Tutorial Theory of Superconductivity Winter Semester 2011/12

Prof. Dr. C. Timm Sheet 1

Problem 1:

For the ideal Bose gas, the logarithm of the grand-canonical partition function is

lnZ =V

λ3g5/2(y)− ln(1− y) (1)

and the particle number is

N =V

λ3g3/2(y) +

y

1− y. (2)

(a) Show that the internal energy is

U =3

2kBT

V

λ3g5/2(y). (3)

(b) Calculate the specific heat per particle,

cV ≡ CV

N=

1

N

(∂U

∂T

)N,V

(4)

both below and above Tc. Show that cV is continuous but not analytical at Tc. What does that implyfor the nature of the phase transition? Sketch cV (T ). The following identity for the derivative of thefunction gn is useful:

g′n(y) =gn−1(y)

y. (5)

It follows easily from the series representation of gn given in the lecture.

Remark: The specific heat for He-4 does not show the temperature dependence found here but ratherdiverges like a weak power law for T → Tc (from both sides). Since the graphical representation ofcV (T ) looks a bit like the letter λ, the superfluid transition in He-4 was called the “λ-transition.” Thediscrepancy between the experiment on He-4 and the exact result for the ideal Bose gas results from theinteractions between He-4 atoms.

Problem 2:

Consider an ideal Bose gas in a three-dimensional harmonic trap realized by the potential

V (r) =m

2(ω1x

2 + ω2y2 + ω3z

2). (6)

Obtain the eigenenergies. Then show that the critical temperature satisfies

kBTc = h̄ω

(N

ζ(3)

)1/3

(7)

for large particle numbers N ≫ 1, where ω := (ω1ω2ω3)1/3. You will probably have to look up an

integral. Compare the result for a uniform ideal Bose gas.

Problem 3:

The Maxwell equations in vacuum read, in Gaussian units,

∇ ·E = 4πρ, (8)

∇ ·B = 0, (9)

∇×E = −1

c

∂B

∂t, (10)

∇×B =4π

cj+

1

c

∂E

∂t. (11)

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To these, we add the London equations,

∂j

∂t=

c2

4πλ2L

E, (12)

∇× j = − c

4πλ2L

B, (13)

where we have assumed that the whole current is a supercurrent.

The goal of this problem is to convince yourself that electromagnetic waves can penetrate a supercon-ductor but with an altered dispersion relation ω(k). Assume that the charge density vanishes everywhereρ ≡ 0.

(a) Write the above six coupled equations in Fourier space, which essentially amounts to substituting∇ → ik, ∂/∂t → −iω. Convince yourself that the solutions are transverse waves.

(b) Eliminate B.

(c) Express the supercurrent density j in terms of the electric field E in two ways. (Check that the re-maining, third equation involving j is also satisfied.) Obtain the dispersion relation ω(k) from comparingthe two expressions. Sketch and discuss the result. How does it fit together with the Meißner-Ochsenfeldeffect?

Remark: The plasma frequency in a superconductor is defined as ωp = c/λL, which should appear in yourresults. We have here simplified things by taking the vacuum Maxwell equations. In real superconductors,the dielectric function at the wave frequency, ϵ(ω), would also enter.

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