Some Nuclear Properties — Question 7

Question 7

A negative reaction Q value implies more than a simple energy deficit

A projectile of mass m strikes a stationary target of mass $M=4m$. A reaction has $Q=-2.00\,\mathrm{MeV}$. Work to leading nonrelativistic order in $|Q|/(Mc^2)$. The available center-of-mass kinetic energy is $E_{\mathrm{cm}}=EM/(m+M)$.

1. Find the laboratory threshold energy.

2. At incident energy 3.00 MeV, find the final kinetic energy available in the center-of-mass frame.

Question 7 - Solution

A negative reaction Q value implies more than a simple energy deficit

(a) Method and calculation

$$E_{{\mathrm{cm}},\mathrm{th}}+Q=0\Rightarrow\boxed{E_{\mathrm{th}}=-Q(1+m/M)}.$$$$\boxed{E_{\mathrm{th}}=2.00(1+1/4)=2.50\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$E_{\mathrm{cm}}=3.00(4/5)=2.40\,\mathrm{MeV}.$$$$\boxed{K_{f,\mathrm{cm}}=E_{\mathrm{cm}}+Q=.400\,\mathrm{MeV}}.$$

Physical check

Some laboratory energy must remain as translational motion of the center of mass. That momentum constraint raises the threshold above the magnitude of Q.