energies (ε) reachable by the FEFG in 3D, and the magnitude of the
chemical potential (μ).
a) Show that μ=εF at T=0 K in the FEFG model.
Tips: atT=0K,the Gibbs energy G = Nμ,
where N is the number of particles; on the other
hand, at T = 0 K, this expression may be compared
with G = E + pV ,
where E, p and V are the total energy,
pressure and volume of the system. E at T = 0 K
can then be readily calculated by integrating DOS(ε)·fFD ·ε.
Further, calculate the pressure by choosing an appropriate
thermodynamic relation, e.g., p = −(∂E/∂V );
the anticipated result
is Nμ = E + pV = NεF,
i.e. μ = εF.
b) Further, assume that this FEFG is heated up to T
> 0 K. Continue assuming that μ = εF, i.e.,
not changing with T. Plot fFD as a function
of ε/εF for T = 0.01 TF, 0.1 TF,
0.5 TF, 1.0 TF and 1.2 TF. Compare the result
with literature data. Up to what temperatures, approximately,
is the assumption of μ = εF reasonable?
c) Now investigate the true temperature dependence
of μ and plot μ/εF as a function of T/TF.
Tips: the total number of FEFG particles (N) is not changing with
temperature. In other words, the integral of DOS at T = 0
K and the integral of DOS·fFD at T > 0 K are
both equal to N; from here it only remains to evaluate the
integral of DOS·fFD numerically, and to plot μ/εF as
a function of T/TF.
d) Replot fFD as a function
of ε/εF for T = 0.01 TF, 0.1 TF,
0.5 TF, 1.0 TF and 1.2 TF using the μ
values found in part c
e) Make an estimate for the electronic heat capacity,
taking into account that only a fraction of the electrons – those
in the vicinity of εF – may contribute to an increase in
the total energy due to heating. Explain why.
please write neatly, and explain your answers
