35\. FILE 35. COVARIANCES FOR ENERGY DISTRIBUTIONS OF SECONDARY
PARTICLES

**35.1. Introduction**

File 35 contains covariance matrices for the energy distribution of
secondary neutrons given in File 5. The data in File 5 are given in the
Laboratory system, and are expressed as normalized probability
distributions.

Since there is usually very fragmentary information with which to
construct the data given in File 5, the uncertainties in the secondary
distributions are highly correlated as a function of incident neutron
energy. It is therefore proposed that only a few (at most 4 or 5)
covariance matrices be used in each MT value in file 35 to cover the
complete incident energy range. Each covariance matrix applies to the
complete secondary energy distributions for the broad incident energy
range specified, regardless of how these secondary energy distributions
are specified, or broken down into various components, in File 5. In
this first attempt at dealing with covariance matrices of secondary
energy distributions, no covariances between the different incident
energy ranges will be allowed. Also, no covariances linking different
materials or reaction types are allowed. Furthermore, no covariances
with information in other files, for instance File 3 and v\_(E) in File
1 are allowed in File 35.

**35.2. Formats**

Each subsection covers a covariance matrix for an incident neutron
energy range, and the complete incident energy range is covered by the
NK subsections.

A new type of LB subsection is defined (LB=7), since in File 35 it is
natural to specify the covariance matrices as absolute covariances
rather than relative covariances. The LB=7 subsection is similar to an
LB=5 subsection, but with entries that are absolute rather than
relative.

The following quantities are defined:

+----------+-----------------------------------------------------------+
| **NK**   | Number of subsections                                     |
+----------+-----------------------------------------------------------+
| **E~1~** | Lowest incident neutron energy to which the covariance    |
|          | matrix in the subsection applies.                         |
+----------+-----------------------------------------------------------+
| **E~2~** | Highest incident neutron energy to which the covariance   |
|          | matrix in the subsection applies. The value of E~2~ in a  |
|          | subsection becomes the value of E~1~ in the next          |
|          | subsection.                                               |
+----------+-----------------------------------------------------------+
| **LS=1** | Flag indicating that the covariance data matrix F~k,k~,   |
|          | is symmetric.                                             |
+----------+-----------------------------------------------------------+
| **LB=7** | Flag indicating that the elements of the covariance       |
|          | matrix F~k,k~ are absolute.                               |
+----------+-----------------------------------------------------------+
| **NT**   | Total number of entries in the list. NT=\[NE\*(NE+1)\]/2. |
+----------+-----------------------------------------------------------+
| **NE**   | Number of entries in the array {E~k~}                     |
+----------+-----------------------------------------------------------+
| **       | Array containing outgoing neutron energies, and defining  |
| {E~k~}** | NE‑1 energy intervals for outgoing neutrons. The value of |
|          | E~1~ in the array must be the lowest outgoing neutron     |
|          | energy possible at E~1~; E~NE~ in the array must be the   |
|          | highest outgoing neutron energy possible at E~2~ and      |
|          | represented in File 5.                                    |
+----------+-----------------------------------------------------------+
| **{F     | covariance matrix. The F~k,k~\'s are ordered by rows,     |
| ~k,k~}** | starting from the diagonal term                           |
|          |                                                           |
|          | {F~k,k~} F~1,1~, F~1,2~; F~2,2~, F~2,3~, \..., \...;      |
|          | F~NE--1,NE--1~ .                                          |
+----------+-----------------------------------------------------------+

The structure of a section of File 35 is as follows:

\[MAT,35,MT/ ZA, AWR, 0, 0, NK, 0\] HEAD

> \<subsection for k=1\>
>
> \<subsection for k=NK\>

\[MAT,35, 0/ 0.0, 0.0, 0, 0, 0, 0\] SEND

The structure of a subsection is:

\[MAT,35,MT/ E~1~, E~2~, LS, LB, NT, NE/{E~k~},{F~k,k~}\] LIST
(LS=1,LB=7)

![](media/image1.wmf)In terms of the dimensionless operators defined in
Section 33.2, the covariance between two points on the unity‑normalized
probability distribution is:

where E~1~ E E~2~, and the P values are defined so that only the term in
the sum for

> E~k~ E~I~ E~k+1~ and E~k~ E~I~ E~K+1~ is non‑zero.

**35.3. Procedures**

Because probability distributions must remain normalized to unity,
covariance matrices in File 35 must satisfy a constraint in addition to
being symmetric. This constraint is that the sums of the elements in any
row of the matrix, therefore also of any column, must be zero.
Therefore, if one were to divide the outgoing energy of a distribution
into n energy ranges, it would only be necessary to give the covariance
matrix of the probabilities in n‑1 energy ranges. The remainder of the
full covariance matrix could be obtained from the above constraint.

There is a procedural objection to the above suggestion. It is that, for
the covariance file, if no information is provided in the file, this
does not imply anything as far as the uncertainties in the data are
concerned. Therefore, it is necessary to provide redundant information
in File 35 for each covariance matrix of a secondary energy distribution
and to require that each covariance matrix in File 35 meets the above
constraint. The sum of each row or column should be \<10^-5^ on the
evaluator\'s energy grid. \[Note that this condition must also be met
with processed data on the user\'s energy grid\].

If F~k,k~ are the covariance elements defined above for the normalized
spectral yields Y~k~ on the evaluator\'s grid {E~k~}, but the above
constraint has not been applied, corrected values ˆF~k,k~ may be
obtained from the following relation.

![](media/image2.wmf)

The secondary energy distribution uncertainty analysis using the
\"hot‑cold\" technique of Gerstl, et al., (see References) can be based
on spectral uncertainty data stored in this form. In this case the
covariance matrix is a 22 matrix for each incident energy range.

**REFERENCES**

1\. S. A. W. Gerstl, \"Uncertainty Analyses for Secondary Energy
Distributions\", *A Review of the Theory and Application of Sensitivity
and Uncertainty Analysis: Proceedings of a Seminar‑Workshop, August
22‑24, 1978*, C. R. Weisbin, et al., Eds, Radiation Shielding
Information Center report, Oak Ridge National Laboratory
**ORNL/RSIC‑42** (1978) p. 219.

2\. S. A. W. Gerstl, R. J. LaBauve, and P. G. Young, *A Comprehensive
Neutron Cross‑Section and Secondary Energy Distribution Uncertainty
Analysis for a Fusion Reactor*, Los Alamos Scientific Laboratory report
**LA‑8333‑MS** (1980).