# 7.3. BONAMI: Resonance Self-Shielding by the Bondarenko Method

*U. Mertyurek, M. L. Williams, and K. S. Kim*

ABSTRACT

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. New spatially dependent self-shielding capability has been incorporated into BONAMI for SCALE 6.3.

Acknowledgments

The authors express gratitude to W. A. Wieselquist 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.

## 7.3.1. Introduction

BONAMI (**BON**darenko **AM**PX **I**nterpolator) is a SCALE
module that performs resonance self-shielding calculations based on the
Bondarenko method [BONAMIIB64]. 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 [BONAMIWWCD15], 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 [BONAMIGC62]. 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 conventional BONAMI approach in SCALE 6.2 introduces significant biases of reactivity and intra pin self-shielded cross sections for multiple fuel rings with uniform or nonuniform temperature profile. New spatially dependent resonance self-shielding method [BONAMIKHW21] has been developed and implemented into BONAMI for SCALE 6.3.

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

where \(\sigma^{(r)}_{X,g}\) is the shielded MG cross section for reaction type *X* of
resonance nuclide *r* in group *g*; \(\sigma^{(r)}_{X}(E)\) is a PW cross section; and \(\Phi(E)\) 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 \(\Phi(E)\), 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 \(\sigma_0\) (called “sigma-zero”) and the
Doppler broadening temperature *T*. Hence,

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. (7.3.1). 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

where \(\Sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}\)
, \(\text{S}_{{}}^{\text{(r)}}\text{(E,T)}\) are the macroscopic total XS and
elastic scattering source for *r*, respectively; and \(\Sigma _{\text{t}}^{\text{(j)}}\text{(E,T)}\),
\(\text{S}_{{}}^{\text{(j)}}\text{(E,T)}\) 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

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,

where \({{\Phi }_{\infty }}\) 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

The IR approximation was proposed in the 1960s for scatterers with
slowing-down properties intermediate between those of NR and WR
scatterers [BONAMIGC62]. 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
that

A value of \(\lambda = 1\) reduces Eq. (7.3.7) to the NR expression, whereas math:lambda = 0 reduces the
equation to the WR expression. Fractional math:lambdas 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 math:lambda 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. (7.3.7) into Eq. (7.3.3) 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

where the background cross section of *r* in the homogeneous medium is
defined as

Although Eq. (7.3.8) provides the flux spectrum as a function of the background
cross section \(\sigma \,_{0}^{(r)}(u,T)\) it is not in a form that can be
preprocessed when the MG library is generated, because the energy variation of
\(\sigma \,_{0}^{(r)}(E,T)\) must be known. If the total cross sections
of the background nuclides in Eq. (7.3.9) have different energy variations, the shape of
\(\sigma \,_{0}^{(r)}(E,T)\) depends on their relative concentrations-which
are not known when the MG library is processed.
However, if the cross sections in Eq. (7.3.9) are independent of energy,
so that the background cross section is *constant*,
Eq. (7.3.8) can be solved for any arbitrary value of \(\sigma \,_{0}^{(r)}\)
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, \(\sigma \,_{0}^{(r)}(E,T)\quad \to \ \ \ \sigma \,_{0,g}^{(r)}\),
where

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. (7.3.10) 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

In this case, the background cross section in Eq. (7.3.9) for nuclide *r* is the
group-dependent expression,

An equation similar to Eq. (7.3.12) 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 sections.

Assuming that \(\sigma \,_{0}^{(r)}\) 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. (7.3.8) is represented by the NR approximation,
\({{\text{S}}^{\text{(r)}}}(\text{E,T})\) to \(\Sigma _{\text{p}}^{\text{(r)}}C(E)\).
In this case, Eq. (7.3.8) can be solved analytically to obtain the following
expression for the flux spectrum used to process MG data as a function of \(\sigma \,_{0}^{(r)}\):

where C(E) includes is an arbitrary constant multiplier that cancels from Eq. (7.3.1).

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 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;

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. (7.3.13) 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,

where \(\Phi (E;\,\,\sigma \,_{0}^{(r)}\,,T)\) is the flux for a given value
of \(\sigma \,_{0}^{(r)}\) 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

and infinitely dilute cross sections defined as,

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^{-3} to ~10^{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 \(\sigma\)_{0} and nuclide temperature.
Several options are available in BONAMI to compute the background cross
section, based on Eq. (7.3.10) and Eq. (7.3.12) in the preceding section. The options are
specified by input parameter “**iropt**” and have the following
definitions:

iropt = 0 => NR approximation with Bondarenko iterations:

Background cross sections for all nuclides are computed using Eq. (7.3.12) with math:lambda = 1; therefore,

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 \(\sigma \,_{0}^{(r)}\) in Eq. (7.3.10). 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.

iropt = 1 => IR approximation with no resonance interference (potential cross sections):

Background cross sections for all nuclides are computed using Eq. (7.3.10). No Bondarenko iterations are needed.

iropt t = 2 => IR approximation with Bondarenko iterations, but no resonance scattering:

Background cross sections for all nuclides are computed using Eq. (7.3.12) with the scattering cross section approximated by the potential value; therefore,

Since the background cross section for each resonance nuclide includes the shielded absorption cross sections of all other nuclides, Bondarenko interactions are performed.

iropt = 3 => IR approximation with Bondarenko iterations:

Background cross sections for all nuclides are computed using the full IR expression in Eq. (7.3.12). Bondarenko interactions are performed.

Computation of the background cross sections in BONAMI generally requires group-dependent values for the IR parameter math:lambda. 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. (7.3.3). 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” [BONAMILam66].
Equivalence
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

where,

\(\hat{\sigma }_{0}^{(r)}\) = background cross section of r in the absorber lump of the heterogeneous system;

\(\sigma \,_{0}^{(r)}\) = background cross section defined in Sect. 7.3.2.1 for an infinite homogeneous medium of the absorber body mixture;

\(\sigma _{esc}^{(r)}\) = microscopic escape cross section for nuclide

r, defined as

\({{\Sigma }_{esc}}\) = macroscopic escape cross section for the absorber lump defined in BONAMI as

where

\(\bar{\ell }\) = average chord length of the absorber body = \(4\ \ \,\times \ \frac{volume}{surface\ \ area}\);

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 \(\bar{\ell }\) 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 \({{\Sigma }_{T}}\bar{\ell }\) 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 [BONAMILHJ65] 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 \(\sigma\)_{0} and
temperature from the tabulated values in the library. Fig. 7.3.1 shows
a typical variation of the f-factor vs. background cross sections for
the capture cross section of ^{238}U in the SCALE 252 group
library.

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, [BONAMIDYB77] arc-tangent fitting, [BONAMIKid74] and an approach developed by Segev [BONAMISeg81]. 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 [BONAMIGre82]. 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. (7.3.25):

where

Fig. 7.3.2 illustrates the expected behavior of Eq. (7.3.22) caused by varying the powers in a panel.

By allowing the power *q* to vary as a function of independent
variable \(sigma\), 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.

Fig. 7.3.3 and Fig. 7.3.3 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 \(\sigma\)_{0} interpolations, the method is not
significantly more time-consuming than the alternative schemes for most
applications.

## 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.

**Data Block 1**

0$ Logical Unit Assignments [4]

masterlib- input master library (Default = 23)

mwt-not used

msc-not used

newlib-output master library (Default = 22)

1$ Case Description [6]

cellgeometry-geometry description

0 homogeneous

1 slab

2 cylinder

3 sphere

numzones-number of zones or material regions

3. mixlength-mixing table length. This is the total number of entries needed to describe the concentrations of all constituents in all mixtures in the problem.

ib-not used

crossedt-output edit option0 no output (Default)

1 input echo

2 iteration list, timing

3 background cross section calculation details

4 shielded cross sections, Bondarenko factors

issopt-not used

iropt-resonance approximation option0 NR (Default) (Bondarenko iterations)

1 IR with potential scattering

2 IR with absorption and potential scattering (Bondarenko iterations)

3 IR with absorption and elastic scattering (Bondarenko iterations)

bellopt-Bell factor calculation option0 Otter 1 Leslie (Default)

escxsopt-escape cross section calculation option0 consistent

1 inconsistent (Default)

2* Floating-Point Constants [2]

1.

bonamieps-convergence criteria for the Bondarenko iteration (Default = 0.001)2.

bellfact-geometrical escape probability adjustment factor. See notes below on this parameter (Default = 0.0).T Terminate Data Block 1.

**Data Block 2**

3$ Mixture numbers in the mixing table [mixlength]

4$ Component (nuclide) identifiers in the mixing table [mixlength]

5* Concentrations (atoms/b-cm) in the mixing table [mixlength]

6$ Mixtures by zone [numzones]

7* Outer radii (cm) by zone [numzones]

8* Temperature (k) by zone [numzones]

9* Escape cross section (cm

^{-1}) by zone [numzones]10$ Not used

11$ Not used

12* Temperature (K) of the nuclide in a one-to-one correspondence with the mixing table arrays.

13* Dancoff factors by zone [numzones]

14* Lbar (\(\bar{\ell }\)) factors by zone [numzones]

T Terminate Data Block 2.

This concludes the input data required by BONAMI.

### 7.3.4.1. Notes on input

In the 1$ array, *cellgeometry* specifies the geometry. The geometry
information is used in conjunction with the 7* array to calculate mean
chord length *Lbar* if it is not provided by the user in the 14* array.

*numzones*, the number of zones, may or may not model a real situation.
It may, for example, be used to specify *numzones* independent media to
perform a cell calculation in parallel with one or more infinite medium
calculations. The geometry description in 1$ array applies only to mean
chord length calculations unless it is provided in 14*.

In the 2* array, *bonamieps* is used to specify the convergence expected
on all macroscopic total values by zone, that is, each \({{\Sigma }_{t}}(g,j)\) in group g and
zone j is converged so that

The “Bell” factor in the 2* array is the parameter used to adjust the
Wigner rational approximation for the escape probability to a more
correct value. It has been suggested that if one wishes to use one
constant value, the Bell factor should be 1.0 for slabs and 1.35
otherwise. In the ordinary case, BONAMI defaults the Bell factor to zero
and uses a prescription by Otter [BONAMIOtt64] to determine a cross-section
geometry-dependent value of the Bell factor for isolated absorber
bodies. It uses a prescription by Leslie^{6} to determine the
Dancoff factor–dependent values of the Bell factor for lattices, which
are much more accurate than the single value. The user who wishes to
determine the constant value can, however, use it by inputting a value
other than zero.

The 3$, 4$, and 5* arrays are used to specify the concentrations of the constituents of all mixtures in the problem as follows:

Entry 3$ (Mixture Number) 4$ (Nuclide ID) 5* (Concentrations)

1

2

.

.

.

.

mixlength

.

Because of the manner in which BONAMI references the nuclides in a calculation, each nuclide in the problem must have a unique entry in the mixing table. Thus one cannot specify a mixture and subsequently load it into more than one zone, as can be the case with many modules requiring this type of data.

The 12* array is used to allow varying the temperatures by nuclide within a zone. In the event this array is omitted, the 12* array will default by nuclide to the temperature of the zone containing the nuclide.

The mixture numbers in each zone are specified in the 6$ array. Mixture numbers are arbitrary and need only match up with those used in the 3$ array.

The radii in the 7* array are referenced to a zero value at the left boundary of the system.

In the event the temperatures in the 8* array are not bounded by temperature values in the Bondarenko tables, BONAMI will extrapolate using the three temperature points closest to the value. For example, a request for 273 K for a nuclide with Bondarenko sets at 300, 900, and 2,100 K would use the polynomial fit from those three temperature points to extrapolate the 273 K value.

The escape cross sections in the 9* array allow a macro escape cross
section (\(\Sigma _{e}^{input}\)) to be specified by zone. (This array can be ignored if
Dancoff factors are provided.) If the Dancoff factor for a zone is
specified as -1 in the input, then the user-specified escape cross
section is used in calculating the background cross sections \(\sigma\)_{0}
as follows:

## 7.3.5. Sample Problem

In most cases, the input data to BONAMI are simple and obvious because the complicated parameters are determined internally based on the options selected. The user describes his geometry, the materials contained therein, the temperatures, and a few options.

This problem is for a system of iron-clad uranium (U^{238} –
U^{235} ) fuel pins arranged in a square lattice in a water pool.

Our number densities are

Fuel:

\({{N}_{{}^{235}U}}\) = 1.4987 × 10

^{-4}\({{N}_{{}^{238}U}}\) = 2.0664× 10

^{–2}

Clad:

\({{N}_{{}^{56}Fe}}\) = 9.5642× 10

^{-5}

Water:

N

_{H}= 6.6662 × 10^{-2}N

_{O}= 3.3331 × 10^{-2}

For the problem, we choose *iropt* = 1 (IR approximation with scattering
approximated by math:lambda\(\Sigma\) _{p}) and *crossedt* = 4 for the most detailed
output edits. An 8-group test library is used for fast execution and a
short output file.

The XSProc/CSAS1X SCALE sequence input file, the corresponding i_bonami FIDO input file created by the sequence under the temporary working directory, and an abbreviated copy of the output from this case follows.

```
=csas1x
Assembly pin
test-8grp
read comp
' fuel
u-235 1 0 1.4987e-4 297.15 end
u-238 1 0 2.0664e-2 297.15 end
' clad
fe-56 2 0 9.5642e-5 297.15 end
' coolant
h 3 0 6.6662e-2 297.15 end
o 3 0 3.3331e-2 297.15 end
end comp
' ====================================================================
read celldata
latticecell squarepitch pitch=1.26 3 fuelr=0.405765 1
cladr=0.47498 2 end
moredata iropt=1 crossedt=4 end moredata
end celldata
' ====================================================================
end
```

FIDO input i_bonami

```
-1$$ a0001
500000
e
0$$ a0001
11 0 18 1
e
1$$ a0001
1 3 5 0 4 1010
1 -1 -1
e
2** a0001
1.00000E-03 0.00000E+00
e
t
3$$ a0001
1 1 2 3 3
e
4$$ a0001
92235 92238 26056 1001 8016
e
5** a0001
1.49870E-04 2.06640E-02 9.56420E-05 6.66620E-02 3.33310E-02
e
6$$ a0001
1 2 3
e
7** a0001
4.05765E-01 4.74980E-01 7.10879E-01
e
8** a0001
2.97150E+02 2.97150E+02 2.97150E+02
e
9** a0001
1.11870E+00 4.15813E+00 1.78119E-01
e
10$$ a0001
92235 92238 26056 1001 8016
e
11$$ a0001
0 0 0
e
13** a0001
2.71260E-01 5.20852E-01 9.24912E-01
e
14** a0001
8.11530E-01 1.38430E-01 4.71798E-01
e
15** a0001
0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
e
16$$ a0001
2 2 2
e
17$$ a0001
0 0 0 0
e
t
```

```
program verification information
code system: SCALE version: 6.2
program: bonami
creation date: unknown
library: /home02/u2m/Workfolder/sampletmp
test code: bonami
version: 6.2.0
jobname: u2m
machine name: node22.ornl.gov
date of execution: 04_dec_2013
time of execution: 21:43:54.38
```

```
1
BONAMI CELL PARAMETERS
---------------------------------------------
Bonami Print Option : 4
BellFactor : 0
Bondarenko Iteration eps : 0.001
Resonance Option : 1
Bell Factor Option : LESLIE
Escape CrossSection Option : INCONSISTENT
CellGeometry : 2
MasterLibrary :
Number oF Neutron Groups : 8
First Thermal Neutron Group : 5
__________________________________________
Processing Zone : 1
Mixture Number : 1
Number Of Nuclides : 2
Dancoff Factor : 0.27126
Lbar : 0.81153
Escape Cross Section Input : 1.1187
Material Temeprature : 297.15
Processing Nuclide : 92235 Number Density : 0.00014987
Processing Nuclide : 92238 Number Density : 0.020664
Bondarenko Iterations
iteration Nuclide Group MaxChange Selfsig0 Effsig0
1 92235 0 0 0 0
1 92238 0 0 0 0
Total number of Bondarenko Iterations : 1
Max Change in Group : 0
Group Eff Macro Sig0 Escape Xsec
1 0.2351032 0.9075513
2 0.2351032 0.9075513
3 0.2351032 0.9075513
4 0.2351032 0.9075513
5 0.2351032 0.9075513
6 0.2351032 0.9075513
7 0.2351032 0.9075513
8 0.2351032 0.9075513
---------------------------------------------------
```

```
Shielding Nuclide 92235
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
1 1 7612.71875 7.19131 0.999998 7.19129
1 2 7612.71875 10.2521 0.999616 10.2481
1 3 7612.71875 24.9361 1.00241 24.9963
1 4 7612.71875 75.1109 1.05902 79.5436
1 5 7612.71875 56.0286 1.00205 56.1434
1 6 7612.71875 198.645 1.0008 198.805
1 7 7612.71875 347.945 1.00024 348.028
1 8 7612.71875 761.257 1.0066 766.282
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
2 1 7612.71875 3.71448 0.999999 3.71448
2 2 7612.71875 7.63235 0.99935 7.62739
2 3 7612.71875 11.841 0.999444 11.8345
2 4 7612.71875 11.5408 1.00561 11.6055
2 5 7612.71875 12.5449 1.00001 12.545
2 6 7612.71875 14.2501 1.00007 14.2511
2 7 7612.71875 14.8125 1.00003 14.8128
2 8 7612.71875 15.1274 1.00015 15.1297
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
18 1 7612.71875 1.21846 0.999996 1.21846
18 2 7612.71875 1.40834 1.0002 1.40862
18 3 7612.71875 8.92885 1.00132 8.94062
18 4 7612.71875 39.2086 1.06274 41.6686
18 5 7612.71875 32.7026 1.00105 32.737
18 6 7612.71875 153.511 1.00089 153.647
18 7 7612.71875 285.775 1.00026 285.848
18 8 7612.71875 636.445 1.00655 640.611
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
102 1 7612.71875 0.060296 1 0.0602962
102 2 7612.71875 0.317627 1.00352 0.318746
102 3 7612.71875 4.16593 1.01325 4.22113
102 4 7612.71875 24.3615 1.07832 26.2695
102 5 7612.71875 10.781 1.00749 10.8618
102 6 7612.71875 30.8844 1.00074 30.9073
102 7 7612.71875 47.3579 1.0002 47.3671
102 8 7612.71875 109.685 1.00781 110.542
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
1007 1 7612.71875 0 0 0
1007 2 7612.71875 0 0 0
1007 3 7612.71875 0 0 0
1007 4 7612.71875 0 0 0
1007 5 7612.71875 12.5448 1.00001 12.5449
1007 6 7612.71875 14.2501 1.00007 14.2511
1007 7 7612.71875 14.8125 1.00003 14.8129
1007 8 7612.71875 15.1278 1.00015 15.13
---------------------------------------------------
```

```
Shielding Nuclide 92238
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
1 1 44.0034676 7.33815 0.999983 7.33803
1 2 44.0034676 10.3566 1.00418 10.3999
1 3 44.0034676 15.0517 0.976844 14.7032
1 4 44.0034676 15.951 0.983793 15.6925
1 5 44.0034676 9.43867 1.00002 9.43887
1 6 44.0034676 10.1008 1.00008 10.1015
1 7 44.0034676 10.7744 1.00004 10.7748
1 8 44.0034676 12.2124 1.00145 12.2301
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
2 1 44.0034676 4.0228 0.999974 4.0227
2 2 44.0034676 9.05886 1.00575 9.11093
2 3 44.0034676 14.0213 0.979923 13.7398
2 4 44.0034676 11.9032 0.98795 11.7598
2 5 44.0034676 8.86555 0.999984 8.86541
2 6 44.0034676 9.24452 1.00002 9.24471
2 7 44.0034676 9.2797 1.00002 9.27987
2 8 44.0034676 9.3077 1.00009 9.30853
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
18 1 44.0034676 0.376356 1.00001 0.376361
18 2 44.0034676 0.000528746 1.00019 0.000528845
18 3 44.0034676 0.000308061 0.966052 0.000297603
18 4 44.0034676 4.75014e-06 0.967842 4.59738e-06
18 5 44.0034676 2.60878e-06 1.00006 2.60893e-06
18 6 44.0034676 5.27139e-06 1.00071 5.27512e-06
18 7 44.0034676 9.3235e-06 1.00018 9.32514e-06
18 8 44.0034676 1.81868e-05 1.00588 1.82937e-05
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
102 1 44.0034676 0.0554327 1.00006 0.0554359
102 2 44.0034676 0.17972 0.978628 0.175879
102 3 44.0034676 1.03011 0.934934 0.963087
102 4 44.0034676 4.04777 0.971568 3.93268
102 5 44.0034676 0.573119 1.0006 0.573462
102 6 44.0034676 0.856257 1.00068 0.856839
102 7 44.0034676 1.49471 1.00017 1.49497
102 8 44.0034676 2.90465 1.00586 2.92168
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
1007 1 44.0034676 0 0 0
1007 2 44.0034676 0 0 0
1007 3 44.0034676 0 0 0
1007 4 44.0034676 0 0 0
1007 5 44.0034676 8.86549 0.999984 8.86535
1007 6 44.0034676 9.24445 1.00002 9.24463
1007 7 44.0034676 9.27974 1.00002 9.27992
1007 8 44.0034676 9.30769 1.00009 9.30852
Zone Calculation is completed in 0 seconds
BONAMI CELL PARAMETERS
---------------------------------------------
Bonami Print Option : 4
BellFactor : 0
Bondarenko Iteration eps : 0.001
Resonance Option : 1
Bell Factor Option : LESLIE
Escape CrossSection Option : INCONSISTENT
CellGeometry : 2
MasterLibrary :
Number oF Neutron Groups : 8
First Thermal Neutron Group : 5
__________________________________________
Processing Zone : 2
Mixture Number : 2
Number Of Nuclides : 1
Dancoff Factor : 0.520852
Lbar : 0.13843
Escape Cross Section Input : 4.15813
Material Temeprature : 297.15
Processing Nuclide : 26056 Number Density : 9.5642e-05
Bondarenko Iterations
iteration Nuclide Group MaxChange Selfsig0 Effsig0
1 26056 0 0 0 0
Total number of Bondarenko Iterations : 1
Max Change in Group : 0
Group Eff Macro Sig0 Escape Xsec
1 0.0003553244 3.487286
2 0.0003553244 3.487286
3 0.0003553244 3.487286
4 0.0003553244 3.487286
5 0.0003553244 3.487286
6 0.0003553244 3.487286
7 0.0003553244 3.487286
8 0.0003553244 3.487286
---------------------------------------------------
```

```
Shielding Nuclide 26056
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
1 1 36461.8672 3.07957 1.00005 3.07972
1 2 36461.8672 4.68958 1.00091 4.69382
1 3 36461.8672 7.85712 0.999843 7.85589
1 4 36461.8672 12.0029 1 12.0029
1 5 36461.8672 12.3689 1.00001 12.369
1 6 36461.8672 12.8598 1.00003 12.8602
1 7 36461.8672 13.5237 0.999906 13.5224
1 8 36461.8672 15.0714 0.99949 15.0637
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
2 1 36461.8672 2.26476 1.00047 2.26583
2 2 36461.8672 4.6817 1.0009 4.68592
2 3 36461.8672 7.81457 0.999813 7.81311
2 4 36461.8672 11.9143 1 11.9143
2 5 36461.8672 12.0468 1.00001 12.0469
2 6 36461.8672 12.065 1.00002 12.0653
2 7 36461.8672 12.0887 1.00005 12.0893
2 8 36461.8672 12.2042 1.00013 12.2057
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
102 1 36461.8672 0.00206393 1.0015 0.00206702
102 2 36461.8672 0.00787763 1.0035 0.00790524
102 3 36461.8672 0.0425504 1.00623 0.0428155
102 4 36461.8672 0.0885525 1 0.0885529
102 5 36461.8672 0.322101 1.00002 0.322109
102 6 36461.8672 0.794804 1.0002 0.79496
102 7 36461.8672 1.43496 0.998734 1.43314
102 8 36461.8672 2.86723 0.996792 2.85803
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
1007 1 36461.8672 0 0 0
1007 2 36461.8672 0 0 0
1007 3 36461.8672 0 0 0
1007 4 36461.8672 0 0 0
1007 5 36461.8672 12.0468 1.00001 12.0469
1007 6 36461.8672 12.065 1.00002 12.0653
1007 7 36461.8672 12.0887 1.00005 12.0893
1007 8 36461.8672 12.2042 1.00013 12.2057
Zone Calculation is completed in 0 seconds
```

```
BONAMI CELL PARAMETERS
---------------------------------------------
Bonami Print Option : 4
BellFactor : 0
Bondarenko Iteration eps : 0.001
Resonance Option : 1
Bell Factor Option : LESLIE
Escape CrossSection Option : INCONSISTENT
CellGeometry : 2
MasterLibrary :
Number oF Neutron Groups : 8
First Thermal Neutron Group : 5
__________________________________________
Processing Zone : 3
Mixture Number : 3
Number Of Nuclides : 2
Dancoff Factor : 0.924912
Lbar : 0.471798
Escape Cross Section Input : 0.178119
Material Temeprature : 297.15
Processing Nuclide : 1001 Number Density : 0.066662
Processing Nuclide : 8016 Number Density : 0.033331
Bondarenko Iterations
iteration Nuclide Group MaxChange Selfsig0 Effsig0
1 1001 0 0 0 0
1 8016 0 0 0 0
Total number of Bondarenko Iterations : 1
Max Change in Group : 0
Group Eff Macro Sig0 Escape Xsec
1 1.494705 0.1593803
2 1.494705 0.1593803
3 1.494705 0.1593803
4 1.494705 0.1593803
5 1.494705 0.1593803
6 1.494705 0.1593803
7 1.494705 0.1593803
8 1.494705 0.1593803
---------------------------------------------------
```

```
Shielding Nuclide 1001
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
1 1 4.33502197 2.98905 0.999485 2.98751
1 2 4.33502197 9.87269 0.999169 9.86448
1 3 4.33502197 19.9332 0.999972 19.9326
1 4 4.33502197 20.4672 0.998926 20.4453
1 5 4.33502197 21.1735 1.00001 21.1736
1 6 4.33502197 26.1886 0.99995 26.1873
1 7 4.33502197 35.0621 0.999821 35.0558
1 8 4.33502197 54.9507 0.997361 54.8057
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
2 1 4.33502197 2.98901 0.999485 2.98747
2 2 4.33502197 9.8726 0.999168 9.86439
2 3 4.33502197 19.9315 0.999972 19.9309
2 4 4.33502197 20.4556 0.998926 20.4336
2 5 4.33502197 21.1321 1.00001 21.1322
2 6 4.33502197 26.0865 0.99995 26.0852
2 7 4.33502197 34.8778 0.999829 34.8718
2 8 4.33502197 54.5786 0.997396 54.4365
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
102 1 4.33502197 3.56422e-05 1.00003 3.56433e-05
102 2 4.33502197 8.96827e-05 0.998165 8.9518e-05
102 3 4.33502197 0.00171679 0.999794 0.00171643
102 4 4.33502197 0.0116042 1.00002 0.0116044
102 5 4.33502197 0.0413709 1.00001 0.0413714
102 6 4.33502197 0.102043 1.00007 0.10205
102 7 4.33502197 0.184322 0.998389 0.184025
102 8 4.33502197 0.372079 0.992163 0.369163
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
1007 1 4.33502197 0 0 0
1007 2 4.33502197 0 0 0
1007 3 4.33502197 0 0 0
1007 4 4.33502197 0 0 0
1007 5 4.33502197 21.1312 1.00005 21.1322
1007 6 4.33502197 26.0871 0.999927 26.0852
1007 7 4.33502197 34.8779 0.999826 34.8718
1007 8 4.33502197 54.5802 0.997367 54.4365
---------------------------------------------------
```

```
Shielding Nuclide 8016
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
1 1 45.7377472 2.36081 0.980071 2.31376
1 2 45.7377472 3.95917 0.995755 3.94236
1 3 45.7377472 3.84394 1 3.84394
1 4 45.7377472 3.85289 1 3.85291
1 5 45.7377472 3.85531 1.00002 3.85537
1 6 45.7377472 3.8648 1.00006 3.86502
1 7 45.7377472 3.89139 1.00014 3.89194
1 8 45.7377472 4.01909 1.00036 4.02056
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
2 1 45.7377472 2.34636 0.979939 2.29929
2 2 45.7377472 3.95907 0.995756 3.94226
2 3 45.7377472 3.84393 1 3.84393
2 4 45.7377472 3.85288 1 3.8529
2 5 45.7377472 3.85529 1.00002 3.85535
2 6 45.7377472 3.86474 1.00006 3.86496
2 7 45.7377472 3.89128 1.00014 3.89183
2 8 45.7377472 4.01888 1.00037 4.02035
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
102 1 45.7377472 0.00010111 1.0002 0.00010113
102 2 45.7377472 0.000101635 0.991292 0.00010075
102 3 45.7377472 7.10826e-06 1.00036 7.11084e-06
102 4 45.7377472 7.2788e-06 1.00001 7.27885e-06
102 5 45.7377472 2.38519e-05 1.00003 2.38526e-05
102 6 45.7377472 5.83622e-05 1.00019 5.83734e-05
102 7 45.7377472 0.000105195 0.998729 0.000105061
102 8 45.7377472 0.000209981 0.996663 0.00020928
mt Group sig0 infDiluted Xsec f-factor shielded Xsec
1007 1 45.7377472 0 0 0
1007 2 45.7377472 0 0 0
1007 3 45.7377472 0 0 0
1007 4 45.7377472 0 0 0
1007 5 45.7377472 3.85523 1.00003 3.85535
1007 6 45.7377472 3.86483 1.00003 3.86496
1007 7 45.7377472 3.89119 1.00016 3.89183
1007 8 45.7377472 4.01794 1.0006 4.02035
Zone Calculation is completed in 0 seconds
module: BonamiM has terminated after a cpu usage of 0.0100 seconds
```

References

- BONAMIDYB77
W. J. Davis, M. B. Yarbrough, and A. B. Bortz. SPHINX, a one-dimensional diffusion and transport nuclear cross section processing code. Technical Report Westinghouse Advanced Reactors Division report WARD-XS-3045-17, Westinghouse, 8 1977.

- BONAMIGC62(1,2)
Rubin Goldstein and E. Richard Cohen. Theory of resonance absorption of neutrons.

*Nuclear Science and Engineering*, 13(2):132–140, 1962. Publisher: Taylor & Francis.- BONAMIGre82
N. M. Greene. Method for interpolating in Bondarenko factor tables and other functions. Technical Report, Oak Ridge National Laboratory, Oak Ridge, TN (USA), 1982.

- BONAMIIB64
Igor Ilich Bondarenko.

*Group constants for nuclear reactor calculations*. Consultants Bureau, 1964.- BONAMIKid74
R. B. Kidman. Improved F-factor interpolation scheme for 1DX. In

*Transactions of the American Nuclear Society*, volume 18. 1974.- BONAMIKHW21
Kang Seog Kim, Andrew M. Holcomb, and William A. Wieselquist. Spatially Dependent Resonance Self-Shielding Capability for Non-uniform Temperature Profile in SCALE-6.3 XSProc-BONAMI. In

*M&C 2021*. 2021.- BONAMILam66
John R. Lamarsh.

*Introduction to nuclear reactor theory*. Volume 3. Addison-Wesley Reading, Massachusetts, 1966.- BONAMILHJ65
D. C. Leslie, J. G. Hill, and A. Jonsson. Improvements to the Theory of Resonance Escape in Heterogeneous Fuel: I. Regular Arrays of Fuel Rods.

*Nuclear Science and Engineering*, 22(1):78–86, 1965. Publisher: Taylor & Francis.- BONAMIOtt64
John M. Otter. Escape Probability Approximations in Lumped Resonance Absorbers. Technical Report, Atomics International. Div. of North American Aviation, Inc., Canoga Park …, 1964.

- BONAMISeg81
M. Segev. Interpolation of resonance integrals.

*Nuclear Science and Engineering*, 79(1):113–118, 1981. Publisher: Taylor & Francis.- BONAMIWWCD15
Dorothea Wiarda, Mark L. Williams, Cihangir Celik, and Michael E. Dunn. AMPX: A Modern Cross Section Processing System for Generating Nuclear Data Libraries. Technical Report, Oak Ridge National Laboratory, Charlotte, NC (USA), 9 2015.