Nuclear magnetism of solid helium-three

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Nuclear magnetism of solid helium-three
Britton, Charles Valentine, 1947-
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v, 72 leaves : ill. ; 28 cm.


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Magnetization ( jstor )
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Dissertations, Academic -- Physics -- UF
Physics thesis Ph. D
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Thesis--University of Florida.
Bibliography: leaves 68-71.
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by Charles Valentine Britton.

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To PouM?
aild Pat


The author wishes to express his sincere appreciation

to his advisor, Professor F. Dwight Adams, for his most

helpful advice and encouragecmnt during the course of this


Special thanks are due Dr. Donovan M. Bakalyar and

Dr. Ephraim B. Flint for their participation in much of

this work. The skillful machining and cheerful technical

advise of William E. Steegcr are gratefully acknowledged.

Finally the author would like to express his heartfelt

appreciation to his wife and daughter for their constant

support through the years.






I INTRODUCTION . . . . . . . .

A Hamiltonian for Solid 3He . . . .
Thermodynamic Quantities . . . . .
Early Exchange Measurements . . . .
Failure of the HNN Model . . . . .
Summary and Objective . . . . .

II APPARATUS . . . . . . . .

High Field Pressure C ll . . . .
Low Temperature Susceptibility Cell .

III PROCEDURE . . . . . . . .

Pressure Measurements . . . . .
Magnetization Measurements . . . .


Exchange Parameters ExtracLed from Pressure
Data . . . . . .
Magnetic Susceptibility ....
Comparison With Theoretical Efforts .
Concluding Rcmarks . . . . . .


A PRESS RE DATA . . . . . . .



REFERENCES . . . . ..........


S. 20

S. 20

S. 30

S. 30
S. 34

. 41

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy



Charles Valentine Britton

December 1977

Chairman: E. Dwight Adams
Major Department: Physics

Nuclear magnetism in solid 3He has been investigated

through measurements of the exchange pressure in high mag-

netic fields and measurements of the magnetic susceptibility

along the melting curve from 20 mK to temperatures approaching

the ordering transition.

The field dependence of the exchange pressure has

been accurately determined and the temperature dependence

verified. From these measurements the value of the lowest

order magnetic exchange parameter has been determined with

increased accuracy.

The magnetic susceptibility has been measured along

the melting curve. For temperatures above 4 mK the behavior

can be expressed in the Curic-Weiss form with 6 = -3.7 mK.

Below 4 mK however, the susceptibility rises abruptly to

values approximately twice those of the Curl-Weiss law.



The relative simplicity of solid 3He has made it an

attractive material for theoretical study and as low tern-

perature techniques have accumulated to test theories and

lay groundwork for new hypotheses.

The single, unpaired neutron in the 3He nucleus gives

rise to a nuclear spin I = 1/2 and a magnetic moment

p = 1.07x1026 J/T. The nuclear spin system is completely

randomized at high temperatures, but as the thermal energy

is decreased the spin interactions will begin to predominate

and at some temperature Tc the solid will become magnetically

ordered. The magnetic dipole interaction was considered by

Pomeranchuk (1950) to produce an ordering temperature on
-7 itrcir
the order of 10-7 K. However, there is another interaction

in 3He that is many times stronger than the magnetic dipole

inLeraction. Bernandes and Primakoff (1960) showed that the

quantulm-Liachanical exchange energy, which arises from anti-

synmmetrizing the Fermi particle wavetunctions, is much larger

in 31e than in other materials. The small mass of the atom

leads to a large zero-point motion and thus a large overlap

of -th atomic wavefunctions. This overlap is a measure of

the exchanieo probability for neighboring atori and gives; the

exchange energy its high value. The estimates of Bernandes

and Primakoff were shown to be considerably too high, but

their work was the beginning of a great theoretical effort

and a stimulus to experimental work already begun on ex-

change-related phenomena in solid 3He.

During the ten years following the work of Bernandes

and Primakoff a wide range of experiments produced data

suggesting that an antiferromagnetic ordering of the nuclear

spins would occur near 2 mK for the solid near melting pres-

sure. These data were interpreted consistently by means of

the Heisenberg Nearest-Neighbor (1INN) model (see next sec-

tion) with rather small experimental errors involved.

The adequacy of the HNN model, however, was brought

into doubt by measurements of the exchange pressure of solid

He in relatively high magnetic fields. The results of Kirk

and Adams (1971, 1974) require an exchange constant smaller

than previously deduced, by a factor of two. Several ex-

periments since then have also revealed inconsistencies with

the HNN model. An ordering of solid 3He has actually been

observed near 1 mK rather than 2 mK (Halperin et al., 1974;

Kummer et al., 1975, 1977).

The discrepancy between the HNN model and the recent

high-field data and low-tempcrature data has been the stimulus

for many theoretical papers during the intervening years.

However, at present, there is no simple model which is

consistent with all the nexisting experimental data.

The experiments described in this dissertation were

designed to provide further information on two of the

previous areas of most basic conflict. Pressure measure-

ments in high fields have been extended to somewhat lower

temperatures and particular attention has been paid to

determining, with increased precision, the field dependence

of the data. The magnetic susceptibility of solid 3He

formed at melting pressure in a Pomeranchuk cell has been

measured, using a pulsed NMR technique, down to tempera-

tures approaching the ordering transition. Preliminary

reports of this work have been made by Flint et al. (1977)

and Bakalyar et al. (1977).

A llariltonian for Solid 3He

As a basis for understanding current theories of exchange

and for interpreting the results presented here, a

generalized Hamiltonian will be presented.

The Hamiltonian for solid 3He may be written as a sum

of contributions from lattice vibrations (HL), the magnetic

dipole interaction (HD), the exchange interaction (H ), and

the Zeci ni energy in a magnetic field (HZ),

H = HL HD + HX + H (1.1)

The chrdct-eristi.c temperature for lattice vibrations

in sol .id He is, el : 20 1, and contri hbuti.ons from the phonons

are negqliible belo .'w 100 mi. As Pomeraonch'u:; noted, the

magnetic dipole interaction will be important only in the I K

region. In the temperature range between 1 mK and 100 mK

only the exchange and Zeeman terms arc relevant. Thus we


H = HX + H, (1.2)

An early model for treating the exchange term is the

Heisenberg Hamiltonian,

H = -2 z Jij (1.3)
with the sum being over all pairs of spins in the system.

This treatment reduces the many-body Fermion problem to

a sum over two-body spin interactions. The exchange energies

J.. are thus one half the difference in energy between the
singlet state antiparallell spins) and the triplet state

(parallel spins) which are formed when the exchange inter-

action breaks the degeneracy of the ground state of the ith

and jth spin 1/2 particles.

2J.. = E E (1.4)
13 s t

The Heisenberg Nearest Neighbor (IlNN) model follows from

Eqn. (1.3) by limiting the summation to first neighbors,

II = -2J E (1.5)
Since all lattice sites are equivalent, the constant J has

replaced ij and has been removed from the summation.


In this approximation we see that the type of magnetic

ordering expected as kT -* JI is determined by the sign of J.

For J>0 the energy of the nearest neighbor pair is minimized

by parallel spin alignment which would produce a ferro-

magnetically ordered solid, whereas for J<0 the energy of

the pair is minimized by antiparallel spin alignment which

would produce an antiferromagnetically ordered solid.

The magnitude of J determines the temperature at which

the transition to the ordered state occurs, and the sign of

J determines the nature of the ordered state.

When an external magnetic field is applied, considera-

tion must be given to the Zeeman energy,

HZ -= i (1.6)

th 3
where 1i is the magnetic moment of the i He atom.

While the nearest neighbor pair exchange model was used

for many years to describe consistently experimental proper-

ties of solid 3He, recent results have shown it to be in-

adequate. Hamiltonians now being considered include some

combination, or all, of the following,

i) pair exchange of nearest neighbors Jnn'

ii) pair exchange of next nearest neighbors Jnnn'

iii) pair exchange of third nearest neighbors J nnn'

iv) triple exchange (around a triangle consisting of

two nearest neighbor legs and one next-nearest

neighbor leg) J3'

v) quadruple exchange in a planar array (around a

rhombus consisting of all nearest neighbor sides)

vi) quadruple exchange in a folded array (around a

tetrahedron consisting of all nearest neighbor

sides) J4F"

Other triple and quadruple exchange processes are

possible but calculations with the many-body wave functions

indicate that the importance of terms not listed above can

be neglected (McMahan and Guyer, 1973; McM;ihan and Wilkins,


Thermodynamic Quantities

Rather than specify which combination of exchange terms

should be included in Hx, we shall consider a high tempera-

ture expansion of the partition function Z derived from an

exchange Hamiltonian of undetermined form, plus the Zeeman

Hamiltonian. This approach was suggested by McMahan (1974)

and yields the expansion

1 1 m Trace (lx+H1n
n In Z = log I

SIn 2 + 3 2 2 -1 3 3 + ... (1.7)
2 xx 2 xxx

1 2 1 [1 + .
+ (AISB) [1- 4"J + + I (3011B) [1 ...
2- xzz 12

where the moments of II are defied by

Trace H H
J2 x x (1.8)
xx Trace H1H1

Trace H H H
J x (1.9)
xzz Trace HIH H '1

n m

Trace Hx. .H Hz. ..II
J (1.10)
x...x z...z Trace HB . 1 H .H

n m n m

with I1 the Hamiltonian of a Heisenberg near neighbor magnet

with exchange constant of magnitude 1. This normalization

of the moments is such that if H is the Heisenberg nearest
neighbor Hamiltonian, all the J's are equal and the expansion

of Z is then equal to that derived by Baker ct al. (1967)

for the spin 1/2 HNN model. These workers give coefficients

in the expansion of in Z up to 10 terms in x = BJ with B=0

and terms up to eighth order in x and y = 8pB for B/O.

The various thermodynamiic quantities of interest are

found from Eqn. (1.7) as follows,

P(T,B) N kT 3 2 2 2 31 3 3 N ( B2
P(T,) V 2 xx -xx 2 xxx xx x V --2

[4y J + ...] (1.11)


y = d(Ln J2 )/d(fn V)
xx xx

yX = d(n 3 )/d (n V)
xxx xxx

y = d(Ln J )/d (kn V)
xzz xzz

2 2 2
X = (kT/V) (a2.n Z B2)V,

N-T [1 + 4J + 12J2 + ...] (1.12)
k TV xzz xxzz

S = (kT in Z)

S(T,B) n 2- 33 1.3254 4
Nk 2 xx + xxx x +

+ (hP B) [-48J X + ...] + ... (1.13)


C = T(
V aT V,H

C(T,B) 32 2 3 3 4 4
= 30 J 3( J + 5.300 J + ...
Nk xx xxx xxxx

+ (BfB)2 [12J .(1.14)

The various J coefficients can be determined by fitting

experimental data. Specific spin llamiltonians can then be

constructed which provide these J's through Eqn. (1.10).

Early Exchange Measurements
IHNN Modal

The original treatment of exchange in solid 3He was

based on the one parameter leisenberg Nearest Neighbor

approxii;itmtion which consists of all the J's in Eqn. (1.7)

being equal. The first properties to be measured which dis-

played exchange effects we-e-r spin diffusion and relaxation.

NMR measurements of these dynamic processes were possible

at temperatures as high as 1K. The analysis required to

extract values of J from these measurements however, is not

straightforward and the results of even high-precision

measurements have not always been consistent. These high-

temperature NHR experiments are reviewed by Guyer et al.

(1971) and are not considered further here.

With lower temperatures becoming more readily available,

considerable improvements in the precision of J measurements

have occurred through observations of various thermal

equilibrium properties. The exchange dependence of these

measurements is subject to a much more direct interpretation

than of the early NMR measurements.

The first thermodynamic determinations of J[ were by

Panczyk et al. (1967) and by Panczyk and Adams (1969) using

high resolution pressure measurements at constant molar

volume. A range of molar volumes, covering most of the bcc

phase were used, allowing the volume dependence of |Ji to be

measured as well as IJ[ itself. The results of Panczyk et

al. are shown in Figure 1, where AP PV(T) P with Po

determined by extrapolating the l/T dependence such that

P(1/T = 0) = P The agreement of this data with the ex-

pression in Eqn. (1.11) is quite good. The contribution of

the J3 term produces a deviation from the 1/T dependence of

only 2% at the lowest temperatures and is not large enough

to determine the sign of J. Tn these measurements B=-0.


] T [_ .... .. T
A6- 24.02

08 23.72 A

C 23.32

0 =22.84

E 21.64


i Zero-Field Pressure n Pressure diff
ne -rT or various raclar

(Note: 1 atia = 101.3 Pa.) (After Pancyk ond
A\ms, 1969.)A--
i -

-- -- -I ^--- ...-- -- *--' .

0 20 40 6G
lI/T (K'

Fin. 1 Zero-Field Pressure Measurements. Pressure differ-
ence A]T [ ( )T) P W verrsss--- for various nolar
volonres. For v = 2-.02 ci-/-'iolOe the hiqh r"rnnrea-
ture plhionon contribution is shown; for the oliher
volumes only the exchancc cor.tribution is s!howri,.
(Note: 1 atm = 101.3 kPa.) (After Panc?.-yk e.nd
Alicams, ]969.)

Magnetic susceptibility measurements are able to deter-

mine the sign of J however. The measurements of Kirk et al.

(1969) covered a wide range of molar volumes in the bcc phase

and are shown in Figure 2. The data are often interpreted

in the Curie-Weiss form. Taking the first two terms of Eqn.

(1.12), setting 0 = 4J/k, we have

X = C/(T 6) (1.15)

for T>>0. The results of Kirk et al. show clearly that 0

(hence J) is negative, indicating antiferromagnetic ordering.

The magnitudes of J found in this work are consistent with

those taken from PV(T) but their precision is considerably

lower than that of the pressure measurements.

The computations of Baker et al. (1967) for the HNN model

show that the ordering temperature TN = -2.748 J/k = -0.68700.

The magnetic susceptibility data and the pressure data in

zero field indicated that the transition to an antiferromag-

netically ordered phase would be expected to occur at

T 2.0 mK for a solid sample near melting pressure

(V = 24.1 cm /mole).

More recently, the specific heat of bcc i3e has been

measured by Castles and Adams (1973, 1975), by Dundon and

Goodkind (1974) and by Graywall (1977) to sufficiently low

temperatures to extract the exchange contribution. Except

for the results of Dundon and Goodkind, the values of J

found in the iieaisuLciments are in excellent agreement with

those found in the zero-field pressure and magnetic

(SAIwfln X5 1iG JV








o t$





-,-I C7
-4 -1



: 'I
I -H


0^ r4

a w

susceptibility measurements. These results are summarized in

Figure 3 (after Greywall, 1977). This figure illustrates

how consistently the one parameter HNN model was able to

describe the specific heat, pressure and magnetic suscep-


Failure of the HNN Model

The first evidence that the I~WN model was inadequate

in describing the properties of solid 3He was provided by

the pressure measurements of Kirk and Adams (1971,1974)

done in moderately high magnetic fields. The effect of an

externally applied magnetic field may be seen from Eqn. (1.11)

The first term involving the magnetic field dependence con-
tains the factors B 2J The departure from 1/T depen-

dence of the pressure in a magnetic field will thus show

the sign of J and the variation with the strength of the mag-

netic field should show the magnitude. The results of Kirk

and Adamn for a sample of molar volume 23.34 cm /mole are

shown in Figure 4. The data do show a qualitative agreement

with the INN model calculation and do show J to be negative.

The magnitude of pressure change caused by the magnetic field

however, is only 60:w as great as that expected, based on cal-

culations using J values measured in previous experiments.

Higher order terms are not capable of reducing thin discrep-

ancy and the differences involved are many times larger than

the experimental unce-i-tainties.

1 .0 -

0.8 -

o./- O 0

0.6 A

0.5- o

E a

0.4 -

0.3 A-A

23.0 23.5 24.0 24.5

M L A R V OLUM i E (cm )

Fig. 3 HNN J's collected from the literature.
(After GCrywall, l 977-)
(G ) Greywall. 1977; ( n) Castles and Adams, 19~.,"- 5;
(0) Kirh et al., 1969; (0) Panczyk and Adams, 169:
(A ) Richardson at al., 1965.

T- [K-1]

Fic 4 Hih-Field Pressure ent. Presr. essure difference
versus T-1 for v = 23.34 cm3/:cl< in several :magnet'.ic
fields. The da5;hcd curvYs show the calculated bc-
havioi based on the EHNN nodel, eouatio; (1.11)
(Note: H = ; 1 kG 0.3 T; 1 atr = 101.3 kPa.)
(After Kirk and Adams, 1971.)

Measurements of the 3He melting curve by Osheroff

et al. (1972) indicated that the expected magnetic ordering

had not occurred at temperatures well below 2 mK. The

melting curve was expected to become quite flat in the re-

gion of the solid ordering (with a maximum occurring if the

solid molar entropy has become equal to the liquid molar

entropy), but Osheroff et al. found the pressure still to

be rising at temperatures well below 2 mK.

The ordering of the solid has now actually been ob-

served by Halperin et al. (1974), Dundon and Goodhind (1974)

and by Kummer et al. (1975, 1977). These experiments observe

a rapid decrease in the entropy of the solid at a temperature

of 1.1 mK rather than 2 mK as expected from the HNN model.

Summary and Objective

A summary of the experiments just discussed is given in

Table 1 with the various J values determined by fitting Eqn.

(1.7) to the experimental data. This compilation is due to

McMahan (1974) and Halperin (1977). The molar volumes of each

experiment have been scaled to 24.14 cm /mole, assuming that
each J is proportional to v

The necessity of going beyond the HNN model to explain

the experiments done at high fields and those done at low

temperatures is obvious but the ability of any more general-

ized spin interaction Hamiltonian to describe solid 311e must

be based on consijstcnt values for Lhe moments of the Hamil-

tonian as lisicd in Table 1. The values shown in colunpn one


r- 0 E r 0 n -1
r- N -

4 C- 4- U)
H H H r
ci> 12 .1 Cr

1 0,%1 rQ Q -
0- r- 11 0 N

D n *D r --i U) iH

H I I N H 0 00
*0 0 x 4 1 5
12 U 12 Hn 1 1 U 1i

ri i- O i ri

N inN
ca o 0 N-

0 ( o +U
-- x Hi
012 X- H H

44 +1 +-
o u

' D oo o m


O-1 - c o
-I, 1 E + l
r1 Ng r~N- m
E N* * c

U2) X1 a U-U1
12 N5 N i
N- in

and two show that experiments which in principle are

determining the same quantities, yield values differing

by as much as a factor of two. If these experiments can-

not be reconciled, the entire notion of a spin-interaction

Hamiltonian may have to be discarded.

The discrepancy shown in the J column of Table 1

is our concern here. These values of the first magnetic

exchange coefficient are derived from magnetic susceptibil-

ities in the first instance and from pressure measurements

in the second. The three independent sets of susceptibility

measurements which are the basis for the first entry in

column one were reported by Kirk et al. (1969), Sites et al.

(1969), Pipes and Fairbank (1969), and Bernat and Cohen (1973)

Determining the Weiss theta (which determines Jxzz through

Eqns. (1.12) and (1.15)) requires extrapolating from a

sometimes quite high temperature and great care must be taken

that curvature produced by higher order J's in Eqn. (1.12)

does not lead to an overestimate of 101. Sites et al. and

Kirk et al. took steps in their analysis to avoid errors owing

to curvature. These two sets of data claim the smallest error

bars of the current measurements and their values for 0 at

melting pressure are in good agreement. Their analysis would

make it appear unlikely that their values of Jxzz is over-

estimates by any amount approaching tLh factor of two required

to reconcile it wit]h- that from the pressure measurements done

in a magnetic field.

The resolution of the pressure measurements of Kirk and

Adams should inherently be superior to that of the suscep-

tibility measurements. The sensitivity of the gauge employed
by them was 2x106 atm while pressure changes caused by the
applied field were as large as 2x>10 atm. The high preci-

sion of the pressure measurements is somewhat clouded by the

field dependence of the pressure. As pointed out by Gold-

stein (1973) the pressure change due to the applied field

should, for thermodynamic reasons, be proportional to the

square of the applied field strength. The data of Kirk and

Adcans show what seem to be random variations from this be-

havior, approaching 20% in one instance.

The present work was undertaken in an effort to remove

some of the uncertainty surrounding the high field pressure

measurements, and to determine the magnetic susceptibility

of solid 3He in the temperature range approaching the ordered




High Field Pressure Cell

A major difficulty present in studying low temperature

properties of bulk solids is that of thermally coupling

the solid material to the source of cooling and also of

achieving a uniform temperature throughout the sample. The

Kapitza thermal boundary resistance between solid helium

and the surface of the cell increases as 1/T3 as the tem-

perature is lowered, and the thermal conductivity of the

solid decreases as T3 as the temperature is reduced. Hence

it is necessary to construct an experimental cell with a

large surface area to maximize the thermal contact with the

solid, and also to limit the maximum dimension of the solid

to allow all portions to reach thermal equilibrium in a

reasonable length of time.

With these conditions in mind a sample chamber was

designed as shown schematically in Figure 5. The upper por-

tion of the cell was constructed of oxygen-free high conduc-

tivity (OFHC) copper. This copper section is thermally an-
3 4
chored to the mixing chamber of a 3He- He dilution refrigera-

tor by means of well annealed h-avy copper wires. To provide

thermal contact witl. the solid, a brush of approximately



_. 41 C 0 F7 P EI R

I I,,.,

.~'~i~ r ~- 7 ~ i,.^

L\ \ \ \ \ # ,"
,- _~-~-.i ,j.^' i /
:- ~-u-^- ' *-,---'.-- -- -------.

S.-... "--j- -. ,__ t


Fig. 5 High Field Pressure Cell.

2x105 Cu wires 0.025 rmm in diameter was welded to the body

of the cell. The lower portion of the cell consists of a

capacitive strain gauge similar to -those described by Straty

and Adams (1969).

The active element of the strain gauge is the berylium-

copper (Be-Cu) diaphragm 1.07 mm thick and 24.4 mm in dia-

meter which forms the lower surface of the sample chamber.

The center of this diaphragm was epoxied to, but insulated

from, the movable plate of a parallel plate capacitor. The

strength of the sample chamber diaphragm is sufficient to

withstand the working pressure greater than 5 MPa and pro-
duces a deflection of 25x101 m/Pa. A second diaphragm

assembly was constructed, identical to the first except for

the thickness of the diaphragm having been doubled. Use of

this less flexible diaphragm will be discussed later.

The bridge circuit used in making the high resolution

capacitance measurements is diagrammed in Figure 6. The

reference capacitor is a silvered mica capacitor mounted on

the still flange of the dilution refrigerator to minimize

thermal drift. An Optimation 1100 signal generator is used

to drive the General Radio 1493 Precision Decade Transform,r

and to provide the reference signal to the Keithley 840 lock-

in amplifieL. Seven decades of the ratio transformer were

used and an eighth digit was interpolated from the output

signal of the lock-in detector as displayed on a strip-chart

recorder. W-ith this resolution we were able to detect dia-

phragm displacemrnts as small as 5:1- 12 meter, corresponding

to pressure chanigs of 0.2 Pa.






Fig. S Capacitance Bridge.

A dipstick" of approximately 10 c:n volume, filled

with activated charcoal, was used to condense He from the

storage bottle and to produce the necessary pressures with-

in the cell. External pressures were measured with a Heise

gauge which covers the pressure range of interest here.

Carbon thermometers, calibrated against the 3He melting

curve during the same run, were used to determine the sample

temperature. Speer carbon composition resistors, nominally

100 ohms, were ground flat on two sides to expose the carbon

core. The two surfaces were glued in close contact with a

copper heat sink, taking care to avoid electrical contact.

These thermometers were thermally linked to the sample cell

and located outside the high magnetic field region. One such

thermometer was mounted within an Nb3Sn superconducting

cylinder capable of excluding magnetic fields of 2.5 T. This

thermometer should be free of any magneto-resistive effects

which can cause large errors in thermometry.

Magnetic fields were generated by a superconducting

solenoid capable of operating in a persistent mode. Field

strengths were determined from the current using data provided

by the manufacturer.

Low Temperature Susceptibility Cell

The magnetic susceptibility measurements were made in a

Pomeranchuk cell, a modification of that used by Kuminer et al.

(1975,1976). The problem of producing solid 3He at tempera-

tures approaching 1 mK is solved by freezing the required

solid from the liquid while holding the temperature of the

system constant. Thermal equilibrium within the solid is

achieved by limiting the thickness of the solid layer formed.

The NMR region of the Pomeranchuk cell is shown in

Figure 7. The tail-piece, filled with liquid and solid 3He

was machined from Epibond 100A thermosetting epoxy resin.

The NMR radio frequency coil was wound around this epoxy

chamber. In one set of measurements the rf coil was wound

in a sixth-order compensated geometry onto a nylon cylinder

which was slipped over the epoxy extension. The second set

of measurements utilized a simple solenoid wound directly

on the epoxy with the addition of a small counter-wound

length to cancel the rf field just above the sample region.

The static magnetic field is produced by a Helmholtz

pair of superconducting coils, mounted in the helium bath

and operated in a persistent mode.

A pulsed NMR spectrometer (Instruments of Technology

PLM-3) is used to measure the magnetization of the 3He within

the rf coil. The output of the PLM-3 is fed into a Fabri Tek

1062 signal average which has a digital printer to record

the averaged signal.

A resistivity heater is distributed uniformly through-

out the experimental region providing the surface on which

the solid 3He is formed. The Heater consists of a 300 m

length of 92% Pt, 8% W alloy wire, 0.023 mmn in diameter,

wound noninductive:y on a spool centered within the rf coil.

The double winding of the heater are spaced 0.15 mm apart

\ j CELL






Fig. 7 NMR -ection. of Su,:cepti' bi -1 Cell.

and each of the 50 layers of wire is separated from the

adjacent layers by a layer of tissue paper 0.076 mm thick.

This heater provides a surface area of 220 cm so that the

layer of solid formed will be quite thin. The volume of
solid formed in a typical heat pulse is 24 ym producing

a layer of solid only 1 vm in thickness. This assures that

the thermal time constant for returning to equilibrium will

be short (less that 10 sec.).
3 4
Pressures of the Hle and He are measured with capaci-

tive strain gauges of the type described previously. The

gauge which measures the 3He pressure is mounted within the

Pomeranchuk cell body. The 4He gauge is mounted on the

mixing chamber flange of the dilution refrigerator. The 4He

pressure is always less than that required to form solid 4He

so this gauge will give an accurate indication of the 4He

pressure in the cell. The capacitance of the 3He gauge is

measured with a General Radio 1615-A capacitance bridge with

an Ithaco 391A lock-in detector. The 4He capacitance is

measured using a ratio transformer and a reference capacitor

mounted on the still flange of the dilution refrigerator.

The off-balance signal from each capacitance measurement is

recorded on a dual-trace chart recorder.

During the course of a magnetization measurement the 3He

pressure must be held constant. This is done through use of

the feedback regulation system ii lustrated in Figure 8. The

3He capacitance bridge is balancPed at thc desired pressure,

and any error siqnal from the lock-in detector is amplified

u~ ~ m W __~J iT


by the Kepco BOP36-5 operational power supply which adjusts

the current in the 4He pressure bomb, thereby compressing

or decompressing the Pomeranchuk cell as required.



Pressure Measuresments

Prior to forming the solid sample of 3He, it was

necessary to calibrate the capacitive strain gauge and

also the resistors which would serve as thermometers

during the actual data-taking. While the entire refrigera-

tor was maintained at approximately II; (by pumping on the

lie cold plate evaporator), 3He was condensed into the cell

and pressurized using the charcoal filled dipstick. The

strain gauge was calibrated against the Heise gauge in the

external plumbing, covering a range in pressure from 3 MPa

to 5 1Pa. The ratio transformer reading R was related to

the pressure P through the equation

P = a + b/(R + AR) (3.1)

using a least-squares fitting routine. The small correction

AR was varied to minimize the rms deviation between the data

and the fitted curve. The ratio AR/R is usually found to be

about 10-2 The term AR can be considered to result from the

fringing field at the edge of the capacitor plates and/or stray

capacitance in the other parts of the circuit.

Having calibrated the sample cell pressure gauge, a

melting curve sample was formed. At a temperature of 1K

the liquid pressure was set at 3.37 MPa and the sample line

to the external plumbing was closed. The dilution refrigera-

tor circulation was begun and the sample cooled. The pres-

sure remained quite constant until the melting curve was

reached at about 0.75 K. Below this temperature solid forms

as more cooling occurs and the pressure drops. The minimum

pressure P js reached at 0.32 K. The strain gauge reading
corresponding to Pin is found by passing through the minimum

in each direction. The coefficient "a" in Eqn. (3.1) is ad-

justed so that the minimum reading yields a pressure of

2.9316 MPa which is the accepted value of Pmin. This small

correction usually amounts to less than 0.01 lPa.

The 3He in the cell will consist of both solid and liquid

down to the lowest temperatures reached in these measurements

(less than 15 mK). As the cell cools below 0.32 K solid 3He

melts (because of its negative latent heat) and the pressure

rises. The various thermometers are calibrated by maintaining

a constant resistance (i.e., temperature) and measuring the

pressure in the cell. Values of the melting pressure as a

function of temperature are used as tabulated by Trickey et

a (197 ) .

To form an all solid sample of large molar volume, the

cell must he warmed above the pressure minimum so that the

sample capillary will not be blocked by solil. The liquid

pressure in this ca;c was ti-hn raiscd to 4.7 Iil'Pa and the

cell allowed to cool. At ].25 K the melting curve was

reached and solid began to form in the cell. The pressure

dropped rapidly as the melting curve was followed toward

the minimum. At 0.89 K the sample was completely solidified

and the pressure change slows dramatically. By noting the

temperature at which the solid left the melting curve the

molar volume can be determined. In this case the all solid

pressure was 3.66 MPa, the melting curve was left at 0.89 K,

indicating a molar volume of 23.96 cm 3/mole. The solid was

annealed for about one hour at a temperature about 10 mK

below the melting curve, to relieve internal pressure gradients.

Once the annealing process was completed, further temperature

changes were made slowly.

Temperature intervals were chosen equally spaced in 1/T

covering temperatures from 150 mK to 15 mK. Having completed

pressure measurements over the entire temperature range the

magnetic field was changed and the procedure repeated. It

was found that the bulk pressure reading changed slightly when

the magnetic field was changed. This may be because of some

helium slipping into or out of the cell as a result of the

warming that accompanied the rather rapid change of field.

When plotting AP = P(T) P versus l/T a value of P was
o o
chosen such that the curve would extrapolate to AP 0 at

1/T = 0.

The pressures measured as a function of temperature with

this diaphragm were found to be about a factor of three lower

than those found in previous invest igations. It was suspected

that the helium contained within the wire brush might be

frozen in place and not able to move and affect the pressure

measured below the brush. If this volume of helium were not

taking part in the pressure changes, it would be possible that

the small volume change associated with the diaphragm move-

ment would be sufficient to substantially lower the pressure

measured. To determine the volume of helium that was actually

producing the pressure changes, the second diaphragm, having

twice the thickness of the first, was used in repeating the

measurement in zero magnetic field.

To account for pressure changes caused by volume change

associated with diaphragm motion we consider P to be a func-

tion of T and V and write

.3P]? = dP ( + I dV] (3.2)
jcTj1 d +T 1TV V (3.2)

By comparing the pressure measurements in zero field made with

the two diaphragms of differing stiffness we are able to

evaluate the correction factor necessary to convert our pres-

sure data to truly volume independent results.

From the pressure-capacitance calibrations of the two

diaphragms we are able to determine the volume-pressure be-

havior of each. The thicker diaphragm is G.61 times stiffer',

hence the volume change for a given pressure is G.61 times

less than that for the thinner diaphragm. Using the dP/dT

values for the two measul-ments and values of the isothermal

comprxessibility (Strar.y and nAdmr ss,3967) we see that the volume

V of solid helium which is producing the pressure behavior

is only 0.041 cm This corresponds to the free volume in

the cell below the wire brush. Although the volume of the

cell is measured to be 1.38 cm the volume of solid which

is among the wires is evidently frozen in the narrow chan-

nels and is not able to affect the pressure measured below

the brush.
1 dV
For the more flexible diaphragm the quantity k P is

equal to 1.50, so the constant volume pressure values are

2.50 times larger than the pressures measured.

Magnetization Measurements

The magnetization measurements are performed in a

Pomeranchuk cell which produces cooling by compression of a

liquid solid mixture of 3He, with the formation of solid

accompanied by the absorption of thermal energy. (The latent

heat of fusion for 3He is negative in the temperature interval

from 1 mK to 320 mK.) Since, in the cooling process, the

cell will always contain a mixture of liquid and solid, the

melting pressure can be used for thermometry.

While the entire refrigerator is at a temperature of 1K,
3 4
the two chambers of the cell are filled with Ile and He. A

pressuro-capacitance calibration is done for each fluid using

external pressure gauges. The 4He pressures include values

up to 2.5 MPa. The 3He pressures cover a pressure range of

2.8 MPa to 3.4 MPa (this being the change in pressure along

the melting curve below 3'0 mE) The He pressure is then

set at 3.24 MPa and the cell is quickly cooled below the

minimum, thus sealing a sample of 3"Ie in the cell with a

molar volume such that it will be entirely melted at a

temperature near 25 mK. Two days of cooling is necessary

to lower the temperature to this range.

When the liquid in the Pomeranchuk cell has cooled to

about 15 mK,a trial compression is made to locate the tran-

sition to the superfluid A phase. A small additive correc-

tion (about 0.3%) is made to the pressure calibration so

that the A transition which occurs at 2.75 mK corresponds to

3.4344 MPa as reported by Halperin et al. (1974). Other

temperatures are measured relative to this pressure using

the melting curve data of Halperin et al. and of Kurrner et

al. (1975, 1976).2.

The magnetization of the 3He within the NMR coil is

measured by analyzing the free-induction-decay (FID) fol-

lowing 250 kHz pulses. Each pulse lasts 160 psec; a delay

of 320 psec is allowed for the transmitter pulse to decay,

then the FID of the nuclear spins is recorded by the Fabri

Tek signal average. Sixteen such pulses are averaged for

a given measurement, and the FID is extrapolated back to the

center of the applied pulse.

When the temperature at- which a magnetization measure-

ment is to be made is reached,the 3He pressure is held

constant by automatically controlling the 41e compression

rate using the feedback mechanism described previously. When

the heat-leak into the cell appears to be constant (as

evidenced by a linear increase of 'He pressure with Lime),

a series of four magnetization measurements, spaced at one

minute invervals, is made. Following the measurements a cur-

rent pulse is applied to the heater within the NMR coil,

forming a controlled volume of new solid on the surface of

the heater wire. During this heat pulse the He pressure

is maintained constant, and the change in 4He pressure as

recorded on the chart output is a measure of the cell volume

change and hence is proportional to the amount of new solid

formed. A series of magnetization measurements is then made

following the heat pulse. A typical sequence is illustrated

in Figure 9. The magnetization values before the heat pulse

and those after the heat pulse are least-square-fitted to

linear functions and extrapolated to the center of the heat

pulse. The difference between the two extrapolations is AM,

the magnetization of the new solid formed (subject to a small

correction for the liquid magnetization to be discussed later)
The change in 4He pressure (also measured at the center of

the heat pulse) is proportional to the change in cell volume

and hence to the volume of solid formed by the pulse. Thus

the susceptibility of the solid at this temperature is

X E AM/AV measured in arbitrary units.

The increase in magnetization measured after the heat

pulse is actually the magnetitiation of the solid produced,

less the magnetization of the equal volume of liquid which

has been excluded by the solid formed. Thus, the suscep-

tibility of the solid is given by

k'---- :













0 4

O )

*H 01



Xs = + XL. (3.3)

The subscripts S and L refer to solid and liquid respectively

and the susceptibilities are expressed in the same arbitrary

units. The liquid susceptibility XL is constant at these

temperatures (if the very slight pressure dependence is ig-

nored). At pressures near solidification XL is found (Ramm,

et al., 1970) to be equal to the Curie value expected at 180 mK.

For temperatures above 5 mK the solid follows the Curie-Weiss

law so we have

XL 1T80 constant


S T-0

where CLand CS are the Curie constants and the temperatures

are measured in mK. Then,

C ,V 180
XS Cs 1 80 L 1
S= V T-0 (3.5)

where V1 and VS are the molar volumes of the liquid and solid

aL melting pressure.

From Eqns. (3.4) and (3.5) we have

X- L 180 I
L VS T-0 1

whichC can b( evalua'-ed usinq o0ur AM/iV va lues, yielding the

consrtcni correction term ^L which is to be substi tuod into

Eqn. (3.3). When the liquid becomes superfluid in the A

phase, the liquid resonance is shifted away from the solid

resonance frequency and the correction is not applicable.

In the early stages of these runs it was noticed that

on some occasions when several heat pulses were applied a

large scatter in the data was seen. After two or three

pulses, subsequent measurements of X would yield values

consistently lower than those recorded in the first few

measurements during a given compression. This behavior

was most noticeable at temperatures above 5 mK. These

anomalously low values of X could be explained in two pos-

sible ways. If a part of the heat pulse were conducted out

of the NMR region, the amount of solid seen in the magnetiza-

tion measurement wo- <' be less than that recorded by the 4He

pressure change and the resulting X value would be low.

Alternatively a portion of the heater could become heavily

coated with solid following a number of heat pulses and this

region would not be in good thermal contact with the liquid

in the cell. The solid in such an isolated region would

remain hot following a heat pulse and hence cause a low

measurement to be made.

To alleviate this problem only a few small heat pulses

were used during the latter compressions. From an accumula-

tion of data at a given temperature, those yielding the

largest susceptibilities were Laken as correct.

During ore compression a different technique was used.

After compressing to 2.5 mK n record ding the usual initial

series of background magnetizations, heat pulse, and final

series of magnetizations, all performed with the 3He tem-

perature regulated, the regulation was stopped by closing

the 4He pressure line between the external pressure bomb

and the cryostat. Thus the temperature of the cell was

allowed slowly to rise in response to the heat leak present.

In a two-hour period the temperature rose from 2.5 mK to

3.7 mK and magnetizations were recorded at three to five

minute intervals.


1 This possibility was pointed out by W.P. Halperin, D.D.
Osheroff and H. Meyer at the 1977 Sanibel Symposium.

2 The gravitationally induced pressure differential gives rise
to a temperature gradient of only 5 pI/cm.



Exchange Parameters Extracted from Pressure Data

Shown in Figure 10 are the pressure data for a sample

of solid 3e with molar volume V = 23.96 cm /mole as measured

with the thinner diaphragm. The effect of the applied field

is readily seen to be one of lowering the exchange pressure.

From Eqn. (1.11) we see that a decrease in pressure with

increasing field requires that J xz/k be negative indicating

that antiferromagnetic ordering is to be expected. The

dashed curve with no data points indicated is the calculated

behavior for a field B = 4.9 T using the HNN model with a

value of J determined from the B 0 data. The measured

behavior in a field of 4.9 T is shown as the inverted tri-

angles. It is evident that the pressure effect caused by

the magnetic field is about half that predicted by the HNN


Again referring to Eqn. (1.11) we see that a plot of P

versus 1/T in zero field should be a straight line with slope

equal to

2 V xx (k (4.3.)
S7 Yxx I


n 100-


0 10 20 30 40 50 60 70

T- (K!<-)

Fig. 10 Pressure Change with Temperatuire for Various Fields.
Pressure scaJl has not been correct -d to account for
non-constant cell volume.

As discussed previously, the pressures measured were

lowered substantially by the nonconstant volume of the cell.

We must multiply the pressure scale by the factor 2.50 (as

determined in the previous chapter) to arrive at the constant-

volume pressure. When this is done, we find the slope to

be 7.74 PaK. Using a value of yxx of 35 as is suitable for

a molar volume of 23.96 cm3/mole (note that yxx = 23Zn[J /3ZnV)

we calculate a value of J /k = 0.65 mK. This value is in

good agreement with those found in other pressure measure-

ments (Panczyk and Adams, 1969) and also in specific heat

measurements (Castles and Adams, 1973 and Greywall, 1977).

The specific heat measurement of Dundon and Goodkind yield

a value approximately 25% larger than this.

For a quantitative measure of the magnetic field

dependence of the pressure we have plotted the data in

Figure 11 as the change in pressure produced by the applied

field, P(T,0) P(T,B), versus the applied field squared.

The straight lines that result for each temperature are con-

firmation of the B dependence. A value of Jxzz/k can be

calculated from Eqn. (1.11) by noting that the slopes of

these lines are given by

z JVkT2 z ] (4.2)
VkT2 :zz k

The slope of the 1/T = 60 data is 8.93 Pa/T2 after the cor-

rection for constant volume pressure has been applied. The

value Yxzz = 17.5 is used as deduced by Guyer (1977) from



150 -



100 -


0 '^r-^'^- - I- -1-_. 1-_ --1 -
0 8 16 24 32 40 48 56
B2 (T2)

Fig. 11 Field Dependence of the exchange pressure.
Pressure scale has not been corrected to account for
non-constant cell volume.

the data of Kirk and Adams (1971,1974). Thus J z/k is

determined to be -0.34 mK. The other 1/T data yield the

same result verifying the 1/T2 dependence (see Eqn. (1.11))

of the pressure shift caused by the applied field. We

therefore find a value of J about half the value of J ,

in agreement with Kirk and Adams and in strong disagreement

with the HNN model.

The major source of error in determining these two

exchange constants is in the scale factor applied to the pres-

sure in order to correct for the nonconstant volume of the

cell. Values used for the compressibility of solid 3He

used in Eqn. (3.2) for the two diaphragms were taken from

Straty and Adams (1968) and are estimated to be accurate to

about 2%. The accuracy of the dV/dP determinations discussed

in the previous chapter for each cell is about 3%. The

"effective volume" V of the cell involves subtracting the

products of these terms and hence has an estimated error on

the order of 10%.

A more precise statement of the results may be made by

eliminating the effect of the pressure correction. The ratio

(J k)2 /(x/k) is independent of the pressure scale and
xxk / (J~zz/k)
is equal to 1.24 mK. The accuracy of this ratio is essen-

tially limited to the accuracy of yxx and Yxzz

Guyer (1977) uses a value of yxzz equal to one half -yx'

based on assumed similar molar volume dependence (recall

S C- ,nlJ /v while 23.nj ~1/.') and the measure-

ments of Kirk and 'Adcms (1971) over a somewhat limited range

of molar volumes. We must also note that the magnetic field

dependence of these data had a rather large scatter thus

indicating that the value yxzz Yxx/2 could be the source

of substantial errors. By leaving the values of yxx and yx

unspecified the results can be stated as [y xx/ xzz1 /k)

(Jxzz/k) = 2.48 mK which entails experimental error whose
major contribution is the molar volume determination. This

error is less than 2%.

The value of J = -0.34 mK reported here is well with-
in the error bars of the Kirk and Adams results shown in

Table 1 and the reduction in uncertainty is significant.

Magnetic Susceptibility

Results of the magnetization measurements are shown in
Figure 12 with X- plotted versus T. For temperatures of

4 mK and higher the data can be represented reasonably well

by a Curie-Weiss law X = C/(T-O) with 0 equal about -3.7 mK.

Below 4 mK a marked increase in X above the Curie-Weiss value

is observed. In the temperature interval from 4 mK to 2.5 mK

the increase in X is so rapid that a plot of inverse sus-

ceptibility would extrapolate to a positive intersection

with the T axis. Such behavior as is shown in this region

would be expected to occur in a system approaching a ferro-

magnetic region, but the susceptibility does not rise higher

than the Curio law. Below 2.5 ml the inverse susceptibility

again extrapolates to a negative intercept this time with

an intercept of api'oximatcl y -2 m;K rather than the previous

-3.7 miK.






0 0



0 2 4 6 8 10 12

T (r K)

Fig. 12 Cur.ic,-n ss plot of so] id Heliuln tirnee inverse
suscepl t i ability.

As mentioned earlier, this method of plotting suscep-

tibilities is quite subject to errors in extrapolating to

find the T axis intercept. Higher order terms in Eqn. (1.12)

produce curvature which can be quite pronounced at these low

temperatures and can lead to an overestimate of the magni-

tude of the intercept. In an effort to account for these

higher order terms a plot of xT versus 1/T is shown in Figure

13. Also shown on this plot are susceptibilities calculated

from the high-temperature expansion of the HNN model for

various values of J. These curves are calculated using the

ten coefficients given by Baker et al. We see that at the

higher temperatures the data can be fitted equally well by

the curves for J/k. =-0.4 mK and for J/k =-0.6 mK, still in

conflict with the pressure measurement determination of

J xzz/k = -0.34 mK. This could be because the inadequacy of
the HNIN expansion for Jxzz or could be a result of the lack

of convergence for this number of terms at these temperatures.

The low temperature data on this plot are also interesting.

A negative slope indicates an approaching antiferromagnetic

transition (negative Weiss theta) and a positive slope indi-

cates a positive Weiss theta. The positive slope seen be-

tween 1/T = 250 and 1/T = 400 corresponds to the rapid drop

in 1/\ between 4 imK and 2.5 mK. This feature seems to be

reproducible. It was evident in both sets of data, each

taken with a different NMR coil geometry in an attempt to

local i z tha- rf pulse. As another check on the behavior in

this tel.perature iervtl th:- magnetization was monitored at

0 0,

0 0

0 0 -

0 O

0 0


c 0

(9, 1

{/,jDnHCjtI y0





0 r





0 C
o .-





three to five minute intervals while the toreperature drifted

from 2.5 mK up to 3.7 mK (without heat pulses, as explained

in the previous chapter). As the temperature crossed the

superfluid A transit-ion the magnetization increased, cor-

responding to the magnetization of the normal liquid. The

size of this step agreed closely with the magnetization of

the liquid measured at 20 mK with no solid present. Mag-

netizations of the solid measured during this period of

drifting temperature followed the same temperature dependence

as did the susceptibilities recorded in the usual manner,

indicating that the amount of solid in the NMR region was

quite constant. Thus although solid was being formed as

the cell warmed,it was undoubtedly accumulating in the upper

portion of the Pomeranchuk cell rather than in the tail


Further confirmation of this unusual behavior would be


Other investigators have recently reported qualitatively

similar susceptibility results. Bernier and Delrieu (1977)

using a spin echo technique find an increase in susceptib-

ility above the Curic-Weiss law at temperatures below 4 mK.

Prewitt and Goodkind (1977) have measured the susceptibility

of solid cooled by nuclear demagnetization of copper and

also find the same deviation from the Curiu-Weiss law. They

also report a sharp drop in susceptibility seen to occur at

1 .'5 mK which they are unable, to explain by means of a sim-

ple t o sub-lattice mcodel. Norii et al. (1977) also report

measurements performed in a Pomeranchuk cell showing a

similar increase in susceptibility over the expected results.

It should be noted that since the present susceptibility

measurements were performed on thin layers of solid in ther-

mal equilibrium with liquid, the possibility of a process

peculiar to the interface influencing the results must be

considered. Ahonen et al. (1976,1977) have measured the

susceptibility of 3He liquid in geometries with large sur-

face areas and found pronounced ferromagnetic behavior at

temperatures below 2 mK. Sokoloff and Widom (1977) propose

that the thin film of "solid" formed at the boundary of the

liquid, undergoes a long range indirect spin coupling

through exchange of atoms with the liquid. An atom in the

solid can exchange with an atom of opposite spin in the

liquid and at another point in the solid the process could

be reversed, leaving the liquid in its original state but

producing what amountsto a long range exchange interaction

between atoms in the solid. Such a process can be used to

explain the positive Curie-Weiss intercept observed in the

liquid by 7h-onen et al.

While the surface area of the solid in the experiment

being presented here is considerably less than the surface

area of the liquid measurements of Ahonen et al., the pos--

siliility of a mechanism such as this producing the observ-

able effects cannot be ruled out.

The recent su:ceptibi] lity moIea slrcemnts of Previtt and

Goodkind (1977), which confirm the measurements reported

here, were performed on solid cooled without the presence

of a liquid interface. This would indicate that the ob-

served departure from the Curic-Weiss law was not produced

by an effect occurring at the solid-liquid interface. How-

ever, the magnetic phase transition and specific heat

anomaly observed by these investigators are reported to

occur at 1.35 mK as opposed to 1.1 mK as reported by Kummer

et al. and Halperin et al. If this discrepancy does not

arise from an error in thermometry (such an error does seem

possible), it might indicate that the transitions observed

using compressional cooling are significantly affected by

the existence of the solid-liquid interface.

It is also of interest to note here that both Kummer

et al. and Ialperin et al. calculated specific heats from

their entropy measurements. While the differentiation in-

volved reduces the accuracy of the results, it is significant

that both groups found a small specific heat peak at about

3 mK and a larger peak in the specific heat at a tempera-

ture ranging from 0.4 mK to 0.8 mK above the final ordering

of the solid.

Direct measurements of the specific heat of bulk solid

3He cooled by nuclear demagnetization of copper by Dundon

and Goodkind (1974) show similar behavior. Just below 4 mK

a small peak occurs followed by a larger peak at a lower tem-

perature but still above the final ordering temperature.

The higher temperature specific heat featur-' occurs quite

close to the rapid increase in suscept ability reported here.

Another feature of the entropy data of Kummer er al.

noted by Halpern (1977) is that throughout the temperature

range of 1.6 miK to 2.5 mK and for fields less than 0.8 T,

the change in entropy for two given field strengths is ap-

proximately independent of temperature. From the Maxwell

relation (sS/Ba)T = (IM/aT)B this implies that the mag-

netization is approximately linear in temperature. The

present data are consistent with a linear behavior in this

range of temperature, rather that the 1/(T 6) dependence

of the Curie-Weiss law.

Comparison with Theoretical Efforts

Since the HNN model was first seriously brought into

question, by the results of Kirk and Adams (1971), the

Hamiltonian has been generalized to include various exchange

mechanisms such as those listed in the initial chapter.

Triple exchange was proposed by Zane (1972 a,b,c) to explain

the results of Kirk and Adams but failed to explain the

ordering temperature which was soon determined by Halperin

et al. A similar problem was faced by the treatment of Gold-

stein (1973,1974). The possibility of quadruple exchange was

presented by Guyer (1974) and calculations by McMahan and

Wilkins (1975) showed that such an interaction could be impor-

tant. Four spin oxchainge was used by Heatherington and Willard

(1975) to produce a ighagnctic phase diagram similar to that

found e:xpcrimnc-tally by Kummer et al. Recently Roger et al.

(1977) hev, prs;ntecd cilculiat!.i onns showing that the

increased susceptibility below 4 mK could possibly be ex-

plained by including four-spin exchange in the Hamiltonian.

Their approach is essentially a generalization of that

taken by Heatherington and Willard. The Hamiltonian of

Roger et al. includes pair exchange up to third nearest

neighbors and the two cyclic quadruple exchange terms men-

tioned previously, the folded array (F) and the planar


3 F
H=-2 (Jn (i j) 4J4F 4ijk
n=l i< J i (4.3)
-4J4P i vjk

where i = jkZ i j)(1k ) + (Iil) (I (Ij k)k j

Only the (P) four-spin permutation involves third neighbors

(the longer diagonal). Hence Roger et al. assume J3 = J4P/2

to simplify further calculations, such as the partition func-

tion expansion. The results are given here in the notation

adopted previously in the work:

J = J + 3 J + 3 J
xzz 1 4 2 2 J
2 2 3 2 + 3 2' j2 2
xx = 2 2 3 2. 25( 4 + 4P

Values for J and J2 are then taken from previous experi-
xzz xx
mental works. By treaLing J F and J4P as variables, and

using molecular field approximati ons, Roger et al. find (as

did 11hatherincgton and Willard) thit- the phase diagrai'.i of

Summer et al. is produced only when J4F completely predomi-

nates over J4P. (This is in direct contrast with the cal-

culations of McMahan and Wilkins who find the J4P process to

be much more favorable than the J4F process.)

Using the parameters of Hetherington and Willard (in-

cluding J4P J3 0) Roger and Delrieu (1977) evaluate the

leading correction to the Curie-Weiss law and find,

-1 -1 2 1.7 mK
X (T) = N p [T 3.12 mK .. ] (4.5)

i.e., that the magnetic susceptibility increases with respect

to the Curie-Weiss law below 10 mK in agreement with the

results presented here, and the early measurements of Osheroff

(1972 a,b). For temperatures below about 6 mK higher order

corrections are expected to be important, but Roger and Delrieu

argue that the qualitative behavior should be given by this

leading correction even below 6 mK.

Concluding Remarks

Measurements of the magnetic susceptibility and exchange

pressure in magnetic fields have been performed for solid

3He. These measurements are used to evaluate the lowest

order magnetic exchange constant Jxzz

The temperature and magnetic field dependence of the

exchange pressure have bc en verified and the accuracy of

J has been increased considerably.

Measurements of the magnetic susceptibility have been

extended to temperatures approaching the ordering transi-

tion. A high temperature expansion using ten terms was

used in an attempt to determine the constant Jz but the

precision of the data or the lack of convergence for the

expansion does not permit a precise value of Jxzz to be


At temperatures below 4 mK the susceptibility rises

considerably above the Curie-Weiss law before returning to

an antiferromagnetic behavior below 2.5 mK.

Susceptibility behavior increasing above the Curie-Weiss

law has been explained through use of a Hamiltonian involving

large contributions of quadruple exchange, quite similar to

the Iamirn tonian used to explain the magnetic phase diagram

in this temperature range. These theoretical approaches are

both based on a high temperature expansion of the partition

function which is of questionable value as the ordering tem-

perature is approached. A more complete understanding of the

magnetic behavior of 3He will depend on more general theoret-

ical approaches as well as more extensive measurements of

magnetic effect-s both in the ordered phase and at pressures

above the melrting curve.


Constant volume correction has not been applied.

B = 0.0 T
1/ZL(K"1)JP -Pn (Pa)

6.66 25.9

10.05 34.2

15.15 48.4

19.92 64.9

24.87 79.5

30.58 97.0

35.71 113.7

40.48 125.2

46.30 142.8

51.02 156.2

56.18 171.8

57.80 205.8

60.98 190.7

66.22 210.5

70.42 220.9

75.19 236.4

B = 2.0 T

I/iL (K-


















P PCQ_(.a


















B = 2.8 T

4/I.LK 1 P -_o.o-a)

6.66 39.6

10.05 24.3

15.15 40.4

19.92 55.0

24.87 68.4

30.58 84.2

35.71 96.7

40.48 108.1

46.30 121.3

50.63 130.0

51.02 133.8

56.18 145.6

57.80 147.5

58.82 151.2

66.00 169.7

67.34 174.4

68.03 175.6

68.49 177.9
















S= 4.0 T

P_-_ _(Pal
















B = 4.9 T









40 16



5 .1 8




















B = 6.0 T
IL -1
SLT-J 1 ,I_.-. P _P _a)

6.66 31.2

10.05 36.1

15.15 47.2

19.92 55.5

24.87 62.1

30.58 69.8

35.71 74.1

40.48 75.7

46.30 74.3

51.81 74.8

53.33 73.6

56.18 73.4

58.82 72.0

60.98 71.3















B = 6.9 T
















Heat-pulse Method

X(A rb itra r1Units)























LIFII ~ hrb it raryjlnitcl

10.00 686.8

13.00 569.8

15.00 477.7

20.00 400.9

Drifting Up_ n Tenmeratuire

T_(m) y_ (Arbitrary Units)

2.53 1803.4
2.62 1796.8
2.67 1766.5
2.71 1744.3
2.73 1710.9
2.84 1652.1
2.88 1591.8
2.94 1542.5
2.98 1489.8
3.03 1447.2
3.09 1385.6
3.14 1344.9
3.19 1306.8
3.23 1266.7
3.28 1241.0
3.33 1197.6
3.37 1173.3
3.41 1125.0
3.45 1105.2
3.50 1076.0
3.54 1049.2
3.58 1033.4
3.62 1006.0
3.66 994.4
3.71 971.0

T _(E_!


In the heat-pulse experiments discussed here thermal

equilibrium occurred within approximately 10 seconds of the

heat-pulse application. This implies a thermal time constant

on the order of three seconds for a layer of solid 3He with a

thickness of a few microns (approximately 1 pm thickness per


Johnson and Wheatley (1970) report an exchange diffusion

constant, D, equal to 1.2 x 10-1 m2/sec for solid formed in

a Pomeranchuk cell below 10 mK. From this value we may calcu-

late a tine constant

= = 0.75 sec

for a layer of 3 pm thickness.

Another approach to calculating the thermal time constant

is to consider the specific heat and thermal conductivity

measured at higher temperatures with a suitable extrapolation

applied for the low temperatures. The molar specific heat of

solid 31Ie near melting has been measured quite accurately down

to 40 miK by Greyvall (1977). A T extrapolation to 2 mK yields

a specific heat value of 4 J/mole K. The thermal conductivity

of solid 3He has been measured by Bertman, Fairbank, Uhite and

Crooks (1966) to be approximately 2 W/K m at 400 mK which nay
-7 -3
be extrapol]ted to 5 x 10- N1/K m at 2 miK using a T 3emperat ure

dependence. For a molar volume, v, near iol i ng we howve


S=--- 2.9 sec
k v

for a thickness of 3 pm.

While the close agreement of this latter calculation with

our observed time constant is probably fortuitous, we can

safely say that neither approach seriously questions the reason-

ableness of our 10 second equilibrium time.


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Charles Valentine Britton was born on January 18,

1947 in Kingsport, Tennessee, where he graduated from

Debyns-Bennett High School in June, 1965. He then entered

Duke University in Durham, North Carolina, was elected

into Sigma Pi Sigma, physics honor society, and received

the Bachelor of Science degree in physics in June, 1969.

He began graduate studies at the University of Florida in

September of that year. Following service in the United

States Army from Sept. 1970 until June 1972 he has been

pursuing the Doctor of Philosophy Degree at the University

of Florida. He is married to the former Rose Patryce Guthrie

and is the father of a nine-year-old daughter, Rose Patryce.

I certify that I have read this.-study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.

E. Dwight Adams, Chairman
Professor of Physics

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.

Jari.s W. Dufty
Associate Profeop r of Physics

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.

Gary G. Ihas
Assistant Professor of Physics

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.

Robert R. Kallman
Professor of Mathematics

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarrly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.

Thomas A. Scott
Professor of Physics

This dissertation was submitted to the Graduate Faculty of
the Department of Physics in the College of Arts and Sciences
and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of

December 1977

Dean, Graduate School


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