Pre-Quantum Electrodynamics
Higher Momentsems.ca.me.h
- PM 10.2
The next terms in the expansion are obtained similarly: the quadrupole term is \[ \phi_q ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^3} \int_{\cal V} d\tau_s r_s^2 P_2 (\hat{\bf r} \cdot \hat{\bf r}_s) \rho({\bf r}_s) = \frac{1}{4\pi \varepsilon_0} \sum_{a,b = x,y,z} \frac{r_a r_b}{r^5} \int_{\cal V} d\tau_s \frac{1}{2} (3 r_{s,a} r_{s,b} - r_s^2 \delta_{a,b}) \rho ({\bf r}_s) \] and can be rewritten as
\[ \phi_q ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} \tag{p_quad}\label{p_quad} \]
in terms of the quadrupole moment
\[ Q_{ab} = \int_{\cal V} d\tau_s (3 r_{s,a} r_{s,b} - r_s^2 \delta_{a,b}) \rho ({\bf r}_s). \tag{quadmom}\label{quadmom} \]
This is a symmetric, traceless rank \(2\) tensor: \(Q_{ab} = Q_{ba}\) and \(\sum_a Q_{aa} = 0\). It therefore has \(5\) independent components.
Our expansion for the potential, with terms up to quadrupole ones, thus looks like
\[ \phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \left( \frac{Q}{r} + \sum_a \frac{r_a}{r^3} p_a + \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} + ... \right) \tag{p_mdq}\label{p_mdq} \]
This can be carried further if we feel like it, with the octopole, hexadecapole, triacontadipole, hexecontatetrapole, … terms (see info box).
Important property: the leading nonvanishing multipole moment is independent of the chosen location for the origin of the coordinate system (see Jackson Prob. 4.4).
A consistent nomenclature for the multipole expansion?
You all know the terms monopole, dipole and quadrupole, and perhaps also the less frequently used octupole, hexadecapole [16], triacontadipole (or dotriacontapole) [32] and tetrahexacontapole (or hexacontatetrapole) [64].
Well I'm afraid physicists are clearly insufficiently educated in the humanities, and have badly screwed up with this nomeclature. Although these terms sound very fancy and their choice seems to make sense, it doesn't.
Mono- is derived from the Greek monos ('alone'); di- is derived from the Greek dis ('twice'); quadru- is a fake Latin prefix (quadri- would be genuine) meaning 'something to do with the number 4', and octu- is another (fake) Latin prefix (both Greek and Latin have octo/okto for 8, but Greek makes compounds with octo- or octa-, never octu-).
A more consistent nomenclature would be to go either fully Greek or Latin, yielding:
| Greek-inspired | Latin-inspired | |
|---|---|---|
| 1 | monopole | unipole |
| 2 | dipole | duopole |
| 4 | tetrapole | quadrupole |
| 8 | octopole | octopole |
Irrespective of whether you have a predilection for the Greek or Latin version, you can go wild and ask how this could generalize. The way to do this is not uniquely defined; here is a set of possibilities for more terms than you might ever (hopefully) need:
| 16 | hexadecapole | sexdecapole |
| 32 | triacontadipole | trigentiduopole |
| 64 | hexecontatetrapole | sexagintiquadrupole |
| 128 | hecatonikosioctopole | viginticentioctopole |
| 256 | diacosipentecontahexapole | ducentiquinquagintisexapole |
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Created: 2024-02-27 Tue 10:31