1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Cyclotron Resonance (Solid State Physics)

  1. Apr 7, 2007 #1

    cepheid

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

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

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

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

    where mt is the transverse mass parameter and ml is the longitudinal mass parameter. A surface on which [itex] \epsilon(\mathbf{k}) [/itex] is constant will be a spheroid. Use the equation of motion with [itex] \mathbf{v} = \hbar^{-1} \nabla_{\mathbf{k}} \epsilon [/itex] to show that [itex] \omega_c = eB/(m_l m_t)^{1/2}c [/itex] 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

    [tex] \hbar \frac{d\mathbf{k}}{dt} = -\frac{e}{c}\mathbf{v} \times \mathbf{B} \ \ \ \textrm{cgs} [/tex]


    [tex] \hbar \frac{d\mathbf{k}}{dt} = -e\mathbf{v} \times \mathbf{B} \ \ \ \textrm{SI} [/tex]

    3. The attempt at a solution

    [tex] \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) [/tex]​

    [tex] = \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) [/tex]​

    Apply the equation of motion with

    [tex] \mathbf{B} = B_x \hat{x} + B_y \hat{y} [/tex]

    [tex] \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}) [/tex]

    Right so, um how am I supposed to proceed to compute such a cross product? :uhh:
     
    Last edited: Apr 7, 2007
  2. jcsd
  3. Apr 7, 2007 #2

    mjsd

    User Avatar
    Homework Helper

    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
    [tex]\hat k_i \times \hat x_j[/tex]
    at the moment I can't recall what this is... but i think [tex]\hat k_i \perp \hat x_i[/tex]
    should try the momentum space representation of the position operator as a guide to convert them, perhaps?
     
  4. Jun 10, 2008 #3
    Did you ever figure this out?
     
    Last edited: Jun 10, 2008
  5. Jun 10, 2008 #4
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Cyclotron Resonance (Solid State Physics)
  1. Solid state physics (Replies: 1)

Loading...