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On the stability of crystal lattices VIII. Stability of rhombohedral Bravais lattices*

Published online by Cambridge University Press:  24 October 2008

H. W. Peng  and
S. C. Power
H. W. Peng
Affiliation:
The UniversityEdinburgh Communicated by M. Born
S. C. Power
Affiliation:
The UniversityEdinburgh Communicated by M. Born
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Extract

The main purpose of the paper is an investigation of the stability of a certain class of Bravais lattices, namely, those with a rhombohedral cell of arbitrary angle. The potential energy is assumed to consist of two terms, each proportional to a reciprocal power of the distance. In the continuous series of lattices obtained by changing the rhombohedral angle, there are included the three cubic Bravais lattices, the simple (s), the face-centred (f) and the body-centred (b) lattices. It is shown that (f) and (b) correspond to a minimum of the potential energy, and (s) to a maximum. A method for calculating the potential energy for the intermediate rhombohedral lattices is developed, and, with the help of a certain characteristic function, it is shown by numerical calculation that the (f) lattice corresponds to the absolute minimum of potential energy, and that no extrema, other than (f), (s) and (b), exist. In the last section, the case of a compound (non-Bravais lattice) is considered, and it is shown that the equilibrium and stability conditions for the law of force assumed can be divided into one set for change of volume, and an independent set for change of shape.

We take this opportunity of expressing our sincere thanks to Prof. Born for his interest in our work, and for much valuable advice.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 38 , Issue 1 , January 1942 , pp. 67 - 81
Copyright
Copyright © Cambridge Philosophical Society 1942

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References

* Note that at these singularities of S n, the function f(x) (1·15) is finite. Since S n has a pole of order ½n, has a pole of order ½ similarly, has a pole of order ½ Therefore, is finite, being equal to the quotient of the residues at the point.