2. (a) (b) (c) Describe the key concepts of the quantum mechanical theory that give rise...

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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) 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)