Question 2 - 2+2+4+2+4=14 marks A large paramagnetic crystal of N atoms is placed in a magnetic field. The atoms are distinguishable from their location in the crystal and can occupy two energy levels as a result of the interaction with the magnetic field (i.e. each atom has spin s = 1/2). The energy gap between these levels is E. Define U/N to be the average energy per atom in the limit of large N (i.e. N ). (a) What is the maximum possible value of U/N for the system? (b) If the system is placed in equilibrium with a heat bath, what is U/N for the most probable configuration of the system? Be sure to justify your answer. (c) Paramagnetic crystals have potential applications in quantum communication technology where a single photon emitter is required. If a single atom in the crystal changes state by emitting a photon of energy e into the heat bath, the population changes from n, n1 + 1 and N2 + N2 -- 1 where n, and ny are the populations in the lower and upper energy states respectively. In terms of nz, determine the entropy change of the crystal for the single photon emission process. Explain the significance of the sign of your result. Be sure to state any approximations you make. (d) Determine the entropy change of the reservoir for the absorption of the photon. (e) Assume the photon emission process is reversible and that a photon emitted from the bath can return the system to the original configuration. Hence determine the population ratio of nz as a function of temperature T