Photons Have Momentum — Question 6

Question 6

A required recoil fraction imposes a minimum incident photon energy

A photon scatters from a resting free electron. Let $\epsilon=E/(m_ec^2)$. For scattering from an initially resting free electron, use $E\prime=E/[1+(E/m_ec^2)(1-\cos\theta)]$, $\lambda\prime-\lambda=\lambda_C(1-\cos\theta)$, and $\lambda_C=h/(m_ec)$.

1. Find the largest possible fraction of incident energy transferred to the electron.

2. Find the minimum incident energy that permits at least 80 percent transfer, using $m_ec^2=511\,\mathrm{keV}$.

Question 6 - Solution

A required recoil fraction imposes a minimum incident photon energy

(a) Method and calculation

$$\frac{K_e}{E}=1-\frac1{1+\epsilon(1-\cos\theta)}.$$$$\theta=\pi:\quad\boxed{(K_e/E)_{\max}=\frac{2\epsilon}{1+2\epsilon}}.$$

(b) Interpretation and further calculation

$$\frac{2\epsilon}{1+2\epsilon}\ge.800\quad\Rightarrow\quad\epsilon\ge2.$$$$\boxed{E_{\min}=1022\,\mathrm{keV}},\quad\text{attained at backscattering}.$$

Physical check

At finite incident energy the scattered photon retains positive energy; complete transfer is only an asymptotic limit.