1. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 1 Nuclear Level Densities and Spin Distributions Dorel Bucurescu National Institute of Physics and Nuclear Engineering, Bucharest, Romania Till von Egidy Physik Department, Technische Universität München, Germany

2. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 2 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. - photon strength function Nuclear level densities increase exponentially and can be directly determined (measured) for a limited number of nuclei & excitation energy range: - by counting the observed excited states at low excitations. - by counting the number of neutron resonances observed in low-energy neutron capture; level density close to E x = B n; Level densities were investigated for 310 nuclei between F and Cf (complete level schemes from ENSDF; n-resonance density from RIPL-2). Nuclear Level densities

3. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 3 Formulae for Level Densities a, E 1 T, E 0

4. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 4 Nuclear Temperature at low excitation energy Thermodynamical definition of temperature T = dE / d log  (E) Integration with T = constant yields  (E) = e (E – Eo) / T / T  The agreement of this formula with experimental  (E) shows  that T = const. at low energy in spite of increasing energy.  This is similar to melting ice where temperature is constant during heating.  T ~ E/n ex ~ A -2/3 indicates that the nucleus is melting from the surface.

5. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 5 Experimental Cumulative Number of Levels N(E) Resonance density is included in the fit

6. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 6 Fitted parameters a and E 1 as function of the mass number A a ~ A

7. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 7 Fitted parameters T and E 0 as function of the mass number A T ~ A -2/3 ~ 1/surface, degrees of freedom ~ nuclear surface

8. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 8 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 from mass tables are important. - pairing energies: Pp, Pn, Pa (deuteron pairing) - shell correction: S(Z,N) = M exp – M liquid drop, M = mass S - 0.5 Pa for even-even nuclei S´ = S for odd-mass nuclei S + 0.5 Pa for odd-odd nuclei - derivative dS(Z,N)/dA (calc. as [S(Z+1,N+1)-S(Z-1,N-1)]/4) TvE & DB, PRC72(2005)044311; C73(2006)049901(E)

9. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 9 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 [S d (A+2,Z+1)-S d (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

10. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 10 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.

11. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 11 Proposed Formulae for Level Density Parameters BSFG a = A 0.90 ( 0.1848 + 0.00828 S´) [MeV -1 ] 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 = A -2/3 / ( 0.0571 + 0.00193 S´) [MeV] 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

12. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 12 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 TvE & DB, PRC72(2005)044311; C73(2006)049901(E)

13. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 13 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 TvE & DB, PRC72(2005)044311; C73(2006)049901(E)

14. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 14 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 DB & TvE, PRC72(2005)067304

15. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 15 Spin Distribution and Spin Cut-off Parameter  2 f(J,  ) = exp( -J 2 /2  2 ) - exp ( - (J+1) 2 /2  2 )  2 is expected to depend on mass A, level density parameter a, temperature T, moment of inertia  deformation  and excitation energy E.  2 = g t; a =  2 g /6 ; t = (U/a) 1/2 ; = 0.146A 2/3 Gilbert, Cameron :  2 = 0.0888 a t A 2/3 Dilg, Vonach et al.:  2 = 0.0150 t A 5/3 Iljinov et al. :  2 = T  ħ 2 ; R = r 0 A 1/3 ;  = 2 / 5 M R 2 Rauscher, Thielemann, Kratz:  2 =  rigid  ħ 2 sqrt(U/a), U=E –  Huang Zhongfu et al.:  2 = 0.0073 A 5/3 ( 1+ (1+4aU) 1/2 )/2a Which formula is correct? Which dependence on A, a, T, , E? What is the influence of the shell structure? Rigid Moment of inertia  rigid ?

16. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 16 Experimental determination of  2 Fit to the experimental spin distributions in level schemes 310 nuclei from 18 F to 251 Cf Complete low energy level schemes, E < ~2-3 MeV (no dependence of  2 on E considered). exp ? 8116 levels and 1556 spin groups. n calc (J) /  J n k (J) = f(J,  ) /  J f(J,  in each nucleus k    k  J  n k (J) - n calc (J)  2  n k 2 ]  k  J  n k (J) - F k  f(J,  )  2  n k 2 ]  n k (J) = number of levels in spin J group of nucleus k F k =  J  n k (J) /  J f(J,  ),  n k = n k 0.25, chosen in order to obtain  2 ≈ 1 in fit

17. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 17 Fits of the spin distribution Even-even nuclei: Strong even-odd spin staggering, J = 0 strength This is largely independent of A. f ee (J,  ) = f(J,  ) (1 + x) + 0.227(14) for even spin, x = - 0.227(14) for odd spin, + 1.02(9) for J = 0. This correction factor was always applied to even-even nuclei.

18. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 18 {

19. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 19 Results of the fits for  2 General Ansatz :  2 = p 1 A p 2 X p 3 X = a, T,  Available information is too little to determine energy dependence: no E dependence. X = a, level density parameter: p 3 = - 0.25(12),  2 = 1.04 X = T, temperature of LD: p 3 = 0.36(14),  2 = 1.04 X =    quadr. deformation : p 3 = 0.14(5),  2 = 1.10 No  2 improvement by additional dependencies!   2 = 2.61(21) A 0.277(18),  2 = 1.04  Surprising weak dependence on A !

20. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 20

21. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 21 Fit of   in different mass groups without dependence on mass A   nuclei all even-even odd odd-odd 18 F – 60 Co 6.8(2) 6.8(3) 6.1(3) 7.3(4) 59 Ni – 100 Tc 7.8(3) 8.1(7) 6.9(4) 10.5(12) 100 Ru – 148 Pm 8.6(3) 8.1(5) 7.9(4) 13.3(20) 145 Sm – 198 Au 12.0(4) 11.5(5) 10.5(5) 16.7(14) 199 Hg – 251 Cf 12.1(6) 10.6(8) 13.6(10) 12.3(15) all 9.1(2) 8.9(3) 8.6(2) 10.5(5)

22. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008

23. 23

24. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 24

25. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 25

26. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 26

27. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 27 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 level density parameters on mass number A, deuteron pairing energy Pa and shell correction S(Z,N): a, T : from A, Pa, S ; (a ~ A 0.90, T ~ A -2/3 ) backshifts: from Pa, dS/dA  These formulas calculate level densities also for other nuclei only from ground state masses given in mass tables (Audi, Wapstra).  Spin cut-off parameter σ 2 was determined from 310 nuclei with 8116 levels below about 3 MeV.  σ 2 varies only with mass, ~A 0.3.  Fit is not improved by additional dependence on a, T or β.  Even-even nuclei have strong even-odd spin staggering and enhancement of spin J = 0.  Agreement with theory: Alhassid et al., P.R.L.99(2007)162504; Kaneko & Schiller, P.R. C75(2007)044304

28. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 28

29. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 29 ERRORS depend on: -Missing or wrongly assigned levels; -Slightly different distributions (e.g., var. with A, type of nucleus); -Artificial cutting of maximum energy; -Odd-even differences.

30. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 30 STATISTICS: 310 nuclei 8116 levels, 1556 spin groups; 42 nuclei with more than 40 levels (21 with more than 50). (~ 5 levels/spin group: therefore many groups have less than 5 levels). Problem: the errors assigned to the number of levels in spin groups. Example: 124 Te (60 levels up to 3.0 MeV): # of levels Error Spin N N 1/2 N 1/4 --------------------------------------------- 0 5 2.2 1.5 1 9 3.0 1.7 2 23 4.8 2.2 3 15 3.9 2.0 4 8 2.8 1.7 We assume completeness of the level schemes of 5 – 8 %, therefore 3 – 5 levels missing or too much. How to distribute this error on the spin group? The error N 1/4 is more realistic: increases only slowly with nr. of levels, therefore groups with many levels have more weight. Also, it provides χ 2 ≈ 1.0

31. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 How to determine the variation of σ 2 with energy? ► Count resonances: - but, for even-even targets, get only spin 1/2 states; with p-capture, 3/2. - for odd-A targets, one gets 2 spin values, i.e. one spin ratio. Is that meaningful? ► From isomeric ratios.

32. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008  Should re-run the level density fits with new spin distribution (N.B.: the set of levels will be somewhat different from the old one: less, or more levels, and sometimes different spin regions)  What to use for the spin distribution in the resonance region ?  Will new staggering formula for even-even nuclei modify the level density parameters of the even-even nuclei? ???

33. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 DISTRIBUTIONS shown are not “real” ones, but just the available ones added all together. Real distributions, only : - in individual nuclei, - for groups of nuclei with same spin window.

34. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008

35. 35 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 a fit to the individual level spacings S i Level spacing correlation: Chaotic properties determine fluctuations about the averages and the errors of the LD parameters.

36. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 36 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

37. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 37 233 Th: Example of a complete low-energy level scheme

38. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 38

39. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 39 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 This formula reduces the shell effect very slowly: 10% at 10 MeV, 50% at 30 MeV.

40. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 40 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

41. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 41 Comparison of calculated and experimental resonance densities

42. D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 42 Comparison with previous level density calculations A. S. Iljinov et al. and T. Rauscher et al. have the following differences: 1. Data Set: Iljinov: Low levels, resonance densities, reaction data. Rauscher: Only resonance densities. We: Low levels and resonance densities. 2. Backshift: Iljinov: Fixed  = c 12 A -1/2, c = 0, 1, 2 for o-o, odd, e-e. Rauscher: Fixed  ½  n  p  from mass tables. We: Independly fitted and calculated with deuteron pairing. 3. Formulas: Iljinov: Different formulas, also rotation and vibration. Rauscher: Ignatyuk‘s formulas We: CT, BSFG, Ignatyuk, different shell and pairing corr. 4. Fit Procedure: Iljinov: Calc. a for each data point, global fit of ã(A). Rauscher: Global fit of ã(A) to resonance densities. We: First individual fit of a,E 1, ã,E 2 and T,E 0, then f (A).