nuclear level densities: energy distribution of all the excited levels: challenge to our
DESCRIPTION
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: - PowerPoint PPT PresentationTRANSCRIPT
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Systematics of Level Density Parameters
Till von Egidy, Hans-Friedrich WirthPhysik Department, Technische Universität München, Germany
Dorel BucurescuNational Institute of Physics and Nuclear Engineering, Bucharest,
Romania
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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.
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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 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.
Experimental Methods
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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 usuallyadded semi-empirically.
Two formulas (models) are investigated:
Back shifted Fermi gas (BSFG) model: parameters a , E1
Constant Temperature (CT) model: parameters T , E0
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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 (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.
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233Th: Example of a complete
low-energy level scheme
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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 fit to 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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Formulae for Level Densities
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Experimental Cumulative Number of Levels N(E)Resonance density is included in the fit
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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 A
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Fitted parameters T and E0 as function of the mass number AT ~ A-2/3 ~ 1/surface, degrees of freedom ~ nuclear surface
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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) = 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)
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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
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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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Various parameters to
explain the level density
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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
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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
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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
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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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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
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Comparison of calculated and experimental resonance densities
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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
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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 ~ A0.90
- 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 E1 and E0 :- T = 5.53 a –0.773 , E0 = E1 – 0.821
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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 .
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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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Energy Spin Nr. ofrange window levels
n binding Spin Density energy (per MeV)
Sample of
input data
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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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-4
-2
0
2
4even-even
(Pn+P
p)/2
12/A1/2
-Pn/3 (odd-Z)
-Pp/3 (odd-N)
BSFG
odd-A
-(Pn+P
p)/2
-12/A1/2
odd-odd
0 50 100 150 200 250-4
-2
0
2
4E1
(M
eV
)
0 50 100 150 200 250
A0 50 100 150 200 250
-Pd/2
-Pd
+Pd/2
dS(Z,N)/dA
dS(Z,N)/dA
dS(Z,N)/dA
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