Exchange Striction

We therefore extend the model by the exchange striction introducing two ion interactions in the Hamiltonian, which depend on the strain tensor $\bar \epsilon$ and on the distance between two ions:

$\displaystyle \hat H=\hat H_{\rm tot} -\frac{1}{2} \sum_{ij,\alpha\beta} {\... ... R_{j} - \mathbf R_i)) \hat \mathcal I_{\alpha}^i \hat \mathcal I_{\beta}^{j}$ (129)

and expand the two ion interaction ${\mathcal J}_{\alpha\beta}(\mathbf R_{ij})$ in a Taylor expansion for small strain:

	$\displaystyle {\mathcal J}_{\alpha\beta}(\bar \epsilon,\mathbf R=(1+\bar a)\mathbf R_{ij})$	$\textstyle =$	$\displaystyle {\mathcal J}_{\alpha\beta}(\bar \epsilon=0,\mathbf R_{ij}) + (\fr... ...athbf R_{ij})^{\alpha'}}{\partial \epsilon_{\beta'}}) \epsilon_{\beta'} + \dots$
		$\textstyle =$	$\displaystyle {\mathcal J}_{\alpha\beta}(0,\mathbf R_{ij})+ \sum_{\alpha'\gamma... ...'\gamma}\mathbf R_{ij}^{\gamma}}{\partial \epsilon_{\beta'}}) \epsilon_{\beta'}$	(130)

Note that the second part of equation (138) describes the exchange striction Hamiltonian, if inserted in equation (137).

$\displaystyle \hat H=\hat H_{\rm tot} -\frac{1}{2} \sum_{ij,\alpha\beta} {\... ...}) \epsilon_{\beta'} \hat \mathcal I_{\alpha}^i \hat \mathcal I_{\beta}^{j}$ (131)

$\displaystyle \sum_{\beta=1,..,6} c^{\alpha\beta} \epsilon_{\beta}$	$\textstyle =$	$\displaystyle \sigma_{\alpha}+ \sum_{i,\delta=1,2,3} G_{mix}^{\alpha\delta}(i)\langle u_{i}^{\delta} \rangle$	(132)
	$\textstyle +$	$\displaystyle \sum_{i,\gamma=1,...} G_{\rm cfph}^{\alpha\gamma}(i) \langle O_{\gamma}(\hat \mathbf J_i) \rangle$
	$\textstyle +$	$\displaystyle \frac{1}{2}\sum_{ii',\delta,\delta'\alpha',\gamma=1,..,3} (\frac{... ...pha}}) \langle \hat \mathcal I_{\delta}^i \hat \mathcal I_{\delta'}^{i'}\rangle$