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<spanid="id1"></span><h1>BONAMI: Resonance Self-Shielding by the Bondarenko Method<aclass="headerlink"href="#bonami-resonance-self-shielding-by-the-bondarenko-method"title="Permalink to this headline">¶</a></h1>

<p><em>U. Mertyurek and M. L. Williams</em></p>

<p>ABSTRACT</p>

<p>BONAMI is a module of the SCALE code system that is used to perform

Bondarenko calculations for resonance self-shielding. BONAMI obtains

problem-independent cross sections and Bondarenko shielding factors from

a multigroup (MG) AMPX master library, and it creates a MG AMPX working

library of self-shielded, problem-dependent cross sections. Several

options may be used to compute the background cross section values using

the narrow resonance or intermediate resonance approximations, with and

without Bondarenko iterations. A novel interpolation scheme is used that

avoids many of the problems exhibited by other interpolation methods for

the Bondarenko factors. BONAMI is most commonly used in automated SCALE

sequences and is fully integrated within the SCALE cross section

processing module, XSProc.</p>

<p>Acknowledgments</p>

<p>The authors express gratitude to B. T. Rearden and M. A. Jessee for

their supervision of the SCALE project and review of the manuscript. The

authors acknowledge N. M. Greene, formerly of ORNL, for his original

development of and contributions to the BONAMI module and methodology.

Finally, the authors wish to thank Sheila Walker for the completion and

publication of this document.</p>

<divclass="section"id="introduction">

<spanid="id2"></span><h2>Introduction<aclass="headerlink"href="#introduction"title="Permalink to this headline">¶</a></h2>

<p>BONAMI (<strong>BON</strong>darenko <strong>AM</strong>PX <strong>I</strong>nterpolator) is a SCALE

module that performs resonance self-shielding calculations based on the

Bondarenko method <aclass="bibtex reference internal"href="Material%20Specification%20and%20Cross%20Section%20Processing%20Overview.html#ilich-bondarenko-group-1964"id="id3">[IlichB64]</a>. It reads Bondarenko shielding factors

(“f-factors”) and infinitely dilute microscopic cross sections from a

problem-<em>independent</em> nuclear data library processed by the AMPX

system <aclass="bibtex reference internal"href="XSLib.html#wiarda-ampx-2015"id="id4">[WWCD15]</a>, interpolates the tabulated shielding factors to appropriate

temperatures and background cross sections for each nuclide in the

system, and produces a self-shielded, problem-dependent data set.</p>

<p>The code performs self-shielding for an arbitrary number of mixtures

using either the narrow resonance (NR) or intermediate resonance (IR)

approximation <aclass="bibtex reference internal"href="#goldstein-theory-1962"id="id5">[GC62]</a>. The latter capability was introduced in SCALE 6.2.

BONAMI has several options for computing background cross sections,

which may include Bondarenko iterations to approximately account for the

impact of resonance interference for multiple resonance absorbers.

Heterogeneous effects are treated using equivalence theory based on an

“escape cross section” for arrays of slabs, cylinders, or spheres.

During the execution of a typical SCALE computational sequence using

XSProc, Dancoff factors for uniform lattices of square- or

triangular-pitched units are calculated automatically for BONAMI by

numerical integration over the chord length distribution. However, for

non-uniform lattices—such as those containing water holes, control rods,

and so on—the SCALE module MCDancoff can be run to compute Dancoff

factors using Monte Carlo for an arbitrary 3D configuration, and these

values are then provided in the sequence input.</p>

<p>The major advantages of the Bondarenko approach are its simplicity and

speed compared with SCALE’s more rigorous CENTRM/PMC self-shielding

method, which performs a pointwise (PW) deterministic transport

calculation “on the fly” to compute multigroup (MG) self-shielded cross

sections. With the availability of IR theory in BONAMI, accurate results

can be obtained for a variety of system types without the computation

<spanid="id6"></span><h2>Bondarenko Self-Shielding Theory<aclass="headerlink"href="#bondarenko-self-shielding-theory"title="Permalink to this headline">¶</a></h2>

<p>In MG resonance self-shielding calculations, one is interested in

calculating effective cross sections of the form</p>

<spanclass="eqno">(301)<aclass="headerlink"href="#equation-eq7-3-1"title="Permalink to this equation">¶</a></span>\[\sigma^{(r)}_{X,g} = \frac{\int_{g}\sigma^{(r)}_{X}(E)\Phi(E)\text{dE}}{\int_{g}\Phi(E)\text{dE}} ,\]</div>

<p>where <spanclass="math notranslate nohighlight">\(\sigma^{(r)}_{X,g}\)</span> is the shielded MG cross section for reaction type <em>X</em> of

resonance nuclide <em>r</em> in group <em>g</em>; <spanclass="math notranslate nohighlight">\(\sigma^{(r)}_{X}(E)\)</span> is a PW cross section; and <spanclass="math notranslate nohighlight">\(\Phi(E)\)</span> is the PW

weighting function, which approximates the flux spectrum per unit of

energy for the system of interest. PW cross section values are known

from processing evaluated data in ENDF/B files; therefore, resonance

self‑shielding depends mainly on determining the problem-dependent flux

spectrum <spanclass="math notranslate nohighlight">\(\Phi(E)\)</span>, which may exhibit significant fine structure variations as a

result of resonance reactions.</p>

<p>The essence of the Bondarenko method is to parameterize the flux

spectrum corresponding to varying degrees of self-shielding, represented

by the background cross section parameter <spanclass="math notranslate nohighlight">\(\sigma_0\)</span> (called “sigma-zero”) and the

Doppler broadening temperature <em>T</em>. Hence,</p>

<spanclass="eqno">(303)<aclass="headerlink"href="#equation-eq7-3-3"title="Permalink to this equation">¶</a></span>\[\left( \Sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}\ +\sum\limits_{j\ne r}

<spanclass="eqno">(304)<aclass="headerlink"href="#equation-eq7-3-4"title="Permalink to this equation">¶</a></span>\[\text{S}^{(j)}(\text{E,T}) \rightarrow \Sigma^{(j)}_{p}C(E) \text{for j = a NR-scatterer nuclide}\]</div>

<p>where C(E) is a slowly varying function representative of the asymptotic

(i.e., no absorption) flux in a homogeneous medium, which approximates

the flux between resonances. In the resolved resonance range of most

important resonance absorbers, the asymptotic flux per unit energy is

<spanclass="eqno">(306)<aclass="headerlink"href="#equation-eq7-3-6"title="Permalink to this equation">¶</a></span>\[\text{S}^{(j)}(\text{E,T}) \rightarrow \Sigma^{(j)}_{s}(\text{E,T})\Phi(\text{E,T}) ;

<p>A value of λ=1 reduces <aclass="reference internal"href="#equation-eq7-3-7">(307)</a> to the NR expression, whereas λ=0 reduces the

equation to the WR expression. Fractional λ’s are for IR scatterers.

Since the type of scatterer can change with the energy, the IR lambdas

are functions of the energy group as well as the nuclide. The λ values

represent the moderation “effectiveness” of a given nuclide, compared to

hydrogen. The AMPX module LAMBDA was used to compute the IR parameters

on the SCALE libraries. (See AMPX documentation distributed with SCALE)

Substituting <aclass="reference internal"href="#equation-eq7-3-7">(307)</a> into <aclass="reference internal"href="#equation-eq7-3-3">(303)</a> and then dividing by the absorber number

density <em>N(r)</em> gives the following IR approximation for the infinite

<spanclass="eqno">(309)<aclass="headerlink"href="#equation-eq7-3-9"title="Permalink to this equation">¶</a></span>\[\sigma _{0}^{\text{(r)}}\text{(E,T)}\ \ =\ \ \frac{1}{{{\text{N}}^{\text{(r)}}}}\,\,\sum\limits_{j\ne r}{\left( \Sigma _{\text{a}}^{\text{(j)}}(\text{E,T})+\lambda _{\text{g}}^{\text{(j)}}\,\Sigma _{\text{s}}^{\text{(j)}}(\text{E,T})\,\, \right)}\]</div>

<p>Although <aclass="reference internal"href="#equation-eq7-3-8">(308)</a> provides the flux spectrum as a function of the background

cross section <spanclass="math notranslate nohighlight">\(\sigma \,_{0}^{(r)}(u,T)\)</span> it is not in a form that can be

preprocessed when the MG library is generated, because the energy variation of

<spanclass="math notranslate nohighlight">\(\sigma \,_{0}^{(r)}(E,T)\)</span> must be known. If the total cross sections

of the background nuclides in <aclass="reference internal"href="#equation-eq7-3-9">(309)</a> have different energy variations, the shape of

<spanclass="math notranslate nohighlight">\(\sigma \,_{0}^{(r)}(E,T)\)</span> depends on their relative concentrations—which

are not known when the MG library is processed.

However, if the cross sections in <aclass="reference internal"href="#equation-eq7-3-9">(309)</a> are independent of energy,

so that the background cross section is <em>constant</em>,

<aclass="reference internal"href="#equation-eq7-3-8">(308)</a> can be solved for any arbitrary value of <spanclass="math notranslate nohighlight">\(\sigma \,_{0}^{(r)}\)</span>

as a parameter. This obviously occurs for the special case in which nuclide

<em>r</em> is the only resonance nuclide in the mixture; i.e., the background materials

are nonabsorbing moderators for which the total cross section is equal to the potential

cross section. In this case, <spanclass="math notranslate nohighlight">\(\sigma \,_{0}^{(r)}(E,T)\quad \to \ \ \ \sigma \,_{0,g}^{(r)}\)</span>,

<spanclass="eqno">(310)<aclass="headerlink"href="#equation-eq7-3-10"title="Permalink to this equation">¶</a></span>\[\sigma \,_{0,g}^{(r)}\,\,=\quad \frac{1}{N_{{}}^{(r)}}\sum\limits_{j\,\ne \,i}{\ N_{{}}^{(j)}\,\lambda _{g}^{(j)}\sigma \,_{p}^{(j)}}\]</div>

<p>If the mixture contains multiple resonance absorbers, as is usually the

case, other approximations must be made to obtain a constant background

cross section.</p>

<p>The approximation of “no resonance interference” assumes that resonances

of background nuclides do not overlap with those of nuclide <em>r</em>, so

their total cross sections can be approximated by the potential values

within resonances of <em>r</em> where self-shielding occurs. In this

approximation, the expression in <aclass="reference internal"href="#equation-eq7-3-10">(310)</a> is also used for the background

cross section.</p>

<p>Another approximation is to represent the energy-dependent cross

sections of the background nuclides by their group-averaged (i.e.,

<spanclass="eqno">(312)<aclass="headerlink"href="#equation-eq7-3-12"title="Permalink to this equation">¶</a></span>\[\sigma _{0,g}^{\text{(r)}}\ \ =\ \ \frac{1}{{{\text{N}}^{\text{(r)}}}}\,\,\sum\limits_{j\ne r}{\left( \Sigma _{\text{a,g}}^{\text{(j)}}+\lambda _{\text{g}}^{\text{(j)}}\,\Sigma _{\text{s,g}}^{\text{(j)}}\, \right)}\]</div>

<p>An equation similar to <aclass="reference internal"href="#equation-eq7-3-12">(312)</a> is used for the background cross sections of

all resonance nuclides; thus the self-shielded cross sections of each

resonance absorber depend on the shielded cross sections of all other

resonance absorbers in the mixture. When self-shielding operations are

performed with BONAMI for this approximation, “Bondarenko” iterations

are performed to account for the inter-dependence of the shielded cross

sections.</p>

<p>Assuming that <spanclass="math notranslate nohighlight">\(\sigma \,_{0}^{(r)}\)</span> is represented as a groupwise-constant

based on one of the previous approximations, several methods can be used to

obtain a parameterized flux spectrum for preprocessing Bondarenko data in the MG

libraries. In the simpliest approach, the scattering source of the resonance

nuclide <em>r</em> in <aclass="reference internal"href="#equation-eq7-3-8">(308)</a> is represented by the NR approximation,

<spanclass="math notranslate nohighlight">\({{\text{S}}^{\text{(r)}}}(\text{E,T})\)</span> to <spanclass="math notranslate nohighlight">\(\Sigma _{\text{p}}^{\text{(r)}}C(E)\)</span>.

In this case, <aclass="reference internal"href="#equation-eq7-3-8">(308)</a> can be solved analytically to obtain the following

expression for the flux spectrum used to process MG data as a function of <spanclass="math notranslate nohighlight">\(\sigma \,_{0}^{(r)}\)</span>:</p>

<spanid="id9"></span><h3>Self-Shielded Cross Section Data in SCALE Libraries<aclass="headerlink"href="#self-shielded-cross-section-data-in-scale-libraries"title="Permalink to this headline">¶</a></h3>

<p>The AMPX code system processes self-shielded cross sections using the

flux expressions described in the preceding section. For MG libraries in

SCALE-6.2 and later versions, the NR approximation in <aclass="reference internal"href="#equation-eq7-3-13">(313)</a> is used to

represent the flux spectrum for nuclides with masses below A=40, since

the NR approximation is generally accurate for low-mass nuclides and/or

high energies. The standard AMPX weight functions are used to represent

C(E) over the entire energy range for all nuclides with A<40, except for

hydrogen and oxygen which use a calculated C(E) from CENTRM. The NR

approximation with a calculated C(E) function is also used to represent

the spectrum above the resolved resonance range for nuclides with A>40;

but in the resolved resonance range of these nuclides, AMPX processes

shielded cross sections with flux spectra obtained from CENTRM

calculations using either a homogeneous or heterogeneous model.

Regardless of the method used to obtain the flux spectrum, the

parameterized shielded cross sections for absorber nuclide “r” are

<spanid="id10"></span><h3>Background Cross Section Options in BONAMI<aclass="headerlink"href="#background-cross-section-options-in-bonami"title="Permalink to this headline">¶</a></h3>

<p>To compute self-shielded cross sections for nuclide <em>r</em>, BONAMI first

computes the appropriate background cross section for the system of

interest and then interpolates the library Bondarenko data to obtain the

f-factor corresponding to this σ<sub>0</sub> and nuclide temperature.

Several options are available in BONAMI to compute the background cross

section, based on <aclass="reference internal"href="#equation-eq7-3-10">(310)</a> and <aclass="reference internal"href="#equation-eq7-3-12">(312)</a> in the preceding section. The options are

specified by input parameter “<strong>iropt</strong>” and have the following

definitions:</p>

<olclass="loweralpha simple">

<li><p>iropt = 0 => NR approximation with Bondarenko iterations:</p></li>

</ol>

<p>Background cross sections for all nuclides are computed using <aclass="reference internal"href="#equation-eq7-3-12">(312)</a> with

<spanclass="eqno">(317)<aclass="headerlink"href="#equation-eq7-3-17"title="Permalink to this equation">¶</a></span>\[\sigma _{0}^{\text{(r)}}\ =\ \frac{1}{{{\text{N}}^{\text{(r)}}}}\,\,\sum\limits_{j\ne r}{\Sigma _{\text{t,g}}^{\text{(j)}}} .\]</div>

<p>Since the background cross section for each nuclide depends on the shielded

total cross sections of all other nuclides in the mixture,

“Bondarenko iterations” are performed in BONAMI to obtain a consistent set of

shielded cross sections. Bondarenko iterations provide a crude method of

accounting for resonance interference effects that are ignored by the

approximation for <spanclass="math notranslate nohighlight">\(\sigma \,_{0}^{(r)}\)</span> in <aclass="reference internal"href="#equation-eq7-3-10">(310)</a>. The BONAMI

iterative algorithm generally converges in a few iterations. Prior to

SCALE-6.2, this option was the only one available in BONAMI, and it is still the default for XSProc.</p>

<olclass="loweralpha simple"start="2">

<li><p>iropt = 1 => IR approximation with no resonance interference

(potential cross sections):</p></li>

</ol>

<p>Background cross sections for all nuclides are computed using <aclass="reference internal"href="#equation-eq7-3-10">(310)</a>. No

Bondarenko iterations are needed.</p>

<olclass="loweralpha simple"start="3">

<li><p>iropt t = 2 => IR approximation with Bondarenko iterations, but no

resonance scattering:</p></li>

</ol>

<p>Background cross sections for all nuclides are computed using <aclass="reference internal"href="#equation-eq7-3-12">(312)</a> with

the scattering cross section approximated by the potential value;

<spanid="id11"></span><h3>Self-Shielded Cross Sections for Heterogeneous Media<aclass="headerlink"href="#self-shielded-cross-sections-for-heterogeneous-media"title="Permalink to this headline">¶</a></h3>

<p>Equivalence theory can be used to obtain shielded cross sections for

heterogeneous systems containing one or more “lumps” of resonance

absorber mixtures separated by moderators, such as reactor lattices. It

can be shown that if the fuel escape probability is represented by the

Wigner rational approximation, the collision probability formulation of

the neutron transport equation for an absorber body in a heterogeneous

medium can be reduced to a form identical to <aclass="reference internal"href="#equation-eq7-3-3">(303)</a>. This can be done for

an “equivalent” infinite homogeneous medium consisting of the same

absorber body mixture plus an additional NR scatterer with a constant

cross section called the “escape cross section” <aclass="bibtex reference internal"href="#lamarsh-introduction-1966"id="id12">[Lam66]</a>.

Equivalence

theory states that the self-shielded cross section for resonance

absorber <em>r</em> in the heterogeneous medium is equal to the self-shielded

cross section of <em>r</em> in the equivalent infinite homogeneous medium;

therefore the f-factors that were calculated for homogenous mixtures can

also be used to compute self-shielded cross sections for heterogeneous

media by simply interpolating the tabulated f-factors in the library to

<spanclass="eqno">(319)<aclass="headerlink"href="#equation-eq7-3-19"title="Permalink to this equation">¶</a></span>\[\hat{\sigma }_{0}^{(r)}\quad =\quad \sigma _{0}^{(r)}\ +\ \ \,\sigma _{esc}^{(r)}\]</div>

<p>where,</p>

<blockquote>

<div><p><spanclass="math notranslate nohighlight">\(\hat{\sigma }_{0}^{(r)}\)</span> = background cross section of r in the absorber lump of the heterogeneous system;</p>

<p><spanclass="math notranslate nohighlight">\(\sigma \,_{0}^{(r)}\)</span> = background cross section defined in <aclass="reference internal"href="#id7"><spanclass="std std-ref">Parameterized Flux Spectra</span></a> for an infinite homogeneous medium of the

absorber body mixture;</p>

<p><spanclass="math notranslate nohighlight">\(\sigma _{esc}^{(r)}\)</span> = microscopic escape cross section for nuclide <em>r</em>, defined as</p>

<spanclass="eqno">(320)<aclass="headerlink"href="#equation-eq7-3-20"title="Permalink to this equation">¶</a></span>\[\sigma _{esc}^{(r)}\quad =\quad \frac{{{\Sigma }_{esc}}}{{{N}^{(r)}}}\]</div>

<blockquote>

<div><p><spanclass="math notranslate nohighlight">\({{\Sigma }_{esc}}\)</span> = macroscopic escape cross section for the absorber lump defined in BONAMI as</p>

<spanclass="eqno">(321)<aclass="headerlink"href="#equation-eq7-3-21"title="Permalink to this equation">¶</a></span>\[{{\Sigma }_{esc}}\quad =\quad \,\frac{(1\quad -\quad c)A}{\bar{\ell }\ \,\ \left[ 1\quad +\quad \left( A\quad -\quad 1 \right)c \right]}\]</div>

<p>where</p>

<blockquote>

<div><p><spanclass="math notranslate nohighlight">\(\bar{\ell }\)</span> = average chord length of the absorber body = <spanclass="math notranslate nohighlight">\(4\ \ \,\times \ \frac{volume}{surface\ \ area}\)</span>;</p>

<p>A = Bell factor, used to improve the accuracy of the Wigner rational approximation;</p>

<p>c = lattice Dancoff factor, which is equal to the probability that a neutron escaping from one

absorber body will reach another absorber body before colliding in the intervening moderator.</p>

</div></blockquote>

<p>Values for the mean chord length <spanclass="math notranslate nohighlight">\(\bar{\ell }\)</span> are computed in BONAMI for slab,

sphere, and cylinder absorber bodies. In the most common mode of operation where

BONAMI is executed through the XSProc module in SCALE, Dancoff factors for

uniform lattices are computed automatically and provided as input to BONAMI.

For nonuniform lattices—such as those containing water holes, control rods,

etc.—it may be desirable for the user to run the SCALE module MCDancoff to

compute Dancoff factors using Monte Carlo for an arbitrary 3D configuration.

In this case the values are provided in the MORE DATA input block of XSProc.

The Bell factor “A” is a correction factor to account for errors caused by use

of the Wigner rational approximation to represent the escape probability from a

lump. Two optional Bell factor corrections are included in BONAMI. The first uses

expressions developed by Otter that essentially force the Wigner escape

probability for an isolated absorber lump to agree with the exact escape

probability for the particular geometry by determining a value of A as a function of

<spanclass="math notranslate nohighlight">\({{\Sigma }_{T}}\bar{\ell }\)</span> for slab, cylindrical, or spherical

geometries. Since the Otter expression was developed for isolated bodies,

it does not account for errors in the Wigner rational approximation due to

lattice effects. BONAMI also includes a Bell factor correction based on a

modified formulation developed by Leslie <aclass="bibtex reference internal"href="#leslie-improvements-1965"id="id13">[LHJ65]</a> that is a function of the Dancoff factor.</p>

</div>

</div>

<divclass="section"id="interpolation-scheme">

<spanid="id14"></span><h2>Interpolation Scheme<aclass="headerlink"href="#interpolation-scheme"title="Permalink to this headline">¶</a></h2>

<p>After the background cross section for a system has been computed,

BONAMI interpolates f-factors at the appropriate σ<sub>0</sub> and

temperature from the tabulated values in the library. <aclass="reference internal"href="#fig7-3-1"><spanclass="std std-numref">Fig. 202</span></a> shows

a typical variation of the f-factor vs. background cross sections for

the capture cross section of <sup>238</sup>U in the SCALE 252 group

<pclass="caption"><spanclass="caption-number">Fig. 202 </span><spanclass="caption-text">Plot of f-factor variation for <sup>238</sup>U capture reaction.</span><aclass="headerlink"href="#id23"title="Permalink to this image">¶</a></p>

</div>

<p>Interpolation of the f-factors can be problematic, and several different

schemes have been developed for this purpose. Some of the interpolation

methods that have been used in other codes are constrained

Lagrangian, <aclass="bibtex reference internal"href="#davis-sphinx-1977"id="id15">[DYB77]</a> arc-tangent fitting, <aclass="bibtex reference internal"href="#kidman-improved-1974"id="id16">[Kid74]</a> and an approach developed by

Segev <aclass="bibtex reference internal"href="#segev-interpolation-1981"id="id17">[Seg81]</a>. All of these were tested and found to be inadequate for use

with the SCALE libraries, which may have multiple energy groups within a

single resonance. BONAMI uses a unique interpolation method developed by

Greene, which is described in <aclass="bibtex reference internal"href="#greene-method-1982"id="id18">[Gre82]</a>. Greene’s interpolation method

is essentially a polynomial approach in which the powers of the

polynomial terms can vary within a panel, as shown in <aclass="reference internal"href="#equation-eq7-3-25">(325)</a>:</p>

<p><aclass="reference internal"href="#fig7-3-2"><spanclass="std std-numref">Fig. 203</span></a> illustrates the expected behavior of <aclass="reference internal"href="#equation-eq7-3-22">(322)</a> caused by varying

the powers in a panel.</p>

<p>By allowing the power <em>q</em> to vary as a function of independent

variable σ, we can move between the various monotonic curves on the

graph in a monotonic fashion. Note that when <em>p</em> crosses the

<em>p</em> = 1 curve, the shape changes from concave to convex, or vice versa.

This shape change means that we can use the scheme to introduce an

inflection point, which is exactly the situation needed for

<pclass="caption"><spanclass="caption-number">Fig. 203 </span><spanclass="caption-text">Illustration of the effects of varying “powers” in the Greene interpolation method.</span><aclass="headerlink"href="#id24"title="Permalink to this image">¶</a></p>

</div>

<p><aclass="reference internal"href="#fig7-3-3"><spanclass="std std-numref">Fig. 204</span></a> and <aclass="reference internal"href="#fig7-3-3"><spanclass="std std-numref">Fig. 204</span></a> show typical “fits” of the f-factors using

the Greene interpolation scheme for two example cases. Note, in

particular, that since this scheme has guaranteed monotonicity, it

easily accommodates the end panels that have the smooth asymptotic

variation. Even considering the extra task of having to determine the

powers for temperature and σ<sub>0</sub> interpolations, the method is not

significantly more time-consuming than the alternative schemes for most

<pclass="caption"><spanclass="caption-number">Fig. 204 </span><spanclass="caption-text">Use of Greene’s method to fit the σ<sub>0</sub> variation of Bondarenko factors for case 1.</span><aclass="headerlink"href="#id25"title="Permalink to this image">¶</a></p>

<pclass="caption"><spanclass="caption-number">Fig. 205 </span><spanclass="caption-text">Use of Greene’s method to fit the σ<sub>0</sub> variation of Bondarenko factors for case 2.</span><aclass="headerlink"href="#id26"title="Permalink to this image">¶</a></p>

</div>

</div>

<divclass="section"id="input-instructions">

<spanid="id19"></span><h2>Input Instructions<aclass="headerlink"href="#input-instructions"title="Permalink to this headline">¶</a></h2>

<p>BONAMI is most commonly used as an integral component of XSProc through

SCALE automated analysis sequences. XSProc automatically prepares all

the input data for BONAMI and links it with the other self-shielding

modules. During a SCALE sequence execution, the data are provided

directly to BONAMI in memory through XSProc. Some of the input

parameters can be modified in the MOREDATA block in XSProc.</p>

<p>However, the legacy interface to execute stand-alone BONAMI calculations

has been preserved for expert users. The legacy input to BONAMI uses the

FIDO schemes described in the FIDO chapter of the SCALE manual. The

BONAMI input for standalone execution is given below, where the MOREDATA

input keywords are marked in bold.</p>

<pclass="centered">

<strong>Data Block 1</strong></p><p>0$ Logical Unit Assignments [4]</p>