**Физика Низких Температур: ***Том ***44**, Выпуск **2** (Февраль 2018), c. 154-167
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### Thermodynamics of dilute ^{3}He –^{4}He solid solutions with hcp structure

**K.A. Chishko**

B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Nauky Ave., Kharkiv 61103, Ukraine

E-mail: chishko@ilt.kharkov.ua

*Received June 4, 2017, revised August 8, 2017*

**Аннотация**

To interpret the anomalies in heat capacity *C*_{V} (*T*) and temperature-dependent pressure *P*(*T*) of solid hexagonal close-packed (hcp) ^{4}He we exploit the model of hcp crystalline polytype with specific lattice degrees of freedom anddescribe the thermodynamics of impurity-free ^{4}He solid as superposition of phononic and polytypic contributions.The hcp-based polytype is a stack of 2D basal atomic monolayers on triangular lattice packed with arbitrary long (up to infinity) spatial period along the hexagonal c axis perpendicular to the basal planes. It is a crystal with perfect ordering along the layers, but without microscopic translational symmetry in perpendicular direction (which remains, nevertheless, the rotational crystallographic axis of third order, so that the polytype can be considered as semidisordered system). Each atom of the hcp polytype has twelve crystallographic neighbors in both first and second co-ordination spheres at any arbitrary packing order. It is shown that the crystal of such structure behaves as anisotropic elastic medium with specific dispersion law of phonon excitations along c axis. The free energy and the heat capacity consist of two terms: one of them is a normal contribution (with *C*_{V} (*T*) ~ *T* ^{3}) from phonon excitations in an anisotropic lattice of hexagonal symmetry, and another term (an “excessive” heat) is a contribution resulted by packing entropy from quasi-one-dimensional system of 2D basal planes on triangular lattice stacked randomly along c axis without braking the closest pack between neighboring atomic layers. The excessive part of the free energy has been treated within 1D quasi-Ising (lattice gas) model using the transfer matrix approach. This model makes us possible to interpret successfully the thermodynamic anomaly (heat capacity peak in hcp ^{4}He) observed experimentally.

PACS: 67.80.B– Solid ^{4}He;

PACS: 05.70.–a Thermodynamics;

PACS: 61.72.Nn Stacking faults and other planar or extended defects.

**Ключевые слова: **polytype, packing entropy, quantum solids.

**Published online:** *December 25, 2017*