# what is the magnetic quadrupole operator?

To find magnetic or electrical moments in quantum theory we must calculate the expectation value of an appropriate operator. the dipoles operator are similar and is easy to find but the magnetic quarupole moment, I couldn't find.

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The magnetic quadrupole moment tensor is given by $$m_{ij}=\left\langle \frac{2}{3}\left(\mathbf{r}\times\mathbf{J}\right)_i r_j \right\rangle,$$ in analogy with the magnetic dipole moment vector $$m_i=\left\langle \frac{1}{2}\left(\mathbf{r}\times\mathbf{J}\right)_i \right\rangle.$$ The magnetic field at a point $\mathbf{R}$ is then, up to quadrupole order, $$B_i(\mathbf{R})=\frac{\mu_0}{4\pi}\left[ m_j \frac{R_iR_j-3R^2\delta_{ij}}{R^5} +\frac{3m_{jk}}{2R^7}\left(5R_iR_jR_k -R^2(R_i\delta_{jk}+R_j\delta_{ki}+R_k\delta_{ij}) \right) \right].$$ This link has what looks like a good exposition of the subject (including the formulae above) but for the life of me I can't figure out what book it's a part of. The list of references looks pretty comprehensive too.

There's a few red flags here, though, which I can't quite see through. I am unsure whether a symmetric + traceless magnetic quadrupole moment is possible, and the expression above definitely has commutation issues in the quantum case, which you should be careful with. On the other hand, the electron spin fortunately doesn't contribute to anything above dipole order, for symmetry reasons, so you're dealing only with the position coordinates and orbital angular momentum where needed.

The current $\mathbf{J}$ is proportional to the particle's velocity and this may or may not be proportional to the canonical momentum (in which case, for example, $\mathbf{m}\propto\mathbf{L}$), depending on what gauge you're working in. Be careful!

Finally, magnetic quadrupoles are really close to the edge of our measurement capabilities, and certainly nothing above the E3/M2 terms has been observed experimentally as far as I'm aware. Electric octopole transitions are very strongly forbidden, with lifetimes of seconds or even years (!), but they have been used to great effect in precision metrology experiments on trapped Ytterbium ions (see e.g. this answer). In general, though, unambiguous detection of effects at this order requires transitions which are forbidden at both the E1 and the E2/M1 orders, and those usually turn out to be hard to address.

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It's not true that higher multipolarities have never been measured. They've been seen in yrast traps, which are states that can only decay by the emission of high-multipolarity photons, because the only states lower in excitation energy have much lower spins. These states tend to be very long-lived. E.g., there's a 12+ state in 52Fe that decays by emission of an E4 photon. –  Ben Crowell May 19 at 3:27
@BenCrowell Thanks for the info. Have you got a reference? –  Emilio Pisanty May 19 at 12:53
That was just the first example of an yrast trap I found by googling on "yrast trap." –  Ben Crowell May 20 at 14:58