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I Nanoscience


									Statistical properties of nuclei: beyond the mean field

            Yoram Alhassid (Yale University)

 • Introduction

 • Beyond the mean field: correlations via fluctuations
     The static path approximation (SPA)
     The shell model Monte Carlo (SMMC) approach.

 • Partition functions and level densities in SMMC.

 • Level densities in medium-mass nuclei: theory versus experiment.

 • Projection on good quantum numbers: spin, parity,…

 • A theoretical challenge: the heavy deformed nuclei.

 • Conclusion and prospects.

  Statistical properties at finite temperature or excitation energy:
  level density and partition function, heat capacity, moment of inertia,…

• Level densities are an integral part of the Hauser-Feshbach theory
 of nuclear reaction rates (e.g., nucleosynthesis)

• Partition functions are required in the modeling of supernovae and stellar

• Study the signatures of phase transitions in finite systems.

 A suitable model is the interacting shell model: it includes both shell effects
 and residual interactions.

 However, in medium-mass and heavy nuclei the required model space is
 prohibitively large for conventional diagonalization.
            Beyond the mean field: correlations via fluctuations
 Non-perturbative methods are necessary because of the strong interactions.
 Mean-field approximations are tractable but often insufficient.
 Correlation effects can be reproduced by fluctuations around the mean field

 Gibbs ensemble          at temperature T can be written as a superposition
 of ensembles   of non-interacting nucleons in time-dependent fields

                                   (Hubbard-Stratonovich transformation).

 • Static path approximation (SPA): integrate over static fluctuations of the
   relevant order parameters.

 • Shell model Monte Carlo (SMMC): integrate over all fluctuations by
   Monte Carlo methods.
   Lang, Johnson, Koonin, Ormand, PRC 48, 1518 (1993);
   Alhassid, Dean, Koonin, Lang, Ormand, PRL 72, 613 (1994).

Enables calculations in model spaces that are many orders of magnitude larger.
                               Level densities
 Experimental methods: (i) Low energies: counting; (ii) intermediate
 energies: charged particles, Oslo method, neutron evaporation; (iii)
 neutron threshold: neutron resonances; (iv) higher energies: Ericson

 Good fits to the data are obtained using the
 backshifted Bethe formula (BBF):

  a = single-particle level density
  D = backshift parameter.
  But: a and D are adjusted for each nucleus
       and it is difficult to predict r
SMMC is an especially suitable method for microscopic calculations: correlation
effects are included exactly in very large model spaces (~10 29 for rare-earths)
          Partition function and level density in SMMC
              [H. Nakada and Y.Alhassid, PRL 79, 2939 (1997)]
Partition function: calculate the thermal energy                    and
integrate                    to find the partition function
Level density: the average level density is found from          in the saddle-
point approximation:

S(E) = canonical entropy;        C = canonical heat capacity.
                Medium mass nuclei (A ~ 50 -70)
            [Y.Alhassid, S. Liu, and H. Nakada, PRL 83, 4265 (1999)]

• Complete fpg9/2-shell, pairing plus surface-peaked multipole-multipole
interactions up to hexadecupole (dominant collective components).

                                               SMMC level densities are well fitted
                                               to the backshifted Bethe formula
                                                         Extract       and D

                                                  •      is a smooth function of A.

                                                  • Odd-even staggering effects
                                                  in D (a pairing effect).

• Good agreement with experimental data without adjustable parameters.
• Improvement over empirical formulas.
                Dependence on good quantum numbers
                              (i) Spin projection
        [Y. Alhassid, S. Liu and H. Nakada, Phys. Rev. Lett. 99, 162504 (2007) ]
        Spin distributions in even-odd, even-even and odd-odd nuclei

  Spin cutoff model:

• Spin cutoff model works very well
  except at low excitation energies.

• Staggering effect in spin for
 even-even nuclei.
                        Moment of inertia
Thermal moment of inertia can be extracted from:

  Signatures of pairing correlations:

  • Suppression of moment of inertia at low excitations in even-even nuclei.
  • Correlated with pairing energy of J=0 neutrons pairs.
                                   A simple model
    [Y. Alhassid, G.F. Bertsch, L. Fang, and S. Liu; Phys. Rev. C 72, 064326 (2005)]

Model: deformed Woods-Saxon potential plus pairing interaction.

(i) Number-parity projection : the major odd-even effects are described
    by a number-parity projection

•   Projects on even (h = 1) or odd (h = -1) number of particles.
                   is obtained from             by the replacement

                             (negative occupations !)

(ii) Static path approximation (SPA)

• include static fluctuations of the pairing order parameter.
iron isotopes (even-even and even-odd nuclei)

        • Good agreement with SMMC

        • Strong odd/even effect
                          (ii) Parity projection

 H. Nakada and Y.Alhassid, PRL 79, 2939 (1997);
 PLB 436, 231 (1998).

            even at neutron resonance
 energy (contrary to a common assumption
 used in nucleosynthesis).

           A simple model (I)
 Y. Alhassid, G.F. Bertsch, S. Liu, H. Nakada
 [Phys. Rev. Lett. 84, 4313 (2000)]

• The quasi-particles occupy levels with parity
according to a Poisson distribution.

     is the mean occupation of                  Ratio of odd-to-even parity level
  quasi-particle orbitals with parity           densities versus excitation energy.
                        A simple model (II)
                       [H. Chen and Y. Alhassid]

• Deformed Woods-Saxon potential plus pairing interaction.
• Number-parity projection, SPA plus parity projection.

                                                   Ratio of odd-to-even parity
      Ratio of odd-to-even parity
                                                   level densities
      partition functions
  Extending the theory to higher temperatures/excitation energies
                [Alhassid, Bertsch and Fang, PRC 68, 044322 (2003)]
• It is time consuming to include higher shells in the fully correlated calculations.
 We have combined the fully correlated partition in the truncated space with the
independent-particle partition in the full space (all bound states plus continuum)

                         • BBF works well up to T ~ 4 MeV
Extended heat capacity (up to T ~ 4 MeV)

                                                Experiment (Oslo)
         Theory (SMMC)

 • Strong odd/even effect: a signature of pairing phase transition
      A theoretical challenge: the heavy deformed nuclei
       [Y. Alhassid, L. Fang and H. Nakada, arXive:0710.1656 (PRL, 2008)]

• Medium-mass nuclei:

 small deformation, first excitation ~ 1- 2 MeV in even-even nuclei.

• Heavy nuclei:

 large deformation (open shell), first excitation ~ 100 keV, rotational bands.

• Conceptual difficulty:
 Can we describe rotational behavior in a truncated spherical shell model?

• Technical difficulties:
 Several obstacles in extending SMMC to heavy nuclei.
                     Example:   162Dy   (even-even)
 • Model space includes 1029 configurations ! (largest SMMC calculation)

•      versus     confirms rotational
character with a moment of inertia:

(experimental value is           ).

 • SMMC level density is in excellent
   agreement with experiments.

Rotational character can be reproduced
in a truncated spherical shell model !
          Even-odd and odd-odd rare-earth nuclei
                   (Ozen, Alhassid, Nakada)

• A sign problem when projecting on odd number of particles (at low T)
 • Fully microscopic calculations of statistical properties of nuclei are now
 possible by the shell model quantum Monte Carlo methods.
 • The dependence on good quantum numbers (spin, parity,…) can be
 determined using exact projection methods.
 • Simple models can explain certain features of the SMMC spin and parity

 • SMMC successfully extended to heavy deformed nuclei: rotational
 character can be reproduced in a truncated spherical shell model.


• Systematic studies of the statistical properties of heavy nuclei.

• Develop “global” methods to derive effective shell model interactions.
DFT -> configuration-interaction shell model map

 Correlation energies in N=Z sd shell nuclei
                Medium mass nuclei (A ~ 50 -70)

We have used SMMC to calculate the statistical properties of nuclei in the
iron region in the complete fpg9/2-shell.
• Single-particle energies from Woods-Saxon potential plus spin-orbit.

• The interaction includes the dominant components of realistic
effective interactions: pairing + multipole-multipole interactions
(quadrupole, octupole, and hexadecupole).

   Pairing interaction is determined to reproduce the experimental gap
  (from odd-even mass differences).

  Multipole-multipole interaction is determined self-consistently and

 • Interaction has a good Monte Carlo sign.
                      Parity distribution
  Alhassid, Bertsch, Liu and Nakada, Phys. Rev. Lett. 84, 4313 (2000)

 The distribution to find n particles in single-particle states with
 parity    is a Poisson distribution:

For an even-even nucleus:

 Where                                is the total Fermi- Dirac

  occupation in all states with parity
                                  Occupation distribution of the even-parity
                                  orbits (  ) in

  • Deviations from Poisson
  distribution for T < 1 MeV
    (pairing effect)

The model should be applied for
the quasi-particles:
        Thermal signatures of pairing correlations: summary

 Nanoparticles (D/d =1) versus nuclei
                            Heat capacity                    Experiment (Oslo)

      Spin susceptibility            Moment of inertia

• Pairing correlations (for D/d ~1) manifest through strong odd/even effects.
         Extending the theory to higher temperatures

[Y.Alhassid., G.F. Bertsch, and L. Fang, Phys. Rev. C 68, 044322 (2003)]

It is time consuming to include higher shells in the Monte Carlo

We have combined the fully correlated partition in the truncated
space with the independent-particle partition in the full space (all
bound states plus continuum):
(i) Independent-particle model
• Include both bound states and continuum:

                                              • Truncation to one major
                                              shell is problematic for T > 1.5
                                              • The continuum is important
                                              for a nucleus with a small
                                              neutron separation energy
Thermal energy vs. inverse temperature

 • Ground-state energy in SMMC
 has additional ~ 3 MeV of correlation
 energy as compared with
 Hartree-Fock-Boguliubov (HFB).

  • Results from several experiments
  are fitted to a composite formula:
  constant temperature below EM
  and BBF above.

 • SMMC level density is in excellent
   agreement with experiments.
   Experimental state density

 • An almost complete set of levels
 (with spin) is known up to ~ 2 MeV.

 (i) A constant temperature formula is
 fitted to level counting.

 (ii) A BBF above EM is determined by
  matching conditions at EM
        A composite formula

(iii) Renormalize Oslo data by fitting
 their data and neutron
resonance to the composite formula

 The composite formula is an excellent
 fit to all three experimental data.

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