Harmonic generation

Authier, A.; Zarembowitch, A.

doi:10.1107/97809553602060000630

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. 1.3, p. 96

Section 1.3.7.4. Harmonic generation

A. Authier^a ^* and A. Zarembowitch^b

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and ^bLaboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France
Correspondence e-mail: aauthier@wanadoo.fr

1.3.7.4. Harmonic generation

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The coordinates in the medium free of stress are denoted either a or $[{\bar X}]$ . The notation $[{\bar X}]$ is used when we have to discriminate the natural configuration, $[{\bar X}]$ , from the initial configuration X. Here, the process that we describe refers to the propagation of an elastic wave in a medium free of stress (natural state) and the coordinates will be denoted $[a_{i}]$ .

Let us first examine the case of a pure longitudinal mode, i.e. $[u_{1} = u_{1}(a_{1},t)\semi \quad u_{2} = u_{3} = 0. ]$

The equations of motion, (1.3.7.7) and (1.3.7.9), reduce to $[\rho u_{1}'' = (\lambda + 2\mu){\partial^{2} u_{1} \over \partial a_{1}^{2}} + [3(\lambda + 2\mu) + 2(l + 2m)] {\partial u_{1}\over \partial a_{1}}{\partial^{2} u_{1}\over \partial a_{1}^{2}} ]$ for an isotropic medium or $[\rho u_{1}'' = c_{11}{\partial^{2} u_{1}\over \partial a_{1}^{2}} + [3c_{11} + c_{166}] {\partial u_{1}\over \partial a_{1}}{\partial^{2}u_{1}\over \partial a_{1}^{2}} ]$ for a cubic crystal (most symmetrical groups) when a pure longitudinal mode is propagated along [100].

For both cases, we have a one-dimensional problem; (1.3.7.7) and (1.3.7.9) can therefore be written $[\rho u_{1}'' = K_{2}{\partial^{2}u_{1}\over \partial a_{1}^{2}} + [3K_{2} + K_{3}] {\partial u_{1}\over \partial a_{1}}{\partial^{2}u_{1}\over \partial a_{1}^{2}}. \eqno(1.3.7.10) ]$

The same equation is also valid when a pure longitudinal mode is propagated along [110] and [111], with the following correspondence: $[\eqalign{[100]\ K_{2} &= c_{11}, \quad K_{3} = c_{111}\cr [110]\ K_{2} &= {c_{11} + c_{12} + 2c_{44}\over 2}, \quad \ K_{3} = {c_{111} + 3c_{112} +12 c_{166} \over 4}\cr [111]\ K_2 &= {c_{11} + 2c_{12} + 4c_{44}\over 3},\cr K_{3} &= {c_{111} + 6c_{112} + 12c_{144} + 24c_{166} + 2c_{123} + 16c_{456}\over 9}.\cr} ]$ Let us assume that $[K_{3} \ll K_{2}]$ ; a perturbation solution to (1.3.7.10) is $[u = u^{0} + u^{1},]$ where $[u^{1} \ll u^{0}]$ with $[\eqalignno{u^{0} &= A \sin (ka - \omega t) &(1.3.7.11)\cr u^{1} &= Ba \sin 2(ka - \omega t) + C a \cos 2(ka - \omega t). &(1.3.7.12)\cr} ]$

If we substitute the trial solutions into (1.3.7.10), we find after one iteration the following approximate solution: $[u = A \sin (ka - \omega t) - {(kA)^{2}(3K_{2} + K_{3})\over 8\rho c^{2}} a \cos 2(ka - \omega t), ]$ which involves second-harmonic generation .

If additional iterations are performed, higher harmonic terms will be obtained. A well known property of the first-order nonlinear equation (1.3.7.10) is that its solutions exhibit discontinuous behaviour at some point in space and time. It can be seen that such a discontinuity would appear at a distance from the origin given by (Breazeale, 1984) $[L = - 2{(K_2)^{2}\over 3K_{2} + K_{3}}\rho \omega u_{0}', ]$ where $[u_{0}']$ is the initial value for the particle velocity.

References

Breazeale, M. A. (1984). Determination of third-order elastic constants from ultrasonic harmonic generation. Physical acoustics, Vol. 17, edited by R. N. Thurston, pp. 2–75. New York: Academic Press.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.3, p. 96