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Some properties of the vibrational spectrum of a lattice

Published online by Cambridge University Press:  24 October 2008

M. Blackman
M. Blackman
Affiliation:
Trinity College
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Extract

The vibrations of the particles which constitute a crystal can be represented in terms of normal vibrations subject to the condition that the energy of the crystal can be expressed as a homogeneous function of the second order in the displacements. This has been shown by Born and by Waller. It has been proved by Debye and, more generally by Born, that the density of the normal vibrations varies as ν2 when the frequency is very small. The spectrum has recently been worked out for a Born-v. Kármán lattice in the two- and three-dimensional cases. Otherwise practically nothing is known about the spectrum in individual cases, except perhaps in isolated cases where the frequency branches split up and the frequency spectrum shows certain gaps.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 33 , Issue 1 , March 1937 , pp. 94 - 103
Copyright
Copyright © Cambridge Philosophical Society 1937

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References

* Atomtheorie des festen Zustandes (Teubner, 1923), §§ 14–19.Google Scholar

† Uppsala Arsskrift (1925).

‡ Ann. Physik, 39 (1912), 789.Google Scholar

§ Blackman, M., Proc. Roy. Soc. 143 (1935), 384.CrossRefGoogle Scholar

∥ To be published shortly.

For details see: Born, M. and Kármán, Th. v., Phys. Zeits. 3 (1913), 297;Google ScholarDehlinger, W., Phys. Zeits. 15 (1914), 276.Google Scholar

* The wave-length must be large compared with the lattice distance.

* This presupposes that p ≠ 0. For p = 0 the proof is very simple.

* In the two-dimensional case where the second order terms can be worked out explicitly, it can be shown (unpublished work) that the result is true in the whole of the corresponding region.

† I am indebted to Dr J. H. C. Thompson for drawing my attention to this point.

‡ Physica, 2 (1935), 698.Google Scholar

* This statement may also be put in the form: Tendency of the θD value to fall or rise.