A New Look at Energy — Question 7

Question 7

Fixed-target and collider thresholds differ because of center-of-mass motion

Two incident particles each have mass m. A reaction requires total final rest mass $3m$; ignore any additional selection rules. At threshold there is no final relative kinetic energy in the center-of-momentum frame.

1. For one target initially at rest, use total energy and momentum to find the projectile threshold kinetic energy.

2. For equal and opposite incident beams, find the total incident kinetic energy needed and compare.

Question 7 - Solution

Fixed-target and collider thresholds differ because of center-of-mass motion

(a) Method and calculation

$$\mathcal S=(E+mc^2)^2-(pc)^2=2m^2c^4+2mc^2E.$$$$\mathcal S_{\mathrm{th}}=9m^2c^4\quad\Rightarrow\quad E_{\mathrm{th}}=\tfrac72mc^2,\quad\boxed{K_{\mathrm{th}}=\tfrac52mc^2}.$$

(b) Interpretation and further calculation

$$P_{\mathrm{collider}}=0:\quad E_{\mathrm{total}}=3mc^2,\quad\boxed{K_{\mathrm{total}}=mc^2}.$$$$\boxed{K_{\mathrm{fixed}}/K_{\mathrm{collider,total}}=5/2}.$$

Physical check

The fixed-target laboratory retains unavoidable bulk motion at threshold, so less of its input energy is available as final rest energy.