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## Homework Statement

I am trying to understand how to obtain the estimated detection rate of a WIMP of mass [itex]100~GeV[/itex] into a Germanium detector of [itex]1kg[/itex] and detection efficiency of [itex]P_{eff}=70 \% [/itex].

## Homework Equations

If you have at hand that the cross section is given [itex] \sigma = \mu_R G_F^2 [/itex] (I think something is wrong with my units here, maybe I should have [itex]\mu_R^2[/itex]) where [itex]\mu_R = \frac{m_N m_{wimp}}{m_{wimp}+m_N}[/itex] the reduced mass, and [itex]m_N[/itex] the mass of the detection medium nucleus.

## The Attempt at a Solution

If I use that the probability of interaction in a width [itex]dx[/itex] in my material with [itex]N[/itex] Germanium atoms is:

[itex]dW = \sigma N dx [/itex]

I have that the probability of detection is [itex]D_{etection-rate}= P_{eff} \times W = P_{eff} \times \sigma N L[/itex]

Where [itex]L[/itex] is the path taken within the detector for the particle to interact.

In 1kg of Germanium I have [itex]N = \frac{1~kg}{m_N} [/itex] atoms.

So:

[itex]D_{etection-rate}= P_{eff} \times \frac{1~kg}{m_N} \times L \times \mu_R G_F^2 [/itex]

My problem is that I don't understand how to get rid of this "L"...