Cyclotron Resonance (Solid State Physics)

AI Thread Summary
The discussion revolves around deriving the cyclotron resonance frequency for a spheroidal energy surface in solid state physics. The energy surface is defined by the equation involving transverse and longitudinal mass parameters. Participants express confusion about calculating the cross product between coordinate space vectors and momentum space vectors in the context of the equation of motion. Suggestions include using the momentum space representation of the position operator to facilitate the calculation. The conversation highlights the complexity of the topic and the challenges faced by students in understanding these concepts.
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Homework Statement



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.

Homework 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}

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? :rolleyes:
 
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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?
 
Did you ever figure this out?
 
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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.
 
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