Multipole expansion
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A multipole expansion is a series expansion of a potential, usually in powers (or inverse powers) of the distance to the origin, as well as functions describing the angular dependence. In principle, a multipole expansion provides an exact description of the potential and generally converges under two conditions: (1) if the sources (e.g., charges) are localized close to the origin and the point at which the potential is observed is far from the origin; or (2) the reverse, i.e., if the sources (e.g., charges) are located far from the origin and the potential is observed close to the origin. In the first (more common) case, the coefficients of the series expansion are called exterior multipole moments or simply multipole moments whereas, in the second case, they are called interior multipole moments. The zeroth-order and first-order terms in the expansion are called the monopole and dipole moment, respectively, while the higher terms are denoted as quadrupole, octupole, etc. moments. In practice, the series expansion is usually truncated to the first non-zero term and evaluated at either very large distances (exterior multipole expansion) or very short distances (interior multipole expansion).
Although rarely done, the potential at a position 
within a charge distribution can be computed by combining interior and exterior multipoles. The potential at a given point P is the sum of two components: (1) the exterior multipole potentials of the charges closer to the origin than P; and (2) the interior multipole potentials of sources further from the origin than P.
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[edit] Applications of multipole expansions
Multipole expansions are widely used in problems involving gravitational fields of systems of masses, electric and magnetic fields of charge and current distributions, and the propagation of electromagnetic waves. A classic example is the calculation of the exterior multipole moments of atomic nuclei from their interaction energies with the interior multipoles of the electronic orbitals. The multipole moments of the nuclei report on the distribution of charges within the nucleus and, thus, on the shape of the nucleus. Truncation of the multipole expansion to its first non-zero term is often useful for theoretical calculations.
Multipole expansions are also useful in numerical simulations, and form the basis of the Fast Multipole Method [1] of Greengard and Rokhlin, a general technique for efficient computation of energies and forces in systems of interacting particles. The basic idea is to decompose the particles into groups; particles within a group interact normally (i.e., by the full potential), whereas the energies and forces between groups of particles are calculated from their multipole moments. The efficiency of the fast multipole method is generally similar to that of Ewald summation, but is superior if the particles are clustered, i.e., if the system has large density fluctuations.
[edit] Multipole expansion form
When determining the potential due to some charge distribution, we have that the total potential V at some point is the sum of the monopole, dipole, quadrupole, etc. potential terms. What we then have is:
V(r)=Vo+V1+V2+... =[(1/4πεor)QTotal] + [(r·p)/(4πεor3)] + [(1/8πεor5)ΣxixjQij] +...
Here, Vo is the monopole potential, V1 is the potential due to the dipole term, and so on. The term Qij is the Quadrupole Tensor and is calculated by:
Qij=∫(3xixj - r2δij)ρ(r)dτ
Here, ρ(r) is the volume charge density of the charge distribution. If instead of a volume charge there is only a surface or line charge density, then those would be used. §J.O.
[edit] Examples of multipole expansions
There are many types of multipole moments, since there are many types of potentials and many ways of approximating a potential by a series expansion, depending on the coordinates and the symmetry of the charge distribution. The most common expansions include:
- Axial multipole moments of a
potential; - Spherical multipole moments of a potential; and
- Cylindrical multipole moments of a
potential
Examples of potentials include the electric potential, the magnetic potential and the gravitational potential of point sources. An example of a potential is the electric potential of an infinite line charge.
[edit] General mathematical properties
Mathematically, multipole expansions are related to the underlying rotational symmetry of the physical laws and their associated differential equations. Even though the source terms (such as the masses, charges, or currents) may not be symmetrical, one can expand them in terms of irreducible representations of the rotational symmetry group, which leads to spherical harmonics and related sets of orthogonal functions. One uses the technique of separation of variables to extract the corresponding solutions for the radial dependencies.
