Linearization and Unionization

All ENDF-format evaluations define their cross sections over at least part of the energy range using tabulations. In many cases, part of some of these tabulations specify that a nonlinear interpolation scheme is to be used to obtain intermediate values. NJOY wants to get everything in linear form for several reasons:
  • the method used for Doppler broadening requires linearly interpolated piece-wise functions;
  • the ACE files for use with MCNP require linear-linear functions;
  • linear functions are easy to average into multigroup form;
  • linear functions are easy to plot; and
  • summation cross sections, such as the total cross section, total inelastic, or total fission, cannot be properly defined as the sum of their parts except when given as linear.
The classic example of the last point is the thermal energy range, where the zero-degree elastic cross section is usually constant with energy (and uses linear-linear interpolation), but the radiative capture cross section follows a 1/v law and uses log-log interpolation. There is no simple "graph-paper" interpolation scheme that can represent the sum of the elastic and capture cross sections to the same accuracy as the original parts. However, if both the elastic and the capture are linearized to a given accuracy on the same "union grid," their sum is also linearized to the same accuracy on this grid.

RECONR solves this problem as follows. It first skips the total cross section and goes to the elastic cross section (MF=3, MT=2). It starts with a list of "nodes" that may include energy values given by the user, may include energies derived from the resonance parameters, and certainly includes the thermal value .0253 eV. It then adds in any energy points given in the elastic section of File 3, and if necessary, it adds in points on the 1, 2, 5, 10, ... grid for each energy decade. Now it looks at each "panel" described by a pair of these nodes and decides whether to divide the panel in half. It will divide the panel if the energy step is too large to represent the 1/v function to within the specified accuracy, or if the interpolation scheme is nonlinear, it will divide the panel if the difference between the specified interpolated function and the linear interpolate is too large. It continues subdividing and checking until the desired accuracy has been achieved over the entire energy range.

The reason for checking the 1/v accuracy for elastic scattering is that Doppler broadening tends to add a 1/v component to the elastic cross section, and this grid will help to handle that correctly in BROADR, even when the radiative capture reaction is missing or extraordinarily small.

Once the elastic cross section has been linearized, the code moves on to the next reaction, perhaps (n,2n), and determines whether any additional points have to be inserted in the union grid to represent that reaction to within the desired tolerance. RECONR continues in this way until all the reactions have been checked. The result is therefore a union grid able to represent all the cross sections with linear interpolation, and that implies that it can also represent the sums of those cross sections with the same accuracy.

The following input deck will carry out the linearization and unionization process for H-1 from ENDF/V-VI (assuming that the appropriate ENDF file has been mounted on unit 20):

        20 21
        '1-H-1 from ENDF/B-VI'/
        125 1/
        .01/ 1% linearization
        '1-H-1 from ENDF/B-VI'/

For a complete description of these input lines, see the RECONR Input Instructions. HINT: use the right-hand button on your browser to bring the link up in a separate window. You can then view the input example and the input instructions at the same time. An examination of the sections of File 3 for MT=1, 2, and 102 will illustrate the kind of energy grids produced and the unionization. Note that the standard .0253 eV value and energies on the 1, 2, 5, 10, 20, ... sequence were included.


23 January 2013 T-2 Nuclear Information Service