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.. _SOURCERER:

Sourcerer: Deterministic Starting Source for Criticality Calculations
=====================================================================

*D. E. Peplow, A. M. Ibrahim, K. B. Bekar, C. Celik, and B. T. Rearden*

The Sourcerer sequence in SCALE deterministically computes a fission
distribution and uses it as the starting source in a Monte Carlo
eigenvalue criticality calculation. Using a reasonably accurate starting
source, developed from the Denovo discrete-ordinates code through the
DEVC sequence, Sourcerer improves the KENO/CSAS Monte Carlo calculation
in two ways. First, the number of skipped generations required to
converge the fission source distribution in the KENO solution is
reduced. Second, for problems with loosely coupled fissionable areas,
the reliability of the final eigenvalue
(:math:`k_{\mathrm{\text{eff}}}`) is increased. Several convergence
diagnostic capabilities available in the KENO codes help the user better
measure when the fission source actually convergences.

.. highlight:: scale

Introduction
------------

Monte Carlo eigenvalue calculations have been used in evaluating
critical and sub-critical systems for decades. Calculations are
typically done iteratively – starting with a set of fission neutrons,
transporting them through the geometry until they leak or are absorbed,
and then tabulating the fission sites for the next iteration. Each
iteration corresponds to a generation in a chain reaction, and the
eigenvalue, :math:`k_{\mathrm{\text{eff}}}`, is the ratio of the number
of fissions in one generation to the number in the previous generation.
Once the fission source distribution (the eigenfunction) has converged,
many generations are simulated to obtain more estimates of
:math:`k_{\mathrm{\text{eff}}}` with lower statistical uncertainty.

Two common questions that concern practitioners are (1) how many
generations (skipped generations) are required before the fission source
distribution is sufficiently converged that generational estimates of
:math:`k_{\mathrm{\text{eff}}}` can be included in the final average of
the eigenvalue (the active generations) and (2) has the fission source
converged to the correct distribution, such that the final value of the
eigenvalue will be correct? For most calculations, the final value of
the :math:`k_{\mathrm{\text{eff}}}` eigenvalue is all that matters, so
the convergence of the generational value of
:math:`k_{\mathrm{\text{eff}}}` is used to determine the number of
skipped generations. Fluxes and reaction rates computed during the
active generations are more sensitive to the entire fission distribution
and should only be accumulated when the fission distribution is
sufficiently converged – which is not necessarily as soon as the
generational value of :math:`k_{\mathrm{\text{eff}}}` has converged. To
address this concern, tools such as Shannon entropy :cite:`shannon_mathematical_1948` can be used to
measure the convergence of the fission source distribution
eigenfunction :cite:`ueki_stationarity_2003,ueki_stationarity_2005`.

For the second question, the reliability in the final
:math:`k_{\mathrm{\text{eff}}}` eigenvalue depends on whether the
fission source converges to the correct distribution. Because no tool
currently exists in SCALE to verify that the fission distribution is
correct, models are often run for many generations using many histories
per generation to ensure that the result does not change. Another
approach to verify eigenvalue accuracy is to run several clones of the
same problem, but each starting with different random numbers seed, and
ensure that they all predict the same value for
:math:`k_{\mathrm{\text{eff}}}`. Addressing this concern relies heavily
on the engineering judgment of the practitioner.

Initial studies :cite:`ibrahim_acceleration_2011` have shown that the use of a
starting fission distribution that is similar to the true fission
distribution can both reduce the number of skipped generations required
for fission source convergence and significantly improve the reliability
of the final :math:`k_{\mathrm{\text{eff}}}` result. A recent study :cite:`ibrahim_hybrid_2013`
focusing on criticality calculations of a spent nuclear fuel cask showed
that the chance of a low eigenvalue result due to undersampling from an
unconverged source was dramatically reduced when using a deterministic
starting source. In that study, a cask holding 24 assemblies was
examined using a uniform starting source, a deterministic starting
source with loose convergence criteria, and a deterministic starting
source with tight convergence criteria. Multiple clones of KENO were run
(with different random number seeds) for different values of skipped
cycles. The number of clones that gave an incorrect result for
:math:`k_{\mathrm{\text{eff}}}` was then tabulated. The results from
that study, presented in Figure 2.4.1, show that using a deterministic
starting source significantly increases the
:math:`k_{\mathrm{\text{eff}}}` reliability.

The Sourcerer sequence in SCALE uses the solution from the Denovo :cite:`evans_denovo_2010`
discrete-ordinates code (through the DEVC sequence) as that starting
fission source distribution in a CSAS/KENO Monte Carlo :cite:`goluoglu_monte_2011` calculation.
For challenging criticality safety analyses, such as as-loaded spent
nuclear fuel transportation packages with a mixed loading of low- and
high-burnup fuel, even a low-fidelity deterministic solution for the
fission source should be more accurate than the typical starting guesses
of uniform or cosine shape over the fissionable regions. The Sourcerer
sequence is fairly automated and uses an input very similar to standard
CSAS (KENO V.a or KENO-VI) inputs, along with a short description of the
mesh and other parameters for the Denovo calculation.

.. _fig2-4-1:
.. figure:: figs/Sourcerer/fig1.png
  :align: center
  :width: 500

  Fraction of failure to agree with the reference *k*\ :sub:`eff` value for KENO calculations with different starting sources (Figure 4 from Ref. 5).

Capabilities
------------

The Sourcerer sequence calls a series of other sequences and utilities
in SCALE – most importantly DEVC (for Denovo) and one of the CSAS
sequences. Because DEVC can only use KENO-VI geometry, the utility
c5toc6 geometry converter is used for KENO V.a geometries. The utility
dso2msl is used to convert the Denovo spatial output (\*.dso file) into a
mesh source lite (\*.msl) file that can be read as a starting source in
KENO. All of the steps in Sourcerer are described in :numref:`tab2-4-1`.

The sequence can be terminated at several points throughout the
calculation using the ``“parm=”`` control on the ``“=sourcerer”`` line that
starts the sequence. This capability can be used to stop the sequence
between steps to ensure that the problem is progressing correctly or
used to make a single source distribution that can be used with many
variants of the final CSAS problem. Also when running several versions
of a problem, if a file that is normally created by the Sourcerer
sequence is supplied, then that step will be skipped, thus saving time.
Note that files that use the name of the input file (*input*.inp) will
be copied back to the user’s working directory automatically when the
sequence finishes. Files that do not use the input file name can be
copied back to the user’s working area with an extra ``“=shell”`` directive
in the ``*input*.inp`` input file.

The Sourcerer sequence can be used with KENO V.a or KENO-VI geometries.
Either multi-group (MG) or continuous energy cross-section libraries can
be used for the final CSAS calculation. Denovo only uses multi-group
libraries, and self-shielding can be done like any MG sequence in SCALE.
For efficient calculations, the user should understand the basics of
Denovo eigenvalue calculations, how to use macromaterials, and how to
use the KENO convergence metrics.

.. _tab2-4-1:
.. table:: Steps in Sourcerer for an input file named *input*.inp.
  :align: center

  +-----------------+-----------------+-----------------+-----------------+
  | Step            | Module/Task     | Creates file    | To stop after   |
  +=================+=================+=================+=================+
  | 0               | Check user      |                 | ``parm=check``  |
  |                 | input           |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  |                 |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  | 1               | ``c5toc6`` – For| ``input.geom0…``|                 |
  |                 | KENO V.a        |                 |                 |
  |                 | sequences, the  | ``00.inp``      |                 |
  |                 | geometry is     |                 |                 |
  |                 | translated into |                 |                 |
  |                 | KENO-VI         |                 |                 |
  |                 | geometry.       |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  |                 |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  | 2a              | Create Denovo   | ``xkba_b.inp``  | ``parm=deninp`` |
  |                 | input and AMPX  |                 |                 |
  |                 | cross sections  | ``ft02f001``    |                 |
  |                 |                 |                 |                 |
  |                 |                 | ``input.mmt``   |                 |
  +-----------------+-----------------+-----------------+-----------------+
  |                 |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  | 2b              | ``devc`` – Deno\| *input*.dso     | ``parm=denovo`` |
  |                 | vo eigenvalue   |                 |                 |
  |                 | calculation to  |                 |                 |
  |                 | compute a       |                 |                 |
  |                 | fission source  |                 |                 |
  |                 | distribution    |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  |                 |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  | 3               | ``dso2msl`` –   | ``input.msl``   |                 |
  |                 | Convert the     |                 |                 |
  |                 | fission source  |                 |                 |
  |                 | distribution    |                 |                 |
  |                 | file into a     |                 |                 |
  |                 | mesh source     |                 |                 |
  |                 | lite file       |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  |                 |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  | 4a              | Create the CSAS |                 | ``parm=csasinp``|
  |                 | input           |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  |                 |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  | 4b              | ``csasX`` - Run |                 |                 |
  |                 | the specific CS\|                 |                 |
  |                 | AS sequence usi\|                 |                 |
  |                 | ng the mesh sou\|                 |                 |
  |                 | rce lite as the |                 |                 |
  |                 | starting source |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+

Using DEVC/Denovo
~~~~~~~~~~~~~~~~~

Some discussion is required about the extent and level of detail needed
in the grid geometry that will be used in the Denovo calculation and the
mesh-based starting source. When using discrete-ordinates transport
(S:sub:`N`) methods alone for solving radiation transport problems, a
good rule of thumb is to use mesh cell sizes on the order of a mean-free
path of the particle. For complex problems, this could lead to an
extremely large number of mesh cells, especially when considering the
size of the mean-free path of the lowest energy neutrons.

In Sourcerer, the goal is to use the S\ :sub:`N` calculation for a quick
estimate. Accuracy is not paramount—just getting an approximation of the
overall shape of the true fission source distribution will benefit the
CSAS Monte Carlo calculation. With a more accurate starting source,
fewer skipped generations may be required. At some point there is a time
trade-off where calculating the starting source guess requires more time
than the saved skipped generations would have used. Large numbers of
mesh cells, as a result of using very small mesh sizes, for S\ :sub:`N`
calculations also use a great deal of computer memory.

Because the S\ :sub:`N` calculation is only used to establish the
initial distribution of source neutrons, the runtime and memory
requirements for Sourcerer can be reduced by using larger/coarser mesh
cell sizes than is typical for a stand-alone S\ :sub:`N` analysis. Some
general guidelines to consider when creating a mesh for the Denovo
eigenvalue calculation/mesh-based starting source are as follows.

-  All fissionable areas of the geometry and areas where neutrons can
   reasonably affect the eigenvalue should be included in the mesh.

-  More detail should be used in the fissionable areas.

-  Mesh planes should be placed at significant material boundaries.

-  Neighboring cell mesh sizes should not be drastically different.

Convergence Metrics in KENO
~~~~~~~~~~~~~~~~~~~~~~~~~~~

KENO provides several tools that can be used to examine the convergence
of :math:`k_{\mathrm{\text{eff}}}` and the fission source distribution.
These include tools based on Shannon Entropy\ :sup:`1,3` and tools based
on the mesh tally metrics described in :cite:`kiedrowski_statistical_2011`. Running Sourcerer
should accelerate Shannon Entropy convergence for KENO calculations, and
Sourcerer/KENO users should always check that the calculation Shannon
Entropy has converged before active generations begin.

Sequence Input
--------------

The input file for a Sourcerer calculation is similar to a CSAS input,
as shown in :numref:`tab2-4-2`. There are two major differences between
Sourcerer and CSAS: the beginning/end syntax of the CSAS input and the
presence of the “read detSource” block for specifying the deterministic
starting source. The CSAS input appears as one block in the Sourcerer
sequence – instead of “=csasXX” use “read csasXX” and instead of “end
data” use “end csasXX”. If any parm= parameters are required for the
CSAS sequence, they can be listed as “read csasX parm=(…)”. Because
Sourcerer runs both the DEVC and CSAS sequences, which will most likely
use different cross-section data libraries (coarse group for S\ :sub:`N`
and fine group or continuous energy for Monte Carlo), the library for
DEVC is listed in the new “read detSource” block, along with other
parameters used by the Sourcerer sequence.

One important note about the geometry in Sourcerer is that for KENO V.a
geometries, an outer boundary region should be added for an accurate
internal conversion to a KENO-VI geometry prior to ray tracing. Also
note that when using CSAS methods with the search capability, the Denovo
mesh must encompass any changes to the size of the geometry.

Parameters used for building the deterministic starting source and
establishing the CSAS sequence are specified in the detSource block. The
library name for the Denovo calculation and the grid for the Denovo
calculations are required. Many optional parameters are available for
controlling the Denovo solver and applying boundary conditions (in the
``eigenValParams`` sub-block). The grid geometry is defined in a sub-block,
or the keyword ``“gridGeometryID=`` \ *n*\ ” can be used to point to a grid
geometry defined in its own input block. The use of macromaterials to
construct a more representative mesh model from the Monte Carlo geometry
is controlled with the ``“mmSubCell=”`` and ``“mmTolerance=”`` parameters in the
macromaterial sub-block.

The overall layout of the detSource block is shown in :numref:`tab2-4-3`. The
more common keywords for the eigenValParams sub-block are shown in
:numref:`tab2-4-4` and :numref:`tab2-4-5`. A full list of the Denovo parameters
appears in Appendix A. Macromaterials are explained in detail in the
DEVC manual, and a list of keywords is given in :numref:`tab2-4-6`.

.. list-table:: Input file for a Sourcerer calculation (and differences with a CSAS input file, where black text is the same as CSAS and green text is new for Sourcerer sequence).
  :name: tab2-4-2
  :align: center

  * - .. image:: figs/Sourcerer/tab2.png

.. list-table:: The detSource block.
  :name: tab2-4-3
  :align: center

  * - .. image:: figs/Sourcerer/tab3.png

.. list-table:: Common Denovo parameters in the eigenValParams sub-block.
  :name: tab2-4-4
  :align: center

  * - .. image:: figs/Sourcerer/tab4.png

.. list-table:: Boundary conditions in the eigenValParams sub-block.
  :name: tab2-4-5
  :align: center

  * - .. image:: figs/Sourcerer/tab5.png

.. list-table:: Macromaterial sub-block input.
  :name: tab2-4-6
  :align: center

  * - .. image:: figs/Sourcerer/tab6.png

Sequence Output
---------------

In addition to the data contained in the main Sourcerer text output
file, many other files are created containing the intermediate data used
by the sequence. These files are listed in :numref:`tab2-4-7`. Some of the
files produced can be viewed using the Java Mesh File Viewer, which is
distributed with SCALE.

Note that files that use the name of the input file (*input*.inp) will
be copied back to the user’s working directory automatically when the
sequence finishes. Files that do not use the input file name can be
copied back to the user’s working area with an extra ``“=shell”`` directive
in the *input*.inp input file.

Instructions on how to use the Java Mesh File Viewer to view the various
output files listed in :numref:`tab2-4-7` as well as how to use the
macromaterial table file are located in the DEVC manual and the Mesh
File Viewer help file, which is accessible through the Help/Help menu.

.. _tab2-4-7:
.. table:: Files created by Sourcerer for an input file named *input*.inp.

  +-----------------+-----------------+-----------------+-----------------+
  | **Filename**    |                 | **Viewer**      | **Description** |
  +=================+=================+=================+=================+
  | Output Summary  |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | *input*.out     |                 | main text       |
  |                 |                 |                 | output file,    |
  |                 |                 |                 | contains        |
  |                 |                 |                 | results summary |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | *input*.msg     |                 | messages file   |
  +-----------------+-----------------+-----------------+-----------------+
  |                 |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  | Geometry        |                 |                 |                 |
  | Conversion      |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | i_c5toc6        |                 | input file for  |
  |                 |                 |                 | c5toc6 module   |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | *input*.geom000 |                 | KENO-VI version |
  |                 | ….inp           |                 | of a KENO V.a   |
  |                 |                 |                 | geometry, if    |
  |                 |                 |                 | applicable      |
  +-----------------+-----------------+-----------------+-----------------+
  |                 |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  | Denovo          |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | i_devc          |                 | input file for  |
  |                 |                 |                 | DEVC sequence   |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | xkba_b.inp      | V\ :sup:`a`     | input file for  |
  |                 |                 |                 | Denovo – if     |
  |                 |                 |                 | this file is    |
  |                 |                 |                 | renamed to have |
  |                 |                 |                 |                 |
  |                 |                 |                 | a \*.dsi        |
  |                 |                 |                 | extension       |
  |                 |                 |                 | (Denovo simple  |
  |                 |                 |                 | input), it is   |
  |                 |                 |                 | viewable        |
  |                 |                 |                 |                 |
  |                 |                 |                 | in the Mesh     |
  |                 |                 |                 | File Viewer     |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | ft02f001        |                 | AMPX formatted  |
  |                 |                 |                 | cross sections  |
  |                 |                 |                 | for Denovo      |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | *input*.dso     | V               | Denovo fission  |
  |                 |                 |                 | source          |
  |                 |                 |                 | distribution    |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | *input*.mmt     | V               | macromaterial   |
  |                 |                 |                 | table, use with |
  |                 |                 |                 | \*.dso or       |
  |                 |                 |                 | \*.dsi file     |
  +-----------------+-----------------+-----------------+-----------------+
  |                 |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  | Mesh Source     |                 |                 |                 |
  | Conversion      |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | i_util          |                 | input file for  |
  |                 |                 |                 | dso2msl utility |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | *input*.msl     | V               | mesh source     |
  |                 |                 |                 | lite file, the  |
  |                 |                 |                 | starting source |
  |                 |                 |                 | distribution    |
  +-----------------+-----------------+-----------------+-----------------+
  |                 |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  | CSAS            |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | i_csasXX        |                 | input file for  |
  |                 |                 |                 | the final       |
  |                 |                 |                 | CSASXX sequence |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | *input*.fission | V               | mesh tally of   |
  |                 | Source.3dmap    |                 | fission source  |
  |                 |                 |                 | distribution    |
  |                 |                 |                 | from KENO       |
  +-----------------+-----------------+-----------------+-----------------+
  |                 | *input*.kenoNuB |                 | text file       |
  |                 | ar.txt          |                 | containing      |
  |                 |                 |                 | value of nu-bar |
  +-----------------+-----------------+-----------------+-----------------+
  | :sup:`a`\ V – c\|                 |                 |                 |
  | an be displayed |                 |                 |                 |
  | with the Mesh F\|                 |                 |                 |
  | ile Viewer.     |                 |                 |                 |
  +-----------------+-----------------+-----------------+-----------------+

Sample problems
---------------

In addition to the sample problems described in this section (with input
files included with SCALE), the reader is referred to the paper by
Ibrahim et al. for a detailed study using a real used nuclear fuel
transport and storage canister containing assemblies with a range of
initial enrichments and burnups.

Jezebel
~~~~~~~

Consider the Jezebel critical assembly [PU-MET-FAST-001 in Volume I of the *International Handbook of
Evaluated Criticality Safety Benchmark Experiments*, NEA/NSC/DOC(95)03,  Organisation for Economic
Co-operation and Development, Nuclear Energy Agency (OECD-NEA), September 2012]. This is a very simple
problem (a single sphere) to solve with CSAS and can be useful as a way to demonstrate the Sourcerer sequence.

Input file
^^^^^^^^^^

The standard CSAS inputs for Jezebel are shown below using both KENO V.a and KENO-VI geometries.

.. list-table::
  :align: center

  * - KENO V.a geometry
    - KENO-VI geometry
  * - ::

        =csas5
        Jezebel
        v7-252n
        read composition
            pu-239 1 0 0.037047   end
            pu-240 1 0 0.0017512  end
            pu-241 1 0 0.00011674 end
            ga     1 0 0.0013752  end
        end composition
        read parameters
            gen=110 npg=1000 nsk=10
        end parameters
        read geometry
            global unit 2
                sphere  1 1  6.38493      .


        end geometry
        end data
        end
    - ::

        =csas6
        Jezebel
        v7-252n
        read composition
            pu-239 1 0 0.037047   end
            pu-240 1 0 0.0017512  end
            pu-241 1 0 0.00011674 end
            ga     1 0 0.0013752  end
        end composition
        read parameters
            gen=110 npg=1000 nsk=10
        end parameters
        read geometry
            global unit 2
                sphere  51  6.38493
                media 1 1 51 vol=1090.3277
            boundary 51
        end geometry
        end data
        end

The above inputs can be easily changed into the following Sourcerer
inputs (with geometry additions in brackets and extra Sourcerer input in
braces).

.. list-table::
  :align: center

  * - KENO V.a geometry
    - KENO-VI geometry
  * - ::

        {=sourcerer}
        {read} csas5
        Jezebel
        v7-252n
        read composition
            pu-239 1 0 0.037047   end
            pu-240 1 0 0.0017512  end
            pu-241 1 0 0.00011674 end
            ga     1 0 0.0013752  end
        end composition
        read parameters
            gen=110 npg=1000 nsk=10
        end parameters
        read geometry
            global unit 2
                sphere  1 1  6.38493
                [cuboid 0 1 7.0 -7.0 7.0 -7.0 7.0 -7.0]

        end geometry
        {end csas5}
        {read detSource}
            {dLibrary="v7-27n19g"}
            {gridGeometry 7}
                {xLinear 14 -7.0 7.0}
                {yLinear 14 -7.0 7.0}
                {zLinear 14 -7.0 7.0}
            {end gridGeometry}
        {end detSource}
        end data
        end
    - ::

        {=sourcerer}
        {read} csas6
        Jezebel
        v7-252n
        read composition
            pu-239 1 0 0.037047   end
            pu-240 1 0 0.0017512  end
            pu-241 1 0 0.00011674 end
            ga     1 0 0.0013752  end
        end composition
        read parameters
            gen=110 npg=1000 nsk=10
        end parameters
        read geometry
            global unit 2
                sphere  51  6.38493
                media 1 1 51 vol=1090.3277
            boundary 51
        end geometry
        {end csas6}
        {read detSource}
            {dLibrary="v7-27n19g"}
            {gridGeometry 7}
                {xLinear 14 -7.0 7.0}
                {yLinear 14 -7.0 7.0}
                {zLinear 14 -7.0 7.0}
            {end gridGeometry}
        {end detSource}
        end data
        end

With either of these variations of the Jezebel problem, the fission
source distribution can be tallied by KENO and saved to a mesh tally
(\*.3dmap) file by adding the following to the input:

::

  read parameters

      cds=1
  end parameters

  read gridGeometry 1
      title="Mesh for collecting fission source distribution"
      xLinear 28 -7.0 7.0
      yLinear 28 -7.0 7.0
      zLinear 28 -7.0 7.0
  end gridGeometry

Note that the mesh grid used for the KENO mesh tally can be different
from the mesh grid used by Denovo to create a starting source in the
Sourcerer sequence. Also note that more total histories (more particles
per generation or more active generations) would be required to produce
a KENO fission source tally with low relative uncertainties in every
voxel.

Output file
^^^^^^^^^^^

The results for the standard CSAS calculations and the Sourcerer results
are shown in :numref:`tab2-4-8` for calculations with the 252-energy-group and
continuous-energy cross sections.

.. _tab2-4-8:
.. table:: Eigenvalue results for the Jezebel problem.
  :align: center

  +----+---------------------------+-----------------+----------------+
  |    | Sample Problem            | CSAS            | Sourcerer      |
  +====+===========================+=================+================+
  | 1. | KENO V.a geometry, v7-252 | 1.0045 ± 0.0017 | 1.0054 ±0.0018 |
  +----+---------------------------+-----------------+----------------+
  | 2. | KENO-VI geometry, v7-252  | 0.9998 ± 0.0018 | 1.0007 ±0.0020 |
  +----+---------------------------+-----------------+----------------+
  | 3. | KENO V.a geometry, ce_v7  | 1.0058±0.0027   | 1.0026 ±0.0017 |
  +----+---------------------------+-----------------+----------------+
  | 3. | KENO-VI geometry, ce_v7   | 0.9990 ±0.0023  | 1.0041 ±0.0016 |
  +----+---------------------------+-----------------+----------------+

The Denovo fission source provides a reliable starting source that is
similar to the actual fission source distribution computed by KENO
(using ``npg``\ =250000), as shown in :numref:`fig2-4-2`.

.. _fig2-4-2:
.. figure:: figs/Sourcerer/fig2.png
  :align: center
  :width: 700

  Fission source distribution from Denovo (left) and KENO (right).

Other variations
^^^^^^^^^^^^^^^^

To increase the Denovo calculation speed (but decrease the fidelity of
the fission source result), the discretization in angle (``quadrature=`` or
``polarsPerOct=/azimuthsPerOct=``) can be coarsened and/or the tolerance
parameters can be loosened. To reduce the amount of memory required by
Denovo, the number of Legendre moments in the scattering cross-section
expansion can be reduced (``legendre=``). Using macromaterials can help
increase the fidelity of the Denovo calculation with only a small
increase in model setup time. Macromaterials do not impact the Denovo
run time. Denovo diagnostic messages can be turned on and will print to
the messages file.

.. list-table::
  :align: center

  * - Macromaterials (higher fidelity)
    - Looser tolerances (faster Denovo)

  * - ::

        read detSource



          macromaterial
            mmSubCell=3
            mmTolerance=0.001
          end macromaterial
        end detSource

    - ::

        read detSource



          eigenValParams



            tolerance=1.0e-2
            kTolerance=1.0e-3
          end eigenValParams
        end detSource

.. list-table::
  :align: center

  * - Screen messages
    - Higher fidelity (slower Denovo)
  * - ::

        read detSource



          eigenValParams



            diagnostics=1
            output=1
            diagnosticLevel=1
          end eigenValParams
        end detSource

    - ::

        read sequence



          eigenValParams



            quadType=2
            polarsPerOct=4
            azimuthsPerOct=4
            legendre=3

          end eigenValParams
        end sequence




.. bibliography:: bibs/Sourcerer.bib