Important formulas¶

GIMIC key equation¶

The GIMIC working equation for the evaluation of the current density susceptibility tensor

$$ {\cal J}^{B_{\tau}}_{\gamma}(\mathbf{r}) = \sum_{\mu\nu} {D_{\mu\nu}} \frac{\partial\chi^*_\mu(\mathbf{r})}{\partial B_{\tau}} \frac{\partial h}{\partial m^K_{\gamma}} \chi_\nu(\mathbf{r}) + \sum_{\mu\nu} {D_{\mu\nu}} \chi^*_\mu(\mathbf{r}) \frac{\partial h}{\partial m^K_{\gamma}} \frac{\partial \chi_{\nu}(\mathbf{r})}{\partial B_{\tau}} + $$ $$ \sum_{\mu\nu} \frac{\partial D_{\mu\nu} }{\partial B_{\tau}}\chi^*_{\mu}(\mathbf{r}) \frac{\partial h}{\partial m^K_{\gamma}}\chi_\nu(\mathbf{r}) -\epsilon_{\gamma\tau\delta} \big [ \sum_{\mu\nu} D_{\mu\nu}\chi^*_{\mu}(\mathbf{r}) \frac{\partial^2 h}{\partial m^K_{\gamma} \partial B_{\delta}} \chi_{\nu}(\mathbf{r})\big ] $$

Anisotropy of the induced current (ACID)¶

The formula for the ACID method

$$ \Delta {\cal J}{^2}(\mathbf{r}) = \frac{1}{3} \big [ \big({\cal J}_x^x(\mathbf{r}) -{\cal J}_y^y(\mathbf{r})\big )^{2} + \big({\cal J}_y^y(\mathbf{r}) - {\cal J}_z^z(\mathbf{r})\big )^{2} + \big({\cal J}_z^z(\mathbf{r}) - {\cal J}_x^x(\mathbf{r})\big)^{2}\big] + $$ $$ \frac{1}{2} \big [ \big ( {\cal J}_x^y(\mathbf{r}) + {\cal J}_y^x(\mathbf{r})\big )^{2} + \big ( {\cal J}_x^z(\mathbf{r}) + {\cal J}_z^x(\mathbf{r}) \big )^{2} + \big ( {\cal J}_y^z(\mathbf{r}) + {\cal J}_z^y(\mathbf{r}) \big )^{2} \big ] $$