Spherical Multipole Moments - Interaction Energies of Spherical Multipoles

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 \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r} \rho_{2}(\mathbf{r}) \Phi_{1}(\mathbf{r})

The potential of the first (central) charge distribution may be expanded in exterior multipoles

\Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{l=0}^{\infty} \sum_{m=-l}^{l} Q_{1lm} \left( \frac{1}{r^{l+1}} \right) \sqrt{\frac{4\pi}{2l+1}} Y_{lm}(\theta, \phi)

where represents the exterior multipole moment of the first charge distribution. Substitution of this expansion yields the formula

U = \frac{1}{4\pi\varepsilon} \sum_{l=0}^{\infty} \sum_{m=-l}^{l} Q_{1lm} \int d\mathbf{r} \ \rho_{2}(\mathbf{r}) \left( \frac{1}{r^{l+1}} \right) \sqrt{\frac{4\pi}{2l+1}} Y_{lm}(\theta, \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 = \frac{1}{4\pi\varepsilon} \sum_{l=0}^{\infty} \sum_{m=-l}^{l} Q_{1lm} I_{2lm}^{*}

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.

Read more about this topic:  Spherical Multipole Moments

Famous quotes containing the words interaction and/or energies:

    “UG [universal grammar] may be regarded as a characterization of the genetically determined language faculty. One may think of this faculty as a ‘language acquisition device,’ an innate component of the human mind that yields a particular language through interaction with present experience, a device that converts experience into a system of knowledge attained: knowledge of one or another language.”
    —Noam Chomsky (b. 1928)

    “Railroad iron is a magician’s rod, in its power to evoke the sleeping energies of land and water.”
    —Ralph Waldo Emerson (1803–1882)