Cooper pairs in atomic nuclei
G. G. Dussel1, S. Pittel2, J. Dukelsky3, and P. Sarriguren3
Departamento de Fisica J. J. Giambiagi, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Bartol Research Institute and Department of Physics and Astronomy,
University of Delaware, Newark, DE 19716 USA
Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain
(Dated: Received October 26, 2018)
We consider the development of Cooper pairs in a self-consistent Hartree Fock mean field for the
even Sm isotopes. Results are presented at the level of a BCS treatment, a number-projected BCS
treatment and an exact treatment using the Richardson ansatz. While projected BCS captures
much of the pairing correlation energy that is absent from BCS, it still misses a sizable correlation
energy, typically of order 1 MeV . Furthermore, because it does not average over the properties of
the fermion pairs, the exact Richardson solution permits a more meaningful definition of the Cooper
wave function and of the fraction of pairs that are collective.
PACS numbers: 21.60.-n, 03.75.Ss, 02.30.Ik, 74.20.Fg
The first breakthrough in the derivation of a micro-
scopic theory of superconductivity was the demonstra-
tion by Cooper [1] in 1956 that bound pairs could be
produced in the vicinity of the Fermi surface for an ar-
bitrarily small attractive interaction. This was followed
soon thereafter by the development of the BCS theory
[2], in which superconductivity was described as the con-
densation of a set of correlated pairs averaged over the
whole system. Soon after the BCS paper, Bohr, Mottel-
son and Pines [3] suggested that a similar phenomenon
could explain the large gaps in the spectra of even-even
nuclei. Since then, the BCS theory has been widely used
to describe superconductivity in condensed matter and
nuclear systems. Moreover, the concept of Cooper pairs
as strongly overlapping objects that go through a con-
densation process at the superconducting transition is
central in the interpretation of the superconducting phe-
nomenon. However, it is not easy to define the Cooper
pair wave function from the mean field BCS theory, and
most frequently it has been related to the pair correlator.
By using the exact solution of the BCS Hamiltonian
given by Richardson in the sixties [4], it was recently
shown [5] that the Cooper pair wave function in a super-
conducting medium has a precise definition. The unique
form of its wave function transforms from a Cooper reso-
nance in the weak coupling BCS region to a quasi bound
pair in the Bose-Einstein condensed (BEC) phase. More-
over, the Richardson solution gives a clear prescription
for evaluating the fraction of correlated pairs as compared
with Yang’s definition [6], providing a more accurate de-
scription of the condensation phenomenon.
The subject of Cooper pairing in atomic nuclei has
come under renewed focus recently in the context of the
mean-field Hartree-Fock-Bogolyubov approach [7, 8]. In
this work, we also explore the role of Cooper pairs in
mean-field treatments of atomic nuclei, comparing the
traditional number-nonconserving BCS approach with a
projected BCS approach and the exact Richardson treat-
ment. We show that substantial differences in correlation
energies arise when pairing is treated exactly for the same
pairing strength, and that interesting differences emerge
in some conceptual properties of the paired system.
We begin by detailing the differences between the three
approaches, focusing on a pairing Hamiltonian with con-
stant strength G acting in a space of doubly-degenerate
time-reversed states (k, k̄),
ck −G
′ck′ , (1)
where ǫk are the single-particle energies for the doubly-
degenerate orbits k, k̄.
Cooper studied the problem of adding a pair of
fermions with an attractive pairing interaction on top of
an inert Fermi sea (FS). He showed that the pair eigen-
state is
|ΨCooper〉 =
2ǫk − E
|FS〉 , (2)
where E is the energy eigenvalue. It turns out that E is
negative for any attractive value of G, implying that the
Cooper pair is bound and that the FS is unstable against
the formation of bound pairs. Cooper suggested [1] that
a theory considering a collection of bound pairs on top
of an effective FS could explain superconductivity.
The BCS approach follows a somewhat different path,
defining instead a variational wave function as a coherent
state of pairs properly averaged over the whole system,
|ΨBCS〉 = e
Γ† |0〉 , (3)
where Γ† =
is the coherent pair. The
BCS wave function breaks particle-number conservation.
Though errors due to the nonconservation of particle
number are negligible in the thermodynamic limit, they
can be important in finite systems such as atomic nuclei.
Indeed, Bohr, Mottelson and Pines [3] noted already in
1958 the importance of taking into account finite size ef-
fects in its application to nuclei. To accommodate these
http://arxiv.org/abs/0704.1594v1
effects, the number-projected BCS formalism (PBCS) [9]
assumes a condensed state of pairs of the form
|ΨPBCS〉 =
|0〉 , (4)
where M is the number of pairs and Γ† has the same
form as in BCS. We would like to emphasize here that Γ†
should not be confused with the operator that creates a
Cooper pair since its structure contains an average over
the correlated pairs close to the Fermi energy and the
free fermions deep inside the Fermi sphere.
The Richardson ansatz [4] for the exact solution of the
pairing Hamiltonian (1) follows closely Cooper’s original
idea. For a system with 2M particles, it involves (in the
ν = 0 sector) a product of M distinct pairs of the form
|Φ >=
Γ†α| 0〉 , Γ
2ǫk − eα
. (5)
The eα, called pair energies in analogy with the Cooper
wave function (2), are in general complex parameters,
which are obtained by solving the set of coupled non-
linear Richardson equations
2ǫk − eα
β( 6=α)=1,M
eβ − eα
= 0 . (6)
The energy eigenvalues are obtained by summing the
lowest M pair energies of each independent solution
The key point to note upon inspection of the Richard-
son pair (5) is that a pair energy close to a particular
2ǫk, i.e. close to the energy of an unperturbed pair, is
dominated by this particular configuration and thus de-
fines an uncorrelated pair. In contrast, a pair energy
lying sufficiently far away in the complex plane produces
a correlated Cooper pair.
As mentioned before, the BCS coherent pairs, with
amplitudes zk = vk/uk, cannot be interpreted as Cooper
pairs since they mix correlated and uncorrelated pairs
over the whole system. Indeed, it has been shown [5] that
only in the extreme BEC limit are all pairs bound and
condensed, and amenable to description by the two ap-
proaches. Usually the structure of the Cooper pair is as-
signed to the pair correlator 〈BCS| c
|BCS〉 = ukvk.
However, if the BCS state represents a fraction of cor-
related pairs within a Fermi sea of free uncorrelated
fermions, the pair correlator cannot guarantee that it
picks up the two fermions from the same pair. The pair
correlator is another averaged property over the set of
correlated pairs.
In what follows we explore the structure of pairing cor-
relations in the even Sm isotopes, from 144Sm through
158Sm. The results are based on a series of self-consistent
deformed Hartree Fock+BCS calculations. The calcula-
tions make use of the density-dependent Skyrme force,
SLy4, and treat pairing correlations using a pairing force
with constant strength G.
The calculations are carried out in an axially sym-
metric harmonic oscillator space of 11 major shells (286
doubly-degenerate single-particle states). This basis in-
volves oscillator parameters b0 and axis ratio q, opti-
mized in order to minimize the energy in the given space.
The strength of the pairing force for protons and neu-
trons is chosen in such a way as to reproduce the ex-
perimental pairing gaps in 154Sm (∆n = 0.98 MeV ,
∆p = 0.94 MeV ), extracted from the binding energies
in neighboring nuclei. We obtain Gn = 0.106 MeV and
Gp = 0.117 MeV . Once we have fitted this reference
strength, we determine the pairing strengths appropri-
ate to the 142−158Sm isotopic chain by assuming a 1/A
dependence. These calculations provide an excellent de-
scription of the properties of the even Sm isotopes.
We then use the results at self-consistency to define
the HF mean field and consider the alternative number-
conserving PBCS and exact Richardson approach to treat
the pairing correlations within this mean field. We ig-
nore the issue of whether the mean field should be self-
consistently modified in these other approaches. In this
way we are able to directly compare the three approaches
to pairing with the same pairing Hamiltonian, which is
the focus of this investigation.
As is well known that the numerical solution of the
Richardson equations (6) involves instabilities due to sin-
gularities arising at some critical values of the pairing
strength G. There have been two recent works that
study these critical regions of parameter space [10] and
propose ways to overcome the singularities [11]. While
these methods alleviate the numerical divergences, thus
allowing for an interpolation method to cross the criti-
cal regions, some problems still persist and we have thus
chosen to use a different approach. Since the singulari-
ties arise as crossings of real pair energies eα with the
unperturbed single-pair energies 2ǫk in the denomina-
tors of (6), we start the numerical procedure at strong
coupling (G = 1 MeV ) with complex single-particle en-
ergies, obtained by adding a small arbitrary imaginary
component. In this way, the singularities are avoided in
the evolution of the system from strong coupling almost
to the G = 0 limit. To obtain the exact solution at the
physical value of G, we then let the imaginary parts go to
zero starting with the solution already obtained for that
G value. The method seems to work for any distribution
of single-particle energies.
A principal focus of our investigation is on the pairing
correlation energy, defined as
EC = 〈Φcorr|H | Φcorr〉 − 〈Φuncorr|H | Φuncorr〉 , (7)
where |Φcorr〉 is the correlated ground-state wave func-
tion and |Φuncorr〉 is the uncorrelated Hartree Fock Slater
determinant obtained by filling all levels up to the Fermi
energy. This quantity reflects the additional energy that
derives from the inclusion of pairing.
Table 1 summarizes our results for the pairing corre-
lation energy in table 1 for all the even Sm isotopes un-
der consideration. Note that the calculations include the
semi-magic nucleus 144Sm, for which the BCS calculation
leads to a normal solution with no pairing correlation en-
ergy. In contrast, the projected BCS calculation leads to
substantial pairing correlations in the ground state. That
number projection is critical in mean-field treatments of
semi-magic nuclei is well known from other calculations
[12]. The exact treatment of pairing leads to a further
lowering of the energy of the ground state of the system,
by 0.3 MeV .
In the calculations other than 144Sm, the effect on the
pairing correlation energy of the exact solution is more
pronounced. While PBCS gives a significant lowering of
the energy of the system due to number projection, it
misses about 1 MeV of the full correlation energy of an
exact treatment. Considering the extensive recent efforts
to carry out systematic microscopic calculations of nu-
clear masses using mean-field methods [13], we feel that
this effect may be quite meaningful. It is not clear that a
renormalization of the strength of the pairing interaction
can accommodate these important corrections.
The results obtained for the Sm isotopes are consis-
tent with studies performed in ultrasmall superconduct-
ing grains [14]. The quantum phase transition from a
superconducting to a normal metal predicted by BCS
and PBCS completely disappears after fully including the
pairing fluctuations by means of the exact solution of the
BCS model. Moreover, the PBCS wave function displays
a strange behavior in the transitional region as compared
with the smooth behavior of the exact wave function [15].
Table I: Pairing correlation energies associated with the
BCS, PBCS and exact Richardson treatments of pairing
for the even Sm isotopes. All energies are given in MeV
Mass EC(Exact) EC(PBCS) EC(BCS)
142 -4.146 -3.096 -1.107
144 -2.960 -2.677 0.
146 -4.340 -3.140 -1.384
148 -4.221 -3.014 -1.075
150 -3.761 -2.932 -0.386
152 -3.922 -2.957 -0.637
154 -3.678 -2.859 -0.390
156 -3.716 -2.832 -0.515
158 -3.832 -2.824 -0.717
A second important feature of Cooper pairing is the
condensate fraction, namely the fraction of pairs of the
whole system that are correlated. Analysis of the off-
diagonal long-range order (ODLRO) that characterizes
superconductors and superfluids led Yang [6] to a defini-
tion of the condensate fraction, λ, in terms of the single
macroscopic eigenvalue of the two-body density matrix.
For a homogeneous system of two spin fermion species in
the thermodynamic limit, λ is given by
d3r1d
r2 |〈ψ↓ (r1)ψ↑ (r2)〉| =
k . (8)
This definition is not appropriate for finite Fermi sys-
tems, however, where several eigenvalues of the two-body
density matrix are of the same order. We modify it,
therefore, by excluding from the two-body density ma-
trix the amplitude of finding two uncorrelated fermions.
More specifically, our prescription for finite systems is to
evaluate the matrix elements of the operator
M(1−M/L)
k,k′=1
ck̄′ck〉 − 〈c
ck〉〈c
ck̄′〉 ,
where L is the total number of doubly-degenerate, canon-
ically conjugate pair states k, k̄.
In BCS approximation, the modified Yang prescription
leads to a condensate fraction
λBCS =
M(1−M/L)
. (10)
We have calculated this quantity for the BCS solutions
obtained for 154Sm as a function of the pairing strength
G and plot the results as the smooth curve in figure 1.
An alternative prescription for the condensate fraction
from the exact Richardson solution was proposed in [5]
and shown to more properly reflect the properties of a su-
perfluid system as it undergoes the crossover from BCS
to BEC. In particular, this new prescription gives a fully
condensed state at the change of sign of the chemical
potential where the whole system becomes bound. This
prescription, however, requires knowledge of the proper-
ties of the precise Cooper pairs in the problem, not an
average over the whole system as provided by the BCS
or PBCS approximations (3,4). The Richardson ansatz
(5) is ideally suited for this as it provides an exact wave
function for each individual Cooper pair. One has to sim-
ply distinguish which pairs are correlated and which are
not. As previously discussed, a correlated pair is char-
acterized by a pair energy eα that is far enough away in
the complex plane from any particular 2ǫk. We therefore
propose the following practical definition for the conden-
sate fraction. It is the fraction of pair energies which in
the complex energy plane lie further from any unperturbed
single-pair energy, 2ǫk, than the mean single-particle level
spacing.
We now return to a discussion of the condensate frac-
tion, as plotted in figure 1 for 154Sm as a function of
the pairing strength G. In addition to the results based
on the pair correlator, as discussed earlier, we also plot
(in the sawtooth curve) the results that derive from the
exact Richardson solution using the prescription just de-
scribed. To illustrate how these latter results emerge,
we show in figure 2 the associated pair energies for four
0.0 0.2 0.4 0.6 0.8
G (MeV)
Exact
FIG. 1: The modified Yang prescription for the BCS treat-
ment of pairing (smooth curve) and the alternative prescrip-
tion discussed in the text (sawtooth curve) for the exact
Richardson treatment. G0 = 0.106 MeV denotes the physical
value of the pairing strength and ǫ1 = µ denotes the strength
at which the whole system binds.
values of G in 154Sm, ranging from the physical value
of G = 0.106 MeV to a fairly strong pairing strength
of G = 0.4 MeV . In 154Sm the mean level spacing be-
tween the Hartree Fock single-particle levels is roughly
0.5 MeV , both around the Fermi surface and far from
it. For G = 0.106 MeV , most of the pair energies lie
very near the real axis and quite close to at least one un-
perturbed single-pair energy, 2ǫk. Two of them (which
form a complex conjugate pair) extend about 1 MeV
in the complex plane, while another two are marginally
collective, lying roughly 0.5 MeV from the closest 2ǫk.
The two most collective pairs, denoted C1 in the figure,
each have a real energy of −15.55MeV , which is roughly
twice the energy of the single-particle levels just below
the Fermi surface. This suggests that the first pairs that
become collective are indeed those built out of the va-
lence orbits. As G increases, we see a gradual increase
in the number of collective pairs, which form an arc in
the complex plane. As can be seen from figure 1, by a
pairing strength of roughly 0.5 MeV all of the pairs of
the system are correlated giving a condensate fraction of
1, even though the BEC regime has not yet been reached.
The BEC limit is realized when the chemical potential µ
crosses the lowest single-particle energy ǫ1 at G = 0.788
for 154Sm. At this point all pairs are bound. However,
the revised Yang prescription (9) fails to predict a com-
plete condensate at this point, in the same way as it fails
to do so in the homogeneous case [5].
The Richardson prescription for Cooper pairs also
gives rise to a different interpretation of their internal
structure. In figure 3, we compare the square of the wave
function for the most correlated Cooper pairs in 154Sm,
i.e. those whose pair energies lie farthest from any un-
-80 -60 -40 -20 0 20 40 60 80
-1.0 -0.5 0.0 0.5 1.0
-40 -20 0 20 40
-20 -15 -10 -5 0 5 10 15 20
Imaginary Part
G=0.4
2C2C1 C1
G=0.106
Imaginary Part
G=0.3
G=0.2
FIG. 2: Pair energies (in MeV ) for the exact Cooper pairs
that emerge from four calculations of the 154Sm isotope. G =
0.106 MeV is the physical value of the pairing strength. In
that panel, we denote the most collective pairs as Ci, for
subsequent notational purposes.
38 40 42 44 46 48 50 52
FIG. 3: Square of the wave function of the most collective
Cooper pairs in 154Sm (denoted C1, C2, C3, C4, and C5) and
the pair correlator (BCS) versus the single-particle levels.
perturbed single-pair energy, with the square of the pair
correlator wave function obtained from the BCS calcula-
tion. All wave functions are plotted versus the order of
the single-particle states to make clear the relevant mix-
ing of configurations in each pair. The pair labels in the
figure (C1 through C5) refer to corresponding labels in
the upper left panel of figure 2. C1 refers to the two most
collective pairs, namely those that are farthest from any
unperturbed single-particle pair. Being complex conju-
gate pairs, both have exactly the same absolute square
of their wave function and thus we only show one in the
figure. C2 refers to the next two most collective pairs,
which as noted earlier are marginally collective according
to our prescription. C3 refers to the next two most col-
lective pairs after C2, which according to the prescription
given above involve perturbative mixing of configurations
and are not collective. C4 and C5, the following pairs in
descending collective order, have real pair energies and
involve almost pure single-particle configurations.
¿From the figure, we see that the pair correlator wave
function is quite spread over several single-particle con-
figurations and is peaked between at the 47th single-
particle level, just beyond the Fermi energy (154Sm has
46 neutron pairs). In contrast, the most highly corre-
lated Cooper pair wave function C1 is somewhat narrower
(less collective) and is peaked slightly within the Fermi
sphere. The less-collective Cooper pairs, C2 through C3,
are peaked progressively further inside the Fermi sphere
and are progressively narrower. From this figure, we con-
clude that the size of even the most collective Cooper
pairs in coordinate space will be larger than the size of
the pair correlator, as was already demonstrated in the
weak coupling BCS regime of cold atomic gases [5]. Re-
cent investigations [7, 8] on the size of the pair correlator
in spherical nuclei have concluded that it is unexpectedly
small in the nuclear surface (2− 3 fm). The present cal-
culations would suggest that the actual size of the few
highly collective Cooper pairs is larger than the typical
size of the pair correlations in the nuclear medium. Fur-
thermore, as is also evident from the figure, less bound
pairs get progressively closer to a particular 2ǫk and the
corresponding Cooper pair wave function is less collec-
tive, i.e. more narrow in energy space, and peaked at
this particular configuration.
In this work, we have studied the role of Cooper pair-
ing in atomic nuclei, focusing on a realistic description
of the even Sm isotopes. We assume that the mean field
is given by the self-consistent HF solution from coupled
HF+BCS calculations, and then consider how the effects
of pairing on that mean field would be modified at several
levels of improved treatment. We consider both the pro-
jected BCS approximation and an exact treatment based
on Richardson’s solution of the pairing problem. Sev-
eral important points emerged. On the one hand, even
though PBCS approximation gives a significant gain in
binding energy over ordinary BCS, it still fails to capture
a sizable component, typically of order 1 MeV . This
might have important implications in efforts to derive
nuclear masses from a microscopic approach. Second, we
discussed a new and improved prescription for identify-
ing the fraction of the pairs in a nucleus that are collec-
tive, which can only be realized when the properties of
the various Cooper pairs in the problem are treated sep-
arately. This new prescription suggests that a slightly
larger number of pairs are collective when compared to
the more usual prescription based on Yang’s definition
of the condensate fraction. Furthermore, it suggests that
the few collective Cooper pairs that arise in real nuclei,
being individually less collective than the pair correlator,
would be spatially more spread out.
The Richardson solution, as generalized in ref. [16],
can be obtained for integrable pairing hamiltonians only.
It is possible, however, to use the Richardson ansatz (5)
in a variational treatment of general non-integrable pair-
ing hamiltonians. The pair energies would play the role
of variational parameters within a generalized Pfaffian
pairing wave function [17], making it possible to treat
pair correlations in a more precise manner for realistic
nuclear systems.
We acknowledge fruitful discussions with N. Sand-
ulescu, P. Schuck and W. Nazarewicz. This work was
supported in part by the Spanish DGI under grants
FIS2005-00640 and FIS2006-12783-C03-01, in part by the
US National Science Foundation under grant # 0553127,
and in part by UBACYT X-053, Carrera del Investigador
Cient́ıfico and PIP-5287 (CONICET-Argentina). One of
the authors (SP) wishes to acknowledge the generous sup-
port and hospitality of the CSIC in Madrid where much
of his contribution to the work was carried out
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http://arxiv.org/abs/nucl-th/0701086
http://arxiv.org/abs/nucl-th/0610061

  We consider the development of Cooper pairs in a self-consistent Hartree Fock
mean field for the even Sm isotopes. Results are presented at the level of a
BCS treatment, a number-projected BCS treatment and an exact treatment using
the Richardson ansatz. While projected BCS captures much of the pairing
correlation energy that is absent from BCS, it still misses a sizable
correlation energy, typically of order $1 MeV$. Furthermore, because it does
not average over the properties of the fermion pairs, the exact Richardson
solution permits a more meaningful definition of the Cooper wave function and
of the fraction of pairs that are collective.

Cooper pairs in atomic nuclei
G. G. Dussel1, S. Pittel2, J. Dukelsky3, and P. Sarriguren3
Departamento de Fisica J. J. Giambiagi, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Bartol Research Institute and Department of Physics and Astronomy,
University of Delaware, Newark, DE 19716 USA
Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain
(Dated: Received October 26, 2018)
We consider the development of Cooper pairs in a self-consistent Hartree Fock mean field for the
even Sm isotopes. Results are presented at the level of a BCS treatment, a number-projected BCS
treatment and an exact treatment using the Richardson ansatz. While projected BCS captures
much of the pairing correlation energy that is absent from BCS, it still misses a sizable correlation
energy, typically of order 1 MeV . Furthermore, because it does not average over the properties of
the fermion pairs, the exact Richardson solution permits a more meaningful definition of the Cooper
wave function and of the fraction of pairs that are collective.
PACS numbers: 21.60.-n, 03.75.Ss, 02.30.Ik, 74.20.Fg
The first breakthrough in the derivation of a micro-
scopic theory of superconductivity was the demonstra-
tion by Cooper [1] in 1956 that bound pairs could be
produced in the vicinity of the Fermi surface for an ar-
bitrarily small attractive interaction. This was followed
soon thereafter by the development of the BCS theory
[2], in which superconductivity was described as the con-
densation of a set of correlated pairs averaged over the
whole system. Soon after the BCS paper, Bohr, Mottel-
son and Pines [3] suggested that a similar phenomenon
could explain the large gaps in the spectra of even-even
nuclei. Since then, the BCS theory has been widely used
to describe superconductivity in condensed matter and
nuclear systems. Moreover, the concept of Cooper pairs
as strongly overlapping objects that go through a con-
densation process at the superconducting transition is
central in the interpretation of the superconducting phe-
nomenon. However, it is not easy to define the Cooper
pair wave function from the mean field BCS theory, and
most frequently it has been related to the pair correlator.
By using the exact solution of the BCS Hamiltonian
given by Richardson in the sixties [4], it was recently
shown [5] that the Cooper pair wave function in a super-
conducting medium has a precise definition. The unique
form of its wave function transforms from a Cooper reso-
nance in the weak coupling BCS region to a quasi bound
pair in the Bose-Einstein condensed (BEC) phase. More-
over, the Richardson solution gives a clear prescription
for evaluating the fraction of correlated pairs as compared
with Yang’s definition [6], providing a more accurate de-
scription of the condensation phenomenon.
The subject of Cooper pairing in atomic nuclei has
come under renewed focus recently in the context of the
mean-field Hartree-Fock-Bogolyubov approach [7, 8]. In
this work, we also explore the role of Cooper pairs in
mean-field treatments of atomic nuclei, comparing the
traditional number-nonconserving BCS approach with a
projected BCS approach and the exact Richardson treat-
ment. We show that substantial differences in correlation
energies arise when pairing is treated exactly for the same
pairing strength, and that interesting differences emerge
in some conceptual properties of the paired system.
We begin by detailing the differences between the three
approaches, focusing on a pairing Hamiltonian with con-
stant strength G acting in a space of doubly-degenerate
time-reversed states (k, k̄),
ck −G
′ck′ , (1)
where ǫk are the single-particle energies for the doubly-
degenerate orbits k, k̄.
Cooper studied the problem of adding a pair of
fermions with an attractive pairing interaction on top of
an inert Fermi sea (FS). He showed that the pair eigen-
state is
|ΨCooper〉 =
2ǫk − E
|FS〉 , (2)
where E is the energy eigenvalue. It turns out that E is
negative for any attractive value of G, implying that the
Cooper pair is bound and that the FS is unstable against
the formation of bound pairs. Cooper suggested [1] that
a theory considering a collection of bound pairs on top
of an effective FS could explain superconductivity.
The BCS approach follows a somewhat different path,
defining instead a variational wave function as a coherent
state of pairs properly averaged over the whole system,
|ΨBCS〉 = e
Γ† |0〉 , (3)
where Γ† =
is the coherent pair. The
BCS wave function breaks particle-number conservation.
Though errors due to the nonconservation of particle
number are negligible in the thermodynamic limit, they
can be important in finite systems such as atomic nuclei.
Indeed, Bohr, Mottelson and Pines [3] noted already in
1958 the importance of taking into account finite size ef-
fects in its application to nuclei. To accommodate these
http://arxiv.org/abs/0704.1594v1
effects, the number-projected BCS formalism (PBCS) [9]
assumes a condensed state of pairs of the form
|ΨPBCS〉 =
|0〉 , (4)
where M is the number of pairs and Γ† has the same
form as in BCS. We would like to emphasize here that Γ†
should not be confused with the operator that creates a
Cooper pair since its structure contains an average over
the correlated pairs close to the Fermi energy and the
free fermions deep inside the Fermi sphere.
The Richardson ansatz [4] for the exact solution of the
pairing Hamiltonian (1) follows closely Cooper’s original
idea. For a system with 2M particles, it involves (in the
ν = 0 sector) a product of M distinct pairs of the form
|Φ >=
Γ†α| 0〉 , Γ
2ǫk − eα
. (5)
The eα, called pair energies in analogy with the Cooper
wave function (2), are in general complex parameters,
which are obtained by solving the set of coupled non-
linear Richardson equations
2ǫk − eα
β( 6=α)=1,M
eβ − eα
= 0 . (6)
The energy eigenvalues are obtained by summing the
lowest M pair energies of each independent solution
The key point to note upon inspection of the Richard-
son pair (5) is that a pair energy close to a particular
2ǫk, i.e. close to the energy of an unperturbed pair, is
dominated by this particular configuration and thus de-
fines an uncorrelated pair. In contrast, a pair energy
lying sufficiently far away in the complex plane produces
a correlated Cooper pair.
As mentioned before, the BCS coherent pairs, with
amplitudes zk = vk/uk, cannot be interpreted as Cooper
pairs since they mix correlated and uncorrelated pairs
over the whole system. Indeed, it has been shown [5] that
only in the extreme BEC limit are all pairs bound and
condensed, and amenable to description by the two ap-
proaches. Usually the structure of the Cooper pair is as-
signed to the pair correlator 〈BCS| c
|BCS〉 = ukvk.
However, if the BCS state represents a fraction of cor-
related pairs within a Fermi sea of free uncorrelated
fermions, the pair correlator cannot guarantee that it
picks up the two fermions from the same pair. The pair
correlator is another averaged property over the set of
correlated pairs.
In what follows we explore the structure of pairing cor-
relations in the even Sm isotopes, from 144Sm through
158Sm. The results are based on a series of self-consistent
deformed Hartree Fock+BCS calculations. The calcula-
tions make use of the density-dependent Skyrme force,
SLy4, and treat pairing correlations using a pairing force
with constant strength G.
The calculations are carried out in an axially sym-
metric harmonic oscillator space of 11 major shells (286
doubly-degenerate single-particle states). This basis in-
volves oscillator parameters b0 and axis ratio q, opti-
mized in order to minimize the energy in the given space.
The strength of the pairing force for protons and neu-
trons is chosen in such a way as to reproduce the ex-
perimental pairing gaps in 154Sm (∆n = 0.98 MeV ,
∆p = 0.94 MeV ), extracted from the binding energies
in neighboring nuclei. We obtain Gn = 0.106 MeV and
Gp = 0.117 MeV . Once we have fitted this reference
strength, we determine the pairing strengths appropri-
ate to the 142−158Sm isotopic chain by assuming a 1/A
dependence. These calculations provide an excellent de-
scription of the properties of the even Sm isotopes.
We then use the results at self-consistency to define
the HF mean field and consider the alternative number-
conserving PBCS and exact Richardson approach to treat
the pairing correlations within this mean field. We ig-
nore the issue of whether the mean field should be self-
consistently modified in these other approaches. In this
way we are able to directly compare the three approaches
to pairing with the same pairing Hamiltonian, which is
the focus of this investigation.
As is well known that the numerical solution of the
Richardson equations (6) involves instabilities due to sin-
gularities arising at some critical values of the pairing
strength G. There have been two recent works that
study these critical regions of parameter space [10] and
propose ways to overcome the singularities [11]. While
these methods alleviate the numerical divergences, thus
allowing for an interpolation method to cross the criti-
cal regions, some problems still persist and we have thus
chosen to use a different approach. Since the singulari-
ties arise as crossings of real pair energies eα with the
unperturbed single-pair energies 2ǫk in the denomina-
tors of (6), we start the numerical procedure at strong
coupling (G = 1 MeV ) with complex single-particle en-
ergies, obtained by adding a small arbitrary imaginary
component. In this way, the singularities are avoided in
the evolution of the system from strong coupling almost
to the G = 0 limit. To obtain the exact solution at the
physical value of G, we then let the imaginary parts go to
zero starting with the solution already obtained for that
G value. The method seems to work for any distribution
of single-particle energies.
A principal focus of our investigation is on the pairing
correlation energy, defined as
EC = 〈Φcorr|H | Φcorr〉 − 〈Φuncorr|H | Φuncorr〉 , (7)
where |Φcorr〉 is the correlated ground-state wave func-
tion and |Φuncorr〉 is the uncorrelated Hartree Fock Slater
determinant obtained by filling all levels up to the Fermi
energy. This quantity reflects the additional energy that
derives from the inclusion of pairing.
Table 1 summarizes our results for the pairing corre-
lation energy in table 1 for all the even Sm isotopes un-
der consideration. Note that the calculations include the
semi-magic nucleus 144Sm, for which the BCS calculation
leads to a normal solution with no pairing correlation en-
ergy. In contrast, the projected BCS calculation leads to
substantial pairing correlations in the ground state. That
number projection is critical in mean-field treatments of
semi-magic nuclei is well known from other calculations
[12]. The exact treatment of pairing leads to a further
lowering of the energy of the ground state of the system,
by 0.3 MeV .
In the calculations other than 144Sm, the effect on the
pairing correlation energy of the exact solution is more
pronounced. While PBCS gives a significant lowering of
the energy of the system due to number projection, it
misses about 1 MeV of the full correlation energy of an
exact treatment. Considering the extensive recent efforts
to carry out systematic microscopic calculations of nu-
clear masses using mean-field methods [13], we feel that
this effect may be quite meaningful. It is not clear that a
renormalization of the strength of the pairing interaction
can accommodate these important corrections.
The results obtained for the Sm isotopes are consis-
tent with studies performed in ultrasmall superconduct-
ing grains [14]. The quantum phase transition from a
superconducting to a normal metal predicted by BCS
and PBCS completely disappears after fully including the
pairing fluctuations by means of the exact solution of the
BCS model. Moreover, the PBCS wave function displays
a strange behavior in the transitional region as compared
with the smooth behavior of the exact wave function [15].
Table I: Pairing correlation energies associated with the
BCS, PBCS and exact Richardson treatments of pairing
for the even Sm isotopes. All energies are given in MeV
Mass EC(Exact) EC(PBCS) EC(BCS)
142 -4.146 -3.096 -1.107
144 -2.960 -2.677 0.
146 -4.340 -3.140 -1.384
148 -4.221 -3.014 -1.075
150 -3.761 -2.932 -0.386
152 -3.922 -2.957 -0.637
154 -3.678 -2.859 -0.390
156 -3.716 -2.832 -0.515
158 -3.832 -2.824 -0.717
A second important feature of Cooper pairing is the
condensate fraction, namely the fraction of pairs of the
whole system that are correlated. Analysis of the off-
diagonal long-range order (ODLRO) that characterizes
superconductors and superfluids led Yang [6] to a defini-
tion of the condensate fraction, λ, in terms of the single
macroscopic eigenvalue of the two-body density matrix.
For a homogeneous system of two spin fermion species in
the thermodynamic limit, λ is given by
d3r1d
r2 |〈ψ↓ (r1)ψ↑ (r2)〉| =
k . (8)
This definition is not appropriate for finite Fermi sys-
tems, however, where several eigenvalues of the two-body
density matrix are of the same order. We modify it,
therefore, by excluding from the two-body density ma-
trix the amplitude of finding two uncorrelated fermions.
More specifically, our prescription for finite systems is to
evaluate the matrix elements of the operator
M(1−M/L)
k,k′=1
ck̄′ck〉 − 〈c
ck〉〈c
ck̄′〉 ,
where L is the total number of doubly-degenerate, canon-
ically conjugate pair states k, k̄.
In BCS approximation, the modified Yang prescription
leads to a condensate fraction
λBCS =
M(1−M/L)
. (10)
We have calculated this quantity for the BCS solutions
obtained for 154Sm as a function of the pairing strength
G and plot the results as the smooth curve in figure 1.
An alternative prescription for the condensate fraction
from the exact Richardson solution was proposed in [5]
and shown to more properly reflect the properties of a su-
perfluid system as it undergoes the crossover from BCS
to BEC. In particular, this new prescription gives a fully
condensed state at the change of sign of the chemical
potential where the whole system becomes bound. This
prescription, however, requires knowledge of the proper-
ties of the precise Cooper pairs in the problem, not an
average over the whole system as provided by the BCS
or PBCS approximations (3,4). The Richardson ansatz
(5) is ideally suited for this as it provides an exact wave
function for each individual Cooper pair. One has to sim-
ply distinguish which pairs are correlated and which are
not. As previously discussed, a correlated pair is char-
acterized by a pair energy eα that is far enough away in
the complex plane from any particular 2ǫk. We therefore
propose the following practical definition for the conden-
sate fraction. It is the fraction of pair energies which in
the complex energy plane lie further from any unperturbed
single-pair energy, 2ǫk, than the mean single-particle level
spacing.
We now return to a discussion of the condensate frac-
tion, as plotted in figure 1 for 154Sm as a function of
the pairing strength G. In addition to the results based
on the pair correlator, as discussed earlier, we also plot
(in the sawtooth curve) the results that derive from the
exact Richardson solution using the prescription just de-
scribed. To illustrate how these latter results emerge,
we show in figure 2 the associated pair energies for four
0.0 0.2 0.4 0.6 0.8
G (MeV)
Exact
FIG. 1: The modified Yang prescription for the BCS treat-
ment of pairing (smooth curve) and the alternative prescrip-
tion discussed in the text (sawtooth curve) for the exact
Richardson treatment. G0 = 0.106 MeV denotes the physical
value of the pairing strength and ǫ1 = µ denotes the strength
at which the whole system binds.
values of G in 154Sm, ranging from the physical value
of G = 0.106 MeV to a fairly strong pairing strength
of G = 0.4 MeV . In 154Sm the mean level spacing be-
tween the Hartree Fock single-particle levels is roughly
0.5 MeV , both around the Fermi surface and far from
it. For G = 0.106 MeV , most of the pair energies lie
very near the real axis and quite close to at least one un-
perturbed single-pair energy, 2ǫk. Two of them (which
form a complex conjugate pair) extend about 1 MeV
in the complex plane, while another two are marginally
collective, lying roughly 0.5 MeV from the closest 2ǫk.
The two most collective pairs, denoted C1 in the figure,
each have a real energy of −15.55MeV , which is roughly
twice the energy of the single-particle levels just below
the Fermi surface. This suggests that the first pairs that
become collective are indeed those built out of the va-
lence orbits. As G increases, we see a gradual increase
in the number of collective pairs, which form an arc in
the complex plane. As can be seen from figure 1, by a
pairing strength of roughly 0.5 MeV all of the pairs of
the system are correlated giving a condensate fraction of
1, even though the BEC regime has not yet been reached.
The BEC limit is realized when the chemical potential µ
crosses the lowest single-particle energy ǫ1 at G = 0.788
for 154Sm. At this point all pairs are bound. However,
the revised Yang prescription (9) fails to predict a com-
plete condensate at this point, in the same way as it fails
to do so in the homogeneous case [5].
The Richardson prescription for Cooper pairs also
gives rise to a different interpretation of their internal
structure. In figure 3, we compare the square of the wave
function for the most correlated Cooper pairs in 154Sm,
i.e. those whose pair energies lie farthest from any un-
-80 -60 -40 -20 0 20 40 60 80
-1.0 -0.5 0.0 0.5 1.0
-40 -20 0 20 40
-20 -15 -10 -5 0 5 10 15 20
Imaginary Part
G=0.4
2C2C1 C1
G=0.106
Imaginary Part
G=0.3
G=0.2
FIG. 2: Pair energies (in MeV ) for the exact Cooper pairs
that emerge from four calculations of the 154Sm isotope. G =
0.106 MeV is the physical value of the pairing strength. In
that panel, we denote the most collective pairs as Ci, for
subsequent notational purposes.
38 40 42 44 46 48 50 52
FIG. 3: Square of the wave function of the most collective
Cooper pairs in 154Sm (denoted C1, C2, C3, C4, and C5) and
the pair correlator (BCS) versus the single-particle levels.
perturbed single-pair energy, with the square of the pair
correlator wave function obtained from the BCS calcula-
tion. All wave functions are plotted versus the order of
the single-particle states to make clear the relevant mix-
ing of configurations in each pair. The pair labels in the
figure (C1 through C5) refer to corresponding labels in
the upper left panel of figure 2. C1 refers to the two most
collective pairs, namely those that are farthest from any
unperturbed single-particle pair. Being complex conju-
gate pairs, both have exactly the same absolute square
of their wave function and thus we only show one in the
figure. C2 refers to the next two most collective pairs,
which as noted earlier are marginally collective according
to our prescription. C3 refers to the next two most col-
lective pairs after C2, which according to the prescription
given above involve perturbative mixing of configurations
and are not collective. C4 and C5, the following pairs in
descending collective order, have real pair energies and
involve almost pure single-particle configurations.
¿From the figure, we see that the pair correlator wave
function is quite spread over several single-particle con-
figurations and is peaked between at the 47th single-
particle level, just beyond the Fermi energy (154Sm has
46 neutron pairs). In contrast, the most highly corre-
lated Cooper pair wave function C1 is somewhat narrower
(less collective) and is peaked slightly within the Fermi
sphere. The less-collective Cooper pairs, C2 through C3,
are peaked progressively further inside the Fermi sphere
and are progressively narrower. From this figure, we con-
clude that the size of even the most collective Cooper
pairs in coordinate space will be larger than the size of
the pair correlator, as was already demonstrated in the
weak coupling BCS regime of cold atomic gases [5]. Re-
cent investigations [7, 8] on the size of the pair correlator
in spherical nuclei have concluded that it is unexpectedly
small in the nuclear surface (2− 3 fm). The present cal-
culations would suggest that the actual size of the few
highly collective Cooper pairs is larger than the typical
size of the pair correlations in the nuclear medium. Fur-
thermore, as is also evident from the figure, less bound
pairs get progressively closer to a particular 2ǫk and the
corresponding Cooper pair wave function is less collec-
tive, i.e. more narrow in energy space, and peaked at
this particular configuration.
In this work, we have studied the role of Cooper pair-
ing in atomic nuclei, focusing on a realistic description
of the even Sm isotopes. We assume that the mean field
is given by the self-consistent HF solution from coupled
HF+BCS calculations, and then consider how the effects
of pairing on that mean field would be modified at several
levels of improved treatment. We consider both the pro-
jected BCS approximation and an exact treatment based
on Richardson’s solution of the pairing problem. Sev-
eral important points emerged. On the one hand, even
though PBCS approximation gives a significant gain in
binding energy over ordinary BCS, it still fails to capture
a sizable component, typically of order 1 MeV . This
might have important implications in efforts to derive
nuclear masses from a microscopic approach. Second, we
discussed a new and improved prescription for identify-
ing the fraction of the pairs in a nucleus that are collec-
tive, which can only be realized when the properties of
the various Cooper pairs in the problem are treated sep-
arately. This new prescription suggests that a slightly
larger number of pairs are collective when compared to
the more usual prescription based on Yang’s definition
of the condensate fraction. Furthermore, it suggests that
the few collective Cooper pairs that arise in real nuclei,
being individually less collective than the pair correlator,
would be spatially more spread out.
The Richardson solution, as generalized in ref. [16],
can be obtained for integrable pairing hamiltonians only.
It is possible, however, to use the Richardson ansatz (5)
in a variational treatment of general non-integrable pair-
ing hamiltonians. The pair energies would play the role
of variational parameters within a generalized Pfaffian
pairing wave function [17], making it possible to treat
pair correlations in a more precise manner for realistic
nuclear systems.
We acknowledge fruitful discussions with N. Sand-
ulescu, P. Schuck and W. Nazarewicz. This work was
supported in part by the Spanish DGI under grants
FIS2005-00640 and FIS2006-12783-C03-01, in part by the
US National Science Foundation under grant # 0553127,
and in part by UBACYT X-053, Carrera del Investigador
Cient́ıfico and PIP-5287 (CONICET-Argentina). One of
the authors (SP) wishes to acknowledge the generous sup-
port and hospitality of the CSIC in Madrid where much
of his contribution to the work was carried out
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http://arxiv.org/abs/nucl-th/0701086
http://arxiv.org/abs/nucl-th/0610061
