Mass and energy dependence of nuclear spin distributions

Transcription

1 Mass and energy dependence of nuclear spin distributions Till von Egidy Physik-Department, Technische Universität München, Germany Dorel Bucurescu National Institute of Physics and Nuclear Engineering, Bucharest, Romania T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 1

2 Extraction of Level Densities from Experimental Level Schemes f() Spin distribution 11 Sn MeV Level scheme E x [MeV] 3 1 Integrated level density (h) N - cumulative nr. of levels T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1

3 T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 3 Formulas for Level Densities / 1) ( / ), ( σ σ σ + = e e f ) ( ) ( ), ( E f E ρ ρ = 5/ 1 1/ ) ( ) ( 1 1 E E a e E E a BSFG = ρ Back-shifted Fermi Gas Formula : free parameters a, E 1 Spin Distribution with σ = spin-cutoff parameter : Constant Temperature Formula : free parameters T, E T E E CT e T ( ) / 1 = ρ

4 Spin Distribution and Spin Cut-off Parameter σ f(,σ) = exp( - /σ ) - exp ( - (+1) /σ ) σ is expected to depend on mass A, level density parameter a, temperature T, moment of inertia Θ, deformation β and excitation energy E. σ = g <m >t; a = π g / ; t = sqrt(u/a); <m > =.1A /3 Gilbert, Cameron : σ =.888 a t A /3 Dilg, Vonach et al.: σ =.15 t A 5/3 Iljinov et al. : σ = T Θ / ħ ; R = r A 1/3 ; Θ = / 5 M R Rauscher, Thielemann, Kratz: σ = Θ rigid /ħ sqrt(u/a), U=E δ Huang Zhonfu et al.: σ =.73 A 5/3 ( 1+sqrt(1+aU))/a Which formula is correct? Which dependence on A, a, T, Θ, β, E? What is the influence of the shell structure? Rigid Moment of inertia Θ rigid? T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1

5 Experimental Cumulative Number of Levels N(E) Resonance density is included in the fit T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 5

6 Methods for determination of nuclear level density and spin distribution - Determination of total nuclear level density ρ(e): fit D(E)=1/ ρ(e) to exp. spacings S i and average n-resonance spacing D res (PRC 7(5)311); 31 nuclei from 18 F to 51 Cf BSFG and CT model parameters. - Determination of average low-energy spin-cutoff parameter σ: fit to experimental spin distributions (counting of levels in spin groups in each nucleus) (PRC 78(8)5131R): 811 levels in 155 spin groups - simple A-dependence: σ =.1 A.8 ; - even-odd spin staggering of spin distribution in even-even nuclei. - Determination of energy and A-dependence of σ: new method of moments of nuclear level schemes in the (I,E x )-plane (PRC 8(9)531); 7 levels in 7 nuclei (with > 18 levels/known spin) σ =.391 A.75 (E-.5 Pa ).31 T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1

7 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,xnypzαγ), 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 (199); ENSDF database. Neutron resonance density: RIPL- database; T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 7

8 New method to determine σ(a,e) with moments in the (E,) plane of level schemes Experimental moments with levels i of nucleus k : Used moments: k,exp M = ( m. n 3 3 3,,, E, E, E, E, E, E Calculated moments: χ of the fit: χ k, calc σ m, n k E, 3 = k m= 1 n= [( M M k,exp m, n i m i M = ρ ( E,, ) k, calc m, n E m ) / n i E ) n dm 7 nuclei with at least 18 levels with known π (7 levels) k ] T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 8

9 Main result of our investigation σ = A ( E.5Pa ' ).31 The energy is back-shifted by the pairing energy deuteron pairing Pa, calculated from experimental mass values: Pa' k 1 = [ M( A +, Z + 1) M( A, Z) + M( A, Z 1)] T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 9

10 Spin staggering of spin distribution for even-even nuclei: x f ee (, σ ) = (1 + x) f (, σ ) = +.7,.7, + 1., even odd zero spin spin spin PR C78(8)5131(R) f(,σ) Cd E x = - 3. MeV σ=3.7(38) 8 The spin staggering is largely independent of mass A T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 1

11 Comparison of our new formula with experimental results f(,σ) 1 σ=.8(83) 8 1 E = - 3 MeV E = - 8 MeV E = 8-11 MeV E = 11-1 MeV σ=.8(1) σ=.18(17) Exp.: from discrete levels 5 E = 3 - MeV 5 E = MeV σ=.87(38) 8 σ=3.15() T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf σ=.(19) σ=.7(5) σ=3.(5) E = - 3 MeV E = 3 - MeV E = MeV σ=.15() present statist. mech. calc. rigid body σ 3 1 σ 3 1 σ eq. (1) eq. (17) eq. (18) E [MeV] Ne Al 7 Al 11

12 Comparison of our new formula with experimental results σ 8 51 V Cr 5 Cr eq. (1) eq. (17) eq. (18) Mn 55 Fe 59 Ni 1 Ni Cu 3 Zn E [MeV] Exp: from evaporated particle angular distributions T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 1

13 Comparison of our new formula with experimental results f(,σ) MeV MeV σ =.15(7) σ =.19(5) MeV MeV σ =.7(5) σ = 3.3(5) MeV MeV σ = 3.5(53) σ =.7(33) 1 5 σ 8 eq. (1) eq. (17) eq. (18) E [MeV] 18 Er 15 Gd 158 Gd T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 13

14 Comparison of new formula with experiment and calculations 3 8 Si Mg σ 1 Exp., level counting Exp., (α,n) ang. distr. present statist. mech. calc. rigid body Exp., level counting present E [MeV] T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 1

15 New determination of level density parameters a and E 1 for BSFG BSFG 3 even-even odd-a odd-odd 5 a [MeV -1 ] E 1 [MeV] A a = ( S') A S ' S +.5Pa'.89 E = Pa' = shell effect S = M exp -M Weiz T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 15

16 New determination of level density parameters T and E for CT CT 3 even-even odd-a odd-odd.5 T [MeV] E [MeV] T = A / /( S' ) E = ' Pa T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 A 1

17 Correlation of level density parameters 3 T = 5.1 a T [MeV] 1.8 T = 5.1a a [MeV -1 ] E [MeV] - E = E1.3 - E = E E 1 [MeV] T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 17

18 Conclusion This very simple formula is a challenge for theoretical calculations σ = A ( E.5Pa ' ).31 Valid for nuclei from F to Cf up to about 15 MeV Published in: Physical Review C8, 531 (9) T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 18

19 T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 19

20 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. T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1

21 33 Th: Example of a complete low-energy level scheme T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 1

22 T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1

23 shell correction S(Z,N) = M exp M liquid drop Macroscopic liquid drop mass formula (Weizsäcker):.M. Pearson, Hyp. Inter. 13(1)59 E nuc /A = a vol + a sf A -1/3 + (3e /5r )Z A -/3 + (a sym +a ss A -1/3 ) = (N-Z)/A; A = N+Z [ E nuc = -B.E. = (M nuc (N,Z) NM n ZM p )c ] From fit to 1995 Audi-Wapstra masses: a vol = MeV; a sf = 17.3 MeV; a sym = 7.7 MeV; a ss = -5. MeV; r = 1.33 fm. T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 3

24 Nuclear Temperature at low excitation energy Thermodynamical definition of temperature T = de / d log ρ(e) Integration with T = constant yields ρ(e) = e (E Eo) / T / T The agreement of this formula with experimental ρ(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 -/3 indicates that the nucleus is melting from the surface. T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1

25 Pa [MeV] - - even-even odd-a odd-odd - Shell correction S [MeV] A T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 5

26 BSFG with energy-dependent a (Ignatyuk) a(e,z,n) = ã [1+ S (Z,N) f(e - E ) / (E E )] f(e E ) = 1 e γ (E - E ) ; γ =. MeV -1 ã =.187 A.9 E = E 1 This formula reduces the shell effect very slowly: 1% at 1 MeV, 5% at 3 MeV. T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1

27 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.. Backshift: Iljinov: Fixed Δ = c 1 A -1/, c =, 1, for o-o, odd, e-e. Rauscher: Fixed Δ = ½ (Δ n + Δ 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.. 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 and T,E, then f (A). T. von Egidy, D. Bucurescu, Mass and energy dependence of spin distributions, Rossendorf 1 7