11.2. Overview

In this section, a qualitative discussion of cross sections is provided, along with various associated cross section operations. In most cases, the discussion is limited to neutron cross sections while most observations apply to other particles.

11.2.1. What are Cross Sections?

Cross sections are parameters that characterize how particles interact with matter. CE or point cross sections characterize a particle traveling at a particular speed in a particular direction. Upon interaction with a target nucleus, the particle can change speed and/or direction. The total cross section quantifies the probability that a neutron will interact with a target nucleus. The total cross section is the sum of the cross sections that define the probability of having any one of several types of interactions. In the case of neutrons, there may be a hundred or more possible reactions that can be categorized as either absorption or scattering interactions.

If a particle is absorbed by a target nucleus, a new compound nucleus is formed that subsequently decays by emitting an energetic particle(s) such as an $(\mathrm{n}, \gamma)$ reaction. In neutron transport codes, fission is typically treated as an absorption reaction, and the neutron multiplicity data (i.e., $\overline{\mathcal{V}}$ ) are used to determine the number of secondary neutrons produced by the fission event. With regard to scattering, a particle may interact with the nucleus without penetrating the nucleus. In this type of collision, the particle scatters in a billiard-ball fashion with the nucleus, and the reaction is referred to as potential scattering. The remaining types of scattering events are elastic and inelastic scattering. During an elastic scattering reaction, the particle is absorbed by the nucleus and subsequently re-emitted, leaving the target nucleus in the ground state. As a result, the kinetic energy is conserved for the elastic scattering reaction. For an inelastic scattering reaction, the incident particle is absorbed by the nucleus, and a compound nucleus is formed. The excited nucleus subsequently decays by emitting an energetic particle, thereby leaving the nucleus in an excited state. Unlike the elastic collision, the kinetic energy of the inelastic scattering reaction is not conserved.

A cross section by itself is not enough information to use in a particle transport calculation. The total cross section and its components provide enough information to describe the possible interactions; however, determining the energy and angle of secondary particles requires a knowledge of the reaction kinematics, or the physics must be presented in a structured manner (e.g., tables of energy and angle distributions of secondary particles) to determine the exiting energy and angle.

The AMPX code system can generate MG libraries, whose uses are described in the next section, as well as CE libraries.

11.2.2. The MG Approach

In the 1940s and most of the 1950s, nuclear analyses were performed without point cross section data because the data simply did not exist. Due to the limited computing resources of the time period, transport calculations that use CE cross sections were not feasible.

Indeed, many calculations were very simple and would not be described as either point or MG calculations. For these calculations, the cross sections were assigned a single value that was selected as the value for the most important energy range for a specific process of a particular isotope. As an example for a thermal reactor calculation, the fission cross section for $^{235}$U would be the average value in the thermal region while the fission cross section for $^{238}$U would be set to zero.

In the early 1950s, the group diffusion method was developed. This method combines slowing-down theory for energy degradation with diffusion theory for the spatial variation of the neutron flux [ampx-WW58]. In the treatment, the full energy range is divided into a number of energy intervals (or groups), and the neutron is assumed to suffer the average number of elastic scattering collisions required to slow down through an energy group prior to emerging in the next energy group. The number of energy groups and associated group constants that were used in this early method is unknown; however, approximate methods were most likely used to obtain the group constants.

The simplest of the group diffusion methods is the one-group diffusion theory method that served as the basis for many early nuclear analyses. Two-group diffusion theory is well documented and may represent the limit of the number of energy groups that could be handled without computer resources.

The MG approach became prominent in the 1950s, and many MG cross section libraries were introduced, including the 6- and 16-group Hansen–Roach libraries [ampx-HR63, IntroHR90], the 26-group Russian Bondarenko library [ampx-Bon64], the 30-group THERMOS library [ampx-Hon61], and the 99-group GAM-II library [ampx-JD63]. Interestingly, the growth of the MG methods paralleled the advancements in computer computational capabilities. The development of the numerical computational methods in parallel with advances in computer capabilities has made the hand calculation methods more feasible.

Four components are required to create a MG cross section library:

point cross section data,

MG energy structure,

flux or weighting spectrum, and

analytic description or tabular description of reaction kinematics.

The subsequent sections discuss the four components needed to generate a MG cross section library.

11.2.2.1. MG Structure

The inherent assumption of the MG approach is that an energy quadrature can be defined so that the cross section variation in an energy group can be adequately represented by a single average value. Years of calculation experience have revealed that the inherent assumption of the MG approach is valid for many conditions; however, there are some configurations where a particular MG structure does not yield acceptable results. In practice, MG structures are typically classified as either a fine or broad group structure. A fine group structure has a sufficient number of energy groups that can be used to analyze a wide variety of problems. Conversely, a broad group structure has a sufficient number of groups that can be used for a specific class of problems as determined by the developer of the cross section library.

The choice of a group structure and associated weighting spectrum are strongly correlated. In an ideal situation, the importance of the weighting spectrum should be minimized with a larger number of groups because the cross section variation should decrease as the energy band becomes smaller. However, experience with some isotopes has revealed that an ideal group structure does not exist because of the rapid cross section variation as a function of energy.

Fine group structures may require between 100 and 30,000 energy bins, and there are several fine group libraries with 200 to 500 energy groups. These fine group libraries can be used to treat a wide variety of problems; however, some specific energy regions and isotopes/nuclides demand additional treatments for extreme cross section variations as a function of energy.

For many applications, a broad group library is directly applicable and suitable for production analyses. Furthermore, some two-dimensional (2D) or three-dimensional (3D) transport calculations with a fine group library can be prohibitive even with current computational capabilities, and a broad group library must be used. Broad group libraries are generated or collapsed from a fine group library and are generally applicable to a limited class of problems. The techniques used to weight the fine group data and subsequently produce a broad group library require a weighting spectrum similar to the spectra of the intended class of problems (e.g., a light-water reactor spectrum).

As noted above, there is not a single fine group energy structure that can be used for all problems that eliminates the need for a weighting spectrum. The selection of a fine group structure is based on the following criteria:

a. thresholds of important reactions for specific isotopes/nuclides in the group structure (e.g., if resonance integrals are frequently calculated, the 0.625 eV boundary should be included to account for the cadmium cutoff value),

b. maintaining the capability to reproduce a previously defined broad group library from the new general purpose library (i.e., retrofitting capability),

experience with previous cross section evaluations,

d. treatment of important resonances (e.g., isolate the 6.67 eV $^{238}$U resonance into one or more energy groups) is required.

e. treatment of dips or windows in the cross section data (For shielding applications, the low cross section values can provide an energy range for neutron streaming; there is a cross section dip below the 25 keV resonance for $^{56}$Fe.), and

f. treatment of resonances at low energies. (For example, $^{239}$Pu and $^{241}$Pu have resonances in the thermal energy region, and most resonance treatments assume free-atom elastic scattering to calculate the slowing-down effects. Moreover, these resonances are in an energy region where the proper slowing-down kinematics should be used. Therefore, enough groups should be selected to explicitly calculate the effects of the low energy resonances.)

11.2.2.2. Weighting Spectrum Selection

Selection of an appropriate weighting spectrum is crucial for collapsing point cross section values to a MG cross section library. MG parameters are characterized by three classes of information:

1. group-averaged parameters such as the average value of a fission or elastic scattering cross section for a specific energy group;

2. group-to-group scattering parameters that define the particle scattering between energy groups. (These MG parameters describe the scattering transfer as a function of scattering angle. Regarding the angle dependence, the parameters are provided in a matrix form so that the individual terms are the coefficients of a Legendre fit to the scattering terms. As an important note, if a cross section value is flat within an energy group, the scattering distribution and associated terms of the scattering matrix depend on the weighting spectrum. If the weighting spectrum is higher at the low end (in energy) of an energy group, the out-of-group transfer is emphasized. Conversely, if the weighting spectrum is higher at the upper end (in energy) of the energy group, the within-group transfer is emphasized.); and

3. additional MG parameters such as the fraction of fission neutrons born in energy group g, $\chi_g$, or the average number of fission neutrons produced in energy group g, $\bar{v}$ .

Each of the above parameters is directly affected by the selection of the weighting spectrum, and the use of a fine group structure can reduce the effects of the weighting spectrum on the individual parameters.

Although the selection of a weighting spectrum for generating MG cross sections is important, there is an additional cross section processing consideration that may reduce the potential problems associated with the generation of a MG library using an inapplicable weighting spectrum. In the resonance region, the cross section resonances significantly affect the spectrum that a nuclide will “see.” In particular, the resonance of a nuclide will “shield” a nuclide from “seeing” a 1/E flux that would normally be present in the absence of the resonance in the slowing-down region. As a result, there are a variety of procedures that can be used to account for the effects associated with the resonances in the slowing-down region. The treatment of the shielding effects in the resonance region is referred to as “resonance self-shielding.”

Most fine-group libraries (at least in AMPX) are generated based on a weighting spectrum that is constructed by splicing 4 simple spectra together. At low energies (arbitrarily chosen to be below 0.125 eV), the weighting function is a Maxwellian spectrum which has a flux shape that assumes the neutron scatters into a region with a free gas scatterer that has no absorption. The Maxwellian flux spectrum has the form:

(11.2.1)\[\phi(E)=M(E)=E e^{-\frac{E}{k T}} ,\]

where

E = energy,

k = Boltzmann constant, and

T = temperature in Kelvin.

In the slowing down range, 0.125 < E < E_cut, and the weighting spectrum is assumed to be $\phi(E)=1 / E$. The cutoff energy E_cut for the slowing down range must be selected and is typically 83 keV by default in the AMPX modules. In the region, E_cut < E < 10⁷ eV, where fission neutrons are born, the following fission spectrum is used:

(11.2.2)\[\phi(E)=\chi(E)=E^{0.5} e^{-\frac{E}{\theta}}\]

where

$\theta$ = temperature of the fission spectrum (e.g., 1.2 $\times$ 10⁶ eV).

For energies above 10⁷ eV, the particles are considered to be in another slowing down region; hence, the spectrum is assumed to have a l/E shape. For fusion applications, the user may want to include a “fusion peak” that is tied to the 1/E spectrum in the 20 MeV range.

In addition, AMPX allows nuclide and temperature dependent flux spectra.

11.2.2.3. Point Cross Sections

The ENDF/B evaluations provide CE representations for many cross sections; however, there are energy regions where the point cross section representation is incomplete and may be zero. For example, the tabulated CE data for 235U are zero from 10$^{-5}$ eV through 2.5 $\times$ 10⁴ eV. There is a logical explanation for the data deficiency. In particular, the example data range for ²³⁵U spans energy ranges known as the resolved and unresolved resonance regions, and the ENDF/B evaluation does not provide CE cross section data in the resonance region. However, the ENDF/B evaluation provides detailed information about the resonance structure in the resonance region, and a processing code must be able to reconstruct the CE cross section representation from the parameters in the resonance region.

11.2.2.3.1. Resolved Resonance Region

In the resolved resonance region, the evaluation also specifies a mathematical set of formulae that are to be used in conjunction with the resonance parameters to calculate the cross section values as a function of energy. Six different resolved resonance formalisms are available in the Version 6 formats of the ENDF/B system; however, only five formalisms are currently in use:

the single level Breit–Wigner (SLBW) representation,
the multilevel Breit–Wigner (MLBW) representation,
the Adler–Adler (AA) representation, and
the Reich–Moore (RM) representation.
the full Reich–Moore (RM) representation.

These formalisms are mentioned in order of increasing complexity and are documented in the ENDF/B procedures manual [ampx-Her09], and are discussed in detail in the POLIDENT user guide [ampx-DG00].

In the resolved resonance region, neutron resonances are described with parameters that define specific characteristics of each resonance (e.g., the resonance energy, size, spin, angular momentum, etc.). Some evaluations may specify more than a thousand resonances. Unfortunately, the ENDF/B formats do not provide an energy mesh or guidance for defining an energy mesh for resonance reconstruction. Due to the lack of specific guidance, many mesh generation schemes have been employed in a variety of codes. The various mesh generation schemes range from very crude approaches that define equal energy bins around the peaks of a resonance to more efficient and accurate expressions that characterize the resonance shape. The more sophisticated methods define a mesh by maintaining a specific ratio of successive cross section values as a fixed constant (e.g., 0.95). (See the discussion of such a technique in the POLIDENT manual [ampx-DG00]). Still, other approaches use adaptive methods that ensure the energy mesh is refined to a point where cross sections can be interpolated within the mesh panels to a user specified tolerance (e.g., POLIDENT module). Techniques to determine the energy mesh within some specified tolerance are expensive and generally require more CPU time than the actual cross section calculation. The increase in CPU time is to be expected since these methods inherently require the cross sections to be calculated as the mesh is being determined.

When cross section resonances are generated, temperature effects must also be taken into account. Doppler broadening is used to account for changes in the resonance structure due to an increase in temperature at thermal energies. During Doppler broadening, the overall cross section of the nucleus does not change as the medium increases in temperature; rather, the effective cross section as seen by the neutron changes with temperature. In particular, at 0 Kelvin, a neutron at an energy, E, “sees” the resonance like the line shape that is calculated from an analytic expression. The cross section is a function of the relative velocity between the neutron and the nucleus. As the temperature of the medium increases, the nucleus experiences thermal motion that is assumed to vary according to a Maxwellian distribution in temperature. Because the motion of the target nucleus causes the neutron to see a distribution of nucleus velocities, the relative velocity and cross section values are functions of the distribution of nucleus velocities. At 0 Kelvin, the cross section peak is the maximum value, and there is no mechanism that would make the cross section value higher than the peak value. Moreover, a neutron at energy E “sees” a cross section value that is averaged over the distribution of relative velocities. The averaging procedure over the distribution of relative velocities is the basis for Doppler broadening. As the temperature of the medium increases, there is a wider distribution of relative velocity values, and the effective peak cross section value will decrease. At energies away from the resonance peak (i.e., “wings” of the resonance), the effective cross section values increase with the associated temperature increase.

11.2.2.3.2. Unresolved Resonance Region

The unresolved region is an energy regime where the effects of resonances must be treated; however, the resonances are so closely spaced in energy that it is either impossible or impractical to resolve the individual resonance parameters. As a result, the unresolved resonance parameters are averages of resolved resonance parameters over energy.

Resonances fit into families divided according to the angular momentum ($\ell$ value) of the nucleus and the angular momentum of the resonance (j-state). The unresolved resonance data in an ENDF/B evaluation are statistical parameters derived from the resonances in each family that can be resolved. These resolved resonances statistically predict how the resonances are spaced in the particular family, and they also the characteristics of the resonances (e.g., the relative size of the fission or elastic scattering component of the resonance).

All of the unresolved data are presented statistically and can be sampled using Monte Carlo techniques to determine ladders of the resolved resonance parameters. Subsequently, the unresolved data can be used in exactly the same fashion as the resolved resonance parameters determined from the measurement and evaluation procedures. It is difficult to prove that the ladders of resonances adequately represent the statistical behavior described in the unresolved resonance data. To alleviate this problem, the AMPX module PURM generates pairs of resonance or levels surrounding the energy of reference given in the ENDF evaluated data files.

Several codes employ a method developed by R. N. Hwang at Argonne National Laboratory for calculating unresolved cross sections as a function of energy [ampx-Hwa87]. In the unresolved resonance region, the resonance parameters are provided for the SLBW formalism, and the resonance widths are distributed according to a chi-squared distribution with a specified number of degrees of freedom. Flux weighted cross section values can be calculated over an evaluator-specified energy interval using the unresolved resonance parameters. In the averaging process, the method by Hwang makes use of the narrow resonance (NR) approximation, and the resulting expressions for the average cross section values can be expressed in terms of fluctuation integrals that are also defined in terms of the Doppler broadening $\psi$ and $\chi$ resonance line shape functions. Despite the simplifications from the NR approximation, the resulting expressions are quite complicated and involve integration of the resonance widths over the evaluator specified chi-square distributions. The method by Hwang makes use of special Gaussian-like quadratures that permit integration over the probability distributions and the ultimate calculation of averaged point cross sections as a function of energy. These averaged point value curves are very smooth functions in energy, and no attempt is made to actually determine the extreme variation in the cross sections that actually are in the real cross sections. This technique is employed in AMPX modules PRUDE and POLIDENT to processes data in the unresolved energy range and form the smooth average cross section functions for selected temperatures and values of a background cross section. The background cross section values are used in the NR approximation to account for other nuclides being mixed in with the nuclide at different concentrations.

11.2.2.4. Scattering Kinematics

The development of a MG library requires CE cross section data coupled with an appropriate energy group structure and weighting spectrum. In the preceding sections, brief discussions are provided for each of these components; however, a MG library cannot be complete without group-to-group scattering matrices. As expected, the point data, energy-group structure, and weighting spectrum must be used to calculate the terms in a scattering matrix; however, additional information must be provided to describe the physics of a particle collision with a target nucleus. The additional information can be referred to as scattering kinematics.

Aside from a specification of the differential cross section as a function of energy and angle, kinematics information is not provided in an ENDF/B evaluation for two body processes (e.g., elastic or discrete level inelastic scattering). All of the parameters necessary to use in the kinematics equations (e.g., the mass ratio and Q values of the inelastic levels) are given in the files, but the evaluation assumes that the cross section processing code accounts for the conservation of energy and momentum in the calculation of transfer matrices.

The kinematics for systems of three or more bodies also must conserve energy and momentum; however, for these cases, there is no general solution since the equations are intractable. Moreover, the kinematics must be presented in some abstract manner in an ENDF/B evaluation (i.e., either as a tabular or analytic fit to some experimental behavior or to some simplified model for the nuclear process). The matrices for multibody interactions can be given in three different ways in the ENDF/B evaluations:

1. The process is assumed to scatter isotropically in the laboratory system. Note that the laboratory system must be used because the transformation from the center-of-mass to the laboratory system would require a solution of the kinematics expressions. Also, the secondary energy distribution of particles is given as a tabular or an analytic distribution.

2. The angular variation of the differential cross section is expressed in a tabular or analytic format that is assumed to be independent of the secondary energy distribution expressed in item one.

3. The angular and energy variation of the scattering distributions are dependent and are presented in a tabular or analytic manner or a combination of both.

AMPX uses procedures developed explicitly for the current release of the system. Furthermore, these new procedures are based on a preference to have a single collection of subroutines with the flexibility and generality to permit the calculation of transfer matrices for any kind of reaction (e.g., two-body or multibody processes for any particle type such as neutrons, gamma-rays, etc.). These procedures are described in the sections of this document that describe the Y12 module.

11.2.3. CE Libraries

Many codes can now use CE data directly without the need to collapse to a MG first. While the self-shielding sequences in SCALE involving the CENTRM module [ampx-Wil11] use point-wise cross section data in the resolved resonance range, CE libraries are also used for criticality methods [ampx-ORN11]. The CE libraries contain the same elements as the MG libraries, and except in the case of kinematics data, the point-wise data are simply stored without the need of a flux. The kinematics data are converted to probability distributions. For each incident energy, a cumulative probability distribution with respect to all possible exit angles is calculated based on the information given in ENDF. For each exit angle, a conditional cumulative probability is also given with respect to the exit energy.

11.2.4. The Modular Concept

The introduction of this manual describes AMPX as a modular programming system. The modular concept isolates a single computing function into a single independent code and provides a communication channel between codes using a standard interface or file. In the limit to this approach, a code would require little or no user interaction. Moreover, the code would perform a single function with an input and an output file specification. There are obvious penalties associated with the limit, because it would require the selection of a formidable number of modules to complete some tasks.

The popular UNIX operating systems prevalent on many modern computing platforms use a similar modular structure. UNIX commands are equivalent to codes with an input and output specification. These specifications can originate from a previous command, or they can be passed to a subsequent command. UNIX commands are generally associated with a single function, but various commands have many options and methods for redirecting output. In addition, the UNIX command execution sequences can be written into scripts that can be re-executed by simply invoking the script.

AMPX uses a similar modular concept. Each module can have options that are specified as part of the input, and ideally, these input options are minimized. A module may require one or more standard input files (e.g., MG or point cross section libraries) and may produce one or more standard output files (e.g., MG or point cross section libraries). The input to AMPX is equivalent to a UNIX script. The modules (commands) are selected, and the options are given in the same stream as input data to the code. Each code can use default locations for required input/output files, or a user can override the default values as part of the input data. Files to be retained can be named and stored using a special module, referred to as “SHELL,” that allows the user to insert UNIX commands into the AMPX execution sequence. As with UNIX scripts, the AMPX code selection and input data information can be retained for subsequent re-execution or modification to perform other similar tasks.

11.2.4.1. Execution Sequences

This section describes the procedure for constructing an AMPX execution sequence in a schematic fashion. Also, a description of the input data specifications is provided for the system’s driver module.

The AMPX driver module is a variation of the driver module used in SCALE, providing a convenient method for selecting codes collected together in a standard program library, introducing new codes, or importing other codes from standard program libraries. A code is selected by the simple command:

=CODENAME

For example, if the Y12 module is desired for execution, the command “=Y12” would be entered, followed by the input data for the Y12 module. The input data can be written in any format that the code uses. In previous versions of AMPX, all modules required an input scheme known as “FIDO” (described in Sect. 11.10). Many of the codes in AMPX still use the FIDO input structure, but newer codes use the keyword free-format driven input, the fixed input scheme, or whatever the code developer prefers. An execution sequence follows the pattern:

=module_name
input options for the module
end
=next_module_name
input options for the next module
end
=next_module_name
.
.
etc.

Note that each module is specified with an = sign followed by the name of the module, with no space between the first character of the module name and the equal sign. The input options for the module are specified on the lines following the name of the module. The termination of a module input is specified with an end card starting in column 1. The remaining modules in the sequence are specified in a similar manner as the first module.

A sample AMPX execution script would have the following form:

=shell
ln -fs /home/centrm_libraries/endfb6/broaden/u235 ft33f001
end
=pickeze
0$$ 33 34
1$$ 1 1 1 1 0 0 e t
2$$ 9228
3$$ 3
4$$ 1
5** 0.
t
end
=charmin
single to plot in=34 out=35
end
=shell
cp ft35f001 /home/u235.tot
end

In the above example, the PICKEZE and CHARMIN modules are specified in the input file. Note that the SHELL specification is used to permit the execution of UNIX commands during the AMPX execution sequence. In the first SHELL command, the u235 data file is linked to Logical Unit 33 (i.e., ft33f001) in the AMPX working directory prior to executing any AMPX module. Subsequently, the PICKEZE and CHARMIN modules are used to process the data. Based on the CHARMIN input specifications, the output is written to Logical Unit 35. When AMPX is executed on a particular machine, the code system is executed in a temporary working directory on the computing platform. If the user wants to keep a data file produced by a specific module, the SHELL command should be used to copy the file to a desired location as determined by the user. In the above example, the data file produced by CHARMIN (i.e., ft35f001) is copied to the /home directory and renamed u235.tot.

The installation procedure will have put a command file ampxrte into the installation directory, which takes the above input as a command line argument.

Several points are noted below:

Since AMPX executes in a temporary directory, all external files accessed during the run must be given using the absolute path. The script sets an environment variable, RTNDIR, which points to the current working directory. Thus files in that directory can also be referenced as ${RTNDIR}/localFile, where localFile is the name of the desired file.
AMPX uses logical unit numbers to identify files. Logical units are bound to files ftnnf001, where nn denotes the logical unit number to use. The number format is always two digits long.
Libraries to be used with SCALE must be in big-endian format. By default, AMPX produces data files in native format. However, if the environment variable SCALEXS is set to yes, then the following logical unit numbers are written or read in big-endian format: 60–70, 80–89.