- #1
- 5,197
- 38
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})
\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?
Last edited:
