treams.lattice.dsumsw1d_shift

treams.lattice.dsumsw1d_shift(l, m, k, kpar, a, r, i) = <ufunc 'dsumsw1d_shift'>

Fast summation of spherical functions on a 1d lattice with out of lattice shifts

Computes

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

directly for one expansion value i. Sum i from 0 to a large value to obtain an approximation of the sum value. In a the lattice vectors are given as rows. The lattice is along the z axis.

Parameters:

• l (integer) – Degree \(l \geq 0\)

• m (integer) – Order \(|m| \leq l\)

• k (float or complex) – Wave number

• kpar (float) – Tangential wave vector component

• a (float) – Lattice pitch

• r (float, (3,)-array) – Shift vector

• i (integer) – Expansion value

Returns:

complex