Question

1. Consider an ensemble of systems each made up of a large number of magnetic moments that are equal in magnitude. You apply a magnetic field of intensity H to the system on your laboratory bench. Each of the j magnetic moments in your laboratory system becomes oriented with respect to this magnetic field. The angle of orientation is vi for the jth magnetic moment of the oth system. Then the energy (Hamiltonian) of the jth magnetic moment in the oth system is Hej = cos ;, and the Hamiltonian for the oth system is then H. Σμcos 97H, j For shorthand we can write ju cos v. the Hamiltonian is = mo, which is the magnetization of the oth system. Then H. = m. H. a) Using the standard (Gibbs) notation O = kBT, write the canonical partition function for the en- semble. It is convenient to use Ho = m,H. b) Obtain the system magnetization in terms of the partition function. Explain what this is in terms of ensemble averaged quantities.