Multipole expansion

A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, $\mathbb {R} ^{3}$ . Similarly to Taylor series, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real- or complex-valued and is defined either on $\mathbb {R} ^{3}$ , or less often on $\mathbb {R} ^{n}$ for some other $n$ .

Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space.^[1]

The multipole expansion is expressed as a sum of terms with progressively finer angular features (moments). The first (the zeroth-order) term is called the monopole moment, the second (the first-order) term is called the dipole moment, the third (the second-order) the quadrupole moment, the fourth (third-order) term is called the octupole moment, and so on. Given the limitation of Greek numeral prefixes, terms of higher order are conventionally named by adding "-pole" to the number of poles—e.g., 32-pole (rarely dotriacontapole or triacontadipole) and 64-pole (rarely tetrahexacontapole or hexacontatetrapole).^[2]^[3]^[4] A multipole moment usually involves powers (or inverse powers) of the distance to the origin, as well as some 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 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.

Expansion in spherical harmonics

Most commonly, the series is written as a sum of spherical harmonics. Thus, we might write a function $f(\theta ,\varphi )$ as the sum

f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }\,C_{\ell }^{m}\,Y_{\ell }^{m}(\theta ,\varphi )

where

Y_{\ell }^{m}(\theta ,\varphi )

are the standard spherical harmonics, and

C_{\ell }^{m}

are constant coefficients which depend on the function. The term

C_{0}^{0}

represents the monopole;

C_{1}^{-1},C_{1}^{0},C_{1}^{1}

represent the dipole; and so on. Equivalently, the series is also frequently written^[5] as

f(\theta ,\varphi )=C+C_{i}n^{i}+C_{ij}n^{i}n^{j}+C_{ijk}n^{i}n^{j}n^{k}+C_{ijk\ell }n^{i}n^{j}n^{k}n^{\ell }+\cdots

where the

n^{i}

represent the components of a unit vector in the direction given by the angles

\theta

and

\varphi

, and indices are implicitly summed. Here, the term

C

is the monopole;

C_{i}

is a set of three numbers representing the dipole; and so on.

In the above expansions, the coefficients may be real or complex. If the function being expressed as a multipole expansion is real, however, the coefficients must satisfy certain properties. In the spherical harmonic expansion, we must have

C_{\ell }^{-m}=(-1)^{m}C_{\ell }^{m\ast }\,.

In the multi-vector expansion, each coefficient must be real:

C=C^{\ast };\ C_{i}=C_{i}^{\ast };\ C_{ij}=C_{ij}^{\ast };\ C_{ijk}=C_{ijk}^{\ast };\ \ldots

While expansions of scalar functions are by far the most common application of multipole expansions, they may also be generalized to describe tensors of arbitrary rank.^[6] This finds use in multipole expansions of the vector potential in electromagnetism, or the metric perturbation in the description of gravitational waves.

For describing functions of three dimensions, away from the coordinate origin, the coefficients of the multipole expansion can be written as functions of the distance to the origin, $r$ —most frequently, as a Laurent series in powers of $r$ . For example, to describe the electromagnetic potential, $V$ , from a source in a small region near the origin, the coefficients may be written as:

V(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }C_{\ell }^{m}(r)\,Y_{\ell }^{m}(\theta ,\varphi )=\sum _{j=1}^{\infty }\,\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }{\frac {D_{\ell ,j}^{m}}{r^{j}}}\,Y_{\ell }^{m}(\theta ,\varphi ).

Applications

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 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. the system has large density fluctuations.

Multipole expansion of a potential outside an electrostatic charge distribution

Consider a discrete charge distribution consisting of $N$ point charges $q i$ with position vectors $r i$ . We assume the charges to be clustered around the origin, so that for all i: $r i < r max$ , where $r max$ has some finite value. The potential $V (R)$ , due to the charge distribution, at a point $R$ outside the charge distribution, i.e., $| R | > r max$ , can be expanded in powers of $1/ R$ . Two ways of making this expansion can be found in the literature: The first is a Taylor series in the Cartesian coordinates $x$ , $y$ , and $z$ , while the second is in terms of spherical harmonics which depend on spherical polar coordinates. The Cartesian approach has the advantage that no prior knowledge of Legendre functions, spherical harmonics, etc., is required. Its disadvantage is that the derivations are fairly cumbersome (in fact a large part of it is the implicit rederivation of the Legendre expansion of $1 / | r - R |$ , which was done once and for all by Legendre in the 1780s). Also it is difficult to give a closed expression for a general term of the multipole expansion—usually only the first few terms are given followed by an ellipsis.

Expansion in Cartesian coordinates

Assume $v (r) = v (- r)$ for convenience. The Taylor expansion of $v (r - R)$ around the origin $r = 0$ can be written as

v(\mathbf {r} -\mathbf {R} )=v(\mathbf {R} )-\sum _{\alpha =x,y,z}r_{\alpha }v_{\alpha }(\mathbf {R} )+{\frac {1}{2}}\sum _{\alpha =x,y,z}\sum _{\beta =x,y,z}r_{\alpha }r_{\beta }v_{\alpha \beta }(\mathbf {R} )-\cdots +\cdots

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