1. Oslo, May 21-24, 2007 1 Systematics of Level Density Parameters Till von Egidy, Hans-Friedrich Wirth Physik Department, Technische Universität München, Germany Dorel Bucurescu National Institute of Physics and Nuclear Engineering, Bucharest, Romania

2. Oslo, May 21-24, 2007 2 Nuclear level densities: Energy distribution of all the excited levels: challenge to our theoretical understanding of nuclei; Important ingredient in related areas of physics and technology: - all kinds of nuclear reaction rates; - low energy neutron capture; - astrophysics (thermonuclear rates for nucleosynthesis); - fission/fusion reactor design.

3. Oslo, May 21-24, 2007 3 Nuclear level densities can be directly determined (measured) for a limited number of nuclei & excitation energy range: - by counting the number of neutron resonances observed in low-energy neutron capture; level density close to E x = B n ; - by counting the observed excited states at low excitations. Problem: how to predict (extrapolate to) level densities of less known, or unknown nuclei far from the line of stability, for which there are no experimental data. Experimental Methods

4. Oslo, May 21-24, 2007 4 Microscopic models: complicated and not reliable. Practical applications: most calculations are extensions and modifications of the Fermi gas model (Bethe): in spite of complicated nuclear structure – only two empirical parameters are necessary to describe the level density. Shell and pairing effects, etc., are usually added semi-empirically. Two formulas (models) are investigated: Back shifted Fermi gas (BSFG) model: parameters a, E 1 Constant Temperature (CT) model: parameters T, E 0

5. Oslo, May 21-24, 2007 5 Heuristic approach We determine empirically the two level density parameters by a least squares fit ( T. von Egidy, D. Bucurescu, Phys.Rev.C72,044311(2005), Phys.Rev.C72,067304(2005), Phys.Rev.C73,049901 ) to : - complete low-energy nuclear level schemes (E x < 3 MeV) and - neutron resonance density near the neutron binding energy. 310 nuclei between 19 F and 251 Cf Empirical parameters: complicated variations, due to effects of shell closures, pairing, collectivity (neglected in the simple model) ; try to learn from this behaviour.

6. Oslo, May 21-24, 2007 6 233 Th: Example of a complete low-energy level scheme

7. Oslo, May 21-24, 2007 7 Level densities: averages Average level density ρ(E): ρ(E) = dN/dE = 1/D(E) Cumulative number N(E) Average level spacing D Level spacing S i =E i+1 -E i D(E) determined by fit to individual level spacings S i Level spacing correlation: Chaotic properties determine fluctuations about the averages and the errors of the LD parameters.

8. Oslo, May 21-24, 2007 8 Formulae for Level Densities

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10. 10 Experimental Cumulative Number of Levels N(E) Resonance density is included in the fit

11. Oslo, May 21-24, 2007 11 Fitted parameters a and E 1 as function of the mass number A

12. Oslo, May 21-24, 2007 12 Fitted parameters T and E 0 as function of the mass number A T ~ A -2/3 ~ 1/surface, degrees of freedom ~ nuclear surface

13. Oslo, May 21-24, 2007 13 Precise reproduction of LD parameters with simple formulas: We looked carefully for correlations between the empirical LD parameters and well known observables which contain shell structure, pairing or collectivity. Mass values are important. - shell correction: S(Z,N) = M exp – M liquid drop, M = mass - S´ = S - 0.5 Pa for e-e; S´ = S for odd; S´ = S + 0.5 Pa for o-o - derivative dS(Z,N)/dA (calc. as [S(Z+1,N+1)-S(Z-1,N-1)]/4) - pairing energies: P p, P n, Pa (deuteron pairing) - excitation energy of the first 2 + state: E(2 1 + ) - nuclear deformation: ε 2 (e.g., Möller-Nix)

14. Oslo, May 21-24, 2007 14 Definition of neutron, proton, deuteron pairing energies: [ G.Audi, A.H.Wapstra, C.Thibault, “The AME2003 atomic mass evaluation”, Nucl. Phys. A729(2003)337 ] P n (A,Z)=(-1) A-Z+1 [S n (A+1,Z)-2S n (A,Z)+S n (A-1,Z)]/4 P p (A,Z)=(-1) Z+1 [S p (A+1,Z+1)-2S p (A,Z)+S p (A-1,Z-1)]/4 P d (A,Z)=(-1) Z+1 [S d (A+2,Z+1)-2S d (A,Z)+S d (A-2,Z-1)]/4 (S n, S p, S d : neutron, proton, deuteron separation energies) Deuteron pairing with next neighbors: Pa (A,Z)= ½ (-1) Z [-M(A+2,Z+1) + 2 M(A,Z) – M(A-2,Z-1)] M(A,Z) = experimental mass or mass excess values

15. Oslo, May 21-24, 2007 15 shell correction shell correction S(Z,N) = M exp – M liquid drop Macroscopic liquid drop mass formula (Weizsäcker): J.M. Pearson, Hyp. Inter. 132(2001)59 E nuc /A = a vol + a sf A -1/3 + (3e 2 /5r 0 )Z 2 A -4/3 + (a sym +a ss A -1/3 )J 2 J= (N-Z)/A; A = N+Z [ E nuc = -B.E. = (M nuc (N,Z) – NM n – ZM p )c 2 ] From fit to 1995 Audi-Wapstra masses: a vol = -15.65 MeV; a sf = 17.63 MeV; a sym = 27.72 MeV; a ss = -25.60 MeV; r 0 = 1.233 fm.

16. Oslo, May 21-24, 2007 16 Various parameters to explain the level density

17. Oslo, May 21-24, 2007 17 Proposed Formulae for Level Density Parameters BSFG a A -0.90 = 0.1848 + 0.00828 S´ E 1 = -0.48 –0.5 Pa + 0.29 dS/dA for even-even E 1 = -0.57 –0.5 Pa + 0.70 dS/dA for even-odd E 1 = -0.57 +0.5 Pa - 0.70 dS/dA for odd-even E 1 = -0.24 +0.5 Pa + 0.29 dS/dA for odd-odd CT T -1 A -2/3 = 0.0571 + 0.00193 S´ E 0 = -1.24 –0.5 Pa + 0.33 dS/dA for even-even E 0 = -1.33 –0.5 Pa + 0.90 dS/dA for even-odd E 0 = -1.33 +0.5 Pa - 0.90 dS/dA for odd-even E 0 = -1.22 +0.5 Pa + 0.33 dS/dA for odd-odd

18. Oslo, May 21-24, 2007 18 BSFG with energy-dependent „a“ (Ignatyuk) a(E,Z,N) = ã [1+ S´(Z,N) f(E - E 2 ) / (E – E 2 )] f(E – E 2 ) = 1 – e – γ (E - E 2 ) ; γ = 0.06 MeV -1 ã = 0.1847 A 0.90 E 2 = E 1

19. Oslo, May 21-24, 2007 19 a = A 0..90 (0.1848 + 0.00828 S’) E 1 = p 1 - 0.5Pa + p 4 dS(Z,N)/dAE 1 = P 2 - 0.5Pa + p 4 dS(Z,N)/dA E 1 = p 3 + 0.5Pa + p 4 dS(Z,N)/dA

20. Oslo, May 21-24, 2007 20 ã= 0.1847 A 0.90 E 2 = p 1 - 0.5Pa + p 4 dS(Z,N)/dA P 2 - 0.5Pa + p 4 dS(Z,N)/dA P 3 + 0.5Pa + p 4 dS(Z,N)/dA

21. Oslo, May 21-24, 2007 21 T = A -2/3 /(0.0571 + 0.00193 S´) E 0 = p 1 - 0.5Pa + p 2 dS(Z,N)/dAE 0 = p 3 – Pa + p 4 dS(Z,N)/dA E 0 = p 1 + 0.5Pa + p 2 dS(Z,N)/dA

22. Oslo, May 21-24, 2007 22 Comparison of calculated and experimental resonance densities

23. Oslo, May 21-24, 2007 23 Experimental Correlations between T and a and between E 0 and E 1 a ~ T -1.294 ~ A (-2/3) (-1.294) = A 0.863 This is close to a ~ A 0.90

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25. Oslo, May 21-24, 2007 25 CONCLUSIONS -New empirical parameters for the BSFG and CT models, from fit to low energy levels and neutron resonance density, for 310 nuclei (mass 18 to 251); -Simple formulas are proposed for the dependence of these parameters on mass number A, deuteron pairing energy Pa, shell correction S(Z,N) and dS(Z,N)/dA : - a, T : from A, Pa, S, a ~ A 0.90 - backshifts: from Pa, dS/dA - These formulas calculate level densities only from ground state masses given in mass tables (Audi, Wapstra). -The formulas can be used to predict level densities for nuclei far from stability; - Justification of the empirical formulas: challenge for theory. -Simple correlations between a and T and between E 1 and E 0 : - T = 5.53 a –0.773, E 0 = E 1 – 0.821

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28. Oslo, May 21-24, 2007 28 Aim (i) New empirical systematics (sets) of level density parameters; (ii) Correlations of the empirical level density parameters with better known observables; (iii) Simple, empirical formulas which describe main features of the empirical parameters; (iv) Prediction of level density parameters for nuclei for which no experimental data are available.

29. Oslo, May 21-24, 2007 29 Completeness of nuclear level schemes Concept in experimental nuclear spectroscopy: “All” levels in a given energy range and spin window are known. A confidence level has to be given by experimenter: e.g., “less than 5% missing levels”. We assume no parity dependence of the level densities. Experimental basis: (n,γ), ARC : non-selective, high precision; (n,n’γ), (n,pγ), (p,γ); (d,p), (d,t), ( 3 He,d), …, (d,pγ), … β-decay; (α,nγ), (HI,xnypzα γ), HI fragmentation reactions; * Comparison with theory: one to one correspondence; * Comparison with neighbour nuclei; * Much experience of the experimenter. Low-energy discrete levels: Firestone&Shirley, Table of isotopes (1996); ENSDF database. Neutron resonance density: RIPL-2 database; http://www-nds.iaea.org

30. Oslo, May 21-24, 2007 30 Energy Spin Nr. of range window levels n binding Spin Density energy (per MeV) Sample of input data

31. Oslo, May 21-24, 2007 31 Previous systematics of the empirical model parameters (BSFG): a - well correlated with the “shell correction” S(Z,N): [ S(Z,N) = ΔM = M exp – M macroscopic ] Gilbert & Cameron (Can. J. Phys. 43(1965)1446): a/A = c 0 + c 1 S(Z,N) E 1 (the ‘back shift’ energy) - generally, assumed to be simply due to the pairing energies : P n – neutron pairing energy, P p – proton pairing energy. Up to now – no consistent systematics of this parameter. (e.g., A.V.Ignatyuk, IAEA-TECDOC-1034, 1998, p. 65)

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