Dynamics of the 1st order phase transition between

the nuclear ordered phases of solid 3He

Takayoshi Tanakaa, Hideaki Itoa, Yutaka Sasakia,b,*, Takao Mizusakia,b

aDepartment of Physics, Graduate School of Science, Kyoto University, Kyoto 606 8502, JapanbResearch Center for Low Temperature and Materials Sciences, Kyoto University, Kyoto 606 8502, Japan

Abstract

Dynamics of the 1st order phase transition between the U2D2 and the high field phases (HFP) was studied by field-cycling method between

these phases by using ultra low temperature magnetic resonance imaging (ULT-MRI). Single Crystal of U2D2 3He was produced at the bottom

of compressional cell in superfluid 3He-B at about 0.5 mK. Domain distribution in the U2D2 crystal was examined by ULT-MRI. We have

measured the NMR signal intensity to extract the time-evolution of the HFP, after the static magnetic field was swept quickly through the critical

field BC1 and was stayed at BZBC1CDB. The volume concentration of the U2D2 decreased exponentially in time during the early stage of the

phase transition. The rate constant depended positively on DB. After the phase transition to the HFP was completed, the static field decreased

through BC1 and was fixed at BZBC1KDB. The observed rate constant was similar to the value in the opposite direction with identical DB. This

exponential evolution and DB dependence of its rate suggest that the early stage of the phase transition is controlled by the nucleation process.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: A. Magnetic materials; D. Nuclear magnetic resonance (NMR); D. Phase transitions

1. Introduction

The nuclear magnetism of solid 3He is considered to be

an ideal system to study fundamental properties of the

magnetism and physics of the highly correlated many body

system. Since the first experimental confirmation of the

nuclear ordering in bcc solid 3He, various investigations

have been made to clarify the nature of this remarkable

quantum crystal [1]. There are two phases in the external

magnetic field. The low field phase has been identified to be

the U2D2 phase, and the high field phase (HFP) is

considered to be the canted normal antiferromagnetic

(CNAF) phase. The U2D2 spin structure has a uniaxial

symmetry, whose axis [̂ is one of the (100) axes in the bcc

crystal lattice. On the other hand, the spin structure of the

HFP has cubic symmetry and has a large magnetization

along the applied field. Due to the large difference in

magnetization between the U2D2 phase and the HFP, the

nature of the phase transition was identified as the 1st order.

0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jpcs.2005.05.037

* Corresponding author.

E-mail address: sasaki@scphys.kyoto-u.ac.jp (Y. Sasaki).

For the case of the 1st order phase transition, a

supercooling phenomenon occurs across a critical point,

and a nucleation barrier between metastable phase and

stable phase plays an important role. In the ideal case, where

this barrier is homogeneous throughout the system, a seed of

stable phase may appear in any place. Following the

formation of the first seed, the path of the phase evolution

can be classified into two categories. When the growth of

the seed is rapid enough, the phase transition proceeds by

expanding the first seed into the entire system. However,

when the growth of the seed is not rapid enough, new seeds

will appear one after another at different locations in the

system. This would generate a mist-like distribution of the

stable phase in the matrix of the metastable phase.

In most of the real situation, it is unlikely to have

homogeneous nucleation barrier throughout the system, and

heterogeneous nucleation must be taken into account [2]. In

this case, a seed of the stable phase may appear at particular

nucleation sites. However, even in this case, as far as there

are a number of possible nucleation sites, the path of phase

evolution can also be categorized into the same two classes

as the homogeneous case.

In the case of the U2D2-HFP transition, there is also an

interesting phenomenon, called as the memory effect [4],

where the magnetic domain distribution in U2D2 phase is

Journal of Physics and Chemistry of Solids 66 (2005) 1475–1477

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T. Tanaka et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1475–14771476

memorized after transforming into the HFP and transform-

ing back to the U2D2 phase by changing the applied

magnetic field. Since spin symmetries in both phases are

different, it is unlikely that the nuclear spin system in the

HFP plays a role in remembering the anisotropy axis [̂ in the

U2D2 phase. To understand this phenomenon, it is also

important to study how the order parameter of the stable

phase develops during this 1st order phase transition.

10–1

2. Experimental procedure

A seed crystal of the U2D2 3He was produced at the

bottom of the 4 mm diameter cylindrical polycarbonate cell,

which contained superfluid 3He-B at 0.5 mK and was

attached below the metallic compressional cell and heat

exchangers. After growing a seed crystal into a single

crystal of desired size, MRI measurement [3] was performed

to check the distribution of magnetic domains in the crystal.

Typically only three large magnetic domains appear at the

bottom of the sample cell with flat domain boundaries

between them. Then an external magnetic field B was swept

up just below the 1st order transition critical field, BC1. After

stabilizing B, a small controlled amount of excess magnetic

field, DBhBKBC1, was applied. A successive measurement

of free induction decay (FID) after a small tipping angle

pulse was performed to measure the amount of the HFP to

be developed in time. The large difference in magnetization

of two phases and a large dipole frequency shift in the U2D2

phase enabled us to distinguish two phases during phase

transition process. After establishing new equilibrium in the

HFP, the reverse process was measured in the same manner.

A typical example of the time evolution of signal

intensity of FID during the phase transition from the U2D2

phase to the HFP is shown in Fig. 1. The vertical axis

dmhjMHFPKMobsj/jMHFPKMU2D2j indicates the

normalized change in the NMR signal intensity Mobs from

0 200 400 600 8000.1

1

time[sec]

δm

Fig. 1. Time evolution of the new phase in the case of U2D2 to HFP at

0.63 mK and DBZ10 mT. The vertical axis dmhjMHFPKMobsj/jMHFPK

MU2D2j indicates how much U2D2 was left after DB was applied.

the equilibrium value MHFP in the HFP, where MU2D2 is the

equilibrium value in the U2D2 phase. This dm gives the

ratio of the volume of the U2D2 phase and the total volume

of the crystal. In the case of a phase transition from the HFP

to the U2D2 phase, this dm is defined as jMU2D2KMobsj/

jMHFPKMU2D2j, so that the dm denotes for the volume

concentration of the metastable phase for both direction of

the transition. Since the observed time constants are much

longer than the measured longitudinal spin relaxation time

in the nuclear ordered solid 3He, which is less than 1 s, we

think that the ratio of the volumes of two phases can be

obtained from this measurement. As shown in Fig. 1, the dm

relaxes exponentially in time with two rate constants. Here

we name them as g1 for the first stage and g2 for the second

stage. These rates depend on temperature T, excess

magnetic field DB, and the size of crystals. However, the

measured rates do not depend on the direction of the phase

transition with identical DB. For small crystals of a few mm3

at low temperature, only single rate constant was observed.

At higher temperatures, g2 decreases to as low as 10K5 sK1.

While g1 was insensitive to a size of crystal, g2 decreased

with increasing a volume of a crystal. We have not yet

determined precise volume and temperature dependences of

g2. The fast rate, g1, depended on DB as shown in Fig. 2.

The dotted line indicates a naive fit with ln g1fK1/(DB)2,

which is a probable dependence due to homogeneous

nucleation model with surface tension. The dashed line

indicates another naive fit with ln g1fK(BcKDB)a, where

ax1, which represents for a typical instability model for a

heterogeneous nucleation. As can be seen, both model

would not fit well, especially in the low DB regime.

The MRI measurement was performed simultaneously

during the phase transition. The time evolution of the

distribution of one of the domains in the U2D2 phase during a

1086420

10–3

10–2

∆B [mT]

γ 1[se

c –1

]

Fig. 2. DB dependence of g1 obtained from a crystal of 5 mm3. The symbol

6 is for the phase transition from the U2D2 phase to the HFP, and 7 for

the phase transition from the HFP to the U2D2 phase. Open symbols are

obtained for a temperature range between 0.55 and 0.60 mK. Grey symbols

are for a temperature range between 0.60 and 0.65 mK. Solid symbols are

for a temperature range between 0.65 and 0.70 mK. See text for the lines.

Fig. 3. Time evolution of domain distribution obtained by ULT-MRI. Each

image, which is a projection to a plane parallel to the vertical axis of the

sample cell, represents for the distribution of one of the domains in the

U2D2 phase at (1) 200 s, (2) 3200 s after crossing BC1 by DBZ1 mT to the

low field side and (3) after achieving thermal equilibrium in the U2D2

phase at TZ0.59 mK. We should note that these images are a little distorted

by the influence of temperature distribution across the sample.

T. Tanaka et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1475–1477 1477

phase transition process from HFP with DBZ1 mT is shown

in Fig. 3. This crystal was filling the lower half of a cylindrical

sample cell, and was connected to liquid through the interface

on top of it. The image was shown as a projection to a plane

parallel to the vertical axis of the sample cell. The image (1)

was obtained soon after the end of the first stage. As can be

seen from the figure, the U2D2 phase appeared from the

liquid-solid interface side of the crystal and growing into the

inside of the crystal during the second stage. This indicates that

the U2D2 seeds appear, namely HFP signal disappears, during

the first stage at the liquid-solid interface, and then U2D2

domain expand into the crystal during the second stage. We

should note that the total shape of a crystal is conserved before

and after the transition. This indicates that the appearance of

the U2D2 phase proceeds, not by creating a new crystal with

different spin structure, but by changing the spin structure in

the existing crystal. We should also note that these images are a

little distorted by the influence of temperature distribution

across the sample, which is caused by the release of latent heat

during phase transition. However, the main feature, that is the

U2D2 phase appears from liquid-solid interface side, could be

trusted. We should also note that the MRI measurement was

performed only in the limited cases of early termination of the

first stage followed by much slower growth during the second

stage. So far we do not know if the same behavior could be

observed under different conditions of phase transition.

As reported earlier [5], the NMR spectrum under a

vertical field gradient was showing consistent behavior, that

was, the spectrum for the stable phase evolves from the

liquid solid interface side of the crystal. The same behavior

was observed for the reversed direction of the field gradient,

as well. This suggests that the location of the first seed is not

controlled by the direction of magnetic field gradient nor by

the possible temperature gradient across the crystal, but by

the geometry of the interface. Moreover, this time evolution

suggests that new phase appears not only from one place at

the interface but from many place at the interface.

3. Discussion

The exponential time evolution and positive dependence

of g1 on DB suggest that this process is governed by a

nucleation rate of the stable phase. On the other hand, MRI

information suggests that this nucleation is occurring at the

interface between liquid and solid. Then a possible scenario

for the first stage is an appearance of many seeds of stable

phase on the interface, followed by a fast enough but limited

growth of these seeds. The fast process terminates at some

point and the slow process, possibly governed by thermal

transport, takes over. Based on this understanding, we may

argue that the phase transition is initiated by heterogeneous

nucleation, occurring at the liquid-solid interface. Actually

the estimated barrier height for the homogeneous nucleation

can be as large as 102 K at DBZ5 mT [6], if we assume the

surface tension between U2D2 phase and HFP per atom

x10K1!Jx10K1 mK, and chemical potential difference

per atom between two phases x(magnetization difference

per atom) !DB. This large value suggests that it is not

possible to do phase transition within a reasonable amount

of time. However, since we do observe a phase transition, it

must be handled by much smaller energy barrier, possibly

assisted by the liquid-crystal interface. Since this scenario is

still too naive, further studies are necessary to conclude.

Acknowledgements

This work was supported by the Grant-in-Aid for

Scientific Research from the JSPS and the Grant-in-Aid

for the 21st Century COE ‘Center for Diversity and

Universality in Physics’ from the MEXT of Japan. One of

the authors (TT) would like to thank for his JSPS Research

Fellowship for Young Scientists-DC1.

References

[1] E.R. Dobbs, Chapter 32 of Helium Three, Oxford University Press,

Oxford, 2001 (and the references therein).

[2] S. Balibar, T. Mizusaki, Y. Sasaki, J. Low Temp. Phys. 120 (2000) 293.

[3] Y. Sasaki, et al., J. Low Temp. Phys. 138 (2005) 911.

[4] T. Ueno, et al., J. Low Temp. Phys. 127 (2005) 1.

[5] T. Tanaka, et al., J. Low Temp. Phys. 138 (2005) 847.

[6] The value given in T. Tanaka et al., J. Low Temp. Phys. 138 (2005) 847

had an error.