Multipole expansion - Citizendia

A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial 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. In Mathematics, truncation is the term for limiting the number of digits right of the Decimal point, by discarding the least significant ones The function being expanded may be complex in general. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Multipole expansions are very frequently used 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 electromagnetic field is a physical field produced by electrically charged objects. A gravitational field is a model used within Physics to explain how gravity exists in the universe 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. For example, the initial term — called the zero-th, or monopole, moment — is a constant, independent of angle. The following term — the first, or dipole, moment — varies once from positive to negative around the sphere. In physics there are two kinds of dipoles ( Hellènic: di(s- = two- and pòla = pivot hinge An electric dipole is a Higher-order terms (like the quadrupole and octupole) vary more quickly with angles. 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

Most commonly, the series is written as a sum of spherical harmonics. In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of Thus, we might write a function $f (θ,φ)$ as the sum

$f(\theta,\phi) = \sum_{l=0}^\infty\, \sum_{m=-l}^{l}\, C^m_l\, Y^m_l(\theta,\phi).$

Here, $Y^m_l(\theta,\phi)$ are the standard spherical harmonics, and $C^m_l$ are constant coefficients which depend on the function. The term $C^0_0$ represents the monopole; $C^{-1}_1,C^0_1,C^1_1$ represent the dipole; and so on. Equivalently, the series is also frequently written^[2] as

$f(\theta,\phi) = C + C_i n^i + C_{ij}n^i n^j + C_{ijk}n^i n^j n^k + C_{ijkl}n^i n^j n^k n^l + \ldots$

Here, each $n i$ represents a unit vector in the direction given by the angles $θ$ and $φ$ , and indices are implictly summed. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational 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

$C_l^m = (-1)^m C^{m\ast}_l\ .$

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. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually ^[3] This finds use in multipole expansions of the vector potential in electromagnetism, or the metric perturbation in the description of gravitational waves. In Vector calculus, a vector potential is a Vector field whose curl is a given vector field In Physics, a gravitational wave is a Fluctuation in the Curvature of Spacetime which propagates as a wave, traveling outward from

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 $r$ . In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms For example, to describe the electromagnetic potential, $V$ , from a source in a small region near the origin, the coefficients may be written as:

$V(r,\theta,\phi) = \sum_{l=0}^\infty\, \sum_{m=-l}^{l}\, C^m_l(r)\, Y^m_l(\theta,\phi)= \sum_{j=1}^\infty\, \sum_{l=0}^\infty\, \sum_{m=-l}^{l}\, \frac{D^m_{l,j}}{r^j}\, Y^m_l(\theta,\phi) .$

1 Applications of multipole expansions
2 Multipole expansion of a potential outside an electrostatic charge distribution
- 2.1 Expansion in Cartesian coordinates
- 2.2 Spherical form
3 Interaction of two non-overlapping charge distributions
4 Examples of multipole expansions
5 General mathematical properties
6 See also
7 References
8 External links

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 gravitational field is a model used within Physics to explain how gravity exists in the universe Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. 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 Fast Multipole Method (FMM is a mathematical technique that was developed to speed up the calculation of long-ranged forces in the N-body problem. 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. Ewald summation is a method for computing the interaction energies of periodic systems (e 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 in the Cartesian coordinates x, y and z, while the second is in terms of spherical harmonics which depend on spherical polar coordinates. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of 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). Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. 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 of an arbitrary function v(r-R) around the origin r = 0 is,

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

with

$v_\alpha(\mathbf{R}) \equiv\left( \frac{\partial v(\mathbf{r}-\mathbf{R}) }{\partial r_\alpha}\right)_{\mathbf{r}= \mathbf0}\quad\hbox{and}\quad v_{\alpha\beta}(\mathbf{R}) \equiv\left( \frac{\partial^2 v(\mathbf{r}-\mathbf{R}) }{\partial r_{\alpha}\partial r_{\beta}}\right)_{\mathbf{r}= \mathbf0}$

If v(r-R) satisfies the Laplace equation

$\left(\nabla^2 v(\mathbf{r}- \mathbf{R})\right)_{\mathbf{r}=\mathbf0} = \sum_{\alpha=x,y,z} v_{\alpha\alpha}(\mathbf{R}) = 0$

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

$\sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} r_\alpha r_\beta v_{\alpha\beta}(\mathbf{R}) = \frac{1}{3} \sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} (3r_\alpha r_\beta - \delta_{\alpha\beta} r^2) v_{\alpha\beta}(\mathbf{R}),$

where δ_αβ is the Kronecker delta and r² ≡ |r|². In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two Removing the trace is common, because it takes the rotational invariant r² out of the second rank tensor. Consider now the following form of v(r-R),

$v(\mathbf{r}- \mathbf{R}) \equiv \frac{1}{|\mathbf{r}- \mathbf{R}|},$

then by direct differentiation it follows that

$v(\mathbf{R}) = \frac{1}{R},\quad v_\alpha(\mathbf{R})= -\frac{R_\alpha}{R^3},\quad \hbox{and}\quad v_{\alpha\beta}(\mathbf{R}) = \frac{3R_\alpha R_\beta- \delta_{\alpha\beta}R^2}{R^5}.$

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

$q_\mathrm{tot} \equiv \sum_{i=1}^N q_i, \quad P_\alpha \equiv\sum_{i=1}^N q_i r_{i\alpha}, \quad \hbox{and}\quad Q_{\alpha\beta} \equiv \sum_{i=1}^N q_i (3r_{i\alpha} r_{i\beta} - \delta_{\alpha\beta} r_i^2)$

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,

$4\pi\varepsilon_0 V(\mathbf{R}) \equiv \sum_{i=1}^N q_i v(\mathbf{r}_i-\mathbf{R})$

$= \frac{q_\mathrm{tot}}{R} + \frac{1}{R^3}\sum_{\alpha=x,y,z} P_\alpha R_\alpha + \frac{1}{6 R^5}\sum_{\alpha,\beta=x,y,z} Q_{\alpha\beta} (3R_\alpha R_\beta - \delta_{\alpha\beta} R^2) +\cdots$

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

$\sum_{\alpha} v_{\alpha\alpha} = 0 \quad \hbox{and}\quad \sum_{\alpha} Q_{\alpha\alpha} = 0.$

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

$V(\mathbf{R}) = \frac{1}{4\pi \varepsilon_0 R^3} (\mathbf{P}\cdot\mathbf{R})$ ,

the electric dipolar potential field. In physics there are two kinds of dipoles ( Hellènic: di(s- = two- and pòla = pivot hinge An electric dipole is a

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,

$V(\mathbf{R}) \equiv \sum_{i=1}^N \frac{q_i}{4\pi \varepsilon_0 |\mathbf{r}_i - \mathbf{R}|} =\frac{1}{4\pi \varepsilon_0} \sum_{\ell=0}^\infty \sum_{m=-\ell}^{\ell} (-1)^m I^{-m}_\ell(\mathbf{R}) \sum_{i=1}^N q_i R^{m}_\ell(\mathbf{r}_i),$

where $I^{-m}_{\ell}(\mathbf{R})$ is an irregular solid harmonics (which is a spherical harmonics function depending on the polar angles of R and divided by R^l+1) and $R^m_{\ell}(\mathbf{r})$ is a regular solid harmonics (a spherical harmonics times r^l). See also Laplace expansion of determinant. In physics the Laplace expansion of a 1/ r - type potential is applied to expand Newton's In Physics and Mathematics, the solid harmonics are solutions of the Laplace equation in Spherical polar coordinates. In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of We define the spherical multipole moment of the charge distribution as follows

$Q^m_\ell \equiv \sum_{i=1}^N q_i R^{m}_\ell(\mathbf{r}_i),\qquad -\ell \le m \le \ell.$

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

A spherical harmonics depends on the unit vector $\hat{R}$ . In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of (A unit vector is determined by two spherical polar angles and conversely. ) Thus, by definition, the irregular solid harmonics can be written as

$I^m_{\ell}(\mathbf{R}) \equiv \sqrt{\frac{4\pi}{2\ell+1}} \frac{Y^m_{\ell}(\hat{R})}{R^{\ell+1}}$

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

$V(\mathbf{R}) =\frac{1}{4\pi \varepsilon_0} \sum_{\ell=0}^\infty \sum_{m=-\ell}^{\ell} (-1)^m I^{-m}_\ell(\mathbf{R}) Q^m_\ell$

$= \frac{1}{4\pi\varepsilon_0} \sum_{\ell=0}^\infty \left[\frac{4\pi}{2\ell+1}\right]^{1/2}\; \frac{1}{R^{\ell+1}}\; \sum_{m=-\ell}^{\ell} (-1)^m Y^{-m}_\ell(\hat{R}) Q^m_\ell, \qquad R > r_{\mathrm{max}}.$

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 appear as coefficients in the 1/R expansion of the potential. Spherical multipole moments are the coefficients in a Series expansion of a Potential that varies inversely with the distance R to a source i

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. In Physics and Mathematics, the solid harmonics are solutions of the Laplace equation in Spherical polar coordinates. The l = 0 term becomes

$V_{\ell=0}(\mathbf{R}) = \frac{q_\mathrm{tot}}{4\pi \varepsilon_0 R}\qquad\hbox{with}\quad q_\mathrm{tot}\equiv\sum_{i=1}^N q_i.$

This is in fact Coulomb's law again. ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form For the l = 1 term we introduce

$\mathbf{R} = (R_x, R_y, R_z),\quad \mathbf{P} = (P_x, P_y, P_z)\quad \hbox{with}\quad P_\alpha \equiv \sum_{i=1}^N q_i r_{i\alpha}, \quad \alpha=x,y,z.$

Then

$V_{\ell=1}(\mathbf{R}) = \frac{1}{4\pi \varepsilon_0 R^3} (R_x P_x +R_y P_y + R_z P_z) = \frac{\mathbf{R}\cdot\mathbf{P} }{4\pi \varepsilon_0 R^3} = \frac{\hat{R}\cdot\mathbf{P} }{4\pi \varepsilon_0 R^2}.$

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

$Q_{z^2} \equiv \sum_{i=1}^N q_i\; \frac{1}{2}(3z_i - r_i^2),$

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 molecules, and recall that a molecule by definition consists of electrons (negative point charges) and nuclei (positive point charges). In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by The total electrostatic interaction energy U_AB between the two distributions is

$U_{AB} = \sum_{i\in A} \sum_{j\in B} \frac{q_i q_j}{4\pi\varepsilon_0 r_{ij}}.$

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

$\mathbf{R}_{AB}+\mathbf{r}_{Bj}+\mathbf{r}_{ji}+\mathbf{r}_{iA} = 0 \quad\Longrightarrow\quad \mathbf{r}_{ij} = \mathbf{R}_{AB}-\mathbf{r}_{Ai}+\mathbf{r}_{Bj} .$

We assume that the two distributions do not overlap:

$|\mathbf{R}_{AB}| > |\mathbf{r}_{Bj}-\mathbf{r}_{Ai}| \quad\hbox{for all}\quad i,j.$

Under this condition we may apply the Laplace expansion in the following form

$\frac{1}{|\mathbf{r}_{j}-\mathbf{r}_i|} = \frac{1}{|\mathbf{R}_{AB} - (\mathbf{r}_{Ai}- \mathbf{r}_{Bj})| } = \sum_{L=0}^\infty \sum_{M=-L}^L \, (-1)^M I_L^{-M}(\mathbf{R}_{AB})\; R^M_{L}( \mathbf{r}_{Ai}-\mathbf{r}_{Bj}),$

where $I^M_L$ and $R^M_L$ are irregular and regular solid harmonics, respectively. See also Laplace expansion of determinant. In physics the Laplace expansion of a 1/ r - type potential is applied to expand Newton's In Physics and Mathematics, the solid harmonics are solutions of the Laplace equation in Spherical polar coordinates. The translation of the regular solid harmonic gives a finite expansion,

$R^M_L(\mathbf{r}_{Ai}-\mathbf{r}_{Bj}) = \sum_{\ell_A=0}^L (-1)^{L-\ell_A} \binom{2L}{2\ell_A}^{1/2}$

$\times \sum_{m_A=-\ell_A}^{\ell_A} R^{m_A}_{\ell_A}(\mathbf{r}_{Ai}) R^{M-m_A}_{L-\ell_A}(\mathbf{r}_{Bj})\; \langle \ell_A, m_A; L-\ell_A, M-m_A| L M \rangle,$

where the quantity between pointed brackets is a Clebsch-Gordan coefficient. In Physics and Mathematics, the solid harmonics are solutions of the Laplace equation in Spherical polar coordinates. In Physics, the Clebsch-Gordan coefficients are sets of numbers that arise in Angular momentum coupling under the laws of Quantum mechanics. Further we used

$R^{m}_{\ell}(-\mathbf{r}) = (-1)^{\ell} R^{m}_{\ell}(\mathbf{r}) .$

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

$U_{AB} = \frac{1}{4\pi\varepsilon_0} \sum_{\ell_A=0}^\infty \sum_{\ell_B=0}^\infty (-1)^{\ell_B} \binom{2\ell_A+2\ell_B}{2\ell_A}^{1/2} \,$

$\times \sum_{m_A=-\ell_A}^{\ell_A} \sum_{m_B=-\ell_B}^{\ell_B}(-1)^{m_A+m_B} I_{\ell_A+\ell_B}^{-m_A-m_B}(\mathbf{R}_{AB})\; Q^{m_A}_{\ell_A} Q^{m_B}_{\ell_B}\; \langle \ell_A, m_A; \ell_B, m_B| \ell_A+\ell_B, m_A+m_B \rangle.$

This is the multipole expansion of the interaction energy of two non-overlapping charge distributions which are a distance R_AB apart. Spherical multipole moments are the coefficients in a Series expansion of a Potential that varies inversely with the distance R to a source i Since

$I_{\ell_A+\ell_B}^{-(m_A+m_B)}(\mathbf{R}_{AB}) \equiv \left[\frac{4\pi}{2\ell_A+2\ell_B+1}\right]^{1/2}\; \frac{Y^{-(m_A+m_B)}_{\ell_A+\ell_B}(\widehat{\mathbf{R}}_{AB})}{R^{\ell_A+\ell_B+1}_{AB}}$

this expansion is manifestly in powers of 1/R_AB. The function Y^m_l is a normalized spherical harmonic. In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of

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 Mathematical study of potentials is known as Potential theory; it is the study of Harmonic functions on Manifolds This mathematical In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point 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 The most common expansions include:

Axial multipole moments of a $\frac{1}{R}$ potential;
Spherical multipole moments of a $\frac{1}{R}$ potential; and
Cylindrical multipole moments of a $\ln \ R^{ }$ potential

Examples of $\frac{1}{R}$ potentials include the electric potential, the magnetic potential and the gravitational potential of point sources. Axial multipole moments are a Series expansion of the Electric potential of acharge distribution localized close to the origin along one Cartesian axis Spherical multipole moments are the coefficients in a Series expansion of a Potential that varies inversely with the distance R to a source i Cylindrical multipole moments are the coefficients in a Series expansion of a Potential that varies logarithmically with the distance to a source i At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field The magnetic potential provides a mathematical way to define a Magnetic field in Classical electromagnetism. Potential energy can be thought of as Energy stored within a physical system An example of a $\ln \ R^{ }$ potential is the electric potential of an infinite line charge. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field

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. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, two Vectors are orthogonal if they are Perpendicular, i One uses the technique of separation of variables to extract the corresponding solutions for the radial dependencies. In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an

References

^ Edmonds, A. Multipole moments are the Coefficients of a Series expansion of a Potential due to continuous or discrete sources (e Axial multipole moments are a Series expansion of the Electric potential of acharge distribution localized close to the origin along one Cartesian axis Spherical multipole moments are the coefficients in a Series expansion of a Potential that varies inversely with the distance R to a source i Cylindrical multipole moments are the coefficients in a Series expansion of a Potential that varies logarithmically with the distance to a source i Quadrupole magnets consist of groups of four Magnets laid out so that in the Multipole expansion of the field the dipole terms cancel and where the lowest significant R. . Angular Momentum in Quantum Mechanics. Princeton University Press.
^ Thompson, William J. . Angular Momentum. John Wiley & Sons, Inc. .
^ Thorne, Kip S. (April 1980). "Multipole Expansions of Gravitational Radiation". Reviews of Modern Physics 52 (2): 299. doi:10.1103/RevModPhys.52.299. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.

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