treams.lattice.dsumsw3d

treams.lattice.dsumsw3d(l, m, k, kpar, a, r, i) = <ufunc 'dsumsw3d'>

Direct summation of spherical functions on a 3d lattice

Computes

\[D_{lm}(k, \boldsymbol k_\parallel, \Lambda_3, \boldsymbol r) = \sum_{\boldsymbol R \in \Lambda_3} h_l^{(1)}(k |\boldsymbol r + \boldsymbol R|) Y_{lm}(-\boldsymbol r - \boldsymbol R) \mathrm e^{\mathrm i \boldsymbol k_\parallel \boldsymbol R}\]

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.

Parameters:

• l (integer) – Degree \(l \geq 0\)

• m (integer) – Order \(|m| \leq l\)

• k (float or complex) – Wave number

• kpar (float, (3,)-array) – Wave vector

• a (float, (3,3)-array) – Lattice vectors

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

• i (integer) – Expansion value

Returns:

complex