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Compton Scattering Using Newtonian Physics

  1. Jan 30, 2010 #1
    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,
     
  2. jcsd
  3. Jan 31, 2010 #2
    \[
    \begin{gathered}
    {\text{Conservation of momentum: }}p_1 + p_{e_1 } = p_2 + p_{e_2 } \hfill \\
    {\text{assume electron starts from rest }}p_1 = p_2 + p_{e_2 } \hfill \\
    {\text{Conservation of energy: }}E_1 + E_{e_1 } = E_2 + E_{e_2 } \hfill \\
    E_1 = p_1 c \hfill \\
    E_2 = p_2 c \hfill \\
    E_{e_1 } = 0 \hfill \\
    E_{e_2 } = \frac{1}
    {2}m_0 v^2 = \frac{1}
    {{2m_0 }}p_e ^2 \hfill \\
    \hfill \\
    p_1 c = p_2 c + \frac{1}
    {{2m_0 }}p_e ^2 {\text{ }} \hfill \\
    \Leftrightarrow 2m_0 c(p_1 - p_2 ) = p_e ^2 \hfill \\
    {\text{Sub in }}p_e ^2 = p_1 ^2 + p_2 ^2 - 2p_1 p_2 \cos \Theta {\text{ from cosine law}} \hfill \\
    \Rightarrow 2m_0 c(p_1 - p_2 ) = p_1 ^2 + p_2 ^2 - 2p_1 p_2 \cos \Theta \hfill \\
    \Leftrightarrow (2m_0 c(p_1 - p_2 ))^2 = (p_1 ^2 + p_2 ^2 - 2p_1 p_2 \cos \Theta )^2 \hfill \\
    \Leftrightarrow 4m_0 ^2 c^2 (p_1 ^2 - 2p_1 p_2 + p_2 ^2 ) = p_1 ^4 + 2p_1 ^2 p_2 ^2 - 4p_1 ^3 p_2 \cos \Theta + p_2 ^4 - 4p_2 ^3 p_1 \cos \Theta + 4p_1 ^2 p_2 ^2 \cos ^2 \Theta \hfill \\
    \Leftrightarrow 4m_0 ^2 c^2 p_1 ^2 - 8m_0 ^2 c^2 p_1 p_2 + 4m_0 ^2 c^2 p_2 = p_1 ^4 + 2p_1 ^2 p_2 ^2 - 4p_1 ^3 p_2 \cos \Theta + p_2 ^4 - 4p_2 ^3 p_1 \cos \Theta + 4p_1 ^2 p_2 ^2 \cos ^2 \Theta \hfill \\
    \end{gathered}
    \]
     
  4. Jan 31, 2010 #3
    Fixed your latex!
     
  5. Jan 31, 2010 #4

    vela

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    Once you get to

    [tex]2m_0c(p_1-p_2) = p_1^2+p_2^2-2p_1p_2\cos\theta[/tex]

    don't square the equation. Instead, divide by [itex]2p_1p_2[/itex]. 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.
     
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