Spherical multipole moments explained

In physics, spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance to a source, i.e., as  Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential.

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 throughout, e.g., the vector has coordinates where is the radius, is the colatitude and is the azimuthal angle.

Spherical multipole moments of a point charge

The electric potential due to a point charge located at

is given by$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash frac\; =\backslash frac\; \backslash frac.$$where }\ \left|\mathbf - \mathbf \right| 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$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\; \backslash left(\backslash frac\; \backslash right)^\backslash ell\; P\_\backslash ell(\backslash cos\; \backslash gamma)$$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 (Fig. 2)$$\backslash cos\; \backslash gamma\; =\; \backslash cos\; \backslash theta\; \backslash cos\; \backslash theta\text{'}\; +\; \backslash sin\; \backslash theta\; \backslash sin\; \backslash theta\text{'}\; \backslash cos(\backslash phi\; -\; \backslash phi\text{'})$$

Substituting this equation for

into the Legendre polynomials and factoring the primed and unprimed coordinates yields the important formula known as the spherical harmonic addition theorem$$P\_\backslash ell(\backslash cos\; \backslash gamma)\; =\; \backslash frac\; \backslash sum\_^\backslash ell\; Y\_(\backslash theta,\; \backslash phi)\; Y\_^(\backslash theta\text{'},\; \backslash phi\text{'})$$where the functions are the spherical harmonics. Substitution of this formula into the potential yields$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\backslash left(\backslash frac\; \backslash right)^\backslash ell\backslash left(\backslash frac\; \backslash right)\backslash sum\_^\backslash ell\; Y\_(\backslash theta,\; \backslash phi)\; Y\_^(\backslash theta\text{'},\; \backslash phi\text{'})$$

which can be written as$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\; \backslash sum\_^\backslash ell\; \backslash left(\backslash frac\; \backslash right)\backslash sqrt\; Y\_(\backslash theta,\; \backslash phi)$$where the multipole moments are defined$$Q\_\; \backslash \; \backslash stackrel\backslash \; q\; \backslash left(r\text{'}\; \backslash right)^\backslash ell\; \backslash sqrt\; Y\_^(\backslash theta\text{'},\; \backslash phi\text{'}).$$

As with axial multipole moments, 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$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\backslash left(\backslash frac\; \backslash right)^\backslash ell\backslash left(\backslash frac\; \backslash right)\backslash sum\_^\backslash ell\; Y\_(\backslash theta,\; \backslash phi)\; Y\_^(\backslash theta\text{'},\; \backslash phi\text{'})$$which can be written as$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\; \backslash sum\_^\backslash ell\; I\_\; r^\backslash ell\backslash sqrt\; Y\_(\backslash theta,\; \backslash phi)$$where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics$$I\_\; \backslash \; \backslash stackrel\backslash \; \backslash frac\; \backslash sqrt\; Y\_^(\backslash theta\text{'},\; \backslash phi\text{'})$$

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$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\backslash frac\backslash left(\backslash frac\; \backslash right)\backslash sum\_^\backslash ell\; Y\_(\backslash theta,\; \backslash phi)\; Y\_^(\backslash theta\text{'},\; \backslash phi\text{'})$$

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:$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\; \backslash sum\_^\backslash ell\; \backslash left(\backslash frac\; \backslash right)\backslash sqrt\; Y\_(\backslash theta,\; \backslash phi)\backslash ,,$$where the general multipole moments are defined $$Q\_\; \backslash \; \backslash stackrel\backslash \; \backslash int\; d\backslash mathbf\text{'}\; \backslash rho(\backslash mathbf\text{'})\; \backslash left(r\text{'}\; \backslash right)^\backslash ell\; \backslash sqrt\; Y\_^(\backslash theta\text{'},\; \backslash phi\text{'}).$$

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:$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\; \backslash sum\_^\backslash ell\; I\_\; r^\backslash ell\; \backslash sqrt\; Y\_(\backslash theta,\; \backslash phi),$$with the interior multipole moments defined as $$I\_\; \backslash \; \backslash stackrel\backslash \; \backslash int\; d\backslash mathbf\text{'}\; \backslash frac\backslash sqrt\; Y\_^(\backslash theta\text{'},\; \backslash phi\text{'}).$$

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$$U\; \backslash \; \backslash stackrel\backslash \; \backslash int\; d\backslash mathbf\; \backslash rho\_2(\backslash mathbf)\; \backslash Phi\_1(\backslash mathbf).$$

The potential

of the first (central) charge distribution may be expanded in exterior multipoles$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\; \backslash sum\_^\backslash ell\; Q\_\backslash left(\backslash frac\; \backslash right)\backslash sqrt\; Y\_(\backslash theta,\; \backslash phi)$$where represents the exterior multipole moment of the first charge distribution. Substitution of this expansion yields the formula$$U\; =\; \backslash frac\; \backslash sum\_^\; \backslash sum\_^\backslash ell\; Q\_\backslash int\; d\backslash mathbf\; \backslash \; \backslash rho\_2(\backslash mathbf)\backslash left(\backslash frac\; \backslash right)\; \backslash sqrt\; Y\_(\backslash theta,\; \backslash phi)$$

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$$U\; =\; \backslash frac\; \backslash sum\_^\; \backslash sum\_^\backslash ell\; Q\_\; I\_^$$

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 azimuthal 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$$P\_\backslash ell(\backslash cos\; \backslash theta)\; \backslash \; \backslash stackrel\backslash \; \backslash sqrt\; Y\_(\backslash theta,\; \backslash phi)$$the exterior multipole expansion becomes$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\backslash left(\backslash frac\; \backslash right)P\_\backslash ell(\backslash cos\; \backslash theta)$$where the axially symmetric multipole moments are defined$$Q\_\backslash ell\; \backslash \; \backslash stackrel\backslash \; \backslash int\; d\backslash mathbf\text{'}\; \backslash rho(\backslash mathbf\text{'})\; \backslash left(r\text{'}\; \backslash right)^\backslash ell\; P\_\backslash ell(\backslash cos\; \backslash theta\text{'})$$In the limit that the charge is confined to the -axis, we recover the exterior axial multipole moments.

Similarly the interior multipole expansion becomes$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\; I\_\backslash ell\; r^\backslash ell\; P\_\backslash ell(\backslash cos\; \backslash theta)$$where the axially symmetric interior multipole moments are defined$$I\_\backslash ell\; \backslash \; \backslash stackrel\backslash \; \backslash int\; d\backslash mathbf\text{'}\; \backslash frac\; P\_\backslash ell(\backslash cos\; \backslash theta\text{'})$$In the limit that the charge is confined to the

-axis, we recover the interior axial multipole moments.

See also

• Solid harmonics
• Laplace expansion
• Multipole expansion
• Legendre polynomials
• Axial multipole moments
• Cylindrical multipole moments

Notes and References

1. Book: Jackson, John David . Classical electrodynamics . 1999 . Wiley . 978-0-471-30932-1 . New York.