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On the stability of crystal lattices. I

Published online by Cambridge University Press:  24 October 2008

Max Born
Affiliation:
The UniversityEdinburgh

Extract

The stability of lattices is discussed from the standpoint of the method of small vibrations. It is shown that it is not necessary to determine the whole vibrational spectrum, but only its long wave part. The stability conditions are nothing but the positive definiteness of the macroscopic deformation energy, and can be expressed in the form of inequalities for the elastic constants. A new method is explained for calculating these as lattice sums, and this method is applied to the three monatomic lattice types assuming central forces. In this way one obtains a simple explanation of the fact that the face-centred lattice is stable, whereas the simple lattice is always unstable and the body-centred also except for small exponents of the attractive forces. It is indicated that this method might be used for an improvement of the, at present, rather unsatisfactory theory of strength.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1940

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References

(1)Voigt, W.Lehrbuch der Kristallphysik (Leipzig, 1910).Google Scholar
(2)Born, M.Atomtheorie des festen Zustandes (Leipzig, 1923).CrossRefGoogle Scholar
(3)Zwicky, F.Phys. Z. 24 (1923), 131.Google Scholar
(4)Niggli, P.Z. f. Kristallogr. 74 (1930), 375.Google Scholar
(5)Goldschmidt, V. M.Fortschr. d. Mineral. 15 (1931), 73.Google Scholar
(6)Born, M. and Goeppert-Meyer, M.Handbuch der Physik (Berlin, 1933), 24, pt. 1 (2nd ed.), article 4, p. 733.Google Scholar
(7)Blackman, M.Proc. Roy. Soc. A, 148 (1935), 365; 149 (1935), 117; 159 (1937), 416; 164 (1938), 62; Proc. Cambridge Phil. Soc. 33 (1937), 94.CrossRefGoogle Scholar
(8)Herzfeld, K. and Lyddane, R. H.Phys. Rev. 54 (1938), 846.CrossRefGoogle Scholar
(9)Born, M.J. Chem. Phys. 7 (1939), 591.CrossRefGoogle Scholar
(10)Kellermann, W.Phil. Trans. Roy. Soc. (in the Press).Google Scholar
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