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Basic Research on Crystal Growth and Surface Physics of Solid Helium under Microgravity

Yuichi Okuda Department of Condensed Matter Physics, Tokyo Institute of Technology Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan Takao Mizusaki Department of Physics, Graduate School of Science, Kyoto University Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan Makio Uwaha Department of Physics, Nagoya University Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan

P(bar) hcp solid

bcc solid

Liquid helium is regarded as a quantum uid which is melted by the quantum uctuation (zero point motion) even at T = 0. Liquid 4 He becomes superuid at T =2.17 K at saturated vapor pressure, which is the manifestation of BoseEinstein condensation, and liquid 3 He, which is a ferminonic system, sets into the BCS-type superuid at around 1 mK. The most outstanding feature of these quantum systems in the low temperature would be the fact that they are extremely pure. All foreign substances are frozen and excluded out of the system. And the thermodynamic parameters, such as temperature and pressure, to specify the system can be controlled very accurately and precise measurements can be performed. They are ideal systems to do a basic research about strongly interacting bosonic and fermionic systems. If we apply an external pressure up to 25 bar for liquid 4 He and 35 bar for liquid 3 He, they are forced into the solid phase due to their strong repulsive interactions. Then we have nice quantum solids which coexist with the quantum uids at very low temperature. There should be a naive physics in the interface between the quantum solid and the quantum uid. Here we would like to focus on the basic physics of crystal shape and crystal growth together with the unique new phenomena which are observed on the interface of the quantum solids.

25 TR3 TR2 TR1

normal fluid

0 0

lambda line


Figure 1: Phase diagram of 4 He in low temperature.

As can be seen from Fig.1, the melting curve is at below 1 K. This means there is almost no latent heat involved in the liquid/solid transition, as is derived from the Clausius-Clapeyron relation. No heat diusion problem comes in, which makes the crystal growth very dicult to analyze. And furthermore the melt of solid 4 He is superuid and the mass transport proceeds very easily. Then the crystal growth of solid 4 He in low temperature can proceed with enormous speed. The growth velocity is proportional to the chemical potential dierence and the coefcient, which is called as growth coecient, is huge; something like 1010 times larger than the ordinary solid case. The crystal forms its equilibrium shape within a very short time. It can 1

be possible to obtain the microscopic information, such as the step-step interaction, from the direct observation of the crystal shape. One of the spectacular examples of the interface of solid 4 He is that the crystal forms horizontal surface under the gravity to minimize the gravitational energy, similar to a liquid. And even more, if some perturbation is applied, the horizontal interface oscillates with small damping, which is called a crystalliza tion-melting wave. This wave is a very unique wave only observable on the quantum solid surface in coexsistence with superuid. Steps and kinks, which play essential roles in the physics of an interface, are regarded here as quasi-particles on the interface. Their numbers are uctuating due to thermal and quantum uctuation even at low temperature. A step can be a line-defect on the 2-dimensional plane and a kink be a point-defect on the 1dimensional step. They are interesting quantum objects which are only realized on the surface of the quantum solids. It is already realized that the 4 He interface is a new and ideal system to study any crystalrelated physics. So far extensive results are accumulated [1] and the eld is still progressing[2]. Present authors also contributed in this eld; for example, (1)sound induced melting of solid[3], (2)negative crystal observed in solid 4 He[4], (3)crystal growth and melting of 3 He in U2D2 ordered solid[5], (4)quantum nucleation of solid[6, 7], and so on. But the precise and thorough research will not be completed without the experiment under the microgravity.

the crystal shape (Wuls theorem) not applicable. Such gravity eect may be eliminated by the experiment with a tiny crystal. But it does not work out for the helium crystal. Due to a larger quantum uctuation, the step on the solid He gets rather wide, though the concept of step is still meaningful. Owing to this eect, the facet size with respect to the total size of the crystal is very small compared with the classical solids, even if temperature is well below the roughening transition temperature. Then we do need a bigger crystal. The experiment with a bigger crystal is dicult on Earth because of the bigger gravity eect. Another benet to carry out experiment under microgravity is that the crystal can be studied, which is free from any contact with the wall. We can essentially access to all crystallographic orientations. This is totally impossible on earth.

Proposed objectives

We would like to propose the following experiments for both 4 He and 3 He. Chemically these species are similar, but in low temperature they behave quite dierently due to the dierence of the quantum eect. Of course 4 He is studied in more detail and easier to be studied, but the importance of the 3 He is clear. The quantum eect is larger than 4 He and 3 He carries a nuclear spin which should play an important role on the crystal growth. But crystal surface details are not well understood at the moment. Comparison between 3 He and 4 He results should be very instructive and important.

Microgravity experiment rel- 3.1 Equilibrium crystal shape evance

As mentioned above, the very precise and basic measurement for crystal growth and surface physic is possible with the 4 He crystal, but unfortunately, it is found that the gravity obscures the experimental results. The pressure gradient in gravity both in the liquid and the solid hampers the precise measurement of pressure. And not only the liquid compresses the solid to deform, but the crystal itself is easily deformed by its own weight. Such a deformation makes the true equilibrium shape obscure and makes the rst principle of getting 2

The transition region between a facet and an adjacent curved part of a crystal is known as a vicinal surface, which is shown in Fig. 2. The vicinal surface is only meaningful for the angle less than 3 deg, where steps are well separated by at terraces. This surface is an ideal system to realize steps and kinks as quasi-particles on the surface and the growth rate of this surface is determined by the dynamics of those quasiparticles. The functional form of this surface manifests the detailed step-step interaction. If the interaction is dominated by the d2 term, the prole of

the surface z(x) is proportional to x3/2 , where d is the distance between the steps. It is believed that this form holds for general crystals. But it is not yet conrmed experimentally. The stringent test of this theory can be done with the equilibrium crystal shape (ECS) of 4 He crystal. The obsevation of ECS with undeformed crystal is essential to determine the step-step interaction. Experimental results on earth are dierent among dierent groups. The exponent of x is ranging between 1.2 and 1.8. The result is controversial. Main reason is that the crystal is deformed by gravity[8]. The rigorous test of the true functional form of the 4 He vicinal surface is only possible under microgravity.

d W

Figure 2: Vicinal surface inclined by a small angle of 1 with respect to a high symmetry facet. In helium crystal a step is not so sharp as the classical ones.


Roughening transition

roughening transition of c-facet is rather clear, the other two facets were not observed in a equilibrium condition. They were observed only in the growing process. There is an open issue in roughening transition of 4 He. In the recent experiment[9], 4 He cyrstal was cooled down to 2 mK, but no other roughening transition and no sign of freezing of kinks were observed down to that temperature. According to eq.(1), most low index planes have to be faceted at 2 mK. The ground state of the crystal surface is expected to be the faceted state by classical theory. But for the quantum crystal, it is not clear whether it is faceted or rough at T = 0. Andreev and Parshin was the rst to point out the possibility of the quantum roughening [10]. They argued quantum uctuation may wash out the surface ordering even at T =0. Contrary to this, the renormalization theory [11] predicted that the crystal lattice periodicity eventually takes over whatever quantum uctuation is. At least for the high Miller index planes, the roughening transition temperature is extremely low especially for solid helium. The concept of quantum roughening is a very important problem which must be tested by experiment under microgravity. However, as long as the experiment is performed on Earth, the denite conclusion may not be obtained. If roughening transition is dened as the diverging point of the height-height correlation function, the gravity kills the transition. In real situation, the gravity will bring a smearing eect on the roughening transition. The roughening transition experiment should be done in space.

It is regarded that roughening transition is classied as the Kosterlitz-Thouless (KT) transition, which is the transition of the 2-dimensional 3.3 Instability of the quantum surface (2D) XY-type universality class. The following of solid relation between the transition temperature, TR and the stiness of the surface at TR is known. Since the solid-superuid interface of He is very sensitive to an external perturbation, pro2 (1) cess of crystallization and melting propagates as kTR = h2 , a wave if the solid surface is not faceted. Surwhere h is the lattice period perpendicular to face tension is the restoring force to make the the surface, and is the stiness of the surface. interface at to recover the equilibrium shape Roughening transition was clearly observed from the perturbed shape. It is eective only on [0001] face of 4 He at TR1 =1.23 K and the rig- at small wavelengths and the gravity is responorous test whether it is KT transition was done sible for maintaining the at interface at long for the rst time. The transition was identied wavelengths (this is why we observe a horizontal interface in He). Thus without gravity it is by observing the curvature of that plane. Following to the c-facet, other two facets (a- not possible to stabilize the solid-liquid interface facet and s-facet) was observed at the transi- against some external perturbations. This is clearly seen in the dispersion of the tions 0.9 K and 0.36 K, respectively. Though the 3

crystallization wave. When the solid encounters 0 a non-hydrostatic uniaxial stress xx the wave frequency is [12]
2 0 q = q 2 2q(xx )2 2 1 P E l q +(s l )g (s l )2


where P is the Poisson ratio and E the Young modulus. If g = 0, the frequency becomes imaginary, the wave uctuation is amplied and the interface is unstable for a small wave number q. Similarly, another instability is predicted [13] if superuid at the interface ows with a velocity v. The interface becomes unstable when the ow velocity exceeds the critical value vc = 4 g l l s l

mass and 2) the crystal growth is governed by nuclear spins in solid and the transport of spins either in liquid or solid. One of spectacular predictions of 3 He is the existence of the magnetic crystallization wave below 0.1 mK[14]. Those unique natures of the 3 He crystal can be studied in detail only under mcrogravity as well as in the 4 He case. Preliminary results of crystal growth and melting of nuclear-ordered solid 3 He was obtained down to 0.5 mK[5]. We found a large dierence in growth and melting coecients, that was attributed to the dierence of crystal growth rates between a faceted surface in grow process and a rough surface in melting. We conrmed that the crystal growth (melt) was completely governed by nuclear spins and thus was sensitive to magnetic eld and the nuclear spin structure.



Low temperature cryostat

Without gravity the critical velocity vanishes, and any tangential ow at the surface makes the interface unstable. The experimental implication is that under microgravity the surface of 4 He crystal is macroscopically unstable with a very weak uniaxial stress or very slow superuid ow. In principle the former instability with stress is universal for all solids, but in practice is observable in a macroscopic scale only in solid 4 He which coexists with superuid. The latter instability with ow, if observed, is very new phenomenon, in which superuidity plays an essential role.


Crystal growth of nuclear-spinordered solid 3 He

There have not been much studies for the case of solid 3 He since liquid is the Fermi liquid (rather viscous liquid) and the entropy difference between solid and liquid stays constant to be R ln 2 down to very low temperatures because of the paramagnetic nature of the nuclear spin in solid and Fermi liquid condensation in liquid 3 He. The growth rate for solid 3 He is extremely slow above 10 mK in comparison with 4 He, so that it sometimes takes many days to get to be in equilibrium. In order to set up the similar situation for 3 He as for 4 He, we have to cool the sample much below the nuclear-spin ordered temperature of the solid (TN =0.93 mK) where the latent heat of the solid/liquid is very small and the liquid is superuid. New features of the crystal growth of solid 3 He are that 1) quantum nature in 3 He is much stronger than in 4 He due to a smaller atomic 4

For the experiment proposed above we need low temperature of 1 K-50 mK for a long period (one month). The dilution refrigerator is a common apparatus to get such low temperature, but it is too dicult to operate it in space. So we would like to develop so called a continuous adiabatic demagnetization refrigerator. The idea is simple. The two sets of common adiabatic demagnetization stage is operated alternatively to get low temperature below 100 mK continuously. Key technology would be a cooling substance and heat switch technology. Chrome potassium alum salt and superconducting heat switch may be the best combination for the temperature. The joint project on development of mK refrigerator is planned with a domestic cryogenic group.


Acoustic microscope technology

Of course, optical measurement is essential for the experiment to monitor how the crystal is under microgravity. In addition to the optical measurement, we would like to propose to exploit the acoustic microscope technology to investigare the 4 He crystal surface. Sound wave is a very useful tool to probe the elastic properties of the helium surface. Using a concave acoustic lens which is shown in Fig. 3 schematically, the sound wave can be focussed on the crystal surface at a very narrow spot. As the sound wave propagates in superuid, we can

use a very high frequency without any dissipation. And the wave length could be as small as, or even smaller than, that of the visible light easily[15]. By scanning the lens in X-Y-Z directions, the reected signal can construct a three-dimensional elastic image of the crystal. Using a tungsten tip nucleator of the crystal seed as shown in Fig. 3, we will be able to get the desired crystal orientation in the microgravity environment.
Acoustic lens

structure should be dominant factor for crystal grow and equilibrium shape of the crystal, it is essential to obtain a magnetic information of nuclear spin in space. We developed MRI microscope applicable at ultra-low temperatures [16, 17] and we had achieved the 2 dimensional space resolution of the MRI microscope to be 10 micron and the ultimate resolution could be improved to be in one macron size. We expect the MRI microscope will be a key tool to study crystal growth of 3 He and new physics will be explored in the crystal growth of 3 He under the microgravity.






Figure 3: Schematic diagram of acoustic microscope investigating the vicinal surface, shape and elastic property.

The surface of the solid helium provides a new and unique system for the fundamental understanding of the crystal growth and surface physics. But the essential result will be obtained only if the proposed experiments are performed under microgravity. We have to overcome a few technical problems. But we believe it will open up a new eld and contribute the fundamental condensed matter physics. This kind of project should be proceeded with the international collaboration. We are already promoting the joint research with foreign groups. Finally we would like to thank the cordial support from NASDA and Mitsubishi Research Institute for the present work.


Double demagnetization and MRI microscope

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In order to carry the project of the crystal growth and surface physics in 3 He, we have to develop two technologies. One of them is the refrigerator operative under micro gravity, which is able to cool systems down to sub-mK temperature range. This would be possible to develop the double demagnetization refrigerator, where the 1st stage should be the reciprocally-operated electronic demagnetization stage as described in section 4.1 and the 2nd stage should be the nuclear demagnetization refrigerator. This refrigeration technology in space is a challenging one which Japanese research team should develop and oer to the world of low temperature community. Second technique which is important to study the problem is the development of the nuclear magnetic resonance imaging (MRI) microscope. In this research, nuclear spin and its ordered 5

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