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 effects and 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 (Hubbard-Stratonovich transformation). 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. 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: 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)] Level density : the average level density is found from in the saddle- point approximation: S(E) = canonical entropy ; C = canonical heat capacity. Partition function : calculate the thermal energy and integrate to find the partition function

[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  Medium mass nuclei (A ~ ) 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, (2007) ] 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

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. A simple model [Y. Alhassid, G.F. Bertsch, L. Fang, and S. Liu; Phys. Rev. C 72, (2005)] (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 level densities versus excitation energy. 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. H. Nakada and Y.Alhassid, PRL 79, 2939 (1997); PLB 436, 231 (1998). is the mean occupation of quasi-particle orbitals with parity

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 parity level 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, (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: (PRL, 2008)] Several obstacles in extending SMMC to heavy nuclei. Conceptual difficulty: Technical difficulties:

Example : 162 Dy (even-even) Model space includes configurations ! (largest SMMC calculation) versus confirms rotational character with a moment of inertia: with (experimental value is ). 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 fpg 9/2 -shell. Single-particle energies from Woods-Saxon potential plus spin-orbit. Medium mass nuclei (A ~ ) 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 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:

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, (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 ( 66 Cr).

SMMC level density is in excellent agreement with experiments. Results from several experiments are fitted to a composite formula: constant temperature below E M and BBF above. Ground-state energy in SMMC has additional ~ 3 MeV of correlation energy 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 E M is determined by matching conditions at E M 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.