Multipole expansion

In physics, multipole expansion is a method for solving Poisson's equation in three spatial dimensions , in which the solution function is developed as a Laurent series . The expansion coefficients of this Laurent series are called multipole moments . It is mainly used in electrostatics and magnetostatics , but can be generalized to any other area of ​​physics where Poisson's equation occurs.

The motivation of multipole expansion is to consider the behavior of electrical potential and magnetic vector potential (or any other potential such as gravitational potential ) at a great distance from charges or currents. For this purpose it is assumed that these potential-inducing charges or currents are restricted to only a small area of ​​space, and that Green's function of the Laplace operator , which occurs in Poisson's equation, is developed as a Taylor series .

Basics

The Poisson equation can be broadly described as

${\displaystyle \Delta \phi ({\vec {r}})=-f({\vec {r}})}$

write, where the Laplace operator is a density and a potential (the minus is convention). The formal solution to this equation is: ${\displaystyle \Delta }$ ${\displaystyle f}$ ${\displaystyle \phi }$

${\displaystyle \phi ({\vec {r}})={\frac {1}{4\pi }}\int \mathrm {d} ^{3}{\vec {r}}'{\frac {f({\vec {r}}')}{|{\vec {r}}-{\vec {r}}'|}}}$

If localized in a volume , the fraction in a Taylor series can be expanded in um for locations that are far outside this volume : ${\displaystyle \rho ({\vec {r}})}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle r\gg r'}$ ${\displaystyle {\vec {r}}'}$ ${\displaystyle {\vec {r}}'=0}$

${\displaystyle {\frac {1}{\left|{\vec {r}}-{\vec {r}}'\right|}}=\sum _{n=0}^{\infty }{\frac {1}{n!}}\left({\vec {r}}'\cdot {\vec {\nabla }}'\right)^{n}\left.{\frac {1}{\left|{\vec {r}}-{\vec {r}}'\right|}}\right|_{{\vec {r}}'=0}}$

This means that the nabla operator only acts on the crossed out coordinates and not on . After the leads have been created, they are evaluated at that point . Forming gives: ${\displaystyle {\vec {\nabla }}'}$ ${\displaystyle {\vec {\nabla }}}$ ${\displaystyle {\vec {r}}'}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {r}}'=0}$

${\displaystyle {\frac {1}{\left|{\vec {r}}-{\vec {r}}'\right|}}=\sum _{n=0}^{\infty }{\frac {1}{n!}}\left(-{\vec {r}}'\cdot {\vec {\nabla }}\right)^{n}{\frac {1}{r}}}$

From dimensional considerations it follows that every term in the Taylor series in leads to a term in the main part of the Laurent series in . In other words, as the distance from the volume under consideration increases, the higher orders of the multipole moments become more and more negligible, since they decrease more and more. ${\displaystyle {\vec {r}}'}$ ${\displaystyle r^{-1}}$ ${\displaystyle r}$

The exact form of the expansion and the multipoles depend on the coordinate system in which they are viewed.

Cartesian multipole expansion

In the case of the Cartesian multipole expansion, the expansion is carried out in Cartesian coordinates . There is

${\displaystyle {\vec {r}}'\cdot {\vec {\nabla }}=r'_{i}\partial _{i}}$,

where Einstein's summation convention is used. Then for a -th order addend a -th order tensor , namely : ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \textstyle \prod _{k=1}^{n}\partial _{i_{k}}{\frac {1}{r}}}$

${\displaystyle {\begin{aligned}{\frac {1}{\left|{\vec {r}}-{\vec {r}}'\right|}}&={\frac {1}{r}}-r'_{i}\partial _{i}{\frac {1}{r}}+r'_{i}r'_{j}\partial _{i}\partial _{j}{\frac {1}{r}}+{\mathcal {O}}(r'^{3})\\&={\frac {1}{r}}+r'_{i}{\frac {r_{i}}{r^{3}}}+{\frac {1}{2}}r'_{i}r'_{j}{\frac {3r_{i}r_{j}-r^{2}\delta _{ij}}{r^{5}}}+{\mathcal {O}}(r'^{3})\end{aligned}}}$

The symbol represents the so-called Kronecker delta . ${\displaystyle \delta _{ij}}$

The formal solution of Poisson's equation can be represented using the identity as follows: ${\displaystyle \phi ({\vec {r}})}$ ${\displaystyle r'_{i}r'_{j}r^{2}\delta _{ij}=r'^{2}r_{i}r_{j}\delta _{ij}}$

${\displaystyle {\begin{aligned}\phi ({\vec {r}})&={\frac {1}{4\pi }}{\bigg [}{\frac {1}{r}}\underbrace {\int f({\vec {r}}')\,\mathrm {d} ^{3}{\vec {r}}'} _{\text{Monopol-}}+{\frac {r_{i}}{r^{3}}}\underbrace {\int \mathrm {d} ^{3}{\vec {r}}'\,r'_{i}f({\vec {r}}')} _{\text{Dipol-}}+{\frac {1}{2}}{\frac {r_{i}r_{j}}{r^{5}}}\underbrace {\int \mathrm {d} ^{3}{\vec {r}}'\,\left(3r'_{i}r'_{j}-r'^{2}\delta _{ij}\right)f({\vec {r}}')} _{\text{Quadrupolmoment}}+\dots {\bigg ]}\\&={\frac {1}{4\pi }}\left[{\frac {1}{r}}q+{\frac {r_{i}}{r^{3}}}p_{i}+{\frac {1}{2}}{\frac {r_{i}r_{j}}{r^{5}}}Q_{ij}+\dots \right]\end{aligned}}}$

Spherical multipole expansion

In the spherical multipole expansion, it is not developed in the individual coordinates, but in the distance. To do this, the term is rewritten in spherical coordinates . It is

${\displaystyle {\vec {r}}'\cdot {\vec {\nabla }}'=r'\partial _{r'}}$

and

${\displaystyle {\frac {1}{|{\vec {r}}-{\vec {r}}'|}}={\frac {1}{r}}{\frac {1}{\sqrt {1+{\frac {r'^{2}}{r^{2}}}-2{\frac {r'}{r}}\cos(\theta -\theta ')}}}}$.

Since this is the generating function of the Legendre polynomials , the expansion can thus be given as closed: ${\displaystyle P_{l}}$

${\displaystyle {\frac {1}{|{\vec {r}}-{\vec {r}}'|}}=\sum _{l=0}^{\infty }P_{l}(\cos(\theta -\theta ')){\frac {r'^{l}}{r^{l+1}}}={\frac {1}{r}}+\cos(\theta -\theta '){\frac {r'}{r^{2}}}+{\frac {1}{2}}(3\cos ^{2}(\theta -\theta ')-1){\frac {r'^{2}}{r^{3}}}+{\mathcal {O}}(r'^{3})}$

With the help of the addition theorem for spherical surface functions , the Legendre polynomial can be written as a sum over spherical surface functions and thus decoupled into and : ${\displaystyle \cos(\theta -\theta ')}$ ${\displaystyle Y_{lm}}$ ${\displaystyle \theta }$ ${\displaystyle \theta '}$

${\displaystyle P_{l}(\cos(\theta -\theta '))={\frac {4\pi }{2l+1}}\sum _{m=-l}^{l}Y_{lm}^{*}(\theta ',\varphi ')Y_{lm}(\theta ,\varphi )}$

Substituting it into the equation for leads to: ${\displaystyle \phi }$

${\displaystyle \phi ={\frac {1}{4\pi }}\sum _{l=0}^{\infty }\sum _{m=-l}^{l}{\sqrt {\frac {4\pi }{2l+1}}}Y_{lm}(\theta ,\varphi ){\frac {1}{r^{l+1}}}\int \mathrm {d} ^{3}{\vec {r}}'{\sqrt {\frac {4\pi }{2l+1}}}Y_{lm}^{*}(\theta ',\varphi ')f({\vec {r}}')r'^{l}}$

The spherical multipole moment is then defined as ${\displaystyle q_{lm}}$

${\displaystyle q_{lm}=\int \mathrm {d} ^{3}{\vec {r}}'{\sqrt {\frac {4\pi }{2l+1}}}Y_{lm}^{*}(\theta ',\varphi ')f({\vec {r}}')r'^{l}}$.

By comparing the coefficients you can see that the term corresponds to the monopole moment, the term to the dipole moment et cetera. ${\displaystyle l=0}$ ${\displaystyle l=1}$

conversion

The conversion between Cartesian and spherical multipole moments is carried out by expressing the spherical surface functions in Cartesian coordinates. For the monopoly moment one obtains

${\displaystyle q_{00}=q}$

and for the three dipole moments

${\displaystyle q_{10}=p_{3}\quad q_{1\pm 1}={\frac {\mp p_{1}+\mathrm {i} p_{2}}{\sqrt {2}}}}$.

The conversion is nontrivial for higher moments, since terms occur in the spherical multipole expansion, but the corresponding tensor has components. Since the number of degrees of freedom must be independent of the coordinate system, you can see that not all Cartesian multipole moments are independent of each other. Among other things, the quadrupole tensor is symmetrical and non- marking, which restricts the degrees of freedom. Since the number of spherical multipole moments increases only linearly and that of the Cartesian multipole moments exponentially, it is not useful to specify the Cartesian multipole moments for higher moments. ${\displaystyle 2l+1}$ ${\displaystyle 3^{l}}$

Applications

Electrostatics

In electrostatics , the Poisson equation for the potential can be derived from the first Maxwell equation . In the Coulomb calibration it is

${\displaystyle \Delta \phi =-{\frac {\rho }{\varepsilon _{0}}}}$

with the electrical potential , the (electrical) charge density and the electrical field constant . The first three moments of the electrostatic potential are the total charge , the electric dipole moment and the quadruple moments . ${\displaystyle \phi }$ ${\displaystyle \rho }$ ${\displaystyle \varepsilon _{0}}$ ${\displaystyle Q}$ ${\displaystyle {\vec {p}}}$ ${\displaystyle Q_{ij}}$

Magnetostatics

In magnetostatics , the Maxwell equations in Coulomb calibration lead to Poisson equations for the vector potential ${\displaystyle {\vec {A}}}$

${\displaystyle \Delta {\vec {A}}=-\mu _{0}{\vec {j}}}$

with the electric current density and the permeability of the vacuum . The magnetic monopole disappears, because in a spatially localized current distribution just as much flows in as out. The leading order term is therefore the magnetic dipole moment . To simplify the tensor structure in the dipole moment, the identity ${\displaystyle {\vec {j}}}$ ${\displaystyle \mu _{0}}$

${\displaystyle r_{i}\int \mathrm {d} ^{3}{\vec {r}}'\,r'_{i}j_{n}({\vec {r}}')=-{\frac {1}{2}}\varepsilon _{lkn}r_{l}\int \mathrm {d} ^{3}{\vec {r}}'\,\varepsilon _{ijk}r'_{i}j_{j}}$

be used. So that

${\displaystyle {\vec {A}}=\mu _{0}{\frac {{\vec {\mu }}\times r}{r^{3}}}+{\mathcal {O}}(r^{-3})}$

with the magnetic dipole moment

${\displaystyle {\vec {\mu }}={\frac {1}{2}}\int \mathrm {d} ^{3}{\vec {r}}'\,{\vec {r}}'\times {\vec {j}}({\vec {r}}')}$.

Gravity

In gravitation it turns out that no negative masses exist as charges. Nevertheless, formally gravitational multipoles can be defined. Starting with the Poisson equation from Newton's law of gravitation

${\displaystyle \Delta \Phi =-4\pi G\rho }$

with the gravitational constant and the mass density , the gravitational monopole is the total mass and the gravitational dipole is the center of mass . ${\displaystyle G}$ ${\displaystyle \rho }$ ${\displaystyle M}$ ${\displaystyle {\vec {r}}_{S}}$

literature

• T. Fließbach : Electrodynamics . Spectrum Academic Publishing House, ISBN 3-8274-2021-0 .
• JD Jackson : Classical Electrodynamics . de Gruyter Verlag, ISBN 3-11-018970-4 .