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I am looking at (elementary) theory of superconductivity. In particular, I am looking at the calculation showing that a (however small) attractive interaction makes the Fermi sea unstable.

Kittel's "Introduction to solid state physics" (7 ed) sketches this calculation in Appendix H. I'm more or less happy of Kittel's version, which seems to follow quite closely the original derivation by Cooper.

The same subject is treated in chapter 9 of Plischke and Bergersen's "Equilibrium Statistical Physics", although in 2nd quantization.

I found a couple of errors in their derivation, which however happen to cancel out to produce the same result as in Kittel (except perhaps for a qualitatively irrelevant 1/2 factor).

I can follow most of their calculations, but there's a point I am not really getting.

The eigenvalue equation on their trial state produces the following equation

[tex]

0 = [E- 2 \epsilon(\mathbf{k})] \alpha_{\mathbf{k}} + v \sum_{\mathbf{q}}\alpha_{\mathbf{k}+\mathbf{q}},\qquad \epsilon_F \leq \epsilon(\mathbf{q}) \leq \epsilon_F + \hbar \omega_D

[/tex]

where [tex]\epsilon(\mathbf{q})[/tex] is the free particle energy and the k's are outside the Fermi sphere (actually the book has a minus in front of v and the 2 in front of [tex]\epsilon[/tex] is missing).

Now the key point is to introduce a "constant" that allows to solve for the energy E. The book takes the continuum limit and sets

[tex]

\sum_{\mathbf{q}}\alpha_{\mathbf{k}+\mathbf{q}} = \int_0^{\hbar \omega_D} d \epsilon \rho(\epsilon) \alpha(\epsilon) = \Lambda

[/tex]

and here comes my problem. Is this exact or there's an underlying, undiscussed assumption?

Actually I can see two assumptions, the second of which I find rather disturbing

1) the coefficients [tex]\alpha_{\mathbf{k}}[/tex] actually depend only on the energy (scalar) and not on the momentum (vector).

2) [tex]\Lambda[/tex] does not depend on [tex]\mathbf{k}[/tex].

Accepting the above equation one gets

[tex]

\alpha(\epsilon) = \frac{v \Lambda}{2 \epsilon(\mathbf{k})-E}

[/tex]

which gives

[tex]

1 = v \int_0^{\hbar \omega_D} d \epsilon \frac{\rho(\epsilon)}{[2 \epsilon(\mathbf{k})-E}

[/tex]

Solving this for E gives basically the final result as in Kittel.

But, as I say, the point 2 above puzzles me a lot. Is this a further assumption? What is its justification?

Is this some kind of truncated self-consistence?

The derivation by Kittel (Cooper) seems to be immune from this point, but I suspect it might be hidden in the assumption made about the matrix element.

Can someone give me an hint?

PS I have 2nd edition of Plischke's book. I'm trying to get hold of the 3rd edition, which could contain a more detailed discussion or, at least, a smaller number of errors...

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# Cooper pairs - 2nd quantization

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