Multipole expansion - AbsoluteAstronomy.com

A multipole expansion is a mathematical series

Series (mathematics)

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

representing a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original function. The function being expanded may be complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

in general. Multipole expansions are very frequently used in the study of electromagnetic

Electromagnetic field

An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

and gravitational field

Gravitational field

The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

s, 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.

The multipole expansion is expressed as a sum of terms with progressively finer angular features. For example, the initial term — called the zero-th, or monopole, moment

Moment (mathematics)

In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

— is a constant, independent of angle. The following term — the first, or dipole

Dipole

In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

, moment — varies once from positive to negative around the sphere. Higher-order terms (like the quadrupole

Quadrupole

A quadrupole or quadrapole is one of a sequence of configurations of—for example—electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.-Mathematical...

and octupole) vary more quickly with angles.

Expansion in spherical harmonics

Most commonly, the series is written as a sum of spherical harmonics

Spherical harmonics

In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

. Thus, we might write a function

as the sum

Here,

are the standard spherical harmonics, and

are constant coefficients which depend on the function. The term

represents the monopole;

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

Here, each

represents a unit vector in the direction given by the angles

and

, and indices are implicitly summed. Here, the term

is the monopole;

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

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

While expansions of scalar

Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

functions are by far the most common application of multipole expansions, they may also be generalized to describe tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s of arbitrary rank. This finds use in multipole expansions of the vector potential

Vector potential

In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....

in electromagnetism, or the metric perturbation in the description of gravitational wave

Gravitational wave

In physics, gravitational waves are theoretical ripples in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as...

s.

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,

-- most frequently, as a Laurent series

Laurent series

In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...

in powers of

. For example, to describe the electromagnetic potential,

, from a source in a small region near the origin, the coefficients may be written as:

Applications of multipole expansions

Multipole expansions are widely used in problems involving gravitational field

Gravitational field

The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

s of systems of mass

Mass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

es, electric and magnetic field

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

s 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

Fast Multipole Method

The fast multipole method is a mathematical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem...

of Greengard

Leslie Greengard

Leslie F. Greengard is an American mathematician, doctor of medicine and computer scientist. He is co-inventor of the fast multipole method in 1987, recognised as one of the top-ten algorithms of the 20th century....

and Rokhlin

Vladimir Rokhlin (American scientist)

Vladimir Rokhlin is mathematician and professor of computer science and mathematics at the Yale University. He is co-inventor of the fast multipole method in 1987, recognised as one of the top-ten algorithms of the 20th century.-Short biography:Vladimir Rokhlin was born on August 4, 1952 in...

, 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

Ewald summation

Ewald summation, named after Paul Peter Ewald, is a method for computing the interaction energies of periodic systems , particularly electrostatic energies. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an...

, but is superior if the particles are clustered,
i.e., if 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

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

in the Cartesian coordinates x, y and z, while the second is in terms of spherical harmonics

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

Adrien-Marie Legendre

Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

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 some dots.

Expansion in Cartesian coordinates

The Taylor expansion

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

of an arbitrary function v(r-R) around the origin r = 0 is,

with

If v(r-R) satisfies the Laplace equation

then the expansion can be rewritten in terms of the components of a traceless Cartesian second rank tensor

Tensor

where δ_αβ is the Kronecker delta and r² ≡ |r|².
Removing the trace is common, because it takes the rotational invariant r² out of the second rank tensor.

Example

Consider now the following form of v(r-R),

then by direct differentiation it follows that

Define a monopole, dipole and (traceless) quadrupole by, respectively,

and we obtain finally the first few terms of the multipole expansion of the total potential, which is the sum of the Coulomb potentials of the separate charges,

This expansion of the potential of a discrete charge distribution is very similar to the one in real solid harmonics given below. The main difference is that the present one is in terms of linear dependent quantities, for

Note

If the charge distribution consists of two charges of opposite sign which are an infinitesimal distance d apart, so that d/R >> (d/R)², it is easily shown that the
only non-vanishing term in the expansion is

,
the electric dipolar potential field.

Spherical form

The potential V(R) at a point R outside the charge distribution, i.e., |R| > r_max, can be expanded by the Laplace expansion

Laplace expansion (potential)

In physics, the Laplace expansion of a 1/r - type potential is applied to expand Newton's gravitational potential or Coulomb's electrostatic potential. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the interelectronic repulsion.The Laplace...

where

is an irregular solid harmonic (defined below as a spherical harmonic

Spherical Harmonic

Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...

function divided by R^l+1) and

is a regular solid harmonic (a spherical harmonic times r^l). We define the spherical multipole moment of the charge distribution as follows

Note that a multipole moment is solely determined by the charge distribution (the positions and magnitudes of the N charges).

A spherical harmonic

Spherical Harmonic

depends on the unit vector

. (A unit vector is determined by two spherical polar angles.) Thus, by definition, the irregular solid harmonics can be written as

so that the multipole expansion of the field V(R) at the point R outside the charge distribution is given by

This expansion is completely general in that it gives a closed form for all terms, not just for the first few. It shows that the spherical multipole moments

Spherical multipole moments

Spherical multipole moments are the coefficients in a series expansionof a potential that varies inversely with the distance R to a source, i.e., as 1/R...

appear as coefficients in the 1/R expansion of the potential.

It is of interest to consider the first few terms in real form, which are the only terms commonly found in undergraduate textbooks.
Since the summand of the m summation is invariant under a unitary transformation
of both factors simultaneously and since transformation of complex spherical harmonics to real form is by a unitary transformation, we can simply substitute real irregular solid harmonics and real multipole moments. The l = 0 term becomes

This is in fact Coulomb's law

Coulomb's law

Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...

again.
For the l = 1 term we introduce

Then

This term is identical to the one found in Cartesian form.

In order to write the l=2 term, we have to introduce short-hand notations for the five real components of the quadrupole moment and the real spherical harmonics. Notations of the type

can be found in the literature. Clearly the real notation becomes awkward very soon, exhibiting the usefulness of the complex notation.

Interaction of two non-overlapping charge distributions

Consider two sets of point charges, one set {q_i } clustered around a point A and one set {q_j } clustered around a point B. Think for example of two molecule

Molecule

A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

s, and recall that a molecule by definition consists of electrons (negative point charges) and nuclei (positive point charges).
The total electrostatic interaction energy U_AB between the two distributions is

This energy can be expanded in a power series in the inverse distance of A and B.
This expansion is known as the multipole expansion of U_AB.

In order to derive this multipole expansion, we write r_XY = r_Y-r_X, which is a vector pointing from X towards Y. Note that

We assume that the two distributions do not overlap:

Under this condition we may apply the Laplace expansion

Laplace expansion (potential)

in the following form

where

and

are irregular and regular solid harmonics, respectively.
The translation of the regular solid harmonic gives a finite expansion,

where the quantity between pointed brackets is a Clebsch-Gordan coefficient. Further we used

Use of the definition of spherical multipoles Q^m_l and covering of the summation ranges in a somewhat different order (which is only allowed for an infinite range of L) gives finally

This is the multipole expansion of the interaction energy of two non-overlapping charge distributions which are a distance R_AB apart. Since

this expansion is manifestly in powers of 1/R_AB. The function Y^m_l is a normalized spherical harmonic

Spherical Harmonic

Examples of multipole expansions

There are many types of multipole moments, since there are many types of potential

Potential

*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...

s
and many ways of approximating a potential by a series expansion

Series expansion

In mathematics, a series expansion is a method for calculating a function that cannot be expressed by just elementary operators . The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function...

, depending on the
coordinates and the symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

of the charge distribution. The most common expansions
include:

Axial multipole moments
Axial multipole moments
Axial multipole moments are a series expansionof the electric potential of acharge distribution localized close tothe origin along oneCartesian axis,denoted here as the z-axis...

of a potential;
Spherical multipole moments
Spherical multipole moments
Spherical multipole moments are the coefficients in a series expansionof a potential that varies inversely with the distance R to a source, i.e., as 1/R...

of a potential; and
Cylindrical multipole moments
Cylindrical multipole moments
Cylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as \ln \ R...

of a potential

Examples of

potentials include the electric potential

Electric potential

In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

,
the magnetic potential

Magnetic potential

The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...

and the gravitational potential of point sources. An example
of a

potential is the electric potential

Electric potential

In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

of an infinite line charge.

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

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

of the rotational symmetry group

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

, which leads to spherical harmonics and related sets of orthogonal functions. One uses the technique of separation of variables

Separation of variables

In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....

to extract the corresponding solutions for the radial dependencies.

Expansion in spherical harmonics

Applications of multipole expansions

Multipole expansion of a potential outside an electrostatic charge distribution

Expansion in Cartesian coordinates

Spherical form

Interaction of two non-overlapping charge distributions

Examples of multipole expansions

General mathematical properties

See also