Calculation of Nuclear Level Densities Near the Drip Lines

1. Calculation of Nuclear Level Densities Near the Drip Lines Shaleen Shukla Steven M. Grimes

2. Motivation • Nuclear Level Densities (NLDs) are inputs in various fields/processes • Design of radioactive beam Accelerators • Microscopic details of fission • Nucleosynthesis • Nuclear medicine • Aim: To explore the nuclei near Drip Lines • Knowledge of NLDs of nuclei near the drip lines is very poor. • Focus on nuclei with A between 40 and 100. • Earliest exploration of NLD by Bethe (1936) S. Shukla: SNP 2008, Athens, Ohio

3. Bethe form raises questions! • Assumes constant single particle spectrum • Non-interacting fermions • Has not been tested for higher excitation energies (above tens of MeVs) • Weidenmuller[!] questioned the effect of wide states on level density and Grimes[#] analyzed their consequence. Bethe form: !: H A Weidenmuller, Phys Let. 10, 331 (1964) #: S M Grimes, Phys. Rev. C 42, 2744 (1990) S. Shukla: SNP 2008, Athens, Ohio

4. This work • Analyses effect of actual single-particle spectrum incorporating width information of the levels. • Two-body interaction • Higher Excitation energies • Nuclei with relatively large A S. Shukla: SNP 2008, Athens, Ohio

5. Calculation of NLDs CODE: EIGENVALUE Compute single-particle levels using effective potential. Bound & Unbound proton and neutron levels Effective potential Single-particle levels used as inputs in many-body codes CODE: SPECTRUM1 Uses spectral distribution methods and moments to calculate the density of states. CODES: XLEV Uses combinatorix to generate the density of states. NLD CODE: CRUNCHER Uses second quantization, computes moments of Hamiltonian for a distribution that reproduces the single-particle states. Uses LANCZOS method. CODE: RHOTHERM Uses thermodynamics to generate the density of states. Computational Procedure S. Shukla: SNP 2008, Athens, Ohio

6. Single Particle Spectrum (EIGENVALUE) • New Code written that uses modified Woods-Saxon Potential Figure taken from http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/shell.html S. Shukla: SNP 2008, Athens, Ohio

7. EIGENVALUE (Bound & Unbound levels) • Bound Levels: Solve above Eq. for negative energies • Unbound Levels: • Phase-shift approach: look for ‘jumps’ of p in phase shift • Derivative: Locate ‘spikes’ in S. Shukla: SNP 2008, Athens, Ohio

8. EIGENVALUE (Bound & Unbound levels) • Uses averaging of the two approaches to locate unbound levels. • Extremely small probability of missing a very narrow or wide resonance. • Generates accurate shell structure. • Output from EIGENVALUE compared with WSPOT & FOP S. Shukla: SNP 2008, Athens, Ohio

9. Level Density Codes • Spectral Distribution methods • SPECTRUM1 • HIGHLIGHTS • Includes definite single particle widths • Written from scratch • CRUNCHER • HIGHLIGHT • Includes two-body Interaction • Direct counting procedure • XLEV • Thermodynamic procedure • RHOTHERM S. Shukla: SNP 2008, Athens, Ohio

10. SPECTRUM1 • Patterned on earlier work by Grimes[#] • Non-interacting Hamiltonian • Number of basis states used: • and are calculated using propagator theorem “The expectation value of a n-body operator will be a n+1 order polynomial in the particle number.” • Delta function used for approximating Hamiltonian #S. M. Grimes, Proc. of Advisory Group Meeting on Basis and Applied Problems of Nuclear Level Densities, BNL (1983). S. Shukla: SNP 2008, Athens, Ohio

11. SPECTRUM1 • Definite widths of single particle levels considered • Also counts the number of particles in a level • Very important for width selection • Two types of calculation • All widths • Narrow widths (< 0.25 MeV) S. Shukla: SNP 2008, Athens, Ohio

12. CRUNCHER • Many-body Schrodinger Eq. • Two-body int.: Yukawa type interaction • Diagonalization of Hamiltonian yields eigenvalues S. Shukla: SNP 2008, Athens, Ohio

13. XLEV (Combinatorial Approach) • Formulated by Jacquemin and Kataria [#] • One body Hamiltonian • Relies on the recursive relation • Here, and # C. Jacquemin and S. K. Kataria, Z. Phys. A324, 261 (1986). S. Shukla: SNP 2008, Athens, Ohio

14. RHOTHERM (Thermodynamic Approach) • Developed by Grimes [#] • BCS Hamiltonian • Logarithm of grand partition function • State Density is • Gap Eq./Saddle point Approx. used to calculate state density # S. M. Grimes et. al, Phys. Rev. C16, 2373 (1974). S. Shukla: SNP 2008, Athens, Ohio

15. Results S. Shukla: SNP 2008, Athens, Ohio

16. Comparison of different methods S. Shukla: SNP 2008, Athens, Ohio

17. Comparison of different methods S. Shukla: SNP 2008, Athens, Ohio

18. Comparison of different methods S. Shukla: SNP 2008, Athens, Ohio

19. Analysis (Stable Nucleus) • Use Bethe formula to fit up to 30 MeV • Level density parameter varies linearly with A Slope All Widths: 0.0923 Narrow Widths: 0.0944 S. Shukla: SNP 2008, Athens, Ohio

20. Analysis for nuclei off stability line • Values of ‘a’ varied by as much as 35% for unstable nuclei compared to stable ones. • Level Density parameter defined as per [#] • Z0 is atomic number for b-stable nucleus # S. I. Al-Quraishi, S. M. Grimes, T. N. Massey and D. A. Resler, Phys. Rev. C67, 015803 (2003). S. Shukla: SNP 2008, Athens, Ohio

21. Summary • Calculated NLDs using various methods. • Consistency checks • Comparison with adopted values for 40Ar and 40Ca • Consistency between various methods at low energies • First calculations with ‘Narrow Widths’ in SPECTRUM1 • NLDs obtained for ‘A’ between 40 and 100 • Linear variation of ‘a’ with ‘A’ for stable nuclei • Preliminary analysis of ‘a’ for nuclei away from stability shows some agreement with results obtained by Al-Quraishi et al. S. Shukla: SNP 2008, Athens, Ohio

22. Thank You! S. Shukla: SNP 2008, Athens, Ohio

23. A=40 (Stable and n-rich) SPECTRUM1 S. Shukla: SNP 2008, Athens, Ohio

24. A=40 (Stable and p-rich) SPECTRUM1 S. Shukla: SNP 2008, Athens, Ohio

25. A=40 (CRUNCHER) S. Shukla: SNP 2008, Athens, Ohio

26. A=90 (SPECTRUM1) S. Shukla: SNP 2008, Athens, Ohio