Question

Calculate the Fermi energy level (EF) in eV for the conduction electrons in trivalent
aluminium. The atomic weight and density of aluminium is 27 g/mol and 2.7 g/cm3
,
respectively. Avogadro’s number NA = 6.02 x 1023 atoms/mol.

Answer 1

The equation for the Fermi energy of valence electrons starts with the equation for the Fermin energy of an electron gas:\[E _{f}=(3\pi ^{2})^{\frac{ 2 }{ 3}}\frac{ ħ^{2} }{ 2m _{e} }\left( \frac{ I }{ V } \right)^{\frac{ 2 }{ 3 }}\]where ħ is Planck's constant divided by 2π; me is the mass of an electron I is the number of electrons; and V is the volume. You are however given atomic weight, density, and Avogadros number. From that we can get the following expression for the volume of an atom:\[V _{atom}=\frac{ M }{ \rho NA }\]where M is the atomic weight; ρ is the density of the material; and NA is Avogadro's number. Now let Iv be the number of valence electrons per atom and substitute Vatom for V in the first equation and you end up with:\[E _{f}=\left( 3\pi ^{2} \right)^{\frac{ 2 }{ 3 }}\frac{ ħ }{ 2m _{e} }\left( \frac{ I _{val}\rho NA }{ M } \right)^{\frac{ 2 }{ 3 }}\]where Ival is the number of valence electrons per atom; and, of course, Ef is the Fermi energy of the valence electrons.