close

Вход

Забыли?

вход по аккаунту

?

The simplified description of dipole radiative strength function

код для вставкиСкачать
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 пЃ§-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 пЃ§-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 пЃ§-decay and photoabsorption.
Two types of strength functions
For gamma- emission process
fEпЃ¬ пЂЅ
Гі f
С–
2 пЃ¬ пЂ«1
EпЃ§
DС–
average level
spacing
photoabsorption
cross-section
For photoabsorption ( E 1)
f E1 пЂЅ
partial gamma-decay
width
пЂ­1
EпЃ§
3(пЃ° c )
пЃі пЃ§ ( EпЃ§ )
2
CLOSED-FORM MODELS
Standard Lorentzian (SLO)
[D.Brink. PhD Thesis(1955); P. Axel. PR 126(1962)]
2
EпЃ§ Р“ r
f пЂЅ f ~
пѓћ 0
2
2 2
2
( EпЃ§ пЂ­ E r ) пЂ« EпЃ§ Р“ r
EпЃ§ п‚® 0
Р“r
Р“ r пЂЅ const ( E пЃ§ ) ~ 5 M eV (T пЂЅ 0)
EпЃ§
Er
Enhanced Generalized Lorentzian (EGLO)
[J.Kopecky , M.Uhl, PRC47(1993)]
[S.Kadmensky, V.Markushev, W.Furman, Sov.J.N.Phys 37(1983)]
f пЂЅ
EпЃ§ Р“ ( EпЃ§ )
( EпЃ§ пЂ­ E ) пЂ« EпЃ§ Р“ пЃ§ ( EпЃ§ )
2
2
r
2
2
2
пЂ«
0.7 Р“ ( E пЃ§ пЂЅ 0)
3
Er
Infinite fermi- liquid
(two-body
dissipation)
f пѓћ const п‚№ 0 [ E пЃ§ п‚® 0]
E пЃ§ пЂ« 4пЃ° T f
2
E пЃ§ пЂ« 4пЃ° T f
2
Р“ ( EпЃ§ ) пЂЅ Р“ r
Tf пЂЅ
U пЂ­ EпЃ§
a
2
2
2
EпЃ§
;
пѓ— K ( EпЃ§ )
K (EпЃ§ ) п‚®
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 пЂЅ f пЂЅ 8 .6 7 4 пѓ— 1 0
K GFL пЂЅ
Er
пЂ­8
пѓ—пЃі rпЃ‡ r
пЂЁ
пЂЅ 1пЂ« F
E0
K GFL пѓ— EпЃ§ пЃ‡ m
1
1
пЂЁE
3
пЂ©
пЂЁ
пЃ‡ dq пЂЁ E пЃ§
пЂ©
2
2
2
пЃ§
пЂ­E
2
r
пЂ©пЂ« K
пЂЁ1 пЂ« F
1 2
пЃ‡ m пЂЅ пЃ‡ coll пЂЁ E пЃ§ , T f пЂ© пЂ« пЃ‡ dq пЂЁ E пЃ§
пЃ‡ coll п‚є C coll E пЃ§ пЂ« 4пЃ° T f
2
1
0
3
GFL
пЂ©
1 2
пѓ©пѓ« пЃ‡ m E пЃ§ пѓ№пѓ»
пЂЅ 0.63
пЂ©
пЂ©
пЂ«
пЂЅ C dq E пЃ§ пЃў 2
1пЂ«
E2
EпЃ§
-” fragmentation” component
s 2 пЂЅ E 2 пЃў 2 пЂЅ 217.16 A
2
2
пЂЅ C dq
EпЃ§ пЃў 2 пЂ« EпЃ§ s2
2
2
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)]
Р“ (J i , EпЃ§ ) пЂЅ
пЃ®

,J f
пЃ„Z ,пЃ„N ,M i ,пЃ„пЃ® i
f
dР“ if
dE пЃ§
/ N Ji
N Ji пЂЅ пЃІ ( E , N , Z , J i )( 2 J i пЂ« 1) пЃ„ E пЃ„ Z пЃ„ N
пѓџ
microcanonical ensemble
пѓџ
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 пЂЁ EпЃ§ , T пЂ© пЂЅ
пЂЅ 8.674 пѓ— 10
пЂ­8
1
1 пЂ­ exp пЂЁ пЂ­ E пЃ§ T f
s пЂЁпЃ· , T f
пЂ©пЂЅпЂ­
1
пЃ°
пЂ©
EпЃ§
пѓ¦
s пѓ§пЃ· пЂЅ
,T f
пѓЁ
Im пЃЈ пЂЁ пЃ· , T f
пЂ©
пѓ¶
пѓ· , M eV
пѓё
пЂ­3
,
,
Approximation of strong collective state for response function
Im пЃЈ пЂЁ пЃ· , T f
пЂ©п‚µ
EпЃ§ пЃ‡ пЃ§ пЂЁпЃ· , T f
пЂЁE
2
пЃ§
пЂ­E
2
r
пЂ©
2
пЂ©
пЂ« пѓ© пЃ‡ пЃ§ пЂЁпЃ· , T f пЂ© EпЃ§ пѓ№
пѓ«
пѓ»
2
п‚· MLO1 - no restriction on multipolarity of the deformation of Fermisurface
пЃ‡пЃ§
пЂЁ пЃ· , T пЂ© пЃЂ пЃўпЃ§ c пЂЁ пЃ· , T пЂ©
Er пЂ« E0
2
пЂЁE
2
r
пЂ­E
2
0
2
пЂ© пЂ« пЂЁпЃ§ пЂЁ
пЃ· ,T
c
пЂ© пЃ·пЂ©
п‚є пЃ‡ M LO1
2
пЃ§ c пЂЁ пЃ· ,T пЂ© пЂЅ 2 /пЃґ c пЂЁ пЃ· ,T пЂ© .
Doorway state approach for collisional relaxation time пЃґ c
пЃґ c пЂЁ пЃ· ,T пЂ©
пЂЅ bпЂЁ пЃ· пЂ«U пЂ©, b пЂЅ
пЃ‡пЃ§
SMLO
Er F
4пЃ° пЃЎ
пЂЁ пЃ· , T пЂ© пЂЅ a ( EпЃ§
,пЃЎ пЂЅ
9
2
пЃі
free
16 m
пЂЁ np пЂ©
,F пЂЅ
пЃі
пЃі пЂЁ np пЂ©
free
пЂЁ np пЂ©
пЂ« U ); a пЂЅ пЃ‡ r (T пЂЅ 0) / E r
п‚· MLO2,MLO3
approximation of independent sources of dissipation for width
пЃ‡пЃ§
пЂЁ пЃ· ,T пЂ© пЂЅ
пЃґ c пЂЁ пЃ· ,T пЂ©
пЂ«
пЃґ s пЂЁ пЃ· ,T пЂ©
п‚є пЃ‡ M LO 2 ,3 ,
пЃґs
пЂЅ ksпЃ‡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
пЃґ c пЂЁ пЃ· ,T пЂ©
ks п‚є ks пЂЁ
п‚є
F
пѓ© пЂЁ пЃ· 2пЃ°
пЃЎ пѓ«
пЂ©
2
2
пЂ«T пѓ№
пѓ»
пѓ¬
пѓЇks пЂ« пЂЁ ks пЂЁ 0 пЂ© пЂ­ kr пЂ© пЂЁ пЃ· пЂ­ Er пЂ© Er ,
пЃ·пЂ©пЂЅпѓ­
пѓЇ
пѓ®ks пЂЁ0 пЂ©, пЃ· п‚і 2 Er .
пЃ· пЂј 2Er ;
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
пЃЈ (пЃ· ) пЂЅ
1
пЃў

qпЂЅn, p
aq
 dr rY10 ( rˆ ) q ( r ,  )
EXTERNAL FIELD
V q ( r , t )    ( t ) a q rY10 ( rˆ )
aqпЂЅn пЂЅ 2 Z / A , aqпЂЅ p пЂЅ пЂ­2 N / A
DENSITY VARIATION
пЃ¤пЃІ q ( r , пЃ· ) пЂЅ
2
h
3
пЃ¤ fq (r , p,пЃ· )
пѓІ dpпЃ¤
f q ( r , p , пЃ· ) пЂ« пЃ¤ ( r пЂ­ R ) пЃІ 0пЃ¤ R q (пЃ± , пЃЄ , пЃ· ).
- change of phase-space distribution function
due to linearized V-L kinetic equation with bondary condition
on moving surface
S E P A R A B L E R E S ID U A L IN T E R A C T IO N
u q q  ( r , r  )   q q   rr Y1 M ( r )Y1 M ( r  ),  q q    ( F0 , F0 )
пЂЄ
'
m
C O L L E C T IV E R E S P O N S E F U N C T IO N
W IT H M O V IN G -S U R F A C E
пЃЈ (пЃ· ) пЂЅ пЃЈ (пЃ· ) пЂ« пЃЈ s (пЃ· )
пЃЈ s (пЃ· )
-
SURFACE COM PONENT
C O L L E C T IV E R E S P O N S E F U N C T IO N
W IT H F IX E D -S U R F A C E (F S A m eth o d )
пЃЈ (пЃ· ) пЂЅ

пЃЈ q (пЃ· )
qпЂЅn, p
пЃЈ
пЃЈ q (пЃ· ) пЂЅ
0
q (
пѓ¬пѓЇ
пѓ¦ пЃ« q п‚ўq п‚ў пЃ« qq п‚ў пѓ¶ пѓјпѓЇ
0
пЃ· ) пѓ­1 пЂ­ пЃЈ q п‚ў (пЃ· ) пѓ§ 2 пЂ­
пѓ·пѓЅ
a
a
a
п‚ў
п‚ў
q
q
пѓЁ q
пѓё пѓѕпѓЇ
пѓ®пѓЇ
пѓ¦ пЃЈ q0 (пЃ· ) пЃЈ q0п‚ў (пЃ· ) пѓ¶ пЃ« qq пЂ­ пЃ« qq п‚ў 0
0
1 пЂ­ пЃ« qq пѓ§
пЂ«
пЃЈ q (пЃ· ) пЃЈ q п‚ў ( пЃ· )
пѓ·пЂ«
2
2
2 2
a qп‚ў пѓё
a q a qп‚ў
пѓЁ aq
2
2
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 пЂЅ const ;
пЃЈ
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
пЃЈ
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
EпЃ§
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 пЃ§ for 174Yb.
The E1 gamma-decay strength function
versus mass number; U=Sn; EпЃ§=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,
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
пЃЈ
2
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)
Р“r=0.026*Er1.91 (MeV)
Relative deviation of RSF within
different models and MLO1
пѓ©
1
пЃіпЂЅпѓЄ
пѓЄNm ax
пѓ«
N m ax

i пЂЅ1
пѓ© пЃі пЃ§ , abs ( Ai , M odel ) пЂ­ пЃі пЃ§ , abs( Ai , M L O 1)
пѓЄ
пЃі пЃ§ , abs ( Ai , M L O 1)
пѓЄпѓ«
пѓ№
пѓє
пѓєпѓ»
2
пѓ№
пѓє
пѓє
пѓ»
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/
Документ
Категория
Презентации
Просмотров
9
Размер файла
1 415 Кб
Теги
1/--страниц
Пожаловаться на содержимое документа