On the division of nuclear charge in fission

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On the division of nuclear charge in fission
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United States. Atomic Energy Commission. MDDC ;
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Clinton National Laboratory
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*^-V 1' ,/ftfA't' C-//".
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MDDC 1125


UNITED STATES ATOMIC ENERGY COMMISSION







ON THE DIVISION OF NUCLEAR CHARGE IN FISSION



by
R. D. Present


Clinton National Laboratory /







Date of Manuscript: January 13, 1947
Date Declassified: March 5, 1947



Reprinted by permission from
PHYSICAL REVIEW





Technical Information Division, Oak Ridge Directed Operations
Oak Ridge, Tennessee

































































































































































i/ii






VOLUME 72. NUMBER I


On the Division of Nuclear Charge in Fission*

R. D. PRESENT
University of Tennessee, Knoxville, Tennessee
(Received February 3, 1947)


The theory of the most probable charge number of a
mission fragment of given mass number is examined. Two
hypotheses previously suggested are (1) that the charges
divide in the same ratio as the masses and (2) that the
most probable charges correspond to minimum energy of
two droplets in contact. These hypotheses predict results
at variance with each other and with preliminary experi-
ments. In this paper the actual division of charge is
calculated for the final configuration of spheres in contact
on the basis of a general nuclear model in which the charge
distribution in the nucleus is non-uniform. The tendency


1. INTRODUCTION

THE division of nuclear charge in fission de-
pends on the extent to which the proton
and neutron densities and spatial distributions
are altered during the fission process.
Feenberg' and Wigner2 independently have
studied the effect of electrostatic repulsion of the
protons on the nuclear particle density for nor-
mal nuclei. The tendency of the protons to
spread outward is accompanied by a tendency
of the neutrons to follow the protons because of
the exclusion principle and well-known properties
of nuclear forces. Whether the neutron density
at the surface of the nucleus is greater or less than
at the center depends on the compressibility of
nuclear matter. Wigner has made the "liquid-
drop" assumption of incompressibility or uni-
form total density, and in the case of uranium
his method gives the result that the proton-
neutron ratio at the surface exceeds the same
ratio at the center by 36 percent. Feenberg ob-
tains a lower result (21 percent for uranium) by
a more elaborate calculation in which the nuclear
compressibility is taken explicitly into account.
While in the former method the proton density
is 21 percent larger at the surface than at the

This document is based on work performed under Con-
tract No. W-35-058-eng-71 for the Manhattan Project, and
the information covered therein will appear in Division IV
of the MPTS as part of the contribution of the Clinton
Laboratories. References 8 and 11 are to Plutonium Project
Reports.
SE. Feenberg, Phys. Rev. 59, 593 (1941).
E. Wigner, University of Pennsylvania Bicentennial
Conference (1941).


of the protons to spread outward results in the smaller
Iragment of an asymmetric fission having a greater proton-
neutron ratio than the larger fragment. In the most
probable division mass ratio 2:3) the most probable
partners (of odd mass number) should both have chain
lengths equal to 3.6. In the case of a 1:2 mass ratio, the
probable chain lengths of the light and heavy partners
should be 2.5 and 4.1, respectively. Data on the inde-
pendent fractional yields of particular chain members are
found to lie near a smooth (error) curve when plotted with
the aid of the theoretical results.



center, the latter method gives a 49 percent in-
crease in proton density accompanied by a 23
percent increase in neutron density in going to
the surface. The associated corrections to the
binding energy are small, and the corrections to
the nuclear radius are negligible. So far there has
appeared to be no way in which to verify the
existence of these effects. It is suggested here that
there must be a small but possibly observable
effect on the nuclear charge distribution of the
primary fission fragments and the most probable
number of #-decays of a fragment of given mass
number.
If the proton density were uniform throughout
the nucleus, the charges of the fragments would
be in the same ratio as the masses. However, if
the proton density is greater at the surface, the
smaller fragment of an asymmetric fission must
have a greater proton-neutron ratio than the
larger fragment. This effect is enhanced in the
Wigner model by the decrease in neutron density
from center to surface. On the Feenberg model
where the neutron density varies in the opposite
fashion, the effect is a differential one caused by
the proton density varying more steeply than the
neutron density. Since the effect depends on the
relative shift of the proton spatial distribution
with respect to the neutron distribution, the
method of Wigner will give somewhat larger
results.
A rough estimate of the size of the effect can
be made as follows: The critical shape of a divid-
ing uranium nucleus, corresponding to the saddle


PHYSICAL REVIEW


JULY 1. 1917





R. D. PRESENT


KI)


FIG. 1. Diagram of the coordinate system employed.

point of the potential energy versus deformation
parameter hypersurface, is not very different
from a prolate spheroid of eccentricity V3/2 with
a slight flattening at the equator preliminary to
"pinching-in."' We assume that the proton and
neutron densities vary only with the distance
from the center, and use a parabolic fit to the
radial dependence of the density computed by
Feenberg, the total charge being normalized to
that of the undeformed nucleus. The spheroid is
sliced into two unequal parts, and the charge on
each fragment computed. The calculation is an
elementary one and, in view of the more precise
calculation to be described in the next section,
no details will be given here. The results can be
expressed in terms of the charge numbers, Z1
and Z2, of the light and heavy fragments and
their mass numbers, A, and A,, before neu-
tron emission. Setting 7 equal to [Z,/(Zi+Za)]/
[A /(AI+A2s)], one finds that for a 2:3 ratio of
the fragment masses, 7 is 1.010, and for a 1:2
division, y is 1.020. Somewhat larger values are
obtained if Wigner's densities are used, y being
1.014 and 1.023 in the two cases, respectively.
These results cannot be very accurate in view of
(1) our use of the density variation appropriate
to the sphere for the critical spheroid and (2) the
further elongation of the drop beyond the critical
shape accompanied by adurther spreading of
charge. U
I R. D. Present, F. Reines. and J. K. Knipp, Phys. RbI.
70, 557 (1946).


2. ASSUMPTIONS AND FORMULATION
In this section we calculate the charge distri-
bution in the deformed nucleus just before scis-
sion into two unequal fragments. This configura-
tion is approximated by two spheres in contact,
and the proton and neutron densities aie taken
to be functions of the distance from the point of
contact and of the angle with the axis of sym-
mnetry. The nuclear model is that already de-
scribed by Wigner.2 We outline the principal
assumptions of the calculation:
(A) The charge redistributes itself rapidly dur-
ing the deformation time for fission so that the
charge density has its equilibrium value for every
shape of the dividing drop. This follows if the
redistribution of the charge is accomplished via
the meson field and requires no redistribution of
the heavy particles nucleonss).
(B) The total particle density is assumed to be
uniform throughout the volume of the deformed
nucleus. This is the liquid-drop assumption of
incompressibility of nuclear matter.
(C) The decrease of electrostatic energy when
the charge goes to the surface is counterpoised
by an increase in the isotopicc spin energy." The
isotopic spin energy arises from the exclusion
principle and the character of nuclear forces and
is represented in the mass defect formula by the
term or terms in N-Z. (The isotopic spin
Tr=(N-Z)/2.) The mass defect formula pro-
posed by Weizsacker and employed by Bethe
and Bacher, Bohr and Wheeler, and others, con-
tains a term in (N-Z)2. Wigner' has shown that
the potential energy of the nucleus computed
from the symmetrical Hamiltonian contains a
term in (N-Z)/A and has included both linear
and quadratic terms in N-Z in his mass defect
formula. The presence of the linear term con-
siderably reduces-the magnitude of the quadratic
term. Wigner's mass defect formula is used in
the following.
(D) It is assumed that the isotopic spin energy
is a "volume energy," i.e., that it can be ex-
pressed as an energy density, depending on the
local difference in proton and neutron density,
integrated over the volume of the nucleus. The
numerical coefficient of this energy density is
obtained from the empirical coefficients of the
SE. Wigner, Phys. Rev. 51, 947 (1937).






DIVISION OF NUCLEAR CHARGE


(N-Z) and (N-Z)' terms in the mass defect
formula. Since these coefficients are approxi-
mately independent of nuclear size, it is assumed
that the same numerical coefficient can be used
for the configuration of spheres in contact as for
normal separated nuclei."
Figure 1 shows the coordinate system used.
Spherical polar coordinates with origin at the
point of tangency and polar axis along the axis of
symmetry are denoted by s, 4, and p; the same
coordinates with origin taken at the center of
one of the spheres are denoted by r, 0, and p.
R1 and R, are the radii of the two spheres in con-
tact and R the radius of the original nucleus. The
ratio R/IR is denoted by Xi; hence l X3+X?= 1.
Let pz(s, P) be the proton density, p,v(s, 4,) the
neutron density, and p the uniform total particle
density. Z, N, and 4 represent as usual the total
numbers of protons, neutrons and nucleons; and
Z1, Ni, and A\ refer to the numbers in sphere i.
Since the effects to be calculated depend on the
difference in proton and neutron densities, we


introduce the "local isotopic spin" l(s, P) defined
by t=(pv-pz),'2p. The average values pz, P.v,
and i are the values appropriate to a uniform
charge distribution. We have

T = Ai = pff fidr = pfff si )dr
1+2
+Pf f t(Ss, 4)dr., (1)
whence
I ff )d=O0. (2)
i+2
It is necessary to evaluate the change in the
electrostatic self-energy of each sphere and in
the mutual electrostatic energy of the two in
going from uniform to non-uniform charge den-
sity. Since the deviation from uniform density is
small, second-order terms in e=pz-Pz=p(i-t)
will be neglected. Then the change in the self-
energy of each sphere is given by


AE.e=- f1 1pz rd)pz(rb)dr i= 1, 2

-=- f I ed.,+ -errrrr (r dr .dTr
2fffff 3 r,) ra d JJJJ JJ r

ffffff fff
= e9J --dradr, = e e(rJ ) V o(r.)dr,.

where Vo(ra) is the potential of a uniform distribution. Hence,
Z,'"'e'p F 3 1 r,-
AE,*I.- ( ---- (-t,)r,. (3)
R, 2 2 R," ?
Similarly, the change in the mutual electrostatic energy is

SE, ff=fd drT dr2 r pz(r)pz(rs2)- zl'i

r 2)
= I2 d r r |pze(rI)+ Azt(re)
=eJ dr, drI- --

=ef f fdre(.r)i''(rf)+e drTer)\'1'1(r)

S= I (i-lt)dr f (d1 r.-t.)dr
_Z__ -pf -_Z-_z,0 (4)
*A further assumption is that the kinetic energy correction terms arising from the gradients of proton and neutron
densities (Weizsacker terms) can be neglected. Feenberg has included these terms explicitly. However, the effect





R. D. PRESENT


where Vl'2'(ri) is the potential at ri because of
a uniform distribution over sphere 2, and u,
=(sil+R22+2Rasicosjl) is the distance to r,
from the center of sphere 2.
We next consider the evaluation of the isotopic
spin energy from the mass defect formula. The
latter involves the partition quantum numbers
(PP'P") of the ground state, where P is equal
to the isotopic spin Tr, and P' and P" are of the
order of unity.2*4 Neglecting7 the terms in P'
and P", the mass defect formula is found to con-
tain two terms depending on the isotopic spin of
the form: ci(N-Z)/A +c2(N-Z)P/A. These cor-
respond to an energy density gr given by
cGp (N-Z c2p (N-Z)2
ST=--- -- +------
A A / A .4A
cIP PN-PZ PN pZ
= -----+c-p.--
A p p
= (2cp/A)t+4c2pP. (5)
The relation (5) is now assumed to hold when
gT and I are functions of position in the deformed
nucleus. The difference in isotopic spin energy
between the non-uniform and uniform charge
distributions is then given by

AEr= fffr -fffgd
1+2 1+2

= (2cipA) f (tf-)dr
I+?


1+2

=4cpf f fl( -)dr (6)
1t2
in view of Eq. (2). The term c1(N-Z)/A makes
no contribution to the energy difference. The
indirect effect of this term is very considerable,
of these terms is partly taken into account when the isotopic
spin energy is taken from the mass defect formula with its
empirically determined coefficient. One can regard the
Weizsacker terms as expanded in powers of (N-Z)/A and
thus partly included in the other terms when empirical
coefficients are used.
E. Feenberg and E. Wigner, "Reports on press in
physics," Phys. Soc. London 8, 274 (1941).
'One neglects P', P', and P"2 compared to P. In the
case of an even-even nucleus P'=P"=0 in the groui
state. %


however, since its omission from the mass-defect
formula leads to a larger value for c2. In the
WeizsAcker-Bethe formula which contains no
term in (N-Z)/A, the coefficient c2 is approxi-
mately 20 Mlev. The value of c, in Wigner's
formula is about 12.5 Mev.
The spatial variation of t throughout the two
spheres is determined by minimizing the energy
difference AE between the non-uniform and uni-
form charge distributions where
AE=AEr+AEc
=AEr+AEIc+AE2c+AEiCe. (7)
We develop t(s, 4) in a power series in s and cost
and, because of the smallness of the effect, use
only the first terms. Assuming continuity of the
densities and their derivatives at the point of
contact of the spheres, one omits linear terms.
Hence
,(si, ,) = 1+-a 6 (s 'R) +t,(s, 'R) cos#,
i= 1,2. (8)
The parameter B is expressed in terms of ae, a2,
03, and 2i by means of Eq. (2), and the latter are
determined from the minimizing conditions
OAE aAE OAE OAE
=_--=-= =0. (9)
Oa\, Oce 9fi l1
a a aa2 aBI a02
The charge number of sphere 1 is altered from
Z i =ZX1 for uniform density to Zi when the
density is not uniform. Thus

Z,-ZX'= fff (z- iz)d


=pfff (i-)dr, (10)

gives the distribution of charge between the frag-
ments. Equations (1) through (10) suffice for the
calculation of I = [Z/(Z, +Z,) ]/[A ,/(A +A )]
for any ratio of fragment sizes (provided that a,
and 0, are not too large).
3. PRELIMINARY ESTIMATE
A preliminary estimate of the size of the effects
can be simply obtained by replacing Eqs. (8)
and (9) by
OAE
t,=+ ,, -- =0, i=1,2, (8'-9')
a6,






DIVISION OF NUCLEAR CHARGE


corresponding to uniform but different charge
densities for the two spheres. Equations (2), (3),
(4), and (6) reduce to


6 5A' + 6522' = U.
AE.'= -'8/5)rR? Zep-3.5,\,".
AEI.e = (4. 3)rR. Zep- (6, +52)
[,X i3 (Xi + X2)].
AEr = 4c2p (4/3),rRa" (5,2X, + 62? XQ).


In the case of the uranium and plutonium iso-
topes the quantity Ze'p*4rR'*A-' is slightly less
than 50 mMU (milli-mass units). The coefficient
4c2 is slightly greater than 50 mMU. In the
following we take both quantities equal to 5(1
rn MU. Then,


AE 51X2a+8522X23 aIXb+i 52X2
100A 2 5
61+62 X1.2
(15)
6 XI+X2


(12) which is to be minimized subject to (11). l'he
charges are then computed from
Z(3 Z=X 1 6X13A, (16)
giving the following results:


X:AX2 = 2:3
=1:2
=1:3


= 1.0194
=1.0360
=1.0607


For the 1:2 division of uranium AE=-1.16
mMUNI. This simple calculation gives values of 7'
slightly larger than those obtained from the more
accurate calculation described below.


4. DETAILED ESTIMATE
The detailed estimate of the effect is based on the use of (8) to represent the departure from
uniform charge distribution. On substituting (8) into (2), (3), (4), and (6) we find
6 = (3 4rRb) ItcLI+a.L.L:) dlA ,+3..A .)i, (18)


E,' = (Ze p, 2R"') ; (16 '5)ra6h,SR+ 3a,LX, .- a,VNR'-1 3A ,X,M +,D,R-!',
AE12s*= Ze2pR- I (8r6. 3). [X-E2 (XI+ + X (a1+ +a, )R-'- (aXX +P2 +a IXPR ( Q+i3,Q,)R-I),
AEr= -4c2p6R-2.1 (alL I+aL2) ( l 11+3;A 2)


(19)
(20)


+4c2pR-. I* (a2M,1+a2.A11MA)-2(atL3,B,+a~s2B2)+(13,C,+3222C2) (21)


L,=f f sdr= 2r(16 15)R,h,

M,= f f f s,4d,=2rI16'7)R7',

N,-= rf frI si'dTr= 2r(24/35)R/7,

A ,= fff, cos,,drT= 2r(32,35)Rb,

B,= fffs, cos4,dr=2r(128/63)R7,

C,= f ffscos#l,dr ,=2r(64/35)R,7,

D,= fffr, 2s2cos driT=2r(544 '3.5-7)RL',

P = fff( S/)dT2 = 2rR'K(X), Pl= 2rR2*K(X-),

Q= fff(22,' 2) cos2drT2= 2rRiW(A), Q = 2rR 24(.-l1).


(22)


where





R. D. PRESENT


TABLE 1. Calculated numerical values.

A.,':A? al aI i I AE/IOOA
2:3 0.0364 0.0187 0.0254 0.0158 -.000127
1:2 0.0465 0.0153 0.0244 0.0290 -.000145
1:3 0.0658 0.0122 0.0225 0.0493 -.000169


The integrals K(X) and 1I'(X) with X = X,'\A are given by
_2pA f' dx
K(X) = dyy' 2v+v
,.-x (2y.v+y'+ )1I
32 2 X +2\3(6+6X-)
=4\-'+-X-- -- +1 (23)
5 15( +X\)? X / X
f21*3 I .vxd.v
It"() =f d 2vv '- + ) 1d
n ,;2 (2yx+y"+1)*
32 8 2(2 -A)n' 1+(5;, 2)
= --X- X---f-3+ + 2nnml-nm -log (n+m) (24)
9 9 9(l+AX)X 48(1+ A)!'2

where m= (+2),/' and n =2X-I'(l + .)1. The evaluated integrals are substituted into (18), (19), (20),
and (21) to give

6= (8,5)(aiXi'+a2Xs) (48, 35)(jiXN 5+3 #X5), (25)

AE, IOOA = S/'2 (,5)(X1.5+X2) (6, 3)(X132'/X +X) +.1 2 -7)(X1'ai2+ j'a2)

+ (11 '35)(X'ai +X7a.) + (I ,'4)(K(X)XI'a.+K(X-')Xsc,)
-- 64 .,'21)(AhlatJ +7n :o3,)+(48,/35)(X I4l, + X23?2) 1. 256,'3. 5.7)5(XS + X?)

(1,4)(l W') + 1 (t'X-')XW ). (26)

Numerical values are inserted in (25) and (26) and the minimizing conditions 1.9j give four simul-
taneous equations for ai, ao, #3, and 03. Since the determinants are close to vanishing, greater accuracy
is obtained by solving these equations by the method of elimination. When Xi3:123 = 1:2 the minimum
energy occurs at: ai=0.184, az=0.099, 1 =0.140, 1B=0.085, 6=0.0377, and AE= -4.7 mMU for
uranium. The signs of the a's and #'s correspond to a charge density increasing with distance s from
the point of contact and with increasing angle 4 with the symmetry axis. The charges on the frag-
ments are obtained from (10) and (22) which give

Z -ZX1A= p" .(4 3)irR/jlpLial/R'- pA g0l 'R2

= (4 3)rRpR 'p -IX1,6+(8. 5) Oal (48,35)X5si1. (27)

When Xi = ? we find from the numbers given above that y= 1.029. This is to be compared with the
value of 1.036 of the preliminary estimate. If the angular variation of charge density is neglected
(i.e., if 1 =3 = 0), the results are changed to: a,= 0.0466, a = 0.0153, 6 =0.0244, AE= -3.4 mMU,
and y = 1.029. As would be expected, the angular variation has a negligible effect on the division of
charge but leads to an appreciable lowering of the energy. For our present purposes it is sufficiently
accurate to omit all terms in O3 and 3a from the preceding formulae. Figure 2 is a plot of (7- 1) vs.
AX obtained in this way, and Table I gives sonr of the numerical results.






DIVISION OF NUCLEAR CHARGE


DISCUSSION OF RESULTS
The theoretical results are represented ini Fig 2
which enables one to calculate the division of
nuclear charge between fragments of given sizes.
For the range of sizes shown, the higher order
terms in a, and at are negligible, as has previ-
ously been assumed.
The principal sources of uncertainty in these
results are (1) the assumption of uniform total
particle density and (2' the use of Wigner's mass
defect formula with the large term in (,V-Z) '.4.
In both respects our calculation gives an upper
limit for (7-1). Since no definite knowledge of
the compressibility of nuclear matter is at hand,
the first assumption is reasonable. However,
Feenberg's method, in which a roughly estimated
value of the compressibility is used, would give
smaller results probably closer to the truth.
No definite confirmation of Wigner's term in
(N-Z)/A for heavy nuclei is yet at hand. The
size of this term relative to the conventional term
in (N-Z)2/A is a consequence of the symmetrical
Hamiltonian. If this term should prove to be
small or absent (as in the Weizsacker-Bethe
formula) the value of (7 -1) would be consider-
ably reduced. It is possible but unlikely that the
calculated values of (7-1) are too large by as
much as 50 percent.
Figure 2 enables one to predict the most prob-
able initial charge of a fission fragment of given
mass. We distinguish the primary fission frag-
ment nucleus of mass number A, from the fission
product nucleus of mass number A,' where
A,'=Ai-1. Experimental measurements deal
with the product nucleus formed from the'frag-
ment nucleus by neutron emission. Since the
fraction of fragment nuclei emitting more than
one neutron apiece is small, this possibility will
be neglected. Figure 3 represents the charge
number Zd of the fragment or of the product
nucleus plotted against the mass number A,' of
the product nucleus. The charge numbers Z,"1
following from the hypothesis of uniform charge
distribution (7= 1) are shown for comparison. In
a very asymmetric division the Z, and Z,'r'
curves differ by as much as one charge number,
the smaller fragment having a higher, and the
larger a lower, charge than would be expected if
the charges divided in the same ratio as the


masses. Still greater differences would be ob-
tained for divisions more asymmetric than those
shown in Fig. 3. The dots in Fig. 3 represent
stable isotopes of odd mass number. A fairly well
defined curve can be drawn through these points;
such a curve would be nearly identical with a plot
of ZA vs. A in the notation of Bohr and Wheeler.
The periodicities of the dots or of the Za curve
are of course not reflected in the theoretical
curve for Z, (the finer details of nuclear binding
represented by the terms in the Hamiltonian
which depend on the partition quantum numbers
P' and P" have been neglected). Because of this
and also because the theoretical curve gives frac-
tional rather than integral charge numbers, no
exact predictions for particular isotopes can be
made. Nevertheless, some predictions are pos-
sible concerning the average behavior of the
number v of 3-decays of a fragment of given
mass and most probable charge for that mass.
Fragments of the same mass but different charge
are formed independently in the fission process,
and their yield as primary fission products can be
measured in a few instances. It is thus possible to
estimate v for several neighboring odd masses and






Ob

0 .



0'





01or


Fin. 2. Calculated curve-of (y-j1 vJ,. kil.








R. D. PRESENT


6'






40..
M5 *.










30 MM- UNlF DE / l lI
.* 4










o..... uNlrg~u #,
a 0 I
Y s 9. .




MA~ O PRO D U NIFGLM US

0ro BO 90 '00 '0 2?0 .30 I '0 30 '60

FIG. 3. Charge vs. mass of the product nuclei.

compare the result with Fig. 3 where P is the
mean vertical distance from the Zi curve to the
dots. The quantity u will be referred to as the
"probable chain length" for a given mass ratio
of the fragments. The following conclusions can
then be drawn from the theoretical curve for Z,:
(1) For that mass ratio which occurs with great-
est frequency in slow neutron uranium fission
(about 2:3 or 93:141) the probable chain
lengths of the complementary fragments are
both equal to 3.6.
(2) In the case of a 1:2 or 78:156 division, the
probable chain lengths for the light and
heavy partners are 2.5 and 4.1, respectively.
(3) In the case of a 1:3 or 58:176 division (not
shown in Fig. 3), the values of v would be 2.4
and 3.5 for the light and heavy partners.
Decay chains for such an extremely asym-
metric split have not been observed but
might be found in very fast neutron fission.

The foregoing conclusions refer to odd may
numbers A/. The stable isobars of even mass
number have not been included in Fig. 3 because
the dots would scatter so widely that the prob-


able chain length a would undergo large fluctua-
tions from one mass number to the next. The
probable chain lengths for the even mass num-
bers are on the average appreciably smaller than
for the odd mass numbers, since the even mass
dots lie mostly below a curve through the odd
mass dots. Data on even masses will be utilized
below.
The energy correction AE caused by spreading
of the charge is small for all observed modes of
division (cf. Table I), and the difference between
the values of AE for symmetric and the most
probable (2:3) asymmetric fission is of the order
of one Mev. Hence the considerations of this
paper make no appreciable change in fission
calculations involving the energy of two spheres
in contact or a deformed ellipsoidal nucleus.
Wigner and Ways have suggested that the
charge might divide in such a way that the total
decay energy of the two chains would be a mini-
mum. Extending this idea to take into account
the electrostatic repulsion of the fragment nuclei,
Way' has calculated the division of charge by
assuming that the most probable division is that
in which the energy of the two product nuclei in
contact is a minimum. While this assumption is a
reasonable one, it can hardly be expected to be
accurate, particularly in view of the fact that the
most probable mass ratio of the fragments ob-
served in the division of various fissionable nuclei
does not in general correspond to minimum
energy of the product nuclei in contact.9'10 The
results obtained by Way, after minimizing the
total decay plus electrostatic energy using the
Bohr-Wheeler mass defect formula, are nearly
the same as ours for the 1:2 mode but are sig-
nificantly different for the 2:3 mode (the prob-
able chain length for 93 is 3.1 compared to our
value of 3.6).
Some confirmation of these results is provided
by the chemical investigation of fission product
decay chains. It appears from the work of Glen-
denin, Coryell, and others" that the known facts
about decay chains, with mass numbers corre-
SE. Wigner and K. Way, Report CC-3032 (1945);
Plutonium Project Reports 9B, 6.4 (1946).
D K. Way, private communication.
S. Fliigge and G. v. Droste, Zeits f. physik. Chemie
[B], 42, 274 (1939).
u Sumiflrized in Glendenin. Coryell, Edwards, and
Feldman, CL-LEG-No. 1 (1946). Most of the data have
been published in the J. Am. Chem. Soc. 68, 2411 (1946).








DIVISION OF NUCLEAR CHARGE


spending to the most likely modes of fission, are
consistent with the result of equal probable chain
lengths (3.6) for the complementary fragments.
Insufficient data are available, however, to draw
conclusions about the chain lengths in a 1:2 divi-
sion. In the case of a few mass numbers, it has
been found possible to measure the independent
yield of a particular member of the decay chain,
formed directly from the fission process. This is
possible when the preceding member of the chain
has a long half-life or is stable (belonging to a
pair of stable isobars). Assume that the total
yield of the chain is known. Then the fraction of
the chain of a given mass formed with a particu-
lar charge is determined in a few instances. If
these fractions are plotted against Z-Zp, where
Zp is the most probable charge for a fission frag-
ment of the given mass, it is reasonable to expect
the points to lie near a smooth (error) curve sym-
metrical about Z = Zp. Such a plot has been made
by Glendenin, Coryell, Edwards, and Feldman;"
they find that if Zp is determined either from
unchanged charge distribution (ZC.)), or from
minimum energy of nuclei in contact (Way's


method), the resulting points show systematic
deviations from a symmetrical error curve, the
lighter mass points (Br" and Rb"), and the
heavier mass points (Xe1" and Cs'a) appearing
to lie on separate curves displaced by about one
unit of charge in both cases. On the other hand,
if the values of Zp are obtained by an empirical
postulate of equal probable chain lengths for
complementary fragments, all points appear to
lie near a smooth curve. We have replotted their
data taking Zp equal to the theoretical value Z,
from Fig. 3. The resulting points lie reasonably
close to a single error curve with some scatter but
no systematic deviation, i.e., there is no evidence
of separate curves for the lighter and heavier
fission products. Some scatter is to be expected in
view of the neglect of finer details of nuclear bind-
ing in the theoretical treatment; however, all the
points lie within one-fifth of a charge unit of the
best drawn error curve symmetrical about Z =Zi.
This can be considered as a preliminary check on
the theory; evidently more experimental points,
corresponding to a greater range of mass numbers,
are needed to provide definitive confirmation.




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