Consider a spin-1/2 paramagnet of N spins, labeled by si, i = 1, ..., N. The...

 

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Consider a spin-1/2 paramagnet of N spins, labeled by si, i = 1, ..., N. The spins take two values, Si=1. The paramagnet is placed in magnetic field B so that the energy of a single spin is Esi -Bsi (for i 1,..., N), where is the magnetic moment of each spin. It is in thermal equilibrium with a reservoir at temperature T. The magnetization is M = µS = u(S₁+S2+...+SN), where S is the total spin. = = fl 1. Find the Boltzmann partition function of the paramagnet as a function of T, N, B (and μ and k the Boltzmann constant, of course). 2. Find the average energy (E) of the paramagnet and use it to deduce the average magnetization (M) (M). Sketch the behaviour of Mmax as a function of the temperature, and discuss the behaviour in the limits of low, T0, and high, T→ ∞, temperatures. Here, |Mmax| is the absolute value of the maximal possible value that the magnetization can take. 3. Explain qualitatively, using the T, B dependence of (M), how the paramagnet can be used for "magnetic cooling". Consider a spin-1/2 paramagnet of N spins, labeled by si, i = 1, ..., N. The spins take two values, Si=1. The paramagnet is placed in magnetic field B so that the energy of a single spin is Esi -Bsi (for i 1,..., N), where is the magnetic moment of each spin. It is in thermal equilibrium with a reservoir at temperature T. The magnetization is M = µS = u(S₁+S2+...+SN), where S is the total spin. = = fl 1. Find the Boltzmann partition function of the paramagnet as a function of T, N, B (and μ and k the Boltzmann constant, of course). 2. Find the average energy (E) of the paramagnet and use it to deduce the average magnetization (M) (M). Sketch the behaviour of Mmax as a function of the temperature, and discuss the behaviour in the limits of low, T0, and high, T→ ∞, temperatures. Here, |Mmax| is the absolute value of the maximal possible value that the magnetization can take. 3. Explain qualitatively, using the T, B dependence of (M), how the paramagnet can be used for "magnetic cooling".