International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 2.4, pp. 349-350
Section 2.4.2. Elastic waves^{a}Laboratoire des Verres, Université Montpellier 2, Case 069, Place Eugène Bataillon, 34095 Montpellier CEDEX, France |
The fundamental equation of dynamics (see Section 1.3.4.2 ), applied to the displacement u of an elementary volume at r in a homogeneous material is Summation over repeated indices will always be implied, and T is the stress tensor. In non-piezoelectric media, the constitutive equation for small strains S is simply The strain being the symmetrized spatial derivative of u, and c being symmetric upon interchange of k and , the introduction of (2.4.2.2) in (2.4.2.1) gives (see also Section 1.3.4.2 ) One considers harmonic plane-wave solutions of wavevector Q and frequency ω,For small compared with the wavelength , the total derivative can be replaced by the partial in (2.4.2.3). Introducing (2.4.2.4) into (2.4.2.3), one obtains where is the unit vector in the propagation direction, is the unit tensor and , where is the phase velocity of the wave. This shows that is an eigenvector of the tensor . For a given propagation direction , the three eigenvalues are obtained by solving To each there is an eigenvector given by (2.4.2.5) and an associated phase velocity
The tensor is symmetric upon interchange of the indices () because . Its eigenvalues are real positive, and the three directions of vibration are mutually perpendicular. The notation indicates a unit vector. The tensor is also invariant upon a change of sign of the propagation direction. This implies that the solution of (2.4.2.5) is the same for all symmetry classes belonging to the same Laue class.
For a general direction , and for a symmetry lower than isotropic, is neither parallel nor perpendicular to , so that the modes are neither purely longitudinal nor purely transverse. In this case (2.4.2.6) is also difficult to solve. The situation is much simpler when is parallel to a symmetry axis of the Laue class. Then, one of the vibrations is purely longitudinal (LA), while the other two are purely transverse (TA). A pure mode also exists when belongs to a symmetry plane of the Laue class, in which case there is a transverse vibration with perpendicular to the symmetry plane. For all these pure mode directions, (2.4.2.6) can be factorized to obtain simple analytical solutions. In this chapter, only pure mode directions are considered.
In piezoelectric crystals, a stress component is also produced by the internal electric field E, so that the constitutive equation (2.4.2.2) has an additional term (see Section 1.1.5.2 ), where e is the piezoelectric tensor at constant strain.
The electrical displacement vector D, related to E by the dielectric tensor , also contains a contribution from the strain, where is at the frequency of the elastic wave.
In the absence of free charges, , and (2.4.2.9) provides a relation between E and S, For long waves, it can be shown that E and Q are parallel. (2.4.2.10) can then be solved for E, and this value is replaced in (2.4.2.8) to give Comparing (2.4.2.11) and (2.4.2.2), one sees that the effective elastic tensor now depends on the propagation direction . Otherwise, all considerations of the previous section, starting from (2.4.2.6), remain, with c simply replaced by .