Mixing term $H_{mix}$

The mixing term $H_{mix}$ is

$$H_{mix} = \frac{1}{2} \sum_{ij} \frac{c_L(ij)-c_T(ij)}{\vert\vec R_{ij}\vert^2} \vec R_{ij}^T\bar a \vec R_{ij}\vec R_{ij}^T \vec P_{ij}$$

$$+ c_T(ij) \vec R_{ij}^T\bar a^T\vec P_{ij}$$

$$= \frac{1}{2}\sum_{ij} 2\frac{c_L(ij)-c_T(ij)}{\vert\vec R_{ij}\vert^2} \vec R_{ij}^T\bar a \vec R_{ij}\vec R_{ij}^T \vec P_{i}$$

$$+ 2 c_T(ij) \vec R_{ij}^T\bar a^T\vec P_{i}$$

$$= \sum_{ij,\alpha\beta} \frac{c_L(ij)-c_T(ij)}{\vert\vec R_{ij}\vert^2} R_{ij}^{\alpha} a_{\alpha\beta} R_{ij}^{\beta} R_{ij}^{\gamma} P_{i}^{\gamma}$$

$$+ c_T(ij) R_{ij}^{\beta} a_{\alpha\beta} \delta_{\alpha\gamma} P_{i}^{\gamma}$$ (90)

$$= -\sum_{\stackrel{i,\alpha=1,..,6}{ \gamma=1,2,3}} G_{mix}^{\alpha... ...sum_{ij,\alpha\beta} c_T(ij) R_{ij}^{\beta} \omega_{\alpha\beta} P_{i}^{\alpha}$$

Note that without crystal field phonon interaction the mixing term $H_{mix}$ is zero, because the expectation values $\langle P_i \rangle$ are zero in this case. The last term reflects the fact, that the energy is not invariant under rotations in case of transversal springs.

In (95) we made use of the phonon-strain interaction constants $G_{mix}^{\alpha\gamma}(i)$ defined writing explicitely the Voigt notation $\alpha=(\alpha\beta)$ by

$$G_{mix}^{(\alpha\beta)\gamma}(i) = -a_0\sum_{j} \frac{c_L(ij)-c_T(ij)}{\vert\vec R_{ij}\vert^2} R_{ij}^{\alpha} R_{ij}^{\beta} R_{ij}^{\gamma} -$$

$$-a_0\frac{1}{2}\sum_{j} c_T(ij) (R_{ij}^{\beta} \delta_{\alpha\gamma}+ R_{ij}^{\alpha} \delta_{\beta\gamma})$$ (91)

Note a symmetry property: in case of inversion symmetry at the position of an ion, the phonon-strain interaction constants $G_{mix}^{\alpha\gamma}(i)$ will be zero.