MP 11.1: Talk
Thursday, March 24, 2022, 16:35–16:55, MP-H6
Relation between the Cartesian multipole expansion and the spherical harmonic expansion — •Nils Walter Schween and Brian Reville — Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, Heidelberg 69117, Germany
The multipole expansion, which is, for example, used to approximate an
electrostatic potential (or a gravitational potential), has two equivalent
forms. First, it is a Taylor expansion, i.e.
| 4πє0 φ(r) = | ∫ | |
dr′3
= | | | | | | ri1 ⋯ ril
Qi1 ⋯ il |
|
| r2l + 1 |
| .
|
Note the Einstein summation convention. Secondly, it is a spherical harmonic
expansion, i.e.
We show that the relation between these two expansions can be formalised as a
series of basis transformations of spaces of homogeneous polynomials of
increasing degree l. These basis transformations allow us to derive an
algorithm to express the components of the multipole tensors, i.e.
Qi1 ⋯ il, as linear combinations of the spherical
multipole moments qlm for an arbitrary degree l. Since the
spherical multipole moments are
| qlm := | ∫ | Ylm*(θ, ϕ) r′l ρ(r)
dr′3 ,
|
this opens the opportunity to compute Qi1 ⋯ il solving
the above integral instead of performing the derivatives and integrations needed
to compute the multipole tensor components directly.