D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 1 Nuclear Level Densities and Spin Distributions Dorel Bucurescu National Institute

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


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


Formulae for Level Densities

a, E1

T, E0


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.


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


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


Fitted parameters T and E0 as function of the mass number A

T ~ A-2/3 ~ 1/surface, degrees of freedom ~ nuclear surface


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)


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


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.


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


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)


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)


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


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 ?


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


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.


{


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 !



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)







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



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.


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


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.


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?

???


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.



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.


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


233Th: Example of a complete

low-energy level scheme



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.


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


Comparison of calculated and experimental resonance densities


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).

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D.Bucurescu, T. von Egidy, Level density, spin distribution-Ohio, July 2008 1 Nuclear Level Densities and Spin Distributions Dorel Bucurescu National Institute