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phys4144 - 1 PHYS 414 Homework 5 This homework due...

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1 PHYS 414: Homework 5 This homework due Wednesday Feb 22 1. Two spins, σ 1 and σ 3 , are held at fixed positions - a and + a . They do not interact with each other. A third spin σ 2 is placed between them at position x , and is free to move along the x -axis. Each spin can be only up or down. All the spins are in thermal contact with a reservoir at temperature T . The third spin σ 2 interacts with the first two spins, σ 1 and σ 3 , through the Hamiltonian H int = 1 σ 2 ( x + a ) 3 + 2 σ 3 ( a - x ) 3 where σ 1 = ± 1, σ 2 = ± 1, and σ 3 = ± 1. (a) (2 points) Calculate the partition function of this system for a given value of x as a function of k B T , J , a , and x . (b) (2 points) Calculate the Helmholtz energy of this system. (c) (3 points) Calculate the force acting on the mobile spin σ 2 at this temperature T . (d) (3 points) The mobile spin is now placed at x = 0. Perform a calculation to show whether it is stable in that location, i.e. whether or not, if given a small displacement from x
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