J. Phys. I France
Volume 1, Numéro 3, March 1991
Page(s) 373 - 382
DOI: 10.1051/jp1:1991139
J. Phys. I France 1 (1991) 373-382

Free-fermion solution for overall equilibrium crystal shape

L. V. Mikheev1 and V. L. Pokrovsky2, 3

1  Institute of Crystallography of the Academy of Sciences U.S.S.R., Leninski Prosp. 59, Moscow 117333, U.S.S.R.
2  Landau Institute for Theoretical Physics, Ul. Kosygina 2, Moscow 117940, U.S.S.R.
3  Institut für Festkörperforschung, forschungszentrum Jülich GmBH, Postfach 1913 D-5170 Jülich, Germany.

(Received 16 July 1990, accepted in final form 8 November 1990)

We generalize the random walk or free-fermion method of Yamamoto, Akutsu and Akutsu to obtain a simple explicit solution for the overall equilibrium crystal shape of a simple cubic crystal at temperatures, T, low with respect to the nearest-neighbour coupling, E. The thermal rounding of the corners and of the edges of the cube appears to be qualitatively different : at the corners the width of the rounded, vicinal surface and its radius of curvature are estimated to be of order $(k_{\rm B}\,T/E) R$, where 2 R is the diameter of the crystal. As the distance l from the corner along an edge increases, the width of the vicinal surface and transverse radius of curvature decrease as exp $(-l/k_{\rm B}\,T/ER)$ reaching an exponentially small value at the middle of the edge. On the other hand, the radius of curvature parallel to an edge grows as exp $(l/k_{\rm B}\,T/ER)$, so that the product of two principal curvatures remains constant up to a numerical factor. A qualitative explanation of the results is presented, based on the strong dependence of the stiffness of the steps on their orientation with respect to the axes of the crystal.

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