Transcript E1 RADIATIVE STRENGTH FUNCTION FOR GAMMA

CLOSED-FORM E1 RADIATIVE
STRENGTH FUNCTIONS
FOR PHOTOABSORPTION AND
GAMMA-DECAY
V. A. Plujko
Taras Shevchenko National University, Kyiv, 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 ~
Гr
( E 
2
Er2 ) 2

0
E Г r2E  0

Г r  const ( E ) ~ 5MeV (T  0)
E
Er
E Г r2
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
•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/]
RSF within modified Lorentzian (MLO)
MLO is based on expression for average gamma-width averaged
on microcanonical ensemble of initial states
[V.A.Plujko (Plyuiko), Sov.J.Nucl.Phys. 52 (1990) 639; Proc. 9th Inter.
Conf. Nucl. Reaction Mechanisms, Varenna, June 5-9, 2000, edited by E.
Gadioli, (Universita degli Studi di Milano, Suppl. N.115, 2000),113]
Г  ( J i ,E ) 

 f ,J f
 Z , N ,M i , i
dГ if
/ N Ji
dE
N Ji   ( E ,N ,Z ,J i )( 2J i  1 ) E  Z  N
dГ if
dE
 d  ( E ) Bif  ( Ei  E f  E ),
Bif    J f M f E f  f Q J i M i Ei i  d   E2  1 ,
2
Q   q a a
/
 /
/
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
8
Axially deformed nuclei - n=2 ;
E r ( E )
n
 r r
r 1


2 2
E  Er
2
r (E  Er )  r
 r ( E )  E 
2
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
The E1 photoabsorption
cross section on 144Nd
The E1 photoabsorption cross section on 144Nd.
Comparisons
of
the
photoabsorption cross sections on
208
Pb.
Panel b shows the low-energy part
of the cross sections. Experimental
data are taken from (a) A. Veyssiere,
H. Beil, R. Bergere, P. Carlos, A. Lepretre,
Nucl. Phys. A159 (1970) 561 and (b) V.V.
Varlamov, M.E. Stepanov, V.V. Chesnokov,
Izvestiya RAN. Seriya Fiz. 67 (2003) 656.
(a)
(b)
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,
f M 1  const ;
P.Oblozinsky, P. (Ed.) (1982) 121.;
Model
2
EGLO
SLO
GFL
MLO1
MLO2
MLO3
SMLO
2.2
22.9
2.6
6.47
6.52
7.16
6.06
(a)
(b)
Dipole strength functions of E1 and M 1
gamma-decay for 160 Dy (а) and 162 Dy
(b): U  Sn . Experimental data are taken
from M. Guttormsen, A. Bagheri, R.
Chankova, J. Rekstad, and S. Siem // Phys.
Rev. C68, 064306 (2003)
Dipole strength functions of E1 and M 1
gamma-decay for 166 Er (а) and 171Yb (b):
U  Sn . Experimental data are taken from E.
Melby, M. Guttormsen, J. Rekstad, A.
Schiller, and S. Siem // Phys. Rev. C63,
044309 (2001) and U. Agvaanluvsan, A.
Schiller, J. A. Becker, L. A. Bernstein, et al. //
Phys. Rev. C70, 054611 (2004)
(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
187.9
159.8
45.8
5.1
5.4
162
Dy
74.3
201.6
55.4
5.2
6.3
119.8
201.1
47.9
3.6
5.0
171
58.6
184.1
31.2
5.6
6.7
172
62.6
292.7
78.3
4.5
5.3
100.5
207.9
51.7
4.8
5.7
166
Er
Yb
Yb
(b)
EGLO
average
(a)
(b)
Dipole strength functions of E1 and M 1
gamma-decay for 97 Mo (а) and 98Mo (b):
U  Sn . Experimental data are taken from
M. Guttormsen, R. Chankova, U.
Agvaanluvsan, E. Algin, L. A. Bernstein,
et al.// Phys. Rev. C71, 044307 (2005)
Dipole strength functions of E1 and M 1
gamma-decay for 148 Sm (c): U  Sn .
Experimental data are taken from S. Siem,
M. Guttormsen, K. Ingeberg, E. Melby, J.
Rekstad, and A. Schiller // Phys. Rev.
C65, 044318 (2002)
Values of  2 deviation of calculated gamma-decay
strength functions from experimental data for nuclei 97 Mo ,
98
Mo , 148 Sm .
Model
EGLO
SLO
GFL
MLO1
SMLO
97
Mo
2.4
494.6
46.9
16.2
12.7
98
Mo
6.539
1656.1
153.6
97.1
78.4
48.9
895.7
45.4
35.0
25.2
148
Sm
Dipole strength functions of E1 and M 1
gamma-decay for
118
Sn (а) and 124Te (b):
U  Sn . Experimental data are taken from
(a)
G.P. Gueorguiev, J.Honzatko,
V.A. Khitrov, C.
Panteleev, A.M. Sukhovoj, // Nucl. Phys. A740, 20
(2004);
and
Vasilieva E.V., Sukhovoj A.M., Khitrov
V.A. // Yad.Fiz., 2001. V. 64. P. 3.
(b)
Dipole strength functions of E1 and M 1
gamma-decay for
138
Ba (а) and
150
Sm (b):
U  Sn . Experimental data are taken from
Vasilieva E.V., Sukhovoj A.M., Khitrov V.A. // Yad.Fiz.,
2001. V. 64. P. 3.
(a)
Values of  2 deviation of calculated gammadecay strength functions from experimental
data for nuclei 118Sn , 138Ba , 150 Sm , 146 Nd ,
124
Te .
EGLO
SLO
GFL
Sn
13.0
44.5
12.7
15.1
12.1
Ba
0.6
109.1
6.9
9.0
8.8
150
Sm
49.4
167.7
22.1
24.2
8.7
146
Nd
25.6
129.2
19.9
17.0
21.9
2.1
195.9
19.7
22.8
18.7
18.1
129.3
16.3
17.6
14.0
Nucleus
118
138
124
Te
average
(b)
MLO1 SMLO
The E1 gamma-decay strength functions versus mass
number for spherical nuclei. Experimental data are taken
from J. Kopecky –file(RIPL).
Model
2
EGLO
SLO
GFL
MLO1
SMLO
5.0
105
5.27
7.55
2.0
Systematics of GDR energies and widths
• Systematics are found on base of resonance parameters, that are obtained
from fitting of experimental data
 Er , |  2 | 0.1,

Er   Er1  2 Er 2  3,  2  1,

 2 Er1  Er 2  3,  2  1.
Er  a1 / A1/ 3  a2 / A1/ 6 (MeV )
r  a3Er1.9 (MeV )
Model
a1
a2
a3
 E2r /  2 BF ,  2r /  2 BF
BF
31.2
20.6
0.026±0.005
1, 1
SLO 35.796±0.008 18.431±0.004 0.02445±0.000025
0.89, 0.81
MLO 36.461±0.006 18.342± 0.003 0.02573±0.00003
0.89, 0.97
Berman B.L., Fultz S.C. // Rev. Mod. Phys. – 1975. – Vol. 47. – P. 713 – 761.
Comparison of systematics with fitting data
Conclusions
• MLO approach with asymmetric shape of the RSF provide unified and
rather reliable simple methods to estimate the dipole RSF both for gammadecay and for photoabsorption over a relatively wide energy interval
ranging from zero to slightly above the GDR peak; EGLO approach is also
rather reliable for description of gamma-decay RSF.
• MLO 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 RSF, so that the contributions of
the different mechanisms responsible for the damping of the collective
states can be further investigated.