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Electron-Phonon Scattering

  1. Apr 15, 2015 #1
    1. The problem statement, all variables and given/known data


    (a) Find fermi temperature and debye temperature. Calculate them for copper.
    (b) Show the scattering wave relation
    (c) What does ##\lambda## mean?

    2014_B6_Q2.png


    2. Relevant equations


    3. The attempt at a solution

    Part(a)
    The fermi temperature and debye temperature is given by:
    [tex]T_F = \frac{\hbar^2 (3n \pi^2)^{\frac{2}{3}}}{2m_e k_B}[/tex]
    [tex] \theta_D = \hbar (6 \pi^2 n)^{\frac{1}{3}} \frac{c}{k_B} [/tex]

    For copper: ##a = 3.5 \times 10^{-10} m##, ##\theta_D = 231 K##, ##\T_F = 5.5 \times 10^4 K##.

    Part(b)
    [tex]k^{'} = (1-\delta)k_F[/tex]
    [tex]E^{'} = (1-\delta)^2E_F[/tex]

    I suppose the phonon gains energy by scattering, so ##E_{ph} = \Delta E = E^{'} - E_F##.
    [tex]E_{ph}= E^{'} - E_F = E_F \left( 1 - (1-\delta)^2 \right)[/tex]
    [tex]k_{ph} = \left(1 - (1-\delta)^2 \right)^{\frac{1}{2}} k_F [/tex]
    [tex]k_{ph} \approx \left( 1 - \frac{1}{2} (1-\delta)^2 \right) k_F[/tex]
    [tex]\frac{k_{ph}}{k_F} \approx \frac{1}{2}(1 + 2\delta) [/tex]

    Substituting in, LHS
    [tex] = \frac{1}{2} \frac{1 + 2\delta}{2\delta} \frac{1}{k_F} [/tex]
    [tex] = \frac{1}{2}(1 + \frac{1}{2\delta}) \frac{1}{k_F} [/tex]
    [tex] \approx \frac{1}{4\delta k_F}[/tex]

    How is this ##\approx \lambda##?

    Part(c)
    Not sure what this "wavelength" means.
     
  2. jcsd
  3. Apr 18, 2015 #2
    Would appreciate any help on this problem, many thanks in advance!
     
  4. Apr 19, 2015 #3
    bumpp
     
  5. Apr 20, 2015 #4
  6. Apr 21, 2015 #5
    Ok, I got this question done. Key is to use the Bragg Condition: ##\vec k^{'} + \vec k_{ph} = \vec k + \vec G##.
     
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