Question 3 A two-dimensional paramagnetic solid consists of a large number, N, of atoms on a crystal lattice. Each atom has a spin which can point in one of six possible directions: +2, +%. +2 where 8, 9, 2 are unit vectors in the X-, y-and z-directions. The crystal is subject to a weak magnetic field B in the x-direction: B = BX and is in equilibrium with the environment at temperature T. (a) If the magnitude of the magnetic moment of each atom is m, make a table showing all possible magnetic moment vectors and the corresponding energy for each orientation. (b) Identify the appropriate ensemble for this system. Justify your answer. (c) Based on your answer to (b), determine the partition function, 21, for one atom in the crystal and hence determine the partition function, Z, of the entire solid. (d) Find the expected value of the x-component of the magnetic moment of a single atom and, hence, the expected value of the magnetic moment and interaction energy of the entire crystal. (e) Determine an expression for the entropy of the entire crystal as a function of temperature. () Use this expression to find the entropy of the crystal in the high temperature limit. Provide a physical interpretation for your result. You may use the limits lim cosh y = 1 and lim sinh y = 0. y 0 y-0 Question 3 A two-dimensional paramagnetic solid consists of a large number, N, of atoms on a crystal lattice. Each atom has a spin which can point in one of six possible directions: +2, +%. +2 where 8, 9, 2 are unit vectors in the X-, y-and z-directions. The crystal is subject to a weak magnetic field B in the x-direction: B = BX and is in equilibrium with the environment at temperature T. (a) If the magnitude of the magnetic moment of each atom is m, make a table showing all possible magnetic moment vectors and the corresponding energy for each orientation. (b) Identify the appropriate ensemble for this system. Justify your answer. (c) Based on your answer to (b), determine the partition function, 21, for one atom in the crystal and hence determine the partition function, Z, of the entire solid. (d) Find the expected value of the x-component of the magnetic moment of a single atom and, hence, the expected value of the magnetic moment and interaction energy of the entire crystal. (e) Determine an expression for the entropy of the entire crystal as a function of temperature. () Use this expression to find the entropy of the crystal in the high temperature limit. Provide a physical interpretation for your result. You may use the limits lim cosh y = 1 and lim sinh y = 0. y 0 y-0
