1. Calculate the approximate ratio of occupied states with energy kT greater than the Fermi energy, to occupied sates with energy kT less than EF for the free electron model (for kT << EF ).
2. In metallic sodium each atom gives up one free electron. The
atomic weight of sodium is 22.99 g and its density is 0.97 g/cm3.
Avogadro's number is 6.023 x 10 23 atoms/ mole. Do the following for low
temperatures.
(a) calculate the EF in the metallic Na.
(b) If the value of vacuum energy Evac (the reference energy
at top of band for an electron that is free or outside of the solid) with
respect to the zero of energy (bottom of occupied band) in the free electron
model of Na is 5.0 eV, calculate the maximum wavelength for photoemission
(i.e. from EF to the Evac reference).
(c) For photoemission caused by 225 nm light, what is the maximum and
minimum kinetic energy for photoemitted electrons (neglecting loss of energy
of electrons by scattering, caused by photoexcitation before emission)?
(d) If the photoemission current is proportional to the product of
the density of occupied initial states and the density of empty final states,
calculate the ratio of photoemission current for a photon energy of 5.5
eV for the case where the initial state is at EF to the case
where the initial state is at one half of EF.
3. If the vacuum level in a metal with atomic weight 60 lies 6 eV above the zero energy (bottom of band) of the free electron distribution, what must the density of a metal be, in which each atom contributes one free electron, if the photon threshold for photoemission is 2 eV?
4. The atom He3 has spin 1/2 and is a fermion.
The density of liquid He3 is 0.081 g-cm-3 near absolute zero.
Calculate the EF energy and the Fermi Temperature TF.
(for hints, see J. Wilks, Properties of liquid and solid helium, Oxford,
1967).
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