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Finding ω_c Given an Energy Surface E(k) in Constant Magnetic Field

  1. Nov 5, 2013 #1
    1. The problem statement, all variables and given/known data
    Consider the energy surface
    E(k)=h2((kx2 +ky2 )/ml+kz2/mt
    where m_l is the transverse mass parameter and m_l is the longitudinal mass parameter. Use the equation of motion:
    h(dk/dt)= -e(vXB) with v=∇k(E)/h to show that ωc=eB/(ml*mt)1/2 when the static magnetic field B lies in the x-y plane


    2. Relevant equations
    a=(1/h)(d2E(k)/dk2)(dk/dt)
    ω=a/v

    3. The attempt at a solution

    I've tried quite a number of things and spent several hours unsuccessfully, but one route I tried was this:
    v= 1/h(∇k(E))=1/h(h2)(kx(i-hat)/ml + ky(j-hat)/ml + kz(k-hat)/mt)
    B is in x-y plane, so B= Bx (i-hat) + By (j-hat)
    so the cross product of v & B = kz*By/(ml) (i-hat) - kz*Bx/(ml) (j-hat) +(ky*Bx - Kx*By)/(mt) (k-hat)

    Now, this is where I am not sure what to do, but I figured I would try the equation:
    a=(1/h)(d2E(k)/dk2)(dk/dt) with the idea that this would be centripetal acceleration = ωv, and this way I could solve for ω_c


    d2E(k)/dk2= h2(((i-hat)+(j-hat))/ml +(k-hat)/mt

    dk/dt=-e/h(vXB)=-e/h*kz*By/(ml) (i-hat) - kz*Bx/(ml) (j-hat) +(ky*Bx - Kx*By)/(mt) (k-hat)

    I doted dk/dt with vXB to get rid of the vectors, and divided by h to get:

    a=e/mtml(Bx*(kz-ky)+By(kx-kz))=ωv

    Although this seems to have somewhat the form I am going for, I have two problems: I don't know how to get rid of all the different k components, and if I divide by the vector v, then I have an answer for ω with components.

    Can someone provide some guidance? Thank you!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 6, 2013 #2
    Or perhaps the method I chose isn't even close...if so could someone please just give me a hint on a better method? I don't know if what I'm doing is just creating a bunch of smoke, or if it's on the correct path. I've been struggling with this problem for days!
     
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