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*^V 1' ,/ftfA't' C//". "3 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 nonuniform. 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 wellknown 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. W35058eng71 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 protonneutron 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 "pinchingin."' 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 liquiddrop 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 NZ. (The isotopic spin Tr=(NZ)/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 (NZ)2. Wigner' has shown that the potential energy of the nucleus computed from the symmetrical Hamiltonian contains a term in (NZ)/A and has included both linear and quadratic terms in NZ in his mass defect formula. The presence of the linear term con siderably reducesthe 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 (NZ) and (NZ)' 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=(pvpz),'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 selfenergy of each sphere and in the mutual electrostatic energy of the two in going from uniform to nonuniform charge den sity. Since the deviation from uniform density is small, secondorder terms in e=pzPz=p(it) 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 (ilt)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(NZ)/A +c2(NZ)P/A. These cor respond to an energy density gr given by cGp (NZ c2p (NZ)2 ST=  + A A / A .4A cIP PNPZ PN pZ = +cp. 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 nonuniform 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(NZ)/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 (NZ)/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 eveneven nucleus P'=P"=0 in the groui state. % however, since its omission from the massdefect formula leads to a larger value for c2. In the WeizsAckerBethe formula which contains no term in (NZ)/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 nonuniform 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? Zep3.5,\,". 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 (millimass 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). [XE2 (XI+ + X (a1+ +a, )R' (aXX +P2 +a IXPR ( Q+i3,Q,)RI), AEr= 4c2p6R2.1 (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 '3.57)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 Xf3+ + 2nnmlnm log (n+m) (24) 9 9 9(l+AX)X 48(1+ A)!'2 where m= (+2),/' and n =2XI'(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 (,VZ) '.4. In both respects our calculation gives an upper limit for (71). 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 (NZ)/A for heavy nuclei is yet at hand. The size of this term relative to the conventional term in (NZ)2/A is a consequence of the symmetrical Hamiltonian. If this term should prove to be small or absent (as in the WeizsackerBethe formula) the value of (7 1) would be consider ably reduced. It is possible but unlikely that the calculated values of (71) 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,'=Ai1. 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 3decays 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 curveof (yj1 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 BohrWheeler 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 CC3032 (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, CLLEGNo. 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 halflife 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 ZZp, 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 onefifth 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 3 1262 08907 9957 I i .1 "i 
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