the operator Q q'(t) = (Qy=tr{o'(t)Q} This is a relationship that describes the dynamics due to the dipolar Hamiltonian decoupled from the fast and trivial dynamics of the Zeeman Hamiltonian. For the hydrogen system the operator is Q = l\ + /*. Taking the trace over the solution of the equation of motion multiplied by Q results in where Â«,(;) = tr^H' DD (tr),[H' DD (t),(ll + Â£)]]
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54 dt = tr by generating the spectral density J from the random spatial functions. The computation of the commutators leads to an expression for the relaxation time in the high temperature regime by identification of terms l =  r ^ 2 /(/ + 1){/ 1 V 0 ) + J (2) (2^ 0 )} where co 0 is the Larmor frequency. A similar calculation leads to the transverse, or spinspin relaxation time. It describes the time dependence of the magnetization amplitude in the xy plane. The initial magnetization is assumed to lie along the xaxis with expectation value (l x )'=tr{l x o} = tr{e>">>l x e iH <'a} The time evolution for the xcomponent of the magnetization to the zero value equilibrium is d(ll + ll) dt = tr o The result of the evaluation leads to the transverse relaxation rate
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55 J=yV/(/ + 1){fJ (0) (0) + ^JÂ«( (yo ) + lJ (2) (2 Q ; 0 )} '2 The crucial difference between T : and T 2 is that T 2 has a contribution at zero frequency. For a correlation time x much shorter than the Larmor frequency all spectral densities become independent of frequency and equal to J w (0). It is possible to determine the ratio between the different orders of spectral densities by keeping in mind that j(o). jd). J( 2) = F (0) 2 :F (d 2 . F ( 2) 2 = 6 . t4 This implies that 7, and T 2 are equal in the case of short correlation times with "white" spectral densities for frequencies far above the Larmor frequency. 3.3.1. Spectral Densities 3.3.1.1. Rotational motion Functional relationships for the spectral densities are needed. They are developed starting from the correlation function g q (T) = FÂ«\t)FÂ«Â»(t+x) In the case of rotational motion the average extends over space elements Q and Q 0 . The average takes place over a function p(Q,f;n 0 ,f 0 ) which defines the probability of taking on the value Q at r and Q 0 at r 0 . This can be alternatively expressed as p{Q,t;Q 0 ,t Q )=P(n,t;n Q ,t 0 )p(Q 1 t)
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56 where P is the conditional probability of taking on Q. starting out with Q 0 . The rotation of a molecule can be compared to a sphere of radius b rotating in a medium of viscosity 77 and is described by a diffusion equation at b 2 s where A s is the Laplacian on the sphere and D s is Stokes' rotational diffusion coefficient 3 Qnr]b The solution to this equation is the probability P(Q,t;Cl 0 ,t 0 ) that the two spins have the orientation Q at time f after having the orientation Cl 0 initially. The initial condition of the problem is â€¢(Q,0) = Â£(QQ 0 ) and the solution is sought for in the form of a spherical harmonics expansion ^(Q,f) = Sc J M VO w (") JM noting that A s Vj M = J(J+1)Yj M . Introducing this into the diffusion equation leads to an equation in the expansion coefficients dr M n
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57 which describes an exponential relationship with time constant b 2 An expansion of the delta function can be given in terms of spherical harmonics which determines the initial coefficient and therefore the total time behavior of the expansion coefficient Cj(t) = Yj M (Q 0 )e~' /T . This supplies the conditional probability P(Q,r;Q 0 ,f 0 ) = XKHW (W"* needed to compute the correlation function using the isotropic probability p(Q 0 ,L) = Â— . Since the random spatial functions An with distance r between the two spins in the molecule are known to be r r V 1 5
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58 it implies that the exponential correlation functions are also established. By Fourier transform, the spectral densities are given by J (0 V) = 24 15r 6 1+o>V 4 r 15/6 1+) = J (2) (rÂ«1 5 a 3 JAcq)~ 47 for cm Â»\ where a is the nearest neighbor separation. The functions are proportional to the density n of particles in the material.
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59 3.4. Motional Narrowing and Second Moment As already mentioned, motion is an important ingredient in understanding the properties of hydrogen. In the case of NMR, the width and the shape of the absorption line are affected by this motion. When spins are in motion the field experienced by a neighboring spin is fluctuating on a time scale x c , the correlation time. These fluctuations are averaged out over a longer observation time. This average is much smaller than the instantaneous local field, thus the motional narrowing. There is an intuitive derivation of the time scales involved. The line width Acq is the inverse of the time t it takes for two spins to dephase by an angle of order unity. It is assumed that the local field difference Aco 0 between two spins stays the same with its sign changing at an average frequency of 1/ r c Â» Ao) 0 . The dephasing angle after time r c is SO = Â±t c Aq) 0 and the mean square angle of this random walk motion is AO 2 = /7<53> 2 = nAo) 0 2 T c 2 where n = tl r c . After a unit change of angle the rate is given by = Aco = Aa) 2 r c
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60 which means that the narrowing is more distinct for shorter correlation times. The effect on 7" 2 is directly related to the correlation time as it is the time it takes for spins to dephase by one radian r =JL = _L_ 2 Ad) Aco 2 r c It also shows the dependence of T 2 on the line width. The value of 7" 2 is a function of motion and can be quantified as a function of the spectral densities. In the case of rapid motion m c Â« 1 all spectral densities are identical while for slow motion m c Â»'\ the dominant contribution originates from J (0) (0). This leads to a ratio of (f x 6 + x 1 + j x 4) / (Â§ x 6) = ^ which states that T 2 is larger, the line width smaller, by a factor of 10/3 for total motional narrowing. The effect of increased motion on the lineshape and the second moment is investigated in the following. For this purpose the correlation function of the magnetization G(t) = tr{M x (t)M x } is calculated. The timedependent magnetization is a result of its equation of motion and leads to G(t) = ^\(E 0 s\M 0 \E' 0 sf exp{iay}exp \jo)(t')dt' where
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61 For random motion the correlation function is averaged over a frequency density f{a>) and collecting the prefactor leads to an expression G(f) = e^\P{a>)e^ rw dco = e^le^ The use of the central limit theorem rewrites the average e ^') = exp(i(jaKOrt'f) = expj \2 j (/ T)(o)(t)co(t T))dr and finally leads to G{t) = e w exp {co 2 )\(tT)gMdT where g is the reduced correlation function with the property g^O) = 1. With this result the two limits of short and long correlation times can be investigated. In the long correlation time limit (
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62 In the short correlation time limit {co 2 )?* Â« 1 it is permissible to neglect r compared to t. This leads to an exponential decay G(0 = e^exp{(^ 2 )rr c } whose Fourier transform indicates a Lorentzian lineshape. The conclusion of this analysis is that the lineshape, in addition to the line width, is an indicator of the dynamic regime the system is in, whether motional narrowing takes place or not. A discussion of the moments of the lineshape follows naturally. The nth moment is defined as M n =j(o)Q) 0 ) n f(a))dcD where f(oo) is the normalized lineshape function which leads to vanishing odd moments if it is symmetric around the Larmor frequency a) 0 . The previously introduced ( Q + S)=if(CD 0 )
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63 and leads to a value of 5 = AV2ln2 = 1.18A. In the case of a Lorentzian n S 2 + (a)co Q ) .where 8 is the HWHH, the moments are diverging if they are taken over all frequencies. A cutoff must be introduced, limiting the frequency to an interval o>a> 0  a which, of course, must be much larger than HWHH. The moments, neglecting terms Â— , are then found to be a M TtCX The large ratio of Â— i T = Â— indicates that this functional dependence should only be tried under this condition. In the Gaussian case the ratio is much smaller with a value of 3. 3.5. Heterogeneous Spin System Relaxation If spins experience different environments their relaxation times are also likely to be different. In the case of zeolite there are at least two distinguishable environments: the surface and the center of the cavity. A group of spins shall be called a spin system. The following discussion is restricted to two different systems a and b. For the moment it shall be assumed that the spin systems, experiencing two different environments, are independent of each other. Each spin system
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64 relaxes therefore independently and has a welldefined relaxation time or, expressed differently, inherent relaxation rate R a and R b . There is also a distinct fraction of magnetization p a and p b associated with each system. At this point the earlier assumption of independence between the spin systems is dropped. The consequence is that the spin systems interact with each other undergoing exchange processes. The exchange process can be a transfer of magnetization which happens by spin diffusion via spin flipflop, or particle diffusion. This interaction between spin systems is a more realistic assumption for physical systems. The outcome for this model is that the measured relaxation rates A + and A" and measured magnetization fractions C + and C~ do not correspond to the physical processes one is interested in. In addition to the previous relaxation rates, there are relaxation rates k a and kÂ„ between the spin systems which are not measured directly. They can only be extracted as a complex byproduct to the overall relaxation. A calculation 8788 will reveal the involved quantities. For simplicity of the following equations, a reduced magnetization is introduced for the longitudinal relaxation (saturation recovery) and the transverse relaxation M 0 The equations of motion for the magnetizations are Bloch equations extended by the additional cross relaxation between spin reservoirs
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65 d(SM a ) dt = R a SM a k a 5M a +k b 5M b = R b 5M b k b 5M b + k a 5M a where the above definitions are employed. These coupled differential equations are solved for the magnetizations in terms of the measured or apparent fractions and relaxation rates 5M, =Cex ''+C*ei '' with complicated expressions for A* = i(R a + R b + k a + k b ) Â± i([R a R b + k a k b f + 4k a k b ) with Cf = SM I0 (^^)Â±(5M I0 5M I0 )^ where subscript 0 indicates the value immediately after the pulse. Another important piece of information is the equation of detailed balance
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66 where p a and p b are the spin fractions of each group. By these means the apparent relaxation rates become dependent on the magnetization fractions of each group by substituting for one of the ks. For known p a and p b , which can be assumed for the case of zeolite from the fraction of molecules in contact with the surface, the real, inherent relaxation rates can be determined. This is achieved by inverting the A + and X~ equations leading to [(VA)C+Pb Ap a rk a ] "a = PbPa which can be plotted versus k a . Knowing the functional form of R a and R b , the functional form of the apparent rate A" can also be generated. The k a for which the calculated value of X corresponds to the experimental value of A" is the exchange rate between the two spin systems. The values of R a and R b can then be determined from the known k a value. The remaining difficulty is to reliably deconvolute two exponential decays. This is even a problem for sophisticated computer routines because of possible local minima in parameter space. As a consequence, initial guesses which are too far away from the true solution lead to a nonconvergent fit. 3.6. Methods of Detection 3.6.1. Continuous Wave (cw) Method In continuous wave mode the system is in a steady state and the spin system therefore in equilibrium with the lattice.
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67 3.6.1.1. Qmeterdetection The Qmeter method relies on a change in impedance as the resonance is approached. It is therefore necessary to study the impedance of a LC circuit. At first, the induction must be determined. The flux d> through a coil which is induced by a circulating current / is defined as
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68 If the circuit is tuned to the resonance condition, co 2 LC = 1, the impedance Zcan (coL) 2 be rewritten in terms of a parallel shunt resistance R = Â— Â— Z = R and the quality factor is expressed as Q = Â— = This can be simplified by assuming Q Â» 1 which is a reasonable assumption for real circuits with Q 100 Z \Anr\Qx\ = /4 4 m\Q% x '] The relative change in Zis AZ Â» AnRQT\x" and is proportional to the change in the detected signal. The dispersion is not measured with this method which is a drawback in itself. It does, however, have the advantage of avoiding the mixing of dispersion and absorption signals. Another problem with the arrangement is that the small signal voltage is superimposed on top of the large carrier voltage. 3.6.1.2. Bridge method The last mentioned problem is eliminated with a bridge arrangement. The carrier voltage V 0 is partially compensated by a bridge or "Tee" so that the detection amplifier does not saturate V = V 0 0i4xQiiz)Vi It is preferable to keep
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69 V Q V\ = aV 0 e x$ > 8V in order to generate a dispersion or absorption signal by having the correct phase on the compensating voltage. This leads to an equation for the voltage V This way, an angle 0 = 0 leads to the Qmeterlike absorption signal but an arrangement in practice is the adjustment of the phase angle and the constancy over time. The appropriate apparatus is a hybrid Tee which has an inverting and a normal input power branch. One branch is connected to a 50 Ohm load, the other is the sample path and is matched to 50 Ohm with an impedance matching device. Scanning through the resonance with a sweeping magnetic field achieves measurable detuning proportional to the response of the sample. Both detection schemes are supplemented with lockin amplification because of signaltonoise (S/N) problems. The lockin acts as a narrow band filter to improve the S/N. To achieve this, it modulates the swept magnetic field H 0 with an audio frequency Q and an amplitude much smaller than the line width to avoid distortion. This modulation leads to a differentiated signal, oscillating at the modulation frequency. This signal is then mixed, i.e. multiplied by, a sinusoid at the same frequency and averaged over time T angle
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70 where n(t) is the noise term, taken to be the Nyquist noise at temperature 0, which is only meaningful as its mean square value With the approximation of long average times (QTÂ» 1) the signal to noise ratio becomes implying that the band width of the lockin is Av = 2/T which improves the S/N by longer integration times at the expense of reduced resolution. 3.6.2. Transient Method 3.6.2.1. Coherent pulses The coherent pulse method opens up the opportunity of exploring line shapes much narrower than those limited by field inhomogeneities. Many pulse sequences are in use. Motionally narrowed line shapes are of special interest in this context. The Hamiltonian for the system in the rotating frame, during the application of a pulse along the xdirection, is given by N ^T4k@R Â°14 k@R
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71 where the H, field is much larger than the field in zdirection. This justifies the approximation H = yHJ x for the short time of the pulse. Applying for a time t means rotating the magnetic moment by an angle 0 = yH,t described by the rotation operator R(Q) = eiH ' = e w ' = e"'Due to the strength of the applied rf field, ideally turns the spins in a negligible time. The pulse sequence described in the following is the original 90, 7180,r sequence proposed by Hahn 89 and is graphically shown in figure 3. The steady field H 0 leads to an excess orientation of spins in zdirection (Figure 3a). After the steady state has been reached, the H r field is switched on for a short time in order to tilt the magnetization by 90 degrees (Figure 3b) from M z into M y : ( M. The same coil producing H, picks up the voltage induced by the magnetization rotating in the xy plane. Without relaxation effects, as expressed by 7", and T 2 , the alternating induced voltage would persist forever. In reality T 2 plays a role but the 7; effect is negligible, considering 7, Â» T 2 for a solid. This gives rise to dephasing which destroys the ideally persisting voltage by phase cancellation (Figure 3c). A free induction decay (FID) results from the superposition of all individual spin signals at slightly different fields. The signal decreases exponentially in the case of a Lorentzian line with a time constant T 2 . This T 2 is
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72 z z i<) it(
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73 a sum of all effects, especially the inhomogeneity of H 0 . It causes some spins to advance, others to lag behind. But this is not what really determines the physics. The goal is to eliminate this unphysical influence by applying a 180 degree pulse The 180 degree pulse produces a mirror image: the slowest spins are ahead of the faster spins (Figure 3d). The /rpulse acts like a time reversal operator. After the same time t between the nl2 and the ^pulse the fastest spins have caught up with the slowest. Since the inhomogeneity of the steady magnetic field has not changed during the time 2t the almost same FID occurs after this time (Figure 3e). The difference is that the voltage has the opposite sign of the rotated magnetization and a decreased amplitude. For the 90,T180 y r sequence the echo, called "solid" echo, has the same sign as the original FID. The reduction in signal amplitude is caused by effects that give insight into the microscopic behavior. Due to collisions during the two time intervals x only a diminished spin echo occurs, e.g. as a result of diffusion effects. T y effects reduce the measured according to 1 1 1 7"an " T t + 27, For T, Â» T 2 , as is the case of solids, this is not a crucial difference. It is important to notice that this pulse sequence is only able to refocus spin and not dipolar components. This means that the echo amplitude, corrected by the spinspin relaxation time, is a measure of the liquidlike fraction of the sample.
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CHAPTER 4 EXPERIMENTAL SETUP 4.1. NMR Setup Many components of the NMR spectrometer are used for pulse as well as continuous wave mode. With the necessary expertise it is therefore possible to switch between the two modes within minutes. This makes the existing apparatus versatile for the different information attainable by NMR. 4.1.1. Pulse Apparatus A diagram with an overview of the complete pulse NMR apparatus is shown in figure 4. The system will be discussed in detail, closely following the diagram. The radio frequency (rf) path originates at the rfgenerator which supplies the ultrahigh frequency (UHF) of 268 MHz continuously. This frequency corresponds to a magnetic field H 0 of 6.3 T for hydrogen according to the Larmor relationship. This magnetic field is in the upper attainable region of the available magnet and is chosen for high resolution and sensitivity, i.e. good signal to noise ratio (S/N), obtainable at high frequencies. The S/N is a function of the magnetic field to the 3/2 power. It therefore pays to work in a high frequency regime as long as no other effects, such as the skin depth are counteracting. Another argument is the short deadtime which is proportional to the reciprocal of the frequency. A continuous rf signal is not desired for the pulse mode. For this reason the pulse generator produces rectangular signals of 74
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75 Crossed Diodes Signal Averager Figure 4. Pulse NMR diagram
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76 variable length T and delay time r between them. Sequences can be programmed from the laboratory computer or selected manually. Four outlets connect four different gates which allow for a phase shift of 0, 90, 180 and 270 degrees. At the gates the rf frequency is blocked and passes only for the time adjusted by the pulse length T. To reduce noise from the rf generator pulse or leakage through the gates in the offstate, crossed diodes are placed after the gates. The diodes do not conduct for voltages less than 0.5 V which is above the noise level. For higher voltages they have no impact. A source of frequencies other than the generated 268 MHz is the rectangular shape of the pulse. A Fourier analysis shows that higher odd multiples of this frequency are present and useful power is lost. This contribution is also diminished by the crossed diodes. Pulse NMR in solidstate physics requires a large amplitude of rf which is associated with a short pulse. The values in the present apparatus are 15 G for the H, field which is equivalent to a ^pulse duration of less than 20 ^isec. In a negligible time the spins have to be tipped over by the desired angle to achieve a quasiinstantaneous situation where time effects are negligible. Another reason for the short pulse is the need for wide coverage of the whole frequency spectrum. These conditions are met by the two high power amplifiers of 40 and 300 Watts. A quarterwavelength line with attached crossed diodes is mounted in parallel. This device serves as noise reduction when the pulses are not turned on. For an understanding of this effect, transmission line theory 90 is necessary. There are special characteristics for A/2 and A/4 lines. In the case of a A/2 line the impedance on one end is simply transformed to the other end, except for the additional resistance of the used cable. For a A/4 line the input and output
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77 impedances Z,, Z 0 and the characteristic cable impedance Z = 50 CI in the standard coaxial case are related to each other by Z,Z 0 =Z 2 . These special cases are a consequence of the general formula for transmission lines Z, _ Z 0 + \Z\ankl Z ~ Z + \Z 0 \ankl 2n where = Â— is the wave number. For high voltages (above 0.5 V) the diodes At conduct and represent a low impedance. For the low impedance at the end of the crossed diodes the resistance at the other end of the A/4 line is very high. This implies that the high voltage level of the frequency corresponding to the wavelength of the spectrometer is not affected. Other frequencies do not match and are dampened proportional to their frequency deviation. For voltage levels below 0.5 V the diodes do not conduct and the net effect is an almost zero resistance at the carrier frequency. This mechanism decreases this frequency noise component but is less efficient at other frequencies. The centerpiece of the pulse spectrometer is the duplexer. Its task is to connect the transmitter, the sample and the receiver, but to simultaneously establish a separation of the transmitter from the receiver. This requirement becomes clear when the power of the transmitter (300 W) is compared to the 0.1 mW power of the sample signal: the highly sensitive receiver would be destroyed by the transmitter power. The duplexer consists of two impedance matching circuits. One transforms the cable impedance of 50 Q to the 3 Q of the series NMR resonance circuit. The other circuit matches the 3 Q to the 400 Q. of the UHF amplifier with an amplification factor of approximately 1000. The lowlevel amplifier is a narrowband, 3 stage solid state amplifier and is another key component of the spectrometer. A fourth stage is designed to
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78 provide variable matching to the input impedance of the incoming signal. The amplifier gain must be high but also robust against selfoscillation. The narrow bandpass is achieved with a resonance circuit for each stage consisting of a fixed coil and a tunable capacitor. To reduce readiness for selfoscillation resistors were soldered across the coils, at the expense of increased band width and reduced gain. The used tunable capacitors were of high mechanical and electrical quality in order to reduce mechanical vibrations and allow for precise fine tuning. The DC power supply to the doublegate transistors (3N21 1, 3N213) was decoupled from the rf path by large inductors. The individual compartments were also carefully separated from each other to prevent feedback between amplification stages. The doublegate transistors were carefully biased in order to operate in a region of high forward admittance. This is the case for about 2 V between the source and gate 2 and zero voltage between the source and gate 1 , which is the signal carrying input. Exact tuning of all resonance circuits is the key to high and stable amplification levels. The entire amplifier was placed into a brass box which was temperature regulated. This seemed necessary due to the temperature dependence of the transistors and the variable temperatures in the laboratory. A temperature regulator with a temperature probe was purchased. A heater was dimensioned to deliver enough power to keep the temperature at 32 centigrades. This value was chosen because it was above any reasonable room temperature. The receiver is protected by two sets of crossed diodes inside the duplexer which shunt the high level transmitter power and limit the input voltage into the amplifier. This could also have been achieved with an additional A/4 line in the main line. 91 Its function is described as follows. For the high transmission levels the diodes would turn on and the upper point of the A/4 line would represent a very high resistance from the transformation of the short
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79 circuit at its end. This would decouple the receiver stage. The small sample signal would only see a negligible resistance and could be processed further on its way to detection. Matching problems prevented this method from being employed. The NMR circuit is tuned to resonance (co 2 LC = 1) at 268 MHz, where the series circuit shows a minimum: Z 2 = r 2 +(1ofLC I coC). This is a desirable property of the series circuit because the small resistance of r = 3 Q only requires a few volts of applied power. An alternative parallel circuit would require several kV for the same power, which would lead to breakdown along the 1.5 m long coaxial cable to the coil. The sample coil must be dimensioned appropriately in conjunction with the tunable capacitor for the applied frequency. The inductance for small coils is determined by the formula 92 L _ N 2 a 2 9a + 10b where 2a is the diameter of the winding, /Vthe winding number and b the length of the coil. As long as b > 0.8a this is a good estimate. The inductance is chosen relatively large to obtain a large quality factor. This is restricted by the corresponding value of the capacitor which should not be below a few pF. The present values are 2.6 pF and 150 nH. The quality factor is about 100. The used coaxial cable is important because it should not introduce excessive losses. A homemade cable was a combination of coppernickel tubing and flexible braid in the lower part of the cryostat. Along the cold finger a commercially available, high quality, rigid coaxial cable was used in order to minimize spurious strain effects. The pulse system is adjusted to the coil. The
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80 inductance is matched with the input capacitance of the duplexer which can also be influenced by the coax cable length. The probe signal is processed in a phase detector after amplification to eliminate the carrier frequency. Preamplification before transformation to low frequencies yields an S/N advantage, because 1/f noise effects are much smaller at high frequencies. The phase detector or mixer receives its reference signal from the rfgenerator and passes only signals which are modulated with this frequency and higher harmonics which are, however, not relevant. By these means the S/N ratio is increased. A phase shifter allows for changing the reference phase. The possibility of detecting absorption and dispersion signals arises. Another amplification by 250 and a signal averager follow. Signalaveraging improves the signal to noise ratio by repeating the experimental procedure and adding up the digitized amplitudes of the signal. The signal improvement increases as the square root of the number of repetitions. The broader the NMR line, the broader the bandwidth of the amplifiers must be in order to avoid omitting parts of the frequency spectrum. A larger bandwidth deteriorates the noise and gain characteristics. The elaborate system at high frequency with the homogeneous magnet and the noise reducing schemes allows for detection of 10 16 spins at 1 K. Before cooldown the NMR system needs to be checked for pulse propagation to the sample coil. An auxiliary coil is placed around the NMR coil and the signal, as a response of the applied pulse, is induced in the pickup coil and detected. The connected oscilloscope displays a voltage of 40 V peak to peak via the weak coupling. This value represented a standard for good operating conditions.
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81 4.1.2. Continuous Wave (cw) Apparatus Many components of the pulse apparatus are also used for cw NMR. For this reason, the modifications and differences compared to the pulse system are described first. The setup is then explained following the complete cw block diagram displayed in figure 5. The entire pulsegenerating equipment including the high power amplifiers, gates and pulse generator is bypassed. Instead of employing a duplexer there are two different methods of how continuous wave data are obtained. Both principles, the Qmeter and the hybrid T have already been discussed in principle. The hybrid T supplies the receiver only with the sample response while the Qmeter is also subject to the input power. The hybrid T is an analogue 93 of the magic T at high frequencies. It works like a bridge and operates with a dummy load, which does, however, waste half of the transmitted power. The rf generator input is split in half and reaches the dummy load and the NMR circuit, which are both matched to 50 Q. As long as the sample is offresonance the hybrid T is matched so that the two paths cancel each other, because the sample branch is out of phase by 180 degrees. The matching is spoiled at resonance and only the NMR change is detectable by the receiver. This is the ideal case; in reality a small deliberate deviation from ideal phase or amplitude matching is adjusted as a signal carrier. The small error picks out the component of the NMR signal in the direction displaying a dispersion or absorption signal, respectively. 94 95 The advantage is an amplification effect and a deliberate choice of signal type instead of a mixture. Another major difference to the pulse apparatus is the modulated sweeping field. The steady field H 0 can be adjusted by a sweep coil in order to ramp the magnetic field through resonance. This is more practical than
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82 RFGenerator Phase Shifter Signal Averager Power Splitter Magnet ? (To I oQ Samp Circuit no Hybrid T 50 \90 \ "Sweep Coil LockIn Ramp Figure 5. Continuous wave (cw) diagram
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83 sweeping the frequency through resonance because all spectrometer components are carefully frequencytuned. The change in the static magnetic field in zdirection is not the only modification. In addition, the auxiliary sweeping field is sinusoidally modulated by 79.3 Hz. This frequency is deliberately chosen as an odd value, such that no other frequency matches it accidentally. The modulation changes the nature of the sample response. Instead of the absorption signal itself, its first derivative is obtained because the sine wave is projected onto the absorption lineshape. Thus, the absorption lineshape is probed yielding the strongest signal for the steepest slopes and therefore monitoring the first derivative of the line. Another change is the desired shift of the measurement to a higher frequency which reduces 1/f noise and allows for using a lockin amplifier. The lockin has the effect of narrowing the bandwidth and improving the signal to noise ratio sharply. Experimentally, the lockin parameters have to be chosen carefully. The time constant is a way to smoothing the recorded signal by integration. It is important, however, not to disguise narrow features with too generous time constants. The lockin is phasesensitive and a phase offset by 90 degrees totally eliminates any existing signal. It is clear that the right phase choice is important for maximizing the signal. The complete cw system can now be understood easily. Again the rf path originates at the frequency generator. The power is split into the signal and reference branch. The signal branch feeds via an adjustable attenuator to control the power input the Qmeter or the hybrid T which leads to the detection of the sample response. The weak signal is amplified in the lowlevel amplifier already described in the pulse mode section. The specialty of the input stage of this amplifier in Qmeter operation is that it performs two tasks at once: it matches the impedance and acts as the resonance capacitor of the tank circuit.
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84 This double task, which is performed by the appropriate tapping off the resonance coil of the first stage, is critical to the observed signal. In hybrid T operation the impedance matching is already provided by two capacitors, one to ground and the other to feed rf. This exterior impedance matching has also been successfully applied for the Qmeter alternative. The signal is mixed with the phase adjustable reference signal at the phase detector or mixer and then fed into another amplifier. The final stages are the lockin and the chart recorder. 4.2. Dilution Refrigerator In order to cool down to temperatures below 4.2 K a dilution refrigerator is used. Its components and the cooling process are described. A dilution refrigerator is operated with a 3 He/ 4 He mixture circulating in a closed gas handling system. The 4 He essentially represents an inert background while 3 He is cycling through the refrigerator. Two pumps drive the gas flow and determine the cooling power with their pumping speed. The gaseous 3 He at room temperature returns from the pumping system, enters the cryostat and condenses at a small bath of liquid helium held at 1 K, the "1K pot". Thermal contact is established by passing the gas through a cupronickel tube which is wound around and then immersed in the bath. The pressure drop of liquid 3 He from the condensation condition (100 Torr) to values corresponding to the still temperature (3 Torr) makes it necessary to introduce a flow limiting impedance. The pressure drop for a circulation rate N is determined by Poiseuille's equation Ap = 77Z/vV 3
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85 where ri is the viscosity, Z the flow limiting impedance, depending on tube parameters / and d, and V 3 the molar volume of 3 He. Without the impedance, the 3 He is in danger of reevaporating which gives rise to a subsequent heat load by recondensation and turbulences. On its way to the mixing chamber the 3 He must be precooled because the dilution cooling mechanism only starts at temperatures below the tricritical point of 0.86 K. Below this temperature the 3 He/ 4 He mixture spontaneously separates into a 3 He rich phase floating on top of a 4 He rich phase due to the mass difference. The 3 He tube is therefore led through the still and various heat exchangers for precooling purposes. Especially the heat exchangers represent an essential means of precooling and increase the efficiency of the refrigerator. Two different kinds are in use: continuous and discrete heat exchangers. 96 The continuous kind consists of an outer tube which guides the inner warmer tube. Instead of truly discrete heat exchangers with single copper blocks, "quasicontinuous" heat exchangers are employed. The heat transfer is ensured by a large surface area of sintered silver. Despite the large contact area the flow impedance is not excessive. The simple continuous counterflow heat exchangers are limited by the Kapitza resistance and viscous heating. The Kapitza resistance is due to the acoustic mismatch between the liquids and the separating metal tube where A is the effective surface area, AT the temperature gradient across the interface and Q the thermal flux. As a consequence the simple heat exchangers fail at temperatures below 100 mK and the more sophisticated designs are needed for lower temperatures. More efficient heat exchangers allow for higher
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86 circulation rates without increasing the temperature. 97 When the 3 He leaves the last exchanger it is colder than the tricritical temperature, once the refrigerator is working, and fills up the concentrated 3 He phase. At this point the dilution cooling process sets in as 3 He quasievaporates from the concentrated phase into the "He phase, or dilute 3 He phase. The cooling power can be quantified Qc=N*[H 3D (T u )H 3C (T M )] where T M is the temperature of the mixing chamber and Hthe enthalpy of dilute and concentrated 3 He. The available cooling power Q M at the mixer is equivalent to the external heat leak in equilibrium and is reduced by the heat load Q L of the incoming, warmer 3 He which has the heat loaded temperature T L q m =q c Ql = n 3 [h 2D (t m )h, c (t m )h, c (t l ) + h 3C (t m )] = N,[H 3D (T U )H 3C (T L )] The calculation is simplified by the assumptions that only 3 He is circulated and no frictional heating arises. It turns out to be important for the performance of the refrigerator to establish a high circulation rate. For further results the involved enthalpies are needed. The dilute phase can be interpreted as an ideal Fermi gas with the specific heat where T F = 0.38 K at x D = 0.064, the maximum 3 He concentration in the 4 He background due to the Pauli principle. For the concentrated phase
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87 CÂ„ = 247 moIeK which is found experimentally. The chemical potential is constant between the concentrated and dilute phase (/i 3C = ^ 3D ) at the coexistence curve The entropy is calculated to be and the enthalpy H 3C = H 3 (0) + T jC 3C dT = H 3 (0) + 1 27 2 Using this information leads to a numerical result for the cooling power 6 M =/N/ 3 (95r M 2 12r L 2 ) Two limits can be investigated. In the case of a single shot experiment or the use of ideal heat exchangers, i.e. when T L =T U , the cooling power is maximal and determined by
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88 In the other limit the cooling power approaches zero if r= 2.83. This again stresses the need for efficient heat exchangers. The mixing chamber, where the cooling power is generated, is thermally linked to the sample cell (Figure 6) by a cold finger consisting of annealed copper bars. The heat contact between the mixing chamber and the cold finger is established by a cake of sintered silver. The cold sintering is performed at 200 atm using a hydraulic press and a specifically machined piston to exert even force on the silver powder. Before sintering, the copper surface must be cleaned with striking solution, applying voltage in reversed polarity, hence reducing any disturbing cations. The so prepared surface is then galvanized with a solution of silver salt as a base for the sintering. After crossing the phase boundary inside the mixer, the 3 He atoms return to the still through the heat exchangers, driven by an osmotic pressure gradient. The temperature in the still must be high enough to establish a sufficient 3 He vapor pressure in order to maintain a certain circulation rate through the system. External power is supplied to the still Qs=N,[H 3 (va P )H 3 (cond)} This implies that the dilution refrigerator provides more cooling power when the still is moderately heated! Almost exclusively 3 He gas leaves the still because the vapor pressure of 4 He is only about 5% of the value for 3 He at these temperatures. Impurities such as air, hydrogen and oil derivates are separated from the 3 He by passing the gas through a 2stage liquid nitrogen trap and then pumping 3 He back into the system. By circulating 3 He, the process is maintained continuously. Without large heat leaks, temperatures of 50 mK can
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89 From Mixing Chamber Plastic Figure 6. NMR cell
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90 be sustained over a long period of time by refilling the 4 K He bath. The 1 K bath is filled continuously by an inlet through a suitable impedance from the 4 K bath. In a dilution process the number of 3 He is independent of temperature but the cooling power decreases with TÂ£. The lower the mixer temperature, the less cooling power is available. This is consistent with a version of the 3rd law of thermodynamics, stating that absolute zero temperature cannot be attained. In an evaporation process the number of molecules removed from the vapor is proportional to the vapor pressure which decreases exponentially. The cooling capacity per molecule, however, remains approximately constant. For lower temperatures the dilution process is more efficient because T 2 >Cexp(1/7") which makes the dilution refrigerator an efficient way of cooling to mK. The presence of impurities poses a problem to the small diameters of the impedances in the closed cycle of the refrigerator . If the system has a leak to air the gas path for helium will start plugging up at nitrogen temperatures. In the present system several leaks occurred during the course of operation. In the external gas handling system three different leaks were discovered and fixed. Another leak at a quasicontinuos heat exchanger inside the vacuum can opened up. Such a leak can cause losing valuable 3 He and prevents from obtaining low temperatures because the 3 He acts as exchange gas to the surrounding 4.2 K helium bath. Also the shaft seals of the mechanical booster and the sealed oil pump had to be exchanged. Another problem are pump oil derivates which can also block the gas pipes. With the problem of contaminated mixture gas was dealt by cycling the gas through a trap immersed in liquid helium. This is the most efficient way of cleaning helium gas from impurities. Another way to prevent clogging in the pipes was to run the refrigerator with a small amount of helium gas upon initial cooldown from room temperature. As a
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91 consequence, the mixture and any impurities are in motion and cycled through the nitrogen trap so that the probability of blocking is minimized. 4.3. Substrates 4.3.1. Zeolite The name "zeolite" is Greek and means "boiling stone". It is a consequence of its property to boil and melt to a glass as it is heated in a flame. Before reaching high temperatures, above 500 centigrades, the zeolite releases water but does not disintegrate. This is in contrast to most other waterbearing crystals and results from the porous structure of zeolite. There are other names associated with zeolites alluding to their unique properties: Molecular Sieve, Solid Solvent or Ion Exchanger. There exists a great variety 98 of naturally occurring as well as synthetic zeolites. All of them are defined "as aluminosilicates with a frame work structure enclosing cavities occupied by large ions and water molecules, both of which have considerable freedom of movement, permitting ion exchange and reversible dehydration." 99 They are grouped into Sodalites (Faujasites), Chabazites, Philipsites, Analcimes and Mordenites. 100 The most commonly used zeolites are the synthetically prepared and commercially available (Union Carbide Corporation, Box 372, South Plainfield, NJ 070880) zeolites X and A of the sodalitefaujasite group. Only their structure will be discussed. The structure 101 102 of zeolite is complicated but completely regular and resembles a "junglegym." Figure 7 provides a simplified view of the structure of zeolite A. 103 The fundamental building block of any zeolite is a tetrahedron of four oxygen ions with a centered aluminium or silicon ion. Each oxygen carries two negative charges, one of them being compensated by Si. The remaining single charge on the oxygen enables it to combine with other Si ions and to
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92 Figure 7. Structure 103 of zeolite A
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93 extend the crystal. If Si is substituted by Al one less charge is neutralized and an additional cation, such as sodium or calcium, is needed. The exchangeable cations attach themselves loosely to the oxygens and act as barriers into the channels connecting the pores. These additional, loosely bound cations are the reason for the facilitated exchange of ions, such as Na by Ca, in zeolites. The structure formed by the tetrahedra is similar to the structural unit in sodalite, the "sodalite unit." It contains 24 (Si, Al) and 36 oxygens and forms a truncated octahedron with one tetrahedron at each corner. To understand this one needs to know that a truncated octahedron consists of eight hexagonal faces, six square faces, 24 vertices and 36 edges. At this point the difference between zeolite type A and X comes into play. For zeolite A the octahedrons assemble along the square faces (Figure 7) while for zeolite X the hexagonal faces connect. This produces a central, truncated cube octahedron with an internal cavity diameter of 9 A for type 5A, consisting of eight sodalite units on a simple cubic lattice. For type X a tetrahedral arrangement of 10 sodalite units, as in diamond, is formed with an internal cavity diameter of 13 A and a quasi spherical shape with four caps cut off. These cages are termed a cages or supercages, in contrast to the /? cages within the initial octahedron of 6.6 A diameter. The p cages are too small to be occupied, except for crystal water, because their connecting channels are only 2.2 A. The s connecting channels between the a cages in type 5A are about 5 A wide, formed by eight bridge oxygens. In type X the connecting channels are 8 A wide, formed by a ring of twelve oxygen ions. Both types of supercages allow for occupation by small molecules. The chemical formulas for the aluminosilicates zeolite A and X are Na 56 [(AI0 2 )JSi0 2 ), x ]x264 H 2 0
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94 and A/a 86 [(>4/O 2 ) 86 (S/O 2 ) 106 ]x264 H 2 O respectively. This formula expresses the similarity between the two types with the difference of an altered AlSi ratio, which leads to the differences in the two lattice structures. The density of zeolite X decreases from 1.97 g/cc to 1.31 g/cc upon dehydration. This is even more prominent for type A with a reduction from 1 .99 g/cc to 1 .27 g/cc. The void volume or porosity is 50% for X and 47% for A. The relative surface area is large with 800 m 2 /g. The symmetry is cubic in both cases. Zeolites can be synthesized. It is thereby possible to create variations that do not exist in nature. The procedure is to prepare a highly reactive aluminosilicate gel which is an aqueous solution of aluminate, sodiumsilicate and sodium hydroxide. The gel crystallizes at temperatures between room temperature and 150 centigrades under atmospheric or elevated pressures. Large numbers of crystallite nuclei are formed which grow from the supersaturated gel. In addition to the fascinating microscopic structure which is worthwhile investigating in itself, zeolite is variably usable 104 in everyday applications. It is used as a water softener exploiting its capability of exchanging Na for Ca. The possibility of customdesigning the width of connecting channels by varying the adsorbed neutralizing cations opens a wide variety of selective filtering. A good example is the process of upgrading gasoline by separating chain from branching hydrocarbons. The chains slip through the channels which is not the case for the less ignitable branching structure.
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95 The ionic character of the aluminosilicates also allows for selective separation of differently polarizable molecules. Polar molecules and saturated hydrocarbons are more readily adsorbed. Zeolites may even be used as carriers of volatile catalysts. The catalysts are trapped in the pores and facilitate the chemical reaction but are not lost in the process. Important to keep in mind is that zeolites are very regular porous materials with a wide spectrum 105 of monodiperse pore sizes connected by i channels of diameters between 3.8 and 10 A according to the chosen type of zeolite. This allows the experimenter to select an appropriate pore dimension with the advantage of monodispersity. The outcome and results of an experiment are therefore more easily relatable to a specific pore radius. 4.3.2. Vvcor Vycor is an extensively used porous substrate. It is a silica glass produced by phase separation of boron in boron glass. The phaseseparated boron forms a connected network which leaves pores behind when leached out. 106 Vycor is commercially available (Corning Glass Works, Houghton Park, Corning, N.Y. 14830) with a wide range of average pore radii. This makes vycor a convenient substrate to use. Interest in vycor, as a material, spurred as experiments revealed a fractal dimension 107 of 1.74 in the range of 30 to 300 A. Average vycor has a mean radius of 30 A and a distribution between 20 0 and 55 A. Its porosity is about 28% with an effective surface area of 200 m 2 /g which is much lower than for zeolite. The density is 1 .5 g/cc. A drawback compared to zeolite is that the pore size distribution is broad and continuous. This makes definite statements about radial dependence difficult. Other problems are reduced interconnectivity due to dead ended pore paths which can act as dead volume in special, dynamic experiments.
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96 4.3.3. Exfoliated Graphite Exfoliated graphite is the classical twodimensional substrate with hexagonal lattice structure. There are several different kinds of exfoliated graphites available, depending on relative surface area and zaxis orientation of the twodimensional graphite flakes. From grafoil (Union Carbide) to papyex (Carbone Lorraine, 3741 Rue JeanJaures, 92231, Gennevilliers) to UCAR ZYX the zaxis distribution is narrowing at the expense of relative surface area. Graphite crystals are intercalated with a strong oxidizing agent such as sulfuric acid, carefully rinsed and rapidly heated. The twodimensional character is imposed on the substrate by rolling it into sheets. The density is 1 g/cc which represents a reduction by a factor of two by this process . The surface area for grafoil is strongly enhanced over graphite with 2030 m 2 /g but decreases to 0.3 m 2 /g for UCAR ZYX. Impurities like Fe, Al, Mn and Si are abundant at levels of 10 to 100 ppm. Outgassing the sheets by heating to 600 centigrades and vacuum pumping can reduce these levels substantially, but impurities still remain a problem for experiments that rely on clean surfaces. A severe problem for highfrequency NMR measurements is the electrical conductivity in the graphite planes. For low temperatures the resistivity decreases to 1.42x10" 5 Qm at low frequencies. 108 This conductivity would still allow for radiofrequency penetration into 0.2 mm sheets regarding the skin depth of 0.15 mm at 270 MHz. ESR and NMR measurements have indicated, however, that the conductivity is considerably enhanced at frequencies as high as 270 MHz such that skin effects expel the rf from the substrate and prevent from obtaining resonance signals for hydrogen on grafoil. 109
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CHAPTER 5 EXPERIMENTAL DATA 5.1 Sample Preparation The goal is to prepare a solid hydrogen sample confined to the cages of zeolite. Several steps have to be undertaken to achieve this. Zeolite 13X is commercially available in pellets and was crushed into a coarse powder to fit into the sample cell with dimensions of 3 mm in length and diameter. This powder had to be outgassed in order to eliminate any adsorbed water, oxygen and other impurities. The crystal water which is part of the zeolite complex is not affected by the outgassing procedure, otherwise the zeolite would "boil" and disintegrate. The outgassing procedure was performed for 10 hours at about 500 centigrades while pumping on the powder, establishing a vacuum of 10" 5 Torr. Loading the sample cell without readsorption of water and oxygen was succeeded at by performing the powder transfer in a dry nitrogen atmosphere. The powder was taken up in one end of a 3 mm hollow pipe and pushed into the sample cell with a pressure of nitrogen at its other end and then compressed with the endcap piston that provided the vacuum seal. Once the sample cell was loaded with zeolite the vacuum can and the dewar were mounted. The sample cell is therefore not directly accessible as is characteristic for a lowtemperature system. The only access is a capillary 97
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98 leading from the external hydrogen gashandling system through the cryostat to the sample cell at the bottom of the dewar. Hydrogen is gaseous at room temperature and contracts substantially upon cooling, thereby liquifying at 20.4 K and solidifying at 13.8 K for the bulk system. This requires a special technique for the preparation of hydrogen samples. Once the cryostat is cooled to 4.2 K with liquid helium, the sample cell must be heated to above the melting point of bulk hydrogen, to about 16 K, in order to avoid blocking the hydrogen filling capillary. This is done by applying a voltage to the hydrogen line heater, which is wrapped around the filling capillary, and adding a small amount of helium exchange gas in the vacuum can in order to establish a stable equilibrium temperature. The heating power on the line and the cooling power of the exchange gas must balance at the desired temperature. Once the system was prepared to these conditions a calibrated amount of ImMole of hydrogen gas was introduced into the capillary by opening a valve that connects the calibration volume with the sample volume. The capillary volume is hereby neglected. This is acceptable because most of the capillary is at a much higher temperature than the sample cell and represents only a small volume fraction due to the small capillary dimensions. The condensed hydrogen sample was then annealed for about 30 minutes at a temperature above the bulk solidification point in order to reduce crystal imperfections. Depending on the experimental requirements the hydrogen line heater was then reduced at different rates and exchange gas added to lower the temperature to equilibrium with the surrounding liquid helium bath. Two different NMR methods at 268 MHz were employed to investigate the hydrogenzeolite system. Continuous wave spectroscopy was used to observe the evolution of the quasistatic orientational order parameter which determines the lineshapes at low temperatures. The molecular dynamics were probed using
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99 coherent pulse NMR techniques. Nuclear spin echo amplitudes, spinspin or transverse relaxation rates T 2 and nuclear spinlattice or longitudinal relaxation times 7", were measured to determine the spectral densities of the motion at low frequencies and high frequencies, respectively. It is important to keep in mind that the total nuclear spin quantum number for parahydrogen is / = 0 and is therefore not detected with NMR. 5.2. Pulse Work "Ordinary" 90, r180, r spin echo techniques were used to monitor the liquid fraction of the hydrogen sample directly. The dipolar broadened lines from the solid component of the sample do not contribute to the ordinary echo, and the echo amplitude, after correction for the T 2 decay, provides a reliable measure of the liquid fraction, rf amplitudes of 15 gauss were used at 268 MHz for high sensitivity and the spectrometer recovery time was 35 //sec. The results of the pulse data have been published in Rail 110 et al. mainly focussing on the supercooling effect. 5.2.1. Transverse Relaxation Time 5.2.1.1. Temperature dependence The sample was cooled to 4.2 K within 115 hours. During cooling, the spinspin relaxation time T 2 was measured by recording the spin echo decay as a function of the time r between the first and second pulse at discrete temperatures. The relaxation can be interpreted as two exponential decays with a faster rate initially. This can be seen from a fit of the echo amplitudes for later r whosefit undercuts the echo amplitudes at early r. The attempt was made to extract the amplitudes and the decay rates of the two involved functions. The quality of the fit depends on the initial guesses used. For a few distinct
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R BOOO 0 2 4 6 8 10 12 14 16 IB 20 TEMPERATURE (KJ Figure 9. Intrinsic transverse relaxation times
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101 temperatures the deconvolution was performed and the two relaxation times are shown in figure 8. Both relaxation times display almost identical dependence on the temperature. This is the reason for using the single, average relaxation time for further discussion. The measured, deconvoluted values were used to determine the physical or intrinsic rates from the heterogeneous spin system relaxation theory described in chapter 3.5. Choosing p bulk = 20 % for the zeolite system allows for determination of T 2 wa ", r 2 bulk and /c b " u 1 lk , the interbath relaxation time, in the graphical fashion already described. The outcome is shown in figure 9. One of the main results is that T 2 b u,k displays a stronger peak than 7 2 wa " which suggests that the peak feature is mainly a result of the bulk component. The very long k^ lk peak value is an indication for negligible interaction between spin systems at the point of critical behavior. In the following discussion average relaxation times are used for reasons mentioned above. The data can reasonably well be fit by a single exponential which has the advantage of stable fitting results. The relaxation times show (Figure 10a) a strong temperature dependence at around 10 K. They peak sharply at this temperature, termed the supercooling transition temperature 7"^, displaying similar values above and below the peak. No effect was observed at 13.8 K, the bulk solidification point. This behavior has been interpreted as supercooling, the suppressed solidification of hydrogen in porous material. The observation of a finite width peak in T 2 instead of a mere drop from liquid to solid values is characteristic for the investigated system. It is explained in terms of the inhomogeneity of the microscopic environment for different hydrogen molecules. The width of the peak represents therefore a superposition of critical regions rather than one distinct collective transition. The interpretation of this data in terms of supercooling will be supported by further data in pulse and continuous wave mode.
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102 CD E F c o (3 X 8 0) Â© o > c CO 1200 1000 E 800 BOO 400 200 m m Sample 2 m Cooling up and down Ortho concentration x = 63.3 % T sc = 9.4 K 8 10 Temperature (K) 12 14 BOO _" 600 400 CD E F CO CD CC CD 2 CD Â« 200 C CO Sample 1 0 Warming up Ortho concentration x = 60.2 % T sc = 9.4 K mm Â« Â» Temperature (K) 10 12 Figure 10. Transverse relaxation time transitions
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103 2000 CO CD E Ic o CO a CD DC CD CD > I 1500 1000 500 Sample 12 Cooling down Ortho concentration x = 73.1 % T SC = 10.8K Â» * 0 1 2 a 4 5 6 7 8 9 10 11 12 13 14 15 IB 17 IB 19 80 Temperature (K) 1000 BOO BOO CO 0) E F c o a CO a CD DC 400 CD E I CO Â£ boo co Sample 12 Warming up Ortho concentration x = 50.8 % T sc = 8.5 K Â« H Â« " Â« Â• * 0 1 2 3 4 B 8 7 B B 10 11 12 13 14 IB IB 17 18 19 20 Temperature (K) Figure 10. continued
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104 5.2.1.2. Orthopara dependence The same measurements have been performed for different orthohydrogen concentrations (Figure 10). This was possible by waiting for the orthopara conversion to proceed until it reached the desired concentration level. The different shapes of the peaks are related to the orthohydrogen contents and the sample preparation time. The important new result of these studies is the ortho concentration dependence of the supercooling transition temperature T^. The supercooling temperatures obtained are between 10.8 and 8.5 K, suppressed from 13.8 K for bulk. The r sc could be fit to a linear function of the parahydrogen concentration (Figure 11). This had not been observed in previous studies 63 and is interesting because it indicates that supercooling is easier to achieve for parahydrogen, the spherically symmetric species. This is not unexpected because of the additional orbital degeneracy and the anisotropic interactions of the orthohydrogen molecules. The 7"^ inferred from the T 2 studies extrapolates to 13.4Â±0.5K at zero parahydrogen concentration, and 3.3+0.8 K for zero ortho content with a slope of 0.10 Â±0.01 K/%. The values of for low ortho contents are especially interesting because they are within the range of the predicted superfluid transition of 6 K. If the entire sample consisted of parahydrogen and the relation between supercooling and parahydrogen contents remained linear down to that region, superfluid hydrogen was a realistic expectation. This region, however, could not be probed using NMR because of the weakness of the NMR signals for very low ortho concentrations. The corresponding orthopara ratios in the sample were obtained from continuous wave studies discussed below.
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105 10 20 30 40 50 PARAHYDROGEN CONCENTRATION (%) Figure 1 1 . Supercooling transition as a function of paraH 2 concentration
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106 5.2.1.3. Hysteresis It has been established by thermally cycling through the peaks that the T 2 transitions are nonhysteretic. The T 2 peak in figure 10a was measured by warming and cooling about the transition temperature and is extremely sharp. It is therefore deduced that this supercooling is purely thermodynamic, and is different from the dynamic supercooling observed by other groups for droplets. For specific heat measurements of hydrogen in porous materials a word of caution is in place. In general, hysteresis is a phenomenon that describes the different behavior of a system when it is alternatively probed upon cooling and heating. Using NMR as a tool of investigation is unproblematic in this concern as it is not based on the application of heat itself. This is, however, not the case for a specific heat measurement. Here, no matter whether the critical region is approached from above or below, the sample's response to a heating pulse is detected. It is therefore not quite justified to speak of probing the transition upon cooling. Instead, heat is always applied and the quality of such data concerning the delicate, entirely heatdependent property of hysteresis is not unquestionable. In the present case of NMR data there is sample dependent sharpness of the T 2 peak and the values of T 2 , but this is not a result of thermal cycling. Another strong argument against hysteresis will be given using the 7* 2 data of hydrogendeuteride in zeolite. 5.2.2. Spin Echo Amplitude Not only the spinspin relaxation times T 2 display a characteristic behavior around 10 K but also the recorded NMR echo amplitudes. The general behavior is a steep increase around 11 K forming a distinct peak at about 10 K and then dropping sharply by a factor of 20. The latter change is also attributed to the liquid/solid transition. This behavior has been observed for many different
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107 samples (Figure 12) which were prepared under slightly different, but controlled conditions, to be discussed below. It was also the initial indication for supercooling in the hydrogenzeolite system. The recorded echo amplitudes must be corrected for the Curie law according to which the magnetization in the high temperature limit is proportional to the inverse temperature. The echoes are therefore multiplied by the temperature at which they were taken. It should be remarked that the initial change in measured echo amplitudes is due to the occurrence of the dramatic peaking of the nuclear spinspin relaxation time T 2 . The final drop of the echo amplitudes upon cooling is then a consequence of the diminished liquid fraction of the hydrogen sample. The average relaxation times were shown in figure 10 for the two samples, but two components could be distinguished which are also attributed to the narrow and broad components of the cw NMR line shape to be discussed later. Although the general features are the same for different samples the shape of this transition varies. The supercooling temperature T sc for the echo amplitude falls in the range of 9.5 to 12.4 K. The width and relative heights of the transition are equally sample dependent. The varying behavior depends on several conditions: the sample preparation which includes the temperature at which the hydrogen gas was condensed and the cooling rate after condensation, the orthohydrogen concentration and whether the sample was warmed or cooled while the echo amplitudes were measured. Two effects are observed. Firstly, the sharpness of the peak decreases as the cooling time increases as seen in figures 12a, 12b, 12d and 12e. The peak becomes broader and the relative height decreases. The discontinuity upon approaching the peak from high temperatures also disappears in this sequence. Secondly, a weak connection seems to exist between a longer cooling time and higher as
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108 600 500 400 300 200 too * Â« \ * M H o H M H Sample 1 Cooling time 1 hour Ortho concentration x = 75 % T sc = 9.56 K 10 12 Temperature (K) 14 16 6000 . 6000 4000 Q. Â£ < o o LU 2000 Sample 3 Cooling time 3 hours Ortho concentration x = 75% jf T sc 12 K * Â« Â« Â« / f * J 1 10 12 14 16 Temperature (K) Figure 12. Echo amplitude transitions
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109 6000 Sample 4 Cooling time 4 hours Ortho concentration # Â«M I M x = 75 % x ^ 4000 E, o c Q. E < O 2000 x: u LU T SC = 10K N M * Â« M M M M \ \ N M M N Â« N Â« M f 10 12 14 Temperature (K) 2000 X > 3 "5. E < o Â£ o LU 1500 1000 500 Sample 10 Cooling time 6 hours Ortho concentration x = 75 % 7T T SC = 10.7K N V M H 8 N Â« Â« N 10 Temperature (K) 15 Figure 12. continued
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110 Sample 12 Cooling Time 15 hours Ortho concentration x = 75 % T SC = 12.4K / 10 15 20 Temperature (K) Figure 12. continued
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111 600 M * M * # Â« Â•St * * Â« Â«Â«Â« Sample 9 Warming time 1 hour Ortho concentration x = 60 % T SC = 10K 10 20 Temperature (K) 0 # Sample 12 Warming time 12 hours ortho concentration x = 51 % T SC = 10.7K Figure 12.continued V 4 6 8 10 12 14 Temperature (K)
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112 demonstrated by figures 1 2a at 9.5 K, 1 2c at 1 0 K, 1 2d at 1 0.7 K and 1 2e at 1 2.4 K. Slower cooling allows the sample to form more homogeneously which leads to a less suppressed solidification. The warming transitions in figure 12f and 12g show a less prominent peak. This is assumed to be the case because parts of the hydrogen sample leave the cell upon warming. An additional reason for the different shapes is that the sample preparation, the temperature and the pressure of condensation, and the filling of the pores are never precisely reproducible. Two exceptional features could be observed. Figure 12b shows the bulk liquid/solid transition which is measured at 13.8 K by the drop in echo amplitude. The explanation for this effect is that bulk hydrogen was still outside the zeolite. This was the only time the pure bulk feature occurred. The other feature is shown in figure 12e. Instead of a single peak the feature rather consists of two or even three peaks. A speculation is that the pores (12 A) exhibit a different transition temperature than the channels (9 A) in zeolite. The transition occurs slowly enough to distinguish between the two different diameters. No other experimental data, however, lead to such a conclusion which makes this a tentative explanation. A more sophisticated analysis removes the T 2 effect on the echoes by extrapolating the echo amplitudes back to "zero" time. The fully corrected echo amplitudes are proportional to the liquid component. Figure 13 shows 7* 2 and the fully corrected echoes taken at an ortho concentration of 73 %. It is obvious how the T 2 peak coincides with the drop of the corrected echo amplitude. This leads to the conclusion that at this temperature the solidification takes place. 5.2.3. Longitudinal Relaxation For more information about the dynamics of the molecules the spinlattice relaxation time 7, was also measured. The chosen method to extract 7, was to
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113 2000 8 12 TEMPERATURE (K) Figure 13. Transverse relaxation time and echo amplitude vs temperature
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114 vary the repetition time of the set of double pulses. The idea is that the spins need time to recover back to the zdirection and less magnetization will be measured when the repetition time is short. This can again be monitored by the resulting spin echo amplitudes and fit to a relaxation time assuming exponential behavior. The problem with this method are the varying amounts of heat introduced into the spin system by the changing repetition times as the echoes are scanned for one relaxation time. The data (Figure 14) are not of high quality due to these problems with the data analysis but do not show real evidence for a transition over the temperature range 3 < T < 15 K. The 7j data for the cooling process (Figure 14a) seem to show a peak at 13 K and a dip at 10.5 K. This could, however, not be confirmed upon warming (Figure 14b). The result is that the relaxation time decreases overall with lower temperature with a substantial scatter, not clearly exhibiting any sharp feature at 10 or 13.8 K, the temperatures in question. This leads to the conclusion that the dynamics of the system is mostly governed by the zerofrequency regime where T 2 has contributions from the spectral densities but not 7j. As the sample is cooled the spectral density changes as a function of frequency. The area under the spectral density is conserved so that an additional contribution at lower frequencies must be taken from higher frequencies as the temperature decreases. The peak in T 2 is a consequence of the zero frequency contribution. Although the spectral density at low frequencies rises, it must, however, dip at zero frequency in order to explain the observed behavior. While the motion of larger clusters close to the critical region shifts towards lower frequencies the zerofrequency contribution even diminishes because no motion at all is only favorable in the solid state. This is the reason why T 2 drops again below the critical region in the solid. The observation that the 7, data does not show evidence of a phase transition supports the interpretation of the peak in T z and the spinecho amplitudes in
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115 100 C/3 E 75 CD E F c o 03 a CD cc c D 3 29 O) C o 50 Cooling down Â« * 12 15 Temperature (K) to E ISO 120 Â© E F c 80 o CO a CD GC 60 c o 3 c o Warming up * Â« 0 2 4 6 10 Temperature (K) Figure 14. Longitudinal relaxation times 12 14
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116 terms of the solidification of the liquid component rather than an abrupt change in the spectral density of the system. 5.3. Continuous Wave Work cw lineshapes were recorded to evaluate the quasistatic orientational order parameter a. The orientational ordering of a small number of quantum rotors with highly frustrated interactions raises a number of questions concerning the nature of the phase transitions and type of order achieved. Answers are expected from the shape and the timeand temperature dependence of the continuous wave NMR line. The results are published in Rail 111 et al. with emphasis on the anomalous ortho para conversion. 5.3.1. OrthoPara Conversion It was initially tried to track the orthohydrogen concentration by measuring the spin echo amplitude overtime. This, however, turned out to be an erroneous method under the given circumstances where the line was too wide for the pulses to cover the entire spectrum satisfactorily. Instead of measuring the changing ortho contents an additional change in the lineshape constituents was folded in. This is verified by the changing spinspin relaxation time over the course of time (Figure 15). The lineshape shifts its relative weights, where the wings gain more importance relative to the center component as orthohydrogen decays away. The echo amplitude is a slightly oscillatory function of the time between the two applied pulses with a frequency of 3 kHz (Figure 16). This nonexponential behavior is possibly a consequence of the different degrees of orientational ordering in the system. Continuous wave NMR was therefore used to track the orthohydrogen concentration. The derivative of the NMR absorption lineshapes is recorded
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117 1000 900 CO ^ 800 CD E 700 BOO c q CO CD cr CD 2 300 CD > CO c CO 500 400 200 100 100 300 200 Time (hours) Figure 15. Transverse relaxation time as a function of time 100 400 100 200 tOO 400 BOO 800 700 MO Time 2 r (/is) Figure 16. Echo amplitude as a function of pulse separation
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118 using standard magnetic field modulation and lockin techniques. The recorded information with such a setup is the differential lineshape. It is integrated in order to obtain the lineshape itself. As the area of the lineshape is a direct measure of the susceptibility of the sample and therefore also of the number of orthohydrogen molecules it is integrated a second time. The differential lineshape is therefore doubly integrated to determine the total nuclear magnetization and thus the orthohydrogen concentration. The way this integration is performed is crucial for the outcome of the quality of the data analysis. In the ideal case, the differential lineshape is perfectly symmetric about its inflection point in terms of amplitude and width. Any deviation in this symmetry affects the integration. It is therefore important to incorporate a measure of symmetry in the integration routine. This was solved by writing a routine that integrated each half of the line separately and compared the results for the two sides. Another advantage of this method is that the baseline correction is performed individually for both sides and also compared with each other as a further measure of data quality. It is thereby guaranteed that no spurious baseline contributions enter into the integration. The orthohydrogen concentration was measured over time for three different samples. The data were taken at 4.2 K up to 1550 hours. After carefully stabilizing the NMR spectrometer by regulating the temperature of the lowlevel amplifier to Â±0.1Â°C, a 1550 hour continuous experiment was performed. The result is shown in figure 17a. The data could be fit with the solution of a bimolecular process 1+ta(0)/
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1 119 400 800 1200 1600 TIKE (Hours) 0 400 BOO 1200 1600 TIKE (Hours) Figure 17. Orthopara conversion behavior
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Figure 17. continued
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121 where S is the signal, x(0) = 75%, which is the initial orthohydrogen contents at room temperature, and t the time. For the first 500 hours the conversion rate constant was determined to be k, =0.425 Â±0.006% /h and for the remaining time k 2 =2.21 Â±0.075% /h. Figure 17b shows the inverse conversion data which are fit with two straight lines. The relative error increases for small x because the signal decreases while the absolute error remains the same. The initial conversion rate is anomalously slow compared to the bulk value of 1 .9%/h at 4.2 K by a factor of approximately 4. Equally remarkable is the increase to an even larger value for long times. In order to confirm these results, two further samples were investigated to exclude amplifier drifts and other destabilizing influences. The conversion rate constants were k g =0.35 Â±0.08% /h and k c =0.54Â±0.13%/h. The errors in these latter experiments were larger due to the short measurement time but entirely consistent with the shortterm conversion of the extensively studied sample. This test was important to establish the stability of the NMR apparatus which was thermally regulated and the gain monitored carefully over the duration of the experiments. The use of an independent standard proton NMR sample in order to monitor the overall gain of the spectrometer was impractical at the UHF frequencies used for high sensitivity studies, and proved to be unnecessary. Assuming a binomial distribution W n for orthohydrogen in an equilibrium orthopara mixture the mean number of orthohydrogen nearest neighbors surrounding an ortho molecule is given by M = f^nW n =Nx where x is the orthohydrogen concentration and N the number of nearest neighbors. As long as the selfdiffusion in the sample is substantial it is capable
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122 of homogenizing the distribution which otherwise deviates from equilibrium as a result of conversion. An explanation for the slow initial conversion rate is the reduced number of nearest neighbors for the molecules on the wall which constitute about 80% of the sample. A simple calculation yields a reduced average number of nearest neighbors. It is assumed that about 20% of molecules are in the center of the pores having the standard 12 nearest neighbors and the remaining 80% having only 5 nearest neighbors due to contact with the curved wall which represents less than the halfspace. This leads to an average number of nearest neighbors of 6.4. This is more than half the value of bulk. This would explain a drop of the conversion rate by a factor of 3.5, since the rate varies as the square of the number of nearest neighbors. The conversion rate is also expected to depend on the hydrogenwall adsorption energy which will be shown to be about 270 Â±50 K using pressure versus temperature data when the sample was taken out. This value for the absorption energy is smaller but comparable to the value of 400 Â±100 K inferred by Monod et al. 112 The adsorption potential inhibits the mobility and the fluctuations of the molecules and reduces the catalytic influence at the wall. The conservation of energy during the conversion requires that the excess rotational kinetic energy (2B, = 171 K) be carried away by phonons. The Debye energy for bulk hydrogen at zero pressure is 120 K, and this requires a twophonon process for conversion in the bulk. One expects that the surface binding reduces the density of surfacephonon states and thus slow the conversion rate involving a molecule near the wall. Theoretical estimates of the phonon states have been carried out, and show that the mechanism is still a twophonon process. The density of phonon states is necessary for a quantitative comparison with the observations. The simple dimensional consideration of 2 versus 3 phonon modes on the wall reduces the density of states by this ratio. It
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123 is intuitively clear that the tortuosity of the porous material also restricts the longwavelength phonons reducing their effect of carrying away energy. These additional influences are difficult to express quantitatively but lead to a further reduction in the conversion rate coming close to the observed factor of 4. While a bimolecular process ("B" in Figure 17c,d) appears to provide the best fit at later times, the relative experimental errors are larger at lower concentrations and an exponential conversion could fit the data. Such a fit, as shown in figure 1 7c, is, however, inconsistent with the behavior at early times because it would numerically dominate the bimolecular rate at all times because any impurityinduced conversion would be constant in time and not be "switchedon" at t = 500 hours. The decay represented by the dashed line A1 of figure 17c corresponds to an exponential decay x(0 = x(0)exp(r/r 1 ) with a decay constant z, =340Â±10 h for T < 500 hours. As shown by the solid line A2 of figure 17d, the decay for late times can be fit by an exponential decay with a time constant r 2 = 573Â±15 h. The behavior observed at late times is similar to the Schmidt result for bulk hydrogen for lower temperatures (1.57 K). He found that diffusion is large enough at 4 < T <12 K to sustain the same conversion constant while for 1.57 K the quadrupole effect overcompensates the reduced diffusion and leads to clustering, thus accelerating the conversion. The enhanced conversion after 500 hours must therefore be explained by the fact that the mean number of nearest neighbors is larger than M 0 =Nx. This may be due to the formation of small clusters whose existence has been inferred from the structure of the cw NMR lineshape and its corresponding quasistatic order parameter. The quantum
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124 diffusion of ortho molecules in the interior of the zeolite pores will, however, be determined by the dipolar interactions as it is in bulk solid hydrogen. The overall influence of the wall potential is therefore expected to raise the temperature for which clustering in pairs or larger units occurs with respect to the bulk because of the stronger anisotropic potentials at the surface and the preferential orientation of the ortho molecules at the surface sites. This has been observed for the temperature dependence of the line width upon cooling. It can explain why clustering effects are observed at temperatures as high as 4 K in these geometries compared to an upper limit of 1.5 K for bulk hydrogen. For lower orthohydrogen concentration, ortho molecules can diffuse more easily as the condition for energy conservation for the process of tunneling is satisfied more probably. 3 This allows for ortho cluster formation at lower ortho concentrations and later times. Clusters of pairs and triplets correspond to a mean number of nearest neighbors of 2 and 3. This is of negligible influence for an original mean value of about 6 orthohydrogen molecules. As soon as the mean number drops to 2 the clusters start to contribute an aboveaverage value. From then on the conversion rate constant increases although the absolute number of converting molecules is still smaller than for earlier times. 5.3.1.1. Impurities The attempt to fit the orthohydrogen concentration (Figures 17c and 17d) using a single exponential fit did not yield satisfactory results. A concentration of approximately 100 ppm of paramagnetic impurities in the zeolite would be needed to generate surface conversion rates comparable to the overall rates observed. This estimate was obtained from a direct comparison of the studies of Cunningham and Johnston 80 of the conversion of hydrogen on paramagnetic impurities adsorbed on porous material. For Fe impurity concentrations x Fa =5*10" 2 on known surface areas, they observed decays of k Fg =1.7% per
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125 sec, while the present value is /f = 0.4% per hour. This leads therefore to an estimate of the required paramagnetic impurity concentrations ^ para ~ *Fe ^ Fe =130 ppm where ii Fe =5.92/i s is the magnetic moment of Fe and fj. g the Bohr magneton. This indicates that magnetic impurities in zeolite do not play an important role in the conversion process, and the sample preparation methods were successful in keeping the surface free from contamination. 5.3.2. Lineshape As discussed in chapter 3 about NMR, diatomic molecules give rise to a special line shape pattern. For axial quadrupoles the order parameter is given by g = Â±(3J] J 2 ). A single crystal sample would result in two NMR absorption lines at q) l Â±Aq)(P,g) with /? the angle between magnetic field and the local molecular symmetry axis, co L being the Larmor frequency. For a powder sample this leads to a Pake doublet with sharp features. If in addition a does not have a fixed value, the line shape becomes a smooth bell shape. For a frustrated system like the one in question it is highly likely to observe lineshapes that display some type of glassy behavior. The width and the specific shape lead to knowledge about the degree of ordering and frustration. The recorded lineshapes are characteristic for glassiness and have a surprisingly large width of about 130 kHz at 4.2 K (Figure 18). There is a dominant center component with a width at half height of 25 kHz and a wing component extending out to the full line width. The time and temperature dependence of the lineshape give further information about the nature and origin of the lineshape.
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126 100 ' . 50 0 50 Frequency (kHz) Figure 18. Time dependence of cw lineshape
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128 100 c < ra c g> 50 100 100 50 0 50 Frequency (kHz) 100 100 50 CO 'c Â€ < CO c 50 100 Sample 5 x = 19% Â•100 50 0 50 Frequency (kHz) 100 ure 18. continued
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129 5.3.3. Time Dependence The lineshape was followed as the sample aged (Figure 18), and it was observed that the narrow, central part of the resonance line, associated with a disordered contribution, with orientationai order parameter
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130 900 BOO 700 600 500 400 300 200 100 Â« Â«N Â« Â« Â« H H II M M Â« M "MM M II #" M Â« M Mi Sample 5 500 1000 Time (hours) 1500 Figure 19. Second moment as a function of time
PAGE 140
131 governed by the wing component while the center component has disappeared. From the pulse data it has been concluded that hydrogen is liquid at these high temperatures, which are even above the bulk melting temperature. This implies that the liquid is strongly constrained by the wall potential and is inhibited in its motion as is expressed by the wide line which is not motionally narrowed at such high thermal excitation. The nonexistence of a sharp transition around 10 K (Figures 20e, 20f, 20g, 20h) and the change in the nature of the line shape implies that the narrow lineshape component of the hydrogen sample leaves the cell first upon warming. This is related to the contribution in the center of the pores which is almost unaffected by the wall potential. The remaining hydrogen is a restricted liquid bound to the surface. This again points to a twocomponent system of two types of environments for the adsorbed hydrogen. Using a dilution refrigerator, the sample was cooled to lower temperatures. The question arose whether the system would orientationally order. This represents a crucial test whether the observed transition in the spinspin relaxation time is supercooling or a severely raised onset temperature of orientational ordering. For the bulk system the onset for ordering is related to the quadrupolar interaction constant of r = 0.8 K and is naturally dependent on the orthohydrogen contents. There is a possibility that this temperature shifts up, aided by the surface potential that locks the molecules in place relative to each other and the rough substrate wall. This potential has an anisotropic component which preferentially binds ortho molecules. Clustering and ordering is expected to occur predominantly at the zeolite surface because of its strong binding potential. The overall influence of the wall potential is therefore expected to raise the temperature for which clustering in pairs or larger units occurs with respect to the bulk because of the stronger anisotropic potential at the surface
PAGE 142
133
PAGE 143
134 60 40 c D 4 < 03 C g> CO 20 20 e) Frequency (kHz) 60 40 B c D Â€ < i CO 20 20 f) Frequency (kHz) Figure 20. continued
PAGE 144
135 60 20 50 50 Frequency (kHz) Figure 20. continued
PAGE 145
136 60 50 0 50 Frequency (kHz) 40 I 50 0 50 Frequency (kHz) Figure 20. continued
PAGE 146
137 and the preferential orientation of the ortho molecules at the surface sites. The occurrence of two conversion constants confirms this expectation. In analogy with the bulk data at 1.57 K and the explanation of the fast rate at later times in terms of clustering at 4.2 K some orientational ordering is present at higher temperatures than for bulk. This effect is, however, unlikely to be the reason for the transition at 10 K because the cw lineshapes do not exhibit any change in width over this temperature range. Indeed, the NMR lines broaden from 130 kHz to 200 kHz upon cooling from 4 Kto 0.94 K (Figure 20a, 20b), which indicates that additional ordering takes place at these characteristic bulk ordering temperatures. This implies that the surface supports the ordering, but not substantially, and that the transitions observed in pulse mode are not due to orientational ordering. This is a strong indication for supercooling at 10 K. The overall temperature behavior leads to the interpretation of a twocomponent system. The lineshapes observed at 4 K are found to consist of two components: (1) a relatively narrow, central (liquidlike) component corresponding to approximately 20% of the spins which is the bulk disordered contribution with a = 0 in the center of the pores; (2) a broad line, 130 kHz in width, originating from small clusters in contact with the surface walls. This is associated with a broad distribution of Pake doublets. It is deduced from a computer simulation for the Pake doublet superposition, discussed in the next chapter, that the order parameters a have a distribution around a mean of about 0.45 with a width of 0.3. In this analysis it is found that the outer wings of the Pake doublet superposition were below noise levels and not observable. The interpretation of this cw NMR result is that at least a certain percentage of ortho molecules are partially ordered as small clusters. The order parameters for an orthohydrogen pair is a = 0.25 and for triplets cr = 0.38. Complete ordering
PAGE 147
138 occurs for a value of unity. The roughness and inhomogeneity of the zeolite surface are expected to lead to such a distribution of a for partially ordered states. 5.4. HydrogenDeuteride The idea for using hydrogendeuteride (HD) is that it is very similar to parahydrogen. In HD, one hydrogen atom is substituted by one neavy hydrogen atom, the deuteron. HD has slightly more mass than hydrogen and therefore slightly less zeropoint motion. It otherwise behaves very similar to parahydrogen because its groundstate is also J = 0 and spherically symmetric. There is only one type of HD because it is heteronuclear, unlike in the case of hydrogen. This is also the reason for an exceptional property of HD: its center of mass and center of rotation do not coincide. The replacement of hydrogen by HD has the advantage that parahydrogen can be simulated. This is of special interest in the case where it is necessary to distinguish between orientational and translational properties. HD does not have orientational degrees of freedom because it is spherically symmetric. Orientational ordering can therefore not take place. This is another valuable test to distinguish between orientational ordering and supercooling at the observed 1 0 K. Four sets of data were taken for HD in zeolite which overlap with the work on hydrogen in zeolite. The longitudinal relaxation time 7, increases for lower temperatures when the sample is cooled (Figure 21) or warmed and is much longer than for hydrogen because of the missing dipolar interaction. Several data points were taken in the critical region but no discontinuity or peak were observed. Similar
PAGE 148
139 30 C/3 E E c g X TO 03 EC 75 c c o 20 19 10 Hydrogen Deuteride Cooling down Â« Â« Â« 0 j 2 3 4 5 6 7 8 9 10 11 12 13 14 19 16 17 18 19 20 Temperature (K) Figure 21 . Longitudinal relaxation time of hydrogendeuteride 4000 2> 3000 X > E. 03 TJ 3 2000 E < o LU 1000 Hydrogen Deuteride Cooling down i / 0 1 2 3 4 6 8 7 8 9 10 11 12 13 14 15 IB 17 18 19 20 Temperature (K) Figure 22. Echo amplitudes of hydrogendeuteride
PAGE 149
140 BOO CO .51 BOO 400 CD E hc o CO X Â«J a; CE CD w La CD U 200 C 2 Hydrogen Deuteride Cooling down T SC = 10.5K Â» N Â« * Â» M H M 0 1 2 3 4 S 6 7 B 9 10 11 12 13 14 15 16 17 IB IB 20 Temperature (K) BOO ^ 700 CO _f BOO CD E j_ 500 c o 09 400 .2 ^ 300 CD e CD > 200 m c 2 100 Hydrogen Deuteride Warming up T sc = 9.8 K Â« * 0 1 2 3 4 S B 7 B 10 11 12 13 14 15 IB 17 18 19 20 Temperature (K) Figure 23. Transverse relaxation times of hydrogendeuteride
PAGE 150
141 caution as in the case of hydrogen applies to this data . The corrected echo amplitudes decrease weakly upon cooling up to 10 K and collapse to small values thereafter (Figure 22). The identical behavior was observed upon warming. The T 2 data recorded over the interesting temperature range show similar behavior as for hydrogen (Figure 23). Again a peak is observable at 10.5 K for cooling (Figure 23a) and 9.8 K for warming (Figure 23b). The peak is only steep on the low temperature side and less distinct for temperatures above the peak. The difference in transition temperatures for cooling and heating is small and can hardly be made responsible for hysteresis. This is especially so because the hysteresis should lead to a lower transition temperature for the cooling procedure rather than a higher temperature. This is another powerful argument against hysteresis. The continuous wave line is of similar shape as for hydrogen with an increased width of 170 kHz. The convincing outcome of this experiment is that the observed effect is indeed supercooling, for HD and hydrogen. Worth mentioning is the fact that the suppression of the melting point is quite substantial in the case of HD because the bulk melting point is 18 K which represents supercooling by an entire 8 K. This is an extension of figure 11 the supercooling transition temperature as a function of parahydrogen because HD simulates 100 % parahydrogen. 5.5 Isochoric Pressure versus Temperature Data Valuable information can be extracted from heating the sample from the solid to the gaseous state while keeping the volume constant. The way the pressure rises is characteristic for the system. This procedure is performed
PAGE 151
142 when all measurements on the sample are completed and it is the last step towards taking out the sample. The hydrogen line heater is gradually increased so that the temperature and the corresponding pressure, measured at the filling line on top of the cryostat, are always in equilibrium. In practice the isochoric condition is not perfectly satisfied. When the sample is heated the sample cell volume becomes connected with the capillary, which had been disconnected by a plug of solid hydrogen. The capillary volume is, however, small compared to the sample volume. There are four distinct regions of pressure observable, slightly shifted for different samples, as the temperature rises (Figure 24). Initially there is no response in pressure until about 8 K where the first bulk material in the hotter capillary and the hydrogen in the center of the pores liquifies and raises the pressure. This continues until about 12 K from where on a mere expansion follows. At this point no more molecules contribute to the pressure but according to the ideal gas law the increasing temperature raises the pressure. The third region, starting at about 18 K, is again characterized by a steep rise in pressure. This is attributed to the desorption of hydrogen molecules off the substrate walls. It was attempted to fit this increase by an exponential ( e \ p = p 0 exp Â— js. V Kl J where E ads is the adsorption energy of hydrogen on the zeolite walls which is extracted from the rise of the pressure curve. There are problems with this analysis in addition to the possibility of nonequilibrium data points. Firstly, it is not clear where to set the baseline of the exponential including the problem of the linear, ideal gas law increase and the choice of the origin for the pressure
PAGE 152
143 1200 o 10 15 20 25 Temperature (K) 10 20 30 40 50 Temperature (K) Figure 24. Isochoric pressure versus temperature data
PAGE 153
144 and temperature. Secondly, as the pressure rises the number of molecules changes continuously. This has an impact on the pressure constant p 0 , which is neglected. The results for the three different samples are 204, 289 and 329 K with errors of several Kelvin. This results in a value of the adsorption energy E ^ =270 Â±50 K and is lower than an earlier estimate of 400 K. Compared to 3QS graphite with a strong adsorption potential of 500 K the higher value for zeolite seems especially high. The last region is again a linear thermal expansion region where all molecules are gaseous, setting in at 22 K. The expansion was followed up to 50 K with pressures of 1200 Torr and demonstrated the high quality of the heat shrinking vacuum seal of KelF on polished brass. Besides the similarities, the samples exhibit some differences. Sample 5 had almost completely converted out to only 5% orthohydrogen. Sample 7 was at x = 50% and sample 8 at x = 70%. This may be the reason for the different onset temperatures of the second steep pressure increase. The absolute pressures are a function of the hydrogen filling, the amount of exchange gas and the preparation conditions. The result from the data, gained from heating the hydrogen sample isochorically, is the existence of a twocomponent system. One contribution is the more volatile bulklike component in the pore center which becomes gaseous and leaves the sample first, leading to an increase in pressure. The other contribution originates from the more strongly bound component on the substrate wall which stays longer in the zeolite and leads to an estimate for the adsorption energy. This is in impressive agreement with the indications gained from the pulse and continuous wave work.
PAGE 154
145 5.6. Zeolite 5A versus 13X Additional studies were carried out using zeolite with smaller dimensions in order to determine the relative significance of the wall behavior. Zeolite 5A was chosen because of its smaller dimensions with 5 A channels and 9 A pores. This would allow for comparison with the previous data since the chemical composition and the microscopic structure is closely related. Remarkable is that virtually all hydrogen molecules have direct contact with the substrate walls. The supercooling transitions observed for hydrogen in zeolite 5A are much sharper. The measured (Figure 25a) and corrected (Figure 25b) echo amplitudes are almost constant until about 13 K and drop to zero at 10 K. The irregularities at 1 1 K are interpreted as fluctuations of the occurring liquid/solid phase transition. T 2 drops from 2 msec above 1 1 K to values <10//sec which is shorter than the recovery time of the spectrometer and echoes cannot be observed anymore. The cw NMR line shapes differ in a controlled fashion. In contrast to zeolite 13X, a narrower wing contribution is observed (Figure 26). The width at half height has changed little (20 kHz) but the line is only 50 kHz wide instead. of 130 kHz below 4 K (Figure 26 b). This confirms the view that orientational ordering is reduced in very small clusters. The line shape broadens from 40 kHz at 5.95 K to 110 kHz at 1.75 K upon cooling (Figure 26) also indicating a higher degree of ordering. The picture arising from the width of the resonance line is that the cluster size decreases with the reduction in available space. The adsorption potential is again extracted from isochoric pressure versus temperature data and is found to be smaller than for 13X, being only 100 K. This complementary data on another type of zeolite (5A) confirms the findings made for zeolite 13 X with the tendency towards more extreme results.
PAGE 155
146 Temperature (K) Figure 25. Echo amplitudes for hydrogen in zeolite 5A
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147 C D n c < CO c ra co 50 40 30 20 10 10 20 30 40 50 100 50 0 50 Frequency (kHz) 100 100 50 i2 c D I 0 c CO 50 100 100 50 0 50 100 Frequency (kHz) Figure 26. Temperature dependence of cw lineshape in zeolite 5A
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148 100 50 C D 4 < 50 100 100 50 0 50 Frequency (kHz) 100 100 CO Â«Â— *c < CO c o> CO 100 100 50 0 50 Frequency (kHz) 100 Figure 26. continued
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CHAPTER 6 COMPUTER SIMULATION From the data described in chapter 5 the system has been deduced to be a twocomponent system. It is therefore desirable to quantify the contributions and the exact order parameters involved in the quasistatic lineshape. The glassiness of the line suggests a superposition of Pake doublets which where discussed in chapter 3. The question is how wide the order parameter distribution is and whether the observed lineshape is reproducible with a combination of a Pake doublet superposition and a Gaussian line. The Pake doublets represent the partially ordered molecules on the substrate walls whereas the Gaussian is the result of the bulklike molecules in the center of the pores. Starting from this hypothesis a computer program was written to simulate the observed lineshape theoretically. The program was written on a personal computer with processor 80386 at 20 MHz with a math coprocessor. The computation time is still about one hour per spectrum. The language used was QUICKBASIC in the framework of LabWindows, a powerful data analysis software program. The task is split into three parts. Program 1, incorporating most of the work, performs the generation and superposition of the Pake doublets. Program 2 generates the Gaussian lineshape while program 3 adds the two components for a plot of the simulated lineshape. Going through program 1 will reveal the problems and solutions to the variable, unbiased and symmetric generation of Pake doublets and their 149
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150 summation. Therefore, a printout of the program, as it was run for the final version of the simulated lineshape, is presented: PROGRAM 1 PAKE DOUBLET SUPERPOSITION DIM shared 1(3001) DIM shared X(3001) DIM shared v(3001) Dim shared funct.filename as string * 30 Dim shared freq.filename as string * 30 Dim shared graph.filename as string * 30 Dim shared f1 as string * 30 c$="c:\LW\3dpake\" print "This program assumes data is written to ";c$ input "Name for Function file*;f1$ a1 %=fmt(funct.filename$,"%s<%s%s",c$,f 1 $) b1%=fmt(freq.filename$,"%s<%s%s%s",c$,f1$,"fr") 'name frequency file with "fr" c1%=fmt(graph.filename$,"%s<%s%s%s",c$,f1$,"g") 'name graph file with "g" oif% = openfile(funct.filename$,0,0,1 ) 'save array ovf% = openfile(freq.filename$,0,0,1 ) 'save array d = 54.2 'solid state constant; for liquid state the constant is 57.68 kHz p = 0 s = 1 CALL Grf Reset (12) z = 1000 ' Half number of sampling points FOR p = 0 to 3001 1 p is the array counter for the function values l(p)0 X(p) = 0 v(p)0 NEXT p ' Introduce a small parameter e to smoothing the divergences
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151 e = 1e2 ' Scan through the order parameter FOR t% = 1 0 to 75 stepl s = t%/100 print "s="; s ' Compute the function values of one wing of the Pake doublet ' The number of function values is proportional to the order parameter ' such that summation of lines is unbiased p = 3/2*z*s FOR l& = 50*z to 100*z step 100/s j=l&/100/z v(p+(zz*s)) = 3*j*d*s X(p+(zz*s)) = (3*1/2)*(1 2*v(p+(zz*s))/(3*d*s))*(1/2)/(1 8*d*s*(1 2*v(p+zz*s)/(3*d*s)+e)) P = P1 NEXT l& 1 Compute the other wing of the Pake doublet p = 3/2*z*s + 1 FOR m& = 50*z to 100*z step 100/s k#=m&/100/z v(p+(zz*s)) = 3*k*d*s X(p+(zz*s)) = (3A1/2)*(1 +2*v(p+(zz*s))/(3*d*s)) A (1/2)/(1 8*d*s*(1 +2*v(p+zz*s)/(3*d*s)+e)) p = p+ 1 NEXT m& ' Summation of the two wing components ' The function values are zero up to where the line starts according to s FORp = 0to z1z*s X(p) = 0 v(p) = v(p) NEXT p 1 Only one wing contributes to the first part of the complete line FOR p = z+1z*s to zz*s/2
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152 X(p) = X(p) v(p) = v(p) NEXT p ' In the center both components contribute FOR p = (zz*s/2) to z X(p) = X(p) + X(p( 1 + z*s) v#(p) = v#(p) NEXT p ' Notice the frequency correction for the line beyond the center point FORp = (z+1)to (z+z*s/2) X(p) = X(p) + X(p+1+z*s) v#(p) = v#(p + 1 + z*s) NEXTp ' Only the other component contributes FOR p = z + 1 + z*s/2 to z + z*s X(p) = X(p + z*s + 1) v(p) = v(p + z*s + 1) NEXTp ' The function values are zero again beyond the line according to s FOR p = z + 1 + z*s to 3*z X(p) = 0 V(p) =v(p + z*s + 1) NEXTp ' Integrate the line in order to normalize by the area Integra = 0 FOR p = 0 to 3000 Integra = Integra + (X(p+1) + X(p)) / 2 * (v(p+1) v(p)) NEXTp print "INTEGRAL of LINE with s=";s;"is";lntegra ' Normalize by Area of Line ' and weight the lines by their order parameter s
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153 FOR p = 0 to 3000 X(p) = X(p)/lntegra If s < 0.25 then s = 0.5s endif X(p) = X(p)/s ' Stronger contribution for pair clusters NEXTp FOR p = 0 to 2000 l(p) = X(p) + l(p) NEXTp ' End the order parameter loop NEXTt% ' Rle the function and frequency values nif% = fmtfile(oif%,"%2001f[w10]\ l() ) nvf% = fmtfile(ovf%,"%2001f[w10]", vO ) cif% = closefile(oif%) cvf% = closefile(ovf%) ' Set up the Graphics CALL SetAxName (0, "FREQUENCY (kHz)") CALL SetTxAlign (1,2) CALL SetAxName (1, "PAKE DOUBLETS") CALL SetTrtle ("3D PAKE DOUBLET SUPERPOSITION") CALL SetAxGridVis(1, 0) CALL SetGrdFrame (1) CALL SetCurv2D (0) CALL SetPointStyle (10) CALL SetXDataType (4) CALL SetYDataType (4) CALL GrfCurv2D (v(), l(), 2001) call saveGraphfile(graph.filename$, 0) ' save ail ports END OF PROGRAM 1
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154 In order to obtain reasonable accuracy and smoothness of the line the arrays are chosen maximally for a 4 megabyte RAM. The function and frequency values were computed and stored in files such that immediate access without recalculation was possible. This was useful considering the long calculation times involved. Starting from the Pake doublet equation where v 0 = ^Â— , it is known that the complete Pake doublet lineshape is a sum of the positive and negative branches. The plussign is valid for v 0 < v + < +2 v 0 and the minussign for 2v 0 < v_ < v 0 . The first difficulty was to guarantee that the frequencies for lineshapes of different order parameters a, here called s, coincide and have the same sampling density. This was necessary for the unbiased summation of different lines. The problem was solved by s dependent function sampling centered around the middle of the total sampling interval, here 2000 (two times z) data points. The order parameter is scanned through from a minimum to a maximum value in steps of 0.01 to get reasonable smoothness of the line. One complete Pake doublet itself consists of two separate wings. These have to be generated individually. Each contribution is stored in 1500 times s array points representing the equal frequency sampling density. The reverse order for the first contribution is necessary so that the line is symmetric for all order parameters s around array 1000. The main complication with the Pake doublets are the divergences at the positive and negative frequency V3 3da
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155 Vn = 3da , where the cosine of 90 degrees becomes zero, corresponding to the infinite contribution from the twodimensional plane. These divergences must be regularized in a fashion that is insensitive to an infinitesimal regularization parameter e. In principal there are four different ways to attempt this task. The correct method is the one where the result is independent of e in the limit of vanishing e. The methods can be separated into introducing the parameter for the frequency itself or in the entire denominator and whether it is real or complex. For a real regularization parameter for the frequency the Pake doublet equation takes on the form a Â± j v \ = Â— 1 oc pi at divergence and for the whole denominator 0*2 ( v) Â— , Â— at divergence For an imaginary regularization parameter only the real part of the expression must be considered 3*3(10 = Re V3 18cfa L v t + ie v l2 + \4v+e 2 l v 2 Q 1/2 V/ 2 + e 2 /v^
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156 where v = 1 Â— which is zero at divergence. The final possibility is g t <(v) = Re\Qdo V3 The interesting property of all expressions is the behavior at divergence. Expressions 1 through 3 have an obvious edependence at the divergence frequency which renders them inept for the regularization procedure because they are highly sensitive to the externally introduced regularization parameter. The last expression, however, has the special property of being zero at the divergence frequencies, and is therefore independent of e. The important difference is that the frequency in the denominator is small compared to the square of e, which is small but finite, and the numerator itself goes to zero independent of the value of e as the divergence is approached. The line converges quickly to its final form hardly changing for e values smaller than 10~ 2 . This value was chosen to incorporate smoothing of the divergences which in practice is a consequence of the ZPM, the temperature effect and the intermolecular interactions. The summation of the two components requires a careful addition of the regions where both are zero, only one contributes and where both parts are nonzero. This care is necessary for shifting the second wing component down to the same frequencies as for the first wing. The lines are then integrated to normalize them by dividing each line by its area. Another issue is the final weighting of the lines with different order parameters. It is conceivable in zeroeth order to weight them evenly and sum
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157 them directly. In first order the lines can be multiplied or divided by the order parameter. The most physical process is to attribute more importance to small order parameters. Small s implies small clusters which are very likely for the small pores of zeolite. A lower cutoff at the value for pairs of hydrogen molecules can be introduced which is equivalent to s = 0.25. The Gaussian line for disordered material covers the remaining narrow contribution. The calculated lines are then added to yield the superposition of Pake doublets. The final part of the program is concerned with plotting the lineshape. The result is shown in figure 27 for constant weighting of a between 0.1 and 0.75. Program 2 is a simple generation of a Gaussian line with varying area and line width. The function values are also stored in files and plotted (Figure 28). This line contributes about 20% to the total lineshape area and has a Gaussian width parameter of 5 kHz. Program 3 takes the calculated values from programs 1 and 2, adds them up and plots them. The lines are somewhat fuzzy in their contour because the number of lines superimposed is still not large enough to yield a completely smooth curve. The line is, however, clearly visible and interpretable. The results are displayed in figure 29a for constant weighting of s between 0.1 and 0.75. The shape is not quite satisfactory for the wings where the experimental line is strongly convex. An attempt was made to remedy this shortcoming. The large values of s from 0.75 to 0.25 were weighted inversely. This introduces the discussed cutoff at the value for pair clusters. The order parameters smaller than 0.25 were weighted inversely after correcting them to 0.5 s. This reduces their importance as well. The reasons are that inverse weighting of small s introduces an unphysical dominance of clusters that are on average smaller than pairs and that the Gaussian represents the smallest order parameters. The result is shown in figure 29b.
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158 PAKE DOUBLET SUPERPOSITION i .2 1 ; ; " 1.0 o.e o.ol 150 100 50 50 100 FREQUENCY (kHz) Figure 27. Pake doublet superposition GAUSSIAN 200 150 100 50 0 50 100 150 FREQUENCY (kHz) Figure 28. Gaussian wavefunction
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PAKE DOUBLETS plus GAUSSIAN 6 ~ 200 FREQUENCY (kHz) Figure 29. Synthesized cw lineshapes
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160 The result of the simulation is that the order parameters cr, i.e. s, follow a distribution around a mean of about 0.45 with a width of 0.3. The important conclusion in this analysis is that the outer wings of the Pake doublet superposition are below the noise level of the experiment and therefore not observable experimentally. This explains why the line is by far not as wide as a fully ordered hydrogen line with a line width of about 330 kHz. On the other hand it shows why the line is broad with some components close to perfect ordering (s = 1). The interpretation of this cw NMR result is that at least a certain percentage of ortho molecules is partially ordered in small clusters. The roughness and inhomogeneity of the zeolite surface is expected to lead to such a distribution of a for partially ordered states. The simulated line is compared with the experimental line and resembles it closely in width and appearance.
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CHAPTER 7 CONCLUSION In chapter 5 the measured data were presented and explanations given for the origin of the behavior of hydrogen in zeolite. In contrast, the present chapter will reorganize the data in terms of physical effects. The data are used to develop ideas about the system and its behavior as a whole. The conclusion chapter is in that sense paralleling the introductory chapter in that I review the open questions about confinement for quantum solids. This chapter will try to give answers to the raised questions as far as they could be resolved during the course of the measurements. The studies mainly focussed on Nuclear Magnetic Resonance data. The hydrogenzeolite system was probed in coherent pulse and continuous wave (cw) modes at 268 MHz. These two methods complemented each other with their information about dynamical properties, the quasistatic order parameter a and the lineshape. Additional information was gained from pressure data as a function of temperature. 1) Substrate Potential and TwoComponent System The substrate potential that acts on the adsorbed molecules constitutes a major difference to a three dimensional system. Its impact is very dominant concerning the behavior of the hydrogen molecules. An important implication of the data is the existence of a twocomponent system which has been attributed to the surface potential. This manifests itself in all experiments performed. The 161
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162 relaxation process exhibits two relaxation rates for the pulse data. The temperature dependence of the lineshape shows the disappearence of the center component upon warming while the broad contribution remains. Also, as the sample ages the narrow center component decays faster than the wings. A final indication is the isochoric pressure versus temperature data. The fashion in which the hydrogen leaves the sample clearly demonstrates two independent contributions. The adsorption energy of hydrogen on the zeolite surface was determined to be 270 Â±50 K from the slope of the second sieep increase in pressure. The explanation for the twocomponent system is the substrate wall potential. It binds the molecules in contact with the wall much more strongly than molecules in the pore center. 2) ZeroPoint Motion The impact of the zeropoint motion was not measured directly. From the behavior of other parameters it is, however, possible to draw conclusions of its importance. From the previously mentioned property of a twocomponent system it follows that the ZPM is not large enough to compensate for the attraction of the surface potential. If that was the case the molecules on the walls would have enough energy to be similar in their behavior to the molecules in the bulklike environment. This implies that the ZPM is relatively weak compared to the substrate potential. Another indication for the importance of the ZPM is the width of the transition. The fluctuations related to the ZPM widen the transition. In the liquid/solid transition of this experiment the transition width is large. This width is, however, interpreted as a result of the system heterogeneity, instead of an excessive ZPM contribution. The environments experienced by the hydrogen molecules are various due to different degrees of contact with the substrate walls as a consequence of the nonspherical pore
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163 shape. The critical region is therefore spread out over 1 to 3 degrees. The reason is that different sample contributions experience the critical region at different temperatures. The displayed transition is a superposition of individual transitions. This is in distinct contrast to the threedimensional system where the critical region is very narrow. 3) Orientational Ordering The expectation that quadrupolar ordering takes place at higher temperatures than for bulk originates from the assistance of the ordering process by the surface potential. It is expected that the surface potential reduces fluctuations and locks the molecules in place before this happens for an unconfined system. The peak in T 2 could be interpreted as the onset of orientational ordering. This was, however, rejected for several reasons. The line width was unaffected by a change in temperature around the transition temperature. It was still broad at temperatures as high as 15.8 K which implies the existence of nonzero order parameters for these temperatures. The remarkable line width of over 100 kHz at 16 K must be attributed to partial ordering. Another piece of information is the line broadening from 4.2 to 0.8 K which evidences additional ordering as the bulk ordering temperature is approached. It is important to note, however, that even for the lowest temperatures investigated no complete ordering is achieved in contrast to the bulk samples with
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164 a Hydrogen in zeolite 5A with pores of only 8 A was also investigated. Similar but more extreme results were obtained in agreement with the previous data. The obtained lineshapes are 50 kHz wide at 4 K. The small width is related to a maximum order parameter of 0.3, indicating small clusters. The picture that arises from the line widths measured for hydrogen in bulk, zeolite 13X and zeolite 5A is that the cluster size decreases with the smaller geometry. This is an intuitively obvious statement. 4) Melting Transition and Supercooling When a system shows unusual orientational properties the logical extension is to investigate the translational transition. The pulse data revealed a strong peak in the transverse relaxation time T 2 at 10 K. A similar peaking was also observed for the nuclear spin echoes. This behavior was interpreted in terms of supercooling and implies that hydrogen in zeolite solidifies about 5 degrees below its bulk melting point. Two features are especially remarkable about the supercooling transition. Firstly, the T 2 was found to be nonhysteretic which emphasizes a purely energetically driven process. This result is strongly backed by the T 2 peaks of HD which were almost identical upon cooling and warming. In addition, their variation is such that the temperature difference is reversed, compared to what would be consistent with hysteresis. Secondly, the supercooling transition temperatures exhibit a linear parahydrogen concentration dependence of O.10Â±0.01 K/%. This implies that parahydrogen can be supercooled more easily than orthohydrogen. Supercooling can be explained using two alternative theories: nucleation theory or the raised free energy for the solid state because of grain boundaries. In the grain boundary model a thin layer wets and solidifies on the zeolite surface. The temperature is depressed because grain boundaries in the solid
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165 state cost additional energy and the solid state becomes favorable only at lower temperatures. In the framework of nucleation theory the surface is not wetted. Instead, the solid develops from a fluctuation in the center of the pore which grows to its equilibrium size, the pore diameter of zeolite. The twocomponent system found in the present case suggests a combination of these models. A monolayer of hydrogen is in direct contact with the surface, wetting it. Its state is different from the bulk case. This can be seen from the high temperature lineshape which is extremely wide. Although probably liquid, hydrogen is still strongly confined to the substrate wall by the attraction potential. The remaining material in the center of the pore stays liquid until it solidifies according to nucleation theory. The immediate solidification on the surface layer is inhibited by the grain boundary energy. An alternative to supercooling is the dimensionality effect. From a molecular dynamics study by Brougthon et al., it was shown that a solid melts in layers. The outermost layer melts at 72 % of the bulk melting temperature. This leads to a melting onset of just below 10 K. This is in close agreement with the experiment and would imply that the melting in the pore is taking place in a twodimensional fashion from the layer in contact with the wall. The bulk contribution is small and therefore negligible. Instead of supercooling it would represent a twodimensional melting point. This seems unlikely, however, because of the surface interaction which does not allow for viewing the molecules on the surface as a free, twodimensional layer. The investigation of the T 2 transit'on in terms of supercooling was a major part of the performed experiment. Several indications, however, confirm this assumption. One indirect argument is that an orientational ordering transition can be excluded, as already discussed in the previous section about orientational ordering. The direct argument is an experiment performed with
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166 hydrogen deuteride (HD) instead of hydrogen. HD is in a pararotational groundstate and does not have orientational degrees of freedom, in this sense imitating parahydrogen. HD is, however, detectable with NMR. Echo amplitudes, T 2 and 7", data are in beautiful agreement with the hydrogen data and therefore completely rule out an interpretation of the 10 K peak in terms of orientational ordering. The lack of irregularities at 13.8 K, which is the bulk melting point, also points toward suppressed melting for the confined system. For one echo amplitude transition a discontinuity was observed which is a further confirmation because it represents the bulk effect in addition to the suppressed transition at 10 K. The ostentatious coincidence of the T 2 peak with the drop in corrected echo amplitude is a final confirmation of the liquid/solid transition at this temperature. The measured transition at 8.5 K is the largest value of supercooling in the literature achieved for hydrogen. 5) Orientational Glass The delayed and inhibited solidification has another implication. When the solid finally prevails, it incorporates a high density of grain boundaries and strain. Under these conditions it is easily conceivable that the order parameter is widely distributed as a consequence of the rough substrate surface. This is exactly what the continuous wave data show: a glasstype lineshape with a width of 130 kHz. This shape represents a wide line for disordered hydrogen and a narrow line for completely ordered hydrogen. It is a combination of broad wings with a narrow component in the center. This becomes obvious upon warming from 4.2 K when the narrow line disappears but the broad contribution remains without a sharp transition at 10 K. The understanding of the lineshape found its manifestation in a successful simulation which is a result of the sum of a superposition of Pake doublets with a distribution of the order parameter
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167 between 0.1 and 0.75 and a narrow Gaussian line. A distribution of the order parameter is a classical indication for an orientational glass. Its frustration is the incomplete orientational ordering which represents the failure of adopting the minimum energy. 6) OrthoPara Conversion It is necessary to know the orthopara hydrogen conversion rate in order to characterize the constantly changing orthopara ratio of the sample. Any data must be labelled by its concentration because of its strong dependence on the orthopara contents. The confined environment is expected to cause changes in the rate. The conversion parameters were extracted from the time dependence of the lineshape area. The conversion presents itself as a bimolecular process with two different constants for early and late times. For the first 500 hours the conversion is characterized by k, =0.425 Â±0.006% /h which is only a fourth of the bulk value. This behavior is explained by the reduced number of nearest neighbors. The molecules on the wall have less than half the number of nearest neighbors because the curved wall represents more than half of a molecules' space on the wall. The surface potential reduces the fluctuating magnetic stray field originating from a molecule by restricting its motion. Another effect on the conversion rate is the reduced density of phonon states with the dimension due to the decrease from 3 to 2 phonon modes. For later times the conversion accelerates to k 2 = 2.21 Â± 0.075% /h and is explained in terms of orthohydrogen clustering effects which make conversion more efficient due to more neighbors and the decreased separation between molecules. The cw lineshape changes over time in the sense that the narrow component decays Â• faster. This is consistent with the explanation of nearest neighbors and a bulklike environment for the molecules in the center of the pores.
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168 7) Superfluid Hydrogen The ambitious perspective of the project was to search for superfluid molecular hydrogen. The idea was to supercool hydrogen below the region of superfluidity which is predicted to be below 6 K. The additional supercooling from the small pore diameter of zeolite gave rise to the hope of achieving such extreme values. The results express some encouragement. The supercooling temperature is fit by a linear function of the parahydrogen concentration with a dependence of 0.10 Â±0.01 K/%. This implies more efficient supercooling for parathan orthohydrogen. Although the lowest measured liquid/solid transition at 8.5 K is still several degrees short of the predicted superfluid temperature this method has some potential. The reason is that if the linear interpolation of the supercooling temperature to parahydrogen concentration relationship held up to 100 % parahydrogen, the predicted liquid/solid transition temperature was at 3.3 Â±0.8 K. This could be within the reach of superfluidity. The NMR method is unfortunately inept to make a statement about such concentrations because parahydrogen by itself is not detectable. The surface tension a LS was mentioned to be a crucial parameter for successful supercooling in nucleation theory. This value was found to be greater than 1 erg/cm by Tell 63 et al. for hydrogen in vycor with a radius of 27 A. This a LS would theoretically be large enough for successful supercooling down to the superfluid transition. This value for a LS must be corrected downward, however, by the present zeolite experiment with the result of one fourth the radius but only twice the temperature suppression. This renders supercooling more difficult to achieve than expected by the earlier measurements. The question is, however, if it makes sense at all to expect superfluidity for a system that only comprises about 20 to 30 molecules per pore. It is
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169 doubtful whether collective behavior such as Bose condensation is possible for such a system. The performed experiment on hydrogen confined to the pores of zeolite explains a variety of questions and makes a contribution to the general understanding of properties that are related to a system characterized by confinement. Further investigations are necessary to shed light on the mechanism of the suppressed solidification, the understanding tor properties of small numbers of particles, where statistical mechanics do not apply in the traditional sense, and the existence of superfluid hydrogen.
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BIOGRAPHICAL SKETCH Markus Rail was born on the 20th of July 1964 in Sindelfingen, Germany, as the son of Albrecht and Doris Rail. After the regular 13 years of Gymnasium (high school), he received the Abitur (Baccalaureat) as valedictorian in 1983 from the Albert Einstein Gymnasium, Boeblingen. Two years of undergraduate studies at the University of Stuttgart were completed with the Physics Vordiplom (Bachelor of Science) in October of 1985. An intermediate 15 months of mandatory military service followed. He continued his studies at the University of Stuttgart for two more semesters before transferring to the Graduate School at the University of Florida in the fall of 1987. There he received a Master of Science in physics in the fall of 1988 with Prof. N. S. Sullivan and extended his stay to pursue the degree of Doctor of Philosophy. 176
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a thesis for the degree of Doctor of Philosophy. Neil S. Sullivan, Chairman Professor of Physics I certify that I have read this study and that in my opinion h conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a thesis for the degree of Doctor of Philosophy. (I > E. Raymond Andrew Graduate Research Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a thesis for the degree of Doctor of Philosophy. Charles" F. wooper Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a thesis for the degree of Doctor of Philosophy. David B. Tanner Professor of Physics
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a thesis for the degree of Doctor of Philosophy. John Kflauder J P/rofe^sor of Physics and Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a thesis for the degree of Doctor of Philosophy. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a thesis for the degree of Doctor of Philosophy. This dissertation was presented to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1991 xfqsepn H. Simmons (Professor of Material Science and Engineering Dean, Graduate School
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