# Homework Help: Finding ω_c Given an Energy Surface E(k) in Constant Magnetic Field

1. Nov 5, 2013

### The Head

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. Nov 6, 2013

### The Head

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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