Title: Dispersion of 5 MHz zero sound in superfluid ³He near T subscript c in magnetic fields /
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Title: Dispersion of 5 MHz zero sound in superfluid ³He near T subscript c in magnetic fields /
Physical Description: x, 176 leaves : ill. ; 28 cm.
Language: English
Creator: Berg, Robert Frank, 1955-
Publication Date: 1983
Copyright Date: 1983
 Subjects
Subject: Low temperatures   ( lcsh )
Superfluidity   ( lcsh )
Liquid helium   ( lcsh )
Magnetohydrodynamics   ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1983.
Bibliography: Bibliography: leaves 171-175.
Statement of Responsibility: by Robert Frank Berg.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00098262
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000479227
oclc - 11795783
notis - ACP5951

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DISPERSION OF 5 MHZ ZRO SOUND
IN SUPERFLUID He
NEAR T IN MAGNETIC FIELDS








By

ROBERT FRANK BERG


A OISSERTATION PRES.EN TD TO THE GRADUATE SC-HOOL
OF ThE Uill.-EER~IT OF FLORIDA IN
PARTIAL FULFILLMENT OF THE kEQUIREMENITS
FOP THE DEGREE OF DOCTOR OF PHILOSOPHY



UN1lERSITY OF FLORIDA

i983















ACKNOWLEDGEMENTS


It is only with the skillful guidance of my research advisor,

Professor Gary Ihas, that I have been able to pursue and complete a

program of doctoral research at the University of Florida. He has been

both mentor and friend, offering critical advice and encouragement

whenever it was needed. I hope that his investment of many long nights

and days in the laboratory with me has proved as rewarding to him as it

has to me. I also owe a great debt to Professor Hugh G. Robinson of

Duke University for the year of training and support I received in his

laboratory a.nd for his understanding and encouragement upon my transfer

cO) Floria .

In adjdition ct professor Ihas, many people associated with the

Department of Pr' ics at U.F. have contributed their time and energy to

help me. Chief ima.n1 theo is Professor Dwight Adams, whose advice and

masertal assistance have proved invaluable on many occasions. Conversa-

cioni Cilth iand ass3Lstnce from Dr. Kurt Uhlig, Dr. Vijay Samalam, Greg

l.aas, John iPolle, nT..n Yi-Hua, Greg Spencer, and Brad Engel have been

an enjoyable parL of my stay here at U.F. Brad Engel also helped in

eating the later porch on of the data for this thesis. The expert

Ct:h.ical assistance of Christian Fombarlet, Don Sanford, James

kobiinson, san the men in the machine shop under Harvey Nachtrieb has

been unfailing. Sheri Hill is thanked for the skillful typing of this

manu Sript.









Three women in my life have been of utmost importance to me.

My mother, Dr. Ernestine Berg, whose example and attitude led me to an

appreciation of science, has always believed in me. My aunt, Dr.

Pauline Hilliard, has given her support during my stay in Gainesville,

especially during the last five months. My wife, Dr. Carol Emerson, is

owed thanks for her patience and understanding during these years and

also for leading me into low temperature physics in the first place.

Funding for this project has been supplied by the Division of

Sponsored Research at U.F. and the National Science Foundation (DMR-

8006929 and DMR-8306579).
















TABLE OF CONTENTS


ACKNOWLEDGEMENTS...................... .............. ............ ii

TABLE OF CONTENTS ..............................................iv

LIST OF FIGURES....................................................vi

ABSTRACT ...................................... ... .. ........... ... ix

SECTION 1 INTRODUCTION...................................... 1


SECTION 2
2.1
2.2
2.3
2.4
2.5

2.6
2.7


SECTION 3
3.1
3.2
3.3
3.-
3.5
3.6
3.
3. .
3.9
3. 10
3.11
3.12
3. 13
3.

SECTION -
1.1
-.2
4.3

.5


PERTINENT THEORY OF 3He......................... 15
General References............................... 15
The Normal Fermi Liquid..........................15
Superfluid Pairing...............................18
Comparison of the ABM and BW States...............20
Strong Coupling and the Finite
Field Phase Diagram ............................ .23
NMR in the A and Al Phases ............ .........27
Sound Propagation in the Normal
Fermi Liquid and the ABM-State....................32

APPARATUS........................................ 51
Large-Scale Features.............................51
Dilution Refrigerator............................55
Nuclear Cooling Stage............................59
Bundle Construction.............................61
Thermal Isolation and Heat Leaks.................65
Superfluid-Handling Apparatus....................69
Sinter Cell Tests for the He Heat Exchanger.....71
He Heat Exchanger Construction and Performance..77
Compressor ......................................... 81
Pressure Measurement and Control.................84
Sound Cell Contents..............................88
Magnetic Fields..................................91
3He NMR Electronics..............................97
Ultrasound Electronics...........................99

TECHNIQUE................................... ... 103
Refrigeration..................................103
Thermom try........................ ...... ......... 104
NMR in He ...................................... 112
Ultrasound Signals..............................114
Data Acquisition...............................118









SECTION 5 DATA REDUCTION.................................121
5.1 Data Examination.................................121
5.2 Temperature.....................................121
5.3 Sound Amplitude and Velocity....................125

SECTION 6 RESULTS AND INTERPRETATIONS.....................127
6.1 Assumptions.....................................127
6.2 Attenuation.....................................140
6.3 Phase Velocity Changes..........................144
6.4 Metastability at the AB Transition..............146
6.5 Summary and Future Work.........................148

APPENDIX A PHASE-LOCKED LOOP ANALYSIS...................... 150

APPENDIX B DATA REDUCTION PARAMETERS.......................154

APPENDIX C DATA ACQUISITION PROGRAM........................155

APPENDIX D DATA REDUCTION PROGRAM ..........................161

APPENDIX E CITED MANUFACTURERS.............................169

REFERENCES ...................................................171

BIOGRAPHICAL SKETCH ...........................................176















LIST OF FIGURES


FIGUP TITLE PAGE

SP, log T phase diagram of 3He...................... 5

2 P,T,B phase diagram of 3He......................... ...6

3 A and Al phases in exaggerated P,T plot............. 7

First sound velocity cI(P)......................... 10

5 *(t) and c(T) for 20 MHz sound ................... ....11

o In sicu 3He NMR thermometry........................14

7 M;R snifts in the A and Al phases..................31

ilr.entum perturbations of CO and C ...............36

9 Frequency regimes for sound in 3He.................39

1,. ClappLng and flapping modes...................... 45

IL T he'i retical attenuation, Im(x) ....................49

12 Theoretical velocity shift, Re(x) ................50

13 Cryostat and dewar .................................52

I -Solenoid and vacuum jacket ......................54

15 Dilution refrigerator performance ..................58

In Bundle welding setup...............................64

17 Time-dependent heat leak to bundle.................68

13 Heca exchanger plus mounted containers.............70

iq Sin.re test cell....... ........................... 75

20 Sincer test cell results...........................76

'L Heat e\chanper details..............................78








22 3He compressor.....................................83

23 Pressure and volume electronics .................86

24 Pressure control loop..............................87

25 Sound cell contents................................89

26 Static field control...............................94

27 NMR lineshape......................................95

28 Static field perturbation test.....................96

29 NMR electronics.................................... 98

30 Ultrasound electronics............................100

31 Slope of vA(T), f(P).............................. 110

32 Gap coupling parameter 6(P) measurements..........111

33 Field tuning of NMR peak..........................113

34 Detected sound pulse appearance...................116

35 Linearity of attenuation measurements.............117

36 Data acquisition cycle............................120

37 Deconvolution of temperature from NMR.............124

38 Data plots: 2 bar, 14 mT....................... 128

39 Data plots: 2 bar, 42 mT.........................129

40 Data plots: 9 bar, 14 mT.........................130

41 Data plots: 9 bar, 42 mT.........................131

42 Data plots: 18 bar, 14 mT........................132

43 Data plots: 18 bar, 29 mT........................133

44 Data plots: 18 bar, 42 mT........................134

45 Data plots: 22 bar, 14 mT........................135

46 Data plots: 22 bar, 42 mT........................136

47 Data plots: 31 bar, 14 mT ........................137

48 Data plots: 31 bar, 29 mT........................138









49 Data plots: 31 bar, 42 mT........................139

50 Scaled attenuation maxima.........................142

51 Locations of attenuation maxima................... 143

52 Velocity shifts scaled to 9 bar...................145

53 AB transition hysteresis.......................... 147

54 Phase-locked loop.................................153


viii















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



DISPERSION OF 5 MHZ ZERO SOUND
IN SUPERFLUID He
NEAR T IN MAGNETIC FIELDS

By

Robert Frank Berg

December 1983


Chairman: Professor Gary G. Ihas
Major Department: Department of Physics

Measurements of the attenuation and phase velocity changes

near Tc of 5 MHz ultrasound have been made in the A and Al superfluid

phases of 3He at 2.05, 9.15, 18.18, 21.53, and 31.12 bar. All measure-

ments were performed in the low fields of either 14.3, 23.6, or 42

millitesla with the field direction parallel to the sound direction.

The finite field is important in four ways. First, it enabled the

anisotropic superfluid phases to be observed at arbitrarily low pres-

sures. Second, it caused a definite orientation of the order para-

meter -vector with respect to the sound direction. Third, it allowed

observation of the narrow Al phase. And fourth, the exploitation of the

He NMR shift in the superfluid gave precise, in situ thermometry for

the sound measurements. This is perhaps the first extensive use of

superfluid He as its own thermometer.








The attenuation peak heights and the drops in the phase ve-

locity below Tc scale with pressure according to the predictions of the

collisionless theory for zero sound in the A-phase, while the absolute

attenuations observed are about one-quarter of that expected based on

this theory. The location of the attenuation maximum corresponds to the

expected clapping mode maximum at the lowest pressure but is signifi-

candcl colier ctin that predicted at all higher pressures.
















SECTION 1

INTRODUCTION



Much of the progress made in man's general understanding of

the physical world has come about by conceptually separating the effects

peculiar to our immediate environment from the more universal aspects of

nature. The force of friction is such a "particular" effect -- realiza-

tion of this was necessary before a Newton could construct a valid set

of mechanical laws. Likewise, the fact that the vacuum is a more funda-

mental physical environment than a mixture of 80% nitrogen and 20%

oxygen at 1.01 bar pressure has spurred the unceasing development of

ever better vacuum pumps. Our usual environment, at a temperature of

about 300 K above absolute zero, can be interpreted as an ocean of

thermal excitations which affects all physical measurements made in

it. Many physical phenomena are very insensitive to room temperatures

but some of course are not. Indeed, many of nature's characteristics

are completely drowned out at 300 K and were only discovered when the

necessary low temperatures were reached. It is this ignorance of

nature's hidden beauties at low temperatures that has led physicists to

build ever better refrigerators.

The condensation of some form or another of matter has often

served as a milestone for low temperature physics. Examples are the

liquifaction of air (-80 K) in the latter part of the nineteenth center.,

and the liquifaction of helium (4.2 K) in 1908. If one -ener3lizes the









concept of condensation to include ordering of any kind, then the exist-

ence of superfluidityy" can be viewed as a condensation in momentum

space. Such condensations of 4He atoms ("helium-II") and conduction

electrons ("superconductivity") were seen after the first liquifaction

of helium.

In general, a superfluid system contains constituent particles

(atoms or electrons) which are correlated over a distance much longer

than the average interparticle distance. This "connectedness" can cause

strange effects on a macroscopic scale, such as flow with absolutely

zero friction. Although some macroscopic effects of superfluids can be

tied together with a purely phenomenological theory (e.g. the London

eqj cions for superconductivity), a microscopic theory represents a

deeper underscandlng. A successful superfluid microscopic theory did

not eisi unciil Bardeen, Cooper, and Schrieffer (BCS) (1957) and

:)goilubo'v ', I.'3) derived the now standard "BCS" pairing picture for

upercoinducting electrons. This theory of electrons in a metallic

lattlce, based on the inherently quantum mechanical nature of fermions,

soon caused speculation about superfluidity in other Fermi systems. The

chief such candidate was liquid 3He, the rare isotope of helium which

has a njclrer spin of 1/2, and the early 1960's saw theoretical predic-

tions )i superfluidity in 3He (cf. Anderson and Morel, 1961; Balian and

Wertn imer 1'9n3),.

The actual critical temperature Tc for the transition from the

normal O:, the superfluid state is not easy to calculate so it was not

.Jncll Its i(ccidental) discovery in 1971 (Osheroff et al., 1972a) that

superrfljid 3He made itself known in the laboratory. The decade of the

1970's has seen an order of magnitude decrease in the routine minimum









temperatures of very low temperature laboratories. As the phase diagram

of Figure 1 shows, the temperature range of .3 to 3 mK contains five

different phases of 3He: the paramagnetic and nuclear spin-ordered

solid phases as well as the "normal" and (two) superfluid liquid

phases. The designations "A" and "B" of the two types of superfluid are

of historical importance only. The A-phase is located in the triangle

with its lower vertex at the "polycritical" point. If even a modest

magnetic field is applied to these superfluid states, the zero-field

picture of Figure 1 must be generalized to the three dimensional plot of

Figure 2. Just above zero field, the triangle of A-phase extends itself

to interpose itself between the normal and B-phase at all pressures.

This region of A-phase expands until, at fields on the order of 6

kGauss, the B-phase is entirely suppressed. Finite magnetic fields also

cause the appearance of the "Al" phase, nature's only magnetic super-

fluid. This part of the phase diagram has the shape of a very narrow

wedge inserted between the A and normal phases. To see the field

effects more clearly, Figure 3 takes a slice of the phase diagram at

nonzero field and distorts the temperature axis to exaggerate the widths

of the A and Al phases. The dotted lines show the shape at zero

field. Notice how the warmest superfluid transition temperature Tcl is

raised above the zero field transition at T .

This thesis is a description of a series of 5 MHz ultrasound

measurements made in the A and Al phases at pressures of 2.05, 9.15,

18.18, 21.53, and 31.12 bar. The application of finite magnetic fields

of 14.3, 28.6 and 42 millitesla caused the slivers of A and Al phases to

extend below the polycritical point thus allowing measurements spanning

most of the pressure difference between zero and melting pressures. All








































40-4 1


CC4
:- C

li rl
C3C















4-4 0













C.

- 4 4
: C
































>l W
04


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CC c C34-1

1-fl








0. 4 rJ
0. 014.










0
O





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CLC


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L I

Lz.





- o -





0 0o 0n

(joq) 3jnSS3Jdd





6















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C-4
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/ N
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SNca

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cc


I--



w
a-

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o)H








of the measurements reported here were done at temperatures within 2% of

Tc since most of the important features of 5 MHz ultrasound dispersion

in the superfluid occur in this narrow range. Also, for temperatures

near Tc, the Ginzburg-Landau (e.g. see Landau and Lifshitz, 1969, ch.

14) theory which describes second-order phase transitions in terms of an

"order parameter" expansion is applicable, facilitating interpretation

of experimental results. The apparatus for this experiment were origi-

nally designed for an ultrasound investigation of single crystals of

magnetically ordered solid 3He. Problems of heating and uncontrolled

solid growth redirected efforts to a liquid experiment for which the

.ppar aE-i aer-e ;ui.c d.

Some of ct,, earlisct Iri.estigations of He used ultrasound at

5, 1, a~rd 25 NHz iPa.-ilson et ai., 1973) and 10 MHz (Lawson et al.,

I)73, 1974-). The qOal[ctf.-Je features observed in these and later

fneasurenents are .; sharp pea". In che attenuation just below T accom-

pariled t': a small ur. -ujift decriese in the sound velocity. These

fe.;tures ire a.1. undlljerct:..id co result from excitation of resonant modes

of the superfilull.1 order paranieter which, due to its complex tensor form,

has mode- of a tl'pe unique in nature.

The propga.iiortn of sound in the superfluid phase should first

be placed in the perspective of sound propagation in the normal liquid

pr.ase. Tr.i velocit;, of 6o:.uld more than doubles as the pressure is

raised froi zero to elting presiare (see Figure 4 constructed from data

tabulated h:. Wheitle:', 1975). A~ the temperature is lowered the veloc-

itCy .nd a3cenuation under-o changes connected with the finite lifetime

i of the norr.,al liquid evcicltions ("quasiparticles") characteristic of

iis:c.latri: behavior (Piadrnic. 1'980). There is an attenuation peak








centered where the sound frequency is comparable to T (wrT-) accompanied

by a small upward shift in the sound velocity, as shown in Figure 5 for

20 MHz sound at 29 bar (after Ketterson et al., 1975). The slopes on

the cold and warm sides of the attenuation peak are proportional to T2

and T-2 respectively. Sound propagation in He at frequencies and

(cold) temperatures such that uT>>1 is termed zero sound and represents

the propagation mode predicted by Landau (1957) on the basis of the

normal Fermi fluid kinetic equation in the collisionless limit. The

sharp changes in attenuation and velocity just below the superfluid

transition temperature Tc at -2.5 mK can be seen in Figure 5 also.

The effects of strong fields on the attenuation of A-phase

sound at multiples of 14.7 MHz at low pressure have been measured re-

cently in France (Avenel et al., 1981; Piche et al., 1982). These

measurements, chiefly concerned with nonlinear effects well below Tc,

find that the clapping resonance occurs at the predicted temperature at

low pressures, in agreement with the data reported here.

Using low fields to orient the superfluid, Ketterson et al.

(1975) performed a series of 20 MHz ultrasound velocity and attenuation

measurements at pressures between 17 and 28 bar which has served as a

qualitative benchmark for my experiment. The measurements described

here also used an orienting field but this field was strong enough and

the temperature resolution fine enough to examine the A and Al phases

down to 2 bar.

The sound attenuation associated with the Al phase was studied

by Lawson et al. (1974) using fields up to 10 kGauss. They observed a

splitting of the characteristic superfluid attenuation peak which was

linear in the applied field. The Cornell group (Lawson et al., 1975)




























0
o .,
0 -
4-4
Un

U3
o a




w


I I I I 1
0 0 0 0 0
0 0 0 0
T ro C\j
(3S/Wu) '3





11





6 2

2
af
et
4
0
x

2-







100



50
E

m


z 20
0
0 ------------ L









S10-
I--
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2 5 10
TEMPERATURE (mK)


F i: .L' F L .:t i. T a *l' T f.n Lr l'Jn : ri- ,- --., r.J


I








also has reported an observation of the velocity drop in the Al phase.

There are rapid drops at both Te and Tc2. To my knowledge there is no

published theory explaining the dispersion of sound in the Al phase.

The choice of 5 MHz transducers represents a compromise. At

temperatures near Tc at 31 bar (-2.7 mK), Tr-3 and the zero sound is not

quite fully "developed." Near Tc at 2 bar (-1.3 mK), eT-10 and the

propagation mode is essentially pure zero sound. Simply using higher

frequency transducers will insure pure zero sound at all pressures but

the energy of a sound quantum then becomes comparable to that of a

thermal fluctuation. For 5 MHz, Ht/k = 0.24 mK. Thus 5 and 10 MHz are

reasonable frequencies for zero sound transducers if one wishes

hm/kT <<1.
c
A few words should be mentioned about the techniques used for

measuring sound attenuation, sound velocity, and 3He temperature. The

sound attenuation measurements were based on a conventional method: use

of a boxcar integrator to sample and average the transient signal in-

duced in a piezoelectric transducer by a short pulse of sound sent

across the sample cell. To measure phase velocity changes, the system

was configured 3; a phase-locked loop. This technique is less conven-

tional than the attenuation measurement and may even be unique to this

laboratory.

The dt:co.ery of large temperature-dependent nuclear magnetic

resonance i~~ ( ) frequency shifts in the superfluid phase (Osheroff et

al., 1972), besdes stimulating much more experimental and theoretical

w.ork in thit area, also gave promise that the 3He sample -itself could be

used a3 a nterm.imeter (e.g. see Richardson, 1977). To date this has not

been done, perhaps because of a lack of certainty about the necessary









calibration data. A few ancillary calibrations made in the course of

these experiments support the theoretical picture of the magnitidue of

these shifts (Leggett, 1974a) as well as corroborate most of the exist-

ing NMR shift calibrations. The net result is a basis for thermometry

in the A and Al phases accurate to ~10% but with a precision achieved in

magnetic fields of modest homogeneity which is comparable to LCMN ther-

mometry. The real advantage is a truly in situ thermometer, as Figure 6

illustrates. This shows the raw sound and NMR data on a run badly

disturbed by a time-dependent heat leak caused by a "touch" in one of

the cryostat vibration isolation mounts. After deconvolution, the sound

attenuation vs. temperature data fall on a curve similar to those of

"quiet" runs but with an inhomogeneous data point distribution along

that curve, where certain temperature regions were retraced due to heat

spikes followed by cooling.

The results of this series of measurements are given in the

last section chiefly in comparison against the collisionless theory of

sound dispersion in the superfluid A-phase. The amplitudes of the peak

attenuations and the velocity drops scale according to the predictions

of this theory, but at the higher pressures, the temperature locations

of the attenuation peaks are colder than predicted. A comparison with

theory using the numerical computation scheme of W61fle and Koch (1978)

is probably required to see if a real discrepancy exists.










5'.
5.



'S
0 ^









aIIN


+


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,














SECTION 2

PERTINENT THEORY OF 3He



2.1 General References

Current theoretical understanding of the superfluid phases of

3He rests on generalizations of the BCS theory of fermion pairing origi-

nally applied to metallic superconductivity (Bardeen, Cooper,

Schrieffer, 1957). Theoretical reviews of superfluid 3He have been

written by Leggett (1975), Anderson and Brinkman (1978), and Wblfle

(1979) and experimental reviews by Wheatley (1975, 1978) and Lee and

Richardson (1978). Shorter but more recent reviews can be found in the

proceedings of the 1981 "Low-Temperature-16" (Physica 107-110, W.G.

Clark, ed.) and 1983 "Quantum Fluids and Solids" (AIP Conference Pro-

ceedings No. 103, E.D. Adams and G.G. Ihas, eds.) conferences. What

follows here is not intended as a review but only as a reminder of those

theoretical aspects of He especially important for understanding the

experimental results presented in this dissertation. For purposes of

consistency, the notation will follow that of Wblfle's review where

possible.

2.2 The Normal Fermi Liquid

Above the superfluid transition temperature of -2 mK but

significantly below the Fermi temperature of ~1 K, liquid 3He is well

described by Landau's "Fermi liquid" theory (Landau, 1956 and 1957).

The starting point for this theory is a gas of fermions of density n and









particle energies Eko depending on momentum p=lk and spin projection o.

At low temperatures this gas is nearly completely condensed into its

ground state: the Fermi sphere in momentum space of radius pF. The

Fermi energy EF and momentum pF= kF are related to the number density n

by the following relations.


kF3
n = "k ()
kuo 37

H2k2
ko 2m* (2)


The quantity m* is an effective mass several times greater than the bare

atI:.lc iiasa and is necessary for the connection to experimental

results. Similarly, the energies eko refer to excitations called quasi-

particlen, consisting of the collective motion of many 3He atoms but

still obeyirig the Pauli exclusion principle.

A consequence of the nearly condensed state of quasiparticles

is that the full three-dimensional momentum space picture can be reduced

c. the t.-o-dimensional one of excitations near the Fermi surface. Also,

the heat capacity and static magnetic susceptibility are then simply

proportional to the density of states near the Fermi surface.


3n mPF
NF = 1-EF= 2 3



This Is postlble because the quasiparticle distribution function is that

for termlons


n(e) = lexp[(e-u)/kT + 1}-1









so that at low temperatures the distribution differs from a step func-

tion centered at the chemical potential p only in the neighborhood

of u. Most properties of normal 3He can be described if one imagines

that the quasiparticles are coupled by a weak effective interaction so

that a deviation in the quasiparticle distribution from that of the

ground state changes the quasiparticle energy by



6 eko (r,t) = 1 fko k'O'6nk'o'(rt) (4)
k'o'


An angular decomposition of the interaction yields the "Landau para-

meters" F, Fa


-1 m a
k k'o' = NF P (k*k')(F 6 + F oa*') (5)
1=0 '


Here, the superscripts s and a stand for symmetric and antisymmetric

respectively and P (cos9) are the Legendre polynomials. The first few

Landau parameters are in principle related to simple measurements of the

molar heat capacity CN, speed of (first) sound cl, and magnetic tempera-

ture T (a constant derived from low temperature absolute susceptibility

measurements). The pertinent relations are




3 CNn
NF 3 RT (6)
F 7kRT


m C lj2 PF
F! = 3( -1, VF m*
F


s m*
F = 3( -- 1)
1 mn








a 3 kT m*
S 1 (9)
0 2
PF

The value of F can be obtained from spin diffusion measurements

(Leggett and Rice, 1968).

2.3 Superfluid Pairing

If the quasiparticle interaction is the least bit attractive,

at a sufficiently low temperature, the Fermi liquid picture must be

supplemented to account for the inevitable instability of quasiparticles

at the Fermi surface for forming bound pairs of opposite moment.

Calculation of this critical temperature T from first principles is a

difficult problem but the pair interaction can be characterized in terms

of the Landau parameters. Doing so reveals that this interaction is

repulsive in the s-wave orbital angular momentum state (X=0) but attrac-

tive in the p-state (1=). The repulsion in the s-state might be

expected from knowledge of the hard-core repulsion between He atoms.

Pairing in the p-state, the basis of the most successful models of

superfluid 3He, necessarily must, according to the Pauli exclusion

principle, be accompanied by a total spin S=l.

At the transition temperature, the quasiparticle energy spec-

trum is modified by the appearance of an energy gap and becomes



Ek + +) 1/2 (10)



where E k- + Ek is now the quasiparticle energy measured from the chemi-

cal potential (u = EF at T=0) and Ak is the off-diagonal mean field

ac 'cli on the Cooper pair (ko,-ko').









Ak Ak = Vkk'gk)o, (11)



Equation (11) is mathematically analagous to equation (4) except that

now Vkk, is the attractive interaction leading to the formation of pairs

and gk',o' is the probability amplitude of a Cooper pair (k'o,-k'o').

Here akoo, is a complex 2x2 matrix in spin space. The actual magnitude

of the energy gap is given by



IAk = [(1/2) Tr (k) 1/2 (12)



At nonzero temperatures, the probability distribution in momentum space

for fermions leads to the particular form for the famous BCS self-

consistency "gap equation."


tanh (Ek,/2kT)
koa'= k Vkk' Ak'o' o 2 (13)


The gap equation uniquely defines the gap parameter for con-

ventional superconducting metals, where Z=0. For iI1, though, several

model states can be created. The fact that more than one superfluid 3He

state actually exists requires the careful consideration of the various

possible theoretical models. It is now widely, though not universally,

believed that the two zero-field states of superfluid 3He are well

described by the so-called ABM and BW models. The ABM model (cf.

Anderson and Morel, 1961; Anderson and Brinkman, 1973) correspojnds to

the experimental A-state and the BW model (cf. Balian and Uerchmier,

1963) corresponds to the experimental B-state.







In order to discuss the various model states in detail it is

convenient to handle the three independent components of the 2x2 sym-

metric matrix Ako' in the form of a vector d(k).


-d + id2 d 1 (14)
"koa, = (14)
d3 dI + id2



The square of the energy gap can be written in terms of d(k) as


+ *
(i = (d*d ) oo, + i(dxd )r-To, (15)



where T is the "vector" of Pauli spin matrices.

The dj's, which deal with the spin part of the superfluid wave

fun,:tin, specify the amplitudes of the gap parameter for those eigen-

states of the pair spin operator having eigenvalue zero. The connection

to the orbi.al part of the wave function is made by decomposing the dj's

along the momentum axes.


3
d (k) d. k (16)
j Ja a


The ~.I, s=l model states of 3He are fully described by the tensor order

parameter d

2.- Coparison of the ABM and BW States

Specification of a particular form for the complex

tensor d. leads to a state which can be compared with other model

states by means of their free energies described as expansions in the

order parameter. This Ginzburg-Landau expansion near Tc must involve









only combinations of dja which are gauge invariant and invariant with

respect to rotations in spin and position (orbital) space. There are

only six such combinations: one second-order and five fourth order. To

write down these free energy invariants it is useful to first define an

order parameter normalized to the rms angular average of the energy gap

over the Fermi surface, A(T).


d.
d a (17)
Ja 3 A(T)


with



d. d. = 1 (18)
Ja


and



A(T) = [(l/4)fdi(1/2)Tr( + )1/2 (19)



In terms of the dja, the invariants are (using Einstein summation, cf.

Mermin and Stare, 1973; Brinkman and Anderson, 1973)



o ja ja
1 = dja d = 1 (20)


^ 2
I = d d. ( (21)
1 ja jci


S= d da dj d = 1 2)
2 = d ia jdi dj


I =d d d d 23)
3 ia ja i13 jB







^* ^*
1 = di dj dja di. (24)



S= d d d d (25)
5 in 18 ja i(


The free energy difference between the normal and superfluid states can

now be written as


F F -F
s n

= NF/2 (1-T/T)A 2 (T) (26)
5
+ (21/80)(3)NF(TkTc)24(T) )2 4I
i=1


In the BCS weak coupling limit, the values of the Bi are



S2 = a3 = 4 = -5 = -251 = 1 (27)



A class of states discovered by Balian and Werthamer (1963)

minimizes the free energy for the weak coupling values in Equation

.r.). The BW states are those unitary states given by



dj A(T) eiR. (28)



where ; is an arbitrary phase and R. is an arbitrary real orthogonal

(i.e. rotation) matrix. The BW state is the only iO state with an

Isocropic energy gap.


I|k(T) a(T)









Since the BW state is the weak coupling minimum energy state,

the identification of the experimentally highly anisotropic A-state with

the ABM model state (Anderson and Brinkman, 1973) has thus been pur-

chased at the cost of abandoning the simple weak coupling picture of

3He, at least in those regions of the phase diagram where the A-state is

more stable than the B-state. Deferring this problem for a moment, the

ABM state can be characterized by the following unitary order parameter:



1/2 ^ ^
d. = (3/2) A(T)d (n +im ) (30)



where d is an arbitrary unit vector in spin space and a and a are arbi-

trary unit vectors in position space constrained only by n*m = 0 .

These latter two vectors define the "angular momentum" vector by their

cross-product



I 5 nxm (31)



which lies along the axis of the anisotropic energy gap



IAk(T)I = (3/2)1/2(T) sine9 (32)



where cosO = k*L.

2.5 Strong Coupling and the Finite Field Phase Diagram

The connection made by Anderson and Brinkman (1973) between

the experimental A-phase and the theoretical ABM model state used the

idea of "spin fluctuation feedback." As Leggett succinctly states, "the

basic physical idea of the Anderson-Brinkman theory is that the torma-









tion of the superfluid state modifies the pairing interaction between

quasiparticles, and that the precise nature of the modification depends

on the particular kind of superfluid state formed" (Leggett, 1975, p.

373). Spin fluctuation exchange, though perhaps the most important

feedback mechanism, i; only one of several "strong coupling" effects

whichh mliht mo.dif: the pairing interaction. Quantitative spin-

fluct.iartoo or "par3-agnon" calculations have been performed by Brinkman

ea al. 1974i, K roda ',1975), Tewordt et al. (1979a,b), and Tewordt et

al. ilU?5, 1973). Instead of using the paramagnon approach, Rainer and

Serene 1.197.'1, Serene and Rainer (1979), and Sauls and Serene (1981)

haje expanded the corrections to the Bi in powers of (kT /EF ) and used

the "s-p" approximation for quasiparticle scattering for their calcula-

tions. Le.in an.l Vall (1979) have combined both the paramagnon and s-p

4catcering approaches. The net result of these various calculations is

a modirtcation of the ei from their weak coupling (kTc / + 0) BCS

values.

Cc.npirison or the strong coupling calculations just mentioned

Wich eprimenrt is obviously important but is complicated by the fact

thac experiOencal measurements yield only combinations of the Bi. Even

though experiment contradicts the earlier predictions for some of the

b IcombLnacions (~'Ilperin et al., 1976), enough measurements to define

all five at any single pressure have not yet been performed. Most of

the useful measurement- which can be performed toward this end exploit

the existence of the A, phase in a finite magnetic field. As shown by

Figure 3, cne normal to A-phase transition is split by a magnetic field

into two transitions at Tel and Tc2. This splitting, a natural conse-

quence of the fact tht 3 He forms a spin-I superfluid, can be viewed as









a relative shift of the Fermi energies of the "spin-up" and "spin-down"

components of the quasiparticles (cf. Ambegaokar and Mermin, 1973). The

size of the interposed "Ai" phase is proportional to the dependence of

the pair coupling constant 11 on EF and the applied field H.


T -T dT jH
cl c2 [ ) o (33)
Tc dEF (1 + Fa)


Measurements involving the A, phase can best be explained by

first considering the Ginzburg-Landau form for the free energy differ-

ence of a system containing only paired parallel spins (no t+ component)

as written in Wheatley's (1975) exposition of Takagi's (1974) work.


F Fn = (NF/2 ((t-nh)A 2 + (t+nh)A 2

+ [B/2(kTc)2]( 4 + A4) (34)

L6/(kTc)21 2 A 2


This equation is similar to Equation (26) except that the energy gaps

for the two spin populations are now separately considered and the

effect of an applied field is included. The reduced temperature t and

field h in Equation (34) are defined by


t 5 (T Tc)/T
c c (35)
h y$H/(2kTc)


where Y is the gyromagnetic ratio for 3He. The parameters n, B and 6

are to be determined experimentally but with the restrictions :)0 and

6<1. The two parameters B and 6 can be related to the B. (Legett,

1975) by





26


S= (7c(3)/16 2)(82 + 84) (36)


-(8 +4+28 )
6 ( 24 (37)
2 4)


where the value of the Riemann zeta function is c(3) 1.202.Experi-

mental measurements of the characteristic BCS heat capacity jump at T

give information on the i The three such independent measurements are

the following:


C C
A N 24 1 3 (38)
C 14 C(3) B2 4 5 2 2 -6)
c--2+- 4 [+-J5 24 B(1-6)

CA N 24 1 3 (39)
CN 14 (3) 2(62+84) 4128

C CN 24 3
C_ 1[i4 ](3) 3(81+82)+(83+84+&5) (40)
CN 14 (3) 3(61 2 3 4 5) (0)


Other experimental quantities yield additional information. There is a

mioi magnetic susceptibility increase in the ABM state of about 1%.


XA XN 2(1+F)
(41)
XN 26(1+6)1)


The transition temperatures are given by



t = ih (42)


h(1-6) (43)
t2 (1+6)3)


so that








2nh
t -t = (+h (44)
1 2 (1+6)


By defining an additional parameter to measure the substate with spin

projection zero, Equation (31) can be generalized to include the BW

state. Levin and Valls (1977) did this to obtain the transition temper-

ature between the A and B phases in a finite field below the polycriti-

cal point. Their rather complicated expression for tAB, which is

proportional to h2, will not be quoted here but it can be used to give

additional information on the Bi since the extra parameter characteriz-

ing the B-phase can be estimated by measuring the slope of the B-phase

susceptibility as a function of temperature.

Finally, the perpendicular NMR shifts in the A,-phase and A-

phase well below t2 give an independent measure of 6.


(dv /dt)A (-6
4(45)
(dv /dt)
L A

This will be discussed further in the next section.

2.6 NMR in the A and Al Phases

The magnetic characteristics of superfluid 3He differ dramati-

cally from those of the normal liquid. The susceptibility of the normal

liquid is an essentially temperature-independent constant equal to that

of a noninteracting Fermi system renormalized by the Fermi liquid para-

meter F0


Y2 2NF/4
XN a
N 1 + Fa








The nuclear magnetic resonant frequency w is simply proportional to the

magnetic field H, a la Larmour:



wL = YH (47)



This picture changes below the superfluid transition at Tc,

where the NMR A-phase resonance is seen to shift away from its normal

liquid value according to the relation



2 = (H)2 + A = (1 T/T ) (48)
=(yH) A c



Such a shift in the resonant frequency is possible only if non-spin-

confer.inm forces are invoked. In order to explain the A-phase fre-

quenc, -titif first seen by Osheroff et al. (1972), Leggett (1973,

197.3~ niodified the precession equations for the nuclear spins S by the

addition of a dipolar torque RD.



= S x H+ R (49)



The magnitude of the nuclear dipolar coupling between two adjacent 3He

3t~rm in the liquid is extremely weak. In temperature units it is

~- 'K and could be readily neglected at typical experimental tempera-

cures, ahictl are some 10 times higher, except for the correlations

intrinsic to the superfluid state. This causes the effective dipolar

coupling energy to be multiplied by the number of condensed Cooper pairs

in the system. The tendency for the pair wave function to seek its most

energetically favorable orientation is thus not "drowned out" by thermal

fluctuations.








For the ABM state, the equilibrium orientation caused by RD

aligns the d vector along the (orbital) angular momentum vector 1. More

generally, on sufficiently short time scales typical of NMR experiments,

the motion of the d vector is given by



d = yd x (H YS/x) (50)



Calculation of the dipolar torque RD and the resulting shift

in the NMR frequency has been done for the various unitary phases of

superfluid 3He (Leggett, 1974a). In general the "longitudinal" fre-

quency (so called because of parallel-ringing magnetic resonance experi-

ments) is a product of a function of pressure f(P) and (1 T/Tc).



A2 = (2 2 (n2)2f(P)(1 T/T ), T < T (51)
A A c c


where, for the ABM state,



f(P) = (/l10),2Np(1 + F0)[Zn(1.14 ec/kTc) 2 (kTc 2(AC/CN) (52)



In Equation (52) c is an energy integral cutoff, estimated to be about

0.7 K by Leggett (1972). Most strong coupling effects are absorbed into

the experimentally measured ratio of the jump in the heat capacity at T

to the normal heat capacity (AC/CN). Here is an angular average of

the square of the quasiparticle renormalization factor and is expected

to be on the order of unity. The quantity Rk which is averaged repre-

sents a modification to the spin current carried by particles of a free

gas to give that of the Fermi liquid (Leggett, 1965; Leggett and Rice,

1968).








Rk m (53)



The nuclear magnetic resonance frequency also exhibits a shift

in the Al phase which can be easily understood in terms of the Ginzburg-

Lindau equation '. Tal.3gi i1475) and Osheroff and Anderson (1974)

nave discussed ho- the snift firegency VA depends on temperature by

starting uith the beha.,Lor of the two (different) order parameters for

the up and dowon-pin components for the liquid. It will be useful to

define a new temperature scale linear in T but with its zero at Te2 and

its "degree" equal to the Jidth of the Al phase.


T2 T
U c2(54)
I T (54)
cl c2


Minimizirn the free energy of Equation (34) results in the

follohiL.g behavior of the gap magnitudes as a function of temperature.

S = U < -1 (normal liquid) (55)

U = M l U) -1 < U < 0 (Al-phase) (56)

EBU, = A + BU < U (A-phase) (57)



Here, It Ls arbitrarily assumed that n >0, thus favoring an up-spin Al

phase. The constants A and B are related to the free energy parameters

B and 6 by



A = tI t2)(kT) 2/8 (58)


B A. 1I 6).











3.0




2.5


0

4*-
C
S1.5-




1.0 -




0.5-




0 II
-2 -1 0 I 2 3

U= (.T,2-T)/(T,-Tc2)


TRANSVERSE NMR SHIFTS NEAR Tc


FIGURE 7. NMR shifts in the A and Al phases








A third constant C relates the perpendicular resonance shift to the

average of the two gaps.



A = C[(A4 + At)/2]2 (60)



The resulting shifts in the Al and A phases are then


2
nA = (c/4)A(l = U) -1 < U < 0 (61)



2 = (C/4)[A + 2BU + 2(A + BU) 1/2(BU)1/2] 0 < U (62)
VA (



"Deep" into the A phase the shift can be well approximated by a linear

form


2
:A = (C/4)[2A + 4BU] U > 1 (63)



The above theory gave an excellent fit to the NMR measurements

uf Osheroff and Anderson (1974) done at the melting pressure in fields

of .49 and .74 Tesla. By taking the ratio of the slopes of the NMR

snEit vs. temperature in the Al and "deep" A phases, they obtained a

alue of 6 = 0.25 5%. Figure 7 is a line drawing of this NMR shift

plotted in units of AC against the doubly reduced temperature U. The

particular slope ratio chosen for this drawing is 5.33 corresponding

to 6 = 0.25.

'.7 Sound Propagation in the Normal Fermi Liquid and the ABM-State

One manifestation of the rich behavior of superfluid 3He due

to its complex tensor order parameter is in how the liquid disperses









sound. Many of the collective modes possible in the superfluid phases

can be directly excited by sound waves of the appropriate frequencies.

In order to give a qualitative explanation of sound experiments in the

A-phase I will, for the most part, quote the simpler calculations expos-

ited by Wl1fle (1978) in his review on sound propagation in superfluid

3He.

At temperatures high enough or frequencies a low enough sound

propagation obeys the classical connections to compressibility K,

density p, and viscosity T. The velocity cl of such a hydrodynamic

mode, ordinary or "first" sound, is given by



2 1
c =1- (64)



and the attenuation by



2m-
a 3 n. (65)
3c p


The Fermi liquid compressibility is


2
K = 2F (66)
p (1+Fs)


independent of temperature while the viscosity has a T-2 temperature

dependence.



1 2 -2 (67)
S PVF T (67)








The velocity of first sound in 311e, taken from Wheatley (1975), is

plotted in Figure 4.

In general, transport properties such as viscosity can be

calculated only if one uses the general form of Landau's Fermi liquid

theory which includes the effects of quasiparticle collisions. For the

normal Fermi liquid, the mean quasiparticle lifetime between collisions

T is proportional to T-2. The dynamic properties of the normal Fermi

liquid can be placed in a context which can be generalized to the super-

fluid phases by writing down the kinetic equation for the distribution

function of the quasiparticles. Starting with the Fermi distribution

for static thermal equilibrium



fk = [exp(e /kT) + 11-1 (68)



the linear deviation cased by the perturbing (sound) field of wave

vector 1 and frequency m, 6fk(r,t), can be written in terms of its

Fourier components



6fk(q,u) = fd3rdt[fk(r,t) fk]exp[i(q.r at)] (69)



The evolution of these Fourier components is governed by the kinetic

eqijat ion


0 00 0
S6fk 06fk 0 06C ekf 0 = -i (6f') (70)
k +i k k + k kf k r(


This important equation uses the following notation: "0" indicates

equilibrium quantities, "" means kq, and dek is the change in the









quasiparticle energy caused by both the direct gain in the external
ext
field 6ext and the shift in the distribution function Sfk, namely



k = 6ext + Tr, fkk 6fk, (71)



The term on the right hand side of Equation (70) is the collision inte-

gral which operates on the deviation of the distribution function from

local equilibrium.



6f = 6fk SEk (72)



What happens if the (sound) excitation period is much shorter

than the quasiparticle lifetime, i.e. wT >> 1? This is the "collision-

less" limit, equivalent to setting the dissipation term of Equation (70)

to zero. Landau (1957) showed that new propagation modes ("zero sound")

may then exist, corresponding to anisotropic oscillations of the Fermi

sphere (cf. Figure 8), with a separate mode for each -component of the

quasi-particle interaction. 3He is such that there is only one propaga-

tion mode not strongly damped (longitudinal zero sound), corresponding

to a nearly spherical perturbation of the Fermi surface.

Equation (70) can be solved for its resonances by using the

conservation laws for the number and current densities 6n and j, which

are defined by



6n(q,w) = 6 k(q,w) (73)
ko


j(q,w) = 1 (k/m)fk (q,). (7.i
ko








FIRST SOUND




/

i







UNPERTURBED
FERMI LEVEL

ZERO SOUND







I


MOMENTUM


PERTURBATION S


FtTiir:E Momentum perturbations of c and c1









These conservation laws are the respective continuity equations



w6n = q-j (75)



j = ) (k/m)(vFk.q)6fk (76)
ko


which are obtained from the kinetic equation by multiplying it by I or k

respectively and then summing over all k.

In the collisionless, zero-sound limit the velocity and atten-

uation are


c c1 2(1 + 2 m* VF
5--)_-_ (F, = 0) (77)
0 5(1 + Fg) 1

2 m* ( VF )2 1(78)
0 15 m(c C)


For 3He, the first and zero sound velocities differ by only ~1%. This

is so because the hard core repulsion leads to a large value of FO,

especially at high pressure. The zero sound attentuation has a T2

dependence, opposite that for first sound.

The general dispersion relation for sound in normal liquid 3He

including the transition region T '-1 has been calculated by various

means (Rudnick, 1980; Wl5fle, 1976b) and requires handling of the dissi-

pative term in Equation (70). Wl6fle's approach is to construct an

approximate collision integral using the k-dependent quasiparticle

relaxation rates. His scheme allows generalization of the kinetic

equation to the superfluid states. Figure 5 from W6lfle (1978) includes

the result of such a normal fluid calculation fitted to the data of








Ketterson et al. (1975). The data points, which are well represented by

the theoretical fit, are not shown.

The complicated nature of the superfluid He model states, as

compared to the Fermi liquid theory, can not allow a description of

superfluid sound propagation based solely on the scalar kinetic equation

(70). This was first verified experimentally (Lawson et al., 1973;

Paulson et al., 1973) by the dramatic changes seen in the velocity and

attenuation of sound just below Tc for both the A and B phases. The

general features are a sharp peak in the attenuation with a width ~1% Tc

and a steep decrease in the velocity starting at Tc. There is also some

theoretical and experimental evidence for order parameter induced atten-

uation changes just above Tc (Emery, 1975, 1976; Paulson and Wheatley

1978; Samalam and Serene, 1978; Pal and Bhattacharyya, 1979). This

small effect will be neglected here.

Simply by knowing the characteristic times of 3He, one might

estimate when changes in the sound dispersion will take place as the

frequency is varied. (Varying temperature changes the characteristic

ries, also leading to dispersion.) Figure 9 shows the various regimes

defined by these (temperature-dependent) characteristic times, with

1 dividing the hydrodynamic and collisionless regimes. The frequency

defined by the superfluid energy gap defines an additional region. For

3 sufficiently warm temperature (T1 on Figure 9), the gap frequency is

less than 1/T and the intervening regime can be called "gapless"; the

energy gap is not well defined due to quasiparticle collisions. At

older r temperatures (T2), the region above I/T but well below A(T )/f is

called by W6lfle (1978) macroscopicc," signifying that the order para-

meter is in a local equilibrium state not significantly perturbed by the

s -und frequnci.'.











3






0
o) z



0 0
o z -


C- -

S a
S 0 t









(E 0
0
0
c_ o





00 .
- .
o







w

i 0







Calculations of the dispersion of pure zero sound (no quasi-

particle collisions) in the ABM state have been done by Wl6fle (1973,

1975a,b, 1976a,b,c), Ebisawa and Maki (1974), and Serene (1974).

Wblfle's approach, originally due to Betbeder-Matibet and Nozieres

(1969), starts with the kinetic equation (70), drops the collision term,

and generalizes the distribution function to a matrix form appropriate

for a superfluid description. This results in expressions for the

perturbations of the order parameter which are induced by the sound

which in turn can be related to the expected sound dispersion.

Cooper pairing of quasiparticles of opposite moment in the

superfluid state causes a new type of long range order in the system.

The quasiparticle (diagonal) distribution function



fkoo' (r,t) = gf 3exp(iqr) (79)



is cnen joined by an off-diagonal distribution proportional to the

eipecctcion value of the pairing



gkoo'(r,t) 3exp(iqr) (80)
gkao' (iqT k_,ak + a
+


where a ,ak is a creation (annihilation) operator acting on a state of

monenctui k and ki = k q/2.

The quantity gk is zero in the normal state and cannot even be

,bserve directly in the superfluid. Rather, changes in gk are seen by

their couple g' to fk. By defining a matrix distribution function in

particle-hole space








nk(rt) = fk (81)

+ 1 f

and an analogous energy matrix



k(r,t) := k k (82)



the linearized response to a sound wave (q,w) in the collisionless limit

can be written as the matrix analogue of Equation (70).


0 -0 -0 -0
k k+ 6nk + 6, F_-- n- + nk+ 6k = 0 (83)


The quantity Ink E nk an is the linear deviation of the distribution

function from its equilibrium value, which is

-0
-0 1 k k
k = + tanh(2Jj. (84)



As before, energies are measured from the Fermi surface



Ek = k/2m* (85)



and the superfluid energy spectrum includes the effect of the energy gap



Ek + k 1/2 (86)



The perturbations in the components of the energy matrix sk

are related to the perturbations of the two distribution functions.








6k = kk6fk' + 6Ekt (87)


6koo= gkaok'o'6gkFoo, (88)


In analogy to the normal Fermi liquid equation (4), the equilibrium

energy gap is related to the off-diagonal distribution function by the

self consistency equation



A1oo' i, gkok'ok1'g'o' (89)


thus defining gkok'O'"

The matrix kinetic equation (83) is a system of linear inte-

gral equations which can be solved for the case where the interactions

fr, and 7kk, are assumed to be independent of the magnitude of

k (|kl = kF). W6lfle's (1976b) approach is to rewrite Equation (83) by

defining sectorss" composed of combinations of 6fk, 6gk, 6Sk, and 6Ak

and mulciply them by appropriate matrices which contain only equilibrium

quantLties for their entries. Solving the resulting matrix equation is

then more straightforward, leading to explicit expressions

for ':"k and 6 k.

'-K ....- = Nf f )(i XK,)6r, 2 (6k' + 66_kj)


(hw + h2') ( + +
+ 41 2 k'6 k k6k (90)


NF gd ?k'__
': + gkk'k'6k1' =2 4 kk' 2 c ( + ')6Ek'
k' I ak
I 2. 2 4 2 +k (91)
T- $ 4' )Ak, + Ak) ,6A + ,6K, ,}k








For the above two equations, the following abbreviations were used:



n = A2k-q/m* (92)



Ok = (1/2Ek) tanh(Ek/2kT) (93)

Sdek dOk
Dk(q,,T) -4AkL[2 :- LW2o k k n2 dk] (94)


Dk a (h-)2(ha)2 4E2] 2[(ha)2 4-E] (95)



Equations (90) and (91) were arrived at assuming that the gap

parameter A o, is unitary but otherwise are not specific to any model

state. Specialization to the ABM state requires the particular form of

the ABM gap, namely Equation (30) inserted into Equation (14).



koo' = i/37 A(T)[(n + in)*k][d*("T2)o,] (96)


Now, by a particular choice of coordinates, the perturbation of the gap

parameter can be written in terms of the azimuthal components of the

first spherical harmonic Y lm


+1 3
k-' i6 (8 1/2Y m(k) 6dj (rj 2)o (97)
m=-1 j =1


Using these forms for the gap and its perturbation induced by the sound

in the general equations (90) and (91) results in an expression for each

of the three expansion parameters 6d, d0, d_l. Collective modes

(resonant oscillatory distortions) of the order parameter show up as

poles of the 6d in the complex frequency plane.
m








The equilibrium (no sound) order parameter is proportional to

Y11 (k) as can be seen by choosing coordinates such that n = x

and a = y in Equation (96). It turns out that there is no pole asso-

ciated with 6dI.

There is one pole associated with 6d; excitation of this

mode amouits to a distortion of the equilibrium gap parameter by an

amount proportional to Y l- (k). In the limit of T near T the. temper-

3aure dependence of tnis mode's frequency is (W6lfle, 1975)


hwl [(. 5)(2V 3)]1/2 /37- A(T) (98)

S1.23236 0(T)



where -,.,T) = *3.2 -i,) is the maximum magnitude of the ABM gap.

The mAl mode is often referred to as the "clapping" mode as the

Yl-1 (k) oscillations results tn a movement of the n and a vectors

to;arjs and away from each other as.Figure 10 shows.

The equation for od. yields two modes. These modes are some-

times called "normal flapplnz" and "super flapping." Figure 10 illu-

strates normal flapping: a and a "flap" up and down, thus rocking

the e-.'ector back and forth away from its equilibrium position. Near

T., the temperature dependence of the two flapping frequencies are

given by



0^'l (- -T) [1 (28/374) i(3)(Ao(T)/kTc)] (99)



n,,.) = 2 .() [1 (28/54 )((3)(6 (T)/kTC)] (100)
sf _01 0 c
















"CL APPING"




















"FLAPPING"








m


A
n


FIGURE 10. Clapping and flapping modes


Ar^
A s^







The above gap resonance features, as well as those due to

breaking of Cooper pairs (Mw = 2A), show up in the general expressions

for the sound attenuation and velocity. The function 50 containing this

dispersion information can be written in a form emphasizing the impor-

tance of the relative orientation of the sound vector q and the gap

axis a (Serene, 1974):


5 = s cos (8) + c 2sin (B)cos (8) + sin 4() (101)
c i


where cosd a q*1 and F a 0. The component functions are



S= (45/4) { } (102)

50 2 (45/2) {3 = (45/8) {3sin Ocos 2 / (103)
c
- 2(A /fl)2<(X + )cos2 >]}


0 = (45/16) {(3/2)2/ (104)

-(1/4)2/l[
]


Here
a (1/41) fd2A is the angular average and

cos a k*t

S d k 2 d (105)
tdEk Ek dE(0k)


tanh(Ek/2kT) lA 2
A dek 2 ]2E (106)
k [E (Lw/2) 2E k
[k









For T near T ,



S= i [(O tanh( sin2 1 + 0 -)2 (107)
4 [(m1/2A ) 2 sin2 /2 k c


For zero sound of low enough frequency, X simplifies to (Wm/kT) times a

function of (A0/ H ) only. This makes it possible to express the sound

attenuation and velocity in terms of a universal function defined by



x = (2kT/5%w)O0 (108)



The sound attenuation is then


K2
a = Im(x) (109)
c0F kT


and the shift in the sound velocity from its zero sound value cO is



c Re(x) (110)
CO0 F kT


Figures 11 and 12 from Wl6fle (1978) show what the imaginary

and real parts of the components of the universal function X(A0 /) look

like. Although pairbreaking at AO/XW = 1/2 is implicitly included, the

dominant features are the various gap resonance modes. In particular,

Xi shows a strong, sharp peak in the attenuation with a corresponding

feature in the velocity shift. The overall anisotropy of dispersion is

of course due to the anisotropy of the order parameter itself. The

maximum value of the attenuation for any orientation is proportional to

the prefactor of Equation (109).










2
a -- (111)
max s
c0 F0 c


So far, the effects of quasiparticle collisions have been

ignored. A good accounting of their effects, which should include the

background T2 attenuation behavior and a broadening of the collective

modes peaks, might start by adding a dissipative term to the right hand

side of the matrix kinetic equation (83). This is a generalization of

the scalar collision integral. Wl6fle and Koch (1977) were able to

construct an approximate form of this dissipative term that was numeri-

cally tractable. Their calculations, involving some 30 double integrals

on angle and energy, give fair agreement with the attenuation data of

Paulson ec al. (1977) and the attenuation and velocity data of Ketterson

et al. (1975). In particular, the clapping attenuation peak is broaden-

ed and the corresponding "derivative-like" structure in the velocity

shut is no longer present.

The presentation of the preceding pages, chiefly concerned

tich the theory of collisionless sound in superfluid 3He, is intended to

ser.e as i basis for comparing the experimental results to be described

later in this thesis. In our experiment the magnetic field was parallel

to the so.nd vector, thus orienting the -vector of the superfluid

perpendicularly to the sound (except for a small fraction next to the

container boundaries). Thus only xI of Equations (101) through (111)

w111 be considered for comparing the sound attenuation and velocity

data.



















































) 1.0
C AT)/h. )2 a I-T/Tc


FIGURE 11. Theoretical attenuation, Im(x)















































(CoTJ/hw)2 a I-T/ T


Fl.I.IFC 12. Theoretical velocity shift, Re(x)















SECTION 3

APPARATUS



The apparatus for performing these experiments necessarily

include the refrigerator, constructed before my tenure at the University

of Florida. Since construction details have not been reported previous-

ly, I will do so here. Much of the general discussion of refrigeration

principles follows Lounasmaa (1974).

3.1 Large-Scale Features

The cryostat and most electronic systems rest in a 24 m3

copper screen room. Electronic filtering for the 125 VAC lines in-

cludes, besides the usual low pass network, provision for elimination of

the occasional bursts at 3510 Hz used for campus clock synchroniza-

tion. All vacuum pumps and the dilution refrigerator gas handling board

are outside the screen room as well as the computer and signal average

for data acquisition. The signal average has analog electrical connec-

tions to the screen room electronics via three 1-MHz low-pass filters.

Figure 13 indicates the gross dimensions of the cryostat and

dewar. The "super-insulated" dewar (Cryogenics Associates, 62 liters)

and cryostat are suspended from a triangular aluminum plate whi:h L[

supported at its three corners by optical-bench air mounts. For dJmag-

netization of copper nuclei, a custom-built, American Magnetics,

niobium-titanium superconducting 8 tesla magnet is used. This magnet,

with its compensating coils, is mounted directly on the 10-liter vacuum










SUPPORT/
TAB L E



BAFFL ES-




PUMPING
LINES, ETC.








VACUUM
JACKET





30 cm


VAPOR-COOLED
CURRENT LEAD




SUPERCONDUCTING
CURRENT LEAD


WINDING


8 TESLA
SOLENOID


CRYOSTAT


AND DEWAR


FIGURE 13. Cryostat and dewar















Figure 14

8 Tesla superconducting solenoid and some of the contents of the
vacuum jacket.



A 1 K pot

B still

C continuous concentric heat exchanger

D discrete heat exchangers

E mixing chamber

F flexible heat link

G squeeze connection to bundle flange

H copper demagnetization bundle

I thermal shields

J vacuum jacket wall

K indium heat switch

L vertical Helmholtz coil pair

M compensation windings

N 8 tesla solenoid





































10cm


SOLENOID


AND VACUUM JACKET


FIGURE 14. Solenoid and vacuum jacket








jacket and the entire assembly hangs from the pumping line and baffle

assembly in the neck. In order to conduct up to 78 amperes between the

magnet and its external current supply, a two-level system of current

leads is used. The lower portion is a "sandwich" of normal and super-

conducting metal strips which connects to the upper, normal vapor-cooled

portion near the top of the dewar belly. These vapor-cooled leads,

developed especially for low-duty cycle use (Berg and Ihas, 1983),

lowered the overall helium boiloff rate by "30% when they replaced the

leads supplied by the magnet manufacturer. The average consumption of

liquid helium (from a two month period including five magnetization

cycles) is now 15.5 liters per day, including transfer losses.

Figure 14 shows the outlines of the contents of the vacuum

jacket, consisting of three chief portions: the dilution refrigerator,

the copper bundle, and the experimental volume.

3.2 Dilution Refrigerator

Over the last two decades, dilution refrigeration has emerged

from nonexistence to become the method of choice for cooling scientific

experiments to temperatures significantly below 1 K. The power of this

technique can be briefly explained by comparison with the older techni-

ques of 3He evaporative cooling. Both 3He evaporative and dilution

cooling use the binding energy for extracting single atoms from the bulk

concentrated phase. Both can be operated in a continuous, circulating

mode for an indefinite length of time and are essentially unaffected by

the presence of magnetic fields. The evaporative cooling power of a

single atom is its liquid binding energy L and the rate of 3He atom

extraction is proportional to the vapor pressure, which is exponential








in temperature. Thus, the overall cooling power goes as



Q(evaporation) L exp(- (112)
kTj (112)



and in practice the highest possible pumping speeds give a minimum

temperature of about 0.3 K for a pure He cryostat,

Dilution refrigeration works because of the nonzero solubility

of 3He in 4He at arbitrarily low temperatures, about 6.4% below 40 mK.

Cooling, obtained by transferring He atoms from the pure into the dilute

phase in a "mixing chamber," is driven by an osmotic gradient, which in

turn is produced by pumping 3He vapor from a "still" which is at a

temperature high enough to give a reasonable circulation rate. Even

though the cooling power per 3He atom decreases with temperature as T2

the circulation rate, n, is independent of mixing chamber temperature so

chat the oJerall power for dilution cooling is



Q(dilution) ~ n3 T2 (113)



If careful account is taken of factors such as thermal isolation and

liicodu heating, the practical limit of a dilution refrigerator is

decerminid by the quality of the heat exchangers between the incoming

and outgoing liquid streams. Within the past few years, temperatures

belou 3 mK hare been achieved by groups using fine silver powder heat

exchangers (e.g. Frossati, 1978; Oda et al., 1983).

Cur Jilution refrigerator is of conventional (pre-1978) de-

sign. The I K pot, of 270 cm3 volume, liquifies the incoming He. A

"pickup" capillary, with an impedance dictated by the heat load on the








pot, replenishes the He supply from the 4.2 K bath. A manually oper-

ated valve allows the pot to be filled in about 10 minutes at the time

of first cooldown or if the pickup impedance becomes blocked. Such

blocking did occur during this quite extended experiment and daily use

of the manual fill valve was required. Careful use of the valve re-

sulted in negligible entropy increase of the experiment, even during the

1 mK runs.

The 3He still (100 cm3) contains a "film burner" designed for

a He purifier (Kirk and Adams, 1974) and reduces the 4He content of the

vapor removed to about 1%.

The heat exchanger system between the still and the mixing

chamber consists of a continuous exchanger followed by six step ex-

changers. The continuous exchanger, 1.5 m in length, is simply a thin-

walled copper-nickel tube of 1.8 mm inside diameter which contains the

dilute stream and an inner capillary of 0.51 mm inside diameter for the

concentrated stream. Each of the six step exchangers is a pair of

copper tubes silver-soldered together and packed with copper powder

(-200 mesh, C-110, U.S. Bronze). Bulk flow is through a central longi-

tudinal tube created by a greenedd" steel wire during the sintering at

900C. The volume available to the helium ranges from 2 cm to 4 cm in

each of the six exchangers.

The 30 cm3 copper mixing chamber contains copper powder (-400

mesh) sintered in place at 8500C.

Figure 15 characterizes the performance of our dilution refri-

gerator by plotting its cooling power as function of mixing chamber

temperature for three different circulation rates. The circulation

rates of 17, 25, and 36 micromoles per second correspond to still heater












































t-
0o



(U
0



E
o


E
U

- N rO
II II II
O~ <])


0


(M ') 83MOd


< 0


4 o
s


0
ro





C)
u
1-


4
E

(U



r.

oE




O -
6





E .2







Li













0


0 N1-100


I h







powers of 130, 360, and 700 microwatts respectively. Film burner power

was 140 microwatts for each curve. The minimum mixing chamber tempera-

ture in the unloaded state is about 14 mK although temperatures as low

as 10 mK lasting for a few hours have been observed. These latter inci-

dents have always been associated with the lowering of the magnetic

field at the experiment or the copper bundle and are attributed to

demagnetization of the copper nuclei in the mixing chamber by the fring-

ing fields of these magnets. This effect is verified by the comparable

heating seen at the mixing chamber when one of these fields is raised at

the same speed used when mixing chamber cooling was observed.

3.3 Nuclear Cooling Stage

Achieving refrigeration sufficient to study the superfluid

phases or the magnetically ordered solid phase of 3He usually requires

adiabatic nuclear demagnetization cooling. Although cooling to these

temperatures is possible with Pomeranchuk compression or demagnetization

of the electronic paramagnetic salt cerous magnesium nitrate (CMN), the

former is limited to melting curve pressure (34.3 bar at 3 mK) and both

lose cooling power just below the ordering temperatures of the refriger-

ants, about 1 mK for both 3He and CMN.

Nuclear cooling can be simply understood by considering the

thermodynamics of a noninteracting nuclear dipole system placed in a

magnetic field, B. The Zeeman interaction energy for each dipole is



m = -pgmB (11-)



where m is the magnetic quantum number, g is the nuclear gyrom gnetic

ratio, and v = e(/2Mc is the nuclear magneton. For n moles, the

partition function is









I nN
Z = [ exp --J] (115)
m=-I



Here I is the nuclear spin and No is Avogadro's number. The expressions

for entropy and magnetization derived from Z can be simplified by ex-

panding in the parameter pgIB/kT. If the effect of the dipolar field is

included, the entropy, S, magnetic susceptibility, x, and heat capacity,

Cg, can be written as


2 2
A (B2 + b2)
S nR tn (21+1) + b (116)
21 T

A




B T2
X (117)




oT
o


where
R = ideal gas constant

u = free space permeability

2 2
N = N Io1+1) uu2g 2/3k = molar nuclear Curie constant

b" dipolar interaction


Under ideal conditions of no heat leaks or nonnuclear heat capacities,

the final temperature, Tf, after reducing the field from Bf to Bi, is

related. to the initial temperature by


B2 +2 1/2
T f T (119)
f 2 + b2 i
1.
3-


and the heat capacity is unchanged.










CB(Tf) CB(TI) (120)



The construction of the nuclear cooling stage for our cryostat

is similar to that described by Muething (1979). The two major differ-

ences between the Ohio State Univeristy nuclear stage and ours at the

University of Florida are the wire diameter and the 3He heat exchanger;

these differences will be discussed in more detail. As was done at OSU,

copper was chosen for the demagnetization material. Copper gives the

advantages of high thermal conductivity, no superconducting transition,

ready availability in wire form, and low cost. Its disadvantages are

its low Curie constant (A = 4.04x10 K/mole) and its relatively large

Korringa constant of 1.1 sec-K.

3.4 Bundle Construction

Since the construction details of our bundle differ somewhat

from those of the Ohio State University bundle, the building process

will be described here. The gross features of the final product are a

close-packed cylindrical copper wire bundle 5.6 cm in diameter and 40 cm

long welded to an 11 cm diameter copper plate or flange. This flange

has an array of holes for clamping devices, such as the 3He heat ex-

changer, to the bundle.

The wire chosen for the bundle was a commerical grade, coated

magnet wire (Essex Corporation) whose special, high temperature insula-

tion (Allex) allowed the bundle to withstand temperatures of 300'C tor

extended periods of time. Thus, the wires were protected during the

welding operation and the slight flowing of the Allex during annealing

could be used to bond the bundle together without the use of epox',








suspected as the cause of large time dependent leaks (Konter et al.,

1977). The wire chosen was (#24 AWG, 0.51 mm diameter) based on the

measured 4.2 K resistivity of the wire material, p = 5x10- ohm-m. The

net eddy current heating power in a changing magnetic field, B, is


4 *2
P N B2 (121)
8p



where N is the number of wires of length X and radius r. For a typical

"cold" rate of 3x104 Tesla/sec this gives P = 10 nW, an acceptable

level for this process.

Preparation of the wire for bundle construction commenced by

winding some 200 turns onto a 27 cm diameter light aluminum drum. Two

cuts through the wires along the axis of the drum gave two tresses of

wires 42 cm long. About forty such tresses were made. Allex is resist-

ant to most chemical stripping agents so, in order to remove 4 cm of

insulation from one end of each tress, the tresses were dipped in a bath

of molten sodium hydroxide (NaOH) contained in a stainless steel

beaker. Immediately afterwards, the wires were bathed in a weak solu-

tion of acetic acid to neutralize any residue of the strong base.

Finally, they were rinsed twice with distilled water in an ultrasonic

cleaner.

Collecting the wires together for welding was begun by

Etraigncening the tresses and carefully laying them together in a

trough. A hose clamp then drew the bare ends into a close-packed array

5.1 cm in diameter. A 5.1 cm i.d., 5.6 cm o.d. cylindrical OPHC copper

collar iith an interior bevel on one end was forced around the wires by

alternately pounding on the collar and sliding the hose clamp down








towards the center of the wires. The protruding 3 mm of base wire

provided the fill metal during welding. The bundle of wires was immedi-

ately clamped into a support jig, designed so that most of the bundle

could be lowered into a glass nitrogen dewar (see Figure 16). The dewar

was filled with liquid nitrogen until only the OFIC collar and the

stripped ends were not immersed. Then the tops of the wires and collar

were housed in a glove box continuously purged by helium gas drawn from

evaporating liquid helium. An arc welder operating at five kilowatts in

the D.C. mode with an argon gas shielded tip fused the wire tops to-

gether. After correcting minor nonuniformities of this first weld by

filing, the bundle flange, an 11 cm diameter plate with a central hole

to mate to the bundle, was slid over the fused wire tops and came to

rest on the collar. The two pieces were then welded together using the

glove box and other procedures as before, after which the entire as-

sembly was ready for wrapping.

The bundle flange was now fitted with a bearing race so that,

by inserting the bearing balls, the flange with its protruding wires was

left free to turn with respect to this support. The next step was to

attach two #24 Allex insulated copper wires at the top of the bundle,

which, as it rotated, would be wrapped along its length by these

wires. The two wires were fed from opposite sides by a spring and brake

assembly, which used feedback to give a very constant and balanced

tension. As the wrap progressed, the bundle was struck sharply and

simultaneously with two hammers in an opposing fashion all around the

perimeter of the bundle. This compressed the wires of the bundle to-

gether so that, when the wires were eventually cut and tied off, the

final diameter was < 5.6 cm and quite uniform.























0-
r>





Zu

L I

m m
LJ "








The wrapped bundle and flange structure was removed from the

rotating support and inserted into a clamp designed to straighten the

bundle and preserve its alignment with respect to the flange. Heating

for five days at 275*C in a slightly flowing 4He atmosphere annealed the

copper and caused the Allex insulation to flow slightly and thus bond

the individual wires together. It was possible to avoid completely

using epoxy and still have a quite rigid structure.

The last step was to trim the bundle to the appropriate length

using a "glass wheel" (as used for cutting stainless steel tubing). The

cut end showed only a few dislocations in the hexagonal close-packed

array of 9000 wires.

3.5 Thermal Isolation and Heat Leaks

Effort expended on refrigeration for a millikelvin cryostat

must be complemented with measures to prevent the flow of heat into or

generation of heat within the final cooling stage. All material connec-

tions between stages of different temperatures were constructed to

minimize undesired heat flow along these connections. The load bearing

supports between stages are 1 cm diameter AGOT graphite rods. All

electrical connections and helium carrying capillaries were heat sunk at

the 1 K pot, the still, the mixing chamber, and the copper bundle.

Electrical heat sinks usually relied on a thin layer of cigarette paper

soaked with Stycast 2850FT (Emerson and Cuming) as an electrically

insulating thermal conductor. Capillaries were thermally anchored b.

soldering 10 to 30 cm of the line around a metal post and bolting the

post to the refrigerator. Silver solder was used at the lower stages to

avoid problems with soft solder superconductivity. The lowest cempera-

ture capillary connections were made with 300 micron (12 mil) o.d.








copper-nickel tubes. Most electrical connections were made with copper-

nickel wire except for the connections to the lowest temperature stage,

which were made with Nb-Ti superconducting wires (the copper cladding

was removed from all but the end connections of each length of wire).

Two nested thermal shields anchored at the still and mixing

chamber reduce the influx of heat due to blackbody radiation. The upper

portion of these shields is of welded sheet copper and the lower "tail"

section is a cylinder of longitudinally aligned copper wires ("coil

foil") coated with a hardened slurry of epoxy (Stycast 1266, Emerson and

Cuming) and finely ground coconut charcoal. This design (Tanner et al.,

1977) has a high infrared absorption coefficient while minimizing the

heat load due to eddy current heating during changes of the copper

bundle field. The bottoms of the shields are aligned with each other,

the vacuum jacket, and the copper bundle by a series of teflon "bicycle

wheels" with nylon mono-filament spokes. The influx of infrared radia-

tion to the lower temperature stages was also reduced by placing tabs of

copper foil at the vacuum jacket pumping port and over unused feed-

through holes at the various refrigerator stages.

rhe heat link between the bundle and the mixing chamber con-

3sisc of the heat switch, the flexible link, and the squeeze connec-

tors. The squeeze connectors allow quick mounting and removal of the

bundle from the cryostat, thus allowing the mounting of the experiment

to be chiefly a table-top procedure. To establish the squeeze connec-

tion, 3 g-:.ld-plated copper collar, welded to the flexible link, is

pressed over a matching gold-plated copper post welded to the bundle

flange. A thick nylon ring is then pressed around this assembly. As

the cryostat cools, contraction of this nylon ring provides the strong








squeezing force necessary (Muething et al., 1977). The flexible link is

two bundles of copper wires each about 1 cm2 in cross section. At its

upper termination is a heat switch of conventional design which consists

of two copper posts of semicircular cross section separated by Vespel

(DuPont) rods and surrounded by a small persistable superconducting

solenoid. The superconducting link is a strip of indium about 1 cm long

and about 10 mm2 in cross-section (March and Symko, 1965). Indium is

readily available in pure form, is easily fabricated and attached to the

refrigerator by virtue of its low melting point, and has a relatively

low critical field of 28 millitesla.

Despite these various thermal isolation measures, the heat

leak to the bundle was frustratingly high at 50 to 100 nanowatts for the

first several weeks of the run during which the data for this thesis

were taken. Tests showed that no nearby electrical equipment and, with

the possible exception of the dilution refrigerator mechanical pump, no

nearby mechanical equipment were implicated. The heat leak due to

electrical and capillary connections to the bundle was expected to be

< 1 nanowatt. As the data on Figure 17 show, the heat leak to the

bundle eventually dropped to about 3 nanowatts or about 50 picowatts per

mole of refrigerant. Since we knew of no changes external to the cryo-

stat which could have caused this drop in the heat leak, we surmised

that the depletion of some energy reservoir in the bundle had been

taking place during this time (see Pobell, 1982).

Our primary suspect for the time-dependent heat leak is ortho-

para conversion of hydrogen molecules. The associated energy, transi-

tion rate, and equilibrium concentrations of this process are well known

for the bulk solid (Silvera, 1980). Such a heat leak as a function of

time would be






































20 50 100 200 500 1000 2000 5000
TIME AFTER COOLDOWN (HOURS)


HEAT LEAK TO BUNDLE


FIGURE 17. Time-dependent heat leak to bundle









-3
3.8x10 n
(Kt+1/X())2


where t is the time in hours of cooldown from 300 K (essentially infi-

nite temperature as far as the ortho concentration is concerned) to "0

K" and n is the number of moles of hydrogen involved. In the denomina-

tor X(0) is the high-temperature limit of ortho concentration (3/4) and

K is the rate constant (0.019 hour- ). The curve plotted onto the data

of Figure 17 is Equation (122) for 3.5x10-4 mole of hydrogen. Ignoring

the residual heat leak of about 3 nanowatts, the fit is rather good.

About 5x103 mole of hydrogen gas was used as an exchange gas for cool-

ing the cryostat down to 10 K and, although it should have ended up in

solid form at the bottom end of the vacuum jacket wall, we cannot be

certain of this. Also, a recent report by (Mueller et al., 1983) cites

the discovery of pockets of hydrogen gas in copper used for construction

of a refrigerator. Without some form of microscopic analysis to prove

otherwise, such a possibility for the bundle wires or the body of the

3He heat exchanger is only speculative.

3.6 Superfluid-Handling Apparatus

At the heart of the cryostat is the experimental region out-

lined schematically by Figure 18. The "anchor" to the bundle flange is

the 3He heat exchanger and on this structure the compressor, strain

gauge, and cell body sit. Demountable struts raise the body of the heat

exchanger to center the cell body in the magnetic field. These two OFHC

copper pieces bolt to both the bundle flange and the heat exchanger; the

mating surfaces are gold-plated. Also bolted to the bundle flange are a

carbon resistance thermometer, a resistance heater, a heat sink for the










COMPRESSOR

STRAIN GAUGE,


HEAT,
EXCHANGER


LIQUID


3HE


BODY


CONTAINERS


FIGURE 18. Heat exchanger plus mounted containers








12 nongrounded connections to the experimental electrical devices, and a

capillary heat sink for the 4He line to the compressor. The six ground

connections needed were made by soldering these wires to a metal tab

bolted to the bundle flange which in turn was electrically connected to

the upper part of the cryostat through both Cu-Ni capillaries and a

superconducting wire.

The advantage given by this nuclear cooling stage design which

allows the bundle, heat exchanger, and cryostat to be connected to one

another by simple bolt and squeeze connections is that quick "turn-

around" times are possible. While one experiment is being run, another

can be constructed and tested on a separate heat exchanger. The practi-

cal realization of this advantage naturally requires the existence of at

least two heat exchangers; for this reason, the exchanger described

below was designed and built.

3.7 Sinter Cell Tests for the 3He Heat Exchanger

Various materials and preparation schemes have been employed

to build millikelvin heat exchangers. Although these schemes were

nicely reviewed by Harrison (1979) in the more general context of the

Kapitza resistance problem, I decided to make a direct experimental

comparison of several types using consistent methods of thermometry,

heat input, and cell design. Recently, mechanical and electrical mea-

surements of submicron copper and silver powder sinters were performed

to further understand their effectiveness as heat exchangers (Robertson,

et al., 1983).

Until recently, copper has been the most commonly used mate-

rial for low-temperature heat exchangers due to its high bulk thermal

conductivity and its availability in the forms of foils, wires, flkRes,









and powders. The copper powder we used (Vacuum Metallurgical) in these

tests was nominally 0.03 micron in particle diameter. However, electron

micrographs showed a characteristic diameter closer to 0.07 micron. The

black appearance of this powder is apparently due to the presence of a

large (60-80%) copper oxide volume fraction. This estimate is based on

the 16% mass loss that occurred upon heating a sample to 450*C in a

hydrogen atmosphere.

In order to build an oxide-free copper exchanger, we tried two

approaches. The first was to reduce the powder at low temperatures

until a 15' mass loss was achieved. The resulting powder, which had

suffered a little particle size increase (.08 to .14 micron diameters)

was t'en "cold-pressed" (at room temperature) into a copper cell. The

second approach was to first press the raw powder into the cell and then

heat ic in a reducing atmosphere to obtain a 15% mass loss. A third

approach, uncried by us, is to "presinter" the powder before pressing it

Inc cthe cell for sintering. The "presintering" might more appropri-

acely be calledd "oxide reduction."

In recent years, due to its low nuclear heat capacity and

reliti.e cleanliness, silver powder, especially the .07 micron "Japanese

p:wder", i(Vacuum Metallurgical) has been the most popular material for

milllkelvin heat exchangers. Since a sample of our silver powder suf-

ferel less than 1% mass loss upon heating to 300C, we decided to cold-

press the raw powder into a silver-plated copper cell with no heat

treatment whatsoever.

Palladium, with its high paramagnetism, showed promise as a

heat exchanger material for 3He samples, as verified by foil measure-

ments of Avenel et al. (1973). However, to our knowledge, only two








palladium powder exchangers have ever been built. Our palladium test

cell was sintered according to the recipe devised at Ohio State

University, (see Muething, 1979) which employs carbon monoxide and

hydrogen as reducing agents, and helium to flush out the hydrogen before

cooldown to prevent its absorption by the palladium.

A total of five cells, whose particulars are described in

Table 1, were built. A cross-section of a sinter test cell is shown in

Figure 19. The chamber in the oxygen-free copper body was cut out by

spark etching and abrasive cleaning was necessary to remove the result-

ing residue on the walls. Final cleaning consisted of an acid bath dip

and heating in a hydrogen atmosphere to remove surface oxides.

For economical thermometry, 0.5 watt, 220 onm "old" Speer

carbon resistors were ground to a 1.6 mm thickness and attached to leads

with silver paint. These thermometers, calibrated against the He

melting curve, worked well down to 5 mK when immersed directly in the

liquid 3He.

Thermal time constants of the sinter cells were measured by

applying a current pulse to the Cu-Ni heater sufficient to cause an

initial 5 to 10% temperature rise over the outside temperature, which

was that of the temperature-regulated mixing chamber. After the passage

of a small signal transient, attributed to the response of the thermo-

meter bridge electronics to the heater pulse and the finite response

time of the carbon thermometer, the He temperature decayed approxi-

mately exponentially to that of the mixing chamber. The thermal time

constants obtained were converted to thermal boundary resistances using

the interpolated and extrapolated heat capacity data of Greywall iid

Busch (1982). For comparison purposes, these resistances are plotted








normalized to sinter volume in Figure 20. Although a normalization with

respect to sinter surface area would be more interesting theoretically,

the sinters were too small to allow accurate area measurements. The

multiplication of the resistances by the temperature in Figure 20 demon-

strates the T-1 dependence of boundary resistance often seen before in

this regime.

Although the tests for all of the cells were not at the same

pressure, data taken in the silver powder cell at 4.6, 14.6, and 27.6

bar gave resistances within 25% of their averaged values. Thus a

direct comparison using Figure 20 should be valid to at least this

uncerctanty.

The best results were obtained with the copper powder reduced

and sintered after packing (cell #5) and with cold-pressed silver powder

fill. fhe palladium (#4), which did not do as well, is interesting,

since Lf surface area is estimated by average particle diameter, it is

actually abo.t three times better than cells #1 and #5. Cold-pressed

fine copper powder whether oxidized or not (cells #2 and #3), does very

poorly beiow 5:' mK.

By far, the best exchanger "per unit difficulty" is the cold-

pressed silver powder cell. For this reason I built a full-scale 3He

heat exchanger of this type.
















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Table 1: Descriptions of sinter test cells

#1. 0.07 micron Ag powder (Vacuum Metallurgical) cold-pressed at 850

bar into Ag-plated cell. 43% packing fraction. 4.6 bar 3He pres-

sure.

#2 0.03 micron Cu powder (Vacuum Metallurgical) was first reduced

(165-180C in He+H2 atmosphere for 120 minutes; this caused 15.3%

mass loss) then cold-pressed at 1110 bar. 43% packing fraction.

14.6 bar 3He pressure.

#3 Untreated 0.03 micron Cu powder (Vacuum Metallurgical) cold-pressed

at 1200 bar. 34% packing fraction. 14.6 bar 3He pressure.

#4 1 micron Pd powder (Leico) packed at 1110 bar then sintered accord-

ing to OSU recipe (Muething, 1979) using CO (810-860C for 2 hours)

and H2 (570-600*C for 5 hours) and He "flushouts" while cooling.

52% packing fraction. 3.2 bar 3He pressure.

#5 0.03 micron Cu powder (Vacuum Metallurgical) packed at 855 bar then

sintered at 200-220*C for 230 minutes in H2 atmosphere. This

caused 13% axial shrinkage and 15.5% mass loss. 34% packing frac-

tion. 3.2 bar 3He pressure.



3.8 3He Heat Exchanger Construction and Performance

The gross dimensions of the OFHC copper body of the heat

exchanger and the locations of the cell mounting holes (see Figure 21)

were chosen to duplicate those of the first exchanger for this cryostat,

which used sintered palladium powder. Three independent wells, each

divided into two or three annular sections by 0.8 mm thick cylindrical

copper walls, were cut into the copper body using a spark etch milling



















EXCHANGER BODY CROSS-SECTION


3 cm


HEAT EXCHANGER


TOP FACE


FIGURE 21. Heat exchanger details








machine (EDM). The dividing walls insured that no portion of the sinter

would be more than 1.6 mm from bulk copper. For sealing vessels to the

heat exchanger wells with gold o-rings, a sealing surface much harder

than annealed copper is required. To this end, and also to provide hard

material for tapping the bolt circles for cell attachment, six flat

brass rings were inlaid into grooves cut into the exchanger body at the

upper and lower entrances of the three wells. The best means of doing

this turned out to be first coating the appropriate exchanger and brass

ring surfaces separately with silver solder (Silvaloy 355, Eutectic),

removing all traces of flux, and finally brazing the rings into place in

a hydrogen atmosphere at 700*C using greened stainless steel weights to

prevent "floating" of the rings. This final brazing simultaneously

annealed the copper body. A final, light EDM cut was made to align the

inside diameters of the brass inlays with that of their respective

wells. The protuding inlays were then machined to nearly the level of

the copper body and drilled and tapped with appropriate bolt circles

(#2-56).

At this stage leak tests uncovered two passages between screw

holes tapped into the brass inlays and their respective wells due to

gaps in the solder apparently causedby flux residue. These were suc-

cessfully sealed by injections of Stycast 2850FT (Emerson and Cuming).

Preparation of the exchanger for packing with silver powder

including light scrubbing to remove the EDM residue, followed by an acid

bath dip to remove a partial brass surface on the copper body caused by

diffusion of zinc from the brass inlays during annealing. All surfaces

of the heat exchanger body were then plated with silver using a zomaIer-

cial jeweler's electroplating solution and fine silver anode (Jadow).








Although the depth of the wells probably attenuated the electric field

near the well bottoms, this was apparently compensated by the.electro-

negativity difference of silver and copper and the end result was at

least an optically thick coating of silver on all exchanger surfaces.

After silver plating, the brass sealing surfaces were lapped

to the same level as the copper body and then polished by wet sanding

with fine emery paper. Final cleaning was done in an ultrasonic bath.

Packing the annular sections of the exchanger wells was done

in layers one to three mm thick per pressing in order to assure uni-

formity of volume packing fraction throughout. The nominally 0.07

micron diameter "Japanese" silver powder was packed using a press con-

structed from an inexpensive 1.5 ton automobile hydraulic jack. Pres-

sures applied to the silver varied from 370 to 410 bar. The final

pressing left the 487 gram copper exchanger body holding some 61 grams

of pressed silver powder. The volume packing fraction of the three

wells varied from 46% to 48%.

Figure 21 shows the arrangement of the two sizes of holes which

were drilled into the packed sinter. The two large holes, 2.4 mm in

diameter, anticipate possible future use of the exchanger in flow exper-

iments and greatly enhance hydrodynamic heat flow through the helium

between the top and bottom of each well. The array of small holes, 1.0

mm in diameter, decreases the average distance between bulk liquid

helium and the sinter interior.

Addition of the array of small holes improved the thermal time

constant of the RJe exchanger at 3 mK from about 30 to about 6

minutes. The required density of these holes should be comparable to

nhe characteristic length of the sinter, defined by (see Muething,

1 9 9)










*2 k V
( *) = (123)
h A
K


where


hK Kapitza conductance per unit area

k3 ks
k=
3 k s3+k

k T 3He conductance in the sinter


V,A = sinter volume, area.






Using hK derived from a measurement of Ahonen et al. (1978) and ks found

by normalizing the bulk value by the sinter to bulk ratio for palladium

found by Muething and setting V/A to that of a single spherical particle
*
of 0.07 micron diameter gives 1 = 2 mm.

While the final version of this cold-pressed silver powder He

heat exchanger has not been completely characterized as a function of

helium temperature and pressure its performance has been entirely satis-

factory. Below 10 mK, the thermal time constant for heat flow between

1/2 mole of helium and the copper bundle is about six minutes except at

temperatures much below three millikelvin where the T of the copper

nuclei becomes the important thermal "resistance" (T T is 1.1 sec.K for

copper).

3.9 Compressor

Only two of the three heat exchanger wells were used in this

experiment. Most of the liquid 3He sample was contained in these L-)

wells and in the compressor. This compressor, whose outline is shotn bh








Figure 22, allows the 3He pressure to be precisely controlled, espe-

cially at pressures above 29.3 bar, where solid 3He begins to form in

the warmer regions of the sample fill line.

This arrangement of concentric bellows is not the usual one

for He compressors. The more common figuration is that used in

Pomeranchuk (1950) compressors where the 3He bellows is everted so that

a compression consists of an expansion of the 3He bellows, thus avoiding

inadvertent compression of the solid helium which might form in the

bellows' folds. Since solid growth in the compressor was not desired in

tne first place, the present design saves valuable space in the experi-

mental region.

The two bellows are made of single-ply beryllium copper

(Robert Thaw). The top and bottom beryllium-copper pieces act as caps

for cne 3He container walls of 321 stainless steel. The post inside the

3He bellows acts as a stop and conserves 3He volume while still provid-

ing channels sufficient for keeping the He in thermal equilibrium with

cne rest of the sample. All metal-to-metal seals were made with Stycast

2'35)FT (Emerson and Cuming) to avoid the use of superconducting solder;

these seals held to pressures of at least 37 and 17 bar on the

3He aid "Re sides respectively. Movement of the bellows could be de-

tected by measuring the capacitance between electrodes glued to the

outside of the He brass cap and the inside of the stainless wall.

The position of this compressor determines the total experi-

mental 3He volume. The maximum stroke of 8.6 mm causes a volume change

of 1.,) cr3. Thus, when the He volume in the compressor is increased

fro its mLnimum of 7.6 cm to its maximum of 11.0 cm3, the total exper-

i 3H volume decreases from .4 3 to 10.4 cm3
lsaenti Hvolume decreases from 11.4 cm to 10.4 cm















4HE SPACE IO mm




VOLUME
VAU TRANSDUCER
VACUUM E ELECTRODES
SPA CE

BE-CU
3HE SPACE -, BELLOWS




GOLD EPOXY
O-RING _:SEALS







3HE COMPRESSOR


FIGURE 22. He compressor









As mentioned in the introduction, attempts to grow single

crystals of solid 3He in the sound transducer region failed because of

solid nucleation elsewhere. My chief suspicion is that this solid grew

inside the compressor due to a heat leak associated with "fast"

(~1 bar/hour) compression. Irreproducibility and minor nonmonotonic

behavior of the volume transducer signal suggests that the outer elec-

trode may have become partially unglued from the stainless steel.

Friction due to this defective transducer or to "scraping" of other

parts may be the source of trouble. As part of a pressure regulation

loop there have been no problems with the compressor, however.

3.10 Presiurl [lbaurement and Control

A capacicive strain gauge measured the sample pressure in

iLtu. The design used is similar to the original one first used for 3He

by Scracy and iams '1969). Figure 23 shows a schematic diagram of the

electronic; uIed 'for the pressure and volume measurements. Although

both trani.uceri required a three-terminal capacity measurement, a.total

of only five cryoscat wires was used by putting both signals on a common

line at dirierent frequencies. In both systems, a ratio transformer

primary is driven at the reference frequency of either one or five

kilonerct. The secondary is then adjusted so that the voltage at the

sum point is nalled. The reference capacitors (20 pF) are anchored to

the I K "He pot.

ijbmicron position changes could be seen by the volume trans-

ducer but it had non-monotonic and irreproducible behavior attributed to

a loose electrode. The sensitivity of the pressure transducer was about

10-" bar and worked well up to 36 bar. The settings of the ratio trans-

former for the strain gauge were related to sample pressure through a









calibration made against a 1500 psi (103 bar) Heise Bourdon pressure

gauge. 22 points between 4 and 36 bar were recorded. This procedure

took several hours as each point required usually more than 10 minutes

to change the pressure and wait for pressure equilibrium. The resulting

data were fit to a second-order polynomial with an rms deviation of less

than 0.1 bar which is more than the estimated inaccuracy of the Heise

gauge obtained from 3He melting curve comparisons at the nuclear order-

ing temperature (1.0 mK) and at the melting curve minimum pressure (318

mK).

The pressure measurement system described above was used to

control the sample pressure by the feedback scheme outlined in Figure

24. The error signal of the 3He pressure lock-in amplifier drives a

heater in a 4He "bomb" suspended just beneath, and weakly coupled ther-

mally to, the top of the vacuum jacket. By appropriately fine tuning

the He and 3He amounts the bomb can be caused to operate near the

liquid-vapor critical point at 5.2 K where the expansivity of the

4He is large. Gross adjustments to the sample volume and density are

made by adding or removing gas through the He and 3He fill capillar-

ies. The two control valves are always left shut except during these

adjustments.

The dependence of sound velocity on pressure imposes a re-

quirement on the stability of the pressure during a single run where

changes in phase velocity are measured. Referring to Figure 4 shous

that a velocity precision of Ac/c 0-5 requires AP-10-4 bar In the

region where (dc/dP)/c is greatest. The stability of pressure reglda-

tion, as estimated by observation of the volume transducer output,

indicated that this requirement was met.





































i-l a L> >
< o 0 0
i-I I -

LA LLTII
L 1 1 -






4HE 3HE


PRESSURE
TRANSDUCER
ELECTRONICS


VOLUME
TRANSDUCER
ELECTRONICS


C RYOSTAT

HE
BOMB I-
(5K)
-- SOLID
PLUG
ABOVE
29.3 bar
4HE



i 3HE




T RAI I
GAUGE
P U IO

-PR--ESSURE CONTRO--L LOOP-----
PRESSURE CONTROL LOOP


FIGURE 24. Pressure control loop








3.11 Sound Cell Contents

The cylindrical beryllium-copper cell slung underneath the

middle well of the 3He heat exchanger contains an assembly of parts

bathed in 3He and shown by Figure 25. Nine separate pieces fit inside

the outer sleeve machined from Stycast 1266 (Emerson and Cuming). This

sleeve is a hollow cylinder with four longitudinal slots allowing good

3He thermal conduction and absolute alignment of the radiofrequency (RF)

coils with respect to externally applied fields.

The fraction of the total 3He sample actually probed is only

about 1%: that portion contained between the two sound transducers

separated by the cylindrical spacer. Sound pulses are transmitted and

received by the two 9.53 mm diameter, 5 MHz, gold-plated X-cut quartz

piezoelectric transducers. Ground connections to the transducers' inner

faces are made via silver paint coating the ends of the spacer. Elec-

trical connections to the opposite faces of the transducers are made by

light, nonmagnetic springs, which also hold the transducers in place

against the spacer. A small amount of indium solder holds a connecting

wire (not shown) to the end of each spring.

The machined epoxy spacer defines the sound path length (6.25

.05 mm) and diameter (5.0 mm). These estimates allow for the thermal

contraction of Stycast 1266 (Swift and Packard, 1979). Some 23 mm2 of

radial holes, including those created by the crenelationss" at the

spacer ends, provide thermal conduction paths for the helium inside.

In order to perform NMR measurements on the same sample probed

by the sound, a flattened "saddle-shaped" RF coil was wound on grooves

cut into the spacer. The location of this coil is the reason that the

more usual choice of metal or quartz for a spacer material was not











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L
oO

Q)

N Z


C5 -1 J LLJ
z -- U
<


CO Cf)








made. The coil was constructed by winding half of some 100 to 150 turns

of #44 (45 micron) formvar-insulated copper wire on each side of the

"saddle". At 930 kHz, its inductance was 95 microhenries and its Q was

15. The coil's orientation placed the field it produced perpendicular

to the plane of rotation of the static field.

One interesting note is that a naive calculation of the eddy

current heating induced in the gold transducer coating can give a value

as high as many nanowatts at the RF levels and frequencies routinely

used in this experiment. Apparently the details of the RF field config-

uration and its intersection with the gold film prevent such a disaster

from occurring. The heating seen at RF currents ten times that ordi-

narily used was less than or on the order of one nanowatt.

Not shown in Figure 25 is a miniature heater for nucleation of

solid growth for magnetically ordered solid He experiments. This

hester consisted of a dab (-0.1 mm3) of Stycast 1266 (Emerson and

Coming, containing 0.07 micron silver powder (Vacuum Metallurgical) near

the percolation limit. This slurry was hardened across the cut faces of

a caisted pair of 45 micron formar-insulated copper wires. Heaters of

this tL'p had resistances of several hundred to several thousand ohms

jhichr were stable at low temperatures. Their small size enabled the

locaLion of qucleated solid 3He to be well controlled. Unfortunately,

unpredic.table "barnouts" limited their reliability. For the purposes of

this \liquid) experiment it is sufficient to know that the unused heater

e
and takes up only 2% of the sound path area.

A second RF coil consisting of 220 turns of #44 (45 micron)

formr.ar-Lnsiulated copper wire was wound in a "double solenoid" config-




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