Diracobama2181
- 70
- 3
- 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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