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Charge vibrations in the liquid-drop model of fission
Nuclear Physics A212 (1973) 387--412; (~) North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher
C H A R G E V I B R A T I O N S IN T H E L I Q U I D - D R O P M O D E L OF F I S S I O N R. HOLUB and G. R. CHOPPIN Department o f Chemistry, Florida State Universit.v, Talhthassee, Florida 32306
Received 25 April 1973 (Revised 8 June 19731 Abstract: The two-spheroid liquid-drop model of fission developed by Nix and Swiatecki has been extended to include charge vibrations. Using the Myers-Swiatecki mass formula, the appropriate stiffness constants have been obtained in analytic form. The effective mass, Ma, associated with charge vibrations for fission shapes was deduced from eigenenergies for the giant dipole resonance calculated by Updegraff and Onley, assuming that the eigenenergy is given by h~,)air,ol~ -- h \ K A a / M j ,
where Kaa is the stiffness for charge vibrations. To account for the experimental charge dispersion widths a highly constricted fission shape was chosen. The depertdence of fragment deformation energy on mass increases more rapidly than in the Nix and Swiatecki treatment when the eigenvalues of two new normal modes have similar values; it is concluded that this, together with spherical closed-shell effects, may explain the discrepancy between the Nix and Swiatecki values and experimental data. Various other aspects of the effect of charge vibrations are discussed.
1. Introduction
N o a d e q u a t e theory for the distribution o f nuclear charge in fission has been developed. As a result o f this deficiency, a n u m b e r o f empirical hypotheses such as t h a t o f equal charge distribution ( E C D ) , m i n i m u m potential energy ( M P E ) , a n d unchanged charge density ( U C D ) have been p r o p o s e d by various authors. O f these three only the M P E postulate has been suggested to have even a semi-empirical basis. Nix and Swiatecki 1) m e n t i o n e d the possibility o f a d d i n g an a d d i t i o n a l degree o f f r e e d o m related to charge vibrations in their d e v e l o p m e n t o f the t w o - t o u c h i n g spheroids l i q u i d - d r o p m o d e l o f nuclear fission. Schmitt 2), however, considered that the a d d i t i o n o f such a degree o f freedom w o u l d not affect the results o f any m o d e l in a significant way. Nevertheless, the lack o f the inclusion o f the effect o f charge distrib u t i o n in the l i q u i d - d r o p m o d e l is surprising since the term for the s y m m e t r y energy in the mass f o r m u l a is very strongly d e p e n d e n t on the nuclear c o m p o s i t i o n . The mass f o r m u l a developed by M y e r s a n d Swiatecki 3) successfully a p p l i e d the principal o f the " b u n c h i n g o f levels" to explain shell effects, a n d drew attention to the fact that it is intrinsically m o r e reasonable in a mass f o r m u l a to use a surface energy term which is d e p e n d e n t on the nuclear c o m p o s i t i o n . A mass f o r m u l a with this latter 387
388
R. HOLUB AND G. R. CHOPPIN
feature might be useful in an attempt to include the charge distribution in the liquiddrop model. In a sense, a variation in the surface energy term during fission is the basis of the Brandt and Kelson a) approach to fission theory. A further development that encouraged us to consider the inclusion of charge vibrations in the liquid-drop touching-spheroid model is the fact that the giant dipole resonance has been shown to be a collective as well as a single-particle phenomenon. It is possible to speak about neutron and proton density variations in such a way that the effective mass inside of a nucleus can move against the given stiffness of the nuclear matter, i.e., without changing the overall nuclear shape. Such an effect has been treated by means of fluid dynamics s) to obtain the familiar expression: Edi p ~ 7 0 A - ; MeV, where Ea~p is the giant dipole resonance energy. It has also been shown that this energy decreases substantially as the nucleus deforms to a necked-in fission shape and becomes zero when the fragment neck disappears. Finally, it seemed possible that certain discrepancies in the Nix and Swiatecki ~) (hereafter referred to as NS) liquid-drop model might be eliminated when a charge vibration was included in the theory. These discrepancies were related primarily to the dependence of the deformation energy distribution on fragment mass and to the magnitude of the stiffness of the mass-asymmetry vibrations. As in the NS development, we use no adjustable parameters except for the effective mass in the charge vibration. This effective mass has not been determined rigorously but was derived from the data of Updegraff and Onley 6).
2. U n i t s
The units used in this paper are those of NS; however, it should be noted that the surface energy, C2 A~ot, is composition-dependent (i.e., on N and Z ) through the C2 factor, where
C2 =
a2 [1 - 1.79((No
--Zo)/Ao)2].
(1)
Similarly the fissionability parameter x is (N, Z ) dependent since
c3 zo x = :a_,[1
- 1.79((No-
Zo)/Ao)2 ]A~2 -
(2)
This is a consequence of the dependence of the surface energy term in the MyersSwiatecki mass formula. As a result the more neutron-rich nuclides have a greater fissionability than would be the case without the dependence of the surface energy on The subscript 0 denotes the fissioning nucleus while subscripts I and 2 denotc the fission fragments.
CHARGE VIBRATIONS
389
the composition. This (N, Z ) dependence of x represents only a change of units and, therefore, does not affect the saddle-point shapes calculated as a function of x, although it does affect the particular nuclides which are represented by these shapes. The NS "radius-mass-surface energy" units were used for the calculations of the giant dipole resonance energies, i.e.,
hf2o = 22.44/,/Ao MeV,
(3)
where f2 o is the frequency of the charge vibrations in spherical nuclei (fissioning ones), Oo = [C2 A~/Mo] ½.
(4)
It should be noted that N o, Z o and A o are dimensionless.
3. Potential energy
The basic expression for the potential energy used for all the calculations is: PE = PE1 + PE2 + MCE, where PE, = - C, A, + C2 At f(1) + C3(Z2 /A~Og(1), PE 2 = - C, A 2 + C2 A2~f(2) + C3(ZZz/A~')g(2),
(5)
MCE = (Z, Z2e2/I)[S(1, 2 ) + S ( 2 , I ) - 11, where C1, C2 and C3 are given by Myers and Swiatecki 3) (hereafter referred to as MS), l is the distance between the centers of the two fragments, and S(1, 2), S(2, 1), f(1), f(2), g(1) and g(2) are shape-dependent terms discussed later. MCE stands for the mutual Coulomb energy. PEt and PE2 are simply the potential energies given by the MS mass formula for the fragments. It should be emphasized that the MS mass formula is useful for "small deformations" and that the model of fission based on touching spheroids is valid for nuclei lighter than those in the mass region of radium. The validity of this approximation for heavier nuclei is dependent on the establishment of an equilibrium at the scission point similar to the one assumed for the saddle point. This would be the case only if a significant scission barrier existed as suggested by N6renberg 7). This model of two touching spheroids is a transition state approach and neglects entirely any postsaddle-point dynamical motion. The main problem is to translate these expressions into the system of two touching spheroids which is (as NS have shown) a very good approximation for the real fissioning system at the saddle-point equilibrium. The "translated" expressions follow:
390
R. H O L U B A N D
G. R. C H O P P I N
PE = [ 1 - t c ( T + 2 N o A / A o u ) Z ] { - a l Ao U +az A~o U'tf(1)} + C 3 ( Z o U_ANo)ZA~o U _ ; g 0 ) + 5 [Zo U - A N o ] [ Z o ( 1 -
3
U)+ANo]
lA~o U ~
3 12
+ -
- [S(I, 2 ) + S ( 2 , 1 ) - 1 ]
5 I"0
A o ( l - U)! I + C3[Zo(I - U) + A No]ZAo~(l - U)- ;9(2).
(6a)
In surface energy units eq. (6a) is: a, A/~' {2 - , < ( T + 2 a A ')'- u C2 U /
PE
1
{U~,f(l)II_K(T+
+ 1--KT 2 .
.
.
t
.
.
5
+ 2 x (3(ci +c_,) EU -
Ap][(I
("T 2 a A 2~--U (, ] 1 ,
U))
2aA')2] + ( l _ U ) } f ( 2 ) [ l _ t c ( T U ] d
2aA ]2] t
iTu:
t
- U)+Ap-IES(1, 2 ) + S ( 2 , 1 ) - 1 ]
+ [ U - ap] z u - + g ( 1 ) + [(1 - u) + Ap]2(l - V)-~,q(2)] .
(6b)
The degrees of freedom are A, U, Cl and c, (in the functions f, g, and S). Further: (i) ~, 312:./t o = C 3, where l = c I + c2 (the main semiaxes of two touching spheroids). (ii) T = ( N o - Z o ) / A o. (iii) A = N / N o - A/A o = V - U. The degree o f freedom A is the fractional neutron deviation from unchanged charge density (UCD), which is the "charge vibration" degree of freedom (the " n e u t r o n " vibration, however, is Au = - A z ) . If N/No = A/Ao, U C D describes the charge distribution. When N/No > A/A o, the fragment is neutron-rich which is the usual case for the heavy fragments. (iv) U = AlIA o (the fractional mass as used in NS). The expressions in brackets [ T - 2 N o A / A o ( I - - U ) ] and (T+ ZNoA/Ao U) are obtained from ( N I - Z x ) / A 1 and ( N 2 - Z z ) / A 2 when the appropriate substitutions are made. Also, A1 = - A 2 and UI = 1 - UI = U2 as required by mass and charge conservation. ( v ) f ( 1 ) = l +- - ~2 c q 2- - T o 4s ~ , ,3 where cq is a deformation parameter related to the major semiaxis Cl. This relation, which holds for small ellipsoidal deformations, is derived from a = Roe . . . . ~,.-~),
b = Roe . . . . (r+-~.),
c = Roe . . . . :.
(8)
which is given in MS (p. 29)where a, b and e are semiaxes, a is a deformation parana-
CHARGE
VIBRATIONS
391
eter, 7 is a shape parameter, and Ro is the diameter of the fragment. Since it is usual and justified to limit the fissioning shapes to axially symmetric ones we can drop the parameter 7 (i.e., 7 = 0). The semiaxes a and b (a = b for symmetry) are determined by volume conservation, leaving us with one degree of freedom for each fragment: c = R o e~
[-i.e., ~ = log (c/Ro)].
(9)
The parameter a is related to the distortion parameter e:
~2 = a Z ( l _ ~a).
(10)
Since we work in [Ro] = roA o units the expression (9) must be changed into ~1 = l o g ~
1-~log~
,
(11)
which says for example that when the fractional mass U = ½ (at the saddle point), R~ of the fragment equals 1.0 x U + or R~ = Ro/1.26, where R o is the radius of the fissioning nucleus. When the mass asymmetry is such that all the mass is concentrated in one fragment, i.e., in the original nucleus, cq = log c-!-1 1--~ log cl U~ (vi)
g(l)=
1-1
2
4
S ~ 1 -- T~
-~ = log CI(1-- 7llogcl)
~.
3
Rl •
Similar expressions apply for fragment 2. (vii) The terms [Z o U - A N o ] and [ Z O O - U ) + A N o ] are obtained when both degrees of freedom, U and A, are put into Z~ and Z 2 in eq. (5). (viii) When the mutual Coulomb energy term is changed into the [Ro] , [M0] and [C2 A0~] system of units, a simple rearrangement must be made to change l, the distance between the charge centers of two touching spheroids, into [Ro] units, as indicated in eqs. (6a) and (6b).
(ix)
3;. 2"
S(1, 2) =,,~o (2n + 1 - ~ n + 3)'
s(2, 1) = Z
3 I°
,,=o (2n + 1)(2n + 3)'
where z; A,
(¢1"I-£2) 2
-
-- -
-
(c, +c2) 2
c2
(12)
These expressions were taken from NS and simplified forms are used in our later calculations. The Potential Energy Matrix. In this treatment of the saddle-point fissioning system, represented by small vibrations about an equilibrium system, it is in equilibrium with respect to all degrees of freedom except fission (which has a negative stiffness constant), we limit ourselves to four degrees of freedom, U, A, ct and c2. NS have shown that the degrees of freedom of rotational energy are not coupled with these four.
392
R. HOLUB AND G. R. CHOPPIN
1,50
?~ X=0.8~
I00
so K,,~.(E~b,,Ts) (= .
-50
°t
I
,
i
I.I0
I
I
I
(N/Z ) P
I
I
Fig. 1. Saddle-point stiffness constant for charge vibrations, KAa, as a function of p, the neutronto-proton ratio, of the fissioning nucleus for various fissionability parameters x. The range of p corresponds to that of known nuclei. I
~,×=0.1
//×=o2
o
jjjx=o
(o)
-2
J~,,,#
-~.
__~o.8 ' p(",z)
"
Fig. 2. Saddle-point stiffness constant KAu as a function o f p o f the fissioning nucleus for various fissionability parameters x. The range o f p corresponds to that o f k n o w n nuclei.
M o r e o v e r , it is very difficult to o b t a i n any e x p e r i m e n t a l d a t a on spins o r o r b i t a l a n g u l a r m o m e n t a o f the fission fragments. By c o n t r a s t o u r degrees o f f r e e d o m are subject to direct physical o r chemical measurements. Constrtlction o f the P E m a t r i x is d o n e in a s t a n d a r d w a y by t a k i n g second derivatives o f the P E e q u a t i o n (6b) a n d e v a l u a t i n g these at the saddle point. N S did this numerically from an e x p a n s i o n describing the PE. The i n d i v i d u a l r e a r r a n g e d and simplified second derivatives K u are given b e l o w as analytical expressions a n d are p l o t t e d in figs. 1-4:
Kvv = 2U~(f[/v+~J~'-8-f)+2x
lo(2s-11
3(c1+c2) +U
-+~o
,o,
~,,~7
(-~-g+--3-gu +2~ae'v'j_j ,
(13)
CHARGE VIBRATIONS
393
0.2
03
E~
K~C,(~0UNITE
I
I
I
I0
I
I
p(",~)
I
I
I
15
K~ct
Fig. 3. Saddle-point stiffness constant as a function o f p of the fissioning nucleus for various fissionability parameters x. The range o f p corresponds to that of k n o w n nuclei.
//'/
1.5
,o~.
//
osl ~
,.
/ ~
1.0
05
Fig. 4. Saddle-point stiffness constants as function of the fissionability parameter x. The constant
Kvv is in units of [E~(°)], Kcjcl and Kczc2 in units of [E~°VRu21 and K~c I in units of [E~(°)/R°].
I~
+ 2xpFlO(2S-
= -16~T~U~(fb+~f) l--leT 2
1)
-1 2O-~(g'v+~g)] ,
(14)
J
L3(ct +c2)
(t5)
(
KAa = 32a2tC A~ 1 5 . 6 7 7
2fu~
18.56
K ac ~ = - 81cToU ~fcl
1 -~T
2
)
1 1 -- t~T 2
+ 2xpU-~g'c,
+2xp 2
I
+4U-~g 3 ( c t + c2)
1
,
(16)
(17)
394
R. H O L U B A N D rr}rlt
--
lr i
G. R. C H O P P I N
¢t
gc,c , = U Jc~c-t-xu~gc~c~
V2_(2s_-_1)
5x
+ 6(c, + c2i L(c, + cO 2
. ~1 , + ~. . ......
S',(1, 2)+S~.~(I, 2)
[(~S- 1)
lOx
S L,(1, 2)-t-S'c,(2, l) cl +c2
K .... - 6(c, +c2) L(c, +,.2) 2
(C I _JrC2) 2
2 )+:~,.,,.~, ..... 2 ,1) l J
.... 1 2)] -~'"~'clc2(
'
(18) (19)
'
where No
6-
,
T
No-Zo . . . . . . . . . .
Ao
S(2,1)----4
t
Vc ~ U,
C 1 + C2
,
Zo
(simplified expressions (12) as given in NS),
'("2 2 - ) . # ')i'log ',1 21 3, +;.2I + 2. --;.~ 2~
No
p-
,
Ao
22
1
Cl
C 1 + C2
V
cz
S,t (I, 2) = S,1,(2, I) = .-~-- S(I, 2), ~c~
'i , 2) = Sc,(2, ' So,_( I) = -~Ls(1,2), CCz
82 S'c;~,(I, 2) = S'~'=~(2, 1) = &.2 S(1,2),
S" (1 2 ) = S'~[~2(2, 1 ) -
S(1 2), 6'¢ l UC 2
.~"' ¢1 2) = S,7,~,(2, 1) = -az - - S(i, 2), --c,_c2~
,
0 f(1) aU
~C 2
aU [I + ~2: e ,2- - T ~ 4 7 ,3-] ---- ft~,r 7
11
a~ = log c~ u ~' and similarly for gb, J ~ , f,~u,, J~[c,, etc.
1- U C2
CHARGE VIBRATIONS
395
Similar to NS, the potential energy matrix may be written now as:
AV
=
I(~UA~c1 cc2)
Kvv Kay Kw, - Kvc,
Kay Ka,j K j~,
Kvc, Kac, K ....
- Kgc,
Kc,c2
- Kvc, - K.,c~ Kc,.~ Kc,c,
1&2
In this matrix: Kuc t :
- Kvc 2,
Kac I =
- KAc 2 ,
Kc,c, = Kc:c2.
This is a result of the symmetry of the two touching spheroids, and because c~U = U - ½ , c~ci = c i - c o (Co is the saddle-point deformation taken from NS). All constants, given in the analytical expressions of eqs. (13) to (19) are evaluated at the saddle point. The constants K not containing subscript A are shown in fig. 4. The agreement with the values calculated by NS is proof of the validity of our calculations. The physical meaning of the Kjj is discussed in sects. 4 and 6. 4. Kinetic energy and considerations of charge vibration effective mass
It should be re-emphasized that this work is confined completely to the subspace of touching spheroids and no attempt has been made to investigate overlapping spheroids, fission movement and/or the kinetic energy of the fission movement. The extension to overlapping spheroids, etc., is unlikely to be of value as the potential energy around the saddle point does not vary smoothly 1). The kinetic energy and the effective masses associated with the c 1 and c2 degrees of freedom are taken from NS. A rather important result of introducing the charge vibrations was finding that for the limiting case of a sphere with U = 1 (i.e., the fissioning nucleus) the charge vibration energy E A is ( K~a t t 70 EA = h 'M~cr~! ~ AG MeV-
This relationship adds support to the basic assumption of this paper that the rigorous fluid dynamics treatment is equivalent (or very closely related) to our description of charge vibrations, at least for small vibrations. The above expression follows because for small vibrations 1 2 ~McrfA +½K~AA 2 = E,
which gives the frequency for the charge vibration 09 A =
[ K A.t/ l"~teff] ½.
396
R. H O L U B A N D
G. R. C H O P P I N
The stiffness constants for U = 1 are as follows (in units of surface energy):
K~a(U =
A~o-f
1) = 8~'a 2
1-tt'T 2
Using Green's 8) formula, which behaves more regularly at very low (Ao < 10) masses, gives /(a~(U = 1) =
2Ao 42 94.07__ +4xp2.
(20b)
17.97 The effective mass has been estimated in the following simplistic way. When a small nucleus such as 4He undergoes a "charge vibration", two nucleons must exchange places in order to achieve a dipole vibration. The overall shape does not change during such a vibration and in the case of ~He, half of the mass of the nucleus is engaged, so that the effective mass would be ½ (in units of Mo, the mass of the nucleus). The rigorous fluid mechanical result is hO)di p --
2"06 o R ] / 2 ~oo ' / ( M A4N~Zo) '
where 2K/Ao is a stiffness constant in Green's formula for the symmetry energy and MoA~/4NoZ o is a quantity that can be related to our effective mass (if No = Zo, it is equal to Mo). In these considerations our effective mass would be MCff~ - Ma
-
l 8
Ao~ No Zo "
When these expressions are substituted into eqs. (3) and (4) one gets dipole vibration energies given in fig. 5. The agreement is sufficient to provide strong support for the validity of the approximations of this approach. A difficulty arises when the distortion of the fissioning nucleus is included. The effective mass increases again due to the elongation along the symmetry axis (it must
3C 2C
E,jip(MeV) IC
J6o
260
300
A Fig. 5. D i p o l e resonance energy Eaip as a function o f n u c l e o n n u m b e r A. The solid line represents fluid d y n a m i c s value, the p o i n t s are as calculated in this work.
CHARGE
VIBRATIONS
397
travel "further") and the parallel dipole vibration energy Ed~p II decreases while the energy of the vibration perpendicular to the symmetry axis, Eaj p , (which does not interest us in fission since it does not effect the N/Z ratio of the fragments) increases. According to Danos 8): Edip ± --
0.91 -~-' +0.09,
E~lip I I
b
where c and b are major and minor semiaxes, respectively. The vibration for very constricted shapes is described by Updegraff and Onley 6). The energy Edipl I behaves as shown in fig. 6. While this curve has been calculated for 238U, it is believed the curves for all nuclei have similar shapes with magnitudes increased for the lower masses. The range of saddle-point shapes is indicated in fig. 6 where we see that Ea~, II varies by factor of three depending on the deformation parameter, ft. In this work a value o f f l ~ 1.9 was used.
,o
region ofsoddle points 1 used in lhis work
E dipll(MeV)
05
I0
0 o:> ,8:0
,8=05
I J L5
20
oDco ,8=1.0
,8=15
,8=20
Fig. 6. Parallel dipole resonance energy Edipl I for 238U as a function o f the d e f o r m a t i o n
ft.
We do not know exactly the stiffness KAj for the very constricted shape nor the effective mass; we know it must be related to the Ed~p II" Following the NS model we make the approximation that the potential energy resistance to charge vibration is given by the stiffness constant for the system of two touching spheroids. This approximation allows the calculation of the total charge vibration system, even though, strictly speaking, there is no mass flow between touching separated spheroids. The charge vibration stiffness Kaa is large and, consequently, the velocity of charge vibration (proportional to ,,/Ka.4) is such that there is always time, before the scission occurs, to complete several charge vibrations.
5. Transformation to normal coordinates
To treat the PE and KE matrices properly one must allow for the particular fission problematics since the vibration is not a standard one. As the nucleus moves closer and closer to the saddle point the effective mass for charge vibration increases. If these movements can be described as "out of phase" (i.e., neutrons and protons move
398
R. HOLUB AND G. R. CHOPPIN
"against each other"), the movement corresponding to mass asymmetry vibration can be described as "in phase" 9). Therefore, one assumes the effective masses are o f the same order (i.e., MA = 4My). However, the stiffness constants, K~j and K,, for x = 0.5-0.8, (see figs. 1, 10 and 11) differ by a factor of ~ 3.5x 102 and the velocities by approximately 10. This "time factor" of ~ 10 is assumed, with good reasons, to be crucial in the case of fission since fission is a collective motion to) and as such is "slow" (i.e., << 10 -22 sec) but not so slow as to allow all the mass asymmetry movements to establish equilibrium. In other words, the NS approximation is applied, i.e., that the velocity of mass-asymmetry movements dU/dt = ~r = 0 and as a consequence the effective mass-asymmetry mass is infinite. Taking all the previous considerations into account we divide the system into regular 3 x 3 " m o v i n g " matrices - those including A, c 1 and cz degrees of freedom and that of a 4 x 4 " f r o z e n " PE matrix. When one assumes in the Lagrangian i t ) that 0" = 0, it does not mean that the U-degree of freedom is not coupled to the other PE matrix components, and the nuclear matter must arrive at an asymmetric saddle point with no mass vibrations since there is not enough time to transport all the mass through the constriction even though the system is still affected by the potentials in the nuclear matter undergoing fission. The " m o v i n g " matrices, however, do relate to vibrations. The moving matrices are: [AOc I ¢~c2]
KAc I
Kcicl
K ....
-- KAc I
K ....
K ....
o
6c 1 , L¢~¢2J
o
1o
( M e , * M,)
M,
Lo
Mz
(M~,_ + Mt)J
1, L~c2J
where 6cl = Cl-Co, 6c2 = ¢ 2 - c 0 , Mt is the effective mass for fission movement, Me, = M~2 [effective mass for shape vibrations 11)], and z~ and 661 are time derivatives. The saddle-point deformations c¢01) and C~o2) (fragments 1 and 2) and M s are taken from NS. The K E matrix for the 6c 1 and 6c2 degrees of freedom is given in NS. Diagonalizing this system 13) gives the following quantities: (a) Eigenvalues:
:/'1
=
½(A+ D ) + x / ( ½ ( A - - D ) ) 2+B2, 22 = C,
2 3 = ½ ( A + D ) - - x / ( ½ ( A - - D ) ) Z + B 2, where
A - KAA MA
B -- x:2KA"' \ / M A Mc,
(21)
CHARGE VIBRATIONS
399
C -- K~:, + K~,~ M~ + 2Ml
J//c~
(b) Eigenvectors (normalized): 1
A p --
(~i-
....
K~3),
41 +K 2
s = .d}(&~
1 d'
(K~, +~3),
-
\/1
c~c2),
+
q-K 2
where
~l = AxIMA,
~3
O,
,/= G(~<,,-:c
d~/
--
K
,',
=
The appropriate determinant of the "frozen" matrix is:
i Kuv-2
KAy
Kuc,
-- Kuc
i K.4u t Ku~, -Kw,
KAj K~, --KA~ ,
KA<.,
--KAy,
K .... K~:~
K .... K~::
l
=
O,
(21b)
where 2 ( - K,,) designates the stiffness constant of the normal mode "mass asymmetry" which depends on all the other elements of this PE matrix. Comparison between 21,22 and 23 is shown in figs. 7 and 8. Since A and 2 I, and D and 23 are very similar A and D are not included in figs. 7 and 8. These quantities differ appreciably only when tile constant K increases in value, which happens when 2~ ~ 2 3 (or A ~ D) so that the coupling between A- and d-modes is greatest. • 2.5
X.O.8 X-O.7
~
2.0
j j/x-o.6
G/J/×-o.~
1,5
0.~
~ ~ IO
P(",z)
X-04
×=01
15
Fig. 7. Eigenvalue ),~ for tile normal mode A' as a function of p. The range of O corresponds to that o f known fissioning nuclei.
400
R. H O L U B
AND
G. R. C H O P P I N
4,0
3.C 2 (,flOUN,TS) 2£ X
1
1.0
x Fig. 8. E i g e n v a l u e s 22 a n d 23 for the n o r m a l m o d e s d" anad s as a f u n c t i o n o f the f i s s i o n a b i l i t y p a r a m e t e r .v.
o
x~2. - ~
./x,o2
/ X~0.6
-0.5
'
1.0
'
'
'
'
'
1.5
'
'
p(N,z) Fig. 9. T h e c o u p l i n g c o n s t a n t K as a f u n c t i o n o f p o f t h e f i s s i o n i n g n u c l e u s for v a r i o u s f i s s i o n a b i l i t y p a r a m e t e r s x. T h e r a n g e o f p c o r r e s p o n d s to t h a t o f k n o w n rmclei.
The mass asymmetry K., (m = U - ½ ) is already diagonalized and is independent o f the K E matrix in our approximation. Consequently, Km = K v v -
KJuKAA -- 2KAu Kva KAd+ KZu Ka K~ K aA -- K~d
K e a = \l~2Kt, c,,
Kaa = \./~2Kac,,
The variation of K,, with x and p =
K a = K . . . . - l < c , c . ..
N/Z is shown in figs. 10 and 11.
(22)
CHARGE VIBRATIONS
401
s~
/ /
0'.
/' (0)
Km(EsUNITS) 0 // -05 i
/
/
-]0
f
/ ~t
/
! '
'
'
'
015
'
'
'
'
1.0
x Fig. 10. Saddle-point stiffness constant Km for the normal coordinate system as a function of the fissionability parameter x. The results of our calculations are given by the solid line and the results of NS by the dashed line. The short vertical lines show K,, when charge vibrations are included.
1.0 ~X=08 __.____-----
X=0.7
- -
X=0.6
0.~
~X=0.5 (0}
Km(Es UN,T~
X=04 X=0.5
- 0 , .=
-X-0.2 -I.0
--
J '
'
,.'o
. . . .
I.~
X-OJ
'
'
p(N/z) Fig. 11. Saddle-point stiffness constant K,, for the normal coordinate system as a function ofp of the fissioning nucleus for various fissionability parameters x. These curves correspond to the short vertical lines in fig. 10. The range of p corresponds to that of known nuclei. This peculiar m - n o r m a l m o d e
can be i n c o r p o r a t e d
into the f r a m e o f m o v i n g
matrices as f o l l o w s . W h e n there is a n y d e v i a t i o n in m a s s a s y m m e t r y in the f i s s i o n i n g m a t t e r as it a p p r o a c h e s the saddle p o i n t (the resistance o f stiffness against this is g i v e n by
K,,,),
the s a d d l e - p o i n t characteristics
o f the other c o o r d i n a t e s
(c, = Co,
402
R. HOLUB AND G. R. CHOPPIN
c2 = Co, A = A o = 0) must be changed in their dependence on this nonsymmetric "'new" saddle-point shape which has a certain value of mass asymmetry m s a 0. Other vibrations are not similarly affected as they have enough time to rearrange themselves. This adjustment is made simply by requiring that the overall system of four degrees of freedom be described as a sum of squares (diagonalized) so that 21 [ l x / l ~ K 2 (Ax/MA_Kdx/Mc1)+ X m l 2
q-22 S2 q-')'3
i~/1+1 K2 (KAx/M~+dx/Mc,)+ym
, K~v
=[OVAec, Wl]
K~j
KAc,
K c,
k-Kcc,-Kay,
+Kmm 2
--KAcl
.... Kclc" l
(23)
L~c2J
Kc,o~ K;,c I
where
~U = U-½ = m,
x,/m~, d = \,'~M~(vcl-~c2).
When all the expressions for K,,,, 2~, 23 . . . . etc. are substituted and the factors of identical ~ U x A, ~9c1x #e2 products set equal to each other, the result is the following equations for X and Y, which were used in eq. (23) to describe the changed properties of the "moving" modes whenever m ¢ 0. We get two equations for X and Y:
)q2Xx/M-4Am +232YKx/MaAm X/I + K z (1 + K z
2KAo,
- 2 , 2XK,]'M~, din +23 2Yx/M~,dm _ 2(x/~Kvc,) ' x/1 + K z \/1 + K 2 X = KJvfVM~- KK.vl, i'
,
(24)
21 x/l + K 2
r
(25)
-
23 X/1 + K 2
This completes the transformation. Before the physical consequences can be discussed, however, the excitation energy distribution and related distributions, must be considered. 6. Results and their treatment From the equations of motion, given any set (e.g., ~U, A, 6c~, 6cz) of initial conditions (which are given statistically as multivariant normal distributions whose most probable values and widths depend on nuclear temperature), the final observable
CHARGE VIBRATIONS
403
properties are obtained also as multivariant (four) normal distributions with normal coordinates, which can be transformed into final individual isotopic cross sections (U and A) and the number of neutrons evaporated (i.e., 6cj_ and 6c2). The translational kinetic energy, E t, and excitation energies X, are related by E t + X, +)(2 = Eo + 2X ° , where X ° is the saddle-point excitation energy of fragment i. The translation energy is: Et = 4 [ U - P A I l ( I U)+pA]E o (1 + 2 X , ,.,/Eo)s '
(26)
and 5x
E° - 6(c, +e~)"
(27)
The excitation energies are: X 1 = X°+X1,ss+Xl,d,d'+Xl,a,A'+Xl,,,m,
X2 = X° + X,..s- X,,a,d'- X,.,t,A'- X,,,.m,
(28)
where Jr1 s is the first derivative of X 1 over s [s = ~/~(6c~ + 6c2) and the other terms previously defined are converted to our normal coordinates. The expressions (26) and (28), in spite of their being approximations only (firstorder expansions), have been found by NS to be satisfactorily accurate in the region of small vibrations. To determine this they calculated numerically, from a set of typical initial conditions, using the equations of motion, the final values and compared them to the values from equations corresponding to (26) and (28). The individual derivatives were calculated by increasing stepwise with a computer the expressions for deformation energy. The differences were less than 10 O//o.In our case, however, these expressions were obtained analytically, as were the stiffness constants and, again, the 0.25
0.20
0.15
0.10
0.0~
,
,
1
1
1
1 0.5
1
1
i 1.0
X Fig. 12. The total t r a n s l a t i o n a l kinetic energy E ° as a function of the fissionability p a r a m e t e r x. The results o f our calculations are given by the solid line, the results of NS by the d a s h e d line.
°o5[
404
R. HOLUB AND G. R. CHOPPIN
,/
004
,
0
(0)
/
0,03 I (
/
/
0 02~
,
i
,
,
i
I
1,0
05 X
Fig. 13. The excitation energy X~ ° of a single fragment as a function of the fissionability parameter x. The results of our calculations are given by the solid line, the results of NS by the dashed line. a g r e e m e n t was quite satisfactory. Figs. 12 and 13 show the agreement o f X ° a n d E o with x. R e s t a t i n g , o u r basic c o n s i d e r a t i o n is t h a t the t o t a l d e f o r m a t i o n energy for a n o n viscous f r a g m e n t is c o m p o s e d o f d e f o r m a t i o n energy o f the i n d i v i d u a l fragments plus the e x t r a energy which comes f r o m m u t u a l C o u l o m b energy changes n o t being c o n c e n t r a t e d at one point. It is n o r m a l l y a s s u m e d for a non-viscous b o d y no transfer o f collective m o t i o n into internal excitation occurs. M a t h e m a t i c a l l y for f r a g m e n t 1, a t the saddle p o i n t X0 = u rr3z3/2 4 .3",~+ 1 U - . } y ( t ~ 2 --T~-~Z~
- x zI ~ "' - - r ~ - ~ e 3 ) + [ S ( l , 2 ) - t ] E o.
(29)
W h e n the derivatives over the n o r m a l c o o r d i n a t e s are taken, the d e p e n d e n c e on m o f the whole s a d d l e - p o i n t system must be t a k e n into account. W e k n o w t h a t when m # 0 the d i s t o r t i o n a s y m m e t r y m o d e d increases with m as well as does the Av i b r a t i o n . Therefore, the excitation energy, in a general fashion m a y be written: X, -
---rit
I-,¢ z
T+
[/(I U ] -3
m)-I] '
+zx u -
- ( A + c,,,)p]" [g(1, ,,,)- I]
10x + - -
I V - ( A + C,,,)p][(1 -
3(c, +Cz)
U)+(A +CJ,,)p][S(l, 2,
where f(1, m)- ! ,
w
t
2 2
= log c ' +B;" (1 , U; \ -v-log
4
3
cl+B'm~ ~ U~
] ,
,,,)-I],
(30)
CHARGE VIBRATIONS
405
g(1, m)-- 1 = --3-~ , 2 --ToS0~ 4 3,
so, 2, ,,,) = ~3{(1~.,(,,,) ~,,(,,,) -
2 , ;,(,,)~,
I i)log [l~.,(,.)j -{-}'l("}')l x,~[.,
'
½
[(~, +~',,,)=-
cl + B'mJ '
cL+c2 m =
U-~,
B' = - (Y--KX)x/I+K2 v/2M~,(I + K) c
=
(X+KY)'h+K~
-
"J~O + K) The constants B' and C (figs. 14 and 15) express the changes with the most probable distortion asymmetry and the most probable charge which occur when m ~ 0, respectively. The manner in which B' and C were obtained is covered in the discussion section. The results of taking the derivatives are:
3X, -
U~f.(1)+½xU_~9: ( I ) + E ° I_S{I,2)- 1 + S ' , ( I , 2 )
(3C 1
L
OX' -Eo [S(1'2)-I ac2
8X,_
fiX
8s
~[\/~(ac, +0c2) ]
_
X,.j 8X,
Xl,tt | ~
+ S ; ' ( I ' 2) ,
l (8?, +~X,I
"
gX,_ ?d
8c2 / '
(~X 1 = -(f-1)UiSKTa BA
(31)
]
L cl + c U
\12 ,,.,c,
] ,
£l q-C2
1 . . . . 1 - a'T 2
- f[~(m)U~+ } ( f - 1)U- '~- 8 ~ r a c ( f -
(32)
1 (~X,
~X,.]
\/2 \8c t
(9c2]'
2x(g-l)U
l)U ~
1
_, ~p.
~/~.j
L\---
-
+K \/MaJ
"
•
+ B'm _-- log ~l U ~_
--2
2
4
[1 +y~ - - r ~
( 1 - v ' log c'+"mi+ U~_ ! '
(35)
\/1+
(36)
,/~+K-,
(37)
where
fly(m) = ~
(34)
l -~cT 2
+2xU-~[¼ob(m)+(O - 1)(~- pC)]. ,~, -
(33)
3n
J, m
=
U-½.
406
R. H O L U B A N D G. R. C H O P P I N
20
~
15
B'(%uN,Ts) 0.5
x=ol
'°I oL
i.o
7
x-~3
15
Fig. 14. The q u a n t i t y B' as a f u n c t i o n of p for v a r i o u s fissionability p a r a m e t e r .v. Tile range corresponds to t h a t of k n o w n nuclei.
\
04
C
O2
X=06 :\,
,,
'~"\
X=04
~'~X=O
8
-02
- 04
~X=OI 15
I0
Fig. 15. Tile q u a n t i t y C as a funtion o f p for various fissionability p a r a m e t e r s .v. The ra nge c o r r e s p o n d s to t h a t o f k n o w n nuclei.
Figs. 16-18 show and c o m p a r e those quantities, some of which by NS. Disagreement between our values and those calculated in most cases as can be seen from figs. 12 and 13. In mass calculations where the values depend heavily on ct/, (e.g., for K a and K,, for x = 0.8) the agreement with NS is less most likely due to the simplification
were also calculated by NS is negligible S ( I , 2) and S(2, 1) satisfactory. This is
=
which explicitly is for " s m a l l " deformations s). However, since at x = 0.8 the validity
CHARGE VIBRATIONS
407
of the two-spheroid model is questionable, this omission of not going to higher orders may be excused. The other small discrepancy for X~,a and X~,~ in figs. 16 and 20 is clearly of different origin. Possibly it could be associated with terms involving mutual
oo~
,
0
'
IO
X Fig. 16. The excitation energy derivatives XI. a and X~,~ as a function of the fissionability parameter .v. The results of our calculations are given by the solid lines and the results of NS by the dashed lines.
I
,\\,
X: 0 8\\\, '\
0 05
~.
°t,!,
x.o3 \ x ° o s ~
10
\
15
:(N:.) Fig. 17. The excitation energy derivative XI, j as a function o f p for various fissionability parameters x. The range corresponds to that of k n o w n nuclei.
Coulomb energy expansions as given in NS and in expressions (12), where not using the double precision arithmetic in calculations could account for the discrepancy. Since it is small we did not repeat the calculations using double precision arithmetic.
408
R. HOLUB A N D G. R. CHOPPIN OJ X=0.6
0.05 (0) ×,,~'(Esu.,~s)
\
0
X=0.3~ i
,
i
i
=0.4
\ i
i
i
1.0
i
i
J
1,5
P( z) Fig. 18. The excitation energy derivative X l , a , as a function o f n e u t r o n - t o - p r o t o n n u m b e r ratio p of fissioning nucleus for various fissionability parameters .v. The range corresponds to that of k n o w n
nuclei. 7. D i s c u s s i o n
Before considering the overall role of charge vibrations in fission it is useful to discuss the physical meanings of the quantities introduced in the preceding sections. The stiffness constants given in figs. 1 to 4 may be divided into those determined by bulk properties and those dependent on surface properties. The charge vibration stiffness KaA, the cross term stiffness Kau and the mass-asymmetry vibrations Kvv are the former type while all others fall in the second group. The bulk stiffnesses contain the shape terms f(ci) and #(ci) while the surface stiffness constants contain only the derivatives of these terms (compare equations (13)-(19)) and are generally much smaller, especially with respect to A~a which is about 103 greater. In spite of this the eigenvalues ).i are all of the same approximate size because the effective masses compensate for the disparities between the stiffness constants. This has an important effect on many of the observable fission properties because the compatibility of the individual normal modes makes their effect on each other much more pronounced. Especially crucial is the point where eigenvalues for charge and distortion asymmetry vibration, A' and d', are approximately equal, which happens in our calculations for x -~ 0.6 and p ~ 1.3 when the mode mixing constant K shown in fig. 9 reaches low negative values. Consequently, observables such as C and XI,,, shown in figs. 15, 19 and 20, the number of neutrons evaporated as a function of fragment mass and, to a much smaller degree, the most probable charge of the fragments show an increase which agree with the available experimental data 14, 1 s). Because of the approximate character of the two-spheroid model of fission and because of the simplifications introduced concerning the charge vibration effective
CHARGE VIBRATIONS
~
0.2
409
X=0.8
O. (0)
>
Xt,m(E s UNITS)
I
X=O.T
~....x.o.~ ,,.~..,,.--X~ 0..
0 i
X . 0 . 2X. i, ',O ~ 3 , i' 1.0
l
i
i
i
i 1.5
i
X=0.4 i
P(N,z) Fig. 19. The excitation energy derivative Xl.,,, as a function of O for various fissionability parameters x. The range corresponds to that of known nuclei. 0.2
Bi÷i
01
/t
I ___ 0.5
i
1.0
X Fig. 20. The excitation energy derivative Xl .... as a function of the fissionability paralneter x. The results of our calculations are given by the solid line and the results o f NS by the dashed line. The vertical lines show Xl.,,, derivative when charge vibrations are included. Their lengths correspond to the curves in fig. 19. The experimental points for Bi and Ra fission induced by protons are taken from refs. 14. ts).
mass M~ it is to be expected that the agreement in fig. 20 would not be quantitative. These approximations were justified as our primary goal was an investigation of the importance of charge vibration degree of freedom rather than development of a refined theory. It is obvious that in the region of heavy nuclei a situation must hold where 21 ~ 2a since )~l increases while 23 decreases with increasing x (see figs. 7 and 8). The exact location of the region where 21 ~ 23 is uncertain as the deformability of the fragments depends on shell effects; whenever the stiffness of one fragment increases it has basically the effect of perturbing and increasing the value of 23, which may
4t0
R. HOLUB AND G. R. CHOPPIN
shift the region of 2~ v 23 to heavier nuclei, or which may enlarge that region. Hopefully theoretical X 1.... values may be derived which would agree much better with the experimentally observed ones. [t is possible to extend our speculations. Myers 16, 17) has introduced an expression into the mass formula for neutron skin thickness for the surface region where the neutron density is several times greater than in the nuclear interior. It is easy to imagine that. for instance, those stiffnesses which depend on surface properties would be affected by introducing such a neutron skin term. This may change Kac , constant (fig. 3) into negative values (say hypothetically between - 0 . 5 to - 0 . 8 ) . Such a negative stiffness constant may be generally interpreted as causing instability in the direction of the particular coordinates which in this case would mean increasing elongation and neutron number leading to lower potential energy. Similarly, the constant /*AV, which is negative in most cases (fig. 2), indicates that if the mass split is m > 0 the neutrons have tendency to flow in. This agrees with the well-known experimental fact that the heavier fragment is more neutron-rich than the lighter one. I f the hypothetical large and negative constant K~c , is introduced into expression (22) for the constant K,, a very great change takes place; K,, decreases its value and at a certain point it may even become negative introducing asymmetry in fission without invoking any single-particle effects. Although there would also be important changes in the Xa,,,, quantities, this work cannot investigate these effects in any detail partly due to the lack of an adequate liquid-drop mass formula. However, unpublished results of Nair As) make quantitative conclusions which are very much the same. Nair's treatment is complete and includes all droplet model degrees of freedom (e.g., compressibility and charge density variations). With respect to the above speculations it should be added that our treatment also makes the K,,, constant lower by ~ 10 "~ tbr 0.6 > x > 0.8 (see fig. 10). This and the speculations in the preceding paragraph are related to the findings of Ryce et al. ~o) who point out that if one makes the hypothetical assumption that there is no asymmetry energy in the mass formula (or ~c = 0 in the MS mass formula), asymmetric fission would be favored in this region o f x . However, the asymmetry energy overrides this result (in our framework it is equivalent to saying that KAA is very large, and K,,, is lowered by only a small amount; the small value of the decrease is also due to the fact t h a t / ~ , , , is mostly positive and small). It should be noted, however, that Ryce et al. do not take into account the effect of deformations which our treatment does consider. One important point concerning the possible existence of a scission barrier should be made. The constant Ka~ is a bulk-type constant, i.e., it depends on composition rather than on shape and does not differ much for different shapes from the values given in fig. 1. The effective mass k l A does depend on shape, or more precisely on the magnitude of the constriction of the neck region in the fissioning nucleus 9) (fig. 6). As a restllt of these two observations, one might expect some variations in charge dispersion widths tbr different fissioning nuclei; however, this is in a complete dis-
CHARGE VIBRATIONS
411
agreement with experimental data. One way to overcome this discrepancy is to introduce a scission barrier 7), which would be basically the same for all fissioning nuclei. In conclusion a few minor points should be made. The J(1,a quantity, fig. 17, is usually positive which means that when A > 0, the excitation energy increases due to a lowering of the repulsive energy in the fragment which exceeds the decrease due to increased neutron excess. By contrast X1, v (unlike X1, ~) decreases because, other things constant, the deformation is being removed by adding mass to the deformed fragment at the saddle point. The two new normal modes are d ' and A'. [n the limiting case when K = 0, one gets properties of d- and A-modes as described in NS. The constants B' and C in expressions (30) give the total effect of all appropriate Kij and Mj on c°(m) and A (m) respectively• One can see that if K = 0, B' =
Y
_
Kuc,,
x';'2~4~
B = 2B',
Ka
which is equal to the B-value given by NS, and C -
X
Kau
\,I M A
KA a
These results are reasonable since the greater the coupling between U and c 1 degree of freedom (Kvc,) and the softer (smaller) the distortion-asymmetry stiffness constant K a are, the more affected the nucleus is. The more asymmetrically distorted it is with m > 0, the more neutrons are evaporated from the heavier fragment. The same holds for K~v/KAA, except in this case it concerns charge vibrations. Acknowledgment is extended to R. J. de Meijer and R. Eaker for assistance with some of the calculations. This research was supported by contract AT-(40-1)-1797 of the USAEC.
References 1) J. R. Nix and W. J. Swiatecki, Nucl. Phys. 71 (1965) 1 2) H. W. Schmitt, Proc. Syrup. on the physics and chemistry of fission, Vienna, 1969, paper SM 122/122 3) W• D. Myers and W. J. Swiatecki, University of California Lawrence National Laboratory report UCRL-11980 (1965) 4) A. Brandt and I. Kelson, Phys. Rev. 184 (1969) 1025 5) B. M. Spicer, Advances in Nuclear Physics, vol. 2 (Plenum Press, New York, 1969) 6) W. E. Updegraaff and D. S. Onley, Nucl. Phys. A161 (.1971) 191 7) W. N6renberg, Phys. Rev. C5 (1972) 2020 8) N. Danos, Nucl. Phys. 5 (1957) 23 9) W. J. Swiatecki and M. Blann, unpublished work 10) L. Wilets, Theories of nuclear fission (Clarendon Press, Oxford, 1964)
412
R. H.OLUB AND G. R. CH.OPPIN
l t) H. Margenau and G. Murphy, The mathematics of physicas and chemistry, 2nd ed. (Van Nostrand, New York, 1956) 12) H. Lamb, Hydrodynamics (Dover, New York, 1945) 13) H. Geldstein, Classical mechanics (Addison-Wesley, Mass., 1959) 14) E. Cheifetz, Z. Fraertkel, J. Galin, N. Lefort, J. Peter and X. Tarrago, Phys. Rev. C2 (1970) 256 15t E. Konecny and H. W. Schmitt, Phys. Rev. 172 (1968) 1213 16) W. D. Myers, Phys. Lett. 30B (1969) 45l 17) W. D. Myers, Proc. 1971 Mont Tremblant lnt. Summer School, ed. D. J. Rowe, L. E. H. Trainor, S. S. M. Wong and T. W. Donnelly (University of Toronto Press, 1972) p. 233 181 R. K. Nair, University of Florida, thesis, 1970 [9) S. A. Ryce, R. R. Wyman and A. T. Stewart, Can. J. Phys. 50 (1972) 2217
Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher
C H A R G E V I B R A T I O N S IN T H E L I Q U I D - D R O P M O D E L OF F I S S I O N R. HOLUB and G. R. CHOPPIN Department o f Chemistry, Florida State Universit.v, Talhthassee, Florida 32306
Received 25 April 1973 (Revised 8 June 19731 Abstract: The two-spheroid liquid-drop model of fission developed by Nix and Swiatecki has been extended to include charge vibrations. Using the Myers-Swiatecki mass formula, the appropriate stiffness constants have been obtained in analytic form. The effective mass, Ma, associated with charge vibrations for fission shapes was deduced from eigenenergies for the giant dipole resonance calculated by Updegraff and Onley, assuming that the eigenenergy is given by h~,)air,ol~ -- h \ K A a / M j ,
where Kaa is the stiffness for charge vibrations. To account for the experimental charge dispersion widths a highly constricted fission shape was chosen. The depertdence of fragment deformation energy on mass increases more rapidly than in the Nix and Swiatecki treatment when the eigenvalues of two new normal modes have similar values; it is concluded that this, together with spherical closed-shell effects, may explain the discrepancy between the Nix and Swiatecki values and experimental data. Various other aspects of the effect of charge vibrations are discussed.
1. Introduction
N o a d e q u a t e theory for the distribution o f nuclear charge in fission has been developed. As a result o f this deficiency, a n u m b e r o f empirical hypotheses such as t h a t o f equal charge distribution ( E C D ) , m i n i m u m potential energy ( M P E ) , a n d unchanged charge density ( U C D ) have been p r o p o s e d by various authors. O f these three only the M P E postulate has been suggested to have even a semi-empirical basis. Nix and Swiatecki 1) m e n t i o n e d the possibility o f a d d i n g an a d d i t i o n a l degree o f f r e e d o m related to charge vibrations in their d e v e l o p m e n t o f the t w o - t o u c h i n g spheroids l i q u i d - d r o p m o d e l o f nuclear fission. Schmitt 2), however, considered that the a d d i t i o n o f such a degree o f freedom w o u l d not affect the results o f any m o d e l in a significant way. Nevertheless, the lack o f the inclusion o f the effect o f charge distrib u t i o n in the l i q u i d - d r o p m o d e l is surprising since the term for the s y m m e t r y energy in the mass f o r m u l a is very strongly d e p e n d e n t on the nuclear c o m p o s i t i o n . The mass f o r m u l a developed by M y e r s a n d Swiatecki 3) successfully a p p l i e d the principal o f the " b u n c h i n g o f levels" to explain shell effects, a n d drew attention to the fact that it is intrinsically m o r e reasonable in a mass f o r m u l a to use a surface energy term which is d e p e n d e n t on the nuclear c o m p o s i t i o n . A mass f o r m u l a with this latter 387
388
R. HOLUB AND G. R. CHOPPIN
feature might be useful in an attempt to include the charge distribution in the liquiddrop model. In a sense, a variation in the surface energy term during fission is the basis of the Brandt and Kelson a) approach to fission theory. A further development that encouraged us to consider the inclusion of charge vibrations in the liquid-drop touching-spheroid model is the fact that the giant dipole resonance has been shown to be a collective as well as a single-particle phenomenon. It is possible to speak about neutron and proton density variations in such a way that the effective mass inside of a nucleus can move against the given stiffness of the nuclear matter, i.e., without changing the overall nuclear shape. Such an effect has been treated by means of fluid dynamics s) to obtain the familiar expression: Edi p ~ 7 0 A - ; MeV, where Ea~p is the giant dipole resonance energy. It has also been shown that this energy decreases substantially as the nucleus deforms to a necked-in fission shape and becomes zero when the fragment neck disappears. Finally, it seemed possible that certain discrepancies in the Nix and Swiatecki ~) (hereafter referred to as NS) liquid-drop model might be eliminated when a charge vibration was included in the theory. These discrepancies were related primarily to the dependence of the deformation energy distribution on fragment mass and to the magnitude of the stiffness of the mass-asymmetry vibrations. As in the NS development, we use no adjustable parameters except for the effective mass in the charge vibration. This effective mass has not been determined rigorously but was derived from the data of Updegraff and Onley 6).
2. U n i t s
The units used in this paper are those of NS; however, it should be noted that the surface energy, C2 A~ot, is composition-dependent (i.e., on N and Z ) through the C2 factor, where
C2 =
a2 [1 - 1.79((No
--Zo)/Ao)2].
(1)
Similarly the fissionability parameter x is (N, Z ) dependent since
c3 zo x = :a_,[1
- 1.79((No-
Zo)/Ao)2 ]A~2 -
(2)
This is a consequence of the dependence of the surface energy term in the MyersSwiatecki mass formula. As a result the more neutron-rich nuclides have a greater fissionability than would be the case without the dependence of the surface energy on The subscript 0 denotes the fissioning nucleus while subscripts I and 2 denotc the fission fragments.
CHARGE VIBRATIONS
389
the composition. This (N, Z ) dependence of x represents only a change of units and, therefore, does not affect the saddle-point shapes calculated as a function of x, although it does affect the particular nuclides which are represented by these shapes. The NS "radius-mass-surface energy" units were used for the calculations of the giant dipole resonance energies, i.e.,
hf2o = 22.44/,/Ao MeV,
(3)
where f2 o is the frequency of the charge vibrations in spherical nuclei (fissioning ones), Oo = [C2 A~/Mo] ½.
(4)
It should be noted that N o, Z o and A o are dimensionless.
3. Potential energy
The basic expression for the potential energy used for all the calculations is: PE = PE1 + PE2 + MCE, where PE, = - C, A, + C2 At f(1) + C3(Z2 /A~Og(1), PE 2 = - C, A 2 + C2 A2~f(2) + C3(ZZz/A~')g(2),
(5)
MCE = (Z, Z2e2/I)[S(1, 2 ) + S ( 2 , I ) - 11, where C1, C2 and C3 are given by Myers and Swiatecki 3) (hereafter referred to as MS), l is the distance between the centers of the two fragments, and S(1, 2), S(2, 1), f(1), f(2), g(1) and g(2) are shape-dependent terms discussed later. MCE stands for the mutual Coulomb energy. PEt and PE2 are simply the potential energies given by the MS mass formula for the fragments. It should be emphasized that the MS mass formula is useful for "small deformations" and that the model of fission based on touching spheroids is valid for nuclei lighter than those in the mass region of radium. The validity of this approximation for heavier nuclei is dependent on the establishment of an equilibrium at the scission point similar to the one assumed for the saddle point. This would be the case only if a significant scission barrier existed as suggested by N6renberg 7). This model of two touching spheroids is a transition state approach and neglects entirely any postsaddle-point dynamical motion. The main problem is to translate these expressions into the system of two touching spheroids which is (as NS have shown) a very good approximation for the real fissioning system at the saddle-point equilibrium. The "translated" expressions follow:
390
R. H O L U B A N D
G. R. C H O P P I N
PE = [ 1 - t c ( T + 2 N o A / A o u ) Z ] { - a l Ao U +az A~o U'tf(1)} + C 3 ( Z o U_ANo)ZA~o U _ ; g 0 ) + 5 [Zo U - A N o ] [ Z o ( 1 -
3
U)+ANo]
lA~o U ~
3 12
+ -
- [S(I, 2 ) + S ( 2 , 1 ) - 1 ]
5 I"0
A o ( l - U)! I + C3[Zo(I - U) + A No]ZAo~(l - U)- ;9(2).
(6a)
In surface energy units eq. (6a) is: a, A/~' {2 - , < ( T + 2 a A ')'- u C2 U /
PE
1
{U~,f(l)II_K(T+
+ 1--KT 2 .
.
.
t
.
.
5
+ 2 x (3(ci +c_,) EU -
Ap][(I
("T 2 a A 2~--U (, ] 1 ,
U))
2aA')2] + ( l _ U ) } f ( 2 ) [ l _ t c ( T U ] d
2aA ]2] t
iTu:
t
- U)+Ap-IES(1, 2 ) + S ( 2 , 1 ) - 1 ]
+ [ U - ap] z u - + g ( 1 ) + [(1 - u) + Ap]2(l - V)-~,q(2)] .
(6b)
The degrees of freedom are A, U, Cl and c, (in the functions f, g, and S). Further: (i) ~, 312:./t o = C 3, where l = c I + c2 (the main semiaxes of two touching spheroids). (ii) T = ( N o - Z o ) / A o. (iii) A = N / N o - A/A o = V - U. The degree o f freedom A is the fractional neutron deviation from unchanged charge density (UCD), which is the "charge vibration" degree of freedom (the " n e u t r o n " vibration, however, is Au = - A z ) . If N/No = A/Ao, U C D describes the charge distribution. When N/No > A/A o, the fragment is neutron-rich which is the usual case for the heavy fragments. (iv) U = AlIA o (the fractional mass as used in NS). The expressions in brackets [ T - 2 N o A / A o ( I - - U ) ] and (T+ ZNoA/Ao U) are obtained from ( N I - Z x ) / A 1 and ( N 2 - Z z ) / A 2 when the appropriate substitutions are made. Also, A1 = - A 2 and UI = 1 - UI = U2 as required by mass and charge conservation. ( v ) f ( 1 ) = l +- - ~2 c q 2- - T o 4s ~ , ,3 where cq is a deformation parameter related to the major semiaxis Cl. This relation, which holds for small ellipsoidal deformations, is derived from a = Roe . . . . ~,.-~),
b = Roe . . . . (r+-~.),
c = Roe . . . . :.
(8)
which is given in MS (p. 29)where a, b and e are semiaxes, a is a deformation parana-
CHARGE
VIBRATIONS
391
eter, 7 is a shape parameter, and Ro is the diameter of the fragment. Since it is usual and justified to limit the fissioning shapes to axially symmetric ones we can drop the parameter 7 (i.e., 7 = 0). The semiaxes a and b (a = b for symmetry) are determined by volume conservation, leaving us with one degree of freedom for each fragment: c = R o e~
[-i.e., ~ = log (c/Ro)].
(9)
The parameter a is related to the distortion parameter e:
~2 = a Z ( l _ ~a).
(10)
Since we work in [Ro] = roA o units the expression (9) must be changed into ~1 = l o g ~
1-~log~
,
(11)
which says for example that when the fractional mass U = ½ (at the saddle point), R~ of the fragment equals 1.0 x U + or R~ = Ro/1.26, where R o is the radius of the fissioning nucleus. When the mass asymmetry is such that all the mass is concentrated in one fragment, i.e., in the original nucleus, cq = log c-!-1 1--~ log cl U~ (vi)
g(l)=
1-1
2
4
S ~ 1 -- T~
-~ = log CI(1-- 7llogcl)
~.
3
Rl •
Similar expressions apply for fragment 2. (vii) The terms [Z o U - A N o ] and [ Z O O - U ) + A N o ] are obtained when both degrees of freedom, U and A, are put into Z~ and Z 2 in eq. (5). (viii) When the mutual Coulomb energy term is changed into the [Ro] , [M0] and [C2 A0~] system of units, a simple rearrangement must be made to change l, the distance between the charge centers of two touching spheroids, into [Ro] units, as indicated in eqs. (6a) and (6b).
(ix)
3;. 2"
S(1, 2) =,,~o (2n + 1 - ~ n + 3)'
s(2, 1) = Z
3 I°
,,=o (2n + 1)(2n + 3)'
where z; A,
(¢1"I-£2) 2
-
-- -
-
(c, +c2) 2
c2
(12)
These expressions were taken from NS and simplified forms are used in our later calculations. The Potential Energy Matrix. In this treatment of the saddle-point fissioning system, represented by small vibrations about an equilibrium system, it is in equilibrium with respect to all degrees of freedom except fission (which has a negative stiffness constant), we limit ourselves to four degrees of freedom, U, A, ct and c2. NS have shown that the degrees of freedom of rotational energy are not coupled with these four.
392
R. HOLUB AND G. R. CHOPPIN
1,50
?~ X=0.8~
I00
so K,,~.(E~b,,Ts) (= .
-50
°t
I
,
i
I.I0
I
I
I
(N/Z ) P
I
I
Fig. 1. Saddle-point stiffness constant for charge vibrations, KAa, as a function of p, the neutronto-proton ratio, of the fissioning nucleus for various fissionability parameters x. The range of p corresponds to that of known nuclei. I
~,×=0.1
//×=o2
o
jjjx=o
(o)
-2
J~,,,#
-~.
__~o.8 ' p(",z)
"
Fig. 2. Saddle-point stiffness constant KAu as a function o f p o f the fissioning nucleus for various fissionability parameters x. The range o f p corresponds to that o f k n o w n nuclei.
M o r e o v e r , it is very difficult to o b t a i n any e x p e r i m e n t a l d a t a on spins o r o r b i t a l a n g u l a r m o m e n t a o f the fission fragments. By c o n t r a s t o u r degrees o f f r e e d o m are subject to direct physical o r chemical measurements. Constrtlction o f the P E m a t r i x is d o n e in a s t a n d a r d w a y by t a k i n g second derivatives o f the P E e q u a t i o n (6b) a n d e v a l u a t i n g these at the saddle point. N S did this numerically from an e x p a n s i o n describing the PE. The i n d i v i d u a l r e a r r a n g e d and simplified second derivatives K u are given b e l o w as analytical expressions a n d are p l o t t e d in figs. 1-4:
Kvv = 2U~(f[/v+~J~'-8-f)+2x
lo(2s-11
3(c1+c2) +U
-+~o
,o,
~,,~7
(-~-g+--3-gu +2~ae'v'j_j ,
(13)
CHARGE VIBRATIONS
393
0.2
03
E~
K~C,(~0UNITE
I
I
I
I0
I
I
p(",~)
I
I
I
15
K~ct
Fig. 3. Saddle-point stiffness constant as a function o f p of the fissioning nucleus for various fissionability parameters x. The range o f p corresponds to that of k n o w n nuclei.
//'/
1.5
,o~.
//
osl ~
,.
/ ~
1.0
05
Fig. 4. Saddle-point stiffness constants as function of the fissionability parameter x. The constant
Kvv is in units of [E~(°)], Kcjcl and Kczc2 in units of [E~°VRu21 and K~c I in units of [E~(°)/R°].
I~
+ 2xpFlO(2S-
= -16~T~U~(fb+~f) l--leT 2
1)
-1 2O-~(g'v+~g)] ,
(14)
J
L3(ct +c2)
(t5)
(
KAa = 32a2tC A~ 1 5 . 6 7 7
2fu~
18.56
K ac ~ = - 81cToU ~fcl
1 -~T
2
)
1 1 -- t~T 2
+ 2xpU-~g'c,
+2xp 2
I
+4U-~g 3 ( c t + c2)
1
,
(16)
(17)
394
R. H O L U B A N D rr}rlt
--
lr i
G. R. C H O P P I N
¢t
gc,c , = U Jc~c-t-xu~gc~c~
V2_(2s_-_1)
5x
+ 6(c, + c2i L(c, + cO 2
. ~1 , + ~. . ......
S',(1, 2)+S~.~(I, 2)
[(~S- 1)
lOx
S L,(1, 2)-t-S'c,(2, l) cl +c2
K .... - 6(c, +c2) L(c, +,.2) 2
(C I _JrC2) 2
2 )+:~,.,,.~, ..... 2 ,1) l J
.... 1 2)] -~'"~'clc2(
'
(18) (19)
'
where No
6-
,
T
No-Zo . . . . . . . . . .
Ao
S(2,1)----4
t
Vc ~ U,
C 1 + C2
,
Zo
(simplified expressions (12) as given in NS),
'("2 2 - ) . # ')i'log ',1 21 3, +;.2I + 2. --;.~ 2~
No
p-
,
Ao
22
1
Cl
C 1 + C2
V
cz
S,t (I, 2) = S,1,(2, I) = .-~-- S(I, 2), ~c~
'i , 2) = Sc,(2, ' So,_( I) = -~Ls(1,2), CCz
82 S'c;~,(I, 2) = S'~'=~(2, 1) = &.2 S(1,2),
S" (1 2 ) = S'~[~2(2, 1 ) -
S(1 2), 6'¢ l UC 2
.~"' ¢1 2) = S,7,~,(2, 1) = -az - - S(i, 2), --c,_c2~
,
0 f(1) aU
~C 2
aU [I + ~2: e ,2- - T ~ 4 7 ,3-] ---- ft~,r 7
11
a~ = log c~ u ~' and similarly for gb, J ~ , f,~u,, J~[c,, etc.
1- U C2
CHARGE VIBRATIONS
395
Similar to NS, the potential energy matrix may be written now as:
AV
=
I(~UA~c1 cc2)
Kvv Kay Kw, - Kvc,
Kay Ka,j K j~,
Kvc, Kac, K ....
- Kgc,
Kc,c2
- Kvc, - K.,c~ Kc,.~ Kc,c,
1&2
In this matrix: Kuc t :
- Kvc 2,
Kac I =
- KAc 2 ,
Kc,c, = Kc:c2.
This is a result of the symmetry of the two touching spheroids, and because c~U = U - ½ , c~ci = c i - c o (Co is the saddle-point deformation taken from NS). All constants, given in the analytical expressions of eqs. (13) to (19) are evaluated at the saddle point. The constants K not containing subscript A are shown in fig. 4. The agreement with the values calculated by NS is proof of the validity of our calculations. The physical meaning of the Kjj is discussed in sects. 4 and 6. 4. Kinetic energy and considerations of charge vibration effective mass
It should be re-emphasized that this work is confined completely to the subspace of touching spheroids and no attempt has been made to investigate overlapping spheroids, fission movement and/or the kinetic energy of the fission movement. The extension to overlapping spheroids, etc., is unlikely to be of value as the potential energy around the saddle point does not vary smoothly 1). The kinetic energy and the effective masses associated with the c 1 and c2 degrees of freedom are taken from NS. A rather important result of introducing the charge vibrations was finding that for the limiting case of a sphere with U = 1 (i.e., the fissioning nucleus) the charge vibration energy E A is ( K~a t t 70 EA = h 'M~cr~! ~ AG MeV-
This relationship adds support to the basic assumption of this paper that the rigorous fluid dynamics treatment is equivalent (or very closely related) to our description of charge vibrations, at least for small vibrations. The above expression follows because for small vibrations 1 2 ~McrfA +½K~AA 2 = E,
which gives the frequency for the charge vibration 09 A =
[ K A.t/ l"~teff] ½.
396
R. H O L U B A N D
G. R. C H O P P I N
The stiffness constants for U = 1 are as follows (in units of surface energy):
K~a(U =
A~o-f
1) = 8~'a 2
1-tt'T 2
Using Green's 8) formula, which behaves more regularly at very low (Ao < 10) masses, gives /(a~(U = 1) =
2Ao 42 94.07__ +4xp2.
(20b)
17.97 The effective mass has been estimated in the following simplistic way. When a small nucleus such as 4He undergoes a "charge vibration", two nucleons must exchange places in order to achieve a dipole vibration. The overall shape does not change during such a vibration and in the case of ~He, half of the mass of the nucleus is engaged, so that the effective mass would be ½ (in units of Mo, the mass of the nucleus). The rigorous fluid mechanical result is hO)di p --
2"06 o R ] / 2 ~oo ' / ( M A4N~Zo) '
where 2K/Ao is a stiffness constant in Green's formula for the symmetry energy and MoA~/4NoZ o is a quantity that can be related to our effective mass (if No = Zo, it is equal to Mo). In these considerations our effective mass would be MCff~ - Ma
-
l 8
Ao~ No Zo "
When these expressions are substituted into eqs. (3) and (4) one gets dipole vibration energies given in fig. 5. The agreement is sufficient to provide strong support for the validity of the approximations of this approach. A difficulty arises when the distortion of the fissioning nucleus is included. The effective mass increases again due to the elongation along the symmetry axis (it must
3C 2C
E,jip(MeV) IC
J6o
260
300
A Fig. 5. D i p o l e resonance energy Eaip as a function o f n u c l e o n n u m b e r A. The solid line represents fluid d y n a m i c s value, the p o i n t s are as calculated in this work.
CHARGE
VIBRATIONS
397
travel "further") and the parallel dipole vibration energy Ed~p II decreases while the energy of the vibration perpendicular to the symmetry axis, Eaj p , (which does not interest us in fission since it does not effect the N/Z ratio of the fragments) increases. According to Danos 8): Edip ± --
0.91 -~-' +0.09,
E~lip I I
b
where c and b are major and minor semiaxes, respectively. The vibration for very constricted shapes is described by Updegraff and Onley 6). The energy Edipl I behaves as shown in fig. 6. While this curve has been calculated for 238U, it is believed the curves for all nuclei have similar shapes with magnitudes increased for the lower masses. The range of saddle-point shapes is indicated in fig. 6 where we see that Ea~, II varies by factor of three depending on the deformation parameter, ft. In this work a value o f f l ~ 1.9 was used.
,o
region ofsoddle points 1 used in lhis work
E dipll(MeV)
05
I0
0 o:> ,8:0
,8=05
I J L5
20
oDco ,8=1.0
,8=15
,8=20
Fig. 6. Parallel dipole resonance energy Edipl I for 238U as a function o f the d e f o r m a t i o n
ft.
We do not know exactly the stiffness KAj for the very constricted shape nor the effective mass; we know it must be related to the Ed~p II" Following the NS model we make the approximation that the potential energy resistance to charge vibration is given by the stiffness constant for the system of two touching spheroids. This approximation allows the calculation of the total charge vibration system, even though, strictly speaking, there is no mass flow between touching separated spheroids. The charge vibration stiffness Kaa is large and, consequently, the velocity of charge vibration (proportional to ,,/Ka.4) is such that there is always time, before the scission occurs, to complete several charge vibrations.
5. Transformation to normal coordinates
To treat the PE and KE matrices properly one must allow for the particular fission problematics since the vibration is not a standard one. As the nucleus moves closer and closer to the saddle point the effective mass for charge vibration increases. If these movements can be described as "out of phase" (i.e., neutrons and protons move
398
R. HOLUB AND G. R. CHOPPIN
"against each other"), the movement corresponding to mass asymmetry vibration can be described as "in phase" 9). Therefore, one assumes the effective masses are o f the same order (i.e., MA = 4My). However, the stiffness constants, K~j and K,, for x = 0.5-0.8, (see figs. 1, 10 and 11) differ by a factor of ~ 3.5x 102 and the velocities by approximately 10. This "time factor" of ~ 10 is assumed, with good reasons, to be crucial in the case of fission since fission is a collective motion to) and as such is "slow" (i.e., << 10 -22 sec) but not so slow as to allow all the mass asymmetry movements to establish equilibrium. In other words, the NS approximation is applied, i.e., that the velocity of mass-asymmetry movements dU/dt = ~r = 0 and as a consequence the effective mass-asymmetry mass is infinite. Taking all the previous considerations into account we divide the system into regular 3 x 3 " m o v i n g " matrices - those including A, c 1 and cz degrees of freedom and that of a 4 x 4 " f r o z e n " PE matrix. When one assumes in the Lagrangian i t ) that 0" = 0, it does not mean that the U-degree of freedom is not coupled to the other PE matrix components, and the nuclear matter must arrive at an asymmetric saddle point with no mass vibrations since there is not enough time to transport all the mass through the constriction even though the system is still affected by the potentials in the nuclear matter undergoing fission. The " m o v i n g " matrices, however, do relate to vibrations. The moving matrices are: [AOc I ¢~c2]
KAc I
Kcicl
K ....
-- KAc I
K ....
K ....
o
6c 1 , L¢~¢2J
o
1o
( M e , * M,)
M,
Lo
Mz
(M~,_ + Mt)J
1, L~c2J
where 6cl = Cl-Co, 6c2 = ¢ 2 - c 0 , Mt is the effective mass for fission movement, Me, = M~2 [effective mass for shape vibrations 11)], and z~ and 661 are time derivatives. The saddle-point deformations c¢01) and C~o2) (fragments 1 and 2) and M s are taken from NS. The K E matrix for the 6c 1 and 6c2 degrees of freedom is given in NS. Diagonalizing this system 13) gives the following quantities: (a) Eigenvalues:
:/'1
=
½(A+ D ) + x / ( ½ ( A - - D ) ) 2+B2, 22 = C,
2 3 = ½ ( A + D ) - - x / ( ½ ( A - - D ) ) Z + B 2, where
A - KAA MA
B -- x:2KA"' \ / M A Mc,
(21)
CHARGE VIBRATIONS
399
C -- K~:, + K~,~ M~ + 2Ml
J//c~
(b) Eigenvectors (normalized): 1
A p --
(~i-
....
K~3),
41 +K 2
s = .d}(&~
1 d'
(K~, +~3),
-
\/1
c~c2),
+
q-K 2
where
~l = AxIMA,
~3
O,
,/= G(~<,,-:c
d~/
--
K
,',
=
The appropriate determinant of the "frozen" matrix is:
i Kuv-2
KAy
Kuc,
-- Kuc
i K.4u t Ku~, -Kw,
KAj K~, --KA~ ,
KA<.,
--KAy,
K .... K~:~
K .... K~::
l
=
O,
(21b)
where 2 ( - K,,) designates the stiffness constant of the normal mode "mass asymmetry" which depends on all the other elements of this PE matrix. Comparison between 21,22 and 23 is shown in figs. 7 and 8. Since A and 2 I, and D and 23 are very similar A and D are not included in figs. 7 and 8. These quantities differ appreciably only when tile constant K increases in value, which happens when 2~ ~ 2 3 (or A ~ D) so that the coupling between A- and d-modes is greatest. • 2.5
X.O.8 X-O.7
~
2.0
j j/x-o.6
G/J/×-o.~
1,5
0.~
~ ~ IO
P(",z)
X-04
×=01
15
Fig. 7. Eigenvalue ),~ for tile normal mode A' as a function of p. The range of O corresponds to that o f known fissioning nuclei.
400
R. H O L U B
AND
G. R. C H O P P I N
4,0
3.C 2 (,flOUN,TS) 2£ X
1
1.0
x Fig. 8. E i g e n v a l u e s 22 a n d 23 for the n o r m a l m o d e s d" anad s as a f u n c t i o n o f the f i s s i o n a b i l i t y p a r a m e t e r .v.
o
x~2. - ~
./x,o2
/ X~0.6
-0.5
'
1.0
'
'
'
'
'
1.5
'
'
p(N,z) Fig. 9. T h e c o u p l i n g c o n s t a n t K as a f u n c t i o n o f p o f t h e f i s s i o n i n g n u c l e u s for v a r i o u s f i s s i o n a b i l i t y p a r a m e t e r s x. T h e r a n g e o f p c o r r e s p o n d s to t h a t o f k n o w n rmclei.
The mass asymmetry K., (m = U - ½ ) is already diagonalized and is independent o f the K E matrix in our approximation. Consequently, Km = K v v -
KJuKAA -- 2KAu Kva KAd+ KZu Ka K~ K aA -- K~d
K e a = \l~2Kt, c,,
Kaa = \./~2Kac,,
The variation of K,, with x and p =
K a = K . . . . - l < c , c . ..
N/Z is shown in figs. 10 and 11.
(22)
CHARGE VIBRATIONS
401
s~
/ /
0'.
/' (0)
Km(EsUNITS) 0 // -05 i
/
/
-]0
f
/ ~t
/
! '
'
'
'
015
'
'
'
'
1.0
x Fig. 10. Saddle-point stiffness constant Km for the normal coordinate system as a function of the fissionability parameter x. The results of our calculations are given by the solid line and the results of NS by the dashed line. The short vertical lines show K,, when charge vibrations are included.
1.0 ~X=08 __.____-----
X=0.7
- -
X=0.6
0.~
~X=0.5 (0}
Km(Es UN,T~
X=04 X=0.5
- 0 , .=
-X-0.2 -I.0
--
J '
'
,.'o
. . . .
I.~
X-OJ
'
'
p(N/z) Fig. 11. Saddle-point stiffness constant K,, for the normal coordinate system as a function ofp of the fissioning nucleus for various fissionability parameters x. These curves correspond to the short vertical lines in fig. 10. The range of p corresponds to that of known nuclei. This peculiar m - n o r m a l m o d e
can be i n c o r p o r a t e d
into the f r a m e o f m o v i n g
matrices as f o l l o w s . W h e n there is a n y d e v i a t i o n in m a s s a s y m m e t r y in the f i s s i o n i n g m a t t e r as it a p p r o a c h e s the saddle p o i n t (the resistance o f stiffness against this is g i v e n by
K,,,),
the s a d d l e - p o i n t characteristics
o f the other c o o r d i n a t e s
(c, = Co,
402
R. HOLUB AND G. R. CHOPPIN
c2 = Co, A = A o = 0) must be changed in their dependence on this nonsymmetric "'new" saddle-point shape which has a certain value of mass asymmetry m s a 0. Other vibrations are not similarly affected as they have enough time to rearrange themselves. This adjustment is made simply by requiring that the overall system of four degrees of freedom be described as a sum of squares (diagonalized) so that 21 [ l x / l ~ K 2 (Ax/MA_Kdx/Mc1)+ X m l 2
q-22 S2 q-')'3
i~/1+1 K2 (KAx/M~+dx/Mc,)+ym
, K~v
=[OVAec, Wl]
K~j
KAc,
K c,
k-Kcc,-Kay,
+Kmm 2
--KAcl
.... Kclc" l
(23)
L~c2J
Kc,o~ K;,c I
where
~U = U-½ = m,
x,/m~, d = \,'~M~(vcl-~c2).
When all the expressions for K,,,, 2~, 23 . . . . etc. are substituted and the factors of identical ~ U x A, ~9c1x #e2 products set equal to each other, the result is the following equations for X and Y, which were used in eq. (23) to describe the changed properties of the "moving" modes whenever m ¢ 0. We get two equations for X and Y:
)q2Xx/M-4Am +232YKx/MaAm X/I + K z (1 + K z
2KAo,
- 2 , 2XK,]'M~, din +23 2Yx/M~,dm _ 2(x/~Kvc,) ' x/1 + K z \/1 + K 2 X = KJvfVM~- KK.vl, i'
,
(24)
21 x/l + K 2
r
(25)
-
23 X/1 + K 2
This completes the transformation. Before the physical consequences can be discussed, however, the excitation energy distribution and related distributions, must be considered. 6. Results and their treatment From the equations of motion, given any set (e.g., ~U, A, 6c~, 6cz) of initial conditions (which are given statistically as multivariant normal distributions whose most probable values and widths depend on nuclear temperature), the final observable
CHARGE VIBRATIONS
403
properties are obtained also as multivariant (four) normal distributions with normal coordinates, which can be transformed into final individual isotopic cross sections (U and A) and the number of neutrons evaporated (i.e., 6cj_ and 6c2). The translational kinetic energy, E t, and excitation energies X, are related by E t + X, +)(2 = Eo + 2X ° , where X ° is the saddle-point excitation energy of fragment i. The translation energy is: Et = 4 [ U - P A I l ( I U)+pA]E o (1 + 2 X , ,.,/Eo)s '
(26)
and 5x
E° - 6(c, +e~)"
(27)
The excitation energies are: X 1 = X°+X1,ss+Xl,d,d'+Xl,a,A'+Xl,,,m,
X2 = X° + X,..s- X,,a,d'- X,.,t,A'- X,,,.m,
(28)
where Jr1 s is the first derivative of X 1 over s [s = ~/~(6c~ + 6c2) and the other terms previously defined are converted to our normal coordinates. The expressions (26) and (28), in spite of their being approximations only (firstorder expansions), have been found by NS to be satisfactorily accurate in the region of small vibrations. To determine this they calculated numerically, from a set of typical initial conditions, using the equations of motion, the final values and compared them to the values from equations corresponding to (26) and (28). The individual derivatives were calculated by increasing stepwise with a computer the expressions for deformation energy. The differences were less than 10 O//o.In our case, however, these expressions were obtained analytically, as were the stiffness constants and, again, the 0.25
0.20
0.15
0.10
0.0~
,
,
1
1
1
1 0.5
1
1
i 1.0
X Fig. 12. The total t r a n s l a t i o n a l kinetic energy E ° as a function of the fissionability p a r a m e t e r x. The results o f our calculations are given by the solid line, the results of NS by the d a s h e d line.
°o5[
404
R. HOLUB AND G. R. CHOPPIN
,/
004
,
0
(0)
/
0,03 I (
/
/
0 02~
,
i
,
,
i
I
1,0
05 X
Fig. 13. The excitation energy X~ ° of a single fragment as a function of the fissionability parameter x. The results of our calculations are given by the solid line, the results of NS by the dashed line. a g r e e m e n t was quite satisfactory. Figs. 12 and 13 show the agreement o f X ° a n d E o with x. R e s t a t i n g , o u r basic c o n s i d e r a t i o n is t h a t the t o t a l d e f o r m a t i o n energy for a n o n viscous f r a g m e n t is c o m p o s e d o f d e f o r m a t i o n energy o f the i n d i v i d u a l fragments plus the e x t r a energy which comes f r o m m u t u a l C o u l o m b energy changes n o t being c o n c e n t r a t e d at one point. It is n o r m a l l y a s s u m e d for a non-viscous b o d y no transfer o f collective m o t i o n into internal excitation occurs. M a t h e m a t i c a l l y for f r a g m e n t 1, a t the saddle p o i n t X0 = u rr3z3/2 4 .3",~+ 1 U - . } y ( t ~ 2 --T~-~Z~
- x zI ~ "' - - r ~ - ~ e 3 ) + [ S ( l , 2 ) - t ] E o.
(29)
W h e n the derivatives over the n o r m a l c o o r d i n a t e s are taken, the d e p e n d e n c e on m o f the whole s a d d l e - p o i n t system must be t a k e n into account. W e k n o w t h a t when m # 0 the d i s t o r t i o n a s y m m e t r y m o d e d increases with m as well as does the Av i b r a t i o n . Therefore, the excitation energy, in a general fashion m a y be written: X, -
---rit
I-,¢ z
T+
[/(I U ] -3
m)-I] '
+zx u -
- ( A + c,,,)p]" [g(1, ,,,)- I]
10x + - -
I V - ( A + C,,,)p][(1 -
3(c, +Cz)
U)+(A +CJ,,)p][S(l, 2,
where f(1, m)- ! ,
w
t
2 2
= log c ' +B;" (1 , U; \ -v-log
4
3
cl+B'm~ ~ U~
] ,
,,,)-I],
(30)
CHARGE VIBRATIONS
405
g(1, m)-- 1 = --3-~ , 2 --ToS0~ 4 3,
so, 2, ,,,) = ~3{(1~.,(,,,) ~,,(,,,) -
2 , ;,(,,)~,
I i)log [l~.,(,.)j -{-}'l("}')l x,~[.,
'
½
[(~, +~',,,)=-
cl + B'mJ '
cL+c2 m =
U-~,
B' = - (Y--KX)x/I+K2 v/2M~,(I + K) c
=
(X+KY)'h+K~
-
"J~O + K) The constants B' and C (figs. 14 and 15) express the changes with the most probable distortion asymmetry and the most probable charge which occur when m ~ 0, respectively. The manner in which B' and C were obtained is covered in the discussion section. The results of taking the derivatives are:
3X, -
U~f.(1)+½xU_~9: ( I ) + E ° I_S{I,2)- 1 + S ' , ( I , 2 )
(3C 1
L
OX' -Eo [S(1'2)-I ac2
8X,_
fiX
8s
~[\/~(ac, +0c2) ]
_
X,.j 8X,
Xl,tt | ~
+ S ; ' ( I ' 2) ,
l (8?, +~X,I
"
gX,_ ?d
8c2 / '
(~X 1 = -(f-1)UiSKTa BA
(31)
]
L cl + c U
\12 ,,.,c,
] ,
£l q-C2
1 . . . . 1 - a'T 2
- f[~(m)U~+ } ( f - 1)U- '~- 8 ~ r a c ( f -
(32)
1 (~X,
~X,.]
\/2 \8c t
(9c2]'
2x(g-l)U
l)U ~
1
_, ~p.
~/~.j
L\---
-
+K \/MaJ
"
•
+ B'm _-- log ~l U ~_
--2
2
4
[1 +y~ - - r ~
( 1 - v ' log c'+"mi+ U~_ ! '
(35)
\/1+
(36)
,/~+K-,
(37)
where
fly(m) = ~
(34)
l -~cT 2
+2xU-~[¼ob(m)+(O - 1)(~- pC)]. ,~, -
(33)
3n
J, m
=
U-½.
406
R. H O L U B A N D G. R. C H O P P I N
20
~
15
B'(%uN,Ts) 0.5
x=ol
'°I oL
i.o
7
x-~3
15
Fig. 14. The q u a n t i t y B' as a f u n c t i o n of p for v a r i o u s fissionability p a r a m e t e r .v. Tile range corresponds to t h a t of k n o w n nuclei.
\
04
C
O2
X=06 :\,
,,
'~"\
X=04
~'~X=O
8
-02
- 04
~X=OI 15
I0
Fig. 15. Tile q u a n t i t y C as a funtion o f p for various fissionability p a r a m e t e r s .v. The ra nge c o r r e s p o n d s to t h a t o f k n o w n nuclei.
Figs. 16-18 show and c o m p a r e those quantities, some of which by NS. Disagreement between our values and those calculated in most cases as can be seen from figs. 12 and 13. In mass calculations where the values depend heavily on ct/, (e.g., for K a and K,, for x = 0.8) the agreement with NS is less most likely due to the simplification
were also calculated by NS is negligible S ( I , 2) and S(2, 1) satisfactory. This is
=
which explicitly is for " s m a l l " deformations s). However, since at x = 0.8 the validity
CHARGE VIBRATIONS
407
of the two-spheroid model is questionable, this omission of not going to higher orders may be excused. The other small discrepancy for X~,a and X~,~ in figs. 16 and 20 is clearly of different origin. Possibly it could be associated with terms involving mutual
oo~
,
0
'
IO
X Fig. 16. The excitation energy derivatives XI. a and X~,~ as a function of the fissionability parameter .v. The results of our calculations are given by the solid lines and the results of NS by the dashed lines.
I
,\\,
X: 0 8\\\, '\
0 05
~.
°t,!,
x.o3 \ x ° o s ~
10
\
15
:(N:.) Fig. 17. The excitation energy derivative XI, j as a function o f p for various fissionability parameters x. The range corresponds to that of k n o w n nuclei.
Coulomb energy expansions as given in NS and in expressions (12), where not using the double precision arithmetic in calculations could account for the discrepancy. Since it is small we did not repeat the calculations using double precision arithmetic.
408
R. HOLUB A N D G. R. CHOPPIN OJ X=0.6
0.05 (0) ×,,~'(Esu.,~s)
\
0
X=0.3~ i
,
i
i
=0.4
\ i
i
i
1.0
i
i
J
1,5
P( z) Fig. 18. The excitation energy derivative X l , a , as a function o f n e u t r o n - t o - p r o t o n n u m b e r ratio p of fissioning nucleus for various fissionability parameters .v. The range corresponds to that of k n o w n
nuclei. 7. D i s c u s s i o n
Before considering the overall role of charge vibrations in fission it is useful to discuss the physical meanings of the quantities introduced in the preceding sections. The stiffness constants given in figs. 1 to 4 may be divided into those determined by bulk properties and those dependent on surface properties. The charge vibration stiffness KaA, the cross term stiffness Kau and the mass-asymmetry vibrations Kvv are the former type while all others fall in the second group. The bulk stiffnesses contain the shape terms f(ci) and #(ci) while the surface stiffness constants contain only the derivatives of these terms (compare equations (13)-(19)) and are generally much smaller, especially with respect to A~a which is about 103 greater. In spite of this the eigenvalues ).i are all of the same approximate size because the effective masses compensate for the disparities between the stiffness constants. This has an important effect on many of the observable fission properties because the compatibility of the individual normal modes makes their effect on each other much more pronounced. Especially crucial is the point where eigenvalues for charge and distortion asymmetry vibration, A' and d', are approximately equal, which happens in our calculations for x -~ 0.6 and p ~ 1.3 when the mode mixing constant K shown in fig. 9 reaches low negative values. Consequently, observables such as C and XI,,, shown in figs. 15, 19 and 20, the number of neutrons evaporated as a function of fragment mass and, to a much smaller degree, the most probable charge of the fragments show an increase which agree with the available experimental data 14, 1 s). Because of the approximate character of the two-spheroid model of fission and because of the simplifications introduced concerning the charge vibration effective
CHARGE VIBRATIONS
~
0.2
409
X=0.8
O. (0)
>
Xt,m(E s UNITS)
I
X=O.T
~....x.o.~ ,,.~..,,.--X~ 0..
0 i
X . 0 . 2X. i, ',O ~ 3 , i' 1.0
l
i
i
i
i 1.5
i
X=0.4 i
P(N,z) Fig. 19. The excitation energy derivative Xl.,,, as a function of O for various fissionability parameters x. The range corresponds to that of known nuclei. 0.2
Bi÷i
01
/t
I ___ 0.5
i
1.0
X Fig. 20. The excitation energy derivative Xl .... as a function of the fissionability paralneter x. The results of our calculations are given by the solid line and the results o f NS by the dashed line. The vertical lines show Xl.,,, derivative when charge vibrations are included. Their lengths correspond to the curves in fig. 19. The experimental points for Bi and Ra fission induced by protons are taken from refs. 14. ts).
mass M~ it is to be expected that the agreement in fig. 20 would not be quantitative. These approximations were justified as our primary goal was an investigation of the importance of charge vibration degree of freedom rather than development of a refined theory. It is obvious that in the region of heavy nuclei a situation must hold where 21 ~ 2a since )~l increases while 23 decreases with increasing x (see figs. 7 and 8). The exact location of the region where 21 ~ 23 is uncertain as the deformability of the fragments depends on shell effects; whenever the stiffness of one fragment increases it has basically the effect of perturbing and increasing the value of 23, which may
4t0
R. HOLUB AND G. R. CHOPPIN
shift the region of 2~ v 23 to heavier nuclei, or which may enlarge that region. Hopefully theoretical X 1.... values may be derived which would agree much better with the experimentally observed ones. [t is possible to extend our speculations. Myers 16, 17) has introduced an expression into the mass formula for neutron skin thickness for the surface region where the neutron density is several times greater than in the nuclear interior. It is easy to imagine that. for instance, those stiffnesses which depend on surface properties would be affected by introducing such a neutron skin term. This may change Kac , constant (fig. 3) into negative values (say hypothetically between - 0 . 5 to - 0 . 8 ) . Such a negative stiffness constant may be generally interpreted as causing instability in the direction of the particular coordinates which in this case would mean increasing elongation and neutron number leading to lower potential energy. Similarly, the constant /*AV, which is negative in most cases (fig. 2), indicates that if the mass split is m > 0 the neutrons have tendency to flow in. This agrees with the well-known experimental fact that the heavier fragment is more neutron-rich than the lighter one. I f the hypothetical large and negative constant K~c , is introduced into expression (22) for the constant K,, a very great change takes place; K,, decreases its value and at a certain point it may even become negative introducing asymmetry in fission without invoking any single-particle effects. Although there would also be important changes in the Xa,,,, quantities, this work cannot investigate these effects in any detail partly due to the lack of an adequate liquid-drop mass formula. However, unpublished results of Nair As) make quantitative conclusions which are very much the same. Nair's treatment is complete and includes all droplet model degrees of freedom (e.g., compressibility and charge density variations). With respect to the above speculations it should be added that our treatment also makes the K,,, constant lower by ~ 10 "~ tbr 0.6 > x > 0.8 (see fig. 10). This and the speculations in the preceding paragraph are related to the findings of Ryce et al. ~o) who point out that if one makes the hypothetical assumption that there is no asymmetry energy in the mass formula (or ~c = 0 in the MS mass formula), asymmetric fission would be favored in this region o f x . However, the asymmetry energy overrides this result (in our framework it is equivalent to saying that KAA is very large, and K,,, is lowered by only a small amount; the small value of the decrease is also due to the fact t h a t / ~ , , , is mostly positive and small). It should be noted, however, that Ryce et al. do not take into account the effect of deformations which our treatment does consider. One important point concerning the possible existence of a scission barrier should be made. The constant Ka~ is a bulk-type constant, i.e., it depends on composition rather than on shape and does not differ much for different shapes from the values given in fig. 1. The effective mass k l A does depend on shape, or more precisely on the magnitude of the constriction of the neck region in the fissioning nucleus 9) (fig. 6). As a restllt of these two observations, one might expect some variations in charge dispersion widths tbr different fissioning nuclei; however, this is in a complete dis-
CHARGE VIBRATIONS
411
agreement with experimental data. One way to overcome this discrepancy is to introduce a scission barrier 7), which would be basically the same for all fissioning nuclei. In conclusion a few minor points should be made. The J(1,a quantity, fig. 17, is usually positive which means that when A > 0, the excitation energy increases due to a lowering of the repulsive energy in the fragment which exceeds the decrease due to increased neutron excess. By contrast X1, v (unlike X1, ~) decreases because, other things constant, the deformation is being removed by adding mass to the deformed fragment at the saddle point. The two new normal modes are d ' and A'. [n the limiting case when K = 0, one gets properties of d- and A-modes as described in NS. The constants B' and C in expressions (30) give the total effect of all appropriate Kij and Mj on c°(m) and A (m) respectively• One can see that if K = 0, B' =
Y
_
Kuc,,
x';'2~4~
B = 2B',
Ka
which is equal to the B-value given by NS, and C -
X
Kau
\,I M A
KA a
These results are reasonable since the greater the coupling between U and c 1 degree of freedom (Kvc,) and the softer (smaller) the distortion-asymmetry stiffness constant K a are, the more affected the nucleus is. The more asymmetrically distorted it is with m > 0, the more neutrons are evaporated from the heavier fragment. The same holds for K~v/KAA, except in this case it concerns charge vibrations. Acknowledgment is extended to R. J. de Meijer and R. Eaker for assistance with some of the calculations. This research was supported by contract AT-(40-1)-1797 of the USAEC.
References 1) J. R. Nix and W. J. Swiatecki, Nucl. Phys. 71 (1965) 1 2) H. W. Schmitt, Proc. Syrup. on the physics and chemistry of fission, Vienna, 1969, paper SM 122/122 3) W• D. Myers and W. J. Swiatecki, University of California Lawrence National Laboratory report UCRL-11980 (1965) 4) A. Brandt and I. Kelson, Phys. Rev. 184 (1969) 1025 5) B. M. Spicer, Advances in Nuclear Physics, vol. 2 (Plenum Press, New York, 1969) 6) W. E. Updegraaff and D. S. Onley, Nucl. Phys. A161 (.1971) 191 7) W. N6renberg, Phys. Rev. C5 (1972) 2020 8) N. Danos, Nucl. Phys. 5 (1957) 23 9) W. J. Swiatecki and M. Blann, unpublished work 10) L. Wilets, Theories of nuclear fission (Clarendon Press, Oxford, 1964)
412
R. H.OLUB AND G. R. CH.OPPIN
l t) H. Margenau and G. Murphy, The mathematics of physicas and chemistry, 2nd ed. (Van Nostrand, New York, 1956) 12) H. Lamb, Hydrodynamics (Dover, New York, 1945) 13) H. Geldstein, Classical mechanics (Addison-Wesley, Mass., 1959) 14) E. Cheifetz, Z. Fraertkel, J. Galin, N. Lefort, J. Peter and X. Tarrago, Phys. Rev. C2 (1970) 256 15t E. Konecny and H. W. Schmitt, Phys. Rev. 172 (1968) 1213 16) W. D. Myers, Phys. Lett. 30B (1969) 45l 17) W. D. Myers, Proc. 1971 Mont Tremblant lnt. Summer School, ed. D. J. Rowe, L. E. H. Trainor, S. S. M. Wong and T. W. Donnelly (University of Toronto Press, 1972) p. 233 181 R. K. Nair, University of Florida, thesis, 1970 [9) S. A. Ryce, R. R. Wyman and A. T. Stewart, Can. J. Phys. 50 (1972) 2217