d.bucurescu, t. von egidy, level density, spin distribution-ohio, july 2008 1 nuclear level...
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 20081
Nuclear Level Densities and Spin Distributions
Dorel BucurescuNational Institute of Physics and Nuclear Engineering, Bucharest,
Romania
Till von EgidyPhysik Department, Technische Universität München, Germany
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 20082
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 Ex = Bn;
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
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 20083
Formulae for Level Densities
a, E1
T, E0
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 20084
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/nex ~ A-2/3 indicates that the nucleus is melting from the surface.
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 20085
Experimental Cumulative Number of Levels N(E)Resonance density is included in the fit
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 20086
0
5
10
15
20
25
30 even-even
a (
MeV
-1)
BSFG
odd-A odd-odd
0 50 100 150 200 250-4
-2
0
2
4
E1 (
MeV
)
0 50 100 150 200 250
A0 50 100 150 200 250
Fitted parameters a and E1 as function of the mass number Aa ~ A
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 20087
Fitted parameters T and E0 as function of the mass number A
T ~ A-2/3 ~ 1/surface, degrees of freedom ~ nuclear surface
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 20088
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) = Mexp – Mliquid 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)
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 20089
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[Sd(A+2,Z+1)-Sd(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
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200810
shell correctionshell correction S(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.
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200811
Proposed Formulae for Level Density Parameters
• BSFGa = A0.90 ( 0.1848 + 0.00828 S´) [MeV-1]
E1 = -0.48 –0.5 Pa + 0.29 dS/dA for even-evenE1 = -0.57 –0.5 Pa + 0.70 dS/dA for even-oddE1 = -0.57 +0.5 Pa - 0.70 dS/dA for odd-evenE1 = -0.24 +0.5 Pa + 0.29 dS/dA for odd-odd
• CTT = A-2/3 / ( 0.0571 + 0.00193 S´) [MeV]
E0 = -1.24 –0.5 Pa + 0.33 dS/dA for even-evenE0 = -1.33 –0.5 Pa + 0.90 dS/dA for even-oddE0 = -1.33 +0.5 Pa - 0.90 dS/dA for odd-evenE0 = -1.22 +0.5 Pa + 0.33 dS/dA for odd-odd
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200812
a = A0..90 (0.1848 + 0.00828 S’)
E1 = p1 - 0.5Pa + p4dS(Z,N)/dA E1 = p2 - 0.5Pa + p4dS(Z,N)/dA E1 = p3 + 0.5Pa + p4dS(Z,N)/dA
TvE & DB, PRC72(2005)044311; C73(2006)049901(E)
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200813
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
TvE & DB, PRC72(2005)044311; C73(2006)049901(E)
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200814
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
DB & TvE, PRC72(2005)067304
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200815
Spin Distribution and Spin Cut-off Parameter 2
• f(J,) = exp( -J2/22 ) - exp ( - (J+1)2/22)2 is expected to depend on mass A, level density parameter a,
temperature T, moment of inertia deformation and excitation energy E.
2 = g <m2>t; a = 2 g /6 ; t = (U/a)1/2; <m2> = 0.146A2/3
• Gilbert, Cameron : 2 = 0.0888 a t A2/3 • Dilg, Vonach et al.: 2 = 0.0150 t A5/3 • Iljinov et al. : 2 = T ħ2; R = r0 A1/3 ; = 2/5 M R2 • Rauscher, Thielemann, Kratz: 2 = rigidħ2 sqrt(U/a), U=E – • Huang Zhongfu et al.: 2 = 0.0073 A5/3 ( 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 ?
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200816
Experimental determination of 2
• Fit to the experimental spin distributions in level schemes• 310 nuclei from 18F to 251Cf• Complete low energy level schemes, E < ~2-3 MeV (no dependence of 2 on E considered). exp ?
• 8116 levels and 1556 spin groups.
• ncalc(J) / J nk(J) = f(J,) / J f(J, in each nucleus k
kJ nk(J) - ncalc(J) 2nk2
] kJnk(J) - Fkf(J,) 2nk
2 ]
nk(J) = number of levels in spin J group of nucleus k
Fk = Jnk(J) /J f(J,) ,
nk = nk0.25 , chosen in order to obtain 2 ≈ 1 in fit
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200817
Fits of the spin distribution Even-even nuclei: • Strong even-odd spin staggering, J = 0 strength• This is largely independent of A.
• fee(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.
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200818
{
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200819
Results of the fits for 2
• General Ansatz : 2 = p1 Ap
2 Xp3
• X = a, T , • Available information is too little to determine energy
dependence: no E dependence.
• X = a, level density parameter: p3 = - 0.25(12), 2 = 1.04• X = T, temperature of LD: p3 = 0.36(14), 2 = 1.04
• X = quadr. deformation : p3 = 0.14(5), 2 = 1.10• No 2 improvement by additional dependencies!
2 = 2.61(21) A0.277(18) , 2 = 1.04 Surprising weak dependence on A !
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200821
Fit of in different mass groupswithout dependence on mass A
nuclei all even-even odd odd-odd
18F – 60Co 6.8(2) 6.8(3) 6.1(3) 7.3(4) 59Ni – 100Tc 7.8(3) 8.1(7) 6.9(4) 10.5(12)100Ru – 148Pm 8.6(3) 8.1(5) 7.9(4) 13.3(20)145Sm – 198Au 12.0(4) 11.5(5) 10.5(5) 16.7(14)199Hg – 251Cf 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)
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200827
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 ~ A0.90 , 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 σ2 was determined from 310 nuclei with 8116 levels below about 3 MeV.
σ2 varies only with mass, ~A0.3. 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
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200829
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.
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200830
STATISTICS:
310 nuclei8116 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: 124Te (60 levels up to 3.0 MeV):
# of levels Error
Spin N N1/2 N1/4
--------------------------------------------- 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 levelsmissing or too much. How to distribute this error on the spin group?
The error N1/4 is more realistic: increases only slowly with nr. of levels,
therefore groups with many levels have more weight. Also, it provides χ2 ≈ 1.0
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008
How to determine the variation of σ2 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.
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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?
???
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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.
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200835
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 Si=Ei+1-Ei
D(E) determined by a fit to the individual level spacings Si
Level spacing correlation:
Chaotic properties determine fluctuations about the averages and the errors of the LD parameters.
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200836
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), (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
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200837
233Th: Example of a complete
low-energy level scheme
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200838
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200839
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
This formula reduces the shell effect very slowly: 10% at 10 MeV, 50% at 30 MeV.
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200840
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
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200841
Comparison of calculated and experimental resonance densities
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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 200842
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 npfrom 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,E1, ã,E2 and T,E0, then f (A).