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

• J. H. Wood
Chapter
Part of the The IBM Research Symposia Series book series (IRSS)

## 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)

## Keywords

Programming System Gaussian Elimination Logarithmic Derivative Bloch Function Gradient Computation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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