Mixing term

$\displaystyle \hat H_{mix}$	$\textstyle =$	$\displaystyle \frac{1}{2} \sum_{ij} \frac{c_L(ij)-c_T(ij)}{\vert\mathbf R_{ij}\vert^2} \mathbf R_{ij}^T\bar a \mathbf R_{ij}\mathbf R_{ij}^T \hat \mathbf P_{ij}$
		$\displaystyle + c_T(ij) \mathbf R_{ij}^T\bar a^T\mathbf P_{ij}$
	$\textstyle =$	$\displaystyle \frac{1}{2}\sum_{ij} 2\frac{c_L(ij)-c_T(ij)}{\vert\mathbf R_{ij}\vert^2} \mathbf R_{ij}^T\bar a \mathbf R_{ij}\mathbf R_{ij}^T \hat \mathbf P_{i}$
		$\displaystyle + 2 c_T(ij) \mathbf R_{ij}^T\bar a^T\hat \mathbf P_{i}$
	$\textstyle =$	$\displaystyle \sum_{ij,\alpha\beta} \frac{c_L(ij)-c_T(ij)}{\vert\mathbf R_{ij}\... ...{ij}^{\alpha} a_{\alpha\beta} R_{ij}^{\beta} R_{ij}^{\gamma}\hat P_{i}^{\gamma}$
		$\displaystyle + c_T(ij) R_{ij}^{\beta} a_{\alpha\beta} \delta_{\alpha\gamma}\hat P_{i}^{\gamma}$	(89)
	$\textstyle =$	$\displaystyle -\sum_{\stackrel{i,\alpha=1,..,6}{ \gamma=1,2,3}} G_{mix}^{\alpha... ...{ij,\alpha\beta} c_T(ij) R_{ij}^{\beta} \omega_{\alpha\beta}\hat P_{i}^{\alpha}$

Note that without crystal field phonon interaction the mixing term $\hat H_{mix}$ is zero, because the expectation values $\langle \hat \mathbf P_i \rangle$ 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 explicitly the Voigt notation $\alpha=(\alpha\beta)$ by

	$\displaystyle G_{mix}^{(\alpha\beta)\gamma}(i)$	$\textstyle =$	$\displaystyle -a_0\sum_{j} \frac{c_L(ij)-c_T(ij)}{\vert\mathbf R_{ij}\vert^2} R_{ij}^{\alpha} R_{ij}^{\beta} R_{ij}^{\gamma} -$
			$\displaystyle -a_0\frac{1}{2}\sum_{j} c_T(ij) (R_{ij}^{\beta} \delta_{\alpha\gamma}+ R_{ij}^{\alpha} \delta_{\beta\gamma})$	(90)

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.