Title: Electric-field induced glass phase in molecular solids at low temperatures
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Title: Electric-field induced glass phase in molecular solids at low temperatures
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Language: English
Creator: Pilla, S., 1970-
Publisher: University of Florida
Place of Publication: Gainesville Fla
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Publication Date: 1999
Copyright Date: 1999
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Subject: Glass   ( lcsh )
Physics thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Physics -- UF   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Summary: ABSTRACT: The simple molecular solids of CO and N2 have been the subject of intensive experimental and theoretical studies during the last three decades. The interest in these systems derives from the fact that they serve as prototypes for studying the underlying physics of the simplest glass formers. Dielectric measurements in these systems are particularly important because they directly probe the orientational ordering. Due to strongly hindered molecular rotations in these solids, the relevant time scales for collective molecular reorientation are expected to be in the audio frequency range, but due to the lack of required sensitivity in this range, such measurements have not been carried out in the past. I have developed an AC capacitance bridge with two parts per billion sensitivity in the audio frequency range for measuring the real part of the dielectric constant, which allows us to study the collective orientational properties of these molecular systems. The high sensitivity audio frequency dielectric measurements of solid CO and N2, have revealed unexpected new behaviors for these molecular solids. Strong hysteresis effects have been observed in the high temperature hcp phase of N2 and this is believed to originate from the strong geometrical frustration of the interactions.
Summary: ABSTRACT (cont.): When the N2 samples are annealed in an external ac electric field stronger than ~20 V/m at frequencies ranging from few Hz to few MHz for ~20 hrs. in the 41.5 K < T < 56.5 K temperature range, spontaneous tunneling of the macroscopic polarization states is observed at various random temperatures upon cooling down to 4.2 K, along with strong hysteresis effects. This indicates that N2 has an yet not understood "electric field induced glass phase", whenever the solid samples are annealed as mentioned above. Also, I have observed strong lambda-like behavior of the dielectric constant of solid N2 at ~56 K, which may be interpreted as the thermodynamic glass transition temperature at which N2 undergoes a transition from high temperature orientational liquid (or disordered) phase to field cooled glass phase. Similar experiments on CO show neither the field induced glass phase nor any type of dipolar ordering (neither glass-like nor short range) down to 4.2 K. A simple phenomenological model based on the quantum mechanical treatment of molecular rotation is developed to explain their dielectric behavior. These measurements are extended to N2-Ar mixtures as well, and the traditional glass phase boundary of these mixtures is mapped much more accurately. The field induced glass phase with the associated thermodynamic glass transition temperature is also present up to 50% of Ar concentrations. This indicates that the mechanisms that drive this interesting glass phase are indeed very strong, and new experiments are necessary to understand this peculiar phase.
Thesis: Thesis (Ph. D.)--University of Florida, 1999.
Bibliography: Includes bibliographical references (p. 90-94).
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General Note: Vita.
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ELECTRIC-FIELD INDUCED GLASS PHASE IN MOLECULAR SOLIDS AT
LOW TEMPERATURES












By

S. PILLA













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


1999


























To The Lord















ACKNOWLEDGMENTS


There are a number of people who gave me assistance, guidance, and support

while completing this dissertation; it is my sincere desire to thank them all. I

am most indebted to Professor Neil Sullivan, my research advisor, not only for

his academic guidance and support but most of all for being understanding and

compassionate during critical times in my research life. I express my deepest

gratitude to Professor Khandkar Muttalib for being more than just a co-advisor;

I thank him for being a friend and extending his advice and help on many critical

occasions. Without his perennial optimism and encouragement, this work would

not have been completed. My sincere thanks and high praise go to Professor

Yasu Takano, who served on my supervisory committee, for many enlightening

and useful discussions and for his continued interest in my work. Special gratitude

is extended to Professor Sergei Obukhov and Dr. Bowers Russell for serving as

members of my committee.

In no simple words can I express my deepest gratitude to Dr. Jaha Hamida,

who is my colleague, and the best friend I ever had. She is the most wonderful

person I ever met, and I am very fortunate to work with her. Without her strong

support and help, the difficult work in the lab would have been truly arduous. I

appreciate her love for Indian cuisine and Indian music. She has a special gift of

making every moment a memorable moment; all the conferences I attended along

with her, all the dinner parties we had at her place, are the most memorable.












My sincere thanks go out to the members of the Machine Shop Bill Mal-

phurs, Marc Link, Edward Storch, Theodore Melton, Stephen Griffin, and Robert

Fowler, who helped me in building the Capacitance Bridge even though I had

very poor designs for the parts to be machined. On many an occasion they have

cheered me up with their wonderful sense of humor when I was weighed down

by the experiments. In particular, I thank "Uncle Bob" Robert Fowler for

advising me to get married. I gratefully acknowledge the help of Greg Labbe and

Brian Lothrop for providing me the liquid He on time, even on long week-ends and

lI"li.1.,\ Without the help of the electronics engineers, Larry Phelps and Paul

Strider, this work wouldn't have been complete. I thank them for gold-plating the

capacitor cells, and putting up with all my silly electronics designs. The help of

Dan Roscigno and Chandra ('I. i'reddy, our computer system administrators, are

truly appreciated. Dr. Desirio and Ray Thomas are remembered for lending their

instruments as well as their invaluable help on many occasions. I take this oppor-

tunity to thank the .l] -i, secretaries Connie Philebaum, Joan Raudenbush,

Ed Ricci, Suzan Rizzo, Darline Latimer, and Gloria Armstrong who were most

cheerful in extending their help when ever I was in need. In particular Darline,

Suzan, Connie, and Ed are remembered for their most friendly nature.

Finally, I would be remiss if I did not acknowledge those closest to me, my

family and friends. First, I would like to express my love for my wife Vijaya

and thank her for her support, patience, and understanding, especially during the

long thesis-writing phase, and during long nights I spent collecting data. I cannot

express my sincere gratitude and love for my brother, Ravi Pilla, in simple words.

He has been most influential in my life so far, and his courage, ambition, and












the love for 1l.1-ii .,1 sciences always inspire me. I thank my parents and sisters

who were supportive during the early phase of my research career. My special

gratitude and love goes to my in-laws, Kiran and Subrahmanyam, whose prayers,

love, and affection gave me the strength to complete the work. I truly thank

my friends Kingshuk M.iiniii1.ir, ('li., 1.', Parks, Pietro Stachowiak, Vladimir

Boychev, Garrett Granroth, Lakshmi P. Musunur, and all others for their help.

Finally, I would like to thank Dr. Michael Jones and Dr. Youli Kanev for their

LATEX template which had been most useful in writing this dissertation.

Above all, I thank Almighty God for giving me the peace and joy that surpasses

all human understanding. He is the very reason for my being, and I am grateful

for His abiding presence that helped me tackle every problem in my life with His

Strength and direction, confident that problems are not a sign of His absence, but

only a method of His teaching.














TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . ii

LIST OF FIGURES . . . . . . . . . . . . viii

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

ABSTRACT . . . . . . . . . . . . . .. x

CHAPTERS

1 INTRODUCTION . . . . . . . . . . . . 1
1.1 Intermolecular Potential ........................ 4
1.2 Crystal Structures . . . . . . . . . . . . . 6
1.3 Past Experim ents ... .. .. .. .. ... .. .. .. .. ... .. 6

2 AC CAPACITANCE BRIDGE . . . . . . . . ... 7
2.1 O verview . . . . . . . . . . . . . . . . 7
2.2 Bridge Circuit . . . . . . . . . . . . . . . 10
2.3 Low Capacitance Cable Design .................... 12
2.4 Sample Cell Design ........................... 14

3 AUDIO FREQUENCY DIELECTRIC RESPONSE OF SOLID CO AND
N 2 . . . . . . . . . . . . . ..... . 21
3.1 Overview .. ... ..... .. .. .......... ... .. . ....... 21
3.2 Dielectric Response of Solid CO and N2 ............... 26
3.3 Mean-Field Calculations ........................ 33
3.4 Conclusion . . . . . . . . . . . . . . . . 42

4 ELECTRIC FIELD INDUCED GLASS PHASE IN SOLID N2 . . 44
4.1 Overview . . . . . . . . . . . . . . . . 44
4.2 Experimental Data . . . . . . . . . . . . . 45
4.3 Conclusion . . . . . . . . . . . . . . . . 56

5 DIELECTRIC RESPONSE OF N2-Ar SOLID SOLUTIONS IN THE AU-
DIO FREQUENCY RANGE . . . . . . . . . 58
5.1 O verview . . . . . . . . . . . . . . . . 58
5.2 Experimental Data ........................... 62












5.2.1 Electric Field Induced Glass Phase in N2-30%Ar Samples . 67
5.2.2 Electric Field Induced Glass Phase in N2-Ar Mixtures with
Ar Concentration of 40% and Above . . . . ... 72
5.3 Conclusion . . . . . . . . . . . . . . . . 75

6 CONCLUSION . . . . . . . . . . . . 77

APPENDICES

A NUMERICAL VALUES FOR E(T) OF CO .... . . 80

B NUMERICAL VALUES FOR E(T) OF N2 ... . . 85

REFERENCES . . . . . . . . . . . . . 90

BIOGRAPHICAL SKETCH . . . . . . . . . . 95














LIST OF FIGURES

1.1 Pa3 crystal structure . . . . . . . . . . . .... 2
1.2 P T diagrams for N2 . . . . . . . . . . . ... . 5
1.3 P T diagrams for CO . . . . . . . . . . .... 5

2.1 AC capacitance bridge block diagram .. . . . . . 9
2.2 Equivalent circuit diagram of the ac capacitance bridge . . ... 10
2.3 Low capacitance triaxial cable design .... . . . . 13
2.4 Sample cell design . . . . . . . . . . . . . . 15
2.5 Lock-in output of the ac capacitance bridge . . . . ..... 18
2.6 Background data when both the cells are empty. . . . . ... 19

3.1 Microwave x-band measurements of E(T)' of N2 by Kempinski et al.
[1] . . . . . . . . . . . . . ...... . 22
3.2 Microwave x-band measurements of E(T)' of N2-Ar mixtures by
Kempinski et al. [1] ... . . . . . . . 22
3.3 Audio frequency E(T)' of COO.2-N2 .4-Ar0.4 mixtures by Liu et al.
[2] . . . . . . . . . . . . . . ... . .. . . 24
3.4 Audio frequency E(T)" of CO0.2-N2 .4-Ar0.4 mixtures by Liu et al.
[2] ...... . ............. . . . .... .... . 25
3.5 Dielectric constant of pure CO in the audio frequency range . . 27
3.6 Dielectric constant of pure 14N2 in the audio frequency range . . 28
3.7 Hysteresis curves for E of CO at 1 kHz . . . . . . . . 31
3.8 Variation of the loss angle with temperature for solid CO . . 32
3.9 Various angle parameters used in mean-field calculations . . 34

4.1 E(T) for pure N2 at 100 V/m and 5 kV/m field strengths of the
applied electric field ...... . . . . . . . . 46
4.2 E(T) of pure N2 at 5 kV/m and 1 kHz excitation field . . ... 48
4.3 E(T) of pure N2 annealed at 52 to 55 K for 10 to 13 hrs. at various
field strengths, and cooled immediately . . . . . . . . 51
4.4 E(T) of pure N2 annealed at 41.5 < T < 43 K for 6 to 8 hrs. and
then at 50 < T < 53 K for another 6 to 8 hrs. at various field
strengths, and frequencies ... . . . . . . . . 52
4.5 E(T) of pure N2 near the melting temperature. Electric field strength
is 5 kV/m at 1 kHz . . . . . ...... ... . . 55

5.1 N2-Ar phase diagram . . . . . . . . . . . .... 59












5.2 e(T) of N2-Ar mixtures below 20 K. ....... . . . . . 63
5.3 E(T) of zero field cooled N2-Ar mixtures. . . . . . ...... 64
5.4 N2-Ar phase diagram showing the glass-phase boundary. . . . 66
5.5 e(T) of field cooled N2-30%Ar mixtures. . . . . . . . 68
5.6 e(T) of field cooled N2-30%Ar mixtures near melting temperature. 70
5.7 e(T) of field cooled N2-40%Ar mixtures. . . . . . . . 71
5.8 e(T) of field cooled N2-40%Ar mixtures near melting temperature. 73
5.9 e(T) of field cooled N2-49%Ar mixtures. . . . . . . . 74














LIST OF TABLES


3.1 Values of the molecular parameters for CO and N2 . . . ... 39


A.1 Numerical values for
A.2 Numerical values for
A.3 Numerical values for
A.4 Numerical values for


B.1
B.2
B.3
B.4


Numerical values for
Numerical values for
Numerical values for
Numerical values for


the curves in Fig. 3.5 (0.5 3.0 kHz) .
the curves in Fig. 3.5 (4.0 7.0 kHz) .
the curves in Fig. 3.5 (8.0 12.0 kHz). .
the curves in Fig. 3.5 (14.0 20.0 kHz).

the curve (1) in Fig. 3.6 (0.5 3.0 kHz).
the curve (1) in Fig. 3.6 (4.0 7.0 kHz).
the curve (1) in Fig. 3.6 (8.0 12.0 kHz).
the curve (1) in Fig. 3.6 (14.0 20.0 kHz).















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

ELECTRIC-FIELD INDUCED GLASS PHASE IN MOLECULAR SOLIDS AT
LOW TEMPERATURES

By

S. Pilla

December 1999

Chairman: N. S. Sullivan
IM. i r Department: Physics

The simple molecular solids of CO and N2 have been the subject of intensive

experimental and theoretical studies during the last three decades. The interest in

these systems derives from the fact that they serve as prototypes for studying the

uni'1 ilvi,._ pl' -ii of the simplest glass former. Dielectric measurements in these

systems are particularly important because they directly probe the orientational

ordering. Due to strongly hindered molecular rotations in these solids, the relevant

time scales for collective molecular reorientation are expected to be in the audio

frequency range, but due to the lack of required sensitivity in this range, such

measurements have not been carried out in the past. I have developed an AC

capacitance bridge with two parts per billion sensitivity in the audio frequency

range for measuring the real part of the dielectric constant, which allows us to

study the collective orientational properties of these molecular systems.

The high sensitivity audio frequency dielectric measurements of solid CO and

N2, have revealed unexpected new behaviors for these molecular solids. Strong












hysteresis effects have been observed in the high temperature hcp phase of N2 and

this is believed to originate from the strong geometrical frustration of the interac-

tions. When the N2 samples are annealed in an external ac electric field stronger

than ~20 V/m at frequencies ranging from few Hz to few MHz for ~20 hrs. in the

41.5 K < T < 56.5 K temperature range, spontaneous tunneling of the macroscopic

polarization states is observed at various random temperatures upon cooling down

to 4.2 K, along with strong hysteresis effects. This indicates that N2 has an yet

not understood "electric field induced glass pli,-' when ever the solid samples

are annealed as mentioned above. Also, I have observed strong A-like behavior

of the dielectric constant of solid N2 at ~56 K, which may be interpreted as the

thermodynamic glass transition temperature at which N2 undergoes a transition

from high temperature orientational liquid (or disordered) phase to field cooled

glass phase. Similar experiments on CO show neither the field induced glass phase

nor any type of dipolar ordering (neither glass-like nor short range) down to 4.2 K.

A simple phenomenological model based on the quantum mechanical treatment of

molecular rotation is developed to explain their dielectric behavior.

These measurements are extended to N2-Ar mixtures as well, and the the tra-

ditional glass phase boundary of these mixtures is mapped much more accurately.

The field induced glass phase with the associated thermodynamic glass transition

temperature is also present up to 50% of Ar concentrations. This indicates that

the mechanisms that drive this interesting glass phase are indeed very strong, and

new experiments are necessary to understand this peculiar phase.















CHAPTER 1
INTRODUCTION

Ci.v ,, i .'-1.,' (solidified gases) make up a relatively narrow group of substances

that are gases at room temperature and solids at low temperatures. This group of

solids includes the atomic (i '..' i.. v-1.,l (helium, neon, argon, krypton, and xenon),

the simplest molecular ( i..-1..-1 made up first of all of diatomic molecules (hydro-

gen, nitrogen, carbon monoxide, oxygen, fluorine, chlorine), and then of somewhat

larger molecules (carbon dioxide, dinitrogen oxide, methane, ammonia, and few

others). The study of these simplest molecular solids has great importance in the

present day solid state because they resemble the theoretical solid-state models and

are thus the best testing ground for accurate theoretical treatment. These solids

are ideal objects for studying many fundamental problems of condensed-matter

.l. .- such as phase transitions, isotropic and anisotropic interactions, melting,

glass formation, impurity effects, effect of external stresses such as electric and

magnetic fields, and high pressures. One more reason why ( i v, i v-1. ,l are of fun-

damental interest is the crucial role of quantum effects, which are not only essential

in determining solid-state properties, but are the source of a new crystal state, the

quantum crystal.

The molecular crystals N2, CO, N20O, and CO2 have much in common. All of

these crystals are formed by linear molecules and have the Pa3 structure in the

orientationally ordered low temperature phase. This Pa3 structure unique to these

( i.-.,l1- is a complex structure with four interpenetrating cubic sublattices (Fig.

1.1) and anti-ferro-orientational ordering of the molecules. The anisotropic part of
































Figure 1.1: Pa3 (iv-1.,1 structure


the intermolecular interaction in these crystals is determined by the electrostatic

quadrupole forces as well as by those parts of the valance and dispersion forces

whose angular dependence is similar to that of quadrupoles. Nitrogen, not only

because of being one of the most abundant elements but also because of several

interesting properties, was, and still is, subjected to intense study. Also, as a

classical analog of the simplest molecular solids of H2, D2, and HD; nitrogen, and

nitrogen-argon mixtures have special importance in understanding the molecular

crystals in general.

The interactions in these solids are highly frustrated because the symmetries of

the interactions and that of the lattice geometry are incompatible and as a result

new phenomena occur. The long range ordering is however fragile, and with the

introduction of relatively small amounts of disorder (e.g. by replacing some N2












molecules with non interacting spherical Ar atoms) long range order is lost, and

quadrupolar glass states with randomly frozen orientations are observed. These

systems are believed to be analogous to the spin glasses and are particularly im-

portant because the interactions are short range and well known, and the order

parameters and molecular dynamics can be studied directly. The properties of

these solids were investigated by, among others, raman and infrared spectroscopy,

x-ray, electron-, and neutron-scattering methods, propagation of sound, specific

heat, thermal expansion, and thermal conductivity. However, the molecular ro-

tations in these solids are strongly hindered, and as a consequence the relevant

time scales for collective molecular reorientation are expected to be much larger

than those for an ideal free rotor (~10-12 sec). In fact, glassy behaviors have

been observed for solid N2-Ar mixtures for which the characteristic reorientational

time scale is ~10-4 sec [3]. Measurements in this more interesting audio frequency

range have not been performed due to the lack of required sensitivity. Nary et al.

have carried out dielectric loss measurements of solid CO in the audio frequency

range[4], but to fully ascertain the nature of the dipolar reorientations in the ge-

ometrically frustrated 3 as well as the ordered a-phases, one also needs to study

the dielectric constant in the entire temperature range of the solid phase. More-

over, CO has an intrinsic dipole moment; while this makes it more accessible for

dielectric studies because of the large change in its dielectric loss with temperature

[4, 5, 2], it also makes it more difficult to isolate the much smaller contribution

from the quadrupolar interactions which are responsible for the orientational or-

dering. I have developed a three terminal AC capacitance bridge [6, 7] with two

parts per billion sensitivity for measuring the real part of the dielectric constant












(Chapter 2) which allows us to study the collective orientational properties of solid

quadrupolar molecular systems in the relevant audio frequency range.


1.1 Intermolecular Potential


The fundamental issue in understanding any molecular solid is the form of

the intermolecular potential. Given a proper hamiltonian, one may in principle

compute static properties determined explicitly by the potential such as (iv -1.1

structure, cohesive energy, and P V data as well as lattice frequencies and all

dynamics-dependent properties such as specific heat, thermal expansion, dielectric

constant, etc. However, even for a system as ideal as solid nitrogen, the inter-

molecular potential is not satisfactorily understood. Experimental data exists to

subject any trial potential to critical examination and under this scrutiny all so far

proposed fail in v.ti ivi. degrees. Following preliminary calculations by De Wette

[8] and Jansen and De Wette [9], Kohin [10] performed the first elaborate computa-

tion on the low-temperature low-pressure phase of nitrogen and carbon monoxide

in 1960 and this remained the only theoretical work for almost a decade until fur-

ther experimental research stimulated vigorous theoretical activity [11, 12, 13]. In

Chapter 3 I will discuss more about the inter molecular potential while presenting

my mean-field calculations for calculating the dielectric constant of solid N2 and

CO using Kohin potential.





















































Figure 1.2: P


T diagrams for N2 according Mills et al. (1986) [14]


,. *.
I '!;


......


c.' -.


i.


b ]


I.
'1 Pd


T diagrams for CO according Mills et al. (1986) [14]


Figure 1.3: P












1.2 Crystal Structures


Molecular (i '.., i '..V-..1, are the simplest systems that display phase transitions in

the solid state. Recent investigations at high and superhigh pressures revealed that

these solids exhibit many phases with various orientational structures. The phase

diagrams of N2 and CO shown in Figs. 1.2 and 1.3 are drawn with the data from

x-ray and NMR studies, Raman and IR spectroscopy, P V T measurements,

and numerous theoretical calculations taken into account, an effort of more than

half a century in all. An excellent review of the work is provided by Manzhelii

[15].


1.3 Past Experiments


Nitrogen was first liquefied in the laboratory by Cailletet in December 1877

and solidified by Olszewski in 1883. In 1916 Eucken observed an anomaly in the

heat capacity of the solid which he correctly ascribed to a phase transition (a 3

transition). This transformation, which at equilibrium vapor pressure occurs at

temperature TIa, is one of the salient features of solid nitrogen. Electron diffraction

experiments by Horl (1961), and Venables (1970), x-ray diffraction measurements

by Vegard (1929), Ruhemann (1932) and latter by many more authors, show that

the ( .,-1..1 structure is cubic (a-N2) for T < TO3, and hexagonal (/-N2) for T >

TaO. In 1955 Swenson traced the transition in P T plane and discovered a new

phase (7 N2) at pressures above about 3500 atm (Fig. 1.2). An excellent review

of all the experiments prior to 1975 is given by Scott [16], but the most up-to-date

review can be found in Manzhelii's work published in 1997 [15].















CHAPTER 2
AC CAPACITANCE BRIDGE

2.1 Overview


Low frequency dielectric measurements are very important in the study of cry-

ocrystals and molecular glasses, as studies of the slow relaxation processes in these

systems provide information about their collective interactions. In principle the

dielectric susceptibility can be a very sensitive probe for studying such slow pro-

cesses, while other techniques such as NMR, heat capacity, and thermal conduc-

tivity measurements lack the required sensitivity. In almost all cases, however

the change in the dielectric constant with external influence such as temperature,

DC electric and magnetic fields etc. is very small, requiring a high sensitivity,

high resolution apparatus with long time stability (of the order of few hours) and

good temperature control. The two principal methods for detecting changes in

the dielectric constant are to incorporate a capacitor as a frequency determining

element in an oscillator[17, 18, 19] or to use a capacitance bridge[20, 21] for com-

paring the capacitor with a reference capacitor. An excellent review article by

Adams[6] discusses various techniques involved in achieving the highest sensitivity

in dielectric susceptibility measurements. There has been tremendous improve-

ment in the highest sensitivity achieved over the last two or three decades, though

the technique is fundamentally the same throughout. During the 1970s the highest

sensitivity reported was [22, 23] 10-6 and during the 1980s it was [7] 10-7. With

the help of better lock-in detection as well as signal averaging techniques, a few












low temperature groups [24, 25] have recently reported a sensitivity of 10-8. A

recent progress report by Donnelly's group [26] shows that work is underway to

try to achieve a sensitivity of 10-10 with the help of low temperature preamplifiers

for the study of dielectric behavior of very small samples at mK temperatures.

However, up to the present time the best sensitivity reported is only 10-8. In this

work I report an AC capacitance bridge developed for the dielectric study of bulk

samples at temperatures ranging from liquid He temperatures up to liquid nitrogen

temperatures.

I focus here mainly on the constructional details of the sample cells and low

noise, low capacitance, guarded cables, which we show have been able to enhance

the resolution by at least an order of magnitude to 2 x 10-9 in the real part of the

dielectric constant in the entire temperature range while the loss part has a 2 x 10-7

resolution similar to that reported elsewhere [6, 7]. The novel feature of the bridge

design is the use of a carefully machined dual capacitance cell with cylindrical

capacitors machined into a single block of OFHC copper. One capacitor serves as

the sample cell and the second as a reference cell. With the careful matching of

the capacitances, spurious temperature dependent effects due to thermal expansion

and changes in the electrical conductivity of the cell are annulled automatically to

a high degree. In addition this technique allows us to determine the background

dielectric response to a high accuracy over a wide temperature range (4.2-80 K).

This ability is critical for detecting small dielectric response in field cooling cycles

and non-linear susceptibilities in molecular glasses and related sy-I ,i -[27].




























- - --i - - - -

b ,- c \7 g --
d LOCK-IN

h >




Virgin Ground







Figure 2.1: AC Capacitance Bridge. (a) Ultra low distortion 1 mHz to 200 kHz
function generator (Stanford Research Systems Model DS360), (b) isolation trans-
former (TEGAM Model ST-248A), (c) 0.05 kHz to 20 kHz 7 decade [28] ratio
transformer (TEGAM Model PRT73), (d) 100 Q potentiometer, (e) gain of 1 buffer
amplifier, (f) 11,111.1 2 with 0.01 2/step decade resistor (QuadTech Decade Resis-
tor Model 1433-20), (g) gain of 10 voltage preamplifier (Stanford Research Systems
Model SR550), (h) lock-in amplifier (Stanford Research Systems Model SR510).


















Vi Cx Vo



RO
Rb2R0


x= 0 C5


Figure 2.2: Equivalent Circuit Diagram of the AC Capacitance Bridge. Rt : re-
sistance of the ratio transformer, Rr : Resistance of the tap, Rbl, Rb2 : variable
resistors for balancing the dielectric loss, Ro, R., and Co, C. are inside the cryostat.

2.2 Bridge Circuit


The block diagram and the equivalent circuit diagram are shown in Figs. 2.1

and 2.2. An input voltage V of frequency v is applied across an inductance (ratio

transformer) of 1 H which is in parallel with a reference capacitor Co and an

unknown capacitor Cx. A null reading is obtained at the synchronous detector

such as a lock-in amplifier, by adjusting the tap of the transformer and the decade

resistor RbI. At balance, Vo is zero, and


:, (2.1)
Co 1 x

where x is the transformer setting. The change in the loss angle 6 is given by


A (tan 6) = 2w (Co + C5) AR.


(2.2)












For a cylindrical capacitor of length L, inner conductor radius a, outer conductor

radius b, and filled with a material of dielectric constant e,


S 27eoLe3)
In(b/a)


If both Co and C. are two cylindrical cells made out of the same materials, have

very close geometrical capacitances, are maintained at the same temperature, and

C. is filled with a material of unknown dielectric constant while Co has vacuum,

we have

f (2.4)
1 -Xi

where f is a small correction factor due to differences in the geometrical capacitance

of the cells. This factor can be obtained when both the cells are empty i.e. f =

Xo/(I Xo).

Detailed analysis of the realistic circuit (Fig. 2.1) shows that in situations where

the highest possible resolution is desirable, C3 should be as small as possible and

C4 = C5. The key elements in the circuit are the precision ratio transformer

and the two isolation transformers. For the success of the technique, two coaxial

lines, one for each of the input terminals of the capacitors, neither of which is

grounded, is crucial. In a typical arrangement where both the cells are at low

temperatures, these leads can be as long as 1.5 m. These distributed capacitances

not only influence Eq. (2.1) but also reduce the signal to noise ratio. In general

the changes in the lead capacitances C4, C5, and C3 with the change in liquid He

level in the dewar will lower the long time stability of the bridge. These problems












of distributed capacitances, vibration noise in the leads, condensation of moisture

on the leads outside the dewar are addressed in the present design.


2.3 Low Capacitance Cable Design


As discussed above, the most stringent requirements for achieving very high

sensitivity and excellent stability of the capacitance bridge are, a highly stable

reference capacitor and small lead capacitances. Most of the commercially available

coaxial cables have about 1 pF/cm resulting in ~150 pF for a typical cable 1.5

m long. We designed a low capacitance triaxial cable (C4 and C5) to significantly

reduce these capacitances (Fig. 2.3). By maintaining a good vacuum we achieved a

capacitance of only 25 pF for a 1.5 m long lead. The middle conductor is driven by

a low noise buffer amplifier (gain A = 1) to maintain it's potential close to that of

the inner conductor thereby further reducing the noise induced by this distributed

capacitance. The outer most conductor is grounded so that the load seen by the

buffer amplifier is also stable. The cable used for output lead (i.e., C3 which is

common to both the cells) is also of a similar design except that it does not have a

third conductor and a buffer amplifier because when both cells are balanced, this

lead will be carrying a signal of a few nanovolts only. By using a 12.5 mm inner

diameter stainless steal tube for C3, we achieved a capacitance of 20 pF (including

the connector capacitance) for a 1.5 m long cable. To reduce the noise pick up by

vibrations, a tension of 230 g weight is maintained in the stainless steel wires [29].

A lock-in preamplifier is placed at the top of the dewar where the above lead ends.

"Hand Formable" Cu cables [30] made as short as possible, with insulated Cu shim

foil as the outer conductor in a triax, are used for connecting the input leads to the












Brass B ox -


e d




To Vac. Pump




II



d



j \i h

Figure 2.3: Low Capacitance Triaxial Cable Design. (a) High vacuum BNC post,
(b) Stainless Steal (SS) wire [29] loaded under tension (~230 gwt), (c) small copper
disc soldered to the SS wire, (d) nylon disc, (e) 6 mm ID SS tube, (f) insulated
copper shim foil which is the outer conductor in a triax, (g) very thin (~0.2 mm)
wall copper end cap with the open end soldered to the SS tube, (h) 1 mm diameter
copper post, (i) STYCAST 2850FT seal, (j) small coil of the SS wire to reduce
thermal stress.












buffer amplifiers, isolation transformers etc. Copper braided cables are unsuitable

because one can induce large capacitance changes by accidentally flexing them. By

judiciously placing heat reflectors in the neck of the dewar, moisture condensation

problems were taken care off. We observed no condensation on top of the dewar

even when the cell temperature is as high as 100 K and the dewar is filled to the

brim with liquid He. Vacuum in the lead cables further helped in reducing the

temperature fluctuations due to changes in the liquid He level. All three leads

were connected at the top of the dewar (before they are connected to the vacuum

pump) so that same vacuum is maintained in all three of them even if there were

any leaks in the leads to the gaseous He in the vacuum cans which isolate the cells

from the liquid He bath. To further reduce such effects as building vibrations and

vacuum pump noise, the entire dewar assembly is placed on a 10 ton concrete pad

which is well isolated from the floor.


2.4 Sample Cell Design


Figure 2.4 shows the design used for the capacitance cells (only one cell is

shown). The sample cell and the reference cell are machined out of the same

OFHC block and are gold plated to prevent oxidation. Either cell could be used as

a reference cell and their geometrical capacitances (160 pF) differ by less than 0.1%.

Each cell is a 10 cm long cylindrical shell with 1mm gap, and 2.5 cm inner cylinder

diameter. A ratio of 100 to 1 between length and gap provides good electric field

homogeneity. Outer cylinders form one single copper block so that both cells can

be maintained at the same temperature. In addition this common outer conductor

which usually carries a signal of few nanovolts at null detection, shields the inner


















HI e
f















Or2 \ / Or2
Orl
Figure 2.4: Sample cell design. (a) To low capacitance lead, (b) STYCAST 2850FT
seal, sample fill line, (d) gap > 0.5 mm, (e) Ge thermometer, (f) outer conductor
(gold plated OFHC copper), (g) 1 mm gap, (h) inner conductor (gold plated OFHC
copper), (i) 8 segment wire wound resistor wrapped on a gold plated copper rod,
(j) 0.2 mm thick, 3 mm long lip (Orl, Or2, Or3) three 1 mm thick STYCAST
2850FT O-rings. Or2 is the spacer grounded to within 50 /tm accuracy. Inner
conductor diameter: 2.5 cm, inside length between top and bottom STYCAST
spacers: 10 cm.


Z-- a












conductors which carry large input signals from the nearby ground planes, thereby

reducing the ground noise pickup. These cells are sealed with STYCAST 2850FT

epoxy resin. It is crucial that the copper lip sandwiched between the STYCAST

O-rings be as thin as possible (~0.2 mm) and at least 3-4 mm long so that the lip

can bend as the STYCAST contracts with temperature. It is also important that

an adequate gap (at least 0.5 mm) be left between the innermost O-ring and the

copper block (see figure). We made STYCAST O-rings by pouring fresh STYCAST

with catalyst 24LV into a teflon mold and curing it for more than 24 hrs. The two

O-rings which are used as spacers are in particular very crucial for the homogeneity

(the cylinders need to be concentric) of the electric field. We were able to match

the diameter and thickness (1 mm) to within 50 /tm by using fine emery to grind

the O-rings. The entire cell block is placed on a nylon platform and suspended

from the top flange of the vacuum can with the help of four nylon rods. Care has

been taken to ensure that no part of the STYCAST touches other solid surfaces.

A 1 mm ID, 30% cupro-nickel alloy tube with a wire wound heater around it is

connected to each cell separately for admitting the sample gases. Vacuum jackets

separate these sample fill lines from the liquid He bath. The same STYCAST epoxy

resin is used to seal these tubes at the cell end as well as electrically isolate them

from the cells. To reduce the strain on these seals with thermal contraction, the

tubes are coiled into a two-turn spring before connection to the cells. A separate

heater segment may be used for heating this spring. The capacitor cells designed

as above are tested to withstand pressure differences up to 150 kPa absolute. The

heaters at the center of each cell are divided into 8 segments so that while preparing

the sample one can apply differential heating and cooling from the bottom of the












cell. Apiezon-N grease [31] is used for good thermal contact between heaters and

the cells. These cells are separated from the liquid He bath by two vacuum cans.

An additional wire wound resister on the outer wall of the inner can is used for

temperature control of the entire cell assembly up to 100 K.

A homemade current controller with a thermometer anchored to the inside wall

of the inner can in a negative feedback loop, is used as the temperature controller.

Two additional thermometers located in the wells of the central conductors of the

cells are used to monitor the temperature homogeneity. The heaters in the cells

are unsuitable for sensitive measurements because the leakage currents can induce

large transient electric fields as well as undermine the sensitivity of the instrument.

When not in use these 8 segments are bunched together at the top of the dewar,

disconnected from the power supply, and left to float electrically. A 4He pressure

of a few hundred Pa in the inner can is sufficient to provide the required thermal

contact between the cells and the heater. Using this inner vacuum can with He gas

as the heat exchanger is most crucial for long time temperature stability of such

large capacitor cells. The separation of the cells from the grounded vacuum can

walls is more than 2.5 cm there by reducing C3 (~ 20 pF). With this arrangement,

we are able to screen all external electric field fluctuations and gain long time

-,1,1 iliI v of the instrument.

Figure 2.5 shows the lock-in output at a few representative frequencies when

both cells are empty. The jump in the output voltage (in phase) corresponds to

a change of 1 x 10-7 in x for a 5 V,, input voltage across each cell. The jump in

the out-of-phase output voltage corresponds to a 0.1 Q2 change in Rbl. This data

clearly demonstrates the long time stability and excellent resolution (two parts



















(a)

Input Signal:5V @ 1 kHz
Output: in phase

I I I I I I I I I
2 4 6 8 10 12 14 16 18

(b)


Input Signal' 5 V @ 2 kHz
Output: in phase


0.2
0.1
. 0.0
-0.1
-0.2

0.2
0.1
> 0.0
-0.1
-0.2


0.1

> 0.0

-0.1



0.2
0.0
-0.2
d-0.4
-0.6
-0.8


I I I I I I I I


I I I I I I I


0 2 4 6 8 10 12 14 16 18



Input Signal: 5 V @ 2 kHz
PP
Output: out of phase



0 2 4 6 8 10 12 14 16 18

(d)

Input Signal : 5 V @ 10 kHz
PP
Output: in phase

-I0 2 4 6 10 12 14 16 1
0 2 4 6 8 10 12 14 16 18


t (min)


Figure 2.5: Lock-in output at high dynamic reserve and 0.1 sec. time constant
for various frequencies. For output (a), (b), and the lock-in sensitivity is 500 nV
while 1 jiV for output (d).


I I *















1.00068


1.00066 -


1.00064


1.00062 -


f
S1.00060 -


1.00058 -


1.00056 -


1.00054


1.00052


1.00050 I '
0 10 20 30 40 50 60 70 80
T(K)
Figure 2.6: Background data when both the cells are empty. Excitation potential
is 5 Vpp at 1 kHz. There is no observable hysteresis in the background with thermal
cycling












per billion resolution before signal averaging) of the apparatus. Fig. 2.6 shows the

background data (i.e. the correction factor f) when both cells are empty. Except

for a small kink at -15 K, the background is featureless up to ~80 K. From this

data we note that the cell capacitances are matched to within 0.068%.

In summary, I report the realization of an ultra-high sensitivity susceptometer

based on the well known AC capacitance bridge technique which has a long time

stability of the order of few hours. The resolution achieved was two parts per

billion for the real part of the dielectric constant. This apparatus is most useful

for the study of the orientational behavior of condensable gases such as N2, 02, CO,

C12 etc. at low temperatures ranging from 4.2 to 80 K. With little modification

one may use it at higher temperatures also but with reduced sensitivity.















CHAPTER 3
AUDIO FREQUENCY DIELECTRIC RESPONSE OF SOLID CO AND N2

3.1 Overview


As discussed in Chapter 1, solid nitrogen and solid CO belong to the group of

molecular crystals with linear molecules, which, besides rare-gas solids and that

of H2, D2, and HD, are the simplest (il..-.1, with regard to crystal structure,

lattice dynamics, and other .l r.. -ii ., properties. Also, as a classical analog of the

quantum solids: H2, D2, and HD; nitrogen has special importance in understanding

the molecular (i 1-..-.1 in general. As quadrupole glasses are formed from site

random solid solutions of these simple molecules (like N2, ortho-H2, CN-) in a

host of spherical atoms (Ar, para-H2, Br-, and K+), which are similar to spin

glasses, pyrochlores, spinels, nuclear antiferromagnets etc., the understanding of

these solids is even more important and it is one of the fore front problems in the

present day condensed matter experiments.

Dielectric constant/loss measurements in these systems are particularly im-

portant because they directly probe the orientational ordering. However, only

measurements at microwave frequencies have been performed in the case of solid

N2 and N2-Ar solutions by Kempinski et al. [1] in 1995. Figs. 3.1, 3.2 summarizes

their dielectric constant and loss data. The molecular rotations in these solids

are strongly hindered, and as a consequence the relevant time scales for collective

molecular reorientation are expected to be much larger than those for an ideal free

rotor (~10-12 sec). In fact, glassy behaviors have been observed for solid N2-Ar














1.40


N

Ar /

/1.44
o oo 000






1.42 -_---
20 40 60 80
T.K

Figure 3.1: Real part of permitivity e' versus temperature for pure N2 and Ar with
structural phase transition 3-a for N2 at 34.5 K. Microwave x-band measurements
by Kempinski et al. [1]




1.58


1.54 04


1,50
1.485 -


1.46 0.7
1.441
0 10 20 30 40 50 60 70
T,K

Figure 3.2: Variations of E'(T) in a wide concentration region-no traces of glassy
behavior (lack of cusp maximum) for all concentrations are observed. Microwave
x-band measurements by Kempinski et al. [1]












mixtures for which the characteristic reorientational time scale is ~10-4 sec [3].

Measurements in this more interesting audio frequency range have not been per-

formed due to the lack of required sensitivity. Nary et al. have carried out first

audio frequency dielectric loss measurements of solid N20 and CO [4] in 1982,

but to fully ascertain the nature of the dipolar reorientations in the geometrically

frustrated 3 as well as the ordered a-phases, one also needs to study the dielectric

constant in the entire temperature range of the solid phase. Their work mainly

focused on the molecular reorientation rates in these solids and they concluded

that there was no short-range dipolar order in N2O, but from the observed differ-

ence between the maximum value of the dielectric constant (where the dielectric

loss has a peak) and the theoretical estimate, CO has a significant antiferroelectric

short-range order.

In 1983-84 Liu et al. published their audio frequency dielectric constant/loss

data of N2/Ar/CO and CO/N2 solid solutions [5, 2]. Figs. 3.3, 3.4 show their

dielectric constant and loss data. CO has an intrinsic dipole moment; while this

makes it more accessible for dielectric studies because of the large change in its

dielectric loss with temperature, it also makes it more difficult to isolate the much

smaller contribution from the quadrupolar interactions which are responsible for

the orientational ordering. The conventional wisdom had been that because CO is

very similar to N2 in shape and size, and both interact through electric quadrupole-

quadrupole (EQQ) interactions, its presence will only enhance the dielectric re-

sponse of N2-Ar glass without affecting it otherwise qualitatively. Therefore in the

N2-Ar-CO solid solutions, CO is used as the tracer material to probe the dielectric

response of N2-Ar quadrupolar glasses. With the three terminal AC capacitance

















1600










L580- -



E 0.20 CO
O AO N2
0.40 Ar




L560F- H
o a ..... 50 Hz
S0...... 500 Hz
K....... GK Hz
Pa ------- 5 K Hz


10 20 30
(7K)

Figure 3.3: Real part e' of the dielectric constant of COo.2-N2 0.4-Aro.4 as a func-
tion of temperature at four frequencies. The dielectric dispersion indicates a re-
orientation rate slowing down through audio frequencies. Dielectric constant/loss
measurements by Liu et al. [2]




















0.004 '-




0.20 CO
S04A N
0.40 Ar


0.002 -
S, .- 50 Hz
S--500 H
'. \ t .*- 5 KHz
--,, -20 KHz






0 10 20 30

T(K}
Figure 3.4: Imaginary part E" of the dielectric constant of CO.2-N2 0.4-Ar0.4 as a
function of temperature at four frequencies. Dielectric constant/loss measurements
by Liu et al. [2]












bridge with two parts per billion sensitivity for measuring the real part of the di-

electric constant discussed in Chapter 2, we can study the collective orientational

properties of solid quadrupolar molecular systems in the relevant audio frequency

range with out the use of the dipolar tracers. In the following sections I present

our experimental data followed by the theoretical discussion.


3.2 Dielectric Response of Solid CO and N2


Figures 3.5 and 3.6 show the real part of the dielectric constant e of CO and

N2 as a function of temperature T, in the 0.5 kHz to 20 kHz frequency range.

The sensitivity and reliability of the apparatus can be judged from the observed

jumps in e(T) at the structural phase transition at T,3, which is 61.57 K for CO

and 35.61 K for N2. At this temperature the low T fcc phase, the orientationally

ordered Pa3 structure (or a phase), undergoes a first order transition to a high T

hcp or 3 phase which does not support long range orientational order at any finite

temperature. A second jump in E(T) occurs at the melting transition at T,, which

is at 68.13 K for CO and 63.15 K for N2. Note that while the figures show e up to

three decimal place accuracy, our apparatus has a much higher sensitivity, so the

error bars for e are negligible. However, as shown by the values of To3 and T,,,

the accuracy in the absolute value of temperature is within 0.3 K.

In addition to the expected jumps, we observed several surprising features:

1. For T < 25 K, e(T) for polar CO and non-polar N2 are similar, including the

dip in e(T) near 7.5 K. This shows the absence of any dipolar ordering in

CO down to 4.2 K. Also E(T) for CO did not depend on time (of the order

of several days) which would otherwise indicate a slow thermal relaxation

















- 050 kH

2.00
- 3.00
4.00 "


0
7-
x


0.16



0.14



0.12



0.10


U
\.
%


10 15
T (K)


8s00 | /
0 0
10 0 "
12 0
14 3 "" |
16 0 "I
15, 0 l
20 0 i








I
T .or .




I I



I i


- I !
P T


I I I .
$/
A A


0 10 20 30 40
T (K)


50 60 70


Figure 3.5: Dielectric constant of pure CO in the audio frequency range relative
to E', the value at 4.2 K. E' = 1.4211 at 0.5 kHz. The inset shows the experimental
and the theoretical results for the a phase below the dipolar freezing temperature
Tp in greater detail.


T T

I I I I


Experiment -
Af-






Theory -


-w
w
to


0.08


20 25


0.06



0.04



0.02



0.00



















12.5
4.5 /
o Experiment -
0

10.0 -' /,.



/ Theory
S0 -" ---050 kHz
7.5 00 "" o "
_________ 2.00 "
X 5 1 0 15 20 v 3-00 "
T (K) 400 "
S500 "

5.0 - .00 "
B .00 "
a 9.00 "
10.0 "
12.0 "
2.5 14.0 "
16.0 "
1 B-0 "
.' 20.0 "
S. Theory

0.0 Onset of Order Onset or
Onset of
TI Hysteresis

0 5 10 15 20 25 30 35 40 45 50

T(K)

Figure 3.6: Dielectric constant of pure 14N2 in the audio frequency range relative to
e', the value at 4.2 K on curve (1). e' = 1.4255 at 0.5 kHz. Curve (2) corresponds
to 6e/e' of the same sample heated above T,, and cooled.












towards a dipolar ordered state. On the other hand the dielectric response

differs markedly from the conventional temperature dependence for a non-

polar material described by the Clausius-Mossotti (C-M) equation.

2. For T > 25 K and in the a phase, e(T) for CO is strikingly different from

that of N2. The difference is due primarily to the dipolar contribution from

CO. The sharp rise at Tp for CO can be attributed to the dipolar melting at

Tp, above which dipoles begin to flip. I point out that the results reported

for N2/Ar/CO solid solutions by Liu et al. (Figs. 3.3, 3.4) are qualitatively

indistinguishable from our data for pure CO in this temperature range, ex-

cept for a shift in Tp, where CO was used as a tracer to increase the dielectric

response. This shows the importance of probing molecular quadrupolar sys-

tems without a dipolar tracer.

3. The most surprising feature for N2 occurs in the 3 phase, where it has a highly

anomalous E(T). For thermal cycling below a temperature Th, 42 K, there

is no hysteresis. In this regime, E(T) retraces a unique curve depending on

the initial conditions, including the jump at Ta3. However, if the system is

taken above Th, during a heating cycle (with the AC electric field present), the

observed e(T) follows a different curve during cooling, for which e(T) is higher

than the previous curve. The difference between the two curves depends on

how far above Th, the system was heated. Fig. 3.6 shows a typical hysteresis

curve for a 1 kHz electric field of strength 1 kV/m where the system was

heated to ~48 K along curve (1) and cooled along curve (2). When the

system is heated to a temperature less than 48 K but above Th, and cooled,

the e(T) lies between curves (1) and (2). Once the system is cooled below












Th, the corresponding E(T) curve remains reversible on thermal cycling, as

long as the highest temperature remains below Th. Within 0.1 K accuracy,

To3 remains the same for all curves. In Chapter 4 I will discuss more about

Th, and the effect of the electric field on E(T) of N2. It is interesting to note

that the thermal resistivity data of pure N2 reported by Koloskova et al. in

1973 [32] also has an anomaly close to Th, but no particular attention has

been paid in the literature [15] to this interesting temperature range.

4. If the system is left isolated (in the absence of an AC electric field) at a

temperature above Th for a long time (of the order of several hours), E(T)

retraces the lowest curve, independent of the initial conditions.

5. No hysteresis was observed for CO in the entire a phase (Fig. 3.7). The small

window of temperature between Tp0 and Tm, prevented me from (.,vii ; out

similar measurements in the 3 phase of CO.

6. The variation of the loss angle (A tan 6) for CO are shown in (Fig. 3.8). The

temperatures at which the loss angle has peaks matches very well with Tp,

where varepsilon(T) also has peaks. From Fig. 3.8 it is clear that loss peak

levels off for frequencies above 2 kHz. The loss data for N2 are featureless

for all frequencies and temperatures studied.

7. The frequency dependence of E at low temperatures can be collapsed on a

unique curve when scaled by its value at 4.2 K. This is true for curves (1)

and (2) and all other intermediate curves in Fig. 3.6. In general at 4.2 K, E

increased with increasing frequency for both CO and N2 (-0.06% increase at
























1.65



1.60 ---- Warm Up
SCooling


1.55 -



1.50



1.45 -



1.40 I I I
10 20 30 40 50
T (K)

Figure 3.7: warm-up and cool-down data for e of CO at 1 kHz. No hysteresis can
be observed even at very high resolution and up to 20kHz, indicating that there is
no dipolar ordering in CO.




















0.50 kHz
1.00 ,

3.00
4.00
5.00
6.00

- 8.00
- 9.00
10.0
-s- *12.0
-- -- 14.0 ,
.... 16.0
18.0
20.0


. I I


0 10 20 30 40

T (K)


50 60 70


Figure 3.8: Variation of the loss angle with temperature for solid CO. One can
observe the peak height leveling off for frequencies above 2 kHz.


0.30




0.25




0.20


0.15


0.10




0.05




0.00


,. I












20 kHz from its value at 0.5 kHz), but we do not understand this frequency

dependence. There is no frequency dependence for CO above Tp.

The existence of the strong hysteresis above Th, for pure N2 is unexpected. The

conventional picture [15] is that the 3 phase is an orientationally disordered phase,

which becomes ordered in the a phase with a Pa3 structure. Based on my observa-

tion of the onset of hysteresis at Th, > Tas3, I conjecture that the 3 phase above Th,

has many available configuration states separated by small barriers in the orien-

tational free energy. These states are responsible for the observed hysteresis. On

cooling, orientational order begins to set in at Th,. At this temperature, significant

configuration entropy is lost, and the system is locked in one of the available free

energy minima. On cooling further, partial ordering towards a Pa3 configuration

occurs, and it triggers the structural transition at T,3, where the a phase can in

principle accommodate a complete orientational ordering. As long as the system is

kept below Th,, it does not have access to the other configuration states, and there

is no hysteresis. Once the system is heated above Th,, it can access other configu-

rations and become trapped in a different state upon cooling. The corresponding

e(T) follows a different curve. In Chapter 4 I show that indeed this hysteresis is a

precursor to electric field induced glass phase in nitrogen.


3.3 Mean-Field Calculations


While the 3 phase of N2 is novel, the behavior of e(T) in the a phase is also

unconventional. In order to understand the role of orientational ordering, I have

developed a phenomenological model based on the known temperature dependence

of the orientational order parameter for N2 in the a phase. I find that although













Z A









Figure 3.9: Various angle parameters used in mean-field calculations.


the molecules are large, we need to consider the discrete quantum mechanical

orientational levels to explain the slow increase in e(T) in this regime. In contrast,

the classical treatment gives only a linear temperature dependence. In the present

model, the change in e(T) at TOp to be close to the experimental value requires

50% residual orientational ordering in the j3 phase. This is consistent with our

conjecture that the ordering begins at Th > TOg. The same description holds

also for CO up to the dipolar freezing temperature Tp. Above Tp the dipolar

contribution overwhelms the orientational contribution. The sharp rise at Tp is

associated with the onset of the flipping of the dipoles by 180'. However, the

molecules are not yet free to rotate and are locked in the Pa3 structure. The

subsequent decrease is therefore slower than the characteristic 1/T dependence

expected for free dipoles. I find that a phenomenological model that accounts for

the flipping without free rotation can describe the slower fall off qualitatively and

also predict the jump at Tp. For CO, the drop in e at T0p is consistent with the

change in the order parameter with no residual ordering in the 3 phase. Below I

briefly sketch the phenomenological model for the a phases of CO and N2.












The average potential energy of a diatomic molecule at a site in an a form

crystal can be written as [10]


V = CyP2(cos 0), (3.1)


in which 0 is the angle between the molecular axis (() and the crystal axis (Z)

at the lattice site and P2(cos) = -+ cos2 (0). The quantity y is the average

value of P2(cos 0j) for the neighboring molecules j. C is a function of the molecular

parameters and the lattice constants. It is about -520 K for N2 and -673 K for

CO [10]. The order parameter y is experimentally known for both CO and N2 in

the a phase but little is known about the residual ordering in the 3 phase. If we

write a = ciso+Aa where aiso is the isotropic component and Aac is the anisotropic

component of the electric polarizability of a single molecule in the condensed phase,

the volume polarization can be written as


P = N((a E}} = N((aio E)) + N({{Aa E)), (3.2)


where E = (1 + -Xe)Eext, Xe is the electric susceptibility, N is the number density,

and (( .)) refers to both configurational and thermal average. In the present

experiment at constant volume N is a constant. As shown in Fig. 3.9 if Q2 is the

angle between the external electric field (Ee.t) and Z, the anisotropic contribution

to the single molecule polarization (p) can be written as


p = Aa(1 + Xe)E,,t cos(( 0)(, (3.3)
3











and

p. Er,, = Aa(1 + -)Ee,, cos2( 0). (3.4)
3

From which we obtain


P = oXeEeIt = (1 + Xe)E,,,E. (3.5)
3

where

S= N ((cos2(- 0)))), (3.6)

and
(cos(Q 0) -coCs2( O)e-V/kTdcos O)d(cosQ)
((cos VT kd (cos 0) (cos Q) (3.7)

Using the previously reported values of the order parameter y [11, 12, 13, 15], one

can calculate r and E (= 1 + Xe), using


E = 1+. (3.8)


However these calculated values of E increase linearly with T and do not agree

with the experimental results. Here I would like to discuss briefly about various

theoretical models proposed for the observable dielectric constant E(T) of molecular

crystals formed by H2, D2, N2, CO etc. Several authors have contributed to this

interesting field in the past but I mainly focus on the theoretical discussion of

Heinrichs [33, 34] and Wallace [23]. Heinrichs showed that the classical Lorentz-

field theory is exact in the limit where the correlations due to dipolar fluctuations

are neglected, and he calculated the corrections to the polarizability caused by

quantum dipolar fluctuations. However there does not seem to be a theory that












describes the temperature dependence of e at constant volume. The effect of

nearest-neighbor dipolar fields on the dielectric constant is considered in detail

by Wallace. I have carried out similar calculations for CO and N2 but the effect

turned out to be at least an order of magnitude smaller than the effect of molecular

rotation on the dielectric polarization. Also this effect does not give the correct

temperature dependence in the case of CO and N2 for which E(T) increases with T.

Now let us consider the later effect proposed by Wallace. All the above diatomic

molecules have anisotropic molecular polarizabilities all and aj_ for electric fields

applied parallel and perpendicular to the molecular axis (i). For time averaged

molecular polarizability we have a = lall + -, and for a fixed molecule in an

electric field
1
ac(OE) = c + -6a(3 cos2 OE 1) (3.9)
3

where 6a = all a, aiso = ac, and DOE = Q 0. If no thermal averaging is

required, one can separate the variables as


(3cos2E 1) = (3 cos2 1)(3 cos2 0- 1). (3.10)


It is clear that ((3 cos2 0 1)) is nothing but the long-range order parameter and

in the limit where the molecular rotation is free, this average goes to zero. Also as

Wallace considers it, the average ((3 cos2 Q 1)) = 0 for the particular symmetry

of the quadrupolar solids i.e., for Pa3 ordered solids. This implies that in the

orientationally ordered a-phase the contribution from the molecular rotation is

effectively zero. But in the disordered 3-phase where the rotation is not completely











free, the contribution from ((3cos2 Q- 1)(3cos20 1)) may be significant. And

for powdered samples this contribution would also be zero.

However, one serious objection to the above discussion is that even in the

ordered a-phase of such large molecules as CO and N2, where quantum correlations

can be very strong, one cannot separate the variables as suggested by Wallace. i.e.


((3cosOE 1)) # ((3 cos 1))((3cos 0- 1)). (3.11)


Below I show that when we consider the above averaging with out the separation

of variables along with the thermal averaging over the discrete quantum mechan-

ical orientational levels indeed gives the correct temperature dependence of the

dielectric constant. Table 3.1 summarizes all the important parameters of CO and

N2 used in my calculations.

When the separation of the .,,i.,'.ent rotational energy levels of a rigid rotor

in the potential Eq. (1) are larger than kT, the classical Boltzman distribution

which we employed here may not yield good results. In fact the separation of the

two lowest energy levels (AE/k) is ~ 104 K for CO and ~ 93 K for N2 (setting U

= 1). As suggested in [10] if the potential in Eq. 1 represents the field in which a

diatomic molecule rotates, the Schr6dinger equation for such rigid rotor is

-h2 n9 1 i 2
As s1in 00 + 7 +CbP2(cos0)= E. (3.12)
2A sin0 0\ o ) sin 0 002














Table 3.1: Values of the molecular parameters for CO and N2


Quantity Symbol Units N2 Value CO value
Dipole moment [35, 36] p C m ... 4.10 x 10-31
Quadrupole moment [37, 38, 39] Q C m2 4.89034 x 10-40 8.33 x 10-40
Isotropic polarizability [40, 41] aiso C m2 j-1 1.68 x 10-40 1.86 x 10-40
Anisotropic polarizability [40, 41] Aa C2 m2 j-1 7.75 x 10-41 5.92 x 10-41
Number density [42, 43, 15] N m-3 2.219 x 2.23 x 1028
Nearest-neighbor distance [10] Ro A 4.00 3.98
Nuclear Separation [10] 2d A 1.104 1.128
Moment of Inertia [10] A kg m2 1.417 x 1.479 X 10-46
a-f3 transition temperature [15] T 3p K 35.61 61.57
melting temperature [15] Tm K 63.15 68.13
a crystal field constant [10] C/k K -520 -673
a lattice constant [10] a A 5.667 5.64
a crystal structure [15] Pa3 Pa3
/3 crystal structure [15] P63/mmc P63/mmc
Latent heat of transition [10] AHa_3 J/mol 228.9 633.0


Here A is the moment of inertia of the

q (0, O ) = Q(0)4() yields


rigid rotor. Separation of variables as


d- 2) dQ- 2 - 2 n L 2) =- 0,


(3.13)


where ) = cos 0, 1h2 = ACy + 2AE, and Ah2 = 3ACy. Eq. (10) is a spheroidal

wave equation and the eigensolutions Qm,,(A, w)) are oblate spheroidal wave func-

tions. A is ~ -275 K for N2 and ~ -370 K for CO. For such large values of A, an

expansion of the eigenfunctions in terms of Laguerre polynomials is appropriate.

The eigenfunctions Qm (A, ,) and the eigenvalues Ptim(A, wc) for several of the low-

est rotational states have been calculated using the method of Flammer[44]. For

various (y, T) sets, ((cos2(Q 0))) is calculated first by determining Qim and tml,,











for several of the lowest eigenstates and the expectation values of (cos2( 0)),m
for each of these states. Then

Z ^3(COS(Q_ o)>-E3/kT
((cos 2( 0))) = Yj(Cos2( -j (3.14)


where YZj jejxp(-Ej/kT) is the partition function, 7j is the degeneracy of the

jth state and Ej = -p 4j A-V). The E calculated for N2 using Eqs. (2,4) for

various temperatures, is shown as the dashed curve in Fig. 3.6. This is in better

qualitative agreement with the experiment. In order to obtain the correct jump at

T/O, we need to set y = 0.5 in the 3 phase. The same a, .l-i has been carried

out for CO with the known order parameter, and the resulting E(T) is shown as

the dashed curve in Fig. 3.5(inset). Again the agreement with the experiment is

qualitatively better. We mention two weaknesses of the above model. First, both

N2 and CO show a small dip in E(T) near 7.5 K which remains unaccounted for

in the present model. Second, if one takes the a1go for a free gaseous molecule

[45, 35, 37, 40, 46, 47, 48, 49], and number density N calculated from the lattice

constants [42, 43], the experimental value of the dielectric constant at 4.2 K and

0.5 kHz is about 15% and 11% lower than the calculated value in the case of

CO and 14N2 respectively. Note that in Figs. 3.5 and 3.6 we have taken the

dielectric constant at 4.2 K (E') as the only adjustable unknown fitting parameter

to compare theory with the experiment. There is no further fitting parameter for

the temperature dependence up to TOp for N2 and CO as well.

In the case of CO above Tp, the dipoles are free to flip by 180 and in this

regime the dipolar contribution to the dielectric constant should also be taken into












consideration. Because the molecules are not yet free to rotate and are locked in

the Pa3 structure, in the presence of an external electric field the Hamiltonian may

be written as

H = C P2(cos 0) o|E| f(1 ) cos(Q 0). (3.15)

The first part is the average potential energy of a single molecule as in Eq. (1). E is

the internal electric field, [-o is the intrinsic dipole moment, and f is some function

of the order parameter. Because the molecules are orientationally ordered, the

polarization per unit cell will be modified by the order parameter. Qualitatively

when y is unity (i.e. the molecules are completely ordered) the second term should

be zero. When y is zero (i.e. the molecules are free to rotate) it should simply be

*Io E. In general f can be of the form (1 4)'. Then the volume polarization

will be
P = N((pmo)) = N ,moiE,

f J'of cos(Q O)e-H/kTd(cos Q)d(cos ) (3.16)
J'e-H/kTd(cos )d(cos 0)
This integral can be solved easily for small values of Ee.t by expanding it in Taylor

series and keeping only the first few terms. Now from the C-M equation (E -

1)/(E + 2) = N,,mo/3Eo, one can calculate the static dielectric constant. To this

one needs to add the polarizability contribution also. Using the same value of

the polarizability as used at low T, we find that for f(1 y) = (1 y)1/' the

experimental data can be fit very well up to TaO (the dotted curve in Fig. 3.5). Here

also the the theory correctly predicts the change in E(T) at T"O as well as the sharp

rise at Tp. In particular the predicted change in E(T) at Tp is in excellent agreement

with the extrapolated value of the DC dielectric constant from experimental results












within 0.8%. I have no explanation for the particular exponent in (1- y)1'8, but the

qualitative decrease which is slower than the characteristic 1/T dependence of free

dipoles can be understood within our model. It is generally believed that the 15%

lower experimental value for the polarizability compared to that of a free gaseous

molecule is an indication of antiferroelectric ordering below freezing temperature

for CO [4]. However, I observed no hysteresis in the entire temperature range.

Instead, we can understand the lower value due to the following reasons. Firstly,

due to the hindered motion of the molecules in the a-phase, the polarizability in

the condensed phase will be lower than that for a free molecule. This is consistent

with the fact that there is also ~ 11% reduction in the experimental value for N2.

Secondly the number density N for powdered samples should be smaller than the

one we used here, which is calculated from the lattice constants [42, 43].


3.4 Conclusion


In summary, I have made the first measurements of the dielectric constant

of pure CO and N2 in the audio frequency range and observed many anomalous

features. The low temperature behavior of both solids are very similar, but they

are nevertheless very distinct from the conventional non-polar materials and can

be qualitatively understood in terms of a phenomenological quantum mechanical

model. The small dip in E(T) as well as the small frequency dependence of E' are not

well understood. Further theoretical calculations taking into account the quantum

dipole fluctuations as suggested by Heinrichs may be necessary to understand the

above disparities. A comparison of Figs. 3.3 and 3.4 with Figs. 3.5 and 3.8 shows

that my data are very similar to Liu's data except for a shift in Tp. This strongly












indicates that in CO/N2/Ar samples, most of the dielectric response is due to CO

and adding CO to N2/Ar solid solutions does not reveal the uni'1 ilvi,_- behavior of

the quadrupolar glass. In Chapter 5, I will discuss more on the dielectric behavior

of N2/Ar solid solutions with out using the dipolar tracers. Also, comparison of

Figs. 3.1 and 3.6 shows that, as pointed out in the Overview, the interesting

time scales for the quadrupolar molecular solids are in the audio frequency range.

Further work I have carried out (Chapter 4) on the dielectric response of solid N2

strengthens our conjecture.

The strong hysteresis effects in the /-phase of pure N2 show that the conven-

tional picture of the onset of ordering in these frustrated systems must be modified.

The results are consistent with a departure from ergodicity due to trapping in a

limited region of configuration space. As a consequence of the frustration of the in-

teractions, the configuration space has a rugged landscape with many quasi-equal

low energy minima separated by potential energy barriers. This phenomenon is

common to a wide class of glass former and other frustrated systems with the in-

troduction of disorder. The most significant result of this study is that the glassy

effects are produced in a purely geometrically frustrated system without the intro-

duction of disorder. In Chapter 4 I will discuss the effect of electric field on the

dielectric response of solid N2 and show that the above hysteresis is a precursor to

the electric field induced glass phase, which has not been observed before in any

of the quadrupolar solids.















CHAPTER 4
ELECTRIC FIELD INDUCED GLASS PHASE IN SOLID N2

4.1 Overview


In Chapter 3 I have discussed at length the presence of unexpected hysteresis

in E(T) of pure solid nitrogen whenever we heated the sample above T,, in the

presence of an external uniform electric field. It is interesting to note that ex-

cept in Koloskova's work in 1973 [32], no where else in the literature we find any

reference to this particular temperature. Though it is generally believed that the

orientational ordering in N2 begins at a temperature above Ta [15], no experiment

could pin point the temperature at which the ordering begins. It is theoretically

proved that the high temperature hcp phase cannot support any long range order

but there can still be some short range local ordering. One reason for this lack

of experimental evidence may be that the Q-phase of nitrogen is very complicated

and the experimental sensitivity is poor in the case of heat capacity, thermal con-

ductivity measurements etc. Unfortunately, x-ray and neutron diffraction as well

as NMR measurements fail to probe the orientational nature of the molecules at

these high temperatures. As I pointed it out before, Th, may be the temperature

at which the short range order sets in. Then the next logical step is to explore the

temperature region around Th, much more closely and see if any new results come

up. In the following sections I present the strange results I have obtained for the

e(T) of nitrogen, not only at T,, but all the way up to T,,. To my knowledge it












is for the first time that we are exploring the highly complicated /-phase of pure

nitrogen in great detail.



4.2 Experimental Data


In Fig. 4.1, I have presented the E(T) of pure solid N2 obtained with 100

V/m, and 5 kV/m excitation fields. From here after, all the electric field strengths

referred to are 1kHz peak-to-peak values, unless otherwise specified. In this graph

one can observe sharp kink near Th,, which was absent in the previous Fig. 3.6.

Only when we anneal the sample at 41.5 < T < 43 K for 6 to 8 hrs., we observe this

sharp kink but when the sample was heated past this window of temperature rather

fast (say, in 0.5 hrs.) as in Fig. 3.6, the kink could not be observed, suggesting that

the thermal processes in this window of temperature are rather slow. This could

be one of the reasons why the previous experiments could not detect anything in

particular at this temperature. I would like to emphasize here that due to the

geometry of the cell, the sample reached the equilibrium temperature in less than

5 min. whenever the thermal bath temperature was changed. Also, because He

gas was used for heat exchange and no other metal objects are in contact with the

cell, the sample cell temperature is homogeneous throughout its volume. I have

observed that the kink becomes sharper and the window of temperature narrower

as we increase the field strength. This particular kink is present at all audio

frequencies. For Th < T < 50 K, the E(T) is linear in T and for different runs of

the experiment the slopes are slightly different but nevertheless linear. Both the

lower curves in Fig. 4.1 show the warm-up data. When the sample was heated

past Th, and up to 50 K, the cool-down curves in Fig. 4.2, 3.6 follow different paths,




















20



15



0 10 100 V/m 1 kHz w n
5 kV/m w
v 100 V/m c
"CJI "




0 T T
].ap h

0 10 20 30 40 50 60
T (K)
Figure 4.1: e(T) for pure N2 at 100 V/m and 5 kV/m field strengths of the applied
electric field. See text for details.












and even if we anneal the sample for 6 to 8 hrs. at T,, during cooling, the kink

is absent. That is, the kink is present only during the warm-up cycle. Fig. 4.2

shows a family of cool-down curves (curves 2, 3, 4) which were obtained by field-

cooling the previously zero field-cooled sample from various temperatures between

Th, and 50 K. These curves show the progressive increment in the hysteresis (or

higher polarization) with the maximum temperature from which the sample was

cooled. Once the sample is field-cooled (say, curve 2) and warmed-up again to

temperatures above Th,, the kink we observed for curve 1 (the zero field-cooled

sample) at T,, could not be observed any longer, irrespective of the annealing time

at Th,.

As the curves in Fig. 4.1, 4.2 show, when the temperature is raised further,

up to ~ 55 K, and the sample is annealed once again at 50 < T < 53 K for 10

to 12 hrs., we observe that e(T) increases sharply reaching a maximum at ~ 52

K and then decrease slowly. For 5 kV/m excitation field, there is a kink at ~ 52

K, but for 100 V/m excitation field e(T) reaches a plateau at 52 K. When such a

sample is cooled-down, as the second family of curves in Fig. 4.2 and the upper

curve in Fig. 4.1 show, something most remarkable happens. The warm-up and

cool-down curves are no longer similar even in the a-phase and are very different

from the zero field cooled (or other hysteresis curves) curve. For 4.2 < T < 30 K,

the e(T) decreases linearly with T during warm-up cycle, while it increases linearly

with T during cool-down cycle. Also, I have observed spontaneous tunneling of

polarization states at various random temperatures, indicated by arrows. We can

summarize the experimental facts as follows:

1. Spontaneous tunneling is present only during the cool-down cycle.

































I


AD

A


A'


I QA


- E~OO 0
2~


A


T T
. I I .


10 20


30
T (K)


Figure 4.2: E(T) of pure N2 at 5 kV/m and 1 kHz excitation field. The data clearly
shows gradual progression from completely ordered a-phase to yet unknown electric
field induced glass phase. One can observe spontaneous tunnelling at various
temperatures in the field cooled sample. See text for details.


**--


/0


I I


I I I I I I I I I I I












2. Tunneling is present in both a as well as /-phases.

3. The temperatures at which the tunneling takes place is random.

4. In general for field-cooled samples (upper family of curves in Fig. 4.2), the

change in E at TO is smaller than that for hysteresis curves (lower family of

curves in Fig. 4.2).

5. The TO3 is the same for all samples to with in 0.1 K.

6. Though I observed no kink at Th, in the cool-down lower family of curves in

Fig. 4.2, there is a kink at Th, in the upper family of curves.

7. For 30 K < T < TO3, in the upper family of curves in Fig. 4.2 during

all warm-up cycles, E(T) shows sharp rise with temperature. At large field

strengths (for 5 kV), this change is rather abrupt with strong tendency to

tunnel to higher polarization states, but for small field strengths (20 V/m),

this rise is rather smooth (Fig. 4.4).

8. In the upper family of curves in Fig. 4.2, even if we warm-up the sample

from 4.2 K to just above T,,3 and then cool it down, the warm-up and cool-

down curves are very different, and one could observe spontaneous tunneling

while cooling. Experimentally, I observed sharp change in x value (ratio

transformer setting) in a fraction of a second, even though the rate of change

of temperature is very slow (AT/At < 0.05 K/min). In comparison, the time

required for ca-3 transition is about 10-15 min. Also, I observed that time

required for a-3 transition in the upper curves is relatively longer, about

20-30 min.












9. Though the e shows a decrease (4.2 K value) with thermal cycling in the

upper family of curves in Fig. 4.2, when we anneal the sample once again

at Th, < T < 53 K, the e(T) goes up giving us a similar set of curves. In a

sense, we some how boost the polarization by annealing the sample in the

above temperature range in the presence of the external field.

10. To check the significance of annealing at Th,, I have warmed up a zero-field

cooled sample all the way up to 52-55 K without spending much time at

Th, (< 10 min. including cool-down time), and annealed at these elevated

temperatures for 12 hrs. in the presence of the external excitation field. After

cooling the sample to a-phase (in a couple of hours), I have obtained a-phase

e(T) (Fig. 4.3). These curves show no hysteresis or field cooling effects. Even

if I anneal the sample at 45 K for 10-12 hrs. in the presence of the field, and

cool it down without spending much time at Th,, there was neither hysteresis

nor field-cooling effect (Fig. 4.3). This strongly indicates that the thermal

processes at Th, are most important for the presence of hysteresis as well as

field cooling effects which include spontaneous macroscopic tunnelling of the

polarization state of the sample.

11. Also, as the lower family of curves in Fig. 4.2 show, annealing at Th, produces

only the hysteresis effect, but not the complete field-cooling effects shown by

the upper family of curves in Fig. 4.2. In a sense, hysteresis is only a

precursor to the field induced glass phase in solid nitrogen. To reach this

new phase we need not only anneal the sample at Th, for 4-6 hrs, but also

anneal at temperatures above 50-53 K for 10-13 hrs.









51



4



0 ---OV/m @ 52 K
1 V/m @ 52 K {
3 10 V/m @ 52 K
S50 V/m @52K
100 V/m @52 K
1 kV/m @ 52 K
C1 kV/m @ 45 K
2 2



1 r








0-
I I I I I I





0 5 10 15 20 25 30 35 40

T(K)



Figure 4.3: E(T) of pure N2 annealed at 52 to 55 K for 10 to 13 hrs. at various
field strengths, and cooled immediately. Excitation frequency is 1 KHz for all field
strengths. All the curves show warm-up data but the cool-down data in a-phase
are also very similar. See text for details.









52











4-

-o 0 V/m
--- 5.2 kV/m @ 90 Vl-lz
1 kV/m dc
C 2 2V/m@ 1kHz
0 --...20V/m@1kHz/
S5kV/m @ 1kH-z '
X






-1 -


0 5 10 15 20 25 30 35
T (K)

Figure 4.4: E(T) of pure N2 annealed at 41.5 < T < 43 K for 6 to 8 hrs. and
then at 50 < T < 53 K for another 6 to 8 hrs. at various field strengths, and
frequencies. After annealing the samples are cooled slowly to 4.2 K over next 6 to
8hrs. All the curves show warm-up data only. See text for details.












12. The next step is to find out the excitation field strengths for which solid nitro-

gen shows this remarkable field induced glassy behavior. Fig. 4.4 summarizes

the E(T) in the a-phase of nitrogen, field cooled at various frequencies and

field strengths. All the samples were initially zero field cooled, then care-

fully annealed at 41.5 < T < 43 K for 6 to 8 hrs., and then annealed at

50 < T < 53 K for 10 to 12 hrs. in the respective external fields. After cool-

ing the samples all the way to 4.2 K, a standard 200 V/m excitation field at

1kHz was applied to obtain the warm-up data in the a-phase. This particular

sequence is followed because the AC Capacitance bridge discussed in Chap-

ter 2 is sensitive enough only for excitation fields stronger than 200 V/m,

and in the audio frequency range. Also, as I pointed it out before, once the

sample temperature is below Th,, the excitation field has no observable effect

on the glassy behavior of the samples. I have observed strong field induced

glassy behavior for external electric fields stronger than ~20 V/m (peak-to-

peak). The warm-up a-phase E(T) curves of the samples field cooled in fields

stronger than 20 V/m are very similar as shown in Fig. 4.4. Although E(T)

curve of the sample field cooled in 2 V/m in Fig. 4.4 is different form that

of the zero field cooled sample, and it shows reduced hysteresis (see table),

I observed no spontaneous tunneling for this sample and the E(T) curve is

very different from that for samples field cooled in 20 V/m and above. This

suggests that the threshold field strength above which we observe the glassy

behavior is some where between 2 V/m and 20 V/m. It is interesting to

note that for dc field cooled sample, though we observe hysteresis in E(T)

(see table), we do not observe spontaneous tunneling and linear temperature












dependence at these temperatures, which characterize the complete glassy

nature of the field cooled samples. However, dc field cooled sample exhibits

a slightly different E(T) behavior than that of the zero field cooled sample

(Fig. 4.4), and it's E(T) curve is similar to that of 2 V/m field cooled sample.

To find out the upper limit of the frequency of the electric field for which we

observe the glassy behavior, I field cooled the sample in 5.2 kV/m, 90 MHz

external uniform field, and the observed E(T) curve is shown in Fig. 4.4. I

observed no spontaneous tunneling for this sample, but the hysteresis is still

present and the E(T) curve is closer to that of the zero field cooled sample.

This indicates that the glass states in the solid nitrogen can be excited only

at low frequencies extending up to few MHz. This explains why previous

microwave measurements by Kempinski et al. [1] did not reveal any new

behaviors.

13. I would like to emphasize here that the hysteresis values shown in the various

tables and plots are accurate to within 1%, and for zero field cooled samples,

I observed no hysteresis or shift in the absolute value of E at 4.2 K even

after annealing the sample for long time at 50 to 55 K. This shows that

simple lattice dislocations, voids or vacancies can not explain the presence of

hysteresis in solid nitrogen. The external electric field indeed is responsible

for the not yet understood hysteresis and glassy behavior of solid nitrogen.

14. So far I have considered only the dielectric behavior of nitrogen for temper-

atures below 55 K while the melting temperature is at 63.15 K. The most

surprising behavior of E of N2, which is crucial for understanding all the new

unexpected behaviors discussed above, is observed in the 53 < T < 63 K




























0

I-~


0



-10



35


40 45 50 55
T(K)


T
I h


I


M
. .I


T (K)


*(T) of pure N2 near the melting temperature. Electric field strength


is 5 kV/m at 1 kHz. See text for details.


Figure 4.5:












temperature range (Fig. 4.5). As the temperature of the field cooled sample

was raised slowly to ~57 K, the E shows a massive, spontaneous change at

Tg, but quickly drops with increasing temperature. The A-like behavior at

this temperature, with massive change in E (~ 2.5 x 10-2) which is twice as

large as the maximum shift in E at 4.2 K for field cooled samples, and further

linear decrease of E with temperature up to Tm, shows that at Tg 57 K the

/-phase nitrogen undergoes rotational melting. For T < Tg, in the presence

of the external electric field, the /-phase nitrogen shows remarkable glassy

behavior, while for T > Tg, where it is a rotational liquid, the lattice dislo-

cations, vacancies etc. may be predominant. However, in Chapter 5 we will

see that at these high temperatures close to the lattice melting, the dielec-

tric constant of N2, and N2-Ar mixtures is highly reproducible with thermal

cycling (Tg < T < Tm), strongly indicating that lattice defects (such as

vacancies and dislocations) do not have an effect on their dielectric behavior.


4.3 Conclusion


From the above observations, I conjecture that, when solid nitrogen is annealed

for extended time (of the order of 20 hrs.) in 41.5 < T < 55 K temperature range

in the presence of an external electric field stronger than 20 V/m and at frequen-

cies ranging from few Hz to few MHz, we observe a new glass phase extending

the entire temperature range of the solid phase. The strong hysteresis observed

when solid nitrogen is annealed in the 41.5 < T < 43 K temperature range only,

is a precursor to this as yet not understood glass phase. Simple lattice disloca-









57

tions or vacancies may not be the mechanism which explains this interesting but

nevertheless complicated behavior of nitrogen.















CHAPTER 5
DIELECTRIC RESPONSE OF N2-Ar SOLID SOLUTIONS IN THE AUDIO
FREQUENCY RANGE

5.1 Overview


While ordinary glasses represent one of the oldest materials known to man, a

comprehensive understanding of the formation of glasses has not yet been achieved.

For this reason, interest has become focused on conceptually simpler systems which

are in many ways closely related to the general problem of glasses. The study of the

so-called spin glasses has brought out the key role of the interplay between frustra-

tion and disorder in these compounds. Dilute systems of interacting quadrupoles,

such as ortho-para H2, para-ortho D2, N2-Ar mixtures, and KCN-KBr solid so-

lutions, which incorporate in a simple way both frustration and disorder, have

therefore received a continuously growing interest. Frustration in these systems

arises from the topological impossibility of ensuring the minimum possible energy

for all pairs of neighboring quadrupoles. In the case of N2-Ar mixtures, the in-

teractions are short ranged and well characterized, and due to their large Il-. -ii .,i

size can be treated classically.

The first phase diagram for (N2) -1-Arr, was constructed by Barrett and A, I' r

in 1965 [50] (Fig. 5.1) using x-ray diffraction. These solid solutions undergo

phase transitions to a long range periodic (Pa3, Fig. 1.1) orientationally ordered

phase with a cubic lattice structure for T < 35.6 K. THis long range order is

however fragile with respect to disorder, and for N2-Ar mixtures with x(Ar) > 23%,








59









90- I I I

N Liquid
70- Niquidt Soic-d' -



S50- "
nr DV-N2
D, \


I 30-
f.c.c. h.c.p. f.c.c.

10 A ,i I i -
II IB. I
Ar 20 40 60 80 N2
MOLE PERCENT NITROGEN
Figure 5.1: N2-Ar phase diagram












quadrupolar glass states have been observed [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 1,

2]. In this glass phase, both the principal axes for the molecular quadrupoles and

and the local order parameters evaluated with respect to those axes vary at random

throughout the sample. The quadrupolar glasses have only been observed when

the lattice structure is hcp. In view of the geometrical nature of the frustration

it is important to explore the effect of lattice geometry on the local ordering.

The N2-Ar solid solutions offer a unique opportunity for studying the effect of

lattice geometry on the ordering because both hcp and fcc structures exist for

high substitutional disorder, namely fcc for x(N2) < 57% and hcp for 57 < x(N2)

< 77% (Fig. 5.1). In 1994 Hamida etal. [3, 60] studied samples with x(N2) =

50% because this concentration is sufficiently close to the fcc/hcp phase boundary

that the effects of quenched disorder should be comparable for the two structures.

Their NMR results have shown that contrary to expectations, the orientational

order parameter has a very sharp distribution of values at 50% concentration. In

this new orientational glass state (hashed boundary in Fig. 5.1) the molecular axes

are distributed at random but the mean order parameters evaluated with respect

to those local axes have only a very narrow distribution.

From Fig. 5.1 it is clear that for high concentrations of x(N2) > 77%, the

hcp structure transforms at low temperatures to a cubic, orientationally ordered

structure (Pa3). For x(N2) < 77% this transformation does not occur at all;

it appears possible to pass continuously from the hot, orientationally disordered

solid to zero temperature. A continuous freezing of molecular orientations occurs

instead of the sharp phase transition, giving rise to a ,i qi.i.lrupolar I.'1.-- at low

temperatures. NMR of 15N2 in N2-Ar observed a continuous orientational ordering












[51] from 20 to 10 K for x(N2) = 67%. No slow rotations were found with spin echos

and stimulated echos [51]. Neutron scattering on z(N2) = 72% found only short

range orientational order at low temperatures [54]. The disorder was found to be

dynamic at high temperatures and "frozen-in" at lower temperatures. Specific heat

measurements [53] did not find any discontinuous behavior, nor did they observe

any long thermal relaxations, such as occur in ordinary glasses.

As discussed in Chapter 1 and 3, dielectric spectroscopy can provide a determi-

nation of molecular reorientation rates in polar or non-polar systems. Due to the

lack of required sensitivity of the apparatus in the past, no dielectric susceptibility

experiments were carried out for pure N2-Ar solid solutions. CO with its small

intrinsic dipole moment is better suited for these studies but the phase separation

of CO-Ar solutions prevent one from studying their dielectric behavior systemat-

ically. However, the dielectric spectroscopy of CO/N2/Ar solutions [5, 2] showed

that the reorientation rate of CO in these solid solutions slowed down through

audio frequencies from 15 K to below 4 K. From the temperature dependence of

the mean CO reorientation rate, Liu et al. [2] concluded that co-operative freezing

processes are not involved in slowing of the reorientations, but whether this kinetic

effect is intimately related to the formation of the quadrupolar glass is not known.

No cusp maxima were observed in dielectric measurements of N2-Ar mixtures at

microwave frequencies [1] which could correspond to the orientational glass state.

However, the characteristic reorientational time scale for this glass phase is 10-4

sec. [3] and at microwave frequencies one may not observe the effects of the glass

regime. From the dielectric response of CO and N2 discussed in the previous chap-

ters, it is evident that our apparatus is sensitive enough to probe the interesting












quadrupolar glass phase in N2-Ar mixtures down to 4 K and characterize the phase

boundary very accurately for the first time. Also in this chapter I explore the new

electric field induced glass regime in N2-Ar mixtures.


5.2 Experimental Data


As discussed in the previous section, so far there was no direct demarcation

of the phase boundary of the glass phase in the N2-Ar mixtures. Below I show

from the audio frequency dielectric data, the phase boundary of this mysterious

glass phase. Fig. 5.2 shows the dielectric constant of N2-Ar mixtures for various

Ar concentrations below 20 K. E' is the value of E at 4.2 K on the lower warm-up

curve. All the samples were obtained first by mixing the gases at room temperature

for more than 24hrs and then condensing to liquid phase at least two times (after

each condensation all the sample was brought back to room temperature) and

then finally annealed at solid-liquid transition temperature for 6 to 8hrs. The

dielectric data obtained from multiple solid samples (for the same gas mixture)

clearly showed that phase separation of N2 and Ar did not occur for all the mixtures

studied. For 10% Ar sample no hysteresis was observed in E(T) with thermal

cycling below 35 K and E(T) changed smoothly without any observable jump at

T,,3 (Fig. 5.3). But T,3 is characterized by the change of slope in E(T) at this

temperature.

For 16% Ar sample I observed the hysteresis loop in E(T) at T,3 (Fig. 5.2)

when the sample is thermally cycled between 4.2 K and 30 K. This is the first

direct evidence of the glass phase in N2-Ar mixtures. Fig. 5.2 shows E(T) for

three more samples with increasing Ar concentration. For all the plots in Fig.










63





N -16%Ar
2
S2-



I I I

20 25 30

) 1 -
1 N -30%Ar
2



40- 0 2 1
4 6 8 10 12 14

S1 N -40%Ar
2




I I I I I I
4 6 8 10 12 14

S 1 N -49 %A
W

0- C


4 6 8 10 12 14 16

T (K)


Figure 5.2: E(T) of N2-Ar mixtures below 20 K for 1 kHz and 5 kV/m excitation
field. We can observe the gradual change in the hysteresis loop with increasing Ar
concentration as well as the onset of glass regime characterized by the change of
slope of E(T).










64





14 -


0% Ar
12 10%Ar
16% Ar
L20% Ar
30% Ar
10 -




X

S6



4



2



0



-2 I I I I I
0 10 20 30 40 50 60
T (K)

Figure 5.3: E(T) of zero field cooled N2-Ar mixtures for 1 kHz and 5 kV/m excita-
tion field. We can observe the shift in Too to lower temperatures with increasing
Ar concentration. Only warm-up data are shown. See text for details.












5.2, red up-triangles represent warm-up data and blue down-triangles represent

cool-down data. It is clear from these plots that Tp progressively decreases with

increasing Ar concentration and the hysteresis loop in E(T) becomes open-ended at

4.2 K for x(Ar) > 16%. One can also observe from these plots that the hysteresis

increases (from the separation of red and blue curves at 4.2 K) with increasing

Ar concentration and there is observable change in slope of E(T) (see red curves)

associated with the onset of glass regime. Fig. 5.3 shows warm-up data of E(T)

for various N2-Ar mixtures below 55 K. For the sake of clarity, data for 40% and

49% Ar concentrations are not shown in this plot as they are very similar to that

of 30% Ar data.

Figure 5.4 summarizes the results I obtained from the study of the detailed

dielectric response functions of N2-Ar solid solutions. The purple solid circles

outlining the glass phase up to 49% Ar concentration are from this work (marked

IIB). Red solid line separating fcc phase (marked III) and hcp phase (marked

IIA) is obtained from x-ray diffraction measurements [50]. The area covered by

the purple dashed lines represents the orientational glass phase (from this work).

Our thermometers are accurate to 0.2 K in their absolute value in determining

the Tp0 of pure solid N2. I obtained 35.51 K for Ta0 while the accepted value

is 35.61 K (Table: 3.1). From the observed hysteresis loops as well as from the

accuracy of our thermometers, we may conclude that the phase boundary between

fcc and hcp phases (solid red curve) is not a sharp phase boundary as is generally

thought. The branching of the purple lines for Ar concentration less than 20%

shows the presence of orientational glass regime in the fcc phase as well. Also

from Fig. 5.4, it is clear that the new phase (marked IC) obtained by Hamida


















liquid
S solid -
-- liquid


p -N


f.C.c.
IA


I I I I I 1 .. .1 I I I .j
10 20 30 40 50 60 70 80 90 100
r N
Mole % of N 2


Figure 5.4: N2-Ar phase diagram showing the glass-phase boundary obtained from
the dielectric response of these solid solutions.


90 .


80 ,


60


60












et al. [3] where mean order parameters evaluated with respect to the local axes

have only a very narrow distribution (or Quadrupolarization), is a part of the

orientational glass phase obtained from the dielectric measurements. However, no

effect of the quadrupolarization on the dielectric response functions is observed.

In the following sections I present the effect of the external electric field (ac field)

on the dielectric response of N2-Ar mixtures.


5.2.1 Electric Field Induced Glass Phase in N2-30%Ar Samples

In Chapter 4, I showed that the external ac electric field has a strong effect

on the dielectric response of solid N2 whenever the system is exposed to the field

above a characteristic temperature Th,. Also I showed that at Tg, E of solid N2

exhibits A-like behavior which may indicate rotational melting at this temperature.

Then it is only natural to extend the work to explore this interesting behavior in

N2-Ar mixtures as well. Fig. 5.5 shows the dielectric response of N2-30%Ar field-

cooled in 5kV/m and 1 kHz excitation field. Once the zero field cooled sample is

warmed-up along the red curve (open squares) after exploring the low temperature

orientational glass phase below 25 K (inset) and annealed at 42 < T < 45 K for 6

to 8hrs., I obtained a kink in E(T) at 44 K, similar to the 100% N2 sample. For

all N2-Ar samples up to 49% Ar concentration I studied, only when the sample

temperature is raised above the characteristic temperature Th in the presence of

the field, that we observe the field induced hysteresis. Also, once the field is turned

off and the samples are annealed for 10 to 12 hrs. above Th, and cooled down, we get

back the initial dielectric response (in this case red curve). When the annealed 30%

Ar sample temperature is further raised, at ~49 K, E increased sharply when the





















.1.0
-w

0.5

0.0


5 10 15
T (K)


20 25


30%Arw
30% Arc
30%Arw
30% Arc


0 10 20 30 40 50 60
T (K)

Figure 5.5: E(T) of N2-30%Ar mixtures field cooled in 1 kHz and 5 kV/m electric
field. Spontaneous tunneling of polarization at various random temperatures as
well as hysteresis due to the field effect can be observed. Kink at ~44 K is observed
during the first warm-up. See text for details.












sample is once again annealed for 10 to 12 hrs. at this temperature. This behavior

is very similar to that of the 100% N2 described in Chapter 4. After raising the

temperature further to 52 K and cooled-down, E(T) showed strong hysteresis as

well as spontaneous tunneling of the macroscopic polarization at various random

temperatures (purple open circles). Once again, this behavior is similar to that

of 100% N2, indicating that the processes that lead to this remarkable behavior

in solid N2 are indeed very strong. The orange (plus sign) curve shows the data

obtained during the second warm-up cycle of the field cooled (along purple open

circles) sample. When the temperature is further raised, orange curve merges

with red (open squares) at some elevated temperature and at ~56 K, once again

we observe the A-like behavior in E(T) (Fig. 5.6). Red (open squares) warm-up

curve and orange (open down-triangles) cool-down curve show that above Tg, E(T)

is highly reproducible without any hysteresis. However, once the sample is cooled

below Tg along the orange curve, once again we obtain another A-like cusp at a

slightly lower temperature than Tg (about 1 K) and with reduced peak height. The

resulting polarization is higher than any other curve for this sample. Also, when

compared with the cusp maxima of 100% N2 at Tg, the peak height for 30% Ar

sample is about 1.75 times smaller; i.e. with increasing Ar concentration the cusp

maxima decreases.


















9R -


26


22


20


18


16


14


12


10


8


6


4

3


N -30% Ar
2













o c

v C
0 1 E5

T)
-S


-D

-a
o



-D


5 4 0 45 50 55






T (K)


Figure 5.6: E(T) of N2-30%Ar mixtures field cooled in 1 kHz and 5 kV/m electric
field near melting temperature. A-like cusp in E(T) at Tg as well as small hysteresis
in Tg can be observed. See text for details.
















N2-40%Ar


Ar30% w
Ar40% w
Ar40% c
Ar40% w


I I I I I I I I I I I I


0 5 10 15 20


25 30 35 40 45 50 55 60


T (K)


Figure 5.7: e(T) of N2-40%Ar mixtures field cooled in 1 kHz and 5 kV/m electric
field. Though no spontaneous tunneling of polarization is observed, strong hys-
teresis due to the field effect can be observed. Also no kink at ~44 K is observed
but the change of slope about this temperature is observed. See text for details.












5.2.2 Electric Field Induced Glass Phase in N2-Ar Mixtures with Ar
Concentration of 40% and Above

Figure 5.7 shows the dielectric response of N2-40%Ar samples. For T < 20 K,

the response is very similar to that of 30% Ar sample. Some of the key differences

in the dielectric response from the 30% Ar samples are:

1. Annealing at 42 < T < 45 K for 10 to 12 hrs. did not result in a sharp kink

but a smooth and gradual change in slope can be observed.

2. Further annealing at 49 < T < 52 K for 10 to 12 hrs. did not result in

sharp increase in E(T) as before (Fig. 5.7), but it increased linearly with

temperature.

3. When this sample is cooled from 52 K, strong field induced hysteresis is ob-

served as before, but the characteristic spontaneous tunneling of macroscopic

polarization is not observed (blue open down-triangles in Fig. 5.7).

4. Once this field-cooled sample is warmed-up again along the green open up-

triangles curve, it meets the first warm-up curve at some elevated tempera-

ture (Fig. 5.7). When the sample is further warmed-up, little below Tg (<

0.5 K) E undergoes a sharp jump followed by the A-like cusp at Tg (Fig. 5.8).

5. The cusp maximum at Tg further reduced when compared to that of pure

N2 sample by a factor of 3.5. Above 57 K, E(T) changed linearly up to the

melting temperature, and TM is characterized by a sharp drop in E (Fig. 5.8).

6. Figure 5.9 shows E(T) of N2-49%Ar sample. For this sample also we observe

only the field induced hysteresis but no spontaneous tunneling of the macro-

















N2-40%Ar


I I I


T T
g M
. I I i I I ,I .


50 52 54 56 58 60 62 64

T (K)


Figure 5.8: E(T) of N2-40%Ar mixtures field cooled in 1 kHz and 5 kV/m electric
field near melting temperature. A-like cusp in E(T) at Tg as well as a smaller peak
in E(T) below Tg can be observed. See text for details.

















N -49%Ar
2


A-!I


10 -


0w
0 C

W
17 C


0 10 20 30 40 50 6(


T (K)

Figure 5.9: E(T) of N2-49%Ar mixtures field cooled in 1 kHz and 5 kV/m electric
field. No kink at ~44 K is observed but the change of slope about this temperature
is observed. A-like cusp in E(T) at Tg as well as a smaller peak in E(T) below Tg
can be observed.See text for details.












scopic polarization; i.e. for Ar concentration of 40% and above we may not

observe the characteristic tunneling.

7. Similar to the 40% Ar sample, E of N2-49%Ar sample also shows a sharp

increase ~1.2 K below Tg, then decrease smoothly with temperature. At Tg,

once again we observe that the cusp maximum decreased by a factor of 7

when compared to that of pure N2 sample.

8. The presence of a smaller peak below Tg is rather unique for samples with

Ar concentration of 40% and above. We can also observe that with increase

in Ar concentration, this peak moves to the lower temperatures while Tg

remains the same.



5.3 Conclusion


From the above discussion it is clear that the thermal processes that lead to

the strange but very interesting electric field induced glass phase in N2 are very

strong and further work is necessary to characterize this l---pl.-, The data

clearly show that for T > Tg, E of all the samples show no hysteresis indicating

that the solid is in a "rotational-in' II state; i.e. for Tg < T < TAI, the individual

molecules are completely free to rotate before the lattice meltdown at TM. It is

interesting to note that, solid CO exists in the /3-phase over a temperature range

of ~6.5 K (Table: 3.1) below melting temperature TM and for N2, Tg is ~6.2 K

below TAM. If our conjecture is true, then Tg for CO coincides with its T3, and this

may explain why we observe no electric field induced hysteresis in the case of CO

as well. This is in confirmation with our mean-field theory developed in Chapter









76

3, where the theory required that the residual ordering in CO above TaO be zero to

correctly predict the jump in e at TaO. However, for the same model we required

that the residual ordering be about 50% above To in the case of N2. Solid 02 is

another example of frustrated quadrupolar solid which has a rich phase diagram

due to its (much stronger than the electric quadrupole moment) intrinsic magnetic

dipole moment. Dielectric study of this remarkable solid may help us understand

this new effect in the quadrupolar solids in general.















CHAPTER 6
CONCLUSION

In this thesis I presented the first audio frequency dielectric constant measure-

ments of CO and N2 (Chapter 3). In Chapter 2 I presented the details of construc-

tion of the ultra high sensitivity AC Capacitance Bridge with two parts per billion

sensitivity for measuring the real part of the dielectric constant in the 4.2 < T <

100 K temperature range. This apparatus has at least an order of magnitude better

sensitivity than any other suceptometer built before. In Chapter 4 I discussed at

length the effect of external electric field on the orientational ordering of nitrogen.

From the observed hysteresis in the dielectric constant data, I showed that nitrogen

exhibits strong field cooling effects in the entire solid phase. The A-like behavior of

the dielectric constant at ~56 K, is interpreted as the thermodynamic glass tran-

sition temperature below which nitrogen has an yet not understood "field induced

glass pli,-' which extends the entire solid phase of nitrogen. A simple mean-field

theory is proposed (Chapter 3) to explain the low temperature a-phase dielectric

behavior of CO and zero field cooled N2. In Chapter 5 I presented the dielectric

constant data of N2-Ar solid solutions. From the observed dielectric response I

mapped the phase boundary of the traditional N2-Ar glass much more accurately

than before. I have also shown that along with the traditional N2-Ar glass phase,

these mixtures have the field-induced glass phase up to 60% of N2 concentration.

In this chapter I will review the major results and provide suggestions for future

experimental work.












From the mean-field calculations it is apparent that quantum mechanical ori-

entational levels play a crucial role in shaping not only the order parameter but

also the dielectric constant of pure CO and N2. Because the dielectric constant

is very sensitive to the orientational behavior of the molecules, from the observed

hysteresis effects as well as the spontaneous tunnelling of macroscopic polarization

state of the crystallites in a powdered sample in the low temperature a-phase sug-

gest that there might still be few unknown terms in the hamiltonian. This fact was

voiced by many authors [16] earlier and it is unfortunate that even for a system

as ideal as nitrogen, theoretical understanding of the interaction potential is very

limited. The present results strongly suggest that it is worth while to consider a

trial potential which includes interaction terms with external electric fields.

The strong A-like cusp at ~56 K as well as spontaneous tunnelling at various

temperatures suggest that there might be associated changes in the heat capacity

of the sample. Though the change in the energy is very small (~10 nJ/mole) if we

consider only the orientational part, it is hard to imagine that at these tempera-

tures the orientational degrees of freedom are decoupled from those of the lattice.

If they are indeed strongly coupled, there must be an observable change in the

heat capacity of the samples. In the present experimental configuration, because

of the large amount of the metal in the sample cell (see Chapter 2, Cell Design),

whose heat capacity increases rapidly as we reach liquid nitrogen temperatures, I

could not observe the temperature excursion at the cusp maximum. But, a thin

wall (~0.2 mm) cylindrical shell (similar to the present cell) calorimeter may be

constructed easily and by subtracting the heat capacity of the zero field cooled

nitrogen sample from that of the field cooled one (same sample), obtained in two









temperature scans in an adiabatic calorimetry method, one may observe any small

difference in the heat capacity due to field cooling. Such an experiment, if success-

ful, can prove beyond doubt that indeed we have a new thermodynamic phase of

nitrogen (similar to the pressure induced phases) due to field cooling. Presently

such an experiment is ui-,. i, .v.


Another experiment that can be carried out with relative ease is CW NMR

of field cooled "N2 samples. I have constructed a high sensitivity, low rf level

marginal oscillator for this purpose. The sample cell consists of two circular silver

electrodes 1.5 cm diameter (10 p/m thick silver layer thermally deposited on STY-

CAST 2850 base 8 mm thick) separated by 1 cm long fiber glass cylinder. On the

outer surface of the cylinder a 20 turn NMR coil is wound closely which covers the

entire length of the cylinder. For zero field cooled samples at 4.2 K we observe

Pake doublets (ordered samples) [3]. However if the samples are field cooled and

we have a glass phase, we should expect the line broadening as for N2-Ar glass

phase. This experiment may give us an independent confirmation of the existence

of the glass phase. Presently the NMR work is underway. With the same cell I

am considering to carry out the pulsed NMR experiments also. Other experiments

which require dedicated setup are x-ray and neutron diffraction experiments which

can tell us about the crystal structure of this interesting new phase. Simple lat-

tice dislocations, grain-boundaries, voids, vacancies etc. may not be the driving

mechanisms for the emergence of this phase.












APPENDIX A
NUMERICAL VALUES FOR E(T) OF CO










Table A.I: Numerical values for the curves in Fig. 3.5 (0.5 3.0 kHz).

0.5 kHz 1.0 kHz 2.0 kHz 3.0 kHz
T e T e T e T e


4.32
5.43
7.12
7.98
9.87
12.21
14.31
16.64
18.73
21.08
21.3
22.46
23.41
24.35
25.25
26.34
27.43
28.68
29.9
31.1
32.52
33.78
35.24
37.08
37.9
39.44
41.5
45.5
48.79
52.22
55.72
58.33

58.52
61.19
63.42
65.43

67.84
70.28


1.4211246
1.4211085
1.421092
1.4210895
1.4210953
1.4211318
1.421186
1.4212788
1.4213752
1.4215056
1.4215253
1.4216749
1.4220711
1.4238753
1.4306318
1.4594426
1.527139
1.5913219
1.6044059
1.6009877
1.5933339
1.," 483
1.5787421
1.5696058
1.5659465
1.5593571
1.5515994
1.5397623
1.". ;-"i
1.5278295
1.5239669
1.5219329

1.4745417
1.4729108
1.4743504
1.4746131

1.5133278
1.511'"V417


4.32
5.43
7.09
7.96
9.93
12.28
14.37
16.69
18.66
21.11
21.31
22.34
23.43
24.36
25.25
26.35
27.43
28.67
29.89
31.09
32.52
33.78
35.23
37.09
37.88
39.45
41.52
45.53
48.77
52.24
55.81
58.23

58.53
61.2
63.36
65.37

67.89
70.21


1.421136
1.4211211
1.4211054
1.421103
1.4211105
1.4211473
1.421202
1.4212956
1.421: -.*,
1.4215189
1.4215404
1.4215924
1.4217823
1.422322
1.4244187
1.4345493
1.4677178
1.5432088
1.5905215
1.5979834
1 i.' I i"-.
1.5s,._'ss
1.5787844
1.5695986
1.5659816
1.5593478
1.5515199
1.5396833
1.5329272
1.5277818
1.5.2 i i,',
1.521894

1.4744277
1.4723402
1.4740843
1.47439

1.5136143
1.5099551


4.31
5.42
7.07
7.95
9.94
12.26
14.38
16.72
18.6
21.1
21.38
22.08
23.44
24.36
25.26
26.35
27.43
28.67
29.89
31.09
32.53
33.78
35.22
37.09
37.88
39.45
41.55
45.56
48.78
52.25
55.88
58.18

58.53
61.22
63.32
65.3

68.14
70.15


1.4211447
1.4211305
1.4211173
1.4211143
1.421121
1.4211586
1.4212143
1.4213085
1.421 ')4;
1.421529
1.4215511
1.421602
1.4217137
1.421'",25
1.4225284
1.42 .'i ;4
1.4376539
1.4810886
1.5516026
1.5870761
1.5910222
1.5S<'71>3
1.57874<,-
1.5695736
1.565999
1.5593405
1.5514598
1.5396448
1.5 2'-s73
1.5277463
1.5238
1.5218539

1.4743706
1.4719041
1.4738432
1.474215

1.5135291
1.5100703


4.31
5.53
7.04
7.95
9.96
12.26
14.37
16.71
18.62
21.1
21.41
22.19
23.44
24.37
25.26
26.35
27.43
28.67
29.88
31.1
32.55
33.78
35.22
37.11
37.88
39.46
41.57
45.58
48.79
52.24
55.94
58.11

58.51
61.37
63.29
65.22

68.39
70.04


1.4211454
1.4211306
1.4211169
1.4211149
1.4211233
1.4211582
1.421214
1.4213127
1.4214027
1.4215315
1.4215576
1.4216137
1.4216986
1.4218179
1.422129
1.4235301
1.42' I 'II.7
1.4534 ',)1
1.5129489
1.5699145
1.5877497
1.-,2.'722
1.5786273
1.5695489
1.565999
1.5593163
1.551: ,-
1.5395746
1.5328168
1.5277114
1.5237521
1.5218224

1.474 ''i
1.4717842
1.4737352
1.4739642

1.5132201
1.510193










Table A.2: Numerical values for the curves in Fig. 3.5 (4.0 7.0 kHz).

4.0 kHz 5.0 kHz 6.0 kHz 7.0 kHz
T e T e T e T e


4.31
5.52
7.03
7.96
9.88
12.29
14.34
16.68
18.63
21.08
21.79
22.34
23.45
24.37
25.27
26.35
27.43
28.67
29.88
31.11
32.55
33.78
35.23
37.11
37.89
39.48
41.58
41.58
45.6
48.81
52.23
55.98
58.06

58.46
61.51
63.28
65.16


68.51 1.5130734
69.97 1.510(2-4


1.4211437
1.4211276
1.4211133
1.4211122
1.4211187
1.4211559
1.4212094
1.421305
1.4213987
1.4215321
1.4215838
1.4216225
1.421694
1.4217913
1.422013
1.422'".
1.4267507
1.4432413
1.4 1'11-'1 S
1.5547689
1.-~.4 ;-' ;
1.5845144
1.5784703
1.5695432
1.5659766
1.5592602
1.5513242
1.5513242
1.5394973
1.5327434
1.5276699
1.5237048
1.5217858

1.4741737
1.4719638
1.4735093
1.4737582


4.31
5.52
7.02
7.96
9.9
12.31
14.32
16.65
18.64
21.08
21.66
22.49
23.45
24.37
25.27
26.35
27.43
28.67
29.88
31.11
32.55
33.78
35.23
37.11
37.89
39.48
41.61
41.61
45.62
48.82
52.21
56.
58.01

58.4
61.68
63.28
65.11

68.59
69.91


1.4211261
1.4211124
1.4210981
1.4210959
1.4211033
1.4211397
1.4211935
1.4212974
1.4213899
1.4215209
1.4215596
1.4216162
1.4216799
1.4217652
1.4219302
1.4225664
1.4250525
1.4362739
1.47-2
1.5363214
1.5791409
1.5 '' "
1.5782578
1.5695324
1.5659403
1.5592057
1.5512134
1.5512134
1.5 ',42v'
1.5326397
1.5276131
1.523656
1.5217797

1.4741038
1.4722559
1.4734678
1.4736162

1.5129957
1.5103512


4.31
5.52
7.01
7.97
9.92
12.32
14.31
16.67
18.64
21.08
21.76
22.56
23.45
24.37
25.27
26.35
27.43
28.67
29.88
31.11
32.56
33.78
35.23
37.1
37.89
39.49
41.63
41.63
45.65
48.82
52.18
56.01
58.

58.34
61.92
63.29
65.


68.68 1.5129481
69.84 1.5104481


1.4211046
1.4210937
1.4210794
1.4210778
1.4210908
1.42113
1.4211805
1.4212788
1.4213707
1.4215019
1.4215504
1.4215997
1.4216596
1.4217381
1.4218807
1.4223548
1.4241898
1.43258
1.4611259
1.5212974
1.5738696
1.581963
1.5 7 I 1", i
1.5695214
1.565'' isS
1.5591635
1.55113
1.55113
1.5393248
1.5 2-'4,
1.5275519
1.5236216
1.5217837

1.474115
1.4726956
1.473377
1.47 i


4.31
5.51
6.99
7.97
9.96
12.33
14.31
16.7
18.6
21.12
21.8
22.59
23.45
24.37
25.27
26.35
27.43
28.67
29.88
31.11
32.56
33.78
35.22
37.1
37.9
39.49
41.65
41.65
45.65
48.82
52.17
56.02
57.99

58.27
62.06
63.31
64.93

68.71
69.78


1.4210956
1.4210',2
1.4210672
1.4210652
1.421071
1.4211016
1.4211077
1.4212568
1.421355
1.4214997
1.4215488
1.4215924
1.4216477
1.4217247
1.4218516
1.4222406
1.4 -' i,.' -';
1.4303871
1.4539446
1.5096661
1.5687433
1.5806062
1.5778752
1. 1,' 4 -."
1.5658615
1.5591124
1.5510328
1.5510328
1. -. i'i, '-4
1.5325309
1.5274922
1.5235798
1.5217944

1.4741765
1.4730683
1.4733756
1.4732826

1.5130163
1.5105659










Table A.3: Numerical values for the curves in Fig. 3.5 (8.0 12.0 kHz).

8.0 kHz 9.0 kHz 10.0 kHz 12.0 kHz
T e T e T e T e


4.31
5.51
6.99
7.97
9.94
12.32
14.32
16.7
18.58
21.12
21.79
22.62
23.45
24.37
25.27
26.35
27.43
28.67
29.88
31.11
32.56
33.78
35.21
37.1
37.9
39.49
41.66
45.66
48.8
52.16
56.01
57.97

58.17
62.18
63.32
64.85

68.76
69.78


1.4210681
1.4210491
1.4210348
1.4210327
1.4210386
1.4210726
1.4210803
1.421 '-
1.4213114
1.421471
1.4215106
1.4215689
1.4216242
1.4216964
1.421812
1.4221396
1.42- ''
1.428642
1.44,17 1,
1.4982214
1.5626441
1.57 \' i13
1.577601
1.5694321
1 .I" I 7'I1 ,.
1.5590597
1.5509412
1.5391783
1.532478
1.5274132
1.5235412
1.5218217

1.4743401
1.4735378
1.473502
1.4733254

1.51:;- .s,
1.511 17 '>7


4.31
5.51
6.98
7.97
9.94
12.31
14.33
16.68
18.6
21.12
21.62
22.64
23.45
24.37
25.26
26.34
27.42
28.67
29.88
31.11
32.56
33.78
35.2
37.1
37.9
39.5
41.68
45.68
48.79
52.17
56.
57.97

58.19
62.32
63.33
64.75

68.79
69.79


1.4210362
1.4210339
1.4210098
1.4210084
1.4210127
1.4210583
1.4211142
1.4212099
1.4213021
1.4214421
1.4214676
1.4215293
1.4215754
1.4216562
1.4217645
1.4220405
1.4229749
1.4272954
1.44 -'"
1.4876003
1.5556953
1.5767789
1.5772025
1.5693687
1.5657347
1.5589941
1.5508313
1.5390969
1.5324232
1.5273107
1.5234982
1.5218277

1.4744801
1.4739041
1.4735852
1.4732781

1.5133876
1.5111759


4.31
5.51
6.98
7.97
9.96
12.29
14.35
16.68
18.66
21.12
21.53
22.64
23.46
24.37
25.26
26.34
27.42
28.67
29.88
31.11
32.57
33.78
35.19
37.1
37.9
39.51
41.69
45.68
48.77
52.18
55.99
57.97

58.11
62.44
63.33
64.68

68.82
69.83


1.421028
1.4210085
1.421-i' nL2
1.421-11-14
1.421_'ii'I'
1.4210383
1.4210971
1.4211863
1.42-' 1 ,
1.4214983
1.421452
1.4215174
1.4215674
1.4216459
1.4217503
1.4219995
1.4227942
1.4265027
1.4402769
1.4804824
1.550009
1.5749318
1.57,1 ',1,
1.5693109
1.3',3 ,2
1.5589298
1.5507515
1.5390308
1.5323945
1.5272505
1.5234755
1.5218845

1.4743306
1.4740476
1.4733492
1.4730031

1.5128063
1.5113431


4.31
5.51
6.96
7.97
9.94
12.31
14.37
16.67
18.64
21.08
21.59
22.65
23.46
24.37
25.26
26.34
27.42
28.67
29.88
31.11
32.57
33.78
35.2
37.14
37.91
39.52
41.72
45.72
48.75
52.21
55.96
57.9

58.17
62.57
63.33
64.6

68.82
69.93


1.4211'i'I,
1.4211'i s '
1.4209712
1.42-,'1,'73

1.4210063
1.4210662
1.4211571
1.4212525
1.4213715
1.4214094
1.4214679
1.4215315
1.4216171
1.421723
1.4219065
1.4225038
1.4251957
1.4353544
1.4674427
1.-, ',~ 49
1.5701369
1.5758757
1. ,' in' 1
1.5655849
1.558848
1.5506153
1.5388471
1.5323384
1.5271529
1.523434
1.5219476

1.47 ;".'i41
1.4735876
1.4724682
1.4721458

1.5117934
1.5107349










Table A.4: Numerical values for the curves in Fig. 3.5 (14.0 20.0 kHz).

14.0 kHz 16.0 kHz 18.0 kHz 20.0 kHz
T e T T e T


4.31
5.51
6.95
7.97
9.56
12.34
14.44
16.67
18.59
21.1
21.74
22.66
23.46
24.37
25.26
26.34
27.42
28.67
29.88
31.11
32.57
33.78
35.2
37.13
37.91
39.53
41.74
45.74
48.74
52.25
55.93
57.73

57.97
62.64
63.33
64.48

68.77
70.04


1.4203053
1.4208976
1.42w's-15
1.4208778
1.4 -'11 4
1.4209103
1.42'" '21
1.4210702
1.4212
1.4213054
1.4213566
1.4214205
1.4214804
1.42155
1.4216473
1.4218099
1.4223042
1.424421
1.4325041
1.4589106
1.5257243
1.5652871
1.5748735
1.5''\'ITS
1.5654552
1.5587201
1.5504478
1.5387112
1.5322812
1.5270068
1.5233834
1.5219486

1.4731731
1.4733622
1.4719176
1.4715499

1.5112087
1.5101695


4.31
5.5
6.94
7.97
9.95
12.3
14.44
16.68
18.56
21.13
21.67
22.59
23.45
24.36
25.26
26.34
27.42
28.67
29.88
31.11
32.57
33.78
35.2
37.17
37.91
39.53
41.78
45.76
48.73
52.32
55.89
57.76

57.98
62.8
63.32
64.42

68.74
70.19


1.4207873
1.4207807
1.4207701
1.4207668
1.422'141.'
1.420793
1.4208485
1.4209192
1.4210042
1.4211485
1.4211946
1.4212573
1.4213283
1.4213817
1.421452
1.4216079
1.4220478
1.4237507
1.4302463
1.4521779
1.5147993
1.5597374
1.5735122
1.:, ,' is -
1.5652048
1.553 4'n,-s
1.5501004
1.5384004
1.5320488
1.5267582
1.7.2 WiT' 7
1.5217965

1.47- ',I
1.4733666
1.471423
1.4711242

1.510825
1.5097075


4.31
5.5
6.93
7.97
9.98
12.34
14.42
16.68
18.57
21.1
21.58
22.37
23.46
24.36
25.26
26.34
27.42
28.66
29.88
31.11
32.57
33.78
35.21
37.23
37.92
39.55
41.82
45.76
48.72
52.41
55.86
57.7

58.
62.97
63.29
64.35

68.71
70.35


1.4207336
1.420726
1.4207106
1.4207085
1.4207107
1.4207411
1.4207914
1.4-,1' 72
1.420966
1.4211803
1.4212005
1.421 ;i.',
1.4213049
1.4213576
1.4214227
1.4215942
1.4219377
1.4233009
1.4-2'->418
1.441 .'is ',, I
1.5047474
1.5538839
1.5721478
1.567 I,4
1.5650904
1.55b8 ',4
1.549914
1.5 ', ',I -'
1.5320115
1.5266595
1.5232414
1.5218406

1.4726675
1.4735787
1.4711007
1.4708637

1.5106444
1.509369


4.31
5.5
6.93
7.97
9.94
12.35
14.44
16.7
18.58
21.04
21.73
22.19
23.46
24.36
25.26
26.34
27.41
28.66
29.87
31.11
32.57
33.78
35.21
37.25
37.92
39.55
41.85
45.77
48.72
52.6
55.84
57.66

58.01
63.12
63.24
64.29

68.67
70.5


1.4 2'11 i,1''-'
1.4204547
1.4204454
1.4204429
1.420474
1.420529
1.4205709
1.4207678
1.4209178
1.4210865
1.421124
1.4211331
1.421176
1.4212665
1.4213708
1.4215959
1.4218341
1.4230253
1.4275611
1.4434742
1.4964196
1.5484746
1.5707375
1.5675437
1.5648403
1.5581828
1.549689
1.". 2-'.-2 )
1.5320195
1.5265699
1.5231948
1.5217926

1.4724215
1.473693
1.47061
1.4703718

1.5094217
1.5076513












APPENDIX B
NUMERICAL VALUES FOR E(T) OF N2













Table B.1: Numerical values for the curve (1) in Fig. 3.6 (0.5 3.0 kHz).


0.5 kHz 1.0 kHz 2.0 kHz 3.0 kHz
T e T e T e T e


4.15
5.71
7.39
9.22
10.26
11.57
13.53
14.08
15.35
18.16
20.85
23.3
25.53
27.43
29.43
31.69
33.28
34.7
35.22
35.39

35.56
37.92
39.8
42.08
45.21
48.41
52.74


1.4255284
1.4255078
1.425499
1.425519
1.4255596
1.4256396
1.4257938
1.4258344
1.4259568
1.4262499
1.4266067
1.4270166
1.427482
1.42793
1.4284901
1.42"' 1 ',
1.42989
1.4305932
1.4309241
1.4311297

1.434 '7 ',
1.435299
1.4362049
1.4375529
1.4400587
1.4431659
1.4459464


4.15
5.71
7.39
9.22
10.26
11.57
13.53
14.08
15.33
18.16
20.84
23.3
25.53
27.43
29.43
31.69
33.28
34.71
35.22
35.38

35.57
37.93
39.8
42.08
45.2
48.41
52.79


1.4255496
1.4255302
1.4255225
1.4255431
1.4255843
1.4256649
1.4258196
1.42's,.Ill
1.4259815
1.4262764
1.4266349
1.427044 i
1.4275091
1.4279588
1.4-2',184
1.4292841
1.4299199
1.4306275
1.4309536
1.431154

1.4344079
1.4353369
1.4362399
1.437588
1.4400957
1.4432076
1.4460045


4.15
5.71
7.39
9.22
10.25
11.57
13.53
14.08
15.32
18.16
20.86
23.3
25.52
27.43
29.43
31.69
33.28
34.71
35.22
35.38

35.56
37.93
39.79
42.08
45.18
48.4
5 2.\


1.4255743
1.425556
1.4255502
1.4255713
1.4256137
1.4-'-0.' i4 i
1.4 ", 4'-' -
1.4258897
1.4260103
1.426307
1.4266661
1.4270755
1.4275398
1.4279912
1.42 -',514
1.4293184
1.4299541
1.4306635
1.43,i V'i -
1.4311912

1.4344447
1.4353773
1.4362 ,
1.4376272
1.4401355
1.44;J,;
1.441.11, 1 .


4.15
5.71
7.39
9.22
10.26
11.57
13.52
14.08
15.31
18.16
20.86
23.3
25.51
27.43
29.42
31.69
33.28
34.71
35.22
35.37

35.56
37.93
39.79
42.08
45.17
48.4
52.89


1.4255961
1.4255766
1.4255713
1.4255961
1.4256361
1.4257178
1.425872
1.4259115
1.4260321
1.4263305

1.42 1- i V ,-
1.42711'i',

1.4275639
1.4- 1184
1.4-2~ 756
1.429344;
1.4299813
1.4306901
1.4310139
1.4312154

1.434473
1.4354057
1.4363088
1.4376557
1.4401635
1.44 ;-'46
1.4461199













Table B.2: Numerical values for the curve (1) in Fig. 3.6 (4.0 7.0 kHz).


0.5 kHz 1.0 kHz 2.0 kHz 3.0 kHz
T e T e T e T e


4.15
5.71
7.39
9.22
10.26
11.57
13.52
14.08
15.3
18.16
20.86
23.3
25.5
27.43
29.42
31.69
33.28
34.71
35.22
35.37

35.56
37.93
39.79
42.08
45.16
48.4
52.93


1.4256166
1.4255996
1.4255919
1.425619
1.4256631
1.4257408
1.4258956
1 .4 "'"' i
1.4260515
1.4263529
1.4267109
1.4271191
1.4275857
1.4->11419
1.42 ,.12-7
1.4293685
1.4300049
1.4307125
1.4310393
1.4312379

1.4344'\'.
1.435433
1.4363361
1.4376831
1.4401903
1.4433139
1.4461642


4.15
5.7
7.39
9.22
10.27
11.56
13.52
14.08
15.29
18.16
20.86
23.29
25.5
27.43
29.42
31.69
33.28
34.71
35.22
35.37

35.56
37.94
39.79
42.08
45.15
48.4
52.98


1.42-0. ',,4
1.4256231
1.425619
1.4256408
1.4 -'". .1', ',
1.4257632
1.4259179
1.4259562
1.426078
1.42638
1.4 -'.7 I
1.427142
1.4276111
1.42' 1, ..1
1.4 -', ', 1
1.4293927
1.4300309
1.4307421
1.431063
1.4312604

1.4345306
1.4354573
1.436364
1.4377086
1.4402141
1.44 i 461
1.4462127


4.15
5.7
7.39
9.22
10.27
11.57
13.53
14.08
15.29
18.16
20.85
23.29
25.5
27.43
29.42
31.69
33.28
34.71
35.22
35.37

35.56
37.94
39.79
42.08
45.14
48.41
53.02


1.425672
1.4256455
1.4256414
1.4256749
1.4257149
1.4257' -'-'
1.4259409
1.425978
1.4261027
1.4264047
1.4267621
1.4271644
1.4276352
1.4-'"11 )
1.42- ,', 29
1.4294169
1.4300557
1.4307681
1.4310925
1.431284

1.4345573
1.4354917
1.4363913
1.4377377
1.4402481
1.44 i I i
1.446254


4.15
5.7
7.39
9.21
10.27
11.57
13.53
14.08
15.29
18.17
20.85
23.29
25.49
27.43
29.42
31.69
33.28
34.71
35.22
35.37

35.56
37.94
39.79
42.08
45.13
48.4
53.06


1.4256967
1.4-'". 1 7,
1.4-'"., '\I '
1.4257002
1.4257432
1.4258191
1.4259691
1.426005
1.4261369
1.4264347
1.4267969
1.4271992
1.4276641
1.4281204
1.4 -.s47
1.4294476
1.4300829
1.4308029
1.4311197
1.4313159

1.4345917
1.4355184
1.4363974
1.4377657
1.4402749
1.44341
1.4463018













Table B.3: Numerical values for the curve (1) in Fig. 3.6 (8.0 12.0 kHz).


0.5 kHz 1.0 kHz 2.0 kHz 3.0 kHz
T e T e T 1 Te


4.15
5.7
7.39
9.21
10.27
11.57
13.53
14.08
15.29
18.19
20.83
23.28
25.5
27.43
29.42
31.69
33.28
34.71
35.22
35.36

35.56
37.94
39.79
42.08
45.12
48.41
53.09


1.4.-'- IIIs
1.4257149
1.4257067
1.4257343
1.4257749
1.42S->14
1.4 'i' 'I S
1.4260286
1.4261651
1.426463
1.4 2' '22S'
1.4272269
1.4276924
1.4281563
1.4287113
1.4294736
1.4301112
1.4308301
1.4311492
1.4313455

1.434-'22-'
1.4355582
1.4 ',.4">54
1.4377954
1.440313
1.4434381
1.4463503


4.15
5.71
7.39
9.21
10.27
11.57
13.53
14.08
15.28
18.19
20.83
23.28
25.5
27.44
29.41
31.69
33.28
34.71
35.22
35.36

35.56
37.95
39.79
42.08
45.12
48.41
53.12


1.4257573
1.4257455
1.4257437
1.4257626
1.4258085
1.4258791
1.426038
1.4260739
1.4262093
1.4265001
1.4 11'io.,' '
1.4272681
1.4277225
1.4281852
1.4287496
1.4295061
1.4301549
1.4308744
1.4311812
1.4313821

1.4346622
1.4356068
1.4364916
1.4378251
1.4403618
1.44 W4S'58
1.446409


4.15
5.71
7.39
9.21
10.27
11.57
13.53
14.08
15.29
18.2
20.83
23.28
25.5
27.44
29.41
31.69
33.28
34.71
35.21
35.36

35.56
37.95
39.78
42.08
45.11
48.42
53.16


1.4257991
1.4257861
1.4257838
1.4258038
1.4258444
1.4259179
1.426081
1.4261233
1.4262628
1.42-2'.2ys
1.4269076
1.4273188
1.4277525
1.4282212
1.4287844
1.4295332
1.430185
1.4311'"'12
1.4312214
1.4314152

1.4341.'i ;
1.4356543
1.4365261
1.4378726
1.4403958
1.44 ,. 18
1.4464742


4.15
5.71
7.39
9.2
10.27
11.57
13.54
14.08
15.29
18.2
20.85
23.28
25.49
27.44
29.41
31.69
33.28
34.71
35.21
35.36

35.56
37.95
39.78
42.08
45.1
48.42
53.2


1.4258703
1.425 ',11I
1.42 '-,7'i
1.4258756
1.4 '".'i' iI
1.4 "I' ,i
1.4261692
1.4262281
1.426337
1.426619
1.4 o'"is 77
1.4274048
1.42-7 -.-1
1.4-'s"i4 i
1.4 -' ,_2')
1.42',11 i.',
1.4302955
1.4310169
1.431287
1.431525

1.4347653
1.435751
1.4365926
1.4379529
1.4405071
1.4436106
1.4465718




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