Exchange Striction

If the two ion interactions are strain dependent, an additional contribution to the strain tensor has to be considered in the equations (110) to (112). The resulting contribution is commonly called exchange striction, because predominantly the magnetic exchange interactions will contribute.

We therefore extend the model by the exchange striction writing the Hamiltonian

  $$\hat H=\hat H_{\rm tot} -\frac{1}{2} \sum_{ij,\alpha\beta} {\... ... R_{j} - \mathbf R_i)) \hat \mathcal I_{\alpha}^i \hat \mathcal I_{\beta}^{j}$$ (105)

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

	$${\mathcal J}_{\alpha\beta}(\mathbf R=(1+\bar a)\mathbf R_{ij})$$	$=$	$${\mathcal J}_{\alpha\beta}(\mathbf R_{ij}) + \frac{\partial {\mat... ...mathbf R_{ij})^{\alpha'}}{\partial \epsilon_{\beta'}} \epsilon_{\beta'} + \dots$$
		$=$	$${\mathcal J}_{\alpha\beta}(\mathbf R_{ij})+ \sum_{\alpha'\gamma=1... ...a'\gamma}\mathbf R_{ij}^{\gamma}}{\partial \epsilon_{\beta'}} \epsilon_{\beta'}$$	(106)

Note that the second part of equation (114) describes the exchange striction Hamiltonian, if inserted in equation (113).

  $$\hat H=\hat H_{\rm tot} -\frac{1}{2} \sum_{ij,\alpha\beta} {\... ...}} \epsilon_{\beta'} \hat \mathcal I_{\alpha}^i \hat \mathcal I_{\beta}^{j}$$ (107)

In section 12.10 we show, how the position derivatives of the exchange parameters can be calculated.

Equation (112) is extended for the case of exchange striction:

	$$\sum_{\beta=1,..,6} c^{\alpha\beta} \epsilon_{\beta}$$	$=$	$$\sum_{i,\delta=1,2,3} G_{mix}^{\alpha\delta}(i)\langle u_{i}^{\delta} \rangle$$	(108)
		$+$	$$\sum_{i,\gamma=1,...} G_{\rm cfph}^{\alpha\gamma}(i) \langle O_{\gamma}(\hat \mathbf J_i) \rangle$$
		$+$	$$\frac{1}{2}\sum_{ii',\delta,\delta'\alpha',\gamma=1,..,3} \frac{\... ...lpha}} \langle \hat \mathcal I_{\delta}^i \hat \mathcal I_{\delta'}^{i'}\rangle$$