# Cyclotron Resonance (Solid State Physics)

1. Apr 7, 2007

### cepheid

Staff Emeritus
1. The problem statement, all variables and given/known data

Cyclotron resonance for a spheroidal energy surface. Consider the energy surface

$$\epsilon(\mathbf{k}) = \hbar^2 \left( \frac{k_x^2 + k_y^2}{2m_t} + \frac{k_z^2}{2m_l} \right)$$​

where mt is the transverse mass parameter and ml is the longitudinal mass parameter. A surface on which $\epsilon(\mathbf{k})$ is constant will be a spheroid. Use the equation of motion with $\mathbf{v} = \hbar^{-1} \nabla_{\mathbf{k}} \epsilon$ to show that $\omega_c = eB/(m_l m_t)^{1/2}c$ when the static magnetic field B lies in the xy plane.

2. Relevant equations

Dynamics of Bloch Electrons

The equation of motion for an electron subject to the periodic potential of a crystal lattice is

$$\hbar \frac{d\mathbf{k}}{dt} = -\frac{e}{c}\mathbf{v} \times \mathbf{B} \ \ \ \textrm{cgs}$$

$$\hbar \frac{d\mathbf{k}}{dt} = -e\mathbf{v} \times \mathbf{B} \ \ \ \textrm{SI}$$

3. The attempt at a solution

$$\mathbf{v} = \hbar^{-1} \nabla_{\mathbf{k}} \epsilon(\mathbf{k}) = \hbar^{-1} \left( \hat{k}_x \frac{\partial}{\partial k_x} + \hat{k}_y \frac{\partial}{\partial k_y} + \hat{k}_z \frac{\partial}{\partial k_z} \right) \hbar^2 \left( \frac{k_x^2 + k_y^2}{2m_t} + \frac{k_z^2}{2m_l} \right)$$​

$$= \hbar \left( \hat{k}_x \frac{k_x}{m_t} + \hat{k}_y \frac{k_y}{m_t} + \hat{k}_z \frac{k_z}{m_l} \right)$$​

Apply the equation of motion with

$$\mathbf{B} = B_x \hat{x} + B_y \hat{y}$$

$$\frac{d\mathbf{k}}{dt} = -\frac{e}{c}\left( \hat{k}_x \frac{k_x}{m_t} + \hat{k}_y \frac{k_y}{m_t} + \hat{k}_z \frac{k_z}{m_l} \right) \times (B_x \hat{x} + B_y \hat{y})$$

Right so, um how am I supposed to proceed to compute such a cross product? :uhh:

Last edited: Apr 7, 2007
2. Apr 7, 2007

### mjsd

so, you have problem with mixing coordinate space vectors with momentum space vectors. you can still do this as long as you know what is
$$\hat k_i \times \hat x_j$$
at the moment I can't recall what this is... but i think $$\hat k_i \perp \hat x_i$$
should try the momentum space representation of the position operator as a guide to convert them, perhaps?

3. Jun 10, 2008

### Aaronse_r

Did you ever figure this out?

Last edited: Jun 10, 2008
4. Jun 10, 2008

### Aaronse_r

I just had this on a test, but didn't get it done so as you can imagine i was curious about it. Also, just realized you put this on here a year ago so nevermind.