BONAMI: Resonance SelfShielding by the Bondarenko Method¶
+U. Mertyurek and M. L. Williams
+ABSTRACT
+BONAMI is a module of the SCALE code system that is used to perform +Bondarenko calculations for resonance selfshielding. BONAMI obtains +problemindependent cross sections and Bondarenko shielding factors from +a multigroup (MG) AMPX master library, and it creates a MG AMPX working +library of selfshielded, problemdependent 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.
+Acknowledgments
+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.
+Introduction¶
+BONAMI (BONdarenko AMPX Interpolator) is a SCALE +module that performs resonance selfshielding calculations based on the +Bondarenko method [IlichB64]. It reads Bondarenko shielding factors +(“ffactors”) and infinitely dilute microscopic cross sections from a +problemindependent nuclear data library processed by the AMPX +system [WWCD15], interpolates the tabulated shielding factors to appropriate +temperatures and background cross sections for each nuclide in the +system, and produces a selfshielded, problemdependent data set.
+The code performs selfshielding for an arbitrary number of mixtures +using either the narrow resonance (NR) or intermediate resonance (IR) +approximation [GC62]. 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 +triangularpitched units are calculated automatically for BONAMI by +numerical integration over the chord length distribution. However, for +nonuniform 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 selfshielding +method, which performs a pointwise (PW) deterministic transport +calculation “on the fly” to compute multigroup (MG) selfshielded 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.
+Bondarenko SelfShielding Theory¶
+In MG resonance selfshielding 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 problemdependent 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 selfshielding, represented +by the background cross section parameter \(\sigma_0\) (called “sigmazero”) 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 +selfshielding. 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 +selfshielded 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 (10). 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.
+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 Dopplerbroadened 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 slowingdown 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 +slowingdown source of background nuclide j gives
+The IR approximation was proposed in the 1960s for scatterers with +slowingdown properties intermediate between those of NR and WR +scatterers [GC62]. 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 λ=1 reduces (16) 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 (16) into (12) 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 (17) 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 (18) 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 (18) are independent of energy, +so that the background cross section is constant, +(17) 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 selfshielding occurs. In this +approximation, the expression in (19) is also used for the background +cross section.
+Another approximation is to represent the energydependent cross +sections of the background nuclides by their groupaveraged (i.e., +selfshielded cross) values; thus
+In this case, the background cross section in (18) for nuclide r is the +groupdependent expression,
+An equation similar to (21) is used for the background cross sections of +all resonance nuclides; thus the selfshielded cross sections of each +resonance absorber depend on the shielded cross sections of all other +resonance absorbers in the mixture. When selfshielding operations are +performed with BONAMI for this approximation, “Bondarenko” iterations +are performed to account for the interdependence of the shielded cross +sections.
+Assuming that \(\sigma \,_{0}^{(r)}\) is represented as a groupwiseconstant +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 (17) is represented by the NR approximation, +\({{\text{S}}^{\text{(r)}}}(\text{E,T})\) to \(\Sigma _{\text{p}}^{\text{(r)}}C(E)\). +In this case, (17) 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 (10).
+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 selfshielded 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.
+SelfShielded Cross Section Data in SCALE Libraries¶
+The AMPX code system processes selfshielded cross sections using the +flux expressions described in the preceding section. For MG libraries in +SCALE6.2 and later versions, the NR approximation in (22) is used to +represent the flux spectrum for nuclides with masses below A=40, since +the NR approximation is generally accurate for lowmass 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 selfshielded cross sections in the master library, +AMPX converts them to Bondarenko shielding factors, also called +ffactors, 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, withingroup scattering, and +elastic scatter. Recent SCALE libraries include ffactors at ~10–30 +background cross section values (depending on nuclide) ranging from +~10^{−3} to ~10^{10} barns, which span the range of +selfshielding conditions. Typically the ffactor 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.
+Background Cross Section Options in BONAMI¶
+To compute selfshielded 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 +ffactor corresponding to this σ_{0} and nuclide temperature. +Several options are available in BONAMI to compute the background cross +section, based on (19) and (21) 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 (21) with +λ=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 (19). The BONAMI +iterative algorithm generally converges in a few iterations. Prior to +SCALE6.2, 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 (19). 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 (21) 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 (21). Bondarenko interactions are performed.
+Computation of the background cross sections in BONAMI generally +requires groupdependent 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.
+SelfShielded 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 (12). 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” [Lam66]. +Equivalence +theory states that the selfshielded cross section for resonance +absorber r in the heterogeneous medium is equal to the selfshielded +cross section of r in the equivalent infinite homogeneous medium; +therefore the ffactors that were calculated for homogenous mixtures can +also be used to compute selfshielded cross sections for heterogeneous +media by simply interpolating the tabulated ffactors in the library to +the modified sigmazero 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 Parameterized Flux Spectra 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 [LHJ65] that is a function of the Dancoff factor.
+Interpolation Scheme¶
+After the background cross section for a system has been computed, +BONAMI interpolates ffactors at the appropriate σ_{0} and +temperature from the tabulated values in the library. Fig. 44 shows +a typical variation of the ffactor vs. background cross sections for +the capture cross section of ^{238}U in the SCALE 252 group +library.
+ +Interpolation of the ffactors 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, [DYB77] arctangent fitting, [Kid74] and an approach developed by +Segev [Seg81]. 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 [Gre82]. 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 (34):
+where
+Fig. 45 illustrates the expected behavior of (31) 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 ffactors.
+ +Fig. 46 and Fig. 46 show typical “fits” of the ffactors 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_{0} interpolations, the method is not +significantly more timeconsuming than the alternative schemes for most +applications.
+ + +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 selfshielding +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 standalone 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 option
+++0 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 option
+++0 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 option
+++0 Otter +1 Leslie (Default)
+ +
escxsopt—escape cross section calculation option
+++0 consistent
+1 inconsistent (Default)
+
2* FloatingPoint 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/bcm) 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 onetoone 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.
+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 [Ott64] to determine a crosssection +geometrydependent 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 userspecified escape cross +section is used in calculating the background cross sections σ_{0} +as follows:
+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 ironclad 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 λΣ_{p}) and crossedt = 4 for the most detailed +output edits. An 8group 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
+test8grp
+read comp
+' fuel
+u235 1 0 1.4987e4 297.15 end
+u238 1 0 2.0664e2 297.15 end
+' clad
+fe56 2 0 9.5642e5 297.15 end
+' coolant
+h 3 0 6.6662e2 297.15 end
+o 3 0 3.3331e2 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$$ a0001
+ 1 3 5 0 4 1010
+ 1 1 1
+e
+2** a0001
+ 1.00000E03 0.00000E+00
+ e
+t
+ 3$$ a0001
+ 1 1 2 3 3
+e
+ 4$$ a0001
+ 92235 92238 26056 1001 8016
+e
+5** a0001
+ 1.49870E04 2.06640E02 9.56420E05 6.66620E02 3.33310E02
+ e
+ 6$$ a0001
+ 1 2 3
+e
+7** a0001
+ 4.05765E01 4.74980E01 7.10879E01
+ e
+8** a0001
+ 2.97150E+02 2.97150E+02 2.97150E+02
+ e
+9** a0001
+ 1.11870E+00 4.15813E+00 1.78119E01
+ e
+10$$ a0001
+ 92235 92238 26056 1001 8016
+e
+11$$ a0001
+ 0 0 0
+e
+13** a0001
+ 2.71260E01 5.20852E01 9.24912E01
+ e
+14** a0001
+ 8.11530E01 1.38430E01 4.71798E01
+ 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 ffactor 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 ffactor 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 ffactor 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 ffactor 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 ffactor 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 ffactor 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 ffactor 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 ffactor 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 4.75014e06 0.967842 4.59738e06
+ 18 5 44.0034676 2.60878e06 1.00006 2.60893e06
+ 18 6 44.0034676 5.27139e06 1.00071 5.27512e06
+ 18 7 44.0034676 9.3235e06 1.00018 9.32514e06
+ 18 8 44.0034676 1.81868e05 1.00588 1.82937e05
+
+mt Group sig0 infDiluted Xsec ffactor 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 ffactor 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 : 9.5642e05
+
+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 ffactor 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 ffactor 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 ffactor 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 ffactor 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 ffactor 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 ffactor 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 ffactor shielded Xsec
+ 102 1 4.33502197 3.56422e05 1.00003 3.56433e05
+ 102 2 4.33502197 8.96827e05 0.998165 8.9518e05
+ 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 ffactor 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 ffactor 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 ffactor 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 ffactor 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 7.10826e06 1.00036 7.11084e06
+ 102 4 45.7377472 7.2788e06 1.00001 7.27885e06
+ 102 5 45.7377472 2.38519e05 1.00003 2.38526e05
+ 102 6 45.7377472 5.83622e05 1.00019 5.83734e05
+ 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 ffactor 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
+

+
 DYB77 +
W. J. Davis, M. B. Yarbrough, and A. B. Bortz. SPHINX, a onedimensional diffusion and transport nuclear cross section processing code. Westinghouse Advanced Reactors Division report WARDXS304517 (August 1977), 1977.
+
+ GC62(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.
+
+ Gre82 +
N. M. Greene. Method for interpolating in Bondarenko factor tables and other functions. Technical Report, Oak Ridge National Lab., TN (USA), 1982.
+
+ Kid74 +
R. B. Kidman. Improved Ffactor interpolation scheme for 1DX. Trans. Amer. Nucl. Soc, 1974.
+
+ Lam66 +
John R. Lamarsh. Introduction to nuclear reactor theory. Volume 3. AddisonWesley Reading, Massachusetts, 1966.
+
+ LHJ65 +
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.
+
+ Ott64 +
John M. Otter. Escape Probability Approximations in Lumped Resonance Absorbers. Technical Report, Atomics International. Div. of North American Aviation, Inc., Canoga Park …, 1964.
+
+ Seg81 +
M. Segev. Interpolation of resonance integrals. Nuclear Science and Engineering, 79(1):113–118, 1981. Publisher: Taylor & Francis.
+
+ WWCD15 +
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 Lab.(ORNL), Oak Ridge, TN (United States), 2015.
+
+ IlichB64 +
Igor ́Ilích Bondarenko. Group constants for nuclear reactor calculations. Consultants Bureau, 1964.
+
+