Title: Magnetic ordering of bcc solid 3He at melting pressure
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00099324/00001
 Material Information
Title: Magnetic ordering of bcc solid 3He at melting pressure
Physical Description: xi, 134 leaves : ill. ; 28 cm.
Language: English
Creator: Tang, Yi-Hua, 1946-
Publisher: s.n.
Copyright Date: 1987
Subject: Solid helium   ( lcsh )
Helium -- Magnetic properties   ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Statement of Responsibility: by Yi- Hua Tang.
Thesis: Thesis (Ph. D.)--University of Florida, 1987.
Bibliography: Bibliography: leaves 130-133.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00099324
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 - 001020630
oclc - 17886309
notis - AFA2044


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To my wife, Yolanda


I would first like to express my most sincere and deepest

appreciation to my research advisor, Professor E. Dwight Adams, for his

constant and skillful guidance throughout the entire course of this

work. It is he who led me into this fascinating field of ultra-low

temperature physics.

Special thanks are given to Dr. Kurt Uhlig, who was involved in the

early work of this research. I owe a great debt to him, not only for his

valuable advice in the period we worked together, but also for what he

has built for this experiment.

I also like to thank Dr. Donald Bittner and Greg Haas for their

helpful assistance for taking and analyzing the data from the


Professor Gary Ihas and his group members, Brad Engel and Dr. Greg

Spencer, gave their very friendly support to this work, including some

equipment used in the experiment. The discussion with Professors Pradeep

Kumar and Neil Sullivan, and Dr. Douglas Osheroff in the process of this

work were very fruitful. The cryogenic service from Don Sanford and

Christian Fombarlet was very satisfactory for keeping the experiment

running successfully.

Mrs. Karen Teele is thanked for her help of editing this manuscript

on the department's word processor.


Finally, I would like to thank my wife Yolanda for her under-

standing and patience, as well as my parents and parents-in-law for

their taking good care of my two sons.

This work was supported by the National Science Foundation under

Grant No. DMR-8312959.



ACKNOWLEDGEMENTS............. .....................................iii

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

LIST OF TABLES....................................................vii

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


CHAPTER 1 INTRODUCTION.........................................1

1.1 General Remarks on Solid 3He.........................1
1.2 Heisenberg Nearest Neighbor Model and Its Failure....2
1.3 Magnetic Ordering and Magnetic Phase Diagram..........8
1.4 Multiple Exchange Models and Their Consequences.....16
1.5 Purpose of This Work................................27

CHAPTER 2 APPARATUS............................................30

2.1 General Description.................................30
2.2 Dilution Refrigerator...............................31
2.3 Nuclear Demagnetization .............................36
2.4 Thermal Isolation...................................39
2.5 as Handling System.................................40
2.6 He Sample Cell and Strain Gauge....................43
2.7 Pt Pulsed NMR Thermometer...........................47
2.8 Superconducting Magnets..............................51
2.9 Electronics for Pressure Measurement................52
2.10 Electronics for NMR Thermometer.....................54
2.11 Electronics for Superconducting Magnets.............56

CHAPTER 3 EXPERIMENTAL TECHNIQUES.............................59

3.1 Refrigeration.......................................59
3.2 Pressure Measurement................................62
3.3 Thermometry.........................................65
3.4 Data Acquisition....................................70

L .

CHAPTER 4 DATA REDUCTION...................................... 72

4.1 Temperature Smoothing...............................72
4.2 Pressure versus Time and
Pressure versus Timperature.........................75
4.3 Entropy of Solid He................................76

CHAPTER 5 RESULTS AND DISCUSSION..............................81

5.1 General Features for Different
Order Phase Transition....... .. . ...........81
5.2 Ordering Transitions of Solid 1He
at Melting Pressure................................83
5.3 Magnetic Phase Diagram.............................102
5.4 Spin Wave for High-Field Phase.....................105
5.5 Concluding Remarks .................................111

PRESSURE MEASUREMENT.............................. 114


APPENDIX C MAGNET RAMP PROGRAM................................119

APPENDIX D TEMPERATURE PROGRAM................................122

APPENDIX E DATA ACQUISITION PROGRAM...........................123

APPENDIX F ENTROPY PROGRAM..... ...............................127

REFERENCES........................................................ 130

BIOGRAPHICAL SKETCH...............................................134



Table Page

1. Comparison of u2d2 model with experimental results.........25

2. Operating parameters of DRP-43.............................36

3. Ordering temperature and entropy discontinuity of the
first-order transition below 0.390 T.......................87

4. Transition temperature for fields
from 0.400 T to 0.410 T .................................... 88



Figure Page

1.1 Pressure measurement in high-field phase....................7

1.2 u2d2 structure.............................................12

1.3 Magnetic phase diagram of solid 3He (experimental).........15

1.4 Compact exchages in bcc lattice............................22

1.5 SCAFJ and SSQAF structures.................................23

1.6 Phase diagram for two-parameter model (theoretical)........26

2.1 Dewar and cryostat .........................................32

2.2 Experimental arrangement.................................34

2.3 Gas handling system of the dilution refrigerator...........41

2.4 3He gas handling system....................................44

2.5 3He sample cell............................................46

2.6 Principle of pulsed NMR thermometry........................49

2.7 Electronics for pressure measurement.......................53

2.8 Electronics for Pt pulsed NMR thermometer..................55

2.9 Electronics for superconducting magnets....................58

3.1 Calibration of 3He cell (center gauge).....................63

3.2 Calibration of strain gauge (bottom gauge).................64

3.3 NMR tuning curve ...........................................67

4.1 Fit of temperature as function of time.....................73

4.2 Melting pressure of 3He versus cell temperature
at B = 0.373 T.............................................77


5.1 Features of different order phase transition....

5.2 P(T) and dynamic behavior of P(t)..........................84

Melting pressure

Melting pressure

Melting pressure

Melting pressure

Melting pressure

Melting pressure

Melting pressure

Melting pressure

Molar entropy of

Molar entropy of

Molar entropy of

of 3He at B

of 3He at B

of 3He at B

of 3He at B

of 3He at B

of 3He at B

of 3He at B

of 3He at B

solid 3He at

solid 3He at

solid 3He at

0.266 T.....................89

0.390 T.....................91

0.495 T..................... 92

0.465 T..................... 94

0.400 T.....................95

0.402 T.....................96

0.404 T.....................97

0.410 T.....................98

= 0.266 T..................99

= 0.373 T.................100

- 0.495 T.................101

5.14 Specific heat of solid 3He at B = 0.495 T.................103

5.15 Magnetic phase diagram of solid 3He (modified)............104

5.16 AT versus magnetic field..................................106

5.17 Comparison of pressure versus temperature for 0.495 T
between free-spin system and solid 3He...................109

5.18 Behavior of free-spin 1/2 system at an effective field
Beff = 1.96 T.............................................110

........ 82

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



Yi-hua Tang

August, 1987

Chairman: Professor E. Dwight Adams
Major Department: Physics

From high-resolution measurement of melting pressure of solid 3He

in applied magnetic fields, ranging from 0.266 T to 0.495 T, the entropy

as a function of temperature was determined by making use of the

Clausius-Clapeyron equation. The entropy discontinuity was found at the

transitions between the paramagnetic phase (PP) and the low-field phase

(LFP), as well as between the low-field phase and the high-field phase

(HFP), indicating that the nature of these transitions was first order.

No entropy discontinuity was seen at the transition from the paramag-

netic phase to the high-field phase, indicating that it was not first

order, but possibly a second-order transition or X transition.

The magnetic phase diagram of solid 3He was reestablished based on

the data from this experiment and work by other authors. The

intersection of the first-order transition line and PP-HFP transition

line was determined at B = 0.396 T and T = 0.90 mK.


By fitting the data of pressure as a function of temperature in the

high-field phase at B = 0.495 T, the spin-wave velocity was found to be

6.2, 7.8, or 8.9 cm/sec for one, two or three modes, respectively. As

indicated by the T of the melting pressure, the high-field phase was

magnetically ordered, rather than just a fully polarized paramagnetic




1.1 General Remarks on Solid 3He

Solid 3He is one of the quantum crystals (3He, 4He, H2, HD, D2,

etc.) which show quantum effects macroscopically. Because of the small

mass of 3He atom and relatively weak potential between the atoms

(principally the Van der Waals potential) there is a large zero-point

motion of solid 3He atoms. The 3He atoms are not well localized and

their wave function is very broad. For example, the amplitude of zero-

point motion is about 30% of the nearest neighbor separation at melting

pressure (Roger, Hetherington, and Delrieu, 1983). When a first atom

moves around, second atom nearby may occupy the lattice vacancy created

by the first atom. The similar process could happen between the second

atom and third atom, and so on, causing a "ring" exchange interaction

(McMahan, and Wilkins, 1975).

On the other hand, the nucleus of 3He atom is a spin 1/2 fermion.

The exchange of atom positions is accompanied by the exchange of nuclear

spins of these atoms. The exchange process in space among the 3He atoms

also induces an effective interaction between nuclear spins. For

classical solids the nuclear exchange interaction is always much smaller

than the nuclear magnetic dipolar interaction, which may cause a nuclear

ordering at about 10-7 K. But the exchange interaction in solid 3He

causes the nuclear magnetic ordering at 10-3 K (see, for example, Cross

and Fisher, 1985). In fact, solid 3He is the unique example which shows

the nuclear magnetic ordering due to exchange of whole atom positions.

The nuclear spin system of solid 3He provides a regime with very

rich magnetic properties. The interaction between experimental and

theoretical research on this subject has much improved our knowledge

about this interesting magnetic system. Nevertheless, there are still

ambiguities and disagreements in both of the experiments and theories.

Furthermore, research on the magnetism of solid 3He may be useful in

understanding other physical systems, such as two dimensional Wigner

crystal of electrons, metallic glasses, solid hydrogen and deuterium,

and solid He. The problem of magnetism of solid 3He still attracts a lot

of attention from experimentalists and theorists.

1.2 Heisenberg Nearest Neighbor Model and Its Failure

There are several basic interaction energies in solid 3He. The

excitation energy of the electron from the closed shell is of order 105

K. The characteristic energy of the phonon interaction in the solid is

about 10 K. The nuclear dipole interaction has an order of 10-7 K, much

lower than the temperature which the experiments could reach. The most

interesting interaction in solid 3He, then, is the exchange interaction.

The Hamiltonian of solid 3He in an applied magnetic field could be

written as

H = Hph + Hex + Hz + Hd (1.1)

where Hph is the part for phonon interaction, Hex for exchange

interaction, Hz for Zeeman interaction and Hd for magnetic dipole

interaction. In the temperature range of interest the Hamiltonian can be

simplified to the spin Hamiltonian Hs

Hs = Hex + Hz (1.2)

Bernardes and Primakoff (1959) first showed that the exchange

interaction in solid 3He was many times larger than the nuclear dipole

interaction. But the prediction of the nuclear ordering temperature was

near 0.1 K, much higher than what the later experiments indicated.

The obvious theory of the exchange process was the Heisenberg

nearest neighbor model. Simply assuming that the exchange energy J is

the same for all the nearest neighbors, we can write the exchange

Hamiltonian as

Hex = -2J I. Ij (1.3)

where Ii and I. are the nuclear spins for 3He atoms labelled by i and j,

respectively. The summation is for all the nearest pairs in solid 3He. A

negative J will give an antiferromagnetic structure (Ii and Ij anti-

parallel) in order to minimize the ground state energy of the system. A

positive J will give a ferromagnetic structure (Ii and Ij parallel) for

the same reason.

The Zeeman Hamiltonian in an applied magnetic field can be written

H =- Ei B
z i

= Yi E Si* B (1.4)

where Si and p are the spin and magnetic moment of the atom labelled by

i, B is the applied magnetic field, and Y is the gyromagnetic ratio.

The partition function of solid 3He in an applied magnetic field is

Z = Tr exp [ -B ( H + H ) ] (1.5)
ex z

where B is 1/kBT, and kB stands for Boltzmann constant. The free energy

can be obtained by expanding the partition function in power series

of B (Roger, Hetherington and Delrieu, 1983)

F = lnZ

2 3
-- { [ln2 + e e-
2 3 228

S[ Y6B8 2 2
+ I[ 2 (1 + 8 + a2-- + + (16)

The parameters in the expansion, e2, e3 etc., are related to the

exchange energy J. This expansion is under the assumption that the

exchange rate is small compared to k T/6 The entropy is given by

S=- ( F )
S( e2 /8 +3 /12 (1.7)
NkB (ln2 e2 2 /8 + e B3 /12 + *** ) (1.7)

The specific heat is

C = T ( )
v T v

NkB 2
- (e2 e *** ) (1.8)

The susceptibility is

0 kB a2F
X v aB2 T

T + A/T (1.9)

0 R i 2
where the Curie constant C = ( 2 ) the permeability of free
B 2k
space, R the gas constant, v the molar volume, and -' the Zeeman
splitting in mK per tesla. The pressure of solid 3He in a magnetic field


P -,( -)2
av T
_r _V 2
= (e2 8/8 e3 82/24 + .. ) +

+ (Y6BB/2)2 (6'/2 + a28/16 + ***) (1.10)

where the primes denote the derivatives with respect to molar volume v.

The first reliable measurement of the coefficient e2 in zero

magnetic field and high temperature range 13 < T < 100 mK was done by

Panczyk and Adams (1970). From the pressure measurement they found

accurate values of 3e2/3v. After the integration with respect to the

molar volume, 2 was determined as 5.14 (v/24)36.26 (mK)2 where v is in

cm3/mole. For the melting pressure at this temperature range v = 24.2

cm3/mole, that gives e = 6.95 (mK)2

Greywall (1977) and Hebral et al. (1979) determined e2 by using

specific heat measurements at high temperature range 20 < T < 50 mK. The

value of e2 from these experiments was in good agreement with the value

obtained by Panczyk and Adams (1970).

Kirk, Osgood and Garber (1969) performed the first experiment to

yield the sign of the exchange energy by measuring the magnetic

susceptibility. The results were consistent with the Curie-Weiss law,

giving a negative value of 6 for Equation (1.9), and thus a negative

value of J. Also, the exchange energy J obtained from this experiment

was in good agreement with the result of Panczyk and Adams (1970).

In the period before 1975 there were many theoretical papers based

on the Heisenberg Nearest Neighbor Antiferromagnetic model (HNNA), which

gave a quite satisfactory agreement with the experiments performed in

the high-temperature range.

The first important experiment which contradicted the prediction of

HNNA model was the pressure measurement of solid 3He in an applied

magnetic field by Kirk and Adams (1971). The result gave J = 0.392 mK

at molar volume 23.34 cm3/mole, which qualitatively confirmed the

antiferrromagnetic behavior of solid 3He with a negative exchange

energy, but quantitatively only produced about half of the field

dependence predicted by the theory using the value J obtained from the

early experiment (Figure 1).

Another breakdown of the HNNA model was the experiment of the

susceptibility measurement. According to HNNA model the magnetization M

of solid 3He would follow Curie-Weiss law as a first-order approximation

0. -


C.3 1 x_--< H= 60 kG-

10 15 20 25 30 35 40 45
T- [K-']

Figure 1.1 Pressure measurement in high-field phase
(After Kirk and Adams, 1971)

Pressure difference versus 1/T for molar volume v = 23.34 cm3/mole
in different magnetic fields. The dashed lines are the theoretical
behavior based on HNNA model. The solid lines are the fits for the
experimental data. (Note: H = B, 10 kG = 1 T, 1 atm. = 101.3 kPa.)

M = T (1.11)

Prewitt and Goodkind (1977) measured the static nuclear magnetization of

solid 3He as a function of temperature and molar volume through its

nuclear ordering temperature by using the SQUID technique. They reported

that the magnetization below 5 mK increased with decreasing temperature

more rapidly than the Curie-Weiss behavior displayed at higher

temperatures. Another result found by these authors was that the

magnetization decreased rapidly to 40% of its maximum value and became

temperature independent at the lower temperatures. Bernier and Delrieu

(1977) used pulsed NMR technique and obtained similar results to those

of Prewitt and Goodkind (1977) above the transition temperature.

Thouless (1965) first pointed out the important role of multiple

exchange mechanism for understanding the magnetic properties of solid

3He. He settled the question of the sign for different exchanges, that

is antiferromagnetic for nearest neighbor exchange, ferromagnefic for

three-particle ring exchange. He also gave the form for four-spin ring


Guyer and his collaborators (1969-1975) published a series of

papers to discuss the three-spin and four-spin exchange process (Guyer

and Zane, 1969; Zane, 1972; McMahan and Guyer, 1973; Mullin, 1975; and

Guyer, Mallin and McMahan, 1975). By 1974 Guyer was convinced that the

HNNA model was not adequate to describe the magetism of solid 3He.

1.3 Magnetic Ordering and the Magnetic Phase Diagram

In the past decade the experimental technique for obtaining the

ultra-low temperature below 1 mK has been greatly improved. That has

provided experimentalists a possibility to probe directly the nuclear

magnetic ordering of solid 3He.

Halperin et al. (1974) first observed that the entropy of solid 3He

along the melting curve decreased by 80% in an interval of 100 uK at

temperature of 1.17 mK, suggesting a first-order phase transition. This

result also demonstrated the failure of HNNA model which predicted that

a second-order transition of nuclear ordering would happen at about

2 mK.

Kummer et al. (1975) first reported the determination of the

entropy of solid 3He in an applied magnetic field, and therefore a

portion of the magnetic phase diagram of solid 3He. They found that at

low fields below 0.41 T the ordering occurred over a very narrow tempe-

rature interval, and the ordering temperature was depressed by the

field. But above 0.41 T the character of the ordering suddenly changed

with reduction in entropy occurring more gradually, and the ordering

temperature increased with increasing applied magnetic fields. These

authors interpreted the low-field ordering representing a magnetic phase

transition. They suggested that the ordered phase was probably an anti-

ferromagnetic or spin-flop phase. For the ordering in higher fields they

interpreted as paramagnetic ordering of spins by the applied field,

rather than a phase transition.

Prewitt and Goodkind (1980) repeated their susceptibility

measurements (1977) in magnetic fields up to 0.58 T. Below 0.4 T they

obtained the results more or less the same as that in the experiment of

1977, showing that the ordering was first order and that the ordered

state was antiferromagnetic. At the high fields above 0.4 T they found

that 1/X dropped below the low-field curve at a well defined

temperature, with the temperature departure rising when the applied

magnetic field was increased. The authors considered that the behavior

of 1/X reflected the second-order phase transition between the

paramagnetic phase and the high-field phase. In the high-field phase the

magnetization approached to a saturated value which was much higher than

the magnetization in the low-field phase.

Nothing about the magnetic structure of the high-field phase and

the low-field phase was known until the NMR experiments were performed

by Adams, Schubert, Haas and Bakalyar (1980), and Osheroff, Cross and

Fisher (1980) at almost the same time but independently.

Adams et al. (1980) used a Pomeranchuck cell and worked in fields

up high to 2.98 T. The 3He sample in the NMR coil was a polycrystalline

mixture of ordered and disordered solid at or near the transition

temperature. In the ordered phase below 0.41 T they found resonant peaks

in the spectrum below and above the Larmor frequency YB0 The magnitude

of the frequency shifts indicated a large anisotropy energy, which was

incompatible with the cubic magnetic lattice of most theoretical models

available at that time (see, for example, Hetherington and Willard 1975;

Okada and Ishikawa 1978; or Roger et al., 1977). For the high fields

above 0.41 T Adams et al. investigated the behavior of the melting

pressure, magnetization and frequency shift as functions of time. From

0.43 T to 2.98 T Adams et al. observed a frequency shift of about 3 kHz

independent of the fields. This implied a constant magnetization in

high-field phase equal to 0.55 Msat, where Msat was the saturation

magnetization of solid 3He. The magnetic structure could not be

determined by these data alone. But from the result of this experiment

the high-field phase was quite close to that predicted by Roger, Delrieu

and Hetherington as a "spontaneously spin-flopped state," which has two

simple cubic sublattices with orthogonal magnetizations. Adams et al.

(1980) concluded from the rapid onset of the frequency shift that a

second-order phase transition was occurring at the high-field phase.

Osheroff et al. (1980), on the other hand, were able to grow single

crystals in the ordered phase and study their properties over a broad

temperature range below the ordering temperature. In principle, NMR

experiments probe the energy dependence of spatially uniform spin

rotations. The authors observed large frequency shifts above the Larmor

frequency at the low-field phase, with a zero-field resonant frequency

near zero temperature of O/2w = 825 kHz. The large frequency shifts

implied a large anisotropy of dipole energy in the low-field phase, and

a breakdown of the cubic symmetry in the magnetic structure of solid

3He, in agreement with the results reported by Adams et al. (1980).

Furthermore, Osheroff et al. analyzed the spectrum of their

antiferromagnetic resonance and determined the stringent constraints on

possible sublattice structures. By using single crystals which could

support only three sets of resonsnces, Osheroff et al. found three

domains with only three possible spin orientations. The direction of

anisotropy was along one of the three principal axes [001], [010],

[100]. There were two modes observed in each domain, coupled with the

existence of a single non-zero resonance at zero field. This implied
that the dipole anisotrpy had the form E a (l'd) where 1 and d were

anisotropy axes in real space and spin space (Figure 1.2).

The quasihydrodynamic equation of motion for spin can be written as

(Halperin and Saslow, 1977)


Figure 1.2 The u2d2 structure
Magnetically, the u2d2 structure has tetragonal symmetry. d and
1 are the spin and space anisotropy axes, respectively.

S = YS x B + ED /an

2 -1
=B Y X S (1.12)

where S is spin, B the applied field, n change in orientation of the

spin ordering in terms of small rotations about three coordinate

axes, ED the dipole enegy on these rotations, and X the susceptibility

tensor. The first equation is the torque equation. Notice that the

dependence of the dipole energy on small rotations E D() puts an

additional torque aED/an on the total spin. The second equation is the

kinematic relation between angular velocity n and angular momentum. The

total energy of spin in a field is

E = ED(n) + Y2 S S S-B (1.13)

The solution of equation (1.12) under the condition of minimizing the

total energy E was found by Osheroff et al. (1980) as

2 = 1 1 j 2 + [(r2 02 + 4 2 2 cos2]112 (.1)
2 I L 0 w 0 ) + 4 W-L 0

where wL specifies the Larmor frequency, n0 the antiferromagnetic

resonance frequency at zero field, 0 the angle between the magnetic

field and the anisotropic axis in solid 3He associated with the

sublattice structure. The resonant spectrum obtained by Osheroff et al.

(1980) was in good agreement with the equation (1.14).

The classic antiferromagnetic models of bcc lattice, such as

NAF, SCAFI, SCAFI could not produce frequency shift because these

structures have a magnetic cubic symmetry. The results of Osheroff et

al. show that the original magnetic cubic symmetry for the low-field

phase was broken. Osheroff et al. (1980) proposed a possible magnetic

structure for the low-field phase, u2d2 structure, which has a

tetragonal magnetic symmetry (Figure 1.2). In the u2d2 structure the

spin orientation alternates to the opposite direction every two

successive planes.

Theoretical calculation for u2d2 structure (Cross, 1982; Cross and

Fisher, 1985) gave the value 0 /2w of 1230 kHz, where factor was the

renormalizatlon factor due to zero point spin fluctuations. Compared to

the experimental value 0 /2w of 825 kHz, factor was about 0.67, quite

bit smaller than the value 0.85 that derived from the spin-wave

calculation on simple antiferromagnetic states (Anderson, 1952; Kubo,

1952). More complicated structure umdm (m is an integer bigger than 2)

could be also possible for the results of NMR experiments, but the

longer sequences umdm made the agreement progressively worse. Whether

the spin structure is u2d2 or umdm is still not certain. And the

quantitative agreement between the experiment and theory for u2d2

structure is also an open problem. Nevertheless, the NMR experiments

have greatly improved our knowledge about the magnetic structure of

solid 3He.

NAF: Normal antiferromagnetic phase with two simple cubic
ferromagnetic sublattices with opposite magnetization.
SCAFII: Simple cubic antiferromagnetic phase with two simple cubic
antiferromagnetic sublattices with parallel magnetization.
SCAFJ: Simple cubic antiferromagnetic phase with two simple cubic
antiferromagnetic sublattices with orthogonal magnetization.


*8 -- -l-l-l- -- i-- -7 -




0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4
T (mK)

Figure 1.3 Magnetic phase diagram of solid 3He (Experimental)

PP: paramagnetic phase, LFP: low-field phase, HFP: high-field

The dashed line is from the NMR results of Adams et al. The solid
line is a fit to the NMR results of Osheroff et al. The dot-dashed line
is the tradition line determined by Prewitt and Goodkind from the
measurements of static susceptibility. The possible magnetic structures
for low-field phase and high-field phase are also shown in the figure.

Figure 1.3 is the magnetic phase diagram of solid 3He based on the

work of Adams et al. (1980), Osheroff (1982), and Prewitt and Goodkind

(1980). All the authors agreed that the transition between the

paramagnetic phase and LFP, or between LFP and HFP were first order. But

the order of the transition between the paramagnetic phase and HFP was

not clear. Osheroff et al. considered that transition should be first

order too. Adams et al. and Prewitt et al. found that was a second-order


1.4 Multiple Exchange Models and Their Consequences

1.4.1 The Symmetry Property of the Multiple Exchange Processes

After the experiments of Kummer et al. (1975) and of Halperin et

al. (1974, 1978), it was believed that the HNNA model was not

appropriate. There were several models proposed, such as spin glass

model (Beal-Monod, 1977), vacancy models (Sokoloff and Widom, 1975a,

1975b; Andreev, Marchenko and Meierovich, 1977; and Heriter and Lederev,

1977, 1978), multiple exchange models (Hetherington and Willard, 1975)

etc. Some of these models are proved not suitable to solid 3He. The

multiple exchange phenomenological models were a success, and the only

ones which can account for most of the experimental results at the

present time.

The exchange rate J is much smaller than the zero-point motion

frequendy w0 (the attempt frequency). The typical value of J/w0 is 10-

from experiment. Therefore, each exchange process happens rather

separately, and the total exchange Hamiltonian can be written as the sum

of individual exchange terms

H = E J P (1.15)
ex {p p p

where Pp is a permutation operator for a transposition p, Jp is the

exchange frequency corresponding to Pp.

For a fermion system like solid 3He the wave function has to be

antisymmetric under interchange of two 3He atoms (both of spin and

spatial coordinates). Generally, we have

P(R) ) = (-1)p (1.16)

where P(R) is the permutation operator for spatial coordinates, P() is

the permutation operator for spins, (-1)P is negative for a permutation

involving an odd number of interchanges (even atoms), (-1)P is positive

for a permutation involving an even number of interchanges (odd atoms).

Suppose there are only even permutations Pe; then the total wave

function i must satisfy

P ) = + 1 (1.17)

Since the orbital wave function of the ground state 0r is nodeless from

the general argument of quantum mechanics (Courant and Hilbert 1937),

then O0r must be symmetric with respect to all allowed permutations

p(R) r (1.18)

From Equations (1.17), (1.18) we have

p(a) +
e X = + X0 (1.19)

where X0 is the spin wave function at the ground state, and a symmetric

function corresponding to a ferromagnetic state.

By a similar argument, odd permutations will give an antisymmetric

spin wave function, which corresponds to an antiferromagnetic state.

The general conclusion, as shown by Thouless (1965), is that even

permutations lead to ferromagnetism and odd permutations lead to

antiferromagnetism. This symmetry property is an important criterion for

determining what kind of multi-exchange interaction takes place in

different magnetic phases of solid 3He.

1.4.2 The different exchange processes and their consequence for bcc

solid 3He

The Lennard-Jones interaction between 3He atoms shows a very deep

repulsive potential and very weak attractive potential. The 3He atom,

therefore, could be treated as a hard core sphere with diameter

a, = 2.14 A. The hard core may cause two spin exchange to have a

tremendous large effective potential compared to ring exchange of

nearest or even next nearest neighbors. That is the basic physical

argument of the multiple exchange models. We consider only the most

compact 3-particle and 4-particle cyclic exchange in the following


Three-particle cyclic exchange. Any permutation operator may be

expressed as the product of pairwise permutation, which in turn can be

written in terms of Pauli matrices as

P (1 + a )
ij 2 1 j


The 3-particle exchange process is shown in Figure 1.4a. The

operator for 3-particle cyclic exchange (i j k) is, then

P = P P
Pijk = j jk
= (1 + a "- a ) (1 + o ak) (1.21)

By using the relation

we have

(a i oa) (ak oi) = j ak+ i i (a x ak)

Pj = 1 + aj + a. + G i +
ijk 4 1* j k kk 1
+ i .* (a. xa )]
1 3 k


(1 .23)

There is also a conjugate exchange (i j k) which has the form

Pijk =1 [1 + oi. a. + 0. j o + k- -
ijk 4 1 ( 3 k k i
I a. (U. x 0 )]
1 j k


The sum of Equations (1.23) and (1.24) gives the 3-particle cyclic

exchange operator as

P + P
ijk ijk
= (1 + o1 j + j k + 5 ak + (1.25)

The Hamiltonian for 3-particle exchange is

T -1
He = J z (Pik + Pi ) (1.26)
ex t < jjk> k 1jk

which could be reduced to the sum of pairwise spin interaction according

to Equation (1.25). So, the 3-particle exchange alone leads to the same

universality class as the pairwise interaction (i.e. Heisenberg

interaction). We would expect that 3-particle exchange process may cause

only a second-order phase transition the same as the Heisenberg nearest

neighbor interaction. There is no doubt about the existence of the

first-order phase transition between the paramagnetic phase and the low-

field phase, as all the experimental results have shown. Also, the 3-

particle exchange process favors ferromagnetism on contrast to the

antiferromagnetic property of the low-field phase.

Four-particle cyclic exchange. The 4-particle cyclic permutation

operator Pijkl can be written as

Pijkl Pijk Pi (1.27)

By using Equations (1.20) (1.25), the 4-particle exchange operator can

be expressed as

P + P -1
ijkl ijkl
1 (0 + i j G (1.28)

where E is taken over the six distinct couples among the four
particles (i, j, k, 1), and

Gijkl = oi* a ) (a ka ) +

+ (oai o1) (o* oa) (aoi oa) (oa o) (1.29)

The operator Gijkl can not be reduced further to Heisenberg pairwise

operator. It is this operator which leads to a first-order phase


Roger, Hetherington and Delrieu (1983) studied the 4-particle

exchange processes in great detail by using a computer to minimize the

free energy of the system. They calculated two kinds of most compact 4-

particle cyclic exchange process involving four first neighbors. One is

folded 4-particle exchange, the other is planar 4-particle exchange

(Figure 1.4b). It turns out from their calculation that for folded 4-

particle exchange the magnetic structure should be the simple cubic

antiferromagnetic (SCAFi, Figure 1.5a). Because of the cubic symmetry of

SCAFI structure, the anisotropic component of the dipole-dipole

interaction between the nuclear magnetic moments vanishes. This

contradicts the experimental result of a very high antiferromagnetic

resonance frequency which requires a large dipolar anisotropy.

Therefore, the SCAFI structure, induced by folded 4-particle exchange

alone, is not adequate for describing the magnetic structure of the low-

field phase. For the planar 4-particle exchange Roger et al. found a

stable structure with minimum free energy, simple square antiferromag-

netic phase (SSQAF, Figure 1.5b). Although the SSQAF structure has a

large dipolar anisotropy, which produces two degenerate longitudinal




------- 2'




Figure 1.4 Compact exchanges in bcc lattice

(a) The most compact 3-particle cyclic exchange in a bcc lattice.

(b) The most compact 4-particle cyclic exchange in a bcc lattice.
(1-2-3-4) represents a folded exchange. (1-2'-3'-4') represents a planar



/ I I /

4- /1

Figure 1.5 SCAFJ and SSQAF structures
(a) Magnetic structure of simple cubic antiferromagnetic
phase (SCAFJ), which consists of two interpenetrating simple cubic
antiferromagnetic structures with perpendicular magnetizations.
(b) Magnetic structure of simple square antiferrromagnetic phase
(SSQAF). The lines [100] are ferromagnetic. Perpendicular to [100] there
is a planar structure with two interpenetrating simple square
antiferromagnetic lattices.

antiferromagnetic resonance modes and one transverse mode, the model

still cannot represent the observed phase. According to the theoretical

calculation of Roger et al. (1983), the transverse mode has resonance

frequency 1.4 MHz, much larger than the observed value of 825 kHz in the

experiment of Osheroff et al. (1980). The longitudinal modes will

separate into two modes in a magnetic field, giving a more complicated

spectrum than that observed by Osheroff et al. (1980).

Two-parameter model. Since none of the single parameter models can

explain the experimental data satisfactorily, it is natural to think of

multi-parameter models. The philosophy is to find the Hamiltonian with

the fewest parameters which could account for the experimental data.

The most successful model in the two-parameter model proposed by

Roger et al. (1980, 1983) includes 3-particle exchange Jt and planar 4-

particle exchange Kp. The Hamiltonian of this model can be written as

Hex J e (P + k P -1)
ijk ijk

K E (P ijk+ P (1.30)
lp ) jkl ijkl>

where the sum is taken over all distinct 3 or 4-particle exchanges.

According to symmetry property of the ground state, Jt and Kp both are

negative. The structure was solved by computer simulation. The ground

state is an up-up-down-down phase (u2d2) under the condition IJtl >

0.251Kpl, in agreement with the model proposed by Osheroff et al. (1980).

Roger et al. (1983) evaluated the values of Jt and Kp by using the

coefficient e2 and spin-wave velocity v, which can be calculated exactly

with 4-spin exchange. Compared with the experimental values of e2 and v,

the Jt was found to be -0.13 mK, and Kp -0.385 mK.

The other properties could be calculated from the set of Jt and Kp.

Table 1.1 lists a comparison between the results calculated by Roger et

al. (1983) and experimental values. They are in quite good agreement.

Table 1

Comparison of u2d2 model with experimental results

Experimental results Calculated results
Jt=-0.13mK, Kp=-0.385mK

Susceptibility c/5.9 < X < c/5.2 X = c/5.08
(Morii et al., 1978)

Zero-field resonance Q = 825 kHz 800 < Q0 S 860 kHz
frequency (8sheroff and Yu, 1980)

Mean spin-wave o = 8.4 0.4 cm/sec 8 cm/sec
velocity (Osheroff and Yu, 1980)

Figure 1.6 shows the phase diagram based on two-parameter model

(Jt' Kp). There is a first-order transition between the low-field phase

and the paramagnetic phase at low magnetic fields, and a first-order

transition between u2d2 phase and the high-field phase at about 1.6 T

below 1 mK. Also the phase diagram shows a second-order transition

between the high-field phase and the paramagnetic phase at high fields.

These features are qualitatively consistent with the experimental phase

diagram shown in Figure 1.2.

There is another first-order line ending at an intersection point

and extending upward. This is consistent with the results reported by

3- 15 HFP

1lo } i /

2" //

t7 T(mK)

I -


0 I

0 i I i
0 0.5 I 1.5 2

Figure 1.6 Phase diagram of two-parameter model (theoretical)

The phase diagram is based on two-parameter model of Roger et al.
(1983) with Jt = -0.13 mK and Kp = -0.385 mK. The solid lines are the
first-order tradition lines. The dashed line predicts a second-order
tradition. The dot-dashed line in the insert is also a second-order
tradition line predicted by Heisenberg nearest neighbor
antiferromagnetic model. The proposed magnetic structures for the low-
field phase and high-field phase are also shown in the figure.

Osheroff (1982), although it is not clear experimentally how and whether

this line ends. The theoretical calculation also predicts that the

second-order line bends over at very high magnetic fields of about 16 T.

No experiment has reached such high fields yet.

There are some other models calculated by Reger et al. (1983) such

as a three-parameter model including 3-particle exchange (Jt), both

planar and folded 4-particle exchanges (Kp, Kf); or a three-parameter

model including 2-particle (JNN), 3-particle (Jt) and planar 4-particle

(Kp) exchanges. These models do not fit the experimental results as well

as two-parameter model (Jt,Kp).

Physically, an exchange process depends on the free space available

without causing a drastic potential increase between particles. A

puzzling question is why different exchange processes could happen with

comparable rates at the same molar volume of solid 3He. Unless there is

an yet unknown mechanism which can yield many exchange processes at

comparable rates, it is unlikely that several different exchange

processes will happen in the same system with comparable rates. A

candidate theory to explain the different multiple exchanges having

comparable rates is "zero point vacancies" which is a finite

concentration of vacancies existing in the ground state. These vacancies

may play an essential role in the exchange mechanism (see discussion of

Cross and Fisher, 1985, for detail).

1.5 Purpose of This Work

Most 3-dimensional localized magnetic systems show only small

quantum effects that can be accounted for by the spin-wave perturbation

theory (Anderson, 1952; Kubo, 1952). Only solid 3He provides a highly

quantum 3-dimensional magnetic system, which we expect to understand

from first principle of quantum mechanics. In the past ten years

experimental and theoretical physicists made great efforts to build up a

picture of the magnetism of solid 3He both in experimental and

theoretical views and to explain this system in a consistent way. Up to

today there are still questions, uncertainties and disagreements in this

field. It will be always controversial until the physics about it is

completely understood and no longer attractive to physicists.

Although the multiple exchange models seem to account best for most

of the experimental data, there are still quantitative discrepancies

between the experimental phase diagram and the theoretical phase

diagram. For theorists, more accurate calculation beyond the mean field

approximation is needed to verify the correctness of the multiple

exchange models. In the experimental phase diagram the region around the

intersection of phase lines is quite ambiguous. The following are the

main discrepancies among the theories and experiments.

1. The multiple exchange models (2-parameter, 3-parameter) predict

a first-order phase line, extending upward and ending at an intersection

point. The theory also predicts a second-order phase line adjacent to

the previous one (first-order line) but bending over at high magnetic

fields (see Figure 1.6). None of the experiment performed before found

both of these two lines. Either a first-order phase line or a second-

order phase line was reported.

2. Experimentally, the transition between the paramagnetic phase

and the high-field phase has been reported as first order (Osheroff,

1982; Cross, 1982), second order (Prewitt and Goodkind, 1980; Kummer et

al., 1975; Adams et al., 1980), or A transition (Uhlig et al., 1984). It

raises a question: Is the high-field phase a totally polarized

paramagnetic phase or a new phase? Roger, Hetherington and Delrieu

suggested that the high-field phase was a canted normal antiferro-

magnetic phase (CNAF). This state has a broken symmetry transverse to

the magnetic field and is separated from the paramagnetic phase by a

second-order phase line.

The purpose of this work is to probe the region around the

intersection point of different phase boundaries in the phase diagram,

to confirm the order of the transition between the paramagnetic phase

and the high-field phase, and to determine whether the high-field phase

is an ordered phase or just paramagnetic ordering by the applied field.

We hope that the information from this work would remove the ambiguity.

A sensitive way of finding the entropy of solid 3He and then the

order of the transition is by using the melting pressure and the

Clausius-Clapeyron equation

dP s 1 (13
dT V V
s 1

where Ss, S1, Vs and VI are the solid and liquid entropies and molar

volumes, respectively. The method used in this experiment is to make

high-resolution measurement on the melting pressure at different applied

magnetic fields P(T,B) in the region 0.2 < B < 0.5 T, spanning the low-

and high-field phases, and in a temperature range going well below the

transition temperature. A specially designed sample cell serves for

pressure measurement. A pulsed NMR thermoneter measures the temperature

of the cell.



2.1 General Description

The magnetic ordering transition of solid 3He at melting pressure

happens at about 1 mK. A dilution refrigerator, combined with nuclear

demagnetization technique, is able to cool 3He at melting pressure down

to below 1 mK without difficulty.

In order to perform an experiment at such a low temperature several

general experimental principles have to be followed (see, for example,

Lounasmaa, 1974). In this experiment the cryostat and most of the

electronic instruments rest in a 40.5 m3 (5 m long x 3 m wide x 2.7 m

high) copper screen room which is grounded to the earth at one single

point. Low pass filters for 0-60 Hz, 120 V AC power line and high

frequency filters for eliminating the burst at 3150 Hz used for campus

clock synchronization are installed to give a "clean" power supply for

all the instruments inside the screen room. All vacuum pumps and the gas

handling board of dilution refrigerator are outside the screen room as

well as the computer and the data acquisition unit. Great care has been

taken to isolate the cryostat from vibration sources. For example, the

dewar is suspended from a triangular aluminium plate which is supported

at its three corners by pneumatic isolation mounts, model XL-A made by

NRC (Newport Research Corporation). Bellows are put in pumping lines

between the cryostat and pumps. These methods are proved to be very
effective for reducing the heat leak down to the order of 10 W.

Figure 2.1 shows the gross dimensions of the dewar and cryostat.

The dilution refrigerator, the PrNi5 bundle for nuclear demagnetization

and experimental volume are inside the vacuum can, as shown in Figure

2.2. The average consumption of liquid helium is about 22 liters per

day, including transfer losses.

A home-made superconducting magnet for the nuclear demagnetization

is made of NbTi matrix wire. The maximum field at the center of this

magnet can go up to 5 T. A commercial NbTi superconducting magnet, made

by NALORAC provides a magnetic field for NMR thermometer and 3He sample

with high homogeneity of 0.3 ppm at the experimental region.

2.2 Dilution Refrigerator

A dilution refrigerator has become standard equipment for

experimental research below 1 K since the principle of the dilution

refrigeration was suggested by London (1951).

A remarkbale property of 3He and He mixture was discovered by

Edwards et al. (1965). It is of most importance that the equilibrium

concentration of 3He in the dilute phase of 3He and 4He mixture happens

to be finite. Even at absolute zero temperature this concentration of

3He is still 6.4%. Below 0.5 K liquid 4He is effectively in its quantum

mechanical ground state due to its zero nuclear spin and superfluid

properties. There are practically no phonon or roton excited in liquid

4He. Therefore liquid 'He has no entropy or heat capacity for the

cooling process. In contrast to 4He, 3He atom with nuclear spin I=1/2

obeys Fermi-Dirac statistics. Landau's theory of Fermi liquid applies to

Current Lead


Vaccum Can Pumping Line

Superconducting Magnet
for Nuclear Demog.

30 cm

'Support Table

-IK Pot Pumping Line

3He Pumping Line

-Vaccum Con


Jucting Magnet
and 3He

Figure 2.1 Dewar and cryostat

liquid 3He in the concentrated phase (Wilkes, 1967). The entropy and

heat capacity of liquid, as a normal Fermi liquid, is proportional to

temperature T near absolute zero degree, providing a significant cooling

power in the temperature range which is of interest in dilution


The main part of the dilution refrigerator used in this work is a

commercial one, model DRP-43 made by former SHE company. The 3He-4He

solution used in the refrigerator contains 1.35 moles of 3He and 3.64

moles of 4He, equivalent to 50 cm3 of 3He and 100 cm3 of 4He. Basically,

all the 4He is contained in the dilution refrigerator unit. The 3He in

the dilution refregirator is about 1.1 moles at a condensing pressure of

100 torr. The rest of the 3He is circulating by the pumping system.

Figure 2.2 shows the dilution refrigerator unit. Incoming 3He gas

condenses into liquid at the 1 K pot, which keeps at about 1.3 K by

pumping He in the pot. The liquid 3He then comes into the still, which

cools the liquid 3He further down to 0.5-0.7 K. A heater inside the

still is designed for adjusting the 3He flow and cooling power. After

passing through a continuous heat exchanger and three discrete heat

exchangers, the liquid reaches mixing chamber, the coldest part in the

dilution refrigerator unit. The cooling process in the mixing chamber

happens at the phase boundary due to 3He atom transferring from 3He

concentrated phase into the dilute phase. The cooling power Q is

proportional to the 3He circulation rate n3 and the square of the mixing

chamber temperature (Q a 3 T2). Table 2.1 lists some operating

parameters of the dilution refrigerator used in this experiment. The

typical 3He circulation rate is about 0.5 mmole/sec. It takes 24 to 36

Figure 2.2 Experimental arrangement

A 1K pot

B still

C continuous heat exchanger

D discrete heat exchangers (three stages)

E mixing chamber

F Tin heat switch between the dilution refrigerator and
nuclear demagnetization stage

G vespel supporting rods

H thermal link (Cu) to the nuclear demagnetization stage

I superconducting magnet for the nuclear demagnetization

J PrNi5 nuclear demagnetization stage

K superconducting magnet for NMR thermometer and 3He sample

L thermal link (pure Ag) to 3He sample cell and strain gauge

M Pt pulsed NMR thermometer

N 3He sample cell

0 vacuum can

P strain gauge

Q thermal shield linked on mixing chamber

R thermal shield linked on still

10 cm

hours to precool the nuclear demagnetization stage (0.6 moles of PrNi5)

down to about 8 mK in a magnetic field of 4 T.

Table 2

Operating parameters of DRP-43

Heat load of the mixing chamber (mW) 0.15 0.4
Still power (mW) 20 20
3He circulation rate (mmole/sec) 0.75 0.83
Mixing chamber temperature (mK) 20 95

2.3 Nuclear Demagnetization Stage

Magnetic ordering of solid 3He at melting pressure occurs at about

1 mK. In order to perform the experiment at such a low temperature

cooling methods other than the dilution refrigerator are necessary.

Pomeranchuk cooling and nuclear demagnetization are the commonly used

techniques to reach the temperature range of millikelvin or sub-


The principle of nuclear demagnetization was proposed by Gorter

(1934). Because the nuclear magnetic moments are about 2000 times

smaller than the electronic magnetic moments, 2000 times larger values

of Bi/Ti are required in order to obtain the same entropy reduction in

the nuclear system than in the electronic system, where Bi stands for

the initial magnetic field and Ti stands for the precooling temperature.

Nuclear demagnetization became practical only because of the succeccful

development of the dilution refrigerator and superconducting magnet.

For a non-interacting nuclear dipole system in a magnetic field the

partition function is


m=+I nN0
Z = [ exp ( ngnmB/kBT)] (2.1)

where I is the nuclear spin, pn the Bohr nuclear magneton, gn the

nuclear Lande g-factor, m the magnetic quantum number, B the external

magnetic field, n the number of moles of the nuclear specimen, No

Avogadro's constant, kB Boltzmann constant.

The entropy of the nuclear dipole system, therefore, is

x x (2I+1)x (2I+1)x
S=nR coth(- coth ---2 +
21 21 21 21

+ ln [ sint(2- x / sinh ] ( 2.2)

where x -- pngnmB/kBT and R is the gas constant.

Under the condition of the adiabatic demagnetization B/T must be a

constant, which gives

T = T. (2.3)

where the subscript i refers to the initial state and f refers to the

final state.

Considering the internal dipolar interaction, Equation (2.3) should

be modified to

2 2
B + b
f B + 2 21 Ti (2.4)

where b represents the effective dipolar field (see Lounasmaa, 1974).

The heat capacity and the magnetic susceptibility can be written as

SA (B2 + b2) (25)
CB 22

X = (Curie's law) (2.6)

where u0 = free space permeability,
nNi(i+1 2 2
A = nNO ) n gn /3kB (2.7)

The method of hyperfine enhanced nuclear demagnetization is also

widely used for reaching millikelvin temperature (Altshuler, 1966;

Andres and Bucher, 1968). A large hyperfine field can be induced by

moderate external fields in singlet ground state ion with high Van Vleck

susceptibility. The enhancement factor is defined by a = 1+K where K is

the Knight shift. The range of the factor a is about from 10 to 200,

giving a large effective field. All the formulas of Equation (2.2) to

(2.7) are still effective with the replacement of B by aB.

The nuclear demagnetization stage in this experiment uses 0.6 moles

of PrNi5 as refrigerant; PrNi5 has Pr+5 ion which can produce a

hyperfine enhanced magnetic field under the interaction of an external

magnetic field. The factor a of PrNi5 is 16.4. The nuclear ordering

temperature of PrNi5 is near 0.4 mK, which is about the temperature

limitation by nuclear demagnetization of PrNi5 (Kubota et al., 1980). An

important difference between the hyperfine nuclear enhanced

demagnetization and the brute force nuclear demagnetization on copper is

the much shorter nuclear spin-lattice relaxation times T1 that are

encounted in hyperfine enhanced demagnetization materials; T1 is usually

so short (of order 10 us at 1 K) that it can not easily be observed

experimentally. So the electrons are always in local thermal equilibrium

with the rare earth nuclei.

It is worth mentioning that some materials other than copper, PrNI5

etc. are very promising for being nuclear refrigerants. For example,

indium is a very suitable material for the first demagnetization stage

in a two stage demagnetization oryostat because of its large Curie

constant, small Korringa constant, high filling factor for making the

nuclear demagnetization stage and its mechanical property for easy

handling (Tang et al., 1985).

2.4 Thermal Isolation

In order to stay at millikelvin temperatures for a long time to do

the experiment, it is particularly important to prevent heat flow into

the experimental region and to reduce the heat leak down to a level of


The section 2.1 has described several effective ways to cut down

the heat leak from the mechanical vibration and electromagnetic

radiation. The other principal source of the heat leak is from the

thermal conduction through supports or residual gas in the vacuum can.

It is a common practice to anchor heat shields, supports, tubes,

electric leads to intermediate cooling stages. In our cryostat two heat

shields are anchored to the still and the mixing chamber, respectively.

These heat shields are made of 0.08 cm thick oxygen free high conductive

copper (OFHC copper). The bottoms of the shields are aligned with each

other by a series of spacers made of brass ring and thin vespel pieces

as isolated parts. These two heat shields effectively absorb the

blackbody radiation from the 4 K helium bath.

Tubes reaching the low temperature region for 3He sample and

coaxial cables are made of copper-nickel which has poor thermal

conductivity and quite good electrical conductivity at low temperatures.

These tubes are anchored at all the different temperature stages from

1 K pot, still, first step heat exchanger to mixing chamber. Advance

wire (copper-nickel alloy) is used for the electrical leads. For coaxial

cables Apiezon N grease is introduced into the tubes to improve the

thermal contact between leads and tubes in order to reduce the heat leak

along the leads.

The residual exchange gas in the vacuum can is possibly a serious

source of the heat leak. It has been reported that the ortho-para

conversion of hydrogen molecules causes a time dependent heat leak up to

100 nW (see, for example, Berg, 1983). This heat leak will decrease to

an acceptable level after a couple of weeks from the start of cooldown.

In order to avoid this ortho-para conversion of hydroden we use

1000 ptorr He gas as the exchange gas, which is pumped out for about 6

hours at temperature of 10 K. After a couple of nuclear demagnetization

the heat leak stabilizes at 1 nW.

2.5 Gas Handling System

Figure 2.3 shows the gas handling system for the dilution

refrigerator. The general feature of the system could be referred to

Lounasmaa's book (1974). We only mention several special points for our


The filter F1 in the path of 9B3 boost pump is designed for

preventing oil vapor of 9B3 and other materials of the cracked oil from

backstreaming into the 3He circulation system. The filter is a

cylindrical container (9.5 cm in diameter, 15 cm high) full of molecular

sieves (Linde Adsorbents, Type 13X, 1/16 pellets). In order to prevent





- ag

o a


00 a

>, i^

00 4- CD CD 00

-'H 00 to 0 C) r 4-4 1
to > 0 T C 0 CC
C0 to fl *Ht bo to D
C5 0 'H1 C
2 .3 Do 00 C '-4
D D C CD 0 C C DO a,
C '-4 0 r Cl) 3
C" a a a co aL to
(C' -l E > E0 C CC (
(u o e n) o c

0. a. i 00 a O C S
23 DC CD 0.
to CC to ta o s x c
CC-a > (0 to 01 01 C 4-4
D a co a a a a ao o
0. 4-) 4-4 4-C 4-4 4-4 4-) 4> 4- a-

M r
"-4. . . . . .... ..a
0)i 01 a CO 0- CO F

oil cracking the heater power of 9B3 is reduced to 60% of the recom-

mended value. The liquid N2 cooled traps F3 and F4, located in the 3He

condensed line after the mechanical pump ED660, have a similar function

as filter Fl. Only one of the F3 and F4 is in use. The other is ready

for replacement when one is going to be blocked. Uhlig et al. (1983) has

described a special design of the traps F3 and F4. With the help of

these measures the cryostat has been operated properly for 10 months

without blocking.

Figure 2.4 is the diagram of 3He gas handling system. By using the

dipstick along with liquid He storage dewar, it is convenient to

introduce 3He from the tank at room temperature into the sample cell and

strain gauge in the cryostat. The dipstick is made of a stainless steel

tube (1.27 cm in diameter, 20 cm long) filled with charcoal. It can

easily raise the pressure in the cells up to 10 MPa.

2.6 3He Sample Cell and Strain Gauge

In the study of liquid and solid He a measurement of pressure can

provide very fundamental information about the sample. The partition

function Z containing the Hamiltonian of the microscopic system gives

all the thermodynamic quantities. The pressure is related to the

partition function Z by

P = kBT (lnZ/aV)T (2.8)

The capacitance strain gauge, proposed first by Straty and Adams

(1969), measures the sample pressure in situ. A sample cell and a strain

gauge used in this experiment are designed specifically to operate at



= t


1 mK temperature range and pressure along the melting curve of 3He with

a sensitivity AP 0.1 Pa or AP/P 3x108.

The 3He sample cell contains three main pieces as shown in Figure

2.5. The part B is a flexible diaphragm which under the pressure change

gives the different capacitance for measurement. The part A and B form a

sample chamber. The 3He is introduced through a filling capillary into

the chamber (not shown in the figure). An indium "0" ring seals the gap

between A and B very well at the melting pressure or even higher

pressure over 10 MPa.

The design of the diaphragm geometry is based on the formula given

by Straty and Adams (1969)

S15P 3/2
AP =2E ( _____ 1 C (2.9)
3a S C

S = 15a2 /4t2 (2.10)
y m

where a is the diaphragm radius, t the diaphragm thickness, Pm the

maximum pressure in the measurement, Sy the yield stress of the

material, E the modulus of elasticity of the material, 1 the distance

between two plates of the capacitor, AP the pressure sensitivity

required, AC/C the relative sensitivity of the capacitance bridge.

The sample cell sits at the center of magnetic field. The cell is

designed to meet a number of special requirements of measuring pressure

as a function of temperature and magnetic field with high precision. The

cell body is constructed of sterling silver. The gauge is made of

beryllium copper. Eight 0.76 mm diameter pure silver wires are welded to

the cell body to provide additional contact with the packed silver

A-bottom view


A 3




Figure 2.5 3He sample cell
A: heat exchanger, B: diaphragm and movable plate, C: fixed plate
holder, 1: holes, 2: Ag wires for thermal contact, 3: Ag packed
powder, 4: indium seal, 5: capacitor plates, 6: Pt NMR thermometer

powder. Another eight 1 mm holes drilled along the length of the silver

powder and a central 5 mm diameter hole give a great contact between the

liquid 3He and silver powder. The powder is packed from 3.44 g

700 A commercial silver manufactured by Vacuum Metallurgical Co. LTD.

(Japan) at 25 MPa and left unsintered. The filling factor after the

packing is 40.6%. The surface area of the power is 7.8 m 2. The open

space of the cell is 0.49 cm3 and the pore space of the packed silver is

0.84 cm3. All these means have greatly improved the thermal contact

between the demagnetization stage and helium sample in the cell.

A strain gauge located out of the magnetic field provides a

comparison with the melting pressure in the sample cell sitting at the

center of the field. The gauge is made of beryllium-copper. Only 0.61 g

silver powder are packed in the gauge. The gauge has been heat-treated

at 350 OC for 30 minutes to improve the mechanical stability and thermal

property. The surface area of the sintered powder is 0.55 m 2.

2.7 Pt Pulsed NMR Thermometer

For nuclear paramagnets, such as platinum or copper, their nuclear

susceptibility follows Curie's law

nuc T (2.11)
nue T

for the temperature higher than the ordering temperature, where nu is

the nuclear susceptibility, Cnuc is the Curie constant for the nuclear

spin system, and Tn is the nuclear spin temperature. By measuring the

nuclear susceptibility the temperature Tn can be determined through

Equation (2.11). Because the nuclear ordering temperature is of

microkelvin range, this sort of thermometry is very suitable for any low

temperature yet achieved in bulk matter.

There are several advantages of using platinum as the material of

the thermometer. Firstly, platinum has relatively long spin-spin

relaxation time -2 (1 ms) which simplifies the observation and recording

of the NMR signal. Secondly, the spin-lattice relaxation time T1 follows

Korringa's relation (1950)

1T = K (2.12)

where T is the conduction electron temperature, and K is the Korringa

constant. Because the Korringa constant of platinum is very small

(0.0296 sec*K, by Aalto et al., 1972), the T1 is very short even at 1 mK

temperature range. This results in rapid equilibrium between the nuclear

spin temperature Tn and conduction electron temperature Te. Thirdly,

there is only one magnetic isotope 195Pt, which removes the beat

structure of the NMR signal.

The pulsed NMR technique, shown in Figure 2.6, is used in this work

to measure the nuclear susceptibility of 195Pt. In a steady applied

field BO the nuclear magnetization N is directed parallel to BO; M may

be tipped through an angle 8 by applying a small field B1 perpendicrlar

to BO. The tipping angle depends on how long the pulse BI lasts. The

pulse could be chosen as a finite burst of sinusoidal field B1sinwt.

After the tipping pulse the nuclear magnetization processes around the

steady field B0 with Larmor frequency.


W0 = YB0



Figure 2.6 Principle of pulsed NMR thermometry

where Y is the gyromagnetic ratio of 195Pt. The processing magnetization

induces in a receiver coil an oscillating signal which dies away as the

nuclear spins return to thermal equilibrium. This free induction decay

(FID) is characterized by time T2 the spin-spin relaxation time.

The NMR thermometer must be constructed with finely devided

material because the rf field only penetrates a skin depth into a metal

sample. The skin depth can be written as

d = ( ) (2.14)

where p is the electrical conductivity of the material. v0 is the

magnetic permeability in vacuum, and v is the rf frequency. The brush is

made of platinum wire of 25 pm diameter, guaranteeing the sample is

totally penetrating by rf signal. The highest working frequency of our

NMR thermometer is 4.5 MHz, corresponding to a skin depth of 45 Pm for


The procedure for making Pt brush was as follows: (1) Use 200 Pt

wires of 25 um diameter (99.999% purity) to make a bundle. (2) Insert

the Pt bundle into the slit of the platinum block holder. (3) Weld the

bundle to the platinum block holder in an argon gas environment. (4)

Insulate the wires 4-5 times with Acrylic Spray Coating (Krylon No.

1303A) on the Pt wires of the bundle. (5) Bundle the Pt wires tightly

together to get a high filling factor. Cut the bundle end to give a

length of 1 cm.

The rf coil, made of 50 unm diameter copper wire, has two layers,

each of which is 60 turns. The total length of the coil is 5 mm, and the

diameter of the coil is 2 mm. After winding the first layer Stycast 1266

was used to glue it down, and allowed to dry. Then the second layer was

wound, and the Stycast was used to fix the position of the coil. The

inductance of the coil is 6.4 pH Finally, the Pt brush was placed into

the rf coil, and a slight amount of GE vanish was used to prevent

vibration. The NMR thermometer was screwed to the silver rod (thermal

link) near the 3He sample cell.

2.8 Superconducting Magnets

There are two superconducting magnets: one produces the magnetic

field for the nuclear demagnetization, the other produces the field on

the 3He sample cell and Pt NMR thermometer.

The magnet for the nuclear demagnetization is a home-made one,

built by Yokio Morii in 1982. NbTi superconducting wire "Supercon 279E-

9B1A" was used for the winding of the solenoid. The main solenoid is

15.24 cm long, 5.4 cm inner diameter. There are 32 layers, total 10416

turns. In order to obtain high homogeneity in the demagnetization

region, a pair of compensation coil were added at the ends of the main

solenoid. The field-current ratio of this magnet is 0.0790 T/Amp. The

maximum field is 5 Tesla.

The other superconducting magnet providing the field at the 3He

cell and NMR thermometer is a commercial one, made by Nalorac Cryogenic

Corporation. The maximum field is 4.7 Tesla. Four superconducting shims

are used to provide high field-homogeneity, which is typically 0.3 ppm

in a 10 mm diameter and 15 mm long sample region.

The main field for the nuclear demagnetization and the NMR field

should be put in opposite so that they will be in the same direction in

the reversed demagnetization region. Decreasing the demagnetization

field will cause decreasing the total field in the region where there is

lots of copper and produce a supplemental demagnetization effect.

Superconducting switches were installed on both magnets and all the

shims to facilitate extremely low drift operation. When the supercon-

ducting magnet is working under persistent mode, the trapped field is

found to decay with an initial time constant of 10 hours due to the

process of flux creep (File and Mills, 1963). The superconducting switch

can be thermally actuated by controlling the current in a heater which

is close to the switch.

2.9 Electronics for Pressure Measurement

The capacitance of the strain gauge and 3He cell are measured by

using two sets of capacitance bridge. Through the calibration of

capacitance versus pressure performed in advance, the 3He melting

pressure could be determined accurately.

Figure 2.7 shows the schematic diagram of the electronics for the

pressure measurement. A lock-in amplifier (PAR5204) serves as oscillator

and detector. It supplies a sine wave of 2.4 kHz, 8 Vpp to the bridge

through an isolation transformer (Model Gertsch ST-200C). The ratio arms

of the bridge are the transformer windings of AC Ratio Standard, Model

1011A, from Eaton Corporation. The AC Ratio Standard consists of seven

transformer windings and seven rotary switches. The ratio accuracy of

the standard is based upon the use of a toroidal autotransformer which

is not affected by age or environmental conditions. The resolution of the

Ratio Standard is 1x10-7. The Cp in Figure 2.7 is the capacitance of

the 3He cell. The Cref is a fixed reference capacitor which is made of

beryllium-copper and heat-treated at 350 OC in an H2 environment for 2

* .-0 a,
0I (


hours. The capacitance of the reference capacitor is about 50 pF,

comparable to that of the capacitance of the sample cell. The output

from the bridge is amplified by the lock-in and recorded continuously on

a chart recorder. Also, the output from the lock-in is recorded by the

HP3421A Data Acquisition Unit and stored in the HP9845B computer along

with the NMR signal every 10 to 30 minutes.

The other capacitance bridge for the strain gauge out of the

magnetic field is basically the same as that for 3He cell, except that

the reference voltage used is a sine wave of 3.5 kHz and 5.4 Vpp

2.10 Electronics for Pt Pulsed NMR Thermometer

Figure 2.8 is the schematic diagram for the Pt pulsed NMR

thermometer. The HP3325A synthersizer sends a continuous sine wave to an

electronic gate controlled by a burst pulse timer. The frequency of the

sine wave is set at the Larmor frequency of 195Pt at the magnetic field

applied on the thermometer and 3He cell. The burst pulse timer, made by

the electronic shop of the Physics Department, is triggered by the Sync.

output of the HP3325A. The burst pulse timer has two functions. One is

to provide a pulse, the width of which is

t = nT (2.15)

where n is the number of cycles of the sine wave, and T is the period.

The pulse opens and closes the electronic gate. The tipping angle 0 of

the magnetization M could be adjusted by setting different number of

cycles n. The other function of the burst pulse timer is to trigger the

Nicolet 204-A digital oscilloscope which stores the free induction decay

- -


1~--J----- -, -

I-------_ j

of the magnetization. The burst pulse timer can work in two modes,

automatically or manually.

The signal output from the electronic gate is a sine wave of n

cycles. The signal passes through a RF power amplifier, a suitable

attenuator and a tuning tank to the rf coil inside the cryostat. This rf

signal produces the small magnetic field BI perpendicular to the applied

field BO (see Figure 2.6). The L, C, and C2 compose a broad band tuning

circuit. The peak of the frequency characteristic curve of the tuning

circuit should be adjusted to the Larmor frequency corresponding to B0.

The L is the inductance of the rf coil, which is the transmitter coil as

well as the receiver coil. The C1 is a variable capacitance for

adjusting the resonance frequency to the Larmor frequency. The C2 is the

capacitance of the coaxial cable in the cryostat. The highest frequency

for this NMR thermometer is 4.5 MHz, which is limited by the existence

of C2. Two silicon diodes in the tuning tank compose a gate which allows

the excitation pulse to the resonance circuit and FID signal to the

preamplifier. A resistor of 470 kn is used only for tuning CI to obtain

the maximum output from the tuning tank at the Larmor frequency.

The FID signal, amplified by a low noise amplifier (16db) and a

multi-stage amplifier (45db), is stored in the memory of Nicolet 204-A.

The time integral of the FID is to be calculated by the HP9845B, and

then stored in a data file for analysis later.

2.11 Electronics for Superconducting Magnets

The superconducting magnet for the nuclear demagnetization and the

NMR magnet share a main power supply Kepco ATE6-100M. For most of the

time in the experiment the main power supply serves as the current

source for the nuclear demagnetization magnet. Only when the magnetic

field on the 3He cell and NMR thermometer needs to be changed, is the

main power supply switched to the NMR magnet.

Figure 2.9 shows the electronics for the superconducting magnets. A

resistor of 0.01 0 and a voltmeter are used to monitor the current

through the magnet. The magnetic field can be easily calculated by the

calibration done before.

The HP41-CV calculator serves as a digital data bus, which produces

digital data within a certain time. The Kepco programmer SN488-122 is

used as the interface between HP41-CV and ATE6-100M. The SN488-122

converts the digital data from the bus to an analog voltage output,

which serves as the control signal input for ATE6-100M. In response to

the control signal the ATE6-100M is programmed to the magnitude

initially commanded by the data bus controller.

Another supplementary power supply is available for the persistent

heat switch of the NMR magnet. Anytime when the NMR field needs to be

changed, the heater is energized by about 150 mA into its 5 0

resistance. The superconducting wire which is close to the heater

changes to the normal state. After a necessary adjustment for the field,

the heater is turned off, and the magnet is put back to the persistent


Appendix C gives the HP41-CV program for producing digital data.



C; 0
@3 6,
So) S





'00 t
4*c U
in0 C






3.1 Refrigeration

The techniques for refrigeration used in the experiment are

dilution refrigeration and nuclear demagnetization. An SHE DRP-43

dilution refrigerator runs continuously to keep the cryostat as cold as

about 5 mK. Unlike the dilution refrigeration, the nuclear demagneti-

zation is a one shot operation. The period of one cycle lasts about 2 to

3 weeks in this experiment.

The procedure for cooling down the cryostat from room temperature

(300 K) is as follows. Before the precooling by liquid N2, all the

traps, the 3He cell and strain gauge should be pumped out for 12 hours

or longer at room temperature. The electrical leads for all the

components, including the capacitance gauges, NMR coil, superconducting

magnets, heat switches, carbon resistors, and heaters have to be checked

out at room temperature. Leak checking of the vacuum can, 3He cell,

strain gauge and circulation system of the dilution refrigerator must be

done carefully. If everything goes normally, then it is ready to precool

to liquid N2 temperature (77 K). The vacuum can is filled with dry N2

gas to 1000 ltorr as exchange gas. The dewar used in the experiment is

superinsulated and has no liquid N2 jacket. The cryostat is precooled

with liquid N2 for 5 to 8 hours. After the innermost parts of the dewar

reach 77 K, the remaining liquid N2 should be transferred out. The

electrical test and leak test are repeated at the liquid N2 temperature.

Exchange gas 4He is then put into the vacuum can at a pressure of

1000 utorr. Liquid He is transferred slowly into the dewar at the

beginning in order to use 4He efficiently. When the temperature reaches

about 12 K, the transfer process should be interrupted for about 4 hours

for pumping the residue 4He exchange gas from the vacuum can. After that

the transfer is resumed until the dewar is full of liquid He.

Under normal operating conditions, the average consumption of

liquid He is 22 liters per day. Liquid He is transferred every three

days on schedule.

After the first time of transfer, liquid He in the 1 K pot is

pumped by Kinney pump (see Figure 2.3). The temperature of the 1 K pot

maintains at about 1.3 K. The 1 K pot fills automatically due to the

siphon effect.

The next step is to condense the 3He/4He mixture into refrigerator.

At first the gas storage tanks are opened to the backside of the

mechanical pump ED660. When a part of mixture gas goes into circulation,

and the pressure of the tanks drops down to about 200 mtorr, the mixture

gas may be pumped with the mechanical pump ED660 into the circulation

until all the gas is condensed into the refrigerator. This process takes

about 4 to 6 hours.

The 3He circulation rate is controlled by a heater in the still.

The heater, made of Karma wire with 0.076 mm diameter has a resistance

of 533 ohms at room temperature. Usually the still power is set at 1 mW,

giving 0.75 mmole/sec 3He circulation rate. This power could be

increased to 2 mW after magnetizing the demagnetization stage in order

to speed up the precooling process. It takes 2 days to cool the mixing

chamber and demagnetization stage down to 7 mK in a field of 4 T.

There are two different modes for the nuclear demagnetization. The

fast mode reduces the current in the magnet by 13 mA per step, and the

slow mode reduces the current by 1.3 mA per step. The time interval

between steps is determined by the operator. The nuclear demagnetization

is typically done in several different rate. To start with the

demagnetization a relatively fast rate is chosen. For example, 0.1

mT/sec is recommended. After the field is reduced from 4 T down to about

1.2 T, the rate is slowed down to about 0.05 mT/sec. With this rate the

liquid 3He in the cell goes through the superfluid transition. The

demagnetization is hold after the superfluid transition of liquid 3He

shows up on the chart recorder. It takes about an hour to reach thermal

equilibrium for the 3He in the cell. The NMR thermometer is calibrated

in the vicinity of 3He superfluid transition. As the temperature goes

lower, the thermal time constant gets longer. On the other hand, ramping

the field down would cause heating due to eddy currents in metal in the

field region. Because of these two reasons the demagnetization rate

should be reduced as the temperature is lowered. Except reducing the

demagnetization rate at lower temperature there are several pauses of 2

to 3 hours for each in the demagnetization process to ensure the melting

3He in good thermal equilibrium with the demagnetization stage. When the

temperature is close to the ordering transition of solid 3He, the

demagnetization rate is controlled at 5x10- mT/sec, 200 times slower

than the initial rate. After slowly cooling through the ordering

transition, the demagnetization is stopped. The cryostat warms up

because of the heat leak which is typically 1 nW. The warming rate

varies from 2.3 uK/hr to 5.8 pK/hr, depending on how large the field

left on the demagnetization stage. The cool down-warm up process usually

is repeated twice or even three times at the different warming rate to

see the effect on the pressure changing as a function of time. Each cool

down-warm up cycle takes a few days to a week.

When the cell is warmed up to about 10 mK which corresponds to

almost the maximum entropy of solid 3He, Rln2 per mole, the demagneti-

zation stage needs to be magnetized again.

3.2 Pressure Measurement

The calibration of capacitance for the 3He cell and the strain

gauge as a function of pressure has to be done at rather high

temperature before the 3He in the cell or gauge reaches the melting

curve. The suitable temperature to do so is around 1 K.

Figure 3.1 and Figure 3.2 are the results of the calibration for

the 3He cell and the strain gauge. The pressure is measured by a HEISE

pressure gauge which has an accuracy of 1x10 Pa. The capacitance is the

reading of the ratio transformer, rather than the actual capacitance

value of the cell or gauge. We fit the data in a parabolic function,

having a form

P = AO + A1c + A20 (3.1)

where AO, A1, A2 are the fit parameters, and c is the capacitance.

The parameter A0 is adjusted for fitting the A transition of

superfluid 3He. The capacitance at the A transition could be determined

very accurately on the chart. Having the capacitance corresponding to





u fo
\ c

imm N
CEl \

-m u

CD \ 44 C 1

e u

ck c
u -


n a-

N Lo Ln V, en u m 0
em m e e cn cen e m ru ru

(Jag) aJnssaJd



G) /-0

N m
01 a

S \


rv U

mL 0 0
m \ muc

+ m m m m

01 0
-Q \

m 3'
mU \ N

S e in\" 3

e en Le e e Cu Cu \O

(Jeg) aJflSS Jd

the A transition from the chart and the pressure at the transition based

on Greywall's measurement (1985), the constant term in Equation (3.1)

can be readjusted to be A0 under the assumption that parameters A, and

A2 remain unchanged. After this adjustment a new calibration of pressure

versus capacitance is used for determining the melting pressure,

P = AO'+ A c + Ac2 (3.2)

In a magnetic field the A transition of superfluid 3He splits into

two separate transitions (Gully et al., 1973). For example, the

separation of two transitions at 0.404 T in our experiment is 26 uK.

This splitting increases with the magnetic field. The adjustment for the

parameter AO is based on the center point between the two transitions.

The output of the lock-in amplifier for the 3He cell and strain

gauge are recorded on the chart recorder continuously and by the data

acquisition unit HP3421A, along with the NMR signal at certain time

intervals (see Figure 2.7). The output of the lock-in could be easily

converted into the capacitance using the settings of the lock-in and the

chart recorder. Then, by using the Equation (3.2) the melting pressure

can be determined.

3.3 Thermometry

3.3.1 Setting the Electronics

The Pt pulsed NMR thermometer used in this experiment is attached

on the silver rod near the 3He cell located at the center of the

magnetic field provided by Nalorac superconducting magnet. The ratio of

the resonant frequency to the magnetic field for Pt metal is 8.786

MHz/T. This ratio is different from the value of 9.153 MHz/T for Pt

nucleus because of the Knight shift.

Once the magnetic field on the 3He cell is decided, the frequency

of Pt NMR thermometer is also determined by the relation

v = 8.786 B0 (MHz) (3.3)

where B0 is the static magnetic field on the 3He cell. The capacitor C1

in the tuning tank is adjusted so that the peak of the tuning curve is

at the Pt resonant frequency under this field. The tuning procedure is

as follows (see Figure 2.6). A continuous sine wave from HP3325A is sent

directly to the tuning tank through a resistor of 470 kU. The response

to the different frequency describes a characteristic tuning curve, such

as shown in Figure 3.3. For a lower resonant frequency corresponding to

a lower magnetic field the tuning curve is flatter than that for a

higher resonant frequency. This is because a larger capacitance C1

causes more dissipation and reduces the Q value of the tuned circuit.

For example, the Q is equal to 19.4 for a field of 0.430 T, and is 16.5

for a field of 0.266 T.

The amplitude of the rf signal should be chosen carefully in order

to avoid saturation of the NMR signal at low temperature. For the

temperature range from 0.5 mK to 10 mK, which this experiment is most

interested in, the voltage of 800 to 900 mV (peak to peak) proved

suitable for the rf signal.

It is important to realize that the tipping pulse causes heating to

the nuclear spin system. Since in the Boltzmann distribution a reduction





iS S
S (S
(4 -.

s~iauri AjuJ;;qa) ^.ndin,

of magnetization from M to Mcose is equivalent to a temperature from T

to T/cose, the temperature change for a small tipping angle 6 is

AT T- T Te2 (3.4)
cose 2

Because the heat capacity of a nuclear dipole system Cn is proportional

to 1/T2, the heat introduced by the tipping pulse is inversely

proportional to the temperature as follows

AT C a 2/T (3.5)

A small tipping angle 6 should be used in order to reduce this heating.

The tipping angle is determined by the length and amplitude of the rf

pulse as follows

e = 2n B- (3.6)

where n is the number of cycles, BI is the amplitude of rf magnetic

field, BO is the applied external field. In our experiment n was 75 and

the peak to peak voltage was about 880 mV which corresponds to 4x10-5T

for 81. For the external field B0 from 0.266 T to 0.495 T covered in

this work, the tipping angle is from 2.70 to 5.

3.3.2 Calibration

The NMR thermometer is calibrated against the superfluid transition

of liquid 3He. According to Greywall's temperature scale (1985) this

transition happens at 2.708 mK and melting pressure 34.338 bar. Appendix

D is the program for calculating the temperature from melting pressure

based on Greywall's scale (Greywall, 1985).

In fact, the calibration is performed in the vicinity of the

superfluid transition of 3He. By measuring the temperature from the

melting pressure and the susceptibility from the NMR signal, the Curie

constant could be determined by

C T (3.7)

This calibration is repeated at several points near the A transition

point and the average Curie constant is taken for final use.

Because the field for the nuclear demagnetization and the NMR field

are not totally isolated from each other, the field change at the demag-

netization stage slightly shifts the NMR frequency. The ratio of the NMR

frequency shift to the field change at the demagnetization stage is

about -10 kHz/T. The total shift of the NMR frequency in the whole de-

magnetization process from 4 T to almost zero field is 40 kHz, appro-

ximately 1% of the NMR frequency. Whenever the field at the demagneti-

zation stage is changed, the rf frequency needs to be adjusted to match

the Larmor frequency. The NMR frequency could be adjusted by using

another rf signal to beat against the NMR signal or by sweeping the rf

frequency to find the maximum NMR signal.

Data are taken on both cool down and warm up. But the data from the

warm up is more reliable because the field at the demagnetization stage

and NMR frequency are not changing in the warm up process. The Curie

constant obtained at warm up is used for calculating the temperature.

3.4 Data Acquisition

The information provided in the experiment is recorded continuously

by a three-pen chart recorder and by a HP9845B computer digitally point

by point.

The chart recorder describes the pressure change in the 3He cell

and the strain gauge as a function of time. In the process of nuclear

demagnetization the chart recorder also records the current in the

magnet. The HP9845B computer collects the data from HP3421A data acqui-

sition unit and the Nicolet digital oscilloscope 204-A. There are two

modes for the data taking, automatic and manual. In the automatic mode

the burst pulse timer sends a pulse to trigger the NMR circuit at

intervals which are adjustable from a few minutes to an hour. This mode

is useful for monitoring the temperature and pressure change over a long

time period of a couple of days in a cool down or warm up process. In

the manual mode the burst pulse timer sends a pulse only at the

operator's request. This mode is often used in calibrating the NMR

thermometer. The FID signal is stored in the memory of Nicolet 204-A.

After the NMR circuit is trigger, the HP9845B will command the Nicolet

204-A to transfer the data of the FID signal to the computer and

calculate the time integral of the FID. Also the HP9845B asks the

HP3421A data acquisition unit to scan the lock-in outputs which could be

converted to the capacitance of the 3He cell or the strain gauge

according to the lock-in settings. All the information about the Pt

susceptibility, lock-in outputs, and the time provided by the clock

HP98035A is permanently stored onto a magnetic tape for future analysis

and printed out as a reference.

The data acquisition is controlled by a program "TEMP5A", which is

shown in Appendix E.



Data have been taken in magnetic fields of 0.266, 0.373, 0.390,

0.400, 0.402, 0.404, 0.410 and 0.495 T over a period of 10 months. For

the fields of 0.266, 0.373 and 0.495 T solid 3He has been cooled down to

about 0.5 mK, and then warmed up to 10 or 20 mK. Each cycle of cool down

and warm up took about one month. For the other fields we only concen-

trated on the region of the ordering transition.

4.1 Temperature Smoothing

Because the scatter in Pt susceptibility measurement is around

0.3%, the individual points are not used for determining the

temperature. Instead, the warming rate within a certain time is first

found by fitting the temperature-time relation. This relation is linear

in a short time period of one day or even longer. But, for the warm up

covering 2 or 3 days the best fit would be parabolic. The temperature

assigned to each datum is determined by the fit function T(t) based on

time t, which is recorded on a magnetic tape along with the other

information by the HP9845B computer. This procedure gives smooth

temperatures required for taking the derivative dP/dT from P(T). Figure

4.1a is a linear fit in a 40 hour period for the field 0.266 T. The

warming rate is approximate 2.9 pK/hr. Figure 4.1b shows a better fit by

using a parabolic function for the same field and time interval as








E- -o


(CW") I

N 1a

0>WU) I



rA C




L n




Figure 4.1a. For calculating the temperature within a long time period,

a parabolic fit between time and temperature should be used. For a short

time period about 1 day a linear fit is usually used to calculate the


4.2 Pressure as a Function of Time or Temperature

To obtain the pressure as a function of time or temperature is

quite straight forward. A data file contains the lock-in outputs and the

time when the data are taken. The capacitance of the 3He cell or strain

gauge could be converted from the lock-in output by using the

sensitivity AC/AV which is the ratio of capacitance change relative to

the lock-in output. Then the pressure corresponding to that capacitance

is calculated by Equation (3.2).

In a long time period the ratio transformer has to be adjusted in

order to trace all the pressure change on the chart recorder. It happens

sometimes that this adjustment of the ratio transformer could cause a

small discontinuity on the pressure-time curve. A constant shift for the

piece which does not match the whole curve usually gives a satisfactory

smooth curve of pressure versus time. This procedure is also necessary

for taking the derivative dP/dT from P(T).

From the pressure as a function of time P(t), obtained from the

above procedure and the fit of T(t) it is easy to convert the P(t) into

P(T), the pressure as a function of temperature. Because of the Kapitza

resistance between the 3He sample and the cell body, there is a

temperature discrepancy between 3He and the cell body (or the nuclear

demagnetization stage). The thermal time constant is about 1 hour around

1 mK. The warming rate is controlled at 2 to 6 pK/hr. The temperature of

3He thus lags about 2 to 6 pK behind the cell body. On the other hand

this temperature gradient will cause a small error for taking dP/dT in

the vicinity of the ordering temperature, which is of little consequence

here since we make no use of the temperature dependence of solid entropy

derived from dP/dT.

After the ordering transition happens, the temperature of the

nuclear demagnetization stage and 3He cell body keep the same warming

rate as before because the heat capacity of the nuclear demagnetization

stage is very large and the latent heat absorbed by solid 3He at the

transition is very small.

Figure 4.2 shows the pressure versus cell temperature at 0.373 T.

The pressure is measured relative to the pressure of A transition PA.

Figure 4.2a covers a wide temperature range up to 10 mK. The detail

around the ordering transition is shown in Figure 4.2b, where the dots

are the data points and the dashed line illustrates the behavior of P(T)

during the transition.

The results for other fields will be shown in the next chapter.

4.3 Entropy of Solid 3He

Based on the pressure as a function of temperature obtained from

the previous analysis the entropy of solid 3He can be calculated by

applying the Clausius-Clapeyron equation

S -S
dP s 1
SV (1.31)

From Equation (1.31)

n n

1 40


L ji

L- n Ln

I 0 1 f

/ D

m aL

ii .ri t
-I ----- I ---- I __________ l __________ G




cc /



1 I


"O 0

-4 C

Q3 0
n c

u l -

--4 C


0 I


a ,
^j a~L

C 3 l c
,- PH)-
tmu -uL-

) 0)-

0 a) .0
en -) -

01 4-.C0

.-1 ) aj

-0 4 4)'-
Ci 0C^
3 U r a

-4 C 3*
^ ^Ju
n ^' a u
cn *^ f.

clyQ ) a
3 ^1 -C
pj ; -^ L|4_

*i- E

(d 001) Wd-d

s (V V ) + S (4.1)
5 dT s 1 1

where (-V1) is -1.31 cm3/mole at temperature range 0-20 mK (Soribner

et al., 1969; Halperin et al., 1978). Liquid 3He has very small entropy

in its Fermi liquid regime (Wheatley, 1975; Varma et al., 1976)

S = YRT for T < T (4.2)
1 c

where Y 4.6 K-1 near the melting volume, TF is the Fermi temperature,

which is around 1 K depending on the pressure (Leggett, 1975; Wheatley,

1975). At about 2.7 mK liquid 3He undergoes a superfluid transition and

has an even lower entropy. For the purpose of calculating the entropy of

solid 3He, the liquid entropy below the superfluid transition can be


The derivative dP/dT is taken by

dP ( + N/2) = P(N + I) P(I) (4.3)
dT T(N + I) T(I)

where P(I), T(I) refer to the pressure and temperature of the Ith datum

point; N is an even integer. A suitable N is chosen based on the density

of the points. Usually N is 6 to 10, corresponding to a time interval of

about an hour. Before taking the derivative, the pressure values are

smoothed by a subroutine in a program "SPLOT" (see Appendix F). The

value of dP/dT by Equation (4.3) corresponds to the temperature of

[T(N+I) + T(I)]/2.


We have tried another way to find the derivative dP/dT. First, fit

N points by a polynomial, then take the derivative analytically. This

method does not improve the scatter of the entropy for solid 3He


The method based on Equation (4.3) has been used for calculating

the entropy of solid 3He at different magnetic fields. But it is not

applicable for the data points around the ordering transition because

the temperature of 3He is not in thermal equilibrium with the cell body.

Appendix F gives the program for calculating dP/dT and plotting the

entropy S(T).



5.1 General Features for Different Order Phase Transition

A first-order phase transition is defined by a discontinuity of the

first-order derivative of the Gibbs function at the phase transition.

For a second-order phase transition the first-order derivatives of the

Gibbs function remain unchanged, but the second-order derivatives of the

Gibbs function undergo finite changes. Figure 5.1 shows the main

features of the Gibbs function, entropy (the first-order derivative of

Gibbs function respect to temperature), and specific heat (proportional

to the second derivative of Gibbs function) for the first-order and the

second-order phase transition. The superfluid transition of liquid 3He

is an example of first-order transition. An example of second-order

transition is the transition from superconductor state to normal state

in zero magnetic field.

The most interesting higher-order phase transition is lambda

transition which is accounted for by the fact that the shape of specific

heat versus temperature curve resembles the Greek letter lambda (Figure

5.1c). The transition from ordinary liquid 4He (He I) to superfluid

liquid (He II) is a typical example among many lambda transitions.





Figure 5.1 Features of Gibbs function (G), entropy (S), and
specific heat (C) for different order phase transition
(a) first-order transition
(b) second-order transition
(c) X transition




1 \

5.2 Ordering Transition of Solid 3He at Melting Pressure

It is important to understand the dynamical behavior of P(t) as

distinct from P(T) in order to indicate the order of the various

transitions in this experiment. Because of the very large heat capacity

of the nuclear demagnetization stage and small latent heat absorbed by

solid 3He, the warming rate of the cell body is unaffected by the

entropy taken up by solid 3He in the phase transition. The pressure

measured by the 3He cell reflects the temperature at the liquid-solid

interface. Thus during a first-order transition there is a sloping

platean in P(t), as the interface warms up while the interior of the

solid undergoes the transition (Figure 5.2a). The shape of the plateau

region depends on the warming rate. A slow warm up has a flatter plateau

than that for a fast warm up. A second-order transition with a specific

heat discontinuity will produce a discontinuity in dP/dt although dP/dT

is continuous, implying that the entropy of solid 3He is continuous

(Figure 5.2b). A rapid change in dP/dt will occur in the case of

a A type transition (Figure 5.2c). This "enhancement" in the behavior of

dP/dt relative to dP/dT is well known from the observation of the

superfluid transition in liquid 3He.

Without analyzing the data the basic feature of solid 3He ordering

transition could be seen clearly on the chart.

Figure 4.2 shows the pressure versus cell temperature at a field of

0.373 T. Figure 4.2a covers the temperature range up to 10 mK. Figure

4.2b shows the detail around the transition. The plateau region reflects

the nature of first-order transition. The cell temperature is propor-

tional to the time scale. The point A marks the onset of the first-order

transition. Between points A and B a temperature gradient AT 10 iK

(a) Ist order

latent heat


(b) 2nd order






(c) X


T time

Figure 5.2 P(T) and dynamic behavior of P(t) for
different order phase transition

first-order transition
second-order transition
X transition

accumulates due to the latent heat absorbed by the solid 3He in the

phase transition. During this interval and a few hours afterwards the

3He and thermometer are far from equilibrium. The behavior of P(T) in

the vicinity of the transition is illustrated schematically by a dashed

line. Elsewhere we convert P(t) to P(T) by using the measured warming

rate dT/dt.

The other runs at magnetic fields of 0.266 T and 0.390 T have a

similar plateau region. Figure 5.3a is the pressure measurement covering

up to 10 mK for 0.266 T. Figure 5.3b and Figure 5.4 are the details of

the transition region for 0.266 T and 0.390 T.

Figure 5.5 shows the pressure versus cell temperature (or time) at

a field of 0.495 T. The detail around the transition is shown in Figure

5.5b where the point C marks the transition point. A plateau region of

P(t) has not been seen within the resolution of this experiment,

implying that the transition is not first order. The another run at the

magnetic field of 0.465 T shows similar behavior to that of 0.495 T

(Figure 5.6). For the HFP-PP transition, dP/dt always show the rapid

change characteristic of a A transition. This is particularly apparent

at fields not too near 0.400 T, so that the transition region is

broader. A similar appearance would be also caused by a second-order

transition with a temperature gradient. We conclude that the HFP-PP

transition is not first order for 0.4 < B < 0.495 T. However, we cannot

exclude a transition with a small entropy discontinuity, AS/Rln2 S 0.05,

which would produce an indiscenrnible plateau in P(t).

In the warm up at the magnetic field of 0.400 T there are two

transitions which display different features (Figure 5.7). The

transition at lower temperature shows a plateau although this plateau

lasts for a relatively short time compared with the plateau at low

field. The other transition happening at higher temperature has a

similar behavior to that at higher field of 0.495 T. This indicates that

the transition at lower temperature is a first-order transition, the

other one at higher temperature is not a first-order transition. Several

other runs at the fields 0.402, 0.404, 0.410 T also have the same

feature as 0.400 T. Figure 5.8 through Figure 5.10 show P(T) for these

three fields.

To summerize what we have seen, we understand

1. The transition is first order only when one of the phases is

LFP, for example, the transition between the LFP and PP at low fields or

the transition between LFP and HFP.

2. The HFP-PP transition is not first order for 0.4 < B < 0.495 T.

It is possibly a second-order transition or a A transition.

Upon taking the derivative dP/dT and substituting all the other

values for the quantities in the Clausius-Clapeyron equation as

discussed in section 4.3, we have found the entropy of solid 3He as a

function of temperature. Figure 5.11 shows the molar entropy of solid

3He for 0.266 T. The entropy increases abruptly at the ordering

temperature, and then increases gradually, finally reaches its maximum

value Rln2 at high temperature around 10 mK. The insert shows the detail

of the entropy change near the transition where there is an entropy

discontinuity AS = 0.35 Rln2. Figure 5.12 gives the result for 0.373 T.

It shows a similar feature as Figure 5.11 for 0.266 T, except the

ordering transition happens at a little lower temperature, and the

entropy discontinuity is about ASs = 0.26 Rln2. When the field goes

higher this entropy discontinuity at first-order transition gets

smaller. At 0.390 T, ASs = 0.23 Rln2. Osheroff and Yu (1980) reported

that AS = 0.44 Rln2 for the field 0.014 T. Our result and Osheroff's

result are qualitatively consistent. Table 3 lists the ordering

temperature and the entropy discontinuity of the first-order transition

below 0.400 T from our measurement.

Table 3

Ordering temperature and entropy discontinuity of
the first-order transition below 0.390 T

B (T) Ordering temperature (mK) AS /Rln2

0.014* 1.028 0.443
0.266 1.003 0.35
0.373 0.943 0.26
0.390 0.895 0.23

It is difficult to make an accurate entropy analysis for the case

which covers two transitions from 0.400 T to 0.410 T. Table 4 lists the

transition temperatures and the interval between these two transitions

for the field from 0.400 T to 0.410 T.

* From Osheroff and Yu (1980)

Table 4

Transition temperatures for fields 0.400 T 0.410 T

B (T) Tcl (nK) Tc2 (mK) AT (UK)

0.400 0.875 0.905 30
0.402 0.885 0.930 45
0.404 0.880 0.945 65
0.410 0.830 0.930 100

The interval between two transitions is from 30 to 100 uK. It takes

about 10 hours to one day to go through two transitions if the warming

rate is about 3 pK/hr. Because the 3He in the cell and the thermometer

are not in good thermal equilibrium when the first-order transition

happens and few hours later, the method described in section 4.3 for

calculating the entropy is not suitable for this situation.

Nevertheless, the nature of these two transitions is quite clear from

the trace of pressure as a function of time or cell temperature. An

approximate value of the entropy discontinuity of the first-order

transition at 0.400 T is 0.12 Rln2. This small entropy decrease is

because most of the entropy has already removed by ordering in the high-

field phase (see next section).

There is no entropy discontinuity for the fields of 0.465 T and

0.495 T. Figure 5.13 shows the entropy versus cell temperature at

0.495 T, the insert is the detail of the entropy change. The ordering

temperature T. = 1.03 mK. The entropy of solid 3He increases gradually

before the temperature reaches Tc. Once the temperature passes through

T the slope of the entropy curve decreases greatly. If we take the

derivative dSs/dT, we can find the specific heat for solid 3He by using



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