Question

The maximum Compton shift in wavelength occurs when a photon isscattered through 180^\circ .

Answer 1

Answer: $90\°$

Explanation:

The Compton Shift $\Delta \lambda$ in wavelength when the photons are scattered is given by the following equation:

$\Delta \lambda=\lambda_{c}(1-cos\theta)$     (1)

Where:

$\lambda_{c}=2.43(10)^{-12} m$ is a constant whose value is given by $\frac{h}{m_{e}c}$, being $h$ the Planck constant, $m_{e}$ the mass of the electron and $c$ the speed of light in vacuum.

$\theta)$ the angle between incident phhoton and the scatered photon.

We are told the maximum Compton shift in wavelength occurs when a photon isscattered through $180\°$:

$\Delta \lambda_{max}=\lambda_{c}(1-cos(180\°))$     (2)

$\Delta \lambda_{max}=\lambda_{c}(1-(-1))$    

$\Delta \lambda_{max}=2\lambda_{c}$     (3)

Now, let's find the angle that will produce a fourth of this maximum value found in (3):

$\frac{1}{4}\Delta \lambda_{max}=\frac{1}{4}2\lambda_{c}(1-cos\theta)$      (4)

$\frac{1}{4}\Delta \lambda_{max}=\frac{1}{2}\lambda_{c}(1-cos\theta)$      (5)

If we want $\frac{1}{4}\Delta \lambda_{max}=\frac{1}{2}\lambda_{c}$, $1-cos\theta$   must be equal to 1:

$1-cos\theta=1$   (6)

Finding $\theta$:

$1-1=cos\theta$

$0=cos\theta$  

$\theta=cos^{-1} (0)$  

Finally:

$\theta=90\°$    This is the scattering angle that will produce $\frac{1}{4}\Delta \lambda_{max}$