# Single neutron scattering events and conservation of angular momentum

My question is very simple. I have been reading papers about excitations in a spin-gapped system. In these experiments, the energy transfer $\Delta E$ of a scattered neutron is measured and related to $n$ triplet excitations. In the simplest case, the neutron excites one singlet (a pair of coupled electrons) to one triplet ($n=1$). There are higher excitations (measured from $\Delta E$ of one neutron!) where $n=2$, $n=3$, etc.

My question is if the system begins in a ground state of all singlets (as I believe is assumed), how is it that angular momentum is conserved? I can see how $S_z$ can be conserved - let the incoming neutron have $S_z = 1/2$, and the outgoing neutron have $S_z = -1/2$ while the triplet has $S_z = 1$ so $S_z$ initial = $S_z$ final, but total angular momentum in this case (in the context of measuring $J_{tot}^2$ for example) is clearly not conserved.

Furthermore, for $n=2$, $3$, etc, it especially unclear to me how total angular momentum could be conserved, and it seems that $S_z$ would need to be 0 for all but one of the excited triplets.

I think it's pretty clear I am thinking about this in the wrong way, but I am having trouble finding my mistake.

Thanks!

the energy transfer ΔE of a scattered neutron is measured and related to n triplet excitations

Note the crucial word:"scattered". In scattering angular momenta are involved, not only energy and momentum but also angular momentum is conserved in the interaction. Though the link is about elastic scattering, the analysis of the incoming particle as a plane wave holds for all scatterings:

Assuming that the incident plane wave is in the z direction and hence :

$$\phi_{inc}(\vec{r}) = \exp (i k r \cos \theta) \tag{1}$$

we may express it in terms of a superposition of angular momentum eigenstates , each with a definite angular momentum number l :

$$e^{i\vec{k}\cdot\vec{r}} = e ^{ikr \cos\theta} = \sum_{l=0}^\infty i^l (2l + 1) j_l(kr)P_l(\cos\theta) \tag{2}$$

So all angular momenta are available to the interaction and it depends on the final state which initial state angular momentum was the one involved in the interaction, so that angular momentum is conserved.

• On that note, can you comment on how the neutron spin state before and after? It seems that it could be in either state before, and either state after, if the linear combination of angular momentum states in the plane wave can contribute the angular momentum to the system. Thanks much! – RonManatee Apr 24 '17 at 19:29