Pre-Quantum Electrodynamics
Higher Momentsems.ca.me.h
The next terms in the expansion are obtained similarly: the {\bf quadrupole term} is \[ V_{\mbox{\tiny quad}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^3} \int_{\cal V} d\tau_s r_s^2 P_2 (\hat{\bf r} \cdot \hat{\bf r}_s) \rho({\bf r}_s) = \sum_{a,b = x,y,z} \frac{r_a r_b}{r^5} \int_{\cal V} d\tau_s \frac{1}{2} (3 r_{s,a} r_{s,b} - r_s^2 \delta_{a,b}) \rho ({\bf r}_s) \] and can be rewritten as
\[ V_{\mbox{\tiny quad}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} \]
in terms of the {\bf quadrupole moment}
\[ Q_{ab} = \int_{\cal V} d\tau_s (3 r_{s,a} r_{s,b} - r_s^2 \delta_{a,b}) \rho ({\bf r}_s). \]
This is a symmetric rank \(2\) tensor, \(Q_{ab} = Q_{ba}\). Moreover, it is traceless, \(\sum_a Q_{aa} = 0\). It therefore has \(5\) independent components.
Our expansion for the potential thus looks like
\[ V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \left( \frac{Q}{r} + \sum_a \frac{r_a}{r^3} p_a + \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} + ... \right) \]
This can be carried further if we feel like it, with the {\bf octopole}, {\bf hexadecapole}, {\bf triacontadipole}, {\bf hexecontatetrapole}, … terms (see info box).
\paragraph{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).
{\bf A consistent nomenclature for the multipole expansion?}\\\\ You all know the terms {\bf monopole}, {\bf dipole} and {\bf quadrupole}, and perhaps also the less frequently used {\bf octupole}, {\bf hexadecapole} [16], {\bf triacontadipole} (or {\bf dotriacontapole}) [32] and {\bf tetrahexacontapole} (or {\bf hexacontatetrapole}) [64]. Physicists are clearly insufficiently educated in the humanities: these terms sound very fancy and their choice seems to make sense, but it doesn't. {\bf Mono-} is derived from the Greek {\it monos} ('alone'); {\bf di-} is derived from the Greek {\it dis} ('twice'); {\bf quadru-} is a fake Latin prefix ({\it quadri-} would be genuine) meaning 'something to do with the number 4', and {\bf octu-} is another (fake) Latin prefix (both Greek and Latin have {\it octo/okto} for 8, but Greek makes compounds with {\it octo-} or {\it octa-}, never {\it octu-}).
A more consistent nomenclature would be to go either fully Greek {\it or} Latin,
yielding:
\begin{tabular}{r|ll}
& Greek-inspired & Latin-inspired
\hline
1 & monopole & unipole
2 & dipole & duopole
4 & tetrapole & quadrupole
8 & octopole & octopole
\end{tabular}
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:
\begin{tabular}{r|ll}
16 & hexadecapole & sexdecapole
32 & triacontadipole & trigentiduopole
64 & hexecontatetrapole & sexagintiquadrupole
128 & hecatonikosioctopole & viginticentioctopole
256 & diacosipentecontahexapole & ducentiquinquagintisexapole
\end{tabular}
Created: 2022-02-08 Tue 06:55