Question

Consider a canonical ensemble of ? paramagnetic particles of magnetic moment ? in an external magnetic field B and find

a. An expression for the average magnetic moment per particle

b. The total magnetization.

c. Demonstrate the Curie’s law of paramagnetism that predicts that at high temperature the magnetization of the material is proportional to the ratio between the external field and the temperature

Consider a canonical ensemble of N paramagnetic particles of magnetic moment μ in an external magnetic field B and find An expression for the average magnetic moment per particle The total magnetization. Demonstrate the Curie's law of paramagnetism that predicts that at high temperature the magnetization of the material is proportional to the ratio between the external field and the temperature a. b. c.

Answer 1

u vector = g mu_B J vector E = -g mu_s J vector middot B vector = -mu vector middot B vector Partition function Q_1 (beta) = sigma^J_m = -J exp (- beta E) = sigma^J_m = -J exp (beta g mu_B m_B) [as beta parallel to z axis so take z component of J vector] = sigma^J_m = -J exp (xm) [take x = beta g mu_B B] = e^-jx + e^(-j + 1) x + ...+ e^jx = e^-jx [1 + e^x + ... + e^2jx] = e^-jx [e^(2j + 1) x - 1/e^x - 1] [a (n^n - 1)/n - 1 = > sum of G.P] = e^(j + 1) x - e^-jx/e^x - 1 = e^(j + 1/2) x - e^- (j + 1/2) x/e^x/2 - e^-x/2 = sin h (J + 1/2) x/sin hx/2 Partition function z = Q_1 (beta) = sin h [(J + 1/2) g mu_B B/k_B T]