1. A Recursive Method to Calculate Nuclear Level Densities Models for nuclear level densities Level density for a harmonic oscillator potential Simple illustrations Extension to general potentials Piet Van Isacker GANIL, France

2. Models for nuclear level densities « An Attempt to Calculate the Number of Energy Levels of a Heavy Nucleus » (Bethe 1936): Statistical analysis of Fermi gas of independent particles. Numerous extensions: eg back shift. « Theory of Nuclear Level Density » (Bloch 1953); « Influence of Shell Structure on the Level Density of a Highly Excited Nucleus » (Rosenzweig 1957): ‘Exact’ counting methods in single-particle shell model. Numerous extensions (Zuker, Paar, Pezer,... ). « Nuclear Level Densities and Partition Functions with Interactions » (French & Kota 1983): Effects of residual interaction via spectral distribution method. « [] Level Densities [] in Monte Carlo Shell Model » (Nakada & Alhassid 1997); « Estimating the Nuclear Level Density with the Monte Carlo Shell Model » (Ormand 1997): ‘Exact’ shell-model calculations.

3. Level density in a harmonic oscillator Question: How many (antisymmetric) states with an energy E t exist for A particles in an isotropic HO? Answer: Given by the number of solutions of Solution: c 3 (A,Q) calculated recursively through

4. Solution method We need the number of solutions of Rewrite as Introduce new unknowns Hence we find the recurrence relation:

5. Harmonic oscillator with spin Simple numerical implementation: c 3 (A,Q) can be calculated to very high excitation. Example: The number of independent Slater determinants for A=70 (s=1/2) particles at an excitation energy of 30 h w is spin=1/2; deg=2*spin+1; c[d_,aa_,qq_]:=c[d,aa,qq]= Sum[c[d,aa-aap,qq-qqp-aa+aap]*c[d-1,aap,qqp], {aap,0,aa},{qqp,qqmin[d-1,aap],qq-aa+aap-qqmin[d,aa-aap]}]; c[d_,aa_,qq_]:=Binomial[deg,aa]/; d==0 && qq==0; c[d_,aa_,qq_]:=1/; aa==0 && qq==0; c[d_,aa_,qq_]:=0/; aa==0 && qq!=0; c[d_,aa_,qq_]:=0/; qq<qqmin[d,aa];

6. Comparison with Fermi-gas estimate Fermi-gas estimate (Bethe; cfr Bohr & Mottelson): Correspondence:

7. Leonhard Euler L Euler in Novi Commentarii Academiae Scientiarum Petropolitanae 3 (1753) 125: Tables for the ‘one-dimensional oscillator’ problem.

8. Enumeration of spurious states Only states that are in the ground configuration with respect to the centre-of-mass excitation are of interest. c 3 (A,Q) includes all solutions. Let us denote the physical solutions as This is found by substracting from c 3 (A,Q) those states that can be constructed by acting with the step- up operator for the centre-of-mass motion. Hence:

9. Harmonic oscillator with isospin Question: How many states with an energy E exist for N neutrons and Z protons in a HO? Answer: Given by the number of solutions of Solution: c 3 (N,Z,Q) can be calculated recursively or through

10. Shell effects Fermi-gas estimate (Bethe; cfr Bohr & Mottelson): The quantity c 3 (N,Z,Q) can be evaluated for closed as well as open shells => effects of shell structure on level densities. Example: Comparison of 16 O and 28 Si.

11. Anisotropic harmonic oscillator So far: independent particles in a spherical HO => interaction effects (eg deformation) are not included. The analysis can be repeated for an anisotropic HO with different frequencies w 1, w 2 and w 3. Example: Axial symmetry with Energy is determined by Q 12 and Q 3 : Number of configurations c 3 (N,Z,Q 12,Q 3 ) from: Calculated recursively from:

12. Anisotropic harmonic oscillator Cumulative number of levels up to energy E: Example: Prolate & oblate. Normal & superdeformed.

13. Anisotropic harmonic oscillator Example 1: 38 Ar for  2 =  0.2. Example 2: 56 Fe for  2 =  0.2.

14. Extension to general potentials Assume single-particle levels with energies  n and degeneracies  n with n=1,2,… Question: How many A-particle states with energy E? Answer: Given by the number c(A,E) of solutions of Solution: c(A,E)  c(0,A,E) with c(i,A,E) calculated recursively through

15. Conclusions Versatile approach to compute level densities of particles in a harmonic oscillator potential which includes spin, isospin, deformation... (but without residual interactions). Extension to a general potential [cfr. (micro)canonical partition function for Fermi systems, S.Pratt, PRL 84 (2000) 4255]. Perspectives (general potential) Systematic use in combination with Hartree-Fock calculations (eg for astrophysics). Spurious fraction of states can be estimated. Effects of the continuum can be included. Inclusion of interaction effects?