What Are the Calculations Behind Electron-Phonon Scattering in Copper?

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Homework Statement

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

Homework Equations

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.

 

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

 

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bumpp

 

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bumpp

 

[unscientific]

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