Application of Houston's Method to the Sum of Plane Waves over the Brillouin Zone. II. Body‐Centered Cubic Lattice
M. Miąsek, H. Dodziuk
Apr 1, 1967
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Journal of Mathematical Physics
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Key Takeaway Key Takeaway: The method can be applied to the sum of plane waves over the Brillouin zone in body-centered cubic lattices, providing accurate results for small r values in solid-state physics problems.
Abstract
In a previous paper we considered the function χ(r) ≡ (1/N) Σk eik·r, where the sum runs over the first Brillouin zone of a crystal, and its expansion into series of Cubic Harmonics Kj:χ(r)=∑j=0∞gj(r)×Ki(θ,φ). Houston's method was used in order to find the radial functions gj(r) for several values of j, for χ(r) given for the simple cubic and the face‐centered cubic lattices. In this paper, the same considerations are applied to χ(r) given for the body‐centered lattice. gj(r), with j = 0, 2, 3, are calculated in the region of small r which is assumed as 0 ≤ r ≤ 2a, where a is the lattice constant. In most of the problems of solid‐state physics, where the function χ(r) occurs, it is satisfactory to know its values only for small r, usually not larger than 2a. The function g0(r) is calculated using 3‐, 6‐, and 9‐term expansion formulas, g2(r) and g3(r) using only 3‐ and 6‐term formulas. Comparing gj(r) obtained from the formulas with different number of terms it is established that, for r in the region 〈0, ...