Gradients of E($\overrightarrow k$) from the APW Determinant

Abstract

Because our method of determining ${\nabla _k}\left( {E\left( {\overrightarrow k } \right)} \right)$ was partially determined by the programming system we use for the 1937 APW method,^1,2 we first briefly describe the methods we used to determine E$\left( {\overrightarrow k } \right)$. In this scheme, the E$\left( {\overrightarrow k } \right)$ are taken as the zeroes of the so-called APW determinant $D\left( {E,\overrightarrow k } \right)$:

$$D\left( {E\left( {\overrightarrow k } \right),k} \right) = 0$$

((1))

The matrix M, of which D is the determinant, has as its elements³

$${\left( {H - E} \right)_{ij}} = {\overrightarrow k _i}\cdot{\overrightarrow k _j}\left[ {\delta \left( {{{\overrightarrow k }_i},{{\overrightarrow k }_j}} \right) - \frac{{4\pi R_s^2}}{\Omega }\frac{{{j_1}\left( {\left| {{{\overrightarrow k }_j} - {{\overrightarrow k }_i}} \right|{R_s}} \right)}}{{\left| {{{\overrightarrow k }_j} - {{\overrightarrow k }_i}} \right|}}} \right] - \left[ {\delta \left( {{{\overrightarrow k }_i},{{\overrightarrow k }_j}} \right) - \frac{{4\pi R_s^2}}{\Omega }\cdot\frac{{{j_1}\left( {\left| {{{\overrightarrow k }_j} - {{\overrightarrow k }_i}} \right|} \right.{R_s}}}{{\left| {{{\overrightarrow k }_j} - {{\overrightarrow k }_i}} \right|}}} \right] + \frac{{4\pi }}{\Omega }\sum\limits_\ell {\left[ {\left( {2\ell + 1} \right){P_\ell }\left( {\cos {\theta _{ij}}} \right){j_\ell }\left( {{k_i}{R_s}} \right){j_\ell }\left( {{k_j}{R_s}} \right)} \right]R_s^2\frac{{{{u'}_\ell }\left( {{R_s},E} \right)}}{{{u_\ell }\left( {{R_s},E} \right)}}}$$

((2))