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Unformatted text preview: 1 Physics 510. Final Exam Problem 1 (5 pts.) Consider a system of N particles whose spin-spin interaction depend on their positions: H = 1 2 X i 6 = j J ( | r i r j | ) s i s j . By summarizing over all spin con f gurations ( s i = 1 ) , determine the free energy (e f ective Hamiltonian) of the system as a function of particle positions r i . Assume that J is small enough, so that exp ( H/T ) can be expanded in powers of H/T . Based on your result, what is the behavior of the e f ective two-particle potential in the case when J ( r ) 1 /r 3 (this correspondes to the dipole-dipole interaction of the magnetic moments)? Solution: Z = T r exp 1 2 T X i 6 = j J ( | r i r j | ) s i s j ' T r 1 1 2 T X i 6 = j J ( | r i r j | ) s i s j + 1 8 T 2 X i 6 = j,k 6 = l J ( | r i r j | ) J ( | r k r l | ) s i s j s k s l Here T r corresponds to summation over all spin con f gurations: T r (1) = X s 1 s 2 ....s N 1 = 2 N T r ( s i...
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