MAGNETIC ORDERING OF bce SOLID 3He
AT MELTING PRESSURE
BY
YIHUA TANG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987
To my wife, Yolanda
ACKNOWLEDGMENTS
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 ultralow
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
experiment.
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.
iii
Finally, I would like to thank my wife Yolanda for her under
standing and patience, as well as my parents and parentsinlaw for
their taking good care of my two sons.
This work was supported by the National Science Foundation under
Grant No. DMR8312959.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS............. .....................................iii
TABLE OF CONTENTS.................................................. v
LIST OF TABLES....................................................vii
LIST OF FIGURES..................................................viii
ABSTRACT.............................................................x
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 HighField Phase.....................105
5.5 Concluding Remarks .................................111
APPENDIX A RELATION BETWEEN SPINWAVE VELOCITY AND
PRESSURE MEASUREMENT.............................. 114
APPENDIX B MELTING PRESSURE OF A FREESPIN SYSTEM.............116
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
vi
LIST OF TABLES
Table Page
1. Comparison of u2d2 model with experimental results.........25
2. Operating parameters of DRP43.............................36
3. Ordering temperature and entropy discontinuity of the
firstorder transition below 0.390 T.......................87
4. Transition temperature for fields
from 0.400 T to 0.410 T .................................... 88
vii
LIST OF FIGURES
Figure Page
1.1 Pressure measurement in highfield 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 twoparameter 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
viii
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 freespin system and solid 3He...................109
5.18 Behavior of freespin 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
MAGNETIC ORDERING OF bcc SOLID 3He
AT MELTING PRESSURE
By
Yihua Tang
August, 1987
Chairman: Professor E. Dwight Adams
Major Department: Physics
From highresolution 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
ClausiusClapeyron equation. The entropy discontinuity was found at the
transitions between the paramagnetic phase (PP) and the lowfield phase
(LFP), as well as between the lowfield phase and the highfield 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 highfield phase, indicating that it was not first
order, but possibly a secondorder 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 firstorder transition line and PPHFP 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
highfield phase at B = 0.495 T, the spinwave 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 highfield phase was
magnetically ordered, rather than just a fully polarized paramagnetic
phase.
CHAPTER 1
INTRODUCTION
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 zeropoint
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 107 K. But the exchange interaction in solid 3He
causes the nuclear magnetic ordering at 103 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 107 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)
i
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
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
2k
splitting in mK per tesla. The pressure of solid 3He in a magnetic field
is
P F
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 CurieWeiss 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 hightemperature 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 CurieWeiss law as a firstorder approximation
O
0. 
0.64
0.5
C.3 1 x_< H= 60 kG
10 15 20 25 30 35 40 45
T [K']
Figure 1.1 Pressure measurement in highfield 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 CurieWeiss 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
threeparticle ring exchange. He also gave the form for fourspin ring
exchange.
Guyer and his collaborators (19691975) published a series of
papers to discuss the threespin and fourspin 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
ultralow 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 firstorder phase transition. This
result also demonstrated the failure of HNNA model which predicted that
a secondorder 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 lowfield ordering representing a magnetic phase
transition. They suggested that the ordered phase was probably an anti
ferromagnetic or spinflop 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 lowfield 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 secondorder phase transition between the
paramagnetic phase and the highfield phase. In the highfield phase the
magnetization approached to a saturated value which was much higher than
the magnetization in the lowfield phase.
Nothing about the magnetic structure of the highfield phase and
the lowfield 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
highfield 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 highfield phase was quite close to that predicted by Roger, Delrieu
and Hetherington as a "spontaneously spinflopped 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
secondorder phase transition was occurring at the highfield 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 lowfield phase, with a zerofield resonant frequency
near zero temperature of O/2w = 825 kHz. The large frequency shifts
implied a large anisotropy of dipole energy in the lowfield 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 nonzero resonance at zero field. This implied
2
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)
bliA
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
1
E = ED(n) + Y2 S S SB (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 WL 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 lowfield
phase was broken. Osheroff et al. (1980) proposed a possible magnetic
structure for the lowfield 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 spinwave
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.
15
*8  lll  i 7 
s
.2
01
LFP \\ PP
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: lowfield phase, HFP: highfield
phase.
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 dotdashed line
is the tradition line determined by Prewitt and Goodkind from the
measurements of static susceptibility. The possible magnetic structures
for lowfield phase and highfield 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 secondorder
transition.
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 (BealMonod, 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 zeropoint 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)
e
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)
eO
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 multiexchange 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 LennardJones 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 3particle and 4particle cyclic exchange in the following
analysis.
Threeparticle 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
1
P (1 + a )
ij 2 1 j
(1.20)
The 3particle exchange process is shown in Figure 1.4a. The
operator for 3particle 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
(1.22)
(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
(1.24)
The sum of Equations (1.23) and (1.24) gives the 3particle cyclic
exchange operator as
1
P + P
ijk ijk
= (1 + o1 j + j k + 5 ak + (1.25)
The Hamiltonian for 3particle 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 3particle exchange alone leads to the same
universality class as the pairwise interaction (i.e. Heisenberg
interaction). We would expect that 3particle exchange process may cause
only a secondorder phase transition the same as the Heisenberg nearest
neighbor interaction. There is no doubt about the existence of the
firstorder 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 lowfield phase.
Fourparticle cyclic exchange. The 4particle cyclic permutation
operator Pijkl can be written as
Pijkl Pijk Pi (1.27)
By using Equations (1.20) (1.25), the 4particle exchange operator can
be expressed as
1
P + P 1
ijkl ijkl
1 (0 + i j G (1.28)
pairs
where E is taken over the six distinct couples among the four
pairs
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 firstorder phase
transition.
Roger, Hetherington and Delrieu (1983) studied the 4particle
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 4particle exchange, the other is planar 4particle 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 dipoledipole
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 4particle exchange
alone, is not adequate for describing the magnetic structure of the low
field phase. For the planar 4particle 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
(a)
3
24
 2'
4'
3'
(b)
Figure 1.4 Compact exchanges in bcc lattice
(a) The most compact 3particle cyclic exchange in a bcc lattice.
(b) The most compact 4particle cyclic exchange in a bcc lattice.
(1234) represents a folded exchange. (12'3'4') represents a planar
exchange.
(a) SCAFI
/7
/ I I /
4 /1
/
/
(b) SSQAF
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).
Twoparameter model. Since none of the single parameter models can
explain the experimental data satisfactorily, it is natural to think of
multiparameter 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 twoparameter model proposed by
Roger et al. (1980, 1983) includes 3particle exchange Jt and planar 4
particle exchange Kp. The Hamiltonian of this model can be written as
T
Hex J e (P + k P 1)
ijk ijk
P
K E (P ijk+ P (1.30)
lp ) jkl ijkl>
where the sum is taken over all distinct 3 or 4particle 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 upupdowndown 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 spinwave velocity v, which can be calculated exactly
with 4spin 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)
Zerofield resonance Q = 825 kHz 800 < Q0 S 860 kHz
frequency (8sheroff and Yu, 1980)
Mean spinwave o = 8.4 0.4 cm/sec 8 cm/sec
velocity (Osheroff and Yu, 1980)
Figure 1.6 shows the phase diagram based on twoparameter model
(Jt' Kp). There is a firstorder transition between the lowfield phase
and the paramagnetic phase at low magnetic fields, and a firstorder
transition between u2d2 phase and the highfield phase at about 1.6 T
below 1 mK. Also the phase diagram shows a secondorder transition
between the highfield 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 firstorder line ending at an intersection point
and extending upward. This is consistent with the results reported by
3 15 HFP
010
1lo } i /
HEISENBERG MODEL
2" //
t7 T(mK)
ScNAF
LFP
I 
PARA
0 I
0 i I i
0 0.5 I 1.5 2
T(mK)
Figure 1.6 Phase diagram of twoparameter model (theoretical)
The phase diagram is based on twoparameter model of Roger et al.
(1983) with Jt = 0.13 mK and Kp = 0.385 mK. The solid lines are the
firstorder tradition lines. The dashed line predicts a secondorder
tradition. The dotdashed line in the insert is also a secondorder
tradition line predicted by Heisenberg nearest neighbor
antiferromagnetic model. The proposed magnetic structures for the low
field phase and highfield 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
secondorder 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 threeparameter model including 3particle exchange (Jt), both
planar and folded 4particle exchanges (Kp, Kf); or a threeparameter
model including 2particle (JNN), 3particle (Jt) and planar 4particle
(Kp) exchanges. These models do not fit the experimental results as well
as twoparameter 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 3dimensional localized magnetic systems show only small
quantum effects that can be accounted for by the spinwave perturbation
theory (Anderson, 1952; Kubo, 1952). Only solid 3He provides a highly
quantum 3dimensional 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 (2parameter, 3parameter) predict
a firstorder phase line, extending upward and ending at an intersection
point. The theory also predicts a secondorder phase line adjacent to
the previous one (firstorder 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 firstorder phase line or a second
order phase line was reported.
2. Experimentally, the transition between the paramagnetic phase
and the highfield 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 highfield phase a totally polarized
paramagnetic phase or a new phase? Roger, Hetherington and Delrieu
suggested that the highfield 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
secondorder 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 highfield phase, and to determine whether the highfield 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
ClausiusClapeyron 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
highresolution 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 highfield 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.
CHAPTER 2
APPARATUS
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 060 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 XLA made by
NRC (Newport Research Corporation). Bellows are put in pumping lines
between the cryostat and pumps. These methods are proved to be very
_Q
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 homemade 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 FermiDirac statistics. Landau's theory of Fermi liquid applies to
Vaporcooled
Current Lead
Baffles
Vaccum Can Pumping Line
Superconducting Magnet
for Nuclear Demog.
30 cm
'Support Table
IK Pot Pumping Line
3He Pumping Line
Vaccum Con
Sample
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
refrigeration.
The main part of the dilution refrigerator used in this work is a
commercial one, model DRP43 made by former SHE company. The 3He4He
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 0.50.7 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 DRP43
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
millikelvin.
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 noninteracting nuclear dipole system in a magnetic field the
partition function is
37
m=+I nN0
Z = [ exp ( ngnmB/kBT)] (2.1)
m=I
where I is the nuclear spin, pn the Bohr nuclear magneton, gn the
nuclear Lande gfactor, 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)
1
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)
1
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 spinlattice 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
nanowatts.
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 coppernickel 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 (coppernickel 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 orthopara
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 orthopara 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
cryostat.
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
L
0
44
CC
0.
C1
DO
0r
4
m
CD
0.
C
c
0
43
:3
.4
oC
 ag
o a
:3
DC
00 a
>44
>, i^
DO
00 4 CD CD 00
'H 00 to 0 C) r 44 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
CCa > (0 to 01 01 C 44
D a co a a a a ao o
0. 4) 44 4C 44 44 4) 4> 4 a
DOY
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
S
uo
= t
di
L.
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
Abottom view
6
\2
A 3
B
C5
cm
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 berylliumcopper. Only 0.61 g
silver powder are packed in the gauge. The gauge has been heattreated
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
Cnuc
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 spinspin
relaxation time 2 (1 ms) which simplifies the observation and recording
of the NMR signal. Secondly, the spinlattice 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.
(2.13)
W0 = YB0
Bo
MCose
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 spinspin 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
1/2
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
platinum.
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 45 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 homemade 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 fieldcurrent 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 fieldhomogeneity, 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 lockin 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 ST200C). 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 1x107. The Cp in Figure 2.7 is the capacitance of
the 3He cell. The Cref is a fixed reference capacitor which is made of
berylliumcopper and heattreated at 350 OC in an H2 environment for 2
* .0 a,
0I (
C
amoCO'
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 lockin and recorded continuously on
a chart recorder. Also, the output from the lockin 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 204A digital oscilloscope which stores the free induction decay
 
!
II
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
multistage amplifier (45db), is stored in the memory of Nicolet 204A.
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 ATE6100M. 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 HP41CV calculator serves as a digital data bus, which produces
digital data within a certain time. The Kepco programmer SN488122 is
used as the interface between HP41CV and ATE6100M. The SN488122
converts the digital data from the bus to an analog voltage output,
which serves as the control signal input for ATE6100M. In response to
the control signal the ATE6100M 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
mode.
Appendix C gives the HP41CV program for producing digital data.
58
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CHAPTER 3
EXPERIMENTAL TECHNIQUES
3.1 Refrigeration
The techniques for refrigeration used in the experiment are
dilution refrigeration and nuclear demagnetization. An SHE DRP43
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 downwarm 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
downwarm 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
63
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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 lockin 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 lockin could be easily
converted into the capacitance using the settings of the lockin 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
m
m
m
C4
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)
n
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
Bi
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 4x105T
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 threepen 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 204A. 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 204A.
After the NMR circuit is trigger, the HP9845B will command the Nicolet
204A 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 lockin outputs which could be
converted to the capacitance of the 3He cell or the strain gauge
according to the lockin settings. All the information about the Pt
susceptibility, lockin 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.
CHAPTER 4
DATA REDUCTION
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 temperaturetime 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
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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
temperature.
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 lockin outputs and the
time when the data are taken. The capacitance of the 3He cell or strain
gauge could be converted from the lockin output by using the
sensitivity AC/AV which is the ratio of capacitance change relative to
the lockin 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 pressuretime 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 ClausiusClapeyron equation
S S
dP s 1
SV (1.31)
From Equation (1.31)
01
n n
o
1 40
E
0
L ji
E
L n Ln
I 0 1 f
I II
L I
/ D
I
m aL
t.
ii .ri t
I  I  I __________ l __________ G
.4'.
'1'
/'
cc /
IO
en
r.
en
1 I
CD
*,
"O 0
00
CC
4 C
Q3 0
n c
.c
u l 
*J
C C
4 C
CDI
C)
0 I
WIH
me
C OL
a ,
^j a~L
C 3 l c
4c
, PH)
tmu uL
) 0)
0 a) .0
en ) 
Ciat...
01 4.C0
C'I
Cit.)
.1 ) aj
0 4 4)'
Ci 0C^
3 U r a
4 C 3*
^ ^Ju
n ^' a u
cn *^ f.
vJr CDQ
clyQ ) a
3 ^1 C
pj ; ^ L4_
*i E
(d 001) Wdd
dP
s (V V ) + S (4.1)
5 dT s 1 1
where (V1) is 1.31 cm3/mole at temperature range 020 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 K1 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
neglected.
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.
80
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
significantly.
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).
CHAPTER 5
RESULTS AND DISCUSSION
5.1 General Features for Different Order Phase Transition
A firstorder phase transition is defined by a discontinuity of the
firstorder derivative of the Gibbs function at the phase transition.
For a secondorder phase transition the firstorder derivatives of the
Gibbs function remain unchanged, but the secondorder derivatives of the
Gibbs function undergo finite changes. Figure 5.1 shows the main
features of the Gibbs function, entropy (the firstorder derivative of
Gibbs function respect to temperature), and specific heat (proportional
to the second derivative of Gibbs function) for the firstorder and the
secondorder phase transition. The superfluid transition of liquid 3He
is an example of firstorder transition. An example of secondorder
transition is the transition from superconductor state to normal state
in zero magnetic field.
The most interesting higherorder 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.
T
T
/
T
T
Figure 5.1 Features of Gibbs function (G), entropy (S), and
specific heat (C) for different order phase transition
(a) firstorder transition
(b) secondorder transition
(c) X transition
T
T
(a)
T
1 \
T
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 liquidsolid
interface. Thus during a firstorder 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 secondorder 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 firstorder transition. The cell temperature is propor
tional to the time scale. The point A marks the onset of the firstorder
transition. Between points A and B a temperature gradient AT 10 iK
(a) Ist order
latent heat
time
(b) 2nd order
\
P
T
Cdiscont.
P
time
(c) X
P P
T time
Figure 5.2 P(T) and dynamic behavior of P(t) for
different order phase transition
firstorder transition
secondorder 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 HFPPP 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 secondorder
transition with a temperature gradient. We conclude that the HFPPP
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 firstorder transition, the
other one at higher temperature is not a firstorder 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 HFPPP transition is not first order for 0.4 < B < 0.495 T.
It is possibly a secondorder transition or a A transition.
Upon taking the derivative dP/dT and substituting all the other
values for the quantities in the ClausiusClapeyron 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 firstorder 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 firstorder transition
below 0.400 T from our measurement.
Table 3
Ordering temperature and entropy discontinuity of
the firstorder transition below 0.390 T
B (T) Ordering temperature (mK) AS /Rln2
s
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 firstorder 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 firstorder
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
r'
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v
4i 
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en Z
m aM
Ct
Sn
ru E(
IVJ i^ ^
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0 0 0 9 0 0 c m 0
0s ln 0 n s in M
S "N N mn
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(2d 081) "dd
