MICROSCOPIC CALCULATION OF SYMMETRY PROJECTED NUCLEAR LEVEL DENSITIES Kris Van Houcke Ghent University In collaboration with S. Rombouts, K. Heyde, Y.

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MICROSCOPIC CALCULATION OF SYMMETRY PROJECTED NUCLEAR LEVEL DENSITIES Kris Van Houcke Ghent University In collaboration with S. Rombouts, K. Heyde, Y. Alhassid May 24, 2007

Back-shifted Bethe formula (semi-empirical) H. Bethe, Phys. Rev. 50, 332 (1936); Rev. Mod. Phys. 9, 69 (1937). T. von Egidy and D. Bucurescu, Phys. Rev. C 72, (2005 ). HF+BCS (shell, pairing, deformation effects) or HFB P. Demetriou and S. Goriely, Nucl. Phys. A695, 95 (2001). Shell Model Monte Carlo (pairing+quadrupole) S.E. Koonin, D.J. Dean and K. Langanke, Phys. Rep. 278, 1 (1997). W.E. Ormand, Phys. Rev. C, 56, R1678 (1997). H. Nakada and Y. Alhassid, Phys. Rev. Lett. 79, 2939 (1997). Y. Alhassid, G. F. Bertsch, S. Liu, and H. Nakada, Phys. Rev. Lett. 84, 4313 (2000). Calculating NLD

Saddle-point approximation 56 Fe sd+pf+g 9/2 +sd “smoothing procedure” for thermodynamics

Spin cut-off model (uncorrelated randomly coupled spins) projection in HF+BCS or HFB parity projection in SMMC: strong parity dependence in total NLD ( 56 Fe, 60 Ni, 68 Zn) Y. Alhassid, G. F. Bertsch, S. Liu, and H. Nakada, Phys. Rev. Lett. 84, 4313 (2000). Recent experiments: no parity dependence for J=2 states in 90 Zr (E x = 8-14 MeV) 58 Ni Y. Kalmykov et al., Phys. Rev. Lett. 96, (2006). Recently: projection on angular momentum in SMMC Sign problem except for J=0 Y. Alhassid, S. Liu, and H. Nakada, nucl-th/ Angular momentum and parity dependence

The sign problem in quantum Monte Carlo QMC = a statistical evaluation of the partition function, based on a decomposition of Markov chain of configurations (Metropolis algorithm): biggest restriction: the sign problem, model dependent there are roughly two main categories:  auxiliary-field decompositions  path-integral decompositions

Decompositions auxiliary-field method (SMMC) Hubbard-Stratonovich: our starting point:

A pairing model SU(2) algebra ε j from Woods-Saxon potential Formulate the pairing problem in quasi-spin formalism “Nuclear Superfluidity – Pairing in Finite systems”, D.M. Brink and R.A. Broglia, Cambridge University Press (2005). simulate directly in quasi-spin basis (“spin chain”), with recently developed “canonical loop QMC algorithm” No sign problem for all J, and odd particle numbers !! S. Rombouts, K. Van Houcke and L. Pollet, Phys. Rev. Lett. 96, (2006). K. Van Houcke, S. Rombouts and L. Pollet, Phys. Rev. E 73, (2006). J is a good quantum number !!

Some simulation results Iron region 10 and 11 neutrons in a pf + sdg valence space

Projection on spin Iron region 10 and 11 neutrons in a pf + sdg valence space 56 Fe pf valence space

Distribution in J Signature of isovector pairing effective moment of inertia: spin cut-off parameter in semi-classical approach:

Reduction of effective moment of inertia + increasing pairing correlation energy Nuclear Superfluidity

Signatures of nuclear superfluidity in the NLD 56 Fe valence space: sd + pf + g 9/2 + sd

Back-shifted Bethe parameters 56 Fe ( MeV) SMMC ( MeV): 55 Fe ( MeV) 57 Fe ( MeV) A.V. Voinov et al., Phys. Rev. C. 74, (2006).

Heat capacity 56 Fe valence space: sd + pf + g 9/2 + sd Signatures of the Pairing phase transition

Angular momentum + parity projection in 56 Fe strong parity dependence!!! We consider sd + pf + g 9/2 + sd valence space pf + g 9/2 valence space is too small for study of parity dependence

Conclusions and outlook We solved general isovector pairing model at finite T via QMC Include angular momentum + parity projection without sign problem Signatures of pairing (nuclear superfluidity) can be found in the angular momentum distribution of the NLD Even projected on J, there is still a strong parity dependence in the NLD K. Van Houcke, S. Rombouts, K. Heyde and Y. Alhassid, nucl-th/ v1. Future: use deformed mean-field + pairing study the parity dependence

Motivation a crucial ingredient for the calculation of scattering cross sections: Fermi’s golden rule: (Hauser-Feshbach, Weisskopf-Ewing …) angular momentum distribution + parity dependence:  statistical nuclear reaction rates (thermal stellar reactions)  determination of NLD from measured proton or neutron resonances

The level density a fundamental property of a many-body system calculate Z(β) (has smooth behavior), and inverse Laplace Z(β) via Monte Carlo statistical error saddle-point approximation: