Dipole-Dipole Interaction in Simple Lattices

Abstract

Derivation of the electric dipole-dipole interaction as an electric hyperfine interaction Am.A mathematical model, the spherical model, of electric dipole-dipole interaction in simple lattices is discussed. The partition function for the system, assuming just nearest neighbor interaction, is evaluated for one-and three-dimensional simple lattices. Only the three-dimensional lattice exhibits a phase transition. The transition is discussed. The transition behavior in the presence of an external field is also analyzed, and the electric susceptibility is computed.

On the Green Function for the Anisotropic Simple Cubic Lattice

“…The series (2.1) is convergent provided that |z| ≤ 1/(2 + α) 2 . An explicit expression for µ 2n (α) can be derived from (2.2) in terms of a terminating generalised hypergeometric series.…”

“…where −π < arg(w − α) < 0 and C (2) 0 (α) ≡ 1, provided that α = 1. When α = 1 the expansion (8.1) breaks down because the branch-point singularities at w = α and w = 2 − α are confluent in the limit α → 1.…”

“…It can be shown that (5.6) is valid for all points z ∈ R 0 (α) which are in the cut z plane. When 0 < α ≤ 2 we can use (5.6) with z = 1/(u − i ) 2 to calculate G R (u) and G I (u) for √ 4 + α 2 ≤ u < ∞, while for α > 2 it is also possible to use (5.6) to determine G R (u) and G I (u) in the additional interval 0 < u ≤ √ α 2 − 4. If z ∈ R 0 (α) we can modify the formula (5.6) by replacing the hypergeometric function 2 F 1 ( where the upper and lower signs are valid for Im(z) > 0 and Im(z) < 0, respectively.…”

“…We shall now suppose that the variable z = 1/w 2 is allowed to trace out any path P which lies in a complex plane that is cut along the real axis from z = 1/(2 + α) 2 to z = +∞. It is found that the modular functions k 2 ± (α, z) map the path P into two associated paths which do not cross the straight line formed by the real interval [1, ∞).…”

“…Next Schwarzian transformation theory is used to reduce both these second-order differential equations to the standard Gauss hypergeometric differential equation. From this result it is deduced that wG(α, w) = 2 where {l, m, n} denotes a set of integers, w = u + iv is a complex variable in the (u, v) plane, and α is a real parameter in the interval −∞ < α < ∞, plays an important role in many lattice statistical problems which involve the simple cubic lattice with partially anisotropic nearest-neighbour interactions [1][2][3][4][5][6][7][8]. We can assume, without loss of generality, that l ≥ m ≥ 0 and n ≥ 0.…”