Spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, i.e., as 1/R. Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential.
Contents
- Spherical multipole moments of a point charge
- General spherical multipole moments
- Note
- Interior spherical multipole moments
- Interaction energies of spherical multipoles
- Special case of axial symmetry
- References
For clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density
Spherical multipole moments of a point charge
The electric potential due to a point charge located at
where
This is exactly analogous to the axial multipole expansion.
We may express
Substituting this equation for
where the
which can be written as
where the multipole moments are defined
As with axial multipole moments, we may also consider the case when the radius
which can be written as
where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics
The two cases can be subsumed in a single expression if
General spherical multipole moments
It is straightforward to generalize these formulae by replacing the point charge
where the general multipole moments are defined
Note
The potential Φ(r) is real, so that the complex conjugate of the expansion is equally valid. Taking of the complex conjugate leads to a definition of the multipole moment which is proportional to Ylm, not to its complex conjugate. This is a common convention, see molecular multipoles for more on this.
Interior spherical multipole moments
Similarly, the interior multipole expansion has the same functional form
with the interior multipole moments defined as
Interaction energies of spherical multipoles
A simple formula for the interaction energy of two non-overlapping but concentric charge distributions can be derived. Let the first charge distribution
The potential
where
Since the integral equals the complex conjugate of the interior multipole moments
For example, this formula may be used to determine the electrostatic interaction energies of the atomic nucleus with its surrounding electronic orbitals. Conversely, given the interaction energies and the interior multipole moments of the electronic orbitals, one may find the exterior multipole moments (and, hence, shape) of the atomic nucleus.
Special case of axial symmetry
The spherical multipole expansion takes a simple form if the charge distribution is axially symmetric (i.e., is independent of the azimuthal angle
the exterior multipole expansion becomes
where the axially symmetric multipole moments are defined
In the limit that the charge is confined to the
Similarly the interior multipole expansion becomes
where the axially symmetric interior multipole moments are defined
In the limit that the charge is confined to the