Spherical multipole moments are the coefficients in a
series expansion
In mathematics, a series expansion is an expansion of a function into a series, or infinite sum. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division) ...
of a
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
that varies inversely with the distance R to a source, ''i.e.'', as 1/''R''. Examples of such potentials are the
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
, the
magnetic potential Magnetic potential may refer to:
* Magnetic vector potential, the vector whose curl is equal to the magnetic B field
* Magnetic scalar potential
Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism analogous to electr ...
and the
gravitational potential
In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric po ...
.
For clarity, we illustrate the expansion for a
point charge
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
, then generalize to an arbitrary charge density
. Through this article, the primed coordinates such as
refer to the position of charge(s), whereas the unprimed coordinates such as
refer to the point at which the potential is being observed. We also use
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
throughout, e.g., the vector
has coordinates
where
is the radius,
is the
colatitude
In a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between a right angle and the latitude. Here Southern latitudes are defined to be negative, and as a result the colatitude is a non- ...
and
is the
azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematicall ...
al angle.
Spherical multipole moments of a point charge
The
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
due to a point charge located at
is given by
where
is the distance between the charge position and the observation point and
is the angle between the vectors
and
. If the radius
of the observation point is greater than the radius
of the charge, we may factor out 1/''r'' and expand the square root in powers of
using
Legendre polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
This is exactly analogous to the
axial multipole expansion.
We may express
in terms of the coordinates of the observation point and charge position using the
spherical law of cosines In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.
Given a unit sphere, a "sphe ...
(Fig. 2)
Substituting this equation for
into the
Legendre polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
and factoring the primed and unprimed coordinates yields the important formula known as the
spherical harmonic addition theorem
where the
functions are the
spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a ...
. Substitution of this formula into the potential yields
which can be written as
where the multipole moments are defined
As with
axial multipole moments
Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the ''z''-axis. However, the axial multipole expansion can also be applied ...
, we may also consider the case when the radius
of the observation point is less than the radius
of the charge. In that case, we may write
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
and
are defined to be the lesser and greater, respectively, of the two radii
and
; the potential of a point charge then takes the form, which is sometimes referred to as
Laplace expansion
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an matrix as a weighted sum of minors, which are the determinants of some submatrices of . Spec ...
Exterior spherical multipole moments
It is straightforward to generalize these formulae by replacing the point charge
with an infinitesimal charge element
and integrating. The functional form of the expansion is the same. In the exterior case, where
, the result is:
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 ''Y''
ℓm, 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. In the interior case, where
, the result is:
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
be centered on the origin and lie entirely within the second charge distribution
. The interaction energy between any two static charge distributions is defined by
The potential
of the first (central) charge distribution may be expanded in exterior multipoles
where
represents the
exterior multipole moment of the first charge distribution. Substitution of this expansion yields the formula
Since the integral equals the complex conjugate of the interior multipole moments
of the second (peripheral) charge distribution, the energy formula reduces to the simple form
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
azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematicall ...
al angle
). By carrying out the
integrations that define
and
, it can be shown the multipole moments are all zero except when
. Using the mathematical identity
the exterior multipole expansion becomes
where the axially symmetric multipole moments are defined
In the limit that the charge is confined to the
-axis, we recover the exterior
axial multipole moments
Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the ''z''-axis. However, the axial multipole expansion can also be applied ...
.
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
-axis, we recover the interior
axial multipole moments
Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the ''z''-axis. However, the axial multipole expansion can also be applied ...
.
See also
*
Solid harmonics In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions \mathbb^3 \to \mathbb. There are two kinds: the ''regular solid harmonics'' R^m_\ell(\mathbf), wh ...
*
Laplace expansion
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an matrix as a weighted sum of minors, which are the determinants of some submatrices of . Spec ...
*
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, \R^3. Similarly ...
*
Legendre polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
*
Axial multipole moments
Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the ''z''-axis. However, the axial multipole expansion can also be applied ...
*
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. Such potentials arise in the electric potential of long line charges, and the anal ...
Electromagnetism
Potential theory
Moment (physics)