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.

Introduction

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 collapse

• Study the signatures of phase transitions in finite systems.

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

However, in medium-mass and heavy nuclei the required model space is prohibitively large for conventional diagonalization.

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

Beyond the mean field: correlations via fluctuations

He G U D (Hubbard-Stratonovich transformation).

/H Te

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

• 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 fluctuations.

5/ 4 21/ 412

xa E

x xE a E e

a = single-particle level density parameter.

= backshift parameter.

and it is difficult to predict But: a and are adjusted for each nucleus

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

SMMC is an especially suitable method for microscopic calculations: correlationeffects are included exactly in very large model spaces (~1029 for rare-earths)

Partition function and level density in SMMC

[H. Nakada and Y.Alhassid, PRL 79, 2939 (1997)]

Level density: the average level density is found from in the saddle-point approximation:

S(E) = canonical entropy; C = canonical heat capacity.

2

1

2

S E

T CE e

( ) lnS E Z E 2 /C E

ln / ( )Z E ( )Z Partition function: calculate the thermal energy and integrate to find the partition function

( )Z

( )E H

[Y.Alhassid, S. Liu, and H. Nakada, PRL 83, 4265 (1999)]

• is a smooth function of A.

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

• Good agreement with experimental data without adjustable parameters.

• Improvement over empirical formulas.

SMMC level densities are well fitted to the backshifted Bethe formula

Extract and

a

a

Medium mass nuclei (A ~ 50 -70)

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

Spin distributions in even-odd, even-even and odd-odd nuclei

(i) Spin projection [Y. Alhassid, S. Liu and H. Nakada, Phys. Rev. Lett. 99, 162504 (2007) ]

2( 1) / 2

3

2 1

2 2J JJ J

e

Spin cutoff model:

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

• Staggering effect in spin for even-even nuclei.

Dependence on good quantum numbers

22

IT

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.

Moment of inertia

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 ( = 1) or odd ( = -1) number of particles.

ˆ11

2i NP e

A simple model [Y. Alhassid, G.F. Bertsch, L. Fang, and S. Liu; Phys. Rev. C 72, 064326 (2005)]

1/1 kE Tk kf f e

(negative occupations !)

• include static fluctuations of the pairing order parameter.

(ii) Static path approximation (SPA)

is obtained from by the replacement

iron isotopes (even-even and even-odd nuclei)

• Good agreement with SMMC

• Strong odd/even effect

(ii) Parity projection

Ratio of odd-to-even parity leveldensities versus excitation energy.

even at neutron resonanceenergy (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.

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

/ tanhZ Z f

is the mean occupation of quasi-particle orbitals with parity

f

A simple model (II)[H. Chen and Y. Alhassid]

• Deformed Woods-Saxon potential plus pairing interaction.

Ratio of odd-to-even parity partition functions

Ratio of odd-to-even paritylevel densities

• Number-parity projection, SPA plus parity projection.

• 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)

Extending the theory to higher temperatures/excitation energies

[Alhassid, Bertsch and Fang, PRC 68, 044322 (2003)]

• BBF works well up to T ~ 4 MeV

Extended heat capacity (up to T ~ 4 MeV)

• Strong odd/even effect: a signature of pairing phase transition

Theory (SMMC)Experiment (Oslo)

A theoretical challenge: the heavy deformed nuclei

• 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.

Can we describe rotational behavior in a truncated spherical shell model?

[Y. Alhassid, L. Fang and H. Nakada, arXive:0710.1656 (PRL, 2008)]

Several obstacles in extending SMMC to heavy nuclei.

• Conceptual difficulty:

• Technical difficulties:

Example: 162Dy (even-even)

• Model space includes 1029 configurations ! (largest SMMC calculation)

• versus confirms rotational character with a moment of inertia:

with(experimental value is ).

2J T

135.5 3.3I MeV 137.2MeV

2 2J IT

Rotational character can be reproduced in a truncated spherical shell model !

• SMMC level density is in excellent agreement with experiments.

Even-odd and odd-odd rare-earth nuclei (Ozen, Alhassid, Nakada)

• A sign problem when projecting on odd number of particles (at low T)

Conclusion

• 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 distributions.

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

Prospects

• Systematic studies of the statistical properties of heavy nuclei. • Develop “global” methods to derive effective shell model interactions.

Correlation energies in N=Z sd shell nuclei

DFT -> configuration-interaction shell model map

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.

Medium mass nuclei (A ~ 50 -70)

• Interaction has a good Monte Carlo sign.

• 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 renormalized.

Parity distributionAlhassid, 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:

( )!

nff

P n en

For an even-even nucleus:

_

_

( )( )tanh

( ) ( )n odd

n even

P nZf

Z P n

Where is the total Fermi- Dirac

occupation in all states with parity

f

( )

1

1 aEaf

e

Occupation distribution of the even-parityorbits ( ) in 9/ 2g 60Ni

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

The model should be applied for the quasi-particles:

1

1 aa Ea a

f fe

Nanoparticles (

versus nuclei

Spin susceptibility Moment of inertia

Heat capacity

Thermal signatures of pairing correlations: summary

Experiment (Oslo)

• Pairing correlations (for ~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 approach.

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 MeV.

• The continuum is important for a nucleus with a small neutron separation energy (66Cr).

• SMMC level density is in excellent agreement with experiments.

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

and BBF above.

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

Thermal energy vs. inverse temperature

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 neutronresonance to the composite formula

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