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Nuclear level densities: Energy distribution of all the excited levels: challenge to our

Systematics of Level Density Parameters Till von Egidy, Hans-Friedrich Wirth Physik Department, Technische Universit ät München, Germany Dorel Bucurescu National Institute of Physics and Nuclear Engineering, Bucharest, Romania. Nuclear level densities:

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Nuclear level densities: Energy distribution of all the excited levels: challenge to our

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  1. Systematics of Level Density Parameters Till von Egidy, Hans-Friedrich WirthPhysik Department, Technische Universität München, GermanyDorel BucurescuNational Institute of Physics and Nuclear Engineering, Bucharest, Romania

  2. Nuclear level densities: • Energy distribution of all the excited levels: challengeto 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. Experimental Methods Nuclear level densities can be directly determined (measured) for a limitednumber of nuclei & excitation energy range: - by counting the number of neutron resonances observed in low-energy neutron capture; level density close to Ex = Bn; - 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.

  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 twoempiricalparameters are necessary to describe the level density. Shell and pairing effects, etc., are usuallyadded semi-empirically. Two formulas (models)are investigated: Back shifted Fermi gas (BSFG) model: parameters a , E1Constant Temperature (CT) model: parameters T , E0

  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 : • - completelow-energy nuclear level schemes (Ex < 3 MeV) • and • - neutron resonance density near the neutron binding energy. • 310 nuclei between 19F and 251Cf • Empirical parameters: complicated variations , due to effects of shell closures, pairing, collectivity (neglected in the simple model) ; • try to learn from this behaviour.

  6. 233Th: Example of a complete low-energy level scheme

  7. Level densities: averages Average level densityρ(E): ρ(E) = dN/dE = 1/D(E) Cumulative number N(E) Average level spacing D Level spacing Si=Ei+1-Ei D(E) determined by fit to individual level spacings Si Level spacing correlation: Chaotic properties determine fluctuations about the averages and the errors of the LD parameters.

  8. Formulae for Level Densities

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

  10. Fitted parameters a and E1 as function of the mass number A

  11. Fitted parameters T and E0 as function of the mass number AT ~ A-2/3 ~ 1/surface, degrees of freedom ~ nuclear surface

  12. Precise reproduction of LD parameters with simple formulas: • We looked carefully for correlationsbetween the empirical LD parameters and well known observables which contain shell structure, pairing or collectivity. Mass values are important. • shell correction: S(Z,N) = Mexp – Mliquid 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: Pp , Pn , Pa (deuteron pairing) • excitation energy of the first 2+state: E(21+) • nuclear deformation: ε2 (e.g., Möller-Nix)

  13. Definition of neutron, proton, deuteron pairing energies:[G.Audi, A.H.Wapstra, C.Thibault, “The AME2003 atomic mass evaluation”, Nucl. Phys. A729(2003)337] Pn(A,Z)=(-1)A-Z+1[Sn(A+1,Z)-2Sn(A,Z)+Sn(A-1,Z)]/4 Pp (A,Z)=(-1)Z+1[Sp(A+1,Z+1)-2Sp(A,Z)+Sp(A-1,Z-1)]/4 Pd (A,Z)=(-1)Z+1[Sd(A+2,Z+1)-2Sd(A,Z)+Sd(A-2,Z-1)]/4 (Sn, Sp, Sd : 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

  14. shell correctionS(Z,N) = Mexp – Mliquid drop Macroscopic liquid drop mass formula (Weizsäcker): J.M. Pearson, Hyp. Inter. 132(2001)59 Enuc/A = avol + asfA-1/3 + (3e2/5r0)Z2A-4/3 + (asym+assA-1/3)J2 J= (N-Z)/A; A = N+Z [ Enuc = -B.E. = (Mnuc(N,Z) – NMn – ZMp)c2 ] From fit to 1995 Audi-Wapstra masses: avol = -15.65 MeV; asf = 17.63 MeV; asym = 27.72 MeV; ass = -25.60 MeV; r0 = 1.233 fm.

  15. Various parameters to explain the level density

  16. Proposed Formulae for Level Density Parameters • BSFG a A-0.90 = 0.1848 + 0.00828 S´ E1 = -0.48 –0.5 Pa + 0.29 dS/dA for even-even E1 = -0.57 –0.5 Pa + 0.70 dS/dA for even-odd E1 = -0.57 +0.5 Pa - 0.70 dS/dA for odd-even E1 = -0.24 +0.5 Pa + 0.29 dS/dA for odd-odd • CT T-1 A-2/3 = 0.0571 + 0.00193 S´ E0 = -1.24 –0.5 Pa + 0.33 dS/dA for even-even E0 = -1.33 –0.5 Pa + 0.90 dS/dA for even-odd E0 = -1.33 +0.5 Pa - 0.90 dS/dA for odd-even E0 = -1.22 +0.5 Pa + 0.33 dS/dA for odd-odd

  17. BSFG with energy-dependent „a“ (Ignatyuk) a(E,Z,N) = ã [1+ S´(Z,N) f(E - E2) / (E – E2)] f(E – E2) = 1 – e –γ (E - E2) ; γ = 0.06 MeV -1 ã = 0.1847 A0.90 E2 = E1

  18. a= A0..90 (0.1848 + 0.00828 S’) E1 = p3 + 0.5Pa+ p4dS(Z,N)/dA E1 = p1 - 0.5Pa + p4dS(Z,N)/dA E1 = P2 - 0.5Pa+ p4dS(Z,N)/dA

  19. ã= 0.1847 A 0.90 E2 = p1 - 0.5Pa+ p4dS(Z,N)/dA P2 - 0.5Pa+ p4dS(Z,N)/dA P3 + 0.5Pa+ p4dS(Z,N)/dA

  20. T = A-2/3/(0.0571 + 0.00193 S´) E0 = p1 - 0.5Pa+ p2dS(Z,N)/dA E0 = p3 – Pa+ p4dS(Z,N)/dA E0 = p1 + 0.5Pa+ p2dS(Z,N)/dA

  21. Comparison of calculated and experimental resonance densities

  22. Experimental Correlations between T and a and between E0 and E1 • a ~ T-1.294 ~ A(-2/3) (-1.294) = A0.863 • This is close to a ~ A0.90

  23. 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 numberA, deuteron pairing energy Pa, shell correctionS(Z,N) and dS(Z,N)/dA: • a, T : fromA, Pa , S , a ~ A0.90 • - backshifts: fromPa , 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 E1 and E0: • T = 5.53 a –0.773 , E0 = E1 – 0.821 CONCLUSIONS

  24. 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 .

  25. Completenessof 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), (3He,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

  26. Energy Spin Nr. of range window levels n binding Spin Density energy (per MeV) Sample of input data

  27. Previous systematics of the empirical model parameters (BSFG): a - well correlated with the “shell correction” S(Z,N): [ S(Z,N) = ΔM = Mexp – Mmacroscopic] Gilbert & Cameron (Can. J. Phys. 43(1965)1446): a/A = c0 + c1 S(Z,N) E1(the ‘back shift’ energy) - generally, assumed to be simply due to the pairing energies : Pn– neutron pairing energy, Pp – 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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