# Selection rules for atomic emission: why doesn't the photon carry orbital angular momentum

The standard treatment of light-matter interaction is often accompanied by a discussion of selection rules for electronic transitions via a dipole coupling $\hat r \cdot \vec E$ (see for example Griffith's discussion of time-dependent perturbation theory). This discussion shows, via symmetry arguments or explicit computations, that non-vanishing matrix elements exist only for states separated by $\Delta m = \pm 1,0$. This in turn is said to be understandable in terms of the photon being a spin 1 particle.

This is all well and good, but I would expect that photons can also carry orbital angular momentum, not just spin. Therefore, I would expect many more transitions to be possible, where the extra angular momentum is stored in orbital degrees of freedom. Why is this not the case?

This is in fact what you get if you consider emission by a classical source $J_\mu$, in which context the fact that the photon has spin is accounted for by using vector spherical harmonics for the radiation, rather than the spherical harmonics of Laplace's equation. Those vector harmonics carry orbital, as well as spin, degrees of freedom, and the emitted waveform carries the same total angular momentum numbers as the source.

Photons can carry orbital angular momentum. See this wiki. The orbital angular momentum is related to the fact that the light field is not radially symmetric in a plane transverse to the dominant wavevector.

When we consider the interaction between light and an atom, for example, we often work in the dipole approximation. This is effectively an approximation which is to say the light field is entirely uniform over the volume of the atom. The light field being uniform over the volume of the atom means that we can approximate it as a plane wave. We can often make this approximation because an atom has a linear dimension of $\approx 1$ nm whereas the wavelength of light, and thus the spatial variation of it, has a linear dimension of $\approx 1$ $\mu$m. Thus the atom is much smaller than the spatial variation of the light field so it often sees an effectively uniform field.

That is to say, the spatial variation of an optical field due to non-zero orbital angular momentum of light is on too large of a scale for an atom to notice.

However, we can relax the dipole approximation and allow for fields with spatial variation across the volume of the atom. One way to think about this is as follows. Recall that in a given state the electron in an atom occupies some volume of space given by the atomic orbital. In the dipole approximation the electric field of the light field puts a uniform force on the electron cloud. This force can change the pattern of the electron cloud to move it into a different spatial configuration. However, since the force is uniform there are restrictions on which states can be achieved by such an electric field. These restrictions are exactly the "dipole selection rules".

However, if we relax the dipole approximation we can now allow for a field which varies spatially across the volume of the atom. This means that the electron cloud is pulled in different direction in different places throughout the volume of the atom. This means new states are achievable from a given initial state. That is, the selection rules are different. The first approximation between the dipole approximation is the quadrapole approximation and there are of course octopole and high pole approximations. There are also levels of approximation which consider the effect of magnetic fields on the motion of the electron.

In any case, to consider the effect of the orbital angular momentum of light on the electronic state of an atom it would be necessary to decompose the light in to higher order moments beyond the dipole approximation. Because you are beyond the dipole approximation there are now different selection rules which apply.

The reason you likely haven't heard about much of this is 1) it is isn't often dealt with and 2) it is rare that you must work outside of the dipole approximation. In fact, because of the size mismatch between the wavelength of light and an atom it would be difficult to transfer orbital angular momentum of light into electronic angular momentum. Perhaps it could be possible in a highly excited Rydberg state in which the electron cloud now has a dimension more comparable to the wavelength of light which might excite it.

• Gamma rays could have the right size-- – S. McGrew Jul 15 '18 at 0:52
• It is true that higher energy (shorter wavelength) photons could have the right size to drive such transitions, however, their energies would be too high to interact strongly with the electron. That is, they would drive it at the wrong frequencies. It is true that I have restricted my attention to the alkali atoms and the transitions I am familiar with. Those transitions are in the range of 200 nm to 800 nm or so. – jgerber Jul 15 '18 at 0:54
• This is intriguing. I looked up the transition energy of inner-shell electrons in uranium [hal.archives-ouvertes.fr/hal-00601993/document]: ~4500 ev or ~.27 nm. The size of an atom is 0.1 to 0.5 nm, so we're in the right ballpark. – S. McGrew Jul 15 '18 at 1:37