Nuclear-charge distributions describe the dispersion of yields with mass and atomic numbers (A and Z) of ~1000 primary fission products from each of the many fission reactions (yields are for the fission products after prompt-neutron emission and before beta decay). However, only a small fraction of the yields have been measured, and theoretical models are not sufficiently advanced to give reliable yield estimates. Therefore, two empirical models have been proposed that include mathematical functions derived from the available experimental data (Wahl, 1988).

While the ZP and A’P models give different perspectives on the nuclear-charge distribution from fission, comparisons of the results have been helpful in improving both models. The ZP model treats the dispersion of fractional independent yields [FI] with Z for each A of primary fission products; the A’P model considers the dispersion of independent yields [IN] with A’ for each Z of primary fission products, where A’ is the average mass number of the precursor primary fragments that give products with A by prompt-neutron emission (A’ = A+ VA).

Experimental independent yields can be supplemented by yields calculated from the ZP or A’P models for the fission reaction of interest, prior to dividing amongst the isomeric states (Madland and England, 1977). The spins of the states need to be defined for both treatments, and the isomeric cumulative yields can be calculated by summation of the independent yields based on the spins and branching fractions to the isomeric states.

(a) Zp model

The ZP model has been used for charge distribution calculations in the two major fission yield evaluations (US ENDF/B-VI and UKFY2 (and UKFY3)). ZP-model parameters are defined as FZ, FN, az and AZ, and are either constant or linear functions of A’ in each of the regions considered (as shown in Fig. 11). Fractional independent yields are expressed in terms of these parameters:

FI(A, Z) = (0.5) [F(A)] [N(A)] [erf(V) — erf—

Z (A) — Zp (A) + 0.5
V2 oy(A)

Z (A) — Zp (A) — 0.5
V2 [oz (A1)]

in which Zp (AH ) = A’H [ZF / AF ] + AZ(A’H )

O. Fig. 10 Various Gaussian curves fitted to mass distributions of fission processes:

249 252

thermal-neutron fission of Cf, and spontaneous fission of Cf

98 A. L. NlChOlS

and Zp (Al ) = A[Zp / Ap ] — AZ(A’Hc)

where ZF and AF are the atomic number and mass number of the fissioning nuclide, Ah and Al are the mass numbers for the heavy and light fission peaks respectively, A’Hc = Ap — A’l, and

N(A) is the normalisation factor to achieve ЦРІ) = 1.00 for each A, which has to be implemented because P(A) destroys the inherent normalisation of the Gaussian distributions; values of N(A) seldom deviate by more than 10% from unity.

All of the parameters AZ(A/), <jz (A/), PZ (A’) and PN (A/) depend on A’, and the

region in which A’ falls. Apart from distributions close to symmetry, all of the parameters are constant except AZ, which has a small negative slope (AZ is the displacement of ZP from an unchanged charge distribution):

AZ = {Zp — A'[Zf /Af]}h = {A'[Zf /Af] — Zp}L

Near to symmetry, the width parameter [oz] changes abruptly twice to a lower value, the even-odd proton and neutron factors [FZ and FN] equal 1.0, and the AZ function undergoes a zig-zag transition from positive values for light fission products to negative values for heavy fission products. The ZP model has been modified to include parameter slopes vs. A’ and slope changes in the wing regions.

Values of the ZP-model parameters have been determined for each of twelve fission reactions (229Th thermal, 232Th fast, 233U thermal, 235U thermal, 238U fast, 238U 14

MeV neutrons, 238Np thermal, 239Pu thermal, 241Pu thermal, 242Am thermal, 249Cf thermal, and 252Cf sf) by the method of least squares. However, all of the model parameters can only be determined by this method for the thermal fission of 235U. Other fission reactions require the following equation to estimate ZP-model parameter values [PM]:

PM = P(1) + P(2)(Zf — 92) + P(3)(Af — 236) + P(4)(E* — 6.551)

Fig. 11. Zp functions for thermal fission of 235U

Solid lines calculated by L. S., red. %2 = 3.6(0.7) Dashed lines from systematics, red. %2 = 6.6(0.6)

where E* is the excitation energy above the ground state of the fissioning nuclide. P(i) values were calculated by the method of least squares from the PM values that could be determined by least squares for individual fission reactions. Fig. 12 shows the adopted parameter values as points and the derived functions as lines.

Checks have been made on the validity and usefulness of the ZP-model parameters obtained from the PM calculations for each of the 12 fission reactions investigated. The reduced X values were within a factor of approximately 2 of those determined by the method of least squares with variation of as many parameters as possible. Further details of these studies can be found in IAEA-CRP (2000).

b) A’p model

Studies are underway to determine whether a simplified A’P model can be used with only five Gaussian functions to represent the element yields, rather than the many needed for the association of one model parameter with each element yield (IAEA — CRP, 2000). Fig. 13 shows the results of calculations for the thermal-neutron fission of 235U close to symmetry (Z = 42-50).

The data include independent yields for individual elements or for complementary element pairs when the data are available from radiochemical and on-line mass — separator measurements. Solid-black symbols represent 43Tc yields, while open symbols represent the yields for heavy fission products. The Gaussian width parameters [o = oA] are close to the average global parameter of 1.50 for 47Ag and the 50Sn-42Mo pair. However, the values of o are considerably larger for 48Cd and 43Tc-49In pair, implying that these radionuclides could be formed by more than one process. The data for 48Cd and the 43Tc-49In pair can be represented better by two Gaussian functions (symmetric and asymmetric), each with a o value of 1.50 and peak separation of about 4 A’ (Fig. 14). Significant deviations occur for the curve of the 43Tc-49In pair, underlining the need for a re-evaluation of these data.