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Unformatted text preview: 1 Physics 510. Midterm Exam Problem 1 (5 pts.) Consider a system composed of two coupled spins, s 1 , 2 = 1 , subjected to magnetic f eld B . Its Hamiltonian is given by H = Js 1 s 2 B B ( s 1 + s 2 ) . a) Find the Helmholtz free energy, and the average magnetic moment, m = B h s 1 + s 2 i , for an arbitrary temper- ature and magnetic f eld. b) Find the linear magnetic susceptibility, = ( m/B ) B , as a function of temperature. Solution: a) There are four states of the system: s 1 = s 2 = 1 ; s 1 = s 2 = 1 and two states with s 1 = s 2 . Therefore, F = T log exp J T exp 2 B B T + exp 2 B B T + 2 exp J T = T log exp 2 J T cosh 2 B B T + 1 J log 2 m = F B = 2 B sinh (2 B B/T ) cosh (2 B B/T ) + exp (2 J/T ) b) B 0 : m = F B = 1 1 + exp (2 J/T ) 4 2 B B T therefore ( T ) = 4 2 B T 1 1 + exp (2 J/T ) Problem 2 (5 pts.) As an alternative to Van der Waals and lattice gas models, one can describe the gas-liquid transition within an...
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