# Klein-Nishina Cross-Section
Motivation: Reviewing Scattering probabilities for [[Compton Scattering]]
The Klein-Nishina Formula yields the differential cross-section (the "likelihood" and angular distribution) of photons scattered from a free electron.
![[kn-plot.png]]
The differential cross section formular:
$
\frac{d\sigma}{d\Omega} = \frac{1}{2}r_e^2 \left(\frac{\lambda}{\lambda'} \right)^2 \left[\frac{\lambda}{\lambda'} + \frac{\lambda'}{\lambda} - \text{sin}^2(\theta) \right]
$
where:
- $r_e$ is the classical electron radius (~2.82 fm or $7.94 \times 10^{-30}$ m$^2$)
- $\lambda / \lambda'$ is the ratio of the wavelengths of the incident and scattered photons
- $\theta$ is the scattering angle (0 for undeflected)
Note that $E_{\gamma'} / E_\gamma$ = $\lambda / \lambda'$
## Script to make the plot
```
clear
clc
b = 10^-28;
meshVal = 1000;
e = [0.001 0.1 0.511 1 2.6 10];
theta = linspace(0,2*pi,meshVal);
re2 = 7.94*10^-30;
sigma = zeros(length(e),meshVal);
for i = 1:length(e)
A = 1 ./ (1+ e(i)*(1 - cos(theta)));
sigma(i,:) = 1/2 * re2 * A.^2 .* (A + (1./A) - sin(theta).^2) ./b;
polarplot(theta,sigma(i,:), 'LineWidth', 1.5)
hold on
end
hold off
legend('1 keV', '100 keV', '511 keV', '1 MeV', '2.6 MeV', '10 MeV')
title('Klein-Nishina Scattering Angle Distribution')
```