NMR studies of the orientational behavior of quantum solid hydrogen films adsorbed on boron nitride


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NMR studies of the orientational behavior of quantum solid hydrogen films adsorbed on boron nitride
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xv, 200 leaves : ill. ; 29 cm.
Kim, Kiho, 1962-
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Physics thesis, Ph. D   ( lcsh )
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Thesis (Ph. D.)--University of Florida, 1996.
Includes bibliographical references (leaves 189-199).
Statement of Responsibility:
by Kiho Kim.
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SCopyright 1996


Kiho Kim

To My Parents


To My Wife


I would like to thank Professor Neil S. Sullivan, my adviser, not only for his stim-

ulating suggestions and valuable advice, but also for his patient encouragement. He

has taught and trained me in numerous aspects of low temperature physics, including

the theoretical background and experimental skills of NMR.

I would like to thank Professors G. Ihas, H. V. Rinsvelt, S. Hershfield, and W. S.

Brey for serving on my supervisory committee. My special gratitude goes to Professor

G. Ihas for his advice on various low temperature problems and for his encouragement.

In addition, I would like to dedicate this dissertation to my parents and my wife

who have supported me with their precious daily prayers. I am truly indebted to my

parents very much, which my meager writing abilities cannot express, by their love

and encouragement throughout my life. My special love and gratitude go to my wife,

Jungwon Shin, who has always been supportive, encouraging, and sacrificing herself

with our two lovely sons, Jooho and Juneho.

I would like to express special thanks to all of the members of the physics machine

shop, especially to "uncle" Robert Fowler, and to the members of the physics elec-

tronic shop for their support on various projects. Also, the cryogenic engineers, Greg

Labbe and Brian Lothrop are thanked for preparing liquid helium and providing it

on time.

I would like to express my gratitude to Dr. J. Hamida, and three graduate col-

leagues, E. Genio, S. Pilla and P. Moyland, for their helpful discussions and experi-

mental help. My special thanks go to Dr. J. R. Bodart who cooperated with me on

my crucial first project and helped me in the early stages of this project.

Finally, I would like to thank my undergraduate Professors, J. H. Kim, C. T.

Ahn and I. J. Choi, who had encouraged and supported me. Also, my undergraduate

colleagues and friends, Dr. J. M. Han, Dr. S. S. Jung, and I. H. Huh, are thanked

for their encouragement and friendship.

"God is love" (1 John 4:16)


ACKNOWLEDGEMENTS ..................................................... iv

LIST OF TABLES ..................................................... ix

LIST OF FIGURES .............................................. x

ABSTRACT ...................................... ............................. xiii


1 INTRODUCTION ................................................. 1

1.1 Overview of the Properties of Hydrogens ...................... 1
1.2 Chronicle of Researches on Hydrogens in 3D ................. 11
1.3 Chronicle of Researches on Hydrogens in 2D ................... 13
1.4 Boron Nitride as a New Substrate ............................. 19
1.5 The Dipole-Dipole Interaction ................................. 21
1.6 Intermolecular Interactions ............. ...................... 21
1.7 Orientational Properties of Hydrogen .......................... 24
1.8 Order Parameters ........................................... 29
1.9 The Orientational Ordering in 3D ........................... 31
1.10 Two-Dimensional Orientational Ordering Model ............... 33
1.11 Hydrogen Molecules on a Surface .............................. 36
1.12 Liquid Hydrogen: Supercooling and Superfluidity .............. 39
1.13 Theoretical NMR Line Shapes ................................. 42

2 THE ENHANCED NMR SPECTROMETER ....................... 50

2.1 Overview ........................................................ 50
2.2 The UHF CW-NMR Spectrometer ........................... 52
2.3 The NMR Signal Processing .................................. 58
2.4 Conclusion ......................... .... ................... 59

3 THE NMR SAMPLE CELL ....................................... 60

3.1 The NMR Sample Cell Structure ............................. 60
3.2 Summary .......................... ......................... 63

4 THE EXPERIMENTAL STUDIES .................................. 65

4.1 The Sample Preparation ....................................... 65
4.2 The NMR Data Processing .................................... 67
4.3 Data Processing ........................................... 69

5 STUDY OF 12-LAYER HYDROGEN FILMS ....................... 70

5.1 Overview ........................................... .... 70
5.2 Results .......................................................... 74
5.3 Discussion ............................................... 80
5.4 Conclusion ................... ............................. 91

6 STUDY OF BILAYER HYDROGEN FILMS ....................... 92

6.1 Overview ......................................... ....... 92
6.2 Results ........................................... ...... 96
6.3 Discussion ............................................... 99
6.4 Conclusion ...............................................114


7.1 Overview ................................................115
7.2 Results ................ .................................118
7.3 Discussion ............. ............. ..................... 131
7.4 Conclusion ..................................................... 133


8.1 O verview ....................................................... 134
8.2 R results ............................................ ............ 136
8.3 Discussion ............... ..................... ........... 156
8.4 Conclusion ...................... ............................. 160

9 SUMMARY AND FUTURE RESEARCH ...........................161


A THE DILUTION REFRIGERATOR .................................164

A.1 Overview ............... ......... .. .................... 164
A.2 Liquid mixtures of 3He and 4He .............................. 165
A.3 Principle of the dilution refrigerator ........................... 169
A.4 Osmotic Pressure ................................ ............ 171
A.5 Refrigeration Capacity ............... .......................... 174

B OPERATIONAL MANUAL .................................... ..... 175

B.1 Cool down procedures .................. ......................... 175
B.2 Gas handling system ............................................ 184

LOW TEMPERATURES .........................................186

REFERENCES .................................................................189

BIOGRAPHICAL SKETCH ...................................................200


Table Page

1.1 Parameters (r7, c, o) in L-J potential........... .......... 4

1.2 The ortho-para designation.................................. 5

1.3 Electrical quadrupole-quadrupole interaction energy .............. 22

1.4 Definition of the operator equivalents n" and Om ................. 28

A.1 Properties of pure and saturated dilute 3He. ................... 168

C.1 Some relevant properties of Solid hydrogen ..................... 187

C.2 Heats of transformation of selected hydrogen isotopes at the triple points. 187

C.3 Specific volumes of selected hydrogen isotopes at the triple points. 187

C.4 Triple and Critical Points of selected hydrogen isotope ............ 188


Figure Page

1.1 Form of the Lennard-Jones potential .......................... 3

1.2 The registered triangular lattice of H2 on BN. ................... 15

1.3 The layered structure of (a) boron nitride and (b) graphite. ........ 19

1.4 Four orientational geometries ................................ 23

1.5 The molecular rotational energy levels for an isolated H2 molecule .... 25

1.6 The phase diagram for bulk hydrogen. ......................... 32

1.7 2D model of quadrupolar orientational ordering. .................. 34

1.8 The two-sublattice phases ................................... 35

1.9 A phase diagram proposed by MFT ................ .......... 38

1.10 The axis definitions. The Oz axis is the local axially symmetric axis... 41

1.11 Theoretical NMR absorption line shapes ....................... 43

1.12 Energy level diagram in a strong magnetic field .................. 45

1.13 Energy level diagram in a strong magnetic field for hindered rotors... 46

1.14 Theoretical NMR absorption line shapes for hindered rotors ........ 47

2.1 Block Diagram of UHF CW-NMR Hybrid-Tee Bridge Spectrometer... 53

3.1 Cross section of the superfluid 4He leak-tight Kel-F Sample Cell. .... 61

4.1 Typical NMR line shapes ................................... 67

5.1 Typical CW NMR line shapes for 0.57
5.2 Typical CW NMR line shapes for near X=0.38 .................. 76

5.3 Typical CW NMR line shapes for X<0.25 and 1.01

5.4 The temperature dependence of the separation of the peaks A ......

5.5 The temperature dependence of the heights A ..................

5.6 The temperature dependence of the separation of the peaks B. ......

5.7 The temperature dependence of the heights B ..................
























An interpretation of the complex many peaked structure ........... 86

A phase diagram for a 12-layer film of H2 on BN ................. 88

Schematic representation of the orientational ordering in thick films .. 89

The phase diagram for bilayer films of quantum rotors ............ 95

The bilayer NMR spectra ................................... 97

The conversion rate k vs. the number of layers on BN ............. 98

Typical CW NMR line shapes for 0.55
6v vs T (K) and X.......................................... 103

|la=6v/3d vs T-1 for 56% ortho-H2. ............... .......... 105

The NMR line shape for a co-existing region .................... 106

The evolution of the NMR line shape for a glass regime ........... 107

Typical CW NMR line shapes for X<0.55 ...................... 109

A schematic representation of the pinwheel phase ................ 110

A schematic representation of the quadrupolar glass ordering ....... 111

A schematic representation of the co-existing phase ............... 112

Typical derivative NMR line shapes ......................... 119

The phase diagram for a monolayer film ....................... 120

Two NMR line structure: (a) pinwheel and (b) herringbone ........ 121

The NMR line shapes for a pinwheel structure ................... 123

The temperature dependence of the pinwheel NMR spectra ........ 124

bv vs T (K) and X for the pinwheel NMR structure .............. 125

The NMR line shapes for the herringbone phase ................. 127

6v vs T (K) and X for a herringbone structure .................. 129

7.9 The temperature dependence of the herringbone NMR structure..... 130

7.10 Schematic representation of the orientational order phase .......... 132

8.1 The phase diagram for submonolayers hydrogen film .............. 137

8.2 The submonolayer NMR spectra ............................. 139

8.3 The NMR line shapes for a pinwheel structure ................... 141

8.4 The NMR line shapes for a herringbone structure at high X ........ 142

8.5 The T-dependent herringbone NMR spectra .................... 144

8.6 The X-dependent herringbone NMR spectra .................... 146

8.7 The NMR line shapes for the herringbone structure at high T ...... 147

8.8 The NMR line shapes for a herringbone structure at low X ......... 148

8.9 6v vs X and T ........................................... 150

8.10 The NMR spectra for a transition from a glass to the herringbone phasel51

8.11 The NMR line shapes for an origin of the CF induced ordered phase. 151

8.12 The NMR line shapes for a crystal field induced ordered phase ...... 153

8.13 6v vs X and T for a C.F. ordering ............................ 154

8.14 The NMR spectra for a transition from the C.F. to the glass phase 155

8.15 Schematic representation of the orientational ordered phases ....... 158

A.1 The phase separation diagram for liquid 3He and 4He mixtures. ..... 166

A.2 The basic components of the dilution refrigerator. ................. 167

A.3 The pressure developing across a superleak. ................... .. 171

B.1 The gas handling systems for emptying the tanks ................. 183

B.2 The gas handling systems for running the fridge backwards ......... 183

B.3 The gas handling systems for running the fridge forwards........... 184

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



Kiho Kim

August 1996

Chairman: Neil S. Sullivan
Major Department: Physics

NMR operating at 268 MHz has been employed to perform systematic studies of

the orientational behavior of molecular hydrogen films (from 12-layer to submono-

layer coverages) adsorbed on hexagonal boron nitride (BN). The interest in exploring

the orientational behavior of ultra-thin films (and thick films) of hydrogen molecules

is to determine the influence of restricted geometries on the orientational ordering of

quantum rotors. The effects of geometrical constraints are expected to be dramatic

because of the frustration introduced in the interactions between rotors by the sub-

strate. Also, the interest in these experiments is to test for the existence of the various

orientationally ordered phases predicted by mean field theory and the Monte Carlo

simulations and to search for the existence of the quadrupolar glass ordering in two

dimensions (2D) which was seen in three dimensions (3D) for ortho concentrations


In NMR studies of a 12-layer film of molecular hydrogen physisorbed on hexago-

nal boron nitride (BN) for ortho-H2 concentrations 0.23

centration (X) and temperature (T). Partial orientational ordering for this system

does not occur until the ortho concentration falls below X=0.66 at T 0.4 K. An

examination of the proton line shape using CW N.MR reveals a two component struc-

ture for high ortho concentrations (0.55
and a narrow central line. We believe that the central line is seen due to the fact

that in a random mixture of ortho- and para-molecules in higher layers and the local

surroundings of different ortho-molecules are not the same and that the broad wings

are seen due to the ordered layers (the first two layers) near the substrate. The most

unusual feature occurs at low ortho concentrations 0.25
We interpret this feature as associated with a quadrupolar glass ordering, originating

from the intermediate layers. The result also implies a strong effective crystal field,

IV_ k7.8 K.
NMR studies of bilayer films were carried out for ortho concentrations 0.30
and temperatures 0.05
in terms of X and T from clear structural changes of the NMR, spectra. A region

characterized by spin loss from Curie's law has been observed at temperatures below

T-0.7 K for ortho concentrations 0.4
been identified from NMR line shapes and is discussed. An orientationally ordered

pinwheel phase was observed just above the glass regime. As a result of the reduced

number of nearest neighbors, the ortho-para conversion rate is considerably reduced

with respect to the bulk (k~0.2%/h).

NMR studies of monolayer hydrogen films were carried out for ortho concen-
trations 0.32
mapped out from the observation of the clear structural changes of the NMR spectra.

Two distinct orientationally ordered phases are seen: an expected pinwheel structure

and an expected herringbone structure. The main peak splitting of the pinwheel

spectra increases from 33 kHz to 61 kHz as the ortho concentration X decreases. The

main peak separation of the herringbone spectra remains constant (-53 kHz), but the

shoulder separation increases from 158 kHz to 188 kHz as X decreases. The transition

from the pinwheel phase to the herringbone phase is continuous, indicating at most

a weak first order transition.

Submonolayer hydrogen films have also been studied for ortho-H2 concentrations

mapped out in terms of X and T from clear structural changes of the NMR spectra,

including the first observations of an orientational quadrupolar glass ordering and the

new crystal field induced ordered phase in 2D. The transitions from one phase to the

others will be discussed in terms of the corresponding NMR line shapes. As a result

of the reduced number of nearest neighbors and the surface induced potential energy

barriers, the ortho-para conversion is hindered, and the rate is very slow (k~-0.09%/h).


1.1 Overview of the Properties of Hydrogens

Although all solids show quantum effects to a certain degree, solids made of the

lightest elements (3He, 4He, H2, D2, etc.) have special properties of purely quantum

mechanical origin unobserved in other solids. For this reason, they are known as

quantum crystals. The main property of the quantum crystals is that the constituent

atoms (molecules) are only weakly localized with respect to quantum zero point mo-

tion (ZPM). The particles have only small mass and the cohesive forces (typically

Van der Waals) are very weak. Therefore, a particle can be visualized as one oscillat-

ing about the minimum of a shallow potential well. The well, being weak, has only

a small curvature and the amplitude of the excursions about the mean position is

very large compared to that for a heavy "classical solid" such as Kr. The frequency

of the motions determines the average quantum kinetic energy. Therefore, there is

a large amplitude of ZPM and the deviations about the equilibrium position are a

sizable fraction of the nearest neighbor separation. The important consequences of

ZPM are characterized by correlations, exchange, and anharmonicity of the phonon


Hydrogen has been extensively studied and its electronic excitation spectra were

fundamental to the establishment of quantum mechanics in the early part of the

twentieth century. The hydrogen atom (H) is the simplest atomic species that exists

in nature and composes almost 75% of the matter in the universe. The molecular

hydrogens (H2, D2, T2, HD, etc) form the simplest of all molecular solids. Hydrogen

forms diatomic molecules. The three isotopes (H, D and T) combine to give six

hydrogen molecules: H2, HD, HT, D2, DT and T2. This sequence is in order of

increasing molecular weight. The three "homonuclear" hydrogens are H2, D2, and T2,

and the three "heteronuclear" hydrogens are HD, HT and DT. The hydrogens H2,

HD and D2 are not radioactive whereas the tritiated hydrogens, HT, DT and T2 have

been very unpopular because of their radioactivity, making it difficult to study these

elements at low temperatures. Above room temperature, gaseous molecular hydrogen

consists of three parts ortho and one part para as governed by spin statistics.

The combination of the light mass, small moment of inertia, weak interactions, and

the quasi-metastability of the ortho-para species results in fascinating low tempera-

ture behavior that can be understood to a large extent from the consideration of first

principles. In addition, H2 and N2 provide good probes of gas surface interactions and

two-dimensional phases (quantum mechanically and classically, respectively) because

the microscopic interactions of H2 and N2 molecules are very well known.

Central to all of the condensed phase properties is the intermolecular interaction

which is well-known for H2 from elementary calculations. The solid hydrogen has

therefore provided an almost ideal experimental testing ground for the theory of both

isotropic and anisotropic interactions. Moreover, ortho-hydrogen is a particularly

good subject for NMR because among the materials it is the second most sensitive

element for NMR.

In the solid state, molecular hydrogen has a number of features which distinguish

it from other molecular solids. Most important is that even in the solid, the free

rotor state that describes the rotational motions of an isolated molecule is almost

undistorted by the interactions with neighbors. Thus, the solid can be visualized as

an ensemble of molecules all translationally localized at lattice sites but freely rotating

at high temperatures. This remarkable state is a consequence of the large molecular

rotational constant (small moment of inertia) and the weak anisotropic forces which








-1 I I I I
0.0 0.5 1.0 1.5 2.0


Figure 1.1: Form of the Lennard-Jones potential which describes the interaction of
two inert gas atoms. The minimum occurs at r/a=21/6"1.12. Note how steep the
curve is inside the minimum, and how flat it is outside the minimum. The value of
V at the minimum is -c; and V=0 at r=a.

Table 1.1: Parameters (?7, e, a) in L-J potential.

are a result of the almost spherical molecular charge distribution and the relatively

large intermolecular nearest neighbor distances (~3.79A) in the zero pressure solid.

Interactions between hydrogen molecules (H2, HD and D2) are weak Van der Waals
forces, approximated by the Lennard-Jones (6-12) potential shown in Figure 1.1,

V(r) = -4[() ()] (1.1)
r r

where e is the well depth and a the hard core radius of the molecules. The param-

eters e and a are almost the same for the three hydrogens, derived from nearly the

same polarizability, e/kB_37 K and a -2.93 A. Due to their light mass and weak

intermolecular interaction, quantum ZPM is very important and causes the three hy-

drogens to have very different thermal properties at low temperatures. Most notably,

bulk H2 has the lowest triple point in nature at 13.80 K, followed by HD at 16.60 K,

and D2 at 18.69 K. The heliums (3He and 4He) have even weaker interactions due to

fully closed electron shells, and their ground states at T=0 K are superfluid under

normal pressure.

Gas 7 j (K) a (A) Tt (K) in 3D T,(K) in3D
3He 0.241 10.22a 2.556a 3.3102
4He 0.181 10.22a 2.556a 5.1899
H2 0.0764 37.0 2.928 13.804(e)b 33.796(e)b
13.956(n) 33.19(n)
HD 0.0382 37.0 2.928 16.60 35.91
D2 0.0382 37.0 2.928 18.69(e) 38.262(e)
18.73(n) 38.34(n)
Ne 0.00843 36.7 2.788 24.553 44.40
Ar 8.75 x10-4 119.80 3.405 83.806 150.70
Kr 2.69x 10-4 164.0 3.624 115.763 209.5
Xe 1.04x10-4 230.4 3.921 161.391 289.72

Table 1.2: The ortho-para designation.
molecule and I Nuclear weight
spin of nucleon I mo J 'i (Imot)' gj Designation
Hydrogen State 0 Even para
Tritium Symmetry AS S AS 1
IN=1 State 1 Odd ortho
Symmetry S AS AS 3
Deuterium State 1 Odd para
IN=1 Symmetry AS AS S 3
State 0, 2 Even ortho
Symmetry S S S 6

The quantum degrees of freedom are usually characterized by the de Boer param-

eter, which is defined as

r= (1.2)

where e and a are the parameters in the Lennard-Jones potential, Equation 1.1. In

Table 1.1, we list the quantum parameters, the bulk triple point (Tt) and critical

point temperatures (T,) of selected gases for reference and comparison. Data for

the rare gases are from [Cra77], and data for the hydrogen from [Sou86]. The data

with superscript a is from [Das75], and e and n stand for equilibrium and normal

hydrogens, respectively.

Due to requirements on the symmetry of the wave function, molecular hydrogen

has two species: para and ortho. The former is characterized by an even rotational

quantum number J, the latter by an odd value for J. The allowed combinations

for an anti-symmetric total wave function require consideration of only the latter

two, and are given in Table 1.2. Allowed combinations of the nuclear-spin state

and the rotational states for hydrogen, deuterium, and tritium, and the ortho-para

designations are shown in Table 1.2. Antisymmetric states are designated by AS and

symmetric by S. Imol is the total molecular nuclear spin and J the rotational quantum

number. Transitions between these states are forbidden and as a result samples

of almost pure para or ortho species can be prepared. Because of the existence of

the metastable ortho and para species of the homonuclear molecules H2 and D2,

at temperatures below 20 K, solids composed of these molecules can be regarded as

mixtures of molecules in the spherically symmetric J=0 and the elongated J=1 states,

whereas in solid HD all the molecules are normally in the J=0 state. Since crystals

can be grown with any desired ortho-para concentration ratio, the homonuclear solids

offer a unique opportunity to study anisotropic interactions and molecular orientation

phenomena. Furthermore, the ortho-para transitions induced by the interaction of

the nuclear moments of the molecules with the molecular fields give rise to rotational

diffusion and ortho-para conversion processes which lead to interesting time effects

in the thermal and spectroscopic properties. For homonuclear molecules, conversion

requires a simultaneous change of parity of the spin and orbital states. On the surface

of Grafoil, the conversion rate is very slow (0.40%/h for pure ortho-H2 and 0.07%/h

for pure D2) [Kub85]. The rate equation governing conversion in the solid or liquid

is the second order:
dt- -kc (1.3)

with the solution
1 1
= kt, (1.4)
co(t) co(0)

where k is the rate constant. An isolated molecule is stable and will not convert;

the perturbation that causes the transition arises from interactions with neighboring

molecules. In particular, the nuclear spin transition requires a magnetic field gradient.

Classically speaking, the field must vary over the dimension of the molecule to create a

torque on the proton magnetic moments to reorient them from parallel to antiparallel.

Quantum mechanically, a uniform field has no matrix elements between the singlet

and triplet spin state, but a field gradient can mix these state and allow transitions.

Therefore, an I=1 ortho-H2 molecule will be perturbed by the dipolar magnetic field of

another ortho-H2 molecule. This field can arise both from the nuclear spin magnetic

moment and the rotational magnetic moment. A J=0, I=0 para molecule has no

magnetic moment and cannot cause conversion.

At zero pressure hydrogen has a large compressibility. This is a characteristic

of quantum solids. The lattice is expanded so that the particles do not sit in the

minimum of the attractive wells of their neighbors. Without this expansion the

ground state energy would be raised substantially since the large ZPM results in an

overlap of the molecular hard cores.

At low temperatures hydrogen molecules undergo a disorder-order phase transition

in which the molecules align along certain crystalline directions. This orientationally

ordered state is the area of main concern in this dissertation. This state has been

studied by an assortment of techniques : x-ray and neutron diffraction, NMR, Raman

and infrared absorption, etc.

The underlying physics of the glass state remains as one of the most outstanding

unsolved problems of contemporary condensed matter physics [Han91, Ric89, Cus87,

Lev95, Lie86, Pol76, Aus75, Som81, Bla89, Gol85]. The nature of the glass transition

has proven resistant to the traditional methods of both solid state and liquid state

models. New theoretical concepts, such as frustration and replica-symmetry breaking,

have been introduced to describe the glass transition. Because of the topological

disorder, lattice-based treatments are not relevant and the integral equations and

diagrammatic approaches of liquid theory are inappropriate for the quenched (frozen-

in) disorder that characterizes glasses. For this reason there is intense interest in

new well-characterized prototypical glass systems for which there is a clear picture of

the nature of the frustration and quenched disorder that appear to be common to all

glasses. The orientational glasses [Bin92, Est82, Pre82, War83, Ham94, Gro85, Vol92,

Bie91, Har85] (solid ortho-para H2 [Sul87, Sul88, Was82, Li88, Lin90, Ens95], solid

N2-Ar [Est82, Pre82, War83, Ham94], and KCN-KBr [Hoc90, Mic88, Woc92, Bos91,

Wat95, Kai95, She91] mixtures) in which electric quadrupole moments are frozen in

random configurations at low temperatures provide a unique opportunity to meet

this need because the physics of the frustration and bond disorder is particularly well

understood. The local ordering can be probed directly by experimental methods over

very large time scales, and the effects of lattice geometry and restricted dimensions

on the frustrated interactions can be systematically explored.

A better understanding of the basic physics of orientational glasses can be expected

to have far reaching applications in very different physical problems (spin glasses, or-
ganic glasses, neural networks, optimization problems, protein folding, metallic and

structural glasses) because all of these systems have a number of universal proper-

ties, a disordered glass-like low-temperature state with strong local correlations but

no long range order, and extremely slow polydispersive relaxation processes below

characteristic temperature ranges. There is a strong analogy between the spin glasses

[Cho86, Bin86, Hei83, Tou77] in which magnetic dipole moments become frozen in

random orientations, and the freezing of the electric quadrupole moments in the ori-

entational glasses. Frustration in different forms occurs in all of these systems and

systematic studies of the low temperature dynamics are needed to test the theoretical

models of the glass transition.

There is intense interest in the properties of H2 as a confined quantum solid

or quantum liquid in restricted geometries adsorbedd films [Das89, Vee86, Bie91,

Zep90, Mar93, Ma89, Liu93, Lah89], suspended droplets [Mar87], and porous materi-

als [Tel83, Ral91, Bre90]) for which new phenomena have been predicted. Substantial

supercooling of liquid H2 below the bulk freezing point has been observed in very small

geometries (to 7.9 K for 55% para-H2 in zeolites [Ral91]), and new superfluids are pre-

dicted [Gin72, Mar92] if the liquid can be sufficiently cooled. Liquid para-H2 (orbital

angular momentum J=0, total nuclear spin I=0) is expected to undergo Bose-Einstein

condensation at Ts6 K, and liquid ortho-H2 would form a new anisotropic superfluid

with no known counterpart for T<1.5 K.

In recent years, many attempts were made in a hope to observe Bose-Einstein
condensation of hydrogen as predicted by theoretical calculations of Ginzburg et al.

[Gin72]. In addition to supercooling in very small geometries, it has been suggested

[Mar92, Gin72] that the top layers of thin films of H2 on suitable substrates could

be supercooled close to the expected transition for superfluidity, i.e. a Kosterlitz-

Thouless 2D transition with TKT -1 K for a surface density as,0.08 A-2. The

thermal wavelength of H2, AT=12.3T-1/2 A, is much larger than the nearest neighbor

separation for T< 6 K, and delocalization effects including quantum exchange [Bie91]
are expected to dominate the properties of the H2 surface [Liu92, Wie90, Gin72]. Path

integral Monte Carlo simulations of H2 surfaces [Bie91] show that roughly half of the

molecules participate in quantum exchange for temperatures 0.5
exchange also involving molecules in the underlying H2 layer. The Monte Carlo calcu-

lations indicate that superfluidity is possible for low adlayer coverages near as-0.02

A-2 for T~0.6 K provided the film remains liquid. The lowest temperatures reached

for liquid H2 films are 5.74 K for H2/D2/graphite [Liu92] and 5.96 K for H2/graphite


Recent quasi-elastic neutron scattering studies [Zep90, Bie91] have shown that
highly mobile quasi-liquid layers are formed for HD multilayers on MgO substrates.

The last adlayer appears to be a hexagonal lattice surface-melted layer that wets

the solid-vapor interface, with the HD molecules jumping from site-to-site with jump

times T = x 10-11 s [Zep90]. The jump rates are thermally activated with an

activation energy of 16.1 K, which is much smaller than the vacancy activation energy

in the bulk solid [Ebn75, Sul95]. It is not known if true quantum exchange without

thermal activation occurs at the lowest temperatures for which the adlayers are fluid

or if quantum exchange occurs in the underlying layers or between layers.

This dissertation investigates a variety of the two-dimensional orientational or-

dered phases of ortho-H2 (the J=1 molecular species) physisorbed as a V x /3 regis-

tered solid on boron nitride (BN), which were predicted by mean field theory [Har79]

and the Monte Carlo simulations [Osh79]. One of the main goals is to find a phase dia-

gram for molecular hydrogen in a two-dimensional system in terms of ortho-hydrogen

concentration and temperature. Another goal is to look for the superfluid transi-

tion in high coverages of hydrogen films of this system at temperatures around 6

K. Here, NMR studies have been carried out to examine the orientational ordering

of ortho molecules (with orbital angular momentum J=l) in 12-layer, two-, mono-

and submono-layer films of H2 adsorbed on BN. The effects of the restricted geometry

and substrate interactions on the ordering make these systems interesting to examine.

Carefully grown films [Cur65, Bos67, Sch85] form h.c.p. structures, and the increased

quantum ZPM in outer layers [Wag94] and the frustration of the h.c.p. structure may

inhibit long range ordering. Periodic ordering has not been observed for bulk hexag-

onal structures. At high temperatures, the molecules behave as free quantum rotors,

but at low temperatures the anisotropic electric quadrupole-quadrupole (EQQ) and

substrate interactions are predicted to result in orientationally ordered structures.

This system is of particular interest because the orientational ordering corresponds

to that of a 2D-lattice of strongly frustrated interacting quantum quadrupoles. New

2D orientational glass configurations are expected [Hol91] for strong quenched site-

disorder at low quadrupole concentrations. Here this will be tested for a particularly

well-characterized 2D system. Kubik et al. [Kub85] observed a pinwheel ordering for

ortho-H2 (X[J=1]=0.90) on exfoliated graphite below 0.6 K, corresponding to one

of the phases predicted by mean field theory [Har79] and Monte Carlo calculations

[Osh79]. The nature of the phase transition, however, was not clear and no ordering

was observed for para-D2 down to 0.3 K. The increased quantum fluctuations in 2D

could inhibit the orientational ordering in monolayers or lead to a new ordered phase.

1.2 Chronicle of Researches on Hydrogens in 3D

The work of Nakamura [Nak55] had its origin in a major development in the inves-

tigation of the solid hydrogens by showing that the electric quadrupole-quadrupole

interaction between hydrogen molecules plays a key role in orientational ordering.

The application of NMR spectroscopy was pioneered by Hatton et al. [Hat53] in

1949, and by Reif et al. [Rei53] in 1953. The work of these authors on the splitting

and shape of the NMR line due to the nuclear spins of the J=l molecules confirmed

the quenching of the rotational angular moment at low temperatures, which was

shown by Tomita et al. [Tom55] to be of a cooperative nature. This conclusion was

consistent with the specific heat measurements carried out in 1954 by Hill et al.

[Hil54], who were the first to observe a A-type specific heat anomaly indicative of a

phase transition, and also found that the transition temperature decreases linearly

with decreasing concentration X of the J=l species. The quenching of the rotational

angular moment and the dominance of the EQQ interaction were also confirmed by

the interpretation of the inelastic neutron scattering data of Elliott et al. [E1167] in

1969. Of great importance for the further development in the investigation of the

solid hydrogens was the successful preparation of nearly pure J=1 solids by Depatie

et al. [Dep68] in 1968 by the technique of preferential adsorption. The mean-field

theory of the order-disorder transition for X=l, assuming EQQ interactions and a fcc

lattice, was formulated by James et al. [Jam67] and the effect of J=O impurities on

the transition was treated by Sullivan et al. [Sul76] in 1976. The investigation of the

NMR properties, including spin-echo and spin-lattice relaxation processes, has been

successfully pursued by Meyer et al. [Mey79] in an extensive series of experiments,

by Gaines et al., and by others. The corresponding theoretical development has been

carried out by Nakamura et al. and others, and in particular by Harris [Har70]. Some

of the highlights of this NMR work are the discovery of the rotational diffusion in

solid H2 by Amstutz et al. [Ams68] in 1968, and its interpretation by Oyarzun and

Kranendonk in terms of resonant conversion of ortho-para into para-ortho pairs, and

the detection and interpretation of the striking effect of the irreversible ortho-para

conversion process on the NMR spectrum by Berlinsky et al. [Har73] in 1973, who

also extended the theory of the conversion process originally developed by Motizuki

and Nagamija [Mot79] in 1956, and the identification of the separate NMR features

due to the nine inequivalent nearest neighbor pairs of J=1 molecules in solid H2 at

low ortho concentrations by Meyer et al. [Mey79] in 1979.

Several reviews on the properties of H2 are available as a textbook format. Among

them, Farkas's excellent book of 1935 was possibly written too soon [Far35], because

the famous Woolley review became the standard reference textbook for almost three

decades [Woo48]. A concise, more modern work, with considerable heavy hydrogen

data is also available [Mac73, Mac73].

Several extensive reviews fall into the handbook category. The American ones are

by the National Bureau of Standards in Boulder [Rod81, Mcc81, Rdm75]. Computer-

ized lists of all available references are presented, rather than the selected references

listed by reviews in the textbook category. The fluid equation of state, which first

appears in Woolley's work, grows to extensive and detailed proportions. An ex-Soviet

counterpart also exists [Ese71, Kha84].

A recent work by Silvera complements the others nicely [Sil80]. This covers solid-

state thermodynamics, potentials, and ordered rotational moments. It summarizes

the elegant quadrupolar studies of the past two decades. More recent books in the

textbook format have been published by Kranendonk in 1983 [Kra83] and by Souers

in 1986 [Sou86]. The former contains the theoretical analysis of solid-state properties

in terms of free molecules and the intermolecular interactions and provides a self-

contained account of the theoretical interpretation of the main properties of solid

hydrogen. The latter covers the assembled hydrogen properties for a mature area of


1.3 Chronicle of Researches on Hydrogens in 2D

A major part of the theoretical interest in 2D systems has been concerned with

the possibility of phase transitions and the nature of the long-range order that may

occur in these systems. It has been known for forty years [Ons44, Pei36] that the

Ising model has a phase transition in two dimensions, and it is generally believed that

in analogous cases, where the order is described by a single real scalar quantity, there

will in most cases be a phase transition from an ordered state at low temperatures

to a disordered state at high temperatures. In many systems, however, the order is

described by a quantity with more than one degree of freedom.

However, there have been various important results for 2D systems obtained over

the years. Also, there is a great interest in the influence of restricted dimensionality

on the behavior of physical systems, particularly, but not exclusively, in relation to

critical phenomena near phase transitions. The main reason for the interest in 2D

systems is probably that, while they are similar in many respects to three dimen-

sional (3D) systems, the theoretical analysis is somewhat simpler. The geometry of

a plane is simpler and more familiar than the geometry of a volume, integrals are

easier to evaluate, and fewer particles need to be studied in a molecular dynamics or

similar numerical computation. All these would be more true of one dimensional (1D)

systems, but they are known to have peculiar properties not shared by 3D systems.

There are particular theories that can be solved in 2D but not in higher dimensions.

Apart from these theoretical points, the study of these 2D systems on surfaces may

provide valuable information about the nature of orientational interactions in 2D and

a surface interaction that is not readily available from microscopic measurements.

For example, the isotherms for gas adsorption have information about the area and

homogeneity of the substrate surface.

The properties of bulk solid hydrogen at low temperatures have been studied for

many years. The static susceptibility of solid hydrogen at 2 K was measured as early

as 1937 [Laz37] since then. The low temperature orientational properties of hydrogen

have been identified and investigated almost continuously. There have been various

important results for 2D systems obtained over the years, such as Langmuir's theory

of gas adsorption on surfaces [Lanl8], Peierl's argument for the absence of long-range

order in 2D solids and isotropic ferromagnets [Pei34, Pei35], and Onsager's solution

of the 2D Ising model [Ons44]. It is only in recent years that a wide variety of 2D

systems have become available for detailed experimental study so that the relevance

of various theoretical ideas could be assessed. At the same time, there has grown

up a great interest in the influence of dimensionality on the behavior of physical

systems, particularly, but not, exclusively, in relation to critical phenomena near

phase transitions.

Although there have been numerous studies of hydrogen chemisorbed on various

surfaces (particularly metals), only a few physisorbed molecular hydrogen studies

have been performed because of several stumbling blocks to the study of physisorbed

hydrogen; first, as the sub-monolayers of hydrogen on well-known substrates such as

graphite and BN do not solidify until the temperature is reduced to 20 K or less

[Nie77], this eliminates the use of many modern surface analysis techniques such as

low energy electron diffraction (LEED), Auger electron spectroscopy, and field emis-

sion microscopy which are normally designed for operation at higher temperatures;

second, hydrogen, being a very light molecule, scatters electrons and x-rays very

weakly so there are large background signals from the substrate; and third, neutron

diffraction and specific heat measurements suffer from low signal-to-noise ratios and

the presence of heating arising from conversion, respectively. Also, many methods

Ortho-H2 Para-H2

Figure 1.2: The registered triangular lattice of H2 on BN.

used in statistical mechanics, such as Monte Carlo and renormalization group meth-
ods, are rendered complicated and difficult due to the quantum mechanical nature of
hydrogen molecules. The technique of NMR would also be expected to be useful for
the study of adsorbed hydrogen because ortho-hydrogen has a large nuclear magnetic
moment. When atoms of helium or molecules of hydrogen or deuterium are adsorbed
onto a surface, their small mass leads to a large ZPM, so that films of helium or
hydrogen behave very differently from films of heavier molecules. It may be less easy
for the molecules to be attached to particular sites on the substrate so that they may
be regarded, to a first approximation, as moving freely on the surface.

The earliest 2D measurements on H2 were adsorption isotherms [Con61]. Der-

icbourg et al. [Der76] and Daunt et al. [Dau81] performed pressure measurements

in 1976 and in 1981, respectively, but this work has been primarily concerned with

multilayer adsorption. Prior to these, in 1969, Thomy et al. [Tho70] showed struc-

tures arising from 2D phases in the sub-monolayer region by measuring the adsorp-

tion isotherms of several inert gases (most notably Kr). In 1978 Stockmeyer et al.

[Sto78] performed inelastic neutron scattering measurements and in 1980 Mattera et

al. [Mat80] employed molecular beam scattering to measure the energy levels of ad-

sorbed hydrogen molecules. From the molecular beam work it has been possible to

derive a laterally averaged molecule-surface interaction potential. A binding energy

of 483 K for H2 on graphite has been obtained. In 1974 Bretz et al. [Bre74] performed

specific heat measurements, in 1977 Nielsen et al. [Nie77] did elastic neutron scat-

tering experiments and in 1982 Seguin et al. [Seg82] did LEED measurements in an

attempt to study the 2D translational phases. In 1979, Harris et al. [Har79] employed

mean field theory in order to study the orientational behavior of hydrogen molecules

adsorbed on graphite.

The results of the LEED and specific heat measurements, in the low temperature

and low coverage region, show that molecular hydrogen forms a V3 x Vf registered

solid, resulting in hydrogen molecules located at the center of every third carbon ring

(see Figure 1.2). Note that in Figure 1.2, the shapes of ortho and para molecules are

exaggerated to distinguish one from the other and distributed only for the purpose of

presenting the registered solid. This structure has also been observed for 3He, 4He,

N2, Kr, and CH4 adsorbed on graphite.

In this phase, the hydrogen can be modeled by a hexagonal triangular lattice

of interacting quantum quadrupoles. Each molecule also experiences a crystal field

arising from Van der Waals interactions with neighboring molecules and with the

substrate. Harris and Berlinsky [Har79] have used mean field theory to study such a

system and have predicted the existence of a variety of orientationally ordered phases

which depend on the relative strength of the crystal field and the molecular field and

on the temperature. In addition, a Monte Carlo calculation [Osh79] for the analogous

system of classical quadrupoles predicted the existence of two of the phases. If the

crystal field were zero, O'Shea and Klein found ordering into the pinwheel phase. If

it were large enough to force all of the molecular axes to lie in a plane parallel to the

surface, ordering into the herringbone phase was seen.

A system closely analogous to physisorbed H2 is physisorbed N2, which also forms

a v x /3 registered lattice on graphite [Kje76]. The major difference is that the N2

molecules can be treated as classical rotors because of their large moment of inertia.

Adsorbed N2 molecules have been observed to undergo an orientational ordering tran-

sition into the herringbone phase at 28 K. This had been confirmed by specific heat

measurement [Chu77, Mig83], neutron scattering [Eck79], LEED [Die82] and NMR


Beginning in the early 1970's, NMR studies identified the various orientational

phases and produced a phase diagram for bulk hydrogen [Sul72]. Sullivan et al.

[Sul72] observed that below 55% ortho-hydrogen concentration, the ordering is in the

form of a quadrupolar glass. However, no one has seen the 2D quadrupolar glass due

to lack of NMR sensitivity. In 1973, NMR was used to measure the longitudinal and

transverse relaxation times T1 and T2 of ortho-H2 adsorbed on graphitized carbon

black [Rie73]. In 1980, more recent theoretical work by Cowan et al. [Cow80] has

shown that in 2D both T1 and T2 depend strongly on the angle between the applied

static magnetic field Ho and the substrate and that for a powder substrate, the

effective T2 would be considerably shorter than TI, even in the motional narrowing

limit WoTc<1.

In 1978 and 1985, Kubik et al. [Kub78, Kub85] employed NMR to measure the
orientational behavior of sub-monolayers of H2 and D2 adsorbed on graphite from 0.3

K to 12 K with high ortho concentrations and have observed that for 90% ortho-H2,

the splitting of the high temperature NMR doublet increases rapidly with decreasing

temperature near 0.6 K and an additional structure appears. This structure agrees

well with the expected T=0 K spectrum for the orientationally ordered pinwheel

phase predicted by mean field theory. Consequently, the rapid increase in the split-

ting is interpreted as an orientational ordering transition. In 1985, heat capacity

measurement were done by Motteler et al. [Mot85] and Freimuth et al. [Fre85]. In

1989, Ma et al. [Ma89] carried out adsorption isotherms to study the two-dimensional

translational modes identifying the liquid-solid transition.

More recently, in 1990, inelastic neutron diffraction experiments were carried out

by Freimuth et al. [Fre90]. In 1992, adsorption isotherms on graphite as the substrate

were performed for the hydrogen isotopes by Vilches et al. [Vil92]. NMR studies of HD

and D2 adsorbed onto MgO have been performed and orientational ordering identified

at elevated temperatures by Jeong et al. [Jeo93]. Especially in 1993, quasi-adiabatic

heat capacity and volumetric vapor pressure isotherm techniques were used to study

the thermodynamic properties of monolayers H2 adsorbed on HD plated graphite

(H2/HD/Gr) and bilayer HD on bare graphite (HD/HD/Gr) by Liu et al. [Liu93].

They proposed a phase diagram composed of a tilted triple line, a distorted liquid-

vapor coexistence region, and weak heat capacity anomalies at 10.1 K for H2/HD/Gr

and showed the phase existing above Tt in the second HD layer. In 1994, Evans

et al. [Eva95b] carried out the first NMR measurements of hydrogen on BN, and

systematically studied the NMR absorption line shapes for ortho-H2 fractions the

on BN. In both cases a distinct two-component line shape is observed: a central

component ( 35 kHz width) corresponding to rotationally disordered molecules, and a

broad outer wing structure ( 152 kHz) attributed to orientationally ordered molecules

with mean orientational order parameter < a >=0.86.

Boron 0 Carbon
0 Nitrogen
(a) (b)

Figure 1.3: The layered structure of (a) boron nitride and (b) graphite.

1.4 Boron Nitride as a New Substrate

In order to perform 2D studies and to test the effects of the reduced dimensionality

of the system, it is critical to have well-characterized substrates. Hydrogen adsorbed

onto a surface provides a model of the 2D system for this purpose. Knowledge gained

from this relatively simple 2D system may aid in understanding more complicated

systems in reduced dimensions.

In general, two modes exist when a gas is adsorbed on a substrate. Physisorption

occurs when the adsorption potential is rather weak and the adsorbed molecules retain

their molecular form. Chemisorption, however, occurs where the adsorption potential

is rather strong and the adsorbed molecules are dissociated into their atomic forms.

From the results of numerous experimental studies, it has been confirmed that hydro-

gen physisorbs onto graphite and boron nitride. A summary of key data concerning

physical adsorption potentials has been compiled by Vidali et al. [Vid91]. They con-

sidered over 250 gas-surface systems and tabulated the Van der Waals adsorption

coefficient, the equilibrium distance, well depth, and binding energy of the laterally

averaged interaction potential as deduced from analyses of experimental data and


Graphite has been used as a substrate for decades and many types of experiments

have used it because it is available in forms with both a high specific surface area

and a high proportion of uniform basal planar adsorbing surfaces. However, BN has

gained recent interest because of its similar crystal structures. Both BN and graphite

are based on the planar structure. The lattice dimensions of BN are about 5% greater

than those of graphite. With these well known substrates, it is possible to investigate

the effects of binding energy, lattice spacing, and corrugation. The layer structures

of BN and graphite are shown in Figure 1.3.

MgO is another substrate of interest. The crystals of MgO form a cubic structure.

However, it is believed that the strong corrugations of the surface overwhelm the

adatom-adatom interaction and are responsible for the orientational ordering. While

an interesting finding in itself, it indicates that MgO is not a good surface for the

study of frustration and disorder in orientationally ordered systems in two dimensions.

In general, there are at least four methods to characterize a substrate: adsorption

and desorption of gases, electron microscopy (both scanning and transmission), small-

angle x-ray, and neutron scattering, and mercury porosimetry. Of these techniques,

adsorption isotherm is the most extensively used to study substrates, whereas mercury

porosimetry is used to characterize porous materials.

Automated volumetric adsorption isotherm studies of H2, HD and D2 adsorbed

onto BN were carried out to characterize the substrates and to measure the critical

temperatures, the triple points and isosteric heats of adsorption [Eva95]. In con-

trast to HD/MgO [Ma89] and HD/graphite [Liu93], no substep (to within 5% of

a layer) corresponding to liquid-solid coexistence was seen for the second layer of

HD/BN, and only a weak substep was seen near third layer completion. The critical

points Tr were determined from Lahrer plots [Lah89] of the inverse compressibility

x- l=(lnp)/9V as a function of temperature. Evans et al. [Eva95] found that al-
though the adsorption potential for HD on BN is weaker than for MgO and graphite

substrates, a lower Tcr was not observed for BN. The measured isosteric heats of

adsorption qt=kBT2(O(lnP)/9T) N,A yield a binding energy for the second layer of

12010 K, compared to 17710 K for HD/graphite.

1.5 The Dipole-Dipole Interaction

The nuclear dipole-dipole interaction plays a key role in the intramolecular inter-

action between two hydrogen atoms within a hydrogen molecules. Also, the dipole-

dipole interaction is known as one of sources of the NMR line broadening. NMR

lines of a system of spins in an inhomogeneous magnetic field have a certain width

owing to the spread of their Larmor frequencies. A broadening of similar nature can

be produced in imperfect crystals by the coupling of nuclear quadrupole moments

with small electric field gradients having values which differ from one lattice site to

another in a random fashion. In both cases the line width is due to the differences

among the resonance frequencies of the individual spins rather than to interactions

among them, and the corresponding broadening of the line is called inhomogeneous


1.6 Intermolecular Interactions

One of the central problems in the study of molecular hydrogen in the condensed

state is the intermolecular interactions.

In part, the interactions are of interest because they ultimately determine the

equation of state, crystal structures, excitation spectra, etc., but most attention is

focused here because they are fundamental in an orientational ordering of an ensemble

of hydrogen molecules. The low density solid is an almost ideal testing ground for

theoretical interaction potentials. The problem is highly simplified in that, at least

to moderate densities, the interactions can be fairly well described by a sum of pair

Table 1.3: Electrical quadrupole-quadrupole interaction energy normalized to the
coupling constant F.

EQQ Energy/F
Classical Semiclassical
Configuration (Q)=Q (Q)=Q=|Q State Quantum-mechanical
T -12.5 -2 8, 9 -4
X 3.125 4, 5,6, 7 0
H 9.375 4 1 6
L 25 4 1 6

interactions between molecules. Since the gas phase single molecule properties are

almost undistorted when condensed into the solid state, the pair interactions in the

solid can be represented by the interaction between an isolated pair with very small

corrections for environmental effects.

The anisotropic interaction energy depends on the mutual orientation of the axes

of the H2 molecules. In general, an energy surface, or the energy as a function

of molecular orientation and separation, must be determined. Four orientational

geometries commonly used in calculations of the intermolecular interaction energy are

shown in Figure 1.4. Note that only the X geometry [Figure 1.4(c)] is non-coplanar.

The classical quadrupolar energies of the four configurations of Figure 1.4 are given

in Table 1.3 [Sil80]. In this table, the classical case corresponds to quadrupoles

with fixed orientations along the molecular symmetry axes: the semiclassical to the

substitution of Q=(2/5)Q corresponding to the average of Q in the J=1 rotational

state. This reduces the energy by 4/25 and quantum-mechanical refers to the energy

of an isolated pair of molecules, each in the J=1 rotational state. This is of importance

in discussions of the orientationally ordered state of H2. The interaction energy of

a pair of molecules is the lowest in the "T" configuration and highest in the "L"

configuration. Here the two molecules are oriented along two different body diagonals

in a cubic lattice. Calculations have in general been carried out by three techniques:


(a) T

(b) H


(d) L

Figure 1.4: Four orientational geometries commonly used in calculations of the inter-
molecular interaction energy.

self-consistent field (SCF), Heitler-London (HL), and configuration interaction (CI).

SCF and HL are special cases of CI. Discussion of these various methods are beyond

the scope of this dissertation.

The orientational energy of an H2 molecule depends strongly on its rotational

state and environment. Interaction energies arise from a number of sources: overlap

energy including charge repulsion and charge transfer interactions, attractive disper-

sion forces due to molecular polarizability, induction energy due to the interaction of

a permanent multiple moment on one molecule with the polarizability of the other,

and the interactions of permanent multiple moments. In the low density solid, J is

a good quantum number, and we need only consider J=0 and 1. The J=0 state is

spherically symmetric and thus has no orientationally dependent energy. For the J=1

state a number of circumstances can arise. An isolated J=1 molecule has a three-

fold orientational degeneracy (ignoring the intramolecular hyperfine interaction and

nuclear spin). This degeneracy will be partially lifted by the crystal field of a J=0


1.7 Orientational Properties of Hydrogen

The rotational properties of molecular hydrogen in the gaseous, liquid, and low-

pressure solid state can be described by the free rotor model. Also, even on the surface

of BN and graphite, hydrogen molecules can be modeled very well by free rotors. The

free rotor energies Ej are given by

Ej = BjJ(J + 1) (1.5)

where Bj=85.35 K (43.04 K) for H2 (D2) [Sil80]. The difference in the values of

Bj arises from the different moments of inertia. The rotational energy levels of a

hydrogen molecule are shown in Figure 1.5. The same diagram applies to D2, but

(2J+1) J



509.9 K


+ -0

J (2J+1)
5 (11)


844.7 K

1 v(3)
170.5 K

I= 1

Figure 1.5: The molecular rotational energy levels for an isolated H2 molecule.

scaled down about a factor of 2 due to its larger moment of inertia. In Figure 1.5, I

is the total nuclear spin, and numbers in parentheses are the m degeneracies.

Since the J=1 state of H2 (D2) is 171 K (86.1 K) above the J=0 state, one would

expect that only J=0 and 1 states would have any significant population below 20 K.

However, for homonuclear molecules conversion between odd and even J states is very

slow (1.9%/h for bulk hydrogen). 'H nuclei are fermions so the H2 wave function must

be antisymmetric with respect to atomic interchange. A rotational state with even

parity must be combined with a nuclear spin state with I=0, which has odd parity.

An odd rotational state must have I=1. Similar arguments apply to D2 except that

deuterons are bosons, so the rotational and nuclear spin states have the same parity.

The states with the greatest spin degeneracy are labeled ortho; the other are para.

These results are summarized in Table 1.2.

In 2D, the rotational energy spectrum is given by [Eyr44]

Em = Bm2 (1.6)

where m=0, 1, 2, -... The orbital wave functions are symmetric under permu-

tation of the nucleons for Iml even and antisymmetric for Iml odd. The interactions

with the surface do not mix the nuclear spin states which are used to identify the

ortho and para species.

For theoretical treatments of 2D systems such as H2 and D2 physisorbed on BN, we

must include the anisotropic interactions between each H2 molecule and the substrate

in addition to the anisotropic interactions between the molecules. For each pair of

J=1 molecules (many body interactions will be ignored), the anisotropic part of the

Hamiltonian is then

Haniso = H + E HQQ. (1.7)
i i
It is apparent that for nearest neighbors, other pairwise and three-body inter-
actions are negligible in comparison to the EQQ interactions. For widely separated

molecules we must consider other interactions, such as the single-molecule crystal-field

Hamiltonian [Har79, Har70b]

He = Vc(J -), (1.8)

where |VcI is the strength of the crystal field.

It is well-known that orientational interactions between hydrogen molecules are
dominated by the EQQ interaction [Nak55]. The EQQ Hamiltonian can be written

as [Kra59, Har70b]

HEQQ = o70 r C(224; mn)Y2m(wi)Y 2(w) Y4m+n* (Qi). (1.9)
9 mn

C(224;mn) is a Clebsch-Gordan coefficient, YL(w) is a spherical harmonic of rank L,

and wi and ~ij denote the spherical angles of the ith molecular axis (Oi, i) and the

line joining molecules i and j (Oij, Oij), respectively.

Fo = 2(1.10)

is the nearest neighbor quadrupole coupling constant, where eQ is the quadrupole
moment of the molecule and Rij is the intermolecular separation. The values of Fo

for the bulk and a v-3x /3 registered solid on various substrates (graphite, BN, etc.)

S 0.87K for both bulk H2 and D2[Sil80]
Fo = 0.528K(0.510K) for H2 (D2)/graphite[Kub85] (1.11)
0.470K for H2/BN.

Originally, it was felt that the reduction in F was due to ZPM of the molecular
centers and could be accounted for by averaging Yo(fij)/Rj weighted by the pair

distribution function. Harris [Har70] showed that the averaging process changes the

magnitude of the effective EQQ interaction but not its form. The pair distribution

Table 1.4: Definition of the operator equivalents (" and 0m of the spherical harmonics
Y1" and Y2m, respectively.

Operator Moment
o0 =(1/v2) Jz M=(ot )
i=:F (J- 'iJY) M'= ( +1)
EO= (3Jz2-2) Q=(eot)
e I (T 2(JJ+tJJ) Q=(elt
e+2= (JgJ) Q2=( e2t)

function can be obtained because short range correlations needed to minimize hard
core repulsion cause the pair distribution function to fall off more rapidly for small
separations than for large. The main difficulty in the calculation is that the phonon
wave function is not known very well. More recent calculations [Gol79] indicate that

ZPM alone is insufficient to account for the reduction of F. It is also necessary to
take into account Van der Waals contributions of the two body-term interactions.
For the 2D system, ZPM is likely to be even greater than in 3D, leading to greater

In a very qualitative way it is well known that He and HEQQ are competing in-
teractions. For Vc>0, He tends to cause all molecules to be oriented perpendicular
to the surface, where as HEQQ tends to cause pairs to orient in a "tee" configuration
with one molecule lying parallel and the other perpendicular to the line joining the
pair. For Vc<0, He tends to cause all the molecular axes to line up in the plane of the
BN surface, while the EQQ interaction would prefer to have some of these oriented
perpendicular to the plane.
Both the EQQ and antiferromagnetic exchange interactions have some similari-
ties as well as some important differences. They both oppose the alignment of the

molecules that is imposed by either the crystal field or the magnetic field. How-

ever, the lowest energy configuration for the EQQ interaction is a "tee" configura-

tion between molecular pairs while for the antiferromagnetic exchange interaction

the molecules align antiparallel. If the system of quadrupoles cannot attain its low-

est energy configuration for all pairs, this system is said to be "frustrated." The

phenomenon of frustration [Cho86, Bin92, Lie86, 11179, Pol76, Aus75, Som81, She88]

occurs in a wide variety of apparently disparate systems [Bin86, Hei83, Tou77, Bin80]

when the competition between interactions does not lead to simple ordered states

that are favorable to all interactions, or when there is a fundamental geometrical

incompatibility between the symmetry of the interactions and that of the underlying

lattice structure (or a combination of both features). The orientational glasses have

highly frustrated anisotropic interactions, while purely geometrical frustration [Bin92]

occurs for the pyrochlores Tb2Mo207 [Ram94] and CsNiCrF6 [Har94] with antiferro-

magnetic interactions of spins at vertices of corner-connected tetrahedra. Although

the combined effects of frustration and disorder are needed for most glass former,

it is now known that a glass ordering can occur without disorder for very strong

geometrical frustration (compare the spin glass behavior of the B-spinel CsNiFeF5

[Alb82] with the antiferromagnetic ordering of FeF3 [Fer86] for the same lattice struc-

ture). The effects of lattice geometry on frustration are therefore of key importance

in understanding the microscopic origins of glass behavior.

1.8 Order Parameters

For ortho-hydrogen molecules, we need only consider the rotational degrees of

freedom associated with the orbital angular momentum J=1. This is due to two

properties: (i) the anisotropic intermolecular interactions which can lead to admix-

tures of higher J states in the ground state are weak compared to the separation of

the rotational energy levels, and J can therefore be regarded as a good quantum num-

ber; and (ii) for our experimental temperature range, kBT<<[E(J=3)-E(J=1)]=845

K, only the lowest rotational state J=1 is thermally populated. The relevant moments

are the expectation values of the operator equivalents of the spherical harmonics in

the manifold J=l and are given in Table 1.4.

For interacting ortho molecules, we must consider the many-particle crystal wave
function in order to account for collective effects which tend to correlate the orien-

tational degrees of freedom of the ortho molecules. The individual molecular states

cannot be described in terms of pure states, but rather in terms of a single-particle

density matrix pi determined by the quadrupole and dipole moments Qm and M,,

Pi= 3+ E Min'n + Q meOi (1.12)
n=0,,l m=0,l,2

where 13 is the unit matrix.

We should in principle consider eight independent degrees of freedom, but in the
absence of dipolar interactions (diatomic molecules have no permanent electric dipole

moment) the dipole moments are expected to vanish. This is referred to as the
"quenching" of the angular momentum. Of the five remaining degrees of freedom,

three can be used to define local principal axes (Yi, Yj, ii) for the quadrupole, leaving

only two intrinsic quadrupolar degrees of freedom. These intrinsic degrees of freedom

can be represented by two local order parameters, the alignment

a, = V6Qi = (3J,"2 2), (1.13)

and the eccentricity

7i = Qi2 Q-2* = (J2 2). (1.14)

As a result of the condition Trp2<1, these parameters must satisfy the inequal-

ity ai2+3rs2<4. The allowed values of (a, 77) are then constrained to -
Irj <1

The root mean square local order parameter (a2)/2g can be determined directly

from measurements of the second moment [Abr61, Sli90, Fuk81] of the NMR line

shape f(v),

M2 = f (- vo)2f()dv = D2(2 avg. (1.15)

f(v) is the normalized NMR line shape, and D is the strength of the intramolecular

dipole-dipole interaction.

1.9 The Orientational Ordering in 3D

The road to understanding the crystal structures or phase diagram of the solid
hydrogens started back in 1930 with x-ray diffraction study [Kee30] at 4.2 K. In the
1950's and 1960's, active research on the crystal structures of the solid hydrogens

had been carry out by neutron studies, electron diffraction, x-ray diffraction, and

NMR. Finally, in the late 1960's, the crystal structure of solid hydrogen came to

be known. It was found that the phase diagram of bulk, solid hydrogen consists

of three distinct orientational phases below 4 K. In 1965, Clouter and Gush [Clo65]

observed an abrupt change in the infrared spectrum of hydrogen upon cooling, which

is a known transition from h.c.p. to fcc. The ground state is the Pa3 structure with

four molecules per unit cell. The molecular centers sit on the sites of an fcc lattice.
The molecular axes are oriented along the body diagonals. More accurately, it is
the molecular orbitals or the axes of quantization of the molecular angular moment
that orient along the body diagonals. The Pa3 structure can be decomposed into

four interpenetrating simple cubic structures such that on any given sublattice the
axes of the individual molecular orbitals are all parallel. The phase transition from

3.0 Hexagonal /


S- / L
w 2 / Cubic
mr 2.0
I- /

" //
-- 1.0 -/

Disorder I / Long Range
-- Order

s I I I I
0 0.2 0.4 0.6 0.8 1.0

Figure 1.6: The phase diagram for bulk hydrogen.

disordered hcp to Pa3 is evidently driven by the EQQ interaction, since the difference

in energy of the hcp and fcc lattice for isotropic interactions (~10-3 K/molecule) is

much smaller than the EQQ energy (~5 K/molecule). In 1972, Sullivan and Pound

[Sul72] studied H2 by NMR to T-85 mK, and found that for T<0.3 K and X<0.55, a

new phase appeared. This phase is believed to be a quadrupolar glass ordering which

is a randomly distributed orientationally ordered phase. The phase diagram for bulk

hydrogen is shown in Figure 1.6.

1.10 Two-Dimensional Orientational Ordering Model

A triangular lattice of quadrupoles is an example of a system exhibiting frustra-

tion, a key concept in the behavior of glassy systems. Figure 1.7 illustrates one model

of orientational ordering for a 2D system of quadrupoles. A system is frustrated if

it is impossible for every pair of neighboring molecules to achieve their lowest energy

configuration. For a pair of quadrupoles, this is a "tee" (or "T") configuration as

mentioned earlier. Figure 1.7(a) shows four quadrupoles confined to a square lattice.

In this case, the symmetry of the underlying lattice matches the symmetry of the

intermolecular quadrupolar interaction. At zero temperature the system relaxes into

its lowest energy state and each molecule forms a "tee" configuration with each and

every neighboring quadrupole. Figure 1.7(b) shows a system of quadrupoles confined

to a hexagonal (triangular) lattice. Such a lattice symmetry is incompatible with

the intermolecular interaction. At zero temperature, the system is frustrated because

it is impossible for all nearest neighbors on a triangular lattice to be mutually per-

pendicular. Frustration is reduced when the system is diluted. For hydrogen there

will always be some dilution because conversion produces J=0 molecules which have

spherical wave functions. The lowest energy for the entire system may be a herring-

bone or a pinwheel pattern. The two sublattices are shown in Figure 1.8. In the


I __________________________

(b) Frustrated


(c) Disordered

(d) Disordered

Figure 1.7: 2D model of quadrupolar orientational ordering.

\ *. \



Figure 1.8: The two-sublattice phases. (a) The pinwheel phase-the molecular axes of
sublattices 2, 3, and 4 are parallel to the surface, but those of 1 are perpendicular.
(b) The herringbone phase-the molecular axes are parallel to the surface for 1, 2, 3
and 4 sublattices.

pinwheel phase, the molecular axes of sublattices 2, 3, and 4 are parallel to the sur-

face but those of 1 are perpendicular. In the herringbone phase, the molecular axes

are parallel to the surface for sublattices. Figure 1.7(c) and Figure 1.7(d) show the

effect of the addition of a vacancy or a neutral molecule (e.g. a noble gas or para-H2).

Since a neutral molecule does not orientationally interact with any of its neighbors,

these molecules behave as inert diluents; that is, they act to dilute the orientational

interactions among the molecules. Since all of the lattice sites are no longer on the

square lattice of Figure 1.7(c), it is negligible. However, the effect of the substitu-

tion of inert diluents on the triangular lattice of quadrupoles is quite noticeable. In

this case, the remaining quadrupoles may align more completely with respect to each

other. In other words, the degree of frustration decreases as the disorder increases for

a non-square lattice. In addition, if there is no frustration, then the system is stable

against small amounts of random dilution [part (c)]. In the presence of frustration,

random dilution will cause local reorientation of molecules but reorientation of large

numbers of molecules will be blocked by energy barriers at low temperatures. The

system may not be able to achieve the lowest energy configuration, becoming stuck

in a metastable state (a glass), which depends upon the thermal history. Low J=1

concentration adsorbed H2 is a likely candidate for the formation of a quadrupolar

glass ordering. Such a state would be characterized by freezing of the molecules into

random orientations. This has been proposed for bulk H2 when the J=l concentration

falls below 55% [Sul76].

1.11 Hydrogen Molecules on a Surface

Based on the para-rotational (disordered) state, each molecule is in a state of
six-fold symmetry. The Hamiltonian of Equation 1.7 will cause the J=1 state to be

split by an energy A into mjO= and doubly degenerate mj=l states. The mj=O

state has a prolate spherical wave function with its z-axis normal to the substrate.

The mj=l state has an oblate spheroidal wave function. One may think of these

as molecules standing up or lying down on the substrate.

The system can be described by an local orientational order parameter

S- (1 J )avg (1.16)

The brackets ( ) indicate a thermal average. An equivalent expression for a can be

obtained in terms of the fractional population po of the mj=0 state

3 1
S= (po- ).(117)
2 3

As T-+ oo, all three mj states are equally populated, resulting in a-+0. At T=0 K,

a=1 or -1/2 depending upon whether the ground state has mj=0 or mj=l.

In the high temperature limit it is easy to obtain an expression for a since the

EQQ interactions are averaged out by thermal motion (molecular re-orientation).

Neglecting the second term in Equation 1.7, the energies of the mj states are

V3 2
Eo = (YO 3Jj 2 I Y1) = 2V (1.18)
3 3


Ei = ,(Yi 1 3J2 2 Y^') Vc. (1.19)

The energy gap A between the ground and excited states is temperature independent

and equal to I Vc .

The populations of the mj states are related by the Boltzman factor, resulting in

po= {1 +2exp(: A/kBT)}-1. (1.20)

where the upper sign is for V,>0, and the lower for V,<0. In the high temperature

limit, putting A =1 Vc 1, expanding the exponential and retaining only the first order

term gives

1 2
Po = [1 + Vc/k BT]. (1.21)
3 3

T (K) / F

Figure 1.9: A phase diagram proposed by MFT [Har79].

Inserting this into Equation 1.17 gives

v = c (1.22)

The NMR spectrum gives a direct determination of the local order parameter al,

therefore a measurement of the temperature dependence of the spectrum gives |VcI.

If the EQQ interaction is considered, then within the mean field approximation

[Har79] A becomes temperature dependent and is given by

A = K1/ (1.23)

Based on MFT, Harris and Berlinsky proposed a phase diagram which is com-

posed of various orientational orders in terms of the strength of the crystal field and

temperature (See Figure 1.9). However, it is well known by experiment that the mean

field calculation yields poor results in 3D, and it is expected that it will give worse

results in 2D because MFT does not consider the quantum fluctuations.

The molecular field always opposes the crystal field since a and Vc have the same

sign. The crystal field attempts to align all of molecules in the same direction whether

it be standing up or lying down. This alignment competes with the EQQ interaction

which favors a "T" configuration for each pair. The crystal field is determined by the

substrate and the neighboring molecules.

1.12 Liquid Hydrogen: Supercooling and Superfluidity

One of the great challenges of condensed matter physics is the direct observation of

Bose-Einstein condensation (BEC). Immediately after the experimental observation

of superfluidity in helium, London [Lon38] proposed that the macroscopic occupation

of the zero momentum state, or the formation of a condensate, might be responsible.

The original Landau theory of superfluidity [Lan41] did not require the existence of

a condensate, but Bogoliubov [Bog47] showed that for weakly interacting bosons, a

condensate led to the energy spectrum required by the Landau theory, putting the

London proposal on firmer ground.

Bose condensation of liquid helium into the superfluid state has been observed

and studied for some time. That helium alone should posses superfluidity is doubted

by many, and a current search exists for other superfluids.

The transition temperature for an ideal Bose gas can be calculated [Lan69], and

it is given by

1h2 n 2
TA= 3.31 ( )3, (1.24)
MkB g

where n is the concentration, M the molecular mass, and g the degeneracy for a

single particle state (1 for para-H2 and 9 for ortho-H2). In bulk form, the superfluid

transition of helium occurs at 2.17 K, approximately 2 degrees below condensation,

and near the predicted temperature of 3 K.

Hydrogen is the lightest of all elements. While it has a large ZPM, the intermolec-

ular interactions are relatively weak. Thus, molecular para-H2 is a potential candidate

for superfluidity if the right conditions can be achieved. Calculations have been made

by Ginzburg and Sobyanin for para-H2, and they find a superfluid transition tem-

perature of ~6 K. It is reasonable to expect this calculation to be an upper limit on

the actual temperature since the same formula overestimates the helium transition

temperature. Maris et al. have argued that the onset temperature for para-H2 should

be lower than 4.5 K [Mar87].

However, the melting temperature for hydrogen is approximately 14 K and this

precludes the possibility of a superfluid transition in normal bulk hydrogen. Studies

have been performed to try to supercool hydrogen under various conditions. For

example, heat capacity studies of hydrogen in vycor have been performed by Tell et

al., and supercooling to 9.9 K reported [Tel83].

/ Y


Figure 1.10: The axis definitions. The Oz axis is the local axially symmetric axis.

In 2D, a Kosterlitz-Thoules transition (a second order transition: no latent heat

of transformation) for helium films has been found and an onset temperature given


7r h2
Tx = n- (1.25)
2 MkB

where ns is the superfluid film density at the onset temperature. If hydrogen films

were to follow a similar behavior, then an estimate can be made from this equation

for hydrogen films. One such estimate by O. Vilches for a monolayer of hydrogen with

a film density of 0.08 A-2 places the transition temperature around 2 K [Vil92]. The

results of current investigations of helium and hydrogen on surfaces will be useful to

determine the conditions to study hydrogen in reduced dimensions.

In experiments for 12-layer film coverages, we have not seen the expected super-

fluid transition. Currently, thick films (more than 12-layer) are being prepared to

see if the expected superfluid transition exists for thick films of hydrogen molecules

adsorbed on BN.

1.13 Theoretical NMR Line Shapes

In the presence of a large static external magnetic field, the Hamiltonian for the

ensemble of interacting nuclear spins and molecular excitations may be written as

H = HNZ + Hoz + Hso + HDD (1.26)


HNZ = -yhIzHo = -alI (1.27)


Hoz = -fhJzHo E -b(J,) cos0 (1.28)

are the nuclear Zeeman energy and the orbital Zeeman energy of the magnetic moment

associated with molecular rotation in a strong magnetic field, respectively. 7 is the

gyromagnetic ratio. i designates the direction of the magnetic field (see Figure 1.12),

a and b have the values 4.258Ho (0.6536Ho) kHz and 0.6717Ho (0.3368Ho) kHz for H2

(D2) [Har70b, Ram56], respectively.
The spin-orbit coupling Hamiltonian which represents the interactions of the nu-

clear spins with the magnetic field caused by the molecular rotation currents is

Hso = -cI J = -cI,(J,) cos (1.29)

where I is the total nuclear spin, I = 1(1) + J(2), and J the angular momentum. c

has the value 113.9 kHz (8.773 kHz) for H2 (D2) [Har70b, Ram56].

HDD represents the intramolecular dipolar interaction, and is given by

HDD = -5d[3 r)((2) r (1) J(2)] (1.30)

where d has the value 57.67 kHz (25.24 kHz) for H2 (D2). The secular component of

the Hamiltonian for the dipole-dipole interactions can be written in the convenient


-200 -173 -86.5 0 86.5 173 200
(-3d) (-3d/2) (3d/2) (3d)
Frequency (kHz)


1 1 1

-200.3 -68.2 27.3
(-c-3d/4) (-c2/9d-3d/4) (c-3d/4)


Frequency (kHz)

Figure 1.11: Theoretical NMR absorption line shapes for a polycrystalline sample of
solid hydrogen proposed by Reif and Purcell. (a) Shape predicted if angular momen-
tum is quenched ((Jz)=O) (b) Shape predicted if (Jz)O.

form [Su178]

HDD = d(1 Jz)(I2 1)(3cos02 -1). (1.31)

This HDD is responsible for the fine structure of the NMR line shapes. Two eigenstates
exist for HDD; one is the singlet state which represents the spin zero para-H2 molecules
and the other is the triplet state which represents the spin-one ortho-H2 molecules.
As a consequence, the para species do not yield NMR signals because I=0, but act
as inert diluents. However, the ortho species yield NMR signals and are orientable
because they bear quadrupole moments.
The singlet state can be written as

S0) = ( +-)- -+)) (1.32)

and the triplet state as

I 1) = ++), (1.33)

0) = ( +-)+ -+)), (1.34)

I -1) =I --). (1.35)

The corresponding energies of each state can be determined easily, and from the
energy levels we can deduce the two resonant frequencies leading to the famous Pake
doublet spectrum.
Collecting all of the individual terms, the total Hamiltonian becomes

Htot =-al, b(J,) cos cl(J.1) cos0- d( J2 11) 1)(3cos? 2 1). (1.36)
2 2

Iz Strong Field E/h
-1 a- d o (3 cos 8 1)/2

d o (3 cos e 1)

1 -a- d o (3 cos -1)/2

Figure 1.12: Energy level diagram for an isolated ortho molecule in a strong magnetic
field for (J,)=0.

In a strong magnetic field, we consider I quantized [i.e. angular momentum is

quenched ((Jz)=0)] along the z axis and get for the perturbation energy of an ortho-

molecule (J=l, I=1)

h-'Ek = -al + da(3 cos2 1)(1 (1.37)

After calculating the energies of the triplet state, we get the energy level diagram

shown in Figure 1.13. From the energy levels we can deduce the resonance frequencies

v = a (cos2 1). (1.38)

Equation 1.38 depends on the orientation of the reference axes at the position of the

ortho-hydrogen under consideration with respect to the external field Ho. Since this

orientation is different for different crystals in the polycrystalline hydrogen sample,

it it necessary to average Equation 1.38 over the angular factor cos 9 in order to find

the intensity distribution function. A detailed treatment of this can be found in Ref.


Figure 1.13 shows the theoretical NMR line shapes proposed by Reif and Purcell

[Rei53] for a polycrystalline sample of solid hydrogen in 3D in a strong magnetic

Hindered Zeeman Energy


Note that 4 = cos9

Figure 1.13: Energy level diagram for an isolated ortho molecule in a strong magnetic
field for hindered rotors.

V, V2 V3V4 V5 V6

Figure 1.14: Theoretical NMR absorption line shapes of ortho-H2 adsorbed on a
polycrystalline adsorbent as proposed by Dubault and Legrand [Dub74].

field. In a strong magnetic field HNZ+Hoz>Hso+HDD, so we take I quantized

along the z axis. If the crystalline field quenches the rotational magnetic moment of

the molecule, the expectation value of J vanishes in each of the three states of the real

wave functions and the interactions Hoz and Hso in Hto, vanish. Figure 1.13(a) shows

the theoretical NMR absorption line shape for the case of quenching of the orbital

momentum in 3D. The nondegenerate mj=0 state again gives rise to the NMR line

shape of Figure 1.13(a).

On the other hand, if the orbital momentum is not quenched, the degeneracy of

the mj=l state is lifted, and one gets a finite spin-rotational moment interaction.

For this case of non-quantization, the energies become

3 3
E = -al, -b(J,) cos + d(3cos2 1)(1 3Jj)(1 3I) (1.39)

where 0 is the cosine of the angle between Ho and the symmetry axis. The corre-

sponding energy-level diagram for this state is more complicated and involves the

constants b and c as shown in Figure 1.13. The four lines arising in the mj=l

state yield the NMR line shape shown Figure 1.13(b). The resultant line shape then

consists predominantly of the shape characterizing the state lying lower in energy and

superimposed upon it the shape of the higher-lying state weighted by exp(-AE/kBT),

where AE is energy difference between the two states. The interesting features of this

3D NMR line shape [Figure 1.13(b)] are that the outermost shoulder separation is

approximately 400 kHz wide and the splitting of the main peak is ~136 kHz wide.

Note that the ratio of the intensities of Figure 1.13(a) to Figure 1.13(b) is 1: 1.5.

The theoretical NMR spectra for ortho-H2 molecules physically adsorbed on a

surface at low temperatures have been reported by Dubault and Legrand [Dub74]. In

Dubault and Legrand's study, the proposed theoretical NMR line shapes of hindered

rotors in 2D (as shown in Figure 1.13 for the case of Vc<0), resemble that given

by Reif and Purcell [Rei53]. For the selection rule, Am1=l, the four resonance

frequencies are given by

6v = a [c cos 0 (3 cos 02 1)a], (1.40)

where a is the local orientational order parameter, which is discussed below. Fig-

ure 1.13 is drawn taking into account the random angular orientation of the surface

and calculating the distribution of the two doublet widths. The spectra of Figure 1.13

are characterized by two principal points of discontinuity and four secondary points

of discontinuity, the algebraical expressions of which are given as

v = -v6 = -c + -tda, (1.41)

2c2 15
2 = -- = + -da, (1.42)
45da 8

y = -v4 = c + -Cda. (1.43)

As illustrative examples, a=-0.48 is used for Figure 1.13(a), and a=-0.58 for Fig-

ure 1.13(b). Also, Dubault and Legrand pointed out that when Vc is small, the

difference between the rotational energy levels (mj=0 and mJ=l) becomes of the

same order of magnitude as the temperature, and so rapid transitions between these

levels occur, resulting in an averaging of the dipolar interaction which gives rise to

the central line.

If the Zeeman level spacings hv are not negligible in comparison to kBT, the

intensities of the peaks of the spectrum are not equal. The relative intensities are

determined by the difference between population factors of the Zeeman level between

which the transitions occur. Thus,

I1 f(a + (d))=--
2 fa -i = exp(/hv), (1.44)
12 f (a (d)) 2

where I1 is the relative intensity of the peak observed on the low frequency side of

the center of gravity of the spectrum and /3=/kBT.


This chapter provides information on how to improve the NMR signal-to-noise

ratio and how to process NMR signals. In this chapter, a double-tuning technique for

tank-circuit design is discussed for continuous-wave NMR measurements at ultrahigh

frequencies and very low temperatures (in the millikelvin range) where long lossy

transmission lines must be used to reduce the transfer of heat to the sample region.

The technique uses two separate sets of tuning capacitors, one set at low temperatures

to match the tank circuit closely (but not exactly) to a lossy (low thermal conduc-

tivity) transmission line, and a second set to achieve an accurate match to 50 Q at

the input of a quadrature hybrid-tee bridge. The system allows for a stable, tunable,

wide frequency coverage and an improvement in the signal-to-noise ratio by a factor

of about 40 over conventional probe designs. The absorption signal of a monolayer

coverage of hydrogen absorbed on boron nitride (~ 1016 spins with a filling factor of

10-5) has been observed at 50 mK with an applied RF field of 3 mG.

2.1 Overview

In order to study systems in reduced dimensions, such as H2 physisorbed on

graphite, HD on MgO, and H2 in zeolite using CW NMR techniques [Kub78, Jeo93,

Ral91], it is critical to have a high S/N signal/noisee) ratio and a reliable NMR sample

cell in order to observe very dilute nuclear spin densities (,1014 per cubic centimeter).

Furthermore, the interesting physical phenomena in these systems occur at temper-

atures ranging from 50 mK to 20 K, and therefore necessitate special considerations

for S/N optimization at very low temperatures. Cryogenic preamplifiers [Ruu94]

placed near the sample coil have been used to enhance the sensitivity of CW NMR

measurements at low frequencies, but this method is generally rather complicated to

implement and has the added disadvantage of providing a relatively large heat load

on the cryogenic region of the sample and generally prohibits studies below T~ 1 K.

The purpose of this chapter is to present a unique and simple technique for optimizing

the NMR S/N ratio for an ultrahigh-frequency (UHF) quadrature hybrid-tee bridge

(QHT) spectrometer (shown in Figure 2.1) for very low-temperature measurements
(T<0.1 K) by providing a facile and efficient match of the NMR pick-up coil to a

long transmission line extending from the cold finger of a dilution refrigerator. A

Kel-F NMR sample cell of improved design that is superfluid 4He leak tight has been

developed for very low temperatures. This sample-cell design is suitable for studies of

two-dimensional systems with very low spin densities and will be discussed in Chapter


In previous tank-circuit designs, large signal losses occur mainly due to impedance

mismatching at the junction of the cold end of the coaxial transmission line and the

so-called conventional series tank circuit, which consists of a low-temperature NMR

radio frequency coil connected via a 50 Q coaxial transmission line to variable series-

tuning and parallel-matching capacitors located at room temperature. This conven-

tional tank-circuit/transmission-line design has a serious mismatch and a very large

standing-wave ratio that effectively magnifies the line losses. At very low tempera-

tures, this cannot be remedied using R.A. McKay's method [Con93] because we must

use long, low-thermal-conductivity, high-loss transmission lines to reduce the heat

leak from room temperature if we are to attain temperatures T<0.1 K. If no low

temperature matching is used, the Q value for a conventional tank circuit using a 2

m length of high-quality commercial cryogenic (low-thermal-conductivity) cable (loss

factor a= 0.17 nepers/m) turns out to be about 3, leading to a significant decrease in

the S/N ratio. This is to be compared with a Q=52 for our double-tuning technique.

Several methods for improving the sensitivity of pulsed NMR probes are discussed

in Ref. [Con93, Leo83], but these are generally relevant for high-temperature mea-

surements (T>1 K). One such arrangement in Ref. [Con93, Leo83] uses fixed tuning

near the coil. The frequency of the NMR experiments is adjusted to the probe's

resonant frequency, and then the field is readjusted to the new NMR frequency. As

noted in Ref. [Con93], this probe is impractical because it seems like "the tail wagging

the dog". Also, low-loss probes in which incorporate long tuning shafts necessitating

complicated heat sinking and vacuum seals and also suffer from freezing or breaking

of capacitors at the cold end.

Two approaches were undertaken to improve the CW-NMR sensitivity: Firstly, by

accommodating only one series variable capacitor near the NMR signal pick-up coil

[Leo83], and secondly, by placing an additional parallel variable capacitor near the

coil. Without experiencing any problem with the variable capacitors near the sample

coil down to 50mK, we observed that the NMR S/N ratio could be increased by a fac-

tor of five using the former method. In the following section, we will explain a unique

and simple technique for optimizing the NMR S/N ratio using capacitive tuning at

both the low temperature and high temperature ends of the transmission line. The

arrangement has worked successfully down to 50 mK, yielding a S/N improvement of


2.2 The UHF CW-NMR Spectrometer

As an alternative to using cryogenic preamplifiers [Ruu94], we have successfully

improved the NMR S/N ratio by impedance matching the low temperature NMR

tank circuit to the coaxial transmission line using two cryogenic, air-dielectric variable

trimmer capacitors at low temperatures and by using a second combination of variable



uhf Hybrid 4 hf
enerator Tee
368MHz 3 C
r -------- Diode Detector

C Tuning Box
r----"--"- -- -----------
(----------------- ----------- -

Placed inside Low
the dewar Pass Filter

Transmission Line
e= I2X Ii


Ramp SignalPC


Figure 2.1: Block Diagram of UHF CW-NMR Hybrid-Tee Bridge Spectrometer.

air-trimmer capacitors at room temperature for fine tuning. The block diagram of

the UHF NMR spectrometer is shown in Figure 2.1

We used a QHT to form a bridge configuration for CW NMR detection [Sul73],

but 1800 hybrid tees have also been used [Kle67]. The method used to match the

NMR tank circuit to the bridge is of chief concern [Haa94]. The QHT is a four-port

+3 dB strip line decoupler [Kle67, Haa94] which splits the RF excitation applied at

port 1 into equal and isolated quadrature phased voltages at ports 2 and 3. Port 2 of

the magic tee is terminated in a purely resistive 50 Q load, and port 3 is connected

to the series NMR tank circuit via a 50 Q coaxial transmission line of length 2A. If

Vj is the vector voltage at the jth port, then near the impedance matching condition

I V V4 I> | V3 V2 ( (2.1)


I V4 I V3 V2 +47rnQ V3 I (X'sinp X" cos p) (2.2)

where x=X' j X" is the nuclear spin susceptibility of the sample, r7 is the filling

factor, and p is the departure from quadrature of the phases of V3 and V2. Note that

j= /-1T. The quality factor Q is the ratio of the energy stored in the coil to the energy

dissipated in the circuit per RF cycle. If p = nir (amplitude unbalance only), IV41

gives the NMR absorption X", while for p=(n+1)7r/2, the dispersion x' is detected.

The bridge unbalance, IV41, is detected at port 4 by a suitable receiver system. The

voltage measured at port 4 of the QHT is zero when ports 2 and 3 are terminated

with equal impedances since the reflected voltages cancel at the output. In practice,

however, the tank circuit is mismatched slightly in order to provide a small reflected

carrier voltage upon which the NMR signal can be measured. The RF carrier exiting

port 4 is amplified, and the signal voltage modulating the carrier is detected by an RF
diode detector and filtered as the magnetic field is swept through resonance. The use
'Model FSCM12457, Merrimac Research Corporation, West Caldwell, New Jersey.

of a lock-in amplifier to slightly modulate the magnetic field sweep and to detect the

NMR signal allows for the direct observation of the derivative line shape of a system of

nuclear spins. The magnetic field sweep and lock-in amplifier output are interfaced to

a personal computer using a National Instruments AT-MIO-16H-25 digital-to-analog

conversion board (DAC) for automatic data acquisition.

We found that the sensitivity of the so-called conventional tank circuit (which is

the same tank circuit as in Figure 2.1, but without C1 and C2) was inadequate for

thin film studies of molecular hydrogen in zeolite or adsorbed on BN due to the large

impedance mismatch at the cold end of the coaxial transmission line. A large fraction

of the signal voltage is not transmitted from the coil as a result of the mismatch.

The complex reflection coefficient pt for harmonic voltage waves is defined as the

ratio of the phasor value of the reflected voltage wave to the phasor value of the

incident voltage wave at the load terminals [Chi68, Fuk81]

Pt = 1)/( +1), (2.3)
Zo Zo

where Zt is the terminal load impedance and Zo is the characteristic impedance of

the coaxial transmission line. To minimize the reflected signal from the sample coil,

we require ZoZt. In the conventional tank circuit design, the NMR coil located at

the cold end of the transmission line is tuned to resonance and matched to 50 Q at

port 3 using a room temperature capacitive tuning box. In this case the terminal

impedance at the end of the transmission line is Zt=rs+jwL at resonance, where r,

and L are the resistance and inductance of the coil, respectively. It is very difficult

to match the impedance to 50 Qf by adjusting the inductance of an isolated coil. For

example, Iptl for the conventional tank circuit at the cold end of the transmission line

turns out to be 0.96 for a 100 nH coil at 300 MHz.

However, by accommodating two variable air-trimmer capacitors (C1 and C2) near

the RF signal pick-up coil as in Figure 2.1, we can minimize pt by taking advantage

of the relation

Zt = QwL[ 1 ]2 (2.4)

at resonance for the low temperature matched tank circuit2 [Fuk81]. This match-

ing network is used because of its ability to accommodate a large range of complex

impedances. The variable capacitors are used in order to closely tune and match the

tank circuit to a wide range of resonant frequencies in contrast to the inflexible fixed

tuning method discussed by Walton and Conradi [Wal89]. In the procedure to match

the tank circuit to 50 Q, C1 is first adjusted to minimize the bridge unbalance output

at port 4, and then C2 is adjusted to further minimize the unbalance without the room

temperature capacitive tuning box in place. This procedure is repeated until the un-

balance reaches a minimum value. In the end, Zt of the tank circuit becomes nearly

50 Q, resulting in a much smaller reflected voltage at the cold end. As the signal is

transmitted along the transmission line, there is only a small impedance mismatch 50

Q + AZ. Perfect matching (AZ=0) at the cold end is not desirable for two reasons:

firstly, AZ changes appreciably and unpredictably after the assembly is cooled from

room temperature, and secondly, it is important to be able to adjust the amplitude

and phase unbalance carefully when the assembly is cold so that the absorption (or

dispersion) signal can be observed rather than a mixture of both forms. These needs

can be satisfied very simply by allowing for additional fine tuning and impedance

matching by using a second combination of variable air-trimmer capacitors at room

temperature (C3 and C4 in Figure 2.1) to adjust AZ to a pure resistance and thereby

obtain a real carrier voltage in phase with the absorption signal (or alternatively by

selecting a phase unbalance to detect the dispersion signal).

When the vacuum can of the dilution refrigerator was evacuated, the circuit could

be tuned and matched for a wide range of incident RF excitation using the room

temperature tuning box. The change in dielectric constant due to the addition of
2Motorola RF Device Data Volume II, (Q4/88) p7-52.

a small amount of exchange gas, however, made it difficult to tune and match the

circuit under any circumstances, even with the room temperature adjustment. In

addition, since most of the tuning is achieved near the coil and isolated well within

the vacuum can of the dilution refrigerator, tuning drifts seen using the conventional

coil arrangement associated with the changing level of the liquid helium dewar were a

serious concern for the low temperature tank circuit. The addition of the fine tuning

capacitors C3 and C4 enable us to correct for these changes after the probe was cooled.

The NMR spectrometer described above has been found to be remarkably stable and

free from microphonics and quite versatile for fixed field studies. The tuning is also

very stable over the course of NMR measurements lasting for several weeks, and even

during the operation of the dilution refrigerator. However, because this system is

able to detect a signal from 1016 spins with a signal-to-noise ratio exceeding 100, the

construction of a background-free sample cell is of the utmost concern.

Non-magnetic, coaxial air-dielectric trimmer capacitors (C1 and C2) purchased

from Johanson Electronics Co. 3 were used at the cold end of the transmission

line to minimize distortion of the magnetic field. Although these capacitors are not

advertised for cryogenic use, they have been used successfully at temperatures down

to 50 mK and have survived many thermal cycles to room temperature. The part

numbers and capacitance ranges are P/N 5762 and 0.6 6pF for C1 and P/N 5642

and 1 30 pF for C2, respectively. A 1/4-inch diameter 2-turn NMR pick-up coil

was wound compactly around a Kel-F sample cell in order to form the resonant tank

circuit. The inductance L of the coil was 120nH, yielding a Q-value of 52.

Before assembling this system onto the cold finger of our dilution refrigerator,

we tested the probe by measuring the Electron Spin Resonance (ESR) absorption

spectrum at 268 MHz of a DPPH (Diphenyl-Picryl-Hydrazyl) sample at room tem-

perature and at 77 K. A comparison of the conventional tank circuit and new low
3Johanson Manufacturing Co., Rockaway Valley Road, Boonton, NJ 07005

temperature tank circuit using the same RF level H1 showed that the ESR signal

amplitude was about 40 times larger with the latter arrangement. RF levels were

kept low to avoid any saturation in spectrum throughout these experiments.

A low RF excitation of -25 -30 dBm to avoid saturation effects was used to

acquire the data, and the lock-in amplifier time constant and sensitivity were set to 1

second and 50 mV, respectively, throughout these projects. Derivative line shapes for

several layers of H2 measured using the conventional tank circuit typically exhibited

distorted baselines and S/N ratios on the order of 10 even after extensive signal


2.3 The NMR Signal Processing

The CW NMR QHT bridge spectrometer consisted of the magnetic field sweep

and the RF absorption detector.

The magnetic field sweep consisted of four main components; a lock-in amplifier,

a programmable power supply, a summing amplifier and a modulation magnet. An

audio frequency modulation signal was generated by the lock-in amplifier and sent to

a summing amplifier, where it was combined with a linear ramp (generated by an IBM

compatible personal computer) and sent to the current feed of a programmable power

supply. The power supply generated the current for the vertical, superconducting

modulation magnet located between the sample cell and the main magnet.

The radio frequency spectrometer consisted of five main components; an RF signal

generator, a quadrature hybrid tee, an RF amplifier, audio amplifiers, and a lock-

in amplifier. The RF amplifier, with an amplification factor of approximately 250,

detects the imbalance from port 4 of the QHT. An RF diode after the RF amplifier

converts the AC signal into DC and eliminates the need for a mixer and its associated

phase from the experiment. However, the cost associated with this simplificaton is

some loss of information. The DC signal can not be passed through a mixer to separate

the absorption and dispersion signals. After the RF diode and some amplification, a

band-pass filter (10-1000 Hz) was used to separate the audio frequency of the NMR

signal from the RF carrier signal. A second audio amplifier was then used to raise

the signal voltage to an optimal level for the lock-in amplifier. The lock-in amplifier

compared the amplitude, frequency, and phase of the incoming signal with that of

the audio modulation it generated and relayed to the modulation magnet.

One important consequence of this method is that the derivative of the NMR

line shape was actually measured. Such derivative measurement is a consequence

of using the lock-in amplifier as the detecting instrument. One benefit of derivative

measurement is that the derivative of the line shape is more sensitive to small changes

in the line shape than the integrated absorption line shape itself. All integrated

absorption line shapes presented in this thesis have been numerically integrated by a

pc computer.

2.4 Conclusion

We have modified our UHF QHT CW-NMR spectrometer by improving the match-

ing of the tank circuit at the cold end and have tested the method down to 50 mK.

The enhanced signal strength and improved stability achieved using the matched coil

tank circuit have proved to be a significant improvement over conventional coil CW-

NMR techniques. Compared to the use of cryogenic preamplifiers which are relatively

complicated in construction and radiate energy into the low temperature environment

in normal operation, this unique and simple technique is ideal for a number of CW-

NMR studies where high sensitivity is critical, such as for experiments in reduced

geometries and surface studies where small signal intensities are expected.


In this chapter an NMR sample cell and probe design suitable for high-sensitivity

experimental studies, e.g., thin film studies of molecular hydrogen adsorbed on sub-

strates such as hexagonal boron nitride, are discussed. Easily machinable Kel-F

(polychlorotrifluoro ethylene :Fluorothene) is used in the NMR sample cell construc-

tion in order to fulfill two requirements. The sample cell should be superfluid 4He

leak-tight and the material for the sample cell should exhibit a minimal background

proton NMR signal. Using our new sample cell design, we have been able to observe

the signal from monolayer coverages of molecular hydrogen containing 1016 spins ad-

sorbed on the hexagonal BN surface with a signal-to-noise ratio > 100.

3.1 The NMR Sample Cell Structure

A successful new version of the NMR sample cell design [Su191] is discussed in

this chapter for thin film studies of molecular H2 in reduced dimensions at ultra-high

Larmor frequencies that is compact, protonless and superfluid-tight at millikelvin

temperatures. A relatively large sample volume was desired due to the inherent low

sensitivity of a surface measurement. In addition, it is very important that the design

allow for ease of admission and extraction of molecular H2 or other adsorbents via

the warm end of the sample cell relative to the cold finger end which is attached to

the mixing chamber of a high circulation rate dilution refrigerator. The cold finger

provides good thermal contact to the sample at low temperatures via a brush of

Mixing Chamber


Fill line

Fill line adaptor-

Indium Seal

8 0.125 inches O.D.

Bs Indium Seal
Brass -- = -'ii., 1 .^ l
Se Cu hairs

Moon ring Kel-F Chamber Kel-F Coil Holder

Figure 3.1: Cross section of the superfluid 4He leak-tight Kel-F Sample Cell.

copper hairs penetrating the sample cell. Thermal contact is assured at very low

temperatures by condensing 4He in the cell to wet all surfaces.

We were able to meet these requirements in the design of the new NMR sample

cell shown in Figure 3.1. The overall dimensions of the Kel-F are 0.986 inches long ,

1.0 in. o.d. at each end and 0.675 in. o.d. at the middle section of the chamber. The

UHF NMR RF coil consists of two turns of 0.12 in. o.d. copper wire. The ring shape

of the Kel-F coil holder is used to provide mechanical stability in the presence of any

vibrations. Superfluid seals are provided by Indium "O"-rings at each flange, and the

half moon rings (0.108 in. thick) distribute the pressure of the 316 stainless steel 2-56

screws. The capacitors used for tuning the NMR tank circuit were variable trimmer

capacitors 1 (JMC P/N 5762 and 5642) and were found to be both non-magnetic and

stable on thermal cycling to low temperatures. Among the materials available, Kel-F

was chosen for the NMR sample chamber due to its low proton content (estimated

100ppm), thus minimizing the background 'H NMR signal. In addition, it is well

known that Kel-F is easily machinable, has good thermal properties and is durable

at low temperatures [Laq52, Sco59].

In order to provide a reliable vacuum sealed sample cell, the inner diameter of the

cylindrical Kel-F sample chamber was carefully machined to be 0.005 inches larger

than the snouts which extend out of the metal end flanges, one of which is welded

to the Cu cold finger and secured to the cold end of the cell. Sample cells which fit

tightly on the metal snouts consistently developed a leak as the cell was cooled below

77 K. The other end of the sample cell was similarly sealed using a hollow brass flange

into which the fill line adapter is soft soldered. A stainless steel capillary tube fill line

wrapped with heater wire extended from the top of the cryostat and was soldered

into the fill line adaptor. There are lips on both ends of the Kel-F chamber which fit

loosely into grooves on the open-ended flanges of the sample cell where an indium ring
'Johanson Manufacturing Co, Rockaway Valley Road, Boonton, NJ07005

is placed to achieve a superfluid 4He tight seal. Indium rings (0.003 in. in diameter)

were placed in the 0.005 inch deep grooves in the metal flanges. In assembling the

Kel-F chamber and metal parts, we used 16 non-magnetic 316 Stainless Steel 2-56

screws 2 which fasten the flanges via two brass half-moon rings at each end. A small

amount of protonless fluorosilicated lubricant 3 was applied to the indium to facilitate

the assembly. A brush of enamel-free copper wires (0.025mm o.d.) was soft-soldered

to the cold metal snout in order to assure thermal contact to the superfluid 4He which

in turn coated the sample and the metallic surfaces. A larger metallic area is desirable

for cooling to the lowest temperatures, but this leads to a loss in the quality factor of

the NMR coil, and concomitant loss in NMR sensitivity. We therefore chose a brush

of wires as a compromise between these competing factors.

The NMR sample cell was initially leak tested in a helium atmosphere while

being pumped on by a helium mass spectrometer leak detector (MS-17AM, Veeco

Instruments, Inc. Terminal Dr. Plainview, NY 11803) at room temperature and 77

K. A slow rise in the leak rate due to the diffusion of 4He through the Kel-F at

room temperature necessitated leak testing the cell at 77K to confirm a tight seal.

The cell was tested again in the dilution refrigerator by cooling slowly to liquid

nitrogen temperature and introducing He gas into the cell via the gas fill line while

the vacuum can of our dilution refrigerator was being pumped and monitored by the

leak detector. The cell performed excellently even after several thermal cycles and

continued to remain leak-tight well below 100 mK.

3.2 Summary

This design offers a reliable and practical cell for NMR studies of solidified gases

confined to small geometries at temperatures below 0.1 K. The NMR sensitivity
2From B & B Socket Products, Inc., Anaheim, California, USA.
3Fluorolube Grease GR-90: Fisher Scientific, Fair Lawn, NJ, USA.


realized in this design allowed us to observe 1016 protons at 1 K with a signal to noise

ratio of 100 :1 for a lock-in detection time constant of 1 second, and a line width of

~250 kHz.


4.1 The Sample Preparation

Hexagonal boron nitride (BN) has recently gained considerable interest because

of its similarity in crystal structure to graphite, which has been extensively used for

more than two decades as a substrate in experimental studies of physisorbed films.

Both BN and graphite have favorable characteristics, such as high homogeneity and

large crystal sizes, relatively low corrugation in the substrate potential, and com-

patibility with different experimental techniques. The crystal structure dimension of

BN is, however, about 2% greater than that of graphite. BN is more ideal for NMR

measurements because its electrical conductivity is much less than that of graphite,

making it a particularly attractive substrate for use with physisorbed molecular sys-

tems. The powdered forms of BN are commercially available from the Carborundum

company 1, Advanced Ceramics Corporation 2, and Johnson-Matthey 3. We pur-

chased 50 g of powdered BN from Johnson-Matthey because they offered the largest

surface areas (10-15 m2/g) of all the products available.

The as-delivered samples must be carefully cleaned and annealed prior to any

experiment in order to achieve the highest homogeneity and to remove any impurities

associated with the manufacturing process. The procedure used is especially critical

for surface adsorption studies. While the procedures necessary to produce clean
1The Carborundum Company, Boron Nitride Division, 168 Creekside Drive, Amherst, NY 14228-
2Advanced Ceramics Corporation, 11907 Madison Avenue, Lakewood, Ohio 44107.
3Johnson-Matthey, 30 Bond Street, Ward Hill, MA 01835-8099

samples have been well established for exfoliated graphite, many different cleaning

methods and various heat treatments [Ros58, Pie62, Ram73, Reg79, Gri80, Del83,

Boc84, Lah82, Alk91] are commonly used for BN powders. The temperature employed

for different treatments varies from 200' C to 9000 C over the course of 3 hours to 5


Recently, Wolfson et al. [Wol95] reported that annealing a BN sample at 9000 C is

not always sufficient to produce clean samples and fails to remove soluble borate con-

taminants (namely boric oxide and boric acid impurities) which would tend to distort

the homogeneity of the substrate. In order to remove any of these impurities which

may be present in the BN powder, we washed the samples in methanol and rinsed

the powder with de-ionized water. We repeated the washing process three times,

being certain to filter and dry the BN powder each time. Small magnetic filings were

found during the washing process and carefully removed from the powder samples.

We believe the filings resulted from the powdering process during manufacturing.

After producing a clean powder, we immediately placed the BN sample into a

sealed 2 in. diameter quartz tube which was connected to a low pressure pumping

station and placed inside an oven (Thermolyne, Type 21100 Tube Furnace). The

sample was outgassed for 4 hours at 2000 C and then annealed for 24 hours at 9000

C under a vacuum of 10-6 torr. The sample was slowly cooled down to room tem-

perature and kept under vacuum until it was ready to load into the Kel-F NMR

sample cell [Kim96b]. When loading the BN powder into the sample cell, the time

of exposure to the atmosphere was minimized and the sample was evacuated with a

diffusion pump rapidly after assembly. The BN powders were left as a loose uncom-

pressed sample and contained within a Kel-F sample chamber around which a 1/4

in. diameter 2-turn NMR coil was wound. In order to make thermal contact with

the refrigerator a bundle of fine copper wires was inserted into the powder with one

end of the wires soldered to a Cu flange. This brass flange was an integral part of


0 (a) 0.2 (b)

0.2 0.
0.0 ,
0.2 0

-0o.2 -0.2

E -0.4 -.4


^' -o.e -O
-0.8-08 -
_. -0.8
-1. 0 . -'-
-300 -200 -100 0 100 200 300 -300 -200 -100 0 100 200 30C
Frequency (kHz) Frequency (kHz)

Figure 4.1: Typical NMR line shapes. (a) with only the proton background, and (b)
with both the 12-layer hydrogen film and proton background.

a cold finger of a high circulation rate dilution refrigerator. The NMR sample cell

was placed in the vacuum can of the dilution refrigerator and located at the center

of a 7 tesla superconducting magnet of moderate homogeneity (30 x 10-6 over the

NMR sample volume). Gaseous H2 samples were introduced into a sample chamber

at about 20 K through the capillary fill line, onto which a bifilar heater wire was

wound to reduce the chances of a frozen hydrogen blockage. After condensing the

sample, we introduced a sufficient quantity of 4He gas into the chamber to completely

wet all surfaces in order to improve the thermal contact [Ebe94].

4.2 The NMR Data Processing

The NMR absorption signals were observed using a UHF CW NMR quadrature

hybrid tee bridge spectrometer operating at 268 MHz. The matched resonant tank

circuit is located in the low temperature region of the dilution refrigerator and is

connected to the spectrometer via a 2A 50 0 coaxial transmission line of low thermal

conductivity. The bridge spectrometer detects the absorptive component of the NMR

signal by adjusting room temperature fine-tuning capacitors so that there is no error in

phase. The output of the bridge circuit was amplified by a low noise room temperature

narrow band UHF receiver, and the signal voltage modulating the carrier is detected

by an RF diode detector and filtered as the magnetic field is swept through resonance.

The use of a lock-in amplifier to slightly modulate the magnetic field sweep and to

detect the NMR signal allows for the direct observation of the derivative line shapes.

The magnetic field sweep and lock-in amplifier output are interfaced via an IEEE 488

bus to a personal computer using a National Instrument AT-MIO-16H-25 digital-to-

analog conversion board (DAC) for automated data acquisition. The data acquisition

system allows continuous data collections 24 hours a day except during a helium

transfer (about half an hour).

To reduce spurious NMR signals from being introduced into our proton NMR line

shapes, special care was taken to minimize one major source of a 1H background

signal by using Kel-F (known to have a negligible proton content) for the sample

chamber. However, as a result of the enhanced sensitivity of our spectrometer, we

did detect a small residual background signal.

Figure 4.1(a) illustrates a typical derivative NMR line shape for the residual back-

ground signal. The shape of the background signal examined remained constant over

the entire temperature range studied, indicating we have only one source of the back-

ground signal. Figure 4.1(b) shows a typical derivative NMR absorption line shape of

12-layer H2 films/BN which includes the background signal. The line shapes arising

from the molecular hydrogen were therefore determined by subtracting a fixed back-

ground signal (weighted according to the temperature) which was measured under

the same conditions without any hydrogen sample. The 1H NMR absorption line was

recorded by sweeping the magnetic field through resonance with a sawtooth voltage

(125 gauss in amplitude) and a simultaneous AC (210 Hz) modulation of 1 gauss. A

low RF excitation of -30 dBm was used to avoid saturation effects, and the lock-in

amplifier time constant and sensitivity were set to 1 s and 50 mV, respectively. We

estimate an RF field in the coil of about 3 mG, assuming that all of the incident

excitation applied is transmitted to the coil as a result of the improved matching

configuration of the tank circuit.

The NMR line shapes presented here represent the average of a single up- and

down-sweep. The sample temperatures were determined by taking the average start-

ing and ending temperatures for each down- and up-sweep.

4.3 Data Processing

Data processing involved averaging data files (up- and down-sweep) collected in

an IBM PC. The total number of data files reached up to 18,000 for four different

experiments. Due to hindered conversion rates (especially in mono- and submonolayer

coverages), each experimental run lasted 15-80 days, used 12-layer to submonolayer

coverages, and ranged in temperature from 0.05-10 K.

Data analysis programs were written using Mathematica 4 in order to subtract

spectra and integrate line shapes under the same conditions for presentation as func-

tions of ortho concentration and temperature. Also, this analysis program allowed

simultaneous calculations of the second moments (local order parameter), magnetiza-

tion and conversion rates, etc. which are useful to obtain information of the NMR line

shapes. Most importantly, the final analysis involved visually looking for trends in

the NMR line shapes (both derivative and integrated) as a function of temperature,

sample age, and the coverages of H2 adsorbed onto the surface.

Note that the NMR line shapes presented in the following chapters are averages

of a single up- and down-sweep.

4Mathematica, Wolfram Research, Champaign, IL 61821.


This chapter contains the results and interpretations of systematic studies of 12-

layer hydrogen films adsorbed on a hexagonal BN substrate using high sensitivity

Continuous Wave (CW) NMR techniques. The purpose of these experiments is to

examine the nature of the 2D orientational ordering of quantum rotors on a well-

defined substrate [Har79, Kub78, Kul.S5] and to test for the existence of a new 2D

orientational glass for dilute ortho concentrations.

5.1 Overview

The hydrogen molecule is the simplest diatomic molecule that exists in nature and

the properties of molecular hydrogen have been studied extensively since the early

part of the twentieth century. Many detailed studies of both experimental [Kub85,

Sul75] and theoretical [Har70, Har70b, Si180, Har85] work have been reported for the

condensed phases in bulk (3D) demonstrating the fundamental quantum mechanical

nature of hydrogen, but very little is known about its 2D behavior.

In bulk H2 samples, it has been well-established that the electrostatic quadrupole-

quadrupole (EQQ) interactions, VQQ, play the major role among the anisotropic in-

termolecular interactions [Nik55]. At sufficiently high temperatures, kBT>IVQQI in

the bulk solid, the hydrogen molecules form an h.c.p. structure in which molecules

are free to rotate and the mean orientation of each molecule is zero. As the ther-

mal energy is decreased, the intermolecular EQQ interactions become significant and

the molecules adopt orientations to minimize their interaction energy. The inter-

actions are highly frustrated because of the incompatibility of the symmetry of the

quadrupole-quadrupole interaction and the lattice symmetry: the lowest energy for

an isolated pair of axial quadrupoles is a "tee" configuration with molecular axes

mutually perpendicular, and this cannot be satisfied for any 3D close-packed lat-

tice. The frustrated EQQ interactions between ortho molecules (orbital angular mo-

mentum J=1, total nuclear spin I=1) in the cubic lattice lead to an unusual Pa3

antiferro-orientational ordered structure [Sil80, Kra83]. In order to correctly pre-

dict the condensed phases of hydrogen at low temperatures, quantum mechanical

approaches must be used to describe both the translational and rotational degrees of

freedom of the molecules.

In recent years studies of solid hydrogen in restricted geometries adsorbedd films

[Das89, Vee86, Bie91, Zep90, Mar93, Ma89, Liu93], suspended droplets [Mar87], and
porous materials [Tel83, Ral91, Bre90]) have received much attention as a means to ex-

amine the effects of reduced dimensions, geometrical constraints and the substrate in-

teractions on the orientational ordering. Carefully grown films [Cur65, Bos67, Sch85]

form h.c.p. structures, and the increased quantum zero point motion in the outer

layers [Wag94] combined with the frustration of the h.c.p. structure may inhibit long

range ordering. Periodic ordering has not been observed for bulk hexagonal struc-


Many interesting affects are expected to occur in hydrogen layers as a consequence

of the quantum mechanical nature of the films. Liquid para-H2 (J=0, I=0) is expected

to undergo Bose-Einstein Condensation (BEC) at about 6 K, and if supercooled

sufficiently, liquid ortho-H2 (J=1, and I=1) would form a new anisotropic superfluid

below 1 K with no known counterpart [Gin72, Mar92]. Substantial supercooling of the

liquid has been observed down to 8 K in porous zeolite [Ral91], and many groups have

concentrated their efforts in an attempt to observe a Kosterlitz-Thouless transition

into a BEC state [Sil95]. This superfluid transition has not been observed in the

12-layer experiments reported in this study. Further studies, for coverages greater

than 12-layer films, are being prepared to test for superfluidity in 2D because it is

well known that the surface forms a highly mobile quasi-liquid layer with appreciable

quantum exchange of molecules in the free-surface layer for temperatures as low as 6

K [Mar93].

Several groups [Wol95, Eva95] have shown that BN is an excellent substrate for

reduced dimensionality studies. Ar adsorption isotherm experiments at 77 K are com-

monly used to study the homogeneity and strength of the substrate. A good test of

the quality of the substrate preparation is the sharpness of the liquid-solid substep at

about 35 torr for the Ar/BN isotherm at 77 K [Ma89]. A careful characterization of

BN as a substrate has been carried out previously by Evans et al. [Eva95] using auto-

mated volumetric isotherm experiments in this laboratory. In contrast to HD/MgO

and HD/graphite [Liu93], no substep corresponding to liquid-solid coexistence was

seen for the second layer of HD/BN [Eva95] and only a weak substep was seen near

third layer completion. The critical points Tcr were determined from a Lahrer plot

[Lah89] of the inverse compressibility (X- 1= In p/OT). The isosteric heats of adsorp-

tion qst=kBT2(a In P/O T)N,A yield a binding energy of 12010 K for the second

layer on MgO compared to 17710 K for HD on graphite [Liu93]. The BN powders

used for our experiments were from the same manufacturer's batch as used by Evans

et al. [Eva95] for 4-layer and 8-layer film NMR studies [Eva95b]. Different methods

of treatment of the BN from those described by Evans et al. were, however, used in

the present studies.

In this study, we have explored the local orientational ordering for thin films in

order to examine the effect of geometrical frustration in 12-layer hydrogen films for

ortho concentrations 0.23

So (a) T=158mK


.o -

-300 -0 -10 0 10 a2 >
Frequency (kHz)

t0 o X=0.63



I o
0o .

Frequency (kHz)

(e) T=368mK

o .

00 -

"0 -200 -10 0 10 oO -
Frequency (kHz)

o (g) T=546mK



Frequency (kHz)

(no T=158mK


300 -a0 -.o o 100 aoo
Frequency (kHz)

Frequency (kHz)


Frequency (kHz)

(h) T=546mK


Frequency (kHz)

Figure 5.1: Typical CW NMR derivative (a, c, e, g) and corresponding integrated (b,

d, f, g) absorption line shapes for high ortho concentrations 0.57




zo 30o


5.2 Results

In this section, we present averaged derivative and integrated CW NMR absorp-

tion line shapes as functions of temperature and the ortho concentration. Both the

derivative and integrated CW NMR absorption line shapes were visually examined

in order to identify possible trends as the temperature and ortho concentration were


Prior to the NMR studies, we carried out a measurement of a partial adsorption

isotherm of Ar on BN at 77 K to determine the number of molecules adsorbed per

layer. The surface area of the powdered sample can also be determined by noting the

volume admitted in the Ar adsorption at P ~ 34 torr, corresponding to the fluid-

solid transition observed at 1.06 monolayers [Mig93]. Also, from this measurement

and the knowledge of the associated volumes of the H2 gas handling system, we are

able to determine the H2 coverages to within 5% for a given volume. Using the above

method, we determined that a monolayer coverage occurred for a pressure of 75 mbar

in our loading volume. This value corresponds to ~50% of the amount of hydrogen

per layer used by Evans et al. [Eva95], which is consistent with the reduced volume

of our sample cell.

Typical CW NMR derivative and integrated absorption line shapes for high ortho

concentrations 0.57
shapes are a strong central Gaussian peak (FWHM~ 602 kHz) and the appearance

of partial orientational ordering as evidenced by the formation of two wing structures

separated by approximately 3005 kHz. The wings begin to appear for an ortho con-

centration X<0.66 and temperature T<0.4 K, suggesting that orientational ordering

does not occur for high ortho concentrations (X>0.66) even at the lowest temper-

atures. For this concentration range, the outer wings increase in amplitude below

T=300 mK as the ortho concentration X decreases as shown in Figures 5.1(a)-(d).

Above approximately 300 mK the sample temperature plays a major role in the local

orientational ordering of ortho-H2 molecules as shown in Figures 5.1(e)-(h). In par-

ticular, the local order parameter a decreases as the temperature is increased in this

temperature range. For the high ortho concentration, even below 1 K, the intensity

of the outer wing of the NMR derivative line shapes was found to depend strongly

not only on the ortho concentration but also on temperature. This is attributed to

the growth of the local orientational ordering for the outer broad component of the

NMR line shape. In other words, the degree of local orientational ordering is found

to increase as the intensity of the outer wings increases, in contrast to the first NMR

report of hydrogen films (4 and 8 layers) on BN by Evans et al. [Eva95b]. These

new results imply that the orientational ordering occurs predominantly in the first

two layers near the BN substrate for the temperature range explored. The amplitude

of the outer wings increases as the sample temperature and the ortho concentration

decrease at the expense of the amplitude of the central peak, clearly showing that

there is a continuous evolution with more disordered molecules becoming frozen into

orientationally ordered states as the temperature is lowered. Note that for 4- and

8-layer films, the first four layers have been shown to undergo orientational ordering

[Eva95b]. Generally, it was found that as the ortho concentration decreased, the local

orientational ordering occurred at even higher temperatures (as high as T~1.7 K for

X-0.23). Note that the outer peaks of the wings correspond to a mean local order

parameter a=0.840 0.005 for ortho concentrations 0.57
The clear changes observed in the derivative NMR absorption line shapes occur

rapidly for the intermediate ortho concentrations 0.45
line shapes are presented in Figure 5.2. Figures 5.2 (a) and (b) show the emergence

of clear double peaks (marked by A and B) in one side of the outer wings. One peak

component marked by B resembles the line shape expected for a quadrupolar glass

for this fraction of the ortho and para mixtures.


S B T273mK
A B X-0.39

/ B A

-o.. (a)
=0 20 -0oo 0o 100o 2o

Frequency (kHz)

0- T=389mK

0 B A

.OS \ I


Frequency (kHz)

o0. T=159mK
0 X=0.379



-06 (e)

Frequency (kHz)

o4 C T=105mK
X=0.376 C


-o (g)

Frequency (kHz)

Figure 5.2: Typical CW NMR derivative

ortho concentration near X=0.38.

(b) T=272mK


so -


Frequency (kHz)

(d) T=389mK

0 7

Frequency (kHz)

S(f) T=159mK
so ca=0.99

Frequency (kHz)

(h) T=105mK

S\ =0.99
so /

-IO i i i = .9

Frequency (kHz)

and integrated absorption line shapes for


In this interpretation there are three components to the line shape: a central

disordered line attributed to the outer layers, an inner quadrupolar glass line shape

with outer edges in the spectrum at position B (attributed to the intermediate layers),

and a more completely ordered structure with wings extending out to position A due

to the influence of the crystal field on molecules near the substrate. (A detailed

discussion of the fit is given in the following section.)

NMR line shapes for a quadrupolar glass have also been seen in the analysis of

submonolayer data for ortho concentrations very close to this concentration range

and will be discussed later. However, the line shapes are no longer symmetric with

respect to the center of the absorption spectrum. This is believed to be due to partial

RF saturation because the NMR spectra presented are more symmetric than those

shown by Evans et al. [Eva95b], which used higher RF excitation levels for 4- and

8-layer films studies, and possibly due to the difficulties in accurately subtracting the

background spectrum that occurs on the high frequency side of the spectrum. Be-

cause of this background signal, the high frequency side of the difference spectrum is

not fully reliable. Alternatively, the substrate interactions could contain an asymme-

try which would affect the CW NMR line shape. Below approximately T=200 mK

[in Figures 5.2(e) and (g)], one of the double peaks (marked by B) is seen to merge

with the peak marked by A. (A new wing is marked by C.) This behavior is seen

to dominate for ortho concentrations as low as X=0.27. Below X=0.27 the shape of

the outer wings returns to the original high concentration NMR line shape but with

an increase in the peak splitting. The separation of the outer wings at the onset

of the new shape is approximately 3807 kHz for ortho concentrations X<0.25. In

addition, the increasing separation of the outer wings at low ortho concentrations

is accompanied by a broadening of the central Gaussian peak to 805 kHz. This

behavior was also noted by Evans et al. [Eva95b] without detailed observations. We

have further verified from preliminary results of monolayer studies, that the NMR

0o 0 T=1.01K



-o.3 I 1 '
-300 -200 -100 0 la 20 x
Frequency (kHz)

SL. T=1.31K

-o 02 (c)

,00 -20 -t, O, 0 10, 20 ,

Frequency (kHz)

0o T T=1.47K



a300 -200 -100 o 10o 20a 3

Frequency (kHz)

0.2 T=1.50K


a"' (g)



-3o -20 -1o 0 0 2o a 3

Frequency (kHz)

Frequency (kHz)




o -1 1

Frequency (kHz)

(f) A T=1.4750K
(h) X=0.234

o o=0.80
.Uoo -2o -loo o lo =oo

Frequency (kHz)
Frequency (kHz)

Frequency (kHz)

Figure 5.3: Typical CW NMR derivative and integrated absorption line shapes for

low ortho concentrations X<0.25 and temperatures 1.01

200 o 00


splitting increases as the ortho concentration decreases. Note also that the wings of

the NMR spectra appear to increase their separation corresponding to an increasing

a as the ortho concentration decreases. This behavior, however, is to be contrasted

with observations in the bulk where the orientational order decreases as the ortho-H2

concentration is reduced. This means that the nature of the orientational ordering

varies significantly both with the ortho concentration and with temperature. These

results indicate that as the ortho concentration is lowered the substrate interactions

tend to be a major factor in the local orientational ordering. The mean order param-

eter for ortho concentration 0.25
CW NMR line shapes for very low ortho concentrations X<0.24 and temperatures

as high as T=1.5 K. At these very low concentrations, almost all molecules in the

first three layers undergo orientational ordering at temperatures as high as T=1.5 K

[Figure 5.3(e)]. Note that the mean order parameter decreases to &=0.850 0.002

for temperatures T>1.0 K.

The principal feature of the ortho concentration dependence is that the orien-

tationally ordered molecules represented by the outer wings of the absorption line

spectrum become more completely ordered as the sample ages and the disordered

central line becomes less disordered. This behavior is comparable with that observed

for bulk hydrogen for which there is a continuous evolution of order parameters with a

decreasing smoothly as a function of age to a glass state [Sul87, Mey84]. However, no

orientational ordering was observed for our 12-layer system for temperatures T>1.7

K even for low ortho concentrations X<0.23.

Using the intensity of the integrated NMR signals, the ortho-para conversion for

this system was best fitted by a bimolecular decay process with a conversion rate of

1.430.005%/h. This is less than the bulk value of 1.9%/h and greater than the value

(0.9%/h) for the 8-layer film coverages previously reported [Eva95]. This is due to

the reduced number of nearest neighbor interactions for conversion. The conversion

rate per layer may be linearly dependent on the layer coverage.

5.3 Discussion

The relative line shapes are age dependent and the relative intensity of the wings

increases as the ortho concentration and the temperature decrease. This is found to

be contradictory to the report by Evans et al. [Eva95b]. The central Gaussian fraction

decreases as X decreases but the full width at half height increases from 652 kHz

for high X (X>0.57) to 805 kHz for dilute ortho concentrations (X<0.25). Also

the mean order parameter a increases as the ortho concentration decreases. The

broadening of the central line may be expected for lower coverages of H2 (Nfilms < 2).

and may continue to evolve to produce an NMR doublet as reported by Kubik et al.

[Kub78, Kub85].

The general results of our NMR studies of H2 adsorbed on a BN substrate can be

more deeply understood by comparing this report with Ref. [Eva95b]. In the previous

study it was concluded that only the first 4 layers near the substrate of an 8-layer

film of hydrogen undergo orientational ordering below about 1 K (for X=0.70). The

conclusion of the new studies presented in this paper is that only the first two layers

undergo substantial ordering. This indicates that the BN substrate plays a more

appreciable role in the orientational ordering of the molecular hydrogen for 8-layer

coverages than for 12-layer films. In other words, for 12-layer films the anisotropic

interactions between molecules play a more significant role than the BN substrate


The outer layers (Nfijms > 3) of H2 on BN appear to remain orientationally

disordered down to the lowest temperature explored (T=80 mK). The absence of ori-

entational ordering in subsequent layers may be attributed to the increased quantum

zero-point motion in the outer layers of the film.






350 *
I *

0.0 0.2 0.4 0.6 0.8
Temperature (K)

Figure 5.4: The temperature dependence of the separation of the peaks A. The solid
line shows the best fit for a crystal field IVcl|7.8 K.








* *


0S S
S. S

Temperature (K)

Figure 5.5: The corresponding temperature dependence of the heights A of the two
doublet structures.









Se *e e




0.0 0.2 0.4 0.6 0.8
Temperature (K)

Figure 5.6: The temperature dependence of the separation of the peaks B.


0.25 -











** *
* *

'I I I I I I I I I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Temperature (K)

Figure 5.7: The corresponding temperature dependence of the heights B of the two
doublet structures.

The most unusual feature, however, is the appearance of a well-defined doublet

that occurs at low ortho concentrations 0.25
5.5 show the temperature dependence for the separation of the outermost peaks VAA

of the doublet (marked by A in Figure 5.2) and for the height (or amplitude) of

the peak, respectively. Also, the temperature dependence for the separation VBB

and the height of the inner peak (marked by B in Figure 5.2) are illustrated in

Figure 5.6 and Figure 5.7, respectively. Note that above Ti700 mK, the two peaks

are indistinguishable in the NMR line shapes. For 2D systems, one expects the line

shapes to be sensitive to the crystal field Vc resulting from the anisotropic interaction

between hydrogen molecules and the substrate. Vc can be found from the temperature

dependence of the splitting of the NMR spectra. For an axial potential, the NMR

splitting vx(T) is given by

vxx(T) = 3ad(3 cos2 1) (5.1)


a = (3/2)(Po 1/3). (5.2)

p is the angle between magnetic field B = Boi and i axis, and Po is the fractional

population of the mj=l states. For ortho-H2, d is the dipolar constant (57.67 kHz).

Po is given by the Boltzmann factor

Po = (1/3)[1 + 2exp(-A/kBT)]-1 (5.3)


A = Vc (27/2)ra(T), (5.4)

using molecular-field theory for a pure system of J=1 molecules in the absence of

long-range order [Har79]. F is the quadrupole coupling constant for hydrogen on BN.

By solving Equations (6.9)-(6.11) simultaneously one obtains the temperature

dependence of the order parameter a. From the intermolecular separation we es-

timate r=0.47 K, and the best fit to the temperature dependence is obtained for

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