# Confused about the anapole (toroidal) moment

I am a bit confused about the existence of the anapole moment. As far as I understand, in order to fully describe a charge-current distribution, one needs, beside the normal electric and magnetic moments, toroidal moments. In the static regime, the lowest order, toroidal dipole moment, is called anapole moment. I can understand so far. Now, if I understand this well, the anapole moment appears when doing the expansion of the vector potential in term of $$1/R$$ i.e. a term of this expansion is associated with the anapole moment. However, in Griffiths and Jackson (these are the 2 books I used), when they derive the multipole exapnsion of the vector potential, they obtain the magnetic dipole, quadrupole and so on, but it doesn't seem to be any term left behind. Actually, as far as I understand, the anapole moment should appear at the same order as the magnetic quadrupole. So, assuming their derivation is general (and I assume it is as they took a random current distribution), where is the anapole moment term hidden? And where are all the other higher order toroidal moments, too? A second thing I am confused about is the fact that the anapole moment seem to produce parity violation in atoms. How can this be possible, as the anapole moment is a QED phenomena and QED doesn't violate parity. Is the weak interaction involved somehow? Thank you!

## 1 Answer

I'm not going to answer all your questions, but I'll try to clarify a few things. I leave mathematical details to the references cited below. Additionally, I'll talk about classical electrodynamics and not QED.

There is a little bit of confusion about the nomenclature here, although even some published articles are not always coherent between them either. There is no anapole term or moment or mode; the anapole resonance or state, occurs when the dipole (the one we are used to) and the toroidal dipole have identical amplitudes but opposite phases, hence canceling the total dipolar radiation ("ana" meaning "without" in Greek). It is thus an interference effect.

The confusing thing about the toroidal dipole is that it is, as you observed, omitted in most textbook derivations. The reason why is that those derivations are made in the static (or quasistatic) regime, where phase variations of the field across the structure (retardations) are neglected; this is also called long-wavelength limit. When one takes retardations into account and derives the Taylor expansion new terms appear, the most important one being the toroidal moment. And it does not stop to the toroidal moment, there is an endless amount of higher order dipole moment in the expansions, although one could indeed expect them to be negligible in most cases.

If you are familiar with what is often called Mie theory (the theory of the field scattered by a sphere illuminated by a planewave), and the expansion of a scattered field onto vector spherical harmonics, the anapole can appear clearly in the $$a_1$$ coefficient (the so called dipolar coefficient) as a sharp dip into an otherwise broad resonance (see Fig. 1 in [1]). Plotting the $$a_1$$ coefficient in the complex frequency plane reveals multiple poles corresponding to the various terms mentioned above (the imaginary part of the frequency being related to the width of the resonance).

This very recent paper contains some interesting discussions; The figure 1 illustrates what I said about the dipolar Mie coefficient (called "c" in this paper):

A very good and comprehensive reference about all this toroidal stuff is the following book: "Toroidal Multipole Moments in Classical Electrodynamics", by Stefan Nanz. Inside should be the answer to the few questions of yours that I did not directly address. If you do not have access to this book, the same author has published several articles on the matter; I could only find those two on arXiv: https://arxiv.org/abs/1507.00752 and https://arxiv.org/abs/1507.00755