Page.1 Microcanonical level densities of non-magic nuclei Robert Pezer University of Zagreb Croatia Alberto Ventura ENEA, Bologna Italy

Page.2 Outline  Motivation and introduction  SPINDIS computer program  Description of the system  Mean field, pairing  Residual interactions  Experimental results  Iron 56 region

Page.3 Motivation  Thermodynamical physical properties of the finite small system are interesting  Applications: astrophysics, heavy ion collisions, reaction rates...  Spectral properties of the high exctitation energy regime  Interesting subject of treatment of the nontrivial many-body interacting systems  Accumulation of precise experimental data (Oslo group method) that can be addresed properly within the microcanonical ensemble

Page.4 Introduction  Microcanonical thermodynamics is very atractive framework for describing some properties of the atomic nuclei – more specifically nuclear level densities (NLD)  PROBLEM: calculation is hard to performe even for pure mean field Hamiltonian, independent (quasi)particles for higher excitation energies  Complicated combinatorial problem addressed many times in the past  Rare examples of taking residual interactions into account even in some simplified form  Resort: multiplication by some convinient energy dependent factors – “rotate and vibrate by hand”

Page.5 Introduction  Canonical ensemble approach  Powerfull tool for description of systems that can be considered infinite (N/V)  Homogenity, contact with baths  Provide some information even for small finite systems  Use of saddle point approximation – assumption is that the number of degrees of freedom available to the nucleus is infinite  good for total LD (energy distribution) at higher excitation energies OK, but not so good for angular momentum distributions (notably in the vincinity of the yrast line levels) – Gaussian shape overestimate tail

Page.6 Nuclear excitations Energy landscape E exc J Energy - temperature Angular momentum (deformation) Quantum chaos Collective motion Particle-hole Rotation

Page.7 How to deal with this complexity?  Canonical ensemble approach – it is not easy to extract relevant physics even in this approach (for example Nakada Alhassid PRL 79 p2939 (1997), Alhassid at all PRC (2005) )  Microcanonical thermodynamics offer several atractive features  Proper and natural treatment of the conserved quantities  Canonical results easy to obtain once you get microcanonical level densities  Natural way to describe atomic nuclei

Page.8 Which way to go?  How to obtain microcanonical level density?  One road is clever state space transformation (truncation, Monte carlo states sampling), but full many-body dynamics (effective operators etc) – good for low energy and not too heavy nuclei, hard to do for high excitation energy  Another approach is to simplify Hamiltonian, and hope to keep enough residual dynamics, but work in a full state space – this is the road followed in this work  serious disadvantage of the shell model approach to NLD calculation is very large scale of the combinatorial problem involved – this separates into two parts 1.How to efficiently generate microscopic configurations 2.How to efficiently calculate distributions once we have configuration

Page.9 Problem & solution  For both problems SPINDIS algorithm offers solution  in fact it gives more – it also solves pairing problem within the single levels exactly  Computer program has been developed (D.K. Sunko, Comput. Phys. Comm. 101 (1997) 171.) that implement this powerfull mathematical method for effective generation of a full many- body state space (no core) and angular momentum distributions  Sofisticated microscopic configuration generator and distributions calculator  Provides good starting point for further development – subject of this work

Page.10 Grand canonical 56 Fe as an exmple

Page.11 Canonical 56 Fe as an exmple

Page.12 Microcanonical  Fixed number of particles, energy is averaged in arbitrary bins

Page.13 Microcanonical  Fixed number of particles, energy is averaged in smaller bin, oscillations

Page.14 All together  equivalence at high excitation energies

Page.15 Microcanonical vs GC R. Pezer, A. Ventura and D. Vretenar, Nucl. Phys. A 717 (2003) 21.

Page.16 SPINDIS: distributions  fast combinatorial algorithm for calculating the non-collective excitations of nuclei – given as a two component mixture of the neutrons and protons  Provides solution to the multiplicity problem of a single level in terms of a certain polynomials for a full Racah decomposition

Page.17 SPINDIS  Spherical multi-shell model multiplicity problem beeing solved simply by polynomial multiplication  the generating function is the product of the generating functions of individual levels  Fast and effective method for a numerical implementation  Provides full seniority basis generation for neutrons/protons subsystem when applied to atomic nuclei (It is 20th aniversary of the method)  This completes the formalism needed for the effective generation of the full state space that in addition diagonalizes the schematic Hamiltonian:

Page.18 SPINDIS features  Fast – suitable for any nuclei  General – any single particle level scheme is welcome  Each nucleus is treated individually  Effective seniority basis generation  Already include simple (diagonal pairing) residual interaction  Gives number of levels at a given excitation energy, total angular momenta and parity  Problem of residual interaction  How to improve long and short range interaction description

Page.19 SPINDIS & PAIRING  Treatment of the pairing interaction can be improved further by moving to more realistic constant pairing interaction:  Although the Hamiltonian looks simple, it is already nontrivial many-body problem that is not easy to solve  Solution utilised here is based on the landmark work of Richardson who showed that diagonalisation can be reduced to a set of nonlinear coupled equations for a limited number of complex variables (Bethe ansatz)

Page.20 SPINDIS & PAIRING  There are recent advances in providing the numerical solution: Rombouts, Van Neck, Dukelsky PRC (2004) and Dominguez, Esbbag, Dukelsky J. Phys. A (2006)  Already Richardson provided a hint on how to numerically treat this set of equations – cluster transformation after a change of variables SPL

Page.21 SPINDIS & PAIRING & rest  Methods can be readily implemented in SPINDIS but since we need to solve Richardson equations many times we need more effective approach than presented in literature  In the present version we have succeded in developing one that is fast enough for LD calculation  It is optimised variant of the cluster equations that are obtained after a, previously shown, invertible change of variables  We have also included monopole (diagonal) part of the residual interaction of the following form (Volya at al, Phys. Lett. B 509 (2001) p37):

Page.22 How good it is?  Iron region as an example  Input is SPL obtained from Woods-Saxon potential (abeit different from NDT) – same for all the nuclei considered here  Pairing strength is set to standard values from literature (G n,p =0.22,0.2 MeV)  Monopole interaction strength is set to 0.2 MeV  Also we have included gaussian shaped random interaction that smooth unphysical strong oscillations of the LD in controlled way (that have origin in use of spherical mean field)  Same set of parameters for all!  For comparison results from P. Demetriou and S. Goriely, Nucl. Phys. A 695 (2001), are also provided

Page.23 Problem with SPL See also Fig. 16 in S. Rombouts, K. Heyde, N. Jachowicz, Phys. Rev. C 58 (1998) 3295

Page.24 SPL neutrons Number of j-shells : 13 Number of particles : 30 <-- : Fermi level, 2 particles 56 Fe [ i ] : N Lj 2j+1, P, energy [MeV]: [ 1 ] : 1 s1/2 2, 1, [ 2 ] : 1 p3/2 4, -1, [ 3 ] : 1 p1/2 2, -1, [ 4 ] : 1 d5/2 6, 1, [ 5 ] : 2 s1/2 2, 1, [ 6 ] : 1 d3/2 4, 1, [ 7 ] : 1 f7/2 8, -1, [ 8 ] : 2 p3/2 4, -1, <-- [ 9 ] : 2 p1/2 2, -1, [ 10 ] : 1 f5/2 6, -1, [ 11 ] : 1 g9/2 10, 1, [ 12 ] : 2 d5/2 6, 1, [ 13 ] : 3 s1/2 2, 1,

Page.25 SPL protons Number of j-shells : 10 Number of particles : 26 <-- : Fermi level, 6 particles 56 Fe [ i ] : N Lj 2j+1, P, energy [MeV] : [ 1 ] : 1 s1/2 2, 1, [ 2 ] : 1 p3/2 4, -1, [ 3 ] : 1 p1/2 2, -1, [ 4 ] : 1 d5/2 6, 1, [ 5 ] : 1 d3/2 4, 1, [ 6 ] : 2 s1/2 2, 1, [ 7 ] : 1 f7/2 8, -1, <-- [ 8 ] : 2 p3/2 4, -1, [ 9 ] : 1 f5/2 6, -1, [ 10 ] : 2 p1/2 2, -1,

Page.26 Level densities

Page.27 Level densities

Page.28 Level densities

Page.29 Level densities

Page.30 Level densities

Page.31 Level densities

Page.32 Level densities

Page.33 Level densities – parity projected

Page.34 Microcanonical temperature

Page.35 Angular momentum distribution

Page.36 Perspective  Call for standard treatment of the input  Optimised SPL for NLD calculations  Improvement in this sector is of highest importance  Inclusion of the diagonal quadrupole- quadrupole interaction in np chanel  It is already implemented in SPINDIS but not tested enough to draw reliable conclusions yet  Complicated calculations involving CFP  Playing with random interactions seems promising

Page.37 Summary  SPINDIS algorithm has been described and comparison of the results in micro(macro)canonical formalisms provided  Description of the system  Hamiltonian  Richardson equations  Residual interactions  results in iron region  Total NLD and parity projected  Angular momentum distribution  Microcanonical “temperature”