**32. FILE 32, COVARIANCES OF RESONANCE PARAMETERS**

**32.1. General Description**

File 32, MT=151, contains the variances and covariances of the resonance
parameters given in File 2, MT=151. The resonance parameters, used with
the appropriate resonance formulae, provide an efficient way to
represent the complicated variations in the magnitudes of the different
resonant partial cross sections, compared to the use of File 3 alone.
Similarly for File 32, the use of the covariances of the resonance
parameters of individual resonances provides an efficient way of
representing the rapid variation over the individual resonances of the
covariances of the partial cross sections. The covariance data of the
processed cross sections include the effects of both File 32 and File 33
within a given energy region, similar to the way the cross sections
themselves are the sum of contributions from File 2 and File 3.

In the resonance region, the covariances of the partial cross sections
are often characterized by a) \"long‑range\" components that affect the
covariances over many resonances, and b) \"short‑range\" components
affecting the covariances of the different partial cross sections in the
neighborhood of individual resonances. The former often can best be
represented in File 33, while the latter can be given in File 32.

When the material composition is dilute in the nuclide of concern and
the cross sections are to be averaged over an energy region that
includes many resonances, the effects of \"short‑range\" components are
unimportant and the covariances of the averaged cross sections can be
well represented by processing the long‑range components given in File
33. Therefore, the covariances of the cross sections in the unresolved
resonance energy region should be given entirely by means of File 33
unless resonance self‑shielding in this energy region is thought to be
of practical significance for a particular nuclide. For many nuclides
these conditions may also be valid in the high‑energy portion of the
resolved resonance energy region.

In the resolved resonance region it may be necessary to calculate
covariances for the resonance self‑shielding factors to obtain the
uncertainty in the Doppler effect. As another example, one may require
group cross-section covariances where the groups are narrow compared to
the resonance width or where only a few resonances are within a group.
In these cases File 32 should be used. Because this situation may be
important only in the lower energy portion of the resolved resonance
region, File 32 need not include the whole set of resonances given in
File 2. File 33 remains available for use in combination with File 32.

The ENDF‑6 formats for File 32 are structured to maintain compatibility
with those of ENDF/B‑V, and there are new features to permit
representation of covariance components among the parameters of
different resonances. Covariances between resonance energies and widths
are now also allowed. While File 32 was limited in ENDF‑5 to the
Breit‑Wigner representations (LRF=1 or 2), in ENDF‑6 formats covariances
may also be given for the Reich‑Moore (LRF=3) and Adler‑Adler (LRF=4)
formulations. A limited representation is offered for unresolved
resonance parameter covariance data. The conventions are retained that
the cross section covariances for the resonance region are combined from
covariance data in File 32 and File 33, and that relative covariances
given in File 33 apply to the cross sections reconstructed from File 2
plus File 3 when that option is specified in File 2. (There is a partial
exception for LRF=4.) Since the ENDF‑6 formats do not allow a continuous
range of values for the total angular momentum J of a resonance,
covariance data for the resonance spin are no longer recognized.

Note that File 32 formats retain many restrictions. For example, there
is no provision for representing covariances between the parameters of
resonances in two different materials or in two isotopes within the same
elemental evaluation. Since File 32 can become cumbersome if many
resonances are treated, evaluators will do so only for nuclides of
greatest practical importance.

The strategy employed for File 32 is similar to that for smooth cross
sections in that the variance of a resonance parameter or the covariance
between two such parameters can be given as a sum of several components.
Some contributions can be labeled by resonance energy and parameter
type, while others can arise from long‑range covariances among
parameters of the same type for different resonances in the same
isotope. The latter are labeled by energy bands, and the same
uncertainty characteristic is applied to the indicated parameter of all
the File 2 resonances in a given band.

The idea of assigning the same relative covariance to all parameters of
a given type in an energy region has limited validity. One limit arises
because long‑range uncertainties in reaction yields don\'t generally
carry over proportionately to uncertainties in the corresponding
reaction widths. However, gamma‑ray widths are sometimes known only for
resonances at low neutron energy, and then the average of these values
is used for the resonances at higher energy. This situation motivates
formats resembling File 33 except that a relative uncertainty in the
file applies to the indicated parameter (*e.g.*, Γ~γ~) of every
resonance in the indicated energy range. The approach here is to allow
the long range correlations to extend over any energy interval in which
consistent resolved resonance formulations are utilized.

The definitions of common quantities are as given in Chapter 2.

**32.2. Formats**

The format for File 32, MT=151, parallels the format for File 2, MT=151,
with the restriction to LRF=1,2,3 or 4 for LRU=1 (resolved parameters)
and to LRF=1 for LRU=2 (unresolved parameters).

The general structure of File 32 is as follows:

> \[MAT,32,151/ ZA, AWR, 0, 0, NIS, 0\] HEAD
>
> \[MAT,32,151/ ZAI, ABN, 0, 0, LFW, NER\] CONT (isotope)
>
> \[MAT,32,151/ EL, EH, LRU, LRF, NRO, NAPS\] CONT (range)
>
> \<subsection for the first energy range for the first isotope\>
>
> \[MAT,32,151/ EL, EH, LRU, LRF, NRO, NAPS\] CONT (range)
>
> \<subsection for the second energy range for the first isotope\>
>
> \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--
>
> \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--
>
> \[MAT,32,151/ EL, EH, LRU, LRF, NRO, NAPS\] CONT (range)
>
> \<subsection for the last energy range for the first isotope\>
>
> \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--
>
> \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--
>
> \[MAT,32,151/ ZAI, ABN, 0, 0, LFW, NER\] CONT (isotope)
>
> \[MAT,32,151/ EL, EH, LRU, LRF, NRO, NAPS\] CONT (range)
>
> \<subsection for the first energy range for the last isotope\>
>
> \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--
>
> \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--
>
> \[MAT,32,151/ EL, EH, LRU, LRF, NRO, NAPS\] CONT (range)
>
> \<subsection for the last energy range for the last isotope\>
>
> \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--
>
> \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--
>
> \[MAT,32, 0/ 0.0, 0.0, 0, 0, 0, 0\] SEND
>
> \[MAT, 0, 0/ 0.0, 0.0, 0, 0, 0, 0\] FEND

Data are given for all ranges for a given isotope, then for successive
isotopes. The data for each isotope start with a CONT (isotope) record;
those for each range with a CONT (range) record. File segments need not
be included for all isotopes represented in the corresponding File 2.

If the \"range\" record preceding a subsection has NRO0, indicating that
the energy dependence of the scattering radius is given in File 2, the
initial file segment within the subsection has the form:

> \[MAT,32,151/ 0.0, 0.0, 0, 0, 0, NI\] CONT
>
> \<NI sub‑subsections as defined in Chapter 33.2 for the
>
> energy‑dependent covariances of the scattering radius\>

The next record of a subsection (the first record if NRO=0) has the
form:

\[MAT,32,151/ SPI, AP, 0, LCOMP, NLS, 0\] CONT.

If the compatibility flag, LCOMP, is zero, NLS is the number of L‑values
for which lists of resonances are given in a form compatible with that
used in ENDF‑5, as described in Section 32.2.1. If LCOMP, then NLS=0 and
subsection formats are as shown in Sections 32.2.2. If LCOMP, the
covariance matrix is expressed in compact form as described in Sect.
32.3 and 32.4.

**32.2.1 Compatible resolved resonance subsection format.**

This format differs from that used for ENDF‑5 only in that covariances
of the resonance spin are all zero. It is applicable only for resolved
parameters (LRU=1) and for the Breit‑Wigner formalisms (LRF=1 or 2). The
following covariance quantities are defined:

  ------------ ----------------------------------------------------------
  **DE^2^**    Variance of the resonance energy in units eV^2^.

  **DN^2^**    Variance of the neutron width GN in units eV^2^.

  **DNDG**     Covariance of GN and GG in units eV^2^.

  **DG^2^**    Variance of the gamma‑ray width in units eV^2^

  **DNDF**     Covariance of GN and GF in units eV^2^.

  **DGDF**     Covariance of GG and GF in units eV^2^.

  **DF^2^**    Variance of the fission width GF in units eV^2^.

  **DJDN**     (null) covariance of resonance J‑value and GN.

  **DJDG**     (null) covariance of resonance J‑value and GG.

  **DJDF**     (null) covariance of resonance J‑value and GF.

  **DJ^2^**    (null) variance of resonance J‑value.
  ------------ ----------------------------------------------------------

A complete subsection for LCOMP = 0 has the following form:

> \<If NRO is not 0, a control record and NI sub‑subsections for
> energy‑dependent
>
> covariances of the scattering radius\>

\[MAT,32,151/ SPI, AP, 0, LCOMP, NLS, 0\] CONT (LCOMP=0)

\[MAT,32,151/AWRI, 0.0, L, 0, 18\*NRS, NRS/

> ER~1~, AJ~1~, GT~1~, GN~1~, GG~1~, GF~1~,
>
> DE~1~^2^, DN~1~^2^, DNDG~1~, DG~1~^2^, DNDF~1~, DGDF~1~,
>
> DF~1~^2^, DJDN~1~, DJDG~1~, DJDF~1~, DJ~1~^2^, 0.0,
>
> (DJDN~1~,DJDG~1~,DJDF~1~=0.0)
>
> \<three physical records for each resonance through resonance NRS\> \]
> LIST
>
> \<a similar list record for each additional value of L through NLS\>

### 

### Note that in this compatible format no covariance can be given between parameters of different resonances even if they overlap.

**32.2.2 General resolved resonance subsection formats.**

Following the record that starts with the target spin, SPI, if it states
LCOMP=1, the next card of a subsection defines for that isotope and
energy range how many (NSRS) sub‑sections will occur for covariances
among parameters of specified resonances and how many (NLRS)
sub‑sections are to contain data on long‑range parameter covariances.
The complete structure of an LCOMP=1 subsection is as follows:

\<f NRO0, a control record and NI sub‑subsections for energy‑

dependent covariances of the scattering radius\>

> \[MAT,32,151/ SPI, AP, 0, LCOMP, NLS, 0\] CONT (LCOMP=1,NLS=0)
>
> \[MAT,32,151/AWRI, 0.0, 0, 0, NSRS, NLRS\] CONT
>
> \<NSRS sub‑subsections each giving covariances among specified
>
> parameters of enumerated sets of resonances\>
>
> \<NLRS sub‑subsections each giving long‑range covariances
> contributions
>
> for stated energy regions for a particular type of resonance
> parameter\>

# Covariance formats for NSRS‑type sub‑subsections

The formats here differ from the LCOMP=0 formats of Section 32.2.1 in
that covariance between parameters of different resonances appear,
resonance representations LRF=1 to 4 are allowed, there is no
segregation by L‑value, and the number of parameters considered per
resonance is declared in each sub‑subsection primarily to avoid
tabulating zero covariances for fission widths in files concerning
structural materials. The listed resonances must be present in File 2,
but there is no requirement that all resonances be included in File 32
that are given in File 2.

The following paragraphs cover the formats for the NSRS sub‑subsections
for the various (LRF) resonance formulations for resolved parameters
(LRU=1). LRF=1 through 4 are allowed.

[LRF=1,2, LRU=1.]{.underline} These cases have the same formats. All the
resonances for which covariances are to be included are divided into
blocks. Covariances between parameters can only be included for
resonances in the same block. (A given resonance can appear in more than
one block.) A sub‑subsection covers the data for a block. Within each
such block, the parameters are given of those resonances for which
covariances are included, and the upper triangular representation of the
parameter covariance matrix is included for the whole block of
resonances. For each block one specifies the number MPAR of parameters
to be included for each listed resonance in the block, since for most
cases there is, for example, no fission width and the size of the
covariance matrix can therefore be minimized.

The NSRS‑type sub‑sections have the form below for LRF=1 and 2:

> \[MAT,32,151/ 0.0, 0.0, MPAR, 0, NVS+6\*NRB,.NRB/
>
> Parameters for ER~1~, AJ~1~, GT~1~, GN~1~, GG~1~, GF~1~,
>
> each resonance for ...
>
> which there is ...
>
> covariance data. ER~NRB~, AJ~NRB~, GT~NRB~, GN~NRB~, GN~NRB~, GG~NRB~,
> GF~NRB~,
>
> Resonance parameter V~11~, V~12~, \.... V~1,MPAR\*NRB~, V~22~,
>
> variances and \..., V~2,NPAR\*NRB~, V~33~,
>
> covariances for all \..., V~MPAR\*NRB,MPAR\*NB~, 0.0, 0.0\] LIST
>
> resonances listed.

Note that the record first lists all the parameters of resonances in the
group, for positive identification, and then lists the covariance terms.

+-----------+----------------------------------------------------------+
| **MPAR**  | Number of parameters per resonance in this block which   |
|           | have covariance data (In order: ER, GN, GG, GF, GX,      |
|           | indices 1‑5)                                             |
+-----------+----------------------------------------------------------+
| **NVS**   | Number of covariance elements listed for this block of   |
|           | resonances,                                              |
|           |                                                          |
|           | NVS = \[NRBMPAR(NRBMPAR+1)\]/2.                          |
+-----------+----------------------------------------------------------+
| **NRB**   | Number of resonances in this block and for which         |
|           | resonance parameter and covariance data are given in     |
|           | this sub‑section.                                        |
+-----------+----------------------------------------------------------+
| **ER**    | Lab system energy of the k^th^ resonance (in this        |
|           | block).                                                  |
+-----------+----------------------------------------------------------+
| **AJ~k~** | Floating‑point value of the spin for the k^th^           |
|           | resonance.                                               |
+-----------+----------------------------------------------------------+
| **GT~k~** | Total width (eV) for the k^th^ resonance.                |
|           | GT=GN+GG+GF+GX.                                          |
+-----------+----------------------------------------------------------+
| **GN~k~** | Neutron width of the k^th^ resonance.                    |
+-----------+----------------------------------------------------------+
| **GG~k~** | Gamma‑ray width for the k^th^ resonance.                 |
+-----------+----------------------------------------------------------+
| **GF~k~** | Fission width of the k^th^ resonance.                    |
+-----------+----------------------------------------------------------+
| **V~mn~** | Variance (eV^2^) or covariance matrix element, row m and |
|           | column, n m. If j MPAR is the parameter index (j=4 for   |
|           | fission width) for the k^th^ resonance in the block, m = |
|           | j + (k‑1)MPAR. The indexing is the order defined above   |
|           | under MPAR.                                              |
+-----------+----------------------------------------------------------+

[LRF=3, LRU=1.]{.underline} The representation for the Reich‑Moore
resonance formulation is very similar. The listing of the parameters
among which covariance terms are allowed takes the form used in File 2,
namely:

ER, AJ, GN, GG, GFA, GFB.

Similarly, in interpreting the indices for the covariance matrix of the
parameters in a given block of resonances, the order when LRF=3 is ER,
GN, GG, GFA, GFB and the largest possible value of MPAR is 5.

[LRF=4, LRU=1]{.underline}. The Adler‑Adler resonance representation
includes background constants as well as resonance parameters. The
uncertainty in the background constants is to be treated indirectly just
as the smooth background given in File 3. (That is, in the energy region
EL, EU for LRF=4 any relative uncertainty data in File 33 applies to the
sum of the cross section in File 3 and the contribution of the cross
sections computed from the Adler‑Adler background constants. Great care
will be required if this uncertainty representation for Adler‑Adler fits
is used in other than single‑isotopes evaluations.) The inherent
assumption is that covariance data will be detailed for the largest
resonances, and those representing the Adler‑Adler backgrounds. It is
assumed that LI=7.

An LRF=4 NSRS sub‑subsection takes the following form, using the
previous definitions where possible:

> \[MAT,32,151/ 0.0, 0.0, MPAR, 0,NVS+6\*NRB, NRB/

DET~1~, DWT~1~, GRT~1~, GIT~1~, DEF~1~, DWF~1~,

> Parameters for GRF~1~, GIF~1~, DEC~1~, DWC~1~, GRC~1~, GIC~1~,
>
> each reference
> **\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--**
>
> in the block. DET~NRB~, DWT~NRB~, GRT~NRB~, GIT~NRB~, DEF~NRB~,
> DWF~NRB~,
>
> GRF~NRB~, GIF~NRB~, DEC~NRB~, DWC~NRB~, GRC~NRB~, GIC~NRB~,
>
> Resonance parameter V~11~, V~12~, \-\-\-\-\-\-\-\-\-\-\-\--,
> V~1,MPAR\*NRB~, V~22~,
>
> variances for all \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--,
> V~2,NPAR\*NRB~, V~33~,
>
> resonances listed. \-\-\-\-\-\-\-\-\--, V~MPAR\*NRB,MPAR\*NB~, 0.0,
> 0.0\] LIST

The Adler‑Adler parameters of the selected list of resonances are given
in the same redundant style indicated in section 2.2.1.3. For LRF=4 the
maximum value of MPAR is 8 and for a given resonance the covariance
matrix indexing is in the order:

> μ = DET = DEF = DEC, DWT = DWF = DWD = v,
>
> GRT, GIT, GRF, GIF, GRC, GIC

[Format for Long‑Range Covariance Sub‑subsections (LRU=1)]{.underline}.
Here are described the forms that the sub‑subsections may take to
represent long‑range (in energy) covariances among parameters of a given
type. The strategy is to use formats that resemble those for File 33 but
refer to a particular parameter and equally to all resonances of a given
isotope within the indicated energy regions.

Here each sub‑subsection must identify the resonance parameter
considered (Γ~n~, Γ~γ~, *etc*.) via the parameter IDP, indicate the
covariance pattern via a value of LB, and give the energy regions and
covariance components. The following table defines the permitted values
of LB. Note the definition of one LB value not defined in File 33, and
that LB=3 and 4 are not employed in File 32 because of LB=5 is typically
much more convenient. The term COV\[Γ~α~(i),Γ~α~(j)\] is defined as the
covariance between the Γ~α~ parameters for two different resonances
(indexed i and j) in the same nuclide.

> ![](media/image1.wmf)**LB=‑1** Relative variance components entirely
> uncorrelated from resonance‑to‑ resonance but having constant
> magnitude within the stated energy intervals.
>
> ![](media/image2.wmf)**LB=0** Absolute covariance components
> correlated only within each E interval.

![](media/image3.wmf)**LB=1** Relative covariance components correlated
only within each E interval

> **LB=2** Fractional covariance components fully correlated over all E
> intervals with variable magnitude.
>
> ![](media/image4.wmf)
>
> **LB=5** Relative covariance components in an upper triangular
> representation of a symmetric ![](media/image5.wmf)matrix.

Note that the F^(--1)^, F^(1)^, and F^(5)^ parameters have the
dimensions of relative covariances, the F^(0)^ are absolute covariances,
and the F^(2)^ are relative standard deviations.

#### 

#### The S and P are dimensionless operations defined in Section 33.2

The sub‑subsection format for LB=--1, 0, 1, or 2 is:

\[MAT,32,151/ 0.0, 0.0, IDP, LB, 2\*NE, NE/{E~k~,F^(LB)^}\] LIST

#### For LB=5 the requisite format is

\[MAT,32,151/ 0.0, 0.0, IDP, LB, NT, NE/{E~k~},{F~kk~^(5)^}\] LIST
(LB=5)

In the above:

  ----------- -----------------------------------------------------------
  **LB**      Indicator of covariance pattern as defined above.

  **NE**      Number of energies in the parameter table for a given
              sub‑section.

  **NT**      NE\*(NE+1)/2.

  **IDP**     Identification number of a resonance parameter type. This
              index depends on the resonance formulation used, as
              summarized in the table below. This table is consistent
              with the order of parameters used in the NSRS‑type
              sub‑subsections to define the parameter covariances
              included, the first NPAR parameters in the list for the
              given LRF value.
  ----------- -----------------------------------------------------------

+-----------+-------------+-------------+-------------+-------------+
| ##        | LRF         |             |             |             |
+-----------+-------------+-------------+-------------+-------------+
| ## IDP    | **1**       | **2**       | **3**       | **4**       |
+-----------+-------------+-------------+-------------+-------------+
| 1         | E~r~        | E~r~        | E~r~        | E~r~=μ      |
+-----------+-------------+-------------+-------------+-------------+
| 2         | Γ~n~        | Γ~n~        | Γ~n~        | v           |
+-----------+-------------+-------------+-------------+-------------+
| 3         | Γ~γ~        | Γ~γ~        | Γ~γ~        | GRT         |
+-----------+-------------+-------------+-------------+-------------+
| 4         | Γ~f~        | Γ~f~        | Γ~f~        | GIT         |
+-----------+-------------+-------------+-------------+-------------+
| 5         | Γ~x~        | Γ~x~        | Γ~x~        | GRF         |
+-----------+-------------+-------------+-------------+-------------+
| 6         |             |             |             | GIF         |
+-----------+-------------+-------------+-------------+-------------+
| 7         |             |             |             | GRC         |
+-----------+-------------+-------------+-------------+-------------+
| 8         |             |             |             | GIC         |
+-----------+-------------+-------------+-------------+-------------+

For LB=0, ±1, 2, and 5 the energy values in the tables are monotonically
increasing and cover the range EL to EH. For LB=5 the F‑values are given
in the upper triangular representation of a symmetric matrix by rows,
*i.e.* F~11~, F~12~, \..., F~1,NE-1~; F~22~, F~2,NE-1~; F~NE-1,NE-1~

**32.2.3 Unresolved resonance subsection format (LRU=2)**

For the unresolved resonance region a simplified covariance formulation
is permitted. For the purposes of covariance representation, no energy
dependence is specified for the average parameter relative covariances,
even though in file 2 the unresolved region may be represented with
energy‑dependent average Breit‑Wigner parameters using LRF=2. Relative
covariance elements are tabulated, unlike the cases above.

If the evaluator wishes to represent the relative covariance of the
unresolved resonance parameters, the subsection for a given isotope has
the following form.

\[MAT,32,151/ SPI, AP, 0, 0, NLS, 0\] CONT

\[MAT,32,151/ AWRI, 0.0, L, 0, 6\*NJS, NJS/

> D~1~, AJ~1~, GNO~1~, GG~1~, GF~1~, GX~1~,
>
> \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--
>
> D~NJS~, AJ~NJS~, GNO~NJS~, GG~NJS~, GF~NJS~, GX~NJS~\] LIST
>
> \<NJS card images for each L value in successive LIST records\>

\[MAT,32,151/ 0.0, 0.0, MPAR, 0,(NPAR\*(NPAR+1))/2,NPAR/

> RV~11~, RV~12~, \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--,
>
> \-\-\-\-\-\-\-\-\-\--, RV~1,NPAR~,\-\-\-\-\-\-\--, RN~NAPR,NPAR~\]
> LIST

MPAR is the number of average parameters for which relative covariance
data are given for each L and J, in the order D, GNO, GG, GF, and GX,
for a maximum of 5. That is, relative covariance values for the first
MPAR of these are tabulated for each (L,J) combination. If MPAR is given
as 4 when LFW=0 on the CONT (isotope) record, then the four covariance
matrix indices per (L,J) combination represent D,GNO, GG, and GX.

NPAR=MPAR\*(sum of the values of NJS for each L).

The LSSF flag is defined in Section 2.3.1.

RV~ij~, the relative covariance quantities among these average
unresolved parameters for the given isotope. The final LIST record
contains the upper triangular portion of the symmetric relative
covariance matrix by rows.

**32.3. General Description of Compact Covariance Matrix**.

This format was developed in order to provide a means of communicating
the actual resonance-parameter covariance matrix (as determined during
the data-evaluation process) in a compact, legible, and accurate form,
for those situations in which it is not practical to provide the entire
covariance matrix using the LCOMP=1 format. The covariance matrix is
specified as uncertainties plus correlation matrix. The correlation
coefficients (whose values range from -1 to 1) are presented in a
compact representation; each approximated correlation coefficient
differs from the original value by at most 0.005.

If the covariance matrix element connecting parameter number *i* with
parameter number *j* is denoted by *V~ij~*, the uncertainty on parameter
*i* by *D~i~*, and the correlation coefficient by *C~ij~*, then these
quantities are related by

> *D~i~^2^* = *V~ii~* *V~ij~* = *D~i~* *C~ij~* *D~j~*

Values for *C~ij~* range from -1 to +1; values for *D~i~* are always
positive. Note that the diagonal elements of *C~ij~*, those for which
*i* = *j*, are always exactly 1.0 and therefore are never specified
explicitly. Compacting the off-diagonal correlation coefficients is
accomplished as follows:

1\. Drop (set to zero) all values of *C~ij~* between -0.02 and +0.02.

2\. Multiply the remaining coefficients by 100.

3\. Map all positive values greater than K and less than or equal to K+1
to the integer K.

4\. Map all negative values less than -K and greater than or equal to
-K-1 to the integer -K.

The reverse mapping takes the integer to the center of the range. For
example, the positive integer 87 (which corresponds to original
correlation coefficients in the range from 0.87 up to and including
0.88) maps to 0.875. The negative integer -12 (which corresponds to
original correlation coefficients in the range from -0.12 inclusive down
to -0.13 exclusive) maps to correlation coefficient -0.125.

Two types of records are used to read the mapped correlation
coefficients. A CONT record defines NNN (the number of parameters) and
NM (the number of lines of INTG records that follow).

**\[MAT, 32, 151 / 0.0, 0.0, 0, NNN, NM, NM\] CONT**

**\[MAT, MF, MT / II, JJ, KIJ \] INTG**

**\[MAT, MF, MT / II, JJ, KIJ \] INTG**

\< Continue until a total of NM INTG records are read \>

The information stored in the INTG records may be summarized as follows:
Let *i* and *j* represent two of the parameters in the numbering scheme
defined above, with *i* \> *j*. All those correlation coefficients
mapped to non-zero integers *K~ij~* are printed in File 32; zero-valued
*K~ij~* are printed (as blanks or zeros) only when they occur on the
same record with non-zero values. Each line (record) in the file begins
by specifying the location (i.e., by specifying *i* and *j*, with *i* \>
*j*); other *K*=s on the same line correspond to (*i*, *j* +1), (*i*,
*j* +2), \... (*i*, *j* +17) \[so long as *j* + 17 \< *i* \]. If there
are more non-zero *K*=s for the same *i*, they are given on another
line, again beginning with the next non-zero *K*.

The following FORTRAN can be used to read the mapped correlation
coefficients and reconstruct the correlation matrix:

DIMENSION K(18), CORR(n,n)

C Note that n must be greater than or equal to NNN

READ (LIB,10) NNN, NM, NX, MAT, MF, MT, NS

10 FORMAT (33X, 3I11, I4, I2, I3, I5)

C The CONT record: NNN is dimension of CORR(NNN,NNN), and

C NM is the number of lines to follow in the file

DO I=1,NNN

DO J=1,I-1

CORR(J,I) = 0.0

END DO

CORR(I,I) = 1.0

END DO

DO M=1,NM

READ (LIB,20) I, J, (K(N),N=1,18), MAT, MF, MT, NS

20 FORMAT (I5, I5, 1X, 18I3, 1X, I4, I2, I3, I5)

C This is the INTG record

JP = J -- 1

DO N=1,18

JP = JP + 1

IF (JP.GE.I) GO TO 30

IF (K(N).NE.0) THEN

IF (K(N).GT.0) THEN

CORR(JP,I) = (K(N)+0.5)/100.0

ELSE

CORR(JP,I) = - (-K(N)+0.5)/100.0

END IF

END IF

END DO

30 CONTINUE

END DO

Note that only the upper-triangular portion of the correlation matrix
\[CORR(J,I) for J \< I \] is defined by this procedure. The
lower-triangular portion can be found by invoking symmetry \[CORR(I,J) =
CORR(J,I)\].

**32.4 Formats for Compact Covariance Matrix**

The compact format for covariances uses File 32, MT=151, with LCOMP = 2
(with LRF  = 1, 2, 3, and 7) for LRU=1 (resolved-resonance parameters).
The format parallels the format for File 2, MT = 151, with parameter
uncertainties specified along with parameter values. Correlation
coefficients are given separately.

The general structure of File 32 is as given in Section 32.2.

**\[MAT, 32, 151 / ZA, AWR, 0, 0, NIS, 0\]HEAD**

**\[MAT, 32, 151 / ZAI, AWR, 0, LFW, NER, 0\]CONT** (isotope)

**\[MAT, 32, 151 / EL, EH, LRU, LRF, NIS, NAPS\]CONT** (range)

\<subsection for the first energy range for the first isotope\>

**\[MAT, 32, 151 / EL, EH, LRU, LRF, NIS, NAPS\]CONT** (range)

\<subsection for the second energy range for the first isotope\>

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

**\[MAT, 32, 151 / EL, EH, LRU, LRF, NIS, NAPS\]CONT** (range)

\<subsection for the last energy range for the first isotope\>

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

**\[MAT, 32, 151 / ZAI, AWR, 0, LFW, NER, 0\]CONT** (isotope)

**\[MAT, 32, 151 / EL, EH, LRU, LRF, NIS, NAPS\]CONT** (range)

\<subsection for the first energy range for the last isotope\>

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

**\[MAT, 32, 151 / EL, EH, LRU, LRF, NIS, NAPS\]CONT** (range)

\<subsection for the last energy range for the last isotope\>

**\[MAT, 32, 0 / 0.0, 0.0, 0, 0, 0, 0\]SEND**

**\[MAT, 0, 0 / 0.0, 0.0, 0, 0, 0, 0\]FEND**

Formats for the individual subsections differ depending on the value of
LRF, and are discussed in Sections 32.4.1.

**32.4.1 Compact covariance matrix formats for subsections of a
particular energy range and isotope**

In the format descriptions to follow, the notation is as defined in
Section 2. Uncertainties are denoted by parameters whose names begin
with AD@ but are otherwise identical to the parameter of Section 2. For
example, ER represents the resonance energy in the laboratory system, in
units of eV. Therefore, DER represents the uncertainty (square root of
the variance) associated with the resonance energy; DER has the same
units as ER.

Parameters which are not variables (that is, which are not searchable
parameters in the analysis process) do not have uncertainties associated
with them. For example, the total spin of a resonance, denoted by
parameter AJ, has no associated uncertainty DAJ. In File 32, the value
0.0 is given instead of DAJ (for LRF = 1,2,3).

For LRF = 1 and 2, the redundant parameter GT (which is equal to GN +
GG + GF) has no corresponding uncertainty DGT specified explicitly.
Instead, if needed, DGT may be calculated using values for DGN, DGG, and
DGF, and the correlation matrix.

During the evaluation process, not all potential variables are treated
as searchable parameters. For example, if no capture cross section data
were available, the evaluator might choose to set all capture widths GG
to a constant value. The associated uncertainty DGG is then specified in
File 32 as -1.0, indicating that this parameter=s uncertainty is not
known. (The proper procedure to be used in evaluating the effect of
these unvaried parameters on the final covariance matrix remains an open
question, and is not addressed in this document.)

In order to express the correlation matrix as compactly as possible, the
resonance parameters (those which may be varied during the evaluation
process) are implicitly numbered, in the order in which they occur in
the listing of File 2. For LRF = 1, 2, or 3, the non-searchable
parameter AJ is included in the list but is NOT included in this
numbering, nor (for LRF=1 or 2) is the redundant parameter GT.
Parameters whose value is given but whose uncertainty is unknown (as
described in the previous paragraph) are nevertheless included in the
numerical ordering. For completely non-fissile nuclides, fission widths
are not included in the numbering scheme; likewise, for fissile or
fissionable nuclides for which the evaluator chose to use only one
fission width, the second fission width (for LRF = 3) would not be
counted.

Notation used here not previously defined:

**NRSA** Total number of resonances for all L values

**32.4.1.2** **SLBW and MLBW (LRU=1, LRF=1 or 2)**

The structure of the *subsection* (assuming NRO = 0) is:

**\[MAT, 32, 151 / SPI, AP, 0,LCOMP, NLSA, 0\] CONT**

**\[MAT, 32, 151 /AWRI, QX, 0, 0,12\*NRSA, NRS /**

**ER~1~, AJ~1~, GT~1~, GN~1~, GG~1~, GF~1~,**

**DER~1~, 0.0 , 0.0, DGN~1~, DGG~1~, DGF~1~,**

**ER~2~, AJ~2~, GT~2~, GN~2~, GG~2~, GF~2~,**

**DER~2~, 0.0 , 0.0, DGN~2~, DGG~2~, DGF~2~,**

**\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--**

**ER~NRS~, AJ~NRS~, GT~NRS~, GN~NRS~, GG~NRS~, GF~NRS~**

**DER~NRS~, 0.0, 0.0, DGN~NRS~, DGG~NRS~, DGF~NRS~ \] LIST**

**\[MAT, 32, 151 / 0.0, 0.0, 0, NNN, NM, 0 \] CONT**

**\[MAT, MF, MT / II, JJ, KIJ \] INTG**

**\[MAT, MF, MT / II, JJ, KIJ \] INTG**

\< Continue until a total of NM INTG records are read \>

Note that NNN is the total number of unique resonance parameters
included in this listing, not including AJ or GT. For fissile nuclides,
NNN = NRS 4; for non-fissile nuclides, NNN = NRS 3.

**32.4.1.3 Reich Moore Limited (LRU=1, LRF=7)**

The structure of the *subsection* is

**\[MAT, 32, 151/ SPI, AP, LAD, LCOMP, NLS, NLSC\] CONT**

**\[MAT, 32, 151/AWRI, 0, 0, 0, 12\*NRSA, NRSA /**

**ER~1~, AJ~1~, GN~1~, GG~1~, GFA~1~, GFB~1~,**

**DER~1~, 0.0 , DGN~1~, DGG~1~, DGFA~1~, DGFB~1~,**

**ER~2~, AJ~2~, GN~2~, GG~2~, GFA~2~, GFB~2~,**

**DER~2~, 0.0 , DGN~2~, DGG~2~, DGFA~2~, DGFB~2~,**

**\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--**

**ER~NRS~, AJ~NRS~, GN~NRS~, GG~NRS~, GFA~NRS~, GFB~NRS~,**

**DER~NRS~, 0.0, DGN~NRS~, DGG~NRS,~ DGFA~NRS~, DGFB~NRS~ \] LIST**

**\[MAT, 32, 151 / 0.0, 0.0, 0, NNN, NM, 0 \] CONT**

**\[MAT, MF, MT / II, JJ, KIJ \] INTG**

**\[MAT, MF, MT / II, JJ, KIJ \] INTG**

\< Continue until a total of NM INTG records are read \>

Note that NNN is the total number of resonance parameters in this
listing (again not including AJ). For non-fissile nuclides, NNN = NRSA
3. For fissile nuclides, NNN = NRSA 4 if only one fission channel is
used, and NNN = NRSA 5 if both are used.

**32.4.1.3 Reich Moore Limited (LRU=1, LRF=7 )**

The structure of the *subsection* is

**\[MAT,32,151/ 0.0, 0.0, 0,LCOMP, NJS, 0 \] CONT**

**\[MAT,32,151/ 0.0, 0.0, NPP, 0, 12\*NPP, 2\*NPP/**

**MA~1~, MB~1~, ZA~1~, ZB~1~, IA~1~, IB~1~,**

**Q~1~, PNT~1~, SHF~1~, MT~1~, PA~1~, PB~1~,**

**MA~2~, MB~2~, ZA~2~, ZB~2~, IA~2~, IB~1~,**

**Q~2~, PNT~2~, SHF~2~, MT~2~, PA~2~, PB~1~,**

**\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--**

**A~NPP~, MB~NPP~, ZA~NPP~, ZB~NPP~, IA~NPP~, IB~NPP~,**

**Q~NPP~, PNT~NPP~, SHF~NPP~, MT~NPP~, PA~NPP~, PB~NPP~ \] LIST**

**\[MAT,32,151/ AJ, PJ, 0, 0, 6\*NCH, NCH/**

**IPP~1~, L~1~, SCH~1~, BND~1~, APE~1~, APT~1~,**

**IPP~2~, L~2~, SCH~2~, BND~2~, APE~2~, APT~2~,**

**\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--**

**IPP~NCH~, L~NCH~, SCH~NCH~, BND~NCH~, APE~NCH~, APT~NCH~ \] LIST**

**\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--**

**\[MAT,32,151/0.0, 0.0, 0, NRS, 12\*NX, NX/**

**R~1~, GAM~1,1~, GAM~2,1~, GAM~3,1~, GAM~4,1~, GAM~5,1~,**

**GAM~6,1~, \..., GAM~NCH,1~,**

**DER~1~, DGAM~1,1~, DGAM~2,1~, DGAM~3,1~, DGAM~4,1~,DGAM~5,1~,**

**DGAM~6,1~, \..., DGAM~NCH,1~,**

**ER~2~, GAM~1,2~, GAM~2,2~, GAM~3,2~, GAM~4,2~, GAM~5,2~,**

**GAM~6,2~, \..., GAM~NCH,2~,**

**DER~2~, DGAM~1,2~, DGAM~2,2~, DGAM~3,2~, DGAM~4,2~,DGAM~5,2~,**

**DGAM~6,2~, \..., DGAM~NCH,2~,**

**\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--**

**ER~NRS~, GAM~1,NRS~, GAM~2,NRS~, GAM~3,NRS~, GAM~4,NRS~, GAM~5,NRS~,**

**GAM~6,NRS~, \..., GAM~NCH,NRS~**

**DER~NRS~,DGAM~1,NRS~,DGAM~2,NRS~,DGAM~3,NRS~,DGAM~4,NRS~,DGAM~5,NRS~,**

**DGAM~6,NRS~, \..., DGAM~NCH,NRS~ \] LIST**

\<The above two list records are repeated until each of

the NJS *J ^π^* states has been specified.\>

**\[MAT, MF, MT / II, JJ, KIJ \] INTG**

**\[MAT, MF, MT / II, JJ, KIJ \] INTG**

\< Continue until a total of NM INTG records are read \>

For LRF = 7 (unlike other formats), the number of channels may vary from
one spin group to another. The number of resonance parameters NNN is
therefore given by the sum (from 1 to NJS) of ( NCH NRS ).

**32.5 Procedures**

As indicated earlier, it is desirable to utilize File 32 when self
shielding is important or when only a few resonances fall within an
energy group of the processed cross sections. It is believed that in
most cases the covariances the evaluator needs to represent will not use
many of the available File 32 options. One does not expect to find
covariance data in File 32 for all the resonance parameters in File 2.

[Correspondence Between Files 2 and 32. Completeness of File
32.]{.underline}

1\. The overall energy range for a given isotope in File 32 is nested
within the corresponding range of File 2. In either case, there may be
several energy range control records. The following rules apply
separately for each isotope:

a\. The smallest lower range limit EL for files 2 and 32 must agree.

> b\. The highest upper range limit EH for File 32 may be smaller than
> or equal to that for File 2.
>
> c\. In file 32 as in File 2, the energy ranges of the subsections may
> not leave gaps. Subsections with LRU=1 may overlap if consistent
> resonance formulations are referenced (LRF=1 or 2).
>
> d\. An unresolved energy region (LRU=2) may be used in File 32 if one
> is employed in File 2. If one is used, its lower energy range limit
> must equal the corresponding limit for File 2.

2\. In a file 32 LCOMP=0 subsection, any selection of the resonances
shown in File 2 with LRF=1 or 2 must be listed in order of increasing
energy. The resonance energy given in File 32 shall agree with that in
File 2 for the same resonance to a relative tolerance of 10^-5^ to
assume positive identification.

3\. In LCOMP=1, LRU=1 subsections, any selection of the resonances shown
in File 2 for consistent resonance formulations may be listed in
NSRS‑type sub‑subsections. The listed resonances should be in order of
increasing energy within each block, and the blocks should be arranged
by order of increasing energy for the lowest‑energy resonance in the
block. A given resonance may appear in more than one block. The
resonance energy given in File 32 shall agree with that in File 2 for
the same resonance to a relative tolerance of 10^-5^.

4\. The energy identifiers of every NLRS‑type sub‑subsection must cover
the entire energy range (EL,EH) of the subsection. However, any desired
covariance components may be null.

5\. NLRS‑type sub‑subsections may be included in any order in LCOMP=1,
LRU=1 subsections.

[Obtaining Cross Section Covariances From File 32 and File
33.]{.underline}

1\. Outside the combined energy range covered by File 32, whether in
part or all of the energy range covered by File 2, the covariance data
in File 33 refers to the cross sections reconstituted from File 2 plus
File 3 according to the value of the LSSF flag.

2.  In LCOMP=0 subsections a non‑zero variance or covariance involving
    the resonance spin J (AJ) will be treated as null.

3\. When File 32 is present for a resolved resonance range (LRU=1), the
covariances among the effective cross sections in the region are
obtained by combining data from Files 32 and 33.

> a\. A resonance parameter covariance matrix is developed by summing
> all the contributions for each resonance from the File 32 subsections
> with LRU=1.
>
> b\. The resonance parameter covariance file is processed to obtain the
> covariance data of effective cross sections implied by File 32. (This
> data will in general be a function of isotopic dilution and
> temperature.) Covariance for the various isotopes are summed with
> appropriate weights if the evaluation is an \"elemental\" one.

c\. Covariances derived from File 32 are summed with those given in File
33. With one exception, relative covariances given in File 33 apply to
the effective cross sections reconstituted from File 2 and File 3. In
that exception, when the Adler‑Adler (LRF=4) resonance formulation is
employed, any relative covariances in File 33 for that energy region
apply to the sum of the cross section from File 3 with the smooth
background given for the LRF=4 resonance data.

4\. When File 32 is present for the unresolved resonance range (LRU=2),
covariance data for the region are obtained as follows:

> a\. File 33 covariance data for this energy region are taken to
> represent the covariances of evaluated average cross sections for
> \"infinite isotopic dilution.\"

b\. To obtain covariances among effective cross sections for material
dilutions such that uncertainties in self shielding can become
important, the effects of the unresolved resonance parameter covariances
in File 32 and the evaluated average cross section covariances in File
33 are to be combined. A means for this combination was described by
deSaussure and Marable \[Ref. 1\]. (Note that covariances of the
effective cross sections in one test case were not much affected by
uncertainties in average parameters. See B.L. Broadhead and H.L. Dodds
\[Ref. 2\].)

[Example For Mat 3333.]{.underline}

The nucleus of concern in this example has resonances represented by
LRF=2 for the energy range (1,50). Five (5) resonances are given in File
2, one of which lies at negative energy, but the resonance at 15 eV is
considered only relative to long‑range uncertainties. Table 32.1 shows
the File 2 for the example, and Table 32.2 shows the corresponding file
32.

In File 32, there are included three NSRS‑type sub‑subsections and four
NLRS‑type sub‑subsections. Of the former, the first refers only to the
negative energy resonance and indicates the following parameter
covariance contribution for that resonance, in eV^2^

  -------------- -------------- -------------- -------------- --------------
  E~r~           1.0            -1.5           0              -0.2

  Γ~n~           -1.5           4.0            0              -1.0

  Γ~γ~           0              0              0              0

  Γ~f~           -0.2           -1.0           0              0.8
  -------------- -------------- -------------- -------------- --------------

This block is mostly to express the negative correlation between neutron
width and apparent resonance energy for this guessed resonance.

The second sub‑subsection refers only to the 5‑eV resonance, and conveys
the following covariance contribution:

  -------------- -------------- -------------- -------------- --------------
  E~r~           0.510^-6^      0              0              0

  Γ~n~                          110^6^         110^6^         0.210^6^

  Γ~γ~                          110^6^         0.0            2.010^6^

  Γ~f~                          0.210^6^       2.010^6^       410^6^
  -------------- -------------- -------------- -------------- --------------

The third block covers Γ~n~ covariances for the 30 and 40 eV resonances.

The first NLRS sub‑subsection gives the Γ~γ~ uncertainty correlated
among all the resonances. The value is assumed to have been determined
solely by analysis of the 5‑eV resonance. Note that the variance for the
5‑eV resonance generated by the sub‑subsection fills the \"hole\" in the
table just above (with the value 1610^-6^). The second sub‑subsection
expresses the expected resonance‑to‑resonance uncorrelated fluctuation
in Γ~γ~. The third and fourth sub‑subsections cover long‑range
uncertainties in Γ~n~ and Γ~f~ ‑ the example for the former might be
hard for the evaluator to defend for a real nuclide.

**Example 32.1.** **File 2 for Sample of File 32**

99.280+3 2.70 +2 0 0 NIS=1 0333332151 HEAD

99.280+3 1.00 +0 0 LFW=0 NER=1 0333332151 CONT

1\. +0 5.00 +1 LRU=1 LRF=2 NRO=0 NAPS=0333332151 CONT

1.0 +0 0.6 +0 0 0 NLS=1 0333332151 CONT

2.7 +2 0.0 +0 L=0 LRX=0 6\*NRS=30 NRS=5333332151

.1.0 +0 1.5 +0 4.04 +0 3.0 +0 0.04 +0 1.0 +0333332151

5.0 +0 1.5 +0 0.07 +0 0.01 +0 0.04 +0 0.02 +0333332151

1.5 +1 0.5 +0 0.08 +0 0.01 +0 0.04 +0 0.03 +0333332151

3.0 +1 1.5 +0 7.04 +0 1.0 +0 0.04 +0 6.0 +0333332151

4.0 +1 1.5 +0 6.04 +0 5.0 +0 0.04 +0 1.0 +0333332151 LIST

0.0 +0 0.0 +0 0 0 0 0333332 0 SEND

0.0 +0 0.0 +0 0 0 0 03333 0 0 FEND

**Example 32.2**. **Sample File 32**

99.280+ 3 2.70 + 2 0 0 NIS=1 0333332151 HEAD

99.280+ 3 1.00 + 0 0 LFW=0 NER=1 0333332151 CONT

1\. + 0 5.00 + 1 LRU=1 LRF=2 NRO=0 NAPS=0333332151 CONT

1.0 + 0 0.6 + 0 0 LCOMP=1 NLS=0 0333332151 CONT

2.7 + 2 0.0 + 0 0 0 NSRS=3 NLRS=4333332151 CONT

0.0 + 0 0.0 + 0 MPAR=4 16[^1] NRB=1333332151

‑1.0 + 0 1.5 + 0 4.04 + 0 3.0 + 0 0.04 + 0 1.0 + 0333332151

1.0 + 0 ‑1.5 + 0 0.0 + 0 ‑0.2 + 0 4.0 + 0 0.0 + 0333332151

‑1.0 + 0 0.0 + 0 0.0 + 0 0.8 + 0 0.0 + 0 0.0 + 0333332151 LIST

0.0 + 0 0.0 + 0 MPAR=4 16^\*^ NRB=1333332151

5.0 + 0 1.5 + 0 0.07 + 0 0.01 + 0 0.04 + 0 0.02 + 0333332151

0.5 ‑ 6 0.0 + 0 0.0 + 0 0.0 + 0 1.0 ‑ 6 ‑1.0 ‑6333332151

‑0.2 ‑ 6 0.0 + 0 ‑2.0 ‑ 6 4.0 ‑ 6 0.0 + 0 0.0 + 0333332151 LIST

0.0 + 0 0.0 + 0 MPAR=2 22^\*^ NRB=2333332151

3.0 + 1 1.5 + 0 7.04 + 0 1.0 + 0 0.04 + 0 6.0 + 0333332151

4.0 + 1 1.5 + 0 6.04 + 0 5.0 + 0 0.04 + 0 1.0 + 0333332151

2.0 ‑ 3 0.0 + 0 0.0 + 0 0.0 + 0 1.0 ‑ 3 0.0 + 0333332151

‑0.5 ‑ 4 2.0 ‑ 3 0.0 + 0 2.0 ‑ 3 0.0 + 0 0.0 + 0333332151 LIST

0.0 + 0 0.0 + 0 IDP=3 LB=1 2\*NE=4 NE=2333332151

‑2.0 + 0 1.0 ‑ 2 5.0 + 1 0.0 + 0 0.0 + 0 0.0 + 0333332151 LIST

0.0 + 0 0.0 + 0 IDP=3 LB=‑1 2\*NE=8 NE=4333332151

‑2.0 + 0 1.0 ‑ 2 1.0 + 0 0.0 + 0 1.0 + 1 1.0 ‑ 2333332151 LIST

5.0 + 1 0.0 + 0 0.0 + 0 0.0 + 0 0.0 + 0 0.0 + 0333332151 LIST

0.0 + 0 0.0 + 0 IDP=2 LB=5 NT=6 NE=3333332151

1.0 ‑ 5 2.0 + 1 5.0 + 1 1.0 ‑ 4 1.0 ‑ 4 2.0 ‑ 4333332151 LIST

0.0 + 0 0.0 + 0 IDP=4 LB=1 2\*NE=4 NE=2333332151

1.0 ‑ 5 4.0 ‑ 4 5.0 + 1 0.0 + 0 0.0 + 0 0.0 + 0333332151 LIST

0.0 + 0 0.0 + 0 0 0 0 0333332 0 SEND

0.0 + 0 0.0 + 0 0 0 0 03333 0 0 FEND

**REFERENCES**

1\. G. DeSaussure and Marable, ***Nucl. Sci. Eng.* 101**, 285 (1989)

2\. B. L. Broadhead and H. L. Dodds, ***Trans. Am. Nucl. Soc.* 39**, 929
(1981)

[^1]: ^\*^NVS+6\*NRB