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 MDDC - 1125 
 
 I 
 
 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 
 
PHYSICAL REVIEW 
 
 VOLUME 72, NUMBER 1 
 
 JULY 1 , 1947 
 
 On the Division of Nuclear Charge in Fission* 
 
 R. D. Present 
 
 University of Tennessee, Knoxville, Tennessee 
 
 (Received February 3, 1947) 
 
 The theory of ihe most probable charge number of a 
 lission 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 
 
 of the protons to spread outward results in the smaller 
 fragment 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. 
 
 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 Wigner- independently have 
 studied the effect of elect'-ostatic 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 lotal 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. VV-3S-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. 
 
 ' E. Feenberg, Phys. Rev. 59, 593 (1941). 
 
 ■ E. Wigner, University of Pennsylvania Bicentennial 
 Conference (1941). 
 
 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 /3-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 
 
R. D. PRESENT 
 
 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, Z\ 
 and Zi, of the light and heavy fragments and 
 their mass numbers, Ai and Ai, before neu- 
 tron emission. Setting y equal to [_Zi/{Zi+Zi)2/ 
 \^A\/{Ai+A2)2, one finds that for a 2:3 ratio of 
 the fragment masses, y 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 a^further spreading of 
 charge. 
 
 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- 
 metry. The nuclear model is that already de- 
 scribed by Wigner.^ 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 (nucleons). 
 
 (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 "isotopic 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 
 T{ = {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 — Zy. 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 
 
 » R. D. Present, F. Reines, and J. K. Knipp, Phys. Rev. 
 70, 557 (1946). 
 
 * E. VVigner, Phys. Rev. 51, 947 (1937). 
 
DIVISION OF NUCLEAR CHARGE 
 
 3 
 
 (N — Z) and (N—Zy 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, \p, and <p; the same 
 coordinates with origin taken at the center of 
 one of the spheres are denoted by r, d, and ip. 
 Ri and i?2 are the radii of the two spheres in con- 
 tact and R the radius of the original nucleus. The 
 ratio Ri/R is denoted by Xi; hence Xi'+X2^ = l. 
 Let pz{s, ^) be the proton density, pn{s, ^) the 
 neutron density, and p the uniform total particle 
 density. Z, N, and A represent as usual the total 
 numbers of protons, neutrons and nucleons; and 
 Zi, Ni, and Ai refer to the numbers in sphere 1. 
 Since the effects to be calculated depend on the 
 diflFerence in proton and neutron densities, we 
 
 introduce the "local isotopic spin" t{s, ^) defined 
 by / = (p.v — pz)/2p. The average values pz, p.v, 
 and i are the values appropriate to a uniform 
 charge distribution. We have 
 
 Ti = Ai = pJ j JldT = pj J Jtisui'Odu 
 
 1+2 
 
 + pj j Ji{s,,i2)dT2, (1) 
 jjj(t-i)dr = 0. (2) 
 
 whence 
 
 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 f = pz — pz = pii — 
 will be neglected. Then the change in the self- 
 energy of each sphere is given by 
 
 --7//////I 
 
 -////// 
 
 Pz(r 
 
 .)pz(r6)- 
 
 J'ab 
 
 -pz' 
 
 dTadn i = l,2 
 
 pze(r 
 
 —dTadri, 
 
 -i//////^ 
 
 pze(r 
 
 —dTadrb 
 
 = ef f fe{ia)Voir.)dT., 
 
 rab 
 
 where FoCfa) is the potential of a uniform distribution. Hence, 
 
 ^Ei'=- 
 
 ///I 
 
 ■{t-ti)dTi. 
 
 Ri J J J \2 2R: 
 Similarly, the change in the mutual electrostatic energy is 
 
 Pz(r0pz(r2) — pz- 
 
 (3) 
 
 
 -/// 
 
 = Z2<'»e=p 
 
 rfr:€(r:)Fo(°'(r 
 
 Ct-ti)dri 
 
 ■'-/// 
 
 dT,({U)\\^'»{T2) 
 
 an 
 
 -Z/o'e^ 
 
 ■/// 
 
 {i—l->)dTi 
 
 Ui 
 
 (4) 
 
 ' A further assumption is that the kinetic energy correction terms arising from the gradients of proton and neutron 
 densities (VVeizsacker terms) can be neglected. Feenberg has included these terms explicitly. However, the effect 
 
R . D . PRESENT 
 
 where Vt^^^iii) is the potential at Ti because of 
 a uniform distribution over sphere 2, and mi 
 = (5i^+i?2^ + 2i?25iCOs^i)* is the distance to ti 
 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 Tf, and P' and P" are of the 
 order of unity.^'*'* Neglecting' the terms in P' 
 and P", the mass defect formula is found to con- 
 tain two terms depending on the isotopic spin of 
 the lorm:ci{N-Z)/A+C2{N-ZY/A. These cor- 
 respond to an energy density gr given by 
 
 gT- 
 
 Cip 
 
 ' A 
 
 N-Z\ Cip (N-zy 
 
 (^) 
 
 +- 
 
 ClP PN — pz 
 'a' p 
 
 -C2P 
 
 (^)' 
 
 = {2cip/A)t+^C2pt\ (5) 
 
 The relation (5) is now assumed to hold when 
 gT and t 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 
 
 ASr= I I {grdj- I 1 \ grdr 
 
 = {2c,p/A) 
 
 ///<'- 
 
 i)dT 
 
 1+2 
 
 = ic,pj j (t{t-i)di 
 
 (6) 
 
 in view of Eq. (2). The term Ci{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 progress in 
 physics," Phys. Soc. London 8, 274 (1941). 
 
 ' One neglects P', P'\ and P"^ compared to i^. In the 
 case of an even-even nucleus P' = P" = in the ground 
 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 c^ is approxi- 
 mately 20 Mev. The value of Ci 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 A£ between the non-uniform and uni- 
 form charge distributions where 
 
 AE=AEt+AEc 
 
 = A£7-+A£i<^+A£2<^+AEi2<^. (7) 
 
 We develop t{s, ^) in a power series in 5 and cosi/- 
 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 
 
 i = l,2. (8) 
 
 The parameter S is expressed in terms of oci, 0:2, 
 01, and 02 by means of Eq. (2), and the latter are 
 determined from the minimizing conditions 
 
 dAE dAE dAE dAE 
 
 da\ dui d0i 302 
 
 = 0. 
 
 (9) 
 
 The charge number of sphere 1 is altered from 
 Zi""=ZXi' for uniform density to Zi when the 
 density is not uniform. Thus 
 
 Zi-ZX, 
 
 '=/// 
 
 (pz — pz)dT 
 
 "in 
 
 {i-l,)dr, (10) 
 
 gives the distribution of charge between the frag- 
 ments. Equations (1) through (10) suffice for the 
 calculation of 7 = [^i/(2i+Z2)]/[^ 1/(^1+^2)] 
 for any ratio of fragment sizes (provided that a, 
 and 0i 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) bv 
 
 dAE 
 
 t. = i + 8t, = 0, 1 = 1,2, (8'-9') 
 
 d8i 
 
DIVISION OF NUCLEAR CHARGE 
 
 corresponding to uniform but different charge AE di^Xi^ + Si'Xi^ 5iXi' + 52X2' 
 densities for the two spheres. Equations (2), (3) 
 (4), and (6) reduce to 
 
 iOOA 2 5 
 
 Si + Si WXi' 
 
 5xU+s,\,' = u, (11) 7 —rr (15) 
 
 ' 6 \l+A2 
 
 AE.'= -(8/5)7r/?-Ze-p<5,XA (12) ^^.^ieh is to be nuniniized subject to (11). Hie 
 
 A£i./= - [A/i)irK'-Ze-p-{6, + 8-2) charges are then computed from 
 
 Zi=Z\i^-8i\M, (16) 
 
 giving the following results : 
 
 X,':X2' = 2:3 7 = 1-0194 
 
 In the case of the uranium and plutonium iso- =1:2 =1.0360 (17) 
 
 •[Xi'X2V(Xi+X2)], (13) 
 A£r = 4c2p-(4/3)iri?'-(6i2Xi3 + 62'X23). (14) 
 
 lopes the quantity Ze'^p-4TrR--A~^ is slightly less 
 
 1:3 =1.0607 
 
 than 50 mMU (milli-mass units). The coefficient For the 1:2 division of uranium A£=— 1.16 
 
 4c2 is slightly greater than 50 niMU. In the mMU. This simple calculation gives values of 7 
 
 following we take both quantities equal to 50 slightly larger than those obtained from the more 
 
 MiMU. Then, 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 
 
 d = (3/4r/?^) • ( (aiLi + a.L,) - (^,A,+p,A,) ] , (18) 
 
 A£.'^ = (ZeV2i?^)- I -(16/5)ir5XiS;?6_^3a;L.Xr-aiA^./?-=-3/3,yl.X.2+/3.Z>,i?-''}, (19) 
 
 AEu' = Ze'-pR-- 1 -(87r6/3)-CXi^X2V(Xi + X2)] + (aoXi'P2 + aiX2'Pi)i?-^-(/32Xi'(32+/3iX2'(3i)i?-M, (20) 
 
 AEt = -iCipSR-' ■ I (aiLi + a2L2) - {Hi 1 1+/32^2) j 
 
 + 4c2p7^-^-!(ai2A/i + a2W2)-2(ai^,-B, + a2^252) + (/3i^C,+^2^C2)!, (21) 
 
 where 
 
 Li = J j j^i'dr, = 27r(16/15)i^l^ 
 Ml = fj Jsi'dn = 2T(16/7)i?i', 
 Ni= f f ^rl^5,2cfrl = 27^(24/35)i?l^ 
 
 Ai= f f Csi^ cosiidTi = 2Tr(32/35}Ri\ (22) 
 
 5i= r r r5,H-osr/'i(;?ri = 27r(128/63)2?i', 
 Ci= f f r5,^cosVidri = 2ir(64/35)/?i', 
 Di= f f fn'^si' cosiAi(iri = 27r(544/3' • 5 • 7)i?,', 
 P2= f f C {si'/ui)dT2 = 2irRi*K(\), Pi = 2TR;i*K{\-'), 
 
 Q^= f C fis2'/u2)coshdri = 2TRi*W{\), Qi = 2TrR2'W{\-'). 
 
6 
 
 
 R. D. PRESENT 
 Table I. Calculated numerical values. 
 
 
 
 X,>:Xi' 
 
 ai 
 
 at i 
 
 T-l 
 
 AE/IOOA 
 
 2:3 
 1:2 
 1:3 
 
 0.0364 
 0.0465 
 0.0658 
 
 0.0187 0.0254 
 0.0153 0.0244 
 0.0122 0.0225 
 
 0.0158 
 0.0290 
 0.0493 
 
 -.000127 
 -.000145 
 -.000169 
 
 The integrals K{\) and ir(X) with X = Xi/X2 are given by 
 
 dx 
 -+1 
 
 Ki\) = I dvy^ 
 
 *'o "^x !//■.! (2v.v+y-- 
 
 32 2 I /X + 2\' /6+6X-X-\ i 
 
 = 4X-^+— X-= ( ) ■( 1 + 1 , (23) 
 
 5 15(1 +X)- IV X / V X- / I 
 
 15(1+X)- 
 '2'^ /-> xdx 
 
 
 (2>^.V + y2+l)i 
 
 32 8 2(2-X)m» l + (5X/2) 
 
 = X-« — X-^H 1 {2nm^-nm-\og(n + m)\, (24) 
 
 9 9 9(1+X)X^ 48(l+X)5/2 
 
 where OT = (X+2)/X and w = 2X-'(l+X)!. The evaluated integrals are substituted into (18), (19), (20), 
 and (21) to give 
 
 6 = (8/5)(aiXi= + a2X2^)-(48/35)(^iXi5+^,Xo5), _ (25) 
 
 A£/100^ = -5V2-(5/5)(Xi5+X2*)-(6/3)(Xi3X2VXi+X2) + (12/7)(XJai2 + XoV-) 
 + (ll/35)(Xi'ai + Xo'ao) + (l/4)(ii:(X)Xi'«o + K(X-i)X2'ai) 
 -(64/21)(Xi'ai0:+X2'a2/32) + (48/35)(Xi-/3,= + X2W)-(256/3'-5-7)(Xi'|S,+X2'^2) 
 
 -(l/4)(Pr(X)Xi'/32+H^(X-')X2^/3i). (26) 
 
 Numerical values are inserted in (25) and (26) and the minimizing conditions (9) give four simul- 
 taneous equations for ai, a^, jii, and IS^- Since the determinants are close to vanishing, greater accuracy 
 is obtained by solving these equations by the method of elimination. When Xi^ : X2' = 1 : 2 the minimum 
 energy occurs at: ai = 0.184, a2 = 0.099, /Ji = 0.1.40, /32 = 0.085, 5=0.0377, and A£=-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 \p with the symmetry axis. The charges on the frag- 
 ments are obtained from (10) and (22) which give 
 
 Zi-ZXi'=-p5(4/3)7r/?i'+pLia,/i?--p^i/3i//?2 
 
 = (4/3)7r7?V- I -Xi'6-f (8/5)Xi»a,-(48/35)X,*/3il. (27) 
 
 When Xi' = g we find from the numbers given above that 7 = 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 /3,=/32 = 0), the results are changed to: a, = 0.0466, a2 = 0.0153, 5 = 0.0244, A£=-3.4 niMU, 
 and 7 = 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 /3i and jS^ from the preceding formulae. Figure 2 is a plot of (7 — 1) J'^- 
 Xi' obtained in this wa\', and Table I gives some of the numerical results. 
 
DIVISION OF NUCLEAR CHARGE 
 
 DISCUSSION OF RESULTS 
 
 The theoretical results are represented in 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 ai and a-i are negligible, as has previ- 
 ously been assumed. 
 
 The principal sources of uncertainty in these 
 results are (1) the assumption of uniform total 
 l)article density and (2) the use of Wigner's mass 
 defect formula with the large term in {N—Z)/A. 
 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—ZY/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 Ai from the fission 
 product nucleus of mass number Ai where 
 Ai =Ai—\. E.xperiniental 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 Z,- of the fragment or of the product 
 nucleus plotted against the mass number A,' of 
 the product nucleus. The charge numbers Z/" 
 following from the hypothesis of uniform charge 
 distribution (7 = 1) are shown for comparison. In 
 a very asymmetric division the Z, and Zi"" 
 curves difTer 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 
 
 A,' 
 
 1 
 
 1 
 
 t 
 
 t 
 
 1 
 
 00 010 
 
 020 
 
 030 
 
 K - 
 
 040 
 
 050 
 
 Fig. 2. Calculated curve'of (7— 1) vs. \i'. 
 
R. D. PRESENT 
 
 6i 
 
 - 
 
 
 60 
 
 
 " y 
 
 ss 
 
 
 ' ' y 
 
 n 50 
 
 J 
 
 - 
 
 ■'•' / 
 
 3 
 
 J 
 
 3 
 
 : 
 
 •' 
 
 '/ 
 
 L. 40 
 
 J 
 
 3 
 C 
 [ 
 
 yy 
 
 ' /y 
 
 
 5 5S 
 
 .•X 
 
 
 50 
 
 
 ll -NON-UNIFORM DENJlTt 
 
 
 / 
 
 ^""- UNIFORM OCNSITV 
 
 25 
 
 - 
 
 
 SO 
 
 
 
 M4SS OF PRODUCT NUCLEUS a' i 
 
 Fig. 3. Charge vs. mass of the product nuclei. 
 
 compare the result with Fig. 3 where v is the 
 mean vertical distance from the Z,- curve to the 
 dots. The quantity v 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 e.xtremely asym- 
 metric split have not been observed but 
 might be found in very fast neutron fission. 
 
 The foregoing conclusions refer to odd mass 
 numbers Ai. 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 v would undergo large fluctua- 
 tions from one mass number to the next. The 
 probable chain lengths for the oven 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 A£ caused by spreading 
 of the charge is small for all observed modes of 
 division (cf. Table I), and the difference between 
 the values of A£ 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 Way* 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.''" 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, Cor^^ell, and others" that the known facts 
 about decay chains, with mass numbers corre- 
 
 «E. Wigner and K. Way, Report CC-3032 (1945); 
 Plutonium Project Reports 9B, 6.4 (1946). 
 
 • K. Way, private communication. 
 
 "> S. Fliigge and G. v. Droste, Zeits f. physik. Chemie 
 [B], 42, 274 (1939). 
 
 " Summarized 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 
 
 9 
 
 spending to the most likely modes of fission, are 
 consistent with the result of equal probable chain 
 lengths (3.6) fo'^ 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 {Z^^'^), 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 (Xe"' and Cs"^) appearing 
 to He on separate curves displaced by about one 
 unit of charge in both cases. On the other hand, 
 if the values of Z/> 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. 
 
UNIVERSITY OF FLORIDA 
 
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