# Parameters of neutrino interactions in targets?

## Summary:

I am trying to understand which parameters are fundamental for the interaction between neutrinos in a given target.

## Main Question or Discussion Point

The interaction of neutrinos should be proportional to the amount of material that it goes through. If i want to calculate the number of interactions that i could expect when a neutrino beam go through different materials, the first thing that comes to mind is the nuclei of the targets, as the target gets denser, more interactions could be expected, but, this relation is lineal?

I mean, i expect just dependence in the amount of mass, but a read some informal comments about how there is others factors as dependence of the atomic number.

Does anyone have information about that?

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mfb
Mentor
For plausible detector sizes the neutrino beam doesn't get weaker within the range of the detector, so the event rate will be proportional to the amount of sensitive material in the detector.
To find the event rate you need the interaction cross section for each type of atom in your detector. It depends on the type of neutrino and the neutrino energy, too, and there is no simple relationship.

jtbell
Mentor
as the target gets denser, more interactions could be expected, but, this relation is lineal?
By "denser", do you mean more targets (nuclei) per unit volume (larger "number density" of targets), or do you mean "heavier" targets (larger atomic number and/or atomic mass)?

Are you acquainted with the concept of "cross section?" In general, for a "thin" target, such that you can neglect the decrease in intensity of the beam as it progresses through the target, the number of interactions is $$dN = -Nn \sigma \, dx$$ where $N$ is the number of incoming particles (neutrinos in your case), $n$ is the number of targets (nuclei) per unit volume, $dx$ is the thickness of the collection of targets, and $\sigma$ is the "cross section" of a single target, which encapsulates the details of the interaction of a single incoming particle with a single target.

For a "thick" collection of targets, you have to integrate to account for the decrease in intensity of the beam inside the target, which leads to an "exponential absorption law" $$N = N_0 e^{-n \sigma x}$$ Here I use "target" to mean a single nucleus. In this case the effects of "heavier" targets (nuciel) would be encoded in the cross section $\sigma$.

However, when I was an grad student in experimental neutrino physics about 40 years ago, IIRC we thought in terms of the cross section for neutrino interactions on individual protons and neutrons. That is, we thought in terms of a collection of protons and neutrons, not a collection of nuclei. And we would have separate $n$ and $\sigma$ for protons and neutrons.

mfb
Mentor
However, when I was an grad student in experimental neutrino physics about 40 years ago, IIRC we thought in terms of the cross section for neutrino interactions on individual protons and neutrons. That is, we thought in terms of a collection of protons and neutrons, not a collection of nuclei.
That works well at high energies (with accelerators as sources, for example) but doesn't work well at low energies (e.g. neutrinos from the Sun or nuclear reactors) where you have to consider the state of the nucleus.
For neutrinos every detector is a thin detector.