Transcript E1 RADIATIVE STRENGTH FUNCTION FOR GAMMA

Lorentzian-like models of E1
radiative strength functions
V. A. Plujko, O. M. Gorbachenko, E. V. Kulich
Taras Shevchenko National Kyiv University, Ukraine
Content
1. Introduction: average description of the gamma-transitions by the use of
radiative strength function (RSF).
2. Closed-form description of the dipole RSF:
SLO; EGLO; GFL; MLO (SMLO).
3. Determination of the RSF function parameters.
4. Calculations and comparisons with experimental data.
5. Conclusions.
Introduction
Gamma-emission is the most universal channel of the nuclear decay,
because it is, as a rule, realized during emission of any particle or cluster.
The strengths of electromagnetic transitions between nuclear states are much
used for investigations of nuclear models, mechanisms of -decay, width of
the collective excitations and nuclear deformations.
Average strengths of  - transitions are described by radiative strength
functions.
It is very important for decreasing in computing time to use simple
closed-form expressions for -ray strength functions, since these functions in
the most cases are auxiliary quantities required for calculations of other
nuclear reaction characteristics.
The goal of this contribution is to overview and test practical methods for
the calculation of E1 radiative strength functions both for -decay and
photoabsorption.
Radiative strength functions
The photoexcitation strength function

E   3E  c 

E1
2
 
f E1 E
The gamma-decay strength function
Гі  f
і
f E 
2


1
E
Dі
partial gamma-decay
width
average level
spacing
Main closed-form models of E1 RSF
Standard Lorentzian (SLO)
[D.Brink. PhD Thesis(1955); P. Axel. PR 126(1962)]
f  f ~
E Г r2
( E 
2
Er2 ) 2

E Г r2
0
E  0
Г r  const ( E ) ~ 5MeV (T  0)
Enhanced Generalized Lorentzian (EGLO)
[J.Kopecky , M.Uhl, PRC47(1993)]
[S.Kadmensky, V.Markushev, W.Furman, Sov.J.N.Phys 37(1983)]
f 
E Г ( E )
( E  Er2 )2
2
 E Г ( E )
2
2
f  const  0 [ E  0]
Г ( E )  Г r
Tf 
U  E
a
E2  4 T f2
2
E
;

0.7 Г ( E  0)
Er3
Infinite fermi- liquid (twobody dissipation)
E2  4 T f2
 K ( E )
K ( E ) 
empirical factor from fitting exp.
data
Generalized Fermi liquid (GFL) model
(extended to GDR energies of gamma- rays)
[S. Mughabghab, C. Dunford PL B487(2000); Ext.:V.A. Plujko, O.O.Kavatsyuk, Proc. 11th Int.
Symp. Capture Gamma-Ray Spectr. and Related Topics (CGS 11), 2002, 793. ]
8
f  f  8.674 10   r  r
K GFL 

E  Er2
2
Er
 1  F11 3
E0
 m   coll

1 2

 KGFL  m E 
1 
F01
3

2
1 2
 0.63
 E , T f    dq  E 
 coll  Ccoll
 dq

KGFL  E m
 E
2
 E   Cdq E
-” fragmentation” component
 4 2T f2
2

E2
1
 Cdq
E
E2  22  E s2
s2  E2  22  217.16
A2
RSF for gamma-decay
[V.A.Plujko, NPA649 (1999); Acta Phys. Pol. B31 (2000) 435.
V.A. Plujko, S.N. Ezhov, M.O. Kavatsyuk et al ,J.Nucl.Sci Techn. (2000);
Plujko V. A., Kadenko I. M., Kulich E. V., Goriely S. et al
Proc. of Workshop on photon strength functions and related topics, Prague,
June 17-20, 2007, PSF07, 2008; http://arxiv.org/abs/0802.2183]


f E , T  8.674 10
8

1
1  exp  E T f



E

s  
,T f


1
s  , T f   Im   , T f


3
,
MeV
,



Approximation of strong collective state for response function


Im   , T f 

E   , T f


2 2
E  Er
2



   , T f E 


2
• MLO1 - no restriction on multipolarity of the deformation of Fermi-surface

  , T    c   , T 

Er2  E02
Er2

E02
  
c
  ,T   
2
  MLO1
 c   ,T   2 /  c   ,T 
Doorway state approach for collisional relaxation time
c 
2
Er F   9 16m ,
 b   U , b 
,
 free  np 
 ,T 
4 
•SMLO
F
  np 
 free  np 
   ,T   a( E  U ); a  r (T  0) / Er
At U=0, width is similar to that proposed by S.Coriely ( PhL. B436(1998) 10)
RSF for photoabsorption
 
f E1 E  8.674 10
•SMLO
8
E r ( E )
n
 r r
r 1

 
 E  a E
Axially deformed nuclei - n=2 ;

2 2
E  Er
2
r (E  Er )  r
 r ( E )  E 
2
Estimation of experimental data
•
If experimental or evaluated data for some nuclei is absent in data base,
then the total cross section is approximated by the total photoneutron cross
section
 E1    , abs     , sn     ,1nx     ,2nx   ...    , F 
• The total photoneutron cross section is estimated with partial cross sections
  , Nnx     , Nn   , Nn p    , Nn  ...
GDR parameter determination
• The adjustment is performed by the least square method with minimizing
2 
1
N  N par



  theor E ,i   exp E ,i
 
 exp E ,i
i 1

N







2
• Energy dependent errors are used for estimated data :
Spherical nuclei
Deformed nuclei
  E    min  b Er  E
 min  0.1;   E ,min   0.5
  E






 b E1  E , E  E1 ,
 min

  min , E1  E  E2 ,


 b E  E2 , E  E2 .

 min
Photoabsorbtion cross sections and gammadecay strength function
Dipole strength functions of E1 and M 1
gamma-decay for 166 Er (а) and 160 Dy (b):
U  Sn . Experimental data are taken from
(a) E. Melby, M. Guttormsen, J. Rekstad, A. Schiller, and
S. Siem // Phys. Rev. C63, 044309 (2001)
(b) M. Guttormsen, A. Bagheri, R. Chankova, J. Rekstad,
et al. // Phys. Rev. C 68 064306 (2003).
(a)
Values of  2 deviation of calculated gammadecay strength functions from experimental
data for nuclei 160 Dy , 162 Dy , 166 Er , 171Yb ,
172
Yb .
Model
SLO
GFL
MLO1 SMLO
160
Dy
223.6
191.5
52.5
5.7
7.1
162
Dy
98.3
249.5
66.8
4.5
4.5
149.8
248.2
57.7
2.6
2.5
171
70.6
222.8
37.1
4.6
4.4
172
77.3
357.4
92.8
4.3
5.1
123.9
253.9
61.4
4.3
4.7
166
Er
Yb
Yb
average
(b)
EGLO
Systematics of GDR
energies, widths and peak cross section
•
Systematics are found on base of
resonance parameters, that are
obtained from fitting of experimental
data
Er  a1 / A1/ 3  a2 / A1/ 6 (MeV )
 Er , |  2 | 0.1,

Er   Er1  2 Er 2  3,  2  1,

 2 Er1  Er 2  3,  2  1.
Er  a1 (1  a3  I A  ) / A1/ 3  a3 (1  a4  I A  ) / A1/ 6 ( MeV )
2
r  a5Er1.91 (MeV )
2
I  (N  Z ) A
 Sr 2  a6 60 NZ A (MeV·mb)
n
Sr 
 r  r
r 1
Berman B.L., Fultz S.C. // Rev. Mod. Phys. – 1975. – Vol. 47. – P. 713 – 761.
Comparison of systematics with fitting data
Conclusions
• Modefied LOrentzian approach is based on general relations between the
RSF and the nuclear response function. Therefore it can potentially lead to
more reliable predictions among other simple models.
• The energy dependence of the width is governed by complex mechanisms
of nuclear dissipation and is still an open problem.
• Reliable experimental information is needed to better determine the
temperature and energy dependence of the Radiative Strength Function, so
that the contributions of the different mechanisms responsible for the
damping of the collective states can be further investigated.
•The RSF within SLO, GFL and EGLO models for gamma-decay are not
agree with general expression for radiative strengths in heated nuclei which
corresponds to detailed balance principle with the canonical distribution for
initial states.
• In the EGLO expression for RSF includes an additional phenomenological
contributions.
• Gamma-ray energy dependence of widths in expressions within EGLO
and GFL models is introduced formally by substitution of the gamma-ray
energy instead of GDR energy.
[ T. Belgya, O. Bersillon, R. Capote, T. Fukahori, G. Zhigang, S. Goriely, M. Herman, A.V.
Ignatyuk, S. Kailas. A. Koning, P. Oblozinsky, V. Plujko and P. Young. IAEA-TECDOC-1506:
Handbook for calculations of nuclear reaction data: Reference Input Parameter Library-2,
IAEA, Vienna, 2005, Ch.7; http://www-nds.iaea.org/RIPL-2/]