The Nuclear Level Densities in

Closed Shell 205-208Pb Nuclei

Syed Naeem Ul Hasan

Introduction:

Nuclear level density: Bethe Fermi gas model in 1936. For many years, measurements of NLD have been interpreted in the framework of an infinite Fermi-gas model.

Gil. & Cam. CTF, BSFG were later proposed accounting shell effects etc.

Shell Model Monte Carlo (SMMC)

Experimental NLD, Counting of neutron (proton) resonances Discrete levels counting Evaporation spectra OSLO METHOD

Method has successfully been proven for a No. of nuclei.

However in cases where statistical properties are less favorable the method foundation is more doubtful.

A test at the lighter nuclei region has been made already for 27,28Si.

The limit of applicability of method on closed shell nuclei was also required.

Experimental details

MC-35 cyclotron at OCL,38 MeV 3He beam bombarded on 206Pb and 208Pb targets having thickness of 4.707 and 1.4 mg/cm2.Following reactions were studied,

206Pb(3He, 3He´)206Pb206Pb(3He, )205Pb208Pb(3He, 3He´)208Pb208Pb(3He, )207Pb

The particle- coincidences were recorded while the experiment ran for 2-3 weeks.

Oslo Cyclotron Lab

http://www.physics..no/ocl/intro/

CACTUS Concrete wall

Detector ArrangementThe charged ejectiles ----> 8 collimated Si at 45o to the beam.

The -rays detection -----> CACTUS: 28 NaI(Tl) 5”x5”

detection = 15% of 4π.

Particles & -rays are produced in rxns are measured in both particle- coincidence & particle singles mode by the CACTUS multi-detector array.

thickness spectrumGating on particles

Particle - coincidences

Unfolding of coincidence spectra

Extracting Primary- spectra

Raw Data

-spectra calibration

and alignment

Particle Spectra Calibration

Data Reduction

Data Analysis:

3He- -

Coincidence Spectra:

Unfolding:Detector response of 28 NaI detectors are determined 11 energies and interpolation is made for intermediate -energies.

Folding iteration method is used;Unfolded spectrum is starting point, such that, f = R u.

– First trial fn as, uo = r

– First folded spectrum, fo = R uo – Next trial fn, u1 = uo + (r - fo)

– Generally, ui+1 = ui + (r - fi)

– Iteration continues until fi ~ rFluctuations in folded spectra Compton background are subtracted.

Response function of 4 MeV .

Response Matrix

RawRaw

foldedfolded

unfoldedunfolded

Test of method:

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f(Ex, Eγ ) = R (Eγ′ E γ

∑ , ′ E γ ) ⋅u (Ex, ′ E γ )

Primary -matrix:Assumption: The -decay pattern from any Ex is independent of the population mechanismThe nucleus seems to be an CN like system prior to -emission.

Method: The f.g. -spectrum of the highest Ex is estimated by,f1 = f.g. -spectrum of highest Ex bin.g = weighted sum of all spectra.wi = prob. of decay from bin 1 - i.ni = i / j

Assumption: The -decay pattern from any Ex is independent of the population mechanismThe nucleus seems to be an CN like system prior to -emission.

Method: The f.g. -spectrum of the highest Ex is estimated by,f1 = f.g. -spectrum of highest Ex bin.g = weighted sum of all spectra.wi = prob. of decay from bin 1 - i.ni = i / j

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h = f1 − g,

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g = niwi fii

∑

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g = niwi fii

∑

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Multiplicity Normalization :

Algorithm:1. Apply a trial fn wi

2. Deducing hi = fi - g

3. Transforming hi to wji (i.e. Unfold h, make h having same energy

calibration as wi, normalize the area of h to 1).

4. If wji (new) ≈ wj

i (old) then the calculated hi would be the Primary- function for the level Ei, else proceed with (2)

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i =A( f i)

M i

ni =σ 1

σ i

=A( f1)

M1

M i

A( f i)

Some experimental conditions can introduce severe systematic errors, like pile-up effects, isomers etc.

fgh=f-g

Ex ~ 4.5-5.5 MeV.

Testing of an Experimental spectrum:

Brink-Axel Hypothesis;

Transforming;

A, B, are free parameters.A. Schiller et al./ Nucl. Instr. & Methods in Physics Research A

447 (2000) 498-511

Extraction of NLD & GSF:

Ef

fEdiff

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P(Ei ,Eγ )∝ Τ(Eγ )ρ (Ei − Eγ )

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~

= A exp[α(E − Eγ )]ρ (E − Eγ )

T~

= B exp(αEγ )T (Eγ )

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Gamma transition probability

Theoretically

Minimizing;

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=0.0146A5 / 3 1+ 1+ 4a(u − E1)

2a

Nuclear Level Density:

At low Ex; o comparing the extracted NLD to

At Bn deducing NLD from resonance spacing data.

BSFG level density extrapolation

where a = level density parameter,

U = E-E1 and E1 = back-shifted parameter,€

mod el(E) = ηexp(2 aU)

12 2a1/ 4U 5 / 4σ,

208Pb =0.634

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. No of known levels Excitation energy bin

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Sn=

2σ 2

D

1

(I +1)exp(−(I +1)2 /2σ 2) + I exp(−I2 /2σ 2)

⎡

⎣ ⎢

⎤

⎦ ⎥

Experimental NLD

Entropy:

o is adjusted to give S = ln ~ 0 close to ground state band

The ground band properties fullfil the Third law of dynamics:

S(T =0) = 0;

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S(E) = k ln(Ω(E))

Ω(E) =ρ (E)

ρ o

Gamma Strength Function:

The fitting procedure of P(Ei, E) determines

the energy dependence of T(Ei, E) .

The fitting of B must be done here.

Assumptions:– The decay in the continuum E1, M1.

– No of states with = ± is equal

Radiative strength function is,

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f (Eγ ) =1

2π

T(Eγ )

Eγ3

CollaboratorsCollaboratorsMagne Guttormsen University of Oslo

Suniva Siem University of Oslo

Ann Cecilie University of Oslo

Rositsa Chankova University of Oslo

A. Voinov Ohio university, OH, USA

Andreas Schiller MSU, USA

Tom Lønnroth Åbo Akademi, Finnland

Jon Rekstad University of Oslo

Finn Ingebretsen University of Oslo

Thank You