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Acoustic study of disordered liquid 3He in high-porosity silica aerogel

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Title:
Acoustic study of disordered liquid 3He in high-porosity silica aerogel
Creator:
Choi, Hyunchang ( Dissertant )
Lee, Yoonseok ( Thesis advisor )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
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Copyright Date:
2007
Language:
English

Subjects

Subjects / Keywords:
Acoustic attenuation ( jstor )
Aerogels ( jstor )
Heating ( jstor )
Liquids ( jstor )
Low temperature ( jstor )
Magnetic fields ( jstor )
Signals ( jstor )
Temperature dependence ( jstor )
Transducers ( jstor )
Transition temperature ( jstor )
Dissertations, Academic -- UF -- Physics
Physics thesis, Ph. D.
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
The effect of disorder is one of the most interesting and ubiquitous problems in condensed matter physics. A fundamental question, "How does a ground state evolve in response to increasing disorder?", encompasses many areas in modern condensed matter physics, especially in conjunction with quantum phase transitions. We address the same question the low temperature phases of liquid ³He which is of a special interest since , in its bulk form, it is the most well understood unconventional superfluid and the purest substance known to mankind. Unlike conventional s-wave pairing superconductors, the unconventional superconductors are vulnerable to any type of impurity, and this fact has been used to test the unconventional nature of the order parameter in heavy fermion and cuprate superconductors. The aerogel/3He system provides a unique opportunity to conduct a systematic investigation on the effects of static disorder in unconventional superfluids. We have investigated the influence of disorder, introduced in the form of 98% porosity silica aerogel, on the superfluid 3He using various ultrasound techniques. Our primary effort is in understanding the complete phase diagram of this relatively new system. In particular, the high magnetic field region of the phase diagram has not been explored until this work, and the nature of the A-like to B-like transition in this system has not been elucidated. We identified a third superfluid phase emerging in the presence of magnetic fields, which resembles, in many respects, the A1-phase in bulk. Our zero field study of the A-B transition in aerogel revealed that two phases coexist in a narrow window of temperature right below the superfluid transition. Sound attenuation measurements conducted over a wide range of temperatures and pressures show a drastically different behavior than in bulk. In particluar, in the B[beta]-like phase, our results can be interpreted as strong evidence of a gapless superfluid. ( , )
Subject:
3he, 3helium, a1phase, acoustic, aerogel, attenuation, disorder, helium, impurity, sound, superfluid
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Title from title page of source document.
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Document formatted into pages; contains 156 pages.
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Includes vita.
Thesis:
Thesis (Ph.D.)--University of Florida, 2007.
Bibliography:
Includes bibliographical references.
General Note:
Text (Electronic thesis) in PDF format.

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Copyright Choi, Hyunchang. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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7/12/2007

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ACOUSTIC STUDY OF DISORDERED LIQUID 3He INT HIGH-POROSITY SILICA
AEROGEL






















By

HYUNCHANG CHOI


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

2007
































Copyright 2007

by

Hyunchang Choi


































To my family









ACKNOWLEDGMENTS

Most of all, I thank my advisor, Yoonseok Lee for his guidance and encouragement. His

enthusiasm for the physics subj ect and his way of approaching the problems has been very

inspiring for me. He tried to help me improve my strengths and overcome my weaknesses. I

also thank Mark. W. Meisel. His continuous support sustained my confidence as a research

student.

I appreciate the help I received from Jian-sheng Xia, Naoto Masuhara, Carlos L. Vicente

and Ju-Hyun Park. Their expertise and efforts on the research were amazing. Their knowledge

based on their experience with practical problems was crucial to overcome obstacles that I have

encountered. Ju-Hyun Park taught me baby steps at the beginning of my laboratory experience.

He was a model student who spends his time gladly for other students when they need help.

I would also like to thank our collaborators outside and inside the physics department for

their support. All of the aerogel used in my experiments were grown by Mulder' s group at

University of Delaware. The acoustic cavity employed for the cw shear impedance measurement

was made by Guillaume Gervais in Halperin' s group and Ca1.Sro.5RuO4 Samples were prepared

by Rongying Jin in Mandrus group. The experimental cells and maintenance parts were made by

Mark Link and other machinists and consistent supply of liquid 3He by Greg Labbe and John

Graham enabled us to continue the experiment without interruption. Dori Faust did all the paper

work for ordering supplies and more. I could complete my proj ects with their sincere help.

I cannot neglect to say thank you to my lab mates, Pradeep Bhupathi and Miguel Gonzalez,

who spent their valuable time to revise this thesis. From time to time, Byoung Hee Moon

inspired me with his creative ideas. Pradeep made the Melting Curve Thermometer, which was

used as a main thermometer for the attenuation measurement.












TABLE OF CONTENTS


page

ACKNOWLEDGMENTS .............. ...............4.....


TABLE ............ _...... ___ ...............7....


LIST OF FIGURES .............. ...............8.....


AB S TRAC T ............._. .......... ..............._ 1 1..


CHAPTER


1 INTRODUCTION ................. ...............13.......... ......


1 .1 Overvi ew ................. ...............13........... ...
1.2 Pure Liquid 3He .............. ...............14....
1.2. 1 H history ................. ...............14.......... ....
1.2.2 Fermi Liquid ............... ...............16
1.2.3 Superfluid 3He .............. ...............21....
1.3 Superfluid 3He in Aerogel .............. ...............25....
1.4 Liquid in Porous Media .............. ...............28....


2 THE Al PHASE OF SUPERFLUID 3He IN 98% AEROGEL................ ...............35


2. 1 Overvi ew ................. ...............3.. 5......... ...
2.2 Experiments .............. ...............36....
2.3 Results............... ...............42
2.4 Discussion ................. ...............44........... ...


3 THE A PHASE OF SUPERFLUID 3He IN 98% AEROGEL ......____ ....... ...__...........60


3 .1 Overvi ew ................. ...............60........... ...
3.2 Experiments .............. ...............62....
3.3 Results............... ...............62
3.4 Discussion ................. ...............64........... ...


4 ATTENUATION OF LONGITUDINAL SOUND INT LIQUID 3He/98%
AEROGEL ............. ...... .__ ...............76...


4. 1 Overvi ew ................. ...............76........... ...
4.2 Experiments .............. ...............78....
4.3 Results............... ...............8 1
4.4 Discussion ................. ...............88........... ...


5 CONCLUSION............... ...............10











APPENDIX

A ORIGIN SCRIPT FOR MTP CALIBRATION INT THE HIGH FIELD PHASE.................108


A. 1 MPTfukuy amale s sTn. c............ ...... ... ...............111
A.2 MPTfukuyamalow5.c ................. ...............114...............

B PARTS OF THE EXPERIMENTAL CELL FOR ATTENUATION MEASUREMENT...117

C TYPICAL SETTING FOR NMRKIT II............... ...................124

D ORIGIN SCRIPT FOR THE DATA ANALYSIS OF THE ATTENUATION
MEASUREMENT S ............ ..... .__ ...............126...

E TRANSPORT MEASUREMENT ON Ca1.Sro.5RuO4....... ...............1 133

F NEEDLE VALVE FOR DR137 ................. ...............144......... ...

LIST OF REFERENCES ................. ...............148........_......

BIOGRAPHICAL SKETCH ..............._ ...............156........_ ......









TABLE
Table page

2-1 Slopes (mK/T) of the splitting for the Al and A2 transitions ................. ......................59










LIST OF FIGURES


Figure page

1-1 Temperature dependence of sound velocity and attenuation of pure 3He ................... ......3 1

1-2 P-H-T phase diagram of superfluid 3He............... ...............32..

1-3 Gap structures of the superfluid 3He, the B phase and A phase. ................ ................. .33

1-4 Microscopic structure of 98% Aerogel by Haard. ............. ...............33.....

1-5 P-B-T phase diagram of superfluid 3He in 98% aerogel ................. .........................34

2-1 Splitting of the phase transition temperature in fields ................. ......... ................46

2-2 Cut-out views of the experimental cell and the acoustic cavity ................. ................ ...47

2-3 Schematic diagram of the vibrating wire ................. ...............48........... ..

2-4 Arrangement of the experimental cell and the melting pressure thermometer ................. .49

2-5 Schematic diagram of the continuous wave spectrometer ................. .......................50

2-6 Frequency sweep for two different spectrometer settings at 28.4 bars and 3 T.................51

2-7 Zero field acoustic signals at 28.4 bars for different spectrometer settings as a
function of temperature ........._._.._......_.. ...............52....

2-8 For high fields, we used the MPT calibration of Fukuyama' s group ........._.._... ..............53

2-9 Acoustic traces for 3, 5, 7, 9, 11, and 15 T at 33.5 bars on warming .............. .............54

2-10 Acoustic traces for 1, 2 and 3 T.............. ...............55....

2-11 Acoustic traces for 3, 5, 7, 9, 11, and 15 T at 28.4 bars on warming ........._.._... .............56

2-12 Transition temperatures vs. magnetic field for 28.4 and 33.5 bars ............... ...............57

2-13 Degree of splitting for each individual transition vs. magnetic field............... ................58

3-1 The Stanford NMR measurement on superfluid 3He in 99.3% aerogel ................... ..........70

3-2 Cooling and warming traces taken at 28.4 bars and 33.5 bars ................. ............... .....71

3-3 Acoustic traces of tracking experiments ................ ...............72...............

3-4 The relative size of the steps for the supercooled aerogel A-B transition ................... ......73











3-5 Field dependence of the warming A-B transition in aerogel for 28.4 bars ......................74

3-6 The zero field phase diagram of superfluid 3He in 98% aerogel ................... ...............75

4-1 The attenuation of longitudinal sound in liquid 3He/aerogel at 16 bars for 15 IVHz
measured by Nomura et al ........... _...... ._ ...............92..

4-2 Schematic diagram of experimental setup .............. ...............93....

4-3 Temperature determined by IVPT and Pt-NMR thermometer ................. ............... .....94

4-4 Evolution of receiver signals on warming ................. ...............95........... ..

4-5 Velocity of bulk 3He and liquid 3He in aerogel at Tc ............_. ..._... ........._......96

4-6 Signal from receiver. ............_. ...._... ...............97...

4-7 Linearity test .............. ...............98....

4-8 No significant change in attenuation indicates that the heating by the transducer is
negligible............... ...............9

4-9 Receiver signal trace vs. time at 0.4 mK and 29 bars ........._.._ ....._... ........._......99

4-10 The normalized attenuation in the superfluid phase at 12 and 29 bars as a function of
the reduced temperature. ............_. ...._... ...............100...

4-11 Phase diagram at zero field ........._.._ ...... .___ ...............100..

4-12 Absolute attenuations for pressures from 8 to 34 bars are presented as a function of
tem perature .............. ...............101....

4-13 Norm al sized attenuati on i n sup erflui d ................ ...............102...........

4-14 Attenuation vs. pressure ................. ...............103...............

4-15 Zero energy density of states at zero temperature vs. pressure for the unitary and the
Born scattering limits ................. ...............104................

4-16 Attenuation for normal liquid .............. ...............105....

A-1 Origin worksheet, 'Pad'. Input parameters ................. ...............109........... ..

A-2 Label control window. ............. ...............109....

A-3 Origin worksheet, 'PtoT' ................ ...............110......... .....

C-1 The typical N1VRkitlI setting. ........... ......__ ...............125.










D-1 The screen shot of Origin worksheet for data analysis. ................... ............... 127

E-1 A Zero Hield Phase diagram of Ca2-xSrxRuO4 fr0m S. Nakatsuji et al. ............................ 137

E-2 Picture of a test sample .............. .....................138

E-3 Temperature dependence of the resistance in the absence of magnetic Hields. ................139

E-4 Resistance vs. temperature for various levels of excitation at low temperature. .............139

E-5 The excitation dependence of the resistance for each temperature ................. ...............140

E-6 The temperature dependence of the slopes. ...._ ......_____ ......___ ..........14

E-7 Normalized magneto resistance. ............. ...............141....

E-8 Field sweeps below 200 G. .............. ...............141....

E-9 The position of the shoulder and dip structure in Hields vs. temperature. ................... .....142

E-10 Magneto resistance for Hields perpendicular to the plane (blue) and 20o tilted away
from the c-axis (red) at 20mK. ........... ......__ ...............143

E-11 Magneto resistance for low Hields perpendicular to the plane (blue) and 20o tilted
away from the c-axis (red) at 20mK. ............. ...............143....









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

ACOUSTIC STUDY OF DISORDERED LIQUID 3He INT HIGH-POROSITY SILICA
AEROGEL

By

Hyunchang Choi

May 2007

Chair: Yoonseok Lee
Major Department: Physics

The effect of disorder is one of the most interesting and ubiquitous problems in condensed

matter physics. A fundamental question, "How does a ground state evolve in response to

increasing disorder?", encompasses many areas in modern condensed matter physics, especially

in conjunction with quantum phase transitions. We address the same question in the low

temperature phases of liquid 3He, which is of a special interest since in its bulk form, it is the

most well understood unconventional superfluid and the purest substance known to mankind.

Unlike conventional s-wave pairing superconductors, the unconventional superconductors are

vulnerable to any type of impurity, and this fact has been used to test the unconventional nature

of the order parameter in heavy fermion and cuprate superconductors. The aerogel/3He system

provides a unique opportunity to conduct a systematic investigation on the effects of static

disorder in unconventional superfluids. We have investigated the influence of disorder,

introduced in the form of 98% porosity silica aerogel, on the superfluid 3He using various

ultrasound techniques. Our primary effort is in understanding the complete phase diagram of

this relatively new system. In particular, the high magnetic field region of the phase diagram has

not been explored until this work, and the nature of the A-like to B-like transition in this system

has not been elucidated. We identified a third superfluid phase emerging in the presence of










magnetic fields, which resembles, in many respects, the A y-phase in bulk. Our zero field study of

the A-B transition in aerogel revealed that two phases coexist in a narrow window of temperature

right below the superfluid transition. Sound attenuation measurements conducted over a wide

range of temperatures and pressures show a drastically different behavior than in bulk. In

particluar, in the B-like phase, our results can be interpreted as strong evidence of a gapless

superfluid.









CHAPTER 1
INTTRODUCTION

1.1 Overview

The effect of disorder is one of the most interesting and ubiquitous problems in condensed

matter physics. Metal-insulator transitions [Lee85] and the Kondo effect [He69] are two

examples of phenomena in which disorder, in the form of various types of impurities, plays a

fundamental role. The influence of disorder on ordered states such as the magnetic or

superconducting phase has also attracted tremendous interest, especially in systems that undergo

quantum phase transitions [Sac99].

The response of Cooper pairs to various types of impurities depends on the symmetry of

the order parameter [And59, Abr61, Lar65]. The strong influence of a small concentration of

paramagnetic impurities on a low temperature superconductor is in stark contrast to its

insensitivity to nonmagnetic impurities [And59, Abr61]. Unconventional superconductors with

non-s-wave pairing are vulnerable to any type of impurity [Lar65], and this fact has been used to

test the unconventional nature of the order parameter in heavy fermion and cuprate

superconductors [Vor93, Tsu00, Ma04].

Given a great deal of quantitative understanding of the intrinsic properties of superfluid

3He [Vol90], the aerogel/3He system provides a unique opportunity to conduct a systematic

investigation on the effects of static disorder in unconventional superfluids. In this system, a

wide range of impurity pair breaking can be attained by continuously varying the sample

pressure. Furthermore, the nature of the impurity scattering can be readily altered by modifying

the composition of the surface layers, the 4He preplating. We have investigated the influence of

disorder (98% aerogel) on the superfluid 3He using various ultrasound techniques. Our primary

effort is focused on understanding the complete phase diagram of this relatively new system.









Especially, the high magnetic field region of the phase diagram had not been explored until this

work, and the nature of the A-like to B-like transition in this system had not been elucidated. In

this study, three main experimental results are presented along with in-depth discussions.

In the rest of this chapter, a brief description of 3He physics, with an emphasis on the

acoustic properties of Fermi liquids and unconventional BCS superfluids, is presented. High

porosity silica aerogel acting as quenched disorder is also discussed in this chapter. In chapter 2,

the acoustic impedance measurements, in high magnetic fields, which led us to observe the third

superfluid phase, are described. Chapter 3 focuses on the zero field experiment that revealed the

existence of the B RA transition and the coexistence of the two phases on warming. The third

experiment, the absolute sound attenuation measurement by direct sound propagation, is

discussed in chapter 4. Finally in chapter 5, we conclude with a summary of the experimental

results, the physical implications, and a few suggestions for future directions. Various

supplementary materials are collected in appendices. Especially, low temperature transport

measurements on Ca1.Sro.5RuO4

Throughout this thesis, many transition temperatures are mentioned. For example, T, and

T,, indicate the superfluid transition temperature and the AB transition temperature,

respectively. In most of the cases, it is clear whether the transition is in the bulk or in aerogel.

However, in the case where the distinction is warranted, Tc or T are used for the transitions

in aerogel.

1.2 Pure Liquid 3He

1.2.1 History

The isotopes 3He and 4He are the only two stable isotopes in this universe which remain

liquid down to the lowest available temperature. They are light enough that the zero-point









motion overcomes the attractive inter-atomic interaction, which is very weak due to their filled

2s electronic configuration. At low temperatures, the matter-wave duality and the degenerate

conditions two main pillars of quantum world emerge as their de-Broglie wavelength becomes

comparable to the inter-atomic spacing, providing a reason to call them quantum liquids.

It is not surprising that the existence of 4He (product of vigorous nuclear fusion) was first

observed from the visible spectrum of solar protuberances, considering its minute =: 5 ppm

abundance in the Earth's atmosphere. Fortunately (especially to low temperature physicists), a

substantial amount of 4He trapped underground can be found in some natural gas wells. The

much rarer isotope 3He was discovered by Oliphant, Kinsey, and Rutherford in 1933 [Oli33].

Unlike 4He, a reasonable amount of 3He could only be produced artificially by the B-decay of

tritium in nuclear reactors. It is not hard to imagine why the first experiment on pure liquid 3He

was conducted by Sydoriak et al. at Los Alamos Scientific Laboratory [Syd49a, b]. In this work,

the authors proved that 3He indeed condenses into a liquid at saturated vapor pressure, contrary

to the predictions of distinguished theorists such as London and Tisza. Since then, a tremendous

amount of effort has been poured in this subj ect and has lead to discoveries of various low

temperature phases in liquid as well as solid 3He [Vol90].

One of the most remarkable properties that these two isotopes share is the appearance of

superfluid phases in which liquids can flow through narrow capillaries with almost no friction.

The superfluid transition temperature in liquid 3He (- 2 mK) is three orders of magnitude lower

than in liquid 4He (- 2.2 K), reflecting the fundamental difference in quantum statistics. While a

4He atom with zero nuclear spin obeys Bose-Einstein statistics, a 3He atom with spin 1/2 follows

Fermi statistics. A Bose system prefers to condense into the lowest-energy single particle state,

the so called Bose-Einstein condensation (BEC), at a temperature where the wavefunction of the









particle starts to overlap. This BEC usually accompanies the onset of superfluidity as evidenced

in liquid 4He and dilute cold atoms.

The superfluidity of a Fermi system was first discovered in a metal, mercury, in 191 1 by

Kammerlingh Onnes [Kam 11], although the microscopic understanding of the phenomenon did

not come to light until almost a half century later by the theory of Bardeen, Cooper and

Schrieffer (BCS) [Bar57]. A bound pair of two electrons, known as a Cooper pair, with a spin

singlet and s-wave orbital state may be looked upon as a composite boson that is Bose-

condensed. Cooper pairs can be formed with an arbitrarily small net attractive interaction in the

presence of the filled Fermi sea background. In a conventional superconductor it is known that

the attractive interaction is provided by the retarded electron-phonon interaction. For liquid 3He,

however, the origin of the pairing interaction is not clearly understood microscopically.

In his seminal work on Fermi liquid theory [Lan56, Lan57], Landau conceived a

phenomenological theory to provide a theoretical framework for an interacting fermionic many

body system at low temperatures with a specific example, liquid 3He, in his mind. Since then,

liquid 3He has served as a paradigm for a Fermi liquid whose nature transcends the realm of

fermionic quantum fluids.

1.2.2 Fermi Liquid

At the heart of Landau' s Fermi liquid theory is the quasiparticle, a long lived elementary

fermionic excitation near the Fermi surface [Mar00]. The energy levels of the interacting system

have a one to one correspondence to the ones in a Fermi gas without mixing or crossing levels.

The quasiparticle represents the total entity of the bare particle and some effect of the interaction.

Therefore it is expected that the mass of a quasiparticle is different from the bare mass. The

quasiparticle energy should also depend on the configuration of other quasiparticles around, and

this molecular Hield type interaction can be parameterized by a set of dimensionless numbers,









Landau Fermi liquid parameters, { F,", F," ), where 1 indicates angular momentum. More

specifically, suppose that an excited state was created by adding a quasiparticle labeled by k (let

us assume that the spin quantum number is imbedded here) and the energy of the excited state

relative to the ground state is given by (-. Any perturbation in the occupancy of the states 6n,-

near the Fermi sea would cause a change in the excitation energy e,- of the quasiparticle k no

longer identical to 4-. The difference between these two values produces a change in the

molecular field on the quasiparticle, Sc~ via interactions between the quasiparticles,

1~-c;,s 11


where f~, 4, is Landau's interaction function which generates the change in energy of a

quasiparticle with momentum k by the perturbation in the distribution of quasiparticles with

momentum k", and V is the volume. In a perfectly isotropic system like liquid 3He and at low

temperatures, the interaction function should depend only on the angle, 6, between the two

moment with the magnitude kF Then, the Landau interaction function can be decomposed into

symmetric (orbital) and antisymmetric (spin) parts,

ftps = f,"4 + cr o' f y (1-2)

where cr, is the ith' Pauli matrix and s (a) denotes symmetric (antisymmetric). This separation is

possible only when the spin-orbit coupling is negligible; this is justified in 3He since the only

spin-orbit coupling is the weak dipole-dipole interaction (less than CIK). Each interaction

function can be expanded for each angular momentum component in terms of a basis set of

Legendre polynomials. The dimensionless Landau parameters, F," and F," are obtained by









normalizing with the density of states at the Fermi surface, N(0) = m pF ~i2 3 1 F

[Noz64].

These parameters determine various macroscopic properties of the liquid; conversely,

some of these parameters can be determined by various experiments. For example, one finds the

effective mass, m* from the Galilean invariance

m*"/ m =1 +F," / 3. (1-3)

It is worth mentioning that Eq. 1-3 is an exact result without further higher order corrections.

This is the most important Fermi liquid correction that renormalizes the density of states at the

Fermi energy, N(0) Consequently, the heat capacity and the magnetic susceptibility need to be

modified accordingly,


C, =k2T. (1-4)


72 2 N(0)
4 1 + Fo (-5

where y is the gyromagnetic ratio. The magnetic susceptibility also receives a correction through

the spin channel led by Foa. The strength of Fermi liquid theory is in the fact that with the

knowledge of a few (experimentally determined) Landau parameters, most of the physical

properties can be calculated self-consistently since the higher angular momentum components

decrease rather rapidly. In 3He, Fo" 10 90 F," 5 15, and Ff -(0.70 0.75) indicating

that liquid 3He is indeed a strongly interacting (correlated) system with enhanced effective mass

and magnetic susceptibility due to ferromagnetic tendency (minus sign in Fo")

While the thermodynamic properties of a Fermi liquid resemble those of a Fermi gas with

an adequate renormalization through the Fermi liquid parameters, dynamical properties are









unique in the sense that new types of collective modes are predicted to exist in this system

[Lan57]. Ordinary hydrodynamic sound (first sound) in a liquid propagates by restoring its local

equilibrium through scattering processes. Therefore, it requires the sound frequency to be much

smaller than the scattering rate, i.e., or~ << 1, where 0 ~is the sound frequency and r is the

relaxation time. The sound velocity, cy which is determined by the compressibility, r,, and

the sound attenuation, a,, which is dominated by viscous processes, are given by [Vol90]

2 1 1
c,=(pr,) (1+Fo )(1+-F, )v (1-6)
3 3


az= 9,ll (1-7)

where p, r, mi are the mass density, the viscosity and the sound frequency, respectively. At

low temperatures (<< 1 K), the sound velocity at a given pressure is constant in the

hydrodynamic regime [Abe61].

In a Fermi liquid r varies as 1/ T [Wil67]. Therefore, at low enough temperatures where

r becomes much longer than the period of the sound wave, the conventional restoring

mechanism for hydrodynamic sound becomes ineffective. However, Landau realized that new

sound modes emerge in the collisionless regime (mr~ >> 1) if the relevant Fermi interactions are

strong and repulsive enough, and named these excitations as zero sound modes [Lan57]. At zero

temperature, a zero sound mode in a Fermi liquid can be pictured as a coherent oscillation of the

elastic Fermi surface ringing without damping. Normal modes of this spherical membrane are

associated with specific zero sound modes such as longitudinal zero sound (LZS) and transverse

zero sound (TZS). As the temperature rises, incoherent thermal quasiparticle scattering prohibits

the coherent oscillations of the Fermi surface, causing a damping of zero sound. For longitudinal









zero sound, a strong repulsive Fd' value (- 10 90 depending on pressure) plays a major role in

stabilizing this mode with a slightly higher zero sound velocity, c,, compared to that of the first

sound and attenuation, a, ac 1/ t,


c, = c, (1+ 2 n *(1+}F")( F 2 + 0[( F 4]}, (1-8)
15 na c, c,

2 n *(1+' F;")2 v v
a,, =" ( F 3 1+ O[( F 2 ]) (1-9)
15 na vFz 1, 1

Since r ac T the first to zero sound crossover occurs by lowering the temperature at a fixed

frequency with a trade mark of a symmetric temperature dependence on both sides of broad

attenuation maximum (see Fig. 1-1) [Kee63, Abe65, Ket75].

Other than Landau theory, a visco-elastic model can also well describe the first to zero

sound transition. Liquid 3He possesses not only viscosity as a liquid but also elasticity like a

solid [Hal90], and this elastic character is especially pronounced in the collisionless regime

[And75, Cas79, Cas80, Vol84]. Therefore, the existence of zero sound modes (especially the

transverse zero sound mode, as can be seen below) implies the solid-like character of liquid 3He

[Kee63, Kee65, Bet65, Abe66, Ki67, Whe70, Rud80]. From the usual kinetic gas formula, the

viscosity coefficient, r, can be written as

r ~ p < v~ >'/ R = p < v~ > c (1-10)

It is based on momentum exchange between quasiparticles with the Fermi velocity, vF and

quasiparticle life time, r (1 = v~r). For or~ >> 1, the quasiparticles do not have enough time to

collide with each other during a period of sound oscillation and the damping becomes weak. The

velocity and sound attenuation of the longitudinal sound from a visco-elastic model are given by










c, = c, + (co c, ) ,(1-11)
1+0 ri22


c, -c 0, 7i
at=o (1-12)
C c" 1+0~ r
1 a

The crossover from the first sound to zero sound is clearly represented in these expressions.

Transverse zero sound is another collective mode predicted by Landau. However, the strength of

the relevant interaction (Ff ) is marginal in liquid 3He. It turns out that the phase velocity of this

mode is so close to the Fermi velocity that the mode decays effectively into quasiparticle-hole

pairs causing strong damping (Landau damping). The zero sound modes are expected to

disappear in the superfluid state as the gap develops at the Fermi level. As will be discussed in

the following section, it is of great importance that these modes actually are resurrected in the

superfluid by coupling to order parameter collective modes.

1.2.3 Superfluid 3He

The BCS theory immediately invigorated the search for a superfluid transition in liquid

3He. However, many physicists struggled for longer than a decade in frustration coming from

the lack of evidence of the transition and the difficulty in predicting the transition temperature.

Finally in 1971, Osheroff, Richardson and Lee [Os72] observed peculiar features on the melting

line in a Pomeranchuk cell filled with 3He, which indicated some type of transitions. Although

their original paper falsely identified them as transitions associated with magnetic transitions in

the coexisting solid detailed NMR experiments [Osh72a] and the help of tremendous intuition

provided by Leggett confirmed that the features were indeed superfluid transitions arising from

the liquid. In their NMR experiment, they applied a field gradient along the z-axis in order to



SThe title of the paper is "Evidence for New Phases in Solid 3He".









determine the position and identity of the NMR signal source as either solid or liquid. This was

probably the first 1D magnetic resonance imaging experiment, although the work of Paul

Lauterbur in 1973 [Lau73] is officially credited as the first magnetic resonance imaging. Today,

three phases (A, B, A y) in the superfluid have been identified experimentally (Fig. 1-2).

The Cooper pairs in all three phases are spin-triplet p-wave pairing which possesses an

internal structure. The general wave function Y for a spin triplet pair is a superposition of all

spin sub states (Sz = -1, 0, +1) as given by




where y,~, (k), ry7 ,(k) and ry ~(k) are the complex amplitudes for each sub state. The

theoretical Balian-Werthamer (BW) state [Bal63] that corresponds to the B-phase is the most

stable state in zero magnetic field at low pressures. The B-phase is composed of all three

substates and accordingly has smaller magnetic susceptibility since the |I~> + |I> component

is magnetically inert. Because of the pseudo-isotropic gap in this phase as shown in Fig. 1-3

(this is quite unusual for intrinsically anisotropic superfluid), the B-phase exhibits similar

properties to conventional superconductors: for example, an exponential temperature dependence

of the specific heat at low temperatures. On the other hand, the Anders on-B ri nkm an-Morel

(ABM) [And60, And61] state (A-phase) is an equal spin pairing state, constituted by only |ff>

and |I> states. The gap has nodal points at the north and south poles in the direction defined

by the angular momentum of the Cooper pair. At high pressures (P > 21 bars), the ABM state is

stabilized near To owing to the strong coupling between quasiparticles [Vol90, And73]. The

point where normal phase, A-phase and B-phase meet at zero field is called the polycritical point









(PCP). The phase transition between A- and B-phase is first order as evidenced by strong

supercooling of the transition and the discontinuity in heat capacity at the transition.

Magnetic fields have profound and intriguing influences on the phase diagram of pure

superfluid 3He. Depairing of the |I~> + |I> component in the B phase by magnetic fields, an

effect that is similar to the Clogston-Chandrasekhar paramagnetic limit in a superconductor

[Tin96], promotes the growth and appearance of the A-phase at all pressures. As a result, a

magnetic field shifts the first order A-B transition line to a lower temperature in a quadratic

fashion and eventually quenches the B phase around 0.6 Tesla and 19 bars [Tan91]. In addition

to the effect on the A-B transition, a magnetic field induces a new phase, the A y-phase, between

the normal and the A-phase by splitting the second order transition into two second order

transitions. The A y-phase is very unique in the sense that it contains only |ff> pairs (it is a fully

polarized superfluid). A magnetic field generates small shifts in the Fermi levels of up and down

spin quasiparticles by the Zeeman energy and, consequently, results in different superfluid

transition temperatures (particle-hole asymmetry). Therefore, in the A y-phase, only the spin up

component participates in forming Cooper pairs. The splitting in transition temperatures (the

width of the Al phase) is linear in field and symmetric relative to the zero field To in the weak

coupling limit since the two spin states form pairs independently. The width of the A y-phase in

bulk is about 60 CIK/Tesla at melting pressure [Isr84, Sag84, Rem98].

A stable state can be found by searching a state with the lowest free energy. Since the

order parameter of superfluid 3He grows continuously from the transition temperature

(continuous phase transition), the free energy can be expanded in powers of the order parameter

as long as the temperature is near the transition temperature (Ginzburg-Landau theory) [Lan59].

The expansion is regulated by the symmetries of the system, and the coefficients for the









expansion can be determined from microscopic theories. The free energy functional ( f) should

have full symmetry that the higher temperature phase possesses. For example, normal liquid 3He

possesses a full continuous symmetry including time-reversal symmetry. For p-wave pairing

states, f with a gap amplitude A is given by


f = ~f, + ea + -1-A4 p (1-14)


where I, is the fourth-order invariant of the order parameter and

a(T)= -N(0)(1- T/ T). (1-15)

In the weak coupling limit,

P, = P, = P4 5 P = -2P, = 4p,, (1-16)


P, = S(3)N(0)(k, T,) (1-17)


and the BW state is found to be the most stable [Bal63]. One can easily expect that there are

many local free energy minima (including a few saddle points) in this multi-dimensional

manifold space. The hierarchy among these local minima depends on the specific values of the

p, -parameters. For example, as the pressure rises, the P, 's start to deviate from the weak

coupling values mainly due to strong coupling effects (e.g. spin fluctuations). This strong

coupling correction helps the ABM phase win over the BW phase. This effect is responsible for

the A-phase appearing above the polycritical point [And60, 61].

Nonzero angular momentum of the Cooper pair allows pair vibration modes, called order

parameter collective modes (OPCM). These collective excitations are measured as anomalies in

sound attenuation and velocities below To An attenuation peak by pair breaking can also occur

for a frequency above 28A/. The frequency range of ultrasound (10 ~100 MHz) conveniently









matches with the size of the gap and the spectra of various OPCM' s in superfluid 3He. The

thermally excited quasiparticle background also provides an independent channel for sound

attenuation. In particular, sound attenuation from thermally excited quasiparticles decreases

exponentially at low temperature as an isotropic gap opens up in the B-phase.

1.3 Superfluid 3He in Aerogel

High porosity silica aerogel consists of a nanoscale abridged network of SiO2 Strands with

a diameter of ~ 3 nm and a distance of ~ 20 nm. Since the diameter of the strands is much

smaller than the coherence length, 5,(15 80 nm for 34 0 bars [Dob00]) of the superfluid,

when 3He is introduced, the aerogel behaves as an impurity with the strands acting as effective

scattering centers. Although current theoretical models treat aerogel as a collection of randomly

distributed scattering centers, one should not lose sight on the fact that the aerogel structure is

indeed highly correlated. The microscopic structure of 98% porosity aerogel is shown in Fig. 1-

4 [Haa00], where the density is 0.044 g/cm3 and the geometrical mean free path is 100 ~ 200 nm

[Por99, Haa00]. The velocity of longitudinal sound in 98% aerogel was measured as 50 ~ 100

m/s [Fri92].

Early studies using NMR [Spr95] and torsional oscillator [Por95] measurements on 3He in

98% aerogel found substantial depression in the superfluid transition and superfluid density

[Por95, Spr95, Spr96, Mat97, Gol98, All98, He02a, He02b]. The phase diagram of 3He/aerogel

(with mostly 98% porosity) has been studied using a variety of techniques [Por95, Spr95, Mat97,

Ger02a, Bru01, Cho04a, Bau04a]. The phase diagram in P-T-B domain is shown in Fig. 1-5

[Ger02a].

When aerogel is submerged in liquid 3He, a couple of solid layers, which are then in

contact with surrounding liquid, are formed on the aerogel surface. Therefore, the scattering off









the aerogel surface has elements of both potential scattering and spin exchange scattering.

However, the magnetic scattering between the itinerant 3He and localized moments in solid

layers can be turned off by preplating magnetically inert 4He layers on aerogel surface. Although

the early NMR measurement by Sprague et at. concluded that the 4He coating turns the A-like

phase into a B-like phase at ~19 bars [Spr96], the effect of the spin-exchange scattering on the

phase diagram is elusive. Alles et at. provided evidence of the BW state in aerogel from the

textural analysis of NMR line shape [All98]. However, in these early experiments, no evidence

of the A-like to B-like transitions were observed until Barker et at. observed clear transition

features below the aerogel superfluid transition only on cooling while exploiting the

supercooling effect [Bar00b]. Considering the fact that data collecting in NMR experiments

typically done only on warming cycle to minimize the interaction with the demag field, it is not

surprising that the earlier experiments could not observe these transitions. Further studies using

acoustic techniques [Ger02a, Naz05] and an oscillating aerogel disc [Bru01] have confirmed the

presence of the A-B like transition in the presence of low magnetic fields.

The nomenclature for the two superfluid phases in aerogel as the A-like and the B-like

phases heavily relies on the spin structure without direct experimental inputs for the orbital

structure. Through detailed NMR studies, we now believe that the B-like phase at least has the

same order parameter structure as the BW phase. On the contrary, the identification of the A-like

phase is far from clear since there are many possible phases with a similar spin structure

represented by the equal spin pairing [Fom04, Vol06, Fom06, Bar06].

An interesting aspect of 3He in 98% porosity aerogel is that no superfluid transition was

found down to 150 CIK below 6.5 bars [Mat97]. This experimental observation renders us to

contemplate a possible quantum phase transition in this system [Bar99].









The first reports of superfluid transition in 98% aerogel were immediately followed by a

theoretical model, the homogeneous scattering model (HSM), based on the Abrikosov-Gorkov's

theory [Abr6 1] in which the spin exchange scattering off dilute paramagnetic impurities in s-

wave superconductors was treated perturbatively. The isotropic homogeneous scattering model

(IHSM) [Thu98a, Han98] could provide a successful explanation for the observed suppression of

the transition temperatures. In that theory, the ratio of the coherence length, 5,, to the mean free

path, C plays a maj or role as a pair-breaking factor, 5, / Q. The fact that the coherence length in

superfluid 3He can be tuned by pressure makes this system very attractive in probing a wide

range of parameter space of 5, / 8 without changing the density of the impurity, i.e. the mean

free path. Thuneburg was able to provide a better fit to the pressure dependence of To by using

an isotropic inhomogeneous scattering model (IISM) [Thu98a].

One cannot ignore a visually evident fact that aerogel strands are by no means a collection

of isotropic scattering centers nor small enough to be treated by a purely quantum mechanical

scattering theory. Volovik argued that, in the ABM state, the coupling of the orbital part of the

order parameter to the randomly fluctuating anisotropic structure, i.e. aerogel, should prevent the

onset of a long range order, thereby leaving it as a glass state in the long wavelength limit

[Vol96]. Fomin proposed a class of orbitally isotropic equal spin pairing states, so called the

robust phase, as another candidate for the A-like phase [Fom03]. In contrast, the robust phase

has a long range order in the presence of random anisotropy disorder, but as argued by Volovik,

it is not more stable than the ABM state from the free energy point of view [Vol05]. Specific

NMR frequency shifts have been calculated for both scenarios [Fom06, Vol06].









1.4 Liquid in Porous Media

Normal liquid 3He in the hydrodynamic regime might be treated as a classical liquid in

porous media. Liquids in porous media have been studied for both technical and academic

reasons rooted in a broad range of practical applications such as in biological systems and the oil

industry [Zho89, Hai99]. Even though practical structure properties (e.g. permeability) of the

pore can be, in principle, extracted from ultrasonic measurements [War94, Joh94a, Joh94b], it is

appreciated that the motion of fluid in a porous structure is a non-trivial problem and a

theoretical challenge. For the classical hydrodynamic fluid, sound is attenuated by various

mechanisms, including the friction between liquid and pore surface and the squirt of fluid in

narrow cracks [Bio55a, Bio55b, DVor93]. Biot developed a theory for acoustic response for the

friction in the high frequency regime (small viscosity) and in the low frequency regime (large

viscosity). For longitudinal sound, the attenuation is proportional to 1/r in large viscosity


regime and to JTin the other regime [Bio55a, Bio55b]. In Dvorkin et al.'s model of a water

saturated rock, the attenuation from the squirt-flow mechanism is an order of magnitude stronger

than the attenuation from the friction [Dvo93].

In the 1950's, Biot developed a theory for sound propagation in a porous elastic media,

which is filled with a viscous fluid [Bio55a,b]. He began his study from the Poiseuille type flow

which does not have turbulence and is valid at low frequency. Considering the effect of the

relative motion between the fluid and the solid, he found two longitudinal waves (first and

second kind) and one rotational wave. The liquid and the solid tend to move in phase for the first

kind and to move out of phase for the second kind of longitudinal wave. Because the second

kind is highly attenuated, only the first kind is the true wave at lower frequencies. An isotropic

porous media with uniform pore size is considered for the calculation. He predicted that the










shape of the pores is not important for the frequency dependence of the frictional force. At low

frequencies, decrease in viscosity results in more liquid motion and increases the attenuation

quadratically as T2 for the liquid 3He. At higher frequencies, Biot accounted for the deviation

from Poiseuille flow by replacing the static viscosity with a frequency dependent one. In this

regime, only a thin layer of fluid is locked on to the porous surface and a strong attenuation

occurs within this layer [War94]. In this case, the attenuation of the first kind wave is

proportional to 1/ T for the liquid 3He.

In 1994, Warner and Beamish measured the velocity and the attenuation of transverse

sound in a helium filled porous media [War94]. They could study both the low frequency regime

and the high frequency regime. Based on Biot' s model, they could determine the structure

parameters of the porous media tortuosityy, permeability and effective pore size) from their

acoustic measurements.

The dynamic permeability for a number of realistic models with variable pore size was

calculated by Zhou and Sheng [Zho89]. They found that the frequency dependence of the

dynamic permeability did not depend on the models except two scaling parameters. However,

the dynamic permeability of all the models could not be collapsed into a single curve when the

throats connecting the pores were very sharp or the porosity was close to 1 [Joh89].

Yamamoto et al. also studied the effect of pore size distribution theoretically [Yam88].

They found that the velocity and the attenuation of sound are not affected by the pore size

distribution in the low and high frequency limits, but strongly depend on the pore size

distribution in the intermediate frequency range. Their prediction agreed with experimental data

of attenuation in marine sediments [Ham72]. In the intermediate frequency regime near the low

frequency limit, attenuation of the longitudinal sound decreases as pore distribution increases.









Tsiklauri investigated the slip effect on the acoustics of a fluid-saturated porous medium

[Tsi02]. In general, in fluid dynamics, it is assumed that the fluid at the surface of the solid

moves at the same velocity as the solid does, which is known as the "no-slip boundary

condition". But when slippage occurs between the fluid and the solid surfaces, the so called "slip

effect" becomes important, particularly in the case of a highly confined porous media. Biot's

theory can describe acoustics for the scenario of the sloshing motion between a classical

Newtonian liquid and the porous frame assuming no slip effect. Tsiklauri introduced a

phenomenological frequency dependence of the slip velocity and found that it affects the

effective viscosity in the intermediate frequency domain leading to a higher attenuation than

what is predicted by the Biot theory. In the low and high frequency limits, his results agreed

with the predictions of Biot.















a 29.3 be r







100








C


O~

E I I I I I && I 1
I 2 81 0 06
Teprtue(K
Fiur 1-.Tmeauedpnec fsudvlctyadatnaino ue3e[e7] h
soi ie r h rslso i sn E .11 n -2[oa7.Tesprli
trnstoni sgaldbthshrchnena2mK Reitdfguewh
permssio fom ..RahadJB etroi unu lisadSld eie
bySB rceED dm, n ..Dfy(lnmPesNwYr,17)p23
Coyigt(17)bySrigr















































Figure 1-2. P-H-T phase diagram of superfluid 3He [Webli].

























B-Phase A-Phase


Figure 1-3. Gap structures of the superfluid 3He, the B phase and A phase. The space between
inner and outer shell represents a gap amplitude in momentum space.


Figure 1-4. Microscopic structure of 98% Aerogel by Haard (computer simulation) [Haa01i].














30


Figure 1-5. P-B-T phase diagram of superfluid 3He in 98% aerogel (blue line). Phase diagram
for bulk Superfluid 3He is shown as a dotted line [Ger02a]. Reprinted figure with
permission from G. Gervais, K. Yawata, N. Mulders, and W. P. Halperin, Phys. Rev.
B 66, 054528 (2002). Copyright (2002) by the American Physical Society.


rra~-)









CHAPTER 2
THE Al PHASE OF SUPERFLUID 3He INT 98% AEROGEL

2.1 Overview

In pure superfluid 3He, minute particle-hole asymmetry causes the splitting of the

superfluid transition through the Zeeman coupling in magnetic Hields. As a result, the third

phase, the Al phase, appears between the normal and the Az phase (the A phase in magnetic

Shields) [Amb73, Osh74]. In this unique phase, the condensate is fully spin polarized. The Al

phase has been studied by several groups and the width of the phase was found to increase

almost linearly in field by = 0.065 mK/T at the melting pressure [Osh74, Sag84, Isr84, Rem98].

Recently, Gervais et at. [Ger02a] performed acoustic measurements in 98% aerogel up to 0.5 T

and found no evidence of splitting in the transition.

Baramidze and Kharadze [Bar00a] made a theoretical suggestion that the spin-exchange

scattering between the 3He spins in liquid and solid layers on the aerogel surface could give rise

to an independent mechanism for the splitting of the transition. Detailed calculations [Sau03,

Bar03] show that antiferromagnetic (ferromagnetic) exchange reduces (enhances) the total

splitting in low Hields, but one recovers the rate of the particle-hole asymmetry contribution in

high Hields as the polarization of the localized spins saturates. These calculations were

performed with the assumption that the A phase in aerogel is the ABM state. Figure 2-1 presents

the low Hield data from Gervais et at. and the results of the calculation with (green) and without

(red) the anti-ferromagnetic exchange. The comparison between the theoretical calculation and

the results of Gervais et at. suggests that the spin exchange between the localized moments and

the itinerant spins is anti-ferromagnetic.

However, Fomin [Fom04] recently formulated an argument that the order parameter of the

A-like phase in aerogel should be inert to an arbitrary spatial rotation in the presence of the









random orbital field presented by the aerogel structure. This condition enforces a constraint on

the order parameter for equal spin pairs, he named new state satisfying the constraint as a 'robust

phase'. This theory predicts that an A y-like phase would be induced by a magnetic field for a

certain condition (e.g., in the weak coupling limit). This is a new type of ferromagnetic phase

with nonzero populations for both spin proj sections unlike the Al phase where only one spin

component participates in forming Cooper pairs. However, the splitting of the Al and Az

transitions in this case seems to evolve in a different manner compared to bulk 3He. If the A

phase in aerogel is correctly identified as an axial state, then a similar field dependent splitting of

the superfluid transition must exist at least in the high field region since the level of particle hole

asymmetry is affected only marginally by the presence of high porosity aerogel.

2.2 Experiments

The experiment was performed at the High B/T Facility of the National High Magnetic

Field Laboratory located at University of Florida. Figure 2-2 shows the cut-out views of the

sample cell and the acoustic cavity. The main cell body (except the part of heat exchanger) was

originally made out of titanium to reduce the nuclear spin contribution to the heat capacity in

magnetic fields. However, a leaky seal was found repeatedly at the silver-titanium epoxy seal

and forced us to replace the top part with the one made of coin silver. The top part of the cell

body forms a diaphragm so that the cell pressure can be monitored capacitively. We used the

same acoustic cavity that was utilized in the work of Gervais et al. [Ger02a, Ger02b]. The 98%

porosity aerogel was grown inside the acoustic cavity by Norbert Mulders at the University of

Delaware. The cavity is composed of a quartz transverse sound (AC-cut) transducer and a

longitudinal sound (X-cut) transducer separated by a 0.010" diameter stainless steel wire. The

3/8" quartz transducers were manufactured by Valpey-Fisher (currently Boston Piezo-Optics,

Bellingham, MA). The aerogel was grown inside the cavity formed by sandwiching the two









stainless steel wires between the transducers under external force. After the aerogel was grown

in the cavity, the copper wires were attached to the transducer with silver epoxy and the cavity

was placed on top of a Macor holder, since the vigorous chemical reaction does not allow us to

use epoxies or solder before the completion of critical drying process. In this geometry, the

transducers are in contact with both bulk liquid and liquid in the aerogel. Therefore, we detected

the change in the electrical impedance of the transducer caused by both liquids. The acoustic

measurement was performed at the third harmonic resonance, 8.7 MHz, for all the data presented

in this chapter.

The volume of the cell is designed to be less than 1 cm3 to ensure a short thermal

relaxation time. The sample liquid in the cell is cooled by the PrNiS demagnetization stage (DS)

through a 0.9 m-long annealed silver heat link extending below the DS. The transitions in the

pure liquid were confirmed independently by a vibrating wire (VW) placed in liquid near the

ultrasound transducer [Rem98]. The vibrating wire was made out of 0. 1 mm manganin wire bent

into a semi-circular shape with a 3.2 mm diameter. The resonance frequency of the VW was 13

k
the Lorentz force causes VW to oscillate. The amplitude of induced voltage across VW at a

fixed excitation frequency was measured using a Lock-in amplifier. The amplitude of the

induced voltage follows the amplitude and damping of the VW oscillation, which is determined

by the viscosity of the surrounding liquid. The acoustic spectrometer output was recorded

continuously while the temperature of the sample varied slowly. No significant hysteresis was

observed for data taken in both warming and cooling directions. The data presented (if not

specified) were taken on warming and the typical warming rate in our study was 0.1 0.2 mK/h.

2 Actually aerogel was grown around the whole cavity. All aerogel formed outside of the cavity was removed using
tooth pick after the completion of the drying procedure.









Temperature was determined by a 3He melting pressure thermometer (MPT) attached to the

silver heat link right below the cell in the experimental Hield region. Figure 2-4 shows the picture

of the MPT and the sample cell on the silver heat link.

The spectrometer (Fig. 2-5) used in this experiment was made by Jose Cancino, an

undergraduate research assistant [Lee97]. This is a continuous wave (cw) bridge type

spectrometer employing a frequency modulation (FM) technique. The FM signal was produced

by an Agilent E4423B generator and the rf output was set at 11 dBm. The FM excitation signal

was fed into the transducer at its resonance frequency through the first port of a 50 ohm matched

quadrature hybrid (QHB, SMC DQK-3-32S). The third port of the QHB was connected to the

transducer inside the cryostat. The output signal from the fourth port of the QHB was amplified

by a low noise preamplifier (MITEQ, AU-1519) and demodulated by a double balanced mixer

(mini-circuits, ZLW-1-1). This procedure converts the FM modulated rf signal into an audio

signal at the modulation frequency. The demodulated low frequency signal was detected with a

two-channel lock-in amplifier (Stanford Research System, SR530). The impedance mismatch at

the transducer causes a reflected signal to appear on the fourth port. The reflected signal from

the transducer can be nulled by feeding the (amplitude and phase adjusted) FM signal to the

second port of the QHB. The transducer works like a resonator for the FM radio receiver

[Kra80]. If the signal to be transmitted and a sinusoidal carrier are given by Eq. 2-1 and Eq. 2-2,

the frequency modulated signal can be written as Eq. 2-3.

xm, (t) = cos(m,,,t) (2-1)

x, (t) = x, cos(me)t) (2-2)


l (t) = Acos( [rre + mx,,(r)]dr) = A cos[met + Wsin(m,,,)] (2-3)









me~ and me~ are the modulation and carrier frequencies. ma~ is the modulation amplitude that

regulates maximum shift from sc~ [WikO6]. As the FM signal is fed to the resonator, the output

signal from the transducer is proportional to the product of the resonator amplitude, f(mi) and

the amplitude of the input signal, V, (t) The FM signal has a Einite but small oscillation that is

centered around me ,. Within this Einite region, the slope of the resonance line shape, a~i), can

be taken as a constant. Therefore, the output from the transducer, V2, iS given by Eq. 2-4. The

mixer works as a multiplier of two input signals, which are coming from the transducer through

the QHB and from the splitter (mini-circuits, ZFSC-3-1). The multiplied (mixed) signal

(e V, (t)V2 (t)) has two frequency components, low and high frequency parts. The low frequency

part of the signal coming from the mixer is given by Eq. 2-5. Finally, the AC component of this

signal is measured by the low frequency lock-in amplifier, which is proportional to a~i), as

shown in Eq. 2-6.


V2 (t) OCf (mi)V, (t) = [B + a wa sin(imst)]V, (t) (2-4)



V3(t) ocB+a ^ sin(mst) (2-5)


V4 (t) OCa (2-6)

Therefore, the spectrometer detects the slope of the transducer resonance shape, a, at a

given frequency, [Bol71]. In the ideal case, the acoustic response measured by a Lock-in

amplifier should produce zero in the 1f mode and a maximum in the 2f mode at resonance. The

schematic signal shape at each step is shown in Fig. 2-5. In our measurements, the excitation

frequency is Eixed at the resonance, while the 3He is still in the normal fluid. A large ratio of the

modulation amplitude (3 k








discrete nature in frequency spectrum. To be sensitive enough to detect any changes in the slope

a, the modulation amplitude should be chosen such that the magnitude of the deviation from me~

is small compared to the width of the transducer resonance frequency. A crucial part of the

measurement is tuning the spectrometer by adjusting the attenuators (JFW, 50R-028) and the

phase shifter to produce a null condition for maximum sensitivity. By going through this tedious

procedure, better than 1 ppm resolution in frequency shift is achieved.

Figure 2-6 shows the spectrometer outputs obtained by sweeping the frequency through the

transducer resonance for two different spectrometer settings at 28.4 bars and 3 T. Frequency

sweeps in the normal and deep inside of the superfluid are plotted together. By using Set2, we

were able to reduce the asymmetry between the size of maximum and minimum of the acoustic

response. It also shifted the crossing point in the normal and superfluid curves closer to zero.

Figure 2-7 shows how the spectrometer tuning affects the acoustic response during a temperature

sweep. After tuning the spectrometer to shift the crossing point closer to zero, the acoustic

responses at To and in the superfluid became almost the same. It can be noticed that Fig. 2-7 (b)

shows clearer changes in slope at Te and at T,, on warming, as opposed to Fig. 2-7 (a). Later

on, to obtain the clear slope change, the spectrometer was tuned to produce a symmetric

resonance shape in the normal liquid.

At zero field, the Greywall scale [Gre86] was adopted to convert the measured melting

pressure to temperature using the solid ordering transition as a fixed point to establish the

pressure offset. In the presence of magnetic fields, the calibration by the University of Tsukuba

group [Yaw01, FukO3] was employed. In their work, the calibration was given in two separate

regions the paramagnetic phase and the high field phase of solid 3He up to 14.5 T. However,

below 3.5 mK, our region of interest, only the calibration in the high field phase is available.









Consequently, the range of our temperature determination is limited to fields between 7 and 13 T

for the pressures of our work. The dash-dotted lines in Fig. 2-8 represent the high field phase

transition of the solid 3He in the MPT and below the boundary is the region where the calibration

is done. The melting pressure in the high field phase was given by Eq. 2-7 [Yaw01], where the

fourth-order temperature dependence is expected by spin-wave theory and the sixth-order

correction originates from the dispersion correction.

P(T; H) = Po(H) +c4 (H)T4 6 ,(H)T6 (2-7)

We used interpolated sets of coefficients for the fields applied in our work. The width of the

bulk Al phase identified in the acoustic trace was used to fix the pressure offset. At each field,

the calibration curve P(T; H) is vertically adjusted so that the measured melting pressure

interval of the bulk Al phase maps out the correct temperature width at the same experimental

condition (two point calibration method). The pressure offsets for all fields (including zero field)

are around 6 kPa within 10%. The temperature width of the Al phase was obtained using the

results from Sagan et al. [Sag84] and Remeijer et al. [Rem98]. Sagan et al. measured the shift of

transition temperature in fields for pressures ranging from 6 to 29 bars and determined their

splitting ratio (mK/T) from the linear fit. Remeij er et al. conducted their experiments only at the

melting pressure and performed a fit including a quadratic term. For consistency, our own linear

fit from the data of Remeij er et al at the melting pressure and splitting ratio from Sagan et al. for

other pressures are used to obtain the width of Al phase at the pressures studied. By using the Al

width rather than the actual transition temperature as a fixed point, we can circumvent the

possible inconsistency in the absolute temperature scale, which is used in the previous work on

the bulk Al phase. The data points represented by the solid and open circles in Fig. 2-8 were

obtained in this way. For 5 T at 28.4 bars, the aerogel transition temperatures (squares) were









determined by forcing the bulk A2 transition (diamond) on the linear fit for T,2 (Single point

calibration method). We also made an estimation of the aerogel Al transition temperatures

beyond the Hield range that our prescribed calibration method allows, by assuming a constant

warming rate, which is set by the bulk transition temperatures and time interval. The crosses are

obtained in this manner. The agreement between the filled circles and crosses in the overlapping

region is excellent and encouraging. The sensitivity of the melting pressure thermometry rapidly

declines in higher Hields and lower temperatures due to a decrease in solid entropy. For example,

IdP /dTI drops from 3.3 kPa/mK at 2 mK and zero Hield to 0.1 kPa/mK at 15 T for the same

temperature. This intrinsic property of the melting curve, in combination with the enhanced

noise in high fields, renders it practically impossible to make an accurate determination of the

aerogel A2 transition temperatures well below the dotted line where IdP /dTI << 0.1 kPa/mK.

Typical noise in our high field pressure measurement is about + 4 Pa. The Origin script for the

high field MPT calibration is given on appendix A.

2.3 Results

The acoustic traces for six different fields at 33.5 bars are shown in Fig. 2-9. The acoustic

trace is plotted along with that of the VW to compare the transition signatures of the bulk liquid.

The vibrating wire measurement was done in a similar fashion as described by Remeij er et al.

[Rem98] and the amplitude at the resonance is shown in the figure. Two sharp cusps in the VW

trace correspond to the Al and A2 transitions in pure liquid as reported in a previous work

[Rem98]. These features are concurrent with the jumps in the acoustic trace. The transitions in

aerogel are not as sharp as in the bulk. However, the smooth slope changes are quite clear and

similar signatures of the superfluid transition in aerogel have been observed by Gervais et al

[Ger02a]. The field dependent evolution of the transition features is demonstrated in Fig. 2-9.









Below 3 T, we were not able to resolve the double transition features in aerogel while the

features from the bulk can be traced down to zero field, merging into one (for example, see Fig.

2-10). As the field increases, the gap between the two transition features in each liquid widens.

It is important to emphasize that the bulk Az and aerogel Al transitions cross each other around 5

T and continue to move apart in higher fields. A similar behavior was observed at 28.4 bars

(Fig. 2-1 1), but the crossing occurred at a different field, around 7 T. The warming traces of the

acoustic response are shown in Fig. 2-1 1 for several fields at 28.4 bars along with those of the

vibrating wire (amplitude at resonance).

The straight lines in Fig. 2-12 are the results of linear fits to the data points represented by

the open (bulk) and solid (aerogel) circles including zero field results. The slope of each linear

fit is listed in Table 2-1. All data points determined by MPT including zero field data (red

circles) are used for the fitting with equal weights. The asymmetry in the splitting is of special

importance in two ways. First, the asymmetry ratio is a direct measure of strong coupling effects

such as spin-fluctuation [Vol90]. Second, it provides a valid self-consistency check for our

temperature calibration since only the total width of the splitting has been utilized. The

asymmetry ratios are also listed in Table 2-1 where

T(a) -7a).41 a ~)C l7a).42 a~)C) (2-8)


and Tac, is the zero field bulk (aerogel) transition temperature. The bulk slopes and r are in

good agreement, within 8% with previous measurements [Sag84, Rem98]. It is notable that the

Al slopes for the bulk and aerogel are same in order of magnitude, which is consistent with the

theory of Sauls and Sharma [Sau03]. The degree of individual splitting relative to the zero field

transition temperature is plotted in Fig. 2-13. For both pressures, the asymmetry in the aerogel is

consistently smaller, by 22%, than the bulk value for both pressures.









2.4 Discussion

The comparable slopes of the aerogel and bulk transitions suggest that the new phase in

aerogel has the same spin structure as the Al phase in bulk. The asymmetry ratio is related to the

fourth-order coefficients, P,, in the Ginzburg-Landau free energy expansion as defined by

r = -P, /(P, P,4 5 P) [Vol90]. In the weak coupling limit at low pressure, r a 1 and the

strong coupling effect tends to increase this ratio as pressure rises. Within the spin-fluctuation

model, the strong coupling correction factor, 3, can be estimated from

r = (1 + 3) /(1 3) [Vol90]. For 33.5 bars in aerogel, 3 = 0.09 The level of the strong coupling

contribution at this pressure corresponds to that of the bulk at around 15 bars [Sag84, Rem98],

which indicates substantial reduction of the strong coupling effect. The weakening of strong

coupling effects by the presence of impurity scattering has been discussed theoretically [Bar02]

and confirmed experimentally through an independent estimation from the field dependent

suppression of the A-B transition by Gervais et al. [Ger02a]. In Ginzburg-Landau limit and

below the PCP, the width of the A phase in magnetic field can be derived to have the following

form,


ST, B lB
T, B \B, (2

Gervais et al. extract a value of g(P) at 34 bars in aerogel that also matches that of 15 bars in

bulk. It is worth mentioning that Tae at 33.5 bars also falls on T, around 15 bars. However, the

A y-like phase suggested by Fomin [Fom04] based on the robust phase requires a quite different

asymmetry ratio. In this case, the asymmetry ratio

rF = -{ + B/l(P, + Ps)}', (2-10)









where B = 94, + P, + 5P4 + 4P5 and reaches = 0.16 in the weak coupling limit [Bar06]. This

asymmetry ratio is inconsistent with our observation, even when reasonable variations in the /7

parameters are allowed. Fomin adopted a condition, Ay, = A~ in the presence of magnetic

Shields, to preserve an exact isotropy of the robust phase. According to a recent theoretical work

by Baramidze and Kharadze [Bar06], by introducing a small imbalance (Att aA 8) in the

Fomin's model, a more symmetric splitting was achieved.

Sauls and Sharma [Sau03] suggested that the anti-ferromagnetic coupling of 0. 1~0.2 mK

between spins in solid and liquid might be responsible for the suppressed splitting below 0.5 T

observed by Gervais et al. [Ger02a]. Their calculation shows that the slope of the splitting starts

to increase smoothly around 0.5 T (= exchange Hield strength between spins in solid) and

reaches the slope close to that of bulk superfluid in high fields (Fig. 2-1). Our data cannot

confirm this behavior owing to the lack of a low Hield temperature calibration. We point out that

the data points acquired by assuming constant warming rate (Fig. 2-12) characteristically fall

below the linear fit in the low Hield region. This fact, along with the observations made in low

fields by Gervais et al. and us might suggest the presence of antiferromagnetic exchange

coupling between the localized and mobile 3He spins. This brings up an interesting possibility of

a completely new Kondo system.
















2.17 I


2.11






0 ~1x04 2,00 3x104 4xiO4
B(G~au ss)
Figure 2-1. Splitting of the phase transition temperature in fields [Sau03]. Circles represent the
low field data from Gervais et al. [Ger02a] and lines represent the theoretical
predictions by Sauls and Sharma [Sau03]. Green lines are calculated including
antiferomagnetic exchange between liquid and solid layer of 3He and red lines are
calculated excluding it. Without the spin exchange interaction, the Tc splitting is
comparable to that of pure 3He. Reprinted figure with permission from J. A. Sauls
and P. Sharma, Phys. Rev. B 68, 224502 (2003). Copyright (2003) by the American
Physical Society.











Pressure
~ Garge

_ Acoustic Cavity

-- Vib~rating Wire


Titianium body ~

Silver Sinter I

Silver Base ~


(a) Experime ntal Cell


Spacer (~250 pCm)





Aerogel (98?' porosity)


(b) Acoustic Cavity

Figure 2-2. Cut-out views of the experimental cell and the acoustic cavity.


Longitudinal Sound
Transducer




Tra nsverse Sound
Tra nsdu~er






























Resistor (-








Pre-amplifier





Figure 2-3. Schematic diagram of the vibrating wire (VW).

























































Figure 2-4. Arrangement of the experimental cell (top) and the melting pressure thermometer
(bottom) on the silver heat link extending below the nuclear demagnetization stage to
the high field region.



49









































Attenuator atI' n. :1l.:.r.
(F, 50R-0281) (a FW, 50R-0E






2 4



3 Q uadrature Hybrid
(SMC, DOK-3-32S)



Transducer




Figure 2-5. Schematic diagram of the continuous wave spectrometer (modified from Lee' s
design [Lee97]). The arrows represent flow of signal. The signal shape at each step is
shown next to the arrow. The model number of each component is specified in
parenthesis.




















10 --
Set1 (Normal)
--s 0 's----Set1 (Superfluid)

ed~, 's~ 'S Set2 (Normal)
'~~ '; -- Set2 (Superfluid)





-1 '
lil li
8.8 8.9 .69 8.9 866 .9
Frqec Mz

Figur 2-.Feunysepfrtodfeetsetrmtrstig t2. asad3T h
da shdlnsaemaue tlwtmeaue hc sde ntesprli n h
soli lie ar curdi h omlfudpae e1adSt ersn ifrn
attenatio setn inc pcrmtr I e2(e rcs, ecudrdc h
asmmtr in the maiu n iiu .Se2as hfstecrsigpiti h
noma an suefli trcscoert eo

















j Cooling

L T
C aAB
5: 4-

S0 T,



(a) acTo
-4
1.4 1.6 1.8 2.0 2.2 2.4

T (m K)



SWarming T
--- Cooling


cT
02 aAB

0' T
-2 ac

Q aAB
(b) T
-4
1.4 1.6 1.8 2.0 2.2 2.4

T (mK)




Figure 2-7. Zero field acoustic signals at 28.4 bars for different spectrometer settings as a
function of temperature. Panel (b) presents data taken after fine tuning of the
spectrometer to Set2 in Fig. 2-6. In Panel (a) the acoustic response at 1.4 mK is much
larger than at T. In Panel (b) the acoustic responses at 1.4 mK and T, are almost
the same as at Tand the changes in slope at T,, and at T,, on warming are clearer,
compared with those in Panel (a).




















I .II I I I


3.2


P = 28.4 bars /1-

2.8-







2.0 ,



1.6C I


0 2 4 6 8 10 12 14 16

B (Tesla)




Figure 2-8. For high Hields, we used the MPT calibration of Fukuyama's group [Yaw01]. Below
3.5 mK the calibration is given only for the high Hield phase, i.e. in the area under the
dash-dotted line. The sensitivity of the MPT becomes too low to determine
temperature well below the red dotted line. We used the width of bulk Al phase as a
Eixed point for the temperature calibration and obtained transition temperatures in
aerogel (closed circle). See detailed calibration procedure in the text.






















TaA2 T ~jTa
aA1


B = 3 Tesla


2400


2600


2000 2200 2400 2600

Time (min)






aA2

TA2 TA1


TaA1


B =11 Tesla
2000 2200 2400 2600

Time (min)


Time (min)


Time (min)


h
3
ed
a,
o
o
n
v,
a,
rr


2200 2400 2600 2800


1200 1400


Time (min) Time (min)




Figure 2-9. Acoustic traces for 3, 5, 7, 9, 11i, and 15 T at 33.5 bars on warming. The sharp jumps
in the VW trace are identified as the Al and A2 transitions in the bulk liquid. The
acoustic trace also shows two sharp features at the exactly same time positions.

T,1(2) indicates the position of the Al(2) transition in aerogel. The straight lines in the

top-left panel are shown to illustrate the change in slopes at the transition.







































2200 2400 2600 2800
Time (min)


-P = 33.5 bars
B = 3 Tesla \~




-\I TaA2


2400 2600 2800
Time (min)


3000 3200


2







- 2

0

0 O

-2


2400


2600


Figure 2-10. Acoustic traces for 1, 2 and 3 T. Pressures for each plot are labeled in the panels.
Below 3 T, we were not able to resolve the double transition features in aerogel,
while the features from the bulk can be traced at 1 T.


Time (min)
























T aA2 TA1 T

T1 aA1

B = 3Tesla

800 1000 1200

Time (min)









4




0 200 400 600 800

Time (min)


aA2

1 T~
-2 -

B = 9Tesla
2400 2600 2800 3000 3200

Time (min)







|IT



oC T

B =11 Tesla 'A2
1600 1800 2000

Time (min)








TA1

41 aA2 aTaA

P =28.4bars TA2 :1
B = 15Tesla
1400 1600 1800 2000

Time (min)


---


T


ed
4 m
r
(I,
o
O "
cn
o
o
Q


B = 7 Tesla
0 200 400 600

Time (min)


A1-


Figure 2-11. Acoustic traces for 3, 5, 7, 9, 11, and 15 T at 28.4 bars on warming. Each graph
shows the acoustic (lower) trace along with that of the vibrating wire as a function of
time. The sharp jumps in the vibrating wire trace are identified as the Al and A2
transitions in the bulk liquid. The acoustic trace also shows two sharp features at the
exactly same time position of Al(2) transition in aerogel.
















-2 0 2 4 6 8 10 12 14 16

B (Tesla)


IIIIIIIIIIIIIIIII


P = 28.4 bars


,6'


2.8-



2.4-



2.0-


-P = 33.5 bars

-'


2.8



2.4


2.0 t


-2 0 2 4 6 8

B (Tesla)


10 12 14 16


Figure 2-12. Transition temperatures vs. magnetic field for 28.4 and 33.5 bars. Open (solid)
circles are for the bulk (aerogel) transitions determined by the two point calibration
scheme. Open squares for 28.4 bars are obtained by the single point calibration
method. Crosses are based on the constant warming rate. See the text for the
temperature calibration procedures. The solid (dashed) lines are the results of linear
fit for aerogel (bulk).














0.04-



0.03-







-0.03-
P = 28.4 bars

0 2 4 6 8 10 12

B (Tesla)


0.05 ..

0.04 o

i'0.03 -




S-0.0

-0.04 -
P = 33.5 bars

-0.05 ***
0 2 4 6 8 10 12

B (Tesla)



Figure 2-13. Degree of splitting for each individual transition vs. magnetic field. AT is defined
by AT = TA1(2) TC Where Tc is the zero field superfluid transition temperature. AT
is similarly defined for aerogel. For the Aliz transition, AT is positive (negative).
Filled (open) symbols are for the aerogel (bulk).









Table 2-1. Slopes T/ B (mK/T) of the splitting for the Al and A2 transitions in bulk and aerogel at
28.4 and 33.5 bars. The asymmetry ratios, r for bulk and ra for aerogel, are also
listed (see the text for a definition).
Tn/IB T,,2/B Ir T z/B TA2/B r
28.4 (bars) 0.038 -0.026 1.4610.06 0.034 -0.030 1.1310.10
33.5 (bars) 0.043 -0.028 1.5410.11 0.042 -0.035 1.2010. 14









CHAPTER 3
THE A PHASE OF SUPERFLUID 3He INT 98% AEROGEL

3.1 Overview

Early NMR studies of 3He/aerogel show evidence of an equal spin pairing (ESP) state

similar to the bulk A phase [Spr95] and a phase transition to a non-ESP state similar to the bulk B

phase [Bar00b]. A large degree of supercooling was observed in this phase transition (A-B

transition), indicating the transition is first order. Further studies using acoustic techniques

[Ger02a, Naz05] and an oscillating aerogel disc [Bru0 1] have confirmed the presence of the A-B

transition in the presence of low magnetic Hields.

While the effects of impurity scattering on the second order superfluid transition have been

elucidated by these early studies, experiments designed to determine the effects of disorder on

the A-B transition have been rather inconclusive [Ger02a, Bru0 1, Bar00b, Dmi03, Naz04a,

Bau04a]. It is important to emphasize that the free energy difference between the A and B phases

in bulk 3He is minute compared to the condensation energy [Leg90]. Moreover, both phases

have identical intrinsic superfluid transition temperatures. The nature of highly competing

phases separated by a first-order transition is at the heart of many intriguing phenomena such as

the nucleation of the B phase in the metastable A phase environment [Leg90], the profound effect

of magnetic fields on the A-B transition [Pau74], and the subtle modification of the A-B transition

in restricted geometry [Li88]. We expect this transition to be extremely sensitive to the presence

of aerogel and conj ecture that even the low energy scale variation of the aerogel structure would

have a significant influence on the A-B transition.

A number of experiments have been performed with the purpose of systematically

investigating the A-B transition in aerogel [Ger02a, Bru01, Bar00b, Naz04a, Bau04a]. In

experiments by the Northwestern group [Ger02a] using a shear acoustic impedance technique, a










significantly supercooled A-B transition was seen while no signature of the A-B transition on

warming was identified. In the presence of magnetic Hields, however, the equilibrium A-B

transitions were observed and the Hield dependence of the transition was found to be quadratic as

in the bulk. However, no divergence in the coefficient of the quadratic term g(/7) (see Eq. 2-9)

was observed below the melting pressure. This result is in marked contrast with the bulk

behavior that shows a strong divergence at the polycritical point (PCP) [Hah93]. The

Northwestern group concluded that the strong-coupling effect is significantly reduced due to the

impurity scattering and the PCP is absent in this system. Although this conclusion seems to

contradict their ob servation of a supercooled A-B transition even at zero Hield, other theoretical

and experimental estimations of the strong-coupling effect in the same porosity aerogel [Cho04a,

Bar02] seem to support their interpretation. On the other hand, the Cornell group [Naz04a]

investigated the A-B transition in 98% aerogel using a slow sound mode in the absence of a

magnetic field. While the evidence of the supercooled A-B transition was evident, no warming

A-B transition was observed. Nonetheless, they observed a partial conversion from B A phase

only when the sample was warmed into the narrow band (= 25 ptK) of aerogel superfluid

transition. Recently, the Stanford group conducted low field (H = 284 G) NMR measurements

on 99.3% aerogel at 34 bars [Bau04b]. They found a window of about 180 CIK window below

the superfluid transition where the A and B phases coexist on warming with a gradually

increasing contribution of the A phase in the NMR spectrum (Fig.3-1).

For two sample pressures of 28.4 and 33.5 bars, we have observed the A-B transition on

warming in the absence of a magnetic field and have found evidence that the two phases coexist

in a temperature window that is as wide as 100 ptK. This chapter provides a detailed description

of our results and interpretations.









3.2 Experiments

We used the same experimental techniques that were described in Chapter 2. The sample

cell experience a stray Hield of less than 10 G, which arises from the demagnetization magnet.

3.3 Results

The traces of acoustic signal taken near 28.4 bars in zero magnetic Hield are shown in Fig.

3-2 (a). The traces show the acoustic responses between 1.4 and 2.5 mK. The sharp jumps in

the acoustic traces around 2.4 mK mark the bulk superfluid transition and the distinct slope

changes are associated with the superfluid transitions in aerogel (see chapter 2). The signatures

of the supercooled A-B transition in the bulk and aerogel appear as small steps on the cooling

traces. The identification of the step in the acoustic impedance as the A-B transition has been

established by a systematic experimental investigation of Gervais et al [Ger02a]. The cooling

trace (blue) from the normal state of bulk reveals a well defined aerogel transition at 2.0 mK

( T,). The supercooled A-B transitions from the bulk (TAB ) and aerogel (T,, ) are clearly shown

as consecutive steps at lower temperatures. After being cooled through both A-B transitions,

both clean and dirty liquids are in the B phase. On warming the trace follows the B phase and

progressively merges into the A phase (cooling trace) around 1.9 mK. This subtle change in

slope is the signature of the A-B transition on warming. This was the first indication of a

possible A-B transition in aerogel on warming at zero field.

Figure 3-2 (b) at 33.5bar shows similar features, but the supercooled A-B transitions in

aerogel and bulk occur at almost the same temperature. On several occasions, we have observed

that, when the liquid cooled from the normal phase, the supercooled A-B transition in bulk and

aerogel occur simultaneously. Furthermore, the supercooled bulk A-B transition always precedes

the aerogel transition, which might suggest that the aerogel A-B transition is induced by the bulk









transition through proximity coupling. However, we have not carried out a systematic study on

this issue.

In order to test our identification of this merging point as the warming B RA transition in

aerogel, we performed tracking experiments similar to those described by Gervais [Ger02a]. The

sample liquid is slowly warmed from the aerogel B phase up to various points around the feature,

and then cooled slowly to watch the acoustic trace for the signature of the supercooled A-B

transition. During the turn around, the sample stays within 30 CIK from the highest temperature

reached (hereafter referred to as the turn-around temperature) for about an hour. If the warming

feature is indeed the A-B transition, there should be a supercooled signature on cooling only after

warming through this feature. The color coded pairs of the traces in Fig. 3-3 (a) are the typical

results of the tracking experiments for different turn-around temperatures at 28.4 bars. In the

bottom (blue) cooling trace from the normal state, one can clearly see two supercooled A-B

transition steps. The sharper step appearing at 1.7 mK corresponds to the bulk A-B transition.

We find that the size of the step indicating the aerogel A-B transition depends on the turn-around

temperature. We can make a direct comparison of each step size since the supercooled A-B

transitions in aerogel occur within a very narrow temperature range, = 40 IK. Similar behavior

was observed for 33.5 bars as shown in Fig. 3-3 (b).

From the data obtained in the tracking experiments, the relative size of the steps at the

supercooled aerogel A-B transition, is plotted in Fig. 3-4 as a function of the turn-around

temperature. The relative size is the ratio of the step size for each trace and the step size for the

trace cooled down from normal fluid. For both pressures we see narrow temperature regions

(shaded regions in the figure) where the size of the steps grows with the turn-around temperature

until T is reached. For T > T no appreciable change in the step size is observed. This










suggests that only a portion of the liquid in aerogel undergoes the B A conversion on warming

in that region. An inevitable conclusion is that the A and B phases coexist in that temperature

window. A similar behavior has been observed in 99.3% porosity aerogel, although in the

presence of a 284 G magnetic field [Bau04b]. Nazaretski et al. also found the coexistence of the

A and B phases at zero field from low field sound measurements [Naz04b]. It is worthwhile to

note that at 10 G, the equilibrium A-phase width in the bulk below the PCP is less than 1 IK. A

quadratic field dependence in the warming A-B transition is observed in our study up to 2 kG

(Fig. 3-5). The suppression of B-like phase is proportional to B2 as seen in the previous

experiment by Gervais et al. [Ge02a]. Unfortunately, no information on the spatial distribution

of the two phases can be extracted from our measurements.

3.4 Discussion

In Fig. 3-6, a composite low-temperature phase diagram of 3He in 98% aerogel is

reproduced along with our A-B transition temperatures. We plot the lowest temperatures where

the B A conversion is first observed on warming. It is clear that the slope of the A-B transition

line in aerogel has the opposite sign of that in the bulk in the same pressure range. However, in a

weak magnetic field, the slope of the bulk A-B transition line changes its sign from positive

( p < pc where pc represents the polycritical pressure) to negative (p > pc ) (see the dotted line

in Fig. 3-6)3. The first order transition line is governed by the Clausius-Clapeyron relation,


dTBP s A (3-1)


where s and v represent molar entropy and volume for the A and B phase. Since s, < sA, the


3 In the bulk, this behavior persists even in high fields up to the critical field. The sign crossover point gradually
moves down to around 19 bars near the critical field. See Ref. Hah93.









change of the slope indicates the sign change in Av = v, vA It is interesting to ponder why the

strong coupling effects cause this sign change. We find no published results addressing this

issue. Nonetheless, this observation and the fact that no divergence in g(P) has been observed

in aerogel allow us to look at the phase diagram from a different point of view. It appears, that in

effect, the P-T phase diagram is shifted to higher pressures in the presence of aerogel rather than

simply shifted horizontally in temperature. As a result, the PCP has moved to a physically

inaccessible pressure as a liquid, thereby leaving only the weak coupling dominant portion in the

phase diagram. If this interpretation is correct, then we have to face a perplexing question: How

can we explain the existence of a finite region of the A phase at pressures below the PCP at B =

0? We argue that anisotropic scattering from the aerogel structure is responsible for this effect.

Although there is no successful quantitative theoretical account of the A-B transition for p > pc,,

the Ginzburg-Landau (GL) theory presents a quantitative picture for the A-B transition in a small

magnetic field B, (relative to the critical field) for p < pc Under these conditions, the quadratic

suppression of the A-B transition arises from a term in the GL free energy, namely

.f = g B,,A,, Aj B,., (3 -2)

where A,, represents the order parameter of a superfluid state with spin (p) and orbital (i) indices

[Vol90]. The main effect of this term is to produce a tiny splitting in To for the A and B phases.

In the GL limit, the free energy (relative to the normal state) of the A (B) phase is

f,,, = -a /2# 3, and ap,i, is the appropriate combination of 13 parameters that determine

the fourth-order terms in the GL theory. For p < pe, the B phase has lower free energy than the

A phase ( P, < P ). However, when the two phases are highly competing, i.e., P, = Ps even a










tiny splitting in the superfluid transition, 6 = T, TCB(> 0) results in a substantial temperature

region (much larger than m ) where the A phase becomes stable over the B phase.

The simplified representation of the aerogel as a collection of homogeneous isotropic

scattering centers is not sufficient to describe minute energy scale phenomena such as the A-B

transition. The strand-like structure introduces an anisotropic nature in the scattering, e.g.,

p-wave scattering. This consideration requires an additional term in the GL free energy

[Thu98b, Fom04],

fa = alalALA L5a (3-3)

where a is a unit vector pointing in the direction of the aerogel strand. In other words, the

aerogel strand produces a random Hield that couples to the orbital component of the order

parameter. This random orbital Hield plays a role analogous to the magnetic Hield in spin space,

thereby splitting the superfluid transition temperature. If the correlation length of the aerogel

structure, 5, is longer than the length scale represented by 4 the local anisotropy provides a

net effect on the superfluid component, and f, would give rise to the A-B transition, even in the

absence of a magnetic Hield. Detailed free energy considerations indicate that the anisotropy

would favor the cit 11 configuration for the A phase [Rai77], where 1 indicates the direction of

the nodes in the gap. Using the expression for the coupling strength a, calculated in the

quasiclassical theory [Thu98b], we Eind that f, is comparable to f, produced by a magnetic

Hield B, = Irg (Tc / p)1 ~ 1 kG, where 7 is the gyromnagnetic ratio of 3He and is

the mean free path presented by the impurity scattering off the aerogel strand. Since



4 Private conununication with J.A. Sauls.










Be oc 1M [ fn a ( / ) ], where & / represents the anisotropy in the mean free path, only a

fraction of 1% anisotropy is sufficient to produce the observed A phase width. The

inhomogeneity of the local anisotropy over length scales larger than 5, naturally results in the

coexistence of the A and B phase.

A PCP where three phases merge, as in the case of superfluid 3He, should have at least one

first order branch [Fom04, Yip91]. When this branch separates two highly competing phases

with distinct symmetry, the PCP is not robust against the presence of disorder. In general, the

coupling of disorder to the distinct order parameters will produce different free energy

contributions for each phase. Consequently, a strong influence on the PCP is expected under

these circumstances [Aoy05]. It is possible that the PCP vanishes in response to disorder (as it

does in response to a magnetic field) and a region of coexistence emerges. An experiment on

3He-4He mixtures in high porosity aerogel reported a similar disappearance of the PCP [Kim93].

Strikingly similar phenomena have also been observed in mixed-valent manganites where the

structural disorder introduced by chemical pressure produces the coexistence of two highly

competing phases (charge ordered and ferromagnetic phases) separated by a first-order transition

[Dag0 1, Zha02]. A growing body of evidence suggests that the coexistence of the two phases is

of fundamental importance in understanding the unusual colossal magnetoresistance in that

material.

Considering the energy scales involved in the A-B transition and anisotropy, it is not

surprising to see a difference in the details of the A-B transition in aerogel samples even with the

same macroscopic porosity. However, it is important to understand the role of anisotropic

scattering. We propose that the effect of anisotropic scattering can be investigated in a









systematic manner, at least in aerogel, by introducing controlled uniaxial stress, which would

generate global anisotropy in addition to the local anisotropy.

Aoyama and Ikeda have studied the A-like phase in the Ginzburg-Landau regime and

found that the quasi-long-ranged ABM state has a free energy lower than the planar and the

robust states [Aoy05]. Neglecting the inhomogeneous effect of aerogel on the order parameter,

they parameterized the impurity effect only through the relaxation time and found that the

impurity scattering weakens the strong coupling effect. The reduction of the strong coupling,

which is consistent with our result in chapter 2, shrinks the width of ABM phase, T,, T ,.

They also studied an inhomogeneity effect of the impurity scattering, especially with strong

anisotropy, and found the anisotropy increases the width, T,, T, 3. Their expectation on the

anisotropy scattering is consistent with our arguments. They interpreted the existence of the A-

like phase below the bulk PCP [Ger02a, Naz04b, Bau04b] as a lowering of the PCP, and this

effect is explained qualitatively by taking into account the anisotropic scattering on the ABM

state. The pressure of the PCP, which is different from that of bulk, was determined by the

competition between two effects from the reduced r and the anisotropy scattering. Inspired by

our suggestion, Aoyama and Ikeda calculated the effect of global anisotropy induced in

uniaxially deformed cylindrical aerogel of two distinct flavors, compressed and stretched along

the symmetry axis [Aoy06]. For both cases, they found that the ABM state opens up for all

pressures. Furthermore, in stretched aerogel, a sliver of a new phase, the polar phase is predicted

to appear just below Ta,.

Recently, Pollanen et al. performed x-ray scattering experiments on the aerogel and found

that a certain degree of anisotropy exists even in an undeformed aerogel sample and argued that

the global anisotropy could be generated during the growth and drying stages [Pol06]. Now, P.










Bhupathi in our group is pursuing a transverse acoustic measurement on the superfluid 3He in

compressed aerogel. He did cut a commercial 98% aerogel sample and put it on a transducer.

The other side of the aerogel was pressed by a cap for 5% in length or just enough to hold the

aerogel. With 5% uniaxial compression, both the supercooled and stabilized AB transitions in

aerogel could be identified, but not in uncompressed aerogel. We suspect that rough surface of

aerogel establishes large open space on the boundary to the transducer, except at some contact

points. Because of the high attenuation of the transverse sound, the transducer measures the

liquid property very near to its own surface and if the open space is dominant, the acoustic

response will be the same as those from the bulk. It is important to measure the A-like phase for

a sample without compression as a bench mark. For the next trial, aerogel will be grown on a

transducer in situ by the Mulders' group at the University of Delaware.







































30 -


25 -


m 20-




10 -


5 -


0-


bulk \
bulk a
AB

Tea
eo A
ABr


r*


I I I I I I I I I I I


0.8 1


0.2 T .o


0.0-

2.20 2 22 2.24


O


T ""'.







I~~ .


2.26 2.28 2.30 2.32 2.34
Temperature (mK)


2.36 2.38 2.40


2 114 118 18 210
Temperature (mK)


2.2 214


Figure 3-1. The Stanford NMR measurement on superfluid 3He in 99.3% aerogel. (a) The
fractions of A-like phase vs. temperatures. The superfluid 3He in 99.3% aerogel was
warmed up from B phase at low temperature and was cooled down at certain
temperature. The temperatures of data (open circle) represent those turning points.
The fraction was determined from the weight of the NMR line for the A-like phase
[Bau04a]. (b) The phase diagram of superfluid 3He in 99.3% aerogel at 28.4 mT
[Bau04b]. The closed circles represent the AB transition on warming. Reprinted
figure with permission from J.E. Baumgardner, Y. Lee, D.D. Osheroff, L.W.
Hrubesh, and J.F. Poco, Phys. Rev. Lett. 93, 055301 (2004)] and [J.E. Baumgardner
and D.D. Osheroff, Phys. Rev. Lett. 93, 155301 (2004). Copyright (2004) by the
American Physical Society.


























-4 L
1.4


10

8


1.6 1.8 2.0 2.2 2.4
T (mK)


-29=3pas P ="L 33. bas a
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
T (m K)


Figure 3-2. Cooling (blue) and warming (red) traces taken at 28.4 bars (a) and 33.5 bars (b). The
signatures of the aerogel superfluid transitions and the aerogel A-B transitions are
labeled as T, and T, -













I I I I I I


-o5C














20-P = 33.5 bar


Isis
1.0 1.5 2.0 2.5

T (mK)




Figue 33. coutictraes f tackng epermens a 284 brs nd 3.5bar nzromgei
fil.Ec aro amn n usqet oln sclrcdd h unaon


te prtrsaeidctdb tevria iefrec ai.Frcaiytetae r
sitdvrtial n etcllnsaeadd auly h ro sidct h
dieto ftmeatr hnei ie















1.2 -




u 0.81. i-f

II
0.6--


oN 0.4- -

S0.2 C ;' i I 28.4 bar
I / 0'33.5 bar


1.6 1.8 2.0 2.2 2.4 2.6
Turn-around Temperature (mK)



Figure 3-4. The relative size of the steps for the supercooled aerogel A-B transition is plotted as a
function of the turn-around temperature for 28.4 and 33.5 bars. The relative size is
the ratio of the step size for each trace and the step size for the trace cooled down
from normal fluid. The lines going through the points are guides for eyes. The
dashed vertical lines indicate the aerogel superfluid transition temperatures.




















2.2 --














S1.4 I*





Fiue35.Feddeedne ftewamn -Btastini erglfr 28.4 bars, (buecrce

an 33.5 bars (e ice.Tiage ersn Tacfoeahpsur.Tedhd
lines show thae agevaueo Taca ndtedte ie r ierft fT nec
prssr.


















30


20 -




10C -





0.0 0.5 1.0 1.5 2.0 2.5

T (m K)




Figure 3-6. The zero field phase diagram of superfluid 3He in 98% aerogel along with that of the
bulk (dashed lines) [Gre86]. The dotted line is the bulk A-B transition at 1.1 kG
measured by Hahn [Hah93]. The aerogel superfluid transition line shown in blue is
obtained by smoothing the results from Cornell and Northwestern groups [Mat97,
Ger02a]. The two closed circles are the lowest temperatures where the B A
conversion starts on warming, and the points are connected by a dashed line that is a
guide for the eyes.









CHAPTER 4
ATTENUATION OF LONGITUDINAL SOUND IN LIQUID 3He/98% AEROGEL

4.1 Overview

High frequency ultrasound lends us a unique spectroscopic tool for investigating the

superfluid phases of liquid 3He [Hal90, Vol90]. The frequency range of ultrasound conveniently

matches with the superfluid gap, and the exotic symmetry of the superfluid phases allows various

acoustic disturbances to couple to the superfluid component. The sharp pair-breaking edge

occurs at Ami = 2A(T), where 0i is the ultrasound excitation frequency and A(T) represents the

temperature dependent superfluid gap. The coupling of ultrasound to many of the order

parameter collective modes has been demonstrated by strong anomalies in sound attenuation and

velocity [Hal90]. All of these rich ultrasound spectroscopic signatures could be elucidated

because zero sound continues to propagate in the superfluid phase despite the presence of a gap

at the Fermi energy.

In the presence of high porosity aerogel, the mean free path, La = vF a,, preSented by the

98% aerogel is in the range of 100 200 nm as mentioned earlier. This length scale competes

with the inelastic quasiparticle scattering length, 2 = vF 2 and becomes relevant only below

=10 mK. Although impurity scattering has a marginal influence on the thermodynamic

properties of the normal liquid, a significant change in the transport properties, specifically

thermal conductivity, has been predicted at low temperatures [VenOO].

The pair-breaking mechanism from impurity scattering is known to induce impurity bound

states inside the gap and to smooth the square root singularity at the gap edge [Buc81]. In the

presence of severe pair-breaking, the system turns into a so-called gapless superconductor with

the gap completely bridged by the impurity states. The impurity states have a broad influence on









the physical properties in the superfluid states. In superfluid 3He/aerogel, the gapless nature has

been evidenced by recent thermal conductivity and heat capacity measurements [Fis03, Cho04b].

The acoustic properties of liquid 3He in aerogel are also affected by the presence of

impurity scattering [Rai98, Gol99, Nom00, Ger01, Ich01, Hig03, Hig05, Hri01]. The classic

first to zero sound crossover in the normal liquid was found to be effectively inhibited by the

impurity scattering, maintaining carz < 1, where r = rfz, /(zo + r, ) [Nom00]. It was also argued

that an attempt to increase the excitation frequency would face an extremely high damping

[Rai98]. Losing the luxury of having well defined zero sound modes at low temperatures has

hampered investigations on the superfluid phases in aerogel using high frequency ultrasound.

Although the high frequency transverse acoustic impedance technique [Lee99] has been

successful in identifying various transition features [Ger02a, Cho04a, Vic05], only a few

attempts have been made to investigate the superfluid gap structure using conventional high

frequency longitudinal sound by Northwestern group [Nom00] and Bayreuth group [Hri01].

However, the Bayreuth experiment, which used a direct sound propagation technique, suffered

from poor transducer response and observed no suppression of the superfluid transition

temperature in aerogel. We suspect that there might be a thin layer of bulk liquid between the

transducer and aerogel, which complicated the sound transmission through the liquid in aerogel.

No published results are available from this work other than the aforementioned reference.

In 2000, Nomura et al. conducted high frequency sound (14.6 MHz) attenuation

measurements of liquid 3He in 98% aerogel at 16 bars using an acoustic cavity technique

[Nom00]. They found that the crossover from first to zero sound was effectively obscured by the

impurity scattering of aerogel strands below = 10 mK (Fig. 4-1). This behavior was rather easily

understood in the framework of a simple viscoelastic model. However, this approach failed to










explain their results at higher temperatures, where the inelastic scattering between the quasi-

particles is dominant. Furthermore, the sound attenuation in aerogel was found to saturate

around 50 mK rather than follow a 1/ T2 dependence as observed in the bulk liquid. Higashitani

et al. attempted to explain these results by incorporating a collision drag effect [Nom00, Hig03].

The Northwestern group employed a cw method using a short path length acoustic cavity, which

is not adequate in determining the absolute sound velocity or, especially, the attenuation. This

cw method relies on observing interference patterns developed in the cavity, and under

conventional operation this pattern can be generated by sweeping the excitation frequency.

However, with a high Q quartz transducer, this method is not feasible. Therefore, they had to

sweep sample pressure to generate the necessary variations of the sound velocity, and this

approach inevitably accompanies temperature drift. In addition to these difficulties, an auxiliary

assumption had to be made to perform a two-parameter fit in attenuation and the reflection

coefficient at the transducer surface In this work, we present our results of high frequency (9.5

IVHz) longitudinal sound attenuation measurements using a pulsed ultrasound spectroscopic

technique.

4.2 Experiments

Figure 4-2 (a) shows the schematic diagram of the experimental cell used for this study.

The conceptual design of the cell is similar to the one used in the transverse acoustic impedance

experiment described in previous chapters. The experimental cell consists of a pure silver base

with 9 m2 Of silver sinter and a coin silver enclosure. The ceiling of the enclosure forms a

diaphragm, so the cell pressure can be monitored capacitively. Dimensions of each part can be

5 They assumed that the attenuation at 25 bars and 0.6 mK is zero. The reflection coefficient was determined as 0.8,
which satisfies the pressure dependence of attenuation at Tc The attenuation of each pressure was calculated by
the visco-elastic model. The visco-elastic model contains an aerogel mean free path as another parameter, which
was taken as 240 nm in their work.









found in appendix B. Two matched longitudinal LiNbO3 transducers (A and B as indicated in

Fig. 4-2 (a)) are separated by a Macor spacer, which maintains a 3.05 (+ 0.02) mm gap between

them. The aerogel sample was grown in the space confined by the transducers (1/4" diameter,

9.5 MHz resonance frequency) to ensure the contact between the transducer surface and the

aerogel sample. This is an extremely important procedure since a thin layer of bulk liquid or an

irregular contact between the transducer and aerogel would cause an unwanted reflection at the

boundary of aerogel and liquids [Joh94a]. A 1 MHz AC-cut quartz transducer is placed right

above the transducer A to monitor bulk response using a FM modulated cw method as described

in chapter 2 (12.8 MHz, 500 Hz deviation frequency, 100 Hz modulation frequency). A piece of

cigarette paper interrupts the path between the transducer A and the quartz transducer in order to

spoil the back reflection. For transducer B, the irregular surface of silver sinter effectively

diffuses unwanted sound propagation through the bulk liquid. The sample cell is thermally

attached to the gold-plated copper flange welded to the top of the Cu nuclear demagnetization

stage. An MPT and a Pt wire NMR thermometer (PLM-3, Instruments of Technology, Helsinki,

Finland) are located right next to the sample cell on the same flange. The MPT was used for T >

1 mK and the Pt NMR thermometer, calibrated against the MPT, was used for T < 1 mK. Figure

4-3 shows temperature determined by MPT and Pt-NMR thermometer as a function of time for a

complete cooling and warming cycle. The Pt-NMR thermometer is calibrated using an equation,

T = M~ /(As B) where M~, As, B are the normalized Curie constant, the magnetization

determined by NMR and of the background signal, respectively. As was obtained by integrating

the NMR free induction decay for a fixed time span. Two coefficients (M~ and B) were treated

as fitting parameters to match the temperature determined by the MPT. The green line in

Fig. 4-3 was obtained in this way. As a result the thermal gradient between the MPT and the









Pt-NMR thermometer the green line does not match perfectly with the black line. However, by

shifting about 5 minutes in Pt-NMR response one can find an excellent consistency between two

thermometers (red line in Fig. 4-3). The typical warming and cooling rates were 3 CIK/min.

A commercial spectrometer, LIBRA/NMRKIT II (Tecmag Inc., Houston, TX) was

employed for this study (Fig. 4-2 (b)). The same spectrometer was used for acoustic

measurement on pure liquid 3He by Watson at UF [Mas00, Wat00, Wat03]. Typically, the

output level from the NMRKIT II was set to the maximum of 13 dBm and was fed to the

transmitter through a -20 dB attenuator. This input signal, after amplified to an appropriate level

by a power amplifier, was used to excite transducer B. A 1 Cls pulse was generated by the

transducer B (transmitter) and the response of the transducer A (receiver) was detected. The

acoustic signal from the receiver was amplified by a low noise preamplifier, Miteq AU-1114,

and was passed to NMRKIT II. The data acquisition began right after the end of the excitation

pulse. The width of pulse, 1 Cls, is short enough to separate echoes in the low attenuation regime

and wide enough in frequency to cover the resonance spectrum of the transducers. Although a

matched pair of transducer was used, the slight difference in resonance spectra of two

transducers determines the overall shape of the response (see next section). The typical setting

of the LIBRA/NMRKIT-II can be found in appendix C. Origin scripts are used for data analysis

and the details are in Appendix D. Each measurement presented in this work is the result of

averaging 8 pulses responses generated in the phase alternating pulse sequence at 9.5 MHz of

primary frequency. The data was taken every 5 minutes and the interval between pulses for the

average was 4 seconds. All the measurements presented in this work were done at zero magnetic

field.









4.3 Results

Figure 4-4 (a) shows the responses of transducer A at 34 bars in the time domain for

selected temperatures. The primary response, which begins to rise around 8 Cls, grows below the

aerogel superfluid transition ( Tc = 2.09 mK, indicated by an arrow in the figure). As

temperature decreases, a train of echoes starts to emerge and four clear echoes are visible at the

lowest temperature. The time interval between the primary response and the subsequent echo is

twice of that between the excitation pulse and the primary response. For 14 bars, only the first

echo is clear at the lowest temperature due to the higher attenuation (Fig 4-4 (b)). From the time

of flight measurements, the speed of sound was determined to be 350 (110) m/s at 34 bars, which

is approximately 20% lower than in bulk. The velocity did not show temperature dependence

within + 3%. At 9.3 k
22 bars [Gol99]. The coupling between the normal component of the superfluid 3He and the

mass of the elastic aerogel modifies the two-fluid hydrodynamic equation [Gol99]. Two

longitudinal sound modes are found and the velocities of slow mode and fast mode ( vs, vf ) are

given as

c,~c,1/1+pa/p, (4-1)


c, = ca / Pp (4-2)

The velocity of the fast mode calculated from the modified equation agrees well with our result

(with a 5% discrepancy at most) (see Fig. 4-5). Due to the ringing of the transducers, the width

of the received pulses is much broader than the excitation pulse (1 Cls). The location of the rising

edge of the signal depended on the sample pressure, and the receiver signal disappeared when the

sample cell was completely evacuated. These observations assure that the detected signal at the

receiver is from the sound pulse passing through the aerogel/liquid. The step-like shape of the










responses, as shown in Fig. 4-6 (a), is caused by a slight mismatch in the spectra of the

transducers. Figure 4-6 (b) shows a Fourier transformed transmitted signal (black circle) and the

frequency profie of each transducer (blue and red lines). The frequency profie of each

transducer was obtained using a duplexing scheme involving a directional coupler. The green

line in Fig. 4-6 (b) is the result of simple multiplication of the two frequency profiles from the

transmitter and receiver. As can be seen, this profie mimics the spectra of the receiver obtained

by the conventional method used in this experiment. A weighted multiplication would improve

the match and the difference in fine structures may be caused by duplexing the signal.

For most of our measurements, -7 dBm excitation pulses were used (+13 dBm excitation

with -20 dB attenuator). Various levels of excitations ranging from 3 to -27 dBm were tested in

order to check the linearity. Figure 4-7 (a) presents the signals for different excitations taken at

40 mK and 33 bars. The signal shows a mild distortion near the peak area at a 0 dBm excitation

and the peak structure is severely distorted with a -3 dBm excitation due to saturation. However,

the unsaturated part of the response scales linearly with the excitation. The linearity of the

acoustic response was confirmed in the superfluid as well where the sound attenuation is lower.

In Fig. 4-7 (b), the signal is normalized to its peak amplitude and the normalized signal for lower

excitations fall on top of the signal from our typical setting (-20 dB). The signal shapes and the

attenuations did not show any dependence on excitation within 5%, which means the sample and

measurement scheme are in the linear regime. The warming traces with -7 dBm and -13 dBm

excitations overlap well within the experimental error indicating that that heating effect caused

by pulse transmission is negligible (Fig. 4-8).

In Fig. 4-6 (a), two normalized primary responses, taken at 0.4 mK and 2.5 mK and at 29

bars, are displayed for comparison. Despite the difference in their absolute sizes, the normalized










responses have an almost identical shape, which indicates that the spectra of the transducers

remain the same throughout the measurements, and the change in the response is caused by the

change in the acoustic properties of the medium. Therefore, the relative sound attenuation can

be obtained by comparing the primary responses. We have used two different methods to extract

the relative attenuation from the signal before being normalized: one was to use the peak value of

the primary response and the other was to use the area under the curve by integrating the signal

from the rising edge (8 Cls point in Fig. 4-6 (a)) to a Eixed point in time (23 Cls point in Fig. 4-6

(a)). Both methods produce consistent results to within 7%, except at 10 bars (13%). The

absolute attenuation at 0.4 mK and 29 bars was used as a reference point in converting the

relative attenuation into the absolute attenuation. Absolute attenuations for the other

temperatures were calculated by comparing the area of the primary response for each

temperature to that for the standard absolute attenuation.

Because of an electronic glitch, probably from a slip of the relay switch in the NMRKIT II,

the spectrometer sensitivity was not guaranteed to be the same for every run. Therefore, data

were collected in one batch for each pressure at a Eixed temperature, 9 mK. The relative

attenuations for all other pressures were obtained by linking it to the attenuation at 9 mK and 29

bars. Only for 34 bars, the absolute attenuation was calculated independently from its own

echoes. We decided the slope of the signal decay from two points, the peaks of the primary

response and the first echo. To check the uncertainty of this method, we compared it to the slope

determined by the peaks of the primary response, the first echo, and the second echo. The

discrepancy of two methods gives a negligible difference on the standard absolute attenuation (+

0.08 cm )~.









As shown in Fig. 4-9, the peaks of the primary response and the subsequent echoes exhibit

a bona fide exponential decay. With the knowledge of the gap between the transducers, the

absolute value of attenuation was determined. Due to the drastic acoustic impedance mismatch

at the boundary between the transducer and the aerogel/3He sample, the echoes were assumed

perfectly reflected at the interface6

The normalized attenuations for 12 and 29 bars as a function of the reduced temperature

are shown in Fig. 4-10 along with that of pure 3He measured at 29 bars using 9.5 MHz sound

excitation [Hri01]. There are three main contributions to the ultrasound attenuation in pure

superfluid 3He: (1) pair-breaking mechanism, (2) coupling to order parameter collective modes

(OPCM), and (3) scattering of the thermal quasi-particles [Hal90]. All the features mentioned

above have been extensively investigated theoretically and experimentally. At ~ 9.5 MHz, the

contributions from (1) and (2) give rise to a strong attenuation peak as clearly shown in the

figure. These salient features are completely missing in the aerogel, although the superfluid

transition is conspicuously marked by the rounded decrease in attenuation. A similar behavior

was observed by Nomura et al. [Nom00] and this behavior was interpreted as indirect evidence

that the sound propagation remained hydrodynamic even in the superfluid phase [Hig05]. We

have checked the primary responses in the normal liquid (13 mK at 29 bars) and the superfluid (1

mK at 34 bars) using higher harmonics of the transducers (up to 96 MHz). Only the 9.5 MHz

excitation produced a recognizable response in the receiver, thereby confirming that the sound

mode at 9.5 MHz remains hydrodynamic even in the superfluid phase. If the sound at 9.5 MHz

was the zero sound, the attenuation would not show that much frequency dependence. The zero



6 Using the composite density of 3He/98% aerogel at 29 bars, pc = 0.113 g/cm measured sound velocity, 330 m/s,
and the known acoustic impedance value for LiNbO3, Z, = 3.4 x 106 g/s-cm less than 1% loss is expected at the
boundary.









sound regimes are still supposed to exist at high frequencies, but the attenuation of zero sound is

expected to be too high for a quantitative measurement.

On cooling at 29 bars (filled circles in Fig. 4-10), our data show a sharp jump at 1.5 mK

( T /T, = 0.7 ). This feature is the supercooled A aB transition in aerogel. On warming, the

sound attenuation follows the B-phase trace of cooling and then merges smoothly with the

cooling trace of the A-phase around 1.8 mK ( T / T, = 0.9) without showing a clear B RA

transition feature. This observation is consistent with the previous transverse acoustic

impedance measurements described in chapter 3 [Vic05], which proved that this merging point is

where the B RA conversion begins. On cooling, the A to B transition is observed down to 14

bars (Fig. 4-11). Neither an A-B transition feature nor hysteretic behavior was observed for 12

bars and below. Therefore, our attenuation results on warming are for the B phase in aerogel,

except for the 200 CIK window right below the superfluid transition at high pressures. A broad

shoulder feature around T/ T, = 0.6 in the 29 bars trace progressively weakens as the pressure

decreases, and this structure eventually disappears, as can be seen in the 12 bars trace. In Fig. 4-

11, the T,, and the T,, determined in this attenuation measurement are plotted along with

previous results presented in chapter 3 (open circle). Our phase diagram is in good agreement

with the ones mapped by different methods [Mat97, Ger02a, Hal04]. In the B phase with a clean

isotropic gap, sound attenuation, a ac e-a(T) kBT (Where k, is the Boltzmann constant) is expected

from the thermal quasi-particle contribution mentioned above. However, the temperature

dependence of attenuation in aerogel is far from exponential and is not quenched down to

TT, /T 0.2 Furthermore, the sound attenuation approaches a fairly high value for both

pressures at zero temperature, allowing a reasonable extrapolation.









The absolute attenuations on warming for several pressures are plotted in Fig. 4-12. The

attenuation increases as pressure is reduced. The shoulder is less pronounced for low pressure

and disappears below 21 bars completely. Zero temperature attenuations a, are estimated by

assuming a quadratic temperature dependence (thin black line) for the low temperature region -

we do not believe that this specific assumption affects our conclusion since our data already

reached very close to zero temperature limit. The attenuation at lowest temperature decreases as

the pressure is enhanced and it saturates above 25 bars. At the superfluid transition temperature,

the attenuation of 8 bars is very close to that of 10 bars. Due to the high attenuation at this

pressure, the signal from receiver was quite small but still we could detect the temperature

dependence below the superfluid transition.

The normalized attenuations show the shoulder features disappear below 21 bars (Fig. 4-

13(a)). Higashitani et al.'s calculations [Hig06] for several pressures are plotted with our data at

34 bars (Fig. 4-13(b)). Their calculations show more pronounces shoulder features and lower

zero temperature attenuation for the same pressure. The absolute attenuation at the superfluid

transition, ac, and the normalzed zero temperature attenuation, a, /ac are given in Fig. 4-14(a).

All values are taken from warming data except at 8 bars, and it is noteworthy that both a, and ac

increase as the pressure is reduced. Although the dashed line following the normalized zero

temperature attenuation is a guide for eye, a, /ac approaches 1 near 7 bars, which is close to the

critical pressure below which no superfluid transition in 98% aerogel has been observed. The

error bar is estimated from the differences of the attenuations calculated by the peak and the area

of the primary response.

Absolute attenuation measurements allow quantitative analyses. Based on the theory of

Higashitani et al. in which collision drag effect is incorporated, one can get the mean free length









in this system. As shown in Fig. 4-14(b), the whole pressure dependence of ac can be nicely fit

with a single parameter for R = 120 nm. Figure 4-15 shows the lower bounds of the zero energy

density of states at T =0 estimated from a, / ac (details in next section).

For the normal liquid, the low temperature part of the attenuation is quite similar to those

reported by Nomura et al. [Nom00]. However, our results show quite different behavior above

40 mK (Fig. 4-16 (a)). A broad minimum appears around T,= 40 mK for 29 bars and the

attenuation continues to increase with temperature for T > T,. Based on our measurements for

three pressures, it seems that T, decreases with the sample pressure. Interestingly, Normura et

al. expected a similar rise in attenuation at high temperatures by considering the decoupling of

liquid from aerogel, although their experimental results did not follow their prediction. In

general, decoupling of liquid from the hosting porous medium occurs when the viscous

penetration depth, 3 = 29 /pro is of the order of the average pore size. This sloshing motion

of the porous medium and viscous liquid provides an extra damping mechanism. Since the

viscous penetration depth of 3He followsl1/ T, Nomura et al. argued that the condition for

decoupling would be satisfied at a certain temperature. Furthermore, by invoking Biot's theory

[Bio55a], they proj ected a T dependence of attenuation in the high temperature region.

However, our high temperature data can be best fit to ~ T"' for all sample pressures that we

studied. High temperature fits shown as dashed lines in Fig. 4-16 (a) present temperature

dependence with powers of 0.64, 0.73 and 0.73 for 10, 21 and 29 bars, respectively. For the

whole temperature range, the sound velocity remains constant within our experimental

resolution.









4.4 Discussion

Recently, Higashitani et at. [Hig05] calculated the hydrodynamic sound attenuation for this

specific system based on the two-fluid model. Their calculation considered the aerogel motion

generated by the collision drag effect [Ich01, Hig03]. In their theory, the total sound attenuation,

a,, has two contributions, a, = ad IT a, Where ad is the attenuation caused by the friction

between the aerogel and the liquid, and a;, is the hydrodynamic contribution. ad is proportional

to the temperature dependent frictional relaxation time, rp whereas a,, is proportional to the

shear viscosity, r(T). The detailed temperature dependence of the attenuation also requires

knowledge of the superfluid density, ps, in addition to the parameters described above. Despite

the complexity of the model, their calculation provides a reasonably good account for our

observation, such as the broad shoulder structure observed at 29 bars and the monotonic

approach to finite attenuation near zero temperature [Hig05]. The moderate change of

attenuation below To agrees with Higashitani et al.'s prediction based on the friction and the

viscosity contribution (Fig. 4-13(b)). The relatively steep slope of Nomura et at. [Nom00] might

be caused by their assumption that the attenuation at 25 bars and 0.6 mK is close to zero, which

is about 2 cml in our data. The shoulder of the temperature trace at high pressure can be

explained by the combination of the continuously decaying viscosity contribution and the friction

contribution which increases just below To for high pressure with reasonable aerogel mean free

path. However, as pointed our earlier, the shoulder feature is much more pronounced in their

calculation and the a, is lower than our projected values from the measurements. Their

calculation is based on the homogeneous scattering model which tends to overestimate the









frictional damping but to underestimate the viscous damping'. The agreement between our

experimental results and their calculation can be improved by incorporating inhomogeneous

scattering which gives rise to further reduction of the average value of the order parameter. The

extra reduction in the order parameter would be reflected in a larger viscosity and, accordingly, a

larger a, is expected. In addition, an increased normal fluid fraction will weaken the frictional

effect. It is also expected that the effect of anisotropic scattering would enter into play through

various relaxation times. Detailed calculations including these effects are under consideration.

In particular, as the temperature approaches zero, r, 4 0 and ad a 0 Consequently,

the zero temperature attenuation is proportional to the finite shear viscosity, itself related to the

zero-energy density of states, n(0), by the relation, r(0) / r(T,) = n4 (0) for the unitary limits

[Hig05]. Here, n(0) is the zero energy density of states at zero temperature, which is normalized

to the density of states for normal liquid. Regardless of a specific theoretical model, the finite

zero temperature attenuation reflects the existence of a finite zero-energy density of states

induced by impurity scattering. Our results are in concert with the work of Fisher et al. [Fis03]

in which finite thermal conductivity (mostly in the A-phase) in the zero temperature limit was

observed [Sha03]. Indirect evidence for the presence of finite impurity states has also been

suggested from the heat capacity measurement performed by a Northwestern group [Cho04b].

They inferred a linear temperature dependence in the heat capacity below = 1 mK by analyzing

the heat capacity jump at the aerogel transition and the heat capacity in the superfluid phase

down to 1 mK. However, n(0), estimated by the two different experimental techniques, shows

significant discrepancies, especially in the pressure dependence. Because the frictional

SPrivate communication with Higashitani

SA power of 2 is theoretically predicted in the Born limit.









contribution disappears at T= 0, the attenuation ratio a, /ac is the same as the ratio of the

viscosity, if we ignore the frictional contribution at To Therefore, a, / ac in Fig. 4-14(a)

provides the lower bound for the viscosity ratio. The lower bounds of n(0) are estimated from

the viscosity ratio for the unitary and the Born scattering limits (Fig. 4-15). The density of states

estimated from the specific heat measurements by the Northwestern group is about 0.6 for

pressures from 10 to 30 bars [Cho04b].

The unique porous structure could be responsible for the increase of the attenuation at high

temperature (Fig. 4-16 (a)). In other words, there is no well-defined pore size as aerogel has a

completely different topology than other conventional porous materials. Furthermore, the nano-

scale strand diameter is much smaller than any other length scale involved in the 3He/aerogel

system up to 70 mK. In Fig. 4-16 (b), we show several relevant length scales estimated for

98% porosity aerogel: the effective mean free path, Q, including aerogel scattering, the viscous

penetration depth of 3He, 3, the average distance between aerogel strands, R = 20 nm, and the

diameter of aerogel strand, a = 3 nm [Thu98a]. At all temperatures shown, 3 is larger than any

other length scales. However, it is interesting to notice that the attenuation minimum ( T,)

occurs around the temperature where Q crosses R and T, moves to a higher temperature for

lower pressure. For T < T,, C is larger than a and the hydrodynamic description for the motion

between the aerogel and the liquid ceases to be valid. Therefore, Biot' s theory which is basically

a hydrodynamic theory may not be applicable. Only for T > T,, the system enters into a regime

where Biot' s theory can be considered, particularly the intermediate frequency regime for most

of our high temperature data, and a is expected to be T independent. Currently, we do not have

an explanation for the origin of the T" dependence for T > El However, the slip effect









[Tsi02] may give rise to a more temperature dependence owing to the enhanced velocity gradient

of liquid at a solid surface. A distribution of pore sizes [Yam88, Pri93], the squirt mechanism

[Dvo93], and/or the unique topology of the aerogel also need to be considered for a better

understanding.

















I IIII il


I I I I IIII 1


I I I I I I I I


Tea=1l.5 mK P" =16.3 bar


ra :



rSi
5


J 2.
O
~O' a
"
Oi~
i O D

~O
Oo
B i ~
~ i
Q i
i i
i F
I" 'I
i "
B ~
~


E
U


O Qg


I


i
"I

i
i
r


r


I I I


I I I I I I I |
2 4 68
10


I


68


2 4 68
100


T (m K)


Figure 4-1. The attenuation of longitudinal sound in liquid 3He/aerogel at 16 bars for 15 MHz
(circle) measured by Nomura et al. [Nom00]. The attenuation in bulk 3He (dotted
line), given for comparison, shows the first to zero sound transition around 10 mK.
Reprinted figure with permission from R. Nomura, G. Gervais, T.M. Haard, Y. Lee,
N. Mulders, and W.P. Halperin, Phys. Rev. Lett. 85, 4325 (2000). Copyright (2000)
by the American Physical Society.



















Aerogel -cl


LiNbO3 Transducer (B) /
(Transmitter, 9.5 MHz)


Macor Spacer


Silver Sin~ter


(b)

Figure 4-2. Schematic diagram of experimental setup. (a) Experimental cell (b) Pulse
spectrometer.


Capacitor
Diaphragm
AC-cut Quartz Transducer

- LiNbOs Transducer (A)
(Receiver, 9.5 MHz)
















2.0


1 .5 PtNMR (w. time shift)













2000 2500 3000 3500

Time (min)



Figure 4-3. Temperature determined by MPT and Pt-NMR thermometer. With an appropriate
shift in time, the Pt-NMR thermometer shows better match with MPT at high
temperature (red line).

















) II""


101

0 T

0~ c~=~~~~=-~~~ 2A

irel Cs) 10 0







5 ~ --- ~ .~.~-----i
4~ -i--~-i-



3 j




1 ~-i0.5
0 T1.0
0 c,
501.
The~g@100


Figure 4-4. Evolution of receiver signals on warming. (a) For 34 bars, the superfluid transition is
indicated by the arrow. Four echoes can be seen in the trace taken at the lowest
temperature. (b) For 14 bars, only the first echo is observed at the lowest
temperature.














450

400

350

300

250


/2 00
Theory
150 3He in Aerogel

0 5 10 15 20 25 30 35
P (bar)


Figure 4-5. Velocity of bulk 3He [Hal90] and liquid 3He in aerogel at Tc The dashed line is
calculated from the two-fluid model modified by impurities (Eq. 4-1).














25 mK












0 5 10 15 20
Time (cis)


0.5 C


8




0-4


0


-0.4 0.0 0.4


Freq (MHz)
(b)

Figure 4-6. Signal from receiver. (a) Receiver signals normalized to its peak amplitude at 2.5 mK
and 0.4 mK at 29 bars. (b) FFT of the transmitted signal (Balck circles), each
transducer (red and blue) at 0.3 mK and 34 bars. The green line is obtained by the
product of the two FF T (A and B).































0 5 10 15 20


Attenuator
--10 dB
-13 dB
--16 dB
--20 dB -
S-23 dB
-26 dB


16 E


Time (tLs)


Attenuator
-20 dB
-26 dB
-30 dB
S-40 dB


0.8 C


0.4-


0 5 10 15 20 25


Time (CLs)

(b)

Figure 4-7. Linearity test. (a) for 40 mK and 33bars and (b) for 0.3 mK, and 33bars.














29 bars








c2-
a, Attenuator
-20 dB (second run)
26 dB


0 1 2

T (m K)



Figure 4-8. No significant change in attenuation indicates that the heating by the transducer is
negligible.


O 20 40 60 80 100


Time (cls)



Figure 4-9. Receiver signal trace vs. time at 0.4 mK and 29 bars. The primary signal and the
subsequent echoes follow an exponential decay.


















0.8-

~-0.6-

0.4-

0.2-

0.0 l
0.0 0.2 0.4 0.6 0.8 1.0 1.2
TIT,



Figure 4-10. The normalized attenuation in the superfluid phase at 12 (triangle) and 29 (circle)
bars as a function of the reduced temperature. Tc represents the superfluid transition
temperature for the liquids in aerogel or bulk. The attenuation of the bulk is the
measurement by Hristakos [Hri01], i.e. Eska's group.



-A Present Work (Long. Sound)
-0 Present Work (Tran. Sound)
-* Gervais et al. (Tran. Sound)
*-Matsumoto et al. (Torsional) :
-Greywall Bulk
25-+

20C -/"

15-

10~ -




0.0 0.5 1.0 1.5 2.0 2.5
T (mK)



Figure 4-11. Phase diagram at zero field. Our T,, (red triangle) values agree with those
measured in other methods [Mat97, Ger00] (blue data). The AB transition on
warming (red closed circle) and cooling (red star)is also observed in the transmission
measurement. The red open circles are data from chapter 3. The black line
represents bulk Tc [Gre86].




Full Text

PAGE 1

1 ACOUSTIC STUDY OF DISORDERED LIQUID 3He IN HIGH-POROSITY SILICA AEROGEL By HYUNCHANG CHOI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 Copyright 2007 by Hyunchang Choi

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3 To my family

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4 ACKNOWLEDGMENTS Most of all, I thank my advisor, Yoonseok Lee for his guidance and encouragement. His enthusiasm for the physics subject and his way of approaching the prob lems has been very inspiring for me. He tried to help me improve my strengths and overcome my weaknesses. I also thank Mark. W. Meisel. His continuous su pport sustained my confidence as a research student. I appreciate the help I received from Jian-s heng Xia, Naoto Masuhara, Carlos L. Vicente and Ju-Hyun Park. Their expertise and efforts on the research were amazing. Their knowledge based on their experience with practical problems was crucial to overcome obstacles that I have encountered. Ju-Hyun Park taught me baby steps at the beginning of my laboratory experience. He was a model student who spends his time glad ly for other students when they need help. I would also like to thank our collaborators outside and inside the physics department for their support. All of the aeroge l used in my experiments were grown by Mulders group at University of Delaware. The acoustic cavity employed for the cw shear impedance measurement was made by Guillaume Gervais in Halperins group and Ca1.5Sr0.5RuO4 samples were prepared by Rongying Jin in Mandrus group. The experimental cells and maintenance parts were made by Mark Link and other machinists and consistent supply of liquid 3He by Greg Labbe and John Graham enabled us to continue the experiment w ithout interruption. Dori Faust did all the paper work for ordering supplies and more. I could co mplete my projects with their sincere help. I cannot neglect to say thank you to my lab mates, Pradeep Bhupathi and Miguel Gonzalez, who spent their valuable time to revise this thesis. From time to time, Byoung Hee Moon inspired me with his creative ideas. Pradeep made the Melting Curve Thermometer, which was used as a main thermometer for the attenuation measurement.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 TABLE.............................................................................................................................................7 LIST OF FIGURES.........................................................................................................................8 ABSTRACT...................................................................................................................................11 CHAP TER 1 INTRODUCTION..................................................................................................................13 1.1 Overview...........................................................................................................................13 1.2 Pure Liquid 3He................................................................................................................14 1.2.1 History....................................................................................................................14 1.2.2 Fermi Liquid........................................................................................................... 16 1.2.3 Superfluid 3He........................................................................................................21 1.3 Superfluid 3He in Aerogel................................................................................................25 1.4 Liquid in Porous Media....................................................................................................28 2 THE A1 PHASE OF SUPERFLUID 3He IN 98% AEROGEL............................................... 35 2.1 Overview...........................................................................................................................35 2.2 Experiments......................................................................................................................36 2.3 Results...............................................................................................................................42 2.4 Discussion.........................................................................................................................44 3 THE A PHASE OF SUPERFLUID 3He IN 98% AEROGEL................................................ 60 3.1 Overview...........................................................................................................................60 3.2 Experiments......................................................................................................................62 3.3 Results...............................................................................................................................62 3.4 Discussion.........................................................................................................................64 4 ATTENUATION OF LONGIT UDINAL SOUND IN LIQUID 3He/98% AEROGEL..............................................................................................................................76 4.1 Overview...........................................................................................................................76 4.2 Experiments......................................................................................................................78 4.3 Results...............................................................................................................................81 4.4 Discussion.........................................................................................................................88 5 CONCLUSION..................................................................................................................... 106

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6 APPENDIX A ORIGIN SCRIPT FOR MTP CALIBRATI ON IN THE HIGH FIELD PHASE ................. 108 A.1 MPTfukuyamalessTn.c..................................................................................................111 A.2 MPTfukuyamalow5.c.................................................................................................... 114 B PARTS OF THE EXPERIMENTAL C ELL FOR ATTENUATION MEASUREMENT... 117 C TYPICAL SETTING FOR NMRKIT II............................................................................... 124 D ORIGIN SCRIPT FOR THE DATA ANAL YSIS OF THE ATTENUATION MEASUREMENTS..............................................................................................................126 E TRANSPORT MEASUREMENT ON Ca1.5Sr0.5RuO4........................................................133 F NEEDLE VALVE FOR DR137........................................................................................... 144 LIST OF REFERENCES.............................................................................................................148 BIOGRAPHICAL SKETCH.......................................................................................................156

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7 TABLE Table page 2-1 Slopes (mK/T) of the splitting for the A1 and A2 transitions..............................................59

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8 LIST OF FIGURES Figure page 1-1 Temperature dependence of sound velocity and attenuation of pure 3He......................... 31 1-2 P-H-T phase diagram of superfluid 3He............................................................................. 32 1-3 Gap structures of the superfluid 3He, the B phase and A phase......................................... 33 1-4 Microscopic structure of 98% Aerogel by Haard. .............................................................33 1-5 P-B-T phase diagram of superfluid 3He in 98% aerogel.................................................... 34 2-1 Splitting of the phase tran sition tem perature in fields....................................................... 46 2-2 Cut-out views of the experime ntal cell and the acoustic cavity. ........................................ 47 2-3 Schematic diagram of the vibrating wire........................................................................... 48 2-4 Arrangement of the experimental cell and the m elting pressure thermometer.................. 49 2-5 Schematic diagram of the continuous wave spectrometer................................................. 50 2-6 Frequency sweep for two different spect rom eter settings at 28.4 bars and 3 T................. 51 2-7 Zero field acoustic signals at 28.4 bars for different spectrometer settings as a function of tem perature...................................................................................................... 52 2-8 For high fields, we used the MPT calibration of Fukuyam as group................................ 53 2-9 Acoustic traces for 3, 5, 7, 9, 11, and 15 T at 33.5 bars on warming................................ 54 2-10 Acoustic traces for 1, 2 and 3 T......................................................................................... 55 2-11 Acoustic traces for 3, 5, 7, 9, 11, and 15 T at 28.4 bars on warming................................ 56 2-12 Transition temperatures vs. m agnetic field for 28.4 and 33.5 bars.................................... 57 2-13 Degree of splitting for each indi vidual transition vs. m agnetic field................................. 58 3-1 The Stanford NMR measurement on superfluid 3He in 99.3% aerogel.............................70 3-2 Cooling and warming traces taken at 28.4 bars and 33.5 bars........................................... 71 3-3 Acoustic traces of tracking experiments............................................................................ 72 3-4 The relative size of the steps for the supercooled aerogel A-B transition ......................... 73

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9 3-5 Field dependence of the warming AB transition in aerogel for 28.4 bars........................ 74 3-6 The zero field phase diagram of superfluid 3He in 98% aerogel....................................... 75 4-1 The attenuation of longitudinal sound in liquid 3He/aerogel at 16 bars for 15 MHz measured by Nomura et al .................................................................................................92 4-2 Schematic diagram of experim ental setup......................................................................... 93 4-3 Temperature determined by MPT and Pt-NMR thermometer........................................... 94 4-4 Evolution of receiver signals on warming......................................................................... 95 4-5 Velocity of bulk 3He and liquid 3He in aerogel at TC........................................................96 4-6 Signal from receiver....................................................................................................... ....97 4-7 Linearity test......................................................................................................................98 4-8 No significant change in attenu ation indicates that the heating by the transducer is negligible............................................................................................................................99 4-9 Receiver signal trace vs. tim e at 0.4 mK and 29 bars........................................................ 99 4-10 The normalized attenuation in the superfluid phase at 12 and 29 bars as a function of the reduced tem perature................................................................................................... 100 4-11 Phase diagram at zero field..............................................................................................100 4-12 Absolute attenuations for pressures from 8 to 34 bars are presented as a function of tem perature.................................................................................................................... ..101 4-13 Normalized attenuation in superfluid............................................................................... 102 4-14 Attenuation vs. pressure.................................................................................................. .103 4-15 Zero energy density of states at zero te m perature vs. pressure for the unitary and the Born scattering limits....................................................................................................... 104 4-16 Attenuation for normal liquid.......................................................................................... 105 A-1 Origin worksheet, Pad. Input parameters.....................................................................109 A-2 Label control window......................................................................................................109 A-3 Origin worksheet, PtoT................................................................................................. 110 C-1 The typical NMRkitII setting........................................................................................... 125

PAGE 10

10 D-1 The screen shot of Origin worksheet for data analysis. ................................................... 127 E-1 A Zero field Phase diagram of Ca2-xSrxRuO4 from S. Nakatsuji et al. ............................137 E-2 Picture of a test sample................................................................................................... .138 E-3 Temperature dependence of the resistan ce in the absence of magnetic fields. ................ 139 E-4 Resistance vs. temperature for various le vels of excitation at low tem perature.............. 139 E-5 The excitation dependence of the resistance for each tem perature.................................. 140 E-6 The temperature dependence of the slopes...................................................................... 140 E-7 Normalized magneto resistance....................................................................................... 141 E-8 Field sweeps below 200 G...............................................................................................141 E-9 The position of the shoulder and dip structure in fields vs. tem perature......................... 142 E-10 Magneto resistance for fields perp end icular to the plane (blue) and 20o tilted away from the c-axis (red) at 20mK.......................................................................................... 143 E-11 Magneto resistance for low fields pe rpendicu lar to the plane (blue) and 20o tilted away from the c-axis (red) at 20mK................................................................................ 143

PAGE 11

11 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ACOUSTIC STUDY OF DISORDERED LIQUID 3He IN HIGH-POROSITY SILICA AEROGEL By Hyunchang Choi May 2007 Chair: Yoonseok Lee Major Department: Physics The effect of disorder is one of the most in teresting and ubiquitous problems in condensed matter physics. A fundamental question, How does a ground state evol ve in response to increasing disorder?, encompasses many areas in modern condensed matter physics, especially in conjunction with quantum phase transitions. We address the same question in the low temperature phases of liquid 3He, which is of a special interest since in its bulk form, it is the most well understood unconventional superfluid and the purest substance known to mankind. Unlike conventional s-wave pairing superconductors, the unc onventional superconductors are vulnerable to any type of impurity, and this fact has been used to test the unconventional nature of the order parameter in heavy fermion and cuprate superconductors. The aerogel/3He system provides a unique opportunity to conduct a system atic investigation on th e effects of static disorder in unconventional superfluids. We have investigated the influence of disorder, introduced in the form of 98% porosity silica aerogel, on the superfluid 3He using various ultrasound techniques. Our primary effort is in understanding the comple te phase diagram of this relatively new system. In particular, the hi gh magnetic field region of the phase diagram has not been explored until this work, and the nature of the A -like to B -like transition in this system has not been elucidated. We identified a third superfluid phase emerging in the presence of

PAGE 12

12 magnetic fields, which resembles, in many respects, the A1-phase in bulk. Our zero field study of the A-B transition in aerogel revealed that two phase s coexist in a narrow window of temperature right below the superfluid transition. Sound a ttenuation measurements conducted over a wide range of temperatures and pressures show a dras tically different behavior than in bulk. In particluar, in the B -like phase, our results can be interp reted as strong evid ence of a gapless superfluid.

PAGE 13

13 CHAPTER 1 INTRODUCTION 1.1 Overview The effect of disorder is one of the most in teresting and ubiquitous problems in condensed matter physics. Metal-insulator transitions [Lee85] and the Kondo effect [He69] are two examples of phenomena in which disorder, in the form of various types of impurities, plays a fundamental role. The influence of disorder on ordered states such as the magnetic or superconducting phase has also attracted tremendous interest, especially in systems that undergo quantum phase transi tions [Sac99]. The response of Cooper pairs to various type s of impurities depends on the symmetry of the order parameter [And59, Abr61, Lar65]. The strong influence of a small concentration of paramagnetic impurities on a low temperature s uperconductor is in stark contrast to its insensitivity to nonmagnetic im purities [And59, Abr61]. Unconven tional superconductors with non-s-wave pairing are vulnerable to a ny type of impurity [Lar65], and this fact has been used to test the unconventional nature of the order parameter in heavy fermion and cuprate superconductors [Vor93, Tsu00, Ma04]. Given a great deal of quantita tive understanding of the intrin sic properties of superfluid 3He [Vol90], the aerogel/3He system provides a unique opport unity to conduct a systematic investigation on the effects of sta tic disorder in unconventional s uperfluids. In this system, a wide range of impurity pair breaking can be attained by continuously varying the sample pressure. Furthermore, the nature of the impurity scattering can be read ily altered by modifying the composition of the surface layers, the 4He preplating. We have inve stigated the influence of disorder (98% aerogel) on the superfluid 3He using various ultrasound techniques. Our primary effort is focused on understanding the complete phase diagram of this relatively new system.

PAGE 14

14 Especially, the high magnetic field region of the phase diagram had not been explored until this work, and the nature of the A -like to B -like transition in this system had not been elucidated. In this study, three main experimental results are presented along with in-depth discussions. In the rest of this chapter, a brief description of 3He physics, with an emphasis on the acoustic properties of Fermi liquids and unconven tional BCS superfluids, is presented. High porosity silica aerogel acting as quenc hed disorder is also discussed in this chapter. In chapter 2, the acoustic impedance measurements, in high magne tic fields, which led us to observe the third superfluid phase, are described. Chapter 3 focuse s on the zero field experiment that revealed the existence of the BA transition and the coexistence of the two phases on warming. The third experiment, the absolute sound attenuation measurement by direct sound propagation, is discussed in chapter 4. Finally in chapter 5, we conclude with a summary of the experimental results, the physical implications, and a few suggestions for future directions. Various supplementary materials are collected in appendi ces. Especially, low temperature transport measurements on Ca1.5Sr0.5RuO4 Throughout this thesis, many transition temp eratures are mentioned. For example, CT and ABT indicate the superfluid tr ansition temperature and the AB transition temperature, respectively. In most of the cases, it is clear wh ether the transition is in the bulk or in aerogel. However, in the case where the distinction is warranted, aCT or aABT are used for the transitions in aerogel. 1.2 Pure Liquid 3He 1.2.1 History The isotopes 3He and 4He are the only two stable isotope s in this universe which remain liquid down to the lowest availa ble temperature. They are li ght enough that the zero-point

PAGE 15

15 motion overcomes the attractive inter-atomic inter action, which is very weak due to their filled 2s electronic configuration. At low temperatur es, the matter-wave duality and the degenerate conditions two main pillars of quantum world emerge as their de-Broglie wavelength becomes comparable to the inter-atomic spaci ng, providing a reason to call them quantum liquids It is not surprising that the existence of 4He (product of vigorous nuc lear fusion) was first observed from the visible spectrum of sola r protuberances, considering its minute 5 ppm abundance in the Earths atmosphere. Fortunately (especially to low temperature physicists), a substantial amount of 4He trapped underground can be found in some natural gas wells. The much rarer isotope 3He was discovered by Oliphant, Kinse y, and Rutherford in 1933 [Oli33]. Unlike 4He, a reasonable amount of 3He could only be produced artificially by the -decay of tritium in nuclear reactors. It is not hard to imagine why the first experiment on pure liquid 3He was conducted by Sydoriak et al. at Los Alamos Scientific Labora tory [Syd49a, b]. In this work, the authors proved that 3He indeed condenses into a liquid at saturated vapor pr essure, contrary to the predictions of distinguished theorists su ch as London and Tisza. Since then, a tremendous amount of effort has been poured in this subjec t and has lead to disc overies of various low temperature phases in liquid as well as solid 3He [Vol90]. One of the most remarkable properties that th ese two isotopes share is the appearance of superfluid phases in which liquids can flow th rough narrow capillaries with almost no friction. The superfluid transition temperature in liquid 3He ( 2 mK) is three orders of magnitude lower than in liquid 4He ( 2.2 K), reflecting the fundamental diffe rence in quantum statistics. While a 4He atom with zero nuclear spin obeys Bose-Einstein statistics, a 3He atom with spin 1/2 follows Fermi statistics. A Bose system prefers to condense into the lowest-energ y single particle state, the so called Bose-Einstein condensation (BEC), at a temperature where the wavefunction of the

PAGE 16

16 particle starts to overlap. This BEC usually acco mpanies the onset of superfluidity as evidenced in liquid 4He and dilute cold atoms. The superfluidity of a Fermi system was first discovered in a metal, mercury, in 1911 by Kammerlingh Onnes [Kam11], although the microscopic understanding of the phenomenon did not come to light until almost a half centu ry later by the theory of Bardeen, Cooper and Schrieffer (BCS) [Bar57]. A bound pair of two electrons, known as a Cooper pair, with a spin singlet and s-wave orbital state may be looked upon as a composite boson that is Bosecondensed. Cooper pairs can be formed with an ar bitrarily small net attrac tive interaction in the presence of the filled Fermi sea background. In a conventional superconductor it is known that the attractive interaction is provided by the retarded electr on-phonon interaction. For liquid 3He, however, the origin of the pair ing interaction is not clearly understood microscopically. In his seminal work on Fermi liquid theory [Lan56, Lan57], Landau conceived a phenomenological theory to provide a theoretical framework for an interacting fermionic many body system at low temperatures with a specific example, liquid 3He, in his mind. Since then, liquid 3He has served as a paradigm for a Fermi li quid whose nature tran scends the realm of fermionic quantum fluids. 1.2.2 Fermi Liquid At the heart of Landaus Fermi liquid theory is the quasiparticle, a long lived elementary fermionic excitation near the Ferm i surface [Mar00]. The energy leve ls of the interacting system have a one to one correspondence to the ones in a Fermi gas without mixing or crossing levels. The quasiparticle represents the tota l entity of the bare particle and some effect of the interaction. Therefore it is expected that the mass of a quasiparticle is different from the bare mass. The quasiparticle energy should also depend on the configuration of other quasiparticles around, and this molecular field type inter action can be parameterized by a set of dimensionless numbers,

PAGE 17

17 Landau Fermi liquid parameters, {s lF, a lF}, where l indicates angular momentum. More specifically, suppose that an excited state wa s created by adding a qua siparticle labeled by k (let us assume that the spin quantum number is imbe dded here) and the energy of the excited state relative to the ground state is given by k Any perturbation in the occupancy of the states n k near the Fermi sea would cause a change in the excitation energy k of the quasiparticle k no longer identical to k The difference between these two values produces a change in the molecular field on the quasiparticle, k via interactions betw een the quasiparticles, k 1 V f k k 'n k k (1-1) where f k k is Landaus interaction function which generates the change in energy of a quasiparticle with momentum k by the perturbation in the distribution of quasiparticles with momentum k ', and V is the volume. In a perfectly isotropic system like liquid 3He and at low temperatures, the interaction func tion should depend only on the angle, between the two momenta with the magnitude k F Then, the Landau interaction f unction can be decomposed into symmetric (orbital) and antisymmetric (spin) parts, fk ,k fk ,k s 'fk ,k 'a (1-2) where i is the ith Pauli matrix and s (a) denotes symmetric (antisymmetric). This separation is possible only when the spin-orbit coupling is negligible; this is justified in 3He since the only spin-orbit coupling is the weak dipole-dipole interaction (less th an K). Each interaction function can be expanded for each angular moment um component in terms of a basis set of Legendre polynomials. The dimensionless Landau parameters, s lF and a lF are obtained by

PAGE 18

18 normalizing with the density of states at the Fermi surface, F FpmN/1~/*)0(32 [Noz64]. These parameters determine various macros copic properties of the liquid; conversely, some of these parameters can be determined by va rious experiments. For example, one finds the effective mass, m* from the Galilean invariance 3 /1/*1sFmm. (1-3) It is worth mentioning that Eq. 1-3 is an exact result without further higher order corrections. This is the most important Fermi liquid correction that renormalizes the density of states at the Fermi energy, )0(N. Consequently, the heat capacity and the magnetic susceptibility need to be modified accordingly, Tk pm CB F V 2 33 (1-4) aF N0 221 )0( 4 (1-5) where is the gyromagnetic ratio. The magnetic su sceptibility also receives a correction through the spin channel led by aF0. The strength of Fermi liquid theory is in the fact that with the knowledge of a few (experimentally determined ) Landau parameters, most of the physical properties can be calculated self -consistently since the higher angular momentum components decrease rather rapidly. In 3He, 9010 s oF, 15 51sF, and ) 75.070.0( a oF indicating that liquid 3He is indeed a strongly interacting (correlated) system with enhanced effective mass and magnetic susceptibility due to ferromagnetic tendency (minus sign in a oF). While the thermodynamic propertie s of a Fermi liquid resemble those of a Fermi gas with an adequate renormalization through the Fermi liquid parameters, dynamical properties are

PAGE 19

19 unique in the sense that new type s of collective modes are predicted to exist in this system [Lan57]. Ordinary hydrodynamic s ound (first sound) in a liquid pr opagates by restoring its local equilibrium through scattering processes. Therefore, it requires the sound frequency to be much smaller than the scattering rate, i.e., 1 where is the sound frequency and is the relaxation time. The sound velocity, 1c, which is determined by the compressibility, N and the sound attenuation, 1 which is dominated by viscous processes, are given by [Vol90] 2 1 0 1 2 1) 3 1 1)(1( 3 1 )(F s s NvFF c, (1-6) 3 1 2 13 2c (1-7) where , are the mass density, the viscosity and the sound frequency, respectively. At low temperatures (<< 1 K), the sound velocity at a given pressure is constant in the hydrodynamic regime [Abe61]. In a Fermi liquid varies as 2/1T [Wil67]. Therefore, at lo w enough temperatures where becomes much longer than the period of the sound wave, the conventional restoring mechanism for hydrodynamic sound becomes ineffectiv e. However, Landau realized that new sound modes emerge in the collisionless regime (1 ) if the relevant Fermi interactions are strong and repulsive enough, and named these ex citations as zero sound m odes [Lan57]. At zero temperature, a zero sound mode in a Fermi liquid can be pictured as a cohe rent oscillation of the elastic Fermi surface ringing without damping. Normal mo des of this spherical membrane are associated with specific zero sound modes such as longitudinal zero sound (LZS) and transverse zero sound (TZS). As the temperature rises, incohe rent thermal quasiparticle scattering prohibits the coherent oscillations of th e Fermi surface, causing a damping of zero sound. For longitudinal

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20 zero sound, a strong repulsive sF0 value ( 10 90 depending on pressure) plays a major role in stabilizing this mode with a sl ightly higher zero sound velocity, 0c, compared to that of the first sound and attenuation, /10 ]})[())(1( 15 2 1{4 1 2 1 2 5 1 10c v O c v F m m ccF F s (1-8) ]})[(1{)( )1( 15 22 1 3 1 2 2 5 1 0c v O c v v F m mF F F s (1-9) Since 2 T, the first to zero sound cro ssover occurs by lowering th e temperature at a fixed frequency with a trade mark of a symmetric temperature dependence on both sides of broad attenuation maximum (see Fig. 11) [Kee63, Abe65, Ket75]. Other than Landau theory, a visco-elastic mode l can also well describe the first to zero sound transition. Liquid 3He possesses not only viscosity as a liquid but also elasticity like a solid [Hal90], and this elastic character is especially pronounce d in the collisionless regime [And75, Cas79, Cas80, Vol84]. Therefore, th e existence of zero sound modes (especially the transverse zero sound mode, as can be seen be low) implies the solid-l ike character of liquid 3He [Kee63, Kee65, Bet65, Abe66, Ki67, Whe70, Rud 80]. From the usual kinetic gas formula, the viscosity coefficient, can be written as 2 2/12~F Fv v (1-10) It is based on momentum exchange between quasiparticles with the Fermi velocity, Fv, and quasiparticle life time, ( Fvl ). For 1 the quasiparticles do not have enough time to collide with each other during a period of sound oscillation and the damping becomes weak. The velocity and sound attenuation of the longitudina l sound from a visco-elas tic model are given by

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21 22 22 1011 )( cccc, (1-11) 22 2 2 1 101 c cc. (1-12) The crossover from the first sound to zero sound is clearly represented in these expressions. Transverse zero sound is another collective mode predicte d by Landau. However, the strength of the relevant interaction (sF2) is marginal in liquid 3He. It turns out that the phase velocity of this mode is so close to the Fermi velocity that th e mode decays effectively into quasiparticle-hole pairs causing strong damping (Landau dampin g). The zero sound modes are expected to disappear in the superfluid state as the gap develops at the Fermi level. As will be discussed in the following section, it is of great importance that these modes actually are resurrected in the superfluid by coupling to order parameter collective modes. 1.2.3 Superfluid 3He The BCS theory immediately i nvigorated the search for a s uperfluid transition in liquid 3He. However, many physicists struggled for longer than a decade in frustration coming from the lack of evidence of the transition and the di fficulty in predicting the transition temperature. Finally in 1971, Osheroff, Richardson and Lee [Os72] observed peculiar features on the melting line in a Pomeranchuk cell filled with 3He, which indicated some t ype of transitions. Although their original paper falsely identified them as tr ansitions associated with magnetic transitions in the coexisting solid1, detailed NMR experiments [Osh72a] and the help of tremendous intuition provided by Leggett confirmed that the features were indeed superfluid transitions arising from the liquid. In their NMR experiment, they applie d a field gradient along the z-axis in order to 1 The title of the paper is Evidence for New Phases in Solid 3He.

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22 determine the position and identity of the NMR signa l source as either solid or liquid. This was probably the first 1D magnetic resonance imag ing experiment, although the work of Paul Lauterbur in 1973 [Lau73] is officially credited as the first magneti c resonance imaging. Today, three phases (A, B, A1) in the superfluid have been identi fied experimentally (Fig. 1-2). The Cooper pairs in all thr ee phases are spin-triplet p-wave pairing which possesses an internal structure. Th e general wave function for a spin triplet pair is a superposition of all spin substates (Sz = -1, 0, +1) as given by |)()|)(|(|)(1,1 0,1 1,1k k k (1-13) where ) (1,1k ) (0,1k and ) (1,1k are the complex amplitudes for each substate. The theoretical Balian-Werthamer (BW) st ate [Bal63] that corresponds to the B-phase is the most stable state in zero magnetic fi eld at low pressures. The B-phase is composed of all three substates and accordingly has smaller magnetic susceptibility since the | | component is magnetically inert. Because of the pseudo-isotropic gap in th is phase as shown in Fig. 1-3 (this is quite unusual fo r intrinsically anisotr opic superfluid), the B-phase exhibits similar properties to conventional superc onductors: for example, an e xponential temperature dependence of the specific heat at low temperatures. On the other hand, the A nderson-Brinkman-Morel (ABM) [And60, And61] state (A-phase) is an equal spin pairing state, constituted by only | and | states. The gap has nodal poi nts at the north and south po les in the direction defined by the angular momentum of the Cooper pair. At high pressures (P > 21 bars), the ABM state is stabilized near CT owing to the strong coupling between quasiparticles [Vol90, And73]. The point where normal phase, A-phase and B-phase meet at zero field is called the polycritical point

PAGE 23

23 (PCP). The phase transition between Aand B-phase is first order as evidenced by strong supercooling of the tran sition and the discontinuity in heat capacity at the transition. Magnetic fields have profound and intriguing influences on the phase diagram of pure superfluid 3He. Depairing of the | | component in the B phase by magnetic fields, an effect that is similar to the Clogston-Chandrasekhar paramagnetic limit in a superconductor [Tin96], promotes the growth and appearance of the A-phase at all pressures. As a result, a magnetic field shifts the first order A-B transition line to a lower temperature in a quadratic fashion and eventually quenches the B phase around 0.6 Tesla and 19 bars [Tan91]. In addition to the effect on the A-B transition, a magnetic fiel d induces a new phase, the A1-phase, between the normal and the A-phase by splitting the second order transition into two second order transitions. The A1-phase is very unique in th e sense that it contains only | pairs (it is a fully polarized superfluid). A magnetic field generates small shifts in the Fermi levels of up and down spin quasiparticles by the Zeeman energy and, co nsequently, results in different superfluid transition temperatures (particle-hole asymmetry). Therefore, in the A1-phase, only the spin up component participates in forming Cooper pairs. The splitting in transition temperatures (the width of the A1 phase) is linear in field and symmetric relative to the zero field CT in the weak coupling limit since the two spin states form pairs independently. The width of the A1-phase in bulk is about 60 K/Tesla at melting pressure [Isr84, Sag84, Rem98]. A stable state can be found by searching a state with the lowest free energy. Since the order parameter of superfluid 3He grows continuously from the transition temperature (continuous phase transition), the free energy can be expanded in powers of the order parameter as long as the temperature is ne ar the transition temp erature (Ginzburg-Landau theory) [Lan59]. The expansion is regulated by the symmetries of the system, and the coefficients for the

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24 expansion can be determined from microsc opic theories. The free energy functional (f) should have full symmetry that the higher temperature phase possesses. For example, normal liquid 3He possesses a full continuous symmetry including time-reversal symmetry. For p-wave pairing states, f with a gap amplitude is given by 5 1 4 2 1 2 i ii NI ff (1-14) where iI is the fourth-order invariant of the order parameter and ) /1)(0()(CTTNT (1-15) In the weak coupling limit, 0 5 6 1 5 4322 (1-16) 2 2 0))(0()3( 8 7CBTkN (1-17) and the BW state is found to be the most stable [Bal63]. One can easily expect that there are many local free energy minima (including a fe w saddle points) in this multi-dimensional manifold space. The hierarchy among these local minima depends on the specific values of the i -parameters. For example, as the pressure rises, the i s start to deviate from the weak coupling values mainly due to strong coupling e ffects (e.g. spin fluctuations). This strong coupling correction helps the ABM phase win over th e BW phase. This effect is responsible for the A-phase appearing above the polyc ritical point [And60, 61]. Nonzero angular momentum of the Cooper pair allows pair vibrati on modes, called order parameter collective modes (OPCM). These collec tive excitations are measured as anomalies in sound attenuation and velocities below CT. An attenuation peak by pair breaking can also occur for a frequency above /2 The frequency range of ultrasound (10 100 MHz) conveniently

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25 matches with the size of the gap and the spectra of various OPCMs in superfluid 3He. The thermally excited quasiparticl e background also provides an independent channel for sound attenuation. In particular, sound attenuation from thermally excited quasiparticles decreases exponentially at low temperature as an isotropic gap opens up in the B-phase. 1.3 Superfluid 3He in Aerogel High porosity silica aerogel consists of a nanoscale abridged network of SiO2 strands with a diameter of ~ 3 nm and a distance of ~ 20 nm. Since the diameter of the strands is much smaller than the coherence length, 0 (15 80 nm for 34 0 bars [Dob00]) of the superfluid, when 3He is introduced, the aerogel behaves as an im purity with the strands acting as effective scattering centers. Although curre nt theoretical models treat aer ogel as a collection of randomly distributed scattering cent ers, one should not lose sight on the fact that the aerogel structure is indeed highly correlated. The microscopic stru cture of 98% porosity aerogel is shown in Fig. 14 [Haa00], where the de nsity is 0.044 g/cm3 and the geometrical mean free path is 100 ~ 200 nm [Por99, Haa00]. The velocity of longitudinal sound in 98% aerogel was measured as 50 ~ 100 m/s [Fri92]. Early studies using NMR [Spr95] and torsio nal oscillator [Por95] measurements on 3He in 98% aerogel found substantial depression in the superfluid transition and superfluid density [Por95, Spr95, Spr96, Mat97, Gol98, A ll98, He02a, He02b]. The phase diagram of 3He/aerogel (with mostly 98% porosity) has been studied using a variety of t echniques [Por95, Spr95, Mat97, Ger02a, Bru01, Cho04a, Bau04a]. The phase diagra m in P-T-B domain is shown in Fig. 1-5 [Ger02a]. When aerogel is submerged in liquid 3He, a couple of solid layers, which are then in contact with surrounding liquid, are formed on th e aerogel surface. Therefore, the scattering off

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26 the aerogel surface has elements of both potentia l scattering and spin exchange scattering. However, the magnetic scattering between the itinerant 3He and localized moments in solid layers can be turned off by preplating magnetically inert 4He layers on aerogel surface. Although the early NMR measurement by Sprague et al. concluded that the 4He coating turns the A-like phase into a B-like phase at ~19 bars [Spr96], the effect of the spin-exchange scattering on the phase diagram is elusive. Alles et al. provided evidence of the BW state in aerogel from the textural analysis of NMR line shape [All98]. Ho wever, in these early experiments, no evidence of the A-like to B-like transitions were observed until Barker et al. observed clear transition features below the aerogel superfluid tr ansition only on cooling while exploiting the supercooling effect [Bar00b]. C onsidering the fact that data collecting in NMR experiments typically done only on warming cycl e to minimize the interaction with the demag field, it is not surprising that the earlier experime nts could not observe these transitions. Further studies using acoustic techniques [Ger02a, Naz05] and an oscilla ting aerogel disc [Bru01] have confirmed the presence of the A-B like transition in the presence of low magnetic fields. The nomenclature for the two supe rfluid phases in aerogel as the A-like and the B-like phases heavily relies on the spin structure without dire ct experimental inputs for the orbital structure. Through detailed NMR studies, we now believe that the B-like phase at least has the same order parameter structure as the BW phase On the contrary, the identification of the A-like phase is far from clear since there are many possible phases with a similar spin structure represented by the equal spin pairing [Fom04, Vol06, Fom06, Bar06]. An interesting aspect of 3He in 98% porosity aerogel is th at no superfluid transition was found down to 150 K below 6.5 bars [Mat97]. Th is experimental observation renders us to contemplate a possible quantum phase transition in this system [Bar99].

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27 The first reports of superfluid transition in 98% aerogel were immediately followed by a theoretical model, the homogeneous scattering model (HSM), ba sed on the Abrikosov-Gorkovs theory [Abr61] in which the spin exchange scattering off dilute paramagnetic impurities in swave superconductors was treated perturbatively. The isotropi c homogeneous scattering model (IHSM) [Thu98a, Han98] could provide a successful explanation for the observed suppression of the transition temperatures. In that th eory, the ratio of th e coherence length, 0 to the mean free path, plays a major role as a pair-breaking factor, /0 The fact that the coherence length in superfluid 3He can be tuned by pressure makes this system very attractiv e in probing a wide range of parameter space of /0 without changing the density of the impurity, i.e. the mean free path. Thuneburg was able to provide a better fit to the pre ssure dependence of CT by using an isotropic inhomogeneous scattering model (IISM) [Thu98a]. One cannot ignore a visually ev ident fact that aerogel strands are by no means a collection of isotropic scattering centers nor small enough to be treated by a purely quantum mechanical scattering theory. Volovik argued that, in the ABM state, the coupling of the orbital part of the order parameter to the randomly fluctuating anisotropic structure, i.e. aerogel, should prevent the onset of a long range order, thereby leaving it as a glass state in the long wavelength limit [Vol96]. Fomin proposed a class of orbitally isotropic equal spin pairing states, so called the robust phase, as another candidate for the A-like phase [Fom03]. In c ontrast, the robust phase has a long range order in the pr esence of random anisotropy diso rder, but as argued by Volovik, it is not more stable than the ABM state from the free energy point of vi ew [Vol05]. Specific NMR frequency shifts have been calculated for both scenarios [Fom06, Vol06].

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28 1.4 Liquid in Porous Media Normal liquid 3He in the hydrodynamic regime might be treated as a classical liquid in porous media. Liquids in porous media have been studied fo r both technical and academic reasons rooted in a broad range of practical applications such as in biological systems and the oil industry [Zho89, Hai99]. Even though practical structure properties (e.g. permeability) of the pore can be, in principle, extrac ted from ultrasonic measuremen ts [War94, Joh94a, Joh94b], it is appreciated that the motion of fluid in a porous structure is a non-trivial problem and a theoretical challenge. For the classical hydrodynamic fluid, sound is attenuated by various mechanisms, including the friction between liqui d and pore surface and the squirt of fluid in narrow cracks [Bio55a, Bio55b, DVor93] Biot developed a theory for acoustic response for the friction in the high frequency regime (small visc osity) and in the low frequency regime (large viscosity). For longitu dinal sound, the attenuation is proportional to / 1 in large viscosity regime and to in the other regime [Bio55a, Bio55b]. In Dvorkin et al.s model of a water saturated rock, the attenuation fr om the squirt-flow mechanism is an order of magnitude stronger than the attenuation from the friction [Dvo93]. In the 1950s, Biot developed a theory for sound propagation in a por ous elastic media, which is filled with a viscous fluid [Bio55a,b]. He began his study from the Poiseuille type flow which does not have turbulence and is valid at low frequency. Considering the effect of the relative motion between the fluid and the solid, he found two longitudi nal waves (first and second kind) and one rotational wave. The liquid a nd the solid tend to move in phase for the first kind and to move out of phase for the second ki nd of longitudinal wave. Because the second kind is highly attenuated, only the first kind is the true wave at lower frequencies. An isotropic porous media with uniform pore size is considered for the calculation. He predicted that the

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29 shape of the pores is no t important for the frequency dependen ce of the frictional force. At low frequencies, decrease in viscos ity results in more liquid motion and increases the attenuation quadratically as 2 T for the liquid 3He. At higher frequencies, Biot accounted for the deviation from Poiseuille flow by replacing the static viscosity with a frequency dependent one. In this regime, only a thin layer of fluid is locked on to the porous surface and a strong attenuation occurs within this layer [War94]. In this ca se, the attenuation of the first kind wave is proportional to T /1 for the liquid 3He. In 1994, Warner and Beamish measured the ve locity and the attenuation of transverse sound in a helium filled porous media [War94]. Th ey could study both the low frequency regime and the high frequency regime. Based on Biots model, they could determine the structure parameters of the porous media (tortuosity, pe rmeability and effective pore size) from their acoustic measurements. The dynamic permeability for a number of realis tic models with variable pore size was calculated by Zhou and Sheng [Zho89]. They found that the frequency dependence of the dynamic permeability did not depend on the models except two scaling parameters. However, the dynamic permeability of all the models could not be collapsed into a single curve when the throats connecting the pores were very shar p or the porosity was close to 1 [Joh89]. Yamamoto et al. also studied the effect of pore si ze distribution theoretically [Yam88]. They found that the velocity and the attenuation of sound ar e not affected by the pore size distribution in the low and high frequency limits, but strongly depend on the pore size distribution in the intermediate frequency range. Their prediction agreed with experimental data of attenuation in marine sediments [Ham72]. In the intermediate frequency regime near the low frequency limit, attenuation of the longitudinal sound decreases as pore distribution increases.

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30 Tsiklauri investigated the slip effect on th e acoustics of a fluid-sa turated porous medium [Tsi02]. In general, in fluid dynamics, it is a ssumed that the fluid at the surface of the solid moves at the same velocity as the solid does, which is known as the no-slip boundary condition. But when slippage occurs between th e fluid and the solid surfaces, the so called slip effect becomes important, particularly in the cas e of a highly confined porous media. Biots theory can describe acoustics for the scenario of the sloshing motion between a classical Newtonian liquid and the porous frame assuming no slip effect. Tsiklauri introduced a phenomenological frequency dependence of the s lip velocity and found that it affects the effective viscosity in the intermediate frequenc y domain leading to a higher attenuation than what is predicted by the Biot theory. In the low and high frequency limits, his results agreed with the predictions of Biot.

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31 Figure 1-1. Temperature de pendence of sound velocity and attenuation of pure 3He [Ket75]. The solid lines are the results of fit using Eq. 1-11 and 1-12 [Roa77]. The superfluid transition is signaled by th e sharp change near 2 mK Reprinted figure with permission from P.R. Roach and J.B. Ketterson, in Quantum Fluids and Solids, edited by S.B. Trickey, E.D. Adams, and J.F. Du ffy (Plenum Press, New York, 1977) p.223. Copyright (1977) by Springer.

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32 Figure 1-2. P-H-T phase diagram of superfluid 3He [Web1].

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33 Figure 1-3. Gap structures of the superfluid 3He, the B phase and A phase. The space between inner and outer shell represents a gap amplitude in momentum space. Figure 1-4. Microscopic structure of 98% Aerogel by Haard (computer simulation) [Haa01].

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34 Figure 1-5. P-B-T phase diagram of superfluid 3He in 98% aerogel (blue line). Phase diagram for bulk Superfluid 3He is shown as a dotted line [Ger02a]. Reprinted figure with permission from G. Gervais, K. Yawata, N. Mulders, and W. P. Halperin, Phys. Rev. B 66, 054528 (2002). Copyright (2002) by the American Physical Society.

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35 CHAPTER 2 THE A1 PHASE OF SUPERFLUID 3He IN 98% AEROGEL 2.1 Overview In pure superfluid 3He, minute particle-hole asymmetry causes the splitting of the superfluid transition th rough the Zeeman coupling in magnetic fields. As a result, the third phase, the A1 phase, appears betwee n the normal and the A2 phase (the A phase in magnetic fields) [Amb73, Osh74]. In this unique phase, the condensate is fully spin polarized. The A1 phase has been studied by several groups and the width of the phase was found to increase almost linearly in field by 0.065 mK/T at the melting pressure [Osh74, Sag84, Isr84, Rem98]. Recently, Gervais et al. [Ger02a] performed acoustic measurem ents in 98% aerogel up to 0.5 T and found no evidence of splitting in the transition. Baramidze and Kharadze [Bar00a] made a theo retical suggestion that the spin-exchange scattering between the 3He spins in liquid and solid layers on the aerogel surface could give rise to an independent mechanism for the splitting of the transition. Detailed calculations [Sau03, Bar03] show that antiferromagnetic (ferroma gnetic) exchange reduces (enhances) the total splitting in low fields, but one re covers the rate of the particle -hole asymmetry contribution in high fields as the polarization of the localized spins saturates. These calculations were performed with the assumption that the A phase in aerogel is the ABM state. Figure 2-1 presents the low field data from Gervais et al. and the results of the calcul ation with (gr een) and without (red) the anti-ferromagnetic exchange. The co mparison between the theoretical calculation and the results of Gervais et al. suggests that the spin exchange between the localized moments and the itinerant spins is anti-ferromagnetic. However, Fomin [Fom04] recently formulated an argument that the order parameter of the A-like phase in aerogel should be inert to an arbitrary spatial rotation in the presence of the

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36 random orbital field presented by the aerogel structure. This condition enfor ces a constraint on the order parameter for equal spin pairs, he name d new state satisfying the constraint as a robust phase. This theory predicts that an A1-like phase would be induced by a magnetic field for a certain condition (e.g., in the weak coupling limit). This is a new type of ferromagnetic phase with nonzero populations for both spin projections unlike the A1 phase where only one spin component participates in forming Cooper pairs. However, the splitting of the A1 and A2 transitions in this case seems to evolve in a different manner compared to bulk 3He. If the A phase in aerogel is correctly identified as an axia l state, then a similar field dependent splitting of the superfluid transition must exist at least in th e high field region since the level of particle hole asymmetry is affected only marginally by the presence of high porosity aerogel. 2.2 Experiments The experiment was performed at the High B/T Facility of the National High Magnetic Field Laboratory located at Univer sity of Florida. Figure 2-2 shows the cut-out views of the sample cell and the acoustic cavity. The main cell body (except the part of heat exchanger) was originally made out of titanium to reduce the nuc lear spin contribution to the heat capacity in magnetic fields. However, a leaky seal was f ound repeatedly at the s ilver-titanium epoxy seal and forced us to replace the top pa rt with the one made of coin silver. The top part of the cell body forms a diaphragm so that the cell pressure can be monitored capacitively. We used the same acoustic cavity that was utilized in the work of Gervais et al. [Ger02a, Ger02b]. The 98% porosity aerogel was grown in side the acoustic cavity by Norbert Mulders at the University of Delaware. The cavity is composed of a quart z transverse sound (AC-cu t) transducer and a longitudinal sound (X-cut) transduc er separated by a 0.010 diamet er stainless steel wire. The 3/8 quartz transducers were manufactured by Valpey-Fisher (c urrently Boston Piezo-Optics, Bellingham, MA). The aerogel was grown insi de the cavity formed by sandwiching the two

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37 stainless steel wires between the transducers under external force2. After the aerogel was grown in the cavity, the copper wires we re attached to the transducer with silv er epoxy and the cavity was placed on top of a Macor holder, since the vi gorous chemical reaction does not allow us to use epoxies or solder before the completion of critical drying pro cess. In this geometry, the transducers are in contact with both bulk liquid an d liquid in the aerogel. Therefore, we detected the change in the electrical im pedance of the transducer caus ed by both liquids. The acoustic measurement was performed at the third harmonic re sonance, 8.7 MHz, for all the data presented in this chapter. The volume of the cell is de signed to be less than 1 cm3 to ensure a short thermal relaxation time. The sample liquid in the cell is cooled by the PrNi5 demagnetization stage (DS) through a 0.9 m-long anneal ed silver heat link extending below the DS. The transitions in the pure liquid were confirmed indepe ndently by a vibrating wire (VW) placed in liquid near the ultrasound transducer [Rem98]. The vibrating wire was made out of 0.1 mm manganin wire bent into a semi-circular shape with a 3.2 mm diamet er. The resonance frequency of the VW was 13 kHz. As shown Fig. 2-3, an alternating current is fed through VW by the voltage oscillator, then the Lorentz force causes VW to oscillate. Th e amplitude of induced voltage across VW at a fixed excitation frequency was m easured using a Lock-in amplif ier. The amplitude of the induced voltage follows the amplitude and damping of the VW oscillation, which is determined by the viscosity of the surrounding liquid. Th e acoustic spectrometer output was recorded continuously while the temperature of the sample varied slowly. No significant hysteresis was observed for data taken in both warming and cool ing directions. The data presented (if not specified) were taken on warming and the typical warming rate in our study was 0.1 0.2 mK/h. 2 Actually aerogel was grown around the whole cavity. All aerogel formed outside of the cavity was removed using tooth pick after the completion of the drying procedure.

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38 Temperature was determined by a 3He melting pressure thermome ter (MPT) attached to the silver heat link right below the cell in the experi mental field region. Figure 2-4 shows the picture of the MPT and the sample cell on the silver heat link. The spectrometer (Fig. 2-5) us ed in this experiment was made by Jose Cancino, an undergraduate research assistan t [Lee97]. This is a conti nuous wave (cw) bridge type spectrometer employing a frequenc y modulation (FM) technique. The FM signal was produced by an Agilent E4423B generator an d the rf output was set at 11 dB m. The FM excitation signal was fed into the transducer at its resonance freq uency through the first po rt of a 50 ohm matched quadrature hybrid (QHB, SMC DQK3-32S). The third port of the QHB was connected to the transducer inside the cryostat. The output signal from the fourth port of the QHB was amplified by a low noise preamplifier (MITEQ, AU-1519) a nd demodulated by a double balanced mixer (mini-circuits, ZLW-1-1). This procedure conve rts the FM modulated rf signal into an audio signal at the modulation frequency. The demodulated low frequenc y signal was detected with a two-channel lock-in amplifier (S tanford Research System, SR530). The impedance mismatch at the transducer causes a reflected signal to appear on the fourth port. Th e reflected signal from the transducer can be nulled by feeding the (a mplitude and phase adjust ed) FM signal to the second port of the QHB. The transducer works like a resonator for the FM radio receiver [Kra80]. If the signal to be tran smitted and a sinusoidal carrier are given by Eq. 2-1 and Eq. 2-2, the frequency modulated signal can be written as Eq. 2-3. ) cos()(ttxm m (2-1) ) cos()(txtxC C C (2-2) )]sin( cos[))]([cos()(0 1t tAdx AtVm m C t m C (2-3)

PAGE 39

39 m and C are the modulation and carrier frequencies. is the modulation amplitude that regulates maximum shift from C [Wik06]. As the FM signal is fed to the resonator, the output signal from the transducer is proportional to the product of the resonator amplitude, ) ( f, and the amplitude of the input signal, )(1tV. The FM signal has a finite but small oscillation that is centered around C Within this finite region, the slope of the resonance line shape, a(), can be taken as a constant. Therefor e, the output from the transducer, 2V, is given by Eq. 2-4. The mixer works as a multiplier of two input signals which are coming from the transducer through the QHB and from the splitter (mini-circuits ZFSC-3-1). The multiplied (mixed) signal ()()(21tVtV ) has two frequency components, low and hi gh frequency parts. The low frequency part of the signal coming from the mixer is given by Eq. 2-5. Finally, the AC component of this signal is measured by the low frequency lock-in amplifier, which is proportional to a(), as shown in Eq. 2-6. )()]sin( [)()()(1 1 2tVt aBtVftVm m (2-4) )sin( )(3t aBtVm m (2-5) atV )(4 (2-6) Therefore, the spectrometer detects the sl ope of the transducer resonance shape, a, at a given frequency, [Bol71]. In the ideal case, the acoustic response measured by a Lock-in amplifier should produce zero in the 1f mode and a maximum in the 2f mode at resonance. The schematic signal shape at each step is shown in Fig. 2-5. In our measurements, the excitation frequency is fixed at the resonance, while the 3He is still in the normal fluid. A large ratio of the modulation amplitude (3 kHz) and the modulation frequency (400 Hz) were chosen to avoid the

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40 discrete nature in frequency spectrum. To be sensitive enough to detect any changes in the slope a, the modulation amplitude should be chosen such that the magnitude of the deviation from C is small compared to the width of the transducer resonance frequency. A crucial part of the measurement is tuning the spectrometer by adju sting the attenuators (JFW, 50R-028) and the phase shifter to produce a null c ondition for maximum sensitivity. By going through this tedious procedure, better than 1 ppm resolution in frequency shift is achieved. Figure 2-6 shows the spectrometer outputs obt ained by sweeping the frequency through the transducer resonance for two diffe rent spectrometer settings at 28.4 bars and 3 T. Frequency sweeps in the normal and deep inside of the superf luid are plotted together. By using Set2, we were able to reduce the asymmetry between th e size of maximum and mi nimum of the acoustic response. It also shifted the cr ossing point in the normal and superf luid curves closer to zero. Figure 2-7 shows how the spectrometer tuning aff ects the acoustic response during a temperature sweep. After tuning the spectrome ter to shift the crossing point closer to zero, the acoustic responses at CT and in the superfluid became almost the sa me. It can be noticed that Fig. 2-7 (b) shows clearer changes in slope at aCT and at aABT on warming, as opposed to Fig. 2-7 (a). Later on, to obtain the clear slope change, the sp ectrometer was tuned to produce a symmetric resonance shape in the normal liquid. At zero field, the Greywall scale [Gre86] wa s adopted to convert the measured melting pressure to temperature using the solid ordering transition as a fixed point to establish the pressure offset. In the presence of magnetic fields, the calibration by th e University of Tsukuba group [Yaw01, Fuk03] was employed. In their wo rk, the calibration was given in two separate regions the paramagnetic phase an d the high field phase of solid 3He up to 14.5 T. However, below 3.5 mK, our region of interest, only the cali bration in the high field phase is available.

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41 Consequently, the range of our temperature determ ination is limited to fields between 7 and 13 T for the pressures of our work. The dash-dotted lines in Fig. 2-8 repres ent the high field phase transition of the solid 3He in the MPT and below the boundary is the region wher e the calibration is done. The melting pressure in the high field phase was given by Eq. 2-7 [Yaw01], where the fourth-order temperature dependence is expect ed by spin-wave theory and the sixth-order correction originates from the dispersion correction. 6 6 4 4)()()();(THcTHcHPHTPo (2-7) We used interpolated sets of coefficients for th e fields applied in our work. The width of the bulk A1 phase identified in the acoustic trace was used to fix the pressure offset. At each field, the calibration curve ) ;(HTP is vertically adjusted so th at the measured melting pressure interval of the bulk A1 phase maps out the correct temperat ure width at the same experimental condition (two point calibration method). The pressure offsets for al l fields (including zero field) are around 6 kPa within 10%. The temperature width of the A1 phase was obtained using the results from Sagan et al. [Sag84] and Remeijer et al. [Rem98]. Sagan et al. measured the shift of transition temperature in fields for pressures ra nging from 6 to 29 bars and determined their splitting ratio (mK/T) from the linear fit. Remeijer et al. conducted their experiments only at the melting pressure and performed a f it including a quadratic term. For consistency, our own linear fit from the data of Remeijer et al. at the melting pressure an d splitting ratio from Sagan et al. for other pressures are used to obtain the width of A1 phase at the pressures studied. By using the A1 width rather than the actual tr ansition temperature as a fixed point, we can circumvent the possible inconsistency in the absolute temperature scale, which is used in the previous work on the bulk A1 phase. The data points represented by th e solid and open circles in Fig. 2-8 were obtained in this way. For 5 T at 28.4 bars, the aerogel transition temperatures (squares) were

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42 determined by forcing the bulk A2 transition (diamond) on the linear fit for 2 AT (single point calibration method). We also made an estimation of the aerogel A1 transition temperatures beyond the field range that our prescribed ca libration method allows, by assuming a constant warming rate, which is set by the bulk transition temperatures and time interval. The crosses are obtained in this manner. The agreement between the filled circles and crosses in the overlapping region is excellent and encouragi ng. The sensitivity of the melti ng pressure thermometry rapidly declines in higher fields and lowe r temperatures due to a decrease in solid entropy. For example, dTdP / drops from 3.3 kPa/mK at 2 mK and zero field to 0.1 kPa/mK at 15 T for the same temperature. This intrinsic property of the melting curve, in combination with the enhanced noise in high fields, renders it practically impo ssible to make an accurate determination of the aerogel A2 transition temperatures we ll below the dotted line where dTdP / << 0.1 kPa/mK. Typical noise in our high field pressure measurement is about 4 Pa. The Origin script for the high field MPT calibration is given on appendix A. 2.3 Results The acoustic traces for six differe nt fields at 33.5 bars are show n in Fig. 2-9. The acoustic trace is plotted along with that of the VW to comp are the transition signatures of the bulk liquid. The vibrating wire measurement was done in a similar fashion as described by Remeijer et al. [Rem98] and the amplitude at the resonance is s hown in the figure. Two sharp cusps in the VW trace correspond to the A1 and A2 transitions in pure liquid as reported in a previous work [Rem98]. These features are concurrent with th e jumps in the acoustic trace. The transitions in aerogel are not as sharp as in th e bulk. However, the smooth slope changes are quite clear and similar signatures of the s uperfluid transition in aerogel have been observed by Gervais et al. [Ger02a]. The field dependent evol ution of the transition features is demonstrated in Fig. 2-9.

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43 Below 3 T, we were not able to resolve the double transition features in aerogel while the features from the bulk can be traced down to zero field, merging into one (for example, see Fig. 2-10). As the field increases, the gap between th e two transition features in each liquid widens. It is important to emphasize that the bulk A2 and aerogel A1 transitions cross each other around 5 T and continue to move apart in higher fields A similar behavior wa s observed at 28.4 bars (Fig. 2-11), but the crossing occurred at a different field, around 7 T. The warming traces of the acoustic response are shown in Fig. 2-11 for several fields at 28.4 bars along with those of the vibrating wire (amplitude at resonance). The straight lines in Fig. 2-12 are the results of linear fits to the data points represented by the open (bulk) and solid (aerogel) circles including zero field result s. The slope of each linear fit is listed in Table 2-1. All data points determined by MPT including zero field data (red circles) are used for the fitting with equal weights. The asymmetry in the splitting is of special importance in two ways. First, the asymmetry rati o is a direct measure of strong coupling effects such as spin-fluctuati on [Vol90]. Second, it provides a va lid self-consistency check for our temperature calibration since only the total widt h of the splitting has been utilized. The asymmetry ratios are also listed in Table 2-1 where ) /() ()(2)()(1)( )( CaAaCaAa aTTTTr (2-8) and CaT)( is the zero field bulk (aerogel) tran sition temperature. The bulk slopes and r are in good agreement, within 8% with previous measurements [Sag84, Rem98]. It is notable that the A1 slopes for the bulk and aerogel are same in orde r of magnitude, which is consistent with the theory of Sauls and Sharma [Sau03]. The degree of individual splitting relative to the zero field transition temperature is plotted in Fig. 2-13. For both pressu res, the asymmetry in the aerogel is consistently smaller, by 22%, than th e bulk value for both pressures.

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44 2.4 Discussion The comparable slopes of the aerogel and bulk transitions suggest th at the new phase in aerogel has the same spin structure as the A1 phase in bulk. The asymmetry ratio is related to the fourth-order coefficients, i in the Ginzburg-Landau free ener gy expansion as defined by ) /(5425 r [Vol90]. In the weak coupling limit at low pressure, 1r and the strong coupling effect tends to incr ease this ratio as pr essure rises. Within the spin-fluctuation model, the strong coup ling correction factor, can be estimated from )1/()1( r[Vol90]. For 33.5 bars in aerogel, 09.0 The level of the strong coupling contribution at this pressure corresponds to th at of the bulk at around 15 bars [Sag84, Rem98], which indicates substantial reduc tion of the strong coupling e ffect. The weakening of strong coupling effects by the presence of impurity scat tering has been discussed theoretically [Bar02] and confirmed experimentally through an inde pendent estimation from the field dependent suppression of the A-B transition by Gervais et al. [Ger02a]. In Ginz burg-Landau limit and below the PCP, the width of the A phase in magnetic field can be derived to have the following form, 4 0 2 0)(1 B B O B B g T TC AB. (2-9) Gervais et al extract a value of )( g at 34 bars in aerogel that al so matches that of 15 bars in bulk. It is worth mentioning that aCT at 33.5 bars also falls on CT around 15 bars. However, the A1-like phase suggested by Fomin [Fom04] based on the robust phase requires a quite different asymmetry ratio. In this case, the asymmetry ratio 1 51)}/(1{BrF, (2-10)

PAGE 45

45 where 543245 9 B and reaches 0.16 in the weak coup ling limit [Bar06]. This asymmetry ratio is inconsistent with our observa tion, even when reasonable variations in the parameters are allowed. Fomin adopted a condition, in the presence of magnetic fields, to preserve an exact isotropy of the robus t phase. According to a recent theoretical work by Baramidze and Kharadze [Bar06], by introducing a small imbalance ( ) in the Fomins model, a more symmetric splitting was achieved. Sauls and Sharma [Sau03] suggested that th e anti-ferromagnetic coupling of 0.1~0.2 mK between spins in solid and liquid might be resp onsible for the suppressed splitting below 0.5 T observed by Gervais et al. [Ger02a]. Their calculation shows that the slope of the splitting starts to increase smoothly around 0.5 T ( exchange field strength between spins in solid) and reaches the slope close to that of bulk superflu id in high fields (Fig. 2-1). Our data cannot confirm this behavior owing to the lack of a low field temperature calibrati on. We point out that the data points acquired by assuming constant warm ing rate (Fig. 2-12) characteristically fall below the linear fit in the low field region. This fact, along with the observations made in low fields by Gervais et al. and us might suggest the presence of antiferromagnetic exchange coupling between the localized and mobile 3He spins. This brings up an interesting possibility of a completely new Kondo system.

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46 Figure 2-1. Splitting of the phase transition temperature in fields [Sau03]. Circles represent the low field data from Gervais et al [Ger02a] and lines represent the theoretical predictions by Sauls and Sharma [Sau03]. Green lines are calculated including antiferomagnetic exchange betw een liquid and solid layer of 3He and red lines are calculated excluding it. Without the spin exchange interaction, the CT splitting is comparable to that of pure 3He. Reprinted figure with pe rmission from J. A. Sauls and P. Sharma, Phys. Rev. B 68, 224502 ( 2003). Copyright (2003) by the American Physical Society.

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47 Figure 2-2. Cut-out views of the experimental cell and the acoustic cavity.

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48 Figure 2-3. Schematic diagram of the vibrating wire (VW). Digital Voltmeter I F B Resistor (~M ) Voltage controlled oscillator Lock-in amplifier Pre-amplifier

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49 Figure 2-4. Arrangement of the experimental ce ll (top) and the melting pressure thermometer (bottom) on the silver heat link extending below the nuclear demagnetization stage to the high field region.

PAGE 50

50 Figure 2-5. Schematic diagram of the conti nuous wave spectrometer (modified from Lees design [Lee97]). The arrows represent flow of signal. The signal shape at each step is shown next to the arrow. The model nu mber of each component is specified in parenthesis.

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51 Figure 2-6. Frequency sweep for tw o different spectrometer settings at 28.4 bars and 3 T. The dashed lines are measured at low temperature which is deep in the superfluid and the solid lines are acquired in the normal fluid phase. Set1 and Set2 represent different attenuation setting in cw spectrometer. In Set2 (red traces), we could reduce the asymmetry in the maximum and minimum. Set2 also shifts the crossing point in the normal and superfluid traces closer to zero. 8.6888.6908.6928.6948.6968.698 -10 -5 0 5 10 Set1 (Normal) Set1 (Superfluid) Set2 (Normal) Set2 (Superfluid) Acoustic Res ponse (a.u.)Frequency (MHz)

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52 1.41.61.82.02.22.4 -4 0 4 8 12 1.41.61.82.02.22.4 -4 -2 0 2 TaC Warming Cooling Acoustic Response (a.u.)T (mK) TaABTABTaCTABTCTC(a) Warming CoolingTaAB Acoustic Response (a.u.)T (mK) TaAB TAB TaC TABTC (b) Figure 2-7. Zero field acoustic si gnals at 28.4 bars for different spectrometer settings as a function of temperature. Panel (b) presen ts data taken after fine tuning of the spectrometer to Set2 in Fig. 2-6. In Pane l (a) the acoustic respons e at 1.4 mK is much larger than at CT In Panel (b) the acoustic responses at 1.4 mK and CT are almost the same as at CT and the changes in slope at aCT and at aABT on warming are clearer, compared with those in Panel (a).

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53 -20246810121416 1.6 2.0 2.4 2.8 3.2 T (mK) B (Tesla) P = 28.4 bars Figure 2-8. For high fields, we used the MPT calibration of Fukuyamas group [Yaw01]. Below 3.5 mK the calibration is given only for the high field phase, i.e. in the area under the dash-dotted line. The sensitivity of the MPT becomes too low to determine temperature well below the red dotted line. We used the width of bulk A1 phase as a fixed point for the temperature calibration and obtained transition temperatures in aerogel (closed circle). See detailed calibration procedure in the text.

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54 24002600 -2 0 2 4 0 1 2 3 4 160018002000220024002600 -2 0 2 4 -1 0 1 2 20002200240026002800 2 4 6 -2 0 2 4 2000220024002600 2 4 6 8 -1 0 1 2000220024002600 4 6 8 10 -2 -1 0 1 2 12001400 6 8 10 12 -2 0 2 B = 3 Tesla Time (min) TA1TaA1TA2TaA2 B = 5 Tesla Time (min) TA2 TaA1TA1TaA2 B = 7 Tesla Time (min) TA1TA2TaA2TaA1 B = 9 Tesla Time (min) TA1TA2TaA2TaA1 B = 11 Tesla Time (min) TA1TA2TaA2TaA1 Acoustic Signal (a.u.) P = 33.5 bars B = 15 Tesla Time (min) TA1TA2TaA2TaA1 VW Response (a.u.) Figure 2-9. Acoustic traces for 3, 5, 7, 9, 11, and 15 T at 33.5 bars on warming. The sharp jumps in the VW trace are identified as the A1 and A2 transitions in the bulk liquid. The acoustic trace also shows two sharp featur es at the exactly same time positions. )2(1 aAT indicates the position of the A1(2) transition in aerogel. The straight lines in the top-left panel are shown to illustrate the change in slopes at the transition.

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55 24002600280030003200 6 9 12 2200240026002800 -2 0 2 2400 2600 -2 0 2 TA2TaA1 Acoustic Response (a.u.)Time (min) P = 28.3 bars B = 1 Tesla TA1 P = 33.5 bars B = 2 Tesla Acoustic Response (a.u.)Time (min) TA2TaA1TA1 P = 33.5 bars B = 3 Tesla Acoustic Response (a.u.)Time (min) TaA2TA2TaA1TA1 Figure 2-10. Acoustic traces for 1, 2 and 3 T. Pres sures for each plot are labeled in the panels. Below 3 T, we were not able to resolve the double transition features in aerogel, while the features from the bulk can be traced at 1 T.

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56 24002600280030003200 -2 0 2 0 2 80010001200 -6 -4 -2 0 0 2 4 160018002000 -2 0 2 -2 0 2 0200400600800 -6 -4 -2 0 -4 0 4 02004006008001000 -4 -2 0 2 -4 0 4 8 1400160018002000 4 6 0 2 B = 9 Tesla Time (min) TA1TaA1TA2TaA2B = 3 Tesla Time (min) TA1TaA1TA2TaA2B = 11 Tesla Time (min) TA1TaA1TA2TaA2B = 5 Tesla Time (min) TA1TaA1TA2TaA2B = 7 Tesla Time (min) TA1TA2 TaA1TaA2Acoustic Signal (a.u.)Time (min) P = 28.4 bars B = 15 Tesla TA1TaA1TA2TaA2VW Response (a.u.) Figure 2-11. Acoustic traces for 3, 5, 7, 9, 11, and 15 T at 28.4 bars on warming. Each graph shows the acoustic (lower) trace along with that of the vibrating wire as a function of time. The sharp jumps in the vibrati ng wire trace are identified as the A1 and A2 transitions in the bulk liquid. The acoustic trace also shows two sharp features at the exactly same time position of A1(2) transition in aerogel.

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57 Figure 2-12. Transition temperatures vs. magnetic field for 28.4 and 33.5 bars. Open (solid) circles are for the bulk (aerogel) transitions determined by the two point calibration scheme. Open squares for 28.4 bars ar e obtained by the single point calibration method. Crosses are based on the constant warming rate. See the text for the temperature calibration procedures. The solid (dashed) lines are the results of linear fit for aerogel (bulk). -20246810121416 1.6 2.0 2.4 2.8 3.2 -20246810121416 1.6 2.0 2.4 2.8 3.2 T (mK)B (Tesla) P = 28.4 bars P = 33.5 barsT (mK)B (Tesla)

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58 Figure 2-13. Degree of splitting for each i ndividual transition vs. magnetic field. T is defined by C ATTT)2(1 where CT is the zero field superflu id transition temperature. T is similarly defined for aerogel. For the A1(2) transition, T is positive (negative). Filled (open) symbols are for the aerogel (bulk). 024681012 -0.03 -0.02 0.03 0.04 024681012 -0.05 -0.04 -0.03 0.03 0.04 0.05 B (Tesla) T/B (mK/T) P = 28.4 bars P = 33.5 bars T/B (mK/T) B (Tesla)

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59 Table 2-1. Slopes BT /(mK/T) of the splitting for the A1 and A2 transitions in bulk and aerogel at 28.4 and 33.5 bars. The asymmetry ratios, r for bulk and ar for aerogel, are also listed (see the text for a definition). BTA/1 BTA/2 r BTaA/1 BTaA/2 ar 28.4 (bars) 0.038 -0.026 1.46.06 0.034 -0.030 1.13.10 33.5 (bars) 0.043 -0.028 1.54.11 0.042 -0.035 1.20.14

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60 CHAPTER 3 THE A PHASE OF SUPERFLUID 3He IN 98% AEROGEL 3.1 Overview Early NMR studies of 3He/aerogel show evidence of an equal spin pairing (ESP) state similar to the bulk A phase [Spr95] and a phase transition to a non-ESP state similar to the bulk B phase [Bar00b]. A large degree of supercooli ng was observed in this phase transition (A-B transition), indicating the transition is first or der. Further studies using acoustic techniques [Ger02a, Naz05] and an oscillating aerogel disc [Bru01] have confirmed the presence of the A-B transition in the presence of low magnetic fields. While the effects of impurity scattering on the second order superfluid transition have been elucidated by these early studies, experiments designed to determine the effects of disorder on the A-B transition have been rather inconclusive [Ger02a, Bru01, Bar00b, Dmi03, Naz04a, Bau04a]. It is important to emphasize th at the free energy difference between the A and B phases in bulk 3He is minute compared to the condensati on energy [Leg90]. Moreover, both phases have identical intrinsic superfluid transition temperatures. The nature of highly competing phases separated by a first-order tr ansition is at the heart of ma ny intriguing phenomena such as the nucleation of the B phase in the metastable A phase environment [Leg90], the profound effect of magnetic fields on the A-B transition [Pau74], and the su btle modification of the A-B transition in restricted geometry [Li88]. We expect this transition to be extremely sensitive to the presence of aerogel and conjecture that even the low ener gy scale variation of th e aerogel structure would have a significant influence on the A-B transition. A number of experiments have been perfor med with the purpose of systematically investigating the A-B transition in aerogel [Ger02a, Br u01, Bar00b, Naz04a, Bau04a]. In experiments by the Northwestern group [Ger02a] using a shear acoustic impedance technique, a

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61 significantly supercooled A-B transition was seen while no signature of the A-B transition on warming was identified. In the presence of magnetic fields, however, the equilibrium A-B transitions were observed and the field dependence of the transition was found to be quadratic as in the bulk. However, no divergence in the coefficient of the quadratic term ) ( g (see Eq. 2-9) was observed below the melting pressure. This re sult is in marked contrast with the bulk behavior that shows a strong divergence at the polycritical point (PCP) [Hah93]. The Northwestern group concluded that the strong-coup ling effect is significan tly reduced due to the impurity scattering and the PCP is absent in this system. A lthough this conclusion seems to contradict their observa tion of a supercooled A-B transition even at zero field, other theoretical and experimental estimations of the strong-coupling effect in the same porosity aerogel [Cho04a, Bar02] seem to support their in terpretation. On the other hand, the Cornell group [Naz04a] investigated the A-B transition in 98% aerogel using a sl ow sound mode in the absence of a magnetic field. While the evidence of the supercooled A-B transition was evident, no warming A-B transition was observed. Nonetheless, they observed a partial conversion from B A phase only when the sample was warmed into the narrow band ( 25 K) of aerogel superfluid transition. Recently, the Stan ford group conducted low field (H = 284 G) NMR measurements on 99.3% aerogel at 34 bars [Bau04b]. They found a window of about 180 K window below the superfluid transition where the A and B phases coexist on warmi ng with a gradually increasing contribution of the A phase in the NMR spectrum (Fig.3-1). For two sample pressures of 28.4 a nd 33.5 bars, we have observed the A-B transition on warming in the absence of a magnetic field and have found evidence that the two phases coexist in a temperature window that is as wide as 100 K. This chapter provides a detailed description of our results and interpretations.

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62 3.2 Experiments We used the same experimental techniques that were described in Chapter 2. The sample cell experience a stray field of less than 10 G, which arises from the demagnetization magnet. 3.3 Results The traces of acoustic signal taken near 28.4 ba rs in zero magnetic field are shown in Fig. 3-2 (a). The traces show the acoustic responses between 1.4 and 2.5 mK. The sharp jumps in the acoustic traces around 2.4 mK mark the bulk superfluid tr ansition and the distinct slope changes are associated with the superfluid transiti ons in aerogel (see chapter 2). The signatures of the supercooled A-B transition in the bulk and aerogel appe ar as small steps on the cooling traces. The identification of the st ep in the acoustic impedance as the A-B transition has been established by a systematic experi mental investigation of Gervais et al [Ger02a]. The cooling trace (blue) from the normal state of bulk reve als a well defined aerogel transition at 2.0 mK (aAT). The supercooled A-B transitions from the bulk (ABT) and aerogel (aABT) are clearly shown as consecutive steps at lower temperatures. After being cooled through both A-B transitions, both clean and dirty liquids are in the B phase. On warming the trace follows the B phase and progressively merges into the A phase (cooling trace) around 1.9 mK This subtle change in slope is the signature of the A-B transition on warming. This was the first indication of a possible A-B transition in aerogel on warm ing at zero field. Figure 3-2 (b) at 33.5bar shows sim ilar features, but the supercooled A-B transitions in aerogel and bulk occur at almost th e same temperature. On several occasions, we have observed that, when the liquid cooled from the normal phase, the supercooled A-B transition in bulk and aerogel occur simultaneously. Fu rthermore, the supercooled bulk A-B transition always precedes the aerogel transition, which might suggest that the aerogel A-B transition is induced by the bulk

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63 transition through proximity coupling. However, we have not carried out a systematic study on this issue. In order to test our identification of this merging point as the warming BA transition in aerogel, we performed tracking experiments similar to those described by Gervais [Ger02a]. The sample liquid is slowly warmed from the aerogel B phase up to various poi nts around the feature, and then cooled slowly to watch the acousti c trace for the signature of the supercooled A-B transition. During the turn around, the sample st ays within 30 K from the highest temperature reached (hereafter referred to as the turn-around temperature) for about an hour. If the warming feature is indeed the A-B transition, there should be a supercooled signature on cooling only after warming through this feature. The color coded pair s of the traces in Fig. 3-3 (a) are the typical results of the tracking experiments for different turn-around temperatures at 28.4 bars. In the bottom (blue) cooling trace from the normal state, one can clearly see two supercooled A-B transition steps. The sharper step appearing at 1.7 mK corresponds to the bulk A-B transition. We find that the size of the step indicating the aerogel A-B transition depends on the turn-around temperature. We can make a direct compar ison of each step size since the supercooled A-B transitions in aerogel occur within a very narrow temperature range, 40 K. Similar behavior was observed for 33.5 bars as shown in Fig. 3-3 (b). From the data obtained in the tracking experime nts, the relative size of the steps at the supercooled aerogel A-B transition, is plotted in Fig. 3-4 as a function of the turn-around temperature. The relative size is the ratio of th e step size for each trace and the step size for the trace cooled down from normal fluid. For both pressures we see narrow temperature regions (shaded regions in the figure) where the size of the steps grows with the turn-around temperature until aAT is reached. For aATT no appreciable change in the step size is observed. This

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64 suggests that only a portion of th e liquid in aerogel undergoes the B A conversion on warming in that region. An inevitable conclusion is that the A and B phases coexist in that temperature window. A similar behavior has been observe d in 99.3% porosity aerogel, although in the presence of a 284 G magnetic field [Bau04b]. Nazaretski et al. also found the coexistence of the A and B phases at zero field from lo w field sound measurements [Naz 04b]. It is worthwhile to note that at 10 G, the equilibrium A-phase width in the bulk below the PCP is less than 1 K. A quadratic field dependence in the warming A-B transition is observed in our study up to 2 kG (Fig. 3-5). The suppression of B-like phase is proportional to 2 B as seen in the previous experiment by Gervais et al. [Ge02a]. Unfortunately, no information on the spatial distribution of the two phases can be extracted from our measurements. 3.4 Discussion In Fig. 3-6, a composite low-temperature phase diagram of 3He in 98% aerogel is reproduced along with our A-B transition temperatures. We plot the lowest temperatures where the B A conversion is first observed on warming. It is clear that the slope of the A-B transition line in aerogel has the opposite sign of that in the bulk in the same pressure range. However, in a weak magnetic field, the slope of the bulk A-B transition line changes its sign from positive (Cpp where Cp represents the polycritical pressure) to negative (Cpp ) (see the dotted line in Fig. 3-6)3. The first order transition line is gov erned by the Clausius-Clapeyron relation, AB AB ABvv ss dT dP (3-1) where s and v represent molar entropy and volume for the A and B phase. Since ABss, the 3 In the bulk, this behavior persists even in high fields up to the critical field. The sign crossover point gradually moves down to around 19 bars near the critical field. See Ref. Hah93.

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65 change of the slope indica tes the sign change in ABvvv It is interesting to ponder why the strong coupling effects cause this sign change We find no published results addressing this issue. Nonetheless, this observati on and the fact that no divergence in )( ghas been observed in aerogel allow us to look at the phase diagram from a different point of view It appears, that in effect, the P-T phase diagram is shifted to higher pressure s in the presence of aerogel rather than simply shifted horizontally in temperature. As a result, the PCP has moved to a physically inaccessible pressure as a liquid, thereby leaving only the weak coupling dominant portion in the phase diagram. If this interpre tation is correct, then we have to face a perplexing question: How can we explain the existence of a finite region of the A phase at pressures below the PCP at B = 0? We argue that anisotropic scattering from the aerogel structure is responsible for this effect. Although there is no succes sful quantitative theore tical account of the A-B transition for Cpp the Ginzburg-Landau (GL) theory presents a quantitative picture for the A-B transition in a small magnetic field B, (relative to the critical field) for Cpp Under these conditions, the quadratic suppression of the A-B transition arises from a term in the GL free energy, namely BAABgfiizz *, (3-2) where iA represents the order parameter of a superfluid state with spin () and orbital (i) indices [Vol90]. The main effect of this term is to produce a tiny splitting in CT for the A and B phases. In the GL limit, the free energy (rel ative to the normal state) of the A (B) phase is )( 2 )(2/BA BAf and )( BA is the appropriate combination of parameters that determine the fourth-order terms in the GL theory. For Cpp the B phase has lower free energy than the A phase (AB ). However, when the two phases are highly competing, i.e., BA even a

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66 tiny splitting in the superfluid transition, ) 0( B C A CCTTT results in a substantial temperature region (much larger than CT ) where the A phase becomes stable over the B phase. The simplified representation of the aerogel as a collecti on of homogeneous isotropic scattering centers is not sufficient to descri be minute energy scale phenomena such as the A-B transition. The strand-like structure introduces an anis otropic nature in the scattering, e.g., p-wave scattering. This consideration requires an additional term in the GL free energy4 [Thu98b, Fom04], jjii aaAAaf 1 (3-3) where a is a unit vector pointing in the direction of the aerogel strand. In other words, the aerogel strand produces a random field that coupl es to the orbital co mponent of the order parameter. This random orbital field plays a ro le analogous to the magnetic field in spin space, thereby splitting the superfluid transition temper ature. If the correlation length of the aerogel structure, a is longer than the leng th scale represented by 0 the local anisotropy provides a net effect on the superfluid component, and af would give rise to the A-B transition, even in the absence of a magnetic field. Detailed free ener gy considerations indicate that the anisotropy would favor the la configuration for the A phase [Rai77], where l indicates the direction of the nodes in the gap. Using the expression for the coupling strength 1 calculated in the quasiclassical theory [Thu98b], we find that af is comparable to zf produced by a magnetic field 1~/)/(/0 1 Cz eTgB kG, where is the gyromagnetic ratio of 3He and is the mean free path presented by the impurity scattering off the aerogel strand. Since 4 Private communication with J.A. Sauls.

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67 /eB [) /( af], where / represents the anisotropy in the mean free path, only a fraction of 1% anisotropy is su fficient to produce the observed A phase width. The inhomogeneity of the local anisotropy over length scales larger than 0 naturally results in the coexistence of the A and B phase. A PCP where three phases merge, as in the case of superfluid 3He, should have at least one first order branch [Fom04, Yip91]. When this branch separates two highly competing phases with distinct symmetry, the PCP is not robust against the presence of disorder. In general, the coupling of disorder to the di stinct order parameters will produce different free energy contributions for each phase. Consequently, a strong influence on the PCP is expected under these circumstances [Aoy05]. It is possible that the PCP vanishes in response to disorder (as it does in response to a magnetic field) and a region of coexistence emerges. An experiment on 3He-4He mixtures in high porosity aerogel reported a similar disappe arance of the PCP [Kim93]. Strikingly similar phenomena have also been observed in mixed-valent manganites where the structural disorder introduced by chemical pr essure produces the coexistence of two highly competing phases (charge ordered and ferromagnetic phases) separa ted by a first-order transition [Dag01, Zha02]. A growing body of evidence suggest s that the coexistence of the two phases is of fundamental importance in understanding the u nusual colossal magnetoresistance in that material. Considering the energy scales involved in the A-B transition and anisotropy, it is not surprising to see a difference in the details of the A-B transition in aerogel samples even with the same macroscopic porosity. However, it is im portant to understand the role of anisotropic scattering. We propose that the effect of an isotropic scattering can be investigated in a

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68 systematic manner, at least in aerogel, by introducing controlled uniaxial stress, which would generate global anisotropy in a ddition to the loca l anisotropy. Aoyama and Ikeda have studied the A-like phase in the Ginzburg-Landau regime and found that the quasi-long-ranged AB M state has a free energy lowe r than the planar and the robust states [Aoy05]. Neglecting the inhomogeneous effect of aerogel on the order parameter, they parameterized the impurity effect only through the relaxation time and found that the impurity scattering weakens the strong coupling effect. The reduction of the strong coupling, which is consistent with our result in ch apter 2, shrinks the width of ABM phase, aAB aCTT They also studied an inhomogeneity effect of the impurity scattering, especially with strong anisotropy, and found the anisot ropy increases the width, aAB aCTT Their expectation on the anisotropy scattering is consistent with our arguments. They in terpreted the existence of the Alike phase below the bulk PCP [Ger02a, Naz04b, Bau04b] as a lowering of the PCP, and this effect is explained qualitatively by taking into account the anisotropi c scattering on the ABM state. The pressure of the PCP, which is different from that of bulk, was determined by the competition between two effects from the reduced and the anisotropy scattering. Inspired by our suggestion, Aoyama and Ikeda calculated the effect of global anisotropy induced in uniaxially deformed cylindrical aerogel of two distinct flavor s, compressed and stretched along the symmetry axis [Aoy06]. For both cases, th ey found that the ABM state opens up for all pressures. Furthermore, in stretched aerogel, a s liver of a new phase, the polar phase is predicted to appear just below aCT. Recently, Pollanen et al. performed x-ray scattering expe riments on the aerogel and found that a certain degree of anisotr opy exists even in an undeformed aerogel sample and argued that the global anisotropy could be ge nerated during the growth and drying stages [Pol06]. Now, P.

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69 Bhupathi in our group is pursuing a transver se acoustic measurement on the superfluid 3He in compressed aerogel. He did cut a commercial 98 % aerogel sample and put it on a transducer. The other side of the aerogel was pressed by a ca p for 5% in length or just enough to hold the aerogel. With 5% uniaxial compressi on, both the supercool ed and stabilized AB transitions in aerogel could be identified, but not in uncompres sed aerogel. We suspect that rough surface of aerogel establishes large open sp ace on the boundary to the transducer, except at some contact points. Because of the high attenuation of the tr ansverse sound, the transducer measures the liquid property very near to its own surface a nd if the open space is dominant, the acoustic response will be the same as those from th e bulk. It is important to measure the A-like phase for a sample without compression as a bench mark. For the next trial, aerogel will be grown on a transducer in situ by the Mulders group at the University of Delaware.

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70 (a) (b) Figure 3-1. The Stanford NMR measurement on superfluid 3He in 99.3% aerogel. (a) The fractions of A-like phase vs. te mperatures. The superfluid 3He in 99.3% aerogel was warmed up from B phase at low temperature a nd was cooled down at certain temperature. The temperatures of data (open circle) represent t hose turning points. The fraction was determined from the weight of the NMR line for the A-like phase [Bau04a]. (b) The phase diagram of superfluid 3He in 99.3% aerogel at 28.4 mT [Bau04b]. The closed circles represent th e AB transition on warming. Reprinted figure with permission from J.E. Baumgardner, Y. Lee, D.D. Osheroff, L.W. Hrubesh, and J.F. Poco, Phys. Rev. Lett. 93, 055301 (2004)] and [J.E. Baumgardner and D.D. Osheroff, Phys. Rev. Lett. 93, 155301 (2004). Copyright (2004) by the American Physical Society.

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71 1.21.41.61.82.02.22.42.6 -2 0 2 4 6 8 10 1.41.61.82.02.22.4 -4 -2 0 2 Warming CoolingP = 33.5 bars (b) Acoustic signal (a.u.)T (mK) TAB, TaAB TaAB TAB TaC TCTaAB Warming Cooling Acoustic Response (a.u.)T (mK) TaAB TAB TaC TABTC (a) P = 28.4 bars Figure 3-2. Cooling (blue) and warm ing (red) traces taken at 28.4 bars (a) and 33.5 bars (b). The signatures of the aerogel superflu id transitions and the aerogel A-B transitions are labeled as aAT and aABT.

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72 1.0 1.5 2.0 2.5 20 30 40 1.0 1.5 2.0 2.5 -5 0 5 10 (b) Acoustic Signal (a.u.) T (mK) P = 33.5 bars Acoustic Signal (a.u.) T (mK) P = 28.4 bars (a) Figure 3-3. Acoustic traces of tr acking experiments at 28.4 bars and 33.5 bars in zero magnetic field. Each pair of warmi ng and subsequent cooling is co lor coded. The turn-around temperatures are indicated by the vertical lin e for each pair. For clarity the traces are shifted vertically and vertical lines are added manually. The arrows indicate the direction of temperature change in time.

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73 Figure 3-4. The relative size of th e steps for the supercooled aerogel A-B transition is plotted as a function of the turn-around temperature for 28.4 and 33.5 bars. The relative size is the ratio of the step size fo r each trace and the step size for the trace cooled down from normal fluid. The lines going through the points are guides for eyes. The dashed vertical lines indicate the aero gel superfluid transition temperatures.

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74 0.000.010.020.030.04 1.4 1.6 1.8 2.0 2.2 28.4 bars, TaC 28.4 bars, TaAB 33.5 bars, TaC 33.5 bars, TaAB T (mK)B2 (T2) Figure 3-5. Field dependence of the warming A-B transition in aerogel fo r 28.4 bars (blue circle) and 33.5 bars (red circle). Triangles represent aCT for each pressure. The dashed lines show the average value of aCT and the dotted lines are linear fits of aABT in each pressure.

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75 0.00.51.01.52.02.5 0 10 20 30 P (bar)T (mK) Figure 3-6. The zero field pha se diagram of superfluid 3He in 98% aerogel al ong with that of the bulk (dashed lines) [Gre86]. The dotted line is the bulk A-B transition at 1.1 kG measured by Hahn [Hah93]. The aerogel superf luid transition line shown in blue is obtained by smoothing the results from Corn ell and Northwestern groups [Mat97, Ger02a]. The two closed circles ar e the lowest temperatures where the BA conversion starts on warming, and the points are connected by a dashed line that is a guide for the eyes.

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76 CHAPTER 4 ATTENUATION OF LONGIT UDINAL SOUND IN LIQUID 3He/98% AEROGEL 4.1 Overview High frequency ultrasound lends us a unique spectroscopic tool for investigating the superfluid phases of liquid 3He [Hal90, Vol90]. The frequenc y range of ultrasound conveniently matches with the superfluid gap, and the exotic symmetry of the superflu id phases allows various acoustic disturbances to couple to the superfluid component. The sharp pair-breaking edge occurs at ) (2T, where is the ultrasound excita tion frequency and ) (T represents the temperature dependent superfluid gap. The coupling of ultrasound to many of the order parameter collective modes has been demonstrat ed by strong anomalies in sound attenuation and velocity [Hal90]. All of these rich ultrasound spectrosco pic signatures could be elucidated because zero sound continues to propagate in the superfluid phase despite the presence of a gap at the Fermi energy. In the presence of high porosity aerogel, the mean free path, aFav, presented by the 98% aerogel is in the range of 100 200 nm as mentioned earlier. This length scale competes with the inelastic quasipa rticle scattering length, iFiv, and becomes relevant only below 10 mK. Although impurity scattering has a marginal influence on the thermodynamic properties of the normal liquid, a significant chan ge in the transport properties, specifically thermal conductivity, has been predicted at low temperatures [Ven00]. The pair-breaking mechanism from impurity scattering is known to induce impurity bound states inside the gap and to smooth the square ro ot singularity at the gap edge [Buc81]. In the presence of severe pair-breaking, the system tu rns into a so-called gapless superconductor with the gap completely bridged by the impurity states. The impurity states have a broad influence on

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77 the physical properties in the supe rfluid states. In superfluid 3He/aerogel, the gapless nature has been evidenced by recent thermal conductivity a nd heat capacity measurements [Fis03, Cho04b]. The acoustic properties of liquid 3He in aerogel are also a ffected by the presence of impurity scattering [Rai98, Gol99, Nom00, Ger 01, Ich01, Hig03, Hig05, Hri01]. The classic first to zero sound crossover in the normal liqui d was found to be effectively inhibited by the impurity scattering, maintaining 1 where ) /(iaia [Nom00]. It was also argued that an attempt to increase the excitation frequency would face an extremely high damping [Rai98]. Losing the luxury of having well defi ned zero sound modes at low temperatures has hampered investigations on the superfluid phase s in aerogel using high frequency ultrasound. Although the high frequency transverse acousti c impedance technique [Lee99] has been successful in identifying various tran sition features [Ger02a, Cho04a, Vic05], only a few attempts have been made to investigate the superfluid gap structur e using conventional high frequency longitudinal sound by Northwestern group [Nom00] and Bayreuth group [Hri01]. However, the Bayreuth experiment, which used a direct sound propagati on technique, suffered from poor transducer response and observed no suppression of the s uperfluid transition temperature in aerogel. We suspect that there might be a thin layer of bulk liquid between the transducer and aerogel, which co mplicated the sound transmission th rough the liquid in aerogel. No published results are available from this wo rk other than the aforementioned reference. In 2000, Nomura et al. conducted high frequency s ound (14.6 MHz) attenuation measurements of liquid 3He in 98% aerogel at 16 bars using an acoustic cavity technique [Nom00]. They found that the cro ssover from first to zero sou nd was effectively obscured by the impurity scattering of aerogel strands below 10 mK (Fig. 4-1). This behavior was rather easily understood in the framework of a simple viscoelast ic model. However, this approach failed to

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78 explain their results at higher temperatures, where the inelasti c scattering between the quasiparticles is dominant. Furthermore, the s ound attenuation in aerogel was found to saturate around 50 mK rather than follow a 2/1T dependence as observed in th e bulk liquid. Higashitani et al. attempted to explain these results by incorporat ing a collision drag effect [Nom00, Hig03]. The Northwestern group employed a cw method us ing a short path length acoustic cavity, which is not adequate in determining the absolute soun d velocity or, especially, the attenuation. This cw method relies on observing in terference patterns develope d in the cavity, and under conventional operation this pattern can be gene rated by sweeping the ex citation frequency. However, with a high Q quartz tran sducer, this method is not feasib le. Therefore, they had to sweep sample pressure to generate the necessa ry variations of the sound velocity, and this approach inevitably accompanies temperature drift. In addition to these difficulties, an auxiliary assumption had to be made to perform a two-pa rameter fit in attenuation and the reflection coefficient at the transducer surface5. In this work, we present our results of high frequency (9.5 MHz) longitudinal sound atte nuation measurements using a pulsed ultrasound spectroscopic technique. 4.2 Experiments Figure 4-2 (a) shows the schematic diagram of the experimental cell used for this study. The conceptual design of the cell is similar to th e one used in the transverse acoustic impedance experiment described in previous chapters. The experimental cell consists of a pure silver base with 9 m2 of silver sinter and a coin silver enclosure. The ceiling of the enclosure forms a diaphragm, so the cell pressure can be monitore d capacitively. Dimensions of each part can be 5 They assumed that the attenuation at 25 bars and 0.6 mK is zero. The reflection coefficient was determined as 0.8, which satisfies the pressure dependence of attenuation at CT. The attenuation of each pressure was calculated by the visco-elastic model. The visco-elastic model contains an aerogel mean free path as another parameter, which was taken as 240 nm in their work.

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79 found in appendix B. Two matched longitudinal LiNbO3 transducers (A and B as indicated in Fig. 4-2 (a)) are separated by a Maco r spacer, which maintains a 3.05 ( 0.02) mm gap between them. The aerogel sample was grown in the spac e confined by the transducers (1/4 diameter, 9.5 MHz resonance frequency) to ensure the contact between the tran sducer surface and the aerogel sample. This is an extremely important pr ocedure since a thin laye r of bulk liquid or an irregular contact between the transducer and aer ogel would cause an unwanted reflection at the boundary of aerogel and liquids [Joh94a]. A 1 MH z AC-cut quartz transducer is placed right above the transducer A to monitor bulk respons e using a FM modulated cw method as described in chapter 2 (12.8 MHz, 500 Hz deviation freque ncy, 100 Hz modulation fr equency). A piece of cigarette paper interrupts the path between the tran sducer A and the quartz transducer in order to spoil the back reflection. For transducer B, th e irregular surface of s ilver sinter effectively diffuses unwanted sound propagation through the bulk liquid. The sample cell is thermally attached to the gold-plated copper flange welded to the top of the Cu nuclear demagnetization stage. An MPT and a Pt wire NMR thermomete r (PLM-3, Instruments of Technology, Helsinki, Finland) are located right next to the sample cell on the same flange. The MPT was used for T > 1 mK and the Pt NMR thermometer, calibrated against the MPT, was used for T < 1 mK. Figure 4-3 shows temperature determined by MPT and Pt -NMR thermometer as a function of time for a complete cooling and warming cycle. The Pt-NMR thermometer is calibrated using an equation, )/(BAMTS where M SA, B are the normalized Curie constant, the magnetization determined by NMR and of the background signal, respectively. SA was obtained by integrating the NMR free induction decay for a fixe d time span. Two coefficients ( M and B ) were treated as fitting parameters to match the temperature determined by the MPT. The green line in Fig. 4-3 was obtained in this wa y. As a result the thermal gradient between the MPT and the

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80 Pt-NMR thermometer the green line does not match perfectly with the black line. However, by shifting about 5 minutes in Pt-NMR response one can find an excellent consistency between two thermometers (red line in Fig. 4-3). The typi cal warming and cooling rates were 3 K/min. A commercial spectrometer LIBRA/NMRKIT II (Tecmag Inc., Houston, TX) was employed for this study (Fig. 4-2 (b)). Th e same spectrometer was used for acoustic measurement on pure liquid 3He by Watson at UF [Mas00, Wat00, Wat03]. Typically, the output level from the NMRKIT II was set to the maximum of 13 dBm and was fed to the transmitter through a -20 dB attenuator. This input signal, after amplified to an appropriate level by a power amplifier, was used to excite tran sducer B. A 1 s pulse was generated by the transducer B (transmitter) and the response of the transducer A (receiv er) was detected. The acoustic signal from the receiver was amplified by a low noise preamplifier, Miteq AU-1114, and was passed to NMRKIT II. The data acquisi tion began right after th e end of the excitation pulse. The width of pulse, 1 s, is short enough to separate echoes in the low attenuation regime and wide enough in frequency to cover the resona nce spectrum of the transducers. Although a matched pair of transducer was used, the sl ight difference in resonance spectra of two transducers determines the overall shape of the response (see next section). The typical setting of the LIBRA/NMRKIT-II can be found in appendix C. Origin scripts are used for data analysis and the details are in Appendix D. Each measur ement presented in this work is the result of averaging 8 pulses responses generated in the ph ase alternating pulse sequence at 9.5 MHz of primary frequency. The data was taken every 5 minutes and the interval between pulses for the average was 4 seconds. All the measurements pres ented in this work were done at zero magnetic field.

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81 4.3 Results Figure 4-4 (a) shows the responses of trans ducer A at 34 bars in the time domain for selected temperatures. The primary response, wh ich begins to rise around 8 s, grows below the aerogel superfluid transition (CT = 2.09 mK, indicated by an arrow in the figure). As temperature decreases, a train of echoes starts to emerge and four clear echoes are visible at the lowest temperature. The time interval between the primary response and the subsequent echo is twice of that between the excitation pulse and th e primary response. For 14 bars, only the first echo is clear at the lowest temperature due to th e higher attenuation (Fig 4-4 (b)). From the time of flight measurements, the speed of sound was de termined to be 350 () m/s at 34 bars, which is approximately 20% lower than in bulk. The velocity did not show temperature dependence within 3%. At 9.3 kHz, Golov et al. observed velocity enhancement of about 2% below CT at 22 bars [Gol99]. The coupling between th e normal component of the superfluid 3He and the mass of the elastic aerogel modifies th e two-fluid hydrodynamic equation [Gol99]. Two longitudinal sound modes are found and the velo cities of slow mode and fast mode (sv, fv) are given as /1/~1a fcc (4-1) /asascc (4-2) The velocity of the fast mode calculated from the modified equation agre es well with our result (with a 5% discrepancy at most) (s ee Fig. 4-5). Due to the ringi ng of the transducers, the width of the received pulses is much broader than the excitation pulse (1 s). The location of the rising edge of the signal depended on the sample pressure, and the receiver signal disappeared when the sample cell was completely evacuated. These obser vations assure that the detected signal at the receiver is from the sound pulse passing through the aerogel/liquid. The step-like shape of the

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82 responses, as shown in Fig. 4-6 (a), is cause d by a slight mismatch in the spectra of the transducers. Figure 4-6 (b) shows a Fourier tran sformed transmitted signal (black circle) and the frequency profile of each transducer (blue a nd red lines). The frequency profile of each transducer was obtained using a duplexing scheme involving a di rectional coupler. The green line in Fig. 4-6 (b) is the result of simple multiplication of the two frequency profiles from the transmitter and receiver. As can be seen, this profile mimics the spectra of the receiver obtained by the conventional method used in this experiment. A weighted multiplication would improve the match and the difference in fine structures may be caused by duplexing the signal. For most of our measurements, -7 dBm exc itation pulses were used (+13 dBm excitation with -20 dB attenuator). Various levels of excitations ranging from 3 to -27 dBm were tested in order to check the linearity. Figure 4-7 (a) presents the signals fo r different excitations taken at 40 mK and 33 bars. The signal show s a mild distortion near the p eak area at a 0 dBm excitation and the peak structure is severely distorted with a -3 dBm excitati on due to saturation. However, the unsaturated part of the res ponse scales linearly w ith the excitation. The linearity of the acoustic response was confirmed in the superfluid as well where the sound attenuation is lower. In Fig. 4-7 (b), the signal is normalized to its peak amplitude and the normalized signal for lower excitations fall on top of the signal from our typical setting (-20 dB ). The signal shapes and the attenuations did not show any dependence on exc itation within 5%, which means the sample and measurement scheme are in the linear regime. The warming traces with -7 dBm and -13 dBm excitations overlap well within the experimental error indicating that that heating effect caused by pulse transmission is negligible (Fig. 4-8). In Fig. 4-6 (a), two normalized primary respons es, taken at 0.4 mK and 2.5 mK and at 29 bars, are displayed for comparison. Despite the difference in thei r absolute sizes, the normalized

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83 responses have an almost identical shape, whic h indicates that the spec tra of the transducers remain the same throughout the measurements, and the change in the resp onse is caused by the change in the acoustic properties of the medium. Therefore, the relative sound attenuation can be obtained by comparing the primary responses. We have used two different methods to extract the relative attenuation from the signal before be ing normalized: one was to use the peak value of the primary response and the other was to use the area under the curve by integrating the signal from the rising edge (8 s point in Fig. 4-6 (a)) to a fixed point in time (23 s point in Fig. 4-6 (a)). Both methods produce consistent results to within 7%, except at 10 bars (13%). The absolute attenuation at 0.4 mK and 29 bars wa s used as a reference point in converting the relative attenuation into the absolute attenua tion. Absolute attenuations for the other temperatures were calculated by comparing the area of the primary response for each temperature to that for the standard absolute attenuation. Because of an electroni c glitch, probably from a slip of the relay switch in the NMRKIT II, the spectrometer sensitivity was not guaranteed to be the same for every run. Therefore, data were collected in one batch for each pressure at a fixed temperature, 9 mK. The relative attenuations for all other pressures were obtaine d by linking it to the atte nuation at 9 mK and 29 bars. Only for 34 bars, the absolute attenuati on was calculated independently from its own echoes. We decided the slope of the signal decay from two points, the peaks of the primary response and the first echo. To check the uncertain ty of this method, we compared it to the slope determined by the peaks of the primary respon se, the first echo, and the second echo. The discrepancy of two methods give s a negligible difference on the standard absolute attenuation ( 0.08 cm-1).

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84 As shown in Fig. 4-9, the peaks of the prim ary response and the subsequent echoes exhibit a bona fide exponential decay. With the knowledge of the gap between the transducers, the absolute value of attenuation was determined. Due to the drastic acoustic impedance mismatch at the boundary between the tr ansducer and the aerogel/3He sample, the echoes were assumed perfectly reflected at the interface6. The normalized attenuations for 12 and 29 bars as a function of the reduced temperature are shown in Fig. 4-10 along with that of pure 3He measured at 29 bars using 9.5 MHz sound excitation [Hri01]. There are three main cont ributions to the ultrasound attenuation in pure superfluid 3He: (1) pair-breaking mechanism, (2) coup ling to order parameter collective modes (OPCM), and (3) scattering of the thermal quasi-p articles [Hal90]. All the features mentioned above have been extensively investigated theoretically and e xperimentally. At ~ 9.5 MHz, the contributions from (1) and (2) give rise to a strong attenuation peak as clearly shown in the figure. These salient features are completely missing in the aerogel, although the superfluid transition is conspicuously marked by the rounded d ecrease in attenuation. A similar behavior was observed by Nomura et al. [Nom00] and this behavior was interpreted as indirect evidence that the sound propagation remained hydrodynamic even in the superfluid phase [Hig05]. We have checked the primary responses in the normal liquid (13 mK at 29 bars) and the superfluid (1 mK at 34 bars) using higher harmonics of the transducers (up to 96 MHz). Only the 9.5 MHz excitation produced a recognizable response in th e receiver, thereby confirming that the sound mode at 9.5 MHz remains hydrodynamic even in th e superfluid phase. If the sound at 9.5 MHz was the zero sound, the attenuation would not show that much frequency dependence. The zero 6 Using the composite density of 3He/98% aerogel at 29 bars, c = 0.113 g/cm3, measured sound velocity, 330 m/s, and the known acoustic impedance value for LiNbO3, tZ= 3.4 106 g/scm2, less than 1% loss is expected at the boundary.

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85 sound regimes are still supposed to exist at high fr equencies, but the attenuation of zero sound is expected to be too high for a quantitative measurement. On cooling at 29 bars (filled ci rcles in Fig. 4-10), our data show a sharp jump at 1.5 mK (7 .0/ CTT). This feature is the supercooled AB transition in aerogel. On warming, the sound attenuation follows the B-phase trace of cooling and th en merges smoothly with the cooling trace of the A-phase around 1.8 mK (9 .0/ CTT) without showing a clear BA transition feature. This observation is consistent with the previous transverse acoustic impedance measurements described in chapter 3 [Vic05], which proved that this merging point is where the BA conversion begins. On cooling, the A to B transition is observed down to 14 bars (Fig. 4-11). Neither an A-B transition feature nor hysteretic behavior was observed for 12 bars and below. Therefore, our a ttenuation results on wa rming are for the B phase in aerogel, except for the 200 K window right below the superf luid transition at high pressures. A broad shoulder feature around 6 .0/ CTT in the 29 bars trace progressively weakens as the pressure decreases, and this structure eventually disappears, as can be seen in the 12 bars trace. In Fig. 411, the aCT and the aABT determined in this attenuation measurement are plotted along with previous results presented in ch apter 3 (open circle). Our phase diagram is in good agreement with the ones mapped by different met hods [Mat97, Ger02a, Hal04]. In the B phase with a clean isotropic gap, sound attenuation, TkTBe/)((where Bk is the Boltzmann constant) is expected from the thermal quasi-particle contribution mentioned above. However, the temperature dependence of attenuation in aerogel is far from exponential and is not quenched down to 2.0/ CTT. Furthermore, the sound attenuation approaches a fairly high value for both pressures at zero temperature, allo wing a reasonable extrapolation.

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86 The absolute attenuations on warming for severa l pressures are plotted in Fig. 4-12. The attenuation increases as pressure is reduced. The shoulder is less pronounced for low pressure and disappears below 21 bars complete ly. Zero temperature attenuations 0 are estimated by assuming a quadratic temperature dependence (thi n black line) for the low temperature region we do not believe that this specific assumpti on affects our conclusion since our data already reached very close to zero temperature limit. The attenuation at lowest temperature decreases as the pressure is enhanced and it sa turates above 25 bars. At the s uperfluid transition temperature, the attenuation of 8 bars is very close to that of 10 bars. Due to the high attenuation at this pressure, the signal from receiver was quite sm all but still we could detect the temperature dependence below the superfluid transition. The normalized attenuations show the shoulder features disappear below 21 bars (Fig. 413(a)). Higashitani et al.s calculations [Hi g06] for several pressures are plotted with our data at 34 bars (Fig. 4-13(b)). Their calculations show more pronounces shoulder features and lower zero temperature attenuation for the same pressure The absolute attenuation at the superfluid transition, C and the normalzed zero temperature attenuation, 0 /C are given in Fig. 4-14(a). All values are taken from warming data except at 8 bars, and it is noteworthy that both 0 andC increase as the pressure is reduced. Alt hough the dashed line following the normalized zero temperature attenuation is a guide for eye, 0 /C approaches 1 near 7 bars which is close to the critical pressure below which no superfluid tran sition in 98% aerogel has been observed. The error bar is estimated from the differences of th e attenuations calculated by the peak and the area of the primary response. Absolute attenuation measurements allow quantitative analyses. Based on the theory of Higashitani et al. in which collision drag effect is inco rporated, one can get the mean free length

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87 in this system. As shown in Fig. 414(b), the whole pressure dependence of C can be nicely fit with a single parameter for = 120 nm. Figure 4-15 shows the lower bounds of the zero energy density of states at T =0 estimated from C /0 (details in next section). For the normal liquid, the low temperature part of the attenuation is quite similar to those reported by Nomura et al. [Nom00]. However, our results sh ow quite different behavior above 40 mK (Fig. 4-16 (a)). A broad minimum appears around MT= 40 mK for 29 bars and the attenuation continues to increase with temperature for MTT. Based on our measurements for three pressures, it seems that MT decreases with the sample pressure. Interestingly, Normura et al. expected a similar rise in at tenuation at high temperatures by considering the decoupling of liquid from aerogel, although their experimental results did not follow their prediction. In general, decoupling of liquid from the hosti ng porous medium occurs when the viscous penetration depth, /2, is of the order of the averag e pore size. This sloshing motion of the porous medium and viscous liquid provi des an extra damping mechanism. Since the viscous penetration depth of 3He followsT /1, Nomura et al. argued that the condition for decoupling would be satisfied at a certain temperature. Furthe rmore, by invoking Biot's theory [Bio55a], they projected a 2 T dependence of attenuation in the high temperature region. However, our high temperature data can be best fit to ~ 7.0 T for all sample pressures that we studied. High temperature fits shown as dashed lines in Fig. 4-16 (a) present temperature dependences with powers of 0.64, 0.73 and 0.73 for 10, 21 and 29 bars, respectively. For the whole temperature range, the sound velocity remains constant within our experimental resolution.

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88 4.4 Discussion Recently, Higashitani et al. [Hig05] calculated the hydrodynamic sound attenuation for this specific system based on the two-fluid model. Their calculation considered the aerogel motion generated by the collision drag effect [Ich01, Hig03] In their theory, the total sound attenuation, t has two contributions, hdt where d is the attenuation caused by the friction between the aerogel and the liquid, and h is the hydrodynamic contribution. d is proportional to the temperature dependent frictional relaxation time, f whereas h is proportional to the shear viscosity, ) (T The detailed temperature dependence of the attenuation also requires knowledge of the superfluid density, s in addition to the parameters described above. Despite the complexity of the model, their calcul ation provides a reasonably good account for our observation, such as the broad shoulder stru cture observed at 29 ba rs and the monotonic approach to finite attenuati on near zero temperature [Hig05] The moderate change of attenuation below CT agrees with Higashitani et al.s prediction based on the friction and the viscosity contribution (Fig 4-13(b)). The relativel y steep slope of Nomura et al. [Nom00] might be caused by their assumption that the attenuation at 25 bars and 0.6 mK is close to zero, which is about 2 cm-1 in our data. The shoulder of the temp erature trace at high pressure can be explained by the combination of the continuously decaying viscosity contribution and the friction contribution which increases just below CT for high pressure with reasonable aerogel mean free path. However, as pointed our earlier, the s houlder feature is much more pronounced in their calculation and the 0 is lower than our projected values from the measurements. Their calculation is based on the homogeneous scatte ring model which tends to overestimate the

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89 frictional damping but to undere stimate the viscous damping7. The agreement between our experimental results and thei r calculation can be improved by incorporating inhomogeneous scattering which gives rise to fu rther reduction of the average valu e of the order parameter. The extra reduction in the order parame ter would be reflected in a larg er viscosity and, accordingly, a larger 0 is expected. In addition, an increased normal fluid fraction will weaken the frictional effect. It is also expected th at the effect of anisotropic scat tering would enter into play through various relaxation times. Detailed calculations including these effects are under consideration. In particular, as the temp erature approaches zero, 0 f and 0 d Consequently, the zero temperature attenuation is proportional to the finite shear viscosity, itself related to the zero-energy density of states, ) 0(n, by the relation, ) 0()(/)0(4nTC for the unitary limit8 [Hig05]. Here, )0(n is the zero energy density of states at zero temperature, which is normalized to the density of states for normal liquid. Regard less of a specific theoretical model, the finite zero temperature attenuation reflects the existen ce of a finite zero-energy density of states induced by impurity scattering. Our results are in concert with the work of Fisher et al. [Fis03] in which finite thermal conductivity (mostly in the A-phase) in the zero temperature limit was observed [Sha03]. Indirect evidence for the presence of finite impurity states has also been suggested from the heat capacity measurement performed by a Northwestern group [Cho04b]. They inferred a linear temperature depe ndence in the heat capacity below 1 mK by analyzing the heat capacity jump at the aerogel transition and the heat capacity in the superfluid phase down to 1 mK. However, )0(n, estimated by the two different experimental techniques, shows significant discrepancies, especially in the pressure dependence. Because the frictional 7 Private communication with Higashitani 8 A power of 2 is theoretically predicted in the Born limit.

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90 contribution disappears at T = 0, the attenuation ratio C /0 is the same as the ratio of the viscosity, if we ignore th e frictional contribution at CT. Therefore, C /0 in Fig. 4-14(a) provides the lower bound for the viscos ity ratio. The lower bounds of ) 0(n are estimated from the viscosity ratio for the unitary and the Born sca ttering limits (Fig. 4-15). The density of states estimated from the specific heat measurements by the Northwestern group is about 0.6 for pressures from 10 to 30 bars [Cho04b]. The unique porous structure could be responsible for the increas e of the attenuation at high temperature (Fig. 4-16 (a)). In other words, there is no well-defined pore size as aerogel has a completely different topology than other conventional porous materials. Furthermore, the nanoscale strand diameter is much smaller than any other length scale involved in the 3He/aerogel system up to 70 mK. In Fig. 4-16 (b), we show seve ral relevant length scales estimated for 98% porosity aerogel: the effective mean free path, including aerogel scat tering, the viscous penetration depth of 3He, the average distance be tween aerogel strands, R = 20 nm, and the diameter of aerogel strand, a = 3 nm [Thu98a]. At all temperatures shown, is larger than any other length scales. However, it is interest ing to notice that the attenuation minimum (MT) occurs around the temperature where crosses R and MT moves to a higher temperature for lower pressure. For MTT, is larger than a and the hydrodynamic description for the motion between the aerogel and the liquid ceas es to be valid. Therefore, Biots theory which is basically a hydrodynamic theory may not be applicable. Only for MTT, the system enters into a regime where Biots theory can be cons idered, particularly the intermediate frequency regime for most of our high temperature data, and is expected to be T independent. Currently, we do not have an explanation for the origin of the 7.0 T dependence for MTT. However, the slip effect

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91 [Tsi02] may give rise to a more temperature depe ndence owing to the enhan ced velocity gradient of liquid at a solid surface. A distribution of pore sizes [Yam88, Pri93], the squirt mechanism [Dvo93], and/or the unique topology of the aerogel also need to be considered for a better understanding.

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92 Figure 4-1. The attenuation of longitudinal sound in liquid 3He/aerogel at 16 bars for 15 MHz (circle) measured by Nomura et al. [Nom00]. The attenuation in bulk 3He (dotted line), given for comparison, shows the fi rst to zero sound transition around 10 mK. Reprinted figure with permission from R. No mura, G. Gervais, T.M. Haard, Y. Lee, N. Mulders, and W.P. Halperin, Phys. Rev. Lett. 85, 4325 (2000). Copyright (2000) by the American Physical Society.

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93 (a) (b) Figure 4-2. Schematic diagram of experiment al setup. (a) Experimental cell (b) Pulse spectrometer.

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94 2000250030003500 0.0 0.5 1.0 1.5 2.0 MPT PtNMR PtNMR (w. time shift) T (mK)Time (min) Figure 4-3. Temperature determined by MPT and Pt-NMR thermometer. With an appropriate shift in time, the Pt-NMR thermometer shows better match with MPT at high temperature (red line).

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95 Figure 4-4. Evolution of receiver signals on warming. (a) For 34 ba rs, the superfluid transition is indicated by the arrow. Four echoes can be seen in the trace taken at the lowest temperature. (b) For 14 bars, only the first echo is observed at the lowest temperature. (a) (b)

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96 05101520253035 150 200 250 300 350 400 450 Bulk 3He Theory 3He in Aerogel v (m/s)P (bar) Figure 4-5. Velocity of bulk 3He [Hal90] and liquid 3He in aerogel at CT. The dashed line is calculated from the two-fluid model modified by impurities (Eq. 4-1).

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97 (a)0510152025 0.0 0.5 1.0 2.5 mK 0.4 mK Amplitude (normalized)Time (s) (b)-0.4 0.0 0.4 0 4 8 Amp (a.u.)Freq (MHz) BA A B A*B Figure 4-6. Signal from receiver. (a) Receiver sign als normalized to its peak amplitude at 2.5 mK and 0.4 mK at 29 bars. (b) FFT of the transmitted signal (Balck circles), each transducer (red and blue) at 0.3 mK and 34 bars. The green line is obtained by the product of the two FFT (A and B).

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98 (a)0510152025 0 8 16 24 Amplitude (a.u.)Time (s)Attenuator -10 dB -13 dB -16 dB -20 dB -23 dB -26 dB (b)0510152025 0.0 0.4 0.8 Amplitude (normalized)Time (s) Attenuator -20 dB -26 dB -30 dB -40 dB Figure 4-7. Linearity test. (a) for 40 mK and 33bars and (b) for 0.3 mK, and 33bars.

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99 012 0 2 4 6 Attenuation (cm-1)T (mK)Attenuator -20 dB (first run) -20 dB (second run) -26 dB29 bars Figure 4-8. No significant change in attenuation indicates that the heating by the transducer is negligible. 020406080100 2 4 6 8 10 ln(Amp) (a.u.)Time (s) Figure 4-9. Receiver signal trace vs. time at 0. 4 mK and 29 bars. The primary signal and the subsequent echoes follow an exponential decay.

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100 0.00.20.40.60.81.01.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 12 bars, warming 29 bars, warming 29 bars, cooling 29 bars, bulk (Eska's group) / (TC)T / TC Figure 4-10. The normalized attenuati on in the superfluid phase at 12 (triangle) and 29 (circle) bars as a function of the reduced temperature. CT represents the superfluid transition temperature for the liquids in aerogel or bulk. The attenuation of the bulk is the measurement by Hristakos [Hri01], i.e. Eskas group. 0.00.51.01.52.02.5 0 5 10 15 20 25 30 35 Present Work (Long. Sound) Present Work (Tran. Sound) Gervais et al. (Tran. Sound) Matsumoto et al. (Torsional) Greywall Bulk P (bar)T (mK) Figure 4-11. Phase diagram at zero field. Our aCT (red triangle) values agree with those measured in other methods [Mat97, Ger00] (blue data). The AB transition on warming (red closed circle) and cooling (red star)is also observed in the transmission measurement. The red open circles are data from chapter 3. The black line represents bulk CT [Gre86].

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101 0.00.51.01.52.0 0 5 10 15 8 bars 10 bars 12 bars 14 bars 21 bars 25 bars 29 bars 34 bars (cm-1)T (mK) Figure 4-12. Absolute attenuations for pressures from 8 to 34 bars are presented as a function of temperature (on warming except 8 bars). Low temperature attenuation below the lowest temperature data point is obtained using a quadratic extr apolation (thin black line).

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102 0.00.20.40.60.81.01.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.00.20.40.60.81.01.2 0.0 0.2 0.4 0.6 0.8 1.0 Theory (Higashitani et al. ) 10 bars 14 bars 20 bars 25 bars 33 bars Current data 34 bars (T)(TC)T / TC (T) TCT / TC 8 bars 10 bars 12 bars 14 bars 21 bars 25 bars 29 bars 34 bars Figure 4-13. Normalized attenuati on in superfluid. (a) Normalized attenuations are presented for each pressure (on warming except 8 bars). (b) Each line represents the attenuation calculated by Higashitani et al. [Hig06]. Current data for 34 bars are plotted as open circles.

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103 05101520253035 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 051015202530 0 10 20 30 40 0CP (bar) C (cm-1) l = 100 nm l = 120 nm l = 140 nm Current data P (bar)C (cm-1) Figure 4-14. Attenuation vs. pres sure. (a) Attenuation at the superfluid transition temperature (C ) is plotted as a function of pressure (red circles). A ratio of the zero temperature attenuation to C is presented by the black circles. The dashed line is a guide for eye. (b) C for each mean free path is calculated by Higashitani et al. [Hig06] and is presented along with our data.

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104 05101520253035 0.0 0.2 0.4 0.6 0.8 1.0 Unitary Born n (0)P (bar) Figure 4-15. Zero energy density of states at zero temperature vs. pressure for the unitary and the Born scattering limits are estimated from C /0 in Fig. 4-14.

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105 110100 1 10 100 110100 4 6 8 10 12 14 10 21 and 29 bars)10 21 and 29 bars) l Ra Length (nm)T (mK) (b) 10 bars 21 bars 29 bars (cm-1)T (mK) (a) Figure 4-16. Attenuation for normal liquid. (a) Abso lute attenuation above the aerogel superfluid transition for each pressure (circle). The solid black line is the result of a fit based on the theory of Higashitani et al. [Hig05]. The high temper ature part is fit to 7.0 T (dashed line). (b) Comparison of the relevant length scales for 3He in 98% aerogel. : effective mean free path, : viscous penetration depth, R: average distance between aerogel strands, a: average diameter of aerogel strand.

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106 CHAPTER 5 CONCLUSION Superfluid 3He in high porosity aerogel is a syst em in which the effects of static impurities on a p-wave superfluid can be inve stigated in a systematic manner. We performed shear acoustic impedance measurements on 98% porosity aerogel in the presence of magnetic fields up to 15 T at the sample pressures of 28.4 and 33.5 bars. We observed the superfluid transition in the aerogel split into two transitions in the presence of magnetic fields above 3 T for both pressures. The field dependence of each transition is consistent with that of the A1 phase observed in pure liquid with a si gnificantly reduced strong coupling effect. Our results provide the first evidence of the A1-like phase in superfluid 3He/aerogel. For the same experimental setup, we observed the A-B transition on warming in zero magnetic field. Our observations show that the A and B phases in aerogel coexist in a width of same temperature of about 100 K. We propose that differences in the relative stability of the A and B phases arising from anisotropic scattering can acc ount for our observations. Consequently, we propose that the effect of an isotropic scattering can be inve stigated experimentally in a systematic manner, at least in aerogel, by introducing controlled uniaxial stress, which would generate global anisotropy in a ddition to the local an isotropy. Aoyama and Ikedas calculations agree with our interpretation with the anisotr opic scattering [Aoy05]. Fu thermore, in stretched aerogel, a sliver of a new phase, the polar phase, is predicted to appear just below aCT. We also performed measurements on longitudi nal sound attenuation at 9.5 MHz from 8 to 34 bars in an attempt to investigate the orbital st ructure of the superfluid phases in 98% aerogel. Using pulsed ultrasound spectroscopic techniques, in the low attenuation regi me, we were able to detect several echoes, which allowed us to determine the absolute attenuation. No acoustic features associated with the or der parameter collective modes we re observed, and these results

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107 are a signature of the hydrodynamic regime [Hig05]. The finite attenuation at zero temperature provides clear evidence for the existence of zero-en ergy density of states inside the gap induced by impurity scattering. During cooling, the atte nuation does not drop as fast as bulk and a shoulder structure is observed at high pressure. This could be e xplained by the effect of friction between the liquid 3He and the aerogel. In superconductors, tunneling spectroscopy has been widely used in mapping out the gap structure [Wol89]. However, we do not have an equivalent experimental probe in the study of superfluid 3He. In this sense, it is interesting to note that unlike specific heat or thermal conductivity, sound attenuation coul d provide useful information on the profile of the impurity states by exploiting a wide range of excitation frequencies. With the same experimental configuration, the absolute sound attenuation and sound velocity have been measured in the normal phase of liquid 3He in 98% aerogel at 10, 21 and 29 bars up to 200 mK. The attenuation shows a minimum at MT 40 mK for 29 bars and follows a power law, 7.0 T for T > MT. As the pressure is lowered, MT is shifted to higher temperature. Currently, we do not have an expl anation for the origin of the 7.0 T dependence at MTT. As expected from a two-fluid model that incorporat es impurities, the sound velocity is reduced from the bulk value at the same pressure by 20% a nd remains constant for the whole temperature range. Due to the scattering of the aerogel, the sound excitation remains as first sound over the entire range of temperatures and pressures studied, which is identified by the absence of a first to zero sound transition.

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108 APPENDIX A ORIGIN SCRIPT FOR MTP CALIBRATI ON IN THE HIGH FIELD PHASE I programmed MPTfukuyamalessTn.c and MPTfukuya malow5.c with Origin C to calibrate numerically the MPT. These Origin C files shoul d be located in the Origin C folder and are operated with Origin7.0-SR4. MPTfukuyamaless Tn.c calculates a pressure shift for MPT calibration, adP, from the width of bulk A1 phase. MPTfukuyamalow5.c is programmed to calculate temperature for a given MPT pressure using adP. A text object was used to execute the C program. For example, if we click the text object, 1aT , in the worksheet, Pad, it will calculate and show 1AT and adP(Fig. A-1). A label control of the text object needs to be set as shown in Fig. A-2. How to make the object can be found at Objects on Windows in the Origin Help menu. Worksheet PtoT calculates te mperature from MPT pressure, P, when the TP object is clicked. Input parameters are arra nged above the button. In Label Control for TP object, PTfukuyamalow5 replaces MPTfukuyamalessTn.

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109 Figure A-1. Origin worksheet, Pad. Input parameters:21AAPPdP is the value measured by MPT. 21AATdTdT is adopted from Sagan et al. and Remeijer et al. references [Sag84, Rem98]. 640,,ccP are MPT calibration parameters for high field phase [Yaw01]. Outputs: 1AT and adP. adP is the pressure shift for MPT calibration obtained by utilizing dT as a fixed value. Figure A-2. Label control window for the object, 1aT , in Origin worksheet Pad.

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110 Figure A-3. Origin worksheet, PtoT, calculate s temperature, T, from MPT pressure, P. 640,,ccP are MPT calibration parameters for high field phase [Yaw01]. adP is obtained from worksheet Pad.

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111 A.1 MPTfukuyamalessTn.c //////////BEGIN #include //3He melting pressure temperature scale (Fukuyama scale). //P in bar, T in mK. //by Hyunchang Choi, Sept 2003 at UF. //This convert Pressure to Temperatur e (from column named 'P' to 'T') //Scale for 1~250mK //Neel pressure 'Pn' should be given from script(Labtalk). #define MAXIT 300 //Maximum allowed number of iterations. #define Tacc 0.001 //accuracy of Temperature #define Tmin 0.5 //Maximum and minimum of Temperature scale #define Tmax 3.5 void MPTfukuyamalessTn() { char szTemp[100]; //Get active worksheet Name. LT_get_str("%H", szTemp, 100); string worksheetName = szTemp; Dataset dd1(worksheetName+"_Ta1"); dd1.SetSize(1); dd1[0]=rtsafe(Tmin,Tmax,Tacc); //Call root finding function. } float rtsafe(float x1,float x2,float xacc) //from 'Numerical recipes in C' //Using a combination of NewtonRaphson and bisection, find the root of a function bracketed //between x1 and x2. The root, returned as the function value rtsafe, will be refined until //its accuracy is known within +/ -xacc.funcd is a user-supplied routine that returns both the //function value and the first derivative of the function. { int j; float df,dx,dxold,f,fh,fl; float temp,xh,xl,rts; funcd(x1,&fl,&df); funcd(x2,&fh,&df); if ((fl>0.0&&fh>0.0)||(fl<0.0&&fh<0.0)) nrerror("Root must be bracketed in rtsafe"); if (fl==0.0) return x1; if (fh==0.0) return x2; if (fl<0.0){ //Orient the search so that f(x1)<0. xl=x1; xh=x2; }else{ xh=x1;

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112 xl=x2; } rts=0.5*(x1+x2); //Initialize the guess for root, dxold=fabs(x2-x1); //the "stepsize before last," dx=dxold; //and the last step. funcd(rts,&f,&df); for (j=1;j<=MAXIT;j++){ //Loop over allowed iterations. if ((((rts-xh)*df-f)*((rtsxl)*df-f)>0.0) //Bisrct if Newton out of range, || (fabs(2.0*f)>fabs(dxold*df ))) { //or not decreasing fast enough. dxold=dx; dx=0.5*(xh-xl); rts=xl+dx; if (xl==rts) return rts; //Change in root is negligible. }else{ //Newton step acceptable. Take it. dxold=dx; dx=f/df; temp=rts; rts-=dx; if (temp==rts) return rts; } if (fabs(dx)
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113 } //For error message. void nrerror(char *error_text) { printf("%s\n", error_text); exit(1); } //////////END

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114 A.2 MPTfukuyamalow5.c //////////BEGIN #include //3He melting pressure temperature scale (Fukuyama scale). //P in bar, T in mK. //by Hyunchang Choi, Sept 2003 at UF. //This convert Pressure to Temperatur e (from column named 'P' to 'T') //Scale for 1~250mK //Neel pressure 'Pn' should be given from script(Labtalk). #define MAXIT 300 //Maximum allowed number of iterations. #define Tacc 0.001 //accuracy of Temperature #define Tmin 0.1 //Maximum and minimum of Temperature scale #define Tmax 3.5 void MPTfukuyamalow5() { char szTemp[100]; //Get active worksheet Name. LT_get_str("%H", szTemp, 100); string worksheetName = szTemp; Dataset dd1(worksheetName+"_P"); Dataset dd2(worksheetName+"_T"); UINT nRows=dd1.GetSize(); dd2.SetSize(nRows); for(int ii=dd1.GetLowerIndex( );ii0.0&&fh>0.0)||(fl<0.0&&fh<0.0)) nrerror("Root must be bracketed in rtsafe"); if (fl==0.0) return x1; if (fh==0.0) return x2;

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115 if (fl<0.0){ //Orient the search so that f(x1)<0. xl=x1; xh=x2; }else{ xh=x1; xl=x2; } rts=0.5*(x1+x2); //Initialize the guess for root, dxold=fabs(x2-x1); //the "stepsize before last," dx=dxold; //and the last step. funcd(P,rts,&f,&df); for (j=1;j<=MAXIT;j++){ //Loop over allowed iterations. if ((((rts-xh)*df-f)*((rtsxl)*df-f)>0.0) //Bisrct if Newton out of range, || (fabs(2.0*f)>fabs(dxold*df ))) { //or not decreasing fast enough. dxold=dx; dx=0.5*(xh-xl); rts=xl+dx; if (xl==rts) return rts; //Change in root is negligible. }else{ //Newton step acceptable. Take it. dxold=dx; dx=f/df; temp=rts; rts-=dx; if (temp==rts) return rts; } if (fabs(dx)
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116 *f=dd1[7]+dd1[10]-P+ dd1[8]*T^4+dd1[9]*T^6; *df=4*dd1[8]*T^3+6*dd1[9]*T^5; } //For error message. void nrerror(char *error_text) { printf("%s\n", error_text); exit(1); } //////////END

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117 APPENDIX B PARTS OF THE EXPETRIMENTAL C ELL FOR ATTENUATION MEASURMENT

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124 APPENDIX C TYPICAL SETTING FOR NMRKIT II NMRKitII/ LIBRA spectrometer was controlle d by Macintosh computer. The NMRkitII software was commanded by the apple script to measure one sequence of transmitted signal and save it as a file. Labview was programmed to save data every 5 minuites for the MPT, the PtNMR thermometer, the transverse transducer, the cell pressure and the NMRkitII. For each data point, the Labview program cal ls the application written by a pple script and the application orders the NMRkitII software to take and save da ta. The detail of apple script can be found in Watsons thesis [Wat00]. The typical set ting for NMRkitII is given in Fig. C-1.

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125 Figure C-1. The typical NMRkitII setting.

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126 APPENDIX D ORIGIN SCRIPT FOR THE DATA ANALYSIS OF THE ATTENUATION MEASUREMENTS Because the data taken by NMRkitII were huge in file size, an Origin script, main2.ogs was programmed for importing and anal yzing data. It was operated in Origin 6 or Origin 7. The screen shot of the Origin widow for data analysis is shown in Fig. D-1. Buttons to execute the Origin scripts are arranged on a layout, EXEbutton. The butt ons Import Data, Upload LS and Upload Pt are for importing data from th e Labview program, the NMRkitII and the PrNMR thermometer. The button, CalMCT, is for cal ibrating temperature from a given MPT pressure in the spreadsheet MCTcal. The spreadsheet MCTcalUF is the P-T table of MPT UF scale and it is used during the calibrati on procedure. The buttons, Integ LS and Sum Pt are used to obtain the area of each signal from the longitudi nal sound and the PtNMR. The ranges of data points for the area calculation are spec ified in the spreadsheet para.

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127 Figure D-1. The screen shot of Or igin worksheet for data analysis.

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128 1main2.ogs //Hyunchang Choi; //Necessary templates //1fixedFN.otw ; worksheet template; contain two colulm named A and B, both set as Y axis. // Uncheck Rename Worksheet to data file name in ASCII import options. //1MultiGraph.otp; graph template; five layers are arranged for each graph. ////////////////////////////////////////////////////////////////////////////////////////// ////////////////// //MCT Temp Calibration ////////////////////////////////////////////////////////////////////////////////////////// ////////// //////; [MCTcal] type -b Did you check parameters?; %D=MCTcal; N=%(%D,3,1); //Pn; %l=%(%D,3,2); //cal table; win -a %D; loop(i,1,wks.maxrows){ t=%D_Pmct[i]-N; for (k=1; k<500;k++){ s=k*500+1; if (t>%l_dP[s]){ for (j=1; j<=500; j++){ q=(k-1)*500+j; if (t>=%l_dP[q]){ %D_Tmct[i]=%l_TmK[q]; break;};}; break;};};}; type -b save file; ////////////////////////////////////////////////////////////////////////////////////////// ////////////////// //Import main dataset //////////////// //////////////////// //////////////// //////////////////// /////////////////////////////////////; [ImportData] create data -w 1000; getfile *.dat ;open -w %A; wks.col1.name$=fileN; wks.col2.name$=cellPb; wks.col3.name$=cellPa; wks.col4.name$=NewMCTb; wks.col5.name$=NewMCTa; wks.col6.name$=TranSXb; wks.col7.name$=TranSXa; wks.col8.name$=TranSYb; wks.col9.name$=TranSYa; wks.col10.name$=longS; wks.col11.name$=pt; wks.col12.name$=time;

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129 wks.addcol(Smct2); wks.addcol(SGpt); wks.addcol(longSI); wks.addcol(longSv); wks.col13.label$=5fftSmooth; wks.col14.label$=9sgSmooth; wks.col1.type=1; wks.labels(); %w=%h; win -a %w; %(para,2,1)="%B"; %(para,2,2)="%w"; worksheet -s 2 0 2 0; // select column 2 worksheet -p 202 1MultiGraph.otp; // plot selected column layer -s 1;layer -i202 %w_cellPa;layer -a; layer -s 2;layer -i202 %w_NewMCTa;layer -a; layer -s 3;layer -i202 %w_TranSXa;layer -a; layer -s 4;layer -i 202 %w_longS;layer -a; layer -s 5;layer -i202 %w_pt;layer -a; //layer -s 6;layer -d; label -s -d 1300 100 %w; getsave %w.opj; ////////////////////////////////////////////////////////////////////////////////////////// ////////// /////// //Load Long Sound data; ////////////////////////////////////////////////////////////////////////////////////////// ////////// /////// //....................................................; //This script import long9p5-#.dat files, calculate magnitude from x,y //and save it within one worksheet. If open a worksheet for each data file; //origin file become huge and the proccess is too slow.; //Jan13,2006 Hyunchang Choi; //.....................................................; [LoadLS] type -b Did you check parameters?; %F=%(para,2,1); //data folder %M=%(para,2,2); B=%(para,2,3); //beginning file; E=%(para,2,4); //end file; create LSmag -w 1000; wks.col1.type=1; label -s %M; win -t data 1fixedFN.otw long9p5; //Import rename off, col A set as y. loop(ii,B,E){ //loop() is more stable than for()

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130 window -a long9p5; open -w %Flong9p5-$(ii).dat; window -a LSmag; worksheet -i ii f$(ii); LSmag_f$(ii)=sqrt(long9p5_A^2+long9p5_B^2); }; win -cd long9p5; type -b save file; ////////////////////////////////////////////////////////////////////////////////////////// //////////////// //Integrate Long Sound ////////////////////////////////////////////////////////////////////////////////////////// ////////// //////; [IntegLS] type -b Did you check parameters?; //Parameters.......................................... %M=%(para,2,2); //save file name B=%(para,2,3); //beginning file; E=%(para,2,4); //end file; p=%(para,2,5); q=%(para,2,6); //.................................................... create LSmax -w 1000; wks.col1.type=1; worksheet -i 0 fileN; worksheet -i 1 xmax; worksheet -i 2 ymax; worksheet -i 3 integ; worksheet -i 4 x1integ; worksheet -i 5 x2integ; label -s %M; loop(ii,B,E){ integrate LSmag_f$(ii) -b 60 -e 2048;//find Max inbetween 60~2048 of row. x=integ.x0; i=x-p; j=x+q; integrate LSmag_f$(ii) -b i -e j; LSmax_fileN[ii+1]=ii; LSmax_xmax[ii+1]=integ.x0; LSmax_ymax[ii+1]=integ.y0; LSmax_integ[ii+1]=integ.area; LSmax_x1integ[ii+1]=integ.x1; LSmax_x2integ[ii+1]=integ.x2; }; type -b save file; ////////////////////////////////////////////////////////////////////////////////////////// ////////////////

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131 //Load PtNMR data ////////////////////////////////////////////////////////////////////////////////////////// ////////// //////; [LoadPt] type -b Did you check parameters?; //Parameters.......................................... %F=%(para,2,1); //data folder %M=%(para,2,2); //save file name B=%(para,2,3); //begining file; E=%(para,2,4); //end file; //....................................................; //This script import ptnmr#.dat files and save it within one worksheet. ; //Jan13,2006 Hyunchang Choi; //..................................................... create PtNMR -w 2048; wks.col1.type=1; label -s %M; win -t data 1fixedFN.otw Pt; loop(ii,B,E){ //loop() is more stable than for() %B=%F\PtNMR$(ii).dat; win -a Pt; open -w %B; win -a PtNMR; worksheet -i ii f$(ii); copy -a Pt_A PtNMR_f$(ii); }; win -cd pt; type -b save file; ////////////////////////////////////////////////////////////////////////////////////////// //////////////// //Sum PtNMR signal ////////////////////////////////////////////////////////////////////////////////////////// ///////////////; [SumPt] type -b Did you check parameters?; //Parameters.......................................... %M=%(para,2,2); //save file name B=%(para,2,3); //beginning file; E=%(para,2,4); //end file; a=%(para,2,7); //beginning of sum; z=%(para,2,8); //end of sum; //....................................................; create temp -w 10000; worksheet -i 0 abs; worksheet -i 1 sel; set temp_sel -e 10000; //needed for copy command below create ptSum -w 1000; wks.col1.type=1;

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132 worksheet -i 0 fileN; worksheet -i 1 sum; worksheet -i 2 InvSum; wks.col2.label$=$(a)t$(z); wks.labels(); label -s %M; for(ii=B;ii<=E;ii++){ temp_abs=abs(PtNMR_f$(ii)); copy -b a temp_abs temp_sel -b a -e z; sum(temp_sel); //using internal function is much fater than using loop for sum PtSum_fileN[ii+1]=ii; PtSum_sum[ii+1]=sum.total; }; type -b save file;

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133 APPENDIX E TRANSPORT MEASUREMNET OF Ca1.5Sr0.5RuO4 E.1 Overview In-plane electrical transport measurements were performed on Ca1.5Sr0.5RuO4 in the presence of magnetic fields up to 15 T app lied along the c-axis a nd in the direction 20o tilted away from that c-axis. Upon subs tituting Sr with isovalent Ca, Ca2-xSrxRuO4 shows an intriguing phase diagram ranging from p-wave superconductor at x = 2 to a Mott insulator at x 0.2 (Fig. E-1). The x = 0.5 system investigated in this wo rk is reported to be at the boundary between the magnetic metal (x < 0.5) and the paramagnetic metallic phase. A small but distinct increase in resistance was observed at T* 450 mK on warming, which is represented as the red circle in Fig. E-1. In addition, T* decreases with the applied magnetic field, and the feature in resistance disappears around 500 G. Our deta iled magneto-resistance measurements reveal unusual behavior in the low temperature and low magnetic field region that, we believe, is directly related to the resistance anomaly obs erved near 0.5 K in zero magnetic field. Interestingly, all transition temperatures at x = 0.5 are near 1K for the magnetic metal (MM), the structure transition (ST), the spin gl ass (SG), and the Fermi liquid (FL). E.2 Experiments The sample was prepared by Rongying Jin in the Mandrus group at Oak Ridge National Laboratory. The conventional 4wire measurements were performed. Typical sample size was 1 1 2 mm. The sample was mounted on thin sapphi re plate (1.2 mm in thickness, 6.5 mm in diameter) with GE varnish and four gold wires (1 mm in diameter) were positioned and were fixed on the sapphire plate with GE varnish. Te mporarily, the sapphire plate was glued with GE varnish on top of an aluminum plate for convenience to handle. To dry GE varnish it was heated for 2 minute on hot plate with 100 oC. The electrical wires were attached to the sample using a

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134 silver epoxy (EpoTek H20E). After applied betw een the sample and gold wire, it was heated for 5 minutes on hot plate at 300 oC. This procedure consistently results in ~ 1 contact resistance. However, the wire came off easily during cooling or at room temperature and only one sample wired by Juhyun Park, a student of M.W. Meisel, wa s survived through thermal cycles. After all 4 wires were set, the sapphire plate was rem oved from the aluminum plate and glued on the sample stage of the dilution cryostat. A test sa mple shown in Fig. E-2 was prepared by winding gold wire around the sample for secure contact. However, its resistance continuously increased in high fields and did not produce reliable results. The winding of a gold wire might cause too much Joule heating by eddy current. It is crucial to have a low contact resi stance to make low temperature transport measurements especially for highly conducting sa mples. A low sample resistance requires a high excitation (current) to have a reasonabl e S/N. However, the high current causes a significant heating mainly at the contact. Ma eno group used Dupont silv er paint 6838 [Ohm00] instead of EpoTek H20E used in our experiments. It is worth to try to evaporate small gold pads directly on the sample and attach wires on the pads using silver epoxy. This method was suggested by Ian Fisher at Stan ford University who found that th is method provides reliable and low loss contacts for thermal cycling. The transport measurements were conducted w ith a LR700 AC resistan ce bridge typically with 1 V excitation. Most of the data were taken from a dilu tion cryostat at room B137 and a magneto resistance with tilted field was taken at Bay #3 in the Micr okelvin Laboratory. E.3 Results and Disscussion The temperature dependences of resistan ce at zero field are presented for high temperatures (Fig. E-3) and for low temperatures (Fig. E-4). An abnormal drop in resistance was observed near 0.5 K. For 1 V excitation, the plot shows linear dependence in temperature

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135 instead of the quadratic dependence which is charact eristic for the Fermi liquid. In 2D metal, the linear T dependence of R is expected by Moriya et al. when the system is near antiferromagnetic instability [Mor90]. Below 0.3 K, the measurement with 2 V excitation shows consistently higher resistance than the one with 1 V excita tion. The difference between two measurements became more pronounced as temperature lowered i ndicating that there was a substantial heating at this excitation level. To address this issue, we conducted systematic measurements at various excitation levels at several fixed temperatures. The temperature of the sample plate is P-I-D controlled at a constant temperature within 0.5% of the set temperature and the resistance was measured using various excitation levels from 1 3 V. Figure E-5 shows the result of this study. The effect of heating is clearly demonstrated in this plot The resistance measured as the excitation level increased. This is the consistent with our interpretation that the change in the resistance is caused by heating since the sample shows metallic behavior in this temperature range. The resistance shows linear excitati on dependence and the sl opes of the linear dependence at each temperature are plotted in Fig. E-6. One can extract zero excitation sample resistance value by extrapolating to the zero ex citation point for each temperature. The orange line in Fig. E-4 was obtained in this manner and represents an es timated zero heating resistance which shows a linear temperature dependence. The magneto resistance (MR) for a field perpendi cular to the plane is presented in Fig. E-7 for wide range of fields. A detailed low field MR is displayed in Fig E-8. The MR increases with a field and dip around 40 kG. Above 40 kG, the MR decreases and finally crosses the zero at 70 kG. We observed unique feat ures in low fields which have never been reported on this material. The inset shows the details of shar p increases in MR at low fields (shoulder like structures). This shoulder struct ure gets smaller and moves to a lower fileds as T increases, and

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136 the structure completely disappears at around 450 mK where the zero field anomaly in resistance was observed. In the low field region, the MR increases linearly and cr osses over to a general quadratic behavior in higher fiel ds. More interestingl y, unexpected dips in MR were observed below 100 G. As temperature increased, these di ps also moved to zero field and disappeared above 450 mK. The positions in field and temperature of the shoulder structures and dips are plotted in Fig. E-9. As the extended lines sugg est, two features seem to converge to the same temperature where the zero field resistance anom aly was observed. This indicates that both features are coming from the same origin or th at one induces the other as like the structure transition induce the magnetic metal transition [Nak03]. MR with the field applied 20o away from the c-axis was measured and displayed along with the one with the fields pe rpendicular to the plane. While the shoulder structure is still present, the low field dip struct ure is missing. Unfortunately, this project ended prematurely because the sample leads lost contacts in a magnet quench occurred during a field sweep.

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137 Figure E-1. A Zero field Phase diagram of Ca2-xSrxRuO4 from S. Nakatsuji et al. [Nak03]. As the Ca is substituted with Sr, the phase of the sample changes from the antiferromagnetic insulator (AFI) to the magnetic metal (MM) and to the paramagnetic metal (PM). The AFI and MM transitions are represented green line and closed circle. Sr2RuO4 is a superconductor (SC) below 1.1 K. For the low concentration region of Sr, the temperatures for structure transitions are marked as ST (open circle). Their heat capacity and re sistance measurement show that the metal phases are always Fermi liquid below FLT (open square). In the shaded area around x = 0.5, the spin glass properties were observed from their susceptibility measurements. The red circle represents a point where the abnormal step in our resistance measurement was observed during a temper ature sweep. Reprinted figure with permission from S. Nakatsuji, D. Hall, L. Balicas, Z. Fisk, K. Sugahara, M. Yoshioka, and Y. Maeno, Phys. Rev. Lett. 90, 137202 (2003). Copyright (2003) by the American Physical Society. Ca2RuO4 Sr2RuO4SC TFL MM AFI SG PM Ts

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138 Figure E-2. Picture of a test sample. The gold wires was wound the sample for better contact. Later it turned out that the winding l oop caused heating problem (see text).

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139 Figure E-3. Temperature dependence of the re sistance in the absence of magnetic fields. Figure E-4. Resistance vs. temperature for variou s levels of excitation at low temperature.

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140 Figure E-5. The excitation dependence of the resistance for each temperature. Figure E-6. The temperature depend ence of the slopes in Fig E-5.

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141 Figure E-7. Normalized magneto resistance. Figure E-8. Field sweeps below 200 G.

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142 0.0 0.1 0.2 0.3 0.4 0.5 0.01 0.1 1 Ca 1.5 Sr 0.5 RuO 4 B (kG)T (K) Shoulder Dip Figure E-9. The position of the shoulder and dip structure in fiel ds vs. temperature.

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143 050100150 0.6 0.7 0.8 0.9 ramp up (20o) ramp up (0o) R (m)B (kG) Figure E-10. Magneto resistance for fields perpendicular to the plane (blue) and 20o tilted away from the c-axis (red) at 20mK. 0.00.10.20.30.4 0.735 0.740 0.745 0.750 0.755 ramp up (20o) ramp up (0o) ramp down (0o) R (m)B (kG) Figure E-11. Magneto resistance for low fields perpendicular to the plane (blue) and 20o tilted away from the c-axis (red) at 20mK.

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144 APPENDIX F NEEDLE VALVE FOR DR137 As a part of maintenance for the cryostat at B137 in Physics buildi ng (DR137), the needle valve for 1K pot was installed to prevent frequent impedance block.

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155 [Yip91] S. K. Yip, T. Li, and P. Kumar, Phys. Rev. B 43, 2742 (1991). [Zha02] L. Zhang, C. Israel, A. Biswas, R. L. Greene, and A. de Lozanne, Science 298, 805 (2002). [Zho89] M.Y. Zhou and P. Sheng, Phys. Rev. B 39, 12027 (1989). *The contents in this dissertation were or will be published at Journals below.* H.C. Choi, A.J. Gray, C.L. Vicente, J.S. Xia, G. Gervais, W.P. Halperin, N. Mulders, Y. Lee, Phys. Rev. Lett. 93, 145302 (2004). Copyright (2004) by th e American Physical Society. H.C. Choi, A.J. Gray, C.L. Vicente, J.S. Xia, G. Gervais, W.P. Halperin, N. Mulders, Y. Lee, J. Low Temp. Phys. 138, 107 (2005). The original pub lication is available at www.springerlink.com C.L. Vicente, H.C. Choi, J.S. Xia, W.P. Ha lperin, N. Mulders, Y. Lee, Phys. Rev. B 72, 094519 (2005). Copyright (2005) by the American Physical Society. Y. Lee, H.C. Choi, N. Masuhara, B.H. Moon, P. Bhupathi, M.W. Meisel, and N. Mulders, will appear in J. Low Temp. Phys. The original publication is available at www.springerlink.com H.C. Choi, N. Masuhara, B.H. Moon, P. Bhupathi, N. Mulders, M.W. Meisel, and Y. Lee, J. Low Temp. Phys., submitted in July 2006. The original publication is available at www.springerlink.com Y. Lee, H.C. Choi, N. Masuhara, B.H. Moon, P. Bhupathi, M.W. Meisel, and N. Mulders, in preparation. H.C. Choi, N. Masuhara, B.H. Moon, P. Bhupathi, N. Mulders, M.W. Meisel, and Y. Lee, in preparation. *Usages for figures cited from published articles are following regulations as written below.* Readers may view, browse, and/or download ma terial for temporary copying purposes only, provided these uses are for noncommercial pers onal purposes. Except as provided by law, this material may not be further reproduced distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the publisher.

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156 BIOGRAPHICAL SKETCH Hyunchang Choi was born in Seoul, Korea, as the eldest son and has two younger sisters. He spent his early childhood in a rural area of Daejeon City, Korea. During his elementary school years, his family moved back to Seoul. His best hobby was competing on math puzzles with classmates at that time. He also enjoye d making pra-models and assembling simple circuits like a clapping switch. In the th ird year of high school, he recei ved an award as a progressive student. The curiosity for the pr inciple of nature guided him to physics and he began his study of physics at Kyunghee University, Korea, in 1992. Owing to his success in the entrance exam, he was granted a full scholarship for one year. After 2 years of mandatory army service, he decided to continue his study in the USA to learn academics in a more open culture. He joined the Physics Department at the University of Florida in 1999 and joined Professor Lees group in 2002.