# Expansion in solid spherical harmonics on the lattice

I'm interested in calculating scattering processes (e.g. Coulomb scattering of an electron beam by a single ion) in the context of lattice quantum field theory, and wonder if there is something like the expansion of a plane wave in spherical harmonics on the lattice? (I mean in discrete space modeled by a three dimensional, finite, cubic lattice.)

So I am looking for an orthonormal basis for complex valued functions on a finite lattice, where the angular and radial variables would be (approximately?) separated, as this is the case for the solid harmonics.

Thank you for your help!

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Do you think of some expansion as in real space Wannier functions? But unfortunately these functions can not be used explicitely as I remember... In addition you will need the interaction Hamiltonian to describe the scattering in the correct shape. Please be more specific what is scattered on the lattice sites!? – strpeter Mar 6 '13 at 23:12
To be quite specific: I would like to express, on the position basis, the Coulomb scattering process calculated in chapter 10 of Quantum Ethics. For the time being, it is expressed on the momentum basis, and I would like to evaluate the angular distribution using a development on the position basis; in continuous space-time, I would develop the result on the spherical harmonics, since they separate the angular and radial variables, but is there something similar on the lattice, where no spherical symmetry is granted? – Sébastien Fauvel Mar 7 '13 at 7:49
Hi user21566, I'm afraid such a thing won't exist: spherical harmonics in $d$ dimensions arise directly because of the $SO(d)$ symmetry around a point, which is broken to a discrete group on the lattice. But good luck finding an alternative! – Vibert Mar 7 '13 at 9:21
I urge you to look at this paper: arxiv.org/pdf/hep-th/9303048v2.pdf in which the author uses spherical harmonics, but discretizes only the radial coordinate. Apparently, this lattice regularization method gives finite answers in d<=3. – Srivatsan Balakrishnan Dec 7 '14 at 16:49
Thank you Srivatsan for this hint! It looks like Mark Srednicki would have needed something similar in his paper, but as you can see, he hasn't written down this lattice version of the spherical harmonics, he's only using them - without even knowing if they exist, I guess. So the question remains open... – Sébastien Fauvel Dec 8 '14 at 9:18

Spherical harmonics are good basis functions for isotropic continuum systems, for example isolated atoms. For lattices, the plane wave is already the appropriate basis function to be used in expansions. You can of course expand the plane wave into spherical harmonics, but I am not aware of any reason this would be practical in the evaluation of scattering processes.

The key geometrical ingredient in lattice calculations is symmetry. Some general characteristics of the scattering response can be immediately deduced from the point group of the unit cell. The full calculation of scattering cross sections involves coherent sums over lattice points, usually carried out in Cartesian coordinates. These give rise to lattice structure factors, which govern the general characteristics of the scattering spectrum. In non-resonant, quasi-elastic scattering, this is essentially the entire process necessary to obtain the scattering response.

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Thank you for your answer! Maybe I should have been more specific: I'm not interested in scattering by a lattice structure, but on a lattice, i.e. in space modeled itself as a three dimensional, finite, cubic lattice. For instance, by the Coulomb scattering of an electron beam by an ion target. In continuous space-time, the expansion of the incoming plane wave in spherical harmonics helps calculating the expansion of the outgoing wave, which gives direct clues about the angular scattering. But what about the angular distribution on the lattice? Is there any similar "tool" there? – Sébastien Fauvel Mar 4 '13 at 8:56
Ah ok, this explanation -- along with your comment and the edited title -- make things clearer. I guess my answer would then be similar to @Vibert's comment to your question. Perhaps he can be tempted into reformulating the comment as an answer? :) – delete000 Mar 8 '13 at 7:51
Oh come on, don't give up that quickly ;) I remember the solid harmonics are solutions of Schrödinger's hydrogen atom eigenvalue equation, when one tries to diagonalize the squared orbital momentum and its projection on an axis too... What about doing the same in discrete space? – Sébastien Fauvel Mar 8 '13 at 8:08