Compton Scattering Using Newtonian Physics

[rpardo]

Hey guys,

Im trying to derive the following equation

(mc^2) [(1/E2)-(1/E1)]+cos(theta)-[((E1-E2)^2)/(2E1E2)]=1

E1 = Incident Photon's energy
E2 = Scattered Photon's energy
theta= scattering angle
m = mass of electron
c = s

Using conservation of energy,conservation of mass, and the cosine law

I've derived the relativistic equation, but I am really stumped on how to derive this...its been a couple days
Any help at all would be appreciated. I feel like I'm really close but the algebra is the problem

Thanks in advance for your help guys and gals,

 

[rpardo]

\[
\begin{gathered}
{\text{Conservation of momentum: }}p_1 + p_{e_1 } = p_2 + p_{e_2 } \hfill \\
{\text{assume electron starts from rest }}p_1 = p_2 + p_{e_2 } \hfill \\
{\text{Conservation of energy: }}E_1 + E_{e_1 } = E_2 + E_{e_2 } \hfill \\
E_1 = p_1 c \hfill \\
E_2 = p_2 c \hfill \\
E_{e_1 } = 0 \hfill \\
E_{e_2 } = \frac{1}
{2}m_0 v^2 = \frac{1}
{{2m_0 }}p_e ^2 \hfill \\
\hfill \\
p_1 c = p_2 c + \frac{1}
{{2m_0 }}p_e ^2 {\text{ }} \hfill \\
\Leftrightarrow 2m_0 c(p_1 - p_2 ) = p_e ^2 \hfill \\
{\text{Sub in }}p_e ^2 = p_1 ^2 + p_2 ^2 - 2p_1 p_2 \cos \Theta {\text{ from cosine law}} \hfill \\
\Rightarrow 2m_0 c(p_1 - p_2 ) = p_1 ^2 + p_2 ^2 - 2p_1 p_2 \cos \Theta \hfill \\
\Leftrightarrow (2m_0 c(p_1 - p_2 ))^2 = (p_1 ^2 + p_2 ^2 - 2p_1 p_2 \cos \Theta )^2 \hfill \\
\Leftrightarrow 4m_0 ^2 c^2 (p_1 ^2 - 2p_1 p_2 + p_2 ^2 ) = p_1 ^4 + 2p_1 ^2 p_2 ^2 - 4p_1 ^3 p_2 \cos \Theta + p_2 ^4 - 4p_2 ^3 p_1 \cos \Theta + 4p_1 ^2 p_2 ^2 \cos ^2 \Theta \hfill \\
\Leftrightarrow 4m_0 ^2 c^2 p_1 ^2 - 8m_0 ^2 c^2 p_1 p_2 + 4m_0 ^2 c^2 p_2 = p_1 ^4 + 2p_1 ^2 p_2 ^2 - 4p_1 ^3 p_2 \cos \Theta + p_2 ^4 - 4p_2 ^3 p_1 \cos \Theta + 4p_1 ^2 p_2 ^2 \cos ^2 \Theta \hfill \\
\end{gathered}
\]

 

[torquil]

$$\[ \begin{gathered} {\text{Conservation of momentum: }}p_1 + p_{e_1 } = p_2 + p_{e_2 } \hfill \\ {\text{assume electron starts from rest }}p_1 = p_2 + p_{e_2 } \hfill \\ {\text{Conservation of energy: }}E_1 + E_{e_1 } = E_2 + E_{e_2 } \hfill \\ E_1 = p_1 c \hfill \\ E_2 = p_2 c \hfill \\ E_{e_1 } = 0 \hfill \\ E_{e_2 } = \frac{1} {2}m_0 v^2 = \frac{1} {{2m_0 }}p_e ^2 \hfill \\ \hfill \\ p_1 c = p_2 c + \frac{1} {{2m_0 }}p_e ^2 {\text{ }} \hfill \\ \Leftrightarrow 2m_0 c(p_1 - p_2 ) = p_e ^2 \hfill \\ {\text{Sub in }}p_e ^2 = p_1 ^2 + p_2 ^2 - 2p_1 p_2 \cos \Theta {\text{ from cosine law}} \hfill \\ \Rightarrow 2m_0 c(p_1 - p_2 ) = p_1 ^2 + p_2 ^2 - 2p_1 p_2 \cos \Theta \hfill \\ \Leftrightarrow (2m_0 c(p_1 - p_2 ))^2 = (p_1 ^2 + p_2 ^2 - 2p_1 p_2 \cos \Theta )^2 \hfill \\ \Leftrightarrow 4m_0 ^2 c^2 (p_1 ^2 - 2p_1 p_2 + p_2 ^2 ) = p_1 ^4 + 2p_1 ^2 p_2 ^2 - 4p_1 ^3 p_2 \cos \Theta + p_2 ^4 - 4p_2 ^3 p_1 \cos \Theta + 4p_1 ^2 p_2 ^2 \cos ^2 \Theta \hfill \\ \Leftrightarrow 4m_0 ^2 c^2 p_1 ^2 - 8m_0 ^2 c^2 p_1 p_2 + 4m_0 ^2 c^2 p_2 = p_1 ^4 + 2p_1 ^2 p_2 ^2 - 4p_1 ^3 p_2 \cos \Theta + p_2 ^4 - 4p_2 ^3 p_1 \cos \Theta + 4p_1 ^2 p_2 ^2 \cos ^2 \Theta \hfill \\ \end{gathered} \]$$

Fixed your latex!

 

[vela]

Once you get to

$$2m_0c(p_1-p_2) = p_1^2+p_2^2-2p_1p_2\cos\theta$$

don't square the equation. Instead, divide by $2p_1p_2$. Throw in factors of c here and there, and you'll have the cosine term and the mc²(1/E₂-1/E₁) terms that you want. You just have to then figure out how the rest works out.

 

[PJC01]

I appreciate your efforts to derive this equation using Newtonian physics. However, I must inform you that this equation cannot be derived using solely classical mechanics. Compton scattering is a phenomenon that involves the interaction between a photon and an electron, and it can only be fully understood by incorporating the principles of quantum mechanics and special relativity.

The equation you are trying to derive is known as the Compton scattering formula, and it was first derived by Arthur Compton in 1923. It describes the change in energy and direction of a photon after it scatters off an electron. This phenomenon cannot be explained using only classical mechanics, as it involves the wave-particle duality of light and the relativistic effects of fast-moving particles.

To derive this equation, you will need to use the principles of quantum mechanics, such as the conservation of energy and momentum, and the concept of wave-particle duality. You will also need to incorporate the relativistic effects of the electron's motion. I would suggest seeking help from a physicist or referring to textbooks and research papers on Compton scattering for a step-by-step derivation of the equation.

In conclusion, while it is admirable that you are trying to understand this phenomenon using classical mechanics, it is important to recognize its limitations and the need for a more comprehensive approach to fully explain it. I wish you the best of luck in your studies.