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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. 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 fluctuations. Good fits to the data are obtained using the backshifted Bethe formula (BBF): a = single-particle level density parameter. 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: with (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) 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. 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 renormalized. • 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 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). 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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