Question

Problem 4 A monoatomic Boltzmann ideal gas if spin / atoms in uniform magnetic field (B), has in additional to its usual kinetic translational energy, a magnetic energy of tuB per atom, where is u the magnetic moment. It is assumed that the gas is so diluted that the interaction of magnetic moments can be neglected. a) What is the partition function for a canonical ensemble of N such atoms. b) Calculate C, from the partition function. c) Draw the Cy/R as function of kT/uB.

Answer 1

Solution Particles have shin half so, when these are placed in magnetic field the interaction of magnetic momenut with magnetic field B will give two energy states I'M B with spin up and spin down states. 0 Partition function is a Z eep (Ber) (BE ) Ei = I M. B > z = exp(-MB B) + exp(+ MB B) - 2 = 2 cosh (MBB) [as e te = KT cosh X This is single atom partition fonction. For Natoms it is given by ZN = 2 = (2 cash MB B) The average value of energy is given by E = -2 ln(ZN) - 2 (Nlm 2 cosh MBB) 93 - Nu B tanh UBB

- specific heat is given by = (BE) - 2 (- NuB tanh MBB) ( T aT. as B- 1 KT c = KN (ABB)" sech CuB p). Typical curve looks like 3 cules KT/UB