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### Test of Level Density models from Nuclear Reactions

Babatunde M. Oginni

Ohio University

Nuclear Seminar

December 3, 2009

Outline

- Introduction
- - Methods of determining level densities
- - Some level density models
- - Motivations
- - Goals for our study
- The Lithium induced reactions
- - Edwards Accelerator Laboratory
- - Level densities from evaporation of 64Cu
- The A = 82 compound nuclear reactions
- - Wright Nuclear Structure Laboratory
- - Some results
- Summary and Conclusion

Methods of determining NLD (I)

- Counting of levels

E

- Main drawbacks – level resolution & missing levels

- Counting of neutron resonances
- - Main drawback – narrow ranges of excitation energy,
- spin and parity ratio

Methods of determining NLD (III)

- Evaporation from compound nucleus

- Level densities obtained for the residual nuclei

- Main drawback – contributions from other reaction mechanisms

- Ericson fluctuation

- - Level densities obtained for the compound nucleus

Some models of NLD (I)

- Fermi gas model (FG) [*]
- 2 assumptions – nucleons are non-interacting fermions
- -- single particle states are equidistant
- in energy.

- Main challenge is to determine ‘a’ and ‘δ’ accurately for each nucleus

* H. A. Bethe, Phys. Rev. 50, 336 (1936)

Some models of NLD (II)

- Many ideas have been suggested for a:

Al-Quraishi [**]

ROHR [*]

a = 0.071*A + V

V = 1.64 A ≤ 38

V = 3.74 38 < A ≤ 69

V = 6.78 69 < A ≤ 94

V = 8.65 94 < A < 170

a = 0.108*A + 2.4 A ≥ 170

α = 0.1062, β = 0.00051

α = 0.1068, γ = 0.0389

* G. Rohr, Z Phys. A – Atoms and Nuclei 318, 299 – 308 (1984);

** S.I. Al-Quraishi et al, Phys. Rev. C63, 065803(2001).

Some models of NLD (III)

- Constant temperature model (CT) [*]

- Gilbert Cameron Model [**]
- - combine CT and FG models.
- Hartree-Fock-BCS model
- - microscopic statistical model

* A. Gilbert et al, Can. J. Phys. 43, 1248 (1965); ** A. Gilbert et al, Can. J. Phys. 43, 1446 (1965)

Motivations

- Astrophysical applications
- - evaluating reliable reaction rates for the production of nuclei
- Production cross sections of radioactive isotopes
- - help answer some salient questions; FRIB
- Fission Product Yields [*]
- Medical Applications

* P. Fong, Phys. Rev. 89, 332 (1953); P. Fong, Phys. Rev. 102, 434 (1956)

Goals for study

- Better understanding of the NLD problem
- Two main projects were undertaken:
- (1.) 6Li + 58Fe 64Cu; 7Li + 57Fe 64Cu
- * Edwards Accelerator Laboratory, Ohio University,
- Athens, Ohio
- (2.) 18O + 64Ni 82Kr; 24Mg + 58Fe 82Sr; 24Mg + 58Ni 82Zr
- * Wright Nuclear Structure Laboratory, Yale University,
- New Haven, Connecticut

Experiments: particle ID

6Li – induced rxn: 23.5, 37.7, 68.0, 98.0, 142.5 and 157.5 angles

7Li – induced rxn: 37.7, 142.5 and 157.5 angles

- Si detectors were used to detect the charged particles:
- TOF and Energy information.
- helions and tritons
- cannot be differentiated
- from each other!

Experiments: calibration

Charged Particle Energy Calibration

-elastic scattering of 6Li on Gold

-elastic scattering of 7Li on Gold

-elastic scattering of d on Gold

-alpha source of 3 known peaks

- Energy = mean (channel #) + offset

Experiments: Optical Parameters (I)

- The transmission coefficients of the entrance and exit channels and the level
- densities of the residual nuclei are input parameters in the Hauser-Feshbach
- codes that were used in our calculations.
- Most of the optical parameters for the exit channels are well documented in
- the literature [*].
- For the entrance channels, we made use of our elastic scattering distribution.
- The optical parameters for our experiments are given in the table:

- The Coulomb radius parameter used was 1.41 fm

* National Nuclear Data Center

Experiment: Optical Parameters (II)

- We compared our data with results of calculations using the optical
- parameters that were obtained:

Results: Proton angular distribution

- Angular distribution of compound nuclear reaction is expected
- to be symmetric about 90 degree.

Results: Break Up Study (I)

6Li α + d (Q = -1.47MeV)α + n + p (Q = -3.70MeV) 5He + p (Q = -4.59MeV)

7Li α + t (Q = -2.47MeV)α + d + n (Q = -8.72MeV)5He + d (Q = -9.61MeV)6He + p (Q = -9.98MeV)α + 2n + p (Q = -10.95MeV)5He + n + p (Q = -11.84MeV)

- Is the break up a 1-step process or a 2-step process ?

6Li 6Li* … 7Li 7Li* …

Results: Break up study (II)

- Direct break up of 6Li is into alpha and deuteron [1-4] while 7Li breaks
- up into alpha-triton and alpha-deuteron-neutron components [4-6]
- Sequential break up of 6Li* and 7Li* require looking up level schemes

- The dominant contribution to break up
- reaction among the excited levels of 6Li
- is the 3+ level at 2.18 MeV [3, 4,7]

Table from TUNL website

(1.) J. M. Hansteen et al. Phys. Rev. 137, B524 (1965); (2.) K. Nakamura, Phys. Rev. 152, 955 (1966);

(3.) E. Speth et al, Phy. Rev. Lett. 24, 1493 (1970); (4.) K. O. Pfeiffer et al. Nucl. Phys. A 206, 545 (1973);

(5.) D. K. Srivastava et al. Phys. Lett. B, 206, 391 (1988); (6.) V. Valkori et al. Nucl. Phys. A 98, 241 (1967);

(7.) A. Pakou et al. Phys. Lett. B, 633, 691 (2006).

Results: Break up Study (III)

- The low energy levels of 7Li are given in the table below:

Table from TUNL website

- The threshold of emitting proton in sequential break up of 7Li is about 10 MeV; most of
- the break up will be through the α-t and α-d-n components

Results: Break up study (IV)

- In order to better understand our break up process, we use the
- method Goshal [*] showed about compound reactions

- We look at this ratio:

A represent proton cross sections

B could be alpha, deuteron or triton cross sections

* S. N. Ghoshal, Phys. Rev. 80, 939 (1950)

Results: Break up study (V)

- We safely conclude that the protons and high energy alphas at
- backward angles are mostly from compound nuclear reactions.
- Thus we can get NLD information from protons and high energy alphas

p + 63Ni

64Cu

α + 60Co

7Li + 57Fe

6Li + 55Mn

p + 60Co

61Ni

d + 59Co

n + 60Ni

CONCLUSION (II)- B. M. Oginni et al., Phys. Rev. C
- 80, 034305 (2009).

CT with T = 1.4 MeV.

- A. V. Voinov, B. M. Oginni, et al.,
- Phys. Rev. C 79, 031301 (R) (2009).

Calibration of the clover detectors

- We did two types of calibrations:
- energy and the efficiency calibrations

- The idea of the calibration is to
- move from the “known”to the “unknown”
- - So we made use of 152Eu source with known activity

152Eu

- Within the energy range that was considered during the
- experiment, the source has fifteen prominent peaks with
- known emission probabilities

Experimental Idea (I)

- For even-even nuclei, most gamma rays
- pass through the 2+ to 0+ levels.
- Production cross section of the 2+ gamma
- is proportional to the production cross
- sections of the nucleus [*].
- Since we know the even-even nuclei that
- are expected from each reaction, we use
- the gamma level schemes to determine the
- gamma energies associated with each
- residual nucleus.

* R. P. Koopman, PhD Thesis, Lawrence Livermore Laboratory

Experimental Idea (II)

- Not all the 2+ gammas were used in the analysis
- RULES FOR SELECTION
- There must be a noticeable gamma peak at the
- energy corresponding to the 2+ gamma
- Since most of the gammas were produced in
- coincidence! We place a gate on each 2+ gamma
- peak and check for other gammas detected in
- coincidence; the gammas used in the analysis
- had at least one gamma decayed in coincidence.

24Mg + 58Fe

Al - Quraishi

Summary

- I talked about the different methods of determining LDs
- I presented some LD models
- I presented the level densities that we obtained for 63Ni
- and 60Co
- I also presented some results from our A = 82 nuclear
- compound reactions

- A better constraint will be achieved in the Yale experiment
- if both the evaporated particles and gammas are detected
- in coincidence

List of Collaborators

- S. M. Grimes, C. R. Brune, T. N. Massey, A. Schiller, A. V. Voinov
- - Ohio University, Athens, OH
- A. S. Adekola
- Triangle University Nuclear Laboratory, NC
- Z. Heinen
- - Savannah River Site, Aiken, SC
- D. Carter, D. Jacobs, J. O’Donnell
- - Ohio University, Athens, OH
- Andreas Heinz (Yale University)
- - Yale University, New Haven, CT

Taking a peep away from stability!

Al - Quraishi

State & Level density

- Each level of spin J comprises 2J+1 degenerate states with
- different projections of J

where

= state density

= level density

cumulative number of levels

Nuclear Processes in stars and stellar explosions

s-process

(AGB)

Pb (82)

protons

Proton-rich

(SNII)

r-process

(SNII)

Sn (50)

rp process

Novae, SNIa

X-ray bursts

Fe (26)

Heavy-element burning

(Massive stars)

CNO Breakout

C(6)

neutrons

H(1)

Big Bang

W. Tan

NLD

NLD from neutron resonances: Levels are excited by the absorption of

neutrons with zero angular momentum, the number of resonances

in the energy interval is

for target nuclei

for J = 0 target nuclei

F = qvB = (mv^2)/R

R = mv/qB Radius of curvature in a magnetic field

NLD

- Rapid increase in # of levels at high energy is expected from simple
- thermodynamics considerations, from probability arguments and
- from nuclear model calculations

- For the thermodynamics consideration

= entropy

= state density

Fermi-gas level-density expressions

1) Single-particle model, no many-body effects

2) Used in most statistical-model calculations.

Errors

- Two main error types we took into consideration: statistical & systematic
- Statistical error is the square root of the number of counts
- Systematic are mainly uncertainties in target thickness (15%), beam charge
- integration (5%) and solid angles (5%)
- We obtained our overall error by propagating the errors

GC model

- The 3 model parameters, T, Ux, and E0, are determined by the requirement that
- the level density and its derivative are continuous at the matching point, Ux.

{Sum over all Energies and spins}

Calibration (cont’d)

- Since we know what the energy associated with each
- peak is, we look at the spectra from each leaf detector

- To obtain the counts expected, we need to know the
- activity of the source at a certain time, the half-life of the
- source and the emission probabilities for each peak

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