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.. _7-3:
BONAMI: Resonance Self-Shielding by the Bondarenko Method
*U. Mertyurek and M. L. Williams*
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.
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.
.. _7-3-1:
BONAMI (**BON**\ darenko **AM**\ PX **I**\ nterpolator) is a SCALE
module that performs resonance self-shielding calculations based on the
Bondarenko method :cite:`ilich_bondarenko_group_1964`. It reads Bondarenko shielding factors
(“f-factors”) and infinitely dilute microscopic cross sections from a
problem-\ *independent* nuclear data library processed by the AMPX
system :cite:`wiarda_ampx_2015`, 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.
The code performs self-shielding for an arbitrary number of mixtures
using either the narrow resonance (NR) or intermediate resonance (IR)
approximation :cite:`goldstein_theory_1962`. 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.
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
expense of CENTRM/PMC.
.. _7-3-2:
Bondarenko Self-Shielding Theory
In MG resonance self-shielding calculations, one is interested in
calculating effective cross sections of the form
.. math::
:label: eq7-3-1
\sigma^{(r)}_{X,g} = \frac{\int_{g}\sigma^{(r)}_{X}(E)\Phi(E)\text{dE}}{\int_{g}\Phi(E)\text{dE}} ,
where :math:`\sigma^{(r)}_{X,g}` is the shielded MG cross section for reaction type *X* of
resonance nuclide *r* in group *g*; :math:`\sigma^{(r)}_{X}(E)` is a PW cross section; and :math:`\Phi(E)` 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 :math:`\Phi(E)`, which may exhibit significant fine structure variations as a
result of resonance reactions.
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 :math:`\sigma_0` (called “sigma-zero”) and the
Doppler broadening temperature *T*. Hence,
.. math::
:label: eq7-3-2
\Phi \text{(E)}\to \Phi \text{(E;}\,\sigma _{\text{0,g}}^{\text{(r)}}\text{,T)}\ \ \,,\,\ \text{E}\in \text{g}\ ; \text{and} \
\sigma^{(r)}_{X,g} \rightarrow \sigma^{(r)}_{X,g}(\sigma^{(r)}_{0,g},\text{T})
With this approach, it is possible to preprocess MG data for different
background cross sections representing varying degrees of resonance
self-shielding. This allows the MG averaging to be performed during the
original MG library processing, so that BONAMI can do a simple
interpolation on the background cross section and temperature to obtain
self-shielded cross sections. This procedure is much faster than the
CENTRM/PMC method in SCALE, which computes a PW flux spectrum by solving
the neutron transport equation on a PW energy mesh in CENTRM and then
evaluates :eq:`eq7-3-1`. in PMC “on the fly” during a sequence execution.
BONAMI performs two main tasks: (a) computation of background cross
sections for all nuclides in each mixture in the system and (b)
interpolation of shielded cross sections from the library values
tabulated vs. background cross sections and temperature. The BONAMI
calculation is essentially isolated from the computation of the
tabulated shielded cross sections, which is performed by the AMPX
processing code system—the only connection is through the definition of
the background cross section used in processing the library values.
Various approximations can be used to parameterize the flux spectrum in
terms of a background XS, as required by the Bondarenko method. We will
first consider several approaches to representing the flux in an
infinite medium, which lead to different definitions of the background
cross section. BONAMI’s use of equivalence theory to extend the
homogeneous methods to address heterogeneous systems, such as reactor
lattices, is discussed in the following section.
.. _7-3-2-1:
Parameterized Flux Spectra
Several approximations can be applied to the infinite medium transport
equation to parameterize the flux spectrum in terms of a background XS,
as required by the Bondarenko method. The resulting homogeneous spectra
are used in AMPX to process MG cross sections which can also can be
applied to heterogeneous systems (i.e., lattices) by using equivalence
theory; thus the key step is determining approximations that provide
parameterized solutions for homogeneous media. The neutron transport
equation for a homogeneous medium at temperature *T*, containing a
resonance nuclide *r* mixed with other nuclides can be expressed as
.. math::
:label: eq7-3-3
\left( \Sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}\ +\sum\limits_{j\ne r}
{\Sigma _{\text{t}}^{\text{(j)}}\text{(E,T)}} \right)\ \Phi \text{(E,T)}\ \,\,\,=\ \ \,{{\text{S}}^{\text{(r)}}}(\text{E,T})\ \,+\,\sum\limits_{j\ne r}{{{\text{S}}^{\text{(j)}}}(\text{E,T})} ,
where :math:`\Sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}`
, :math:`\text{S}_{{}}^{\text{(r)}}\text{(E,T)}` are the macroscopic total XS and
elastic scattering source for *r*, respectively; and :math:`\Sigma _{\text{t}}^{\text{(j)}}\text{(E,T)}`,
:math:`\text{S}_{{}}^{\text{(j)}}\text{(E,T)}` are the macroscopic total cross
section and elastic source, respectively, for a nuclide *j*. The cross sections in
all these expressions are Doppler-broadened to the temperature of the
medium. The nuclides in the summations (i.e., all nuclides except *r*)
are called background nuclides for the resonance absorber *r*.
The NR approximation can be used to approximate scattering sources of
nuclides for which the neutron energy loss is large compared with the
practical widths of resonances for the absorber materials of interest.
Applying the NR approximation for the scattering source of background
material *j* gives
.. math::
:label: eq7-3-4
\text{S}^{(j)}(\text{E,T}) \rightarrow \Sigma^{(j)}_{p}C(E) \text{for j = a NR-scatterer nuclide}
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
represented as,
.. math::
:label: eq7-3-5
C(\text{E})\ =\ \ \,\frac{{{\Phi }_{\infty }}}{E}\ \ \ ,
where :math:`{{\Phi }_{\infty }}` is an arbitrary normalization constant that cancels from the MG
cross section expression. In the thermal range a Maxwellian spectrum is
used for C(E), and in the fast range a fission spectrum is used. The
SCALE Cross Section Libraries section of the SCALE documentation gives
analytical expressions for C(E) used in AMPX to process MG data with the
NR approximation. AMPX also has an option to input numerical values for
C(E), obtained for example from a PW slowing-down calculation with
CENTRM. This method has been used to process MG data for some nuclides
on the SCALE libraries.
Conversely, the wide resonance (WR) approximation has been used to
represent elastic scattering sources of nuclides for which the neutron
energy loss is small compared with the practical width of the resonance.
This approximation tends to be more accurate for heavy nuclides and for
lower energies. The limit of infinite mass is usually assumed, so the WR
approximation is sometimes called the infinite mass (IM) approximation.
Because of the assumption of IM, there is no energy loss due to
collisions with WR scatterers. Applying the WR approximation for the
slowing-down source of background nuclide *j* gives
.. math::
:label: eq7-3-6
\text{S}^{(j)}(\text{E,T}) \rightarrow \Sigma^{(j)}_{s}(\text{E,T})\Phi(\text{E,T}) ;
\text{for} j = \text{a WR-scatterer nuclide}
The IR approximation was proposed in the 1960s for scatterers with
slowing-down properties intermediate between those of NR and WR
scatterers :cite:`goldstein_theory_1962`. The IR method represents the scattering source for
arbitrary nuclide *j* by a linear combination of NR and WR expressions.
This is done by introducing an IR parameter usually called lambda, such
.. math::
:label: eq7-3-7
\text{S}_{{}}^{\text{(j)}}(\text{E,T)}\,\ \to \ \,\underbrace{\lambda _{\text{g}}^{\text{(j)}}\Sigma _{\text{p}}^{\text{(j)}}\,C(E)}_{\mathbf{NR scatterer}}\ +\ \ (1-\lambda _{\text{g}}^{\text{(j)}})\,\,\underbrace{\Sigma _{\text{s}}^{\text{(j)}}(\text{E,T})\Phi (\text{E,T})}_{\mathbf{WR scatterer}}\ \,\ \,\ ;\,\,\ \ \text{E}\in \text{g}\,\text{.}
A value of λ=1 reduces :eq:`eq7-3-7` 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 :eq:`eq7-3-7` into :eq:`eq7-3-3` and then dividing by the absorber number
density *N\ (r)* gives the following IR approximation for the infinite
medium transport equation in energy group g
.. math::
:label: eq7-3-8
\left( \sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}\ \text{+}\ \sigma _{0}^{\text{(r)}}\text{(E,T) } \right)\,{{\Phi }^{\text{(r)}}}\text{(E,T)}\ \ =\,\ \frac{\text{1}}{{{\text{N}}^{\text{(r)}}}}{{\text{S}}^{\text{(r)}}}\text{(E,T)}\ +\ \frac{\text{1}}{{{\text{N}}^{\text{(r)}}}}\sum\limits_{j\ne r}{\lambda _{\text{g}}^{\text{(j)}}\,\Sigma _{\text{p}}^{\text{(j)}}C(E)\,}
where the background cross section of *r* in the homogeneous medium is
defined as
.. math::
:label: eq7-3-9
\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)}
Although :eq:`eq7-3-8` provides the flux spectrum as a function of the background
cross section :math:`\sigma \,_{0}^{(r)}(u,T)` it is not in a form that can be
preprocessed when the MG library is generated, because the energy variation of
:math:`\sigma \,_{0}^{(r)}(E,T)` must be known. If the total cross sections
of the background nuclides in :eq:`eq7-3-9` have different energy variations, the shape of
:math:`\sigma \,_{0}^{(r)}(E,T)` depends on their relative concentrations—which
are not known when the MG library is processed.
However, if the cross sections in :eq:`eq7-3-9` are independent of energy,
so that the background cross section is *constant*,
:eq:`eq7-3-8` can be solved for any arbitrary value of :math:`\sigma \,_{0}^{(r)}`
as a parameter. This obviously occurs for the special case in which nuclide
*r* 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, :math:`\sigma \,_{0}^{(r)}(E,T)\quad \to \ \ \ \sigma \,_{0,g}^{(r)}`,
.. math::
:label: eq7-3-10
\sigma \,_{0,g}^{(r)}\,\,=\quad \frac{1}{N_{{}}^{(r)}}\sum\limits_{j\,\ne \,i}{\ N_{{}}^{(j)}\,\lambda _{g}^{(j)}\sigma \,_{p}^{(j)}}
If the mixture contains multiple resonance absorbers, as is usually the
case, other approximations must be made to obtain a constant background
cross section.
The approximation of “no resonance interference” assumes that resonances
of background nuclides do not overlap with those of nuclide *r*, so
their total cross sections can be approximated by the potential values
within resonances of *r* where self-shielding occurs. In this
approximation, the expression in :eq:`eq7-3-10` is also used for the background
cross section.
Another approximation is to represent the energy-dependent cross
sections of the background nuclides by their group-averaged (i.e.,
self-shielded cross) values; thus
.. math::
:label: eq7-3-11
\sigma \,_{a}^{(j)}(E,T)\quad \to \ \ \ \sigma \,_{a,g}^{(j)}\ \quad ;\quad \ \ \quad \sigma \,_{s}^{(j)}(E,T)\quad \to \ \ \ \sigma \,_{s,g}^{(j)}\text{ for }E\in g
In this case, the background cross section in :eq:`eq7-3-9` for nuclide *r* is the
group-dependent expression,
.. math::
:label: eq7-3-12
\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)}
An equation similar to :eq:`eq7-3-12` 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
Assuming that :math:`\sigma \,_{0}^{(r)}` 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 *r* in :eq:`eq7-3-8` is represented by the NR approximation,
:math:`{{\text{S}}^{\text{(r)}}}(\text{E,T})` to :math:`\Sigma _{\text{p}}^{\text{(r)}}C(E)`.
In this case, :eq:`eq7-3-8` can be solved analytically to obtain the following
expression for the flux spectrum used to process MG data as a function of :math:`\sigma \,_{0}^{(r)}`:
.. math::
:label: eq7-3-13
{{\Phi }^{\text{(r)}}}\text{(E;}\,\sigma _{0}^{\text{(r)}}\text{,T)}\ \ =\,\ \frac{\sigma _{\text{p}}^{\text{(r)}}\ +\ \,\frac{\text{1}}{{{\text{N}}^{\text{(r)}}}}\sum\limits_{j\ne r}{\,\Sigma _{\text{p}}^{\text{(j)}}\,}\ }{\sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}\ \text{+}\ \sigma _{0}^{\text{(r)}}}C(E)\ \ \,\ \to \ \ \,\frac{C(E)\ }{\sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}\ \text{+}\ \sigma _{0}^{\text{(r)}}}
where C(E) includes is an arbitrary constant multiplier that cancels
from :eq:`eq7-3-1`.
A more accurate approach that does not require using the NR
approximation is to directly solve the IR form of the neutron transport
equation using PW cross sections, with the assumption of no interference
between mixed absorber resonances. The IRFfactor module of AMPX uses
XSProc to calculate the self-shielded flux spectrum for MG data
processing using one of two options:
(a) A homogeneous model corresponding to an infinite medium of the
resonance nuclide mixed with hydrogen, in which the ratio of the
absorber to hydrogen number densities is varied in CENTRM to obtain
the desired background cross section values;
(b) A heterogeneous model corresponding to a 2D unit cell from an
infinite lattice, in which the cell geometry (e.g., pitch) as well
as the absorber number density is varied in CENTRM to obtain the
desired background cross section values.
Both of these models provide a numerical solution for the flux spectrum.
Details on these approaches are given in reference 2.
.. _7-3-2-2:
Self-Shielded Cross Section Data in SCALE Libraries
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 :eq:`eq7-3-13` 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
computed from the expression,
.. math::
:label: eq7-3-14
\sigma _{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{(r)}\,,T)\quad =\quad \,\frac{\int_{g}{\ \ \,\sigma _{X}^{(r)}(E,T)\ \,\Phi (E;\,\,\sigma \,_{0}^{(r)}\,,T)\ dE}}{\int_{g}{\ \,\Phi (E;\,\,\sigma \,_{0}^{(r)}\,,T)\ \,dE}}\quad ,
where :math:`\Phi (E;\,\,\sigma \,_{0}^{(r)}\,,T)` is the flux for a given value
of :math:`\sigma \,_{0}^{(r)}` and *T*.
Rather than storing self-shielded cross sections in the master library,
AMPX converts them to Bondarenko shielding factors, also called
f-factors, defined as the ratio of the shielded cross section to the
infinitely dilute cross section. Thus the MG libraries in SCALE contain
Bondarenko data consisting of f‑factors defined as
.. math::
:label: eq7-3-15
f_{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{{}}\,,T)\quad =\quad \,\frac{\sigma _{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{{}},T)}{\sigma _{\text{X,g}}^{\text{(r)}}(\infty )}\quad ,
and infinitely dilute cross sections defined as,
.. math::
:label: eq7-3-16
\sigma _{\text{X,g}}^{\text{(r)}}(\infty )\quad =\quad \,\sigma _{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{{}}=\infty ,T={{T}_{ref}}) \to \ \ \,\frac{\int_{g}{\ \sigma _{X}^{(r)}(E,{{T}_{ref}})\ C(E)\ \,dE}}{\int_{g}{\ \,C(E)\ \,dE}}\quad .
In AMPX, the reference temperature for the infinitely dilute cross
section is normally taken to be 293 K. Bondarenko data on SCALE
libraries are provided for all energy groups and for five reaction
types: total, radiative capture, fission, within-group scattering, and
elastic scatter. Recent SCALE libraries include f-factors at ~10–30
background cross section values (depending on nuclide) ranging from
~10\ :sup:`−3` to ~10\ :sup:`10` barns, which span the range of
self-shielding conditions. Typically the f-factor data are tabulated at
five temperature values. Background cross sections and temperatures
available for each nuclide in the SCALE MG libraries are given in the
SCALE Cross Section Libraries chapter.
.. _7-3-2-3:
Background Cross Section Options in BONAMI
To compute self-shielded cross sections for nuclide *r*, 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` and nuclide temperature.
Several options are available in BONAMI to compute the background cross
section, based on :eq:`eq7-3-10` and :eq:`eq7-3-12` in the preceding section. The options are
specified by input parameter “\ **iropt**\ ” and have the following
(a) iropt = 0 => NR approximation with Bondarenko iterations:
Background cross sections for all nuclides are computed using :eq:`eq7-3-12` with
λ=1; therefore,
.. math::
:label: eq7-3-17
\sigma _{0}^{\text{(r)}}\ =\ \frac{1}{{{\text{N}}^{\text{(r)}}}}\,\,\sum\limits_{j\ne r}{\Sigma _{\text{t,g}}^{\text{(j)}}} .
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 :math:`\sigma \,_{0}^{(r)}` in :eq:`eq7-3-10`. 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.
(b) iropt = 1 => IR approximation with no resonance interference
(potential cross sections):
Background cross sections for all nuclides are computed using :eq:`eq7-3-10`. No
Bondarenko iterations are needed.
(c) iropt t = 2 => IR approximation with Bondarenko iterations, but no
resonance scattering:
Background cross sections for all nuclides are computed using :eq:`eq7-3-12` with
the scattering cross section approximated by the potential value;
.. math::
:label: eq7-3-18
\sigma _{0}^{\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{p}}^{\text{(j)}}\, \right)}
Since the background cross section for each resonance nuclide includes the
shielded absorption cross sections of all other nuclides, Bondarenko
interactions are performed.
(d) iropt = 3 => IR approximation with Bondarenko iterations:
Background cross sections for all nuclides are computed using the full
IR expression in :eq:`eq7-3-12`. Bondarenko interactions are performed.
Computation of the background cross sections in BONAMI generally
requires group-dependent values for the IR parameter λ. These are
calculated by a module in AMPX during the library process and are stored
in the MG libraries under the reaction identifier (MT number), MT=2000.
.. _7-3-2-4:
Self-Shielded Cross Sections for Heterogeneous Media
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 :eq:`eq7-3-3`. 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” :cite:`lamarsh_introduction_1966`.
theory states that the self-shielded cross section for resonance
absorber *r* in the heterogeneous medium is equal to the self-shielded
cross section of *r* 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
the modified sigma-zero value of
.. math::
:label: eq7-3-19
\hat{\sigma }_{0}^{(r)}\quad =\quad \sigma _{0}^{(r)}\ +\ \ \,\sigma _{esc}^{(r)}
:math:`\hat{\sigma }_{0}^{(r)}` = background cross section of r in the absorber lump of the heterogeneous system;
:math:`\sigma \,_{0}^{(r)}` = background cross section defined in :ref:`7-3-2-1` for an infinite homogeneous medium of the
absorber body mixture;
:math:`\sigma _{esc}^{(r)}` = microscopic escape cross section for nuclide *r*, defined as
.. math::
:label: eq7-3-20
\sigma _{esc}^{(r)}\quad =\quad \frac{{{\Sigma }_{esc}}}{{{N}^{(r)}}}
:math:`{{\Sigma }_{esc}}` = macroscopic escape cross section for the absorber lump defined in BONAMI as
.. math::
:label: eq7-3-21
{{\Sigma }_{esc}}\quad =\quad \,\frac{(1\quad -\quad c)A}{\bar{\ell }\ \,\ \left[ 1\quad +\quad \left( A\quad -\quad 1 \right)c \right]}
:math:`\bar{\ell }` = average chord length of the absorber body = :math:`4\ \ \,\times \ \frac{volume}{surface\ \ area}`;
A = Bell factor, used to improve the accuracy of the Wigner rational approximation;
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.
Values for the mean chord length :math:`\bar{\ell }` 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
:math:`{{\Sigma }_{T}}\bar{\ell }` 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 :cite:`leslie_improvements_1965` that is a function of the Dancoff factor.
.. _7-3-3:
Interpolation Scheme
After the background cross section for a system has been computed,
BONAMI interpolates f-factors at the appropriate σ\ :sub:`0` and
temperature from the tabulated values in the library. :numref:`fig7-3-1` shows
a typical variation of the f-factor vs. background cross sections for
the capture cross section of :sup:`238`\ U in the SCALE 252 group
.. _fig7-3-1:
.. figure:: figs/BONAMI/fig1.png
:align: center
:width: 500
Plot of f-factor variation for :sup:`238`\ U capture reaction.
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, :cite:`davis_sphinx_1977` arc-tangent fitting, :cite:`kidman_improved_1974` and an approach developed by
Segev :cite:`segev_interpolation_1981`. 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 :cite:`greene_method_1982`. 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 :eq:`eq7-3-25`:
.. math::
:label: eq7-3-22
f\left( \sigma \right)\quad =\quad f\left( \sigma {{\,}_{1}} \right)\quad +\quad \frac{\sigma {{\,}^{q(\sigma )}}\quad -\quad \sigma \,_{1}^{q(\sigma )}}{\sigma \,_{2}^{q(\sigma )}\quad -\quad \sigma \,_{1}^{q(\sigma )}}\quad \left( f\left( {{\sigma }_{2}} \right)\quad -\quad f\left( {{\sigma }_{1}} \right) \right)\quad ,
.. math::
:label: eq7-3-23
q\left( \sigma \right)\quad =\quad q\left( \sigma {{\,}_{1}} \right)\quad +\quad \frac{\sigma \quad -\quad \sigma \,_{1}^{{}}}{\sigma \,_{2}^{{}}\quad -\quad \sigma \,_{1}^{{}}}\quad \left( q\left( {{\sigma }_{2}} \right)\quad -\quad q\left( {{\sigma }_{1}} \right) \right)\quad .
:numref:`fig7-3-2` illustrates the expected behavior of :eq:`eq7-3-22` caused by varying
the powers in a panel.
By allowing the power *q* 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 *p* crosses the
*p* = 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
interpolating f-factors.
.. _fig7-3-2:
.. figure:: figs/BONAMI/fig2.png
:align: center
:width: 500
Illustration of the effects of varying “powers” in the Greene interpolation method.
:numref:`fig7-3-3` and :numref:`fig7-3-3` 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` interpolations, the method is not
significantly more time-consuming than the alternative schemes for most
.. _fig7-3-3:
.. figure:: figs/BONAMI/fig3.png
:align: center
:width: 500
Use of Greene’s method to fit the σ\ :sub:`0` variation of Bondarenko factors for case 1.
.. _fig7-3-4:
.. figure:: figs/BONAMI/fig4.png
:align: center
:width: 500
Use of Greene’s method to fit the σ\ :sub:`0` variation of Bondarenko factors for case 2.
.. _7-3-4:
Input Instructions
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.
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.
.. centered:: Data Block 1
0$ Logical Unit Assignments [4]
1. masterlib— input master library (Default = 23)
2. mwt—not used