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Compton effect

  1. Jun 26, 2008 #1
    The Compton effect does make sense conceptually but I'm having trouble with the actual derivation.

    The equation starts out from the principles of conservation of energy and mass: [tex]E_{\gamma}[/tex]+[tex]E_{e}[/tex]=[tex]E_{\gamma'}[/tex]+[tex]E_{e'}[/tex]
    which strangely proceeds into the form: [tex]\ E-E'}+m[/tex]=[tex]\sqrt{p^2+m^2}}[/tex] The [tex]\sqrt{p^2+m^2}}[/tex] actually expands into [tex](E-E'cos\theta)^2{}+m^2[/tex] when the RHS is squared

    So my question is where does the [tex]\sqrt{p^2+m^2}}[/tex] come from and why does it expand to [tex](E-E'cos\theta)^2{}+m^2[/tex]?
     
  2. jcsd
  3. Jun 26, 2008 #2

    jtbell

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    Staff: Mentor

    The relativistic relationship between (rest) mass, momentum and energy is [itex]E^2 = (pc)^2 + (mc^2)^2[/itex]. Your source must be using units that make [itex]c = 1[/itex]. I assume this refers to the energy of the electron after the interaction (because the electron is usually assumed to be stationary before the interaction) so it really should read

    [tex]E_{\gamma} - E^{\prime}_{\gamma} + m = \sqrt {(p^{\prime}_e)^2 + m^2}[/tex]


    I assume the E and the E' refer to the outgoing photon (gamma). I'm sorry, different textbooks use different routes to derive the Compton-scattering equation and I'm too tired to try to guess which route your source uses. Does your source give any more details?
     
  4. Jun 27, 2008 #3

    jtbell

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    Looking at your second question again, that step probably comes from conservation of momentum. Usually in this derivation one lets the direction of the incoming photon be along the x-axis. Then the outgoing electron and photon both have x- and y-components of momentum. The x- and y-components are each conserved.

    Also, for a photon, E = pc which in your units (with c = 1) becomes E = p. So that [itex]E^{\prime} \cos \theta[/itex] is probably the x-component of the outgoing photon momentum: [itex]p_x^{\prime} = p^{\prime} \cos \theta = E^{\prime} \cos \theta[/itex].
     
  5. Jun 27, 2008 #4
    Thank you jtbell for clearing that up. I'm still curious where that radical came from, but for now I'm very satisfied with the explanations. =D
     
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