Amplitudes of lattice vibrations

Eckold, G.

doi:10.1107/97809553602060000638

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 270-271

Section 2.1.2.6. Amplitudes of lattice vibrations

G. Eckold^a ^*

^a Institut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D-37077 Göttingen, Germany
Correspondence e-mail: geckold@gwdg.de

2.1.2.6. Amplitudes of lattice vibrations

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Lattice vibrations that are characterized by both the frequencies $[\omega_{{\bf q},j}]$ and the normal coordinates $[Q_{{\bf q},j}]$ are elementary excitations of the harmonic lattice. As long as anharmonic effects are neglected, there are no interactions between the individual phonons. The respective amplitudes depend on the excitation level and can be determined by quantum statistical methods. The energy levels of a lattice vibration ( $[{\bf q},j]$ ) are those of a single harmonic oscillator: $[E_{n} = (n + {\textstyle{1 \over 2}}) \hbar \omega _{{\bf q},j}, \eqno (2.1.2.45) ]$ as illustrated in Fig. 2.1.2.6. The levels are equidistant and the respective occupation probabilities are given by Boltzmann statistics: $[p_n = {{\exp[- (n + 1/2)(\hbar \omega_{{\bf q},j}/{kT})]}\over {\textstyle\sum _{m = 0}^\infty \exp[- (m + 1/2)(\hbar \omega_{{\bf q},j} /kT)]}}. \eqno (2.1.2.46) ]$ In the quasiparticle description, this quantity is just the probability that at a temperature T there are n excited phonons of frequency $[\omega_{{\bf q}, j}]$ . Moreover, in thermal equilibrium the average number of phonons is given by the Bose factor: $[n_{{\bf q},j} = {1 \over {\exp({{\hbar \omega _{{\bf q},j}}/{kT}}) - 1}} \eqno (2.1.2.47) ]$ and the corresponding contribution of these phonons to the lattice energy is $[E_{{\bf q},j} = (n_{{\bf q},j} + {\textstyle{1 \over 2}}) \, \hbar \omega _{{\bf q},j}. \eqno (2.1.2.48) ]$ The mean-square amplitude of the normal oscillator coordinate is obtained as $[\left\langle {\left| {Q_{{\bf q},j}}\right|^2 }\right\rangle = {\hbar \over {\omega _{{\bf q},j}}}\left({n_{{\bf q},j} + {\textstyle{1\over 2}}}\right). \eqno (2.1.2.49) ]$ At high temperatures ( $[kT\gg\hbar\omega_{{\bf q},j}]$ ), the phonon number, the corresponding energy and the amplitude approach the classical values of $[\eqalignno{n_{{\bf q},j}&\mathrel{\mathop{\kern0pt\longrightarrow}\limits_{T \to \infty }}{{kT}\over {\hbar \omega _{{\bf q},j}}}, & (2.1.2.50)\cr E_{{\bf q},j}&\mathrel{\mathop{\kern0pt\longrightarrow}\limits_{T \to \infty }}3NN_Z kT\hbox{ and} & (2.1.2.51)\cr \left\langle {\left| {Q_{{\bf q},j}}\right|^2 }\right\rangle &\mathrel{\mathop{\kern0pt\longrightarrow}\limits_{T \to \infty }}{{kT}\over {\omega _{{\bf q},j}^2 }}, &}%fd2.1.2.51 ]$ respectively. Note that occupation number, energy and amplitude merely depend on the frequency of the particular lattice vibration. The form of the corresponding eigenvector $[{\bf e}({\bf q},j)]$ is irrelevant.