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CLASSICAL TREATMENT OF THE DEEP INELASTIC COLLISION OF 238U ON 238U INCLUDING DEFORMATION DEGREES OF
FREEDOM
M. Münchow, W. Scheid
To cite this version:
M. Münchow, W. Scheid. CLASSICAL TREATMENT OF THE DEEP INELASTIC COLLISION
OF 238U ON 238U INCLUDING DEFORMATION DEGREES OF FREEDOM. Journal de Physique
Colloques, 1984, 45 (C6), pp.C6-463-C6-471. �10.1051/jphyscol:1984655�. �jpa-00224258�
JOURNAL DE PHYSIQUE
Colloque C6, supplbment au n06, Tome 45, juin 1984 page C6-463
238 238 CLASSICAL TREATMENT OF THE DEEP INELASTIC COLLISION OF U ON U
I N C L U D I N G D E F O R M A T I O N D E G R E E S O F FREEDOM*
M . Miinchow and W . Scheid
I n s t i t u t fur Theoretische Physik, Justus-Liebig-UniversitiEt Giessen, F . R. G.
Rksumg
-
Une description classique de collisions trss in&las- tiques entre des noyaux d&form&s sera prgsentke. Les forces de friction et leurs moments s o n t d 6 r i v 6 s d ' u n m o d 6 l e p a r d o u b l e c o n ~ 0 - lutionquiest dgfini comrne paramstre de puissance et de rayon.Le modsle est utilisCpour ktudier la collision trss inklastique de 2 3 8 ~ - 2 3 8 ~ 5 l'knergie Elab=7.42 MeV/amu et sa dgpendance de l'orientation initiale des noyaux.
Abstract
-
A classical description of deep inelastic collisions between deformed nuclei is presented. The friction forces and their noments are derived from a double-folding model, which is defined with a strength and range parameter. The model is used for studying the deep inelastic collision of 2 3 8 ~on 2 3 8 ~ at Elab'7.42 MeV/amu and its dependence on the initial orientation of the nuclei.I - INTRODUCTION
Three or more heavy final fragments have been observed in several heavy ion reactions /I-3/. For the theoretical interpretation of these reactions one assumes that first an intermediate nuclear system is formed in a deep inelastic collision. The intermediate system breaks most probably into two primary fragments, after a large fraction of the initial kinetic energy and angular momentum of the relative mo- tion is transferred 'to intrinsic excitation energy and angular momen- tum, respectively. The highly excited primary fragments can fission in a second step. The mass and angular distributions of the final fragments depend sensitively on the excitation of the intermediate system and on the relative distance between the primary fragments, when they break in parts or fission. Recently proximity and non-equi- librium effects were found in ternary reactions in the collision of
" ~ r on 1 6 = ~ r and lZ9xe on '"sn /I/. In order to describe the se- yuential fission and proximity effects in the collision of 2 3 8 ~on
3 8 properly, we have studied the influence of the deformation and ~
rotational degrees of freedom on the first step of this reaction, i.e. on the deep inelastic collision.
In this paper we describe the time evolution of deep inelastic colli- sions between deformed nuclei in terms of classical mechanics. In Sect.2 we derive the classical equations of motion for the relative motion and the rotation and deformation degrees of freedom. Sect.3 presents a double-folding model for the friction forces and their mo- ments and Sect.4 their multipole expansion. In Sect.5 we apply this model to the deep inelastic collision of 2 3 8 on ~ 2 3 8 ~at Elab'7.42
MeV/amu and study the dependence of the deformation and total kinetic energy loss on the initial orientation of the intrinsic nuclear axes.
h his
work has been supported by BMFT and GSI (Darmstadt) Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984655JOURNAL DE PHYSIQUE
I1
-
CLASSICAL EQUATIONS OF NOTION FOR DEFORMED NUCLEIFor the collision of deformed heavy nuclei we assume that the time evo- lution of the collective coordinates (first moments) can be described with classical equations of motion including conservative and friction
forces. The chosen model describes 13 d2grees of freedom,namely the re- lative motion with the relative vector R and the rotation and deforma- tion of the nuclear shapes with the quadrupole surface coordinates a 2 ~ ( 1) and a2u(2) of the nuclei 1 and 2, respectively. Instead of the coordinates a2,, (i) we will also use the Euler angles ~ ( ~ ) = { 0 4 & ? 2,3)
,
defining the orientation of the principal axes of the nuclei wlth re- spect to the laboratory system (see Fig.l), and the intrinsic defor- mation parameters
Bi
and yi (i=1,2), describing the 8 - , and y-vibrations.Fig.1 The coordinates defining the orientation of the nuclear principal axes.
Let us introduce the abbreviations qk and pk with k=1,
...
13 for the coordinates and their conjugate momenta. Then the equations of motion, including the friction forces Fk, are derived from the classical Hamil- tonian H:The Hamiltonian contains the kinetic and potential+energies of the chosen degrees of freedom. The potential energy V ( R , O . ~ ~ (I ) ,a2,, (2) ) is written as
The n u c l e a r and Coulomb i n t e r a c t i o n p o t e n t i a l s , VN and VC, a r e c a l c u - l a t e d w i t h a d o u b l e - f o l d i n g model developed f o r e l l i p s o i d a l l y de- formed n u c l e i i n Ref. 4 . The n e x t t e r m s i n Eq. ( 2 ) a r e t h e p o t e n t i a l e n e r g i e s o f t h e 8- and y - o s c i l l a t i o n s f o r t h e s e p a r a t e d n u c l e i . I t i s assumed t h a t t h e n u c l e i have a s t a t i c 6-deformation o f f 3 i o i n t h e i r ground s t a t e due t o s h e l l e f f e c t s . For t h e 2 3 8 ~ + 2 3 8 ~ s y s t e m we
-0.26. The s t i f f n e s s p a r a m e t e r s C g i and C i and t h e masses
! ~ ~ t ~ : t v ! k ? a t i o n s have been t a k e n from t h e r o t a t i o n a x - v i b r a t i o n a l model f i t t e d w i t h t h e s p e c t r u m of 2 3 8 ~/ 5 / . The moments of i n e r t i a a r e t a k e n a s t h e moments o f t h e r i g i d r o t a t o r and depend on t h e i n - t r i n s i c d e f o r m a t i o n p a r a m e t e r s .
I11
-
MODEL FOR DISSIPATIVE FORCES BETWEEN ELLIPSOIDALLY DEFORMED NUCLEI+
I n t h e model o f Tsang / 6 / t h e f r i c t i o n f o r c e dF1, a c t i n g on a volume element o f n u c l e u s 1 w i t h p l d r l n u c l e o n s , i s s e t p r o p o r t i o n a l t o t h e o v e r l a p o f t h e d e n s i t i e s and t h e r e l a t i v e v e l o c i t i e s o f t h e mass f l o w s o f t h e n u c l e i i n t h i s volume e l e m e n t ( f o r t h e c o o r d i n a t e s s e e F i g . 2) :
F i g . 2 Schematic p i c t u r e of t h e c o o r d i n a t e s and
v e l o c i t i e s used f o r d e f i n i n g t h e f r i c t i o n f o r c e .
-f +
where v l and v a r e t h e v e l o c i t y f i e l d s o f t h e n u c l e a r m a t t e r o f n u c l e u s 1 and
8 ,
r e s p e c t i v e l y . Tsang h a s c a l c u l a t e d t h e d i s s i p a t i v ef o r c e s between s p h e r i c a l n u c l e i w i t h i n t h i s model. We e x t e n d T s a n g ' s model by i n t r o d u c i n g a d i s t r i b u t i o n f u n c t i o n w i t h r a n g e ro f o r t h e d i s s i p a t i v e i n t e r a c t i o n between t h e volume e l e m e n t s of t h e n u c l e i . Then t h e f r i c t i o n f o r c e , a c t i n g on t h e c e n t e r of mass o f n u c l e u s 1 due t o t h e motion o f n u c l e u s 2 , i s assumed i n t h e form o f t h e follow- i n g d o u b l e - f o l d i n g i n t e g r a l :
-f -+ -+ + + + +
where v r e l ( r l r r 2 ) = v l ( r l ) - v 2 ( r 2 ) . ( 5 )
C6-466 JOURNAL DE PHYSIQUE
The c o o r d i n a t e s and v e l o c i t i e s a r e d e p i c t e d i n F i g . 2 . The Gaussian d i s t r i b u t i o n f u n c t i o n g i ~ r e s r i s e t o n o n l o c a l c o n t r i b u t i o n s t o t h e
f r i c t i o n a l i n t e r a c t i o n between t h e n u c l e i . I n t h e l i m i t r -+0 t h e f r i c t i o n f o r c e ( 4 ) approaches T s a n g ' s a n s a t z . With t h e f r y c t i o n f o r c e ( 4 ) one can d e s c r i b e n o t o n l y t h e d i s s i p a t i o n o f c o l l e c t i v e e n e r g y i n t o i n t r i n s i c e x c i t a t i o n e n e r g y , b u t a l s o a t r a n s f e r o f c o l l e c t i v e a n g u l a r momentum i n t o i n t r i n s i c a n g u l a r momentum. T h i s has a l r e a d y been r e c o g n i z e d f o r s i m i l a r f o r c e s by Deubler and D i e t - r i c h / 7 / .
The a n s a t z ( 4 ) f o r t h e f r i c t i o n f o r c e h a s t h e advantage t h a t it de- pends o n l y on two f r e e p a r a m e t e r s , t h e s t r e n g t h k and r a n g e r,, which b o t h can b e o b t a i n e d by f i t t i n g t h e e x p e r i m e n t a l l y o b s e r v e d d i s s i p a t i o n of e n e r g y . A f u r t h e r advantage o f ( 4 ) i s t h e p o s s i b i l i t y t o c a l c u l a t e t h e d i s s i p a t i v e f o r c e s and moments a n a l y t i c a l l y i f t h e n u c l e a r d e n s i t i e s pl and p2 a r e g i v e n a s e x p a n s i o n s i n terms o f Gaussian f u n c t i o n s , a l l o w i n g e l l i p s o i d a l l y deformed n u c l e a r s h a p e s . The i n t e g r a l s a r e o f t h e same t y p e a s t h o s e s o l v e d f o r t h e i n t e r - n u c l e a r i n t e r a c t i o n ~ w h e r e we u s e d t h e d o u b l e - f o l d i n g model w i t h a Gaussian two-body i n t e r a c t i o n (for d e t a i l s s e e Ref. / 4 / )
.
I n t h e c a l c u l a t i o n o f t h e d i s s i p a t i v e f o r c e s and moments we assume f o r s i m p l i c i t y , t h a t t h e n u c l e i r o t a t e r i g i d l y , n e g l e c t i n g f r i c t i o n f o r c e s i n t h e 6- and y - v i b r a t i o n s . T h e r e f o r e l t h e r e l a t i v e v e l o c i t y ( 5 ) i s g i v e n by:
+ i
where wl and w2 a r e t h e a n g u l a r v e l o c i t i e s o f t h e n u c l e i 1 and 2 , r e - s p e c t i v e l y .
With E q . ( 4 ) we g e t t h e f r i c t i o n f o r c e s a c t i n g on t h e c e n t e r s o f mass
+ 3 4 - 3 + 3
F1 = -F2 = -k(JR+wl x S-w2 x T)
,
( 7 )and t h e moments o f t h e f r i c t i o n f o r c e s w i t h r e s p e c t t o t h e c e n t e r s of t h e n u c l e i ( i = I , 2 )
where
i A
The s c a l a r f u n c t i o n J , t h e v e c t o r s
5
and T and t h e t e n s o r s 0 . a r e d e f i n e d by d o u b l e - f o l d i n g i n t e g r a l s : I j+ I\
Here, xik i s t h e kth C a r t e s i a n component of ri, and (Bi ) k l t h e e l e - ments o f t h e m a t r i c e s Oi.. A s p o i n t e d o u t , we can a n a l y z i c a l l y s o l v e t h e i n t e g r a l s (1 1 ) and ( j 2 ) w i t h t h e same d e n s i t i e s a s used i n t h e c a l c u l a t i o n o f t h e n u c l e a r i n t e r a c t i o n p o t e n t i a l / 4 / .
I V - MULTIPOLE EXPANSION OF THE FRICTION FORCES AND MOMENTS
The s t r u c t u r e o f t h e f r i c t i o n f o r c e s ( 7 ) and t h e i r moments ( 8 ) be- comes a p p a r e n t w i t h a m u l t i p o l e expansion i n t e r m s o f t h e q u a d r u p o l e d e f o r m a t i o n c o o r d i n a t e s . I n view o f t h e l a t e r a p p l i c a t i o n o f t h e f o r - mulas t o t h e c o l l i s i o n o f 2 3 8 ~on 2 3 8 we s p e c i a l i z e t h e m u l t i p o l e ~
e x p a n s i o n t o t h e c a s e o f a c o l l i s i o n o f two i d e n t i c a l n u c l e i . Using t h e u s u a l n o t a t i o n f o r t h e c o u p l i n g of s p h e r i c a l t e n s o r s o f rank [ k ] , we o b t a i n bhe f o l l o w i n g expansion up t o t e r m s l i n e a r i n t h e quadru- p o l e d e f o r m a t i o n c o o r d i n a t e s a [
1
(1 ) and a [ 21 ( 2 ) :u u
where t h e s p h e r i c a l c o m ~ o n e n t s a r e g i v e n by:
-f +
Here, eq a r e s p h e r i c a l u n i t v e c t o r s and R t h e s o l i d a n g l e o f R. I t i s s t r a ~ g h t f o r w a r d t o e x t e n d t h e formula (14) up t o q u a d r a t i c t e r m s i n t h e d e f o r m a t i o n c o o r d i n a t e s . S i m i l a r e x p r e s s i o n s , s l i g h t l y more complex t h a n (141, can b e d e r i v e d f o r t h e moments o f t h e f r i c t i o n f o r c e s
.
The m u l t i p o l e e x p a n s i o n s of t h e f r i c t i o n f o r c e s and moments and a l s o o f t h e i n t e r n u c l e a r p o t e n t i a l y i e l d a n a l y t i ~ a l e x p r e s s i o n s i n + t h e d e f o r m a t i o n c o o r d i n a t e s and s o l i d a n g l e o f R. I n t h e c a s e o f F1 o n l y a few f u n c t i o n s i n R , namely I. ( R )
,
G ( R ),
K ( R ),
LR( 5 )
and ME ( R ),
have t o b e d e t e r m i n e d i n o r d e r t o g e t t h e dependence o f F1 on t h e 13 coor- d i n a t e s o f o u r model ( u p t o t e r m s l i n e a r i n t h e d e f o r m a t i o n c o o r d i - n a t e s ) . The m u l t i p o l e e x p a n s i o n s p l a y an i m p o r t a n t r o l e f o r t h e nu- m e r i c a l s o l u t i o n o f t h e e q u a t i o n s o f motion ( 1 ) . We have found t h a t t h e f a s t e s t p r o c e d u r e i s t o compute t h e p o t e n t i a l s and d i s s i p a t i v e f o r c e s and moments from t h e i r m u l t i p o l e e x p a n s i o n s . Otherwise t h e i n t e g r a l s , g i v e n i n E q s . ( l l ) and ( 1 2 ) , have t o b e c a l c u l a t e d a g a i n and a g a i n i n each s t e p o f t i m e d u r i n g t h e s o l u t i o n o f E q s . ( l ) , s i n c e t h e s e i n t e g r a l s depend on 13 c o o r d i n a t e s a n d , t h e r e f o r e , can n o t b e s t o r e d e c o n o m i c a l l y i n t h e computer.C6-468 JOURNAL DE PHYSIQUE
238 238"
V
-
DYNAMICAL CALCULATIONS FOR THE SYSTEM U-With the above described model we study the deep inelastic collision of 2 3 8 ~on 2 3 8 ~at an energy Elab=7.42 MeV/amu and compare some re- sults with the experimental data of Freiesleben et al. /8/. In the calculations we assume rotational symmetry of the nuclear shapes about the intrinsic 2'-axis neglecting the y-degrees of freedom. The para- meters of the friction force are set k=5-10-20 MeVfmsec and ro=
2.3 fm.
Fig.3 shows the dependence of the 6-coordinates on the internuclear distance for three initial orientations of the nuclei. In the upper part of the figure the symmetry of the initial orientations yield equal 6-values for both the nuclei (B1=B2)
,
whereas the lower part shows a situation where the 6-coordinates develop differently in time(61#62). In general one obtains a strong dependence of the B-defor- mation on the internuclear distance and the initial orientation of
the intrinsic nuclear axes. However, if a friction force in the 6-de- grees of freedom is included in the calculation, the time-variation of the B-coordinates i s reduced.
0.15
12 16 20 2L 28
R l f m l
Fig.3 The @-deformation as function of the relative coordinate R for Elab=7. 42 MeV/amu and L=200
'h
for three different initial orientations as indicated in the figure.I n F i g . 4 t h e f i n a l k i n e t i c e n e r g y (TKE) i s shown a s a f u n c t i o n o f t h e s c a t t e r i n g a n g l e
eCM
f o r v a r i o u s i n i t i a l o r i e n t a t i o n s o f t h e n u c l e i . The TKE i s c a l c u l a t e d by t h e formulam *
+ + + + + 2
TKE = E CM + / (FjR + M 1 ~ 1 + M 2 ~ 2 ) d t -
1
(Erot(i)+Evdi)) ( 1 5 )--a, i=3
W e n o t i c e a f l u c t u a t i o n o f t h e TKE o f t h e o r d e r o f 70 MeV a b o u t t h e mean TKE a t eCM=1800 a r i s i n g due t o t h e d i f f e r e n t i n i t i a l o r i e n t a - t i o n s o f t h e intrinsic n u c l e a r a x e s . A comparison o f two c u r v e s o f F i g . 4 w i t h t h e e x p e r i m e n t a l Wilczynski-diagram o f Ref./8/ i s p r e - s e n t e d i n F i g . 5 and shows a q u a l i t a t i v e agreement.
800
-
-
>Q
-
IW Y k 700
-
238
u
+ 23aUEL,, = 7.42 MeV1 amu
6003,
60 I 90 I 120 I 150 I 180eCM 1°1
Fig.4 The TKE f o r 2 3 8 ~ + 2 3 8 ~ a s f u n c t i o n o f t h e s c a t t e r i n g a n g l e f o r Elap=7.42 MeV/amu and v a r i o u s i n i t i a l o r i e n t a t i o n s of t h e i n - t r i n s i c n u c l e a r a x e s a s i n d i c a t e d a t t h e c u r v e s .
F i g . 6 shows t h e f i n a l i n t r i n s i c e x c i t a t i o n e n e r g y and c o l l e c t i v e angu- l a r momentum o f t h e p r o j e c t i l e a s f u n c t i o n o f t h e t o t a l a n g u l a r mo- mentum f o r two symmetric i n i t i a l o r i e n t a t i o n s o f t h e n u c l e i . The i n - t r i n s i c e x c i t a t i o n e n e r g y o f n u c l e u s i = l o r 2 i s c a l c u l a t e d by
JOURNAL DE PHYSIQUE
I I I I 1 1 I I I
lm dl6
-
-
-
3 -
-
I-
W Y I-
600
- -
ml
U + U ,1766 MeVFig.5 Comparison of the TKE for two different initial orienta- tions with an experimental distribution measured by Freiesleben et al. /8/.
- Fig. 6
Excitation energy E; and collective angular momentum Irot ( 1 ) of the Uranium pro- jectile after the collision as function of the total an- gular momentum for two ini- tial orientations of the intrinsic nuclear axes as indicated in the figure. Be- cause of the symmetry of the
100 chosen initial orientations
the same values are obtained also for the Uranium target, i
.
e. E:=E$, Irot ( I ) =Irot (2) for the special cases shown in the figure.0 200 400 600 800
I Ihl
The excitation energy is strongly correlated with the TKE as can be recognized from Figs.4 and 6. The final collective angular momentum of the projectile and target nucleus has a minimum for I=600
5
in the case No.1 of the initial orientation. This minimum is caused by a cancellation between the moments of the friction force and Coulomb force.VI
-
CONCLUSIONSA double-folding model is introduced for the calculation of the fric- tion forces and their moments. The frictional interaction between the nuclei has a finite range, which has the consequence that the friction interaction does not conserve the total collective angular momentum. In the same manner as collective energy is dissipated into
intrinsic excitation energy, the moments of-the friction forces can also transfer collective angular momentum into intrinsic one.
The dynamical calculations for the system 2 3 8 ~ - 2 3 8 ~ show significant effects on the initial orientation of the intrinsic nuclear axes. We found an appreciable broading of the TKEL versus the scattering angle due to the rotation and deformation degrees of freedom. The model and the calculations need further refinements, namely the dy- namical treatment of the y- and neck degrees of freedom.
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(1983) 189
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(1982) 381 5. EISENBERG J M and GREINER W, Nuclear Theory I (North-Holland,Amsterdam 1 9 70)
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