J. Phys. I France 1 (1991) 373-382
Free-fermion solution for overall equilibrium crystal shapeL. 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 , 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 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 , 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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