Download Solid State 2- Homework 7 Use the Maxwell equation

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 =
8π
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 ) = a1 M + 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.
β
Write M as M ∝ (Tc − T ) and find β .
∂2F
.
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 
χ =
and bring it to the form

 ∂h  h→0
χ ∝ (T − Tc )γ
for T>Tc
χ ∝ (T − Tc )γ '
for T<Tc
Find γ , γ ' .
f) What can you say about the critical exponents we found: β , γ ? 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.