Elastic scattering of WIMPs off nuclei

[maximus123]

Hello,

My problem is as follows

Suppose we want to design a cryogenic calorimeter for detecting WIMPs, such as neutralinos
(\chi). One can show that if a \chi has an elastic collision with a nucleus of mass m_N in the calorimeter, the kinetic energy transferred to the nucleus is

\Delta E=\frac{m_Nm_{\chi}^2}{(m_N+m_{\chi})^2}v^2(1-\textrm{cos}\theta)
​

where v is the neutralino’s velocity in the lab frame and θ is the scattering angle in the c.m.
frame.

Show that to get the maximum energy transfer for a given m_{\chi}, the nucleus should be
chosen such that m_N = m_{\chi}.

I've tried differentiating to find the maximum and I've tried plotting E against m_{\chi} for a range of values and this did not suggest a maximum at m_N = m_{\chi}. Could someone explain why it is the case that the energy transfer is maximum when these masses are equal?

Thanks a lot.

 

[mfb]

It is a maximum with respect to the nuclei mass (the thing we can change), not with respect to the WIMP mass (which we cannot influence).

 

[maximus123]

Yeah I figured that out, I was being an idiot. Thanks for the help though.