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Solid State 2- Homework 7 1) Comparison between perfect conductivity and perfect diamagnetism: Use the Maxwell equation B t Using the equation above, write an expression for B, for a perfect conductor. Suppose a material is a perfect conductor. Apply an external magnetic field. What is the magnetic field in the material? Explain ! Suppose the material is a perfect conductor only at temperatures below Tc. Start with a sample at T>Tc, apply an external magnetic field and then, keeping the external field constant, decrease the temperature below Tc. What is the magnetic field in the material? Explain ! Repeat parts (b) and (c) for a perfect diamagnet. Is there a difference in the behaviour ? If so, what is it, and how does it happen ? E a) b) c) d) 2) The coexistence of the normal and superconducting states: a) We can use the Helmholtz free energy F(B,T,N) for cases where the magnetic field B inside a material is constant. But when we set the external magnetic field constant, we need to minimize a different energy: X(H,T,N) . Write an expression for X and identify it with a thermodynamic energy you know. b) Now calculate the difference between the energy X in the normal and superconducting states: Xnormal – Xsuperconducting = ? c) Using the definition of the critical field: VH c 2 Fn Fs 8 Find the value of the applied field Ha so that the normal and superconducting states will be in equilibrium. 3. Landau's theory for the (ferromagnetic) 2nd order phase transition: Near the critical temperature Tc, we know that the magnetization M is small. We can therefore write the free energy generally as a power series in M: F ( M ) a1M a2 M 2 a3 M 3 a4 M 4 a) Explain why we can discard the terms in odd powers of M giving us F a (T ) 2 b(T ) 4 M M 2 4 b) Minimize the free energy to get the equilibrium values for M. c) Assume b(T)=b is constant. Draw schematically the free energy in the following two cases: i) a>0 ii) a<0 . What is the fundamental difference between the two cases ? Explain the ferromagnetic phase transition and how it happens. d) Expand a(T) to first order around Tc . Now find the equilibrium values for M. Write M as M (Tc T ) and find . 2 F e) Find the free energy F and from it find the heat capacity Cv k BT . T 2 Plot schematically the heat capacity near the critical temperature. f) Add external magnetic field h to the free energy. Now find the susceptibility m and bring it to the form h h0 (T Tc ) for T>Tc (T Tc ) ' for T<Tc Find , ' . f) What can you say about the critical exponents we found: , ? What properties of the system do they depend on ? Compare them to the real experimental values. g) Assume M can now change slowly in space. Add the necessary term to the expansion of the free energy.