treams.lattice.dsumsw1d

treams.lattice.dsumsw1d(l, k, kpar, a, r, i) = <ufunc 'dsumsw1d'>

Direct summation of spherical functions on a 1d lattice

Computes

\[D_{l0}(k, k_\parallel, \Lambda_1, r) = \sum_{R \in \Lambda_1} h_l^{(1)}(k |r + R|) Y_{l0}(-\boldsymbol{\hat z} (r + R)) \mathrm e^{\mathrm i k_\parallel R}\]

using the Ewald summation.

directly for one expansion value i. Sum i from 0 to a large value to obtain an approximation of the sum value. The lattice is along the z-axis. Therefore only \(m = 0\) contributes.

Parameters:

• l (integer) – Degree \(l \geq 0\)

• k (float or complex) – Wave number

• kpar (float) – Tangential wave vector component

• a (float) – Lattice pitch

• r (float) – In-line shift

• i (integer) – Expansion value

Returns:

complex