Impurity scattering

Mahan, G. D.

doi:10.1107/97809553602060000635

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. 1.8, p. 224

Section 1.8.4.3. Impurity scattering

G. D. Mahan^a ^*

^a Department of Physics, 104 Davey Laboratory, Pennsylvania State University, University Park, Pennsylvania, USA
Correspondence e-mail: gmahan@psu.edu

1.8.4.3. Impurity scattering

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Impurities can be either point defects or extended defects such as dislocations. Here we confine our remarks to point defects. When acoustic phonons of long wavelength scatter from point defects, the process is very much like the Rayleigh scattering of light. The basic cross section varies as the fourth power of the frequency. Equivalently, at long wavelength, it varies as the fourth power of the wave number q of the phonons (see Ziman, 1962): $[{{1}\over{\tau_i(q)}} = n_i q^4 C_i,\eqno(1.8.4.3)]$ where [n_i] is the concentration of impurities and [C_i] is a constant characteristic of the impurity. Of course, this lifetime for the phonon of wave number q must be averaged over all of the wavevectors in the crystal as a function of temperature. This averaging is actually mathematically delicate. At temperatures T less than the Debye temperature $[\Theta]$ , the average value of q is $[q\sim k_BT/(\hbar v)]$ and $[R_i \sim T^4]$ . At high temperatures $[(T\,\gt\,\Theta)]$ , the average of $[1/\tau(q)]$ is a constant, since all values of q in the Brillouin zone are equally accessible. In this limit, $[R_i \sim]$ constant.

References

Ziman, J. M. (1962). Electrons and phonons. Oxford University Press.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.8, p. 224