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midterm exam

# midterm exam - 1 Physics 510 Midterm Exam Problem 1(5 pts...

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1 Physics 510. Midterm Exam Problem 1 (5 pts.) Consider a system composed of two coupled spins, s 1 , 2 = ± 1 , subjected to magnetic fi eld B . Its Hamiltonian is given by H = Js 1 s 2 µ B B ( s 1 + s 2 ) . a) Find the Helmholtz free energy, and the average magnetic moment, m = µ B h s 1 + s 2 i , for an arbitrary temper- ature and magnetic fi eld. b) Find the linear magnetic susceptibility, χ = ( ∂m/∂B ) B 0 , 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 μ J T exp μ 2 µ B B T + exp μ 2 µ B B T ¶¸ + 2 exp μ J T ¶¸ = T log exp μ 2 J T cosh μ 2 µ B B T + 1 ¸ J log 2 m = ∂F ∂B = 2 µ B sinh (2 µ B B/T ) cosh (2 µ B B/T ) + exp (2 J/T ) b) B 0 : m = ∂F ∂B = 1 1 + exp (2 J/T ) 4 µ 2 B B T therefore χ ( T ) = 4 µ 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 empirical virial expansion. It is an expansion of free energy in terms of density

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midterm exam - 1 Physics 510 Midterm Exam Problem 1(5 pts...

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