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Diracobama2181

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- Homework Statement
- Suppose that instead of photons, blackbody radiation were composed

of a single species of neutrinos. The neutrino is a spin-1/2 particle

like an electron, with zero electric charge. Without worrying about

the details of the reactions that neutrinos undergo, suppose that they

can be freely created and destroyed such that they maintain thermal

equilibrium with the walls of a cavity. Treat the neutrinos as a grand

canonical ensemble of free particles of mass m, with chemical potential

µ = −mc2.

a)

Show that the heat capacity per unit volume reduces to the following form at low temperature, where the neutrinos are nonrelativistic and fermion quantum statistics reduce to classical Boltzmann statistics.

$$c_v =\frac{1}{V} \frac{dU}{dT}=\frac{4k_B}{λ^3}e^{βµ} [(βµ)^2-\frac{3}{2}βµ]$$

where

$$λ =(\frac{h^2β}{2πm})^{\frac{1}{2}}$$

- Relevant Equations
- $$<n_i>=\frac{1}{e^{β(\epsilon-µ)}+1}$$

I find that $$U=\int Z \epsilon D(\epsilon) e^{-\epsilon β}d\epsilon=\frac{gV}{(2\pi)^3}\int Z \frac{(\hbar)^2k^2}{2m}k^2 (4\pi)e^{-β\frac{(\hbar)^2k^2}{2m}}dk$$

where g=2s+1=2, $$Z=e^{βµ}$$ and $$D(\epsilon)=\frac{gV}{(2\pi)^3}k^2 4\pi$$ for the density of states

From here, I can use

$$c_v =\frac{1}{V} \frac{dU}{dT}$$. My question is whether I set this up correctly?

Thank you.

where g=2s+1=2, $$Z=e^{βµ}$$ and $$D(\epsilon)=\frac{gV}{(2\pi)^3}k^2 4\pi$$ for the density of states

From here, I can use

$$c_v =\frac{1}{V} \frac{dU}{dT}$$. My question is whether I set this up correctly?

Thank you.

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