Cooper pairs in Atomic Nuclei

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• Cooper pairs in Atomic Nuclei

• In this talk, I will describe some work I have
  been involved in with Jorge Dukelsky (Madrid),
  Billy Dussel (Buenos Aires) and Pedro Sarigurren
  (Madrid) which has focused on the properties of
  Cooper pairs in atomic nuclei. The work was
  recently published in
• Phys. Rev. C 76, 011302(R) (2007)
• An outline of my talk is as follows
• 1. Brief historical overview on pairing.
• 2. Richardsons exact solution of the pairing
  problem
• 3. Some general properties of Cooper pairs
• 4. Our study of Cooper pairs in atomic nuclei

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Historical Overview

• Superconductivity discovered in lab of
  Kammerlingh Onnes (1911)
• Cooper (1956) showed that bound pairs could be
  produced in vicinity of Fermi surface for
  arbitrarily small attractive pairing interaction.
• He then suggested that superconductivity could
  perhaps be described as a collection of pairs of
  this kind outside a Fermi sea.
• Bardeen, Cooper and Schrieffer (1957) presented
  BCS theory whereby superconductivity was
  described as a number-nonconserving condensate of
  these Cooper pairs,

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• Bohr, Mottelson and Pines (1958) soon thereafter
  suggested that a similar phenomenon could explain
  large gaps in spectra of even-even atomic nuclei.
  Emphasized however that finite-size effects would
  be critical in describing such systems with small
  number of particles.
• Dietrich, Mang and Pradal (1964) showed how to
  incorporate number conservation into the BCS
  formalism (called PBCS) by using a trial wave
  function
• Important to note that the operator G that
  enters in both BCS theory and PBCS theory is not
  the same beast as the operator that creates a
  Cooper pair. The BCS operator averages over the
  correlated pairs that are close to the Fermi
  energy and also the free uncorrelated fermions
  that are deep inside the Fermi sea.
• Richardson (1963) showed that for a pure pairing
  Hamiltonian Coopers ideas can be generalized and
  that the associated Schrödinger equation can be
  solved exactly for many particles outside a Fermi
  sea, and in this way provide a precise
  definition of the associated Cooper pairs.

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Exact solution of the pairing hamiltonian

• R.W. Richardson, Phys. Lett. 3, 277 (1963) Phys.
  Rev. 141, 949 (1966)
• Eigenvalue equation
• Proposed ansatz for eigenstates (a generalization
  of the Cooper ansatz)
• Here, ? represents the number of unpaired
  particles, i.e. the seniority, and M is the
  number of Cooper pairs. For the ground state,
  ?0 and ?gt 0gt.
• Note also that each Cooper pair in the exact
  solution is distinct.

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• Exact solutions obtained for the ea by solving
  the Richardson equations

Properties -This is a set of M nonlinear coupled
algebraic equations with M unknowns, the e? ,
which are called the pair energies. -The first
and second terms correspond to the equations for
the one-pair system. The third term contains the
many-body correlations and the exchange
symmetry. -The pair energies are either real or
arise in complex conjugated pairs. -There are as
many independent solutions as states in the
Hilbert space. -The solutions can be readily
classified in the weak-coupling (g ? 0) limit.
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Numerical Solution of Richardson Equations

• The Richardson equations typically solved by
  starting in the weak-coupling (g ? 0), where the
  solutions are simple and known, and evolving to
  the physical of g. The Richardson equations have
  singularities, however, when a pair energy is
  real and equals twice a single-particle energy,
  and this makes it difficult to follow evolution
  of the solutions for all pairs with increasing g
  all the way to the physical value.
• In our recent article PRC76, 011302 ( R) (2007)
  , we proposed a practical way to avoid
  singularity problems
• (a) Start numerical procedure using complex
  single-particle energies, by adding a small
  arbitrary imaginary part to them. In this way,
  singularities avoided as we evolve with g.
• (b) Once we reach the physical value of g, we
  let the imaginary parts all go to zero to get
  physical solutions.
• (c) Method seems to work for any distribution of
  single-particle energies and any strength g and
  enables us to obtain all solutions to Richardson
  equations, for very large number of pairs and
  levels.

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General features of pair correlations

• Pairing responsible for crossover from BCS to BEC
  behavior.
• In BCS regime, Cooper pairs spread out over
  system. In BEC regime, they are spatially
  correlated (quasibound molecules). Smooth
  transition between the two.
• This behavior as a function of g emerges nicely
  from the Richardson solution of the pairing
  hamiltonian J. Dukelsky and G. Ortiz, Phys.
  Rev. A 72, 043611 (2005)

Quasibound molecules Pair resonaces
Free fermions
Quasibound molecules Pair
resonances
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Condensate Fraction BCS Definition

• An important feature of Cooper pairing is the
  condensate fraction, namely the fraction of pairs
  in the whole system that are correlated.
• In BCS, no direct measure of condensate fraction.
  Thus, typically determined using definition due
  to Yang RMP 34, 694 (1962) obtained by
  analyzing the Off Diagonal Long Range Order that
  characterizes all superconductors and
  superfluids. This led to a definition of the
  condensate fraction appropriate to homogeneous
  systems in the thermodynamic limit.
• For finite fermi systems, must modify Yangs
  definition. Appropriate modification for analysis
  of BCS solutions is
• where L is total number of doubly-degenerate
  states and M is number of pairs.

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Condensate Fraction From Richardson solution

• Richardsons solution permits a more precise
  definition of the fraction of collective pairs.
  Namely it is the fraction of the pairs whose pair
  energies are far away in the complex plane from
  any particular single-pair energy, 2ek . From
  the structure of the Cooper pair wave function,
• it is clear that when this is the case the
  Cooper pair will indeed be a coherent linear
  combination of several single-particle levels and
  thus be collective.
• As a practical measure, we define a Cooper pair
  arising from Richardsons solution as collective
  if its pair energy lies more than the mean-level
  spacing away from any unperturbed single-pair
  energy.
• This definition has the feature that it yields a
  fraction of 1 when the chemical potential crosses
  the lowest single-particle energy, as it should.
  As we will soon see, Yangs prescription doesnt.

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Cooper Pairs in the even Samarium isotopes

• We study these various aspects of pairing, as
  reflected in nuclei, in the context of the even
  Samarium isotopes, for which Z62 and N80-96.
• We assume a Skyrme force (SLy4) to define the
  mean field and supplement it by a pure pairing
  force.
• Pairing force strength chosen to reproduce
  experimental pairing gaps in 154Sm (?n0.98 MeV,
  ?p 0.94 MeV) when the problem is treated in
  self-consistent Hartree-Fock BCS approximation.
  Resulting strengths are gn0.10620 MeV and
  gp0.11675 MeV. We then include an isotope
  dependence in the pairing strength, gg0/A, as we
  move through the isotope chain.
• Results at self-consistency used to define the
  axially-symmetric HF mean field. Pairing
  correlations then treated with the three
  different approaches
• 1. Pure BCS approximation
• 2. Projected BCS approximation
• 3. The exact Richardson solution.
• Calculations carried out using 11 active
  harmonic oscillator shells, i.e. 286 active
  doubly-degenerate HF levels.

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Correlation energies (in MeV)
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Condensate Fractions - 154Sm
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Pair energies (in MeV) for 154Sm
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Structure of Cooper Pairs for g0.106 MeV
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Summary

• General conclusions on Cooper pairs
• If you consider the exact wave function for a
  pairing model, you find that the ground state of
  the superconducting/superfluid state consists of
  a fraction of Cooper resonances with different
  sizes and different structures immersed in a
  Fermi sea.
• On the contrary, the mean-field BCS or
  projected-BCS wave function pairs off all
  fermions in a unique paired state.
• Furthermore, the analysis of the exact wave
  function suggests an alternative definition for
  the condensate fraction than the standard Yang
  definition, which seems to better describe the
  transition from BCS to BEC.

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• Conclusions for Cooper pairs in nuclei
• For the finite Sm nuclear systems we studied,
  PBCS improves significantly on BCS, but it is
  still far from the exact solution. Typically,
  PBCS misses of order 1 MeV in binding energy,
  except at a closed shell where it is closer.
  Minimally, we would need to renormalize pairing
  strength to correct this.
• We also find that for the physical pairing
  strength appropriate to nuclei in the Sm
  isotopes, only the outermost few valence nucleons
  are coherently paired.
• Lastly, we see that the associated Cooper pairs
  gradually move further and further into the
  nuclear interior, successively dominated by
  orbits further and further below the Fermi
  energy.

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