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ps7 - Physics 880.06 Problem Set 7 Due Tuesday at 11:59 P M...

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Unformatted text preview: Physics 880.06: Problem Set 7 Due Tuesday, May 31, 2011 at 11:59 P. M. 1. (20 pts.) As stated in class, the anisotropic Ginzburg-Landau free en- ergy density is given by the following expression: f = f n + α | ψ | 2 + β 2 | ψ | 4 + 1 2 m ab vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle parenleftBigg − i ¯ h ∇− e ∗ A c parenrightBigg ⊥ ψ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 2 + + 1 2 m c vextendsingle vextendsingle vextendsingle vextendsingle parenleftbigg − i ¯ h ∇ z − e ∗ A z c parenrightbigg ψ vextendsingle vextendsingle vextendsingle vextendsingle 2 + | B | 2 8 π . (1) Here α and β are scalar coefficients, ψ is the complex scalar order parameter, f is the free energy per unit volume, f n is the normal state free energy density, A is the vector potential, B is the magnetic induction, and m ab and m c are the effective masses. We also assume that the coefficient α has the temperature dependence α = α ′ [ T/T c (0) − 1] , (2) where α ′ is a positive constant and T c (0) is the superconducting tran- sition temperature at zero field. The two Ginzburg-Landau equations obtained from this free energy are αψ + β | ψ | 2 ψ + 1 2 m ab parenleftBigg − i ¯ h ∇− e ∗ A c parenrightBigg 2 ⊥ ψ + + 1 2 m c parenleftbigg − i ¯ h ∇ z − e ∗ A z c parenrightbigg 2 ψ = 0 (3) and J = e ∗ 2 m ab bracketleftBigg ψ ∗ parenleftBigg − i ¯ h ∇− e ∗ A c parenrightBigg ⊥ ψ + c.c. bracketrightBigg + + e ∗ 2 m c bracketleftbigg ψ ∗ parenleftbigg − i ¯ h ∇ z − e ∗ A z c parenrightbigg ψ + c.c. bracketrightbigg ˆ z. (4) (a). (10 pts.) By linearizing Eq. 3 with respect to ψ , and finding the energy of the lowest Landau level, find the transition temperature 1 T c ( B ) for (i) B parallel to the z axis; (ii) B parallel to the x axis....
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ps7 - Physics 880.06 Problem Set 7 Due Tuesday at 11:59 P M...

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