urr.tex

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\chapter{Unresolved Resonance Region}\label{ch:urr}

As a first step towards expansion into the unresolved resonance region, Fritz Fr\"{o}hner's code
FITACS \cite{frohner1989evaluation} has been obtained and inserted into SAMMY. FITACS uses Hauser-Feshbach
theory with width fluctuations. The adjustable parameters are neutron strength functions,
distant-level parameters, average radiation widths (at $E = 0$), and average fission widths (at $E = 0$).
The energy dependence of the radiation widths is specified via the giant dipole model, of the fission
widths via Hill-Wheeler fission barrier transmission coefficients, and of the mean level spacing for
s-waves via the Gilbert-Cameron composite formula. Mean spacings for $l > 0$ are given via the
Bethe formula. Moldauer's prescription is used for partial cross sections. Details of the theory are
presented in Section \ref{sec:equations-for-urr}.

Initially (for release M2 of the code), FITACS was incorporated into SAMMY (as segments
SAMFFF and SAMACS) in a limited fashion only. Internal changes were made, to be consistent
with SAMMY notation and to use dynamic dimensioning of arrays. The M + W version of Bayes'
method has replaced the fitting procedure used in FITACS. Calculation of penetrabilities was
extended to all $l$ values (FITACS had used only s, p, d, and f-waves). The output included files from
which plots can be made. Results were reported in SAMMY.PAR in the same format as is used in
the input file (as well as in more human-legible fashion in SAMMY.LPT).

Subsequently, additional modifications, improvements, and new features have been made in
the SAMMY URR treatment:

\begin{itemize}
    \item Partial derivatives with respect to varied parameters are calculated exactly rather than approximately.
    \item A more efficient integration routine has been written for the Dresner integral, Eq. (VIII A.5) \ref{}.
    \item It is possible to include (and vary, if desired) a normalization for each data set.
    \item Elastic cross section data may be fitted.
    \item There is no limit on the number or type of experimental data sets. Data may be kept in separate files rather than appended to the parameter file.
    \item The output has been modified to conform more closely to SAMMY conventions.
    \item An ``annotated'' PARameter file, including key-word-based input, is the default input option, and the only option available for output. (Files in the original format can still be used for input, but options are limited with that format.)
    \item Different sets of average resonance parameters can be used in different energy ranges.
    \item Output can be produced in ENDF/B format, for both File 2 (resonance parameters) and File 32 (covariance matrices). ENDF files cannot be used for input, because the ENDF format requires a more limited theoretical description than does FITACS/SAMMY.
    \item The fitting procedure can be performed sequentially, in similar fashion as in the resolved resonance region. That is, output PARameter and COVariance files from the fit to one data set may be used as input to another run which fits another data set. [Initially, only simultaneous fitting of all data sets was permitted.]
    \item ``No-Bayes'' runs can be made: cross sections will be calculated from the resonance parameters, but no fitting will be done.
\end{itemize}

\noindent
Additions being considered for future revisions of the code include the following:

\begin{itemize}
    \item Multiple nuclides in the sample
    \item An option to calculate multigroup cross sections and covariances
    \item An option to include integral quantities in the fit
    \item Extensions to the theory
    \item Additional ENDF capability (requiring ENDF format changes)
    \item A link between the resolved resonance parameters and those for the unresolved region, in order to provide more consistent evaluated cross sections
    \item Methodologies for retroactive generation of covariance matrices, similar to that used in the resolved resonance region
\end{itemize}

Input for analysis of data in the unresolved resonance region is described in Section VIII.B\ref{}.
Output is described in Section VIII.C\ref{}. The relationship between ENDF parameters and
SAMMY/URR parameters is discussed in Section VIII.D\ref{}. For an example of the use of SAMMY/URR, see \cite{derrien2000neutron} and test cases 73, 88, 127, 133, 134, 142, and 145.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Equations For Unresolved Resonance Region} \label{sec:equations-for-urr}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The formulae for cross sections in the unresolved resonance region, as implemented in
SAMMY, are presented in this section. The implementation is a modified form of that provided by
Fritz Fr\"{o}hner in his FITACS code \cite{frohner1989evaluation}. (Please note that any mistakes in these formulae are
attributable only to the author of this manual, not to Fr\"{o}hner. The author is indebted to Herve
Derrien for significant contributions both to the development of the code and to the composition of
this section of the manual.)

\vspace{0.5cm}
\noindent
\textbf{Elastic cross section}

The elastic cross section is given as the difference between the total cross section and the
sum of all the non-elastic partial cross sections. The total cross section is given by Eqs. (VIII A.1)
through (VIII A.4), and the non-elastic partial cross sections by Eqs. (VIII A.5) through (VIII A.20).

\vspace{0.5cm}
\noindent
\textbf{Total cross section}

The average total cross section, for a given spin and parity and incident channel $c$, may be
written in the form

\begin{equation} \label{eq:urr-tot-xs}
    \langle \sigma_c \rangle = \frac{2\pi g_c}{k_c^2}\left(1-\text{Re}\left\langle S_{cc} \right\rangle\right) \:,
\end{equation}

\noindent
where, as usual, $g_c$ is the spin factor and $k_c$ is the center-of-mass momentum. The average scattering matrix $\left\langle S_{cc}\right\rangle$ is given by 

\begin{equation} \label{eq:avg-scattering-matrix}
\left\langle S_{cc} \right\rangle = e^{-2i\phi_c}\frac{1-\langle R_{cc}\rangle L_{c}^{0*}}{1-\langle R_{cc}\rangle L_{c}^{0}} \:,
\end{equation}

\noindent
and the average R-matrix can be written in the form 

\begin{equation} \label{eq:avg-r-matrix}
\left\langle R_{cc} \right\rangle = R_c^\infty + i\pi s_c \:,
\end{equation}

\noindent
with parameters defined as follows:

\vspace{0.5cm}
\noindent
$R_c^\infty = $ distant-level parameter (in input quantity);

\vspace{0.2cm}
\noindent
$\phi_c = $ hard-sphere phase shift, generated using matching radius $a$ (an input quantity);

\vspace{0.2cm}
\noindent
$L_c^0 = (S_c-B_c)+iP_c$ (see Section \ref{sec:equations-for-scattering-theory}) with boundary condition $B_c$ chosen such that $S_c - B_c = 0$;

\vspace{0.2cm}
\noindent
$s_c = $ pole strength.

\vspace{0.5cm}
\noindent
The pole strength is defined in terms of input quantities $\tilde{S}_c$ (the strength function, for which we have introduced the tilde to avoid confusion with the shift factor used in the definition of $L_c^0$) and $a_c$ (the R-matrix matching radius) as 

\begin{equation} \label{eq:pole-strength-definition}
s_c = \tilde{S}_c\sqrt{E}/2\rho
\end{equation}

\noindent
where $\rho$ is the center-of-mass momentum $k_c$ multiplied by the channel radius $a_c$. \textbf{Please note that many authors choose to report} $\mathbf{\tilde{S}_c}$ \textbf{in units of $\mathbf{10^{-4}}$!}\cite{mughabghab_atlas_2006}

\vspace{0.5cm}
\noindent
\textbf{Non-elastic partial cross sections}

The non-elastic partial cross sections may be written in terms of transmission coefficients $T_x$ as

\begin{equation} \label{eq:urr-partial-xs}
    \langle \sigma_{ab} \rangle = \frac{\pi g_a}{k_a^2}\frac{T_aT_b}{T} \int_0^\infty dt\; e^{-T_\gamma/T}\; \Pi\left(1+\frac{2}{\nu_c}\frac{T_c}{T}t\right)^{-\nu_c/2-\delta_{ac}-\delta_{bc}} \;,
\end{equation}

\noindent
where the quantities to the left of the integral sign are the Hauser-Feshbach expression, and the integrand is the Moldauer prescription \cite{moldauer_1980} for the width fluctuation correction factor. (A derivation of this expression, including the assumptions under which it is derived, is provided in Section \ref{}.) Here $a$ represents the incident channel and $b$ the exit channel; $\nu_c$ and $T_c$ represent the number of degrees of freedom (multiplicity) and transmission coefficient, respectively, for channel $c$. Subscript $\gamma$ refers to photon channels. $T$ is defined as the sum over all channels:

\begin{equation} \label{eq:urr-tot-trans-coeff}
    T = \sum_c T_c \;.
\end{equation}

The transmission coefficient for neutron channels is given by 

\begin{equation} \label{eq:urr-chan-trans-coeff}
    T_c = 1-|\langle S_{cc}\rangle|^2 = \frac{4\pi P_cs_c}{\left|1-\langle R_{cc}\rangle L_c\right|^2} \;,
\end{equation}

\noindent
where $c$ is an incident channel, $P$ and $L$ are as defined in Section \ref{sec:equations-for-scattering-theory}, and the other quantities are given above. For photon and fission channels, the transmission coefficients for spin $J$ are 

\begin{equation} \label{eq:urr-gam-fiss-trans-coeff}
    T_\gamma = 2\pi \langle \Gamma_\gamma\rangle/D_J \qquad \text{and} \qquad T_f = 2\pi \langle \Gamma_f\rangle/D_J \;,
\end{equation}

\noindent
in which $D_J$ is the mean level spacing for levels with this spin.

The $J$-dependence of the mean level spacing is set in SAMMY/FITACS via the Bethe formula (e.g., \cite{frohner_1978}):

\begin{equation} \label{eq:urr-J-dep-level-spacing}
    (D_J(E))^{-1} = (d(E))^{-1}\left\{e^{\frac{-J^2}{2(\sigma(E))^2}}-e^{\frac{-(J+1)^2}{2(\sigma(E))^2}}\right\} \;,
\end{equation}

\noindent
where $d(E)$ is independent of $J$, and $\sigma$ is the spin cutoff parameter. The spin cutoff parameter is related to the level density parameter $a$ and the energy $E$ by the formula

\begin{equation} \label{eq:urr-spin-cutoff-parameter}
    \sigma^2 = (0.14592)(A+1)^{2/3}\sqrt{a(E+BE-PE)} \;,
\end{equation}

\noindent
in which $BE$ represents the neutron binding energy (an input parameter) and $PE$ the pairing energy (also an input parameter). The value for $a$ is determined from the input quantity $D$, which is the mean level spacing of the $l=0$ resonances at $E=0$; note that $D$ includes both $J=I-i$ and $J=I+i$, where $I$ is the spin of the target nucleus and $i=1/2$ is the spin of the neutron. An expression for the inverse of $D$ can be found from Eq. \ref{eq:urr-J-dep-level-spacing} to be

\begin{equation} \label{eq:urr-level-spacing-at-zero}
\begin{split}
    D^{-1} &= \sum_J(D_J(E=0))^{-1} \\
           &= (d(0))^{-1}\left\{e^{\frac{-(I-\frac{1}{2})^2}{2\sigma^2}}-e^{\frac{-(I+\frac{3}{2})^2}{2\sigma^2}}\right\} \;;
\end{split}
\end{equation}

\noindent
this expression is used to determine the value of $\sigma^2$ and hence the value of the level density parameter $a$.

The energy dependence of the mean level spacing is calculated with the Gilbert-Cameron
composite formula \cite{gilbert_cameron}. Let $E_x$ represent the excitation energy of the compound nucleus; this
energy is equal to the sum of the incident neutron kinetic energy $E$ and the neutron binding energy
$BE$ (which is an input quantity). That is to say,

\begin{equation} \label{eq:urr-excitation-energy}
    E_x = E+BE \;.
\end{equation}

The energy dependence for low excitation energies $E_x < E_0$ , where $E_0$ is a matching
energy, is given by the constant-temperature formula

\begin{equation} \label{eq:urr-constant-temperature-formula}
    D^{-1} \sim C_3\frac{\text{exp}\left[{C_2\sqrt{E_0-PE}}\right]}{(E_0-PE)^{3/2}} \; \text{exp}\left[\frac{E_x-E_0}{2}\left(\frac{C_2}{\sqrt{E_0-PE}}-\frac{3}{E_0-PE}\right)\right] \;.
\end{equation}

\noindent
In the code, the matching energy $E_0$ is set at

\begin{equation} \label{eq:urr-matching-energy}
    E_0 = \left[\frac{5}{2}+\frac{150}{(N+Z+1)}\right]
\end{equation}

\noindent
in unit of MeV, with $N+Z$ being the mass number for the target nucleus. Values of the constants $C_2$ and $C_3$ are given by

\begin{equation} \label{eq:urr-constant-temperature-formula}
    C_2 = \sqrt{4a} \qquad \text{and} \qquad C_3 = \frac{1}{12\sqrt{2aq}} \;,
\end{equation}

\noindent
with $q$ defined as

\begin{equation} \label{eq:urr-constant-temperature-formula}
    q = 0.14592(N+Z+1)^{2/3} \;,
\end{equation}

\noindent
where $N+Z$ is again the mass number for the target nucleus and $a$ is the level density parameter.

At higher energies $E_x>E_0$, the energy dependence of the mean level spacing is calculated
via the Fermi-Gas formula

\begin{equation} \label{eq:urr-constant-temperature-formula}
    D^{-1} \propto C_3\frac{\text{exp}\left[C_2\sqrt{E_x-PE}\right]}{(E_x-PE)^{3/2}} \;.
\end{equation}

Note that the two formulae agree at the matching energy (i.e., at $E_x=E_0$).

Radiation widths $\langle\Gamma_\gamma\rangle$ are assumed to depend only on parity $\pi$ and on $E$. The energy
dependence is calculated with the giant dipole resonance model. Fission widths $\langle\Gamma_f\rangle$ may vary with spin as well as parity and incident neutron energy $E$. Energy dependence is calculated with the Hill-Wheeler fission barrier transmission coefficients \cite{hill1953nuclear}. For a given $J^\pi$, the energy dependence of the fission widths is taken to be

\begin{equation} \label{eq:urr-condensed-hill-wheeler-fission}
    \langle\Gamma_f(E)\rangle = \langle\Gamma_f(0)\rangle \frac{1+\text{exp}\left[E_{HW}/W_{HW}\right]}{1+\text{exp}\left[-(E-E_{HW})/W_{HW}\right]} \;,
\end{equation}

\noindent
where the Hill-Wheeler threshold energy $E_{HW}$ and the Hill-Wheeler threshold width $W_{HW}$ are input quantities. This equation may be written in more ``standard'' notation as 

\begin{equation} \label{eq:urr-standard-hill-wheeler-fission}
    \langle\Gamma_f(E)\rangle = \langle\Gamma_f(0)\rangle \frac{1+\text{exp}\left[2\pi(E_f-BE)/\hbar\omega\right]}{1+\text{exp}\left[-2\pi(E_x-(E_{f}-BE))/\hbar\omega\right]} \;,
\end{equation}

\noindent
where, as above, $E_x$ is the excitation energy of the neutron and $BE$ is the binding energy. Also,
$E_f$ is the fission barrier height, and $\hbar\omega$ the width of the fission barrier.

Finally, a few words regarding the derivation of Eq. \ref{eq:urr-partial-xs} are warranted. That
derivation is based on several assumptions:

\begin{enumerate}
\item  The Moldauer prescription \cite{moldauer_1980} for width fluctuations is used. That is, the width fluctuation
correction factor is introduced to compensate for the non-unity of the ratio

\begin{equation} \label{eq:urr-trans-coeff-ratio}
    \left\langle\frac{T_aT_b}{T}\right\rangle / \frac{\langle T\rangle}{\langle T_a\rangle\langle T_b\rangle} \;.
\end{equation}

\item Partial widths obey a chi-squared distribution with $\nu_c$ degrees of freedom (where the value of $\nu_c$ depends on the number of channels of this de-excitation); averages are therefore weighted
with this distribution. In the Moldauer prescription for width fluctuations, simple channels
have 1 $< \nu_c <$ 1.78; for lumped channels, $\nu_c$ is a function of $T_c$.
\item Channels with the same transmission coefficients may be combined by introducing multiplicities.
\end{enumerate}

The integral of Eq. \ref{eq:urr-partial-xs} is described by Fr\"{o}hner as the ``width fluctuation correction or
Dresner factor.'' One (relatively modest) difference between SAMMY and the original FITACS
coding is the algorithm for calculating the Dresner integral; in SAMMY, the coding has been refined
to increase both speed and accuracy of calculation by using a non-uniform grid designed specifically
for this task. (Note: Prior to release 7 of the code, the Moldauer correction was inadvertently disabled in
code. This has now been fixed.)


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Derivation of Non-Elastic Average Cross Section} \label{sec:derivation-urr-non-elastic-xs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Input For Analysis Of Data In Unresolved Resonance Region} \label{sec:input-for-urr}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Two or more input files are required for analysis in the unresolved resonance region (URR).
The first is comparable to the usual SAMMY INPut file, which may contain as few as three lines:
Card set 1 of Table VIA.1\ref{} (the title line), card set 2 (nuclide name, atomic weight, and energy
range), and (at least) one line for card set 3 (alphanumeric information). Options for alphanumeric
commands in the URR are

\begin{verbatim}
    UNRESOLVED RESONANCE region
    EXPERIMENTAL DATA ARe in separate files
    ANNOTATED PARAMETER file for urr
    NO ANNOTATED PARAMETer file for urr input
    ENDF/B-VI FILE 2 IS wanted
    PUT COVARIANCE MATRIx into endf file 32
    COVARIANCE MATRIX FRom old run is used
    GENERATE FILE 3 POINt-wise cross sections
    DEBUG
    DO NOT SOLVE BAYES Equations
    USE ENERGY LIMITS AS given in the input file
    PRINT PARTIAL DERIVAtives
    INCLUDE MIN \& MAX ENergies when creating endf file
\end{verbatim}

\noindent
The first of these is required, as the SAMMY default is the resolved resonance region (RRR). The
other alphanumeric commands are optional; their effects are described below.

The second file, the URR PARameter file, contains the unresolved resonance parameters. In
the URR, there are several differences from the usual SAMMY conventions: To inform the code
that a parameter is to be varied, FITACS assumes that, if the uncertainty is given as zero for a given
parameter, then that parameter is not varied. (Hence there is no means of providing a default value
for uncertainty.) This procedure is in contrast with the usual SAMMY procedure of assigning a
value (generally 1) to a flag for each varied parameter; in the future, the formats for input to the
FITACS portion of SAMMY will perhaps be modified to conform to SAMMY standards.

SAMMY permits several types of modifications to the original FITACS-style PARameter
file: (1) Experimental data may be kept in separate files. (2) Normalizations can be included (and
varied) for each data set. (3) ENDF File 2 and File 32 can be produced. (4) ENDF File 3 can be
produced. (5) The PARameter file itself may be ``annotated'' in order to be more legible to humans.
(6) Units may be specified for various quantities. (7) Different parameters may be specified in
different energy regions. (8) Direct inelastic and/or direct capture components may be added. (9)
Sequential analyses may be performed. (10) The analysis may be restricted to an energy range
smaller than that for which the data are defined. Options (6) through (9) are available only with the 
annotated PARameter file.

Each of the 10 options is described separately below; details are given in Tables 
\ref{table:old-urr-par-input} and \ref{table:annotated-urr-par-input}. Table \ref{table:energy-ranges} 
provides a guide for the various types of energy ranges encountered during a URR analysis.

\begin{enumerate}
    \item When the INPut file contains the phrase

          \texttt{EXPERIMENTAL DATA ARe in separate files},

          experimental data are kept in separate file(s) rather than included as a portion of the URR
          PARameter file. Files names for individual data sets are given on the lines immediately
          following the INPut and PARameter file names in the interactive input stream. See, for
          example, test case tr073 run y.
    \item Normalizations can be included and varied (i.e., fitted) for each data set. That is, the
          theoretical calculation of the cross section is modified by
          \begin{equation}
              Theory = norm\times\sigma_{calculated},
          \end{equation}

          where $norm$ is given by the formula
          \begin{equation}
              norm = a +bE^c
          \end{equation}
          and $a$, $b$, and $c$ are input parameters, specified in the PARameter file. Note that one set of
          values for a, b, and c is given for each data set. Note also that there is no possibility to
          specify b and c unless

          \texttt{EXPERIMENTAL DATA ARe in separate files}.

          See tr073 for examples.
    \item When output in ENDF File 2 format is wanted, the phrase

          \texttt{ENDF/B-VI FILE 2 IS wanted}
          
          must be present in the INPut file. Also include the command

          \texttt{DEBUG}

          if you wish to create an annotated file SAMMY.NDX. This annotated file contains
          comments that define which parameters' values are given; except for the annotations, this file
          is identical to the SAMMY.NDF.

          One additional SAMMY input file must be provided; the name for this file is given in
          the input stream after the name of the PARameter file (or after the COVariance file if it
          exists) and before the name(s) of any data files. This NDF file provides information
          regarding the specifics of the ENDF file to be created. The NDF file is in key word format,
          and contains only the following parameters:

          \begin{itemize}[label={}]
            \item \texttt{Z = } charge
            \item \texttt{A = } atomic number
            \item \texttt{Mat = } ENDF material number
            \item \texttt{NUmber}  of energy points = number
            \item \texttt{Energy}  number 1 = value of energy-point
          \end{itemize}

          Only the one or two characters in capitals are required; others are optional. The value is
          given following the equal sign. One or the other of \texttt{NU...=} and \texttt{E...=} must be
          present. (If both are present, \texttt{NU...=} will be ignored.)

          The number of energy points specifies at how many equally spaced points per energy
          region the parameter values will be printed into the ENDF file. Values may also be printed
          at Emin and Emax, the limits of the analysis, if the phrase

          \texttt{INCLUDE MIN \& MAX ENergies in endf file}

          is given in the INPut file. For example, for 3 points per region and 5 regions, a total of
          $3 \times 5 + 2 = 17$ sets of values would be given in the ENDF file.

          If, instead of having a certain number of points for each energy range, specific values
          of energy are wanted in the ENDF file, then the alternative \texttt{E...=} should be specified.
          Subsequent energies are given one per line, with or without the key word \texttt{E...=} prior to
          the value. These energy values should be the last entries in this file.

          See test cases tr073 and tr127 for examples. Runs a, b, e, and f of tr127 use
          \texttt{NU...=} key word, while run g uses the \texttt{E...=} key word.

          To also obtain the associated covariance file (ENDF File 32), include the phrase

          \texttt{PUT COVARIANCE MATRix into endf file 32}

          \noindent
          in the INPut file. See test case tr128 runs j and k for examples.

    \item ENDF File 3 output (point-wise cross sections, in file SAMMY.FL3) can be generated when the command

          \texttt{GENERATE FILE 3 POINt-wise cross sections}

          is included in the INPut file. The energy grid for this cross section is as defined by the input
          data sets. If the \texttt{DEBUG} command is also used, an annotated ENDF File 3 (SAMMY.FLX)
          output file is also produced. See test case tr073, runs n through t, for examples.

    \item Two modes, annotated and unannotated, are available for the URR PARameter file:

          The unannotated mode is essentially equivalent to Fr{\"o}hner's original FITACS file
          (which includes both parameters and data). Formats for this file are described in
          Table \ref{table:old-urr-par-input}; all numbers, both integer and real, are specified with F10 formats. To use
          this mode, the INPut file must contain the command

          \texttt{NO ANNOTATED PARAMETer file for urr input}

          Test case tr073 has examples of this input mode.

          The annotated PARameter file is described in detail in Table \ref{table:annotated-urr-par-input}; this is the
          default mode for SAMMY. With this option, some parameters are entered by key word;
          other parameter lists have headings to define which parameters are in the list. See, for
          example, test case tr073 run g, or test cases tr128.

    \item Units may be specified for various energy-related quantities by including the phrase
          ``\texttt{in eV}'', ``\texttt{in keV}'', or ``\texttt{in MeV}'' in the appropriate location in the 
          PARameter file.
          (Note that, as always with SAMMY input, capitalization is irrelevant.) If units are not
          specified, defaults are as given in Table \ref{table:annotated-urr-par-input} (i.e., MeV for 
          binding energy and pairing
          energy, eV for all others). The quantities for which units may be specified are as follows:

          \begin{itemize}[label={}]
              \item excitation energies for inelastic states
              \item binding energy
              \item pairing energy
              \item energy maxima for the different ranges (see (7) below)
              \item energies for direct inelastic contribution (see (8) below)
          \end{itemize}
          
          See in particular tr128 run l (letter ``l'' not number ``one'') for examples.

    \item Different parameter values may be used in different energy ranges; see Table 
          \ref{table:annotated-urr-par-input}, card sets 4-7, for input details. See 
          test case tr128 for examples.

    \item A direct inelastic component may be added to the inelastic and total cross sections, and/or a
          direct capture component added to the capture and total cross sections. These components
          are specified numerically on grids chosen by the user; SAMMY interpolates linearly
          between grid points. See card set 11 of Table \ref{table:annotated-urr-par-input} for details. 
          Examples are in tr088 and tr134.

    \item Although many data sets can be analyzed simultaneously in URR, SAMMY also permits
          sequential runs similar to those used in the RRR. For details, see the description of the
          SAMMY.COV file in the next section, \S\ref{sec:output-for-urr}. For examples, see test case 
          tr073 runs a and g.

    \item The default choice for energy range in the URR is to include all energies for which data are
          available. However, the analysis may be restricted to a smaller energy range by including
          the command

          \texttt{USE ENERGY LIMITS AS given in the input file}

          \noindent
          in the INPut file. See test case tr073 runs j, k, l for examples.
\end{enumerate}


\begin{table}[hbt!]
\centering
\caption{Formats for original PARameter file for treatment of the unresolved resonance region}\label{table:old-urr-par-input}
    \begin{tabular}{p{10mm}|p{10mm}|p{130mm}}
        \hline\hline
        Card Set & Line No. & Description \\ \hline\hline
        1        & 1-4      & First four lines are alphanumeric title \\ \hline\hline
        2        & 1        & Number of iterations, fitting tolerance (essentially delta chi squared). Note that integers are to be specified as real numbers. All formats are F10. \\ \hline
                 & 2        & Mass in amu, radius in Fermi (or use default), neutron binding energy in MeV, pairing energy $PE$ in MeV. Again, formats are F10; note that the energy units are MeV, as opposed to the usual SAMMY standard of eV. \\ \hline\hline
        3        & 1,2,...  & Center-of-mass excitation energy (in eV), spin, and parity for the $n$th target level (beginning with ground state). Repeat as many times as needed. \\ \hline
                 & Last     & (Blank) \\ \hline\hline
        4        & 1        & Strength function $\tilde{S}_c$, uncertainty, distant-level parameter $R_c^\infty$ , uncertainty, radiation width $\langle\Gamma_\gamma\rangle$ in eV, uncertainty, mean level spacing D in eV for $l = 0$ \newline
                              \textit{Note: Some authors choose to list strength function in units of $10^{-4}$\cite{mughabghab_atlas_2006}}  \\ \hline
                 & 2        & Strength function, uncertainty, distant-level parameter, uncertainty, radiation width in eV, uncertainty, for $l=1$ \\ \hline
                 & 3        & Strength function, uncertainty, distant-level parameter, uncertainty, radiation width in eV, uncertainty, for $l=2$ \\ \hline
                 & 4,5,...  & As above, for higher $l$ values as needed \\ \hline
                 & Last     & (Blank) \\ \hline\hline
        5        & 1        & For the lowest $J$ value for $l = 0$, \newline - Average fission width $\langle\Gamma_f\rangle$ (eV) \newline - Degree of freedom $\nu_f$ for fission width distribution \newline - Hill-Wheeler threshold energy $E_{HW}$ \newline - Hill-Wheeler threshold width $W_{HW}$ \newline - Uncertainty on the average fission width \\ \hline
                 & 2,3,...  & Repeat line 1 for each possible value of $J$ for $l = 0$. \\ \hline
    \end{tabular}
\end{table}
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\begin{table}[hbt!]
\centering
    \begin{tabular}{p{10mm}|p{10mm}|p{130mm}}
        \hline
                 & 4,5,...  & Repeat lines 1-3 for each possible value of $J$ for $l = 1, 2,$ ... \newline\newline For a given spin $J$ and parity (even or odd $l$), only one set of values is actually used for $\langle\Gamma_f\rangle$ and the other parameters. Nevertheless, all $J$ and $l$ must be included in this list. Only the values associated with the lowest $l$ value will be used for the calculations; the other values will be ignored. \newline\newline For example, the ground state of $^{235}$U is $7/2^{-}$. \newline - For $l = 0$, $J^\pi = 3^-, 4^-$. \newline - For $l = 1$, $J^\pi = 2^+, 3^+, 4^+, 5^+$. \newline - For $l = 2$, $J^\pi = 1^-, 2^-, 3^-, 4^-, 5^-, 6^-$. \newline - The $3^-$ and $4^-$ values used in the calculations will be those given for $l = 0$. The values given for $J^\pi = 3^-$ and $4^-$ with $l = 2$ will be ignored. \\ \hline
                 & Last     & (Blank) \\ \hline\hline
        6        & 1        & Type of cross-section data (\texttt{TOTAl}, \texttt{CAPTure}, \texttt{FISSion}, or \texttt{INELastic}) \\ \hline
                 & 2        & Uncertainties are \texttt{RELAtive} or \texttt{ABSOlute} \\ \hline
                 & 3        & Energy, cross section, and uncertainty for first data point. Normalization and uncertainty ($a$ and $\Delta a$) for this data set. \\ \hline
                 & 4, \ldots        & Energy, cross section, uncertainty. (Note: if \texttt{RELAtive} then these need to be specified only for first data point; the others are assumed to be the same.) \\ \hline
                 & Last     & (Blank) \\ \hline\hline
        6x       & All      & Repeat card set 6 as many times as needed, in any order \\ \hline\hline
        7        & 1        & The single word ``\texttt{NORMALIZATION}''. (Card set 7 is present only if the command ``\texttt{EXPERIMENTAL DATA ARe in separate files}'' appears in card set 3 of the INPut file.) \\ \hline
                 & 2        & Type of cross section, normalization parameters $a, \Delta a, b, \Delta b, c, \Delta c,$ where the normalization for this data set is given by $norm = a + b E^c$ \\ \hline
                 & 3, etc.  & Repeat Line 2 once for each data set. Normalizations must appear in the same order in which the data sets appear. SAMMY will check to be sure the data types are consistent. \\ \hline\hline
    \end{tabular}
\end{table}

\begin{table}
\centering
\caption{Formats for annotated PARameter file for treatment of the unresolved resonance region}\label{table:annotated-urr-par-input}
    \begin{tabular}{p{10mm}|p{10mm}|p{130mm}}
        \hline\hline
        Card Set & Line No. & Description \\ \hline\hline
        1        & 1,2,...  & Alphanumeric title, as many lines as desired. Printed but otherwise ignored. \\ \hline
                 & Last     & \texttt{-----} (First four characters must be hyphens [minus signs]; this ends the title. Other characters on this line are printed but ignored.) \\ \hline\hline
        2        & 1,2,...  & Key word = Value. Possible keywords here are \newline 
                              \begin{tabular}{l l l} 
                              \underline{Key word} & \underline{Meaning} & \underline{Default} \\ 
                              \texttt{ITErations}  & number of iterations& 3 \\
                              \texttt{TOLerance}   & fitting tolerance   & 0.005 \\
                              \texttt{RADius}      & radius in F         & 1.23 AW1/3+0.8 \\
                              \texttt{AW} (atomic weight) & mass in amu  & (no default) \\
                              \end{tabular} \newline Only the letters in capitals are required; the values may be in any format. \\ \hline
                 & Last     & (Blank) \\ \hline\hline
        3        & 0        & ``\texttt{----}'' An optional line of minus signs may be inserted; this line will be ignored by the code. \\ \hline
                 & 1        & ``\texttt{ELAStic and inelastic states}''. \newline
                              Only the first four characters are necessary, others are optional. \newline
                              Units of excitation energy are eV unless specified anywhere on this line (after the first four characters) as ``\texttt{in eV}'', ``\texttt{in keV}'', or ``\texttt{in MeV}''. \\ \hline
                 & 2,3,...  & Center-of-mass excitation energy, spin, and parity for the $n$th target level (beginning with ground state). Format must be 3F10 (ten characters per number, three numbers on a line, decimal points must be included). \\ \hline
                 & Last     & (blank) \\ \hline\hline
        4        & 0        & ``\texttt{----}'' Optional. \\ \hline
                 & 1,2,...  & Key word = Value. Possible keywords here are \newline 
                              \begin{tabular}{l l l} 
                              \underline{Key word}    & \underline{Meaning}          & \underline{Default} \\ 
                              \texttt{BINding energy} & neutron binding energy (MeV) & (none) \\
                              \texttt{PAIring energy} & pairing energy (MeV)         & (none) \\
                              \end{tabular} \newline Only the letters in capitals are required; the values may be in any format. \newline\newline
                              To override the default units, insert a phrase ``\texttt{in eV}'', ``\texttt{in keV}'', or ``\texttt{in MeV}'' after the key word and before the equal sign. \newline
                              Examples: \texttt{Binding Energy (in eV) = 6536000}. \newline
                              \qquad \texttt{Pairing energy in eV = 1610000}. \\ \hline
                 & Last     & (blank) \\ \hline\hline
        5        & 0        & ``\texttt{----}'' Optional. \\ \hline
                 & 1        & ``\texttt{STReng\qquad del\_s\qquad distnt\qquad del\_d\qquad gamma width\qquad del\_g\qquad bethed}'' \newline
                              (Only first three characters are necessary. This line indicates that strength functions, distant-level parameters, etc., are coming next.) \\ \hline
    \end{tabular}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AN INELEGANT SOLUTION TO LARGE TABLE SIZE %%%%%%%%%%%%%%%%%%%
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\begin{table}[hbt!]
\centering
    \begin{tabular}{p{10mm}|p{10mm}|p{130mm}}
                 \hline
                 & 2        & Strength function $\tilde{S}_c$, uncertainty, distant-level parameter $R_c^\infty$ , uncertainty, radiation width $\langle\Gamma_\gamma\rangle$ in eV, uncertainty, mean level spacing $D$ in eV for $l = 0$. F10 formats. \newline
                              \textit{Note: Some authors choose to list strength function in units of $10^{-4}$\cite{mughabghab_atlas_2006}} \\ \hline
                 & 3        & Strength function, uncertainty, distant-level parameter, uncertainty, radiation width in eV, uncertainty, for $l = 1$ \\ \hline
                 & 4        & Strength function, uncertainty, distant-level parameter, uncertainty, radiation width in eV, uncertainty, for $l = 2$ \\ \hline
                 & 5,6,...  & As above, for higher $l$ values as needed \\ \hline
                 & Last     & (blank) \\ \hline\hline
        6        & 0        & ``\texttt{----}'' Optional. \\ \hline
                 & 1        & ``\texttt{FISsion width fnu ethr wthr del\_fission width}'' \newline
                              (Only first three characters are necessary. This line indicates that fission parameters are coming next.) \\ \hline
                 & 2        & For the lowest $J$ value for $l = 0$, \newline
                              \qquad Average fission width $\langle\Gamma_f\rangle$ (eV) \newline
                              \qquad Degree of freedom $\nu_f$ for fission width distribution \newline
                              \qquad Hill-Wheeler threshold energy $E_{HW}$ \newline
                              \qquad Hill-Wheeler threshold width $W_{HW}$ \newline
                              \qquad Uncertainty on the average fission width \newline
                              \qquad $J,l$  \newline\newline
                              The first line contains the lowest $J$ value associated with $l = 0$. Formats are F10 for everything except the $l$-value, which is I5 (i.e., the right-most column is \# 65). Inclusion of $J$ and $l$ in the input file is optional but recommended. \\ \hline
                & 3,4,...   & Repeat line 2 for each possible value of $J$ for $l=0$ \\ \hline
                & 5,6,...   & Repeat lines 2-4 for each possible value of $J$ for $l = 1, 2, ...$ For a given spin $J$ and parity (even or odd $l$), only one set of values is actually used for $\langle\Gamma_f\rangle$ and the other parameters. Nevertheless, all $J$ and $l$ must be included in this list. Only the values associated with the lowest $l$ value will be used for the calculations; the other values will be ignored. \newline\newline
                For example, the ground state of $^{235}$U is $7/2^-$. \newline
                \qquad For $l = 0$, $J^\pi= 3^-, 4^-$. \newline
                \qquad For $l = 1$, $J^\pi= 2^+, 3^+, 4^+, 5^+$. \newline
                \qquad For $l = 2$, $J^\pi= 1^-, 2^-, 3^-, 4^-, 5^-, 6^-$. \newline
                \qquad The $3^-$ and $4^-$ values used in the calculations will be those given for $l = 0$. \newline
                \qquad The values given for $J^\pi = 3^-$ and $4^-$ with $l = 2$ will be ignored. \\ \hline
                & Last      & (blank) \\ \hline\hline
        7       & 0         & ``\texttt{----}'' optional line \\ \hline

    \end{tabular}
\end{table}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AN INELEGANT SOLUTION TO LARGE TABLE SIZE %%%%%%%%%%%%%%%%%%%
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\begin{table}[hbt!]
\centering
    \begin{tabular}{p{10mm}|p{10mm}|p{130mm}}
                 \hline
                & 1         & Key word = Value. Only one possible keyword is permitted here: \newline 
                              \begin{tabular}{l l l} 
                              \underline{Key word}    & \underline{Meaning}          & \underline{Default} \\ 
                              \texttt{ENErgy maximum} & maximum energy in this region (eV) & (none) \\
                              \end{tabular} \newline Only the letters in capitals are required; the values may be in any format. \newline\newline
                              To override the default units, insert a phrase ``\texttt{in eV}'', ``\texttt{in keV}'', or ``\texttt{in MeV}'' after the key word and before the equal sign. \newline
                              Examples: \texttt{Energy maximum in MeV = 0.15}. \\ \hline\hline
        4-7     & all       & Repeat card sets 4-7, once for each energy region, as many times as needed. Alternatively, repeat only line 1 of card set 7, in which case the starting parameter values are assumed to be identical to those in the previous energy region. \\ \hline
        8       &           & In either case, end with a line saying ``\texttt{END of resonance parameter description}''. \\ \hline
        9       & 0         & ``\texttt{----}'' optional line \\ \hline
                & 1         & Type of cross-section data (\texttt{TOTAl, CAPTure, FISSion,} or \texttt{INELastic}). \newline\newline
                              \textbf{Card set 9 will be omitted from this file if} the command ``\texttt{EXPERIMENTAL DATA ARe in separate files}'' appears in the INPut file. \\ \hline
                & 2         & Uncertainties are \texttt{RELAtive} or \texttt{ABSOlute}. (Only ``\texttt{RELA}'' or ``\texttt{ABSO}'' is needed.) \\ \hline
                & 3         & Energy (eV), cross section (barn), uncertainty (barn if \texttt{ABSOlute},dimensionless if \texttt{RELAtive}) for first data point. \newline
                              Norm and unc ($a$ and $\Delta a$) for this data set. \newline
                              Format is 3F10. \\ \hline
                & 4,5,...   & Energy, cross section, uncertainty \newline
                              (Note: if \texttt{RELAtive}, then need specify only for first data point, rest are assumed to be the same.) \\ \hline
                & Last      & (blank) \\ \hline\hline
        9x      & all       & Repeat card set 9 as many times as needed, in any order \\ \hline\hline
        10      & 0         & ``\texttt{----}'' optional line \\ \hline
                & 1         & ``\texttt{NORMalization}''. [Card set 10 may be present only if INPut file specifies ``\texttt{experimental data are in separate files}''.] \\ \hline
                & 2         & Type of cross section, normalization parameters $a, \Delta a, b, \Delta b, c, \Delta c,$ where the normalization for this data set is given by $norm = a + b E^c$ \\ \hline
                & 3,4,...   & Repeat once for each data set. Note that normalizations must appear in the same order in which the data sets appear. SAMMY will check to be sure the data types are consistent. \\ \hline
                & Last      & (blank) \\ \hline\hline
        10a     & 0         & ``\texttt{----}'' optional line \\ \hline
                & 1         & ``\texttt{EARLier normalization}''. [Card set 10a, an alternative to card set 10, is to be used only if an earlier SAMMY run has produced a covariance file.] \\ \hline
    \end{tabular}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{table}[hbt!]
\centering
    \begin{tabular}{p{10mm}|p{10mm}|p{130mm}}
                 \hline
                & 2         & N1, N2, ... Nlast, in [40I2] format. Here N1 is the ordering of the first data set for this run, as it appeared in previous SAMMY runs; see test case tr145 for illustrative examples. \\ \hline
                & Last      & (blank) \\ \hline\hline
        11      & 0         & ``\texttt{----}'' optional line \\ \hline
                & 1         & ``\texttt{DIRECT Inelastic contribution}'', \textit{or} ``\texttt{DIRECT Capture contribution}''. Note that eight characters (rather than the usual four) are required here. \\ \hline
                & 2         & \texttt{Energy =} value, \texttt{Sigma =} value. Both key words (and both values) must be on the same line. \newline \newline
                              \begin{tabular}{l l} 
                              \underline{Key word}    & \underline{Meaning} \\ 
                              \texttt{Energy}         & Energy in (eV)  \\
                              \texttt{Sigma}          & Direct inelastic cross section (b) at that energy \\
                              \end{tabular} \newline\newline
                              To use different units, insert a phrase ``\texttt{in eV}'', ``\texttt{in keV}'', or ``\texttt{in MeV}'' after the key word ``\texttt{Energy}''. \\ \hline
                & 3,4,...   & Repeat line 2 as many times as required. \newline
                              Note that card set 11 may be omitted if the contribution of the direct inelastic cross section is negligible or unknown. \\ \hline\hline
    \end{tabular}
\end{table}

\begin{table}\label{table:energy-ranges}
1
\end{table}


\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Output From Analysis In Unresolved Resonance Region} \label{sec:output-for-urr}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As with the resolved resonance region, each SAMMY/URR run may produce several output files:

\begin{enumerate}
    \item The SAMMY.LPT file contains details of the calculations.
    \item For runs which involve the solution of Bayes' equations and hence the generation of updated parameter values (a ``Bayes run'' as opposed to a ``no-Bayes run'' which simply calculates cross sections), an output file SAMMY.PAR is provided in the annotated form described in Table \ref{table:annotated-urr-par-input}. (This file, of course, is quite different from the file by the same name produced in a resolved resonance region run.)
    \item For Bayes runs, a COVariance file SAMMY.COV is produced, which contains (among other things) the final parameter covariance matrix; see below for more detail.
    \item Files SAMMY.NDF and SAMMY.N32 contain the URR portion of ENDF File 2 and 32, respectively; these files are produced when the appropriate command is in the INPut file.
    \item Files SAMMY.NDX and SAMMY.N3X contain the same information as in SAMMY.NDF and SAMMY.N32 respectively, but also include comment lines defining the parameters whose values are given. Only the uncertainty values are given in SAMMY.N3X; correlations are printed only in SAMMY.N32.
\end{enumerate}

\noindent
The binary COVariance file SAMMY.COV can be used as input to a subsequent SAMMY
run that uses the same R-matrix parameters but different data sets and different normalizations, in a
similar fashion to sequential runs in the RRR. There are slight differences, however, in the usage of
this COVariance file; no auxiliary program comparable to SAMAMR is required here, as there is
only one type of data-reduction parameter (the normalizations). The user must simply rename and
modify the SAMMY.PAR file to contain the appropriate normalization parameters for the data sets
about to be analyzed. The output PARameter file will contain card set 10a of Table \ref{table:annotated-urr-par-input};
PARameter files to be used as input will use either card set 10 (for new data sets for which the
normalizations have not yet been defined) or card set 10a (to re-use normalizations already defined
for the current data sets). See test case tr145 for examples.

To use an output SAMMY.PAR file as input to a new run without the accompanying COV file, it is necessary to delete the first line of the file; this line reads as follows:

\noindent
\qquad\texttt{COVARIANCE MATRIX FRom old run is used}

\noindent
When the COV file is to be used (for sequential runs), keep this line in the PARameter file.

\noindent
Test cases tr073, tr088, tr127, tr128, tr133, tr145, and others provide sample input and output for URR calculations.