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Difficulty with relativistic units, eV/c^2

  1. Mar 27, 2012 #1
    I'm having some difficulty in working with units of mass at the quantum level. This difficulty most clearly manifests itself when I'm doing a Compton scattering problem.

    Recall that Compton scattering is given by
    [tex]\Delta \lambda =\frac{h}{m_{e}c}(1-cos(\theta ))[/tex]
    and that the rest mass of an electron is 5.11 x 10^5 eV/c^2

    My confusion seems to come about in the denominator. I find myself unsure of whether to multiply by c, c^2, or maybe something else entirely. In any case, for my problem, I keep getting the wrong answer, regardless of my method of attack.

    If anyone could shed any insight on my problem, and also give me a big picture of relativistic units (i.e., here is how to handle relativistic units in calculations), it would be very much appreciated.
     
    Last edited: Mar 27, 2012
  2. jcsd
  3. Mar 27, 2012 #2

    fzero

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    To convert from mass to length, we use

    [tex]\hbar c \sim 197~ \mathrm{MeV\cdot fm}.[/tex]

    So your dimensionful quantity is

    [tex] \frac{h}{m_e c} = \frac{2\pi \hbar c}{m_e c^2}, [/tex]

    which is easily computed from the information given.
     
  4. Mar 27, 2012 #3

    jtbell

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    I like to think not in terms of ##m = 5.11 \times 10^5## eV/c2, but rather ##mc^2 = 5.11 \times 10^5## eV. Before plugging this into an equation, I insert c's so that I have this combination to substitute for. In your example:

    $$\Delta \lambda =\frac{h}{m_{e}c}(1- \cos \theta) = \frac{hc}{m_e c^2}(1 - \cos \theta)$$
    $$\Delta \lambda = \frac{1.24 \times 10^{-6} eV \cdot m}{5.11 \times 10^5 eV} (1 - \cos \theta)$$

    Most tables of constants in textbooks list values for hc in various units, or of course you can construct them yourself from values for h and c. hc comes up often enough that I've memorized it naturally.

    Similarly for momentum: p = 100 keV/c means pc = 100 keV. An electron with that momentum has energy

    $$ E = \sqrt{(mc^2)^2 + (pc)^2} = \sqrt{(511 keV)^2 + (100 keV)^2} = 520.7 keV$$
     
    Last edited: Mar 27, 2012
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