PHYS427fa08_HW03 - Physics 427 Fall 2008 Homework # 3...

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Unformatted text preview: Physics 427 Fall 2008 Homework # 3 September 8, 2008; due September 15, 2008 1. One method of cooling a material to a very low temperature is called adiabatic demagnetization. The invention of this process won the Nobel Prize for two people: P. Debye and W. Giauque (pronounced like "joke"). You can already figure out how it works! a. Looking at the relations given in problem 2.2 of the book, express the entropy as a function of N, τ, and B. b. Suppose an isolated system of N = 1023 spins is in thermal equilibrium at an initial temperature ti = kB × 5K, in an initial magnetic field Bi = 1 Tesla (obtainable in many labs). Now suppose the field is reduced to Bf = 10‐4 Tesla. Assume the field reduction occurs very slowly, so that it does not cause any of the spins to jump from one state to the other. (Such a jump would require an energy of 2mB, which is unavailable, since Fourier analysis shows that a very slowly reduced field would have negligible amplitude at the required angular frequency of 2mB/h.) Show that the entropy of the system would be constant during the field reduction and find the final temperature tf and Tf. Assume that the magnetic moment m of each of the spins is one Bohr magneton, μB = 9.27 × 10‐24 Joule/Tesla (a typical value). c. During the field reduction, how much work is done on the system by the source of the anisotropy field which is preventing the spins from pointing in any direction other than z and −z? d. If the field were reduced to zero, you might conclude that the temperature would also be reduced to zero. For a real live experiment, what assumption have you been making that would break down, making it impossible to reduce B to 0? 2. Kittel and Kroemer: Chapter 3; Problem 6: “Rotation of diatomic molecules” 3. Kittel and Kroemer: Chapter 3; Problem 10: “Elasticity of polymers” ...
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