## Normalized Differential Cross Section





In 1928 Klein and Nishina  applied Dirac’s relativistic theory of
the electron to the Compton effect to obtain improved cross sections. The differential
cross section for photon scattering at angle $\phi$ per unit solid angle and per electron,
corresponding to Equation:
\begin{equation}
    \frac{d{_e\sigma}}{d\Omega_\phi}=\frac{r_0^2}{2}       (1+ \text{cos}^2\phi)
    \label{TH}
\end{equation}
from Thomson’s theory, may be written in the form
\begin{equation}
    \frac{d{_e\sigma}}{d\Omega_\phi}=\frac{r_0^2}{2}\bigg       (\frac{h\nu'}{h\nu}\bigg)^2\bigg       (\frac{h\nu}{h\nu'}+\frac{h\nu'}{h\nu}-\text{sin}^2\phi\bigg)
    \label{TH1}
\end{equation}
where, $ h\nu'=h\nu\frac{1}{1+\varepsilon (1+\text{cos}\phi)}$, $\varepsilon = \frac{h\nu}{m_ec^2}$ and  $m_ec^2$=the energy equivalence of the electron rest mass(511 keV);

r$_0$ = classical electron radius(= 2.8 $\times$ 10$^{-15}$m)

h$\nu$ = incident photon energy

h$\nu'$ = Compton scattered photon energy
and
$\phi$ = polar scattering angle in the photon coordinate
system

The differential solid angle is represented
by $d\Omega = 2\pi \text{sin}\phi d\phi$  assuming all azimuthal
angles are equally probable. This crosssection
can be expressed as cross-section
based on polar angle        ($\phi$) given by the relation:
$\frac{d{_e\sigma}}{d\phi}=\pi r_0^2\bigg(\frac{h\nu'}{h\nu}\bigg)^2\bigg(\frac{h\nu}{h\nu'}+\frac{h\nu'}{h\nu}-\text{sin}^2\phi\bigg)\text{sin}\phi$

If h$\nu$ $\approx$ h$\nu'$
$\frac{d{_e\sigma}}{d\Omega_\phi}=\frac{r_0^2}{2}(2-\text{sin}^2\phi)=\frac{r_0^2}{2}(1+\text{cos}^2\phi)$

The total K-N cross section per electron ($_e\sigma$):
$$
_e\sigma=2\pi \int_{\phi=0}^\pi  \frac{d{_e\sigma}}{d\Omega_\phi} \text{sin}\phi  \text{d}\phi
    \label{KN}
$$

$$
=\pi r_0^2 \int_{0}^\pi \bigg(\frac{h\nu'}{h\nu}\bigg)^2\bigg(\frac{h\nu}{h\nu'}+\frac{h\nu'}{h\nu}-\text{sin}^2\phi\bigg)\text{sin}\phi  \text{d}\phi$$
\begin{equation}
_e\sigma    =2\pi r_0^2 \bigg[\frac{1+\alpha}{\alpha^2} \bigg       (\frac{2       (1+\alpha)}{1+2\alpha}-\frac{ln       (1+2\alpha)}{\alpha}\bigg)+\frac{ln       (1+2\alpha)}{2\alpha}-\frac{1+3\alpha}{       (1+2\alpha)^2}\bigg]
    \label{KN1}
\end{equation}
Where $\alpha=\varepsilon=\frac{h\nu}{m_ec^2}$, in which h$\nu$ is to be expressed in MeV and m$_e$c$^2$ = 0.511 MeV.
The relation between  the electron scattering crosssection and the attenuation coefficient       ($\mu$) is $\mu$=n$_{v}$$\times$$\sigma_e$, where  n$_{v}$ is number of electron per unit volume.
## References: 
### Attix, F. H. (2008). Introduction to radiological physics and radiation dosimetry. John Wiley & Sons.
### https://chem.libretexts.org/Courses/Sacramento_City_College/SCC%3A_CHEM_330_-_Adventures_in_Chemistry_(Alviar-Agnew)/11%3A_Nuclear_Chemistry/11.02%3A_Nuclear_Equations


```python
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
import math
from statistics import stdev
phi=np.arange(180)
Einit=0.8 #Energy in MeV; h\nu

Escat=Einit/(1+Einit/0.511*(1-np.cos(phi*math.pi/180))) # Scattering Energy

dsigma_dphi=math.pi*(2.818e-15)**2*(Escat/Einit)**2*(Einit/Escat+Escat/Einit-np.sin(phi*math.pi/180)**2)*np.sin(phi*math.pi/180)


plt.plot(phi/180,dsigma_dphi/max(dsigma_dphi))
plt.fill_between(phi/180, dsigma_dphi/max(dsigma_dphi), color= 'red', alpha=0.2,hatch='/')
plt.xlabel(r'$\phi$, normalized')
plt.ylabel('$d\sigma$/d\u03A6, normalized')
plt.title('Compton Differential Cross-Section, 800KeV, photons')
plt.xlim(0,1)
plt.ylim(0,1)
plt.show()
print(max(dsigma_dphi))
```


```python
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
import math
from statistics import stdev
phi=np.arange(180)
Einit=8 #Energy in MeV

Escat=Einit/(1+Einit/0.511*(1-np.cos(phi*math.pi/180))) # Scattering Energy

dsigma_dphi=math.pi*(2.818e-13)**2*(Escat/Einit)**2*(Einit/Escat+Escat/Einit-np.sin(phi*math.pi/180)**2)*np.sin(phi*math.pi/180)


plt.plot(phi/180,dsigma_dphi/max(dsigma_dphi))
plt.fill_between(phi/180, dsigma_dphi/max(dsigma_dphi), color= 'yellow', alpha=0.2,hatch='/')
plt.xlabel('\u03A6, normalized') # Unicode Character of Symbol
plt.ylabel('$d\sigma$/d\u03A6, normalized')
plt.title('Compton Differential Cross-Section, 8MeV, photons')
plt.xlim(0,1)
plt.ylim(0,1)
plt.show()
print(max(dsigma_dphi))
```


```python

```

c5c1b5c6d7395661115ac10025a2e415ca79b3a8
ipynb
Jupyter Notebook
Normalized Differential Cross Section.ipynb
Devendra20-20/Nuclear_Physics
335bbf18f990e88ac1898eff8b71c8825b4ecd7f
MIT
Normalized Differential Cross Section.ipynb
Devendra20-20/Nuclear_Physics
335bbf18f990e88ac1898eff8b71c8825b4ecd7f
MIT
Normalized Differential Cross Section.ipynb
Devendra20-20/Nuclear_Physics
335bbf18f990e88ac1898eff8b71c8825b4ecd7f
MIT
2021-01-30T03:51:46.000Z
2021-02-18T22:43:59.000Z
Qwen/Qwen-72B
1. YES
2. YES