Some Properties of Atoms — Question 6

Question 6

Electron impact can excite an atom only above a recoil-adjusted threshold

An electron of mass m strikes a stationary atom of mass M. The atom gains internal energy $\Delta$. At threshold the final electron and atom share one velocity. Assume nonrelativistic kinematics and neglect the change of atomic mass due to excitation.

1. Derive the incident kinetic-energy threshold.

2. Evaluate its excess above $\Delta$ for $M/m=4.00\times10^4$ and $\Delta=8.00\,\mathrm{eV}$.

Question 6 - Solution

Electron impact can excite an atom only above a recoil-adjusted threshold

(a) Method and calculation

$$p_i=mv,\quad V=\frac{mv}{M+m},\quad K_f=\frac{p_i^2}{2(M+m)}=\frac{m}{M+m}K_i.$$$$K_i=\Delta+K_f\Rightarrow\boxed{K_{\mathrm{th}}=\Delta(1+m/M)}.$$

(b) Interpretation and further calculation

$$K_{\mathrm{th}}-\Delta=\Delta m/M=8.00/(4.00\times10^4)\,\mathrm{eV}.$$$$\boxed{K_{\mathrm{th}}-\Delta=2.00\times10^{-4}\,\mathrm{eV}}.$$

Physical check

Threshold requires the smallest final translational energy consistent with momentum conservation. A very heavy target makes the recoil correction small but not exactly zero.