International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 4.1, p. 402   | 1 | 2 |

Section 4.1.2.2. Quantization of normal modes. Phonons

B. T. M. Willisa*

aChemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England
Correspondence e-mail: bertram.willis@chemcryst.ox.ac.uk

4.1.2.2. Quantization of normal modes. Phonons

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Quantum concepts are not required in solving the equations of motion (4.1.2.4[link]) to determine the frequencies and displacement patterns of the normal modes. The only place where quantum mechanics is necessary is in calculating the energy of the mode, and from this the amplitude of vibration [|A_j({\bf q})|].

It is possible to discuss the theory of lattice dynamics from the beginning in the language of quantum mechanics (Donovan & Angress, 1971[link]). Instead of treating the modes as running waves, they are conceived as an assemblage of indistinguishable quasi-particles called phonons. Phonons obey Bose–Einstein statistics and are not limited in number. The number of phonons, each with energy [\hbar\omega_j({\bf q})] in the vibrational state specified by q and j, is given by[n_j({\bf q})=\{\exp[\hbar\omega_j({\bf q})/k_BT]-1\}^{-1}\eqno(4.1.2.6)] and the mode energy [E_j({\bf q})] by[E_j({\bf q})=\hbar\omega_j({\bf q})[n_j({\bf q})+(1/2)].\eqno(4.1.2.7)] Thus the quantum number [n_j({\bf q})] describes the degree of excitation of the mode (j q). The relation between [E_j({\bf q})] and the amplitude [|A_j({\bf q})|] is [E_j({\bf q})=N\omega^2_j({\bf q})|A_j({\bf q})|^2.\eqno(4.1.2.8)] Equations (4.1.2.6[link]) to (4.1.2.8[link] [link]) together determine the value of [|A_j({\bf q})|] to be substituted into equation (4.1.2.3[link]) to give the atomic displacement in terms of the absolute temperature and the properties of the normal modes.

In solving the lattice-dynamical problem using the Born–von Kármán analysis, the first step is to set up a force-constant matrix describing the interactions between all pairs of atoms. This is followed by the assembly of the dynamical matrix D, whose eigenvalues give the frequencies of the normal modes and whose eigenvectors determine the patterns of atomic displacement for each mode.

Before considering the extension of this treatment to molecular crystals, we shall comment briefly on the less rigorous treatments of Einstein and Debye.

References

First citation Donovan, B. & Angress, J. F. (1971). Lattice vibrations. London: Chapman and Hall.Google Scholar








































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