Homework Help: Electron-Phonon Scattering

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1. Apr 15, 2015

unscientific

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?

2. Relevant equations

3. The attempt at a solution

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

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

Part(b)
$$k^{'} = (1-\delta)k_F$$
$$E^{'} = (1-\delta)^2E_F$$

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

Substituting in, LHS
$$= \frac{1}{2} \frac{1 + 2\delta}{2\delta} \frac{1}{k_F}$$
$$= \frac{1}{2}(1 + \frac{1}{2\delta}) \frac{1}{k_F}$$
$$\approx \frac{1}{4\delta k_F}$$

How is this $\approx \lambda$?

Part(c)
Not sure what this "wavelength" means.

2. Apr 18, 2015

unscientific

Would appreciate any help on this problem, many thanks in advance!

3. Apr 19, 2015

unscientific

bumpp

4. Apr 20, 2015

unscientific

bumpp

5. Apr 21, 2015

unscientific

Ok, I got this question done. Key is to use the Bragg Condition: $\vec k^{'} + \vec k_{ph} = \vec k + \vec G$.