A serious lack of suitable decay data for many of the short-lived fission products has necessitated the adoption of theoretical half-lives and mean energies, based on a number of modelling methods that focus on the derivation of beta-strength functions. These models are outlined below.

(a) Gross theory: smooth beta-strength functions

The gross theory of P — decay has been used by Takahashi et al (1973) to predict beta — strength functions, and calculate the half-lives and other relevant decay parameters averaged over the final daughter states. The decay constant and average beta and gamma energies per disintegration can be expressed in terms of the sum of each partial decay to the ith final state of energy є:

a = ІA,

i=0

Ep= І A ^ )Ct

i=0 A

-Ey= єi

i=0

in which C. is the ratio of the average kinetic energy of the beta emission to the ith state to the maximum kinetic energy of the beta decay (Q — є).

The equation for A can be reformulated to include the beta-transition matrix element (Щ, &V):

n

A = ngQl M, f(E>).

2П i=0 n

where f is the integrated Fermi function, and E. is the maximum energy available for the emitted electron; П represents the transition operator, and gn is the coupling constant. A major assumption in the evolution of the gross theory is that the density of the final levels is sufficiently high to replace this summation with an integration function:

10

A = TT ЦЫ! ■ fn(g )|2 f(- Eg + l^lSg.

2K — Q n

in which |Mn(Eg)|2 is the product of the square of the transition matrix and the level density of the final states (also referred to as the beta-strength function, and sometimes denoted by the term Sp(E)). Similar manipulations can lead to gross theory expressions for the mean beta and gamma energies:

E в =-л — ГІІ gnl 2 ■ M n (Eg )| 2 Г mc 2(E -1) pE (- Eg +1 — E )2 F (E) dEdEg,

2П A — Q П 1

1 0 — E +1

Er=—r. j!|g a2. M n (Eg )| 2 mc2 (Q + Eg) J pE (- Eg +1 — E )2 F (E) dEdEg,

2П A — Q П 1

where F is the Fermi function, and p is the momentum of the electron. The beta — strength function is assumed to be smooth in this model; any structural features are deemed to be unimportant in deriving half-lives and mean energies.

The gross theory has been systematically applied to calculate P- decay half-lives, and these data compared with known experimental values (Table 8). Half-lives were predicted within a factor of 5 for 70% of 100 fission products with half-lives less than 1 min, and within a factor of 10 for 90% of the same set of fission products (Yoshida, 1977). Good agreement was also obtained for the mean beta and gamma energies, although the theory failed to predict these decay parameters for 82As and 92Rb (odd-odd nuclei).

Table 8: Mean beta-particle and gamma-ray energies for nuclides with Q > 4.5 MeV: comparison of gross theory calculations with evaluated measurements (Yoshida, 1977) Nuclide Mean beta-i ^article energy (MeV) Mean gamma-ray energy (MeV) Half-life (sec) UK evaluation (1973) US evaluation (1975) Gross theory UK evaluation (1973) US evaluation (1975) Gross theory 74Ga 1.072 1.070 1.350 3.043 3.040 2.471 500 76Ga 1.675 1.680 1.832 2.808 2.810 2.136 27 CO о > 2.468 2.523 2.584 0.554 0.606 0.347 17 82As 3.137 3.211 1.888 0.336 0.288 2.909 23 86Br 1.765 1.775 1.946 3.296 3.318 2.936 59 87Br 2.087 2.136 1.757 1.727 1.726 2.387 56 CO CO £ 2.000 2.083 1.156 0.677 0.674 2.463 1100 90Rb 1.789 1.659 1.673 2.560 2.660 2.814 150 91Rb 1.320 1.334 1.533 2.871 2.733 2.533 59 92Rb 3.714 3.459 2.526 0.260 0.261 2.696 4.5 94y 1.193 1.717 1.039 1.043 0.986 2.417 1100 95y 1.713 1.745 0.968 0.523 0.488 2.111 650 97y 1.612 2.162 2.294 0.935 0.935 1.055 1.1 99Zr 1.586 1.621 1.651 0.794 0.794 0.719 2.4

Yoshida and Nakasima (1981) have used the gross theory to determine mean beta and gamma energies for fission products that undergo high-energy p — decay. Approximately 170 radionuclides with known half-lives and decay schemes were assessed in this manner (Q-values > 3 MeV), and the calculated mean energies were compared with available experimental data. There was a reasonable degree of overlap between theory and experiment for the mean beta energy data, but the mean gamma energies exhibited much less satisfactory agreement. This same approach was also adopted for a significant number of poorly-defined nuclides:

(i) measured half-lives, but unknown decay schemes;

(ii) unknown half-lives and decay schemes (as defined in 1980).

Nuclides in category (i) include some radionuclides that contribute significantly to decay heat at short cooling times (e. g., 89Br, 94Rb, 101Nb, 102Zr, 102Nb, 103Mo, 145Ba and 145La). Yoshida and Katakura (1986) combined a cascade gamma transition model with the gross theory of beta decay in a further attempt to improve predictions of the mean gamma energy data for short-lived fission products.

When all of the theoretical decay data were incorporated into decay-heat assessments, good agreement was observed between experiments and calculation (Figs. 15 and 16). Decay-heat measurements of Dickens et al (1980) have been compared with summation calculations in which theoretical mean energies were included. The original JNDC-FP decay-data library overestimated

Fig. 15. Beta energy emission rate after instantaneous pulse of thermal-neutron fission of 235U

Fig. 16. Photon energy emission rate after instantaneous pulse of thermal-neutron fission of 235U

the beta energy release and underestimated the gamma energy release; when the theoretical data were introduced, the decay-heat calculations reproduced the measurements extremely well.

Tachibana et al (1990) extended the gross theory by modifying the one-particle strength function and introducing the AQ0 term. When the Q-value is relatively small, the P — decay is sensitive to the forbiddenness of the transitions to the low-lying states, and the AQ0 term was used to modify the resulting strength functions of these nuclides (and others). Nakata et al (1995) have also refined the gross theory further for odd-odd nuclei, taking into account the selection rule for the beta transition to the ground state. This refined approach gives half-lives in better agreement with the experimental data than the values obtained by the unmodified gross theory (except for parents with a spin and parity of 1-). Furthermore, Nakata et al (1997) have taken the shell effects of the parent nuclei into consideration to produce the semi-gross theory of P- decay; the one-particle strength functions are dependent on the principal quantum numbers and spin-parity of the initial state of the decaying nuclide. This modification also involves raising the beta strengths to a level defined by the AQ0
term (subsequently referred to as Q0o) when the transitions to low-lying levels are highly forbidden.

Yoshida and Tachibana (2000) adopted the concept of Q00 to reproduce experimental half-lives:

0.25 Me V for even — even parent < 1.0 MeV forodd — A parent

1. 75 MeV forodd — odd parent

in which these Q00-values were subsequently multiplied by a parameter that depends on the even/odd properties of each nuclide to give the Q00-factor. The ratio between the calculated and experimental half-lives is given in Fig. 17 on a logarithmic scale; the best agreement is observed when the Q00-factor is unity. Similar analyses of the beta and gamma decay components indicate that a Q00-factor of approximately 0.4 is most appropriate for the calculation of the mean beta and gamma energies (Figs. 18 and 19).