Orientational and translational properties of hydrogen films adsorbed onto boron nitride

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Orientational and translational properties of hydrogen films adsorbed onto boron nitride
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Evans, Morgan D., 1965-
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Thesis (Ph. D.)--University of Florida, 1994.
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Includes bibliographical references (leaves 116-120).
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Morgan D. Evans.
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ORIENTATIONAL
AND TRANSLATIONAL PROPERTIES
OF HYDROGEN FILMS ADSORBED
ONTO BORON NITRIDE











By


MORGAN D. EVANS


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1994























Copyright 1994

by

Morgan D. Evans











ACKNOWLEDGMENTS


If love was a tree, then my feelings for you would be a forest.
Anon., Cast and preserved in the cement of a UF sidewalk.


A committee chairman plays a special role during the attainment of one's Ph.D.

Not only is he an advisor and mentor, but also a role model. It is in this service that Neil

Sullivan has excelled. He has taught me innumerable things through daily example. This

includes the mundane, like patience and hard work, as well as the truly important, like the

ability to be both theoretically and experimentally driven and resourceful. I have been

many times amazed at his ability to switch back and forth, in a single session, from an in-

depth theoretical discussion of NMR to the experimental details so important for success.

He makes these transitions in such a logical and seamless fashion that the listener can not

help but follow along. I am convinced that his innate abilities come from a deep

understanding of the underlying issues being investigated.

My partner in this crusade, Susan Fournier, has always been supportive. We have

studied together for the last six years at two institutions. Living, writing, defending, and

graduating together has added much more to the overall experience than my meager

writing abilities can express.

The members of the physics machine shop hold a special place in this project. As

my own knowledge concerning what I wanted to measure for each experiment evolved,

they patiently built, modified, then rebuilt successive generations of experimental

apparatus. The machinists gave each project the best of their extensive abilities and

always turned out parts professional parts in a timely fashion. One machinist, Billy

Malphurs, worked with me quite extensively. He often served as consultant on projects;












not only being informative as to what was possible but also providing valuable insight into

the functionality of a project.

The engineers in the electronic shop, Larry Phelps and JeffLegge, provided advice
at a moments notice so essential to keep the pace of the research going. Further, Jeff

helped me design and build the electronics for the adsorption isotherm apparatus.

The cryogenic engineers, Greg Labbe and Brian Lothrop, toiled beyond rightful

expectations to provide the liquid helium that is the life-blood of any low temperature lab.

I often met them at odd hours of the night when they were preparing the helium to be used

in the labs the next day. I was rarely inconvenienced by a lack of helium during my tenure.

Their proficiency and efficiency kept the unit cost down to a level where I always felt that

I could try experiments for purely exploratory reasons.

Finally, I would like to state that the research atmosphere of the department was

quite beneficial to my own attitude during my tenure. Everyone in the department shared

in making the environment most pleasant. In particular, one new post-doc, Jeff Bodart,

read a few versions of this thesis making many comments and determinedly hunting down

typographical errors. Jaha Hamida, a researcher with whom I shared the laboratory
facilities, was always a pleasure to work beside and added a fresh and balanced

perspective to everyday life. The members of my doctoral committee made themselves

available for consultation at a moments notice and provided useful comments that aided

greatly to my research. Two UF physics undergraduates, Nay Patel and Jonathan Hack,

spent many late hours in the laboratory helping me develop the experimental apparatus.

Kimberly Yocum took care of our research group with care, humor, and a special

attention to details which made the daily experience much more efficient and enjoyable for

everyone around her.












TABLE OF CONTENTS



ACKNOW LEDGEM ENTS........................................................................................... iii

TABLE OF CONTENTS ........................................................................................... v

LIST OF TABLES ....................................................................................................... viii

LIST OF FIGURES................................................................................................... ix

ABSTRACT................................................................................................................. xii

CHAPTERS

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

1.1 Quantum M echanical Properties of Hydrogen .................................... ........ ... 1

1.2 Substrates .................... .............. .................................................... 3

1.3 M molecular Orientation....................... ............................................................... 6

1.3.1 Overview ............................................................................................... 5

1.3.2 Dipole-Dipole Interaction................................................................... ....7

1.3.3 Pake Doublet Line Shape.......................................................................... 11

1.4 Orientational Ordering of Hydrogen and Deuterium ......................................... 13

1.4.1 Overview ............................................................................................... 13

1.4.2 Review of Three-Dimensional Results...................... ................. .......... 14

1.4.2.a Order Parameters ............................ .......................................... 14

1.4.2.b Phase Diagram ........................... .................................................... 17

1.4.2.c NM R Line Shapes............................................. ....................... 18

1.4.3 Two-Dimensional Orientational Ordering M odel................................ .... 22






1.5 Studies of Adsorbed Hydrogen ............................................................... 23

1.6 Orientational Ordering in Adsorbed Hydrogen ............................................. 25

1.7 Liquid Hydrogen: Supercooling and Superfluidity ........................................... 27

1.8 Selected Properties of Molecular Hydrogen at Low Temperatures ................ 30



2 ADSORPTION ISOTHERMS ........................................................................... 32

2.1 Volumetric Adsorption Isotherms .............................................. ............ 32

2.2 Sample Preparation................................................. .................................. 34

2.3 Experimental Procedures........................................................................... 42

2.3.1 Computer Automated Valve System and Adsorption Algorithm .............. 44

2.3.2 Temperature Control......................................................................... 45

2.4 Isotherms of the Hydrogen Isotopes.......................................................... 49

2.4.1 Interpretation of the isotherms ........................................... ............ ... 49

2.4.2 HD Isotherm s ...................................................... ............................. 52

2.4.3 H2 and D2 Isotherms ......................................................................... 57

2.5 Compressibility, Critical Temperatures, and Larher's Method.......................... 57

2.6 Isosteric Heat of Adsorption .................................................................... 67

2.7 Sum m ary .............................................................. .................................... 71

3 NMR APPARATUS AND PROCEDURES ....................................... ............ 73

3.1 O overview .............. ...................... .............................................. ............ .. 73

3.2 Experimental Apparatus............................................................................ 77

3.2.1 CW NMR Schematic Diagram ............................................................ ... 78

3.2.2 Tuning of the Tank Circuit ......... ....................................................... 82

3.3 Experimental Procedures................................................................................. 85






4 NMR RESULTS AND DATA ANALYSIS....................................................... 89

4 .1 O overview ......................................... ........................................... ............. .. 89

4.2 D ata Processing ........................................................................................... 94

4.3 H2 CW NMR Line Shapes................................................................................. 96

4.4 D discussion ................................................................................................ 106

5 SUMMARY AND IMPLICATIONS................................................................ 113

REFEREN CE S ..................................................................................................... 116

APPEN D ICES ...................................................................................................... 121

A DETAILS FROM SAMPLE BN693 (June, 1993)........................................... 123

B DETAILS FROM SAMPLE BN1093A (October, 1993)................................. 140

C DETAILS FROM SAMPLE BNI 193 (November, 1993).................................. 153

D DETAILS FROM SAMPLE BN294B (February, 1994).................................... 159

BIOGRAPHICAL SKETCH .................................................................................. 174








LIST OF TABLES


Table page

1-1 Ground States of the Hydrogen Isotopes.............................................................2

1-2 Intermolecular Distance of Adsorbed Monolayers................................................ 6

1-3 Heats of Transformation at the Triple Points of H2, HD, and D2.................... ..... 30

1-4 Specific Volumes at the Triple Points of H2, HD, and D2........................................ 31

1-5 Triple and Critical Points of H2, HD, and D2........................................................... 31

1-6 Quantum Parameters of Selected Quantum Parameters........................................... 31



2-1 Critical Temperatures ofmonolayers of H2, HD, and D2 on BN.............................. 65

2-2 Critical Temperature of HD adsorbed onto three substrates.................................... 66

2-3 Critical Temperature of H2 adsorbed onto three substrates.............................. ...66

2-4 Critical Temperature ofD2 adsorbed onto three substrates.................................. 66

2-5 Triple Point Temperatures of H2, HD, and D2 on selected substrates................... 66

2-6 Binding Energies of Hydrogen isotopes on selected surfaces................................. 66








LIST OF FIGURES

Figure page

1-1 Crystal Structure of Graphite, BN, and MgO.................................................... 3

1-2 Scanning Electron Microscope Image of BN ....................................... ............. 4

1-3 Transmission Electron Microscope Image of BN.......................................... ............ 5

1-4 Intramolecular Axes for Dipole-Dipole Interaction.............................. .............. 8

1-5 Zeeman Splitting of Spin 1/2 Energy Levels...................................... ............ 10

1-6 Pake Doublet Line Shape................................................................................ 12

1-7 Orientational Phase Diagram of Bulk ortho-para H2 Mixtures............................... 17

1-8 NMR Line Shapes of Selected Hydrogen Systems ............................................. 19

1-9 Temperature Dependence of Mean Order Parameter........................................... 21

1-10 Two-Dimensional Model of Quadrupolar Orientational Ordering .......................... 24

1-11 Translational Phase Diagram of HD on Graphite............................ ........... .. 28



2-1 Procedure for Simultaneous Isotherm Measurements............................................ 33

2-2 Adsorption Isotherm of Argon on BN ................................................................. 36

2-3 Compressibility of Argon on BN ........................................................................ 37

2-4 Liquid-Solid Substep of Argon on BN............................................ ............ .... 39

2-5 Adsorption Desorption Cycle for Argon on BN................................ ............ 41

2-6 Schematic of Isotherm Apparatus ............................................................... 44

2-7 Electronic Temperature Controller ............................................................... 46

2-8 Examples of Temperature Control ............................................................... 50

2-9 Phase Diagram of HD on Graphite................................................................ 51

2-10 Adsorption Isotherm of HD on BN.............................................................. 53






2-11 D details ofH D on BN ............................................................................................ 54

2-12 Adsorption Isotherms of H2 on BN............................................................. 58

2-13 Low Precision Adsorption Isotherms of D2 on BN........................................... 59

2-14 High Precision Adsorption Isotherms of D2 on BN ........................................... 60

2-15 Larher Plot of Second Layer HD on BN....................................... ............ .. 62

2-16 Larher Plot of Third and Fourth Layer HD on BN............................................ 63

2-17 Larher Plot of H2 and D2 on BN.................................................................. 64

2-18 Isosteric Heat of Adsorption for HD on BN................................... ........... .. 68

2-19 Isosteric Heat of Adsorption for H2 and D2 on BN............................................ 69



3-1 Schematic of CW NMR Apparatus............................................................... 80

3-2 M odel of NM R Tank Circuit ................................... ..................................... 83

3-3 Tuning Characteristics of Tank Circuit......................................................... 85



4-1 N M R Sam ple Cell ............................................................................................ 91

4-2 Two CW NMR Line Shapes Displaying the Onset of Orientational Ordering.......... 97

4-3 Time Dependence ofCW NMR Line Shape for Eight Layers at 0.2 Kelvin............. 98

4-4 Time Dependence ofCW NMR Line Shape for Eight Layers at 0.6 Kelvin............. 99

4-5 Time Dependence of CW NMR Line Shape for Four Layers................................. 101

4-6 Detail of the Time Dependence ofCW NMR Line Shape for sample
B N 1093A ................................................................................................ ........ 102

4-7 Detail of the Time Dependence ofCW NMR Line Shape for sample BN1193....... 104

4-8 CW NMR Line Shape for one Time and Temperature Sweep ............................. 105

4-9 Concentration Dependence of CW NMR Line Shape for sample BN693............... 108

4-10 Temperature Dependence for Two-Component System ..................................... 109












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



ORIENTATIONAL AND TRANSLATIONAL
PROPERTIES OF HYDROGEN FILMS
ADSORBED ONTO BORON NITRIDE



By

Morgan D. Evans

December, 1994


Chairman: Neil Sullivan
Major Department: Physics


As physics continues to expand its knowledge base, physicists seek new frontiers

to investigate. Quantum-mechanical, two-dimensional systems have proven to be a

subject that is not only rich in new discoveries (e.g., Kousterlitz-Thoules transitions and

new phases of matter) but also filled with exciting predictions (e.g., new superfluids). The

physisorption of a gas onto the surface of a homogenous, spacious (on a molecular scale)

substrate with a low adsorption potential is one physical analogue to the much-analyzed

theoretical two-dimensional system. Hydrogen in reduced dimensions has been found to

have suppressed melting and freezing points, thus possibly permitting the onset of Bose

condensation and a new superfluid phase. The use of boron nitride as an adsorption







substrate allows for the study of physisorbed systems with a lower adsorption potential

than previous studies using similar substrates (i.e., graphite and magnesium oxide).

This dissertation has two parts. The first concerns the translational properties and

adsorption energies of hydrogen adsorbed onto boron nitride. These properties are

investigated through the use of volumetric adsorption isotherm techniques. The data

suggest that the adsorption of hydrogen occurs in a step-wise manner at temperatures

below 20 Kelvin. Changes in the translational phases (vapor-liquid-solid) occur between

10 to 20 Kelvin for the first four adsorbed monolayers. Isotopic effects are investigated

through the use of the three common forms of hydrogen: molecular hydrogen (H2),

deuterium hydride (HD), and deuterium (D2). The critical temperatures of the second,

third, and fourth layers are determined, presented, and compared with the known phase

diagrams of hydrogen isotopes adsorbed onto graphite and MgO. While the adsorption

potential of the hydrogen-boron nitride system is found to be less than that of previously

studied substrates, it does not translate into lower critical temperatures.

The second part of this thesis concerns the orientational ordering of H2 in two-

dimensions as investigated through the use of nuclear magnetic resonance (NMR)

techniques. While ordering in three-dimensional (bulk) systems involves a crystal phase

change and a strong ortho-hydrogen concentration dependence, ordering of hydrogen on

boron nitride is shown to be more subtle. Evidence of partial ordering below 1 Kelvin is

uncovered. This ordering is shown to depend not only on the ortho-hydrogen

concentration but also on the number of adsorbed layers.












CHAPTER 1


INTRODUCTION


Nature non nisi parendo vincitur.
Nature, to be commanded, must be obeyed.
Novum Organum 1620, Sir Francis Bacon 1561 1626



1.1 Quantum Mechanical Nature and Other Properties of Hydrogen


The low temperature properties of hydrogen are dominated by the quantum

mechanical nature of the molecule as evidenced by the large zero point motion, quantum

rotator properties, and the existence of two distinct molecular species. The interaction

between adjacent hydrogen molecules is a weak Van der Waals attraction and can be

modeled by the Lennard-Jones (6-12) potential

V(r)= 4e 12 Equation 1-1


where e and a are the well depth and the hard-core radius, respectively, and are known as

the Lennard-Jones (L-J) parameters. The quantum nature of a molecule can be quantified

by rescaling the Schroedinger equation in terms of the L-J parameters:

r*= and V *V(r) Equation 1-2
Sa e
to derive the de Boer parameter
h
A = Equation 1-3

which describes the importance of the quantum mechanical effects.







In terms of this fundamental parameter measuring the translational degrees of freedom,

hydrogen is second only to helium in its quantum mechanical nature.

The quantum-mechanical nature of the rotational degrees of freedom are even

more pronounced. The homonuclear, diatomic forms of molecular hydrogen, H2, D2, and

T2, must be described as quantum rotators with good quantum numbers, J, for the angular

momentum. In accord with the foundations of quantum mechanics, the Pauli exclusion

principle and indistinguishability, molecular hydrogen forms two distinct molecular

species, ortho-hydrogen and para-hydrogen as outlined in Table 1-1. The ortho-para

terminology refers to the spin state with the largest number of occupants at low

temperatures. At high temperatures, room temperature and above, gaseous H2 converts

into three parts ortho and one part para as governed by spin statistics. This nomenclature

can be difficult to follow for the uninitiated. Some authors in the field, P. Soures

[Sou 86], for example, have therefore suggested dropping the ortho-para nomenclature in

favor of a direct referral to the spin status of the species, eg., "spin 1 hydrogen," and "spin

2 deuterium," etc. Unfortunately, the new names have not been universally adapted.


Table 1-1 Ground States of the Homonuclear, Diatomic Molecular Hydrogen Isotopes.
Ortho H2 Para D2 Para H2 Ortho D2
(and T2) (and T2)
Angular J=1 J= J=0 J= 0
Momentum (odd) (odd) (even) (even)

Nuclear I 1 I=1 I= 0 I= 0,2
Spin (even) (odd) (odd) (even)


In contrast, atomic helium (4He) is spinless and the question of indistinguishability

does not arrise for heteronuclear HD. Selected properties, including A, a, and e of the

hydrogen isotopes are provided at the end of this chapter.







Magnesium Oxide


*nitrogen 0 boron 0 carbon
Hexagonal


Simple Cubic


Figure 1-1. Crystal structure of graphite, BN, and MgO.






1.2 Adsorption Substrates



Of the various substrates used for the study of gases adsorbed onto surfaces, three

are of special concern here: graphite, boron nitride (BN), and magnesium oxide (MgO).

As illustrated in Figure 1-1, the crystal structure of both graphite and boron nitride is

based on a planar structure. Furthermore, the lattice dimensions of BN are only about 5%

greater than those of graphite. These dimensions and others are presented in Table 1-2.

For BN, these planes are stacked into platelets of the order of 0.1-5.0 microns in diameter

and 0.01-0.1 microns thick. Figure 1-2 is a scanning electron microscope photograph' of

the boron nitride used in this study. In contrast to the hexagonal structure of graphite and

BN, MgO has a simple cubic crystal structure with a base distance of 4.23 Angstroms.

Thus, with the use ofjust these three substrates, it is possible to investigate the effects of


1 Image produced at the University of Florida.


Boron Nitride


Graphite





















































Figure 1-2. Photograph of BN from a scanning electron microscope image in the
reflective mode.





5



























4'





























Figure 1-3. Photograph of BN from a scanning electron microscope in the transmission
mode. Note the planes of BN which form the platelet.







binding energy, lattice spacing, corrugation, and substrate lattice symmetry on the

translational and orientational properties of adsorbed films.


Table 1-2. Intermolecular distances, in Angstroms, for adsorbed monolayers and bulk
hydrogen.
Substrate
Graphite BN MgO Bulk HCP
in-plane 4.26 A 4.35 A 4.23 A H2: 3.79 A
out-of-plane 6.71 A 6.66 A 4.23 A D2: 3.61 A





1.3 Molecular Orientation


1.3.1 Overview

Nuclear magnetic resonance (NMR) studies at high frequencies are invaluable as

an unambiguous probe of the molecular dynamics and molecular order parameters.

Specifically, the orientational ordering of a system of spin 1/2 molecules is uniquely

determined from the NMR line shape of that system. At high temperatures, the hydrogen

molecule rotates rapidly compared to the interaction time causing the intramolecular

dipole-dipole interaction to average to zero. At lower temperatures, the rotation of the

molecules slows down and the dipole-dipole interaction becomes measureably significant.

At some low temperature the rotation of the molecule is severely inhibited and the solid

undergoes a phase transition to the orientationally ordered phase. This orientational phase

transition (near 1 K) should not be confused with the translational phase transition (the

liquid-to-solid transition) which occurs at a temperature approximately one order higher in

magnitude. The dipole-dipole interaction is central to the determination of the orientation

of the molecular axis. This interaction lifts the rotational degeneracy of the nuclear







Zeeman ground state allowing NMR to probe the orientation of the molecule. The line
shape can be calculated exactly starting from the full Hamiltonian.



1.3.2 Dipole-Dipole Interaction



The intramolecular interaction between two hydrogen atoms within the hydrogen
molecule is due solely to the nuclear dipole-dipole forces. The Hamiltonian for this system
of two spins in a large magnetic field can be written as the sum of two terms, Ho and HDD.


H = H + HDD
Sy2 2 I 1 2 1 Equation 1-4
=-rhBo(rL+ I)+ I,I -3(j -n)(f -n)]

where I is the proton nuclear spin operator and the angles are defined in Figure 1-4. The

first term, Ho, represents the large Zeeman splitting of the nuclear energy levels caused by
the static magnetic field applied in the Z direction. The second term, HDD, represents the
dipole-dipole interaction and results in a perturbation of the energy levels due to the first
term. It is elucidative to express the dipolar Hamiltonian in terms of spherical harmonics,
namely

HDD = 2 { I 2-I 3I cosO+sinEO(L .cosOQ+ -sin()] Equation 1-5

[i2 cose+sinO(J/2 cosO+J 2. sin ()]}

This expression can be further transformed by combining the x and y components
into raising ( + ) and lowering ( J ) operators,

r 2 Equation 1-
Ho = {1 'I 2" -3[JI,' cose+ sineO(' -e-+ + )]* Equatin
[I/ coso + sinO e(/2 -e-' + 1_2 -e ")\}




















X
Figure 1-4. Axes used to define the dipole-dipole interaction.



In general, the dipolar Hamiltonian can be written as the sum of six terms,


HDD H, +H, + H + H +HE + H
where,


HA =I 12I (1- 3cos2 O)

H,= (1-3cos 0)(JI'I f J)= (1- 3cos2 0)(I' )

H = sinOcosOe-' (Jl JJ) = H, *

HE = S in2' Oe II = H, *
4 +


Equation 1-7







Equation 1-8


In the limit that the Zeeman field splitting is much larger than the dipole-dipole interaction

(i.e., H. >> HDD), then
HA H, ,HB >> Hc,Ho Equation 1-9
and


H,HD >>HE ,HF.


Equation 1-10


Furthermore, HA represents the classical interaction between two static local

dipoles. The second term, HB, represents the primary resonance at the Larmor frequency







and allows a simultaneous reversal of two neighboring spins in opposite directions. The
last four small terms represent off-diagonal matrix elements and admix higher states.
Thus, a good approximation is to drop HC, HD, HE, and HF while retaining the secular
terms, HA and HB.
Two eigenstates exist for this simplified Hamiltonian. Since the singlet state
represents the spin zero para-hydrogen molecules, we can define this state to have zero
Zeeman energy. Physically, these molecules act as inert dilutents and are ignored by the
NMR but do provide other physical effects. The triplet term, however, provides the NMR
signal and can be most easily understood by changing the spin variables into spin up and
spin down operators. By defining the total operators,


Equation 1-11


it can be shown that,


H=-rhBIo lz 3 1zlz- z 4 + -

The eigenstates for this Hamiltonian include the singlet state,

I+ -) -+

and the triplet state,
I+ -> + 1- +>
I1)=I++); 10)= ; ,-1)=1--).


Equation 1-12



Equation 1-13


Equation 1-14


The energies of each level can be determined as


-1 -2 1 1
I =I +I = l+ Iz+I
r = (1-3cos o), I I I'I + i)







E_,=(-11H-1) =

Eo= (01 0) =

E,1=(+11}+1) =


rhBo + r

_r2t F
2r3

- y.o + 44r
4r3


Equation 1-14


0


+1
+1 -------^
Iz base 12

Figure 1-5. Effect of dipolar interaction on Zeeman splitting of spin 1/2 energy levels.


Therefore, two resonant frequencies exist, o' and o", where,



ho'= Eo-E= r hBo- 4r3
ha'=E-E1=hB0-3~2 Fi


Equation 1-16


The form of continuous wave (CW) NMR chosen for use utilizes a fixed frequency and a
variable magnetic field, B, to measure the resonant absorption. In this case the resonances
occur at


Equation 1-17


B= Bo a 1







where a = and B = 0 Thus, for a pair of dipoles, or a single crystal of such pairs,
4r' y

the NMR line shape becomes two lines centered at Bo with a peak-to-peak separation of

2caF.

For a powdered sample, many pairs of dipoles are oriented randomly, and an

average over all possible angles, a "powder average", must be performed. Such an

average yields the familiar Pake Doublet Powder Pattern NMR line shape.



1.3.3 The Pake Doublet Line Shape

The Pake doublet is a very unique line shape that can be calculated exactly. The

determination of the line shape depends upon finding the intensity of each frequency given

a distribution of spins (frequencies). For a polycrystalline (powdered) sample, each crystal

in the powder yields two lines due to the dipole-dipole interaction. The frequency

separation of these lines from the resonance center is given by
v = = + a'(3 cos' -1) Equation 1-18
2x
where a' = ya /2 7.

The intensity of the signal is a function of the frequency of applied radiation and is

related to the angle between the intramolecular axis and the externally applied field (e.g.,

the Zeeman axis)
I(v) dv = P(O)df Equation 1-19


Since the interaction is axially symmetric, the differential solid angle depends only upon

the azimuthial angle, i.e.,
dQ = 2x d(cose) Equation 1-20

For a powdered sample, all angles are equally probable, i.e.,

P(0)= Equation 1-21
47r







Thus, the intensity of the signal can be found as a function of the frequency, i.e.,

I(v) = 1 Equation 1-22
2 dvy
2 d (cose)


I(v)













I v
-2a -a 0 a 2a

Figure 1-6. Pake doublet NMR line shape for a powder sample.


The desired differential can be calculated from Equation 1-18, i.e.,
dv
= 6a'cosQ .
d(cosO)


Thus, the intensity diverges at two specific frequencies, v = a', i.e.,

I
(V 2vz a


Equation 1-23





Equation 1-24


where


2a > v> -a Equation 1-25

as 0 < 0 < -. The "powder average" Pake Doublet line shape is presented in
2
Figure 1-6.






1.4 Orientational Ordering in Hydrogen and Deuterium.



1.4.1 Overview.


Orientational ordering may occur for any system of atoms or molecules which have

anisotropic intermolecular interactions (e.g., electro- or magneto-static potentials of

dipolar, quadrupolar, octopolar, or higher order). In such a system, when the energy of

the random motion associated with the temperature of the lattice decreases below the

interaction energy due to the static potentials, the molecules may then align themselves

with respect to their neighbors. For the case of quadrupoles, as in this study, the lowest

energy state is a "tee" formation between pairs. Complete orientational ordering occurs if

this alignment is both long-term and long-ranged. Some systems may show only partial

ordering for only a short period of time and have a wide distribution of local order

parameters. Such systems are said to reside in an orientational glass state.

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

some time. For example, the static susceptibility of solid hydrogen at 2 K was measured

as early as 1937 [Laz 37]. The low temperature orientational properties of hydrogen have

been identified and investigated almost continuously since then. Specific heat

measurements identified a singularity in high ortho-hydrogen samples indicating the

existence of either a transitional or orientational transition [Hil 54, Ahl 64]. X-ray

diffraction revealed a change in the crystal lattice spacing at a similar low temperature

[Mil 66] as did neutron diffraction [Muc 66] and infrared (IR) absorption spectra studies.

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

produced a phase diagram [Sul 72]. Most bulk properties of hydrogen may be found in a

fairly comprehensive review article by Silvera [Sil 80] as well as in a book by P. Souers

[Sou 86].







The orientational ordering of hydrogen and deuterium in part determines the

crystal structure of the bulk solid. If no orientational ordering is present, as is the case for

a system of pure J=0 molecules (para-hydrogen and ortho-deuterium), then the

constituents form a hexagonal close packed (h.c.p.) crystal lattice structure. However,

this triangular lattice structure is incommensurate with the cubic symmetry of quadrupolar

interactions in J=1 molecules (ortho-hydrogen and para-deuterium). Thus orientational

ordering of samples with J=l molecules drives the lattice into a cubic structure. For pure

J=l systems this occurs at approximately 3 K for hydrogen and 4 K for deuterium and the

resulting lattice structure is three interpenetrating cubic lattices each with orientational

order along the three unique body diagonals [Sou 86, Sil 80]. For the more general case

of mixed J systems the transition temperature drops with decreasing J=l concentration.

Below 55% ortho-hydrogen concentration, the ordering is in the form of a quadrupolar

glass [Sul 72].



1.4.2 Review of Three-Dimensional Results


1.4.2.a Order Parameters

Bulk ortho-hydrogen, a quantum-mechanical molecule with J=1, has a relatively

large electric quadrupole moment and exhibits orientational ordering below 4 K. The low

temperature behavior is dominated by the anisotropic quadrupolar interaction, given by

VQQ = ) r F(a,,n,) Equation 1-26

where A is the lattice parameter, R is the intermolecular distance, F is an angular function

describing the anisotropy of the interaction, and FQ is the quadrupolar coupling constant

F = 6e 2- K. Equation 1-27
25a2







Here eQ is the quadrupole moment of the molecule. The quadrupole-quadrupole
interaction can be expressed either in terms of the spherical harmonics or in terms of
spherical tensor operators. The components of the second rank quadrupolar tensor,

Q =( J25 -(j J J J Equation 1-28

define the parameters which describe the degrees of freedom for the system. Of the five
unique components, three are Euler angles that specify the orientations of the principal
axes of the tensor Qab, and two describe the degree of ordering. The latter two are of
special interest. They are the eccentricity,

S= (J: J ) Equation 1-29
which is zero for an axially symmetric body (a good approximation for hydrogen) and the

alignment,

=Q, = -(3J 2). Equation 1-30

The alignment varies from one, for Jz = 0, to minus one-half, for Jz = +1 or -1.
Since the nuclear spin is uniquely related to the angular momentum, the orientational
degrees of freedom can be studied using NMR techniques that probe the nuclear spin of
the ortho-hydrogen molecule.
The effect on any one hydrogen spin in a pure molecular hydrogen system subject
to a large magnetic field can be readily understood in terms of the order parameters. The
energy levels of each atomic spin are affected by three forces: the external Zeeman
magnetic field, the intramolecular forces due to the other proton within the hydrogen
molecule, and the sum of all of the intermolecular forces. The intermolecular forces on
any one spin are much smaller than the intramolecular forces since the dipole-dipole
interaction decays as the cube of the inverse distance between particles and the spacing
between molecules is about four times the intramolecular distance, i.e.,






(A3 (0.75A
= 7j- ) = 0.8%. Equation 1-31
(AR 3.75A

Of the two remaining interactions, the Zeeman field can be made arbitrarily large
relative to the intramolecular interaction. At high resonance frequencies (tens and
hundreds of Megahertz) the Zeeman interaction dominates. Thus, as described above, the
resulting Hamiltonian can be broken into two parts:
H = H. + HDD Equation 1-32

where the dipolar term acts as a small correction to the Zeeman term and splits the single
resonance, vo = y hBo, into two resonances as mentioned in Section 1.3.2,
v = vo vd Equation 1-33
where [Sul 78],

vd = 3 d[r (3 cos2 0 -1) + 37 sin2 0 cos2 0 Equation 1-34
2 2 J
and
1 h2 v2
d -- = 54.2kHz. Equation 1-35
5 2;r(r')

Note that not only the alignment, s but also the eccentricity, rl, has entered into

the line shape and that the exact value of the dipolar constant, d, depends on the

translational and rotational properties of the hydrogen system under examination.
Ortho-hydrogen (J=1) decays into its para-hydrogen (J=0) ground state at low

temperatures [Sil 80]. In contradistinction to ortho-hydrogen, para-hydrogen has no spin
and no higher moments, including quadrupolar. Thus, this J=0 hydrogen does not yield an
NMR signal and behaves as an inert diluent to the system of interacting J=l hydrogens. A
pure ortho-hydrogen system eventually decays into a disordered system and the effects of
disorder on a fixed sample can be studied by simply aging the sample. A partial list of

some of the other properties of hydrogen is provided at the end of this chapter.






1.4.2.b. Phase Diagram
Continuous wave nuclear magnetic resonance (CW NMR) experiments on three-
dimensional (3D or bulk) systems have been performed for some
time[Sil 80, Sul 78, Har 85]. It was found that the phase diagram of bulk, solid hydrogen
includes three distinct orientational phases below 4 K, as illustrated in Figure 1-7. The
appearance of the nuclear magnetic frequency response, the NMR "line shape," depends







g3.0 /'
Hexagonal /


<2.0- Oc
Cubic
S/ -

W/
1.0 -
Disorder Long Range
SDisoOrder

Gloss
0 0.2 0.4 0.6 0.8 1.0
ORTHO CONCENTRATION


Figure 1-7. Orientational ordering phase diagram for bulk hydrogen.







upon the crystalline state of the sample. Both polycrystalline and single crystal samples

have been studied and much literature exists describing the various subtle properties.

However, since the two-dimensional samples more closely parallel the three-dimensional

polycrystalline samples, polycrystalline line shapes will primarily be mentioned.


1.4.2.c NMR Line Shapes

Solid hydrogen above 3 K shows no signs of ordering. The NMR line shape is

fairly narrow, approximately 50 kHz, and relatively independent of ortho-hydrogen

concentration as illustrated in Figure 1-8 (a). Part (b) displays the distribution of the

alignment where unity represents complete ordering. For such a narrow, disordered line

shape, the distribution resembles a delta function at zero order parameter. The

eccentricity is nearly zero for all systems of hydrogen molecules discussed here.

A second phase exhibits long range ordering and occurs for high ortho-hydrogen

concentrations (X > 55%) and low temperatures (T < 3 K). The resulting line shape is the

familiar Pake Doublet, shown for a polycrystalline sample in Figure 1-8(c), with a

characteristic width of 3D/2, or approximately 150 kHz. The line shape for a single

crystal sample (composed of four sublattices) displays two lines per sublattice whose

frequency separation depends upon the angle between the local crystal axis and the applied

magnetic field. The angular dependent splitting has been studied in detail [Har 85] and

magic angle spinning used to contract the four pairs of lines into one set. The distribution

of order parameters resembles a delta function near unity, as presented in Figure 1-8(d).

The third and most interesting phase, the glass phase, has short-ranged ordering

for intermediate ortho-hydrogen concentrations (20 < X < 55%) and low temperatures

(T < 1K). The CW NMR line shape depends strongly upon both the temperature and the

ortho-hydrogen concentration. Accordingly, there exists a broad distribution of the local







NMR LINE SHAPE


-3d -3d/2 0 3d/2


ORDER PARAMETER DISTRIBUTION
Disordered Phase (b)
P(c)


Long-ranged Ordered Phase
P(ao)


-3d -3d/2 0 3d/2 3d


Quadrupolar (
P(<




o


-3d/2 0 3d/2 3d


Proposed


P(a)


-3d -3d/2 0 3d/2 3d


(d)






-o
1

(0





/o


0 1


0 1


Figure 1-8. NMR line shapes and associated distributions of the order parameter, a, for
selected hydrogen systems.








order parameters and the shape of this distribution is highly temperature and ortho-

concentration dependent. An example of such a distribution and its associated line shape

can be seen in Figure 1-8 (f) and (e), respectively. Parts (g) and (h) of Figure 1-8 are

included here for convenience but will be discussed in Chapter 4.

Figure 1-9 (top) illustrates the variation of the line shape corresponding to the

transition from the ordered to disordered phase. Part (a) displays the ordered Pake

Doublet while part (b) shows the orientationally disordered solid NMR line shape. The

mean order parameter, shown in part (c), exhibits a first order phase transition as it rapidly

changes from near unity to zero.

Figure 1-9 (bottom) illustrates the spectral change for the transition from the

quadrupolar glass to the unordered phase for solid hydrogen. Part (a) displays the glass

CW NMR line shape while part (b) shows the unordered line shape. In this case, the

phase change is weakly second order as the mean order parameter undergoes a more

subtle transition, as displayed in part (c).

This new collective orientational phase, the quadrupolar glass phase, was an

important, exciting, and unexpected discovery. The intermolecular interactions of the

hydrogen system are relatively simple and well understood. Further, NMR provides for a

very fundamental probe of the glass state. Thus, this physically simple, yet theoretically

rich, system has been thoroughly examined and modeled. Since so much is known about

the three-dimensional system, the effects of the reduction in the dimensionality of the

system can be readily studied. Hydrogen adsorbed onto a surface provides a model two-

dimensional system for this purpose. Knowledge gained from this relatively simple two-

dimensional system may aid in the understanding of more complicated systems in reduced

dimensions.

A second phase exhibits long range ordering and occurs at high ortho-hydrogen

concentrations (X > 55%) and low temperatures (T < 3 K). The resulting line shape is the








TOP


T<1K


-3d/2 0 3d/2 3d


T>2K




-3d -3d/2 0 3d/2 3


T (K)


0.1 1.0 4


T 2K


---
-3d -3d/2


0 T>
0 3d/2


T (K)


0.1 1


BOTTOM


Figure 1-9. Temperature dependence of the CW NMR line shape and mean order
parameter for the long-ranged ordered phase (top) and the glass phase (bottom) for bulk
molecular hydrogen.


-3d/2 0


3K



- I
3d








familiar Pake Doublet presented for a polycrystalline sample in Figure 1-8(c), with a

characteristic width of3d/2 (approximately 150 kHz).


1.4.3 Two-Dimensional Orientational Ordering Model


Figure 1-10 illustrates one model of orientational ordering for a simple two-

dimensional system of quadrupoles. Part (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. Part (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 each molecule in the

lattice cannot attain its lowest energy state with respect to all of its neighbors. The lowest

energy for the entire system may be a herringbone or a pinwheel pattern. Parts (c) and (d)

show the effect of the addition of a vacancy or a neutral molecule(e.g., a noble gas or

para-hydrogen). 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

equivalent, the system is said to be disordered. The effect of the inert diluents to a system

on the square lattice of part (c) is negligible. However, the effect of the substitution 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.








Two definitions require extra clarity: zero point motion (ZPM) and disorder.

Zero point motion is the intrinsic quantum energy associated with all matter. While it is

quite small for massive objects, it is a very important property for hydrogen. ZPM exists

for each degree of freedom including rotational, translational, and vibrational motion.

Likewise, the term disorder has multiple uses in the literature. Disorder describes a system

which has substitutional impurity as well as a lack of a type of ordering (e.g., rotational or

orientational).



1.5 Studies of Adsorbed Hydrogen



A distinction may be made in the manner in which a gas is adsorbed onto a surface.

In general, two modes exist for the adsorption of hydrogen onto a surface: physisorbtion

where the adsorption potential is rather weak and the adsorbed hydrogen retains its

molecular form, and chemisorption where the adsorption potential is rather strong and the

adsorbed hydrogen dissociates into its atomic form. As will be discussed later in Chapter

2, hydrogen is physisorbed onto boron nitride.

While no studies have been previously reported concerning molecular hydrogen adsorbed

onto boron nitride, many studies have been reported for hydrogen physisorbed onto other

substrates. A history of the studies of the hydrogen isotopes on graphite and

magnesium oxide has been compiled recently elsewhere [Liu 93]. The following

overview is intended to give more of a flavor of the extent and types of the studies that are

particularly related to this work.

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

have been conducted on adsorbed gases, including low energy electron diffraction (LEED)

and Auger electron spectroscopy (LEED-Auger) [Suz 77], x-ray scattering [Ros 83], and









f7\
1:1
ki
I
~iiL


(a)


-0


(c)


> 0 -
/\/\
/ j.
0---^--(b

(b)


0O



(d)


Figure 1-10. Two-dimensional model of quadrupolar orientational ordering.


I
(ZED


-- ~T-~3







ellipsometry [Vol 89]. Adsorption isotherms with graphite as the substrate have been

carried out for most noble gases [Tho 70, Goo 93], the hydrogen isotopes [Vil 92], and

some hydrocarbons [Alk 92, Ham 83]. NMR experiments have been few but at least

three adsorbed gases have been studied at some level: 3He [Rol 72, Cow 84], H2 and D2

[Kub 78], and '5N2 [Sul 83]. Of particular note is the fact that the translational modes of

hydrogen adsorbed onto graphite have been well studied. Methods of study include heat

capacity [Mot 85, Fre 85], inelastic neutron diffraction [Fre 90], adsorption isotherms

[Liu 93], and computer models [Got 90]. Figure 1-11 presents the phase diagram of HD

on graphite as recently reported [Liu 93].

Magnesium oxide is another substrate of interest. Crystals of MgO form a cubic

structure which has the same underlying symmetry as the quadrupole-quadrupole

interaction. NMR studies of HD and D2 adsorbed onto MgO have been performed

[Jeo 93] and orientational ordering identified at elevated temperatures [Jeo 93]. It is

believed that the strong corrugation of the surface overwhelms the adatom-adatom

interaction and is 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. Adsorption isotherms have also

been carried out to study the two-dimensional translational modes [Ma 89] identifying the

liquid-solid transition.


1.6 Orientational Ordering of Adsorbed Quadrupoles

A system of adsorbed nitrogen is the classical analogue of our adsorbed hydrogen

system. Both gases are homonuclear and diatomic and, therefore, molecular quadrupoles.

Both gases have been shown to form a registered solid phase on graphite, in particular,
one molecule occupying every third carbon ring, i.e., a ( j x d3 ) R 300 triangular net.

While the extremely light mass of the hydrogens makes quantum mechanical behavior very







important, the large mass of molecular nitrogen yields mainly classical behavior. Two

common forms of nitrogen exist: 14N2 which has no nuclear dipole moment but a rather

large nuclear quadrupole moment and "N2 for which the nuclear quadrupole moment is

zero but the nuclear dipole moment is non-zero. Like hydrogen, the nuclei of 'N2 are

identical fermions and thus there exist two molecular species: ortho-nitrogen (total nuclear

spin I = and odd orbital symmetry) and para- nitrogen ( I 0 and even orbital

symmetry). Above 4 K the two species are randomly mixed together in a constant 3:1

ortho- to para- nitrogen ratio. This is in contrast to hydrogen for which the ortho to para

ratio strongly depends upon the temperature with about equal mixtures at 77 K.

Nitrogen adsorbed onto graphite has been studied for some time both

experimentally and theoretically. Low energy electron diffraction (LEED) [Die 82], heat

of adsorption [Pip 83], adsorption isotherms [Liu 93], heat capacity [Cha 84, Chu 77], and

NMR [Sul 83] are some of the various experimental methods used to study this rich

system. The results from these studies confirm that the orientational ordering transition

for the two-dimensional system is reduced to 28 K from the bulk temperature of 35.6 K

[Chu 77]. The NMR study directly probed the local order parameter and determined that

the order-disorder transition is of first order. A number of theoretical methods have been

used to study the orientational properties of N2 and H2 adsorbed onto graphite including

classical Monte Carlo simulations, molecular dynamics simulations, renormalization group

analysis, mean-field-like approximations, universality arguments, ground-state energy

investigations, quasiharmonic lattice dynamics, time-dependent Hartree calculations, and

desity-functional theory of freezing [Mar 93].

An analogous reduction in the temperature for the orientational ordering of the

hydrogen isotopes (2.8 and 3.8 K for ortho-H2 and para-D2 in bulk form, respectively) is

expected for a two-dimensional system of hydrogen. One study [Har 79] modeled the

adsorption of H2 on graphite in the mean field approximation using a two-dimensional

system of interacting quadrupoles in the presence of the substrate field. Herringbone and







other orientational structures were predicted depending upon the strength of the substrate

field relative to the quadrupole-quadrupole interaction. Another study [Kub 85] used

NMR to measure the orientational ordering of both hydrogen and deuterium adsorbed

onto graphite. For 90% ortho-H2, the authors observed a splitting of the NMR absorption

line shape near 0.6 K and the addition of line structure. However, for 90% para-D2 (the

I = 1 and odd orbital symmetry species for deuterium), the authors found no evidence of

orientational ordering down to 0.3 K. Given that the two isotopes orientationally order in

the bulk at similar temperatures, the very different behavior was quite unexpected.



1.7 Liquid Hydrogen: Supercooling and Superfluidity

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 [Lan 69] and is

given by:
h2 ( 2Y3
Tl = 3.31 Equation 1-36



where n is the concentration, M the molecular mass, and g is the degeneracy for a single

particle state (1 for para- and 9 for ortho-hydrogen). In bulk form, the superfliud

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 large zero point motion, the

intermolecular interactions are relatively weak, as can be seen from Table 1-6. Thus,

molecular para-hydrogen is a potential candidate for superfluidity if the right conditions








can be achieved. Calculations have been made by Ginzburg and Sobyanin for para-

hydrogen, and a superfluid transition temperature of 6.5 K reported [Gin 72]. It is

reasonable to expect this calculation to be an upper limit on the actual temperature since


I

Tt=8.44K


e


9
e -

e
e
B


L-^


SS-L-







cu-(
:i:
10: *


i


e-0


:043


*-9


Tc= 11.45K

I .


T(K)


Translational phase diagram of HD on graphite, from [Liu93].


14





12


s?
B




rZ


S











S-V


e





monolayer
. .. .


- -


I I I


"" "'


Figure 1-11.







the same formula overestimates the helium transition temperature. Maris et al. have

argued that the onset temperature for para-hydrogen should be lower than 4.5 K [Mar 83].

However, the melting temperature for hydrogen is approximately 14 K and this

precludes the possibility of a superfluid transition in normal bulk hydrogen. Thus 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 [Tel 83].

In two dimensions, a Kosterlitz-Thoules transition (a second order transition; no

latent heat of transformation) for helium films has been found and an onset temperature

given by:
Ir h2
T1 =- n Equation 1-37
2 Mkb

where n' is the superfliud 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 [Vil 92]. The results of current

investigations of helium and hydrogen on surfaces will be useful to determine the

conditions for which to study hydrogen in reduced dimensions.

NMR is an ideal instrument for the study of translational phase transitions. The

width of the resonance is directly related to the motion of the nuclei under study and can

be measured directly by CW NMR. Pulsed NMR can also be used to measure rates of

motion. For example, NMR has been used to study the supercooling of hydrogen in the

pores of zeolite [Ral 91]. Rall measured both the transverse relaxation time (T2) and the

spin echo amplitude as functions of temperature. Both measures were found to decrease

suddenly at about 10 K. The temperature depended upon the para-hydrogen

concentration of the sample with higher concentrations having lower transition







temperatures. From an extrapolation of the data, Rail estimated that 100% para-hydrogen

could be supercooled to approximately 6 K.

Likewise, the supercooling of hydrogen on the surface of boron nitride can be

studied by NMR. However, both the low number of atoms under study in two dimensions

and the background atomic hydrogen signal precluded using the apparatus as set up for

this thesis for temperatures above approximately 7 K. The richness of this field and the

importance of the possible results suggests that the apparatus should be improved and the

high temperature studies performed. Such improvements are underway.



1.8 Selected Properties of the Hydrogen Isotopes at Low Temperatures



This section is provided to familiarize the interested reader with some of the

relevant properties of molecular hydrogen. The properties of hydrogen at low

temperatures has been studied extensively since the use of bulk liquid hydrogen in bubble

chambers of high energy physics experiments began some years ago. The three-dimesional

properties of the hydrogen family of isotopes listed here may be compared with the two-

dimensional properties discussed elsewhere in the text.




Table 1-3. Heats of transformation of selected hydrogen isotopes at the triple points
[Sou 86].
Isotope
Transformation H2 HD D2
Sublimation 1.028 kJ/mole 1.270 1.470
Vaporization 0.911 1.110 1.270
Fusion 0.117 0.159 0.199









Table 1-4. Specific volumes of selected hydrogen isotopes at the triple points [Sou 86].


H2 HD D2
Solid 23.14 *106 m3/mole 21.80*106 20.40*106
Liquid 26.11*10-6 24.61*106 23.16*106
Vapor 0.0158 0.0194 0.00885


Table 1-5. Triple and critical points of selected hydrogen isotopes [Sou 86].

Triple Temp Triple Press Critical Temp Critical Press
e-H2 13.81 (K) 7030(Pa) 32.98 (K) 1.29 (MPa)
52.74 9677 (Torr)
(Torr)
n-H2 13.96 7200 33.19 1.32
54.01 9903
HD 16.604 12,370 35.91 1.484
92.80 11,133
e-D2 18.69 17,130 38.26 1.650
128.51 12,378
n-Dz 18.73 17,150 38.25 1.665
128.66 12,491


Table 1-6. Quantum parameters (A, e, 0,) and bulk triple and
d etcelesf o ga res [Liu 93]2


critical point temperatures


Gas A e/kb(K) o(A) Tt(3D) (K) T,(3D) (K)
3He 0.555 10.22 2.556- 3.3102
4He 0.481 10.22 2.556 5.1899
H2 0.334 37.0 2.928 13.804 (e) 33.976 (e)
13.956 (n) 33.19(n)
HD 0.273 37.0 2.928 16.60 35.91
D2 0.236 37.0 2.928 18.69 (e) 38.262 (e)
18.73 (n) 38.34 (n)
Ne 0.108 36.7 2.788 24.553 44.40
Ar 0.0386 119.80 3.405 83.806 150.70
Kr 0.0221 164.0 3.624 115.763 209.5
Xe 0.0143 230.4 3.921 161391 289.72


2 Table data is from [Liu 93].


-- --~--











CHAPTER 2


ADSORPTION ISOTHERMS

Nam et ipsa scientia potestas est.
For knowledge itself is power.
Religious Meditations, Of Heresies, Sir Francis Bacon 1561 1626


2.1 Volumetric Adsorption Isotherms



A number of experimental methods have been used in recent years to study the

adsorption of gases onto surfaces, including quartz microbalance [Kri 84, Mis 92,

Tab 92], dielectric measurements [Lah 82], NMR [Cra 94], third sound [Zim 92, Spr 94],

neutron diffraction [Wei 91b], LEED [Wliu 93], and ellipsometry [You 90]. Computer

models have also been built and simulations run [Phi 93]. However, the most

straightforward and common method of observation is the volumetric adsorption

isotherm. The most simple version of this method utilizes only one pressure gauge and a

number of valves at room temperature. An amount of gas is loaded into a known

"dosing" or "loading" volume, also at room temperature, and the pressure of that volume

is measured. The valve between the loading volume and the sample volume is then

opened and the gas is allowed to expand into the sample chamber. The final pressure of

the gas above the substrate is then measured. From these two measurements and the ideal

gas law, the number of atoms that are adsorbed onto the surface can be calculated. The

number of atoms adsorbed onto the substrate at a fixed temperature is then plotted against

the measured final pressure in a graph routinely called an adsorption isotherm curve.








One advantage of volumetric isotherms is their simplicity. The method requires

only the repeated measurement of the gas pressure. For a fixed temperature, the chemical

potential is a monotonic function of the equilibrium vapor pressure of the gas above the

surface. Thus, additional properties of the two-dimensional system can be calculated from

one or more of the adsorption isotherm curves. For example, the two-dimensional

compressibility is directly related to the slope of the adsorption isotherm and the isosteric

heat of adsorption can be calculated from two isotherms. It is usually assumed that the

chemical potential of the film is the two-dimensional chemical potential (interactions

within the film) plus the substrate binding energy [Das 75]. The quantity adsorbed onto

the surface is a measure of the average film density.

More than one isotherm can be measured at once thereby maximizing the amount

of information gained per experimental run. As shown in Figure 2-1, after measuring the

first final pressure (Pa) and calculating the associated number of molecules adsorbed (Na),

the temperature (Ta) can be changed to a new (lower) value (Tb) and the final pressure

measured (Pb). The pressure at the first temperature can then be remeasured (a vs. a') to

check for hysteresis before adding more gas to the system (a' -> c). Note that point (a)

measures adsorption while point (a') measures desorption.


Tb Q
N .- Ta
U (d) ....--------- 0
M ." .-- .
B Nb (b) ..
E "
R a).
Na Q (aQ), Tb < Ta

I I

Pb Pa
PRESSURE

Figure 2-1. Procedure for the simultaneous measurement of two isotherms.










2.2 Sample Preparation


All samples of boron nitride powder used in this thesis come from the single

purchase of 50 grams of boron nitride (99.5% metals basis) from JM Electronics' with lot

number B13B03. The boron nitride sample was pressed through a 50 micron mesh by the

manufacturer and was received in the form of a fine white powder. Boron nitride is also

sold in the form of a hot pressed rod, but this form was not obtained for these

experiments.

As discussed above, since the hydrogen sample adsorbed onto the surface is

relatively small, it was essential that additional forms of hydrogen be minimized on the

boron nitride surface. Just as important, the molecular hydrogen itself is sensitive to any

contaminants with large magnetic moments. Adsorbates with large magnetic molecular

moments can artificially increase the rate of ortho-para hydrogen conversion. Thus, care

was taken to keep the sample free from additional water, oils, and even airborne

contaminants. All equipment was cleaned with rubbing alcohol and then heated in a

vacuum to remove the alcohol. The boron nitride was then loaded into an oven and

heated to 3000 C under diffusion pump vacuum (approximately 10-6 Torr) overnight. This

procedure is similar to that prescribed by Regnier, Thomy, and Duval [Reg 79] in their

comparative adsorption isotherm study of Xenon and Krypton on boron nitride and

graphite.

Samples with various bake-out temperatures and times were investigated using

methane isotherms to gauge the sample heterogeneity and cleanliness. A dirty sample was

expected to have either additional small steps due to the adsorption of certain surface

contaminates (e.g., water), more rounded steps due to inhomogeneity, very few distinct

steps, or a shorter first step due to occupation of adsorption sites by contaminants.


' Johnson Matthey Electronics, 30 Bond Street, Ward Hill, MA 01835.








However, the differences in the isotherms between differently prepared samples was

negligible. In fact, after bake-out, the samples were surprisingly stable. Even short

exposure (hours) to air did not seem to alter the isotherms. Such stability is in sharp

contrast to some other substrates (e.g., magnesium oxide) which readily adsorb water

from the air and must be both manufactured and retained in vacuum in order to insure

sample cleanliness.

An attempt was made to increase the NMR signal by maximizing the surface area

per unit volume. Some boron nitride samples were pressed inside a cylindrical former

during bake-out. The result was a white rod resembling a stick of chalk with a density

approximately 5 times that of the powder form. Unfortunately, upon attempting to

measure an adsorption isotherm at 77 K on the pressed sample, it was discovered that the

equilibrium time increased from tens of minutes to tens of hours. The sample was then

broken into millimeter size pieces and the isotherm repeated. However, the fragmentary

sample still had an equilibrium time of a few hours at 77 K which was deemed to be too

long to be useful. All further experiments were performed on non-pressed samples.

A recent study at Southern Illinois University by Migone et al. [Mig 94a] has

investigated different sample preparations of boron nitride. Samples were prepared by

baking under vacuum at temperatures up to 900 C for several hours. Some samples were

also washed in methanol. Adsorption isotherms were then performed utilizing Argon as

the adsorbate. These isotherms were then analyzed for any additional features and the

clarity of the known steps and substeps. Additionally, the curvature of the beginning of

the first step was monitored. Added curvature along this section of the isotherm, shown

in the Figure 2-2 (top), would indicate the presence of impurities on the surface providing

adsorption sites with an adsorption potential stronger than that of rest of the BN surface

[Mig 94b].

The researchers found that the 77.3 K argon isotherms measured on the dirty BN

samples differed from those measured on the clean samples in two ways. First, an










E-4






2-



V3





o







CQ







C:;
CD
3a

0a


I I I I I I I l I .S I


IA
P'


4vL
9^




Liquid Ae
Phase Jo

A
A
^.X


25 35
PRESSURE


P
F


A


A.

;olid
'hase












,r / BN
77 K


Figure 2-2. Adsorption isotherm of argon on boron nitride taken in December 1993. The
first and part of the second layers (top). Detail of liquid-solid transition, approximately
1.5% ofa monolayer, in the first layer (bottom). The lines serve only as guides to the eye.


1 10
PRESSURE (TORR)


If,








I-.,


5


45
(TORR)


llJlllf


rrrlII


I





























5
PRESSURE


E--4






C)


:2
P^7



STE^


7 9
(TORR)


PRESSURE (TORR)


Figure 2-3. Detail of Argon on boron nitride isotherm taken in June 1994. Note that
neither the isotherm (top) nor the calculated compressibility (bottom) show any sign of a
substep at six Torr as identified by other authors for a dirty sample.


C0.




0,


AU--"
^&a


Ar / BN
77 K


A&
/


ii


i&a
{a


CI)




Cr,


C)9


__________~ _~_____________


" '


a.I








additional compressibility peak of approximately 10 M/N was observed near 6 Torr with a

half-width of about 2 Torr. This low pressure peak was believed to be due to the

inadvertent adsorption of oxygen onto the surface of the boron nitride yielding boric

oxide. The contaminant, as measured by the compressibility peak, could be removed by

heat pumping at 900 C. Second, the liquid-solid substep centered at approximately 36

Torr, which is approximately 1.5 2% of a monolayer for a clean sample, diminished in

size and sharpness for a dirty sample to the point where it was barely measurable from the

background noise.

Similar isotherms have been measured at 77 K on the boron nitride sample used in

this laboratory for hydrogen studies at approximately six month intervals for two years.

The top of Figure 2-2 illustrates an isotherm measured in December 1993 for the first and

part of the second layers of the adsorbed film while the bottom details the liquid-solid

transition. The substep is approximately 1.5% of a monolayer. Figure 2-3 shows a low

pressure detail from a June 1994 argon isotherm as well as the calculated compressibility

of the adsorbed film No substep is evident in this pressure range for this data or any

earlier data at the 0.75% of a monolayer level. Argon measurements were not performed

on untreated or dirty samples. It is possible that repeated exposure to, or flushing with,

hydrogen removes many contaminants, including oxygen, from the surface.

Figure 2-4 presents a detail of an argon isotherm measured in June, 1994. A

substep near 38 Torr is quite prevalent in this data as well as all earlier and later

measurements. Since the size of the substep remains unchanged in time, it is very similar

to the same substep measured six months earlier, the sample quality does not seem to

deteriorate under experimental conditions. The compressibility has a full width at half

maximum of approximately 5 Torr and a height of approximately 35 M/N. This height is

very similar to that reported elsewhere [Mig 93] though somewhat wider.

Finally, the Southern Illinois group reported a rather large (approximately 5%)

hysteresis between adsorption and desorption isotherms for argon on boron nitride. In












S- Ar BN
C/)r.























Ar / N
77 77K



































'2Q 30 40 50 60
PRESSURE (Torr)


Figure 2-4. Detail of Argon on boron nitride isotherm measured in June, 1994. The
substep at 38 Torr (top) signaling the liquid-solid transition and the associated
compressibility peak (bottom) are clearly evident in the data.
s ^
3 : d


















C> --
C- _7"

L D ****'** 3 '** 40 ***g3******1 6
PRSUR Tor

Figre24DealoAgoonbrnntiesoermesediJue194Te







other words, the argon adsorbed at a higher pressure than it desorbed in that region of the

isotherm. Similar hysteresis has been documented for other systems. Lysek et. al.

[Lys 91] have used heat capacity techniques to verify their own adsorption isotherm

studies. They found the surprising result that capillary condensation in graphite foam can

begin at 1.1 layers, much sooner than previously thought. It should be noted that the BN

powder used in this study is much less dense than any of the typical forms of graphite

currently used as substrates.

A similar adsorption-desorption cycle was measured for approximately ten layers

of argon on boron nitride and is presented in Figure 2-5 (top). A pressure dependent

hysteresis was found to be centered around layers three and four. A measure of the

systematic errors associated with the adsorption isotherm experiments may be made from

this cycle. As shown in Figure 2-5 (bottom), the liquid-solid substep mentioned above

occurs at the same pressure along both the adsorption and desorption curves. Thus the

displacement shown in the figure is a measure of the error in the number of atoms

adsorbed onto the surface. The two curves are displaced approximately 0.05 cc-STP after

a total adsorption of 18 cc-STP in 100 steps. Thus, an order of magnitude estimate for

the systematic error in the number of atoms adsorbed is 0.003 % per step.

The measurement error for the argon isotherms was greater than that for hydrogen

for two reasons. First, the time to vapor pressure equilibrium was found to be greater for

argon than for hydrogen. It is believed that this time increase was a direct result of the

more massive argon traveling at a slower speed than the extremely light hydrogen which

required additional time to pressure equilibrate across the capillary. Second, the

apparatus, including the temperature controller, was designed for low temperature work.

Thus a liquid nitrogen bath was used to achieve and maintain the higher temperature.

However, the vapor pressure of the argon above the substrate was found to depend upon

the height of the bath (and hence the temperature) and this height was not stable over the

week of data taking. Thus small fluctuations occurred in the data as the bath was refilled.




41

_ I I I e I I i i U S I I S SU.. .U
E-4-

I Desorption
C-) Ar/BN
77K
C I




..... Early
_Late




i-"3 ..- i 1 Adsorption
C)

0 50 100 150 200
Pressure (Torr)


n Desorption ,





-- -- Early
S- ^ ^ Late







SAdsorption

20 30 40 50 60
Pressure (Torr)
Figure 2-5 Hysteresis loop in the adsorption-desorption cycle (top). The data was
recorded at two different times along the same curve as a measure of residual pressure
drift. The liquid-solid substep may be used as a measure of systematic errors (bottom).








As a practical matter, it is difficult to distinguish between a hysteresis loop with

this particular pressure dependence and a systematic measurement of the final vapor

pressure before equilibrium has been fully established. The computer algorithm used to

gather the data for this thesis contained a built-in redundancy (shown above in Figure 2-1).

Specifically, even for the measurement of a single isotherm (a one temperature isotherm),

the final pressure was recorded twice: once after all criterion were met and again at some

time later (approximately 5 minutes). The isotherm could then be analyzed with either or

both final pressure values and any differences easily compared. This method was found to

be quite useful in determining the equilibrium criterion for the different pressure ranges

during the development of the computer program.

In Figure 2-5, both sets of data are presented with the first, or early, data

presented as a small circle and the second, or later, data shown as an open triangle. Note

that the two symbols are displaced more for the desorption curve than for the adsorption

curve. While this may have physical significance, the nature of the experimental apparatus

also contributes to the effect. Namely, the pressure gauge is located at room temperature

in the calibrated volume which is separated from the sample cell by a long capillary. Thus,

the gauge is more sensitive to the adsorption than the desorption process. Some of the

hysteresis shown in Figure 2-5 (top) may be due to a systematic measurement error, but

this effect is not large enough to account for the entire loop.



2.3 Adsorption Isotherm Apparatus and Procedures

The adsorption isotherm experiment was developed to characterize the boron

nitride for use in the low temperature hydrogen NMR experiment. The isotherm

apparatus started out as a very simple set of valves connected with common copper pipe

mounted on a plywood sheet and a brass sample cell inserted into a liquid nitrogen bath.

This setup was useful for methane and argon isotherms and yielded some published data.








However, the isosteric heat of adsorption could not be measured very precisely as only

single temperature isotherms could be performed. More importantly, taking data by hand

is inherently slow relative to a computer operated design that can take data day and night.

The first, second, and third generations of liquid helium temperature isotherm

sticks were shown to suffer from small temperature gradients across the sample cell. The

first stick had a 100 mil brass sample cell wall. The second generation increased this to

1/4 inch. The third generation replaced the brass (chosen for ease of machining) with

copper which has a larger thermal conductivity. A temperature difference of 100 mK at

14 K was found to be too large for isothermal measurements. Such a temperature

gradient causes one part of the sample to be undergoing a transition before another

portion therefore washing out the finer details of the data.

The final stick utilized a two-wall system analogous to heat capacity measurement

systems. The 1/4 inch copper outer wall serves as an adiabatic shield for the thin-walled,

50 mil, copper sample cell. The space between the adiabatic wall and the sample cell was

then loaded with liquid or solid hydrogen to be used in the experiment. The low

temperature hydrogen storage had three purposes. First, the hydrogen vapor minimized

the temperature gradients across the sample cell even if the adiabatic wall had a small

temperature gradient. Second, the low temperature storage acted as an elemental filter.

The only gases with substantial vapor pressures at hydrogen temperatures are the

hydrogen isotopes themselves (i.e., H2, HD, D2), helium-3, helium-4, and neon. Thus, any

other gas coming from either the hydrogen supply bottle or the supply plumbing would

freeze out in the low temperature hydrogen supply cell. Third, while hydrogen above

100 K consists of 25 percent para-hydrogen (spin 0) and 75 percent ortho-hydrogen,

(spin 1) at lower temperatures the ratio of these two species depends of the temperature.

Noting that the vapor pressure difference between ortho- and para-hydrogen is

measurable, it is obvious that the use of room temperature hydrogen creates an additional

time dependent pressure change on a time scale similar to that of the experiment.










1000 10 To Pumps
Tort Tort


To Gas Supply






To To Gas
Sample Cell Supply Cell

Figure 2-6. Schematic of gas-handling system showing the automated valves. Additional
hand-operated valves (not shown) were used only for safety and pumping out the sample.




2.3.1 Computer Automated Valve System and Adsorption Algorithm


The adsorption isotherm apparatus consisted of five computer operated valves

with manual override, as well as various hand-operated valves. The valves were standard

Nupro2 "HT' series bellows valves. Two pressure gauges were used at room temperature.

The first, by MKS3, had a full-scale reading of 10 Torr while the second, by Druck4, had a

full scale reading of 1000 Torr.

For simplicity, as well as economy, both pressure gauges were located directly on

the loading volume and served the double purpose of measuring the loading and final

pressures. This was made possible by reducing the loading volume which includes the

pressure gauge volume to a minimum, about 3.3 cubic centimeters. These pressures were

recorded by the computer and presented on a strip chart on the computer screen.


2 Nupro Company, 4800 East 345th Street, Willoughby, Ohio 44094.
3 MKS Instruments, Inc., 24 Walpole Park South, Walpole, MA, 02081.
4 Druck Limited, Groby, Leicester, England.








After a waiting period of approximately one minute, the computer began to sample

the pressure above the substrate about once every 5 seconds. The program then

calculated both a first, as well as a second order polynomial fit to the previous 20

measured pressures. For the acceptance of a pressure reading, all orders of the polynomial

fits had to meet certain criterion. Equilibrium was determined when all of the criteria were

met at least one dozen times (more for lower pressures).

The criteria utilized to measure equilibrium were a function of the pressure.

Lower pressure data required longer equilibrium times and thus had tighter criteria.

Equilibrium times varied between 2 hours for equilibrium pressures below 0.1 Torr to

about 5 minutes for pressures above 100 Torr. Argon required longer equilibrium times

than the hydrogen isotopes for a given final pressure. For simultaneous isotherm

experiments, reproducibility was measured for every dose by returning to the first

temperature and remeasuring the final pressure. Hysteresis was never observed in the

data.

2.3.2 Temperature Control

The ability to measure, under the right conditions, the layer by layer growth of the

adatoms on the surface through a pressure measurement is, due to the thermal properties

of the gas, limited by the control of the temperature. For precision isotherms, the strong

temperature dependence of the vapor pressure of a gas must be taken into account.

The temperature controller used for the adsorption isotherm experiments was

modified from an in-house designed and built seven channel temperature control unit.

Figure 2-7 presents the schematic diagram of the main elements of the unit. A silicon

diode served as the thermometer for the controller. The band gap of the diode was

approximately 0.87 Volts at room temperature and 4.5 Volts at 4 K. The band gap in a

semiconductor is a strong (approximately exponential) function of temperature and is due

to the reduction in the number of carrier electrons and holes in the conduction bands. The


















11' a
4,

w 0
* u .








1 ,. z



U- UU
n11I I *































N a. UU,- -
o .



CC



















F 7 t c o 5 c I4rllr
N a.
Fv







-zz
0-9
NS s

Ii.l nl O 0 U













w 0
I 1I IL C






S M









0 o







I E A 0.
02 R N 0 |









0 1 s



F m








particular diode used for the thermometer was chosen from a set of diodes after extensive

tests and measurements of their thermal properties.

The sensitivity of the silicon thermometer was in large part determined by the noise

of the constant current source and the operating temperatures. The stability achieved at

low temperatures was approximately 1 part in 105. At higher temperatures, the stability

was decreased by approximately an order of magnitude. The system was not useful for

temperatures approximately greater than 30 K.

The temperature control problem for the adsorption isotherms were complicated

by the length of the experiments, namely 1-3 weeks. Over this long time period the

cooling power from the cold bath to the sample cell decreased greatly as the liquid helium

reservoir evaporated. Since temperature control is by its very nature a balancing act

between all of the warming and cooling power inputs, such a strong drift in the power of

one of the components had a dramatic effect upon the long-time stability of the

temperature. Active electronic temperature control with the usual feedback systems was

found to be inadequate for the precision required for the adsorption isotherm experiments

and other means of control were searched out.

A solution was found in the use of a two wall dipstick to form a passive

temperature controller. The outer wall of the dipstick was put in contact with the liquid

helium, the "cold bath," in the transfer dewar. The inner wall was thermally isolated from

the outer wall with one important exception. A residual amount of helium gas was placed

between the two walls of the dipstick to act as a thermal exchange gas. This is a common

procedure in low temperature physics. In fact, only helium can exist as a gas in this space

since all other gases would "freeze out" when put in contact with the 4 K bath.

The system was made into a passive temperature controller by the choice of

hydrogen as the exchange gas to provide the thermal conduction between the inner wall of

the dipstick and the outer wall of the supply cell in which the sample cell resided. As the

level of the liquid helium in the transfer dewar lowered with time, the thermal conductance








through the helium exchange gas diminished. This effect caused the pressure of the

hydrogen exchange gas to increase which, in turn, led to an increase in the thermal

conductance of the hydrogen! In short, the decreasing cooling power of the cold bath as a

function of liquid helium height was offset by the increasing cooling power of the

exchange gas.

The use of a two wall dipstick in a dewar and the specific choice of exchange gases

acting to form a passive temperature controller has uses beyond the helium/hydrogen

system described above. If one were studying an argon system in a liquid nitrogen dewar,

for example, then one may choose nitrogen and argon as the thermal exchange gases to

provide a similar passive temperature controller at a very different temperature range.

Hydrogen exchange gas was chosen over helium for this inner space for two more

reasons. The first reason concerned limiting the cooling power of the cold bath while the

second reason involved maintaining the sample cell at a temperature less than any portion

of the fill capillary. The vapor pressure of bulk, solid hydrogen is a strong function of

temperature which is in the millitorr range at approximately 10 K. Thus, the use of

hydrogen as an exchange gas provides a natural thermal conductance limit between the

experiment and the cold bath. The same thermal properties of hydrogen conspire to make

hydrogen, when used as an exchange gas, provide a natural low temperature limit on the

sample cell, approximately 10 K. The thermal conductance between the two walls and

between the inner wall and the experiment was set and controlled by keeping the pressure

of both exchange gases in the millitorr range. In practice, it was found that a helium

exchange gas pressure of approximately 60 millitorr and a hydrogen exchange gas

pressure of 1-200 millitorr worked well.

Thermal uniformity across the sample was achieved in two ways. First, the sample

cell was placed inside the supply can. Second, the walls of the supply can were made

relatively thick, 0. 1 inches, and of oxygen free high conductivity (OFHC) copper. As side

benefits, the hydrogen supply used in the adsorption experiments both achieved ortho-para








equilibrium and most impurities in the sample gas were filtered out. Finally, the supply

can, the hydrogen supply gas, and the sample cell provided enough thermal mass to act as

a filter blocking out the high frequency thermal changes.





2.4 Hydrogen Isotope Isotherms


The adsorption of hydrogen isotopes onto graphite has been extensively studied by

H. Freimuth and H. Wiechert et al. [Fre 90, Wie 91], J.M. Gottlieb and L.W. Bruch

[Got 90], and others [Cha 90], and two-dimensional phase diagrams have been

constructed (see Figure 2-9). All three common hydrogen isotopes, H2, HD, and D2, were

used in this study to allow for isotopic comparison. Two goals existed for the isotherms:

One, to map out the translational phase diagrams (e.g., the critical and triple points and

translational phase transitions); two, to measure the isosteric heat of adsorption.




2.4.1 Interpretation


The adsorbate gas is at first adsorbed onto the surface directly into a solid state

with a very low residual pressure. Build-up of the first layer continues with little change in

the vapor pressure and is marked by the nearly vertical climb in an isotherm plot. The

monolayer is complete near a point historically marked as "B" or "B 1" which occurs soon

after the first vertical rise. The following, more horizontal, nearly linear region extends to

the beginning of the second layer where the adatoms usually adsorb into a two-

dimensional gaseous state. Adsorption into the gaseous state is manifested by a decrease

in the isosteric heat of adsorption (q.t) to a value close to the three-dimensional heat of

condensation (0.91 kJ/mole or 109.6 K for H2). The second and higher layers are marked












coem
t0 _: *I IiIii I I* I I I nIl I Ii!i














n- -- -. **.-. u ---




-*
r c oo0 m o .... -.......
C ',d.~ omaar mrrmsrmr
l. 00 OC '- 2 .. .. .. ---- -,r, ........... .. "rrrm


7- 8
787


792 797
TIME (days)


- I S S


0 00000000000000000000000000000 000000000





*I M*********************lHI l***** *****


19.25K -- l1mK


18.05K llmK

16.97K lO1mK


- 23mK

- 7mK

- 25mK
- 16nmK
- 43mK
- 18mE
- 9mK
- 21mK


10.07K

15.33K

14.70K
14.09K
13.48K
12.86K
12.22K
11.77K


802


I2I i I I ft l i i I
21.67K ImnK
21.10K 3mK


20.07K


- 5mK


18.87K 6miK


17.69K 5mK


16.67K 4mK


15.83K


- 4mK


u- ****- ..C *....L..* .*<..*******-C* 15.14K 15mK :
-- ****** ******************* ********* 14.51K 1- 15mK

o ooooooooooooooooooooooooooooooooooo 13.91K 5xK
.-- -..----.muinu-....-..- n... .... 13.30K 6mnK

769 771 773 775 777 7
TIME (days)

Figure 2-8. Temperature control as a function of time for two experimental runs,
BNHD.3T (top) and BND2.28 (bottom).


79


I I


I l i j i l l l i


0---








E--4
^st


0-a
as


E- "



r- -
E-e



. .
S3








0.250--
(a)
0.225 -

0.200t-

0.1751-


0.150

0.125


r


U. IUU1
6


Hz/MgO- 2nd layer


T.=10.05K


Tt=7.20K
, I I


temperature


(K)


4 6 8 10
temperature (K)




Figure 2-9. Translational phase diagrams of the second layer of H2 on MgO [Ma 89] (top)
and graphite [Sch 89] (bottom).


Io0


I


. II








by similar changes both in the slope of the isotherm and the magnitude of the isosteric heat

of adsorption. After a number of layers, these slope changes fade out and the isosteric

heat of adsorption approaches the three-dimensional heat of condensation.


2.4.2 HD Isotherms

The adsorption of deuterium hydride (HD) was chosen over H2 or D2 for detailed

investigation due to the possibility of comparing these results with the interesting results

of two earlier studies, HD on graphite [Liu 93] and HD on MgO [Ma 89]. Both systems

were not only investigated via adsorption isotherms but also with heat capacity techniques.

Both systems were also further explored via quasielastic neutron scattering (QENS), and

phase diagrams were reported [Wei 91, Vil 94]. Both studies found substeps near the

completion of the second layer indicative of the two-dimensional liquid-solid transition.

However, the size of the substep varied greatly. For HD on MgO the substep was

approximately 11% of a monolayer while the same step was found to be only 2% of a

layer for HD adsorbed onto graphite. The change in entropy associated with the melting

of HD on graphite corresponds to approximately 8.5% of the bulk melting entropy at its

triple point [Vil 94].

Figure 2-10 presents the results of a multiple-temperature (simultaneous)

adsorption isotherm run of HD on BN starting with the end of the first layer. The data

from eleven temperatures are plotted and five steps are resolvable in the lowest

temperature data. Note that the pressures at the highest temperature were remeasured as

described above and no hysteresis encountered. The vapor pressure above the surface

during the formation of the first layer is too low to be measured reproducibly with our

apparatus.

To check for possible substeps, another set of isotherms was collected using finer

steps and more temperatures (17). Over 2500 data points were measured over a period of

































0.1 1 10 100
PRESSURE (TORR)


Figure 2-10. Volumetric vapor pressure isotherms of HD on BN. Five steps are visible in
the low temperature data. The data from all temperatures was recorded simultaneously;
lines are guides to the eye.









.1- ** 11.40 K
*--*** 11 6 1 K
E.,4 U_-4-0 11.90 K
*.4.e 12.23 K
C) -s esease 12.61 K
13.03 K
\I + 13.44 K
**-** 13.85 K
C) i eseM 14.26 K
C Awhba 14.66 K
-~ HD / BN


















1 10
I I










**1- 1.40 K
P-R 1 1.61 K0
4- 11.90 K
.- n 12.23 K
C P Mes* 12.61 K
-0--- 13.03 K
-I 0-0- 13.44 K
*.* 13.85 K
S *Meesw 14.26 K
A- -Ask 14.66 K
HD / BN












"o -- -




0.1 1

PRESSURE (TORR)
Figure 2-11. Details of two layers in the HD isotherms. Note the lack of a substep in the
middle of the second layer (bottom) and a possible substep near the completion of the
third layer (top). Lines are guides to the eye.








22 days. As shown in Figure 2-11 (bottom), no substep was found at the 1% of a

monolayer level for the middle of the second layer of HD adsorbed onto boron nitride.

Some data points (near 2.72 cc-STP and elsewhere) were omitted from this figure due to

temperature controller error.

The top portion of Figure 2-10 focuses on a possible substep near the end of the

third layer, between 4.85 and 4.95 cc-STP. The slope change in this region can not be

attributed to temperature controller error since the pressure changes associated with the

temperature fluctuations measured by the thermometer for this region are an order of

magnitude smaller than the observed pressure deviations. More importantly, the change in

slope is observed over three cycles of data points. Thus, the slope change can not be

attributed to one random event.

Two more items of interest are manifest in the bottom portion of Figure 2-11.

First, an example of a single event error in the data can be seen by the deviation of the line

comprised of filled circles and measured at an average temperature of 11.61 K Second,

the slope of the rise portion of the isotherm is slightly negative for some isotherms. While

an isotherm with a vertical rise (an infinite slope) has a straightforward physical

interpretation (it corresponds to a path through a two phase region) a negatively sloped

isotherm is, in fact, unphysical Note that such negative slopes are found in the data only

for the lowest temperatures and pressures measured and that the noise in the data is due to

the resolution of the low pressure (10 Torr) pressure gauge, 0.001 Torr. It is believed

that the thermomolecular effect (a difference in the equilibrium pressure of a gas across a

capillary with the ends at different pressures) plays some role here. Thus, the means of

achieving such low temperatures, a decrease in the fill line heater, is a possible culprit as

the lack of direct heat may allow some portion of the fill line to become cooler than the

sample cell. The final pressure (measured at room temperature) may be the pressure

corresponding to the thermomolecular temperature (for the capillary diameter and the

particular gas) with the pressure of the sample cell as a lower bound. As the number of








atoms admitted to the sample cell increases (i.e., traveling along the isotherm in the

adsorption direction), the pressure in both the sample cell and the capillary increases

causing the thermal conductance of the system along the capillary to increase and the

temperature gradients to return to normal (linear from hot to cod end).

Much effort was expended in attempting to eliminate cold spots in the fill line. For

example, the line heater was made more uniform and a shield added to the fill line. In

addition, the direction of the cycle of pressure measurements was reversed. The earliest

attempts at measuring multi-temperature isotherms first measured the final pressure

corresponding to the lowest temperature, jumped to measure the pressure corresponding

to the highest temperature, then stepped back to the lowest temperature. After

experimenting with all four possible temperature cycles it was found that a cycle which

measured the pressure at the highest temperature, jumped to the lowest, then stepped back

to the highest produced the most satisfactory data. This change in the temperature cycle

reduced the magnitude of the negative slope approximately 5-7 times compared to the

original procedure. However, this effect marks the lower limit of the experiment at

approximately 0.01 Torr.

However, the best solution involved redesigning the fill line to be inside an

evacuated tube. Theoretically, such an arrangement allows for the most linear

temperature gradient along a tube at two temperatures of any passive method. Active

methods, for example, a series of thermometers and heaters, were not attempted. A new

stick was built but not yet used at the time of this writing because it was determined that

the errors involved in calibrations between two sticks were greater than the loss of this

non-essential information. Thus the plan was laid to finish the current experiments before

the introduction of the new stick. The improved stick would also allow for concurrent

heat capacity experiments which was not possible with the old stick.

The top of the second layer was not investigated as thoroughly as the same area in

the third layer. Further investigations are currently underway.











2.4.3 H, and D2 Isotherms

Figures 2-12, 2-13, and 2-14 exhibit the results of isotherms from hydrogen and

deuterium on boron nitride. For both systems at least four steps are present and the

critical temperatures have been calculated (see Table 2-1). Neither system showed any

sign of liquid-solid subsets at the 2% level nor any hysteric effects.




2.5 Compressibility. Critical Temperatures, and Larher's Method


At temperatures near the critical point, the formation of each layer on a substrate

surface involves the two-dimensional condensation of each layer from a dilute to a dense

adlayer. Such a transition is of first order and must end at a critical point. Below the

critical temperature (e.g., in the liquid-vapor coexistence region), the chemical potential

and, therefore, the vapor pressure is constant for a fixed temperature. Thus, the isotherm

is parallel to the number axis in this region. Since the compressibility of a film is

proportional to the slope of the isotherm, the inverse compressibility vanishes. However,

for real systems, surface inhomogeneity causes isotherms in this region to have a finite

slope [Eck 86]. The inverse compressibility for a film at fixed temperature has been given

by [Das 75] as
2 In P
S= In k, TOn P Equation 2-1


where n is the areal film density. The decrease in the inverse compressibility was first used

by Y. Larher [Lar 79] in his study of argon on CdC12 to determine the critical point of the

film. He found a nearly linear rise in the maximum inverse compressibility as a function of







8.5 I II I II I I I I I 1 I
"- eee 13.34 K
++ 12.69 K
E -Be 12.14 K
*H* 11.68 K
S *- --* 11.38 K
I 11.23 K
S AAAA11.16 K
"+++ 11.00 K
S*** 10.61 K
o e- 13.93 K
6.5
Pd


0













1 10
4.5





H2/BN




Pressure (Torr)

Figure 2-12. Volumetric vapor pressure isotherms of H2 on BN. Five steps are visible in
the low temperature data with the last four shown. The data from all temperatures was
recorded simultaneously; lines are guides to the eye.







12.16K
44iA 13.30K
q eOeee 13.91K
-- 14.18K
I[ -6 14.51K
I -& 15.51K
S0 15.83K
Q a* 16.48K
t- H- 16.48K
4- -~- 16.67K





0








O
2 I 11 111 1 I l 11- I I II 111 -l -
0.1 1 10
PRESSURE (Torr)
Figure 2-13. Low precision volumetric vapor pressure isotherms of D2 on BN detailing
the third step. Data is from two runs; lines are guides to the eye.






3.9-

I AAAM 14.82K 3mK
H- ooo0 /12.10K 9mK
SAA A 12.98K 10mK
0000ooooo 13.91K 2mK

S3.4 -





0 0-
030


A 0
C),

A 0
A 0 A
A 0
A 0
SX A 0
A 0

2.4 -







1.9- I I I


Pressure (Torr)
Figure 2-14. High precision, volumetric adsorption isotherms of D2 on BN detailing the
third step. Data is from one multi-temperature isotherm run. The gap in the data at
approximately 3.5 cc-STP is due to experimental difficulties and has no physical
significance. The listed errors represent the standard deviation of the temperature
associated with each volume/pressure point.








temperature above the critical point and a near zero inverse compressibility below it. The

intersection of these two lines was determined to be the critical temperature. Later work

has shown that Larher's method of critical point determination overestimates the critical

temperature by approximately 4% [Eck 86]. Comparisons between adsorption isotherm

and heat capacity results bear this difference out. The authors further pointed out that the

actual overestimation for a real system depends on the temperature range of the data and

the equation of state of the adsorbed system.

Figure 2-15 presents a Larher plot for the second layer of HD on BN. Two lines

are drawn and for clarity only some of the error bars are exhibited. The horizontal line is

drawn to represent the positive finite slope expected of real systems. In this region, the

data is highly scattered due to apparatus difficulties in measuring low vapor pressures.

The positive sloped line corresponds to the linear region beyond the critical temperature.

At higher temperatures, the linear approximation breaks down. From this plot the critical

temperature is estimated to be at 12.3 0.5 K This temperature is similar to but slightly

higher than the value for HD on graphite of 11.8 0.3 K [Liu 93].

Similar plots for the third and fourth layer of HD on BN, top and bottom of

Figure 2-16, respectively, yield critical temperatures of 12.2 0.5 K and 11.8 0.5 K,

respectively. The critical temperatures of H2 and D2 on boron nitride have been similarly

calculated and representative plots are presented in Figure 2-17. The critical temperatures

are also reported in the tables at the end of this section.

It is important to remember that graphite has been the subject of many

investigations using a number of methods over the last few decades. Further, some studies

have used multiple methods to determine the critical points. For example, the HD on

graphite studies of [Liu 93] used both heat capacity measurements and adsorption

isotherm techniques to determine the critical temperatures. The data presented herein

represents the first study of hydrogen on BN and only adsorption isotherm data is

available. Therefore, the critical temperatures listed here should be viewed as preliminary






SI I I I 1 I I I I I I I I I I I 11 1 I I I I


^AA^ bnhd.623
00000 bnhd.502
t**** bnhd.421
***** bnhd.3t


1 1 1


Layer 2
HD / BN


Tc = 12.3 +-


3 15


I I I I I 17i l l
17


0.5 K


SIII9I I
19


Temperature (K)


Figure 2-15. Larher plot of the second layer of HD on BN. The symbols in the legend
denote different experimental runs. The numeric code represents the month and day of
1993 on which the experimental run was begun.


0


.
T\



d


*
A


0O


,










C:)




0-


a-:


a
c

C


c:

5


I I I Ii. I I I .I I I I- It S-1- F I I I I I I 1 II I 1 1 I -


-i -
Z LLayer 3
HD / BN
Tc = 12.2 +- 0.5 K
- I I l


12.0 12.5 13.0 13.5
Temperature (K)


12 13
Temperature (K)


Figure 2-16. Larher plot of the third (top) and fourth (bottom) layers of HD on BN.


^^^^^ bnhd.502
***** bnhd.3t


.--


II


14


1 .5
1.5










^^^^ BND2.28
O;*tvt*3* BND2.218


i l l l l l l li l l l l l l 1 & l l l


Layer 3
D2 / BN


Tc = 13.9 +- 0.5


l~ ~ ~ 13 T itisa


14-


15


Temperature


III lii I I


16
(K)


"^ -


I I I I I I
110


11


555 I I S i I i a I # I I I I I I .
12 13 14
Temperature (K)


Figure 2-17. Larher plot of the third layer of D2 (top) and the second layer of H2 (bottom)
on BN.


o-





: -
0-
:
5-:


C)12


1 I I I I I 1 II I I I II I I I I I 1 II I I I I I 1 555 I 5 I


***** bnh2.116
^^^^^ bnh2.21
ooooo bnh2.722


Layer 2
H2 / BN


To = 11.5 +- 1


! i |1


r


r I. i r_ .


] I l l


-


i




II



A


13


-










.0





.0







until other methods verify these results. Finally, note that Larher's method tends to

overestimate the critical temperatures, as mentioned above.

Adsorption isotherms have been used to identify the two-dimensional melting line

and triple point. A substep in the adsorption isotherm occurs upon crossing the melting

line. At the triple point, the melting line intersects the condensation line in two-

dimensions just as it does for three-dimensional systems. Thus, this determination of the

triple point makes the isotherm search for the melting line of HD on BN all the more

exciting and important.




Table 2-1. Critical temperatures of monolayers of hydrogen isotopes adsorbed onto BN.


Table 2-2. Critical temperatures of HD adsorbed onto three substrates.


HD Critical Temperature (Kelvin)
Layer 2
Substrate Layer 2 Layer 3 Layer 4 Triple
Temp

BN 12.3 0.5 12.2 0.5 11 8 0.5

Graphitea 11.45'-11.8" 12.1 0.3' 11.8 0.4' 8.44 8.58"

MgOc 10.5 0.3 12.1 0.3 12.4 +0.4 9.95 0.1

a[Liu 92a] on graphite foil, 'heat capacity, isothermss, b[Ebe 94c], c[Vil 91]


Critical Temperature (Kelvin)
Isotope Layer 2 Layer 3 Layer 4

H2 11.5 + 1.0 10.9 0.5

HD 12.3 0.5 12.2 0.5 11.8 0.5

D2 14.1 0.6 13.9 0.5







Table 2-3. Critical temperatures of H2 adsorbed onto three substrates.

H2 Critical Temperature (Kelvin)
Triple Temp
Substrate Layer 2 Layer 3 Layer 4 Layer 2

BN 11.5 + 1.0 10.9 + 0.5

Graphite" 9.31 10.08 5.96

Graphiteb 10.0 0.1 11.0 0.5 6.5 0.1

MgOc 10.05 0.05 10.3 0.3 10.4 + 0.2 7.20 0.05

D2 on Graphite 9.44

a[Wie 91b] from heat capacity, b[Per 92] from isotherms, c[Vil 91]

Table 2-4. Critical temperatures of D2 adsorbed onto three substrates.

D2 Critical Temperature (Kelvin)
Triple Temp
Substrate Layer 2 Layer 3 Layer 4 Layer 2

BN 14.1 0.5 13.9 0.5

Graphite" 11.8 0.2 12.1 0.3 11.8 0.4

Graphiteb 13.10 13.86 11.04

MgOc 12.4 0.4d 13 1d 12.9 0.4 12.1 0.2

'[Liu 92a], b[Wie 91], c[Vil 91], d[Deg 88]

Table 2-5. Triple point temperatures of second layer hydrogen isotopes adsorbed onto
selected substrates.

Triple Point Temperature (Kelvin)
Isotope
Substrate H2 HD D2

Graphite" 5.96 '8.44 8.58d 11.04

MgOb 7.20 0.05 9.9 0.1 12.1 0.2

D2 on Graphitec 5.74 6.25 6.4

HD on Graphited 6.25___

"[Wie 91b], b[Vil 91], '[Liu 92a], d[Liu 93] from 'heat capacity, isothermss










2.6 Isosteric Heat of Adsorption


The isosteric heat of adsorption can be derived directly from a pair of isotherms

and is defined as

qI k T 2 A Equation 2-2


where the film density, Nf, and the substrate area, A, are fixed [Das 75]. In general, this

quantity is both a function of temperature and film density. Usually, one is interested in

the heat of adsorption for the first adatom into the gaseous state of an adlayer. Since the

vapor pressure above the first layer for hydrogen at low temperatures is too low to be

practically measured, the second layer value is often reported as in Table 2-6. The

isosteric heat is a measure of the binding energy between the adatom and the surface. If

the adatom-adatom interactions can be neglected (i.e., at low coverages), then the

isosteric heat is simply the sum of the binding energy, eo and the kinetic energy

q,, = c + a (kb T) Equation 2-3

where a = 1/2 for a gas molecule harmonically bound to an adsorption site and a = 3/2

for a two-dimensional ideal gas on a flat surface [Liu 93].

Figure 2-18 shows the isosteric heat of adsorption for the second, third, and fourth

layers of HD on boron nitride. These values are extracted from two isotherms for which
the temperature difference may be too large ( d T= 0.33, 0.49 K) to be a true differential as

defined in Equation 2-2. However, many features are still apparent in the data. Using

Equation 2-3, the binding energy for the second layer of HD on BN is found to

be eg = 120 10 K. This result is somewhat lower than the value of 160 K reported for

HD on graphite [Liu 93] and much larger than the HD-HD binding energy of 37 K.

Figure 2-18 further manifests that the change in isosteric heat as a function of film density



























00 ++++' bnhd.503 T = 12.06K dT = 0.33K
bnhd.3t T = 12.02K dT = 0.49K

o HD/BN
1 -










Sbnhd.503 T = 12.06K dT = 0.33K
^ oWQee bnhd.3t T = 12.02K dT = 0.49K

HD/BN


02 3 4 5 6 7

Volume Adsorbed (cc-STP)
Figure 2-18. Isosteric heat, q,, for the second, third, and fourth layers of HD on BN for
two runs.






p p p p p p p p p a p a pI I p


H2 / BN


C),
C)
CE

0u

U)


11.28K
11.18K


dT = 0.30K
dT = .05K -


I i 1 I i 1 1
'3

Volurnne


1. I I I I 1 1 i


5

Adsorbed


7

(cc-STP)


Volume Adsorbed (cc-STP)

Figure 2-19. Isosteric heat, q., for the second, third, and fourth layers of H2 (top) and the
second layer of D2 (bottom) on BN. Lines are guides to the eye.


^ "^ T =
T =
*--.-**T =







is smaller by approximately 20% than HD on graphite. These two features are consistent

with the lack of a large melting substep. Finally, note that the change in the heat of

adsorption for the third layer is similar to the second layer (approximately 20%) and that

the fourth layer change (approximately 10%) is much smaller than the previous two layers.

After the fifth layer (not shown), the isosteric heat of adsorption tends to a value of 144 K

which can be compared to the three-dimensional heat of vaporization (134 K) and the heat

of sublimation (153 K).

Figure 2-19 exhibits the isosteric heat of adsorption for H2 and D2 on BN. Note

that relative to HD, the heat is less for H2 and more for D2 as expected.

Vidali has compiled a table of potentials of first layer physical adsorption [Vid 91].

This table lists the adsorption potentials and equilibrium distances for a variety of gases on

a number of substrates, including BN, graphite, and MgO. Sources of information include

the results of thermodynamic (e.g., adsorption isotherms and calorimetric measurements)

and LEED experiments as well as theoretical estimates. While no entries exist for the

hydrogen isotopes adsorbed onto BN, the tabulations for the rare gases is interesting.

Bare graphite adsorbs Ar, Ne, Xe, Kr, N2, and CH4 more strongly than either bare BN or

MgO by 10-20%.


Table 2-6. Binding energy of hydrogen into the second adlayer.
BN Graphite MgO
H2 100 20K 120 10K'
HD 130 10K 177 10K
D2 170 + 10K
a Taken from a graph in [Per 92].


A very simple, phenomenological method of comparing the adsorption potentials

of various substrates involves directly relating the temperatures required for a given

adsorbate vapor pressure. For an HD vapor pressure of 0.1 Torr at the beginning of the

second layer, a temperature of 12.6 K is required for graphite, 12.1 K for MgO, and








11.7 K for BN. This simple method emphasizes the relatively weak adsorption potential

of boron nitride.





2.7 Summary


The use of boron nitride as a substrate for the translational phase study of the

hydrogen isotopes has accomplished two goals: more clues have been added to the search

for superfluid hydrogen in two dimensions and the binding energies have been determined.

Although the precision of the determination of these values from the adsorption isotherms

of this study is less than ideal, the isotopic trends are consistent with expectations and

reinforce the accuracy of each measurement. It is important to note that Larher's method

has been shown to overestimate the critical temperatures.

The binding energy of hydrogen isotopes on boron nitride is less than that on

graphite and MgO. However, this decreased attraction does not translate into lower

critical point temperatures as was hypothesized. Further, the magnitude of the liquid-solid

substep appears to have been suppressed below those of hydrogen on either graphite or

MgO. It may be surmised that the surface potential corrugation, lattice spacing and

substrate geometry play a significant role here. In the search for superfluid hydrogen,

much could be learned about the role of binding energies on critical temperatures through

the investigation of a substrate which shares the geometry of graphite and boron nitride

but possesses a larger adsorption potential than graphite.

If the translational phase diagrams of hydrogens adsorbed onto boron nitride are

similar to those of hydrogens adsorbed onto other substrates, then the triple point

temperatures are probably also higher than on other substrates. However, until a substep

in the isotherms is identified, other methods are necessary to determine the triple points








and to confirm this hypothesis. The investigation of the triple points by heat capacity

techniques will probably have to wait until difficulties associated with the low thermal

conductivity of boron nitride (approximately 100-1000 times less than graphite) are

overcome. One possible solution is to add, in addition to copper wires, an inert thermal

link (i.e., 3He) to the system such as was done for the NMR apparatus.

The use of NMR to map out the translational phase diagram for this system has

two advantages. First, NMR is not only ideally suited to determine translational phase

transitions but also to identify the transitional phases. Second, as opposed to the high

electrical conductivity of graphite, the low conductivity of the boron nitride allows for

unhindered NMR measurements. However, such a search would require even more

sensitivity than that required in the efforts to map out the orientational phase diagram, as

discussed in the following chapters of this thesis, due to the inverse temperature

dependence of the magnetic susceptibility (Curie's Law). However, with the possibility of

superfluid hydrogen in the balance, the search for the triple point of the isotopes may have

large rewards indeed.












CHAPTER 3


NMR APPARATUS AND PROCEDURES

Reading maketh a full man; conference a ready man; and writing an exact man.
Of Studies Essay 50, Sir Francis Bacon 1561 1626



3.1 Overview


Historically, NMR studies have been performed on bulk materials [Pak48,

Sul 67]. Recently, however, new opportunities for the study of two-dimensional systems

have emerged [Har 88, Cow 93]. Even one dimensional systems have begun to be

studied to test the effects of reduced dimensionality [Ave 92]. However, the signal from a

single nucleus is very small and provides a natural limit as to the number of atoms

observable by standard NMR techniques.

An NMR signal is directly proportional to the number of atoms being investigated.

Since the number of atoms in a two-dimensional system is proportional to the number of

atoms in a similar three-dimensional volume to the two-thirds power, or

N2D N30 Equation 3-1

a similar reduction in the NMR absorption amplitude is expected. In other words, the

signal-to-noise ratio of two-dimensional systems is inherently low. Fortunately, hydrogen

is the second most sensitive nucleus, after tritium, in the entire periodic table. 9F and He3

are the third and fourth most sensitive elements, respectively.

The intensity of an NMR signal also depends on the polarization of the nuclei

under study, which in turn depends upon the temperature of the system and the local








magnetic field. This dependence can be seen by starting with the expression for the

polarization of a spin system; i.e.,
-tAw
n+ -n 1- e bT
P --- Equation 3-2
n +n
Sl+e bT

and noting that the denominator is essentially unity. One can then expand the exponential

in the numerator,

hco 1 (hAmo,
P= l-{1- +,2 ) -. ....... Equation 3-3


and obtain an expression for the spin polarization valid for small k6 namely,
kT
Amo
P ~ 0 Equation 3-4
2kbT

Since the NMR frequency, o, of a nucleus is directly related to the magnetic field,

B, by the gyromagnetic ratio, y, through the expression,
oo = yB, Equation 3-5

we then have an expression for the polarization in terms of B and T:
hy B
P -- -. Equation 3-6
2kb T

For small B/T, that is, at high temperatures or low fields, the polarization is

approximately linear in B/T. For large B/T, that is, at low temperatures and high fields,

this linear approximation breaks down and the exponential behavior becomes evident as

the polarization approaches unity. For a field strength of 6.29 Tesla, corresponding to a

resonance frequency of 268 Megahertz for hydrogen, the exponential behavior becomes

apparent for temperatures below approximately 0.5 K as the premise for the linear

approximation of Equation 33 breaks down.

It is important to estimate the signal-to-noise ratio before considering an

experiment lest one attempts a theoretically impossible experiment. According to








Abragam [Abr 61], the signal-to-noise ratio for an NMR measurement can be shown to

be:
S1/2
S =v- V,
N L = Cv)Q ~ B B ) Equation 3-7
N f 2 v o k \ T

where fis the noise from the NMR apparatus excluding the coil; v is the frequency; A v is

the bandwidth; Q is the quality factor of the coil; Zr is the nuclear magnetic susceptibility

of the nucleus being probed; B is the magnetic field; and V, is the volume of the sample.

From this equation, it can be seen that the NMR signal is directly proportional to the
filling factor of the coil, 7 which is usually on the order of unity for a typical three-

dimensional experiment.

However, for two-dimensional experiments such as this one, the fill factor is

approximately 10-4 lower. The small filling factor arises from the fact that the apparent

density of the boron nitride powder is very low and the hydrogen molecules occupy only

the surface of the powdered platelets. The actual number of hydrogen molecules in the

sample cell is determined from the adsorption isotherms discussed earlier. The values of

the variables to be used in Equation 3-7 have been estimated from the experimental

conditions to be:
7 104, v 268MHz, Av = 0.3Hz, Q 50,
Equation 3-8
Bo z 62,900gauss, V, z 3cm', T z 4K.

The nuclear magnetic susceptibility of the system may be calculated from:
N(y ph)2 I(I+ 1)
X = 3kT L7*10-'2 (M.KS. units) Equation 3-9
3kbT

where we have used the fact that N 10'9 spins, y, h = 4.47*10-21 Joules/Tesla, and k T

= 4.14*10-21 Joules at 4 K. While the signal-to-noise ratio may now be calculated from

Equation 3-7, it is more instructional to utilize an equation which explicitly shows the

dependence of the signal-to-noise ratio relative to the experimental NMR standard of the

protons in one cubic centimeter of water at room temperature in a field of one Tesla:









S 106 V 4 /y (N H 3/12 \ 1( T -3/
I(I +1) r (A v)- Equation 3-10
N f V,1 3 y N 10 10 300

where Np is the number of protons in a cubic centimeter of water at room temperature,
7*1022, and Vc is the volume of the coil. Using the values for our two-dimensional

system, we find
S
7*10 / f. Equation 3-11


While at first this may seem more than sufficient, in most cases the theoretical

values are rarely achieved. For comparison, the calculated NMR signal-to-noise ratio for
protons in water at room temperature is approximately 106/f [Abr 61] but ratios of 10'-

103 are usually achieved. We also note that the calculated signal for our two-dimensional

experiment is approximately 1000 times smaller than that of bulk hydrogen experiments,

where the fill factor would be between 0.1 and 1 depending upon the construction of the

sample cell.

Another calculation one can make concerning the experimental feasibility of the

experiment is the determination of the voltage across the NMR coil:
1 -
E = -77J4rC o X B0AQ*10~8 2.8*10-' volts Equation 3-12


where A is the area of the coil and the other parameters are as above. This low voltage

demands the use of high amplification and sensitive measurement techniques. According

to Souers [Sou 86], the limit of sensitivity (i.e., the state of the art) of an NMR signal for

hydrogen at 4 K corresponds to a signal-to-noise ratio of 4:1 for a spin density of 1019

spins per cubic centimeter (bulk hydrogen has a spin density of 1023 per cc). Data

acquired for this thesis yielded a similar signal-to-noise ratio which improved slightly as

the temperature decreased. However, the expected 1/T gain in the signal-to-noise ratio








was realized only for temperatures down to 1 K Below 1 K, the magnetization was found

to be independent of temperature.

As the technology for high magnetic fields improves, we will be able to study ever

smaller systems of spins. Organizations such as the National High Magnetic Field

Laboratory in Tallahassee, Florida, and the European collaborations in Grenoble, France,

continue to produce ever higher fields. Furthermore, recent advances in semiconductor

technology allows for the use of spectrometers immersed in the low temperature

environment. Since thermal noise, also known as Johnson or "shot," noise, is often a

limiting factor in experiments with low signal, operating the core electronics at low

temperatures is a great advantage. Recently, an NQR experiment utilized new GaAs

transistors in a marginal oscillator circuit operating at high milliKelvin temperatures and

hundreds of Megahertz. The oscillator was located inches away from the MnSb sample

inside the resonance coil [And 92b]. For this quantum thermometer experiment, the low

temperature, marginal oscillator spectrometer worked extremely well.




3.2 Experimental Apparatus



The heart of the nuclear magnetic resonance system used for these experiments

consisted of a continuous wave (CW) NMR spectrometer. This arrangement was first

used by Purcell, Torrey and Pound [Pur 46] to measure the first magnetic resonance of

protons in paraffin in 1946. Early NMR experiments using this technique were soon

replaced in most research by pulsed NMR techniques. These pulsed systems have many

advantages under a wide variety of circumstances. However, pulsed systems yield data in

the time domain, which is the Fourier transform of the frequency domain in which the CW

NMR line shape is measured. In a world of infinite signal-to-noise ratios, such as

mathematics, these two domains are equivalent. In the real world of small signals and








large noise it can be advantageous to record the data in the domain of interest. In our

case, the NMR line shape (the resonance response as a function of magnetic field) was

examined to obtain information concerning the orientational ordering in the frequency

domain.





3.2.1 Spectrometer Schematic Diagram


The CW NMR spectrometer utilized for this thesis consisted of two parts: the

magnetic field sweep and the RF absorption detector, connected in theory only through

the spins of the hydrogen nuclei under investigation. The sample cell, detection coil, and

sample can were bolted onto the cold finger of a dilution refrigerator resting inside a

6.7 Tesla superconducting magnet.

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 PC)

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 central tenet of the experimental endeavor concerned the width of the NMR

line. Thus, the calibration of the modulation magnet was critical. Calibration of the

modulation magnet was achieved by varying the frequency of the radio frequency signal

(known to 1 part in 106) and measuring the change in current required to center the NMR

signal in the field sweep. This procedure led to a frequency-to-current ratio of 50 kHz per

amp. The well known gyromagnetic ratio of hydrogen, 42.5759 Megahertz per Tesla, can








then be used to convert current to magnetic field. The current-to-field ratio was found to

be 0.085 Amp per Gauss.

In practice, the much larger '9F NMR signal from the sample holder was used to

obtain a more precise figure. Fortuitously, the gyromagnetic ratio of fluorine,

40.0541 Megahertz per Tesla [CRC 91], is only slightly smaller than hydrogen. Thus, the

fluorine NMR signal required only a slightly larger static magnetic field from the main

superconducting magnet (approximately 59 versus 55 amps) for the same resonant

frequency, 268 Megahertz. Furthermore, the NMR line width of fluorine in Teflon

tetrafluoroethylenee) is similar to the line width of molecular hydrogen. Thus, another

sample holder was manufactured out of Teflon to replace the original KelF sample holder.

While KelF is a commonly used material in low temperature physics, a fluorine NMR

signal was not found under our experimental conditions This lack of signal was at first

surprising since KelF is chemically very similar to Teflon with the exception that one out

of every four fluorine atoms is replaced with a chlorine atom.

The radio frequency spectrometer consisted of five main components: an RF signal

generator; a hybrid tee; an RF amplifier; audio amplifiers; and a lock-in amplifier. The

inherent low absolute signal of a two-dimensional experiment necessitates the use of a

high sensitivity spectrometer. The hybrid, or "magic," tee configuration of a CW NMR

spectrometer is the most sensitive of the various CW NMR configurations and is

illustrated in Figure 3-1. As in most CW NMR arrangements, the NMR signal is a small

perturbation on a larger RF carrier signal. While measuring a small change in a relatively

large signal is inherently difficult, measuring the same small change from a null is much

easier. Thus, the RF energy coming into the hybrid tee from the RF generator is split into

two parts.












CAPACITIVE
TUNING BOX
C,




........... .....=
is-P


TRANSMISSI
LINE
(1 = 3 2 /2)


MODULATION
COILS









80386
PERSONA
COMPUTE










LOCK-IN
AMPLIFIER


CIRCUIT


RF
AMPLIFIER
(250X)


RF
DIODE


AUDIO


(lOx)


AUDIO
AMPLIFIER
(6x)


Figure 3-1 Schematic diagram of the Continuous Wave Nuclear Magnetic Resonance
(CW NMR) spectrometer utilizing a hybrid tee.







One half of the RF energy resonates the NMR coil at port (b) while the phase of

the other half is shifted by 180 degrees and open to a real resistive load at port (c). The

high sensitivity of the hybrid tee is achieved by opposing the approximately 50 ohm

impedance tank circuit with a resistance (real impedance) of 50 ohms. The recombination

of these two signals with opposite phases generally results in a small signal and, in fact, no

signal if the two loads are perfectly matched. Upon sweeping the magnetic field through

resonance, both the inductance and resistance of the NMR coil are altered very slightly by

the nuclear spins inside the coil. This change in impedance alters the delicate balance (null

status) of the hybrid tee, is picked up by the amplifiers, and sent on to the recording

equipment.

In practice, the hybrid tee is tuned just off of the null to allow a constant carrier

signal (voltage) which establishes a very important background reference. Usually, the

hybrid tee is further tuned to measure the change in circuit resistance rather than the

change in inductance. This has the effect of measuring the absorption signal rather than

the dispersion signal. However, this is merely a convention as the two signals are directly

related through the Kramers-Kronig relations. Some experimenters choose to look at the

signal "in quadrature"; that is, to measure both the in-phase and out-of-phase signal

simultaneously. While this method has the advantage that any error in phase can then be

corrected through the use of the Kramers-Kronig relations after the data has been

collected, such fine tuning of the signal measured in this experiment could not be

performed due to the low signal-to-noise difficulty. In our case, sample runs proved that

the inherent added complexity of quadrature measurements were not worth the extra

efforts since the signal-to-noise ratio was not improved significantly.

The RF amplifier, with an amplification factor of approximately 250, picks up the

imbalance from the fourth side of the hybrid tee. 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 simplification is some








loss of information. The DC signal can not be passed through a mixer to separate the

absorption and dispersion signals. Thus, the tank circuit was often tuned to insure

measurement of only the absorption signal. This magic tee/RF diode method may be

compared with the Q-meter method [Fuk 87] which measures only the absorption signal

but has less sensitivity.

After the RF diode and some amplification, a band-pass filter (10-1000Hz) 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. This very special 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. While the field was ramped over three times

the line width, the audio frequency modulation covered only a small portion (five percent

or less) of the line shape per cycle. Although narrow lines may be measured without a

ramp or a lock-in amplifier, the direct observation of wide line shapes is more complex as

the large inductance of the ramping (modulation) magnet limits the sweep width or time.

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 line shape itself. All line shapes

presented here have been numerically integrated by a computer.


3.2.2 Tuning of the Tank Circuit

To analyze the tank circuit in detail, we follow the treatment given by Abragam

[Abr 61]. The tank circuit, with the sample inside the cylindrical coil, has an inductance L,








a volume Vc = Vs / and a current,
i = I cosC) t = Re(I e "' ) Equation 3-13

which produces an RF field H1(t) = 2 H1 cost inside the coil. The flux across the coil,

() = Li, produces a sample magnetization, M. Following the standard procedure, both the

complex susceptibility, and magnetization can be broken into real, 2 and Mx, and

imaginary part, X and My, respectively, where,
Mx = Re(2 x HI e) Equation 3-14

and the susceptibility is vanishingly small except near the resonance. The flux, (, through

the coil is the product of the induction, B = H -47rM, and the area of the coil, A.






L C


r






Figure 3-2 Model tank circuit used to analyze the real tank circuit used in the NMR
experiment.

A simplified tuned tank circuit may be modeled as shown in Figure 3-2. In this

model, all capacitance, including all stray capacitance, is assumed, for simplicity, to be in

parallel with the inductance. At high frequencies (RF and higher) stray capacitance is

omnipresent and comes from any and all pairs of conductors (e.g., the windings of the coil

or between the coil and sample can). In practice, both parallel and series capacitance are

present and were corrected for through the use of a room temperature tuning box which

contained tunable parallel and series capacitors. Thus, the impedance of the low








temperature portion of the tank circuit was not measured but rather tuned at room

temperature to 50 ohms. The tuning characteristics of the tank circuit and the role of the

two capacitors are illustrated in Figure 3-3.

Low temperature capacitors were not utilized due to the complications in tuning

such an arrangement. However, the improvement in signal-to-noise from using low

temperature capacitors (estimated at a factor of 10) can make the tuning difficulties (and

the associated pain) worthwhile. Such an arrangement is currently being pursued as a

refinement of the experiment.
Using the above model, both the shunt resistance,

R = Lco /Ir Equation 3-15

and the quality factor of the circuit,
Q=coL/r = RcoL Equation 3-16

can then be defined. The quality factor, Q, is defined as the ratio of the energy stored to

the energy dissipated per cycle in the LCR circuit. The queue is usually a number between

50 and 1000 for RF circuits, much less than for microwave circuits. The impedance of this

tank circuit model can then be deduced from the combination of the parallel and series

impedance,
1 1
+ jCco Equation 3-17
Z r+jLwo(l+4 r f-)

At resonance, LCo2 = 1, the impedance reduces to

1+j4nQexQ X -'
Z=R + j4fQ X Equation 3-18
l+4 7r- j/Q

This expression can be further reduced by realizing that the second term in the

above numerator is much less than unity so that the quotient can be expanded by the

binomial theorem. The complex impedance then becomes
Z z R[1- j41rc QX]= R[1- 4- ex QX"-jnr f- QX']. Equation 3-19










V Cp





II
Cs




Figure 3-3 Tuning characteristics of the CW NMR tank circuit showing the effect of both
the series and parallel capacitance.



Great care was taken to reduce the presence of additional hydrogen atoms which
would add to the NMR signal and limit the experimental resolution. However, it was not
possible to remove hydrogen below the precision of this experiment; a background signal
resisted all efforts of removal. In practice, the background signal was measured before the
molecular hydrogen sample was introduced into the sample chamber. Furthermore, during
the course of an experimental run, the spin-one ortho-hydrogen species decays into spin-
zero para-hydrogen. Thus, as the experiment progresses, the NMR signal decays towards
the background signal. When the signal from the hydrogen dosed sample cell returned to
the pre-dosed level, the experimental run was ended and the cell allowed to return to room
temperature to remove all of the residual molecular hydrogen.



3.3 Experimental Procedure


The greatest difficulty of the experiment concerned the low signal-to-noise ratio
inherent in any two-dimensional experiment. Even though excellent books exist on the








subject of low temperature experimental procedures [e.g., White, Pobell, and Lousanama],

this section is included to guide future experiments with similar problems. As has been

often bemoaned, "...the identification of the problem is often one-half of the solution."

Although the previous experiment used the same low temperature rig, the NMR

set-up was not adequate for the significantly lower signal-to-noise ratio of a two-

dimensional experiment. The previous experiments on this rig studied three- and quasi-

three dimensional hydrogen systems and thus required less sensitivity. Therefore, problems

which were present, but not detectable with the lower gain required for the earlier

experiments, were uncovered. Furthermore, the preceding experiments examined the

narrower orientationally-disordered phase and many of the problems which were

uncovered resulted from the larger field sweeps. Fortunately, the ortho- to para-hydrogen

conversion rate was the lowest yet recorded which allowed for longer signal averaging

times.

At temperatures below approximately 1 K, the width of the line shape increased

from approximately 10 to 40 gauss. An ideal magnetic field sweep for CW NMR is two

to four times the width of the line shape. Such a baseline to line shape ratio insures that

both the signal and the baseline are well determined. One difficulty in measuring wide line

shapes is background drift. During the experiment, it was discovered that the RF detector

was sensitive to wide sweeps of the magnetic field; that is, the background would follow

the large sweeps in a complex fashion. The undesired linear (and largest) response was

minimized by averaging the signals from both of the two sweep directions. Fortunately,

we were able to automate this procedure and thereby minimize the linear and other odd

terms of the pick-up response. However, the even terms of the pick-up response were still

both present and noticeable. After much investigation into electrical paths other than the

electromagnetic response of the sample, possible ground loops for example, we decided

that the origin of the sweep pick-up was vibrational.








There exists two possible origins of the vibrations: mechanical and

electromagnetic. Mechanically, the resistive leads to the superconducting modulation

magnet produced significant joule heating for currents greater than 3 amps. This heating

had two effects: an increase in the liquid helium bath boil-off rate and the detuning of the

tank circuit. The rate of the boil-off not only creates white noise but also alters the tuning

of the tank circuit since the impedance of the cryogenic coaxial cable is temperature

sensitive even though the cable resides inside an evacuated tube which is part of the

vacuum can. This case was easily identified by the signature dependence of joule heating

upon the currents in the modulation leads. Joule heating is independent of direction and

proportional to the square of the current, that is, the second harmonic of the sweep rate, a

non-linear response. This theory was extensively tested and, indeed, confirmed by

simulating the effect with a non-inductive (real resistance) heater and measuring both the

pressure of the helium vapor above the bath and the background noise out of the lock-in

amplifier. The old leads were removed and new, less resistive leads were installed. Even

so, the noise was still directly proportional to the level of the helium bath in the cryostat

with the lowest noise level occurring at low helium bath levels. Thus, during data

acquisition the bath level was rarely refilled to capacity but rather kept at a low level near

the top of the main magnet near the quench danger zone!

In terms of electromagnetic forces, Ampere's law (the magnetic force on a wire is

proportional to the current in the wire) must be taken into account. For AC currents, the

forces on the wires are vibrational. The large frequency sweep required a large DC

current through the modulation magnet. Thus, the electromagnetic forces were

proportional to the width of the field sweep. Larger DC currents through the modulation

magnet leads interact more strongly with the constant AC magnetic field causing vibration

of the magnet leads. This effect was minimized by further twisting the pair of leads about

each other and taping all leads to rigid surfaces. Although using twisted pairs is standard

practice in low temperature physics, the sensitivity of the experiment required more than








usual care. In fact, even after the improvements, which reduced the noise floor by more

than a factor often, the NMR rig made a great accelerometer and picked up many noises

near the audio modulation frequency of 111 Hz (a frequency found to minimize this

effect), including voices and pumps. Consequently, the quietest data was taken at night in

the automated mode. A definite improvement in the signal-to-noise ratio of the

experiment would be to either replace the analog lock-in with one based on digital

technology or to replace the analog prefilters with digital filters which have a higher

dynamic range. The effect of these combined difficulties was to limit the ramp width

which resulted in a poorly measured background for the widest NMR line shapes.