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Zeitschrift für angewandte Mathematik und Physik

, Volume 49, Issue 6, pp 919–933 | Cite as

Equilibrium form of crystals and stable norm

  • W. M. Senn
Article

Abstract.

The equilibrium form of a crystal is the shape which minimizes the total surface tension for a given volume. The surface tension itself is considered to be a function of the orientation of the crystal face within the crystal lattice. This function of orientation is identified with the stable norm on the second homology classes of a Riemannian 3-torus. Minimizing the total surface tension of the crystal leads to an equilibrium form which corresponds to the dual unit ball of the stable norm. We outline the connection between surface tension and stable norm and interpret the differentiability properties of the stable norm in terms of the crystal shape. The differentiability properties, in particular, prove Sohnke's reciprocity law estimating the size of crystal faces.

Key words. Crystal shape, surface tension, stable norm, dual unit ball, minimal average action, differentiability properties, convexity. 

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Copyright information

© Birkhäuser Verlag, Basel, 1998

Authors and Affiliations

  • W. M. Senn
    • 1
  1. 1.Mathemathisches Institut der Universität Bern, Sidlerstrasse 5, CH - 3012 Bern, e-mail: wsenn@iam.unibe.chSwitzerland

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