PramS.a, Vol. 11, No. 4, October 1978, pp. 441-455, ~) printed in India
On the charge and shape polarization of fission fragments, fusioning nuclei, or scattering heavy ions near contact
R A I N E R W H A S S E *
Department of Physics, Indian Institute of Technology, Kanpur 208 016
*Present and permanent address: Sektion Physik, Universitiit Munchen, Am Coulombwall 1, D-8046 Garching, Germany.
MS received 3 May 1978
Abstract. The polarization of the fragments from binary fission or of scattering or fusioning heavy nuclei is investigated in the liquid drop model. Due to the mutual Coulomb repulsion near contact the fragments may be polarized with respect to their charges (electric dipole moments from inhomogeneous charge distributions) as well as with respect to their shapes (quadrupole and octupole deformations). The lowering of the minimum energy near contact due to charge polarization is in the order of 1 MeV if one takes into account the energy from the giant dipole restoring force derived from the volume symmetry energy in addition to the liquid drop ener- gies. The question whether one obtains prelate or oblate shapes is entirely due to the restriction in deformation space (fixed distance between centers-of mass or between the tips of the fragments).
Keywords. Nuclear fission; fusion; heavy ion scattering; polarization.
1. I n t r o d u c t i o n
I n order to calculate t o t a l kinetic energies o f fission f r a g m e n t s static m o d e l s a t the scission p o i n t were used b y several authors. T h e total e n e r g y o f t h e system o f t w o f r a g m e n t s is m a d e s t a t i o n a r y with respect to all d e f o r m a t i o n c o o r d i n a t e s t a k e n into a c c o u n t b u t one. T h e o n e n o t varied usually is the s e p a r a t i o n c o o r d i n a t e , t h e dist- ance b e t w e e n the f r a g m e n t s , which is fixed either to zero or to a small n u m b e r in t h e o r d e r o f t h e thickness o f the diffuse surface o f the fragments. T h e u n d e r l y i n g m o d e l m a y either be the p u r e liquid d r o p m o d e l (Geilikman 1955, 1959; M a l y a n d N i x 1967; H a s s e 1972; R y c e et al 1972), the liquid d r o p m o d e l w i t h s i m p l e shall effects ( V a n d e n b o s e h 1963; A r m b r u s t e r 1965; Sehmitt 1967, 1969), t h e statistical m o d e l (Shigin 1971), or even the shell m o d e l o f the Nilsson t y p e ( N 6 r e n b e r g 1966, 1969, 1972; D i c k m a n n a n d Dietrich 1959; Jensen and W o n g 1970, 1971 ; H o l u b et al 1974;
Wilkins et a11976; A d e e v et a11976).
I n static m o d e l s the m i n i m u m energy at scission is considered to consist o f t h e t o t a l kinetic a n d excitation energies o f the fragments. Little is k n o w n a b o u t the excitation energy. T h e r e f o r e o n e assumes t h a t at post-seission t h e f r a g m e n t s de- excite b y g a m m a or n e u t r o n emission into their (spherical o r d e f o r m e d ) g r o u n d states. H e n c e t h e excitation energy is equal to the difference b e t w e e n the t o t a l self energy o f t h e system at seission a n d the s u m o f the g r o u n d s t a t e energies o f t h e infi- nitely s e p a r a t e d f r a g m e n t s . T h e remaining C o u l o m b i n t e r a c t i o n e n e r g y at scission c o n s e q u e n t l y is equal to t h e t o t a l kinetic energy o f the f r a g m e n t s at infinity a n d c a n b e c o m p a r e d with experiments.
4 4 1
P . - - 6
442 Rainer W tIasse
Static models, however, have several drawbacks:
(i) The scission energy may in fact not be stationary but unstable with respect to separation. The assumption of adiabaticity of the motion through the scission point, however, implies slow motion and hence makes it plausible to neglect this degree o f freedom. According to recent calculations, even a minimum may exist in the potential energy close to scission due to the balance between Coulomb repul- sion and nuclear attraction (Shigin 1971; N6renberg 1966, 1969, 1972; Hasso 1974), due to curvature effects (yon G-roote and Hilf 1969; Hasse 1971), or due to inhomo- geneous charge distributions (Hasse 1972; Ryce et al 1972). This minimum is well established in heavy-ion potentials (Huizenga 1975).
(ii) Following a~ argument of Swiatecki (1970), only the restriction to a limited number o f deformation coordinates yields reasonable shapes of two tangent frag- ments. If one would remove this restriction one obtains as the minimum energy configuration two infinitely separated bodies connected by an infinitely thin thread.
As pointed out by Stissmann (1970) (also Hasse 1971), however, a small amount of curvature energy prohibits the formation of the thread since in the limit of a vanishing radius the surface energy of the thread goes to zero but the curvature energy be- comes proportional to the thread. Therefore a more reasonable assumption than contact would be to fix the distance between the fragment's centres-of-mass.
The higher multipole moments of the surface may not be excited but rather' frozen in' since the energies associated with them are too high. Hence one has to limit the number o f degrees of freedom and again to distinguish between fixing the centres-of mass distance or the distance between the tips. Using the first constraint one always obtains oblate shapes while the second one gives prelate shapes with large octupole deformations. This is understandable if one takes into account that at scission contact forces (adhesion) resulting from the finite range nature of the nuclear forces act between the tips of the fragments which tend to fix the distance between the two nuclei. For scattering or fusion, however, the distance between the centres-of mass is fixed by the bombarding energy and the result is that the nuclei will flatten off during their approach.
(iii) By means of the adiabaticity assumption one neglects the kinetic energy accu- mulated during the descent from saddle to scission. This energy can be estimated in dynamic calculations (Nix and Swiatecki 1965; Nix 1969; Hasse 1969) and is in the order o f 40 MeV in case of 235U(nth,f). It partitions into about 25 MeV transla- tional kinetic energy which should be added to the Coulomb energy and about 15 MeV vibrational energy which should be part of the excitation energy. These two energies, however, can be considered to be constant after scission has taken place since they do not vary strongly with the geometrical shape of the scission configura- tion.
A static model near contact with a limited number of deformation coordinates therefore seems to be a rather good approximation for the explanation of total frag- ment kinetic energies and the comparison between theory and experiment (Dickmann and Dietrich 1969) led to a confirmation of this assumption.
Nevertheless, the question has often been raised whether at the scission point the Coulomb repulsion may be strong enough to polarize the fragments substantially.
Large electric dipole moments of the individual fragments then would tend to lower the energy by a large amount.
The purpose o f this paper is to shed light onto this question. It will be investigated
Polarization of fission fragments 443 with respect to two aspects, i.e. shape and charge polarization. Shape and charge polarization mean that more mass or charge, respectively, is accumulated in the outer parts of the fragments than in the neck. Both kinds of polarizations are merely caused by the Coulomb repulsion but their strengths are determined by different restoring forces. The surface energy works against the shape polarization and the symmetry energy which also determines the giant dipole energies limits the electric dipole moments.
The resulting dipole moments are in the order of 0.04 e.fm per fragment charge number and give rise to a decrease in energy of about 1 MeV.
2. Shapes, dipole moments and energies
In this chapter we formulate a simple model in order to obtain estimates about the magnitudes of shape and charge polarizations. The scission configuration will be described by two tangent or separated fragments with each having even and odd deformations. The charge polarization will be introduced by the electric dipole moments of the fragments. The total energy is assumed to consist of surface, Cou- lomb self, Coulomb interaction, and giant dipole energies. Following along the lines of Hirschfelder et al (1954) the Coulomb energies will be evaluated by the method of expansion in terms of reduced multipoles. The energy is then made stationary with respect to the electric dipole moment and the shape parameters. For the sake of simplicity we shall restrict ourselves in doing this to two equal tangent fragments with aa and % deformations and the problem is reduced to the minimization with respect to three coordinates.
2.1. Shape parameters
Spherical coordinates (r, 0) and an expansion of the radius vector to the surface in terms of Legendre polynomials are very useful to describe small deviations from sphe- rical shapes. Let each fragment be axially symmetric about the z-axis and let 8 run counterclockwise in fragment 1 and clockwise in fragment 2 as indicated in figure 1, then the shape of each fragment is given by
R ( 0 ) = P-o ~o -1 ( l + ~ )
(])
e(~ ) ~ / ( z )
=XR 0 - I
Fi~llre l. Convention of measuring the pole angles 0 in order to 8~hJeve posit~'e odd shape parameters a=t+~ and dipole moments e. The charge densities are indi- cated by hatching.
444 Rainer W Hasse
with ~ =
~,,P.
(cos 0). ( 2 )The convention of figure 1 was chosen because in case o f two identical but reflected nuclei both have the same positive ~.-values. The parameter 41 is eliminated by means of the fixed center-of mass, % : - - 2 7 % % / 5 and ~o -1 guarantees the volume 4rrRos[3 o f each fragment to be conserved (Swiatecki 1956; Hasse 1971),
1 1 % = 2 % 3 . . . ( 3 )
~o -1 : 1 -- ~ %~" - -
105
I f it will be necessary to distinguish between fragments 1 and 2 as in case o f the mutual Coulomb interaction energy, subscripts (1) and (2) will be used.
2.2.
Charge inhomogeneity
The inhomogeneous charge distributions of the fragments are described by their electric dipole moments E,
where
pp=3Z[4rrRo3
is the average charge density and36 72
= - - ,~ ~ - - - % ~ % . . . ( 5 )
I% 385 3 175
conserves the total charge. The convention mentioned above also implies the same positive ~ to be achieved for two identical but reflected fragments with their charges accumulated at the outer corners. In figure 1, the charge densities are indicated by hatching, The connection of E with the usual electric dipole moment
Q:fdSrpprP1
is
Q=Ze~Ro/5.
2.3.
Surface energy
Among the constituents of the deformation energy of the fragments the surface energy is most easy to obtain
Esurf=a2A'13[1-
~: ( - ~ - - ~ ) ' ] Bsurf, (6) with2 5 a 2 4 92
Bsurf ~ 1 q-~ %= + ~ a -- 105 a2a -- 735 a9%2 . . . . (7) Here, a~ is the surface energy coefficient and the expression in parentheses is the sur- face symmetry correction (Myers and Swiatecki 1966, 1967).
Polarization of fission fragments 445 2.4. Coulomb self energy
In order to obtain the C o u l o m b self energy o f each fragment we follow Lawrence (1967) in first calculating the electrostatic potentials and then integrating over the volume. The potential within the b o u n d a r y o f the drop V(r, O) = f d s r' gl, (r', 0')
I r - - r' I -1 can b e divided into two parts, the potential o f the homogeneously charged
b o d y and a correction due to the inhomogeneity, namely
with
V = (l+/xO¢) V h o m - c Vinhom
,om t-'
(8)
Vinhom = -Pv f dar' (r'/R°) t)1 (cos O') l r -- r ' l -1. (9) Then [r - - r' [ -1 is expanded in terms o f Legendre polynomials o f the argument cos 818, where 019. is the angle between r and r',
0P~ (cos 018 ) ( r k r,_k_ 1 for r' >~r.
(lO)
By using (10) and the relation f d e ' Pk (cos 018 ) = 2rrP k (cos (9) Pk (cos 0') o n e gets with the abbreviation ~" = Aor/R o.
Vhom-- Ze [3 17a ~ °° ]
Ro o L2-2 + k=07 PkP , (11a)
[;
_ Ze 7 t , 1 _ 3 78pt + - ( l i b )
Vinhom ~--~s 1-6
Here Pk and s k are reduced multipoles introduced in appendix A1.
The C o u l o m b self energy ECoul, s = (1/2)/dSrppV now separates into three parts which are, roughly spoken, proportional to c e, ,1 and ~2, respectively.
with
Fhom pinhom, 1 2~,inhom, 2 (12)
ECoul, s = (I'-}-E/z0) z --Coul, s -- " (1 + "/z°) ~Coul, s + E ~Coul, s '
EhOm
Coul, s = ~ PP f dsr 1 - VhomEinhom, l 1 - f ( r )
Coul, s --~Pp dSr Vinh°m + ~ o P t Vhom
E i n h o m ,
Coal, s = 2 1 - P, f d a r - - r p Vinhom .R e
(13)
446 Rainer W I-Iasse
These integrals can be partly evaluated by use of the reduced multipoles. Using E 0 Coul -- - - 3ZZe2/5Ro and x = cos 0 one obtains (cf. appendix A1 for qk, tk)
EhOm Coul, s -- _ _ E 0 Coul Ao-5 [ 1 + ~ Pkqk + ~ 5 f dx(~Z--o~ s - a 4) ]
Coul, s : ECoul (qksk + P~ tk)
+ - 4 dx P1 a + a ~" 2 a8 3 a4
3 2
5 ~ Coul, s : Coul + ~ sk tk
if ( 1 ~ 5 a 8 - 5a4)] (I4)
+-4 dx Px ~ a + ~ - - ~
Since we only deal with small deformations, the Coulomb energy is expanded in terms o f the small quantities ~, %, and a a up to the fourth order. The final result is
0 [ 1 4 a a l O % z 9 2
ECoul, s : ECoul 1 -- ~ %2 __ ~ 2 -- ~-9 -- 735 ~ aaz + a~4
q 2133718 (12 4 0 a , 8 )
311272---~ ~ ~3 ~ + , ~ ~ + ~ - ~ ~ + ~
+ . 3 ( 1 2
128 491 ) ]+ '~ + 7-~ '~'~ + 6 ~ '~' " (15)
2.5 Coulomb interaction energy
The Coulomb interaction energy o f two homogeneously charged bodies with a distance I between their centers of mass (figure 1) is obtained in a similar way.
The expansion equation (I0), however, has to be done in a two center coordinate system. Referring to the book of Hirschfelder et al(1954) and to the report of Nix and Swiatecki (1964), one gets with the abbreviation ~ = Aol/R o
EhOm _ Z(1) Z(2) e 2 ~ oo (m + n) ! q(l)m q(2)n
Coul, i A3 (1)0 A )0 1 ~2 " m,n=O m t n! - m 7n (16)
• l 0) (2)
It is noted hero that a factor of (_)n appearing in Nix and Swiatecki (1964) drops out from (16) since the convention of figure 1 has been used. The subscripts in brackets here refer to the fragment numbers.
Polarization of fission fragments
447By inserting the reduced multipoles and the volume conservation factors and keeping terms up to second order in the deformation coordinates, it reads*
[ (
EhOm _ Z(I) Z(2) e 2 2 3 12 2 8 a(l)3
Coul, i l 1 + 1-2 R(1)0 ~ a(l)2 + ~-~ a(1)2 -~- ~-~
2 3 12 2 2 3
+ R(2)O (~ a(2)2
+~a(2)2+85tz(2)3)]+l-3[R(1)o(3a(1)3
4 3 3 4
+ ~ a(1)2 a(2)3 ) + R(2)o (~ a(2)3 + ~ a(2)2 a(2)3 )]
[ (
4 18 2 18 a(1)3 + a(2)2+ a(2)3 + 1--4 R(1) 0 3"5 a(1) 2 + 7-7 R(2)0 3-5 7-7]
2 2 54 a(2)2 + l _ 5 R(1)0].~ a(1)2 a(1)3
+ R(1)0 Ri2)o ~ ~(1)2
_5 10 _3 _2 18
+ K(2)0 ]-~ a(2)2 a2(3) -~- K(1)0 K(2)0-~" a(l)3 °'(2)2
~2 _3 18 J /_6 JR(l)0 f _ ~ g(1) 3 +/~(1)0 K(2)0 -~ a(1)2 °'(2)3 q- 6 400 2
+ _ 6 4 0 0 2 3 3 1 8 0 ]1
r(2)o 1 ~ a( 2)3 + R(1)° R(2)° ~ ~(1)3 ~(2)3 • (17)
For two equal fragments andA=I/Ro,
(17) simplifies to( 4 8 )
Eh°mcoul, i = 2EOoul A-1 + A-a a2
+ ~ a2z + 211%2
(~ 20 ) A . ( 9 3 30 )
+ :~-~ h + ~i "~ + ~ ~2~ + ~ %~
+ A_ e 1340 as as + A_ ~ 78350 %2 ] . (18)
231 21021
]
If the nuclei are inhomogeneously charged one merely has to make the following replacement in (16)
qm~qm - ~o-ltm.
The charge conservation factor/~0 need not be taken into account here since it contributes only to the higher orders in the deforma- tion coordinates. The Coulomb interaction energy now splits up into the homo- geneous part and the correction _inhom . /~Coul, i' oue to inhomogeneity. The resulting expres- sions are quite lengthy, hence we only give here the result for two equal fragments with quadrupole and oetupole deformations up to the second order*There is some discrepancy between the results of Geilikman (1955, 1959) and eq. (17), namely that the coefficients 18135, 18/77, 10/11, and 400/1001 appear there as half of these values.
448 Rainer IT" Hasse
T a b l e 1. S o m e values of the reduced multipoles used to calculate C o u l o m b self a n d i n t e r a c t i o n energies.
3 s 3 2 4 s 243
q o = l + ~ a , + ~ a : + _ - ~ ' + ~ 3 a s a s + ~ a s a #
1 , 20',
+ ~a, + :--~a22ai+3a~'-}-6as'a.+-~-~asasas
72 3 6 c 2118 4
ql = - - 17""~ a ~ a3 ÷ -,o.~ ass ÷ I ~ a~. a32 + ~ as a4
3 12 s 8
q, = . ~ a , + 3"~as + . ~ - 3 ~"
3 4
18 18 , 1
10 3
qs = ~ ' ~ a ~ a s + ~ a s 4OO
72 s a3_~66 2118 4
to D-5 as as + .,,., . a + 1 ~ ~sas~ + ~ a~ a,
1 2 2 2 s
t, =-5 + ~ a, + ~3 as + :--3ts a:
96 ts = as+ ~ - - S a ~ a 3
18 108 4
142
t , = " + i6-5 ~ a3
24 360 5
t, = ~ as' + ~ a~ s + y~ , , ,
540 18
t~ = ~ "s as + IT3 a5 lOO
t? ~ 4 ~ O'8'
9 78 1_22 8
s o = - - 3 3 as ~a - "IT5 ass as + ~oJ " / + ~ " a s a,
2 11 s 23 ,
sl = .3 as + ~ as + 2To "~
9 54
9 3 18 4
s~ = ~3 a~ - o~'~' - ~ as" + ~ i a4 4
3
Po = YO 1 I 729
a," -}- 3 a3 ' --~- ~ a 4 -.[- 3 as'~ - ~- l-~-S aa" as"
27
3
3 8
P ' = - 3 - 5 " ~ ' - a : + ~ a ,
Polarization o f fission fragments 449
Einhom 0 I__ [A_2(~ 2 5 3 1 )
Coul, i : 2ECoul c + 3 % + 7 %~ + 6-3 %2
(3 32 ) )t_,(36 84 268 )
+ ~-' as + ~ 0`5 + 3-3 0`' + ~ 0``' + ~ as'
q_ A-6 0`3 ~ a t- 3465 ~ - ~ 700---'-~ %'~
--F ~-'~ 64800 as o,z -F A-' 54800 ~,]
7007 9009 % ]
58 6 2 a '~ + A -4 a s + 0`, + e~ A-8 "q- % q- 1 ~ %~ + 315 ' ] ~ ]-~
q_~_5(12 48 1899 ) A _ , (~3 5128 )
% + ~'5 as2 + 2 ~ 0`'' + as + 1--6-i-7 % + ~-7 (930 1 3 6 2 0 ) A-' 4120 % as + A-9 74000 ]1
k539 % ~ + 700"---7 %' + 1001 2702---'-7 as2 . (19) 2.6. Volume symmetry energy
Steinwedel and Jensen (1950) interpreted the volume symmetry energy in nuclear mass formulae as an integral over a local energy density, i.e.
f a'r
e~o~, ~ - a. A ~ , ~ } -~ a. a J v o l u m ~ ~ ,'o / '
(20)
as did Mustafa et al (1971). Here a s is a constant and the volume integration over the local deviation of the neutron density from the proton density gives rise to the restoring force against giant-dipole oscillations.
For the assumed proton density (4), the integral in (20) can be solved to give Evol, s = -~ [(N--Z) z - 2 ( N - - Z ) Z c he-4 ql + 4 Z ' E ~ he-6 ta]. (21) Keeping again only terms up to second order one obtains the volume symmetry energy per fragment
_~[ ~ ( 15 ,31 ,\]
evoJ, s =
( N - - Z ) ' + z , ,, 1 + 2 0`, + 3 %" ~ ~i % }1"(22)
3. Results
The total deformation energy of two equal fragments now becomes
Edef - 2 (Esurf + ECout, s + Evol, s) + ECoul, i (23)
4 5 0 Rainer W Hasse
Table 2. Collection of 3-and 4-symbol brackets
4 6 8
(112) = i-5 (123) = ~ (134) = 6 - 3
12 14 72
(156) = 1 ~ (167) = ~-5 (178) = 4199
4 4 8
(222) = ~ (224) = ~ (233) =
40 10 20
(244) -- 693 (246) -- 143 (255) = 4 ~
28 56 48
(266) -- 175 (268) -- 1105 (288) -- 1615
4 200 40
(334) = ~-~ (336) = 3003 (345) = 1001 "
14 112 336
(356) = 4 - ~ (358) = 2 ~ (367) = 12155
36 40 980
(444) -- 1001 (446) -- 1287 (448) = 21879
560 56 1008
(457) -- 21879 (466) -- 2431 (468) = 46'189
4536 72
(477) 230945 (488) = 4199
160 980 840
(556) -- 7293 (558) = 46189 (567) -- 46189
800 700 1200
(666) = 46189 (668) -- 46189 (688) -- 96577
3500 980
(778) = 289731 (888) -- 96577
10 (145) = 9-9
(235) = 20 42 (257) = 7-i--5
70 (347) = 1-~'7
504 (378) 209995
4 (455) -- 143
2 4 22 8
(1111) = 5 (1113) = ~ (1122) - lO5 (1124) = i-0-5
(1133) 46 40 234 4
= 3 - ~ (1135) -- 693 (1144) = 2079 (1223) = 35
4 284 40 580
(1225) = if'7 (1234) -- 3465 (1236) = 1001 (1245) -- 9009
24 60 40 136
(1333) -- 385 (1335) = 100i (1337) = 1--~-7 (1344) = 3003
6 24 122 482
(2222) -- 35 (2224) = 3-~ (2233) = (3333)
1155 = 5005
a n d t h e f o l l o w i n g s e t o f l i q u i d d r o p c o n s t a n t s i s u s e d , % = 1 8 - 8 M e V , K : 1 " 7 8 2 6 , r o = 1 " 1 2 3 f i n , a n d a~=27"62 M e V . F u r t h e r m o r e , s y m m e t r i c f i s s i o n o f ~8~U i n t o t w o x a s p d f r a g m e n t s w i l l b e c o n s i d e r e d .
I n o r d e r t o g e t a f i r s t g u e s s o f t h e d i p o l e m o m e n t w e w i l l d e a l w i t h t w o s p h e r i c a l n u c l e i . I n t h i s c a s e t h e s t a t i c d i p o l e m o m e n t i s e : 0 . 1 3 2 A - L F o r c o n t a c t , * ¢ = 0 . 0 3 3 a n d t h e t o t a l e n e r g y i s l o w e r e d b y a b o u t 0"9 M e V . C o m p a r e d w i t h a b o u t 1 6 0 M e V o f t o t a l k i n e t i c e n e r g y o f t h e f r a g m e n t s , t h i s a m o u n t i s v e r y s m a l l . T h e r e a s o n f o r t h e d i p o l e m o m e n t b e i n g s o s m a l l i s t h a t t h e g i a n t d i p o l e r e s t o r i n g f o r c e i s t o o s t r o n g t o b u i l d u p a n e s s e n t i a l l o c a l p r o t o n e x c e s s .
*E----0-033 converts i n t o the usual dipole m o m e n t o f Q = 0 " 0 4 Z e fln, where Z is the f r a g m e n t p r o t o n n u m b e r .
Polarization of fission fragments 451 Deformations of the fragments will even lower the dipole moments since by shape polarization automatically more charge is accumulated in the outer corners. How- ever, one has to account for the instability in the separation degree of freedom by superimposing a constraint. This constraint may either be fixing the centers-of mass distance 1 (case 1) or the distance d between the tips of the fragments (case 2), where d is given by
d : - l - - 2 R 0Ao-l(1 + a z + % ~ - % ) .
(24)
Also the dimensionless distance 8 : d/R o will be used.
With these two different constraints the energy is first minimized with respect to c to yield the dipole moment as function of deformation. Then ~ is inserted into the deformation energy which in turn is made stationary with respect to o~ and %. The results are displayed in figures 2 and 3.
Case 1 gives always oblate shapes with largo negative quadxupole moments and very small negative octupolo deformations. In the contrary, for case 2 one obtains prelate shapes with large positive quadrupolo and octupole deformations. All moments and electric dipole moments are rapidly decreasing if the separation of the fragments becomes larger.
The energy decrease due to charge polarization in case 1 amounts to about 1.2 MoV and in case 2 to about 0"7 MeV. In both cases these numbers apply for contact.
Those results are understandable if one considers the difference between fission on the one hand and fusion or scattering on the other hand, the former not being simply the time reversed process o f the latter. In fusion or heavy ion scattering the two par- ticles are bombarded at each other with fixed energies and hence the classical paths of their centers-of mass are prescribed. Therefore case 1 should apply to fusion and scattering and a flattening off of the nuclei is expected during their approach.
® 0-0.2
,u 0
~ 0.1
0 O
E O
- - 0 . 1
~ - 0 2
I I I I I
a 2 o . - - .... ~
313.,. " - - - - . , .
_ 0 3
D
2 . 0 3 . 0 4 , 0
),
Figure 2. M i n i m u m d e f o r m a t i o n s =~, a~ a n d dipole m o m e n t s ~ p l o t t e d v s . the center- of mass distance A. Full curve: fixed distance A between the centers of mass (case 1), d a s h e d c u r v e ; fixed distance 8 between the tips (case 2); A a n d 8 i n units o f R0, h a l l o w square, s i n d i c a t e contact.
452 Rainer W Hasse
3 0 0
2 0 0
I o o
¢ o
II
I I [ . [ i I
o 0 . 5 1-0 1.5 2,0
B
1 0 0 . I
1.5 2 - 0 2 - 5 3 . 0 3.5 4 . 0
Figure 3. Minimum energies in MeV plotted vs. ~ and A. The configuration of two infinitely separated spherical fragments has zero energy. For notations of. the caption of Fig. 2. Solid squares indicate contact.
On the other hand, fission leads through a (more or less well defined) scission point where the surface tension causes adhesion of the tips of the~fragments which tends to keep them together (Krappe and Nix 1974; Hasse 1974; Blocki et a11977). Thus case 2 seems to be a good restriction for the fission process close to scission and, as expected, one obtains prolate shapes with large octupole deformations.
The discussion given above is solely based on the liquid drop model. There the two separated nuclei with their shaxp surfaces interact merely via the Coulomb re- pulsion. Quantum mechanica/ly, however, the nuclei interact also by the tails o f their wave functions (Hasse 1974), which gives rise to an attractive force. Its macro- scopic analogue has also recently been discovered (Blocki et a11977) and is now known as proximity force. As a result, in slow fusion or scattering processes the heavy ions may become prolately deformed rather than oblately when approaching each other.
The question whether the prelate or the oblate family of shapes is lower in energy is meaningless because of the lack of a well defined separation coordinate. A co- ordinate transformation except at points where the energy is stationary can always map an energy contour plot into an entirely different one. Even in this simple example, figure 3 shows that the oblate shapes have less energy compared with the prelate ones if plotted against ~ but the opposite is the case when plotted against 8.
Acknowledgements
This work was started at the Institut far Theoretische Physik der Universitat, Heidel- berg, Germany and was continued at the Sektion Physik der Universit~it Mtinchen, Germany. It was completed at the Department o f Physics o f the Indian Institute
Polarization of fission fragments 453 of Technology, Kanpur, where the author stayed as a visiting professor. He thanks Prof. Y R Waghmare and the other members of the Department for the kind invita- tion and the warm hospitality extended to him. Thanks are also due to the German Foreign Office, Division of Cultural Affairs, for providing the travel grant.
Appendices
A.1. Reduced multipoles
In evaluating the Coulomb self and Coulomb interaction energies of the inhomogene- ously charged fragments four kinds of multipole moments are used. With x = c o s 0 and P~=Pk(x) they read
3 Pk (1 q'- a)3+k
qk = d x
3 f
(1 + a ) '+'t~ --- ~ dx P~ P~ 4+k 3 f dx ~
(1 +~y-k
3 f (1 + a ) s-~
s~ -~=2 dx Pk P1 3--k (A1)
Hereby Pz and s z are merely defined by the series expansions since they originally appear as P2 -- ~ f dx Pz In (1 + a) and s 3 = ~- f dx P3 P1 In (1 q-a), respectively. All integrals run from --1 up to +1.
The integrals in (AI) are all of the type (Jl, J2 . . . . J~r) = f dx PA PA "" Pier"
These, however, are known as N-symbol brackets and can be evaluated by using the technique of Clebsch-Gordan vector coupling coefficients. This method is employed in appendix A.2 to yield the reduced multipoles of table 1. As a first application we mention here that
Ao-X : qo -a/a, /% = Ao-X ql/qo. (A2)
A.2. Computation of N-symbol brackets
The N-symbol brackets used in appendix A.1 are defined as
( J l , J~ . . . . JN) = f__ + ~ dx PA(x) VA(x) ... Pier(x). (AS)
In order to be non-zero they must obey the following' selection rules' (i) The sum o f the arguments is an even number
(ii) One argument must not exceed the sum of all others.
454 Rainer IV Hasse
The values of the 1- and 2-symbol brackets simply follow from the orthogonality relation of the Legendre polynomials,
( j ) = 2 3yo; (A,A) - - -
2 j l + l
aA J,- (A4)
The 3-symbol brackets can be obtained in a simple way by using vector coupling coefficients. Coupling the ' angular m o m e n t a ' j l and j~ with projections m l = m 2 = 0 to the total angular momentum J with M = 0 and then coupling J andja to J ' with M ' = 0 gives (Messiah 1964)
PI, PAPA = ~ f J10, A 0 I J0) z (S0, A01J'0 ) p,,.
J J '
(AS)
Here the Clebsch-Gordan coet~cients are used in obvious notation. By integrating the orthogonality relation demands J ' : 0 . In addition, the triangular relation l J--is I <~ J' ~ J-~Ja gives J=Ja and the result is
(Jx,J2, Ja) 2j3q_~ (A 0,A 0 [J3 0)2 - 2 J~ , (A6) where the Wigner-3j symbols are also used.
In evaluating the 4-symbol bracket we first couple jr and A to J, Ja a n d A to J ' and then J and J ' to J'" with all projections equal to zero,
PJ, PAPA PJ,
= ~. (A 0,A 0 I J0) ~- (A 0, A 0 I J'0) ~ (J0, J'01S"0)P,..
(A7)j J ' J "
Again, the orthogonality and triangular relations only allow for J ' t : 0 and J--J" to give
(j,, j,, j,,, j,) = (j, o, j, o I so), (j, o, j, o I Jo),
J
= ~ 2 J+____~l (A, A, J) (A A #).
J
(A8)
The N-symbol brackets are, of course, symmetric in interchanging their arguments.
Hence Jl..J4 may be arranged arbitrarily in order to make use of already calculated 3-symbol brackets.
The Wigner-3j symbols of (A6) can be found in the tables of Rotenberg et al (1959).
Previously Swiatecki (1956) presented a table of 3-symbol brackets. The values of this table together with other 3-and 4-symbol brackets are listed in table 2.
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