Grouping and collapsing data only dependent on incident energy

Data that is only a function of incident particle energy is grouped by performing a Legendre-order-flux weighted averaging of the data between group boundaries. That is, for the l^th Legendre order Q^l of a quantity Q(E) and the l^th Legendre order flux f^l(E), the value of Q^l for group g, Q^l_g, is calculated as,

$${\displaystyle \int_{E_{g+1}}^{E_g} f^l(E) \, Q^l(E) \, dE \over \displaystyle \int_{E_{g+1}}^{E_g} f^l(E) \, dE}$$ (1)

Only the transport correcting cross-sections and the interaction transfer matrix have l > 0 Legendre orders. All other quantities have only the isotropic l = 0 Legendre order; in which case the l-order label is dropped (e.g., $\sigma^{0}_{}$ is written as $\sigma$). For example, consider a cross-section $\sigma$(E) grouped using the 3 groups given by the energy boundaries (20.0, 15.0, 12.3, 0.1 MeV); then the grouped cross-sections are,

$$\sigma_{1}^{} {\displaystyle \int_{15.0}^{20.0} f^0(E) \, \sigma(E) \, dE \over \displaystyle \int_{15.0}^{20.0} f^0(E) \, dE}$$ = (2)

$$\sigma_{2}^{} {\displaystyle \int_{12.3}^{15.0} f^0(E) \, \sigma(E) \, dE \over \displaystyle \int_{12.3}^{15.0} f^0(E) \, dE}$$ = (3)

$$\sigma_{3}^{} {\displaystyle \int_{0.1}^{12.3} f^0(E) \, \sigma(E) \, dE \over \displaystyle \int_{0.1}^{12.3} f^0(E) \, dE}$$ = (4)

Four quantities are not grouped as given by Eq. 1. One is the group speed v_g which is calculated as,

$${1 \over v_g} {\displaystyle \int_{E_{g+1}}^{E_g} { f^0(E) \over v(E) } \, dE \over \displaystyle \int_{E_{g+1}}^{E_g} f^0(E) \, dE}$$ (5)

since it is 1/v_g that appears in transport equations. However, it is v_g which is returned by the access routines. Another quantity not grouped as per Eq. 1 is the group flux f^l_g, which is not weighted,

$$\int_{{E_{g+1}}}^{{E_g}}$$ (6)

The last two quantities not grouped as per Eq. 1 are the transfer and fission matrices. These quantities also depend on outgoing particle energy and are discussed in Section 4.2.

Often the data in a ndfy_{$\bf i$} file are stored with more energy resolution than a problem requires. In this case, the user can request, by calling ndfgroup, that the ndf access routines return the data grouped to a smaller energy group. This smaller energy group must be a subset of the energy group used to generate the data in the ndfy_{$\bf i$} file. (A subset energy group contains only boundaries of the superset group). The example above has six 1-group subsets (20.0, 15.0), (20.0, 12.3), (20.0, 0.1), (15.0, 12.3), (15.0, 0.1), and (12.3, 0.1), four 2-group subsets (20.0, 15.0, 12.3), (20.0, 15.0, 0.1), (20.0, 12.3, 0.1) and (15.0, 12.3 0.1) and itself as a 3-group subset.

Data mapped to a subset group is referred to as collapsed data in this document and the act of mapping the data is called collapsing. To receive collapsed data one must first call the ndf routine ndfgroup (or ndfcgroup). The first two arguments of ndfgroup are the new group boundaries and the number of new groups (labeled ncg is this document). The third argument of this routine is a flux id. The bdfls file is scanned for the requested flux. This flux $\varphi$ is then grouped onto the old groups and the data is collapsed using this flux as a weight for the old group data. The flux is calculated as,

$$\varphi^{l}_{{g'}} \sum_{{g \in g'}}^{} \varphi^{l}_{g} \int_{{E_{g'+1}}}^{{E_{g'}}} \varphi^{l}_{}$$ (7)

where

$$\varphi^{l}_{{g}} \int_{{E_{g+1}}}^{{E_g}} \varphi^{l}_{}$$ (8)

Here, g' is a label for the new groups, $\varphi^{l}_{g}$ is the requested flux grouped (see Eq. 8) and g $\in$ g' means to sum over all g for which the boundaries of g fall inclusively between the boundaries of g'.

Collapsing a quantity Q^l_g is calculated as,

$${\displaystyle \sum_{g \in g'} \varphi^l_g \, Q_g \over \displaystyle \sum_{g \in g'} \varphi^l_g}$$ (9)

In the above example, collapsing the cross-section to the 2-group (20.0, 15.0, 0.1) yields,

$$\sigma_{1}{^\prime} {\sigma_1 \varphi^0_1 \over \varphi^0_1} \sigma_{1}^{}$$ = (10)

$$\sigma_{2}{^\prime} {\sigma_2 \, \varphi^0_2 + \sigma_3 \, \varphi^0_3 \over \varphi^0_2 + \varphi^0_3}$$ = (11)

While collapsing to the 1-group (20.0, 12.3) yields,

$$\sigma_{1}{^\prime} {\sigma_1 \, \varphi^0_1 + \sigma_2 \, \varphi^0_2 \over \varphi^0_1 + \varphi^0_2}$$ = (12)

The speed (and flux as given in Eq. 7) is collapsed differently, so as to be consistent with its previous groupings. Speed is collapsed as,

$${1 \over v_{g'}} {\displaystyle \sum_{g \in g'} { \varphi^0_g \over v_g } \over \displaystyle \sum_{g \in g'} \varphi^0_g}$$ (13)