Photons Have Momentum — Question 5

Question 5

A measured Compton shift determines an angle and an admissibility bound

An incident photon has wavelength $\lambda=\lambda_C$. A scattered component has shift $\Delta\lambda=1.50\lambda_C$. 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 photon scattering angle and scattered energy in units of $m_ec^2$.

2. Determine whether a shift $3\lambda_C$ could arise from one such scattering event.

Question 5 - Solution

A measured Compton shift determines an angle and an admissibility bound

(a) Method and calculation

$$1-\cos\theta=1.50\quad\Rightarrow\quad\boxed{\theta=120^\circ}.$$$$\lambda\prime=2.50\lambda_C\quad\Rightarrow\quad\boxed{E\prime=.400m_ec^2}.$$

(b) Interpretation and further calculation

$$-1\le\cos\theta\le1\quad\Rightarrow\quad\boxed{0\le\Delta\lambda\le2\lambda_C}.$$$$3\lambda_C>2\lambda_C\quad\Rightarrow\quad\boxed{\text{not a single scattering from a resting free electron}}.$$

Physical check

A larger shift might signal multiple scattering or a different initial target state; it fails the stated single-event model.