Oslo, May 21-24, 2007

1

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

Oslo, May 21-24, 2007

2

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.

Oslo, May 21-24, 2007

3

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

Oslo, May 21-24, 2007

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 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

Oslo, May 21-24, 2007

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 :

- 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.

Oslo, May 21-24, 2007

6

233Th: Example of a complete

low-energy level scheme

Oslo, May 21-24, 2007

7

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.

Oslo, May 21-24, 2007

8

Formulae for Level Densities

Oslo, May 21-24, 2007

9

Oslo, May 21-24, 2007

10

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

Oslo, May 21-24, 2007

11

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

Oslo, May 21-24, 2007

12

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

Oslo, May 21-24, 2007

13

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)

Oslo, May 21-24, 2007

14

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

Oslo, May 21-24, 2007

15

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.

Oslo, May 21-24, 2007

16

Various parameters to

explain the level density

Oslo, May 21-24, 2007

17

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

Oslo, May 21-24, 2007

18

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

Oslo, May 21-24, 2007

19

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

Oslo, May 21-24, 2007

20

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

Oslo, May 21-24, 2007

21

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

Oslo, May 21-24, 2007

22

Comparison of calculated and experimental resonance densities

Oslo, May 21-24, 2007

23

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

Oslo, May 21-24, 2007

24

Oslo, May 21-24, 2007

25

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

Oslo, May 21-24, 2007

26

Oslo, May 21-24, 2007

27

Oslo, May 21-24, 2007

28

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 .

Oslo, May 21-24, 2007

29

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

Oslo, May 21-24, 2007

30

Energy Spin Nr. ofrange window levels

n binding Spin Density energy (per MeV)

Sample of

input data

Oslo, May 21-24, 2007

31

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)

Oslo, May 21-24, 2007

32

-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

Oslo, May 21-24, 2007

33