in my previous intuitive understanding, when some particle scatters with a bound state electron and leads to a continuous final state with recoil energy much larger than the initial binding energy, the process should be similar to elastic scattering, since the momentum transfer $q$ is large and one can take the initial state as plane-wave.

However, in some literature on dark matter direct detection based on ionization, the calculation seems not to recover the elastic scattering limit. In these cases, one needs to calculate the ionization form factor

$\sim\langle f |e^{i \vec{q} \cdot \vec{x}}| i \rangle $

with $i$ denoting initial bound electron with discrete energy level, $f$ denoting final state continuum and $e^{i \vec{q} \cdot \vec{x}}$ being the momentum transfer. When the initial state is plane-wave, this leads to momentum conservation $\delta (\vec{p}_i + \vec{q} - \vec{p}_f)$. For bound states like the electrons around the XENON, the atomic factor is related to the spatial overlapping between $\langle f|$ and $e^{i \vec{q} \cdot \vec{x}}| i \rangle$ , which seems to decrease with the recoil energy. This is very surprising to us at first. And then we found the figure 7 of https://arxiv.org/abs/1904.07127, which claims that when the recoil energy is 2 keV, the form factor is dominated by $n=3$ shell, instead of $n=4$ or $n=5$ with more binding electrons. This seems to also hint that the high energy limit of ionization deviates from the elastic scattering.

This is really something confusing to me. Is there are some more intuitive understanding to explain why the high energy limit of ionization deviates from the elastic scattering or some previous calculations/paper?


1 Answer 1


This is just a comment:

Elastic scattering of two particles means that there is no change in energy and only the angles are important in the center of mass system , whether in classical mechanics or quantum interactions.

It is not clear what is the elastic scattering in your question a+b-->a+b, as an ionized atom is not the same as a neutral one. The hint that the limit might not be the elastic limit comes from your quote:

the form factor is dominated

Form factors are used where the crossection (in the real meaning of effective size) of interaction may vary due to the structure of the target. Here is the atomic physics one . The word itself describes that the form/effective-size for the crossection changes depending on the energy of the interaction.

That people are successful in describing low energy interactions, 2 keV is low energy, does not mean that the formula would have to fit high energies, even at the limit. If it does not fit at the limit means that the low energy interaction is important enough to differ a lot from elastic scattering, and that is what dominates the behavior of the formula for low energy.


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