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Magnetic susceptibility and pressure measurements in helium-three nano-clusters

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Magnetic susceptibility and pressure measurements in helium-three nano-clusters
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vii, 139 leaves : ill. ; 29 cm.

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Magnetic permeability ( jstor )
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Thesis (Ph.D.)--University of Florida, 2000.
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Includes bibliographical references (leaves 135-138).
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Printout.
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Vita.
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by Naoki Matsunaga.

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MAGNETIC SUSCEPTIBILITY AND PRESSURE MEASUREMENTS IN
HELIUM-THREE NANO-CLUSTERS











By

NAOKI MATSUNAGA
















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 2000














ACKNOWLEDGMENTS


I would like to express my most sincere gratitude and appreciation to my advisor, Dr. E. Dwight Adams, for his invaluable guidance and financial support during the past several years. His patience and understanding have made my graduate study less stressful.

I would like to thank Dr. Yasu Takano for his help when I was working in the Microkelvin Laboratory. He was always available whenever I needed to consult him.

I would like to thank Drs. G. G. Ihas, Chuck Hooper, and Alexander Angerhofer for serving on my supervisory committee.

Special thanks go to the people in the Microkelvin Laboratory, especially Drs. Jian-Sheng Xia and Volodya A. Shvarts. Volodya helped me in all the stages in my experiment.

Though Dr. Erwin Schuberth of Walther-Meissner Institut stayed here as a visitor to the low-temperature group only for several months in 1999, he spent much time with me to establish the data acquisition scheme in the work.

Many thanks go to the cryoengineers, Greg Labbe and Brian Lothrop; and to the machinists and the electronic engineers who provided outstanding technical support for this work.

Last but not least, I would like to thank my family in Japan for their support throughout my graduate study.

This work was supported in part by the National Science Foundation, Grant DMR-9800712.



ii















TABLE OF CONTENTS


ACKNOWLEDGMENTS ........................ ii

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

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

CHAPTERS

1 INTRODUCTION AND BACKGROUND . ............. . 1
1.1 Quantum Crystal ............................. 3
1.1.1 Nuclear Magnetism of bcc 3He .................... 3
1.1.2 Phase Diagrams of 3He and 4He .................. 16
1.2 Phase Separation in 3He-4He Mixture ................... 18
1.2.1 Regular Solution Theory .................. ..... 18
1.2.2 Calculation of Phase Separation Temperature vs. Pressure and
3He Concentration ........................ 22
1.3 Nuclear Magnetism of 3 He Nano-Clusters . ............... 23

2 EXPERIMENTAL APPARATUS .. ................. . 28
2.1 General Description ............................ 28
2.2 The Cryostat and Dilution Refrigerator ................. 29
2.2.1 Principle of Dilution Refrigeration ................. 29
2.2.2 The 1 K Pot ... ............. ......... ... 30
2.3 Magnetic Refrigerator .......................... 32
2.4 Superconducting Magnets ........................ 36
2.5 3He Melting Pressure Thermometry . .................. 38
2.5.1 The 3He Melting Pressure Thermometer Cell Design .... . 39
2.5.2 Electronics for 3He Melting Pressure Thermometry and Sample
Pressure Measurements . .................. ... 42
2.6 The Experimental Platform on the Nuclear Stage . .......... 47
2.7 The NMR Setup ............... ............ 49
2.8 Pressure Measurement Setup ................. ...... . 53
2.9 Method of Temperature Regulation . .................. 56

3 EXPERIMENTAL PROCEDURES . ................ 58
3.1 Cooldown Procedures ........................... 58
3.2 Calibration of the Mixture Sample Cell and Sample Preparation . . . 59


iii









3.3 Calibration and Preparation of the 3He Melting Pressure Thermometer 64 3.4 Sample Cooling .................... ....... .. 64
3.4.1 Technique and Setup ....................... 64
3.4.2 Phase Separation .................... ... 67
3.4.3 NM R Tuning ....................... .... 67
3.4.4 High Temperature Data Acquisition . .............. 68
3.4.5 Demagnetization and Magnetization of the Nuclear Stage . . . 68
3.5 Temperature Scale ............................ 69
3.6 Data Acquisition ............................. 70
3.6.1 Sample Pressure Measurements . ................ 70
3.6.2 Magnetic Susceptibility Measurements . ............ 74
3.6.3 Spin-Lattice Relaxation Time (Ti) Measurements ...... . 76 3.6.4 Spin-Spin Relaxation Time (T2) Measurements ........ . 76

4 EXPERIMENTAL RESULTS AND DISCUSSION . .......... 79
4.1 Sample Pressures of Interest . .................. .... 79
4.2 Experimental Results and Discussion . ................. 81
4.2.1 Spin-Lattice Relaxation Time (Ti) Measurements ...... . 81 4.2.2 Sample Pressure = 3.54 MPa . ................. 83
4.2.3 Sample Pressure = 3.35 MPa . ................. 88
4.2.4 Sample Pressure = 3.06 MPa .................. 95
4.2.5 Sample Pressure = 2.96 MPa . ................. 107
4.2.6 Sample Pressure = 2.88 MPa . ................. 120

5 MODEL FOR INTERPRETING RESULTS AND SUGGESTED FUTURE
WORK............ .................. 125
5.1 Model for Interpreting the Results and Discussion . .......... 125
5.1.1 Partial M elting .......................... 125
5.1.2 Anomalously Short T1's . .................. .. 127
5.1.3 Magnetic Susceptibility for the 3.54-MPa Sample ...... . 127
5.1.4 Kink in Magnetic Susceptibility and Frequency Shift .... . 128 5.1.5 Two-dimensional-like Magnetic Susceptibility . ........ 128
5.1.6 TN(P) for the Clusters and for Pure Bulk 3He ........ . 128
5.2 Suggestion for Future Work on 3He Nano-Clusters .......... . 129
5.2.1 Magnetic Susceptibility Measurements with a Cold Preamplifier 129 5.2.2 Temperature Scale Improvement . ................ 131
5.2.3 Pressure Measurements of 3He Nano-Clusters . ........ 132
5.2.4 Heat Capacity Measurements . ............. . . . . 133
5.2.5 Field Sweep to Look for a Frequency Shift . .......... 133
5.2.6 Magnetic and Pressure Study of 3He Nano-Clusters with a Larger
Concentration ........................... 133

REFERENCES ............ ................ 135



iv










BIOGRAPHICAL SKETCH ...................... 139






























































V














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 SUSCEPTIBILITY AND PRESSURE MEASUREMENTS IN HELIUM-THREE NANO-CLUSTERS

By

Naoki Matsunaga

December 2000


Chairman: E. Dwight Adams
Major Department: Physics


Simultaneous measurements of pressure, spin-lattice relaxation times (Ti), spinspin relaxation times (T2), and magnetic susceptibility (X) were made in 3He nanoclusters embedded in an hcp 4He matrix, following phase separation of the mixture.

Based on the pressure changes and magnetic susceptibility, three different types of behavior were identified, for which the clusters either remained all solid at the lowest temperature, underwent partial melting upon further cooling after phase separation, or separated with liquid already present. Magnetic measurements were extended from 0.5 mK to above 10 mK for these three different pressure ranges from 2.88 to 3.54 MPa.

In the temperature range of the measurements, where pure bulk 3He would be entirely liquid, the magnetic behavior of the clusters for P < 3.35 MPa indicated 3He solid fraction of about 77, 54, 23, and 19% respectively. The magnetic susceptibility X of the 3.35-MPa sample followed a Curie law to 0.5 mK with , = -5 � 5 AK. For the 3.06- and 2.96-MPa samples, a kink in X was observed at 1.1 mK, which vi









is approximately the magnetic ordering temperature TN that would be expected for pure bulk 3He if it existed at this pressure. However, X was almost constant down to 0.5 mK, with no drop at 1.1 mK, and a frequency shift of z 10 � 10 Hz, below 1.1 mK. Thus if there was magnetic ordering it appears to be quite different than for bulk 3He. For the 2.88-MPa sample, X followed a Curie-Weiss law with a Weiss temperature 9, = 140 pK, indicative of a ferromagnetic tendency, which is similar to that seen in 2D films.

Several spin-lattice relaxation times T were measured for each sample to estimate the necessary time interval between pulses. Anomalously short Zeeman-exchangelattice T's were observed that may indicate distortion of the cluster at the interface with the 4He.

Spin-spin relaxation times T2 were measured in several samples in order to look for possible changes in the spin-spin interactions. No perceptible change was observed in measurements above and below 1.1 mK.

























vii













CHAPTER 1
INTRODUCTION AND BACKGROUND


Magnetic properties of solid 3He resulting from exchange of various numbers of spins have been studied in a variety of situations including the bulk solid phases and two-dimensional (2D) layers. In this work, we have studied the magnetic properties of 3He nano-clusters in phase-separated 3He-4He mixtures confined in a silver sinter, which provides a different geometry for studying exchange processes.

Pure bulk 3He becomes magnetically ordered at temperatures < 1 mK via exchange interactions made possible because of the large zero-point motion of 3He. In nano-clusters, temperatures of possible magnetic ordering, reported by Schrenk et al., were around 1 mK, near that of pure bulk 3He at the melting pressure [1]. They observed a history-dependence in the ordering temperature at pressures below the melting pressure in pure bulk 3He (see Sect. 1.3 for details). Their specific-heat data showed no latent heat at a peak that is a characteristic of the first-order transition seen in pure bulk 3He by Greywall et al. (see Sect. 1.1.1) [2]. Furthermore, integrating their C/T from the lowest temperature to well above the peak, it was found that a substantial fraction of the total spin entropy was not removed, compared with the large entropy removal in the result of Greywall et al. Therefore the magnetic properties of the nano-clusters seemed to be different from those of pure bulk 3He.

Three main questions motivated this work:

1. Do the 3He nano-clusters magnetically order?

2. If so, what is the nature of the transition and the spin configuration?

3. What the exchange processes determine the transition?

1






2

The main task of the work was to look for magnetic ordering in the nano-clusters at pressures below pure bulk 3He melting and to characterize the nature of the ordering, if it exists. Pulsed NMR has been used for the investigation. The most important achievement would be to determine the NMR spectrum in the ordered phase in the nano-clusters, which would provide clear information on the type of ordering.

This dissertation was devoted to studying the magnetic properties of the nanoclusters by pulsed NMR. We have measured the magnetic susceptibility, spin-lattice relaxation times (TI), and spin-spin relaxation times (T2) versus temperatures of 3He nano-clusters embedded in a hcp matrix of 4He, following phase separation of the mixture. Also pressure changes in the sample were studied in this work. Pressure measurements are essential to monitor sample formation, annealing, phase separation, and partial melting, if it occurs. Additionally pressure measurements would give us another approach to study the nano-clusters and an idea of the kinetics of the isotopes in the geometry.

This thesis is arranged into five chapters. In the first chapter, quantum crystal; phase diagrams of pure bulk 3He, pure bulk 4He, and 3He-4He mixtures; the multiple-exchange model, a review of nuclear magnetism of pure bulk 3He; phase separation in 3He-4 He mixtures, calculations of phase separation temperatures; and nuclear magnetism of 3He nano-clusters are discussed, in this order. Chapter two is devoted to the experimental apparatus used in this work. Chapter three describes the experimental procedures, such as calibration of the experimental cells and tuning of the pulsed NMR. The experimental results and discussion are presented in chapter four. Chapter five is a summary of this experiment and future work, followed by an appendix, which discusses a cold preamplifier for pulsed NMR.






3

1.1 Quantum Crystal

In a quantum solid, atoms have a large zero-point motion around their equilibrium sites, moving a large fraction of the distance between neighboring atoms. This has several important consequences:

1. The large zero-point motion causes an anharmonicity such that the conventional

approach for harmonic crystal does not hold.

2. The atoms can change their positions by tunneling.

The last fact is particularly important since it implies that there is a finite overlap of the wavefunctions of neighboring atoms. This provides an atomic exchange process [3] that causes 3He nuclei to magnetically order near 1 mK at melting pressure. (See, for instance, Ref. [4].) For a classical solid the nuclear dipolar interactions would cause ordering at a temperature of - 10-7 K [5].


1.1.1 Nuclear Magnetism of bcc 3He

In this section, nuclear magnetism of solid 3He relevant to this work will be described starting from the multiple-spin-exchange model.


Multiple exchange model

Before the multiple-exchange model was proposed, the Hamiltonian for solid 3He was written with simple pairwise exchange as follows:


H = -2 E JSSj - iB, (1.1) i
where J, S, yp, and B are the quantum mechanical exchange energy, the spin operator, the magnetic moment operator, and the magnetic field, respectively. The first and






4

the second terms represent the exchange Hamiltonian of the two nearest neighbors and the Zeeman Hamiltonian, respectively.
To obtain thermodynamic quantities such as pressure, magnetization, and heat capacity, one must calculate the partition function, Z=tr[exp(-H/kBT)], where kB is the Boltzmann constant. For thermodynamic calculations [6], the free energy F is written as

F = --lnZ, (1.2) where # = 1/kBT. High-temperature series expansions of F for spin = 1/2 by Baker et al. [7] gives

NA 2 03 '1 2)2(l� 2 12
F=- n2+e2 -e3 + + (B)2(l+k+ O+...)+...
8 24 8 8 (1.3)
where NA is Avogadro's number and e2, e3 and ,,, etc. are related to the exchange energy J. For the Heisenberg nearest-neighbor model, e2=12J2, e3=12J3, and 0, = xJ/2kB, where x = number of nearest neighbors, which is 8 for the bcc and 12 for the hcp solid. From this high-temperature expansion of F, pressure, magnetic susceptibility, and specific heat are expressed as

R J 81n IJ 3
P,(T, B) = - lnIJI [3x- 3 2 + +y2(2+12x+52x2+. "')+y4(-1.33-23x+ .)], (1.4)
po (,2F) C
v aB2 T-,+A ...' (1.5) and in zero field
82F_ NAkB
C = -T - (e 2 2 - e33 + *.), (1.6)






5

where x=J/kBT, y=pB/kBT, v is the molar volume, o the permeability of free space, A=9,2-_a2/8kB2, and C = poNA (yh/2)2 /kBv. For bcc 3He, 0,=4J/kB, and A= 4(J/kB)2.

The experiment by Kirk and Adams in 1971 showed for the first time that the simple Heisenberg nearest-neighbor Hamiltonian was inadequate to describe the system [8]. They measured P,(T,B) for B up to 6 T and for temperatures from 20 mK to 100 mK to test the expansion for the Hamiltonian in Eq. 1.1. There was a clear discrepancy between the theory and the data, as seen in Fig. 1.1, which was caused by the y2 terms in Eq. 1.4 that overestimated the pressure change by a factor of two. In 1974 Halperin et al. [9] measured the latent heat along the melting curve of solid 3He using a Pomeranchuk cell and observed an abrupt drop in entropy at a 1.1 mK, indicating a first-order magnetic transition. Again the measurement suggested that the simple Heisenberg nearest-neighbor Hamiltonian, Eq. 1.1, was inadequate, because the Hamiltonian gave a second-order transition at 2 mK. In 1977, Kummer et al. carried out an experiment similar to the one performed by Halperin et al. but in a finite field up to 1.2 T and found that the ordering transition temperature decreased slightly with increasing field up to 0.4 T [10]. With higher fields, the behavior was reversed and an increase in field raised the ordering temperature, causing a kink in the phase diagram. The kink was an indication of a new ordered phase that is known today as the high-field phase (HFP).

Hetherington and Willard [11] proposed cyclic particle-exchange processes among multiple spins to explain the experimental data. Following Dirac [12], the multipleexchange Hamiltonian can be written as :


Hex = E JA(-1)PPn, (1.7) n,a






6













4.0
(10-3atm) J

V= 23.88cm3/mol
3.0


Ap
r H=6T




1.0
(6T)


20 30 40 50 T- (K- )



Figure 1.1: Pressure difference versus T-1 for v = 23.88 cm3/mole in fields up to 6 T. Various symbols for a given B are for different traversals of the temperature region. The solid curves are calculated behavior based on Eq. 1.4. W. P. Kirk and E.D. Adams, Phys.Rev.Lett., 27:393, 1971, ((1971) by American Physical Society.






7

where n is the number of particles in the exchange process and a represents a particular exchange process. Here are several processes for three-particle, four-particle, etc., exchange and P, is the permutation operator. For exchange of an odd number of particles, (-1)P is -1 (ferromagnetic); for an even number of particles, it is 1 (anti-ferromagnetic).


3



O 1=Jnn.

' 4 13=Jt
4=Kp








Figure 1.2: Various exchange processes taking place in bcc solid 3He.


Roger et al. [13] described a simple model to explain why four-particle exchange is important in bcc 3He. In Fig. 1.2 we see that in a (110) plane four atoms in a ring have eight nearest neighbors within the plane. If the space between the atoms expands a little, the four atoms in the ring can undergo a cyclic exchange. However, in two-particle and three-particle exchanges, in order to exchange positions, the atoms encounter the hard cores of neighbors. Following Thouless (1965) [14], Roger et al. wrote the Hamiltonian for up to four-particle exchange processes:


Hex = -Jnn E PGj + Jt E [~[Pik + (Pijk)-1] + Hfour-particle, (1.8) ij ij,k








F P
Hfour-particle = -KF E [Ptkl + (Pi;k)] - Kp E[Pjkl + (P) ] (1.9) ij,k,l ijkl
Here Pis", Pijk', and Pijkl~ are the cyclic permutation operators for two-, three-, and four-particle exchange, respectively, and J, Jt, Kp, and KF stand for nearestneighbor, three-particle, four-particle planar, and four-particle folded exchange processes, respectively (see Fig. 1.2).

For bcc 3He, Roger et al. found that three-particle (Jt) and planar four-particle

(Kp) exchange would give reasonable agreement with existing experimental data. Thus they reduced the exchange Hamiltonian Eq. 1.8 and obtained the Weiss temperature 0, and the coefficient e2 from high-temperature series expansion of F as follows:

0, = 18(Kp - 2Jt) (1.10) kB

and
7 17
e2 = (24)2(J2 - + 64 K ). (1.11) Equation 1.10 shows the competition of three- (ferromagnetic) and four- (antiferromagnetic) particle exchanges. For bcc 3He at the melting density, a fit of the experimental results of the high-temperature expansion coefficient e2 and the mean spin-wave velocity give Jt=-0.13 mK and Kp=-0.385 mK, which yield 9,=-2.25 mK and the ordering temperature (TN) of 1.2 mK [13], [15], [16]. On the presently accepted melting pressure scale, TN has the value of 0.934 mK [17]. A significance of this model is that it gives the U2D2 structure in the ordered low-field phase (see subsequent sections for details), a canted structure for the high-field phase, and a first-order transition between the paramagnetic phase and the high-field phase in the low-field region.

Stipdonk and Hetherington [18] have used three exchange processes J,,, Jt, and K, and obtained a slightly better fit to experimental data.






9

0.8


0.6 0
High-field phase ,

0.4
\ Paramagnetic
phase
0.2
Low-field phase
0.0
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 T (mK)


Figure 1.3: B-T diagram of solid 3He at melting pressure density. From Ref. [19]. Magnetization and heat capacity of bcc 3He

In this section, we present relevant magnetization and heat capacity data in bcc 3He to be compared with the results of this work.

Hata et al. [20] measured the magnetization through TN over a wide range of molar volumes. They formed a sample inside an experimental cell with silver sinter and measured static magnetization with a field of 26 mT using a SQUID. They observed a jump in (inverse) magnetization at TN, as shown in Fig. 1.4, which indicated that the transition was first-order. For each sample, the magnetization started to deviate from a Curie-Weiss law at T s 5TN and dropped to 39% of its maximum magnetization (Mm) at TN, and then stayed constant to the lowest temperature.

Greywall et al. measured the heat capacity from 0.6 mK to 10 mK in fields 0 < B < 1 T [2]. For zero field they found a very sharp peak, with C, changing by nearly two orders of magnitude in a 100 pK interval, as shown in Fig. 1.5. The peak is clear evidence of a latent heat and, hence, a first-order transition. In the result of Greywall et al. integration of their heat capacity showed that the entropy removed







10













4- 24.14 cm3/nmole

..*...
' 3.85



.* * 23.81








. 23.01
..-" .63













0 2 4 6 8 10 TEMPERATURE (mK ) Figure 1.4: Inverse magnetization as a function of temperature for several molar volumes. The deviation from Curie-Weiss law begins at T , 5TN and is shown by the arrows. From Ref. [20]. ()(1988) by Kluwer Academic/Plenum Publishers.








10- - --.-.-.


A
1B Sg



0.1
/


0.001 /
0.1 0.5 I 5 10 T(mK)


Figure 1.5: The heat capacity of bcc 3He at a molar volume of 24.13 cm3/mole near the PP-LFP transition TN. A: zero field. B: 1.0 T. Note there is a latent heat associated with a first-order transition. Compare A to Fig. 1.13. From Ref. [2]. @(1988) Amrican Physical Society. was 0.4 R In2 just in going through the transition, which is to be compared to the entropy removal in the result of Schrenk et al. on 3He nano-clusters (see Sect. 1.3 for details).


Properties of the low field phase (LFP) in bcec solid 3He

Since the magnetic field in this work is less than 0.4 T (see Fig. 1.3), the physical properties of the 3He nano-clusters are to be compared with those in the low field phase (LFP) in bcc solid 3He.

The nature of the LFP was studied by NMR measurements by the Florida group and Osheroff et al. [21], [22]. Both groups observed a very large frequency shift of
- 1 MHz, which indicated that the ordered phase does not have cubic symmetry. Osheroff et al. determined the phase to be consistent with U2D2 as shown in Fig.

1.6.






12













a















A
U2D2 d



Figure 1.6: Spin configuration in the LFP. From Ref. [23]. @(1992) by Kluwer Academic/Plenum Publishers.






13

The U2D2 magnetic structure

In the U2D2 phase, shown in Fig. 1.6, there are two anisotropy axes. The realspace unit vector, 1, which is normal to the ferromagnetic plane, and the spin unit vector, d, to which the sublattice magnetization is either parallel or anti-parallel at zero field. In a finite magnetic field H, d rotates within the plane until it is normal to H. There are usually three domains in a U2D2 single crystal. Because of the low symmetry, the dipole-dipole energy is highly anisotropic, which causes a large frequency shift. For each domain, there are two modes whose frequencies, w+ and w-, are given by the following equations:


()2 [02 + Q02 � _ ( 2)2 + 40o20o2cs20,], (1.12)


where wo = 7H/27r is the Larmor frequency with 7=32.43 MHz/T, Q0 is the antiferromagnetic resonance frequency, which depends upon temperature, and cos0i=11.with h, an unit vector along the magnetic field [21].

Osheroff et al. fitted their data and found that


E cos20, = 1.007 (1.13) i=domain

(see Fig. 1.7), which satisfies the orthnormal condition that implies that the three magnetic domains are orthogonal to each other. Also they determined the frequency 0o over the entire temperature range from T o 0.1 mK to TN as follows:


2 2 = [6.839 - 3.663( ) + 1.452(T ) -1.882( ) (1.14)
1011(Hz) TN TN T

The resonance frequency varies from 525 kHz at TN to 825 kHz at T - 0, which is to be compared to the results in this work later.






14


" 1500 .

o 0.0271
z - a 0.976
,W A 0.0039




500
t





o 0 500 1000 1500 LARMOR FREQUENCY (kHz)


Figure 1.7: Observation of the field dependence of the resonance frequencies w� of NMR spectra for three domains of a single crystal of bcc 3He at 0.487 mK. Solid lines are obtained from Eq. 1.12 with Go=777.7 kHz. From Ref. [23]. @(1992) by Kluwer Academic/Plenum Publishers.

2D films, two-dimensional quantum solids

In this subsection, 2D films are mentioned, to be compared to the results of this work later. In some situations, depending upon the pressure, the 3He at the interface of the nano-clusters may be solid whereas the rest of the cluster remains liquid. In this situation, the clusters would have several solid layers at the interface, which in turn might cause magnetic properties similar to those of 2D films.

Solid 3He adsorbed on a substrate, typically graphite, provides an example of low dimensional magnetism. The second atomic layer of adsorbed 3He shows an interesting evolution from antiferromagnetism to ferromagnetism as a function of its areal density x. Presently, multiple-spin exchange interactions are used to interpret the behavior [13].






15

At low densities, there is competition between odd and even number of exchange processes of 3He atoms in a triangular lattice, producing a highly frustrated magnetic system. On the other hand, at higher areal densities, 3He films exhibit a pure 2D Heisenberg ferromagnetic behavior because of the dominance of three-particle exchange processes [24]. However, no magnetic ordering has been observed in the 2D films.


1.6


1.4




1.0 - a, oan a
1.0


0.8

10 100
T (minK)
Figure 1.8: Magnetic susceptibility of 3He absorbed on graphite for several coverage in the second layer intermediate regime. Dots correspond to the commensurate 4/7 phase at x = 1.62, and diamond to a denser phase (12/19, x = 1.71) where x is the coverage. The lines are fits using a high-temperature series expansion of the multiple-spin-exchange model. Open squares: x = 1.66 for which the fit is obtained by assuming that the 4/7 and 12/19 phases are coexisting. See Ref. [24] for details. @(1998) by Kluwer Academic/Plenum Publishers.


One of the samples studied in this work exhibited behavior similar to that for x = 1.66 in Fig. 1.8 (x=1 corresponds to the areal density of the densest(saturated) first layer of 3He absorbed on a substrate). Our results and a discussion relating to this are given in Chap. 4.






16

Ishida et al. [25] recently observed a peak in their specific heat in 2D 3He on graphite at around 300 tK, which accounts for the missing entropy of the broad maximum near 3 mK observed previously by Greywall and Busch [26]. This may be related to the "missing entropy" in the specific heat of nano-clusters of Schrenk et al. [1].


1.1.2 Phase Diagrams of 3He and 4He

In this section, phase diagrams of pure bulk 3He, pure bulk 4He, and 3He-4He mixtures are presented in order to show the crystallographic structure and to discuss the effect of adding a small concentration of 3He to 4He.


10
Pure bulk 'He melting pressure curve

8
7 U'
.L bcc He



hcp'He
2
Pure bulk 'He melting pressure curve
1 I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
T (K)



Figure 1.9: P-T diagrams for pure bulk 3He and 4He.

Both pure bulk solid 3He and 4He exist in three phases with different crystal structures, namely hexagonal-closed packed (hcp), faced-centered cubic (fcc), and body-centered cubic (bcc). As a result of the zero-point motion, 3He does not solidify until the pressure is about 2.93 MPa. The crystal structure is bcc for low-pressure solid, and hcp for high-pressure solid (hcp and fcc phases are not shown in Fig. 1.9)






17

3.4
S-- x0.05
3.2 - -- x
hcp i
S3.0

2.8 /

2.6 'H

2.4 2.2
0.2 0.6 1.0 1.4 1.8 Temperature (K)


Figure 1.10: Equilibrium P-T phase diagrams for pure 4He and for 4He with 5% 3He concentration (x=0 and 0.05, respectively). The dashed (solid) lines show the pure 4He (4He with 5% of 3He) melting line, A line between normal (I) and superfluid (II), and bcc-hcp boundary. From Ref. [27]. @(1987) American Physical Society. [28]. In 4He there is a small region of bcc phase in the P-T diagram, as shown in Fig. 1.9, and the hcp structure occupies a much larger area. The molar volume of the pure bulk 3He and 4He are quite different, a 24 cm3/mol for 3He, whereas it is - 21 cm3/mol for 4He, near their melting pressures in the temperature range in this work. Thus, this difference in molar volume could have a significant effect on the magnetic properties of the phase-separated 3He nano-clusters, because of the mismatch in molar volume between the nano-clusters and the 4He matrix at the interface.

In 3He-4He mixtures, bcc 4He occupies a larger region in the P-T diagram than in pure bulk 4He, as shown in Fig. 1.10 for a 3He concentration of 5%. As the mixtures are cooled through the bcc-hcp transformation line, the transformation takes place more slowly than in pure 4He since the enhancement of the bcc region shifts the transformation line to lower temperatures. Thus, if the mixtures are cooled too fast for the transformation to take place, the bcc phase could persist as a meta-stable state. As discussed in Chap. 4, meta-stable bcc 4He may have persisted in some






18

low-pressure samples studied in this work, as the samples were cooled after they were annealed.


1.2 Phase Separation in 3He-4He Mixture


1.2.1 Regular Solution Theory

In 1962 Edwards et al. discovered a A-type ordering in the heat capacity of a sample of solid 3He with a 4He molar fraction of 0.277 [29]. They showed that the entropy of the A anomaly, AS=f (C,/T)dT, was the classical entropy of mixing of a regular solution of 4He in 3He [30],


ASm = -R[(1 - x) In (1 - x) + x In x], (1.15)


where x is the molar fraction of 3He or 4He and m stands for mixing. They concluded that the anomaly was the onset of a separation into two phases at a temperature Tp that depended on x.

Edwards et al. extended their measurements over a wide range of concentrations and compared them with the regular solution theory. The regular solution model connects the Gibbs free energy of individual isotopes to that of the mixture via


g(P, T, x) = (1 - x)g4(P, T) + xg3(P, T) - TSm(x) + gE(P, T, x), (1.16)


where x is the concentration of 3He, g4, 93, and g are the Gibbs free energy of 3He, 4He, and mixture, respectively, and gE is the excess free energy per atom. In a regular solution gE has the simple form


gE = Ax(1 - x), (1.17)






19

where A is a function of pressure only [31].

Solving Eqs. 1.15-1.17 below a critical temperature, Te=A/2kB, with the assumption that there is no structural differences between the constituents, one can find the phase separation temperature and the heat capacity, respectively, as follows:


kBT = A 2x) (1.18) In(x-1 - 1)

and

C/k - (1 - 2x)2 (1.19) 4x(1-z)

where x is a function of T given by Eq. 1.18 and T=T/Tc=2kBT/A. Equation 1.18 determines the temperature of phase separation, Tps(x).

The specific heat of the mixture, given by Eq. 1.19, follows a universal curve, as seen in Fig. 1.11, independent of x. These arguments are based on the assumptions that the lattice and nuclear-spin heat capacities are small. In other words, g4 is a function of P only and g3(P,T)=g3(P,O)-kBT ln2. Equations 1.18 and 1.19 fitted the heat capacity and phase separation temperature of Edwards et al. quite closely, which gave A/kB= 0.76 K at 3.6158 MPa [29].

The critical temperature A/2kB and its pressure dependence were investigated by Panczyk et al. [32], and then explained by several theorists [33], [34], [35], [36], [37], [38], [39]. There were two main assumptions in those works as follows. The first was that the isotopes are randomly arranged in the crystal lattice, which is in agreement with classical mixing entropy. The second was that the energy change upon mixing the pure isotopes is mostly the work done in compressing the 3He and expanding 4He, so as to bring the two crystals to the same lattice spacing. These assumptions neglected structural differences in the pure isotopes.






20

As described above, the phase separation line is symmetric about x=1/2. However, Panczyk et al. observed an asymmetry in their experiments on constant-volume pressure measurements (see points in Fig. 1.11) [32]. They measured the isochoric pressure P,(T), and obtained the excess pressure PE(x) directly at Tps(x), where

8gE dT
PE - )T = -2Rx(1 - x)dT (1.20) av dV'

In other words, they measured P,(T) to determine the phase separation curve. The phase separation temperatures obtained by Panczyk et al. indicated that there was an asymmetry when x < 0.3. Mullin took the asymmetry into account and modified the regular solution theory as follows [33]:


gE = Ax(x - 1)(1 + ex), (1.21)


where E - -0.1, according to Mullin's calculation and the measurement of Panczyk et al. With the asymmetry included, the equations for the phase separation curve, the pressure at constant volume and the specific heat, which were derived from the equality of the chemical potentials of 3He and 4He, P3 and pL4, are complicated and had to be solved numerically. The data of Edwards et al. gave e=0�0.006.

Ehrlich and Simmons, using x-ray measurements of Tp, for a series of bcc single crystals with concentrations from x = 0.1 to 0.7, found E=0�0.01 [40]. The asymmetry observed by Panczyk et al. has been attributed to a crystallographic transformation of solid 4He from bcc phase to hcp phase in 4He crystal, which took place slowly in their experiment (see Fig. 1.11 for the points measured by Panczyk et al.).






21








0.6

T (K)

h + b2





0.2 / h+b2

34tm



0 20 40 60 80 100
x(o)


Figure 1.11: The equilibrium phase diagram for hcp and bcc helium at 34 atm, assuming that hcp and bcc mixtures are regular. The hcp, bccz, and bcc2 phases are labeled h, bz, and b2 (1, 2, and hcp stand for 4He-enriched phase, 3He-enriched phase, and hcp 4He, respectively). The bl-b2 equilibrium line is given by Eq. 1.18. It is extended into the h+b2 region for comparison with the h-b2 equilibrium line. The horizontal line represents the temperature at which h, bl, and b2 may be in equilibrium. The points are from (32]. Although the points agree with the regular solution theory well, the measurements did not achieve equilibrium with respect to the bcc-hcp transformation in 4He. Thus, the meta-stable 4He might have persisted. D. O. Edwards and S. Balibar, Phys. Rev. B., 39:4083, 1989. @(1989) by American Physical Society.






22

1.2.2 Calculation of Phase Separation Temperature vs. Pressure and 3He Concentration

In the last section, the expression for the phase separation temperature based on the regular solution (binary solution) was obtained. This allows us to calculate the phase separation temperature as a function of pressure and the initial 3He concentration. These calculations for our sample pressures are to be compared for consistency with the phase separation temperatures observed in the pressure measurements. (See the discussion given for each sample in Chap. 4.) Edwards and Balibar (1988) reviewed extensively the research on isotopic mixtures of 3He and 4He [30]. According to these workers, the phase separation temperatures of the mixtures when Xh << 1 can be calculated as follows,


kBTps - A3 + Ah(1 - 2Xh)
n[(h)-- 1] (1.22) where Xh (h means that 4He is hcp) is an initial concentration of 3He in hcp 4He and T, is the phase separation temperature. Based on empirical data, the parameter Ah is given by
0.43(P - 3.6158)
Ah = 0.76 + 0.43(P - 3.6158) (1.23) where R = 8.31443(J/moleK) is the gas constant [29] and the units for T, v, and P are K, cc/mole, and MPa, respectively. The 3He free-energy difference between the meta-stable and stable crystal structures is expressed as:


A3 = (P - P3)[6v� + 2 3(P - P)] (1.24)


with
Av3 6vg - 6v3
= 3- AP 3.6158 (1.25) AP 10.605 - 3.6158






23

Here PO is the pressure at the bcc-hcp boundary in 3He at T - 0, which was obtained by extrapolating the pressure of the bcc-hcp boundary in pure 3He measured by Straty and Adams to T ~ 0 [28] and found to be 10.605 MPa; vi and 6v3, which were found to be -0.09 and -0.176 cc/mole, are the molar volume differences between bcc and hcp 3He at 10.605 MPa and 3.6158 MPa, respectively; and 03 is a parameter which was obtained by fitting the existing experimental data [32], [41].

Thus one can calculate the phase separation temperature as a function of the initial 3He concentration and the sample pressure, with the result shown in Fig. 1.12. We calculated the phase separation temperatures for our sample pressures in order to check for consistency between our results and the calculations.



0.28
0.26
0.24
" 0.22
0.20
S0.18 D 0.16
I-- - - i- - - I - - - !
0.14 - P = 30 atm

0 .,10 . . . . 2 .
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 'He Concentration (%)


Figure 1.12: Calculation of phase separation temperatures for various pressures, when the initial 3He concentration is small (xh << 1).



1.3 Nuclear Magnetism of 3He Nano-Clusters

As discussed in Sect. 1.2, a homogeneous mixture of 3He and 4He separates into 3He-rich and 4He-rich phases upon phase separation. Below the phase separation






24

temperature, Tps, the 4He-rich matrix has imbedded clusters of almost pure 3He. In the case of nano-clusters, which are about 20 nm in diameter in our packed silver sinters, a significant fraction of 3He would be within a few interatomic distance of the interface between hcp 4He matrix and the 3He nano-clusters. The molar volume of the hcp matrix is about 21 cm3/mole, comparable to that of 3He in the hcp phase (see Sect. 1.1.2). Bulk 3He at the pressures involved has a bcc structure with a much lower density. Thus, there must be a mismatch in density at the interface. However, so far the structure and density profile of the nano-clusters are not known and are likely to be different from that of bulk solid 3He. It is likely that the interface between the two mismatched structures has a significant effect on the properties of the nano-clusters.

Schrenk et al. measured the heat capacity of 3He nano-clusters embedded in a hcp matrix of 4He, following phase separation of the mixture, which had an initial 3He concentration of 1%. They observed a history-dependence in the peak in C,, which they claimed to be TN, at pressures as low as 700 kPa below the melting pressure in pure bulk 3He, as shown in Fig. 1.13 [1]. They found no latent heat that would indicated a first-order transition at the peak in C,. This is in contrast to the strong first-order transition seen in pure bulk 3He by Greywall et al. at TN (see Sect. 1.1.1) [2]. Furthermore, integrating C/T of Schrenk et al. from the lowest temperature to well above the peak, it was found that only 0.3 R ln2 of the spin entropy was removed. In the result of Greywall et al. integration of the heat capacity showed that 0.4 R ln2 of entropy was removed just in going through the transition. This suggests that a substantial fraction of the total spin entropy was not removed in the cooling through the peak in the result of Schrenk et al. Therefore the spins were not completely ordered and there might be an additional peak in the specific heat at a lower temperature. This suggests that the magnetic properties of the nano-clusters may be quite different from those of pure bulk 3He.






25










8
7 oo .

6 1.2
0



4 5 . 0.90 0.95 1.001.1
E o E








2:
-rO





0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
Temperature [mK]


Figure 1.13: History dependence of the heat capacity measurement. The initial concentration of 3He was 1%. This plot shows data taken at P = 3.40 MPa during the warm-up of the sample from different starting temperatures. Open circles: Tmin = 783 pK; solid squares: Tmin = 837 AK; crosses: Tmin = 866 AK; open squares: Tmin = 922 pK; open down-triangles: Tmin = 998 pK. Inset: Dependence of the nuclear magnetic ordering temperature TN of solid 3He clusters on the minimum temperature Tmin to which the sample was cooled before the measurements were started. Open circles: P = 2.80 MPa; crosses: P = 3.10 MPa; open squares: P = 3.36 MPa; solid squares: P = 3.40 MPa. From Ref. [1]. ((1996) American Physical Society.






26



1.4
1.2
1.0


z 0,6
-
0.4
0.2
0.0 .. ......
25 30 35 40 45 so50 55 Pressure [bar]


Figure 1.14: Pressure dependence of the peak temperature of solid 3He clusters in a 4He matrix (circles: Schrenk et al.) in comparison with the results of pure bulk 3He (solid squares: Greywall and Busch [2], crosses: Hata et al. [20]). @(1996) American Physical Society.

Schrenk et al. also plotted the peak temperatures, which they interpreted as TN, versus their sample pressures as shown in Fig. 1.14, in which the peak temperatures are an extension of TN for pure bulk 3He to below the melting pressure of pure bulk 3He. They construed this as evidence that the nano-clusters behaved like bulk helium at a pressure below the melting pressure.

Haley et al. [42] measured melting and freezing of the nano-clusters, both in a silver sinter and an open volume, from 2.96 MPa to 3.48 MPa. They found that melting of the nano-clusters occurred at a higher pressure than bulk solid and there was a substantial hysteresis between the melting and freezing temperatures in the silver sinter and open volumes. These results were similar to those for helium in small pores [43] but differ from those of Schrenk et al. [44] At a sample pressure of 3.40 MPa, the volume changes on melting were less than for bulk solid, indicating that only 8% and 40% of the 3He in the nano-clusters underwent melting in the sinter and in the open volume, respectively.






27

Schrenk et al. had reported a depression of the melting pressure of the nanoclusters relative to that of pure bulk 3He [44]. The pressure change upon freezing indicated that not all of the solid 3He in the nano-clusters melted as they were cooled below the bulk melting temperature. These effects in the melting pressure may be related to those that have been observed in pure 3He or 4He in confined geometries [45]. Pure 3He and 4He in small pores have elevated melting pressures compared to their melting pressures in pure bulk 3He, while a depression of the melting pressure has been reported for 3He on a MgO substrate [46]. Pressure changes upon melting or/and freezing are less than for helium in bulk.













CHAPTER 2
EXPERIMENTAL APPARATUS


In this chapter, the experimental apparatus used in this work is covered in the following order: the dilution refrigerator; the magnetic refrigerator; the superconducting magnets; the 3He melting pressure thermometer (MPT) including the gas handling system; the experimental platform; the NMR setup including the NMR cell design; the pressure measurement setup including the gas handling system; and the temperature regulation method.


2.1 General Description

The temperature range of interest in this work was from below 10 mK down to 0.5 mK. In order to achieve temperatures in this range, the dilution refrigerator cooled the experimental stage down to a 6 mK, which served as a precooling stage for the magnetic refrigerator necessary for lower temperatures.

In order to perform this work at low temperatures, several experimental principles had to be followed [47]. The cryostat and the electronics were located inside a copper rf shielded room ( 5 m long x 3 m wide x 2.7 m high ) which was grounded at a single point. In addition, 0-60 Hz low pass filters were placed in the 120 V AC power lines to avoid power disturbances affecting the instruments inside the screen room. Great care had been taken to prevent mechanical vibrations of the cryostat, which cause a heat leak. The dewar was suspended from a triangular aluminum plate, which was supported at its three corners by pneumatic isolation mounts, model XL-A.' Bellows
'Newport Research Corporation.


28






29

were put in pumping lines between the cryostat and pumps to reduce transmission of vibrations. The dewar was connected to the helium recovery system by a rubber hose for the same purpose. These features reduced the heat leaks to typically 2-3 nW. This corresponded to a warming rate of 2-7 pK/h at low temperatures, allowing us to maintain T < 1 mK for more than a month to complete measurements of a sample.


2.2 The Cryostat and Dilution Refrigerator

The cryostat consisted of the dilution refrigerator unit, the PrNi5 nuclear stage, the gas pumping system, and the floating table. The dilution unit, model DRP-43, was purchased from S.H.E Corp around 1980.2 With a circulation of 0.5 mmole/sec, it cooled the experimental stage down to a temperature of 6 mK without a magnetic field. With a field of 6.5 T on the PrNi5 stage (see Sect. 2.6), it reached 11 mK in two days. Figure 2.1 shows the overall setup of the dilution refrigerator.3 The helium mixture tanks contained 1.35 moles of 3He and 3.64 moles of 4He as specified in the dilution refrigerator manual.4


2.2.1 Principle of Dilution Refrigeration

Dilution refrigeration utilizes a remarkable property of "He and 3He liquid mixtures. Even at absolute zero, phase separation is not complete and 3He has a concentration of 6.4% in the "He-enriched phase. Pumping 3He gas from the still causes 3He in the 3He-enriched phase to diffuse into the 4He-enriched phase in the mixing chamber in order to maintain the equilibrium concentration of 6.4%. At a given temperature, the entropy of the 4He-enriched phase is greater than the entropy of the
2S.H.E Corp, 4174 Sorrento Blvd., San Diego, CA 92121.

3Some portions, which are not used, are omitted for a clarity.

40Operation instructions, model DRP-43 dilution refrigerator unit.






30

3He-enriched phase. Thus 3He passing from the concentrated phase into the dilute phase causes cooling. Therefore, the mixing chamber has the lowest temperature among the various stages of a dilution refrigerator. Heat exchangers are placed between the concentrated phase incoming to the mixing chamber and the dilute phase outgoing from the mixing chamber, so that incoming 3He is precooled before reaching the mixing chamber. More details can be found in numerous books about the subject, e.g. Lounasmaa(1974), Pobell(1991) etc [47], [48].


2.2.2 The 1 K Pot

The operation of the dilution refrigerator relies on a pumped pot of 4He near 1 K in order to condense the re-circulating 3He [49]. The 1 K pot utilizes the principle of vapor cooling that is produced by the difference in enthalpy between gas and liquid 4He. The liquid is vaporized by pumping, which produces the cooling, since the gas has greater enthalpy. This operation should be made in an isenthalpical condition, which was realized by installing a flow impedance to maintain the pressure difference (A P ; 1 atm) between the 4He bath and the pot, which is inside the can of the cryostat. A fine filter5 was installed on the inlet from the bath to prevent impurities such as frozen air from blocking the impedance. A small fraction of the liquid from the 4.2 K 4He bath flows through the impedance into the 1 K pot. At the same time the liquid arriving in the pot is pumped to lower its temperature. As long as the impedance is set properly, which implies that 4He from the bath flows isenthalpically through the impedance, and 4He is continuously supplied from the 4.2 K bath, the temperature of the pot can be maintained at , 1.3 K, serving as a condenser for the dilution refrigerator. The flow rate of liquid 4He through the 1 K pot, which has a volume of about 200 cm3, is - 4 L/day.
5Bekaert Fibre Technologies, 1395 South Marietta Parkway Building 500, Suite 100, Marietta, GA, 30067.








31











Symbols: O0 Valve C--': Needle Valve
Some valves are omitted because there are not in use.
Note that only one of two cold traps is in use at a time.


till Return Line


Flow Meter

Return Li
LargeValve Pressure FreonCooledBaffle 0 StorageMixure

Booster Pump 9B3 - ""









LNTraps
1K Pot Impedance From IVC T
SelKPotHeater 5230 S ae
L lKn ot
Low Temperature Part To Sample Gas Vent I Handling System Safey Valve

1 ent
To Recoveryl

Edwards E2M80 I Welch Mechanical Pump Edwards E2M80HS
Mechanical Pump IVnt
Mechanical PVent Hermetically Sealed Pump In a Crawl Space







Figure 2.1: Gas handling system of the dilution refrigerator. Not to scale.






32

Within the last year, a problem occurred in the 1 K pot impedance. After a couple of weeks of running, blockage in the line occurred, which resulted in termination of the run. In order to solve this problem, we first put a fine filter on the transfer tube in case there were impurities in the storage dewar, such as condensed moisture. Also the plumbing in the proximity of the 1 K pot was disassembled and flushed with trichloroethylene, reassembled, and leak-checked. However, upon cooling again, no noticeable improvement was observed and the blockage still occurred. It was found that rapid pressurization of the pot pumping line to a 20 kPa above atmospheric pressure with helium gas unblocked it by rapid warming. This indicated that the problem existed in the impedance. If it were a problem with the fine filter on the inlet of the 4He pickup line (5 pm pore size) immersed in liquid helium, one could not have managed to unblock it. Since the blockage was in the impedance and was removed by rapid warming of the line, a heater was soldered on the impedance to facilitate unblocking it. The heater, made of three 1.5 kf2 metal-film resistors connected in parallel, worked very well. As a precaution, the heater was turned on periodically for several minutes while monitoring a resistance thermometer that was mounted on the impedance, without heating up the pot itself. If the entire pot had been heated, hydrogen from the pumps trapped in the return line filter of the refrigerator could migrate to the small capillary part and plug it up. As an additional precaution, a cold valve was installed in a line in parallel with the impedance line. (See Fig. 2.1.) Since these changes were made, the impedance-blocking problem has disappeared.


2.3 Magnetic Refrigerator

The temperature range of interest, around 1 mK or below, can not be reached by a dilution refrigerator alone, thus nuclear refrigeration was employed to achieve the temperature range.






33

The cryostat has 1.0 mole of PrNiS as a refrigerant (20 rods), which were soldered with cadmium to 392 copper wires. The top end of the wires were welded to the Cu flange, which was thermally connected to the mixing chamber via a tin heat switch.

If a magnetic field, B, is applied to a nuclear magnetic dipole with a magnetic moment M and a total angular momentum I, there will be 21+1 energy levels with energies Em at temperature T. Then, the partition function, Z, can be written as:

I
Z = e'- ]No, (2.1) m=-I
where Em=,9gBmB and No the number of magnetic dipoles with g nuclear Land6 factor, AB Bohr magneton, 'y gyromagnetic ratio, and m enery level. Once the partition function is obtained, the entropy and other thermodynamic quantities can be calculated as functions of B/T.

Van Vleck paramagnets containing rare earth ions such as Pr3+ have a temperatureindependent electronic susceptibility at low temperatures because they have an electronic singlet non-magnetic ground state of their 4f electronic shells. An external magnetic field induces an electronic magnetic moment that generates a hyperfine field Bint at the 141Pr nucleus, giving an enhanced magnetic field at the nucleus, (Bint+Bexternal). The enhancement factor K = Bint/B+1 m 12 for polycrystalline PrNis. The large internal magnetic field seen by 141Pr results in a substantial reduction in the nuclear spin entropy even at high temperatures and in moderate magnetic fields. For instance, the entropy reduction at T = 11 mK and B =6 T is a 90%, as shown in Fig. 2.2 [48]. Therefore, a moderate dilution refrigerator and superconducting magnet allow one to achieve the condition just mentioned.

The entropy of nuclear spins of PrNi5 can be written as the following:

Rx 1 t (21+ 1) sinh[x 1]
S = [(coth(-) - (2 + 1)coth[ ) n[ 2 ], (2.2)
2 2 2 sinh(f)







34

























A D

10 * 37mT A 02T 07T 0 6T

�L A a
Sa Precooled by
6 mTAr a the dilution refrigerator /




Nuclear Demagnetization e s

0.1 1.0 10 100





Figure 2.2: Nuclear spin entropy of PrNi5 as a function of temperature. The stage is cooled, as shown by the arrows. In this work, the magnetic field applied was a 6.5 T. The initial temperature before the demagnetization of the stage was typically Ti = 11 mK. The lowest temperature of the nuclear stage is - 0.3 mK, because of the magnetic ordering of PrNis. From Ref. [48]. ((1990) Springer-Verlag.






35

where R is the gas constant, x=KlBgB/kBT, K is the hyperfine enhancement factor, MB the Bohr magneton, g the nuclear Land6 factor, B the external field, and T the temperature. For PrNi5, I=5/2. Provided the spin-spin interactions is small compared to the hyperfine interaction and the Zeeman interaction, one can argue that an ideal adiabatic demagnetization/magnetization gives


Tf= BT, (2.3)


where Bi and Bf stand for initial and final external field, respectively. Thermodynamic quantities of non-interacting spins are functions of the ratio between their Zeeman and thermal energies. However, Eq. 2.3 can not be used in a region where the interactions of the nuclear spins is not negligible compared to the Zeeman energy. One has to replace Eq. 2.3 with the following.


Tf = V(KB) + b2 T, (2.4) KBi'

where b is the internal field due to the interactions of the nuclear spins. Nuclear magnetic ordering of PrNi5 occurs around 0.3 mK [50], which is approximately the lowest temperature that can be reached with the stage.

The dilution refrigerator is thermally connected to the nuclear stage via a tin heat switch for removing entropy from the stage during several days of precooling (see the path shown in Fig. 2.2). Then the heat switch is opened (see Sect.2.4), which isolates the nuclear stage from the dilution refrigerator, since tin is a very poor thermal conductor in the superconducting state. From this point, the entropy remains constant except for heating of the stage. Then removing the magnetic field under an adiabatic condition cools the nuclear stage (see Fig.2.2).






36

2.4 Superconducting Magnets

In this work, two superconducting magnets were used. The one that generated the magnetic field for the nuclear demagnetization was manufactured by American Magnetics, Inc.6 It has a field to current ratio of 1058.5 Gauss/Amp and a decay rate of AB/B < 10-4/day in the persistent mode. The other, made by Richard Haley, produces the NMR static field and has a field to current ratio of 224 Gauss/Amp. In order to shield the external field, the outer surface of the coil former for this magnet was coated with Pb, which is superconducting below 7 K. A feature of this magnet is that it has 4 turns of Nb foil on the inside for improving field homogeneity ("superconducting jellyroll") (51].

The electronics for the operation of both magnets is shown in Fig. 2.3. Instead of the Kepco power supply shown in this figure, a HP E3632A DC power supply was used for the NMR magnet.

When the superconducting magnet for the nuclear demagnetization was energized, a DC voltage drop occurred in the stainless steel tubes that were used for the magnet leads inside the 4He bath, because of the resistance of the tube. The reason stainless steel is used instead of Cu is that the stainless steel does not evaporate the 4He liquid in the bath as much as Cu, because of the poor thermal conductivity. In order to prevent the voltage from exceeding the limit in the Kepco power supply and to carry away the heat generated, vapor cooling of the leads was employed. A connection from the bath to the helium recovery system through the stainless steel tubes was made in order to allow 4He vapor to pass through the tubes. Outside the cryostat, a rubber hose was used for the connection to the recovery, which allows one to open and close the connection by a clamp.
6American Magnetics, Inc. P. O. Box 2509, 112 Flint Rd., Oak Ridge, TN, 378312509.








37












Home-Made Ramp Unit - Kepco Power Supply ATE6-100M













ome-Made
current



HP 3465A Multimeter 1na. Heater 100A J (69.3Q) Current monitor










Superconducting Magnet
B=1058.5 (Gauss/A)I(A) (American Magnetics, Inc)
















Figure 2.3: Electronics for the nuclear demagnetization magnet. The magnet for the NMR static field has a similar setup.






38

The operation of the superconducting magnet requires a persistent switch whose function is to keep a current in the superconducting magnet without having current flow from a power supply at all the times. The switch for the demagnetization magnet is located on top of the vacuum can. The switching is made by applying/removing heater power in Fig. 2.3. Bare NbTi wire is located near the heater; thus, applying/removing the heater power makes the NbTi normal-/super- conducting. For additional details of the construction, see Ref. [48], for instance.

A tin heat switch was used to connect and disconnect the nuclear stage to the mixing chamber, because in the superconducting state tin is a very poor thermal conductor since Cooper pairs carry no entropy and the phonon contribution to the thermal conductivity is negligible at low temperatures. Switching is made by applying/removing a small magnetic field to the tin, which makes it normal-/superconducting. Electronics for the heat switch is quite similar to that for the superconducting magnets.


2.5 3He Melting Pressure Thermometry 3He melting pressure thermometry (MPT) employed in this work has several advantages. (1) The pressure change along the melting curve is large, thus, temperature could be determined with a great resolution. (2) There are several easily identifiable fixed points on the curve. (3) A relatively short time is required for measurements. (4) The setup and measurement scheme are relatively simple compared to other thermometers.

By measuring the melting pressure of 3He, the temperature can be determined, since P(T) has been established [17]. Pressure changes cause strain in a diaphram, which moves a capacitor plate, causing a change in capacitance. The pressure is measured as a capacitance or ratio transformer reading in an AC capacitance bridge.






39

2.5.1 The 3He Melting Pressure Thermometer Cell Design

The design of the cell for 3He melting pressure thermometry was like that used by Wen Ni in his high-precision 3He melting pressure measurement [52]. The body was made of 99.99% pure silver.7 A coin-silver flange was hard-soldered to the body with eutectic' silver-copper solder (71.9% Ag -28.1% Cu) at 7960C in a Thermolyne furnace.' A thermocouple was used to monitor the temperature inside the furnace in order not to overheat the cell body. If the temperature exceeded 8000C, the coinsilver flange would start to melt, and the entire body would be ruined. Eight pure silver wires whose diameter was 0.76 mm were diffusion-welded to the cell body as heat exchangers between a sinter and the body, as shown in Fig. 2.4. Then Japanese silver powder,'0 700 A in diameter, was packed inside the cell at pressure of about

2.62 MPa to form a sinter.

Also eight holes were drilled into the sinter to provide enough heat exchange between the liquid 3He and the cell body, as shown in Fig. 2.4. It is important that there be sufficient open volume, which is : 27%, for the solid 3He to grow; otherwise solid 3He formed inside the pores of the silver may cause a long thermal time constant. The initial loading pressure of 3He should be carefully chosen, so that the thermometer has a short time constant of not more than a few minutes, at the lowest temperature. See Chap. 3 for the procedure.

Separately, a diaphragm was constructed of coin silver. Several considerations must be made in the design of the diaphragm and capacitor. The displacement of the
7Surepure Chemetals, Inc, 5-T Nottingham Dr, Florham Park, NJ 07932.

8Eutetic Corporation, 4040 172nd St., Flushing, NY 11358.

9Thermolyne, Type F21100 Furnace, 2555 Kerper Blvd. Bubuque, IA 52001-9990. 'oTokuriki Kagaku, 2-9-12 Kaji-machi Chiyoda-ku Tokyo, 101-0044, Japan.






40

diaphragm is given as
3a4P
y- = (2.5) where P is the pressure, a the radius of the diaphragm, t the thickness, y the displacement, and E the Young's modulus. From this equation it is clear that smaller t and large a give larger y, which provides better resolution. However, one must take the stress and mechanical strength into consideration. The stress, s, can be expressed as
3a2 p
S= 3a2 (2.6) 4t2

Stress in the diaphragm should be kept well below the yield stress. The diaphragm made of coin silver has a thickness of 0.76 mm and a radius of 0.585 cm.

The capacitor plates made of coin silver are also constructed, one of which is attached to the diaphragm and serves as a movable plate while the other disk, which is held close to the movable plate, is used as a fixed plate. An appropriate gap between the plates, about 0.025 mm, was produced by machining the plate(s) rather than using a Mylar sheet as a spacer between them. This procedure not only eliminated possible dielectric capacitance due to the Mylar sheet between the plates but also provided good thermal contact of the upper plate to the nuclear stage. One can consult the extensive review written by Adams for more details [53].

After construction, the 3He melting pressure thermometer cell was tested at room and liquid nitrogen temperatures. Once the cell was made leak-tight, the sensitivity was checked at 77 K with result shown in Fig. 2.5. An Andeen Hagerling ultraprecision capacitance bridge was used in testing.

The gas handling system for pure 3He, depicted in Fig. 2.6, was used to supply/evacuate 3He gas to/from the MPT cell. The 3He gas had a purity of 2 ppm. Wen Ni [52] argued that at 0.4 K 10 ppm of 4He decreased the 3He melting pressure








41







The Stroain Gouge

- Copocitor leads Copacitor Plotes

Diophrogrm

Simple Spoce
Fill Line


Welded


S ilver Wires c-- , ~ M Silver Powder Stainess St. Screws




Figure 2.4: Schematic drawing of the 3He melting pressure thermometer cell.












90

so 1st pressure up o0 o 1st pressure down o 70 2nd pressure down I 60 O a.a
50 0 40 ou oc�
30o o

20 i i i i i I I
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Pressure (MPa)




Figure 2.5: Melting pressure thermometer cell test.






42

by 30 Pa. This corresponds to a change of 10-6 in ratio transformer reading, which was approximately the same as the reproducibility of TN. Therefore 2 ppm of 4He in the 3He gas has almost no effect on the thermometry in the temperature range in this work.
The gas handling system includes a Heise pressure gauge,"la dipstick, and nitrogen cold traps. A dipstick was used in order to draw gas from/to the 3He tank to/from the cell and to control the pressure. The pressure was monitored by the Heise gauge. The nitrogen cold traps were placed in the fill line of the cell and used to trap air before the 3He gas went to/from the cell.


2.5.2 Electronics for 3He Melting Pressure Thermometry and Sample Pressure
Measurements

As explained in Sect. 2.5, a pressure change is detected as a capacitance change. As the diaphragm of the cell deflects due to a pressure change, a change in Cp occurs, causing an off-balance voltage in the bridge. From the off-balance voltage detected by the lock-in amplifier, Cp is obtained, which in turn gives the pressure using the C(P) calibration (see Chap. 3 for the calibration procedures).

Figure 2.7 gives the overall diagram of the capacitance bridge used in the thermometer setup. An Ithaco Dynatrac 391A lockin amplifiers2 generated a sine wave of a 1 kHz applied to the bridge, which included the capacitance (Cp) and a reference capacitance (Crd), through a Gertsch ST-100A isolation transformer.13 "Dresser Instrument Div., Heise Precision Instrument Operation, 153 S. Main St. New town, CT 06740.

12Ithaco, Inc., 735 West Clinton Street, P.O. Box 6437, Ithaca, NY 14851-6437. 13Reconditioned Gertsch ratio transformers are available at Tucker Electronics, www.tucker.com.










43











High Pressure Gauge Low Pressure Partl -r e-ge
0 - 3 psi 0 - 30 psi above ala








Vent
000psi Liquid N2 Trap




Test Bomb Vacuum Pumpl Dipstick - -

O Low Pressure Valve

--[<-High Pressure Valve 'He Melting Pressure High Pressure Vale
Thermometer Cell


'He Gas Handling System
'He - He Mixture Gas Handling System Tet rage 0- r20t Low Paessre G ge- Test Gauge 0 - 2000 psi e - I - P -ssur Gauge
0 0 - 30 psi above armn



LI



Liquid N Trap e-'He Mixture Gas









Dipstick
Paroscientific Pressure Transducer Liquid N, Trap
0 - 900psi


'He - *He Mixture
Experimental Cell


Pressure Intensifier
# In the actual system two metal dewars had been used.
One is in the upper system, the other is in the lower one in this diagram.
'He Gas Cylinder
## Usually two high pressure valves which connect the two systems are closed to
avoid a possible contamination of the pure 'He system.


















Figure 2.6: Gas handling system for pure 3He and 3He-4He mixtures.








44











Gertsch ST-110IOA Transformer


ITHACO 391 rI -Lockin amplifier -atio
Output Input

SCp

GertschAC 1011A Chassis Ratio Transformer I C req


Inside the cryostat




Hewlett Packard 3421A Digitizer
Channel #2
oltec chart-recorder 330



National Instruments
GP-IB Bus extender 11


Fiber optics cable

r----- ------------------------TI
National Instruments
GP-IB Bus extender 1101


L--------------------------------------------j
Outside screenroom # C ref is located on the mixing chamber.















Figure 2.7: Electronics for the 3He melting pressure thermometry.






45

When C, needed to be determined, the setting of the Gertsch AC 1011A ratio transformer was adjusted to balance the bridge, with the lockin serving as a null detector.

For an ideal bridge, the relation is as follows [53]: Vo = ( - 1)C + Cref (2.7)
=i (2.7) V, CP + rf '

where Cref and C, represent the reference and the unknown capacitances, respectively, and x is the ratio transformer reading. The lockin amplifier output of the off-balance voltage was fed to a HP3421A digitizer'4 for data acquisition. The lockin output was also traced on a 3-channel Soltec 330 chart recorder.5 In reality the cable capacitance could be greater than Cf and C,, which would reduce Vo. A result can be found by substituting Cref+Cp+Ccable for the denominator in Eq. 2.7. One might want to use a pre-amplifier to reduce Ccable and gain a higher resolution, depending upon the requirements of the experiment.

The setup in electronics for the sample pressure measurements is quite similar to that for the MPT. There are difference in the instruments, as shown in Fig. 2.8. An Eaton ratio standard and capacitors were isolated from ground by a 1:1 Gertsch ST-200C isolation transformer. The lockin amplifier used in the measurements was an EG&G model 5204,16 which generated a sine wave of ; 800 Hz (variable) applied to the capacitance bridge. Also an Ithaco low noise pre-amplifier model 1201 was added with a gain of 20, because the resolution of the pressure transcucer was not as good as the MPT cell. In addition the high pass and low pass rolloffs of the preamplifier were set at 300 Hz and 3 kHz, respectively. The off-balance voltages of the bridge 14Hewlett Packard, P. O. Box 105005, Atlanta, GA 30348. 15Soltec Corp., 11684 Pendleton St, Sun Valley, CA , 91352. 1'EG&G Princeton Applied Research, P. O. Box 2565, Princeton, NJ 08643-2565.








46











Gertsch ST-200CTransformer Reference -800Hz
PARC 5204 r --Lockin amplifier I I Output nput Cp


Chassis - Ratio Transformer #Ci.aeoei re r.
-----Inside the cryostat Ithaco pre-amplifier Modell201 Gain 20 S1Low Pass Filter 300 Hz Hewlett Packard Input High Pass Filter 3 kHz 3421A Digitizer Channel#3




National Instruments
GP-IB Bus extender 11


Fiber optics cable
# Crefis located on the mixing chamber. r------ - - - - - - - - -


al Instruments I
P-B Bus extender 1101




Outside screemnoom
















Figure 2.8: Electronics for the pressure measurements.






47

were detected by the lockin amplifier and recorded to a computer file via a general purpose interface bus, GP-IB.


2.6 The Experimental Platform on the Nuclear Stage

Initially a copper "cage" was used to mount the cell on the nuclear stage. After several cooldowns, a long time constant for the melting curve thermometer was observed. The loading pressure had been checked, with no indication that the amount of 3He in the cell was a factor.

The annealed cage had been polished, which could have introduced defects in it before mounting the experimental cells. Also soft-soldered spools made of Cu alloy 101 (an oxygen-free high-conductivity copper) were mounted on the stage for heatsinking the fill lines, which may have caused the problem. Further investigation of the cause was not made, however, and it was decided to construct a new experimental platform. First the cage was removed; then a new plate made of copper alloy 101 was heat-treated in an oven at 950C with 10-4 Torr of oxygen gas. Secondly, small capillaries were used for the fill lines to reduce large open volume near the cells. Third, the fill lines were silver soldered to a new spool made of Cu alloy 101. Moreover, the electrical coaxial leads had Apiezon grease placed between the outer and inner conductors, which prevented the inner lead from moving and served as a heat-sink. This reduced any possible noise problem caused by the moving leads. A heat-sink for the leads was heat-treated together with the plate. Only a minimal amount of epoxy was used to glue brass strips to the heat-sink for the electrical connections. Finally the experimental platform was placed on the nuclear stage and no problems were observed during the experiment. The overall arrangement of the nuclear stage is shown in Fig. 2.9.








48











Mixing Chamber Tin heatswitch



Thermal link between magnet
and mixing chamber (Ag wire) Lead shield w/NMR setup inside

Sample Cell
(Simplified for clarity. "He Melting Pressure Thermometer See the detailed pictures in chapter2.)
(MPT) Vespel for thermal isolation

Welded
Rebuilt Experimental Platform
(Annealed, then goldplated.) -Thermal link between PrNi5 and the platform.


Compensation Coil





Superconducting Magnet
(American Magnetics, Inc)
B(T)=0.105851(A)
Decay rate=ABIB=-10/day

PrNis (Cadmium Soldered to 392 Cu wires of 0.8 mm diameter.)








Heatsink for leads and fill lines, electrical leads are not shown in this figure for clarity.










Figure 2.9: Schematic diagram of the nuclear stage. Not to scale.






49

2.7 The NMR Setup

The electronics for the NMR are shown in Fig. 2.10. A Tektronix digital oscilloscope TDS784D'7 and a PLM-418 were both used to obtain the data. The static field for the NMR was generated by the superconducting magnet described in Sect. 2.4. A GP-IB interface bus was used for data acquisition, and connection between the computer and the instruments was made via an optical link. Capacitors for the tank circuit were located in an aluminum box on top of the cryostat.

In the first cell, built by Richard Haley, the NMR coils were wound directly on the body of the cell and were not grounded to the nuclear stage, which may have been responsible for the high minimum temperature encountered by Adams et al. [54]. Therefore a new cell was designed and constructed. The main goals were to reduce the amount of epoxy in the cell and to improve the heat-sinking of the leads going to the cell. The cell had two coils, one a saddle coil and the other a solenoid, noneh of which touched the sample container. The overall setup and the sample container are shown in Figs. 2.11 and 2.12.

The NMR saddle coil has not been used in the current work. However, this was prepared for a situation when a cold preamplifier could be used, with the saddle coil serving as an excitation coil. Grooves for the coil winding were made in the Stycast 126619 coil former, and 20 turns of copper wire (0.038 mm in diameter) were wounded on each side.

Copper wire (0.025 mm in diameter) was used for the NMR solenoid coil, which served both as the pickup and excitation coil (see Fig. 2.12). The coil former, which '7Tektronix, Inc. P.O. Box 1000, Wilsonville, OR, 97070-1000. '8PLM-4, RV-Elektroniikka Oy Picowatt, SF-01510 Vantaa, Finland. "9Deanco, 130 University Blvd., Winter Park, FL 32792.








50









Tuned to 250kHz or 125kHz Hewlett-Packard
PLM-4 Pulse DC Power Supply
NMR Spectrometer E3632A
Preamplifier (Resolution=0.12mA) Wideband I rigger (Integrator Gate)* output














Tektronix Digital Oscilloscope




GP-IB Bus






Cv and Cc are variable and SI cable capacitance respectively.





b C',



Computer Screen
*Timing was determined on the oscilloscope.








Figure 2.10: Electronics for the NMR.








51




















Superconducting NMR solenoid Led superconducting shield






Coin-silver . NMR coils
pressure transducer

... Nb
"superconducting jellyroll"



Silver thermal Silver ther- link with poweder mal link to the heat-exchanger nuclear demagnetization stage





















Figure 2.11: Overall NMR cell setup.






52














































Figure 2.12: Drawing of the 3He-4He mixture sample cell.






53

had three sections to decrease capacitance between layers, was made of Stycast 1266. Each section had 180 turns of winding. A motor-driven coil winding machine with an optical sensor, made by Volodya Shvarts, was used to wind the small coil.
A slight modification was made to improve the performance of the cell by replacing brass mounting screws with BeCu ones. Brass screws may possibly become superconducting, affecting the homogeneity of the static NMR field due to the Meissner effect. Lang had success in using BeCu in a field of 2.5 T for his direct demagnetization of hcp 3He [55], in which he succeeded in demonstrating magnetic ordering for the first time.


2.8 Pressure Measurement Setup

Initially the pressure transducer was a Straty-Adams type gauge attached to an aluminum-oxide ceramics tube that served as a sample container. However, after a couple of tests, a leak was found; thus the ceramics was replaced by Vespel-2220 with the same dimensions (3 mm in diameter and 10 mm in length). Apparently the problem was a crack due to a difference in thermal shrinkage between the ceramics and Stycast 2855FT, which was used to glue the transducer to the ceramics. We built a miniature coin-silver pressure transducer of the Morii-Adams type [53], as shown in Fig. 2.13, and replaced the pressure transducer of Straty-Adams type.

The coin-silver capacitor plates (0.421 mm in diameter and 0.025 mm in the diaphragm thickness in the movable plate) were glued to each other with Stycast 2855FT, with a paper ring (0.076 mm in thickness) placed in between to serve as a spacer and an electrical insulator between the plates. The electrical leads for the 20E. I. Du Pont De Nemours & Company, Du Pont Polymers, Wilmington, DE 19898.






54

capacitor plates were glued with an electrically conductive epoxy H20E-175.21 A large temperature-dependent background capacitance changes in the pressure measurements were observed, which were caused by the Stycast between the plates (see Chap. 3).
A pure silver wire, 0.5 mm in diameter, was welded to the thermal link of the cell to the nuclear stage (see Fig. 2.13), after which the sample container was glued to the silver thermal link. Then for better heat exchange between the sample and the nuclear stage, Japanese silver powder (Tokuriki Silbest, C-8) whose particle size was 70 nm was packed inside the cell with a packing pressure of 800 kg/cm3, which resulted in a filling factor of a 45%. Then a disk of Kimwipe22 was placed on the surface of packed silver in order to electrically isolate the silver power from the diaphragm (see Fig. 2.13). Finally, the pressure transducer was glued to the Vespel-22 container with Stycast 2855FT.

The solid 3He in the small pores of the sinter is held in place because of the small openings and does not transmit pressure to the bulk layer [53]. Thus, one might be concerned that pressure of the clusters in the pores would not be transmitted to the bulk layers near the transducer. Also mechanical properties of a sample cell could cause a non-constant volume condition, which causes a smaller pressure change than that at constant volume. In our case pressure was indeed transmitted to the bulk layer under a constant volume condition, which can be concluded from our experimental results (see Chap. 4 for discussion).

The gas used for the 3He-4He mixtures had a concentration of 0.6% 3He in 4He. The lower part in Fig. 2.6 shows the gas handling system for 3He-4He mixtures, which has a similar construction to the 3He gas handling system, including nitrogen 21Epoxy Technology, 14 Fortune Drive, Billerica, MA 01821 USA. 22Kimberly-Clark Corporation, Roswell, GA 30076-2199.







55















Paper Ring Electrical leads


Stycast 2855




Coin Silver Pressure Transducer Kimwipe Tissue (for isolation of the gauge from silver powder)

Vespel Container Silver Powder (Particle size: 70 nm*, Packing Pressure: 800 kg/cm3, Filling factor: 45%.) Silver Thermal Link Fill Line tycast 2855FT Silver Wire (Welded to the silver thermal link.)


For clarity silver thermal link is shortened.
*Tokuriki Silbest C-8.


Figure 2.13: Heat exchanger and miniature pressure gauge used for this work. Not to scale. The NMR coils are not shown for clarity.






56

cold traps, a pressure gauge, and a dipstick. The pressure was controlled by an intensifierz (see Fig. 2.6) and was monitored by a Paroscientific pressure transducer.24


2.9 Method of Temperature Regulation


Chart Recorder Soltec 330







Resistance Bridge Out in Temperature Controller LR- 110 LR- 130 In out




r---------------- -------- ----- - --R4 1090a Thermometer Heater Mixing Chamber
L----------------------------------------------Inside the Cryostat


Figure 2.14: Electronics for temperature regulation.


Occasionally, the temperature was regulated using the electronics in Fig. 2.14, in order to acquire data or to anneal a sample. This was done by feeding the output of the LR-110 resistance bridge or the MPT off-balance voltage to the LR-130 heater. 23High Pressure Equipment. 1222 Linden Ave, Erie, PA. 24Paroscientific, 4500 NE 148 th Avenue Redmond, WA 98052-5126.






57

Heaters that were 1000 Q wire-wound resistors were located at various points such as the mixing chamber.













CHAPTER 3
EXPERIMENTAL PROCEDURES


In this chapter, the entire experimental procedures are covered step by step in detail from cooldown to data acquisition.


3.1 Cooldown Procedures

The first step for a successful cooldown is to make sure all the parts including the dilution refrigerator are leak-tight. This was carefully checked with a Veeco leak detector, MS-17, at room temperature.' Also electrical connections were checked. After checking these at room temperature, the dewar was raised for a liquid nitrogen transfer. Before starting the transfer, the dewar was pumped out and filled with helium gas to leak-check the vacuum can. Also the dilution refrigerator was leakchecked by introducing the helium mixture gas from the tanks to the plumbing of the refrigerator. The 1 K pot line was flushed with helium gas several times and was kept at a pressure of - 138 kPa of helium gas in order to prevent nitrogen from entering and blocking the flow impedance in the pickup line.

During the transfer of the first 50 L of liquid nitrogen, the helium level in the vacuum can was monitored by the leak detector to leak-check the vacuum can and the 1 K pot as they cooled. Once it was clear that there was no leak in either the can or the pot, 1000 pTorr of dry nitrogen gas was introduced to the vacuum can as exchange gas for cooling, and another 50 L of liquid nitrogen was transferred.
1Veeco, Terminal Drive, Plainview, Long Island, NY.




58






59

After making sure that all of the resistance thermometers indicated that they were at nitrogen temperature, the MPT and the experimental cell were leak-checked by applying pressures with 3He and 3He-4He mixtures, respectively. Then, the remaining liquid nitrogen was transferred back to the storage dewar. It was important to make sure that all of the liquid was removed at the end of the back-transfer, which was accomplished by pumping on the dewar.

Helium gas at a pressure of 1000 pTorr was introduced to the vacuum can as exchange gas, after which the transfer of liquid helium began. During the transfer the resistance thermometer on the mixing chamber was monitored to determine when the temperature was low enough for pumping of the helium exchange gas. When the resistance dropped due to superconductivity in the NbTi thermometer leads, which occurs at 18 K, pumping of the exchange gas began. The vacuum can was pumped for eight to ten hours until the 4He signal became low (, 10-' cc/sec in the Veeco leak detector) and constant. The transfer was then continued until the 100 L transfer dewar was emptied.

After the initial transfer of liquid helium, pumping of the 1 K pot was started. Then, the helium gas mixture for the refrigerator was released from the storage tanks to the exhaust of the mechanical pump (Edwards E2M80HS, a hermetically sealed pump) with the safety solenoid valve manually opened (see Fig 2.1). The gas was condensed and ready for circulation in the refrigerator after - 12 hours.

The consumption rate of liquid helium was about 20 L a day, with a transfer every 3.5 days to keep the bath level above the 1 K pot pickup line.


3.2 Calibration of the Mixture Sample Cell and Sample Preparation

First the tin heat switch was closed in order to bring the PrNi5 stage and the mixing chamber to the same temperature. Pumping of the 1 K pot was stopped and







60


the temperature was regulated at about - 2.5 K, utilizing the setup shown in Fig.

2.14. Then, the mixture, 0.6% 3He in 4He,2 was introduced to the sample cell.

Increasing and decreasing the pressure several times between = 2.78 MPa and a 6.2 MPa "trained" the strain gauge in order to reduce hysteresis. However, a significant hysteresis of the cell was still found, which diminished with the range of pressures used in calibration or testing. There was a z 70 kPa difference between the same "up" and "down" ratio transformer readings. (See Fig. 3.1.) This hysteresis was observed in every cooldown, and thus had to be handled carefully. It may be attributed to the design of the transducer. Fortunately scatter in the down

pressures was small. Thus, a down-calibration was employed for these experiments since the pressure decreased as a sample was cooled along the solid 4He melting pressure curve after the initial pressure was set. (Compare Figs. 3.1 and 3.2.)




6.5
6.0
5.5 - 0 Q_ 5.0
S4.5
- O
4.0 0
3.5
S1ist up-pressure
3.0 - 0 1st down-pressure
* 2nd down-pressure
2.5 I . . i , i i i . t . i . i
0.134 0.136 0.138 0.140 0.142 0.144 0.146 0.148 0.150 0.152 0.154
Ratio



Figure 3.1: Calibration of the mixture cell. Compare up- and down-pressures for ratios.

2The concentration was measured with the Veeco leak detector by the author and W. Ni.






61




3,76 0
3.74
3.72

a
3.70
3.68
IL
3.66
0 1st up-pressure
3.64 � 1stdown-pressure 3 2nd down-pressure
0.12505 0.12510 0.12515 0.12520 0.12525 0.12530 0.12535 0.12540
Ratio



Figure 3.2: Hysteresis check over a pressure change of about 137 kPa.


Then, the cell capacitance was calibrated against the Paroscientific pressure transducer, with a resolution of - 3.4 Pa to establish C(P). The transducer had zero-pressure reading of a -25.5 kPa, 3 which was added to all pressures.

A polynomial equation of the form


P = ao + aiR+ a2R2 a3R3, (3.1)


was used to fit the down-pressure calibration points, where P was the value of the pressure, R the corresponding ratio transformer reading, and the ai's were fitting parameters. (See Fig. 3.3.) The difference between the fitted polynomial curve and the actual down-pressure data is shown in Fig. 3.4. The cell was calibrated between

2.76 MPa and 4.14 MPa.

The sample gas mixture was loaded into the cell at a pressure of a 6.2 MPa, which depended upon the target pressure of the sample. Then, pumping of the 1
3This offset changed slightly; thus it was checked in every run.









62












4.2 P - -658.50755+12822.47031R-83418.59064R2+183152.16729R3
(R.Ratio, P=Pressure)
4.0

3.8

. 3.6

S3.4

3.2
a. I
3.0

2.8

2,6 I I I
0.134 0.135 0.136 0.137 0.138 0.139 0.140 Ratio






Figure 3.3: Calibration of the mixture cell.















300

-20*




. *
E

. -10.

-200 S-300 -t400 S-500

0.134 0.135 0.136 0.137 0.138 0.139 0.140 Ratio






Figure 3.4: Evaluation of the mixture cell calibration.







63

K pot began and the fill line blocked as the sample started to cool. Thereafter the sample solidified under a constant volume while pressure changes were monitored on the chart recorder. The sample reached the temperature of the 4He melting curve and since the mixture was almost pure 4He, it was then cooled along the curve until it deviated from the curve. At this point, when the sample had just become allsolid, the temperature was regulated between 1.7-1.9 K for annealing. Annealing was necessary to eliminate gradients in density in the sample. During annealing the pressure decreased, with the annealing continuing until the pressure drop became dP/dt ~ 300 Pa/hr. (See Fig. 3.5.)



3. 775
3,750
3.725
3.700
CL 3.675
3.650
0 3.625 , 3.600
3.575
3.550
3.525I
0 2 4 6 8 10
Time (h)



Figure 3.5: Pressure versus time during the annealing of the 3.54-MPa sample at ,
1.6 K.


If the pressure was not the one desired or a new sample was needed, the nuclear stage was heated to warm the cell, and pumping of the 1 K pot was stopped in order to remove the blockage in the fill line. Then one could form a new sample by repeating the procedure described above.






64

3.3 Calibration and Preparation of the 3He Melting Pressure Thermometer

Basically the same procedures were taken for calibrating the 3He melting pressure thermometer cell (MPT) as for the calibration of the mixture cell with the temperature regulated at 0.8 K. Before applying a desired pressure to the cell, the diaphragm was trained over the working range between 2.76 MPa and 3.45 MPa in order to reduce hysteresis. Then, calibration of the cell was carried out over the same range. As in the calibration of the mixture sample cell, a power polynomial was used to fit pressure versus ratio (see Fig. 3.6). The cell had much less hysteresis than the mixture cell.

After the calibration, 3He gas was loaded at 3.17 MPa into the cell. It was then cooled through the pressure minimum (Pmin = 2.9333 MPa [9]) of the 3He melting pressure curve. The ratio at Pmin was recorded, to be used as a fixed pressure point for a temporary temperature scale until the nuclear stage reached the solid ordering temperature TN (Niel temperature). Subsequently, TN was used as a fixed pressure and temperature point to provide a more accurate temperature scale at lower temperatures. Upon observing the ratio at Pmin, the temperature was increased to 0.8 K and regulated, whereupon the MPT cell was loaded at a pressure of 3.41 MPa.


3.4 Sample Cooling


3.4.1 Technique and Setup

The MPT and sample cells, which were linked to the nuclear stage, were cooled with the refrigerator. A technique was used to prevent the plug in the MPT cell fill line from slipping during the cooling. First, the nuclear stage was cooled to a temperature slightly above Pmi, = 0.318 mK with the tin heat switch open to let the mixing chamber cool to a slightly lower temperature. Then the tin heat switch was








65




















3.5 /



3.4



3.3 C- 3.2 3.1


0)
3.0 2.9 2.8


0.44 0.45 0.46 0.47 0.48

Ratio







Figure 3.6: Calibration of the melting pressure thermometer cell. The fit of the calibration C(P) is P = 33.08247-245.62655C+609.10729C2-470.684C3, where C = ratio.








66


















10


8 0


n 6

0

4
.



r














-8
S0-*



* *









-8 I1 I I , I
0.44 0.45 0.46 0.47 0.48 Ratio






Figure 3.7: Evaluation of the melting pressure thermometer cell calibration. The fit of the calibration C(P) is P = 33.08247-245.62655C+609.10729C2-470.684C3, where C = ratio.






67

closed in order to connect the nuclear stage thermally to the mixing chamber. In this way the time duration that the MPT was around Pmin was minimized.


3.4.2 Phase Separation

The phase separation of the mixture sample typically occurred around - 170 mK for the 0.6% concentration of 3He, although the temperature depended slightly upon the pressure of the sample. The pressure change due to the phase separation, which was small and proportional to the 3He concentration in 4He, was detected by the pressure transducer on the mixture cell [32].


3.4.3 NMR Tuning

The capacitance in the tank circuit and the NMR static field had to be adjusted in order to optimize the NMR signal. The adjustment was made at - 6.5 mK before taking magnetic susceptibility data. The static field was controlled by the DC current supply as in Fig. 2.10. The procedures are as follows: First, the capacitance in the tank circuit, located on top of the cryostat, was set. Second, the static field was swept with an increment of - 0.2 mA and the NMR signal was recorded for each current. Then, the procedures were repeated for various capacitance values in the tank circuit with the result shown in Fig. 3.8. The total capacitance was determined to be 151 pF, because the 151-pF result had a wider frequency range at the signal peak than the 141-pF result, although the signal levels are quite similar. Thus, even if the current is slightly off-tuned, the signal strength does not decrease much. For 125 kHz resonance frequency the total capacitance in the box was set to 1688 pF for the same reason. From these settings, the cable capacitance was estimated to be about 400 pF, which was equivalent to about 3 m of RG-58 A/U type cable.






68



100

80
o 151pF *
60o 141pF b **






0.330 0.335 0.340 0.345 0,350 0,355 0360
Current(A)



Figure 3.8: Tuning of the NMR made at around 6.5 mK.

3.4.4 High Temperature Data Acquisition

Data were taken at high temperatures reachable by the dilution refrigerator before magnetizing the nuclear stage. During these measurements, the temperature of the MPT was regulated (see Fig. 2.14.) by using the output voltage of the MPT from the lockin as a feedback to the LR-130.4


3.4.5 Demagnetization and Magnetization of the Nuclear Stage

The nuclear stage (see Chap. 2 for the detail) was magnetized to a field of 6.9 T, corresponding to a current of 65 A. It was important that the valves from the vapor-cooled magnet leads were opened to the helium recovery system when current from the Kepco power supply5 exceeded 50 A. Otherwise, a DC voltage in the current
4Linear Research, Inc., 5231 Cushman Place, Suite 21, San Diego, CA 92110.

5Kepco Inc.,131-38 Sanford Avenue, Flushing, NY 11352, phone:718-461-7000.






69

loop, which was mainly generated at the stainless steel leads, would reach 5 V and the Kepco power supply would be shut down at the voltage limiter setting of 5 V.

The stage was precooled for two days to reach around 11 mK for removing most of the entropy (see Fig. 2.2). Then, the heat switch was opened in order to thermally isolate the nuclear stage from the mixing chamber, after which a demagnetization of the stage was carried out. During the demagnetization several temperatures were stabilized in order to measure the relaxation time of the magnetic susceptibility. A typical demagnetization profile was as follows:

6.91 T3.-r. 4.234 T4 -" 2.12 T 4r. 1.06 T -'0.53 T4- 0.32 T

The lowest field to which we demagnetized the nuclear stage depended upon the desired temperature. If the obtained temperature was not low enough, an additional demagnetization to a lower field would be carried out. For example, if a demagnetization of the nuclear stage started when the temperature was 11 mK, at a field around 0.5 T the temperature would reach TN. If the temperature needed to be increased, the nuclear stage was partially magnetized to reach a desired temperature.

The field should be kept above 50 G, the critical field of the cadmium solder, otherwise cadmium becomes a superconductor, preventing it from maintaining thermal contact.


3.5 Temperature Scale

In this work the 3He melting pressure temperature scale of Ni et al. [17] has been used. They established the temperature scale by using 60Co as a primary thermometer and Pt-NMR as a secondary thermometer. Ni's polynomial relations in P(T) relative to the Neel point (PN, TN) are as follows.

For T>TN,
5
P(T) - PN = AnT"; (3.2) n=-4






70

and for T
P(T) - PN = BO - B1T4, (3.3) where the coefficients are A_4=0.78405848, A_3=-4.5968629, A_2=9.5436359, A_1=-10.041752, A0=8.5580332, A1=-4.4431076, A2=1.5962639x10-2 A3=-3.8815407x10-, A4=7.1230765x10-8, A5=-6.0945626x10-", B0=0.1987 kPa, and B1=0.2611 kPa/(mK)4. Here temperatures are in mK and pressures are in kPa.

Signatures of the fixed points are shown in Fig. 3.9, where the output voltages from the Ithaco 391A lockin amplifier are recorded.


3.6 Data Acquisition

In taking data the following information was obtained: temperatures of the nuclear stage, current in the main magnet, fast Fourier transform (FFT) spectra, free induction decay (FID), magnetic susceptibility of the mixture samples, spin-lattice relaxation time (TI), and pressures of the mixture samples. All of these were registered in computer files. The GP-IB interface bus was used for all the data acquisition. At the same time, the MPT pressure, current in the main magnet, and pressure of mixture samples were recorded on the Soltec chart-recorder.


3.6.1 Sample Pressure Measurements

Pressure was recorded typically every 30 seconds. The pressure measurements were useful to detect possible crystallographic change in solid 4He and partial melting in 3He nano-clusters. By measuring temperature and pressure at the same time, it was possible to trace the pressure change as a function of temperature. Possible heating due to the electrical connections to the transducer of the mixture cell was










71









4.0


3.8


3.6


3.4 T A


(a) 3 2

30

Cooling

28 Z


2.6
170 175 180 Time(min)
0.3
0.2
0.1
0.0
-0.1
-0.2 T
" -0 3 .
..3 B Warming
(b) 0
a -0.4
-0.5
0
> -0.6
-0.7
-0.8
-0.9
-1.0
-1.1 l I , i
0.4 06 0.8 1 0 1.2

Time(hr)
1.5



1.0



05 T


c() Warming










0 1000 2000 3000 4000 5000 6000 Time (sec)





Figure 3.9: Fixed points on 3He melting pressure. (a): A-transition, (b): B-transition,

(c): solid ordering transition. Output voltages from the Ithaco 391A lockin amplifier are shown.






72

checked by disconnecting and reconnecting the cables to the pressure transducer and found to be negligible.

The temperature dependence in the transducer was a disturbing factor when the pressure measurements were done. Since the transducer had Stycast between the capacitor plates (see Fig. 2.13), a large temperature dependence was observed. The capacitance caused by the Stycast had to be subtracted from the raw capacitance data as a background. Rather than measure C(T) at fixed pressure, the background was deduced from data at two pressures, 2.88 and 3.54 MPa, as shown in Figs. 3.10 and 3.11.

The high temperature part of the background was taken from the 2.88-MPa sample because of its low phase separation temperature, 145 mK. (See the 2.88MPa sample section in Chap. 4 for details.) Based on Panczyk's work [56], the pressure should be approximately for a range of temperature above Tp,, therefore we concluded the pressure change in this range came solely from the dielectric property of the Stycast.

The low temperature part of the background below 105 mK was taken from the raw capacitance data of the 3.54-MPa sample, because this high pressure sample has a very small contribution from exchange interactions and no partial melting occurred. Therefore P(T) is expected to be constant.

Then the composite background was made by combining the two Figs. 3.10 and 3.11, as shown in Fig. 3.12. In the background subtraction for each sample, adjustment of the point at a 60 mK, where the background had the minimum, to the minimum of the raw data was made.

The pressure data with the background subtracted will be shown in Chap. 4 for each sample.









73










0.131311


0.131310


0.131309


0.131308 S0.131307 0.131306 0.131305 0,131304


0.131303 140 150 160 170 180 190 200 210 220 230

T (mK)






Figure 3.10: Ratio versus T for high temperature portion of background. R = ratio, taken from the 2.88-MPa sample.














0 125220 0.125215 0.125210
n

0.125205 0.125200 0.125195
0 10 20 30 40 50 60 70 80 90 100 110

T (mK)






Figure 3.11: Ratio versus T for the low temperature portion of background. R = ratio, was taken from the 3.54-MPa sample.






74



0.125220 0.125215

0.125210 0.125205

0.125200
S1 =0 2 -I 5 231 2E 6T,3 I28E-T-2925E-,0I.T 51BE 2T" 4 0135E-15T*4 3D41E 18T'
0.125195
0 50 100 150 200 250
T (mK)


Figure 3.12: Background capacitance change due to the dielectric behavior of Stycast.

3.6.2 Magnetic Susceptibility Measurements

A 900 NMR pulse was used to measure the magnetic susceptibility, except for the first two samples (the 3.54- and 3.35-MPa samples), where two tipping angles of 200 and 900 were used. Free induction decay (FID) of a pulsed NMR signal was read from the oscilloscope to the computer. Then, a Labview6 program sampled 500,000 points of the FID with a digitizing rate of 5 M samples/s to get good resolution in the time domain and performed the FFT of the points. For the first two samples, the frequency resolution was 100 Hz, in which only a few points constituted a peak, because the NMR width was a 200 Hz. Later, resolution was improved to 10 Hz as shown in Fig. 3.13 to detect possible frequency shifts due to magnetic ordering. The FFT spectrum over a frequency range of 20 kHz was recorded to a file in the form of a 1D array. Also the first 60,000 points of the FID and the reading of PLM-4, produced by the PLM-4 integrating the first 1 ms of the FID, were recorded for each magnetic susceptibility measurement.
"National Instruments Corp., 6504 Bridge Point Parkway, Austin, TX, 78730-5039.






75



250
Sample Pressure - 2.88 MPa
PLM reading -443
T-0.542 mK
200
:i
150l





0 I I 249000 249500 250000 250500 251000 Frequency(Hz)



Figure 3.13: A typical FFT spectrum. A small peak at 250 kHz was a resonance from the tank circuit, which was subtracted from raw data for analysis by taking magnetization at high temperatures where there was no signal from the 3He clusters. NMR tipping angles

Two NMR tipping angles, about 200 and 900, for pulses were used depending on the temperature range in the 3.54- and 3.35-MPa samples. Because the 3.35MPa sample was the first sample studied, heating due to the 900 pulse and a long relaxation time were concerns. The relationship between the 200 and 900 data needed to combine them was made for the 3.35-MPa sample. A linear relation between the data was found, as shown in Fig. 3.14. In the other samples, 900 pulses were used to obtain the magnetic susceptibilities and measurements of T and T2.


Background magnetization

The NMR electronics and Fermi liquid contributed to the background. By averaging - 50 points measured at a high temperature about or above 100 mK, the background could be determined, since the signal of the 3He clusters was small in this






76




450
400
350
C Fit: PLM (0deg). -.528282.99726PLM(20degs) V 300
250


a 100
50

0 20 40 60 80 100 120 140 160
PLM 20 degrees data



Figure 3.14: Comparison between data (PLM-4 readings) with the two different angles.

temperature range. An example of the background signal is shown in Fig. 3.15. The background did not change from run to run.


3.6.3 Spin-Lattice Relaxation Time (Ti) Measurements

Several relaxation times were measured for each sample in order to estimate the necessary time interval between pulses and also for a better understanding of the 3He nano-cluster. At various temperatures, the equilibrium magnetization Mo was measured. Then after a variable time-period 7, the magnetization M(T) was obtained. Then the time interval was changed to repeat the same sequence. With T and Mo obtained, M(T) could be plotted as a function of time for each temperature.


3.6.4 Spin-Spin Relaxation Time (T2) Measurements

Spin-spin relaxation is caused by the interactions among the nuclear spins. Thus, spin-spin relaxation times T2 were measured using a spin echo technique in several






77



45
40
35
30
25



10


249000 249500 250000 250500 251000
Frequency(Hz)


Figure 3.15: A FFT spectrum taken at 110 mK in the 3.06-MPa sample.

samples in order to look for possible changes in spin-spin interactions. (For spin echo, see for instance [57].) The measurements were carried out manually and the echo amplitude was recorded from the digital oscilloscope on a notebook with the sketch of the echo shape.

To understand T2 measurement better, one may imagine the following situation. Suppose a 900 pulse is applied to a sample in an inhomogeneous magnetic field. If the pulse is intense enough, the entire magnetization will be rotated through an angle in a very short time. However, after a while the total magnetization vector amplitude will decrease because of the field inhomogeneity in the following way. The total magnetization vector is the sum of smaller magnetization vectors, each from a portion of the sample that feels its own local field. Therefore, each magnetization has its own characteristic Larmor frequency depending upon the local field each atom experiences. Some could precess faster, and some slower and as a result the different contributions of the magnetization will get out of phase with each other. The total dephasing time is called T2*, which is the addition of the effects from the field inhomogeneity and the






78

internal field.
1 1 +'= AHo (3.4) T2* T2 The first term of the equation is of interest, because it is an intrinsic spin-spin relaxation time, which reflects the interactions among the nuclear spins.

In this work a standard 900-r-1800 pulse sequence was employed. The scheme is shown in Fig. 3.16.





/2 x






CL





E
0


0
0 500 1000 1500
t 2t Time (a.u.)



Figure 3.16: Scheme of pulse sequence used to measure spin-spin relaxation time T2.













CHAPTER 4
EXPERIMENTAL RESULTS AND DISCUSSION


In this chapter, first, sample pressures of interest for studying the magnetic properties of 3He nano-clusters are described. Then the experimental results and the discussion are presented. The five sample pressures studied in this experiment were 3.54, 3.35, 3.06, 2.96, and 2.88 MPa, which were chosen to span the range from liquid droplets to solid clusters.


4.1 Sample Pressures of Interest


The five sample pressures, as determined below the phase separation temperature, are shown in Fig. 4.1.1 Upon the phase separation, 3He forms clusters, which contain almost pure 3He. Therefore, it might be supposed that the melting pressure of the 3He nano-clusters would be the same as that of pure bulk 3He. For several samples studied in this work, the clusters showed evidence of melting near the 3He melting pressures in Fig. 4.1. However, the melting was incomplete, based on the pressure change and a Curie-law susceptibility with a solid fraction remaining below the melting temperatures (see the results and discussion in this chapter).

In the 3.54-MPa sample, the 3He nano-clusters showed no indication of melting to the lowest temperature in this work, - 0.5 mK, as expected according to the pure bulk 3He melting pressure. For the 3.35-MPa sample, at an intermediate pressure, the clusters underwent partial melting at a temperature around 20 mK, after which the sample did not melt completely, as indicated by the Curie-law behavior of the
'Note: changes in pressure that occur at higher temperatures are not shown in Fig.
4.1.

79







80













solid 3He pure He Melting Curve


~T
Ps
3.54 MPa
3.5

3.35 MPa





33.0
L -2.96 MPa
2.88 MPa



liquid 3He

pure 4He Melting Curve
2.5
I .iI.. . .. . I .....I .... . . * . . . . . .Inel
10 100 1000 Temperature (mK)





Figure 4.1: Sample pressures of interest including pure bulk 3He and 4He melting pressures.






81

susceptibility. In the 3.06- and 2.96-MPa samples, the clusters underwent partial melting just below the phase separation temperature, with a fraction of the 3He remaining as solid at lower temperatures. In the 2.88-MPa sample, the pressure was below the pure 3He melting pressure curve at all temperatures. However, a solid fraction of 3He was found upon phase separation, as indicated by the susceptibility. This will be discussed in detail in Sect. 4.2


4.2 Experimental Results and Discussion


4.2.1 Spin-Lattice Relaxation Time (TI) Measurements

In this work, for each sample except the 2.96-MPa sample, the spin-lattice relaxation time (Ti) was measured at various temperatures. The measurements were necessary in order to determine the minimum time interval between the magnetic susceptibility measurements. Figure 4.3 shows a typical spin-lattice relaxation measurement. The initial Zeeman-exchange-lattice part of the recovery of the signal was only a few tens of seconds long, which was anomalously short compared to that in pure bulk 3He [58] and was also observed by Mikhin et al. [59]. The short T's might indicate distortion of the cluster at the interface with the 4He matrix. The short TI's are consistent with a qualitative model used by Bernier et al. [58] to explain the temperature-independent relaxation time below the phase separation temperature a 100 mK in their bulk 3He samples. They attributed the plateau as a result of the existence of small amount of 4He as lattice defects in bulk 3He, which would serve as pinning sites for dislocation lines in the 3He lattice. The vibration frequencies of the dislocations, which are strongly coupled to the lattice, vary from a few kHz to more than 100 MHz, which overlap the exchange frequencies. Thus, the Zeeman subsystem






82

has a coupling to the lattice, as shown in Fig. 4.2. Further consideration is needed to devise a model for the short T 's in the 3He clusters.

r ---------- -----------Zeeman




Exchange




4He
II




Lattice

3He excitationsl
L------------------------j Long relaxation


Heat Exchanger



Figure 4.2: Schmatic diagram of the various systems of excitation in 3He. Dislocations caused by the 4He matrix could account for the short TI's in this work.


The Zeeman-exchange-lattice part was followed by a very long recovery of the exchange/lattice bath to the heat exchanger, with a time constant of up to a few hours at the lowest temperature [60]. Magnetic susceptibility measurements were spaced several times the longest time constant in order to assure equilibrium. The long time constants, which were obtained by fitting the exchange/lattice-to-heat exchanger part






83

of the relaxation, are shown versus temperature for the samples in Fig. 4.4. For the 2.96-MPa sample, the time intervals were determined based on the T measurements for the 3.06-MPa sample.




0.0
Sample Pressure = 2.88 MPa
-0.5 - --Zeeman-to-Exchange T =1.08 mK

-1.0

-1.5

-2.0

' -2.5

-3.0

-3.5 -Exchange/Lattice-to-Heat Exchanger

-4.0 I I * I * I * I
0 20 40 60 80 100 ,(min)




Figure 4.3: A typical measurement of spin-lattice relaxation.




4.2.2 Sample Pressure = 3.54 MPa Pressure measurements

In the 3.54-MPa sample, partial melting/freezing upon cooling/warming did not occur, as shown in Fig. 4.5. Thus the clusters were all solid. This is as expected based on the melting pressure for pure bulk 3He.







84



















100000t. T2






1000
.. 1000.- o

[m"'-.


100

* 3.54-MPa Sample 0 3.35-MPa Sample
10-- 3.06-MPa Sample U
o 2.88-MPa Sample
------- Fit to 3.54-MPa Sample N

1 10
T(mK)





Figure 4.4: Spin-lattice relaxation time versus temperature for the samples studied.






85

The pressure showed a phase separation temperature of 180 mK, consistent with our calculations in Sect. 1.2.2, which indicated that essentially all of the 3He in the cell had separated in clusters and no 3He was remained dissolved in the solid 4He matrix, and an excess pressure of - 7 kPa, which was approximately a factor of two larger than that obtained by Panczyk et al. [32] for their sample pressure of 3.54 MPa. Ganshin et al. [61] observed an excess pressure of 30 kPa during a rapid cooling, which is similar to that for the 3.54-MPa sample by taking the difference in concentration into account, and 10 kPa during a stepwise cooling for an initial 3He concentration of 2.04%. Thus, the discrepancy between the result of Panczyk et al. and the excess pressure for the 3.54-MPa sample could be by the experimental uncertainty.

As mentioned in Chap. 2, one might be concerned that pressure of the cluster would not be transmitted to the bulk layer near the transducer. We exclude this possibility by the following explanation. Pressure changes with temperature at the constant-volume is related to the pressure changes that we measure by the following equation:
P dP dV
( )v = ( )[1 + VdP] (4.1) fT dT KTVrdP

where 1rT is the compressibility of the sample and Veff the effective volume that causes the pressure change. The term dP/dT is the measured pressure. If Vg is small, the second term of the right-hand of Eq. 4.1 would be significantly greater than 1, which in turn makes the term dP/dT smaller than the left hand side of Eq. 4.1 [53]. In our case, if the pressure measurement had not been at constant volume, the excess pressure would have been much smaller than that observed by other workers [32], [61]. Therefore, the pressure measurements were made under a constant volume condition and pressure was transmitted.







86























-1









0,
-5


-3

C- Phase Separation O
� -4

-5


-6

-7


-8
10 100
T(mK)





Figure 4.5: Pressure change at phase separation in the 3.54-MPa sample, taken on cooling.






87

Magnetic susceptibility measurements

The magnetic susceptibility was measured during warming using both the FFT, which had a resolution of 100 Hz, and the PLM-4. These were consistent with each other down to 1 mK with a similar value of the Weiss temperature. Below 1 mK the sensitivity setting of the oscilloscope was improper, which caused saturated signals in the recording of the FID. Therefore in the FFT the magnetic susceptibility showed a kink, which, as was realized later, was caused by the improper setting. For this reason, we employed only the PLM data, and the results are shown plotted in three different ways in Figs. 4.6, 4.7, and 4.8.

The magnetic susceptibility exhibited a similar behavior to the magnetization results of Hata et al. [20] (see Fig.1.4), which showed a Curie-Weiss law above 5TN and obeyed another Curie-Weiss law below 5TN. Our result showed a Weiss temperature

(0,) of a -0.25 mK and a 0.15 mK obtained by fitting the data above and below 4 mK, respectively, as shown in Fig. 4.6. The magnetization results of Hata et al. exhibited a molar volume dependence of TN as follows,


TN OC V16.5, (4.2)


where v is the molar volume. In their results, 0,, 2TN, which means that 0, obeys the same molar-volume dependence. Then our 0, -0.25 mK for the 3.54-MPa sample corresponds to a molar volume of a 21.3 cm3/mole, which is close to the molar volume of 4He in the temperature range of this work (a 20.95 cm3/mole). Thus, we concluded that the behavior in the magnetic susceptibility might be explained by the influence of the hcp 4He matrix, which has a smaller molar volume (a 21 cm3/mole) than 3He at this pressure.2 As a result of the van der Waals attraction to the 4He
2Pure bulk 3He has a molar volume of 24.12 cm3/mole at 3.54 MPa.






88

surface, the density of the 3He solid in the cluster would be higher at the interface than in the interior. Also the cluster size is only a few atomic spacing, as determined by the concentration of 3He and the silver particle size. For the concentration of 3He, 0.6%, and 70 nm silver particle size, the diameter of the cluster was estimated to be , 20 nm. Three layers of 3He at the interface occupies z 84% of the total volume in the cluster. This indicates that a substantial fraction of 3He atoms are located at the interface. A significantly higher density in the cluster than that of pure bulk 3He at the sample pressure could account for the behavior in the magnetic susceptibility, which is like that of pure bulk solid 3He at a molar volume near that of the hcp 4He matrix.


4.2.3 Sample Pressure = 3.35 MPa

Pressure measurements

The phase separation temperature, 180 mK, was near that for the 3.54-MPa sample, as shown by P(T) in Fig. 4.9. In the 3.35-MPa sample, the clusters underwent partial melting/freezing at , 20 mK (see Fig. 4.9). Hysteresis in temperature upon partial melting/freezing was observed. This behavior was also seen by Haley et al. [42] and may be related to the history dependent heat capacity seen by Schrenk et al. [1] The pressure increase due to the phase separation was similar in size to that in the 3.54-MPa sample. Again, this observation indicated that essentially all of the 3He in the cell had separated into clusters and no 3He was remained dissolved in the solid 4He matrix, as in the 3.54-MPa sample.







89

















0.012
Sample Pressure= 3.54 MPa
0.011

0.010 - ,

0.009 - 1,

0.0080.007 r*

0.006

0.005

, 0.004

0.003 0.002

0.001

0.000 .' ''s' ' '. .
-1 0 1 2 3 4 5 6 7 8 9 10 T (mK)





Figure 4.6: Inverse susceptibility X-v versus T of 3He nano-clusters for the 3.54-MPa sample. A fit is made separately for the data above and below 4 mK, which gives a Weiss temperature 0, of a -0.25 mK and a 0.15 mK, respectively.







90

















1300
1250 Sample Pressure = 3.54 MPa
1200
1150
1100
S.
1050
1000
950 - 0
_ 5 ---------- -------------------- ----900 - S *
C 850 O S
800 �
750 700 650
600 I I I * I
0 2 4 6 8 10 12 T(mK)




Figure 4.7: Magnetic susceptibility times T versus T of 3He nano-clusters in the 3.54MPa sample. A fit is made for the data above 5 mK, which gives : 926.36+0.66571 T(mK).







91


















2200
Sample Pressure = 3.54 MPa
2000 1800

1600

S1400

1200

S1000- Se

- 800

600

400 -*

200 OP
0 __ I * I I * I * I I I
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 1/T (mK1)




Figure 4.8: Magnetic susceptibility versus T- of 3He nano-clusters in the 3.54-MPa sample.







92


















0.0 \ Cooling A Warming

-2.5


" -5.0 A

Partial Melting/Freezing
-7.5


-10.0 Phase Separation


12.5
10 100 T(mK)



Figure 4.9: Pressure change in the 3.35-MPa sample. Partial melting and freezing were observed. Dotted line is the pure bulk 3He melting pressure [17].






93

Magnetic susceptibility measurements

After partial melting, the clusters showed a Curie-like contribution in the magnetic susceptibility, which indicated a solid fraction. By taking the ratio of the Curie constant in this sample to the one in the 3.54-MPa sample (all solid), the solid fraction of this sample was determined to be 77%, as shown in Fig. 4.10. The solid fraction is obtained by taking the ratio of the Curie constant for this sample to the one in the 3.54-MPa sample and multiply the ratio by v(3.35 MPa)/v(3.54 MPa), where v is the molar volume of the 3He in the cluster. The compressibility of hcp solid 4He at a molar volume of - 20.5 cm3/mole is - 0.03 MPa-', which is approximately constant over our sample pressure range. The compressibility is written as follows: dv
Kr- dP (4.3) vdP'

where T is the isothermal compressibility in pure solid 4He, and P the pressure. We define v, and v2 as the molar volumes of the 3.54-MPa sample and the 3.35-MPa sample, respectively. We calculate v2/vl by using Eq. 4.3 as follows:


V = 1 + KTdP = 1.006, (4.4) V1

because K-T a 0.03 MPa-1 and dP - 0.2 MPa. Therefore we obtain the solid fractions for the 3.35-MPa sample by taking the ratio of the Curie constant of the 3.54-MPa sample to the one in the 3.54-MPa sample. This arguement is applied to the samples described later in this chapter.

In this sample the magnetic susceptibility followed the Curie law down to : 0.6 mK with a Weiss temperature 9,=-5 � 5 pK. Thus it was paramagnetic down to the lowest temperature measured, with a very slight antiferromagnetic tendency. However, scatter was too large to conclude it to be antiferromagnetic. This scat-




Full Text

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MAGNETIC SUSCEPTIBILITY AND PRESSURE MEASUREMENTS IN HELIUM-THREE NANO-CLUSTERS By NAOKI MATSUNAGA 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 2000

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ACKNOWLEDGMENTS I would like to express my most sincere gratitude and appreciation to my advisor, Dr. E. Dwight Adams, for his invaluable guidance and financial support during the past several years. His patience and understanding have made my graduate study less stressful. I would like to thank Dr. Yasu Takano for his help when I was working in the Microkelvin Laboratory. He was always available whenever I needed to consult him. I would like to thank Drs. G. G. Ihas, Chuck Hooper, and Alexander Angerhofer for serving on my supervisory committee. Special thanks go to the people in the Microkelvin Laboratory, especially Drs. Jian-Sheng Xia and Volodya A. Shvarts. Volodya helped me in all the stages in my experiment. Though Dr. Erwin Schuberth of Walther-Meissner Institut stayed here as a visitor to the lowtemperature group only for several months in 1999, he spent much time with me to establish the data acquisition scheme in the work. Many thanks go to the cryoengineers, Greg Labbe and Brian Lothrop; and to the machinists and the electronic engineers who provided outstanding technical support for this work. Last but not least, I would like to thank my family in Japan for their support throughout my graduate study. This work was supported in part by the National Science Foundation, Grant DMR-9800712. ii

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TABLE OF CONTENTS ACKNOWLEDGMENTS ii LIST OF FIGURES viii ABSTRACT ix CHAPTERS 1 INTRODUCTION AND BACKGROUND 1 1.1 Quantum Crystal 3 1.1.1 Nuclear Magnetism of bcc ^He 3 1.1.2 Phase Diagrams of ^He and '^He 16 1.2 Phase Separation in ^He-^He Mixture 18 1.2.1 Regular Solution Theory 18 1.2.2 Calculation of Phase Separation Temperature vs. Pressure and ^He Concentration 22 1.3 Nuclear Magnetism of ^He Nano-Clusters 23 2 EXPERIMENTAL APPARATUS 28 2.1 General Description 28 2.2 The Cryostat and Dilution Refrigerator 29 2.2.1 Principle of Dilution Refrigeration 29 2.2.2 The 1 K Pot 30 2.3 Magnetic Refrigerator 32 2.4 Superconducting Magnets 36 2.5 ^He Melting Pressure Thermometry 38 2.5.1 The ^He Melting Pressure Thermometer Cell Design 39 2.5.2 Electronics for ^He Melting Pressure Thermometry and Sample Pressure Measurements 42 2.6 The Experimental Platform on the Nuclear Stage 47 2.7 The NMR Setup 49 2.8 Pressure Measurement Setup 53 2.9 Method of Temperature Regulation 56 3 EXPERIMENTAL PROCEDURES 58 3.1 Cooldown Procedures 58 3.2 Calibration of the Mixture Sample Cell and Sample Preparation ... 59 iii

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3.3 Calibration and Preparation of the ^He Melting Pressure Thermometer 64 3.4 Sample Cooling 64 3.4.1 Technique and Setup 64 3.4.2 Phase Separation 67 3.4.3 NMR Tuning 67 3.4.4 High Temperature Data Acquisition 68 3.4.5 Demagnetization and Magnetization of the Nuclear Stage ... 68 3.5 Temperature Scale 69 3.6 Data Acquisition 70 3.6.1 Sample Pressure Measurements 70 3.6.2 Magnetic Susceptibility Measurements 74 3.6.3 Spin-Lattice Relaxation Time (Ti) Measurements 76 3.6.4 Spin-Spin Relaxation Time (T^) Measurements 76 4 EXPERIMENTAL RESULTS AND DISCUSSION 79 4.1 Sample Pressures of Interest 79 4.2 Experimental Results and Discussion 81 4.2.1 Spin-Lattice Relaxation Time (Ti) Measurements 81 4.2.2 Sample Pressure = 3.54 MPa 83 4.2.3 Sample Pressure = 3.35 MPa 88 4.2.4 Sample Pressure = 3.06 MPa 95 4.2.5 Sample Pressure = 2.96 MPa 107 4.2.6 Sample Pressure = 2.88 MPa 120 5 MODEL FOR INTERPRETING RESULTS AND SUGGESTED FUTURE WORK 125 5.1 Model for Interpreting the Results and Discussion 125 5.1.1 Partial Melting 125 5.1.2 Anomalously Short Ti's 127 5.1.3 Magnetic Susceptibihty for the 3.54-MPa Sample 127 5.1.4 Kink in Magnetic Susceptibility and Frequency Shift 128 5.1.5 Two-dimensional-like Magnetic Susceptibility 128 5.1.6 TAf(F) for the Clusters and for Pure Bulk ^He 128 5.2 Suggestion for Future Work on ^He Nano-Clusters 129 5.2.1 Magnetic Susceptibility Measurements with a Cold Preamplifier 129 5.2.2 Temperature Scale Improvement 131 5.2.3 Pressure Measurements of ^He Nano-Clusters 132 5.2.4 Heat Capacity Measurements 133 5.2.5 Field Sweep to Look for a Frequency Shift 133 5.2.6 Magnetic and Pressure Study of ^He Nano-Clusters with a Larger Concentration 133 REFERENCES 135 iv

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BIOGRAPHICAL SKETCH

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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 SUSCEPTIBILITY AND PRESSURE MEASUREMENTS IN HELIUM-THREE NANO-CLUSTERS By Naoki Matsunaga December 2000 Chairman: E. Dwight Adams Major Department: Physics Simultaneous measurements of pressure, spin-lattice relaxation times (Ti), spinspin relaxation times (T2), and magnetic susceptibility (x) were made in ^He nanoclusters embedded in an hep ^He matrix, following phase separation of the mixture. Based on the pressure changes and magnetic susceptibility, three different types of behavior were identified, for which the clusters either remained all solid at the lowest temperature, underwent partial melting upon further cooling after phase separation, or separated with liquid already present. Magnetic measurements were extended from 0.5 mK to above 10 mK for these three different pressure ranges from 2.88 to 3.54 MPa. In the temperature range of the measurements, where pure bulk ^He would be entirely liquid, the magnetic behavior of the clusters for P < 3.35 MPa indicated ^He solid fraction of about 77, 54, 23, and 19% respectively. The magnetic susceptibility X of the 3.35-MPa sample followed a Curie law to 0.5 mK with = — 5 ± 5 /xK. For the 3.06and 2.96-MPa samples, a kink in x was observed at 1.1 mK, which vi

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is approximately the magnetic ordering temperature Tj^ that would be expected for pure bulk ^He if it existed at this pressure. However, x was almost constant down to 0.5 mK, with no drop at 1.1 mK, and a frequency shift of » 10 ± 10 Hz, below 1.1 mK. Thus if there was magnetic ordering it appears to be quite different than for bulk ^He. For the 2.88-MPa sample, x followed a CurieWeiss law with a Weiss temperature 6^ = 140 /iK, indicative of a ferromagnetic tendency, which is similar to that seen in 2D films. Several spin-lattice relaxation times Ti were measured for each sample to estimate the necessary time interval between pulses. Anomalously short Zeeman-exchangelattice Ti 's were observed that may indicate distortion of the cluster at the interface with the "He. Spin-spin relaxation times T2 were measured in several samples in order to look for possible changes in the spin-spin interactions. No perceptible change was observed in measurements above and below 1.1 mK. vii

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CHAPTER 1 INTRODUCTION AND BACKGROUND Magnetic properties of solid ^He resulting from exchange of various numbers of spins have been studied in a variety of situations including the bulk solid phases and two-dimensional (2D) layers. In this work, we have studied the magnetic properties of ^He nano-clusters in phase-separated ^He-^He mixtures confined in a silver sinter, which provides a different geometry for studying exchange processes. Pure bulk ^He becomes magnetically ordered at temperatures < 1 mK via exchange interactions made possible because of the large zero-point motion of ^He. In nano-clusters, temperatures of possible magnetic ordering, reported by Schrenk et al., were around 1 mK, near that of pure bulk ^He at the melting pressure [1]. They observed a history-dependence in the ordering temperature at pressures below the melting pressure in pure bulk ^He (see Sect. 1.3 for details). Their specific-heat data showed no latent heat at a peak that is a characteristic of the first-order transition seen in pure bulk ^He by Grey wall et al. (see Sect. 1.1.1) [2]. Furthermore, integrating their C/T from the lowest temperature to well above the peak, it was found that a substantial fraction of the total spin entropy was not removed, compared with the large entropy removal in the result of Grey wall et al. Therefore the magnetic properties of the nano-clusters seemed to be different from those of pure bulk ^He. Three main questions motivated this work: 1. Do the ^He nano-clusters magnetically order? 2. If so, what is the nature of the transition and the spin configuration? 3. What the exchange processes determine the transition?

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• ' " 2 The main task of the work was to look for magnetic ordering in the nano-clusters at pressures below pure bulk ^He melting and to characterize the nature of the ordering, if it exists. Pulsed NMR has been used for the investigation. The most important achievement would be to determine the NMR spectrum in the ordered phase in the nano-clusters, which would provide clear information on the type of ordering. This dissertation was devoted to studying the magnetic properties of the nanoclusters by pulsed NMR. We have measured the magnetic susceptibility, spin-lattice relaxation times (Ti), and spin-spin relaxation times {T2) versus temperatures of ^He nano-clusters embedded in a hep matrix of '^He, following phase separation of the mixture. Also pressure changes in the sample were studied in this work. Pressure measurements are essential to monitor sample formation, annealing, phase separation, and partial melting, if it occurs. Additionally pressure measurements would give us another approach to study the nano-clusters and an idea of the kinetics of the isotopes in the geometry. This thesis is arranged into five chapters. In the first chapter, quantum crystal; phase diagrams of pure bulk ^He, pure bulk ^He, and ^He-'^He mixtures; the multiple-exchange model, a review of nuclear magnetism of pure bulk ^He; phase separation in ^He-^He mixtures, calculations of phase separation temperatures; and nuclear magnetism of ^He nano-clusters are discussed, in this order. Chapter two is devoted to the experimental apparatus used in this work. Chapter three describes the experimental procedures, such as calibration of the experimental cells and tuning of the pulsed NMR. The experimental results and discussion are presented in chapter four. Chapter five is a summary of this experiment and future work, followed by an appendix, which discusses a cold preamplifier for pulsed NMR.

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3 1.1 Quantum Crystal In a quantum solid, atoms have a large zero-point motion around their equilibrium sites, moving a large fraction of the distance between neighboring atoms. This has several important consequences: 1. The large zero-point motion causes an anharmonicity such that the conventional approach for harmonic crystal does not hold. 2. The atoms can change their positions by tunneling. The last fact is particularly important since it implies that there is a finite overlap of the wavefunctions of neighboring atoms. This provides an atomic exchange process [3] that causes ^He nuclei to magnetically order near 1 mK at melting pressure. (See, for instance, Ref. [4].) For a classical solid the nuclear dipolar interactions would cause ordering at a temperature of ~ 10~^ K [5]. 1.1.1 Nuclear Magnetism of bcc ^He In this section, nuclear magnetism of solid ^He relevant to this work will be described starting from the multiple-spin-exchange model. Multiple exchange model Before the multiple-exchange model was proposed, the Hamiltonian for solid ^He was written with simple pairwise exchange as follows: H=-2Y,JSiSi-'£f^iB, (1.1) i
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the second terms represent the exchange Hamiltonian of the two neaxest neighbors and the Zeeman Hamiltonian, respectively. To obtain thermodynamic quantities such as pressure, magnetization, and heat capacity, one must calculate the partition function, Z=tr[exp{—H/kBT)], where /cb is the Boltzmann constant. For thermodynamic calculations [6], the free energy F is written as F=-^lnZ, (1.2) where /3 = l/^sT. High-temperature series expansions of F for spin = 1/2 by Baker et al. [7] gives ln2 + ea^ + • • • + ^(7^5^)'(1 + ksO^P +^ + -•) + (1.3) where A^^ is Avogadro's number and 62, 63 and 9yj etc. are related to the exchange energy J. For the Heisenberg nearest-neighbor model, e2=12J^, e3=12J^, and 9y, = xJ/2kB, where x = number of nearest neighbors, which is 8 for the bcc and 12 for the hep solid. From this high-temperature expansion of F, pressure, magnetic susceptibility, and specific heat are expressed as Pv{T, B) = ^£^^[3x-^a:2^. • +y\2+l2x+b2x-'^)+y'{-1.33-23x+•)], (1.4) /io (d''F\ _ C and in zero field C, = -T^ = ^ie,P^-e,P^ + ...), (1.6)

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5 where x=J/kBT, y=iJ,B/kBT, v is the molar volume, /xo the permeability of free space, A=0^^-a2/8/i;B^, and C = HqNa {'jh/2f /ksv. For bcc ^He, 9y,=iJ/kB, and A= AiJ/ksY. The experiment by Kirk and Adams in 1971 showed for the first time that the simple Heisenberg nearest-neighbor Hamiltonian was inadequate to describe the system [8]. They measured Py{T,B) for 5 up to 6 T and for temperatures from 20 mK to 100 mK to test the expansion for the Hamiltonian in Eq. 1.1. There was a clear discrepancy between the theory and the data, as seen in Fig. 1.1, which was caused by the y"^ terms in Eq. 1.4 that overestimated the pressure change by a factor of two. In 1974 Halperin et al. [9] measured the latent heat along the melting curve of solid ^He using a Pomeranchuk cell and observed an abrupt drop in entropy at w 1.1 mK, indicating a first-order magnetic transition. Again the measurement suggested that the simple Heisenberg nearest-neighbor Hamiltonian, Eq. 1.1, was inadequate, because the Hamiltonian gave a second-order transition at 2 mK. In 1977, Kummer et al. carried out an experiment similar to the one performed by Halperin et al. but in a finite field up to 1.2 T and found that the ordering transition temperature decreased slightly with increasing field up to 0.4 T [10]. With higher fields, the behavior was reversed and an increase in field raised the ordering temperature, causing a kink in the phase diagram. The kink was an indication of a new ordered phase that is known today as the high-field phase (HFP). Hetherington and Willard [11] proposed cyclic particle-exchange processes among multiple spins to explain the experimental data. Following Dirac [12], the multipleexchange Hamiltonian can be written as : (1.7)

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6 4.020 30 ^~~AQ 50 Figure 1.1: Pressure difference versus T"^ for v = 23.88 cm^/mole in fields up to 6 T. Various symbols for a given B are for different traversals of the temperature region. The solid curves are calculated behavior based on Eq. 1.4. W. P. Kirk and E.D. Adams, Phys. Rev. Lett., 27:393, 1971, ©(1971) by American Physical Society.

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7 where n is the number of particles in the exchange process and a represents a particular exchange process. Here are several processes for three-particle, four-particle, etc., exchange and P„ is the permutation operator. For exchange of an odd number of particles, (—1)^ is —1 (ferromagnetic); for an even number of particles, it is 1 (anti-ferromagnetic) . Figure 1.2: Various exchange processes taking place in bcc solid ^He. Roger et al. [13] described a simple model to explain why four-particle exchange is important in bcc ^He. In Fig. 1.2 we see that in a (110) plane four atoms in a ring have eight nearest neighbors within the plane. If the space between the atoms expands a little, the four atoms in the ring can undergo a cyclic exchange. However, in two-particle and three-particle exchanges, in order to exchange positions, the atoms encounter the hard cores of neighbors. Following Thouless (1965) [14], Roger et al. wrote the Hamiltonian for up to four-particle exchange processes: Hex = -Jnn X] -^i + ^[Pijk + {Pijk)~^] + -f^four-particle , (1-8) ij i,j,k

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8 F P f^four-particle = -Kp XI i^ijkl + (^ijkl) ^] ~ ^P^l^ijkl + {P^jkl) ] O-'^) i,j,k,l ijkl Here Pij'^, Pijk'^, and Fijfc/'^ are the cyclic permutation operators for two-, three-, and four-particle exchange, respectively, and J„„, Jj, Kp, and Kp stand for nearestneighbor, three-particle, four-particle planar, and four-particle folded exchange processes, respectively (see Fig. 1.2). For bcc ^He, Roger et al. found that three-particle ( Jj) and planar four-particle {Kp) exchange would give reasonable agreement with existing experimental data. Thus they reduced the exchange Hamiltonian Eq. 1.8 and obtained the Weiss temperature and the coefficient 62 from high-temperature series expansion of F as follows: 18 e^ = —{Kp-2Jt) (1.10) Kb and e, = {2Anj? \KpJ, + f^Kl). (1.11) Equation 1.10 shows the competition of three(ferromagnetic) and four(antiferromagnetic) particle exchanges. For bcc ^He at the melting density, a fit of the experimental results of the high-temperature expansion coefficient 62 and the mean spin-wave velocity give Jt=— 0.13 mK and Kp=—Q.Z%h mK, which yield ^„,=— 2.25 mK and the ordering temperature (T^r) of 1.2 mK [13], [15], [16]. On the presently accepted melting pressure scale, Tjv has the value of 0.934 mK [17]. A significance of this model is that it gives the U2D2 structure in the ordered low-field phase (see subsequent sections for details), a canted structure for the high-field phase, and a first-order transition between the paramagnetic phase and the high-field phase in the low-field region. Stipdonk and Hetherington [18] have used three exchange processes J„„, J(, and Kp and obtained a slightly better fit to experimental data.

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9 0.8 0.6 --0.4 n 0.2 " 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 ' T (niK) Figure 1.3: B-T diagram of solid ^He at melting pressure density. From Ref. [19]. Magnetization and heat capacity of bcc ^He In this section, we present relevant magnetization and heat capacity data in bcc ^He to be compared with the results of this work. Hata et al. [20] measured the magnetization through Tat over a wide range of molar volumes. They formed a sample inside an experimental cell with silver sinter and measured static magnetization with a field of 26 mT using a SQUID. They observed a jump in (inverse) magnetization at T^, as shown in Fig. 1.4, which indicated that the transition was first-order. For each sample, the magnetization started to deviate from a Curie-Weiss law at T w bT^ and dropped to 39% of its maximum magnetization (Mmax) at T^, and then stayed constant to the lowest temperature. Greywall et al. measured the heat capacity from 0.6 mK to 10 mK in fields 0 < B < 1 T [2]. For zero field they found a very sharp peak, with C„ changing by nearly two orders of magnitude in a 100 /xK interval, as shown in Fig. 1.5. The peak is clear evidence of a latent heat and, hence, a first-order transition. In the result of Greywall et al. integration of their heat capacity showed that the entropy removed I 1 vT High-field phase > 2 — <-• ( 1-^ — 7 Paramagnetic phase Low-field phase J L _J L_

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10 r T I I r • cm^/mole 23 01 : — 1 1 1 1 1 < 1 1 c ) 2 U 6 8 10 TEMPERATURE (mK) Figure 1.4: Inverse magnetization as a function of temperature for several molar volumes. The deviation from CurieWeiss law begins at T 5Tn and is shown by the arrows. Prom Ref. [20]. ©(1988) by Kluwer Academic/ Plenum Publishers.

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11 10 1 O.I 0.001 0.1 0.5 I 5 10 r(inK) Figure 1.5: The heat capacity of bcc ^ He at a molar volume of 24.13 cm^/mole near the PP-LFP transition Tj^. A: zero field. B: 1.0 T. Note there is a latent heat associated with a first-order transition. Compare A to Fig. 1.13. From Ref. [2]. ©(1988) Amrican Physical Society. was 0.4 R ln2 just in going through the transition, which is to be compared to the entropy removal in the result of Schrenk et al. on ^He nano-clusters (see Sect. 1.3 for details). Properties of the low field phase (LFP) in bcc solid ^He Since the magnetic field in this work is less than 0.4 T (see Fig. 1.3), the physical properties of the ^He nano-clusters are to be compared with those in the low field phase (LFP) in bcc solid ^He. The nature of the LFP was studied by NMR measurements by the Florida group and Osheroff et al. [21], [22]. Both groups observed a very large frequency shift of ~ 1 MHz, which indicated that the ordered phase does not have cubic symmetry. Osheroff et al. determined the phase to be consistent with U2D2 as shown in Fig. 1.6.

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12 Figure 1.6: Spin configuration in the LFP. Prom Ref. [23]. ©(1992) by Kluwer Academic/Plenum Publishers.

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13 The U2D2 magnetic structure In the U2D2 phase, shown in Fig. 1.6, there are two anisotropy axes. The real— space unit vector, which is normal to the ferromagnetic plane, and the spin unit vector, d, to which the sublattice magnetization is either parallel or anti-parallel at — # — * zero field. In a finite magnetic field H, d rotates within the plane until it is normal to H. There are usually three domains in a U2D2 single crystal. Because of the low symmetry, the dipole-dipole energy is highly anisotropic, which causes a large frequency shift. For each domain, there are two modes whose frequencies, a;+ and are given by the following equations: {oo^ = ^[wo' + Qo' ± \/{ioi-niy + Aujo''no''cosWi], (1.12) where ojq = is the Larmor frequency with 7=32.43 MHz/T, is the antiferromagnetic resonance frequency, which depends upon temperature, and cos9i=li-h — with h, an unit vector along the magnetic field [21]. Osheroff et al. fitted their data and found that cos% = 1.007 (1.13) i=domain (see Fig. 1.7), which satisfies the orthnormal condition that implies that the three magnetic domains are orthogonal to each other. Also they determined the frequency over the entire temperature range from T" ~ 0.1 mK to Tjv 3S follows: = [6.839 3.663(^; + 1.452(^)' 1.882(^n. (1.14) The resonance frequency varies from 525 kHz at T^r to 825 kHz at T ^ 0, which is to be compared to the results in this work later.

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14 LARMOR FREQUENCY (kHz) Figure 1.7: Observation of the field dependence of the resonance frequencies o;"^ of NMR spectra for three domains of a single crystal of bcc ^He at 0.487 mK. Solid lines are obtained from Eq. 1.12 with fio=777.7 kHz. Prom Ref. [23]. ©(1992) by Kluwer Academic/Plenum Publishers. 2D films, two-dimensional quantum solids In this subsection, 2D films are mentioned, to be compared to the results of this work later. In some situations, depending upon the pressure, the ^He at the interface of the nano-clusters may be solid whereas the rest of the cluster remains liquid. In this situation, the clusters would have several solid layers at the interface, which in turn might cause magnetic properties similar to those of 2D films. Solid ^He adsorbed on a substrate, typically graphite, provides an example of low dimensional magnetism. The second atomic layer of adsorbed ^He shows an interesting evolution from antiferromagnetism to ferromagnetism as a function of its areal density x. Presently, multiple-spin exchange interactions are used to interpret the behavior [13].

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15 At low densities, there is competition between odd and even number of exchange processes of ^He atoms in a triangular lattice, producing a highly frustrated magnetic system. On the other hand, at higher areal densities, ^He films exhibit a pure 2D Heisenberg ferromagnetic behavior because of the dominance of three-particle exchange processes [24]. However, no magnetic ordering has been observed in the 2D films. 1.6 1.4 O 1.2 1.0 0.8 10 100 T(mK) Figure 1.8: Magnetic susceptibility of ^He absorbed on graphite for several coverage in the second layer intermediate regime. Dots correspond to the commensurate 4/7 phase at a; = 1.62, and diamond to a denser phase (12/19, x = 1.71) where x is the coverage. The lines are fits using a high-temperature series expansion of the multiple-spin-exchange model. Open squares: x = 1.66 for which the fit is obtained by assuming that the 4/7 and 12/19 phases are coexisting. See Ref. [24] for details. ©(1998) by Kluwer Academic/Plenum Publishers. One of the samples studied in this work exhibited behavior similar to that for x = 1.66 in Fig. 1.8 {x=l corresponds to the areal density of the densest (saturated) first layer of ^He absorbed on a substrate). Our results and a discussion relating to this are given in Chap. 4.

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16 Ishida et al. [25] recently observed a peak in their specific heat in 2D ^He on graphite at around 300 /iK, which accounts for the missing entropy of the broad maximum near 3 mK observed previously by Grey wall and Busch [26]. This may be related to the "missing entropy" in the specific heat of nano-clusters of Schrenk et al. [!]• 1.1.2 Phase Diagrams of ^He and ^He In this section, phase diagrams of pure bulk ^He, pure bulk ^He, and ^He-'*He mixtures are presented in order to show the crystallographic structure and to discuss the effect of adding a small concentration of ^He to ^He. Figure 1.9: P-T diagrams for pure bulk ^He and ^He. Both pure bulk solid ^He and ^He exist in three phases with diff'erent crystal structures, namely hexagonal-closed packed (hep), faced-centered cubic (fee), and body-centered cubic (bcc). As a result of the zero-point motion, ^He does not solidify until the pressure is about 2.93 MPa. The crystal structure is bcc for low-pressure solid, and hep for high-pressure solid (hep and fee phases are not shown in Fig. 1.9)

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17 T»mperoture(K) Figure 1.10: Equilibrium P-T phase diagrams for pure "He and for "He with 5% ^He concentration (x=0 and 0.05, respectively). The dashed (solid) lines show the pure "He ("He with 5% of ^He) melting line, A line between normal (I) and superfluid (H), and bcc-hcp boundary. Prom Ref. [27]. ©(1987) American Physical Society. [28]. In "He there is a small region of bcc phase in the P-T diagram, as shown in Fig. 1.9, and the hep structure occupies a much larger area. The molar volume of the pure bulk ^He and "He are quite different, « 24 cm^/mol for ^He, whereas it is « 21 cm^/mol for "He, near their melting pressures in the temperature range in this work. Thus, this difference in molar volume could have a significant effect on the magnetic properties of the phase-separated ^He nano-clusters, because of the mismatch in molar volume between the nano-clusters and the "He matrix at the interface. In 3He-"He mixtures, bcc "He occupies a larger region in the P-T diagram than in pure bulk "He, as shown in Fig. 1.10 for a ^He concentration of 5%. As the mixtures are cooled through the bcc-hcp transformation line, the transformation takes place more slowly than in pure "He since the enhancement of the bcc region shifts the transformation line to lower temperatures. Thus, if the mixtures are cooled too fast for the transformation to take place, the bcc phase could persist as a meta-stable state. As discussed in Chap. 4, meta-stable bcc "He may have persisted in some

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18 low-pressure samples studied in this work, as the samples were cooled after they were annealed. 1.2 Phase Separation in ^He-'*He Mixture 1.2.1 Regular Solution Theory In 1962 Edwards et al. discovered a Atype ordering in the heat capacity of a sample of solid ^He with a ''He molar fraction of 0.277 [29]. They showed that the entropy of the A anomaly, A5'=/ {Cy/T)dT, was the classical entropy of mixing of a regular solution of ^He in ^He [30], ASm = -R[il-x)\n{l-x) + x\nx], ' (1.15) where x is the molar fraction of ^He or ^He and m stands for mixing. They concluded that the anomaly was the onset of a separation into two phases at a temperature Tps that depended on x. Edwards et al. extended their measurements over a wide range of concentrations and compared them with the regular solution theory. The regular solution model connects the Gibbs free energy of individual isotopes to that of the mixture via g{P, T, x) = {lx)g,{P, T) + xg,{P, T) TS^ + gE{P, T, x), (1.16) where x is the concentration of ^He, g^, g^, and g are the Gibbs free energy of ^He, ^He, and mixture, respectively, and gs is the excess free energy per atom. In a regular solution gE has the simple form gs Ax{l x), (1.17)

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19 where ^ is a function of pressure only [31]. Solving Eqs. 1.15-1.17 below a critical temperature, Tc=A/2kB, with the assumption that there is no structural differences between the constituents, one can find the phase separation temperature and the heat capacity, respectively, as follows: and (1.19) — r 4x{l-i) where x is a function of T given by Eq. 1.18 and t=T /Tc=2kBT / A. Equation 1.18 determines the temperature of phase separation, Tps(x). The specific heat of the mixture, given by Eq. 1.19, follows a universal curve, as seen in Fig. 1.11, independent of x. These arguments are based on the assumptions that the lattice and nuclear-spin heat capacities are small. In other words, is a function of P only and gz{P,T)^gz{P,Q)-kBT ln2. Equations 1.18 and 1.19 fitted the heat capacity and phase separation temperature of Edwards et al. quite closely, which gave A/kB= 0.76 K at 3.6158 MPa [29]. The critical temperature A/2kB and its pressure dependence were investigated by Panczyk et al. [32], and then explained by several theorists [33], [34], [35], [36], [37], [38], [39]. There were two main assumptions in those works as follows. The first was that the isotopes are randomly arranged in the crystal lattice, which is in agreement with classical mixing entropy. The second was that the energy change upon mixing the pure isotopes is mostly the work done in compressing the ^He and expanding '*He, so as to bring the two crystals to the same lattice spacing. These assumptions neglected structural difl'erences in the pure isotopes.

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20 As described above, the phase separation line is symmetric about x=l/2. However, Panczyk et al. observed an asymmetry in their experiments on constantvolume pressure measurements (see points in Fig. 1.11) [32]. They measured the isochoric pressure Pv{T), and obtained the excess pressure Pe{x) directly at Tps{x), where In other words, they measured Pv{T) to determine the phase separation curve. The phase separation temperatures obtained by Panczyk et al. indicated that there was an asymmetry when x < 0.3. Mullin took the asymmetry into account and modified the regular solution theory as follows [33]: where e ~ -0.1, according to Mullin's calculation and the measurement of Panczyk et al. With the asymmetry included, the equations for the phase separation curve, the pressure at constant volume and the specific heat, which were derived from the equality of the chemical potentials of ^He and "He, and //4, are complicated and had to be solved numerically. The data of Edwards et al. gave e=0±0.006. Ehrlich and Simmons, using x-ray measurements of Tps for a series of bcc single crystals with concentrations from a; = 0.1 to 0.7, found e=0±0.01 [40]. The asymmetry observed by Panczyk et al. has been attributed to a crystallographic transformation of solid "He from bcc phase to hep phase in "He crystal, which took place slowly in their experiment (see Fig. 1.11 for the points measured by Panczyk et al.). (1.20) gs = Ax{x + ex) (1.21)

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21 Mofm I I I I I I I 0 20 40 60 80 100 X (%) Figure 1.11: The equilibrium phase diagram for hep and bee helium at 34 atm, assuming that hep and bee mixtures are regular. The hep, bcci, and bcc2 phases are labeled h, bi, and ba (1, 2, and hep stand for "He-enriched phase, ^He-enriched phase, and hep "He, respectively). The bi-ba equilibrium line is given by Eq. 1.18. It is extended into the h+h2 region for comparison with the h-b2 equilibrium line. The horizontal line represents the temperature at which h, bi, and ba may be in equilibrium. The points are from [32]. Although the points agree with the regular solution theory well, the measurements did not achieve equilibrium with respect to the bcc-hcp transformation in "He. Thus, the meta-stable "He might have persisted. D. O. Edwards and S. Balibar, Phys. Rev. B., 39:4083, 1989. ©(1989) by American Physical Society.

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22 1.2.2 Calculation of Phase Separation Temperature vs. Pressure and ^He Concentration In the last section, the expression for the phase separation temperature based on the regular solution (binary solution) was obtained. This allows us to calculate the phase separation temperature as a function of pressure and the initial ^He concentration. These calculations for our sample pressures are to be compared for consistency with the phase separation temperatures observed in the pressure measurements. (See the discussion given for each sample in Chap. 4.) Edwards and Balibar (1988) reviewed extensively the research on isotopic mixtures of ^He and ^He [30]. According to these workers, the phase separation temperatures of the mixtures when << 1 can be calculated as follows, ^'^^ ln[(x.)-i-l] ' (1-22) where Xh (h means that "He is hep) is an initial concentration of ^He in hep "He and Tps is the phase separation temperature. Based on empirical data, the parameter Ah is given by . , 0.43(F3.6158) Ah = 0.76 + '—^ L, (1.23) where R = 8.31443(J/moleK) is the gas constant [29] and the units for T, v, and P are K, cc/mole, and MPa, respectively. The ^He fre^energy difference between the meta-stable and stable crystal structures is expressed as: A3 = (P I^)[Svl + i^3(P P^)] (1.24) with _ Avs _ 6vl dv3 AP 10.605 -3.6158" ^^-^^^

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23 Here P3 is the pressure at the bcc-hcp boundary in ^He at T ~ 0, which was obtained by extrapolating the pressure of the bcc-hcp boundary in pure ^He measured by Straty and Adams to T ~ 0 [28] and found to be 10.605 MPa; Sv^ and Sv^, which were found to be -0.09 and -0.176 cc/mole, are the molar volume differences between bcc and hep ^He at 10.605 MPa and 3.6158 MPa, respectively; and is a parameter which was obtained by fitting the existing experimental data [32], [41]. Thus one can calculate the phase separation temperature as a function of the initial ^He concentration and the sample pressure, with the result shown in Fig. 1.12. We calculated the phase separation temperatures for our sample pressures in order to check for consistency between our results and the calculations. 0.28 0.26 0.24 5" 0.22 3 0.20 0.18 E O 0.16 I0.14 0.12 0.10 Figure 1.12: Calculation of phase separation temperatures for various pressures, when the initial ^He concentration is small {xh << 1). 1.3 Nuclear Magnetism of ^He Nano-Clusters As discussed in Sect. 1.2, a homogeneous mixture of ^He and ^He separates into ^He-rich and ^He-rich phases upon phase separation. Below the phase separation 26atm 30 atm 34atm 35 atm 1 1 1 -0— P = -^p = -0— P = I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ^He Concentration (%)

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24 temperature, Tps, the '^He-rich matrix has imbedded clusters of almost pure ^He. In the case of nano-clusters, which are about 20 nm in diameter in our packed silver sinters, a significant fraction of ^He would be within a few interatomic distance of the interface between hep '^He matrix and the ^He nano-clusters. The molar volume of the hep matrix is about 21 cm^/mole, comparable to that of ^He in the hep phase (see Sect. 1.1.2). Bulk ^He at the pressures involved has a bcc structure with a much lower density. Thus, there must be a mismatch in density at the interface. However, so far the structure and density profile of the nano-clusters are not known and are likely to be different from that of bulk solid ^He. It is likely that the interface between the two mismatched structures has a significant effect on the properties of the nano-clusters. Schrenk et al. measured the heat capacity of ^He nano-clusters embedded in a hep matrix of ''He, following phase separation of the mixture, which had an initial ^He concentration of 1%. They observed a history-dependence in the peak in C„, which they claimed to be Tn, at pressures as low as 700 kPa below the melting pressure in pure bulk ^He, as shown in Fig. 1.13 [1]. They found no latent heat that would indicated a first-order transition at the peak in C„. This is in contrast to the strong first-order transition seen in pure bulk ^He by Greywall et al. at (see Sect. 1.1.1) [2]. Furthermore, integrating C/T of Schrenk et al. from the lowest temperature to well above the peak, it was found that only 0.3 R ln2 of the spin entropy was removed. In the result of Greywall et al. integration of the heat capacity showed that 0.4 R ln2 of entropy was removed just in going through the transition. This suggests that a substantial fraction of the total spin entropy was not removed in the cooling through the peak in the result of Schrenk et al. Therefore the spins were not completely ordered and there might be an additional peak in the specific heat at a lower temperature. This suggests that the magnetic properties of the nano-clusters may be quite different from those of pure bulk ^He.

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25 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Temperature [mK] Figure 1.13: History dependence of the heat capacity measurement. The initial concentration of ^He was 1%. This plot shows data taken at F = 3.40 MPa during the warm-up of the sample from different starting temperatures. Open circles: T^in = 783 /xK; solid squares: T„,i„ = 837 //K; crosses: Tmin = 866 //K; open squares: T^in = 922 /iK; open down-triangles: T^j^ = 998 fxK. Inset: Dependence of the nuclear magnetic ordering temperature of solid ^He clusters on the minimum temperature Tmin to which the sample was cooled before the measurements were started. Open circles: P = 2.80 MPa; crosses: P = 3.10 MPa; open squares: P = 3.36 MPa; sohd squares: P = 3.40 MPa. From Ref. [1]. ©(1996) American Physical Society

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26 1.6 14 1.2 1.0 ^ 0.8 2 06 0.4 0.2 0.0, Figure 1.14: Pressure dependence of the peak temperature of solid ^He clusters in a ^He matrix (circles: Schrenk et al.) in comparison with the results of pure bulk ^He (solid squares: Greywall and Busch [2], crosses: Hata et al. [20]). ©(1996) American Physical Society. Schrenk et al. also plotted the peak temperatures, which they interpreted as T^, versus their sample pressures as shown in Fig. 1.14, in which the peak temperatures are an extension of for pure bulk ^He to below the melting pressure of pure bulk ^He. They construed this as evidence that the nano-clusters behaved like bulk helium at a pressure below the melting pressure. Haley et al. [42] measured melting and freezing of the nano-clusters, both in a silver sinter and an open volume, from 2.96 MPa to 3.48 MPa. They found that melting of the nano-clusters occurred at a higher pressure than bulk solid and there was a substantial hysteresis between the melting and freezing temperatures in the silver sinter and open volumes. These results were similar to those for helium in small pores [43] but differ from those of Schrenk et al. [44] At a sample pressure of 3.40 MPa, the volume changes on melting were less than for bulk solid, indicating that only 8% and 40% of the =*He in the nano-clusters underwent melting in the sinter and in the open volume, respectively. Pressure [bar]

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27 Schrenk et al. had reported a depression of the melting pressure of the nanoclusters relative to that of pure bulk ^He [44]. The pressure change upon freezing indicated that not all of the solid ^He in the nano-clusters melted as they were cooled below the bulk melting temperature. These effects in the melting pressure may be related to those that have been observed in pure ^He or ^He in confined geometries [45]. Pure ^He and ^He in small pores have elevated melting pressures compared to their melting pressures in pure bulk ^He, while a depression of the melting pressure has been reported for ^He on a MgO substrate [46]. Pressure changes upon melting or/and freezing are less than for helium in bulk.

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CHAPTER 2 EXPERIMENTAL APPARATUS In this chapter, the experimental apparatus used in this work is covered in the following order: the dilution refrigerator; the magnetic refrigerator; the superconducting magnets; the ^He melting pressure thermometer (MPT) including the gas handling system; the experimental platform; the NMR setup including the NMR cell design; the pressure measurement setup including the gas handling system; and the temperature regulation method. 2.1 General Description The temperature range of interest in this work was from below 10 mK down to 0.5 mK. In order to achieve temperatures in this range, the dilution refrigerator cooled the experimental stage down to «i 6 mK, which served as a precooling stage for the magnetic refrigerator necessary for lower temperatures. In order to perform this work at low temperatures, several experimental principles had to be followed [47]. The cryostat and the electronics were located inside a copper rf shielded room ( 5 m long x 3 m wide x 2.7 m high ) which was grounded at a single point. In addition, 0-60 Hz low pass filters were placed in the 120 V AC power lines to avoid power disturbances affecting the instruments inside the screen room. Great care had been taken to prevent mechanical vibrations of the cryostat, which cause a heat leak. The dewar was suspended from a triangular aluminum plate, which was supported at its three corners by pneumatic isolation mounts, model XL-A.^ Bellows ^Newport Research Corporation. 28

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29 were put in pumping lines between the cryostat and pumps to reduce transmission of vibrations. The dewar was connected to the helium recovery system by a rubber hose for the same purpose. These features reduced the heat leaks to typically 2-3 nW. This corresponded to a warming rate of 2-7 /iK/h at low temperatures, allowing us to maintain T < 1 mK for more than a month to complete measurements of a sample. 2.2 The Cryostat and Dilution Refrigerator The cryostat consisted of the dilution refrigerator unit, the PrNis nuclear stage, the gas pumping system, and the floating table. The dilution unit, model DRP-43, was purchased from S.H.E Corp around 1980.^ With a circulation of 0.5 mmole/sec, it cooled the experimental stage down to a temperature of 6 mK without a magnetic field. With a field of 6.5 T on the PrNis stage (see Sect. 2.6), it reached 11 mK in two days. Figure 2.1 shows the overall setup of the dilution refrigerator.^ The helium mixture tanks contained 1.35 moles of ^He and 3.64 moles of ^He as specified in the dilution refrigerator manual.'* 2.2.1 Principle of Dilution Refrigeration Dilution refrigeration utilizes a remarkable property of ^He and ^He liquid mixtures. Even at absolute zero, phase separation is not complete and ^He has a concentration of 6.4% in the '*He-enriched phase. Pumping ^He gas from the still causes ^He in the ^He-enriched phase to diffuse into the He-enriched phase in the mixing chamber in order to maintain the equilibrium concentration of 6.4%. At a given temperature, the entropy of the "* He-enriched phase is greater than the entropy of the 2S.H.E Corp, 4174 Sorrento Blvd., San Diego, CA 92121. ^Some portions, which are not used, are omitted for a clarity. ^Operation instructions, model DRP-43 dilution refrigerator unit.

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30 ^He-enriched phase. Thus ^He passing from the concentrated phase into the dilute phase causes cooling. Therefore, the mixing chamber has the lowest temperature among the various stages of a dilution refrigerator. Heat exchangers are placed between the concentrated phase incoming to the mixing chamber and the dilute phase outgoing from the mixing chamber, so that incoming ^He is precooled before reaching the mixing chamber. More details can be found in numerous books about the subject, e.g. Lounasmaa(1974), Pobell(1991) etc [47], [48]. 2.2.2 The 1 K Pot The operation of the dilution refrigerator relies on a pumped pot of ^He near 1 K in order to condense the re-circulating ^He [49]. The 1 K pot utilizes the principle of vapor cooling that is produced by the difference in enthalpy between gas and liquid ''He. The liquid is vaporized by pumping, which produces the cooling, since the gas has greater enthalpy. This operation should be made in an isenthalpical condition, which was realized by installing a flow impedance to maintain the pressure difference (A P 1 atm) between the ''He bath and the pot, which is inside the caji of the cryostat. A fine filter^ was installed on the inlet from the bath to prevent impurities such as frozen air from blocking the impedance. A small fraction of the liquid from the 4.2 K "He bath flows through the impedance into the 1 K pot. At the same time the liquid arriving in the pot is pumped to lower its temperature. As long as the impedance is set properly, which implies that "He from the bath flows isenthalpically through the impedance, and "He is continuously supplied from the 4.2 K bath, the temperature of the pot can be maintained at « 1.3 K, serving as a condenser for the dilution refrigerator. The flow rate of liquid "He through the 1 K pot, which has a volume of about 200 cm^ is ^ 4 L/day. ^Bekaert Fibre Technologies, 1395 South Marietta Parkway Building 500 Suite 100, Marietta, GA, 30067.

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Symbols: 9 Q Valve O : Needle Valve Some valves are omitted because there are not in use. Note that only one of two cold traps is in use at a time. sun r— o Large Valve Freon Cooled Baffle Booster Pump 9B3 -O-Ch 1 0 LN^Traps IK Pot Impedance T t~ I L» |lKPot H eater 523n I LjijKpw _[ Low Temperature Part From IVC ~l Vent To Recoveryl Edwards E2M80 Mechanical Pump To Sample Gas Handling System o Welch Mechanical Pump Vent Edwards E2M80HS Hermetically Sealed Pump In a Crawl Space Figure 2.1: Gas handling system of the dilution refrigerator. Not to scale.

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32 Within the last year, a problem occurred in the 1 K pot impedance. After a couple of weeks of running, blockage in the line occurred, which resulted in termination of the run. In order to solve this problem, we first put a fine filter on the transfer tube in case there were impurities in the storage dewar, such as condensed moisture. Also the plumbing in the proximity of the 1 K pot was disassembled and flushed with trichloroethylene, reassembled, and leak-checked. However, upon cooling again, no noticeable improvement was observed and the blockage still occurred. It was found that rapid pressurization of the pot pumping line to « 20 kPa above atmospheric pressure with helium gas unblocked it by rapid warming. This indicated that the problem existed in the impedance. If it were a problem with the fine filter on the inlet of the ''He pickup line (5 //m pore size) immersed in liquid helium, one could not have managed to unblock it. Since the blockage was in the impedance and was removed by rapid warming of the line, a heater was soldered on the impedance to facilitate unblocking it. The heater, made of three 1.5 kfi metal-film resistors connected in parallel, worked very well. As a precaution, the heater was turned on periodically for several minutes while monitoring a resistance thermometer that was mounted on the impedance, without heating up the pot itself. If the entire pot had been heated, hydrogen from the pumps trapped in the return line filter of the refrigerator could migrate to the small capillary part and plug it up. As an additional precaution, a cold valve was installed in a line in parallel with the impedance line. (See Fig. 2.1.) Since these changes were made, the impedance-blocking problem has disappeared. 2.3 Magnetic Refrigerator The temperature range of interest, around 1 mK or below, can not be reached by a dilution refrigerator alone, thus nuclear refrigeration was employed to achieve the temperature range.

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33 The cryostat has 1.0 mole of PrNig as a refrigerant (20 rods), which were soldered with cadmium to 392 copper wires. The top end of the wires were welded to the Cu flange, which was thermally connected to the mixing chamber via a tin heat switch. If a magnetic field, B, is applied to a nuclear magnetic dipole with a magnetic moment // and a total angular momentum /, there will be 2/+1 energy levels with energies em at temperature T. Then, the partition function, Z, can be written as: Z = e-^r°, (2.1) where em=gnBTnB and iVo the number of magnetic dipoles with g nuclear Lande factor, hb Bohr magneton, 7 gyromagnetic ratio, and m enery level. Once the partition function is obtained, the entropy and other thermodynamic quantities can be calculated as functions of B/T. Van Vleck paramagnets containing rare earth ions such as Pt^+ have a temperatureindependent electronic susceptibility at low temperatures because they have an electronic singlet non-magnetic ground state of their 4/ electronic shells. An external magnetic field induces an electronic magnetic moment that generates a hyperfine field at the ^^^Pr nucleus, giving an enhanced magnetic field at the nucleus, (5int+5externai)The enhancement factor K = Bi„t/5+l « 12 for polycrystalline PrNis. The large internal magnetic field seen by ^^^Pr results in a substantial reduction in the nuclear spin entropy even at high temperatures and in moderate magnetic fields. For instance, the entropy reduction at T = 11 mK and B =6 T is « 90%, as shown in Fig. 2.2 [48]. Therefore, a moderate dilution refrigerator and superconducting magnet allow one to achieve the condition just mentioned. The entropy of nuclear spins of PrNig can be written as the following: ^=f[(cotM|)-(..,eotM^,).M!lfflj,

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34 15 O £ V10 a g c ^ 5 0\ •« o 37mT a 0.2 T 07T // 6T 6mT Precooled by / ' the dilution refrigerator./ A //' Nudear Demagnetization Heat switch is opened 1 L 10 10 Temperature T [mK] 100 Figure 2.2: Nuclear spin entropy of PrNis as a function of temperature. The stage is cooled, as shown by the arrows. In this work, the magnetic field applied was ^ 6.5 T. The initial temperature before the demagnetization of the stage was typically = 11 mK. The lowest temperature of the nuclear stage is ^ 0.3 mK, because of the magnetic ordering of PrNis. From Ref. [48]. ©(1990) SpringerVerlag.

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35 where R is the gas constant, x=K/jLBgB/kBT, K is the hyperfine enhancement factor, the Bohr magneton, g the nuclear Lande factor, B the external field, and T the temperature. For PrNi5, 7=5/2. Provided the spin-spin interactions is small compared to the hyperfine interaction and the Zeeman interaction, one can argue that an ideal adiabatic demagnetization/magnetization gives where Bi and Bf stand for initial and final external field, respectively. Thermodynamic quantities of non-interacting spins are functions of the ratio between their Zeeman and thermal energies. However, Eq. 2.3 can not be used in a region where the interactions of the nuclear spins is not negligible compared to the Zeeman energy. One has to replace Eq. 2.3 with the following. where h is the internal field due to the interactions of the nuclear spins. Nuclear magnetic ordering of PrNis occurs around 0.3 mK [50], which is approximately the lowest temperature that can be reached with the stage. The dilution refrigerator is thermally connected to the nuclear stage via a tin heat switch for removing entropy from the stage during several days of precooling (see the path shown in Fig. 2.2). Then the heat switch is opened (see Sect. 2.4), which isolates the nuclear stage from the dilution refrigerator, since tin is a very poor thermal conductor in the superconducting state. Prom this point, the entropy remains constant except for heating of the stage. Then removing the magnetic field under an adiabatic condition cools the nuclear stage (see Fig.2.2). (2.3) (2.4)

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36 2.4 Superconducting Magnets In this work, two superconducting magnets were used. The one that generated the magnetic field for the nuclear demagnetization was manufactured by American Magnetics, Inc.^ It has a field to current ratio of 1058.5 Gauss/Amp and a decay rate of AB/B < IQ-Vday in the persistent mode. The other, made by Richard Haley, produces the NMR static field and has a field to current ratio of 224 Gauss/Amp. In order to shield the external field, the outer surface of the coil former for this magnet was coated with Pb, which is superconducting below 7 K. A feature of this magnet is that it has 4 turns of Nb foil on the inside for improving field homogeneity ("superconducting jellyroll") [51]. The electronics for the operation of both magnets is shown in Fig. 2.3. Instead of the Kepco power supply shown in this figure, a HP E3632A DC power supply was used for the NMR magnet. When the superconducting magnet for the nuclear demagnetization was energized, a DC voltage drop occurred in the stainless steel tubes that were used for the magnet leads inside the "He bath, because of the resistance of the tube. The reason stainless steel is used instead of Cu is that the stainless steel does not evaporate the "He liquid in the bath as much as Cu, because of the poor thermal conductivity. In order to prevent the voltage from exceeding the limit in the Kepco power supply and to carry away the heat generated, vapor cooling of the leads was employed. A connection from the bath to the helium recovery system through the stainless steel tubes was made in order to allow "He vapor to pass through the tubes. Outside the cryostat, a rubber hose was used for the connection to the recovery, which allows one to open and close the connection by a clamp. ^American Magnetics, Inc. P. 0. Box 2509, 112 Flint Rd., Oak Ridge, TN, 37831-

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Home-Made Ramp Unit Kepco Power Supply ATE6-100M Home-Made Current Supply 20mA HP 3465A Multimeter 10mn,< lOOA < Heater (69. 341) Current monitor Superconducting Magnet B=1058.5 (Gauss/A)I(A) (American Magnetics, Inc) Figure 2.3: Electronics for the nuclear demagnetization magnet. The magnet for the NMR static field has a similar setup.

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38 The operation of the superconducting magnet requires a persistent switch whose function is to keep a current in the superconducting magnet without having current flow from a power supply at all the times. The switch for the demagnetization magnet is located on top of the vacuum can. The switching is made by applying/removing heater power in Fig. 2.3. Bare NbTi wire is located near the heater; thus, applying/removing the heater power makes the NbTi normal-/superconducting. For additional details of the construction, see Ref. [48], for instance. A tin heat switch was used to connect and disconnect the nuclear stage to the mixing chamber, because in the superconducting state tin is a very poor thermal conductor since Cooper pairs carry no entropy and the phonon contribution to the thermal conductivity is negligible at low temperatures. Switching is made by applying/removing a small magnetic field to the tin, which makes it normal-/superconducting. Electronics for the heat switch is quite similar to that for the superconducting magnets. 2.5 ^He Melting Pressure Thermometry ^He melting pressure thermometry (MPT) employed in this work has several advantages. (1) The pressure change along the melting curve is large, thus, temperature could be determined with a great resolution. (2) There are several easily identifiable fixed points on the curve. (3) A relatively short time is required for measurements. (4) The setup and measurement scheme are relatively simple compared to other thermometers. By measuring the melting pressure of ^He, the temperature can be determined, since P{T) has been established [17]. Pressure changes cause strain in a diaphram, which moves a capacitor plate, causing a change in capacitance. The pressure is measured as a capacitance or ratio transformer reading in an AC capacitance bridge.

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39 2.5.1 The ^He Melting Pressure Thermometer Cell Design The design of the cell for ^He melting pressure thermometry was like that used by Wen Ni in his high-precision ^He melting pressure measurement [52]. The body was made of 99.99% pure silver.''' A coin-silver flange was hard-soldered to the body with eutectic^ silver-copper solder (71.9% Ag -28.1% Cu) at Tge^C in a Thermolyne furnace.^ A thermocouple was used to monitor the temperature inside the furnace in order not to overheat the cell body If the temperature exceeded 800°C, the coinsilver flange would start to melt, and the entire body would be ruined. Eight pure silver wires whose diameter was 0.76 mm were diffusionwelded to the cell body as heat exchangers between a sinter and the body, as shown in Fig. 2.4. Then Japanese silver powder,^" 700 A in diameter, was packed inside the cell at pressure of about 2.62 MPa to form a sinter. Also eight holes were drilled into the sinter to provide enough heat exchange between the liquid ^He and the cell body, as shown in Fig. 2.4. It is important that there be sufficient open volume, which is w 27%, for the solid ^He to grow; otherwise solid ^He formed inside the pores of the silver may cause a long thermal time constant. The initial loading pressure of ^He should be carefully chosen, so that the thermometer has a short time constant of not more than a few minutes, at the lowest temperature. See Chap. 3 for the procedure. Separately, a diaphragm was constructed of coin silver. Several considerations must be made in the desig n of the diaphragm and capacitor. The displacement of the ^Surepure Chemetals, Inc, 5-T Nottingham Dr, Florham Park, NJ 07932. ^Eutetic Corporation, 4040 172nd St., Flushing, NY 11358. ^Thermolyne, Type F21100 Furnace, 2555 Kerper Blvd. Bubuque, lA 52001-9990. i°Tokuriki Kagaku, 2-9-12 Kaji-machi Chiyoda-ku Tokyo, 101-0044, Japan.

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40 diaphragm is given as y = (2.5) where P is the pressure, a the radius of the diaphragm, t the thickness, y the displacement, and E the Young's modulus. Prom this equation it is clear that smaller t and large a give larger y, which provides better resolution. However, one must take the stress and mechanical strength into consideration. The stress, s, can be expressed as s = 4,2 • * (2.6) Stress in the diaphragm should be kept well below the yield stress. The diaphragm made of coin silver has a thickness of 0.76 mm and a radius of 0.585 cm. The capacitor plates made of coin silver are also constructed, one of which is attached to the diaphragm and serves as a movable plate while the other disk, which is held close to the movable plate, is used as a fixed plate. An appropriate gap between the plates, about 0.025 mm, was produced by machining the plate(s) rather than using a Mylar sheet as a spacer between them. This procedure not only ehminated possible dielectric capacitance due to the Mylar sheet between the plates but also provided good thermal contact of the upper plate to the nuclear stage. One can consult the extensive review written by Adams for more details [53]. After construction, the ^He melting pressure thermometer cell was tested at room and liquid nitrogen temperatures. Once the cell was made leak-tight, the sensitivity was checked at 77 K with result shown in Fig. 2.5. An Andeen Hagerling ultraprecision capacitance bridge was used in testing. The gas handling system for pure ^He, depicted in Fig. 2.6, was used to supply/evacuate ^He gas to/from the MPT cell. The ^He gas had a purity of 2 ppm. Wen Ni [52] argued that at 0.4 K 10 ppm of "He decreased the ^He melting pressure

PAGE 48

The Stroln Gauge Capacitor Plates Sample Space 0,5' i aa^vs/v ;a3 Silver Wires Silver Powder K///////] Stainess St. Screws Figure 2.4: Schematic drawing of the ^He melting pressure thermometer cell. 90 80 • 1st pressure up <^ 0 1 St pressure down o 70 2nd pressure down § • 60 a 9 O 50 cP 40 <* 30 o 20 —J — 1 — 1 — — 1 — , — 1 — . — 1 .1.1 0.5 1.0 1.5 20 2.5 3.0 3.5 4.0 Pressure (MPa) Figure 2.5: Melting pressure thermometer cell test.

PAGE 49

42 by 30 Pa. This corresponds to a change of 10~^ in ratio transformer reading, which was approximately the same as the reproducibihty of T^. Therefore 2 ppm of "^He in the ^He gas has almost no effect on the thermometry in the temperature range in this work. The gas handling system includes a Heise pressure gauge, ^^a dipstick, and nitrogen cold traps. A dipstick was used in order to draw gas from/to the ^He tank to/from the cell and to control the pressure. The pressure was monitored by the Heise gauge. The nitrogen cold traps were placed in the fill line of the cell and used to trap air before the ^He gas went to/from the cell. 2.5.2 Electronics for ^He Melting Pressure Thermometry and Sample Pressure Measurements As explained in Sect. 2.5, a pressure change is detected as a capacitance change. As the diaphragm of the cell deflects due to a pressure change, a change in Cp occurs, causing an off-balance voltage in the bridge. From the off-balance voltage detected by the lock-in amplifier, Cp is obtained, which in turn gives the pressure using the C{P) calibration (see Chap. 3 for the calibration procedures). Figure 2.7 gives the overall diagram of the capacitance bridge used in the thermometer setup. An Ithaco Dynatrac 391A lockin amplifier^^ generated a sine wave of 1 kHz applied to the bridge, which included the capacitance (Cp) and a reference capacitance (Cref), through a Gertsch ST-IOOA isolation transformer. 11 Dresser Instrument Div., Heise Precision Instrument Operation, 153 S. Main St. New town, CT 06740. i^Ithaco, Inc., 735 West Clinton Street, P.O. Box 6437, Ithaca, NY 14851-6437. '^Reconditioned Gertsch ratio transformers are available at Tucker Electronics, www.tucker.com.

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High Pressure Gauge / ^ 0 3000 psi TCbw Pressure (jauge "1 )0 30 psi above atni Dipstick Dipstick Pressure Intensifier # In the actual system, two iwelal dewars had been used. One is in the upper system, the other is in the lower one in this diagram. ## Usually two high pressure valves which connect the two systems are closed to avoid a possible contamination of the pure 'He system. *He Gas Cylinder Figure 2.6: Gas handling system for pure ^He and ^He-^He mixtures.

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44 Gertsch ST1 1 OA Transformer ITHACO 391A Locldn amplifier Output Input Reference Chassis Ratio Gertsch AC 1011 A Ratio Transformer Hewlett Packard 3421 A Digitizer Channel #2 1 Trimmer Cp : Cref] I i Inside the cryostat Soltec chart-recorder 330 National Instruments GP-IB Bus extender 110 Fiber optics cable National Instruments GP-IB Bus extender 1 10 Outside screenroom # C ref is located on the mixing chamber. Figure 2.7: Electronics for the ^He melting pressure thermometry.

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45 When Cp needed to be determined, the setting of the Gertsch AC lOllA ratio transformer was adjusted to balance the bridge, with the lockin serving as a null detector. For an ideal bridge, the relation is as follows [53]: Vo ^ {x-l)Cp + xae{ ^2 7) Vj Cp + Cref ' where Cref and Cp represent the reference and the unknown capacitances, respectively, and X is the ratio transformer reading. The lockin amplifier output of the off-balance voltage was fed to a HP3421A digitizer^'^ for data acquisition. The lockin output was also traced on a 3-channel Soltec 330 chart recorder. In reality the cable capacitance could be greater than Cref and Cp, which would reduce VqA result can be found by substituting Cref+Cp+Ccabie for the denominator in Eq. 2.7. One might want to use a pre-amplifier to reduce Ccabie and gain a higher resolution, depending upon the requirements of the experiment. The setup in electronics for the sample pressure measurements is quite similar to that for the MPT. There are difference in the instruments, as shown in Fig. 2.8. An Eaton ratio standard and capacitors were isolated from ground by a 1:1 Gertsch ST-200C isolation transformer. The lockin amplifier used in the measurements was an EG&G model 5204,^^ which generated a sine wave of w 800 Hz (variable) applied to the capacitance bridge. Also an Ithaco low noise pre-amplifier model 1201 was added with a gain of 20, because the resolution of the pressure transcucer was not as good as the MPT cell. In addition the high pass and low pass rolloffs of the preamplifier were set at 300 Hz and 3 kHz, respectively. The off-balance voltages of the bridge i^Hewlett Packard, P. O. Box 105005, Atlanta, GA 30348. i^Soltec Corp., 11684 Pendleton St, Sun Valley, CA , 91352. i«EG&G Princeton Applied Research, P. O. Box 2565, Princeton, NJ 08643-2565.

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Gertsch ST-200C Transformer Reference ~800Hz PARC 5204 Lockin amplifier Output Input Chassis Ratio Eaton Ratio TransforraoTrimmer Hewlett Packard 3421A Digitizer Channel #3 Cp Cref^ Output Ithaco pre-amplifier Modell201 Input Inside the cryostat Gain 20 Low Pass Filter 300 Hz High Pass Filter 3 kHz ' — Soltec chart-recorder 330 National Instruments GP-IB Bus extender 110 Fiber optics cable # Cref is located on the mixing chamber. 1 National Instruments GP-IB Bus extender 110 Outside screenroom Figure 2.8: Electronics for the pressure measurements.

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47 were detected by the lockin amplifier and recorded to a computer file via a general purpose interface bus, GP-IB. 2.6 The Experimental Platform on the Nuclear Stage Initially a copper "cage" was used to mount the cell on the nuclear stage. After several cooldowns, a long time constant for the melting curve thermometer was observed. The loading pressure had been checked, with no indication that the amount of ^He in the cell was a factor. The annealed cage had been polished, which could have introduced defects in it before mounting the experimental cells. Also soft-soldered spools made of Cu alloy 101 (an oxygen-free high-conductivity copper) were mounted on the stage for heatsinking the fill lines, which may have caused the problem. Further investigation of the cause was not made, however, and it was decided to construct a new experimental platform. First the cage was removed; then a new plate made of copper alloy 101 was heat-treated in an oven at QSO^C with 10"^ Torr of oxygen gas. Secondly, small capillaries were used for the fill lines to reduce large open volume near the cells. Third, the fill lines were silver soldered to a new spool made of Cu alloy 101. Moreover, the electrical coaxial leads had Apiezon grease placed between the outer and inner conductors, which prevented the inner lead from moving and served as a heat-sink. This reduced any possible noise problem caused by the moving leads. A heat-sink for the leads was heat-treated together with the plate. Only a minimal amount of epoxy was used to glue brass strips to the heat-sink for the electrical connections. Finally the experimental platform was placed on the nuclear stage and no problems were observed during the experiment. The overall arrangement of the nuclear stage is shown in Fig. 2.9.

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Tin heatswitch Thermal link between magnet and mixing chamber (Ag wire) 'He Melting Pressure Thermometer Rebuilt Experimental Platform (Annealed, then goldplated.) Compensadon Coil Superconducdng Magnet (American Magnetics, Inc) B(T)=0. 105851(A) Decay rate=AB/B=10^/day Mixing Chamber Lead shield w/NMR setup inside Sample Cell (Simplified for clarity. die detailed pictures in chapter2.) Vespel for thermal isolation •Welded Thermal link between PrNi, and the platform. PrNi, (Cadmium Soldered to 392 Cu wires of 0.8 mm diameter.) Heatsink for leads and fill lines, electrical leads are not shown in this figure for clarity. Figure 2.9: Schematic diagram of the nuclear stage. Not to scale.

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49 2.7 The NMR Setup The electronics for the NMR are shown in Fig. 2.10. A Tektronix digital oscilloscope TDS784D^'' and a were both used to obtain the data. The static field for the NMR was generated by the superconducting magnet described in Sect. 2.4. A GP-IB interface bus was used for data acquisition, and connection between the computer and the instruments was made via an optical link. Capacitors for the tank circuit were located in an aluminum box on top of the cryostat. In the first cell, built by Richard Haley, the NMR coils were wound directly on the body of the cell and were not grounded to the nuclear stage, which may have been responsible for the high minimum temperature encountered by Adams et al. [54]. Therefore a new cell was designed and constructed. The main goals were to reduce the amount of epoxy in the cell and to improve the heat-sinking of the leads going to the cell. The cell had two coils, one a saddle coil and the other a solenoid, noneh of which touched the sample container. The overall setup and the sample container are shown in Figs. 2.11 and 2.12. The NMR saddle coil has not been used in the current work. However, this was prepared for a situation when a cold preamplifier could be used, with the saddle coil serving as an excitation coil. Grooves for the coil winding were made in the Stycast 1266^^ coil former, and 20 turns of copper wire (0.038 mm in diameter) were wounded on each side. Copper wire (0.025 mm in diameter) was used for the NMR solenoid coil, which served both as the pickup and excitation coil (see Fig. 2.12). The coil former, which i^Tektronix, Inc. RO. Box 1000, Wilsonville, OR, 97070-1000. 18PLM-4, RV-Elektroniikka Oy Picowatt, SF-01510 Vantaa, Finland. i^Deanco, 130 University Blvd., Winter Park, FL 32792.

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Tuned to 250kHz or 125kHz PLM-4 E^jlse NMR Spectrometer Wideband output Trigger (Integrator Gate)* E'rcamplifier Tektronix Digital Oscilloscope GP-IB Bus Computer Screen *Timing was determined on the oscilloscope Hewlett-Packard DC Power Supply E3632A (Resolution=0.12mA) C^, and Cp are variable and cable capacitance respectively. Figure 2.10: Electronics for the NMR.

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51 Figure 2.11: Overall NMR cell setup.

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52 Figure 2.12: Drawing of the ^He-^He mixture sample cell.

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53 had three sections to decrease capacitance between layers, was made of Stycast 1266. Each section had 180 turns of winding. A motor-driven coil winding machine with an optical sensor, made by Volodya Shvarts, was used to wind the small coil. A slight modification was made to improve the performance of the cell by replacing brass mounting screws with BeCu ones. Brass screws may possibly become superconducting, affecting the homogeneity of the static NMR field due to the Meissner effect. Lang had success in using BeCu in a field of 2.5 T for his direct demagnetization of hep ^He [55], in which he succeeded in demonstrating magnetic ordering for the first time. 2.8 Pressure Measurement Setup Initially the pressure transducer was a Straty-Adams type gauge attached to an aluminum-oxide ceramics tube that served as a sample container. However, after a couple of tests, a leak was found; thus the ceramics was replaced by Vespel-22^° with the same dimensions (3 mm in diameter and 10 mm in length). Apparently the problem was a crack due to a difference in thermal shrinkage between the ceramics and Stycast 2855FT, which was used to glue the transducer to the ceramics. We built a miniature coin-silver pressure transducer of the MoriiAdams type [53], as shown in Fig. 2.13, and replaced the pressure transducer of Straty-Adams type. The coin-silver capacitor plates (0.421 mm in diameter and 0.025 mm in the diaphragm thickness in the movable plate) were glued to each other with Stycast 2855FT, with a paper ring (0.076 mm in thickness) placed in between to serve as a spacer and an electrical insulator between the plates. The electrical leads for the I. Du Pont De Nemours k Company, Du Pont Polymers, Wilmington, DE 19898.

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54 capacitor plates were glued with an electrically conductive epoxy H20E-175.^^ A large temperature-dependent background capacitance changes in the pressure measurements were observed, which were caused by the Stycast between the plates (see Chap. 3). A pure silver wire, 0.5 mm in diameter, was welded to the thermal link of the cell to the nuclear stage (see Fig. 2.13), after which the sample container was glued to the silver thermal link. Then for better heat exchange between the sample and the nuclear stage, Japanese silver powder (Tokuriki Silbest, C-8) whose particle size was 70 nm was packed inside the cell with a packing pressure of 800 kg/cm^, which resulted in a filling factor of ^ 45%. Then a disk of Kimwipe^^ was placed on the surface of packed silver in order to electrically isolate the silver power from the diaphragm (see Fig. 2.13). Finally, the pressure transducer was glued to the Vespel-22 container with Stycast 2855FT. The solid ^He in the small pores of the sinter is held in place because of the small openings and does not transmit pressure to the bulk layer [53]. Thus, one might be concerned that pressure of the clusters in the pores would not be transmitted to the bulk layers near the transducer. Also mechanical properties of a sample cell could cause a non-constant volume condition, which causes a smaller pressure change than that at constant volume. In our case pressure was indeed transmitted to the bulk layer under a constant volume condition, which can be concluded from our experimental results (see Chap. 4 for discussion). The gas used for the ^He-'^He mixtures had a concentration of 0.6% ^He in ^He. The lower part in Fig. 2.6 shows the gas handling system for ^He-^'He mixtures, which has a similar construction to the ^He gas handling system, including nitrogen 2iEpoxy Technology, 14 Fortune Drive, Billerica, MA 01821 USA. 22Kimberly-Clark Corporation, Roswell, OA 30076-2199.

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55 Silver Wire (Welded to the silver thermal link.) For clarity silver thermal link is shortened. *Tokuriki Silbest C-8. Figure 2.13: Heat exchanger and miniature pressure gauge used for this work. Not to scale. The NMR coils are not shown for clarity.

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56 cold traps, a pressure gauge, and a dipstick. The pressure was controlled by an intensifier^^ (see Fig. 2.6) and was monitored by a Paroscientific pressure transducer.^"* 2.9 Method of Temperature Regulation Chart Recorder Soltec 330 Resistance Bridge LR110 Out In Temperature Controller LR 130 In Out T lennome er Mixing Chamber 1090ft Heater I 1 Inside the Cryostat Figure 2.14: Electronics for temperature regulation. Occasionally, the temperature was regulated using the electronics in Fig. 2.14, in order to acquire data or to anneal a sample. This was done by feeding the output of the LR-110 resistance bridge or the MPT off-balance voltage to the LR-130 heater. 23High Pressure Equipment. 1222 Linden Ave, Erie, PA. ^''Paroscientific, 4500 NE 148 th Avenue Redmond, WA 98052-5126.

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57 Heaters that were 1000 Q wirewound resistors were located at various points such as the mixing chamber.

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CHAPTER 3 EXPERIMENTAL PROCEDURES In this chapter, the entire experimental procedures are covered step by step in detail from cooldown to data acquisition. 3.1 Cooldown Procedures The first step for a successful cooldown is to make sure all the parts including the dilution refrigerator are leak-tight. This was carefully checked with a Veeco leak detector, MS-17, at room temperature.^ Also electrical connections were checked. After checking these at room temperature, the dewar was raised for a liquid nitrogen transfer. Before starting the transfer, the dewar was pumped out and filled with helium gas to leak-check the vacuum can. Also the dilution refrigerator was leakchecked by introducing the helium mixture gas from the tanks to the plumbing of the refrigerator. The 1 K pot line was flushed with helium gas several times and was kept at a pressure of ^ 138 kPa of helium gas in order to prevent nitrogen from entering and blocking the flow impedance in the pickup line. During the transfer of the first 50 L of liquid nitrogen, the helium level in the vacuum can was monitored by the leak detector to leak-check the vacuum can and the 1 K pot as they cooled. Once it was clear that there was no leak in either the can or the pot, 1000 //Torr of dry nitrogen gas was introduced to the vacuum can as exchange gas for cooling, and another 50 L of liquid nitrogen was transferred. Weeco, Terminal Drive, Plainview, Long Island, NY. 58

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59 After making sure that all of the resistance thermometers indicated that they were at nitrogen temperature, the MPT and the experimental cell were leak-checked by applying pressures with ^He and ^He-'^He mixtures, respectively. Then, the remaining liquid nitrogen was transferred back to the storage dewar. It was important to make sure that all of the liquid was removed at the end of the back-transfer, which was accomplished by pumping on the dewar. Helium gas at a pressure of 1000 //Torr was introduced to the vacuum can as exchange gas, after which the transfer of liquid helium began. During the transfer the resistance thermometer on the mixing chamber was monitored to determine when the temperature was low enough for pumping of the helium exchange gas. When the resistance dropped due to superconductivity in the NbTi thermometer leads, which occurs at 18 K, pumping of the exchange gas began. The vacuum can was pumped for eight to ten hours until the ^He signal became low (~ 10~® cc/sec in the Veeco leak detector) and constant. The transfer was then continued until the 100 L transfer dewar was emptied. After the initial transfer of liquid helium, pumping of the 1 K pot was started. Then, the helium gas mixture for the refrigerator was released from the storage tanks to the exhaust of the mechanical pump (Edwards E2M80HS, a hermetically sealed pump) with the safety solenoid valve manually opened (see Fig 2.1). The gas was condensed and ready for circulation in the refrigerator after ~ 12 hours. The consumption rate of liquid helium was about 20 L a day, with a transfer every 3.5 days to keep the bath level above the 1 K pot pickup line. 3.2 Calibration of the Mixture Sample Cell and Sample Preparation First the tin heat switch was closed in order to bring the PrNis stage and the mixing chamber to the same temperature. Pumping of the 1 K pot was stopped and

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60 the temperature was regulated at about « 2.5 K, utilizing the setup shown in Fig. 2.14. Then, the mixture, 0.6% ^He in ^He,^ was introduced to the sample cell. Increasing and decreasing the pressure several times between ^2.78 MPa and « 6.2 MPa "trained" the strain gauge in order to reduce hysteresis. However, a significant hysteresis of the cell was still found, which diminished with the range of pressures used in calibration or testing. There was a ?a 70 kPa difference between the same "up" and "down" ratio transformer readings. (See Fig. 3.1.) This hysteresis was observed in every cooldown, and thus had to be handled carefully. It may be attributed to the design of the transducer. Fortunately scatter in the down pressures was small. Thus, a down-calibration was employed for these experiments since the pressure decreased as a sample was cooled along the solid '*He melting pressure curve after the initial pressure was set. (Compare Figs. 3.1 and 3.2.) 6.5 6.0 5.5 (0 Q. 5 0 g 4.5 tn g 40 a. 3.5 3.0 2.5 1st up-pressure O 1st down-pressure • 2nd down-pressure 0.134 0.136 0.138 0.140 0142 0.144 0.146 0.148 0.150 0,152 0.154 Ratio Figure 3.1: Calibration of the mixture cell. Compare upand down-pressures for ratios. ^The concentration was measured with the Veeco leak detector by the author and W. Ni.

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61 0. S 3 1st up-pressure 1 St down-pressure 2nd down-pressure I 1 I i_ 0.12505 0.12510 0,12515 0.12520 0.12525 0.12530 0.12535 0.12540 Ratio Figure 3.2: Hysteresis check over a pressure change of about 137 kPa. Then, the cell capacitance was calibrated against the Paroscientific pressure transducer, with a resolution of ~ 3.4 Pa to establish C{P). The transducer had zero-pressure reading of « -25.5 kPa, ^ which was added to all pressures. A polynomial equation of the form P = ao + aiR + a2B? + a^R^, (3.1) was used to fit the down-pressure calibration points, where P was the value of the pressure, R the corresponding ratio transformer reading, and the Oj's were fitting parameters. (See Fig. 3.3.) The difference between the fitted polynomial curve and the actual down-pressure data is shown in Fig. 3.4. The cell was calibrated between 2.76 MPa and 4.14 MPa. The sample gas mixture was loaded into the cell at a pressure of « 6.2 MPa, which depended upon the target pressure of the sample. Then, pumping of the 1 ^This offset changed slightly; thus it was checked in every run.

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P . 658.50755*12822,47031R-83418.59064RV183152.16729R° (R-Ratio, P-Pressure) « ' 2,6 0.134 0.135 0.136 0.137 0.138 0.139 Ratio Figure 3.3: Calibration of the mixture cell. Figure 3.4: Evaluation of the mixture cell calibration.

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63 K pot began and the fill line blocked as the sample started to cool. Thereafter the sample solidified under a constant volume while pressure changes were monitored on the chart recorder. The sample reached the temperature of the ^He melting curve and since the mixture was almost pure ^He, it was then cooled along the curve until it deviated from the curve. At this point, when the sample had just become allsolid, the temperature was regulated between 1.7-1.9 K for annealing. Annealing was necessary to eliminate gradients in density in the sample. During annealing the pressure decreased, with the annealing continuing until the pressure drop became dP/dt ~ 300 Pa/hr. (See Fig. 3.5.) Figure 3.5: Pressure versus time during the annealing of the 3.54-MPa sample at ~ 1.6 K. If the pressure was not the one desired or a new sample was needed, the nuclear stage was heated to warm the cell, and pumping of the 1 K pot was stopped in order to remove the blockage in the fill line. Then one could form a new sample by repeating the procedure described above. . .

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64 3.3 Calibration and Preparation of the ^He Melting Pressure Thermometer Basically the same procedures were taken for calibrating the ^He melting pressure thermometer cell (MPT) as for the calibration of the mixture cell with the temperature regulated at 0.8 K. Before applying a desired pressure to the cell, the diaphragm was trained over the working range between 2.76 MPa and 3.45 MPa in order to reduce hysteresis. Then, calibration of the cell was carried out over the same range. As in the calibration of the mixture sample cell, a power polynomial was used to fit pressure versus ratio (see Fig. 3.6). The cell had much less hysteresis than the mixture cell. After the calibration, ^He gas was loaded at 3.17 MPa into the cell. It was then cooled through the pressure minimum (P^in = 2.9333 MPa [9]) of the ^He melting pressure curve. The ratio at P^in was recorded, to be used as a fixed pressure point for a temporary temperature scale until the nuclear stage reached the solid ordering temperature T^f (Neel temperature). Subsequently, was used as a fixed pressure and temperature point to provide a more accurate temperature scale at lower temperatures. Upon observing the ratio at P^in, the temperature was increased to 0.8 K and regulated, whereupon the MPT cell was loaded at a pressure of 3.41 MPa. 3.4 Sample Cooling 3.4.1 Technique and Setup The MPT and sample cells, which were linked to the nuclear stage, were cooled with the refrigerator. A technique was used to prevent the plug in the MPT cell fill line from slipping during the cooling. First, the nuclear stage was cooled to a temperature slightly above P^in = 0.318 mK with the tin heat switch open to let the mixing chamber cool to a slightly lower temperature. Then the tin heat switch was

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65 —I 1 1 I I 1 I I 0.44 0.45 0.46 0.47 0.48 Ratio Figure 3.6: Calibration of the melting pressure thermometer cell. The fit of the calibration C{P) is P = 33.08247-245.62655C+609.10729C2-470.684C^ where C = ratio.

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66 Figure 3.7: Evaluation of the melting pressure thermometer cell calibration. The fit of the calibration C(F) is P = 33.08247-245.62655C+609.10729C2-470.684C3, where C = ratio.

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67 closed in order to connect the nuclear stage thermally to the mixing chamber. In this way the time duration that the MPT was around Fmin was minimized. 3.4.2 Phase Separation The phase separation of the mixture sample typically occurred around ^170 mK for the 0.6% concentration of ^He, although the temperature depended slightly upon the pressure of the sample. The pressure change due to the phase separation, which was small and proportional to the ^He concentration in ^He, was detected by the pressure transducer on the mixture cell [32]. 3.4.3 NMR Tuning The capacitance in the tank circuit and the NMR static field had to be adjusted in order to optimize the NMR signal. The adjustment was made at 6.5 mK before taking magnetic susceptibility data. The static field was controlled by the DC current supply as in Fig. 2.10. The procedures are as follows: First, the capacitance in the tank circuit, located on top of the cryostat, was set. Second, the static field was swept with an increment of « 0.2 mA and the NMR signal was recorded for each current. Then, the procedures were repeated for various capacitance values in the tank circuit with the result shown in Fig. 3.8. The total capacitance was determined to be 151 pF, because the 151-pF result had a wider frequency range at the signal peak than the 141-pF result, although the signal levels are quite similar. Thus, even if the current is slightly off-tuned, the signal strength does not decrease much. For 125 kHz resonance frequency the total capacitance in the box was set to 1688 pF for the same reason. From these settings, the cable capacitance was estimated to be about 400 pF, which was equivalent to about 3 m of RG-58 A/U type cable.

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68 0.330 0.335 0.340 0 345 0.350 0.355 0.360 Current(A) Figure 3.8: Tuning of the NMR made at around 6.5 mK. 3.4.4 High Temperature Data Acquisition Data were taken at high temperatures reachable by the dilution refrigerator before magnetizing the nuclear stage. During these measurements, the temperature of the MPT was regulated (see Fig. 2.14.) by using the output voltage of the MPT from the lockin as a feedback to the LR-130.^ 3.4.5 Demagnetization and Magnetization of the Nuclear Stage The nuclear stage (see Chap. 2 for the detail) was magnetized to a field of 6.9 T, corresponding to a current of 65 A. It was important that the valves from the vapor-cooled magnet leads were opened to the helium recovery system when current from the Kepco power supply^ exceeded 50 A. Otherwise, a DC voltage in the current "Linear Research, Inc., 5231 Cushman Place, Suite 21, San Diego, CA 92110. ^Kepco Inc.,131-38 Sanford Avenue, Flushing, NY 11352, phone:718-461-7000.

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69 loop, which was mainly generated at the stainless steel leads, would reach 5 V and the Kepco power supply would be shut down at the voltage limiter setting of 5 V. The stage was precooled for two days to reach around 11 mK for removing most of the entropy (see Fig. 2.2). Then, the heat switch was opened in order to thermally isolate the nuclear stage from the mixing chamber, after which a demagnetization of the stage was carried out. During the demagnetization several temperatures were stabilized in order to measure the relaxation time of the magnetic susceptibihty. A typical demagnetization profile was as follows: 6.91 t':^""4.234 t"-^" 2.12 t'-^^" 1.06 T % 0.53 T'^' 0.32 T The lowest field to which we demagnetized the nuclear stage depended upon the desired temperature. If the obtained temperature was not low enough, an additional demagnetization to a lower field would be carried out. For example, if a demagnetization of the nuclear stage started when the temperature was 11 mK, at a field around 0.5 T the temperature would reach T^f. If the temperature needed to be increased, the nuclear stage was partially magnetized to reach a desired temperature. The field should be kept above 50 G, the critical field of the cadmium solder, otherwise cadmium becomes a superconductor, preventing it from maintaining thermal contact. 3.5 Temperature Scale In this work the ^He melting pressure temperature scale of Ni et al. [17] has been used. They established the temperature scale by using ^°Co as a primary thermometer and Pt-NMR as a secondary thermometer. Ni's polynomial relations in P(T) relative to the Neel point {Pn, T^) are as follows. For T>Tn, P{T) -PN=j: Ar^T-(3.2) n=-A

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70 and for Tl5=-6.0945626xl0-i\ Bo=0.1987 kPa, and Bi=0.2611 kPa/(mK)^. Here temperatures are in mK and pressures are in kPa. Signatures of the fixed points are shown in Fig. 3.9, where the output voltages from the Ithaco 391A lockin ampHfier are recorded. 3.6 Data Acquisition In taking data the following information was obtained: temperatures of the nuclear stage, current in the main magnet, fast Fourier transform (FFT) spectra, free induction decay (FID), magnetic susceptibility of the mixture samples, spin-lattice relaxation time (Ti), and pressures of the mixture samples. All of these were registered in computer files. The GP-IB interface bus was used for all the data acquisition. At the same time, the MPT pressure, current in the main magnet, and pressure of mixture samples were recorded on the Soltec chart-recorder. 3.6.1 Sample Pressure Measurements Pressure was recorded typically every 30 seconds. The pressure measurements were useful to detect possible crystallographic change in solid ^He and partial melting in ^He nano-clusters. By measuring temperature and pressure at the same time, it was possible to trace the pressure change as a function of temperature. Possible heating due to the electrical connections to the transducer of the mixture cell was

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11 L 1 1 0 » 0 6 0 .8 1 0 Time(hr) 1.5 I Time (sec) Figure 3.9: Fixed points on ^He melting pressure, (a): A-transition, (b): B-transition (c): solid ordering transition. Output voltages from the Ithaco 391 A lockin amplifier are shown.

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72 checked by disconnecting and reconnecting the cables to the pressure transducer and found to be negligible. The temperature dependence in the transducer was a disturbing factor when the pressure measurements were done. Since the transducer had Stycast between the capacitor plates (see Fig. 2.13), a large temperature dependence was observed. The capacitance caused by the Stycast had to be subtracted from the raw capacitance data as a background. Rather than measure C{T) at fixed pressure, the background was deduced from data at two pressures, 2.88 and 3.54 MPa, as shown in Figs. 3.10 and 3.11. The high temperature part of the background was taken from the 2.88-MPa sample because of its low phase separation temperature, 145 mK. (See the 2.88MPa sample section in Chap. 4 for details.) Based on Panczyk's work [56], the pressure should be approximately for a range of temperature above Tp^, therefore we concluded the pressure change in this range came solely from the dielectric property of the Stycast. The low temperature part of the background below 105 mK was taken from the raw capacitance data of the 3.54-MPa sample, because this high pressure sample has a very small contribution from exchange interactions and no partial melting occurred. Therefore P{T) is expected to be constant. Then the composite background was made by combining the two Figs. 3.10 and 3.11, as shown in Fig. 3.12. In the background subtraction for each sample, adjustment of the point at « 60 mK, where the background had the minimum, to the minimum of the raw data was made. The pressure data with the background subtracted will be shown in Chap. 4 for each sample.

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73 0.131303 140 150 160 170 180 190 200 210 220 230 T(mK) Figure 3.10: Ratio versus T for high temperature portion of background. R = ratio, taken from the 2.88-MPa sample. 0.125215 0125210 0.125205 0.125200 0.125195 ' — I — L 0 10 20 30 40 50 60 70 80 90 100 110 .. ' T(mK) Figure 3.11: Ratio versus T for the low temperature portion of background R ratio, was taken from the 3.54-MPa sample.

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0 125220 0,125215 0.125210 0.125205 0.125200 0.125195 I L 100 150 T(mK) 200 Figure 3.12: Background capacitance change due to the dielectric behavior of Stycast. 3.6.2 Magnetic Susceptibility Measurements A 90° NMR pulse was used to measure the magnetic susceptibility, except for the first two samples (the 3.54and 3.35-MPa samples), where two tipping angles of 20« and 90« were used. FVee induction decay (FID) of a pulsed NMR signal was read from the oscilloscope to the computer. Then, a Labview« program sampled 500 000 points of the FID with a digitizing rate of 5 M samples/s to get good resolution in the t.me domain and performed the FFT of the points. For the first two samples the frequency resolution was 100 H., in which only a few points constituted a peak' because the NMR width was « 200 Hz. Later, resolution was improved to 10 Hz as' shown in Pig. 3.13 to detect possible frequency shifts due to magnetic ordering The FFT spectrum over a frequency range of 20 kHz was recorded to a file in the form of a ID array Also the first 60,000 points of the FID and the reading of PLM-4 produced by the PLM-4 integrating the first 1 ms of the FID, were recorded for each magnetic susceptibility measurement. '^^^^^^^^^^^^^^^^ Bridge Point Parkway Austin. TX, 78730-5039.

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75 50 A Sample Pressure 2.88 MPa PLM reading 443 T . 0.542 mK 249500 250000 250500 Frequency(Hz) Figure 3.13: A typical FFT spectrum. A small peak at 250 kHz was a resonance from the tank circuit, which was subtracted from raw data for analysis by taking magnetization at high temperatures where there was no signal from the ^He clusters. NMR tipping angles Two NMR tipping angles, about 20° and 90°, for pulses were used depending on the temperature range in the 3.54and 3.35-MPa samples. Because the 3.35MPa sample was the first sample studied, heating due to the 90° pulse and a long relaxation time were concerns. The relationship between the 20° and 90° data needed to combine them was made for the 3.35-MPa sample. A linear relation between the data was found, as shown in Fig. 3.14. In the other samples, 90° pulses were used to obtain the magnetic susceptibilities and measurements of Ti and T2. Background magnetization The NMR electronics and Fermi liquid contributed to the background. By averaging ~ 50 points measured at a high temperature about or above 100 mK, the background could be determined, since the signal of the ^He clusters was small in this

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76 Figure 3.14: Comparison between data (PLM-4 readings) with the two different angles. temperature range. An example of the background signal is shown in Fig. 3.15. The background did not change from run to run. 3.6.3 Spin-Lattice Relaxation Time (Ti) Measurements Several relaxation times were measured for each sample in order to estimate the necessary time interval between pulses and also for a better understanding of the He nano-cluster. At various temperatures, the equilibrium magnetization Mq was measured. Then after a variable time-period r, the magnetization M(r) was obtained. Then the time interval was changed to repeat the same sequence. With r and M, obtained, M(r) could be plotted as a function of time for each temperature. 3 0 3.6.4 Spin-Spin Relaxation Time (Tg) Measurements Spin-spin relaxation is caused by the interactions among the nuclear spins. Thus, spin-spin relaxation times T2 were measured using a spin echo technique in several

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77 E 20 15 10 Z49000 249500 250000 250500 Frequency(Hz) 251000 Figure 3.15: A FFT spectrum taken at 110 mK in the 3.06-MPa sample. samples in order to look for possible changes in spin-spin interactions. (For spin echo, see for instance [57].) The measurements were carried out manually and the echo amplitude was recorded from the digital oscilloscope on a notebook with the sketch of the echo shape. To understand T2 measurement better, one may imagine the following situation. Suppose a 90° pulse is applied to a sample in an inhomogeneous magnetic field. If the pulse is intense enough, the entire magnetization will be rotated through an angle in a very short time. However, after a while the total magnetization vector amplitude will decrease because of the field inhomogeneity in the following way. The total magnetization vector is the sum of smaller magnetization vectors, each from a portion of the sample that feels its own local field. Therefore, each magnetization has its own characteristic Larmor frequency depending upon the local field each atom experiences. Some could precess faster, and some slower and as a result the different contributions of the magnetization will get out of phase with each other. The total dephasing time is called T2*, which is the addition of the effects from the field inhomogeneity and the

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78 internal field. ^ = ^^+nfAHo (3.4) The first term of the equation is of interest, because it is an intrinsic spin-spin relaxation time, which reflects the interactions among the nuclear spins. In this work a standard 90°-r-180° pulse sequence was employed. The scheme is shown in Fig. 3.16. Figure 3.16: Scheme of pulse sequence used to measure spin-spin relaxation time Tj.

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CHAPTER 4 EXPERIMENTAL RESULTS AND DISCUSSION In this chapter, first, sample pressures of interest for studying the magnetic properties of ^He nano-clusters are described. Then the experimental results and the discussion are presented. The five sample pressures studied in this experiment were 3.54, 3.35, 3.06, 2.96, and 2.88 MPa, which were chosen to span the range from liquid droplets to solid clusters. 4.1 Sample Pressures of Interest The five sample pressures, as determined below the phase separation temperature, are shown in Fig. 4.1.1 Upon the phase separation, ^He forms clusters, which contain almost pure ^He. Therefore, it might be supposed that the melting pressure of the ^He nano-clusters would be the same as that of pure bulk ^He. For several samples studied in this work, the clusters showed evidence of melting near the ^He melting pressures in Fig. 4.1. However, the melting was incomplete, based on the pressure change and a Curi^law susceptibility with a solid fraction remaining below the melting temperatures (see the results and discussion in this chapter). In the 3.54-MPa sample, the ^He nano-clusters showed no indication of melting to the lowest temperature in this work, ^ 0.5 mK, as expected according to the pure bulk 3He melting pressure. For the 3.35-MPa sample, at an intermediate pressure, the clusters underwent partial melting at a temperature around 20 mK, after which the sample did not melt complete ly, as indicated by the Curie-law behavior of the iNote: changes in pressure that occur at higher temperatures are not shown in Fig 4.1. °' 79

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80 Figure 4.1: Sample pressures of interest including pure bulk ^He and ^He melting pressures.

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81 susceptibility. In the 3.06and 2.96-MPa samples, the clusters underwent partial melting just below the phase separation temperature, with a fraction of the ^He remaining as solid at lower temperatures. In the 2.88-MPa sample, the pressure was below the pure ^He melting pressure curve at all temperatures. However, a solid fraction of ^He was found upon phase separation, as indicated by the susceptibihty. This will be discussed in detail in Sect. 4.2 4.2 Experimental Results and Discussion 4.2.1 Spin-Lattice Relaxation Time (Ti) Measurements In this work, for each sample except the 2.96-MPa sample, the spin-lattice relaxation time (Ti) was measured at various temperatures. The measurements were necessary in order to determine the minimum time interval between the magnetic susceptibility measurements. Figure 4.3 shows a typical spin-lattice relaxation measurement. The initial Zeeman-exchange-lattice part of the recovery of the signal was only a few tens of seconds long, which was anomalously short compared to that in pure bulk ^He [58] and was also observed by Mikhin et al. [59]. The short Ti's might indicate distortion of the cluster at the interface with the "He matrix. The short Ti's are consistent with a qualitative model used by Bernier et al. [58] to explain the temperature-independent relaxation time below the phase separation temperature ^ 100 mK in their bulk ^He samples. They attributed the plateau as a result of the existence of small amount of ''He as lattice defects in bulk ^He, which would serve as pinning sites for dislocation lines in the ^He lattice. The vibration frequencies of the dislocations, which are strongly coupled to the lattice, vary from a few kHz to more than 100 MHz, which overlap the exchange frequencies. Thus, the Zeeman subsystem

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82 has a coupling to the lattice, as shown in Fig. 4.2. Further consideration is needed to devise a model for the short Ti's in the ^He clusters. Zeeman Exchange ^He Lattice I ^He excitations 1 N/' Long relaxation Heat Exchanger Figure 4.2: Schmatic diagram of the various systems of excitation in ^He. Dislocations caused by the ''He matrix could account for the short Ti's in this work. The Zeeman-exchange-lattice part was followed by a very long recovery of the exchange/lattice bath to the heat exchanger, with a time constant of up to a few hours at the lowest temperature [60]. Magnetic susceptibility measurements were spaced several times the longest time constant in order to assure equilibrium. The long time constants, which were obtained by fitting the exchange/lattice-toheat exchanger part

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83 of the relaxation, are shown versus temperature for the samples in Fig. 4.4. For the 2.96-MPa sample, the time intervals were determined based on the Ti measurements for the 3.06-MPa sample. 0.0 Sample Pressure = 2.88 MPa 0-5 hi , , ^ u T = 1.08 mK T-*— Zeeman-to-Exchange -1.0 -1.5 , 'p -2.0 ^ -2.5 -3.0 '^•^ Exchange/Lattice-to-Heat Exchanger -4.0 I — I 1 1 1 L 20 40 60 80 100 x(min) Figure 4.3: A typical measurement of spin-lattice relaxation. 4.2.2 Sample Pressure = 3.54 MPa Pressure measurements In the 3.54-MPa sample, partial melting/freezing upon cooling/warming did not occur, as shown in Fig. 4.5. Thus the clusters were all solid. This is as expected based on the melting pressure for pure bulk ^He.

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100000co o 3.54-MPa Sample 3.35-MPa Sample 3.06-MPa Sample 2.88-MPa Sample Fit to 3.54-MPa Sample < 1 T(mK) 10 Figure 4.4: Spin-lattice relaxation time versus temperature for the samples studied.

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85 The pressure showed a phase separation temperature of 180 mK, consistent with our calculations in Sect. 1.2.2, which indicated that essentially all of the ^He in the cell had separated in clusters and no ^He was remained dissolved in the solid ''He matrix, and an excess pressure of ^ 7 kPa, which was approximately a factor of two larger than that obtained by Panczyk et al. [32] for their sample pressure of 3.54 MPa. Ganshin et al. [61] observed an excess pressure of 30 kPa during a rapid cooling, which is similar to that for the 3.54-MPa sample by taking the difference in concentration into account, and 10 kPa during a stepwise cooling for an initial ^He concentration of 2.04%. Thus, the discrepancy between the result of Panczyk et al. and the excess pressure for the 3.54-MPa sample could be by the experimental uncertainty. As mentioned in Chap. 2, one might be concerned that pressure of the cluster would not be transmitted to the bulk layer near the transducer. We exclude this possibility by the following explanation. Pressure changes with temperature at the constant-volume is related to the pressure changes that we measure by the following equation: where /cr is the compressibility of the sample and the effective volume that causes the pressure change. The term dP/dT is the measured pressure. If is small, the second term of the right-hand of Eq. 4.1 would be significantly greater than 1, which in turn makes the term dP/dT smaller than the left hand side of Eq. 4.1 [53]. In our case, if the pressure measurement had not been at constant volume, the excess pressure would have been much smaller than that observed by other workers [32], [61]. Therefore, the pressure measurements were made under a constant volume condition and pressure was transmitted.

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86 T(mK) Figure 4.5: Pressure change at phase separation in the 3.54-MPa sample, taken on cooling.

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87 Magnetic susceptibility measurements The magnetic susceptibility was measured during warming using both the FFT, which had a resolution of 100 Hz, and the PLM-4. These were consistent with each other down to 1 mK with a similar value of the Weiss temperature. Below 1 mK the sensitivity setting of the oscilloscope was improper, which caused saturated signals in the recording of the FID. Therefore in the FFT the magnetic susceptibility showed a kink, which, as was realized later, was caused by the improper setting. For this reason, we employed only the PLM data, and the results are shown plotted in three different ways in Figs. 4.6, 4.7, and 4.8. The magnetic susceptibility exhibited a similar behavior to the magnetization results of Hata et al. [20] (see Fig.1.4), which showed a CurieWeiss law above ST^^ and obeyed another CurieWeiss law below bT^. Our result showed a Weiss temperature {dy,) of -0.25 mK and ^ 0.15 mK obtained by fitting the data above and below 4 mK, respectively, as shown in Fig. 4.6. The magnetization results of Hata et al. exhibited a molar volume dependence of Tn as follows, Tn cx v''-\ (4.2) where v is the molar volume. In their results, 9^ « 2Tn, which means that 9^ obeys the same molar-volume dependence. Then our 9^ ^ -0.25 mK for the 3.54-MPa sample corresponds to a molar volume of « 21.3 cm^/mole, which is close to the molar volume of "He in the temperature range of this work («i 20.95 cm^/mole). Thus, we concluded that the behavior in the magnetic susceptibility might be explained by the influence of the hep ''He matrix, which has a smaller molar volume (w 21 cm^/mole) than ^He at this pressure.^ As a result of the van der Waals attraction to the "He 2Pure bulk ^He has a molar volume of 24.12 cmVmole at 3.54 MPa.

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88 surface, the density of the ^He solid in the cluster would be higher at the interface than in the interior. Also the cluster size is only a few atomic spacing, as determined by the concentration of ^He and the silver particle size. For the concentration of ^He, 0.6%, and 70 nm silver particle size, the diameter of the cluster was estimated to be ~ 20 nm. Three layers of ^He at the interface occupies ^ 84% of the total volume in the cluster. This indicates that a substantial fraction of ^He atoms are located at the interface. A significantly higher density in the cluster than that of pure bulk ^He at the sample pressure could account for the behavior in the magnetic susceptibility, which is like that of pure bulk solid ^He at a molar volume near that of the hep ''He matrix. 4.2.3 Sample Pressure = 3.35 MPa Pressure measurements The phase separation temperature, 180 mK, was near that for the 3.54-MPa sample, as shown by P{T) in Fig. 4.9. In the 3.35-MPa sample, the clusters underwent partial melting/freezing at « 20 mK (see Fig. 4.9). Hysteresis in temperature upon partial melting/freezing was observed. This behavior was also seen by Haley et al. [42] and may be related to the history dependent heat capacity seen by Schrenk et al. [1] The pressure increase due to the phase separation was similar in size to that in the 3.54-MPa sample. Again, this observation indicated that essentially all of the ^He in the cell had separated into clusters and no ^He was remained dissolved in the solid ^He matrix, as in the 3.54-MPa sample.

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89 Figure 4.6: Inverse susceptibility versus T of ^He nano-clusters for the 3.54-MPa sample. A fit is made separately for the data above and below 4 mK, which gives a Weiss temperature 9^, of « -0.25 mK and ^ 0.15 mK, respectively.

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90 3_J 1300 1250 1200 1150 1100 1050 1000 950 900 850 800 750 700 650 F600 ff • Sample Pressure = 3.54 MPa 10 12 T( mK) Figure 4.7: Magnetic susceptibility times T versus T of ^He nano-clusters in the 3.54MPa sample. A fit is made for the data above 5 mK, which gives : 926.36+0.66571 T(mK).

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91 2200 2000 1800 1600 1400 1200 1000 _i 800 600 400 200 0 Sample Pressure = 3.54 MPa -1 — I — I — I — I — I I I I -L 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 1/T(mK"') Figure 4.8: Magnetic susceptibility versus T'^ of ^He nano-clusters in the 3.54-MPa sample.

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Figure 4.9: Pressure change in the 3.35-MPa sample. Partial melting and freezing were observed. Dotted line is the pure bulk ^He melting pressure [17].

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93 Magnetic susceptibility measurements After partial melting, the clusters showed a Curie-like contribution in the magnetic susceptibility, which indicated a solid fraction. By taking the ratio of the Curie constant in this sample to the one in the 3.54-MPa sample (all solid), the solid fraction of this sample was determined to be 77%, as shown in Fig. 4.10. The solid fraction is obtained by taking the ratio of the Curie constant for this sample to the one in the 3.54-MPa sample and multiply the ratio by i>(3.35 MPa)/t;(3.54 MPa), where v is the molar volume of the ^He in the cluster. The compressibility of hep solid ''He at a molar volume of « 20.5 cm^/mole is ^ 0.03 MPa~\ which is approximately constant over our sample pressure range. The compressibility is written as follows: where kt is the isothermal compressibility in pure solid ^He, and P the pressure. We define vi and V2 as the molar volumes of the 3.54-MPa sample and the 3.35-MPa sample, respectively. We calculate 1'2/vi by using Eq. 4.3 as follows: — = 1 + KrdP = 1.006, (4.4) Vi because kt « 0.03 MPa~^ and dP « 0.2 MPa. Therefore we obtain the solid fractions for the 3.35-MPa sample by taking the ratio of the Curie constant of the 3.54-MPa sample to the one in the 3.54-MPa sample. This arguement is applied to the samples described later in this chapter. In this sample the magnetic susceptibility followed the Curie law down to ^ 0.6 mK with a Weiss temperature 9yj=-5 ± 5 /xK. Thus it was paramagnetic down to the lowest temperature measured, with a very slight antiferromagnetic tendency. However, scatter was too large to conclude it to be antiferromagnetic. This scat-

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94 ter level might have been caused by the use of different tipping angles during the measurements. 0.95 0.90 0.85 Sample Pressure = 3.35 MPa 0.80 I0.75 0.70 0.65 0.60 o Cooling • Warming o • •9 T( mK) 10 Figure 4.10: Magnetic susceptibility times T versus T of ^He nano-clusters in the 3.35-MPa sample. Dotted line is a fit of all the points. Fit : 0.76938+4.9628E-4 T (mK). To check for thermal equilibrium and reproducibility, data were taken during both cooling and warming and no hysteresis in the magnetization was observed. No ordering was observed contrary to the claim of Schrenk et al. A plausible explanation for lack of ordering in this sample was that the clusters were very small, with most of the solid in a few amorphous surface layers at the interface with the surrounding ^He, as in the 3.54-MPa sample.

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95 4.2.4 Sample Pressure = 3.06 MPa The 3.06-MPa sample showed a surprising result in the magnetic susceptibility (see below). Because of this observation, this sample was studied the most extensively among the samples in this work. This sample provided some insight and raised many questions about the nature of the ^He nano-clusters. Pressure measurements For this sample pressure, the L2-hcpi-bcc2 univariant curve (L2 is ^He-enriched liquid phase) [30], along which two separated solid phases and a liquid phase coexist, lay only slightly below the phase separation temperature, 180 mK. Thus the pressure changes below the phase separation included the excess pressure of the phase separation starting at 180 mK, followed by the three-phase (univariant) portion, in which some melting of the ^He-enriched phase occurred as the sample was cooled. This close following of the phase separation by the partial melting appeared to have resulted in an unique character for this sample. The excess pressure was somewhat larger than for higher-pressure samples, where partial melting did not occur until significantly lower temperatures. The partial melting of the clusters near the phase separation appeared to have allowed them to anneal with more complete phase separation and relaxation of stress. This might oflfer an explanation for the unique behavior of the magnetic susceptibility of this sample. There was a slight break in pressure at ^ 130 mK, which indicates the end of phase separation followed by partial melting. The jump in pressure at ^ 105 mK, as shown in Fig. 4.11, which was reproducible, is not well understood but could indicate the end of partial melting followed by more complete phase separation made possible by the liquid phase.

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96 Figure 4.11: Pressure change upon cooling in the 3.06-MPa sample. Dotted line i the pure bulk ^He melting pressure.

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0.6 0.5 P 0.4 0.3 Sample Pressure = 3.06 MPa / • 5th Warming O 4th Warming 3rd Warming 2nd Warming A 1st Warming 1 . !_ A 1st Cooling _i — . — . . . 1 1 10 T( mK) Figure 4.12: Magnetic susceptibility times T versus T of ^He nano-clusters in the 3.06MPa sample. A fit is made for the data above 1.1 mK, which gives : 0.54924+2.99E-5 T'(mK).

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98 Magnetic susceptibility measurements This sample was the most extensively studied because of a "kink" in the magnetic susceptibility at 1.1 mK, which was approximately the ordering temperature T^v that would be expected in pure bulk ^He, if it existed at 3.06 MPa. The behavior below 1.1 mK was quite different from that for pure bulk ^He, indicating that the ordered phase was not the U2D2 phase. There was no drop in the magnetic susceptibility at 1.1 mK, and it was almost constant down to 0.51 mK. As discussed below, no clear frequency shift was seen in this sample. Since Schrenk et al. reported a history dependence in the heat capacity, we cycled the temperature by varying the minimum temperature to check if there was similar behavior in the magnetic susceptibility [1]. The peak in the magnetic susceptibility was reproducible with no hysteresis or history dependence. By taking the ratio of the Curie constant in this sample to the one in the 3.54MPa sample (all solid), the solid fraction was determined to be 54% (see Sect. 4.2.3 for the discussion), as shown in Fig. 4.12. The high-T susceptibility above the kink gives a small positive 9^, ^ 0.15 mK indicating a ferromagnetic tendency (see Fig. 4.13). / = 250 kHz data . The magnetic susceptibility with the NMR tuned to 250 kHz is shown in Figs. 4.12, 4.13, and 4.14. The magnetic susceptibility was taken with the FFT as well as the PLM from the 3rd warming, since the frequency resolution of the FFT was improved from 100 Hz to 10 Hz, as shown in Figs. 4.13 and 4.14. The results with the PLM-4 and the FFT are qualitatively consistent with each other, with a similar 9^,. However, the magnetic susceptibility obtained from the FFT remained almost constant below 1.1 mK, while the PLM data showed a slight increase. The reason for this difference is not known. However, there could be two possibilities. (1) The different integration times between the PLM and the FFT could cause this since

PAGE 106

99 the PLM-4 integrates the FID over ^ 1 ms, whereas the FFT utilizes ^ 0.1 s in the FID, with our setting. (2) A hne broadening occurred below 1.1 mK that could cause the difference, since the FFT and the PLM-4 give the same result in the magnetic susceptibility above 1.1 mK. A frequency shift of » 20 Hz is suggested by the FFT spectra above and below 1.1 mK, as shown in Fig. 4.15, which was investigated further in the following sections. / = 125 kHz data . In order to look for a clear indication of a frequency shift, we tuned the NMR to 125 kHz. Since the external field was less, a frequency shift, if one existed, might be more pronounced than for the 250-kHz data. There appears to be a frequency shift of w 20 Hz, as seen in Fig. 4.16. Also the magnetic susceptibility was qualitatively consistent with that at 250 kHz. Therefore, we concluded that there was a frequency shift of w 20 Hz, which is to be compared to the frequency shift observed by Osheroff et al. [21] Calculation from Eq. 1.12 with cos^^j = 0 and 1 give four frequency shifts of ^ 528, 0, 567, and 250 kHz, respectively, which indicate that the ordered phase in this experiment is not the U2D2 phase. The signal-to-noise ratio was not as good as for the 250-kHz data (see Fig. 4.16), since the signal strength is proportional to the square of the frequency. However, we could conclude that there was a "kink" in the magnetic susceptibility at 1.1 mK, and, most importantly the magnetic susceptibility decreased at lower temperatures more than for the 250-kHz case. A small peak at 125 kHz was caused by the tank circuit. The Weiss temperature 9^ was « 0.15 mK, the same as that for the 250-kHz data, as shown in Fig. 4.17. Spin-spin relaxation {T2) measurements . Another check for magnetic ordering was made by measuring T2 at temperatures below and above the "kink", since T2 could reflect a line broadening due to a magnetic ordering. The two measurements

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100 0.000045 0.000040 0.000035 0.000030 d 30.000025 \LL LL 0.000020 0.000015 0.000010 0.000005 0.000000 -Sample Pressure = 3.06 MPa • * u • 3rd Warming o 4th Warming 5th Warming 1 1 1 1 1 1 1 ' 1 J — 1 — 1 1 1 4 5 6 T( mK) 10 Figure 4.13: Inverse of magnetic susceptibility as a function of temperature at 3.06 MPa. FFT of free induction decay at / = 250 kHz was used.

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101 220000 200000 180000 160000 140000 d cd 120000 100000 H U_ LL 80000 60000 40000 20000 • Sample Pressure = 3.06 MPa 0.0 go if / 0.2 0.4 0.6 0.8 1.0 1.2 m{ mK' ) 3rd Warming 4th Warming 5tin Warming -1 1 1 L 1.4 1.6 1.8 Figure 4.14: Magnetic susceptibility versus inverse of temperature at 3.06 MPa. FFT of free induction decay at / = 250 kHzwas used.

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102 Figure 4.15: FFT spectra for the 3.06-MPa sample above and below 1.1 mK, which suggested a frequency shift of »i 10 Hz.

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103 124880 124920 124960 125000 125040 Frequency(Hz) Figure 4.16: FFT spectra with 125 kHz for the 3.06-MPa sample. Although the signal-to-noise ratio is poor, a frequency shift of ^ 20 Hz below 1.1 mk is suggested by the data.

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104 0.0010 0.0008 d 0.0006 d 0.0004 0.0002 Sample Pressure = 3.06 MPa Frequency = 1 25kHz -^--J 1 1 1 1 1 1 i L 1 012345678 T(mK) Figure 4.17: Inverse of magnetic susceptibility as a function of temperature. FFT of NMR free induction decay signal at / = 125 kHz was used. Dotted line is a fit of the points above 2 mK. Fit is l/x=-1.54947E-5+1.25502E-4 T (mK).

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105 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 Time(ms) Figure 4.18: Spin-spin relaxation time measurements at temperatures below and above the "kink" in the 3.06-MPa sample.

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106 appeared to be identical within error, as shown in Fig. 4.18. Thus we concluded that there was no line broadening. Spin-lattice relaxation (Ti) measurements . The fast part of the Ti measurements (the Zeeman-to-exchange part) were carried out at temperatures below and above the "kink" to detect any change in relaxation mechanism due to a magnetic ordering. There was no significant difference between the two sets of data, as shown in Fig. 4.19. 0.0 -0.1 -0.2 -0.3 -0.4 2 -0.5 ^ -0.6 -0.7 -0.8 -0.9 -8 -1.0 o« o 0.77mK • 1.18mK J 1 L 0 100 200 300 400 500 600 700 800 900 1000 x(ms) Figure 4.19: Zeeman-to-exchange part of spin-lattice relaxation time measurements at temperatures below and above the "kink" in the 3.06-MPa sample.

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107 4.2.5 Sample Pressure = 2.96 MPa Since a small excess pressure in the 2.88-MPa sample was observed, as discussed in Sect. 4.2.6, additional pressure measurements in a sample whose pressure was closed to 2.88 MPa were made. Also the magnetic susceptibility was measured in this sample that exhibited the "kink" behavior seen in the 3.06-MPa sample. Because of the kink, the possibility of a frequency shift was investigated. Pressure measurements Pressures were measured at various temperatures, particularly around Tps, with the temperature regulated during the measurements, with the scheme in Fig. 2.14. A resistance thermometer, installed at the top of the PrNig stage, was used for the regulation with the output voltage from the LR-llQ providing a feedback to the LR130. The temperatures were determined by the MPT, as the sample was cooled in steps of w 3 mK from 220 raK to 20 mK. At each temperature the equilibrium pressure was measured; then the background capacitance change was subtracted from the raw data points. Phase separation occurred at ^ 180 mK (see Fig. 4.20). The excess pressure was about two thirds of that in the 3.06-MPa sample, in which the phase separation was followed by partial melting during cooling. The behavior in this sample was quite similar to the one in the 3.06-MPa sample, although the excess pressure was smaller than that in the 3.06-MPa sample. One possible explanation for the excess pressure size could be that there was a mixture of hep and a meta-stable bcc "He, since this sample was annealed close to the bcc phase. In the 2.88-MPa sample, the annealing temperature was lower than in this sample. Thus, more meta-stable bcc may have persisted in the 2.88-MPa sample, which could explain the difference in

PAGE 115

108 (TJ Q. J*: CL < 2 0 -2 -4 -6 -8 -10 -12 -14 -16 10 Partial Melting Phase Separation -I I ' 100 T{mK) i % Figure 4.20: Pressure change of the 2.96-MPa sample on cooling. Dotted line is the pure bulk ^He melting pressure [17].

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109 excess pressure between the two samples. It is unclear if this possible bcc-hcp mixture affected the magnetic susceptibility in this sample, as discussed in the next section. Magnetic susceptibility measurements A frequency shift of x 20 Hz is suggested from the FFT spectra, as shown in Fig. 4.27, which was also observed in the 3.06-MPa sample (see Sect. 4.2.4). By taking the ratio of the Curie constant in this sample to the one in the 3.54MPa sample (all solid), the solid fraction was found to be 23% (see Fig. 4.21). The magnetic susceptibility showed the same behavior as for the 3.06-MPa sample, having a "kink" at 1.1 mK and a small positive Weiss temperature 9^^ « 0.15 mK, as shown in Fig. 4.24. However, this sample had a distinctive hysteresis below 1.1 mK. The susceptibility on cooling was greater than that on warming without any history dependence, as shown in Fig. 4.23. The hysteresis as well as the kink (see Fig. 4.25) was clear in x with the FFT, but less clear with the PLM, as shown in Fig. 4.26. This might be related to the line broadening, as discussed above in Sect. 4.2.4, although the T2 results do not exhibit a large change, as shown in Fig. 4.28. This hysteresis was not seen in the 3.06-MPa sample. Since the kink was observed again in this sample, clear frequency shifts in FFT spectra and changes in T2 and the Zeeman-exchange part of Ti below and above the kink temperature were investigated. As in the 3.06-MPa sample, no substantial change in Ti or T2 due to magnetic ordering was observed, as shown in Figs. 4.29 and 4.28. The signal-to-noise ratio in this sample at 125 kHz seems better than that for the 3.06-MPa sample. The magnetic susceptibility with 125 kHz exhibited a small frequency shift of ^ 10 Hz, as shown in Fig. 4.30. A shift of « 20 Hz appeared in the FFT spectra at 125 kHz for the 3.06-MPa sample, but with less precision. This needs to be investigated further with an improved frequency resolution in the FFT.

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110 0.245 0.240 • 0.235 • 0.230 i • • • ^ 0.225 H ^ 0.220 • • o ^ 0.215 • 1st Cooling 0.210 o "0 o 1st Warming 2nd Cooling 0.205 o o 2nd Warming 3rd Cooling 0.200 J . , — , — 1 10 T (mK) Figure 4.21: Magnetic susceptibility times T versus T of ^He nano-clusters in the 2.96-MPa sample.

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Ill 110000 • Sample Pressure: 2.96 MPa 100000 90000 80000 70000 60000 50000 40000 30000 20000 10000 • 1st Cooling O 1st Warming 2nd Cooling _ 2nd Warming • A 3rd Cooling A 3rd Warming 125kHz O J I L -I I ' 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1/T(mK"') 1.4 1.6 1.8 Figure 4.22: Magnetic susceptibility versus T'^ of ^He nano-clusters in the 2.96-MPa sample. Solid down-triangles: the NMR frequency of 125 kHz, which exhibited a similar behavior to that of the 3.06-MPa sample at 125 kHz. All other points were taken at 250 kHz.

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112 Sample Pressure: 2.96 MPa 1 85000 • o 80000 • (a.u.) 75000 o (Idz 70000 L 0 0 o 0 o • 1st Cooling o 1st Warming 65000 2nd Cooling 2nd Warming 3rd Cooling 60000 — ' 1 1 i 1 1 1 1 1 3rd Warming 0-9 1.0 1.1 1.2 1.3 1.4 1/T(mK"') Figure 4.23: Magnetic susceptibility versus T'^ of ^He nano-clusters in the 2.96-MPa sample. There is a hysteresis between cooling and warming.

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113 9 1st Cooling / 0.00012 O 1st Warming 2ncl Cooling 2ncl Warming •I'' / • 0.00010 3rcl Cooling A 3rcl Warming 0.00008 P LL LL 0.00006 A'' 0.00004 0.00002 0.00000 L. L. . I.I.I. 1 1 1 I 1 X ' I I I 2 4 6 8 10 12 T(mK) Figure 4.24: Inverse of magnetic susceptibility versus T of ^He nano-clusters in the 2.96-MPa sample. Dotted line is given by : l/x=-1.49577E-6+1.3614E-5 T (mK), which is obtained by fitting the points above 1.5 mK.

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375 350 325 300 275 250 225 d 200 175 150 125 100 75 50 25 0 ;_Sample Pressure: 2.96 MPa • : X • 1st Cooling A» o 1st Warming A A 2nd Cooling • — 1 — 1 — 1 — 1 — 1 1 1 1 2nd Warminc A 3rd Cooling ^ 3rd Warming — 1 — 1 — 1 — 1 — 1 1 i_ 1.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 1/T(mK"') 114 Figure 4.25: Magnetic susceptibility versus J-^ of ^He nano-clusters in the 2.96-MPa sample. The kink is less pronounced in the data with the PLM. I

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> 115 300 3 250 03 200 Sample Pressure: 2.96 MPa A ft aS o in I i_ On 1st Cooling 1st Warming 2ncl Cooling 2nd Warnning 3rcl Cooling 3rd Warming 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 in"(mK') Figure 4.26: Magnetic susceptibility versus of ^He nano-clusters in the 2.96-MPa sample. The hysteresis was less clear than that with the FFT. This shows the detail of Fig. 4.25 around the kink.

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116 Figure 4.27: FFT spectra of NMR free induction signal at 250 kHz above and below 1.1 mK in the 2.96-MPa sample, which suggested a frequency shift of « 20 Hz.

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Sample Pressure : 2.96 MPa • 0.83 mK Fit of 0.83 mK data ' ' I.I.I I I I . I 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 Time(ms) Figure 4.28: Spin-spin relaxation time measurements at temperatures below and above the "kink" in the 2.96-MPa sample. The NMR was tuned at 250 kHz.

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118 0.00 -0.05 -0.10 -0.15 ^ -0.20 o >, -0.25 ^ -0.30 !e -0.35 -0.40 -0.45 -0.50 O • to. -1 I I I I I I • 0.82 mK o 1.27 mK _L 0 100 200 300 400 500 600 700 800 900 Time(ms) Figure 4.29: Zeeman-exchange part of spin-lattice relaxation time measurements at temperatures below and above the "kink" in the 2.96-MPa sample. The NMR was tuned at 250 kHz.

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119 Figure 4.30: FFT spectra with 125 kHz in the 2.96-MPa sample. The signal-to-noise is better than that for the 3.06 MPa-sample. There appeared to be a frequency shift of «i 10 Hz.

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120 4.2.6 Sample Pressure = 2.88 MPa Since the interesting "kink" was found in the 3. 06MPa sample, a lower pressure sample, a 2.88 MPa, was formed and studied in order to gain a clearer picture of the nature of the behavior. Pressure measurements For this sample, the phase separation temperature was lower than expected, 145 mK, which was substantially lower than in the first three samples. At this pressure and temperature, the sample was in the region of the mixture phase diagram where the ^He phase was already a liquid. Thus the pressure increase should include both the excess pressure due to phase separation and the increase in melting, which would be larger than for phase separation alone. However, this was not observed. The total pressure change (phase separation and melting) was « 7 kPa, which is the same as for phase separation in the 3.54-MPa sample, as shown in Fig. 4.5. A probable explanation for the lower phase-separation temperature and the smaller pressure change is that the ^He matrix was in a meta-stable bcc phase. At higher temperature where this sample was annealed, the crystal structure was bcc, which may persist because of the meta-stability, as shown in Fig. 1.10. In fact, as the sample was cooled after annealing, no evidence (decrease in pressure) of the bcc-hcp transition in ^He was seen. Therefore it appeared that the ^He matrix remained in a meta-stable bcc phase. Then, no structural change would be involved in phase separation from the bcc mixture to separated phases of ^He and ^He, with a small excess pressure, as we observed. Although the P-T diagram for '^He with small concentration of ^He is different than for pure ^He, as shown in Fig. 1.10, the pressure change due to the bcc-hcp transformation in our ^He-^He sample can be estimated by using the phase diagram

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121 for pure "^He. The pressure change due to the bcc-hcp transformation in pure ^He can be calculated for the path AF1+AP2 shown in Fig. 4.31 that connects the beginning and ending states. We have AP = APi + AF2, (4.5) where APi and AP2 are pressure changes along paths 1 and 2 in Fig. 4.31, respectively. The pressure change AFi can be read from the pure "^He melting pressure curve and AFi «i -0.35 MPa. The pressure change AP2 can be calculated from AP. = -i^, (4.6) V Kt where kt 0.0038 atm"^ at 21 cm^/mole) is the isothermal compressibility in pure solid "^He, v is the molar volume, and P the pressure [62]. The pressure change AP2 is calculated to be ^ 0.15 MPa. The total pressure change expected is ^ -0.2 MPa. This is consistent with what would be read off the P-T diagram in Fig. 4.31. Thus if the transformation occurs, it should be noticeable. However, since no such pressure change was observed, we concluded that the meta-stable bcc ^He persisted. Magnetic susceptibility measurements For this sample, the magnetic susceptibility followed a Curie-Weiss law with Weiss temperature 9,^ = 140 /zK, indicative of ferromagnetic tendency similar to that found in 2D ^He films with intermediate coverage [24] (see Fig. 1.8). By taking the ratio of the Curie constant in this sample to the one in the 3.54-MPa sample (all solid) (see the discussion in Sect. 4.2.3), the solid fraction was found to be 19%, as shown in Fig. 4.33. For this solid fraction, all of the ^He would be within 3 layers of the interface with the ^He, if the clusters had the maximum size of 20 nm, as expected. If there were more, smaller clusters, then there would be even fewer than 3 layers

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122 Figure 4.31: Molar volume and melting pressure curve versus temperature for pure ^He. The dotted arrow in the P-T plane is the path expected if the sample undergoes the bcc-hcp transformation. Calculation of expected pressure change through the bcc-hcp transformation in ^He can be done for path 1 and path 2. The total pressure change is AFi+APaThe pressure change in the bcc-hcp transformation can be read from this figure, by comparing the "expected path" and the "2.88-MPa sample path". The points are from [62].

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123 Figure 4.32: Pressure change of the 2.88-MPa sample at phase separation.

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124 present. This could explain the ferromagnetic behavior in this sample. We note that even at this low pressure, Schrenk et al. found a slight peak in the heat capacity at 1 mK, which they interpreted as a magnetic ordering. However, most of the entropy was not removed in going through the peak. Our magnetization data shows no indication of magnetic ordering, which is inconsistent with the result of Schrenk et al. 0.28 0.26 O 0.24 Sample Pressure = 2.88 MPa Curie-Weiss fit w/ 9=140 ^K, C = 0.188 • 1st Cooling O 1st Warming 2nd Cooling 2nd Warming to.22 0.20 0.18 10 T(mK) Figure 4.33: Magnetic susceptibility times T versus T of ^He nano-clusters in the 2.88-MPa sample.

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CHAPTER 5 MODEL FOR INTERPRETING RESULTS AND SUGGESTED FUTURE WORK In this chapter, a model of the ^He cluster, discussion of the results based on the model, and several suggestions for future work will be presented. 5.1 Model for Interpreting the Results and Discussion In this section, we propose a model of the ^He clusters, as shown in Fig. 5.1, and discuss the important results in this work in terms of the model. At the pressures involved in this work, ''He has an hep structure with a molar volume of ^ 21 cm^/mole, whereas ^He has a bcc structure with a molar volume of ^ 24 cm^/mole. The van der Waals attraction would tend to produce the same density in the ^He at the interface with the ^He matrix as that of the ''He matrix. This causes density gradients in the cluster, because the attraction is weaker in the interior. Figure 5.1 (b) illustrates a cluster at intermediate pressures where partial melting has occurred. However, the detailed density profile of the ^He at the interface is not known. 5.1.1 Partial Melting All samples in this work for P < 3.35 MPa were cooled to below the melting temperature where pure bulk ^He would exist only as a liquid. Although the pressure versus temperature indicated some melting of the clusters, it was incomplete as indicated by a CurieWeiss law susceptibility. In our model, the interior of the droplet would undergo melting, while a few layers adjacent to the ^He would remain solid. A 125

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126 Silver Figure 5.1: Model of the ^He nano-clusters. (a): overall configuration of the clusters within the ^He matrix, (b): details of a cluster for intermediate pressures where partial melting occurs. The cluster has solid layers adjacent to the ''He matrix and liquid inside.

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127 solid fraction was observed even in the 2.88-MPa sample that is below Pmin for the pure bulk solid ^He. 5.1.2 Anomalously Short Ti's Anomalously short Ti's for the Zeeman-to-exchange part of the relaxation, followed by a much longer relaxation of the lattice to the heat exchanger, were observed in all the samples except the 2.96-MPa sample, in which the measurements were not made. The short Ti's might be explained by the lattice distortion in the ^He cluster caused by the mismatch in density with the ^He matrix at the interface. As a result, the ^He at the interface would have dislocations, which could couple the exchange and the lattice subsystems, since the frequencies of the dislocation motions and those of the exchange could overlap. However, further investigation and theoretical study are needed for a clear understanding of the mechanism that gives rise to the short Tj's. 5.1.3 Magnetic Susceptibility for the 3.54-MPa Sample The magnetic susceptibility for the 3.54-MPa sample showed a behavior similar to that seen in the pure bulk ^He with a molar volume of ^ 21.3 cm^/mole. This observation is strong evidence for supporting the model, since the molar volume of 21.3 cm^/mole is near that of the hep ^He matrix surrounding the ^He. The van der Waals attraction would influence the density of the ^He at the interface, which in turn would cause density gradients in the ^He cluster, as shown in Fig. 5.1. This high density of the ^He at the interface would account for the lack of magnetic ordering in the 3.54-MPa sample for T > 0.5 mK. Silmilar arguments would apply to the 3.35-MPa sample.

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128 5.1.4 Kink in Magnetic Susceptibility and Frequency Shift A kink was observed in the magnetic susceptibility at 1.1 mK for the 3.06and 2.96-MPa samples. No drop in x was observed at 1.1 mK, which is a characteristic of magnetic ordering of the pure bulk ^He. Instead x stayed almost constant down to ^ 0.51 mK in the data with the FFT.^ This behavior below 1.1 mK could be interpreted as a consequence of density gradients, which are shown in the model. The ^He at the interface shows no ordering, whereas the ^He in the interior might order as seen in the pure bulk ^He. The total magnetization shows the kink but no drop. The FFT spectra taken above and below 1.1 mK exhibited only a slight frequency shift of PS 10 ± 10 Hz relative to the spectra taken above 1.1 mK, suggeting that the ordered phase is not the U2D2, if it exists. 5.1.5 Two-dimensionallike Magnetic Susceptibility The magnetic susceptibility for the 2.88-MPa sample showed a ferromagnetic tendency similar to that seen in the 2D films. In the 2.88-MPa sample, the solid fraction was found to be 19%, for which all of the solid ^He could be within 3 layers of the interface with the "^He resulting in the 2D like behavior. 5.1.6 T^iP) for the Clusters and for Pure Bulk ^He The heat capacity results of Schrenk et al. exhibited a history-dependence in the peak, which they claimed to be Tat, at pressures as low as 700 kPa below the melting pressure in pure bulk ^He. They concluded that of the cluster is just an extension of Tn of pure bulk ^He, as shown in Fig. 1.14. However, no ordering was observed in this work except possibly at intermediate pressures. Within our model a sharp 1 However, the data with the PLM showed a significant increase down to the lowest temperature, which is not understood.

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129 first-order transition as an extension of T/v to pressures below bulk melting would not occur. This is, in fact, consistent with the results of Schrenk et al., in which most of the spin entropy remains. 5.2 Suggestion for Future Work on "'He NanoClusters 5.2.1 Magnetic Susceptibility Measurements with a Cold Preamplifier A cold preamplifier was built and tested by Lucas Hundley in this laboratory. This would allow us to study the magnetic susceptibility up to the phase separation temperatures, since the signal to noise ratio would be improved by a factor of 10. The magnetic susceptibility measurements across the phase separation temperatures could provide important information, such as kinetics, on the nature of the phase separation of the mixtures. The improved signal to noise would allow the Weiss temperature 6yj to be determined with greater precision. The detailed outline of the amplifier is given below. The cold preamplifier is a solid state device^ designed to operate while submerged in the liquid helium bath as close to the NMR measurement coils as physically possible in order to improve the amplifier noise figure. The amplifier is to be mounted so that it is as isolated as possible from magnetic disturbances of the magnetic cooling system. It is fed a signal from the NMR receiving coil via a thin coaxial line. The signal is then amplified, and may be sent to a room temperature impedance-matching circuit via a standard coaxial line, or a diflFerential twisted pair depending upon the setting of the amplifier (wide-band needs no impedance matching and uses a coaxial line). At room temperature, the signal is then run into the lab's existing data acquisition system and analyzed. ^GaAs FETs are used for the amplifier.

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130 A prototype and an active version of the amplifier, which was designed by Kondo et al. [63], have been constructed and tested at room, liquid nitrogen, and liquid helium temperatures. The active version is designed to fit snugly outside of the nuclear demagnetization magnet and to be fully submerged in the liquid helium bath. It can be readily configured as a narrow-band amplifier, or a wide-band amplifier. Ideally, the narrow band configuration would be selected for use in NMR measurement techniques for its advantages of higher gain and lower noise. Unfortunately, the narrow band amplifier involves the use of an impedance matching network between room temperature and the preamplifier. This network involves the use of a miniature inductor in the submerged preamplifier, and this appears to be the source of some problems. The circuit for the narrow-band preamplifier is shown in Fig. 5.2. Vgl Vg2 c= capacitance Transistors are MESFETS: SonvSKIGGA determined by desired Frequency range and subject to change upon installation on new line production, Sony must be contacted directly for more See Lucas Hundley for info NOTE: no longer in Figure 5.2: Circuit of the cold preamplifier (narrow-band). Figure 5.3 shows the resulting gain of a sinusoidal test signal (0.1 Vpp) for a frequency sweep about 250 kHz fed into the amplifier in narrow-band mode at room

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131 Figure 5.3: Test result of the cold preamplifier. temperature, in liquid nitrogen, and in liquid helium. As can be clearly interpreted from the graphs, as the temperature of pre-amp is lowered the resultant frequency increases, the Q of the resonance increases, and the maximum gain increases. When in the wide-band mode, the amplifier produces a fairly constant gain for the entire region of interest and beyond. The wide-band configuration results in an average gain of 5.5 for input frequencies from 100 450 kHz. 5.2.2 Temperature Scale Improvement Magnetic susceptibility measurements in clusters at temperatures below 0.6 mK to search for possible magnetic ordering would be of interest. In order to do so, a new thermometer would be required because the MPT does not have adequate resolution below T^. Pt-NMR in the same field as the ^He-^He sample can be used.

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132 Also pressure or magnetic susceptibility measurements on a high-density solid ^He with a low magnetic ordering temperature [64] would be good methods to measure temperatures, because x for the high-density pure bulk ^He follows a Curie law down to « 0.5 mK for a molar volume of ^ 21 cm^/mole. Olejniczak et al. have used NMR measurement of x of high-density solid ^He to measure the magnetic susceptibility of a pure bulk ^He sample {v = 24.25 cm^/mole). In their experiment they installed two sample cells that are thermally connected, one for the ^He sample of interest, the other with a high-density ^He sample (21 cm^/mole) to serve as a thermometer. They measured x for the two samples simultaneously under a field gradient that allowed the two signals to be distinguished. In their settings, only one spectrometer was needed, which was an advantage of a high-density solid ^He thermometry by magnetic susceptibility measurements as well over the Pt-NMR. A disadvantage of the pressure thermometry in the a high-density pure bulk solid ^He is the resolution, because the pressure changes are small for a high-density solid ^He. On the other hand, a relatively simple setup like that for the MPT can be used in the pressure measurements. 5.2.3 Pressure Measurements of ^He Nano-Clusters Since our pressure transducer has a large temperature-dependent background, higher resolution pressure measurements with a new sample cell that does not have the background problem are particularly necessary to complete projections of univariant surfaces in the P-T plane [65]. Also additional pressure measurements could determine the existence of the meta-stable bcc ^He in low pressure samples. A new sample cell for pressure measurements has been built and tested. The cell has a diaphragm whose thickness is 1.016 mm appropriate for use up to w 6.0 MPa.

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133 Japanese silver powder^ has been packed at 2.585 MPa to yield a packing fraction of 50% with the same pore size as that of the sinter in the existing NMR cell. 5.2.4 Heat Capacity Measurements As mentioned in the conclusions, simultaneous heat capacity measurements are necessary to determine the change in entropy across the kink temperature. This would give a decisive answer as to whether the ^He nano-clusters are magnetically ordered at the kink temperature. 5.2.5 Field Sweep to Look for a Frequency Shift Since there was a discrepancy in the magnetic susceptibility below the kink temperature between the PLM and the FFT for the 3.06and 2.96-MPa samples, magnetic susceptibility measurements by sweeping the NMR static field is suggeted. The measurements could give an answer to the discrepancy, which might be originated from a frequency shift to outside the bandwidth of the PLM, in which we might be able to detect possible satellites in resonance. 5.2.6 Magnetic and Pressure Study of ^He Nano-Clusters with a Larger Concentration Our results indicate that surface effects play a decisive role in the magnetism of the clusters. By increasing the ^He concentration, the surface-to-volume ratio of the clusters would be decreased and a different behavior in magnetic susceptibility and pressure would be expected. For instance the concentration might be increased to ~ 1.5%, which would also have the advantage of increasing the size of the NMR signal. ^Tokuriki Silbest C-8, nominal poresize = 70 nm.

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134 Since Schrenk et al. used 1% ^He, using a larger concentration might give insight into the history-dependent hysteresis that they observed. Since our model proposes density gradients in the cluster, which strongly depend on the cluster size, to be decisive to determine the magnetic properties of the cluster, we suggest the magnetic susceptibility measurements with a small pore size of silver power, since the size of the cluster is detremined by the pore size. The measurements could support our model.

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REFERENCES [I] R. Schrenk, R. Konig, and F. Pobell. Phys. Rev. Lett, 76:2945, 1996. [2] D. S. Grey wall and R A. Busch. Phys. Rev. B., 36:6853, 1987. [3] R. A. Guyer, R. C. Richardson, and L. I. Zane. Rev. Mod. Phys., 43:532, 1971. [4] M. C. Cross and D. S. Fisher. Rev. Mod. Phys., 57:881, 1985. [5] I. Pomeranchuk. Zh. Eksp. Teor. Fiz., 20:919, 1950. [6] R. P. Feynman. Statistical Mechanics, A Set of Lectures, AddisonWesley Publishing Company, 1972. [7] G. A. Baker, Jr., H.E. Hilbert, J. Eve, and G. S. Rushbrooke. Phys. Rev., 164:800, 1967. [8] W. P. Kirk and E. D. Adams. Phys. Rev. Lett, 27:392, 1971. [9] W. P. Halperin, C. N. Archie, F. B. Rasmussen, R. A. Buhrman, and R.C. Richardson. Phys. Rev. Lett, 32:927, 1974. [10] R. B. Kummer, R. M. Mueller, and E. D. Adams. J. Low Temp. Phys., 27:319, 1977. [II] J. H. Hetherington and F. D. C. Willard. Phys. Rev. Lett, 35:1442, 1975. [12] P. A. M. Dirac. Quantum Mechanics, Oxford Univ. Press, 1958. [13] M. Roger, J. H. Hetherington and J. M. Delrieu. Rev. Mod. Phys., 55:1, 1983. 14] D. J. Thouless. Proc. Phys. Soc, 86:893, 1965. [15] D. D. Osheroff and C. Yu. Phys. Lett A., 86:893, 1980. 16] M. F. Panczyk and E. D. Adams. Phys. Rev. A., 1:1356, 1970. 17] W. Ni, J. S. Xia, E. D. Adams, P. S. Raskins, and J. E. McKisson. J. Low Temp. Phys., 99:167, 1995. 18] H. L. Stipdonk and J. H. Hetherington. Phys. Rev. B., 41:200, 1985. 19] J. S. Xia, W. Ni, and E. D. Adams. Phys. Rev. B., 70:1481, 1993. 135

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136 [20] T. Hata, S. Yamasaki, T. Kodama, and T. Shigi. J. Low Temp. Phys., 71:193, 1988. [21] D. D. OsherofF, M. C. Cross, and D. S. Fisher. Phys. Rev. Lett, 44:792, 1980. [22] E. D. Adams, E. A. Schubert, G. E. Haas, and D. M. Bakalyar. Phys. Rev. Lett, 44:789, 1980. [23] D. D. OsherofF. J. Low Temp. Phys., 87:297, 1992. [24] C. Bauerle, A. S. Chen, S. Triqueneaux, Yu. M. Bunkov, H. Godfrin, and M. Roger. J. Low Temp. Phys., 113:259, 1998. [25] K. Ishida, M. Morishita, K. Yawata, and H. Fukuyama. Phys. Rev. Lett, 79:3451, 1997. [26] D. S. Greywall and P. A. Busch. Phys. Rev. Lett, 65:2788, 1990. [27] B. A. FYaass and R. 0. Simmons. Phys. Rev. B., 36:97, 1987. [28] G. C. Straty and E. D. Adams. Phys. Rev., 150:123, 1966. [29] D. 0. Edwards, A. S. McWilhams, and J. G. Daunt. Phys. Lett, 1:218, 1962. [30] D. O. Edwards and S. Balibar. Phys. Rev. B., 39:4083, 1989. [31] A. H. Wilson. Thermodynamics and Statistical Mechanics. Cambridge Univ. Press, England, page 420, 1957. [32] M. F. Panczyk, R. A. Scribner, G. R. Gonano, and E. D. Adams. Phys. Rev. Lett., 21:594, 1968. [33] W. J. Mullin. Phys. Rev. Lett, 20:254, 1968. [34] P. G. Klemens, R. de BruynOubuter, and C. Le Pair. Physica, 30:1863, 1964. [35] I. Prigogine, R. Bingen, and J. Jeener. Physica, 20:383, 1954. [36] I. Prigogine and J. Jeener. Physica, 20:516, 1954. [37] I. Prigogine, R. Bingen, and A. Bellemans. Physica, 20:633, 1954. [38] I. Prigogine. The Molecular Theory of Solutions. North-Holland, Amsterdam, page 400, 1957. [39] R. A. Coldwell-Horsfall. Proceedings of the 9th International Conference on Low Temperature Physics, Colombus, OH, Edited by D.O. Edwards, J. G. Daunt, F. J. Milford, and M. Yaqub, Plenum, NY, page 1110, 1965. [40] S. Ehrlich and R. 0. Simmons. Can. J. Phys., 65:1569, 1987.

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137 [41] I. Iwasa and H. Suzuki. Proceedings of the 4th International Conference on Phonon Scattering in Condensed Matter, Stuttgart, 1983, editted by W. Eisenmenger, K. Lassman, and S. Dottinger, Springer, Berlin, 260, 1984. [42] R. P. Haley and E. D. Adams. Low Temp. Phys., 23:461, 1997. [43] D. N. Bittner, E. D. Adams. J. Low Temp. Phys., 97:519, 1994. [44] R. Schrenk, O. Friz, Y. Fujii, E. Syskakis, and F. Pobell. J. Low Temp. Phys., 84:131, 1991. [45] D. N. Bittner and E. D. Adams. J. Low Temp. Phys., 97:519, 1994. [46] Y. Eckstein, J. Landau, S. G. Lipson, and Z. Olami. Phys. Rev. Lett, 45:22, 1980. [47] 0. V. Lounasmaa. Experimental Principles and Methods Below 1 K, Academic Press, London, New York, 1974. [48] F. Pobell. Matter and Methods at Low Temperatures, SpringerVerlag, 1991. [49] L. E. De Long, 0. G. Symco, and J. C. Wheatley. Rev. Sci. lustrum., 41:147, 1971. [50] M. Kubota, H. R. Folle, Ch. Buchal, R. M. Mueller, and F. Pobell. Phys. Rev. Lett, 45:1812, 1980. [51] D. Hechtfischer. Rev. Sci. lustrum., 52:237, 1987. [52] W. Ni. Ph.D. dissertation, University of Florida, 1994. [53] E. D. Adams. Rev. Sci. lustrum., 64:601, 1993. [54] E. D. Adams, V. A. Shvarts, R. P. Haley, N. Matsunaga, and J. S. Xia. J. Low Temp. Phys., 113:375, 1998. [55] T. Lang, P. L. Moyland, D. A. Sergatskov, E. D. Adams, and Y. Takano. Phys. Rev. Lett, 77:322, 1996. [56] M. F. Panczyk. Ph.D. dissertation. University of Florida, 1968. [57] E. Fukushima and S. B. W. Roeder. Experimental Pulse NMR, A Nuts and Bolts Approach, AddisonWesley Publishing Company, 1991. 58] M. E. R. Bernier and G. Guerrir. Quantum Fluids and Solids, Sanibel Island, Florida, AIP Conference Proceedings, 1983. 59] N. Mikhin, N. Omelaenko, A. Polev, E. Rudavskii, and V. Shvarts. J. Low Temp. Phys., 113:781, 1998.

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138 [60] E. R. Dobbs. Solid Helium Three, Clarendon Press, Oxford, 1991. [61] A. N. Ganshin, V. A. Maidanov, N. F. Omelaenko, A. A. Penzev, E. Ya. Rudavskii, and A. S. Rybalko. Low Temp. Phys., 24:611, 1998. [62] J. Wilks. The properties of liquid and solid helium, Clarendon Press, Oxford, 1967. [63] Y. Kondo, J. H. Koivuniemi, J. J. Ruohio, V. M. Ruutu, and m. Krusius. Czechoslovak J. of Phys., 113:2843, 1996. [64] Z. Olejniczak, W. P. Kirk, A. A. V. Gibson, P. Kobiela, and A. Czermak. Quantum Fluids and Solids, Sanibel Island, Florida, AIP Conference Proceedings, 1983. [65] P. M. Tedrow and D. M. Lee. Phys. Rev., 181:399, 1969.

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BIOGRAPHICAL SKETCH Naoki Matsunaga was born on November 3rd, 1970, in Ise city, Mie prefecture, Japan. In 1990, he entered Nagoya University, Japan. He recieved a B.S. in Physics in March 1994. After spending 18 months as a graduate student at Nagoya University, majoring in experimental low-temperature physics, he moved to Florida to pursue a Ph.D. in the same field of study in August 1995. After a year in the Microkelvin Laboratory working with professor Yasu Takano and Dr. Paul Moyland on nuclear magnetic ordering and negative temperatures in silver, he started working under the supervision of professor E. D. Adams on magnetic susceptibility and pressure measurements in ^He nano-clusters. He earned a Ph.D. in December 2000. 139

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140 General Audience 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 SUSCEPTIBILITY AND PRESSURE MEASUREMENTS IN HELIUM-THREE NANO-CLUSTERS By Naoki Matsunaga December 2000 Chairman: E. Dwight Adams Major Department: Physics Magnetic studies of ^He in various geometries have contributed for understanding the quantum mechanical nature of the nuclear spin interactions, which in turn would help modern society understand the foundation of nature and develop high technology. For instance, recently in the semiconductor industry, the quantum mechanical understanding of the materials becomes critical to improve the performance of computers. Thus, the magnetic studies of ^He are beneficial to the society such as Florida where there is a potential to grow in high technology. In this work, magnetic properties of phase separated ^He nano-clusters in a hep ^He matrix confined in a silver sinter had been studied. This configuration gave a geometry for studying exchange processes among the nuclear spins.

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. E. D. Adams, Chairman Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. isumasa Takano Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Charles m H Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Gar>^. Ihai Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosoplu'. Alexander Angerhof^r Associate Professor of Chemistrv

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This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted ais partial fulfillment of the requirments for the degree of Doctor of Philosophy. December 2000 Dean, Graduate School


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