No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

The full text of this article is not currently available.

Application of Houston's Method to the Sum of Plane Waves over the Brillouin Zone. I. Simple‐Cubic and Face‐Centered‐Cubic Lattices

### Abstract

In certain problems in solid‐state physics, the radial functions *g*_{j} (*r*) in the expansion χ(r) = Σ_{ j=0} ^{∞} *g*_{j} (*r*)*K*_{j} (θ, φ), where χ(r) is a known function and the *K*_{j} 's are Kubic Harmonics, are of interest. This paper deals with the functions χ(r) ≡ *N* ^{−1} Σ_{k} *e* ^{ ik·r}, where the sum runs over the first Brillouin Zone of a crystal. In particular, the functions χ(r) for simple cubic and face‐centered cubic lattices are expanded into series of Kubic Harmonics and the radial functions *g*_{j} (*r*) for several values of *j* are found using Houston's method, in which the expansion into series of Kubic Harmonics contains only a finite number of terms with lowest *j*'s. *g* _{0}(*r*) is calculated using 3, 6, and 9‐term expansion, *g* _{2}(*r*) and *g* _{3}(*r*) using only 3 and 6‐term expansion. Comparing *g*_{j} (*r*) obtained from the formulas with different numbers of terms it is established that for *r* in the region 〈0, 2*a*〉, where *a* is the lattice constant, the 6‐term approximation is very good. In practice, the functions *g*_{j} (*r*) usually occur in integrands, together with atomic orbitals, and the tabulated results are expected to be particularly useful in the study of Wannier functions in the OPW scheme.

© 1966 The American Institute of Physics

Received 22 April 1965
Published online 09 December 2004

Commenting has been disabled for this content