p847ps8 - Physics 847 Problem Set 8 Due Thursday June 4 at...

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Physics 847: Problem Set 8 Due Thursday, June 4 at 11:59 P. M. Each problem is worth 10 pts. unless otherwise specified. 1. Pathria, Problem. (11.2). Do the main part of the problem, plus parts (a) and (d) only. 2. Pathria, Problem (12.8). 3. Pathria, Problem (12.11). 4. (20 pts.) As discussed in class, the Ginzburg-Landau theory for a spatially uniform scalar order parameter takes the form f = f 0 + α ( T T c ) M 2 + bM 4 , (1) where f is the free energy density, f 0 is a constant, T is the temperature, M is the scalar magnetization, and T c is the critical temperature. (This free energy might be appropriate for the Ising model, for example.) Now suppose that M is spatially varying, so that M = M ( x ), where x is a d-dimensional position vector. We will assume that d = 3. Then there is another term in the free energy involving gradients of M . It can be shown that the leading gradient term takes the form C |∇ M ( x ) | 2 , where C is a constant which can be taken as temperature-independent. Thus, the total free energy in this case is F = F 0 + integraldisplay d 3 x bracketleftBig α ( T T c ) M ( x ) 2 + bM ( x ) 4 + C |∇ M ( x | 2 bracketrightBig
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