Photons Have Momentum — Question 7

Question 7

The heavy-target limit suppresses Compton energy transfer

A photon scatters from a resting target of mass M, with $E\prime/E=[1+a]^{-1}$ and $a=[E/(Mc^2)](1-\cos\theta)$. Assume the target retains its internal state.

1. Expand the fractional energy loss for $a\ll1$.

2. At fixed photon energy and angle, compare recoil from targets of masses m and 1836m in this limit and find the relative error of using a instead of the exact loss.

Question 7 - Solution

The heavy-target limit suppresses Compton energy transfer

(a) Method and calculation

$$\frac{E-E\prime}{E}=\frac a{1+a}=a-a^2+O(a^3).$$$$\boxed{\Delta E\simeq\frac{E^2}{Mc^2}(1-\cos\theta)}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta E_{1836m}/\Delta E_m\simeq1/1836}.$$$$\boxed{\frac{a-a/(1+a)}{a/(1+a)}=a}.$$

Physical check

The approximation is controlled by a, including its angular factor, rather than photon energy alone.