Theory of the Phase and Twin Boundaries in Solid Helium and Reversibility of the bcc-hcp Phase Transition

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1 Journal of Low Temperature Physics manuscript No. (will be inserted by the editor) Theory of the Phase and Twin Boundaries in Solid Helium and Reversibility of the bcc-hcp Phase Transition V.A. Lykah E.S. Syrkin Keywords solid 4 He, phase transition, phase boundary, twin boundary, quantum solids Abstract Phase transition bcc-hcp in solid 4 He has martensitic like features. The phase transition is investigated theoretically in the frames of Burgers mechanism of the coherent lattice transformation. The two order parameters theory allows to account the phase volume changing. Thermodynamic parameter of a phase deviation from the phase equilibrium is introduced. The parameter describes analytically the fine structure of 8 boundary and its splitting into two phase boundaries in the reverse phase transition. It is shown the twin boundaries are nucleus of the origin crystal phase which structure can be restored completely. This is the base for explanation of bcc-hcp phase transition asymmetry found in solid 4 He experiments. A new experiment scheme is proposed to inverse the asymmetry of the phase transition. PACS numbers: s, 67.5.du, 67.3.ef, z Introduction The main opposite approaches to description of phase boundary (PB) consist in consideration of completely ordered (coherent) and completely disordered (incoherent) boundaries. For solid 4 He, the representation about a completely disordered bcc-hcp PB is generally accepted. Recent optical investigation 3 of solid 4 He showed multiple formation of newphase nuclei in an initial single crystal during the bcc-hcp phase transition (PT), after which a polycrystal is formed. The experiment revealed a reversible repeated PB displacement with a shift of the phase-equilibrium conditions. This finding, as well as the transformation kinetic 3,4, stimulated discussion of the analogy between a martensitic transformation and PT :Department of Physics and Technology, National Technical University Kharkov Polytechnic Institute, Kharkov, 6, Ukraine Tel.: Fax: lykah@ilt.kharkov.ua : B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences, Kharkov, 63, Ukraine Tel.: syrkin@ilt.kharkov.ua

2 in solid 4 He. In the work 5 we applied the martensite theoretical approaches to the description of the bcc-hcp PB in solid 4 He. The order parameter (OP) was introduced, PB form and energy were found. In the present work the phase transition bcc-hcp in solid 4 He is investigated theoretically. The two order parameters theory allows to account the phase volume changing. The fine structure of 8 boundary and its splitting into two phase boundaries in the reverse phase transition. The experiments on bcc-hcp phase transition asymmetry in solid 4 He is explained. A new experiment scheme is proposed. Models In solid 4 He, the most often observed bcc-hcp coexistence corresponds to pressures p 6 3 bar and temperatures T K 3. Stability of crystal lattice of solid helium was shown in work. Gibbs free energy was considered. It contains three terms: zero vibration energy, atomic interaction one and external one (pv ). Minimum of free energy is mainly defined by zero vibration and external energies. The atomic interaction energy is too small and is not enough to keep 4 He atom in a crystal lattice position. This is solid helium specific. Nevertheless 4 He crystal exist, its stability is approved by phonon existing. Phonon spectra are very similar for bcc lattices in solid 4 He 8 and Zr 9 or Tl. This fact approves free energy potential relief similarity in solid 4 He and Zr, Tl at least near the equilibrium atomic positions.. Main concepts of the martensitic theory (structure) The first actual mutual lattice orientations, Kurdjumov-Sachs and Nishiyama s ones, were found for the f cc-bcc martensitic PT. The basic concepts of the martensitic theory are: (i) martensitic PT has a diffusionless cooperative character; atomic displacements are correlated and do not exceed the unit-cell size; (ii) there is a high degree of coherence between the lattices of the martensitic and initial phases. General treatment to bcc-hcp transformation is Burgers mechanism 4. This mechanism established following relations (orientations) between bcc and hcp lattices: bcc hcp ; bcc hcp transformation can be presented as superposition of two lattice vibrations. The first. Transverse mode T with polarization produces a displacement of planes in direction. It corresponds to N-point of the phonon Brillouin zone. The planes displacement amplitude equals a to reach hcp phase. The second. Two equivalent long-wavelength shears such as the and squeeze the bcc octahedron into a regular hcp one. The angle changes from 7.5 to 6 in the basal plane. Landau free energy expansion describes the lattice transformation with respect of two OP was developed in 6,7. Martensitic PT bcc-hcp exists in Fe alloys and Tl, then the Nishiyama orientation relationship, is valid: bcc hcp ; bcc hcp The Venables 3 and Iizumi microscopic models describe PT bcc-hcp. On this basis, a model of PB was developed in 5 for one OP in double- and triple-well potentials, which can be applied both to metals and solid helium. According to the Venables model 3, the transformation bcc-hcp between the lattices includes two stages: (i) deformation of the lattice from the orthorhombic basis hcp planes with an angle of 6 to an angle of approximately 7 5, (ii) sliding of each second deformed hcp plane from the position H above the center of the triangular

3 3 basis to the position b above the largest side of the triangular basis with formation bcc. The distance l Hb between the H and b positions can be found 4,5 : l Hb a b 6 a h 3 () Here a h is the interatomic distance in hcp lattice and a b is the bcc lattice constant. The displacement of a cell atom between the positions H and b is the most convenient generalized coordinate ξ changing over the optimal trajectory with simultaneous fitting of all parameters (angle, displacement, deformation). It allows to describe the hcp-bcc transformation using a double- or triple-well potential 5 following to Roitburd 8 and Falk 9 treatment.. Triple-well potential model Considering the hcp-bcc transition, we will use the volume energy density in the form of a triple-well potential, which is conventional in the theory of first order phase transitions, to which martensitic transformations belong, [, 4]. The simplest triple-well potential will be written with allowance for the spatially inhomogeneous term 5 : W ξ z v α dξ dz k 6 ξ 6 6 k 4 ξ 4 4 k ξ ; () where α is the dispersion parameter; v is the cell volume; and k 6 k 4 k are phenomenological parameters. In ferroelectrics the parameter ξ is a relative displacement of atomic layers between the equilibrium positions separated by the distance l Hb. The atomic displacement occurs along the straight line between the minima positions (H) - (b) - (H). At ξ, the position b is at the center of the cell cut by the bcc () plane. The positions H and H are located at the centers of the neighboring hcp cells. Hereinafter, the superscript (3) indicates the triple-well model. In the phase equilibrium, the triple-well potential has the same level of the well bottom. Then potential ()() takes the form where W ξ z v α dξ dz k 6 6 ξ ξ ξ 3 ξ 3 3k 4 4k 6 : k The barrier height between the phases per unit volume (h 3 ) is: h 3 W ξ 3 max k4 3 ; (3) 3k 4 6k 6 (4) 6v 6 k 6 ; (5) Variation of the free energy (3)(3) in the variable ξ gives a differential equation. The bound- have to be taken into ary conditions for displacements in the phases ξ z ;ξ 3 account. The equation integration with 7 gives the shape of the phase boundary: ξ ξ 3 exp z l 3 exp z l 3 ; (6)

4 4 Here, ξ 3 l Hb and l 3 l 3 ξ 3 3α k 6 k 4 αk 6 3 (7) is the characteristic phase boundary width. Plus corresponds to the bcc phase location at z. Equations (4), (5), (7) form a system relating the lattice parameters, the height of the atomic barrier per cell h 3 l 3, and the phase boundary width to the parameters k 6 κ 4, and α of the microscopic potential (3). Substitution of (6) into (3) and further integration yields the surface density of the phase boundary energy: W 3 k 6 ξ 3 6 6v l 3 7 h 3 l 3 4v (8).3 Two OP theory Burgers mechanism 4 of bcc hcp PT was used for two OP theory development 6,7. Landau free energy expansion describes the lattice transformation mechanism with respect of two order parameters which represent shuffles (η) and shears (ε): F G η A η B 4 η4 C 6 η6 A ε C η C η 4 ε C 3 η C 4 η 4 ε (9) To reduce free energy from two OP to effective form in terms of one OP, the minimum on shear OP was found F ε. The power expansion of ε in terms of the shuffle OP η ε C A η C C C 4 3 η O η 6 () A This equation defines the lowest free energy path between bcc hcp structures in the shuffle (η) - shear (ε) plane. The path occurs along a valley with a minimum and a saddle point. An effective free energy can be obtained by ε replacing where F e f f G η B B C A ; A A η B η4 C 4 C C 6 C C 6 η6 () C C 3 ; () A For B and C the effective free energy () coincides with () with accuracy of symbols changing. In long wave limit when the wave vector k slightly differs from the N-point wave vector k N for TA mode, the phonon dispersion relation was established as ρ ω A T T G k N k (3) where ρ is the mass density of the material, G is the coefficient of the shuffle gradient, and A A T T with temperature T at which complete phonon softening is occur. However for the first order PT, the phonon softening is not complete, and PT occurs at the transition A

5 5 + / L * Z * Fig. The size L n of a bcc phase nuclear at splitting twin boundary in hcp is shown as distance between two PB, φ 4 6 ξ. temperature T M T, where T M is defined by Eq. (4). The first order PT bcc hcp has the finite entropy change at T M : S 3A B 8C (4) Eqs. (4), (3), (4) for values of OP, phonon spectrum and entropy change at T M gave enough information for calculation of free energy parameters of Ti and Zr Experimental details 4 Results the fit coefficients are functions of temperature, shown in Figs. and respectively. The results of the fitting procedure are summarized in Figs. and ; 5 Conclusions. Two order parameters theory of bcc-hcp phase transition is accommodated to solid helium.. The continual model of the coherent 8 twin boundary in hcp crystal is developed. Their form is found, process of disintegrating of TB and origin on its place a bcc nuclear is analytically shown. 3. The reason of asymmetry of bcc-hcp PT observed in the experiments in solid 4 He ia established. At cooling from a liquid phase a bcc then hcp crystals are formed (threshold formation of grains). In the opposite PT at heating, the topological defects, TB, in a hcp phase break up to pairs of PB with formation of a bcc phase.

6 6.p.p p,,hcp b a c b c,bcc a Liquid T T T Fig. The phase diagram of 4 He. a) Continuous arrows are ways of change of parameters at direct (a) and the opposite (b) bcc-hcp PT. The area of overcooled bcc phases in experiment? is shaded. b) Dotted arrows are suggested ways around the bottom () and top () triple points for reversing asymmetry of hcp bcc PT (the delay area is shaded). 4. The schema of the inverse of PT bcc-hcp asymmetry in solid 4 He in planned experiments is suggested. It is necessary to obtain from liquid helium a hcp phase, then a bcc phase. In suggested experiments two external thermodynamic parameters, pressure and temperature, have be changed. References. J.W. Christian, The Theory of Transformation in Metals and Alloys. Part. Pergamon Press, NY (975). A.N. Gan shin, V.N. Grigor ev, V.A. Maidanov, A.A.Penzev, A.Ya. Rudavskii, A.S.Rybalko and E.V. Syrnikov, JETP Letters, 73, 39, () 3. M. Maekawa, Y. Okumura, and Y. Okuda, Phys. Rev. B 65, 4455, () 4. A.P. Birchenko, E.O. Vekhov, N.P. Mikhin et al Fizika Nizkih Temperatur (Low Temp. Phys.) 3, 47, (6) 5. V.A. Lykah, and E.S. Syrkin, Bull. of the Russ. Acad. of Sc.: Physics, 7, 39, (7) 6. Sanati-free 7. Sanati-wall 8. Roitburd-SS 9. Falk98

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