2. (a) (b) (c) Describe the key concepts of the quantum mechanical theory that give rise to the phenomenon of the quantisation of electron energy in a material. Estimate the electron energy level separation for an object with a characteristic dimension of 1 nanometre AND 1 centimetre. The fractional occupation of energy levels, f(E), is given by the Fermi-Dirac function: (4 marks) (6 marks) f(E) = = 1 exp(E Ef) +1 where = (kBT)-1 and Ef is the Fermi energy. Lithium (atomic number 3) and beryllium (atomic number 4) have Fermi levels of 4.7 eV and 14.3 eV respectively. (i) (ii) For a sample of lithium at a temperature of 300 C, calculate the fractional occupation energy levels at an energy 1% above the Fermi energy. Calculate the temperature at which the energy levels are 1% above the Fermi energy in a beryllium sample that has a fractional occupation exactly one half of that calculated in part (i). (5 marks) (5 marks)