Cooper L.N., Feldman D. (Eds.) BCS: 50 Years
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Superconductivity: From Electron Interaction to Nuclear Superfluidity 101
Fig. 1. The critical r
s
for superconductivity as a function of Z
∗
. The supercon-
ducting elements are denoted by +; the nonsuperconducting elements by •.
considered separately the contributions to the attractive part coming from
normal processes (in which the interacting electron momentum differences
were less than a typical phonon wave vector, k
D
) and the Umklapprocesses
(U processes) in which their momentum difference was q + Q > k
D
, where
Q is a reciprocal lattice vector. For the latter, I further assumed that the
momentum of the phonon being exchanged was always k
D
, which underes-
timated the contribution of U processes to V . I was thus able to obtain a
simple analytic expression for N(0)V that depended only on the effective
ion charge, Z
∗
, and r
s
, the dimensionless interelectron spacing.
Within this simple model, the net attraction coming from N processes
alone was not strong enough to overcome the screened Coulomb repulsion,
so that, for example, jellium would never superconduct. When U processes
were included, as Fig. 1 shows, the model turned out to do quite well at dis-
tinguishing between normal and superconducting elements in the periodic
table and even had predictive power, in that it predicted the superconduc-
tivity of Mo, W, Y, Sc and Pa. I then examined more closely the influence
of periodicity on the matrix elements for electron-phonon interactions, and
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102 D. Pines
concluded that when these are taken into account, the requirement that
V be attractive and the BCS expression for T
C
could together explain the
empirical rules developed by Matthias.
8. BCS and Nuclear Superconductivity in Copenhagen
In the early summer of 1957 I went to Copenhagen to spend a summer at
Niels Bohr’s Institute for Theoretical Physics en route to Paris for a year
at the Ecole Normale Superieure. My colleagues at the ITP were eager to
hear about BCS, and I agreed to give two lectures describing their theory,
for which a preprint was not yet available. Before I gave the first, Aage
Bohr and Ben Mottelson independently gave me a strong heads-up — that
I should be prepared for the possibility that Niels Bohr would interrupt
early and often, as he had done a few weeks earlier during a seminar on
superconductivity by John Blatt, who was, in fact, not able to finish his
planned talk. So I planned accordingly. I began with a brief summary of
what BCS had achieved, emphasizing that their approach was something
quite new and different and that the resulting energy gap in the electronic
spectrum was just what was needed to explain experiment. I also noted that
BCS might be of more general applicability, since it depended solely on a
net attractive interaction between fermions, and that one should therefore
look into the possible superfluidity of nuclear matter and finite nuclei.
Niels Bohr listened attentively, had only a few questions as I went along,
and then after my talk invited me to come to his study for a further dis-
cussion. There a few days later, in between innumerable efforts to light and
relight his pipe, he explained his own unsuccessful attempts to find a way to
explain the existence of a persistent current and questioned me closely as to
how BCS had managed this. I left with the impression that he did not feel
that either BCS or I had explained to his satisfaction the way in which the
persistent current is carried by the condensate, but that overall he appeared
inclined to accept its major results.
During the course of that summer, Aage, Ben and I had a number of pro-
ductive discussions about applying BCS to nuclear superconductivity, and
early that fall I gave a brief talk on BCS and its likely application to nuclei
at an international Nuclear Conference in Rehovoth, Israel that was followed
by remarks by both Aage and Ben.
12
During the following year, we wrote up
our arguments concerning similarities between nuclear and superconducting
behavior and suggested that the energy gap found in nuclei originated in
superfluid correlations brought about by the net attractive nuclear force.
13
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Superconductivity: From Electron Interaction to Nuclear Superfluidity 103
Aage and Ben encouraged Spartak Beliaev to expand upon our approach,
which he did, reporting on his considerable progress in explaining nuclear
energy gaps and reduced inertial moments at Les Houches in the summer of
1958. The Proceedings of that seminal summer school organized by Philippe
Nozieres,
14
are a gold mine for historians of science, since they also contain
Bob Schrieffer’s lecture notes on superconductivity and contributions from
Brueckner, Bohm, me, and many others.
Our work on nuclear superfluidity stimulated Arkady Migdal to propose
that neutron stars, if they existed, would be mainly made of superfluid
hadrons, and Vitale Ginzburg and David Kirshnitz to note that since stars
rotate, the rotational velocity of that superfluid would likely be carried by
vortices. The subsequent discovery of pulsars and their identification as
rotating neutron stars turned out to provide an opportunity to verify the
presence of hadron superfluids in these stars and to observe directly the
pinned crustal neutron superfluid that is responsible for pulsar glitches and
postglitch behavior (Ref. 15 and G. Baym, this volume).
9. Why John Bardeen and Why BCS?
In the years that have passed since 1957, I have had ample opportunity to
reflect on why it was that John Bardeen succeeded in his efforts to develop
a microscopic theory of superconductivity, while theorists such as Landau
and Feynman had failed. The answer lies, I think, in part in his approach
to major problems in physics:
* Focus first on the experimental results via reading and personal contact.
* Develop a phenomenological description that ties different experimental
results together.
* Explore alternative physical pictures and mathematical descriptions with-
out becoming wedded to any particular one.
* Thermodynamic and other macroscopic arguments have precedence over
microscopic calculations.
* Focus on physical understanding, not mathematical elegance, and use the
simplest possible mathematical description.
* Keep up with new developments in theoretical techniques — for one of
these may prove useful.
* Decide on a model Hamiltonian or wave-function as the penultimate, not
the first, step toward a solution.
* Choose the right collaborators.
* DON’T GIVE UP: Stay with the problem until it is solved.
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104 D. Pines
But would BCS have happened if John had stayed at Bell? I doubt it,
because he did not have the resources or support there to attract postdocs
and graduate students and so build a team. Would BCS have happened if
he had moved to any university but UIUC? This is a good question for the
historians. My own feeling is possibly not, since in Urbana, John had the
support and the nurturing atmosphere that played such an important role in
the development of the theory, one that enabled him to attract and inspire
his youthful collaborators, Leon Cooper and Bob Schrieffer, who played such
a key role in developing BCS.
Acknowledgments
In addition to the paragraphs reproduced here from the Bardeen memoir for
the American Philosophical Society, in writing about my work in Urbana I
have drawn upon some material that will also appear in my contribution to
an online text, Physics for the Twenty-first Century, that is to published by
Annenberg Media in early 2011.
References
1. D. Bohm and D. Pines, A collective description of electron interaction: I. Mag-
netic interactions, Phys. Rev. 82, 625 (1951).
2. D. Pines and D. Bohm, A collective description of electron interaction: II.
Collective vs. individual particle aspects of the interactions, Phys. Rev. 85, 338
(1952).
3. D. Pines, Proc. American Philosophical Society 153, 288 (2009).
4. T. D. Lee and D. Pines, Interaction of a non-relativistic particle with a scalar
field with application to slow electrons in polar crystals, Phys. Rev. 92, 883
(1953).
5. T. D. Lee, F. E. Low and D. Pines, The motion of slow electrons in polar
crystals, Phys Rev. 90, 297 (1955).
6. D. Bohm and D. Pines, A collective description of electron interactions: III.
Coulomb interactions in a degenerate electron gas, Phys. Rev. 92, 609 (1953).
7. D. Pines, A collective description of electron interactions: IV. Electron inter-
action in metals, Phys. Rev. 92, 626 (1953).
8. H. Fr¨ohlich, Proc. Roy. Soc. (London) A 215, 291 (1952).
9. J. Bardeen and D. Pines, Electron-phonon interaction in metals, Phys. Rev.
99, 1140 (1955).
10. D. Pines, The collective description of electron interaction in metals, in Les
Electrons dans les Metaux, Tenth Solvay Congress (Institut de Physique Solvay,
Bruxelles, 1955), pp. 9–57.
11. D. Pines, Superconductivity in the periodic system, Phys. Rev. 109, 280 (1958).
September 13, 2010 17:10 World Scientific Review Volume - 9.75in x 6.5in ch6
Superconductivity: From Electron Interaction to Nuclear Superfluidity 105
12. D. Pines, Nuclear superconductivity, in Proc. Rehovoth Conf. Nuclear Structure
(Interscience Press, 1957), pp. 26–27.
13. A. Bohr, B. Mottelson and D. Pines, Possible analogy between the excitation
spectra of nuclei and those of the superconducting metallic state, Phys. Rev.
110, 936 (1958).
14. P. Nozi`eres (ed.), The Many-Body Problem (J. Dunod, Paris, 1959).
15. D. Pines and M. A. Alpar, Nature 316, 27 (1985).
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September 17, 2010 8:58 World Scientific Review Volume - 9.75in x 6.5in ch7
DEVELOPING BCS IDEAS IN THE FORMER
SOVIET UNION
Lev P. Gor’kov
National High Magnetic Field Laboratory, Florida State University,
1800 East Paul Dirac Dr., Tallahassee, FL 32310, USA
gorkov@magnet.fsu.edu
The essay below is an attempt to re-create the wonderful scientific atmo-
sphere that emerged after the basic BCS ideas first arrived in Russia in
1957. It summarizes the most significant contributions to the microscopic
theory of superconductivity by Russian physicists during the next few years
that gave the theory its modern form.
1. Introduction
These brief historical notes are a revised version of a lecture delivered by
the author during the jubilee Symposium “BCS@50” held at University of
Illinois at Urbana-Champaign (UIUC) in October, 2007.
I confine myself to a comparatively short period between the end of 1957
and the early 1960s. It was the time when the underlying ideas and meth-
ods of the microscopic theory of superconductivity itself were under careful
examination in parallel with experimental efforts to test the BCS theory’s
predictions.
1
–
3
The most significant theoretical accomplishments during that early period
were the canonical transformation version of BCS (Bogolyubov), and the
elaboration of superconductivity theory by using Quantum Field Theory
(QFT) methods, as well as the identification of the Cooper pair wave function
as the symmetry order parameter in the superconducting phase (Gor’kov).
The history of these events in Russia, during the days of the former Soviet
Union (USSR), is not known well in the West.
2. Science in the USSR
After the end of the Second World War exact sciences in the USSR re-
mained practically isolated from the West. By 1957 direct contacts between
107
September 17, 2010 8:58 World Scientific Review Volume - 9.75in x 6.5in ch7
108 L. P. Gor’kov
scientists were cut to minimum on both sides. Soviet authorities strictly
limited the Institutes of the Russian Academy’s subscriptions to Western
journals, partially in an attempt to save on foreign currency. New issues of
Physical Review used to arrive after a delay. Acquiring any modern West-
ern experimental equipment was difficult due to restrictions imposed by the
US on the trade with the USSR. While these restrictions had their antici-
pated negative impact by introducing an unnecessary parallelism in research,
ironically, it also strengthened the traditional originality of Russian Science.
Russian Science continued to maintain a leading position in many areas of
physics, theoretical physics and mathematics.
Participation in atomic projects had boosted the prestige of science and
secured a comparative independence for physicists in Soviet society. Science
in the USSR was organized quite differently than in the Western university
system. Research, at least most of non-classified research, was not concen-
trated in universities as in the West, but mainly at the Institutes of the
Academy of Sciences of the USSR. Soviet Physics and Mathematics also
remained sound and healthy due to concerted governmental effort to attract
and educate a whole new generation of young scientists. The pre-war gen-
eration of physicists remained active and at the height of their power; those
included such figures as Petr (Pyotr) Kapitza, Lev Landau, and Igor Tamm,
to name just a few.
Of especial significance for the exact sciences in the USSR was the tradi-
tion of Scientific Schools. In connection with the theory of superconductivity
I mention only two of them: the Landau School and the group headed by
N. N. Bogolyubov.
Among the physicists that formed the Landau School there were many
internationally respected scientists, the heads of theoretical groups at the
Institutes of the Academy of the USSR. Affiliation with the Landau School
had never born an official character. Most common to the Landau School
was a broad area of interests. The School united theorists who shared a
no-nonsense attitude toward physics, and a high level of the professionalism,
including, in particular, an easy use of the mathematical apparatus. The
carefully elaborated educational process played an important role in achiev-
ing all of this. To be affiliated with the Landau School, one had to pass
the exams of the famous Landau “Theoretical Minimum” that comprised
a considerable part of all ten volumes of the current Landau and Lifshitz
Course of Theoretical Physics. As of today, many theorists got through the
school of the “Theoretical Minimum”; the beginning of Landau’s efforts to
educate his disciples ran back to his stay in Kharkov in the 1930s.
September 17, 2010 8:58 World Scientific Review Volume - 9.75in x 6.5in ch7
Developing BCS Ideas in the Former Soviet Union 109
The only official position Landau had held since 1939 was that of the
head of the small theory department at the Kapitza Institute for Physical
Problems in Moscow. For a long time his group consisted of two people:
E. M. Lifshitz, I. M. Khalatnikov, with A. A. Abrikosov admitted around
1950. Three younger staffers, L. P. Gor’kov, I. E. Dzyaloshinskii and
L. P. Pitaevskii, joined the Landau group in the mid-1950s.
The personal style of N. N. Bogolyubov and his interests in Statistical
Physics were of a more mathematical character. In the mid-1950s he was the
head of the Theoretical Department at the Steklov Mathematical Institute
of the Soviet Academy of Sciences in Moscow, while still preserving posts in
Kiev, Ukraine. In 1956 the Soviet Government organized the Joint Institute
for Nuclear Research (JINR) in Dubna, a city not far from Moscow, and
Bogolyubov was one of its co-founders. JINR was designed as an interna-
tional organization open to scientists from countries belonging to the Soviet
Block. The participation of European countries with somewhat more liberal
traditions, such as Poland, Checkoslovakia and Hungary, made it possible
for scientists at JINR to enjoy a relative amount of freedom in their contacts
with the West. Unlike Landau himself and members of his group, scientists
from JINR traveled abroad.
In studies of the basics of superfluidity and superconductivity Soviet
physicists were at the forefront at that time. The Institute for Physical
Problems (The Kapitza Institute) maintained a leadership position in the
area of low temperature physics. Superfluidity of He II had been discovered
in 1938 by Kapitza and defined as the capability of the liquid to flow along
narrow capillaries without viscosity below the lambda-point, T
λ
= 2.19 K.
The notion that superfluidity and superconductivity were two tightly related
phenomena had become common soon after Landau developed the theory of
He II (1940–41).
2.1. Superfluidity and superconductivity
In papers on helium Landau introduced one of the most important paradigms
of modern Statistical Physics.
4
The basic concept was that low temperature
properties of any macroscopic system could be described in terms of a gas
of excitations called quasiparticles (qps). These qps are brought about as
the result of interactions and, generally speaking, do not have much in com-
mon with the properties of free particles of which the system is formed.
The helium qps must obey Bose statistics. Electrons in metals obey Fermi
statistics.
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110 L. P. Gor’kov
Low energy excitations in He II are quantized sound waves, phonons, with
a linear dispersion: E(p) = cp. Landau proceeded, however, with a general
form of the energy spectrum, E(p), which at a higher momentum may have
an arbitrary shape (to account for experimental data Landau later assumed
for liquid He II the spectrum with the so-called “roton” minimum).
4
Consider T = 0. There are no excitations in the liquid as long as the
helium is at rest. Let then helium move along the capillary with a velocity V.
Forming an excitation inside the moving liquid, E(p), with the momentum
p, its energy, E
0
(p), in the reference system where the capillary is at rest,
becomes, in accordance with the Galileo’s law:
E
0
(p) = E(p) + (p · V) (1)
In other words, forming an excitation in a moving liquid may cost less in
energy (let (p · V) = −pV < 0). The slope of the straight line, pV , drawn
in the (E, p)-plane, increases with the velocity increase, V , and the line will
finally touch the spectrum curve, E(p), at some p so that E(p) − pV = 0.
The condition determines the very moment when creation of excitations in
helium becomes energetically possible for the first time, and at higher V
the system would start heating itself (to “dissipate”) by producing the qps
excitations. Hence, the critical velocity, V
cr
, is defined as:
V
cr
= min(E(p)/p) (2)
Although the initial slope, E(p)
∼
=
cp, could guarantee superfluidity, in the
real He II, V
cr
is less than the speed of sound, c, and is determined by the
“roton” part of the spectrum.
4
Bogolyubov contributed to the theory of superfluidity in his famous 1947
paper
5
on the energy spectrum of the non-ideal Bose gas. He showed that
in the presence of a weak repulsive interaction between the Bose particles,
the parabolic energy spectrum of the ideal gas, E(p) = p
2
/2M, undergoes
a fundamental change. Namely, at low momenta, due to the finite com-
pressibility, the energy spectrum becomes linear in p, corresponding to the
emergence of the “phonon” (sound) mode. Therefore, in accordance with
Landau’s arguments above, the non-ideal Bose gas possesses superfluidity.
In his 1947 paper
5
Bogolyubov successfully exploited the very concept of the
Bose-condensate phenomenon, to wit, the existence of the coherent quantum
state with zero momentum occupied by the macroscopic number of parti-
cles. Therefore the creation and the annihilation operators, a
+
0
and a
0
, for