What causes the observed differences between solid H2 and D2?

PHYSICA[ ELSEVIER

Physica B 197 (1994) 13-18

What causes the observed differences between solid H e and De? Horst Meyer Department of Physics, Duke University, Durham, NC 27708-0305, USA

Abstract

Although solid H 2 and D: have many similarities, such as the orientational ordering through intermolecular electrostatic interaction, they also show striking differences. Basic differences in behavior resulting from those in the masses and in the nuclear magnetic moments will be discussed. The contrasting dynamic behavior of both solids during the martensitic transformation driven by the orientational order-disorder transition is described. A possible scenario that might shed some light on this difference in behavior is presented.

1. Preamble

This paper gives me the opportunity in first thanking the kind colleagues who nominated me and the Fritz L o n d o n award committee for having made me one of the recipients of this Prize. It is a great honor to share it with Dennis Greywall and Albert Schmid. With much gratefulness I am thinking of my mentors, first those at the University of Geneva, namely Richard Extermann, Ernst C.G. Stueckelberg and Dominique Rivier in the Institut de Physique and my father Kurt H. Meyer in the Ecole de Chimie, who encouraged me to study Physics and helped me overcome a number of obstacles. Klaus Clusius directed my thesis work at the University of Zurich and in a long letter r e c o m m e n d e d me to get involved in the exciting field of Low Temperature Physics. Sir Francis Simon invited me to work at the Clarendon L a b o r a t o r y in Oxford in 1953. There I did my postdoctoral apprenticeship in Low Temperature Physics on paramagnetic compounds under the direction of A r t h u r H. Cooke, the future War-

den of New College, and a very bright graduate student, Werner Wolf, who became a prominent m e m b e r of the Yale Faculty. In Oxford I became very intrigued with the beautiful experiments by William Fairbank, then at Duke University, on liquid 3He and 3He-4He mixtures, and it became a wish of mine to work there in the future. During my stay at Harvard, made possible by John H. Van Vleck, I joined a research program on ferrimagnetic garnets. I was then fortunate to be invited to join the Duke faculty to succeed William Fairbank, who was leaving for Stanford University. I owe a lot to my graduate students and postdoctoral fellows, without whose contributions I would not have been able to function. F r o m them I received much stimulation and I learned a great deal. There are too many of them to thank here individually and I accept the award on behalf of all of them. The long-term interaction and collaboration I have had with several of them and the long-lasting friendships that developed have been among the greatest joys in my scientific life.

0921-4526/94/$07.00 ~ 1994 Elsevier Science B.V. All rights reserved SSDI: 0921-4526(93)E0448-P

H. Meyer / Physica B 197 (1994) 13-18

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quite different topics, and rather than reviewing research done by my group, I would like to concentrate on a problem that has been of interest to me for some time. I wish to present it at a very elementary level without the many details that can be found in the references quoted below. Since this paper deals with problems in the solid hydrogens which I have discussed so often with Brooks Harris, I wish to dedicate it to him.

First among them is A. Brooks Harris, of the University of Pennsylvania, my first graduate student. With the submission of a paper sometime this fall, 1993 marks the 35th year of our collaboration. Our research started with calorimetry of rare-earth garnets at Harvard, continued at Duke with sublattice magnetization in garnets, molecules trapped in {3-hydroquinone clathrates (enclosure compounds), and then dealt for 25 years with the solid hydrogens. With R o b e r t Behringer I have collaborated for about 22 years. He was also my graduate student, and our research dealt with critical phenomena in fluid helium mixtures near the superfluid transition and near the liquid-vapor critical point. At D u k e , where he is on the faculty, we are still collaborating now on transport phenomena in liquid helium, and I frequently consult him on other p h e n o m e n a near critical points. Also I have collaborated for 13 years with Insuk Yu, a f o r m e r postdoctoral associate and now professor at Seoul National University, and our interaction is still very strong. Our common interest lies in the solid hydrogens. Finally I wish to gratefully acknowledge the crucial help to our projects by Edward Morgan, our technician over 23 years until his retirement, friend and father confessor to most of us. Last but not least I owe sincere thanks to the Agencies that supported jointly and in turns my research at Duke over 34 years, namely the U.S. Army Research Office, the National Science Foundation and the National Aeronautic and Space Administration. The F. L o n d o n award citation covers several

2. The h y d r o g e n s - a short discussion

In many ways, solid H 2 and D 2 have very similar properties [1,2], and these are in general well understood. Table 1 briefly recapitulates well-known characteristics of these two solids, each of which is a mixture of two components, 'ortho' and 'para'. To avoid confusion, let us drop these two words and deal in terms of the species having rotational angular momenta J = 1 and J = 0. Let us define as X the molefraction of the species with J - - 1 , which carries an electric quadrupole moment eQ, while the species with J = 0 has eQ = 0. Thus in the close-packed lattice with 12 nearest neighbors, there is an electric quadrupole-quadrupole (EQQ) interaction energy between neighbors and next-nearest neighbors, proportional to R -5, where R is the intermolecular distance. This interaction leads to a first-order phase transition into a long-range ordered state, where there is a sublattice structure, similar to an antiferromagnetic system [3].

Table 1 Some parameters in solid H 2 and D 2. Here the vertical columns show (1) the rotational angular momentum, (2) the rotational energy of the lowest state, where I r is the momentum of inertia, (3) the nuclear spin with the corresponding fraction of molecules, (4) the lattice constant at 4 K, (5) the molar volume at 4 K, (6) the triple point Ttp, and (7) the electric quadrupole m o m e n t

J (h)

J(J + 1)h2/(21rks) (K)

I (h)

R (A,)

V (cm3/mole)

Tw (K)

Q (au)

U 2

1 0

170 0

1 0

3,790

23.1

13.8

0.485 0

ortho para

DE

1 0

84 0

3.605

19.9

18.6

0.477 0

para ortho

1 2 (5/6) 0(1/6)

H. Meyer / Physica B 197 (1994) 13-18

H o w e v e r a --I alignment, rather than an antiparallel 1'~ one, is favored between nearest electric quadrupoles. Such an alignment can be only partially achieved in 3D, because of the competition between nearest neighbors, and one h a s - even for the pure J = 1 s o l i d - a situation of a 'frustrated' orientational ordering. The firstorder orientational phase transition, which occurs for X ( J = 1 ) > 0 . 5 3 is accompanied by a crystallographic change from H C P to cubic, and it is of a first-order displacive 'martensitic' nature [4], which implies that there is some hysteresis upon thermal cycling. For X < 0 . 5 3 , the transition to a less orientationally ordered s t a t e - a glassy s t a t e - occurs gradually [5]. In this 'quadrupolar glass' [6] the original H C P phase is conserved [7]. Both solid H e and D 2 grown at low pressures undergo such an orientational ordering process and exhibit a very similar behavior, as shown by several N M R studies [5,6,8,9]. However, they strikingly differ in at least two types of properties. These are (a) the dynamics of the martensitic transition, where the H C P - c u b i c - H C P transformation via a thermal cycling is complete in H 2 but in D 2 the cubic phase becomes stabilized after several such cycles [4], and (b) a very puzzling departure of the nuclear susceptibility in HCP, short-range ordering (J = 1)D 2 from Curie's law, while (J = 1)H 2 appears to follow this law [8,10]. H e r e I would like to discuss only the first puzzle and leave the second to a paper with a more technical format [11]. Both problems are not yet solved, but I hope to convey the message that there are very interesting phenomena to be understood in these 'simple' molecular diatomic solids. What can be the mechanisms leading to these quite different types of behavior, and what is their origin? What are the basic differences between H 2 and D 2 besides the quantum statistics? The three differences I can see are very obvious, namely: (1) The mass. For a given interaction potential, the isotopic molecule with the heavier mass leads to a smaller zero-point amplitude, and hence to a smaller lattice constant in the solid.

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Thus the molar volume of solid D 2 is about 15% smaller than that of solid H 2 (there is only a small density dependence on X (J = 1)). (2) The nuclear magnetic moment magnitude. The proton nuclear moment is approximately 7 times larger than that of the deuteron. Besides obvious consequences in the signal intensities during N M R measurements, this difference has a strong impact upon the J = 1 to J = 0 conversion that proceeds via intermolecular nuclear dipolar interaction. This conversion is of two kinds [2]. First, there is the well-known irreversible conversion, where the m o m e n t u m J = 1 and the rotational energy are transferred to the lattice. This leads to the conversion heat production that prevents cooling the solid hydrogens to very low temperatures. This conversion r a t e - a n d also the h e a t - a r e much larger in solid H 2 than in D 2. As a typical value for X = 0.5, d X / d t = - 0 . 5 % / h in H 2 and - 0 . 0 3 % / h in D 2. Second, there is a reversible J = l - J = 0 conversion, whereby angular momentum and rotational energy are transferred from one molecule to its neighbor in the J -- 1, J = 0 mixed crystal. In this process, energy and momentum are conserved. This m o m e n t u m transfer leads to a final state equivalent to that from an exchange of a J = 1 and a J = 0 particle. Its rate depends on the proximity of other J = 1 neighbors that create an electric field gradient, and that tends to slow down the transfer. The phonon density and the n o n r a n d o m directional changes in the intermolecular potential [12] also play a role in this molecular motion that has been called 'quantum diffusion' [2]. Again, because of its larger nuclear moment, the diffusion rate in solid H i is much larger than in solid D 2 where it has not yet been detected [13]. (3) The nuclear spins and the intramolecular interactions. In the H 2 molecule, the intramolecular nuclear interaction energy results only from dipolar interaction between the two protons. In D2, it comes mainly from the coupling of the nuclear quadrupolar moment with the electric field gradient at the nucleus [14]. This difference is most probably relevant in N M R experiments, but not so in the phase transformation about to be discussed.

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H. Meyer / Physica B 197 (1994) 13-18

3. M a r t e n s i t i c t r a n s f o r m a t i o n s

Martensitic transformations are very common and constitute a truly interdisciplinary topic [15]. Most of the experimental interest is centered on materials of technological importance, such as metals, carbon steels, alloys and ceramics that show a displacive phase transformation from one crystalline structure to another. There is a large amount of literature published in the journal Materials Science and Engineering [15], but the subject is also of great interest to physicists [16]. The problem of these transitions is complex, because in the formation of martensite phases, there is no spatial homogeneity on the 10-100 nanometer scale. One deals with mesostructural systems and one cannot use equilibrium statistical mechanics alone. Interface energy and longrange elastic fields are important, and cannot be treated microscopically. There is a frustrated balance between various factors and a small change in interface energy, possibly due to a composition change, can cause an appreciable shift in the transformation temperature [17]. Reference [18] gives an overview and lists a large number of references on martensitic transformations. They show dramatic advances, both experimental and theoretical, in this field in recent years. In comparison with materials of technological importance, molecular crystals have received only a small amount of attention. Among those investigated experimentally a r e 14N 2 [19], 4He [20], the methanes [21] and the hydrogens. These crystals grown at low temperatures are very pure and in this sense constitute a simpler system than alloys or ceramics (alloys of molecular crystals such as Arl_xN2. x have also been investigated [19]). In H 2, D 2 and also i n 14N2, the HCP-to-FCC displacive transformation is driven by the orientational ordering resulting from the E Q Q interactions. In the hydrogens, the transition temperatures Tnc (HCP-cubic) are a function of X [1]. For X = 1, THc(Dz) = 3.9 K and THc(H2) = 2.9 K, while THc(N2) = 35 K. For H 2 and D 2, both from the X-ray diffraction [4] and from acoustic experiments [22,23], the following

behavior has been reported, and is schematically shown in Fig. 1: Upon cooling the single crystal for the first time through the transition, the HCP phase disappears almost completely and is replaced by a FCC phase. This transformation comes through a sliding of net planes perpendicular to the c-axis. (In this ordered phase the molecules with J = 1 are aligned [1] in certain directions so as to form four interpenetrating sublattices, and the structure is then denoted as Pa3. ) Upon warning, H 2 and D 2 behave differently: in solid H 2, the HCP phase is grown back entirely and its axial direction is the same as the original one. This process can be repeated with the same result. By contrast in D 2, only a fraction of the HCP phase reappears upon warming, and a fraction of the cubic phase remains. Upon recycling through the transition, the cubic phase above the orientational ordering transition becomes progressively stabilized. Acoustic meaD2 1oo

%hcp

..........,5

)

Q

If® I __

ib Q)

]Y Ii,

~TC l TB

T

P

IT C IT B

T

TA

J L

100 %cubic

D ITc tTB .....

T

ITc ITB

T

Cooling Warming

Fig. 1. Schematic percentage of H C P and of F C C (cubic) before, during and after the thermal cycles t h r o u g h the Martensitic transformation in H z and in D z. T h e temperatures TA, T B and T c are m e n t i o n e d in the text. T h e circled n u m b e r s (i) label the ith thermal cycle.

H. Meyer / Physica B 197 (1994) 13-18

surements [23] indicate that it remains stable until roughly 8 K, when it disappears in favor of another close-packed phase. The H C P phase is only recovered along the melting curve. For N2, just as for HE, a thermal cycle through the transition also leads from H C P through cubic back to H C P [19]. Acoustic measurements have shown that in solid H 2 the received signal amplitude drops to zero when H C P is transformed into cubic, and its original value is recovered when the crystal is heated through the transition back to HCP [22,23]. The reason for this large sound attenuation in cubic H 2 is not understood. By contrast, the drop of the signal amplitude for D 2 during the HCP-to-cubic passage is less drastic [23]. When the cubic phase is stabilized through multiple thermal cycles, the acoustic amplitude shows a change at the o r d e r - d i s o r d e r transition temperature with only a very small hysteresis. M. D e v o r e t - who has worked on the solid h y d r o g e n s - and I had a discussion about the possible mechanisms for these striking differences, and a scenario based on the difference in interface energy between H C P and cubic domains came up. It is as follows. Consider a single H C P crystal being cooled through the transition. Small platelets of the cubic phase are being nucleated at the temperature Tg in Fig. 1. These platelets are formed each from displacements in groups of say 100 molecules, and perpendicular to the original c-axis. As T decreases to TB, the dimensions of the platelets forming the new phase increase. While one [1 1 1] axis in each platelet will be parallel to the c-axis of the HCP crystal, the other three [1 1 1] platelet axes have different directions. At T = Tc, the transition is nearly completed and the large zones of the new cubic phase fill most of the space except for a thin H C P 'skin' between them. Now here comes the possible difference between H E and D 2. For H 2, the H C P skin entirely surrounds each zone of the new phase. Therefore when the temperature is increased, the 'skin' acts as a memory for the regrowth of the H C P phase and the process moves backwards. The same H C P crystal as the original one is recovered when the thermal cycle is completed. By contrast, in D 2 the cubic-

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H C P - c u b i c interface energy is larger than in H 2 and is such that by completion of the cooling through the transition T A - T B - T 0 the H C P skin does not cover entirely the surfaces between the regions of the new phase, but is restricted to some areas so that the new phase (cubic) zones of different orientations directly touch each other. Upon warming, there is no longer a memory effect that permits a single H C P crystal to regrow. The only directional m e m o r y that is conserved is that of the [1 1 1] axis, parallel to the original c-axis. As repeated thermal cycling is carried out, the remaining amount of H C P will gradually disappear, and only the cubic regions will remain. Devoret wondered whether the HCP skin might in fact be the cause of the strong acoustic attenuation in H 2. One might then venture the guess that the reason for the smaller attenuation in cubic D 2 is that the H C P skin is not surrounding the cubic crystals in the same way as in H 2. Why should the interface energy be different for H e and D27 Could it depend on the lattice constant? Experiments [24] were carried out where the martensitic transition in solid H 2, was studied as a function of density. T h e r e was no change observed in the reversible behavior when H E w a s compressed to a molar volume equal to (and below) that of solid D 2 at zero pressure. Hence the difference in the lattice constant between H 2 and D 2 at zero pressure cannot explain the difference in the transformation behavior. Could it then depend in some way on the quantum diffusion that permits molecular motion, and which is much faster in H E than in D2? Could this affect the interface energy, and hence the free energy of the crystal-interface system? The energy difference between the H C P and the FCC lattices in H 2 was calculated [25] to be very small, only of the order of A E / k B = 0.1 K, and is not important in influencing the transition temperatures, which are controlled by the E Q Q interaction. The change in molar volume during the transformation is only of the order of 0.2%, as deduced from thermodynamic measurements [26]. Detailed experiments were carried out to study the width of the c u b i c - H C P (order-disor-

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H. Meyer / Physica B 197 (1994) 13-18

d e r ) t r a n s f o r m a t i o n in H 2 f r o m t h e r m a l c o n d u c tivity a n d N M R e x p e r i m e n t s [27]. T h e c o m p l e t e t h e r m a l cycle o f t h e F C C H C P - F C C t r a n s f o r m a t i o n has b e e n s t u d i e d [20] in solid 4He in a series o f b e a u t i f u l o p t i c a l e x p e r i m e n t s (see also t h e list to r e l e v a n t t h e o r e t ical w o r k in R e f . [20]). H e r e , h o w e v e r , t h e t r a n s i t i o n is n o t d r i v e n b y an o r d e r i n g p r o c e s s (spin o r q u a d r u p o l a r o r d e r i n g ) u n l i k e in t h e solid h y d r o g e n s a n d in N 2. It is to b e h o p e d t h a t t h e n e w a d v a n c e s in X - r a y a n d e l e c t r o n d i f f r a c t i o n , as well as in c o m p u t e r s i m u l a t i o n a n d a n a l y t i c t h e o r y , will b e a p p l i e d to t h e s t u d y o f t h e s e t r a n s f o r m a t i o n s in the hydrogens.

Acknowledgements I a m v e r y g r a t e f u l to J . A . K r u m h a n s l for providing me with a short essence of martensitic t r a n s f o r m a t i o n s a n d with r e f e r e n c e s t h a t will b e u s e f u l to physicists i n t e r e s t e d in such t r a n s i t i o n s . A l s o I a p p r e c i a t e d t h e discussions, s u g g e s t i o n s a n d c o r r e s p o n d e n c e with M. D e v o r e t a n d A . B . Harris.

References [1] I.F. Silvera, Rev. Mod. Phys. 52 (1980) 393. [2] J. Van Kranendonk, Solid Hydrogen (Plenum Press, New York, 1983). [3] A.B. Harris, J. Appl. Phys. 42 (1971) 1574. [4] A.F. Schuch, R.L. Mills and D.A. Depatie, Phys. Rev. 165 (1968) 1032. [5] A.B. Harris and H. Meyer, Can. J. Phys. 63 (1985) 3. [6] N.S. Sullivan, M. Devoret, P.B. Cowan and C. Urbina, Phys. Rev. B 17 (1978) 5016.

[7] J.V. Gates, P.R. Granfors, B.H. Fraass and R.O. Simmons, Phys. Rev. B 19 (1979) 3667. [8] M. Calkins, R. Banke, X. Li and H. Meyer, J. Low Temp. Phys. 65 (1986) 47. [9] M. Calkins, R. Banke, X. Li and H. Meyer, J. Low Temp. Phys. 65 (1986) 90. [10] D. Clarkson, X. Qin and H. Meyer, J. Low Temp. Phys. 93 (1993) 119. [11] A.B. Harris, H. Meyer and X. Qin, to be published in Phys. Rev. B. [12] M. Tammaro, M.P. Nightingale and A.E. Meyerovich, Phys. Rev. B 47 (1993) 2573. [13] A.I. Krivchikov, M.I. Bagatskii, V.G. Manzhelii, Ya. Minchina and P.I. Muromtsev, Sov. J. Low Temp. Phys. 15 (1989) 1. [14] N.F. Ramsey, Molecular Beams (Oxford University Press, London, 1956) pp. 235 and 238. [15] L.E. Tanner and M: Wuttig, in: Workshop on FirstOrder Displacive Phase Transformations: Review and Recommendations, Mater. Sci. Eng. A 127 (1990) 137. [16] G.R. Barsch and J.A. Krumhansl, in: Martensite, eds. G.B. Olson and W.S. Owen (American Society of Metals International, Materials Park, OH, 1992) p. 125. [17] S. Kartha, T. Castan, J.A. Krumhansl and J.R. Sethna, Phys. Rev. Lett. 67 (1991) 3630. [18] J.A. Krumhansl and G.R. Barsch, Views from Physics on Martensitic Transformations, to be published in Proc. Intern. Conf. on Martensite 'ICOMAT-92'. [19] H. Klee and K. Knorr, Phys. Rev. B 42 (1990) 3152. [20] K.A. McGreer, K.R. Lundgren and J.E Franck, Phys. Rev. B 42 (1990) 87. [21] R.O. Simmons, Trans. Am. Crystallogr. Assoc. 17 (1981) 17. [22] R. Wanner, H. Meyer and R.L. Mills, J. Low Temp. Phys. 13 (1973) 337. [23] R. Banke, X. Li and H. Meyer, Phys. Rev. B 37 (1988) 7337. [24] X. Qin and H. Meyer, Phys. Rev. B 44 (1991) 4165. [25] L. Nosanow, unpublished calculations (1970). [26] D. Ramm, H. Meyer and R.L. Mills, Phys. Rev. B 1 (1970) 2763. [27] X. Li, D. Clarkson and H. Meyer, Phys. Rev. B 43 (1991) 5719; D. Clarkson and H. Meyer, Fiz. Nizk. Temp. 19 (1993) 487 [Sov. J. Low Temp. Phys. 19 (1993) 343].