Transcript Document

The simplified description of dipole
radiative strength function
V.A. Plujko, E.V.Kulich, I.M.Kadenko,
O.M.Gorbachenko
Taras Shevchenko National University
Kyiv, Ukraine
CONTENT
1. Introduction and radiative strength function (RSF)
definitions.
2. Closed-form description of the RSF:
GFL; MLO; SMLO.
SLO;EGLO;
3.
Semiclassical (MSA) and microscopic (HFBQRPA) methods of E1 calculations.
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.
It is very important for decreasing in computing
time to have 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 investigation was to test practical
methods for the calculation of E1 radiative strength
functions both for -decay and photoabsorption.
Two types of strength functions
For gamma- emission process
f E 
Гі  f
і
E2 1Dі
average level
spacing
photoabsorption
cross-section
For photoabsorption ( E1)
f E1 
partial gamma-decay
width
E1
3( c)2
 ( E )
CLOSED-FORM MODELS
Standard Lorentzian (SLO)
[D.Brink. PhD Thesis(1955); P. Axel. PR 126(1962)]
E Г r2
f  f ~
0
2
2
2
2
( E  Er )  E Г r
E  0
Гr
Гr  const (E ) ~ 5MeV (T  0)
E
Er
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  E )  E Г ( E )
2
2 2
r
2
2

0.7 Г ( E  0)
Er3
Infinite fermi- liquid
(two-body
dissipation)
f  const  0 [ E  0]
E2  4 Tf2
Г ( E )  Г r
Tf 
E2  4 T f2
U  E
a
2
E
;
 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)]
f  f  8.674  10   r  r
8
E
K GFL  E  m
2

KGFL

E
2
r

 K GFL  m E 
 1  F 3
12
Er
1

 1  F1 3
E0
1
0
12
 0.63
m  coll  E ,T f   dq  E 

coll  Ccoll E2  4 2T f2
 dq  E   Cdq E  2

E2
1
 Cdq E2  22  E s2
E
-” fragmentation” component
s2  E2  22  217.16 A2
2
Modified Lorentzian approach (MLO)
was obtained using expression for
average gamma-width
[V.A.Plujko et al., NPA649 (1999); J.Nucl.Sci Techn. (2000)]
Г ( J i , E ) 

 f ,J f
Z , N , M i ,  i
dГ if
dE
/ N Ji
N Ji   ( E , N , Z , J i )( 2 J i  1)EZN

microcanonical ensemble

most appropriate for closed systems like nuclei
Gamma-strength within
MLO
MLO-modified Lorentzian approach
[V.A.Plujko et al, NP A649(1999); J. Nucl. Sci Thech. (2002)]
f  E , T  
E

1
 8.674 10
s  
,Tf
1  exp   E T f  
8
s  , T f   
1

Im   , T f


3
,
MeV
,


,
Approximation of strong collective state for response function
Im   , T f  
E   , T f
E
2


2 2
r
E

   , T f  E 
2
 MLO1 - no restriction on multipolarity of the deformation of Fermisurface
Er2  E02
   , T    c   , T 
  MLO1
2
Er2  E02   c   ,T   


 c   ,T   2 / c   ,T  .
Doorway state approach for collisional relaxation time  c
  np 
9 2 16m
Er F
,   free
, F  free
 b   U  , b 
  np 
  np 
4 
 c  ,T 
SMLO
  ,T   a( E  U ); a  r (T  0) / Er
 MLO2,MLO3
approximation of independent sources of dissipation for width
  , T  

 c  ,T   s  ,T 
  MLO 2,3 ,
s
 ks W .
which are the sum of the collisional and fragmentation components.
• MLO2: Doorway state approach for collisional relaxation time
• MLO3: Fermi-liquid approach for collisional relaxation time
 c   ,T 
ks  ks 

F
2
 2   T 2 



ks   ks  0   kr     Er  Er ,
  
ks  0  ,   2 Er .
  2 Er ;
Moving surface approximation (MSA)
based on solving Vlasov-Landau kinetic equation for finite system
with moving surface
V.I.Abrosimov, M. Di Toro, V.M.Strutinsky, NPA562(1993)41;
V.I.Abrosimov,O.I.Davidovskaya Izv.RAN 68(2004)200
 ( ) 
aq  drrY10 (rˆ)q (r ,  )

 q n , p
1
EXTERNAL FIELD
Vq (r , t )   (t )aq rY10 (rˆ)
aq  n  2 Z / A , a q  p   2 N / A
DENSITY VARIATION
q (r , ) 
2
dp f q (r , p, )   (r  R) 0 Rq ( , , ).
3
h
 f q ( r , p,  )
- change of phase-space distribution function
due to linearized V-L kinetic equation with bondary condition
on moving surface
SEPARABLE RESIDUAL INTERACTION
uqq ( r , r )   qq  rr Y1M ( r )Y1M ( r ),  qq   ( F0 , F0' )
m
COLLECTIVE RESPONSE FUNCTION
WITH MOVING-SURFACE
 ( )   ( )   s ( )
 s ( )
- SURFACE COMPONENT
COLLECTIVE RESPONSE FUNCTION
WITH FIXED-SURFACE (FSA method)
 ( ) 
 q ( )

q n , p

  qq

2
a
q
 
 q0 ( ) 1   q0 ( ) 
 q ( ) 

 qq  

aq aq  
2
2
  q0 ( )  q0 ( )   qq   qq 0
0
1   qq 



(

)


 ( )
q
q
2
2
2 2
aq 
aq aq
 aq
The E1 gamma-decay strength
function on 144Nd for U=Bn
The E1 gamma-decay strength function on 144Nd. The
experimental date are taken from Yu.P. Popov, in Neutron
induced reactions, Proc. Europhys. Topical Conf., Smolenice,
1982, Physics and Applications, Vol. 10, P.Oblozinsky, P.
(Ed.) (1982) 121.; f M 1  const;
2
144Nd
EGLO SLO
GFL MLO1
MLO2
MLO3
SMLO
2.2
2.6
6.52
7.16
6.06
22.9
6.47
The E1 photoabsorption
cross section on 144Nd
The E1 photoabsorption cross section on 144Nd.
The E1 gamma-decay strength
function on 90Zr
The E1 gamma-decay strength function on 90Zr. The
experimental date are taken from G.Szeflinska, Z.Szeflinski,
Z.Wilhelmi, NP A323(1979)253; Z.Szeflinski, G.Szeflinska,
Z.Wilhelmi et al, PL 126b(1983)159
2
90Zr
EGLO SLO
GFL MLO1
MLO2
MLO3
SMLO
27.4
22.4
3.48
5.76
15.32
6.07
10.59
The E1 photoabsorption strength
function on 90Zr
The E1 photoabsorption strength function on 90Zr. The
experimental date are taken from A. Lepretre, H. Beil, R.
Bergere, P. Carlos, A. Veyssiere, M. Sugawara; Nucl. Phys.
A175, 609(1971)
The E1+M1 gamma-decay strength function versus energy
E for 114Cd. Experimental data are taken from
E.Vasilieva, A. Sukhovoj, V.A. Khitrov Yad. Fyz. V.64
(2001)
The E1+M1 gamma-decay strength function versus
energy E for 174Yb.
The E1 gamma-decay strength function
versus mass number; U=Sn; E=0.8U
The E1 gamma-decay strength function versus mass number
A. Experimental data are taken
in: http://www-nds.iaea.or.at/ripl/.
from
J.
Kopecky,
2
EGLO
SLO
GFL
MLO1
MLO2
MLO3
SMLO
A<=80
64.9
527.
78.8
89.8
152.
113.
75.2
80<A<=150
6.59
124.
7.61
10.5
32.5
22.8
6.24
A>150
8.97
34.2
4.66
15.5
12.4
11.1
18.6
All nuclei
6.79
62.2
5.6
11.1
18.0
13.6
10.7
Mass number dependence of the
relative deviation of photoabsorption
C-S within SLO and MLO1 models
Er=31.2*A-1/3+20.6*A-1/6 (MeV)
Гr=0.026*Er1.91 (MeV)
Relative deviation of RSF within
different models and MLO1

1

Nmax

   ,abs ( Ai , Model )    ,abs( Ai , MLO1)


  ,abs( Ai , MLO1)
i 1 

N max



2 1/ 2




Conclusions
Numerical studies indicate that the calculations of E1 radiative
strength functions within the closed-form models give similar
results in a range of gamma-ray energies around the GDR peak.
However the results within MLO(SMLO) and EGLO models
are different from SLO model calculations in the low energy
region. In particular, they have asymmetric shape and for E_g
=7 MeV, the calculated RSF values within SLO model are about
two times greater comparing to the ones obtained for
MLO(SMLO) and EGLO models.
The overall comparison of the calculations within different
models and experimental data showed that MLO(SMLO) and
GFL provide the most reliable simple methods for determining
the E1 radiative strength functions over a relatively wide energy
interval ranging from zero to above the GDR peak. The
MLO(SMLO) and GFL are not time consuming calculational
routes and can be recommended for general use; both of them can
be used to predict the photoabsorption cross-sections and to
extract the GDR parameters from the experimental data for
nuclei of middle and heavy weights but collisional component of
the GFL damping width can become negative in some deformed
nuclei.
Microscopic HFB-QRPA(RIPL3) model and semi-microscopic
MSA approach with moving surface seems to be more adequate
for estimation of the RSF in spherical light and medium-mass
nuclei if reliable values of the GDR parameters are not available.
The studies were performed within RIPL-2&3 projects (IAEA
Research Contract #12492); http://www-nds.iaea.org/RIPL-2/