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.