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´)206Pb
 206Pb(3He, )205Pb
 208Pb(3He, 3He´)208Pb
 208Pb(3He, )207Pb
The particle- coincidences were recorded while the
experiment ran for 2-3 weeks.
Oslo Cyclotron Lab
CACTUS
Concrete wall
http://www.physics..no/ocl/intro/
 Detector Arrangement
The 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.
Data Analysis:
Raw Data
Unfolding of
coincidence spectra
Particle Spectra
Calibration
-spectra
calibration
and alignment
Data
Reduction
thickness spectrum
Gating on particles
Particle - 
coincidences
Extracting Primary- 
spectra
Coincidence Spectra:
3He-
-
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
Response function
Next trial fn, u1 = uo + (r - fo)
–
–
Generally, ui+1 = ui + (r - fi)
Iteration continues until fi ~ r
of 4 MeV
.
Fluctuations in folded spectra Compton background are subtracted.
Response Matrix
Test of method:
f(E x , E )  R (E , E)  u (E x , E)
E
Raw
unfolded

folded
Primary -matrix:
Assumption:
 The -decay pattern from any Ex is independent of the population
mechanism

The nucleus seems to be an CN like system prior to emission.
h  f.g.
f1  g,
Method: The
-spectrum of the highest Ex is estimated by,
g   n i wi f i
i


f1 = f.g. -spectrum of highest Ex bin.
g = weighted sum of all spectra.
wi = prob. of decay from bin 1 - i.
n i = i /  j
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Multiplicity Normalization :
A( f i )
i 
Mi
Algorithm:
ni 
1.
2.
3.
4.
1 A( f1 ) M i

i
M1 A( f i )
Apply a trial fn wi
Deducing hi = fi -  g
Transforming hi to wji (i.e. Unfold h, make h having same energy
calibration as wi, normalize the area
 of h to 1).
If wji (new) ≈ wji (old) then the calculated hi would be the
Primary- function for the level Ei, else proceed with (2)
Some experimental conditions can introduce severe
systematic
errors, like pile-up effects, isomers etc.
Testing of an Experimental spectrum:
h=f-gf g
Ex ~ 4.5-5.5 MeV.
Extraction of NLD & GSF:
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are needed to see this picture.
 Brink-Axel Hypothesis;
P(Ei ,E )  (E )(Ei  E )
 Transforming;
~
  A exp[ (E  E )](E  E )

~
T  B exp( E )T (E )
A, B,  are free parameters.

A. Schiller et al./ Nucl. Instr. & Methods in Physics Research A
447 (2000) 498-511
fEdiff
Ef
 Gamma transition probability
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 Theoretically
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 Minimizing;
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Nuclear Level Density:
 At low Ex;
o
comparing the extracted NLD to
No. of known levels
Excitation energy bin
 At Bn deducing NLD from resonance spacing208
data.
Pb
=0.634


2 2 
1
 Sn 


D (I  1)exp((I  1) 2 /2 2 )  I exp(I 2 /2 2 )
 BSFG level density extrapolation

mod el (E)  
  0.0146A 5 / 3


exp( 2 aU)
,
1/ 4 5 / 4
12 2a U 
1 1 4a(u  E1)
2a
where a = level density parameter,
U = E-E1 and E1 = back-shifted parameter,
Experimental NLD
S(E)  k ln( (E))
Entropy:
(E)
(E) 
o
 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;
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,
1 T(E )
f (E ) 
2 E 3

Collaborators
Magne 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