31\. FILE 31. COVARIANCES OF THE AVERAGE NUMBER OF NEUTRONS PER FISSION

**31.1. General Description**

For materials that fission, File 31 contains the covariances of the
average number of neutrons per fission, given in File 1. MT=452 is used
to specify $\overline{\nu}$, the average total number of neutrons per
fission. MT=455 and MT=456 may be used to specify the average total
number of delayed neutrons per fission, ${\overline{\nu}}_{d}$, and the
average number of prompt neutrons per fission, ${\overline{\nu}}_{p}$,
respectively.

The average number of neutrons per fission is given as a function of
incident energy for induced fission. This energy dependence may be given
by tabulating the values as a function of incident neutron energy or (if
MT=452 alone is used) by providing the coefficients for a polynomial
expansion as a function of incident neutron energy. Whichever method is
used, the result is that the quantities are specified as a function of
incident neutron energy and in this sense are similar to the data given
in File 3. Therefore, the problems associated with representing the
covariances of the average number of neutrons per fission are identical
to those in File 33.

For spontaneous fission, in the sub-library for a radioactive decay, the
average multiplicities are given by zero‑order terms in polynomial
expansions, and the lack of any energy dependence is recognized in the
formats.

**31.2. Formats**

[Induced fission]{.underline} formats for fission neutron multiplicity
in File 31, MT=452, 455, and 456, are directly analogous to those for
File 33 given in Section 33.2.

[Spontaneous fission]{.underline} formats for neutron multiplicity in
File 31, MT=452, 455 and 456, are modified from those in Section 33.2
because there is no energy dependence to express for $\overline{\nu}$ or
its covariance components. There, NC‑ and NI‑type sub‑subsections for
spontaneous fission neutron multiplicity have no energy variables
present.

\"NC‑type\" sub‑subsections for spontaneous fission $\overline{\nu}$,
MT=452, 255, or 456, have the following structure, using definitions
given in Section 33.2.

[For LTY=0]{.underline}:

\[MAT,31, MT/ 0.0, 0.0, 0, LTY, 0, 0\] CONT (LTY=0)

\[MAT,31, MT/ 0.0, 0.0, 0, 0, 2\*NCI, NCI/ {CI,XMTI}\] LIST

[For LTY=1, 2, or 3]{.underline}:

\[MAT,31, MT/ 0.0, 0.0, 0, LTY, 0, 0\] CONT

\[MAT,31, MT/ 0.0, 0.0, MATS, MTS, 4, 1/

\-\-\-\-\--(XMFS,XLFSS)(0.0, Weight) \] LIST

\"NI‑type\" sub‑subsections for spontaneous fission neutron multiplicity
are allowed with LB=0 and LB=1, and have the structure:

\[MAT,31, MT/ 0.0, 0.0, 0, LB, 2, 1/(0.0,F)\] LIST

where F gives the absolute or relative covariance component depending on
whether LB=0 or LB=1. (See Section 33.2 of this manual for notation.)

**31.3. Procedures**

All procedures given in 33.3 concerning the ordering and completeness of
sections of File 33 apply to sections of File 31: $\overline{\nu}$
(MAT,31,452), ${\overline{\nu}}_{d}$ (MAT,31,455) and
${\overline{\nu}}_{p}$ (MAT,31,456).

$\overline{\nu}(E) = {\overline{\nu}}_{d}(E) + {\overline{\nu}}_{p}$Note
that in File 1 $\overline{\nu}$ (MT=452), ${\overline{\nu}}_{d}$
(MT=455) and ${\overline{\nu}}_{p}$ (MT=456) satisfy the relation:

Therefore, if one of these quantities is \"derived\" in terms of the
other two, it is permissible to use \"NC‑type\" sub‑sections with LTY=0
to indicate that it is a \"derived redundant cross section\". See
section 33.2.1.a. for an explanation of this format.

If a section of File 31 is used with MT=456, there must also be a
section of File 31 with MT=452.

When a section of File 31 for either MT=452, 455 or 456 is used for
induced fission, there must be a section in File 33 for the fission
cross sections, *i.e.*, section (MAT,33,18).

**Note**:

1\. Since ${\overline{\nu}}_{d}$ is much smaller than
${\overline{\nu}}_{p}$, it should never be evaluated by subtracting
${\overline{\nu}}_{p}$ from $\overline{\nu}$.

2\. When a polynomial representation is used to describe the data in
File 1 MT=452, the covariance file applies to the tabular reconstruction
of the data as a function of energy and not to the polynomial
coefficients.

3\. The ENDF‑6 formats do not provide for covariance references between
different sublibraries except by use of File 30. Therefore, it is not
possible to express the covariances between, for example, the
$\overline{\nu}$ for spontaneous fission of ^252^Cf and the
$\overline{\nu}$(E) for the major fissile materials by use of NC‑type
sub‑subsections with LTY=1, 2, and 3.

4\. In ENDF‑6 formats there is no provision to express uncertainty in
MT=455 for the decay constants for the various precursor families.