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VortexLattice

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## Homework Statement

It's easier to post a picture of the problem:

## Homework Equations

In picture, and boson occupation number:

[itex]\left\langle n_k \right\rangle = \frac{1}{e^{\beta E(k)} - 1}[/itex]

Where E is the energy of the state with k and [itex]\beta = 1/k_B T[/itex]

## The Attempt at a Solution

Goal: Find M(T), where M is the magnetization. Now my usual understanding of magnetization is that it's the net number of spins in a chosen direction. I'm most familiar with it from the Ising Model, where to find it, it's (number of spins up) - (number of spins down). I'm not really sure what they mean here though, so I assumed they simply mean the total number of magnons.

If I call the number of modes of wavevector less than k per unit volume

[itex]N(k) = (1/2\pi)^3 (4\pi k^3/3)[/itex]

Then my best guess on how to get the answer is:

[itex]M(T) = V \int_0 ^\infty N(k) \left\langle n_k \right\rangle D(\omega)d\omega[/itex]

Where I use the given dispersion relation to find [itex]dk/d\omega[/itex] and convert [itex]k[/itex] to [itex]\omega[/itex] and vice versa.

But I have a few problems... First off is, what is E in my boson occupation number? In the past we've always used [itex] E = \hbar^2 k^2/2m[/itex], but there is no mention of mass here because the magnons are waves, not particles. Also, the dispersion relation given is in the form of an energy (one that's familiar from waves, [itex]\hbar \omega[/itex]), so maybe that could be E(k)? So I don't know what to do there.

Additionally, I'm not even sure I'm putting these equations together correctly. I'm also a little worried that we don't seem to be using the first equation given (U) at all, though I guess it's just supposed to be an example, and not to be actually used?

Another thing I'm not sure about is the limits of integration of whatever we're integrating. But, the hint shows an integral with limits of (0,inf). If I had to guess, I'd say that we're going to use that trick where we say, because the denominator is so large in this problem (because T << 1, β >> 1), the only non negligible pieces of the integrand are going to be for low k anyway, so we can extend the upper limit to inf.

If anyone could point me in the right direction, that would be great! Thanks!

Edit: So I noticed that [itex]D(\omega) = \frac{dN(k)}{d\omega}[/itex], which leads me to think that they give us N(k) just for illustration porpoises. Also, if I just use [itex]E = \hbar \omega[/itex] and say

[itex]M(T) = \int_0 ^\infty \left\langle n_k \right\rangle D(\omega)d\omega[/itex]

It leaves me with the form of the integral in the hint (power of 1/2 in the numerator). Does this possibly seem right?

thanks!

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