A Summing Up — Question 8

Question 8

Recoil kinematics makes detector sensitivity depend on the target mass

For elastic nonrelativistic scattering of a dark particle of mass m on a stationary nucleus of mass M, use maximum recoil energy $E_R=2\mu^2v^2/M$, with $\mu=mM/(m+M)$. Take $mc^2=50$ GeV, $Mc^2=100$ GeV and v/c=$10^{-3}$.

1. Calculate reduced mass and maximum recoil energy in keV.

2. At fixed m and v, show which target mass maximizes the maximum recoil energy.

Question 8 - Solution

Recoil kinematics makes detector sensitivity depend on the target mass

(a) Method and calculation

$$\boxed{\mu c^2=50(100)/150=33.3333\ {\mathrm{GeV}}}.$$$$\boxed{E_{R,\max}=2(33.3333)^2(10^{-3})^2/100=22.2222\ {\mathrm{keV}}}.$$

(b) Interpretation and further calculation

$$E_R=2m^2v^2\frac{M}{(m+M)^2},\quad\frac{d}{dM}\frac{M}{(m+M)^2}=\frac{m-M}{(m+M)^3}.$$$$\boxed{M=m\ \text{maximizes }E_R,\quad E_{R,\max}=\tfrac12mv^2}.$$

Physical check

At matched masses a head-on collision can transfer all projectile kinetic energy. Total detection rates also depend on nuclear coherence and thresholds.