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

2. Models for nuclear level densities • « An Attempt to Calculate the Number of Energy Levels of a Heavy Nucleus » (Bethe1936): 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 Et exist for A particles in an isotropic HO? • Answer: Given by the number of solutions of • Solution: c3(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: • c3(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 hw 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 Petropolitanae3 (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. • c3(A,Q) includes all solutions. Let us denote the physical solutions as • This is found by substracting from c3(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: c3(N,Z,Q) can be calculated recursively or through

10. Shell effects • Fermi-gas estimate (Bethe; cfr Bohr & Mottelson): • The quantity c3(N,Z,Q) can be evaluated for closed as well as open shells => effects of shell structure on level densities. • Example: Comparison of 16O and 28Si.

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 w1, w2 and w3. • Example:Axial symmetry with • Energy is determined by Q12 and Q3: • Number of configurations c3(N,Z,Q12,Q3) 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 1:38Ar for 2=0.2. • Example 2:56Fe 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?