# Density of states: Ashcroft Mermin, Chapter 9 question no2

Ashcroft Mermin chapter 9 number 2

Hey, I'm just sutck on the problem 2 of the Ashcroft Mermin. I did prove the density of state but I'm having a hard time finding $$k_\text{min}$$ and $$k_\text{max}$$ (for the subquestions).

The question is to calculate $$g(E)$$. We start with :

$$E = \frac{\hbar^2 k^2_\bot}{2m}+h_\pm (k_\parallel)$$

where

$$h_\pm (k_\parallel) = \frac{\hbar^2}{2m} \left\lbrack k^2_\parallel + \frac{1}{2}(G^2 - 2 k_\parallel G) \right\rbrack \pm \left\lbrace \left\lbrack \frac{\hbar^2}{2m}\frac{1}{2} (G^2 - 2 k_\parallel G) \right\rbrack^2 +|W_{\mathbf{G}}|^2\right\rbrace^{1/2}$$

Using:

$$g_n (E) = \int \frac{d \mathbf{k} }{4 \pi^3 } \delta(E- E_n (\mathbf{k}))$$

and doing the integral in polar coordonates, I've obtained:

$$g(E) = \int^{2 \pi}_0 d \theta \int^{k_{\parallel}^{max}}_{k_{\parallel}^{min}} d k_\parallel \int^{\infty}_{-\infty} d k_\bot k_\bot \delta(E- E_n (\mathbf{k}))$$

With some properties of the Delta functions (notably $$\delta(a c ) = \delta (a)/|c|$$ and $$\delta(f(x)) = \delta (x - x_i )/ |\partial f(x) / \partial x|_{x_i}$$) I end up with

$$g(E) = \frac{1}{4 \pi^2} \left(\frac{2m}{\hbar^2} \right) (k_\parallel^{\text{max}}-k_\parallel^{\text{min}})$$

The subquestion is: Show, for the lower band that: $$k_\parallel^\text{min} = - \sqrt{\frac{2 m E}{\hbar^2}} + O(W_G^2)$$

for $$E>0$$ and $$k_\parallel^\text{max} =G/2$$ , if the constant energy surface go through the Bragg plan ($$E_{G/2}-|W_G| < E < E_{G/2}+|W_G|$$). I suspect it has something to do with the integral limits (for the lower bands, they are from $$-\infty$$ and $$G/2$$, for the higher band they are $$G/2$$ to $$\infty$$), but I just can't figure out how to complete the math properly. Then we have to prove that for the higher band, we must find that: $$g_+ (E) = \frac{1}{4 \pi^2} \left(\frac{2m}{\hbar^2} \right) (k_\parallel^{\text{max}}-G/2)$$ Which should be easily done once you give the physic argument that the minima of the upper band is at $$k_\parallel = G/2$$, but I'm not sure it's a complete answer/justification for it.

Thank you :)

• Thank you very much for the corrections Feb 23 at 21:10