1. (i) Paramagnetism: Consider a crystal which has an assembly of N non-interacting ions of spin '/2 in a static uniform applied magnetic field H at a temperature T. Make use of your what you learned in statistical mechanics (a) What is the energy of the system? (b) Find the average magnetic moment and the total magnetization assuming the system is in thermal equilibrium with a temperature T. (c) Find the mean square fluctuation of magnetization ((AM)?) = (M2) - (M)2 . Note the fluctuation in magnetization is AM = M - (M). (d) Show that the mean square fluctuation of magnetization is related to the magnetic susceptibility through the relation ((AM)>) = (25-) x where x = (- a is aH /TV the isothermal susceptibility. (e) Derive an expression for the temperature dependence of heat capacity (CH) under a constant magnetic field. Sketch it as a function of temperature. (f) Derive an expression for the temperature dependence of spin entropy and sketch it. (g) Find expressions for the spin entropy for a strong field (H/T > > 1) and a weak field . (h) The crystal is initially at thermal equilibrium with a reservoir at 7 = 1 K in a magnetic field Hi = 10* Gauss. The crystal is then thermally isolated from the reservoir and the field is reduced to Hy= 100 Gauss. What happens to the crystal?1. (i) Paramagnetism: Consider a crystal which has an assembly of N non-interacting ions of spin '/2 in a static uniform applied magnetic field H at a temperature T. Make use of your what you learned in statistical mechanics (a) What is the energy of the system? (b) Find the average magnetic moment and the total magnetization assuming the system is in thermal equilibrium with a temperature T. (c) Find the mean square fluctuation of magnetization ((AM)?) = (M2) - (M)2 . Note the fluctuation in magnetization is AM = M - (M). (d) Show that the mean square fluctuation of magnetization is related to the magnetic susceptibility through the relation ((AM)>) = (25-) x where x = (- a is aH /TV the isothermal susceptibility. (e) Derive an expression for the temperature dependence of heat capacity (CH) under a constant magnetic field. Sketch it as a function of temperature. (f) Derive an expression for the temperature dependence of spin entropy and sketch it. (g) Find expressions for the spin entropy for a strong field (H/T > > 1) and a weak field . (h) The crystal is initially at thermal equilibrium with a reservoir at 7 = 1 K in a magnetic field Hi = 10* Gauss. The crystal is then thermally isolated from the reservoir and the field is reduced to Hy= 100 Gauss. What happens to the crystal