The algorithm to produce equidistributed correlated angle-energy bins
from
= 3 data is implemented by the routine `iequ3`. The data for both the angle-energy probability densities
(
= 3) and for the angular probability densities
(
= 1) must be present. Furthermore, the lists of
incident energies at which the
= 3 and
= 1 probability densities are given must be identical.

The computation of equiprobable angle-energy is done in three steps. First, the = 1 data is used to make an equidistribution of the cosines using the method described above. For each of these equidistributed cosines we need to create equidistributed secondary energy bins. So, the second step is the creation of secondary energy distributions at these new cosines. The third step is the equidistribution of these energies. The only new step here is the second step, that of making an energy distribution at one of the edges of an equiprobable cosine bin. Its details are as follows.

Consider one of the equidistributed cosines. We bracket it between
two
= 3 (or equivalently,
= 1)
cosines from the ``library'` file. We want to use the (energy,
probability density) data from these bracketing cosines to produce
(energy, probability density) pairs for the new cosine. This is
illustrated in the Fig. 2, which has secondary energy increasing to
the right and cosine increasing upward. Suppose that the new cosine
*C*_{new} lies between the library cosines *C*_{0} and *C*_{1}. We
want probability densities for
*C*_{new}, given probability
densities at one set of energies for *C*_{0} (denoted by dots) and at
another set of energies for *C*_{1} (denoted by crosses). The algorithm
is based on multiple linear interpolation as follows.

**Figure 4.2:** Double interpolation of probability
density.

**1.**- We start by translating and scaling the secondary energy
ranges for the cosines
*C*_{0}and*C*_{1}to the interval 0*E*1, and we scale the probability densities correspondingly. The figure shows the situation after this scaling. There are several reasons for starting with this scaling. For one thing, it ensures proper linear dependence on incident energy for the minimum and maximum energy of the secondary particle. For another, it leads to more accurate interpolation of such physical features as resonance peaks. Finally, it makes the interpolation process clearer. **2.**- We merge these scaled energy lists to produce a master
secondary energy list, denoted by the vertical lines in the figure.
**3.**- For each library cosine
*C*_{0}and*C*_{1}we interpolate in energy to get probability densities at the full merged list of secondary energies. This is interpolation horizontally along the top and bottom of the rectangle in Fig. 2. **4.**- Then at each secondary energy we interpolate vertically in
the figure, to get the probability densities at the equiprobable
cosine
*C*_{new}. **5.**- Now that we have probability densities, we use the method
described in the previous section to get equiprobable secondary
energy bins at
*C*_{new}, but these are still scaled to 0*E*1. **6.**- Finally, we undo the scaling by mapping from
0
*E*1 to the physical energy interval.

**1.**- D. E. Cullen,
``TART95: A coupled neutron-photon Monte Carlo transport code'',
Report UCRL-MA-121319, Lawrence Livermore National Laboratory,
July 1995.
**2.**- R. J. Howerton, R. E. Dye, P. C. Giles,
J. R. Kimlinger, S. T. Perkins, and E. F. Plechaty, ``OMEGA: a
Cray 1 executive code for LLNL nuclear data libraries'', Report
UCRL-50400, Vol. 25, Lawrence Livermore National Laboratory,
August 1983.
**3.**- R. J. Howerton, R. E. Dye, and S. T. Perkins,
``Evaluated nuclear data libraries'', Report UCRL-50400, Vol. 4,
Rev. 1, Lawrence Livermore National Laboratory, October 1981.
**4.**- S. T. Perkins, ``Specification of the all-particle
Monte Carlo data files, MCFYi'', Draft Report, Lawrence Livermore
National Laboratory, July 1994.
**5.**- J. A. Rathkopf, ``Format of binary ENDL-type
libraries'', Report #PD-166, Lawrence Livermore National
Laboratory, 17 October 1988.

Work performed under the auspices of the U. S. Department of Energy by Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.