Termodinamika u malim konacnim sistemima

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

• Motivation and introduction
• SPINDIS computer program
• Description of the system
• Mean field, pairing
• Residual interactions
• Experimental results
• Iron 56 region

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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

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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

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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

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Nuclear excitations

• Energy landscape

Rotation
Quantum chaos
Eexc
Energy - temperature
Particle-hole
J
Angular momentum (deformation)
Collective motion
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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 72 064326 (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

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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
• How to efficiently generate microscopic
  configurations
• How to efficiently calculate distributions once
  we have configuration

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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

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Grand canonical
56Fe as an exmple
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Canonical

• 56Fe as an exmple

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Microcanonical

• Fixed number of particles, energy is averaged in
  arbitrary bins

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Microcanonical

• Fixed number of particles, energy is averaged in
  smaller bin, ltEgt

ltTgt oscillations
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All together

• equivalence at high excitation energies

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Microcanonical vs GC

• R. Pezer, A. Ventura and D. Vretenar, Nucl. Phys.
  A 717 (2003) 21.

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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

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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

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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

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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)

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SPINDIS PAIRING

• There are recent advances in providing the
  numerical solution Rombouts, Van Neck, Dukelsky
  PRC 69 061303 (2004) and Dominguez, Esbbag ,
  Dukelsky J. Phys. A 39 11349 (2006)

SPL

• Already Richardson provided a hint on how to
  numerically treat this set of equations cluster
  transformation after a change of variables

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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)

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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 (Gn,p0.22,0.2 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), 95--108. are
  also provided

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Problem with SPL

• See also Fig. 16 in S. Rombouts, K. Heyde, N.
  Jachowicz, Phys. Rev. C 58 (1998) 3295

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SPL neutrons

• Number of j-shells 13
• Number of particles 30
• lt-- Fermi level, 2 particles 56Fe
• i N Lj 2j1, P, energy MeV
• 1 1 s1/2 2, 1, -38.456
• 2 1 p3/2 4, -1, -30.552
• 3 1 p1/2 2, -1, -28.751
• 4 1 d5/2 6, 1, -21.914
• 5 2 s1/2 2, 1, -18.298
• 6 1 d3/2 4, 1, -18.034
• 7 1 f7/2 8, -1, -12.775
• 8 2 p3/2 4, -1, -8.765 lt--
• 9 2 p1/2 2, -1, -6.675
• 10 1 f5/2 6, -1, -6.476
• 11 1 g9/2 10, 1, -3.354
• 12 2 d5/2 6, 1, -0.595
• 13 3 s1/2 2, 1, -0.381

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SPL protons

• Number of j-shells 10
• Number of particles 26
• lt-- Fermi level, 6 particles 56Fe
• i N Lj 2j1, P, energy MeV
• 1 1 s1/2 2, 1, -34.429
• 2 1 p3/2 4, -1, -26.681
• 3 1 p1/2 2, -1, -24.873
• 4 1 d5/2 6, 1, -18.051
• 5 1 d3/2 4, 1, -14.182
• 6 2 s1/2 2, 1, -14.102
• 7 1 f7/2 8, -1, -8.814 lt--
• 8 2 p3/2 4, -1, -4.401
• 9 1 f5/2 6, -1, -2.526
• 10 2 p1/2 2, -1, -2.304

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Level densities
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Level densities
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Level densities
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Level densities
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Level densities
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Level densities
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Level densities
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Level densities parity projected
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Microcanonical temperature
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Angular momentum distribution
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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

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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