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

In summary, the conversation discusses finding the value of ωc using the energy surface E(k) and the equation of motion h(dk/dt)=-e(vXB) with v=∇k(E)/h. The goal is to show that when the static magnetic field B lies in the x-y plane, ωc=eB/(ml*mt)1/2. The attempt at a solution involves finding the cross product of v and B and using the equation a=(1/h)(d2E(k)/dk2)(dk/dt) to solve for ωc. However, there are difficulties in getting rid of the different k components and dividing by the vector v.
  • #1
The Head
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Homework Statement


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


Homework Equations


a=(1/h)(d2E(k)/dk2)(dk/dt)
ω=a/v

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!

 
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  • #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!
 

1. What is the purpose of finding ω_c in an energy surface in a constant magnetic field?

Finding ω_c, or the cyclotron frequency, is important in understanding the behavior of particles in a magnetic field. It helps to determine the motion and energy of charged particles, which is useful in various fields such as plasma physics, particle accelerators, and condensed matter physics.

2. How is the energy surface E(k) related to ω_c?

The energy surface E(k) is a plot of the energy of particles as a function of their momentum. The cyclotron frequency ω_c is related to the curvature of this energy surface. A higher curvature indicates a stronger magnetic field and therefore a higher ω_c.

3. What factors affect the value of ω_c?

The value of ω_c depends on the strength of the magnetic field, the mass and charge of the particle, and the shape of the energy surface. It is also affected by external factors such as temperature and pressure.

4. How is ω_c calculated from an energy surface?

To calculate ω_c from an energy surface E(k), we need to determine the curvature of the energy surface at the point where the particle's momentum is equal to its cyclotron momentum. This can be done by taking the second derivative of the energy surface with respect to momentum.

5. Can ω_c be experimentally measured?

Yes, ω_c can be experimentally measured using various techniques such as cyclotron resonance, electron spin resonance, and nuclear magnetic resonance. These methods involve applying a varying magnetic field and measuring the frequencies at which particles absorb energy, which can then be used to calculate ω_c.

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