Monte Carlo Transport Overview.html 26.5 KB
Newer Older
Batson Iii's avatar
Batson Iii committed
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487


<!DOCTYPE html>
<html class="writer-html5" lang="en" >
<head>
  <meta charset="utf-8">
  
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  
  <title>Monte Carlo Transport Overview &mdash; SCALE Manual 0.0.1 documentation</title>
  

  
  <link rel="stylesheet" href="_static/css/theme.css" type="text/css" />
  <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
  <link rel="stylesheet" href="_static/custom.css" type="text/css" />

  
  
  
  

  
  <!--[if lt IE 9]>
    <script src="_static/js/html5shiv.min.js"></script>
  <![endif]-->
  
    
      <script type="text/javascript" id="documentation_options" data-url_root="./" src="_static/documentation_options.js"></script>
        <script src="_static/jquery.js"></script>
        <script src="_static/underscore.js"></script>
        <script src="_static/doctools.js"></script>
        <script src="_static/language_data.js"></script>
        <script async="async" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
    
    <script type="text/javascript" src="_static/js/theme.js"></script>

    
    <link rel="index" title="Index" href="genindex.html" />
    <link rel="search" title="Search" href="search.html" />
    <link rel="next" title="Keno: A Monte Carlo Criticality Program" href="Keno.html" />
    <link rel="prev" title="CENTRM: A Neutron Transport Code for Computing Continuous-Energy Spectra in General One-Dimensional Geometries and Two-Dimensional Lattice Cells" href="CENTRM.html" /> 
</head>

<body class="wy-body-for-nav">

   
  <div class="wy-grid-for-nav">
    
    <nav data-toggle="wy-nav-shift" class="wy-nav-side">
      <div class="wy-side-scroll">
        <div class="wy-side-nav-search" >
          

          
            <a href="index.html" class="icon icon-home" alt="Documentation Home"> SCALE Manual
          

          
          </a>

          
            
            
          

          
<div role="search">
  <form id="rtd-search-form" class="wy-form" action="search.html" method="get">
    <input type="text" name="q" placeholder="Search docs" />
    <input type="hidden" name="check_keywords" value="yes" />
    <input type="hidden" name="area" value="default" />
  </form>
</div>

          
        </div>

        
        <div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="main navigation">
          
            
            
              
            
            
              <p class="caption"><span class="caption-text">Reactor Physics</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="Polaris.html">Polaris: 2D Light Water Reactor Lattice Physics Module</a></li>
<li class="toctree-l1"><a class="reference internal" href="PolarisA.html">SCALE 6.3 Polaris Input Format</a></li>
</ul>
<p class="caption"><span class="caption-text">Criticality Safety</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="Criticality%20Safety%20Overview.html">Criticality Safety Overview</a></li>
<li class="toctree-l1"><a class="reference internal" href="CSAS5.html">CSAS5:  Control Module For Enhanced Criticality Safety Analysis Sequences With KENO V.a</a></li>
<li class="toctree-l1"><a class="reference internal" href="CSAS5App.html">Additional Example Applications of CSAS5</a></li>
<li class="toctree-l1"><a class="reference internal" href="CSAS6.html">CSAS6:  Control Module for Enhanced Criticality Safety Analysis with KENO-VI</a></li>
<li class="toctree-l1"><a class="reference internal" href="CSAS6App.html">Additional Example Applications of CSAS6</a></li>
<li class="toctree-l1"><a class="reference internal" href="STARBUCS.html">STARBUCS: A Scale Control Module for Automated Criticality Safety Analyses Using Burnup Credit</a></li>
<li class="toctree-l1"><a class="reference internal" href="Sourcerer.html">Sourcerer: Deterministic Starting Source for Criticality Calculations</a></li>
<li class="toctree-l1"><a class="reference internal" href="DEVC.html">DEVC: Denovo EigenValue Calculation</a></li>
<li class="toctree-l1"><a class="reference internal" href="KMART.html">KMART5 and KMART6: Postprocessors for KENO V.A and KENO-VI</a></li>
<li class="toctree-l1"><a class="reference internal" href="K5C5.html">K5toK6 and C5toC6: Input File Conversion Programs for KENO and CSAS</a></li>
</ul>
<p class="caption"><span class="caption-text">Material Specification and Cross Section Processing</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="Material%20Specification%20and%20Cross%20Section%20Processing%20Overview.html">Material Specification and Cross Section Processing Overview</a></li>
<li class="toctree-l1"><a class="reference internal" href="XSProc.html">XSPROC: The Material and Cross Section Processing Module for SCALE</a></li>
<li class="toctree-l1"><a class="reference internal" href="XSProcAppA.html">XSProc: Standard Composition Examples</a></li>
<li class="toctree-l1"><a class="reference internal" href="XSProcAppB.html">XSProc Standard Composition Examples</a></li>
<li class="toctree-l1"><a class="reference internal" href="XSProcAppC.html">Examples of Complete XSProc Input Data</a></li>
<li class="toctree-l1"><a class="reference internal" href="stdcmp.html">Standard Composition Library</a></li>
<li class="toctree-l1"><a class="reference internal" href="BONAMI.html">BONAMI: Resonance Self-Shielding by the Bondarenko Method</a></li>
<li class="toctree-l1"><a class="reference internal" href="CENTRM.html">CENTRM: A Neutron Transport Code for Computing Continuous-Energy Spectra in General One-Dimensional Geometries and Two-Dimensional Lattice Cells</a></li>
</ul>
<p class="caption"><span class="caption-text">Monte Carlo Transport</span></p>
<ul class="current">
<li class="toctree-l1 current"><a class="current reference internal" href="#">Monte Carlo Transport Overview</a><ul>
<li class="toctree-l2"><a class="reference internal" href="#multigroup-physics">Multigroup Physics</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#continuous-energy-physics">Continuous-energy Physics</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="#geometry-packages">Geometry Packages</a></li>
<li class="toctree-l2"><a class="reference internal" href="#eigenvalue-analysis">Eigenvalue Analysis</a></li>
<li class="toctree-l2"><a class="reference internal" href="#shielding-analysis">Shielding Analysis</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="Keno.html">Keno: A Monte Carlo Criticality Program</a></li>
<li class="toctree-l1"><a class="reference internal" href="KenoA.html">Keno Appendix A: KENO V.a Shape Descriptions</a></li>
<li class="toctree-l1"><a class="reference internal" href="KenoB.html">Keno Appendix B: KENO VI Shape Descriptions</a></li>
<li class="toctree-l1"><a class="reference internal" href="KenoC.html">Keno Appendix C: Sample problems</a></li>
<li class="toctree-l1"><a class="reference internal" href="Monaco.html">Monaco: A Fixed-Source Monte Carlo Transport Code for Shielding Applications</a></li>
</ul>
<p class="caption"><span class="caption-text">Radiation Shielding</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="MAVRIC.html">MAVRIC: Monaco with Automated Variance Reduction using Importance Calculations</a></li>
<li class="toctree-l1"><a class="reference internal" href="CAAScapability.html">MAVRIC Appendix A: CAAS Capability</a></li>
<li class="toctree-l1"><a class="reference internal" href="appendixb.html">MAVRIC Appendix B: MAVRIC Utilities</a></li>
<li class="toctree-l1"><a class="reference internal" href="appendixc.html">MAVRIC Appendix C: Advanced Features</a></li>
</ul>

            
          
        </div>
        
      </div>
    </nav>

    <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap">

      
      <nav class="wy-nav-top" aria-label="top navigation">
        
          <i data-toggle="wy-nav-top" class="fa fa-bars"></i>
          <a href="index.html">SCALE Manual</a>
        
      </nav>


      <div class="wy-nav-content">
        
        <div class="rst-content">
        
          















<div role="navigation" aria-label="breadcrumbs navigation">

  <ul class="wy-breadcrumbs">
    
      <li><a href="index.html" class="icon icon-home"></a> &raquo;</li>
        
      <li>Monte Carlo Transport Overview</li>
    
    
      <li class="wy-breadcrumbs-aside">
        
            
            <a href="_sources/Monte Carlo Transport Overview.rst.txt" rel="nofollow"> View page source</a>
          
        
      </li>
    
  </ul>

  
  <hr/>
</div>
          <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
           <div itemprop="articleBody">
            
  <div class="section" id="monte-carlo-transport-overview">
<span id="mctoverview"></span><h1>Monte Carlo Transport Overview<a class="headerlink" href="#monte-carlo-transport-overview" title="Permalink to this headline"></a></h1>
<p><em>Introduction by B. T. Rearden</em></p>
<p>SCALE Monte Carlo transport capabilities enable criticality safety,
shielding, depletion, and sensitivity and uncertainty analysis <a class="bibtex reference internal" href="#rearden-monte-2014" id="id1">[RPP+14]</a>.
SCALE provides separate Monte Carlo capabilities for eigenvalue
neutronics and fixed-source coupled neutron-gamma calculations, in the
KENO code <a class="bibtex reference internal" href="Sourcerer.html#goluoglu-monte-2011" id="id2">[GPJD+11]</a> and fixed-source coupled neutron-gamma calculations in
the Monaco code <a class="bibtex reference internal" href="#peplow-monte-2011" id="id3">[Pep11]</a> Although the eigenvalue and fixed-source
capabilities are provided in separate codes, many capabilities are
shared between them, including physics and geometry packages. The
foundational features shared between the codes are described below, with
specific implementations provided in subsequent sections. Generally, the
use of the Monte Carlo transport solvers in SCALE are best accessed
through the capability-specific sequences: CSAS and Sourcerer for
criticality safety, MAVRIC for shielding, TRITON for depletion,
TSUNAMI-3D for sensitivity and uncertainty analysis, and MCDancoff for
three-dimensional Dancoff factor calculations.</p>
<div class="section" id="multigroup-physics">
<h2>Multigroup Physics<a class="headerlink" href="#multigroup-physics" title="Permalink to this headline"></a></h2>
<p>The multigroup treatment implemented in SCALE has been in use since the
1960s and provides efficient, effective solutions with superior runtime
performance. Problem-dependent multigroup cross section data are
temperature interpolated and resonance self-shielded by other SCALE
modules before they are used in each Monte Carlo calculation. Without
proper resonance self-shielding, accurate multigroup calculations would
not be possible for thermal or intermediate energy spectrum systems.
After self-shielding has been accomplished and the two-dimensional
expansions have been summed into a Legendre expansion of the total
group-to-group transfer arrays, individual nuclide cross sections are
multiplied by their densities and summed into mixtures. These mixture
cross sections can then be used by the deterministic transport codes for
their calculations. The Monte Carlo codes convert the Legendre expansion
of the transfer arrays into probability distributions for the
group-to-group transfers and for the discrete scattering angles and
probabilities that preserve the moments of the Legendre expansion of
each group-to-group transfer. These transfer probabilities, angles, and
angle probabilities are then transformed so that the new group and angle
of scatter are efficiently selected through two random numbers with only
one multiplication and one addition operation. If the selected new group
is negative, it is reset to positive, and the new direction is chosen
isotropically. If the problem is run with P<sub>1</sub> scattering, the
scattering angle is chosen from a continuous distribution. For higher
order scattering, the polar scatter angle is discrete, and the azimuthal
angle is randomly selected from a uniform distribution. Multigroup
physics is implemented for neutron, photon, and neutron-photon coupled
particle transport modes.</p>
<div class="section" id="continuous-energy-physics">
<h3>Continuous-energy Physics<a class="headerlink" href="#continuous-energy-physics" title="Permalink to this headline"></a></h3>
<p>The continuous energy treatment in SCALE provides high resolution
solution strategies with explicit physics representation. The continuous
energy data represent thermal scattering using free gas and s(α,β), with
explicit point-to-point data provided through the thermal region. The
resolved resonance region is represented by pointwise data where the
energy point density is optimized for each reaction of each nuclide.
Data in the unresolved resonance region are represented by probability
tables, and data above the unresolved region implement pointwise data
with explicit point-to-point representation for secondary particles.
Photon yield data represent each discrete photon. Continuous energy
physics contains non-transport data handling to support various flux,
reaction rate, point detector tallies, and sensitivity analysis. In
addition, continuous energy data are converted from a double
differential data format to a lab format in a process where fast look-up
tables are provided during library generation. In SCALE 6.0–6.1,
calculations are performed only at temperatures available on the data
libraries by selecting the library temperature nearest to the desired
temperature for the calculation. Resonance upscattering techniques are
implemented via the Doppler Broadened Rejection Correction method <a class="bibtex reference internal" href="#hart-implementation-2013" id="id4">[HMGR13]</a>.
With SCALE 6.2, problem-dependent continuous energy cross sections at
the user specified temperature are generated at the beginning of the
calculation. Continuous energy physics is implemented for neutron,
photon, and neutron-photon coupled particle transport modes.</p>
</div>
</div>
<div class="section" id="geometry-packages">
<h2>Geometry Packages<a class="headerlink" href="#geometry-packages" title="Permalink to this headline"></a></h2>
<p>Two variants of KENO provide identical solution capabilities with
different geometry packages. KENO V.a implements a simple and efficient
geometry package sufficient for modeling many systems of interest to
criticality safety and reactor physics analysts. KENO-VI implements the
SCALE Generalized Geometry Package (SGGP), which provides a
quadratic-based geometry system with much greater flexibility in
solution modeling. Monaco implements only the SGGP geometry package.
Both packages are based on solid bodies organized into reusable objects
called <em>units</em> that are constructed of material <em>regions</em>. Units can be
conveniently arranged in rectangular or hexagonal <em>arrays</em> of repeating
units. Additionally, nesting is available so that one unit can contain
another unit as a <em>hole</em>, or an array can be nested inside of a unit,
which itself can be repeated in another array. There is no limit to the
number of nesting levels available, so very complex systems can be
quickly generated.</p>
<p>KENO V.a models are constructed from regions of specific shapes
following strict rules which provide great efficiency in geometry
tracking. Allowed shapes are cubes, cuboids (rectangular
parallelepipeds), spheres, cylinders, hemispheres, and hemicylinders.
These shapes must be oriented along orthogonal axes, and they can be
translated, but they cannot be rotated. A major restriction applied to
KENO V.a geometry is that intersections are not allowed, and each region
of a unit must fully enclose the preceding region. An exception to this
rule is in the use of <em>holes</em> through which many units can be placed
within an enclosing unit. However, there is a runtime penalty in
geometry tracking for this flexibility, so this feature should be used
judiciously. KENO V.a provides <em>rectangular</em> arrays where the outer body
of each unit contained in the array must have a cuboidal shape, and
adjacent faces must have the same dimensions. The entire array must be
fully enclosed by the region in which it is placed.</p>
<p>SGGP is a quadratic-based geometry system that provides predefined
bodies including cone, cuboid, cylinder, dodecahedron, ecylinder
(elliptical cylinders), ellipsoid, hexprism, hopper (truncated pyramid),
parallelepiped, planes, rhombohedron, rhexprism (rotated hexprisms),
sphere, and wedge. Bodies not directly provided with SGGP can be
constructed from quadratic surfaces defined with coefficients entered by
the user. All bodies and surfaces can be rotated and translated to any
orientation and position within their respective unit. SGGP also
provides intersecting regions.</p>
<p>SGGP arrays may be composed of cuboids, hexprisms, rhexprisms, or
dodecahedrons. Like KENO V.a, the faces of adjacent units in an array
must have the same dimensions. An array boundary must be specified for
each array, and only the portion of the array within the boundary is
considered a part of the system. Also, the specified array must fill the
entire volume in the specified array boundary. The array boundary may be
any shape that can be specified using quadratic equations.</p>
<p>The use of holes is more flexible in SGGP than in KENO V.a. Within a
unit, holes cannot intersect other holes or the unit boundary, but they
can intersect region boundaries. The use of holes is not necessary to
build complex geometries; they are used primarily to more efficiently
build complex geometries and improve the tracking efficiency of the
simulation. In SGGP the distance to each surface in the unit must be
calculated after each collision. By moving some of the surfaces in a
unit into another unit that is included as a hole, all the surfaces in
the hole unit except the outer boundary are removed from the containing
unit. The judicious use of holes in SGGP can significantly speed up the
calculation.</p>
</div>
<div class="section" id="eigenvalue-analysis">
<h2>Eigenvalue Analysis<a class="headerlink" href="#eigenvalue-analysis" title="Permalink to this headline"></a></h2>
<p>KENO performs eigenvalue calculations for neutron transport primarily to
calculate multiplication factors and flux distributions of fissile
systems in continuous energy and multigroup modes. Both codes allow
explicit geometric representation with their respective geometry
packages. KENO provides a multigroup adjoint capability which is
especially useful for sensitivity analysis. KENO implements standard
variance reduction techniques such as implicit capture, splitting, and
Russian roulette.</p>
<p>The initial fission source distribution in KENO can be specified with
nine options. These options include the default option of a uniform
distribution throughout the fissile material; an axially varying
distribution input by the user or defined as cos(Z) or
(1-cos(Z)):sup:<cite>2</cite>, where Z is the axial position; several options to
initialize the source at a given position (within a given volume, a
given unit, or a unit at a specified array index); or to specifically
provide the coordinates of each starting point.</p>
<p>KENO approximates the real <em>k­eff</em> variance using an iterative
approach and lagging covariance data between generations <a class="bibtex reference internal" href="#ueki-error-1997" id="id5">[UMN97]</a>. KENO
provides a χ<sup>2</sup> test for the normality of <em>k­eff</em> and provides
plots of <span class="math notranslate nohighlight">\(k_{eff}\)</span> by active and inactive generations. KENO reports a
<em>best estimate</em> of <span class="math notranslate nohighlight">\(k_{eff}\)</span> that is computed as the minimum variance of
<span class="math notranslate nohighlight">\(k_{eff}\)</span> based on generations skipped and generations run.</p>
<p>KENO provides track-length tallies for scalar flux and angular flux
moments needed for sensitivity analysis. Additionally, tallies are
provided for reaction rates, with isotopic tallies available only in CE
calculations. KENO also provides mesh tallies based on a user-input
orthogonal grid.</p>
<p>Matrix <span class="math notranslate nohighlight">\(k_{eff}\)</span> calculations provide an additional method of calculating
the <span class="math notranslate nohighlight">\(k_{eff}\)</span> of the system. Cofactor <span class="math notranslate nohighlight">\(k_{eff}\)</span> and source vectors, which
describe the contribution to the system <span class="math notranslate nohighlight">\(k_{eff}\)</span>from each unit, hole,
or array, are available.</p>
<p>KENO provides plots of <em>k­eff­</em>by generation and average <span class="math notranslate nohighlight">\(k_{eff}\)</span>
for visual inspection of source convergence, followed by a <em>χ2</em>
statistical assessment of convergence. Fission source convergence
diagnostic techniques are implemented in KENO to provide improved
confidence in the computed results and to reduce the simulation time for
some cases. Confirming the convergence of the fission source
distribution is especially useful to avoid the false convergence of
<span class="math notranslate nohighlight">\(k_{eff}\)</span> that can be caused by insufficient sampling of important
portions of the system <a class="bibtex reference internal" href="Sourcerer.html#ueki-stationarity-2005" id="id6">[UB05]</a> KENO source convergence diagnostics rely on
Shannon entropy statistics of the mesh-based fission source data.</p>
<p>Parallel computation capabilities are available in both versions of KENO
to provide reductions in wall clock time, especially for sensitivity
analysis or Monte Carlo depletion on computer clusters. By introducing a
simple master-slave approach via message passing interface (MPI), KENO
runs different random walks concurrently on the replicated geometry
within the same generation. The fission source and other tallied
quantities are gathered at the end of each generation by the master
process, and then they are processed either for final edits or next
generations.</p>
</div>
<div class="section" id="shielding-analysis">
<h2>Shielding Analysis<a class="headerlink" href="#shielding-analysis" title="Permalink to this headline"></a></h2>
<p>Monaco is a fixed-source Monte Carlo shielding code that calculates
neutron and photon fluxes and response functions for specific geometry
regions, point detectors, and mesh tallies. Monaco has variance
reduction capabilities, such as source biasing and weight windows, which
can be automated via the MAVRIC sequence. MAVRIC performs radiation
transport on problems that are too challenging for standard, unbiased
Monte Carlo methods. Monaco provides multiple methods to enter the
radioactive source descriptions. Spatial distribution options include
volumetric sources and mesh sources which can be generated by other
codes such as KENO. Energy distributions can be entered by the user or
imported directly from emission data provided by ORIGEN. Spent fuel
analysis is simplified through direct coupling with the ORIGEN binary
concentration files.</p>
<p id="bibtex-bibliography-Monte Carlo Transport Overview-0"><dl class="citation">
<dt class="bibtex label" id="goluoglu-monte-2011"><span class="brackets"><a class="fn-backref" href="#id2">GPJD+11</a></span></dt>
<dd><p>Sedat Goluoglu, Lester M. Petrie Jr, Michael E. Dunn, Daniel F. Hollenbach, and Bradley T. Rearden. Monte Carlo criticality methods and analysis capabilities in SCALE. <em>Nuclear Technology</em>, 174(2):214–235, 2011. Publisher: Taylor &amp; Francis.</p>
</dd>
<dt class="bibtex label" id="hart-implementation-2013"><span class="brackets"><a class="fn-backref" href="#id4">HMGR13</a></span></dt>
<dd><p>S. Hart, G. Ivan Maldonado, Sedat Goluoglu, and Brad Rearden. Implementation of the Doppler broadening rejection correction in KENO. <em>Trans. Am. Nucl. Soc</em>, 108:423–426, 2013.</p>
</dd>
<dt class="bibtex label" id="peplow-monte-2011"><span class="brackets"><a class="fn-backref" href="#id3">Pep11</a></span></dt>
<dd><p>Douglas E. Peplow. Monte Carlo shielding analysis capabilities with MAVRIC. <em>Nuclear Technology</em>, 174(2):289–313, 2011. Publisher: Taylor &amp; Francis.</p>
</dd>
<dt class="bibtex label" id="rearden-monte-2014"><span class="brackets"><a class="fn-backref" href="#id1">RPP+14</a></span></dt>
<dd><p>Bradley T. Rearden, L. M. Petrie, Douglas E. Peplow, Kursat B. Bekar, Dorothea Wiarda, Cihangir Celik, Christopher M. Perfetti, Ahmad M. Ibrahim, S. W. D. Hart, and Michael E. Dunn. Monte Carlo capabilities of the SCALE code system. In <em>SNA+ MC 2013-Joint International Conference on Supercomputing in Nuclear Applications+ Monte Carlo</em>, 06007. EDP Sciences, 2014.</p>
</dd>
<dt class="bibtex label" id="ueki-stationarity-2005"><span class="brackets"><a class="fn-backref" href="#id6">UB05</a></span></dt>
<dd><p>Taro Ueki and Forrest B. Brown. Stationarity modeling and informatics-based diagnostics in Monte Carlo criticality calculations. <em>Nuclear science and engineering</em>, 149(1):38–50, 2005. Publisher: Taylor &amp; Francis.</p>
</dd>
<dt class="bibtex label" id="ueki-error-1997"><span class="brackets"><a class="fn-backref" href="#id5">UMN97</a></span></dt>
<dd><p>Taro Ueki, Takamasa Mori, and Masayuki Nakagawa. Error estimations and their biases in Monte Carlo eigenvalue calculations. <em>Nuclear science and engineering</em>, 125(1):1–11, 1997. Publisher: Taylor &amp; Francis.</p>
</dd>
</dl>
</p>
</div>
</div>


           </div>
           
          </div>
          <footer>
  
    <div class="rst-footer-buttons" role="navigation" aria-label="footer navigation">
      
        <a href="Keno.html" class="btn btn-neutral float-right" title="Keno: A Monte Carlo Criticality Program" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right"></span></a>
      
      
        <a href="CENTRM.html" class="btn btn-neutral float-left" title="CENTRM: A Neutron Transport Code for Computing Continuous-Energy Spectra in General One-Dimensional Geometries and Two-Dimensional Lattice Cells" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left"></span> Previous</a>
      
    </div>
  

  <hr/>

  <div role="contentinfo">
    <p>
        
        &copy; Copyright 2020, SCALE developers

    </p>
  </div>
    
    
    
    Built with <a href="http://sphinx-doc.org/">Sphinx</a> using a
    
    <a href="https://github.com/rtfd/sphinx_rtd_theme">theme</a>
    
    provided by <a href="https://readthedocs.org">Read the Docs</a>. 

</footer>

        </div>
      </div>

    </section>

  </div>
  

  <script type="text/javascript">
      jQuery(function () {
          SphinxRtdTheme.Navigation.enable(true);
      });
  </script>

  
  
    
   

</body>
</html>