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Homework Help: Unique λ at which X-ray and electron have same energy

  1. Sep 23, 2008 #1
    1. The problem statement, all variables and given/known data
    Derive an equation to show that there is a unique wavelength at which X-ray photons and electrons have the same energy. Calculate this wavelength and energy.

    2. Relevant equations
    Here's what I thought was relevant. There may be others!

    For photons: E = hf = h c / λ

    For electrons:
    λ = h / p (de Broglie)
    K.E. = ½ mv^2
    p = mv

    3. The attempt at a solution
    p = √(2mE)
    Substituting in de Broglie
    λ = h / √(2mE)
    E = h^2 / 2mλ^2

    Equating energies
    (h c / λ)X-ray = (h^2 / 2mλ^2)electron
    Gathering the constants (c, h, 2 and m -- the rest mass of an electron)
    (λ)X-ray = (h / 2cm) * (λ^2)electron

    ... which does not have a unique solution :-(
     
  2. jcsd
  3. Sep 23, 2008 #2

    Borek

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    Staff: Mentor

    Doesn't (λ)X-ray = (λ)electron?
     
  4. Sep 23, 2008 #3
    Thanks Borek :)

    (nice hair!)

    That's what the question asks the answerer to show so I hope it's true! I don't think I've shown that it is so.

    Writing y for (λ)X-ray, x for (λ)electron and lumping the constants together as k my attempt shows, when the X-ray and electron energies are the same,
    y = k x^2

    Mmm ... the more I read the question the more ambiguous it becomes. Perhaps it would help if you could translate the question into unambiguous language.

    Best

    Charles
     
  5. Sep 23, 2008 #4

    Borek

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    Staff: Mentor

    Why do you still use two variables for wavelength, when you should use one?
     
  6. Sep 23, 2008 #5
    Thanks again, Borek.

    I need to show that λ = fe(E) and λ = fp(E) intersect before that is legitimate (subscript e for electron, p for photon).

    With that requirement now clear ...

    For photons:
    λ = hc / E
    = (hc) * (1 / E)

    For electrons:
    λ = sqrt(h^2 / 2mE)
    = sqrt(h^2 / 2m) * sqrt(1 / E)

    Regardless of the constant values, these functions intersect once.

    At the intersect the previous equation becomes (thanks!)
    λ = (h / 2cm) * (λ^2)
    = 2cm / h

    and the rest is trivial.
     
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