DATA FORMATS AND PROCEDURES

FOR THE EVALUATED NUCLEAR DATA FILE

0. ENDF-6 PREFACE

This update to revision 4/01 of "Data Formats and Procedures for the Evaluated Nuclear Data File, ENDF" pertains to the version 6 of the ENDF formats. The seventh version of the ENDF/B library, ENDF/B-VII, uses these formats.

Below is a list of changes to the formats and procedures that appear in this edition. In addition, some typographical error corrections are included. Users of this manual who note deficiencies or have suggestions are encouraged to contact the National Nuclear Data Center.

Major updates to Manual for Revision June 2005

Section Page Update

0.7.3 New record type: INTG (N. Larson)

2. Replacement of LRF=5 and 6 with LRF=7 format for R-Matrix parameters. Miscellaneous corrections. (N. Larson)

32. Addition of LCOMP=2 format for resonance-parameter covariance matrices (N. Larson)

Appendix D Addition of D.1.7, Equations for LRF=7 format. Deletion of Sections D.1.5 and D.1.6. Other miscellaneous corrections. (N. Larson)

Major updates to Manual for Revision 2004

Section Page Update

0. 0.5 Added table of library numbers (NLIB)

0.23-.25 Recommended two-dimensional interpolation procedures (M. Greene)

1. 1.7-.8 Energy dependent delayed-group constants added.

2. Miscellaneous corrections and updates (C. Lubitz).

Updates for 2 channel spins (N. Larson, C. Lubitz).

4. 4.1-.4 Removed elastic transformation matrix (C. Dunford)

Increase to 2000, the number of energies for which angular distributions are given (JEFF)

Increase to 201, the number of angles at which a tabular distribution can be given for an incident energy. (JEFF)

6. 6.3-.4 Allow NA=2 for Kalbach-Mann parameterization (JEFF)

6.6 Equation 6.4 corrected (JENDL)

6.10 Add LTP=15 interpolation for ratio to Rutherford scattering (JENDL)

6.12 Revised equation in laboratory frame for LAW=6 (M. Greene)

7. 7.2,7.4 Corrected incoherent inelastic scattering equations (C. Lubitz)

8. 8.9 Added section for stable nucleus (JEFF)

9. 9.1-.2 Added final product identifier, IZAP (JEFF)

10. 10.1-.2 Added final product identifier, IZAP (JEFF)

Appendix D Replaced Section D.3.1 (C. Lubitz).

Appendix F Corrected limit on number of Legendre coefficients

Appendix G Increased limits of number of incident energies and number of angles per incident energy in File 4, 6 and 14

Increased the number of allowed subsections in File 6

0.1. Introduction to the ENDF-6 Format

The ENDF formats and libraries are decided by the Cross Section Evaluation Working Group (CSEWG), a cooperative effort of national laboratories, industry, and universities in the U.S. and Canada,1 and are maintained by the National Nuclear Data Center (NNDC).

Earlier versions of the ENDF format provided representations for neutron cross sections and distributions, photon production from neutron reactions, a limited amount of charged-particle production from neutron reactions, photo-atomic interaction data, thermal neutron scattering data, and radionuclide production and decay data (including fission products). Version 6 (ENDF-6) allows higher incident energies, adds more complete descriptions of the distributions of emitted particles, and provides for incident charged particles and photonuclear data by partitioning the ENDF library into sub-libraries. Decay data, fission product yield data, thermal scattering data, and photo-atomic data have also been formally placed in sub-libraries. In addition, this rewrite represents an extensive update to the Version V manual.2

0.2. Philosophy of the ENDF System

The ENDF system was developed for the storage and retrieval of evaluated nuclear data to be used for applications of nuclear technology. These applications control many features of the system including the choice of materials to be included, the data used, the formats used, and the testing required before a library is released. An important consequence of this is that each evaluation must be complete for its intended application. If required data are not available for particular reactions, the evaluator should supply them by using systematics or nuclear models.

The ENDF system is logically divided into formats and procedures. Formats describe how the data are arranged in the libraries and give the formulas needed to reconstruct physical quantities such as cross sections and angular distributions from the parameters in the library. Procedures are the more restrictive rules that specify what data types must be included, which format can be used in particular circumstances, and so on. Procedures are, generally, imposed by a particular organization, and the library sanctioned by the Cross Section Evaluation Working Group (CSEWG) is referred to as ENDF/B. Other organizations may use somewhat different procedures, if necessary, but they face the risk that their libraries will not work with processing codes sanctioned by CSEWG.

0.2.1. Evaluated data

An evaluation is the process of analyzing experimentally measured cross-section data, combining them with the predictions of nuclear model calculations, and attempting to extract the true value of a cross section. Parameterization and reduction of the data to tabular form produces an evaluated data set. If a written description of the preparation of a unique data set from the data sources is available, the data set is referred to as a documented evaluation.

1 See page vi for a list of present and former members of CSEWG.

2 ENDF-102 Data Formats and Procedures for the Evaluated Nuclear Data File, ENDF/B-V, BNL-NCS-50496 (ENDF-102), edited by R. Kinsey, 1979. (Revised by B. Magurno, November 1983).

0.2.2. ENDF/B Library

The ENDF/B library maintained at the National Nuclear Data Center (NNDC) contains the recommended evaluation for each material. Each material is as complete as possible; however, completeness depends on the intended application. For example, when a user is interested in performing a reactor physics calculation or in doing a shielding analysis, he needs evaluated data for all neutron-induced reactions, covering the full range of incident neutron energies, for each material in the system that he is analyzing. Also, the user expects that the file will contain information such as the angular and energy distributions for secondary neutrons. For another calculation, the user may only need a minor isotope for determining activation, and would then be satisfied by an evaluation that contains only reaction cross sections.

ENDF/B data sets are revised or replaced only after extensive review and testing. This allows them to be used as standard reference data during the lifetime of the particular ENDF/B version.

0.2.3. Choices of Data

The data sets contained on the ENDF/B library are those chosen by CSEWG from evaluations submitted for review. The choice is made on the basis of requirements for applications, conformance of the evaluation to the formats and procedures, and performance in testing. The data set that represents a particular material may change when (1) new significant experimental results become available, (2) integral tests show that the data give erroneous results, or (3) user's requirements indicate a need for more accurate data and/or better representations of the data for a particular material. New or revised data sets are included in new releases of the ENDF/B library.

0.2.4. Experimental Data Libraries

NNDC maintains a library for experimentally measured nuclear reaction data (CSISRS). In addition to the data, the CSISRS library contains bibliographic information, as well as details about the experiment (standard, renormalization, corrections, etc.).

At the beginning of the evaluation process the evaluator may retrieve the available experimental data for a particular material by direct access to the CSISRS database via the World Wide Web or using the NNDC Online Data Service.3 Alternately, the data may be requested from the NNDC, and transmitted in the form of listings, plots, and/or files, which may be formatted to satisfy most needs.

0.2.5. Processing Codes

Once the evaluated data sets have been prepared in ENDF format, they can be converted to forms appropriate for testing and actual applications using processing codes. Processing codes that generate group-averaged cross sections for use in neutronics calculations from the ENDF library have been written. These codes4 include such functions as resonance reconstruction, Doppler broadening, multigroup averaging, and/or rearrangement into specified interface formats.

3 C.L. Dunford, T.W. Burrows, Online Nuclear Data Service, NNDC/ONL-99/3, periodically updated.

4 D.E. Cullen, The 1996 ENDF Pre-Processing Codes (PREPRO96), report IAEA-NDS-39, Rev. 9, 1996

R. E. MacFarlane, D. W. Muir, The NJOY Nuclear Data Processing System, Version 91, report LA-12740-M, October 1994.

The basic data formats for the ENDF library have been developed in such a manner that few constraints are placed on using the data as input to the codes that generate any of the secondary libraries.

0.2.6. Testing

All ENDF/B evaluations go through at least some testing before being released as a part of a library. Phase 1 testing uses a set of utility codes5 maintained by NNDC and visual inspection by a reviewer to assure that the evaluation conforms to the current formats and procedures, takes advantage of the best recent data, and chooses format options suited to the physics being represented. Phase 2 uses calculations of data testing "benchmarks," when available, to evaluate the usefulness of the evaluation for actual applications.6 This checking and testing process is a critically important part of the ENDF system.

0.2.7. Documentation

The system is documented by a set of ENDF reports (see Section 0.8) published by the National Nuclear Data Center at Brookhaven National Laboratory. In addition, the current status of the formats, procedures, evaluation process, and testing program is contained in the Summary of the Meetings of the Cross Section Evaluation Working Group.

0.3. General Description of the ENDF System

The ENDF libraries are a collection of documented data evaluations stored in a defined computer-readable format that can be used as the main input to nuclear data processing programs. For this reason, the ENDF format has been constructed with the processing codes in mind. The ENDF format uses 80-character records. Parameters are written in the form of FORTRAN variables (that is, integers start with the letters I, J, K, L, M, or N, and parameters starting with other letters represent real numbers). A complete list of all the parameters defined for the ENDF-6 format will be found in Appendix A (Glossary).

0.3.1. Library Organization

Each ENDF evaluation is identified by a set of key parameters organized into a hierarchy. Following is a list of these parameters and their definitions.

Library

NLIB

a collection of evaluations from a specific evaluation group (e.g., NLIB 0=ENDF/B).

Version

NVER

one of the periodic updates to a library in ENDF format (e.g., NVER 6=ENDF/B-VI). A change of version usually implies a change in format, standards, and procedures.

A revision number is appended to the library/version name for each succeeding revision of the data set; for example, ENDF/B-VI.2. There is no parameter for the revision number in the format.

5 C. L. Dunford, ENDF Utility Codes Release 6.11, April 1999. Available on the NNDC Web page.

6 Cross Section Evaluation Working Group Benchmark Specifications, ENDF-202, 1974 (last updated 1991).

Sublibrary

NSUB

set of evaluations for a particular data type, (e.g., 4=radioactive decay data, 10=incident-neutron data, 12=thermal neutron scattering data). (See Table 0.1 for the complete list of sub-libraries).

Format

NFOR

format in which the data is tabulated; tells the processing codes how to read the subsequent data records (e.g., NFOR 6 = ENDF-6).

Material

MAT

the target in a reaction sub-library, or the radioactive (parent) nuclide in a decay sub-library; see Section 0.3.2.

Mod

NMOD

"modification" flag; see Section 0.3.3.

File

MF

subdivision of a material (MAT); each file contains data for a certain class of information (e.g., MF=3 contains reaction cross sections, MF=4 contains angular distributions). MF runs from 1 to 99. (See Table 0.2 for a complete list of assigned MF numbers).

Section

MT

subdivision of a file (MF) ; each section describes a particular reaction or a particular type of auxiliary data (e.g., MT=102 contains capture data). MT runs from 1 to 999. (See Appendix B for a complete list of assigned MT numbers).

0.3.2. Library (NLIB)

A library is a collection of material evaluations from a recognized evaluation group. Each of these collections is identified by an NLIB number. Currently defined NLIB numbers are given in the table below.

NLIB

Library Definition

0

ENDF/B - United States Evaluated Nuclear Data File

1

ENDF/A - United States Evaluated Nuclear Data File

2

JEFF - NEA Joint Evaluated File (formerly JEF)

3

EFF - European Fusion File (now part of JEFF)

4

ENDF/B High Energy File

5

CENDL – China Evaluated Nuclear Data Library

6

JENDL – Japan Evaluated Nuclear Data Library

31

INDL/V – IAEA Evaluated Neutron Data Library

32

INDL/A – IAEA Nuclear Data Activation Library

33

FENDL – IAEA Fusion Evaluated Nuclear Data Library

34

IRDF – IAEA International Reactor Dosimetry File

35

BROND – Russian Evaluated Nuclear Data File (IAEA version)

36

INGDB-90 – Geophysics Data

37

FENDL/A – FENDL activation evaluations

41

BROND – Russian Evaluated Nuclear Data File (original version)

0.3.3. Material (MAT)

A material is defined as either an isotope or a collection of isotopes. It may be a single nuclide, a natural element containing several isotopes, or a mixture of several elements (compound, alloy, molecule, etc.). A single isotope can be in an excited or isomeric state. Each material in an ENDF library is assigned a unique identification number, designated by the symbol MAT, which ranges from 1 to 9999.7

The assignment of MAT numbers for ENDF/B-VI is made on a systematic basis assuming uniqueness of the four digit MAT number for a material. A material will have the same MAT number in each sub-library (decay data, incident neutrons, incident charged particles, etc.).

One hundred MAT numbers (Z01-Z99) have been allocated to each element Z, through Z = 98. Natural elements have MAT numbers Z00. The MAT numbers for isotopes of an element are assigned on the basis of increasing mass in steps of three, allowing for the ground state and two metastable states.8 In the ENDF/B files, which are application oriented, the evaluations of neutron excess nuclides are of importance, since this category of nuclide is required for decay heat applications. Therefore, the lightest stable isotope is assigned the MAT number Z25 so that the formulation can easily accommodate all the neutron excess nuclides.

For the special cases of elements from einsteinium to lawrencium (Z99) MAT numbers 99xx are assigned, where xx = 20, 25, 20, l5, and l2 for elements 99 to 103 respectively, one covers the known nuclides with allowance for expansion.

For mixtures, compounds, alloys, and molecules, MAT numbers between 0001 and 0099 are assigned on a special basis (see Appendix C).

0.3.4. Material modification (MOD)

All versions of a data set (i.e., the initial release, revisions, or total re-evaluations) are indicated using the material "modification" flags. For the initial release of ENDF/B-VI, the modification flag for each material (MAT) and section (MT) carried over from previous versions is set to zero (MOD 0); for new evaluations they are set to one (MOD 1). Each time a change is made to a material, the modification flag for the material is incremented by one. The modification flag for each section changed in the revised evaluation is set equal to the new material modification number. If a complete re-evaluation is performed, the modification flag for every section is changed to equal the new material "modification" number.

As an example, consider the following. Evaluator X evaluates a set of data for 235U. After checking and testing, the evaluator feels that the data set is satisfactory and transmits it to the NNDC. The Center assigns the data set a MAT number of 9228 subject to CSEWG's approval of the evaluation. This evaluation has "modification" flags equal to 1 for the material and for all sections. After the file is released, user Y retrieves MAT 9228 from the Center's files, adds it to his ENDF library as material 9228, and refers to it in later processing programs by this number. Should the evaluation of material 9228 subsequently be revised and released with CSEWG's approval, the material will have a MOD flag of 2. This material would have MOD flags of 2 on each revised section, but the unchanged sections will have MOD flags of 1.

7 The strategy for assigning MAT numbers for ENDF/B-VI is described here; other libraries may have different schemes.

8 This procedure leads to difficulty for the nuclides of xenon, cesium, osmium, platinum, etc., where more than 100 MAT numbers could be needed to include all isotopes.

0.4. Contents of an ENDF Evaluation

As described above, sub-library (NSUB) and material (MAT) specify the target and projectile for a reaction evaluation or the radioactive nuclide for a decay evaluation. MF and MT indicate the type of data represented by a section and the products being defined.

The sub-library distinguishes between different types of data using NSUB = 10*IPART+ITYPE. In this formula, IPART=1000*Z+A defines the incident particle; use IPART=0 for incident photons or no incident particle (decay data), use IPART=11 for incident electrons, and IPART=0 for photo-atomic or electro-atomic data. The sub-libraries allowed for ENDF-6 are listed in Table 0.1.

Table 0.1

Sub-library Numbers and Names

NSUB

IPART

ITYPE

Sub-library Names

NSUB

IPART

ITYPE

Sub-library Names

0

0

0

Photo-Nuclear Data

1

0

1

Photo-Induced Fission Product Yields

3

0

3

Photo-Atomic Interaction Data

4

0

4

Radioactive Decay Data

5

0

5

Spontaneous Fission Product Yields

6

0

6

Atomic Relaxation Data

10

1

0

Incident-Neutron Data

11

1

1

Neutron-Induced Fission Product Yields

12

1

2

Thermal Neutron Scattering Data

113

11

3

Electro-Atomic Interaction Data

10010

1001

0

Incident-Proton Data

10011

1001

1

Proton-Induced Fission Product Yields

10020

1002

0

Incident-Deuteron Data

...

20040

2004

0

Incident-Alpha data

The files (MF) allowed are summarized in Table 0.2, and their use in the different sub-libraries is discussed following.

Table 0.2

Definitions of File Types (MF)

MF

Description

1

General information

2

Resonance parameter data

3

Reaction cross sections

4

Angular distributions for emitted particles

5

Energy distributions for emitted particles

6

Energy-angle distributions for emitted particles

7

Thermal neutron scattering law data

8

Radioactivity and fission-product yield data

9

Multiplicities for radioactive nuclide production

10

Cross sections for radioactive nuclide production

12

Multiplicities for photon production

13

Cross sections for photon production

14

Angular distributions for photon production

15

Energy distributions for photon production

23

Photo- or electro-atomic interaction cross sections

26

Electro-atomic angle and energy distribution

27

Atomic form factors or scattering functions for photo-atomic interactions

28

Atomic relaxation data

30

Data covariances obtained from parameter covariances and sensitivities

31

Data covariances for nu(bar)

32

Data covariances for resonance parameters

33

Data covariances for reaction cross sections

34

Data covariances for angular distributions

35

Data covariances for energy distributions

39

Data covariances for radionuclide production yields

40

Data covariances for radionuclide production cross sections

The following MF numbers have been retired: 16, 17, 18, 19, 20, 21, 22, 24, and 25.

0.4.1. Incident-Neutron Data (NSUB 10)

The procedures for describing neutron-induced reactions for ENDF/B-VI have been kept similar to the procedures used for previous versions so that current evaluations can be carried over, and in order to protect existing processing capabilities. The new features have most of their impact at high energies (above 5-10 MeV) or low atomic weight (2H, 9Be), and include improved energy-angle distributions, improved nuclear heating and damage capabilities, improved charged-particle spectral data, and the use of R-matrix or R-function resonance parameterization.

Each evaluation starts with a descriptive data and directory, File 1 (see Section 1.1). For fissionable isotopes, sections of File 1 can be given to describe the number of neutrons produced per fission and the energy release from fission.

A File 2 is always given. For some materials, it may contain only the effective scattering radius, and for other materials, it may contain complete sets of resolved and/or unresolved resonance parameters.

A File 3 is always given. The required energy range is from the threshold or from 10-5eV to 20 MeV, but higher energies are allowed. There is a section for each important reaction or sum of reactions. The MT numbers for these sections are chosen based on the emitted particles as described in Section 0.5 (Reaction Nomenclature). For resonance materials in the resolved resonance energy range, the cross sections for the elastic, fission, and capture reactions are normally the sums of the values given in File 3 and the resonance contributions computed from the parameters given in File 2. An exception to this rule is allowed for certain derived evaluations (see LRP=2 in Section 1.1). In the unresolved resonance range, the self-shielded cross sections will either be sums of File 2 and File 3 contributions, as above, or File 3 values multiplied by a self-shielding factor computed from File 2. (See Sections 2.3.1 and 2.4.21.)

Distributions for emitted neutrons and other particles or nuclei are given using File 4, Files 4 and 5, or File 6. As described in more detail below, File 4 is used for simple two-body reactions (elastic, discrete inelastic). Files 4 and 5 are used for simple continuum reactions, which are nearly isotropic, have minimal pre-equilibrium component, and emit only one important particle. File 6 is used for more complex reactions that require energy-angle correlation, that are important for heating or damage, or that have several important products, which must be tallied.

If any of the reaction products are radioactive, they should be described further in File 8. This file indicates how the production cross section is to be determined (from File 3, 6, 9, or 10) and gives minimal information on the further decay of the product. Additional decay information can be retrieved from the decay data sub-library when required.

Note that yields of particles and residual nuclei are sometimes implicit; for example, the neutron yield for A(n,2n) is two and the yield of the product A-1 is one. If File 6 is used, all yields are explicit. This is convenient for computing gas production and transmutation cross sections. Explicit yields for radioactive products may be given in File 9, or production cross sections can be given in File 10. In the latter case, it is possible to determine the yield by dividing by the corresponding cross section from File 3. File 9 is used in preference to File 10 when strong resonances are present (e.g., radiative capture).

For compatibility with earlier versions, photon production and photon distributions can be described using File 12 (photon production yields), File 13 (photon production cross sections), File 14 (photon angular distributions), and File 15 (photon energy distributions). Note that File 12 is preferred over File 13 when strong resonances are present (capture, fission). Whenever possible, photons should be given with the individual reaction that produced them using File 12. When this cannot be done, summation MT numbers can be used in Files 12 or 13 as described in Section 0.5.9.

When File 6 is used to represent neutron and charged-particle distributions for a reaction, it should also be used for the corresponding photon distribution. This makes an accurate energy-balance check possible for the reaction. When emitted photons cannot be assigned to a particular reaction, they can be represented using summation MT numbers as described in Section 0.5.9.

Finally, covariance data are given in Files 30-40. Procedures for these files are given in Sections 30-40.

0.4.2. Thermal Neutron Scattering (NSUB 12)9

Thermal neutron scattering data are kept in a separate sub-library because the targets are influenced by their binding to surrounding atoms and their thermal motion; therefore, the physics represented10 requires different formats than other neutron data. The data extend to a few eV for several molecules, liquids, solids, and gases. As usual, each evaluation starts with descriptive data and directory file (see Section 1.1). The remaining data is included in File 7. Either the cross sections for elastic coherent scattering, if important, are derived from Bragg edges and structure factors, or cross sections for incoherent elastic scattering are derived from the bound cross section and Debye-Waller integral. Finally, scattering law data for inelastic incoherent scattering are given, using the S(α,β) formalism and the short-collision-time approximation.

9 Used with IPART=0 only.

10 J.U. Koppel and D.H. Houston, Reference Manual for ENDF Thermal Neutron Scattering Data, General Atomic report GA-8774 (ENDF-269) (Revised and reissued by NNDC, July 1978).

0.4.3. Fission Product Yield Data

Data for the production of fission products are given in different sub-libraries according to the mechanism inducing fission. Currently, sub-libraries are defined for neutron-induced fission product yields, and for yields from spontaneous fission. The format also allows for future photon- and charged-particle-induced fission. Each material starts with a descriptive data and directory file (see Section 1.1). The remaining data is given in File 8, which contains two sections: independent yields, and cumulative yields. As described in Section 8.2, the format for these two sections is identical. Covariance data for File 8 are self-contained.

0.4.4. Radioactive Decay Data (NSUB=4)

Evaluations of decay data for radioactive nuclides are grouped together into a sub-library. This sub-library contains decay data for all radioactive products (e.g., fission products and activation products). Fission product yields and activation cross sections will be found elsewhere. Each material contains two, three, or four files, and starts with a descriptive data and directory file (see Section 1.1). For materials undergoing spontaneous fission, additional sections in File 1 give the total, delayed, and prompt fission neutron yields. In addition, the spectra of the delayed and prompt neutrons are given in File 5. The File 5 formats are the same as for induced fission (see Section 5), and the distributions are assumed to be isotropic in the laboratory system. File 8 contains half-lives, decay modes, decay energies, and radiation spectra (see Section 8.3). Finally, covariance data for the spectra in File 5 may be given in File 35; covariance data for File 8 are self-contained.

0.4.5. Photo-Nuclear (NSUB=0) and Charged-Particle (NSUB10010) Sub-libraries

Evaluations for incident charged-particle and photo-nuclear reactions are grouped together into sub-libraries by projectile. As usual, each evaluation starts with a descriptive data and directory file (see Section 1.1). For particle-induced fission or photo-fission, File 1 can also contain sections giving the total, delayed, and prompt number of neutrons per fission, and the energy released in fission. Resonance parameter data (File 2) may be omitted entirely (see LRP=-1 in Section 1.1).

Cross sections are given in File 3. The MT numbers used are based upon the particles emitted in the reaction as described in Section 0.5. Explicit yields for all products (including photons) must be given in File 6. In addition, the charged-particle stopping power should be given. If any of the products described by a section of File 6 are radioactive, they should be described further in a corresponding section of File 8. This section will give half-life, minimum information about the decay chain, and decay energies for the radioactive product. Further details, if required, can be found in the decay data sub-library.

Angular distributions or correlated energy-angle distributions can be given for all particles, recoil nuclei, and photons in File 6. It is also possible to give only the average particle energy for less important reactions, or even to mark the distribution "unknown." (See 6.2.1.)

Finally, Files 30 to 40 might be used to describe the covariances for charged-particle and photo-nuclear reactions.

0.4.6. Photo-Atomic Interaction Data (NSUB 3)

Incident photon reactions with the atomic electrons11 are kept in a separate sub-library. These data are associated with elements rather than isotopes. Each material starts with a descriptive data and directory file (see Section 1.1), as usual. In addition, the material may contain a File 23 for photon interaction cross sections, and File 27 for atomic form factors.

0.4.7. Electro-Atomic Interaction Data (NSUB=113)

Incident electron reactions with the atomic electrons are also kept in a separate sublibrary. These data are again associated with elements rather than isotopes. Each material starts with a descriptive data and directory file (see Section 1.1), as usual. In addition, File 23 is given for the elastic, ionization, bremsstrahlung, and excitation cross sections, and File 26 is given for the elastic angular distribution, the bremsstrahlung photon spectra and energy loss, the excitation energy transfer, and the spectra of the scattered and recoil electrons associated with subshell ionization.

0.4.8. Atomic Relaxation Data (NSUB=6)

The target atom can be left in an ionized state due to a variety of different types of interaction, such as photon or electron induced ionization, internal conversion, etc. This section provides the data needed to describe the relaxation of an ionized atom back to neutrality. This includes subshell energies, transition energies, transition probabilities, and other parameters needed to compute the X-ray and electron spectra due to atomic relaxation.

The materials are elements. Each material starts with a descriptive data and directory file (see Section 1.1), as usual. In addition, a File 28 is given containing the relaxation data for all the subshells defined in the photo-atomic or electro-atomic sublibraries.

0.4.9. Energy and Angular Distributions of Reaction Products (Files 4, 5, and 6)

Several different options are available in the ENDF-6 format to describe the distribution in energy and angle of reaction products. In most cases, the double differential cross section of the emitted particle in barns/(eV-sr) is represented by

(0.1)

where µ is the cosine of the emission angle,

E is the energy of the incident particle,

E is the energy of the emitted particle,

σ(E) is the reaction cross section,

y(E) is the yield or multiplicity of the emitted particle, and

f(µ,E,E) is the normalized distribution function in (eV-unit cosine)-1.

For simple two-body reactions, the energy of the emitted particle can be determined from kinematics (see Appendix E); therefore,

(0.2)

where ξ is defined by Eq. (E.5) in Appendix E.

11 D.E. Cullen, et al., Tables and Graphics of Photon-Interaction Cross Sections from 10 eV to 100 GeV. Derived from the LLNL Evaluated Photon Data Library (EPDL). UCRL-50400, Vol. 6 Rev. (October 1989)

The distribution function f(µ,E) can be given as a section of File 4 with no corresponding section in File 5, or as a section of File 6 with no corresponding sections in Files 4 or 5. For simple continuum reactions, the full distribution is sometimes given as a product of an angular distribution and an energy distribution:

(0.3)

The angular function is given in File 4, and g(E,E) is given in File 5. This simple continuum format does not allow adequate description of energy-angle correlations, and it can only describe one emitted particle. Emitted photons can be described by this scheme also, but the files used are 14 and 15.

For the more complex reactions, the full distribution function is given in File 6. This file allows for all reaction products to be described, and it allows for energy-angle correlation of the emitted particles.

0.5. Reaction Nomenclature - MT

The following paragraphs explain how to choose MT numbers for particle-induced and photo-nuclear reactions for ENDF-6. A complete list of the definitions of the MT numbers will be found in Appendix B.

0.5.1. Elastic Scattering

Elastic scattering is a two-body reaction that obeys the kinematic equations given in Appendix E. The sections are labeled by MT=2 (except for photo-atomic data, see Section 23). For incident neutrons, the elastic scattering cross section is determined from File 3 together with resonance contributions, if any, from File 2. The angular distribution of scattered neutrons is given in File 4.

For incident charged particles, the Coulomb scattering makes it impossible to define an integrated cross section, and File 3, MT=2 contains either a dummy value of 1.0 or a "nuclear plus interference" cross section defined by a particular cutoff angle. The rest of the differential cross section for the scattered particle is computed from parameters given in File 6, MT=2 (see Section 6.2.6).

0.5.2. Simple Single Particle Reactions

Many reactions have only a single particle and a residual nucleus (and possibly photons) in the final state. These reactions are associated with well-defined discrete states or a continuum of levels in the residual nucleus, or they may proceed through a set of broad levels that may be treated as a continuum. The MT numbers to be used are:

Discrete

Continuum

Discrete+Continuum

Emitted Particle

50-90

91

4

n

600-648

649

103

p

650-698

699

104

d

700-748

749

105

t

750-798

799

106

3He

800-848

849

107

α

By definition, the emitted particle is the lighter of the two particles in the final state.

If the reaction is associated with a discrete state in the residual nucleus, use the first column of numbers. In a typical range, MT=50 leaves the residual nucleus in the ground state, MT=51 leaves it in the first excited state, MT=52 in the second, and so on. The elastic reaction uses MT=2 as described above; therefore, do not use MT=50 for incident neutrons, do not use MT=600 for incident protons, and so on. For incident neutrons, the discrete reactions are assumed to obey two-body kinematics (see Appendix E), and the angular distribution for the particle is given in File 4 or File 6 (except for MT=2). If possible, the emitted photons associated with discrete levels should be represented in full detail using the corresponding MT numbers in File 6 or File 12. For incident charged particles, the emitted particle must be described in File 6. A two-body law can be used for narrow levels, but broader levels can also be represented using energy-angle correlation. Photons associated with the particle should be given in the same section (MT) of File 6 when possible.

If the reaction is associated with a range of levels in the residual nucleus (i.e., continuum), use the second column of MT numbers. For incident neutrons, Files 4 and 5 are allowed for compatibility with previous versions, but it may be necessary to use File 6 to obtain the desired accuracy. When Files 4 and 5 are used, photons should be given in File 12 using the same MT number if possible. For more complicated neutron reactions or incident charged particles, File 6 must be used for the particle and the photons.

The "sum" MT numbers are used in File 3 for the sum of all the other reactions in that row, but they are not allowed for describing particle distributions in Files 4, 5, or 6. As an example, a neutron evaluation might contain sections with MF/MT=3/4, 3/51, 3/91, 4/51, and 6/91. A deuteron evaluation might contain sections with 3/103, 3/600, and 6/600 (the two sections in File 3 would be identical). For a neutron evaluation with no 600-series distributions or partial reactions given, MT=103-107 can appear by themselves; they are simply components of the absorption cross section.

In some cases, it is difficult to assign all the photons associated with a particular particle to the reactions used to describe the particle. In such cases, these photons can be described using the "sum" MT numbers in File 12 or 13 (for neutrons) or in File 6 (for other projectiles).

Some examples of simple single-particle reactions follow.

Reaction

MT

9Be(α,n0)12C

50

Fe(n,nc)Fe

91

2H(d,p0)3He

600

6Li(t,d0)7Li

650

6Li(t,d1)7Li

651

For the purposes of this manual, reactions are written as if all prompt photons have been emitted; that is, the photons do not appear explicitly in the reaction nomenclature. Therefore, no "*" is given on Li in the last example above.

0.5.3. Simple Multi-Particle Reactions

If a reaction has only two to four particles, a residual nucleus, and photons in the final state, and if the residual nucleus does not break up, it will be called a "simple multi-particle reaction." The MT numbers that can be used are:

MT Emitted Particles

11 2nd
16 2n
17 3n
22 na
23 n3a
24 2na
25 3nα
28 np
29 n2α
30 2n2α

32 nd

33 nt

34 n3He

35 nd2α

MT Emitted Particles

36 nt2α

37 4n

41 2np
42 3np

44 n2p

43 npα
108 2α
109 3α
111 2p
112 t2α
113 d2α
114 pd
115 pt
116 dα

For naming purposes, particles are always arranged in ZA order; thus, (n,np) and (n,pn) are summed together under MT=28. In addition, there must always be a residual particle. By definition, it is the particle or nucleus in the final state with the largest ZA. This means that the reaction d+tn+α must be classified as the reaction 3H(d,n)4He (MT=50) rather than the reaction 3H(d,nα) (MT=22). The cross sections for these reactions will be found in File 3, as usual.

This list is not exhaustive, and new MT numbers can be added if necessary. However, some reactions are more naturally defined as "breakup" or "complex" reactions (see below).

For compatibility with previous versions, Files 4 and 5 are allowed in the incident-neutron sub-library. In this case, the particle described in Files 4 and 5 is the first one given under "Emitted Particles" above. At high neutron energies, File 6 is preferred because it is possible to describe energy-angle correlations resulting from pre-equilibrium effects and to give distributions for more than one kind of particle. Using File 6 also makes it possible to give an energy distribution for the recoil nucleus. This distribution is needed in calculating nuclear heating and radiation damage. If Files 4/5 are used, photons should be given in File 12 or 13 using the same MT number when possible. Similarly, if File 6 is used to describe the outgoing particle, the photons should also be given in File 6 under the same MT number, or under MT=3, if necessary. However, it often will be necessary to use the nonelastic MT=3 as described below.

For charged-particle sub-libraries, File 6 must be used for these reactions. Photons should be given in File 6 using the reaction MT number when possible. If the photons cannot be assigned to a particular reaction, the nonelastic MT=3 can be used as described below.

0.5.4. Breakup Reactions

A number of important reactions can be described as proceeding in two steps: first one or several particles are emitted as in the simple reactions described above, then the remaining nuclear system either breaks up or emits another particle. In the nomenclature of ENDF-6, these are both called "breakup reactions." For ENDF/B-V, these reactions were represented using special MT numbers or "LR flags". For ENDF/B-VI, the preferred representation uses File 3 and File 6. The same MT numbers are used as for the simple reactions described above. The cross section goes in File 3 as usual, but a special LR flag is used to indicate that this is a breakup reaction (see below). The yield and angular distribution or energy-angle distribution for each particle emitted before breakup is put into File 6. In addition, yields and distributions for all the breakup products are allowed in File 6. For photo-nuclear and charged-particle sub-libraries, the photons are also given in File 6; but for neutron sub-libraries, the photons may be given in Files 6 or 12-15. This approach provides a complete accounting of particle and recoil spectra for transport, heating, and damage calculations. It also provides a complete accounting of products for gas production and activation calculations. Finally, it does all of this without requiring a large list of new MT numbers.

Some examples of breakup reactions are

Reaction

MT

3H(t,n0)5He(nα)

50

6Li(d,n3)7Be(3Heα)

53

7Li(n,nc)7Li(tα)

91

7Li(t,2n)8Be(2α)

16

7Li(p,d1)6Li(dα)

651

9Be(a,n3)12C(3α)

53

16O(n,n6)16O(α)12C

56

By convention, the particles are arranged in Z, A order in each set of parentheses. This leads to ambiguity in the choice of the intermediate state. For example,

12C(n,n)12C(3α)

12C(n,α)9Be(n2α)

or 7Li(t,2n)8Be(2α)

7Li(t,n)9Be(n2α)

7Li(t,α)6He(2nα)

7Li(t,nα)5He(nα)

The evaluator either must choose one channel, partition the reaction between several channels, or use the "complex reaction" notation (see below). Care must be taken to avoid double counting.

In some cases, a particular intermediate state can break up by more than one path; for example,

6Li(d,p4)7Li(tα) Ex = 7.47 MeV,

6Li(d,p4)7Li(n6Li).

If two channels are both given under the same MT number, File 6 is used to list the emitted particles and to give their fractional yields. The notation to be used for this type of reaction is

6Li(d,p4)7Li(X).

Note that the Q value calculated for the entire reaction is not well defined. Another option is to split the reaction up and use two consecutive MT numbers as follows:

6Li(d,p4)7Li(t,α) Ex = 7.4700 MeV, MT=604,

6Li(d,p5)7Li(n6Li) Ex = 7.4701 MeV, MT=605.

The same proton distribution would be given for MT=604 and 605. The mass-difference Q value is well defined for both reactions, but the level index no longer corresponds to real levels.

The choice between the "simple multi-particle" and "breakup" representations should be based on the physics of the process. As an example, an emission spectrum may show several peaks superimposed on a smooth background. If the peaks can be identified with known levels in one or more intermediate systems, they can be extracted and represented by breakup MT numbers. The remaining smooth background can often be represented as a simple multi-particle reaction.

0.5.5. LR Flags

As described above, the MT number for a simple reaction indicates which particles are emitted. However, complex breakup reactions emit additional particles. The identity of these additional particles can be determined from LR or File 6.

LR

Meaning

0

Simple reaction. Identity of product is implicit in MT.

1

Complex or breakup reaction. The identity of all products is given explicitly in File 6.

22

α emitted (plus residual, if any)

23

3α emitted (plus residual, if any)

24

nα emitted (plus residual, if any)

25

2nα emitted (plus residual, if any)

28

p emitted (plus residual, if any)

29

2α emitted (plus residual, if any)

30

n2α emitted (plus residual, if any)

32

d emitted (plus residual, if any)

33

t emitted (plus residual, if any)

34

3He emitted (plus residual, if any)

35

d2α emitted (plus residual, if any)

36

t2α emitted (plus residual, if any)

39

internal conversion

40

electron-positron pair formation

The values LR=22-36 are provided for compatibility with ENDF/B-V. Some examples of their use:

Reaction

MT

LR

6Li(n,n1)6Li(dα)

51

32

7Li(n,nc)7Li(tα)

91

33

10B(n,n12)10B(d2α)

62

35

12C(n,n2)12C(3α)

52

23

16O(n,n1)16O(e+e-)16O

51

40

16O(n,n6)16O(α)12C

56

22

Note that the identity of the residual must be deduced from MT and LR. Only the first particle is described in File 4 and/or File 5; the only information available for the breakup products is the net energy that can be deduced from kinematics.

The use of LR=1 and File 6 is preferred for new evaluations because explicit yields and distributions can be given for all reaction products.

0.5.6. Complex Reactions

At high energies, there are typically many reaction channels open, and it is difficult to decompose the cross section into simple reactions. In such cases, the evaluation should use MT=5. This complex reaction identifier is defined as the sum of all reactions not given explicitly elsewhere in this evaluation. As an example, an evaluation might use only MT=2 and 5. Sections of File 6 with MT=5 and the correct energy-dependent yields would then represent the entire nonelastic neutron spectrum, the entire proton spectrum, and so on. A slightly more refined evaluation might use MT=2, 5, 51-66, and 600-609. In this case, MT=5 would represent all the continuum neutron and proton emission. The discrete levels would be given separately to represent the detailed angular distribution and two-body kinematics correctly. The notation used for complex reactions is, for example, 6Li(d,X).

0.5.7. Radiative Capture

The radiative capture reaction is identified by MT=102. For neutron sublibraries, the only product is usually photons, and they are represented in Files 6 or 12-15. Note that File 6 or 12 must be used for materials with strong resonances. For charged-particle libraries, simple radiative capture reactions must be represented using File 3 and File 6. In addition, radiative capture followed by breakup is common for light targets; an example is d+t→γ+n+α, which is written as a breakup reaction 3H(d,γ)5He(nα) for the purposes of this format. This reaction is represented using MT=102 with the special breakup flag set in File 3. The gamma, neutron, and alpha distributions are all given in File 6.

0.5.8. Fission

The nomenclature used for fission is identical to that used in previous versions of the ENDF format.

MT

Meaning

Description

18

fission

total

19

f

first chance fission

20

nf

second chance fission

21

2nf

third chance fission

38

3nf

fourth chance fission

452

⎯νT

total neutrons per fission

455

⎯νd

delayed neutrons per fission

456

⎯νp

prompt neutrons per fission

458

components of energy release in fission

Cross sections (File 3) can be given using either MT=18 or the combination of MT=19, 20, 21, and 38. In the latter case, MT=18 is also given to contain the sum of the partial reactions.

0.5.9. Nonelastic Reaction for Photon Production

Whenever possible, the same MT number should be used to describe both the emitted particle and the photons. However, this is usually only possible for discrete photons from low-lying levels, radiative capture, or for photons generated from nuclear models. Any photons that cannot be assigned to a particular level or particle distribution can be given in a section with the nonelastic summation reaction MT=3 in File 6, 12, or 13 (for neutrons) or in File 6 (for other projectiles). As described in Section 0.5.2, MT=4, 103, 104, 105, 106, and 107 can also be used as summation reactions for photon production in Files 12 and 13.

0.5.10. Special Production Cross Sections

A special set of production cross sections are provided, mostly for use in derived libraries.

MT

Meaning

201

neutron production

202

photon production

203

proton production

204

deuteron production

205

triton production

206

3He production

207

a production

Each one is defined as the sum of the cross section times the particle yield over all reactions (except elastic scattering) with that particle in the final state. The yields counted must include implicit yields from reaction names, LR flags, or residual nuclei in addition to explicit yields from File 6. As an example, for an evaluation containing the reactions (n,α) (MT=107), and (n,n3α) (MT=91, LR=23), the helium production cross section would be calculated using:

MT207 = MT107 + 3×MT91.

The cross section in File 3 is barns per particle (or photon). A corresponding distribution can be given using Files 4 and 5, or the distribution can be given using File 6 with a particle yield of 1.0. These MT numbers will ordinarily be used in File 3 of special gas production libraries.

0.5.11. Auxiliary MT Numbers

Several MT numbers are used to represent auxiliary quantities instead of cross sections. The values 151, 451, 452, 454, 455, 456, 457, 458, and 459 have already been mentioned. The following additional values are defined

MT

Meaning

251

µL, average cosine of the angle for elastic scattering (laboratory system). Derived files only.

252

ξ, average logarithmic energy decrement for elastic scattering. Derived files only.

253

γ, average of the square of the logarithmic energy decrement, divided by 2 * ξ. Derived files only.

301-450

Energy release rate parameters (eV-barns) for the reaction obtained by subtracting 300 from this MT; e.g., 301 is total kerma, 407 is kerma for (n,α), etc. Derived files only.

851-870

Special series used only in covariance files (MF=31-40) to give covariances for groups of reactions considered together (lumped partials). See Section 30.

The continuous-slowing-down parameters (MT=251-253) and the heat production cross sections (MT=301-450) are usually used in derived libraries only. A complete list of reaction MT numbers and auxiliary MT numbers is given in Appendix B.

0.5.12. Sum Rules for ENDF

A number of ENDF reaction types can be calculated from other reactions. The rules for these summation reactions follow.

MT

Meaning: components

1

Total cross sections (incident neutrons only): 2, 4, 5, 11, 16-18, 22-26, 28-37, 41-42, 44-45, 102-117.

4

Total of neutron level cross sections: 50-91

18

Total fission: 19-21, 38.

103

Total of proton level cross sections: 600-649

104

Total of deuteron level cross sections: 650-699

105

Total of triton level cross sections: 700-749

106

Total of 3He level cross sections: 750-799

107

Total of alpha level cross sections: 800-849

The nonelastic cross section (MT=3) is only used in connection with photon production. It contains the following MT numbers: 4, 5, 11, 16-18, 22-26, 28-37, 41-42, 44-45, 102-117.

0.6. Representation of Data

0.6.1. Definitions and Conventions

The data given in all sections always use the same set of units. These are summarized following.

Parameters

Units

energies

electron volts (eV)

angles

dimensionless cosines of the angle

cross sections

barns

temperatures

Kelvin

mass

in units of the neutron mass

angular distributions

probability per unit cosine

energy distributions

probability per electron volt

energy-angle distributions

probability per unit cosine per electron volt

half life

seconds

The first record of every section contains a ZA number that identifies the specific material. ZA variants are also employed to identify projectiles and reaction products. In most cases, ZA is constructed by

ZA = 1000.0 * Z + A ,

where Z is the atomic number and A is the mass number for the material. If the material is an element containing two or more naturally occurring isotopes in significant concentrations, A is taken to be 0.0. For mixtures, compounds, alloys, or molecules, special ZA numbers between 1 and 99 can be defined (see Appendix C).

A material, incident particle (projectile), or reaction product is also characterized by a quantity that is proportional to its mass relative to that of the neutron. Typically, these quantities are denoted as AWR, AWI, or AWP for a material, projectile, or product, respectively. For example, the symbol AWR is defined as the ratio of the mass of the material to that of the neutron.12 Another way to say this is that "all masses are expressed in neutron units." For materials which are mixtures of isotopes, the abundance weighted average mass is used.

0.6.1.0. Atomic Masses Versus Nuclear Masses

Mass quantities for materials (AWR for all Z) and "heavy" reaction products (AWP for Z > 2) should be expressed in atomic units, i.e., the mass of the electrons should be included. Mass quantities for incident particles (AWI) and "light" reaction products (AWP for Z2) should be expressed in nuclear mass units. For neutrons, this ratio is 1.00000. For charged particles likely to appear in ENDF/B-VI, see Appendix H.

0.6.2. Interpolation Laws

Many types of ENDF data are given as a table of values on a defined grid with an interpolation law to define the values between the grid points. Simple one-dimensional "graph paper" interpolation schemes, a special Gamow interpolation law for charged-particle cross sections, simple Cartesian interpolation for two-dimensional functions, and two non-Cartesian schemes for two-dimensional distributions are allowed.

12 See Appendix H for neutron mass.

0.6.2.1. One-dimensional Interpolation Schemes

Consider how a simple function y(x), which might be a cross section, σ(E), is represented. y(x) is represented by a series of tabulated values, pairs of x and y(x), plus a method for interpolating between input values. The pairs are ordered by increasing values of x. There will be NP values of the pair, x and y(x), given. The complete region over which x is defined is broken into NR interpolation ranges. An interpolation range is defined as a range of the independent variable x in which a specified interpolation scheme can be used; i.e., the same scheme gives interpolated values of y(x) for any value of x within this range. To illustrate this, see Fig. 0.1 and the definitions, below:

x(n) is the nth value of x,

y(n) is the nth value of y,

NP is the number of pairs (x and y) given,

INT(m) is the interpolation scheme identification number used in the mth range,

NBT(m) is the value of n separating the mth and the (m+1)th interpolation ranges.

The allowed interpolation schemes are given in Table 0.3.

Table 0.3

Definition of Interpolation Types

INT

Interpolation Scheme

1

y is constant in x (constant, histogram)

2

y is linear in x (linear-linear)

3

y is linear in ln(x) (linear-log)

4

ln(y) is linear in x (log-linear)

5

ln(y) is linear in ln(x) (log-log)

6

special one-dimensional interpolation law, used for charged-particle cross sections only

11-15

method of corresponding points (follow interpolation laws of 1-5)

21-25

unit base interpolation (follow interpolation laws of 1-5)

Interpolation code, INT=1 (constant), implies that the function is constant and equal to the value given at the lower limit of the interval.

Note that where a function is discontinuous (for example, when resonance parameters are used to specify the cross section in one range), the value of x is repeated and a pair (x,y) is given for each of the two values at the discontinuity (see Fig. 0.1).

A one-dimensional interpolation law, INT=6, is defined for charged-particle cross sections and is based on the limiting forms of the Coulomb penetrabilities for exothermic reactions at low energies and for endothermic reactions near the threshold. The expected energy dependence is

(0.1)

where T=0 for exothermic reactions (Q>0) and T is the kinematic threshold for endothermic reactions (Q0). Note that this formula gives a concave upward energy dependence near E=T that is quite different from the behavior of the neutron cross sections.

This formula can be converted into a two-point interpolation scheme using

(0.2)

and

(0.3)

where E1, σ1 and E2, σ2 are two consecutive points in the cross-section tabulation.

This interpolation method should only be used for E close to T. At higher energies, non-exponential behavior will normally begin to appear, and linear-linear interpolation is more suitable.

Next consider an energy distribution represented as a two-dimensional function of E and E f(E,E). Using a simple Cartesian interpolation, the function is represented by a series of tabulated functions f(E,E). The simple "graph paper" rules are used for interpolating for f(E) at each El. An additional interpolation table is given for interpolation between these values to get the result at E.

Distributions usually show ridges that cut diagonally across the lines of E and E. An interpolation scheme is required that merges smoothly between adjacent distributions without generating the spurious bumps often seen when interpolation along the Cartesian axes E and E is used.

The first non-Cartesian scheme allowed is the method of corresponding points. Given distributions for two adjacent incident energies, f(Ei,Eik) and f(Ej,Ejk), the interpolation takes place along the line joining the kth points in the two functions. When the E grids are different and the grid points are well chosen, this interpolation scheme is analogous to following the contours on a map. Of course, if the E grids are the same for Ei and Ej, this method is exactly equivalent to Cartesian interpolation. The method of corresponding points is selected by using INT=11-15, where the transformed values follow the interpolation laws INT=1-5, respectively.

The second non-Cartesian interpolation scheme allowed is unit-base interpolation. The spectra at Ei and Ei+1 are transformed onto a unit energy scale by dividing each secondary energy by the respective maximum energy. The interpolation is then performed as in the Cartesian method, and the resulting intermediate spectrum is expanded using the maximum energy obtained by interpolating between the end points of the original spectra. The unit-base option is selected by using INT=21-25, where the transformed values follow the interpolation laws INT=1-5, respectively.

Figure 0.1

Interpolation of a Tabulated One-dimensional Function

Illustrated for the Case NP=10, NR=3

0.6.2.1 Two Dimensional Interpolation Schemes

Three schemes are provided for two-dimensional interpolation: 1) simple Cartesian interpolation, wherein one simply interpolates the function values along constant lines of initial and final energy values, 2) a method called unit-base transform, and 3) the method or corresponding energies.

Figure 0.2

Interpolation between Two-Dimensional Panels

Initial Energy

E’

Secondary Energy

f

Ei

E

Ei+1

Consider Figure 0.2. Here E is the initial energy and there are panels at and . The panels describe the probabilities of scattering from these energies to other energies; e.g., and are generally probability distributions that will integrate to unity, when they are integrated over all . These panels will be presented using the usual tabular schemes for arrays, or may be given in the form of an analytic expression.

For the case of simple Cartesian interpolation, intermediate values are determined by interpolating along lines of constant and as noted earlier. Assume that all interpolation schemes are linear-linear in energy and that one wants to determine a value for a distribution at . The equation for this is:

(0.4)

An examination of the above figure illustrates the major problem with Cartesian interpolation; viz., that the panel at will have features from the lower panel at the low end, and from the upper panel at the high end. This, of course, is reasonable, but the resulting function will tend to have artificial peaks when the distributions shift as a function of energy, as is usually the case.

The unit-base transform was devised to try to reduce the non-physical characteristics of Cartesian interpolation. In this case, the data at the two panels surrounding E are transformed to a unit base where the new functions vary according to a variable that ranges from 0 to 1.

(0.5)

In this case, is the energy of the first sink energy in the panel at and is the last point. An exactly analogous equation is used to define the value of at and at . From here, the interpolation is made using an expression such as shown in Eq. (0.7) except that it is made at a constant value of . Special care must be taken to properly account for the integrals of the panel; i.e.,

(0.6)

which requires , or . The latter radical is the Jacobian that is required for the transformation to the unit-base space and is determined from Eq. (0.8). In other words, when Eqn. (0.7) is used, the interpolation is made at constant values of and the function values must be multiplied by the Jacobians for the respective panels.

Two things can be noted about unit-base transform interpolation: 1) if the end points of two panels are the same, unit-base transform is exactly equivalent to Cartesian interpolation, and 2) the same interpolations can be made without transforming to unit-base space and transforming to the energy space at E. The low energy value of the intermediate panel is simply

(0.7)

A similar expression gives the high energy of the intermediate panel, and simply substitutes the top energies for the two panels in place of the bottom energies.

(0.8)

Here we have assumed the upper panel has M points. The Jacobian that should be used with the value from the bottom panel is , and is determined from Eqn (0.12) shown below.

The secondary energy at that corresponds to an at is calculated from

(0.9)

Here we have dropped the arguments from the values at for clarity. An analogous expression determines the secondary energy at the upper panel, and also the Jacobian for this panel. When the two values are interpolated at these two secondary energies and multiplied by their respective Jacobians, the values are simply interpolated using Eqn. (0.7) (or other appropriate expression, if the interpolation scheme is not linear in energy).

The Method of Corresponding Energies (MCE) is a scheme that was designed to circumvent one of the major problems associated with the unit-base transform approach; viz., that the unit-base transform depends directly on the way the end points of the successive distributions are taken. The MCE approach splits the integrals of distributions into equal integral bins and then interpolates linearly between corresponding bins. (The limits of these bins are the “corresponding energies”.) This is a more physical approach than either Cartesian or unit-base transform interpolation and tends to emphasize the significant portions of the distributions. Perhaps a better way of saying this is that it tends to de-emphasize the insignificant portions of the distributions. For example, if 10 equal integral bins are to be used for the interpolation, the energies where the panels integrate to a tenth of the total integral are determined. Then the energies where the panels integrate to two-tenths of the total integral are determined, etc., until 10 sets of energy boundaries are defined for the bins in all panels. The interpolations between corresponding bins of successive panels are then performed using the unit-base transform approach.

It is important to note that that unit-base transform and MCE will require Jacobians to multiply the function values at successive panels, because a variable transformation is involved, while Cartesian interpolation is all done in real energy space, so that the unmodified function values are used.

0.7. General Description of Data Formats

An ENDF "tape" is built up from a small number of basic structures called "records," such as TPID, TEND, CONT, TAB1, and so on. These "records" normally consist of one or more 80- character FORTRAN records. It is also possible to use binary mode, where each of the basic structures is implemented as a FORTRAN logical record. The advantage of using these basic ENDF "records" is that a small library of utility subroutines can be used to read and write the records in a uniform way.

0.7.1. Structure of an ENDF Data Tape

The structure of an ENDF data tape (file) is illustrated schematically in Fig. 0.3. The tape contains a single record at the beginning that identifies the tape. The major subdivision between these records is by material. The data for a material is divided into files, and each file (MF number) contains the data for a certain class of information. A file is subdivided into sections, each one containing data for a particular reaction type (MT number). Finally, a section is divided into records. Every record on a tape contains three identification numbers: MAT, MF, and MT. These numbers are always in increasing numerical order, and the hierarchy is MAT, MF, MT. The end of a section, file, or material is signaled by special records called SEND, FEND, and MEND, respectively.

Figure 0.3

Structure of an ENDF data tape

Tape Ident

(TPID)

First file

First section

First record (HEAD)

First material

Second file

Second section

Second record

Material (MAT)

File (MF)

Section (MT)

Record (MR)

Last Material

Last file

Last section

Last record

Tape end

(TEND)

Material end (MEND)

File end (FEND

Section end (SEND)

0.7.2. Format Nomenclature

An attempt has been made to use an internally consistent notation based on the following rules.

a) Symbols starting with the letter I, J, K, L, M, or N are integers. All other symbols refer to floating-point (real numbers).
b) The letter I or a symbol starting with I refers to an interpolation code (see Section 0.6.2).
c) Letters J, K, L, M, or N when used alone are indices.
d) A symbol starting with M is a control number. Examples are MAT, MF, MT.
e) A symbol starting with L is a test number.

A symbol starting with N is a count of items.

All numbers are given in fields of 11 columns. In character mode, floating-point numbers should be entered in one of the following forms:

±1.234567±n

±1.23456±nn, where nn 38

depending on the size of the exponent. Both of these forms can be read by the "E11.0" format specification of FORTRAN. However, a special subroutine available to the NNDC must be used to output numbers in the above format. If evaluations are produced using numbers written by "1PE11.5" (that is, 1.2345E±nn), the numbers will be standardized into 6 or 7 digit form, but the real precision will remain at the 5 digit level.

0.7.3. Types of Records

All records on an ENDF tape are one of six possible types, denoted by TEXT, CONT, LIST, TAB1, TAB2, and INTG. The CONT record has six special cases called DIR, HEAD, SEND, FEND, MEND, and TEND. The TEXT record has the special case TPID. Every record contains the basic control numbers MAT, MF, and MT, as well as a sequence number. The definitions of the other fields in each record will depend on its usage as described below.

0.7.4. TEXT Records

This record is used either as the first entry on an ENDF tape (TPID), or to give the comments in File 1. It is indicated by the following shorthand notation:

[MAT, MF, MT/ HL] TEXT

where HL is 66 characters of text information. The TEXT record can be read with the following FORTRAN statements:

READ(LIB,10)HL,MAT,MF,MT,NS

10 FORMAT(A66,I4,I2,I3,I5)

where NS is the sequence number.13 For a normal TEXT record, MF = 1 and MT = 451. For a TPID record, MAT contains the tape number NTAPE, and MF and MT are both zero.

0.7.5. Control Records

0.7.5.1. CONT Records

The smallest possible record is a control (CONT) record. For convenience, a CONT record is denoted by

[MAT,MF,MT/C1,C2,L1,L2,N1,N2]CONT

The CONT record can be read with the following FORTRAN statements:

READ(LIB,10)C1,C2,L1,L2,N1,N2,MAT,MF,MT,NS

10 FORMAT(2E11.0,4I11,I4,I2,I3,I5).

The actual parameters stored in the six fields C1, C2, L1, L2, N1, and N2 will depend on the application for the CONT record.

0.7.5.2. HEAD Records

The HEAD record is the first in a section and has the same form as CONT, except that the C1 and C2 fields always contain ZA and AWR, respectively.

0.7.5.3. END Records

The SEND, FEND, MEND, and TEND records use only the three control integers, which signal the end of a section, file, material, or tape, respectively. In binary mode, the six standard fields are all zero. In character mode, the six are all zero as follows:

[MAT,MF,99999/ 0.0, 0.0, 0, 0, 0, 0] SEND14

[MAT, 0, 0/ 0.0, 0.0, 0, 0, 0, 0] FEND

[ 0, 0, 0/ 0.0, 0.0, 0, 0, 0, 0] MEND

13 Records are sequentially numbered within a given MAT/MF/MT.

14 The SEND record has the sequence number 99999.

[ -1, 0, 0/ 0.0, 0.0, 0, 0, 0, 0] TEND

0.7.5.4. DIR Records

The DIR records are described in more detail in Section 1.1.1. The only difference between a DIR record and a standard CONT record is that the first two fields in the DIR record are blank in character mode.

0.7.6. LIST Records

This type of record is used to list a series of numbers B1, B2, B3, etc. The values are given in an array B(n), and there are NPL of them. The shorthand notation for the LIST record is

[MAT,MF,MT/ C1, C2, L1, L2, NPL, N2/ Bn] LIST

The LIST record can be read with the following FORTRAN statements:

READ(LIB,10)C1,C2,L1,L2,NPL,N2,MAT,MF,MT,NS

10 FORMAT(2E11.0,4I11,I4,I2,I3,I5)

READ(LIB,20)(B(N),N=1,NPL)

20 FORMAT(6E11.0)

The maximum for NPL varies with use (see Appendix G).

0.7.7. TAB1 Records

These records are used for one-dimensional tabulated functions such as y(x). The data needed to specify a one-dimensional tabulated function are the interpolation tables NBT(N) and INT(N) for each of the NR ranges, and the NP tabulated pairs of x(n) and y(n). The shorthand representation is

[MAT,MF,MT/ C1, C2, L1, L2, NR, NP/xint/y(x)]TAB1

The TAB1 record can be read with the following FORTRAN statements:

READ(LIB,10)C1,C2,L1,L2,NR,NP,MAT,MF,MT,NS

10 FORMAT(2E11.0,4I11,I4,I2,I3,I5)

READ(LIB,20)(NBT(N),INT(N),N=1,NR)

20 FORMAT(6I11)

READ(LIB,30)(X(N),Y(N),N=1,NP)

30 FORMAT(6E11.0)

The limits on NR and NP vary with use (see Appendix G). The limits must be strictly observed in primary evaluations in order to protect processing codes that use the simple binary format. However, these limits can be relaxed in derived libraries in which resonance parameters have been converted into detailed tabulations of cross section versus energy. Such derived libraries can be written in character mode or a non-standard blocked-binary mode.

0.7.8. TAB2 Records

The last record type is the TAB2 record, which is used to control the tabulation of a two-dimensional function y(x,z). It specifies how many values of z are to be given and how to interpolate between the successive values of z. Tabulated values of yl(x) at each value of zl are given in TAB1 or LIST records following the TAB2 record, with the appropriate value of z in the field designated as C2. The shorthand notation for TAB2 is

[MAT,MF,MT/ C1, C2, L1, L2, NR, NZ/ Zint]TAB2,

The TAB2 record can be read with the following FORTRAN statements:

READ(LIB,10)C1,C2,L1,L2,NR,NZ,MAT,MF,MT,NS

10 FORMAT(2E11.0,4I11,I4,I2,I3,I5)

READ(LIB,20)(NBT(N),INT(N),N=1,NR)

20 FORMAT(6I11)

For example, a TAB2 record is used in specifying angular distribution data in File 4. In this case, NZ in the TAB2 record specifies the number of incident energies at which angular distributions are given. Each distribution is given in a LIST or TAB1 record.

0.7.9 INTG records

INTG, or INTeGer, records are used to store a correlation matrix in integer format. The shorthand notation is

[MAT, MF, MT / II, JJ, KIJ ] INTG

where II and JJ are position locators, and KIJ is an array of dimension 18.

The INTG record can be read with the following FORTRAN statements:

DIMENSION KIJ(18)

READ (LIB,10) II, JJ, KIJ, MAT, MF, MT, NS

10 FORMAT (2I5, 1X, 18I3, 1X, I4, I2, I3, I5)

See File 32, LCOMP = 2, for details regarding the use of this format.

0.8. ENDF Documentation

1. BNL 8381, ENDF - Evaluated Nuclear Data File Description and Specifications, January 1965, H.C. Honeck.
2. BNL 50066 (ENDF 102), ENDF/B - Specifications for an Evaluated Nuclear Data File for Reactor Applications, May 1966, H.C. Honeck. Revised July 1967 by S. Pearlstein.
3. BNL 50274 (ENDF 102), Vol. I - Data Formats and Procedures for the ENDF Neutron Cross Section Library, October 1970, M.K. Drake, Editor.
4. LA 4549 (ENDF 102), Vol. II - ENDF Formats and Procedures for Photon Production and Interaction Data, October 1970, D.J. Dudziak.
5. BNL-NCS-50496 (ENDF 102), ENDF102 Data Formats and Procedures for the Evaluated Nuclear Data File, ENDF, October 1975, Revised by D. Garber, C. Dunford, and S. Pearlstein.
6. ORNL/TM-5938 (ENDF-249), The Data Covariance Files for ENDF/B-V, July 1977, F. Perey.
7. BNL-NCS-50496 (ENDF 102), Second Edition, ENDF-102 Data Formats and Procedures for the Evaluated Nuclear Data File, ENDF/B-V, October 1979, Edited by R. Kinsey. Revised by B.A. Magurno, November 1983.
8. BNL-NCS-28949 (Supplement ENDF 102), Second Edition, Supplement to the ENDF/B-V Formats and Procedures Manual for Using ENDF/B-IV Data, November 1980, S. Pearlstein.
9. BNL-NCS-44945 (ENDF-102), Revision 10/91, ENDF-102 Data Formats and Procedures for the Evaluated Nuclear Data File ENDF-6, October 1991, Edited by P. F. Rose and C. L. Dunford.

1. FILE 1. GENERAL INFORMATION

File 1 is the first part of any set of evaluated cross-section data for a material. Each material must have a File 1 that contains at least one section. This required section provides a brief documentation of how the data were evaluated and a directory that summarizes the files and sections contained in the material. In the case of fissionable materials, File 1 may contain up to four additional sections giving fission neutron yields and energy release information. Each section has been assigned an MT number (see below), and the sections are arranged in order of increasing MT number. A section always starts with a HEAD record and ends with a SEND record. The end of File 1 (and all other files) is indicated by a FEND record. These record types are defined in detail in Section 0.7.

1.1. Descriptive Data and Directory (MT=451)

This section is always the first section of any material and has two parts:

1) a brief documentation of the cross-section data, and
2) a directory of the files and sections used for this material.

In the first part, a brief description of the evaluated data set is given. This information should include the significant experimental results used to obtain the evaluated data, descriptions of any nuclear models used, a clear specification of all the MT numbers defined to identify reactions, the history of the evaluation, and references. The descriptive information is given as a series of records, each record containing up to 66 characters.

The first three records of the descriptive information contain a standardized presentation of information on the material, projectile, evaluators, and modification status. The following quantities are defined for MF=1, MT=451:

ZA,AWR

Standard material charge and mass parameters.

LRP

Indicates whether resolved and/or unresolved resonance parameters are given in File 2:

LRP=1, no File 2 is given (not allowed for incident neutrons);

LRP=0, no resonance parameter data are given, but a File 2 is present containing the effective scattering radius;

LRP=1, resolved and/or unresolved parameter data are given in File 2 and cross sections computed from them must be added1 to background cross sections given in File 3;

LRP=2, parameters are given in File 2, but cross sections derived from them are not to be added to the cross sections in File 3. The option LRP=2 is to be used for derived files only.

LFI

Indicates whether this material fissions:

LFI=0, this material does not fission;

LFI=1, this material fissions.

NLIB

Library identifier: NLIB= 0 for ENDF/B.

Additional values have been assigned to identify other libraries using ENDF format.

NMOD

Modification number for this material:

NMOD=0, evaluation converted from a previous version;

NMOD=1, new or revised evaluation for the current library version;

NMOD2, for successive modifications.

ELIS

Excitation energy of the target nucleus relative to 0.0 for the ground state.

STA

Target stability flag:

STA=0, stable nucleus;

STA=1 unstable nucleus. If the target is unstable, radioactive decay data should be given in the decay data sub-library (NSUB=4).

LIS

State number of the target nucleus. The ground state is indicated by LIS=0.

LISO

Isomeric state number. The ground state is indicated by LISO=0. LIS is greater than or equal to LISO.

NFOR

Library format.

NFOR=6 for all libraries prepared according to the specifications given in this manual.

AWI

Mass of the projectile in neutron units. For incident photons or decay data sub-libraries, use AWI=0.0.

EMAX

Upper limit of energy range for evaluation.

LREL

Release number.

NSUB

Sub-library number. See Section 0.4 for a description of sub-libraries.

NVER

Library version number; for example, NVER=6 for version VI.

TEMP

Target temperature (Kelvin) for data that have been generated by Doppler broadening. For derived data only; use TEMP=0.0 for all primary evaluations.

LDRV

Special derived material flag that distinguishes between different evaluations with the same material keys (i.e., MAT, NMOD, NSUB): LDRV=0, primary evaluation:

LDRV1, special derived evaluation (for example, a dosimetry evaluation using sections (MT) extracted from the primary evaluation).

NWD

Number of records used to describe the data set for this material. Each record can contain up to 66 characters.

NXC

Number of records in the directory for this material. Each section (MT) in the material has a corresponding line in the directory that contains MF, MT, NC, and MOD. NC is a count of the number of records in the section (not including SEND), and MOD is the modification flag (see below).

ZSYNAM

Character representation of the material charge, chemical symbol, atomic mass number, and metastable state in the form Z-cc-AM with

Z, right justified in col. 1 to 3,

- (hyphen) in col. 4,

two-character chemical name left justified in col. 5 and 6,

- (hyphen) in col. 7,

A, right justified in col. 8 to 10 or blank,

M for the indication of a metastable state in col. 11,

for example, 94-PU-239, 1-H - 2, etc.

ALAB

Mnemonic for the originating laboratory(s) left adjusted in col. 12-22.

EDATE

Date of evaluation given in the form "EVAL-DEC74" in col. 23-32.

AUTH

Author(s) name(s) left adjusted in col. 34-66.

REF

Primary reference for the evaluation left adjusted in col. 2-22.

DDATE

Original distribution date given in the form "DIST-DEC74" in col. 23-32.

RDATE

Date and number of the last revision to this evaluation in col. 34-43 in the form "REV2-DEC74”, where "2" is the revision number IREV; IREV is computer retrievable.

ENDATE

Master File entry date in the form yyyymmdd right adjusted in col. 56-63. The Master File entry date will be assigned by NNDC for ENDF/B-VI.

HSUB

Identifier for the library contained on three successive records.

The first record contains four dashes starting in col. 1, directly followed by the library type (NLIB) and version (NVER). For example,

"---- ENDF/B-VI", followed by MATERIAL XXXX starting in col. 23 where XXXX is the MAT number, and REVISION 2 (starting in col. 45 only if required) where "2" is the revision number IREV.

The second record contains five dashes starting in col. 1, and followed by the sub-library identifier (see Table 0.1). For example,

"----- DECAY DATA,"

"----- PHOTO-ATOMIC INTERACTION DATA," or

"----- INCIDENT NEUTRON DATA."

The third record contains six dashes starting in col. 1 and followed by ENDF-6 where "6" is the library format type (NFOR).

Note: the three HSUB records can be generated by the utility program, STANEF.

MFn

MF of the nth section.

MTn

MT of the nth section.

NCn

Number of records in the nth section. This count does not include the SEND record.

MODn

Modification indicator for the nth section. The value of MODn is equal to NMOD if the corresponding section was changed in this revision. MODn must always be less than or equal to NMOD.

1 In the unresolved region, it is also possible to compute a self-shielding factor from File 2 and multiply it by a complete unshielded cross section in File 3.

1.1.1. Formats

The structure of this section is

[MAT, 1,451/ ZA, AWR, LRP, LFI, NLIB, NMOD]HEAD

[MAT, 1,451/ ELIS, STA, LIS, LISO, 0, NFOR]CONT

[MAT, 1,451/ AWI, EMAX, LREL, 0, NSUB, NVER]CONT

[MAT, 1,451/ TEMP, 0.0, LDRV, 0, NWD, NXC]CONT

[MAT, 1,451/ZSYMAM, ALAB, EDATE, AUTH ]TEXT

[MAT, 1,451/ REF, DDATE, RDATE, ENDATE ]TEXT

[MAT, 1,451/ HSUB ]TEXT

------------------------------------

continue for the rest of the NWD descriptive records

------------------------------------

[MAT, 1,451/ blank, blank, MF1, MT1, NC1, MOD1]CONT

[MAT, 1,451/ blank, blank, MF2, MT2, NC2, MOD2]CONT

------------------------------------

------------------------------------

[MAT, 1,451/ blank, blank, MFNXC, MTNXC,NCNXC,MODNXC]CONT

[MAT, 1, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

1.1.2. Procedures

Note that the parameters NLIB, NVER, NSUB, MAT, NMOD, LDRV, and sometimes TEMP define a unique set of "keys" that identifies a particular evaluation or "material" in the ENDF system. These keys can be used to access materials in a formal data base management system if desired.

The flag LRP indicates whether resolved and/or unresolved resonance parameter data are to be found in File 2 (Resonance Parameters) and how these data are to be used with File 3 to compute the net cross section. For incident neutrons, every material will have a File 2. If LRP=0, the file contains only the effective scattering radius; the potential cross section corresponding to this scattering radius has already been included in the File 3 cross sections. If LRP=1, File 2 contains resolved and/or unresolved resonance parameters. Cross sections or self-shielding factors computed from these parameters are to be combined with any cross sections found in File 3 to obtain the correct net cross section. For other sub-libraries (decay data, incident photons, incident charged particles, fission product yields), File 2 can be omitted (use LRP=-1). A number of processing codes exist which reconstruct resonance-region cross sections from the parameters in File 2 and output the results in ENDF format. Such a code can set LRP=2 and copy the original File 2 to its output ENDF tape. Other processing codes using such a tape will know that resonance reconstruction has already been performed, but the codes will still have easy access to the resonance parameters if needed. The LRP=2 option is not allowed in primary evaluations.

The flag LFI indicates that this material fissions in the context of the present sub-library. In this case, a section specifying the total number of neutrons emitted per fission,⎯ν(E), must be given as MF=1, MT=452. Sections may also be given that specify the number of delayed neutrons per fission (MT=455) and the number of prompt neutrons per fission (MT=456), and that specify the components of energy release in fission (MT=458).

The flag LDRV indicates that this material was derived in some way from another evaluation; for example, it could represent an activation reaction extracted from a more complete evaluation, it could be part of a gas production library containing production cross sections computed from more fundamental reactions, it could represent a reconstructed library with resonance parameters expanded into detailed point-wise cross sections, and so on.

The data in the descriptive section must be given for every material. The first three records are used to construct titles for listings, plots, etc., and the format should be followed closely. The remaining records give a verbal description of the evaluated data set for the material. The description should mention the important experimental results upon which the recommended cross sections are based, the evaluation procedures and nuclear models used, a brief history and origin of the evaluation, important limitations of the data set, estimated uncertainties and covariances, references, and any other remarks that will assist the user in understanding the data. For incident neutron evaluations, the 2200 m/s cross sections contained in the data should be tabulated, along with the infinite dilution resonance integrals for capture and fission (if applicable). For charged-particle and high-energy reactions, the meaning of each MT should be carefully explained using the notation of Section 0.5.

1.2. Number of Neutrons per Fission,⎯ν, (MT=452)

If the material fissions (LFI=1), then a section specifying the average total number of neutrons per fission,⎯ν (MT=452), must be given. This format applies to both particle induced

(1.1)

and spontaneous fission, each in its designated sub-library. Values of⎯ν may be tabulated as a function of energy or coefficients provided for the following polynomial expansion of (E),

where ⎯ν(E) = the average total (prompt plus delayed) number of neutrons per fission produced by neutrons of incident energy E(eV),

Cn = the nth coefficient, and

NC = the number of terms in the polynomial.

MT=452 for an energy-dependent neutron multiplicity cannot be represented by a polynomial expansion when MT=455 and MT=456 are utilized in the file.

1.2.1. Formats

The structure of this section depends on whether values of⎯ν (E) are tabulated as a function of energy or represented by a polynomial. The following quantities are defined:

LNU

Test that indicates what representation of (E) has been used:

LNU=1, polynomial representation has been used;

LNU=2, tabulated representation.

NC

Count of the number of terms used in the polynomial expansion. (NC4).

Cn

Coefficients of the polynomial. There are NC coefficients given.

NR

Number of interpolation ranges used to tabulate values of ⎯ν(E). (See 0.6.2.)

NP

Total number of energy points used to tabulate⎯ν(E).

Eint

Interpolation scheme (see 0.6.2 for details).

⎯ν(E)

Average number of neutrons per fission.

If LNU=1, the structure of the section is

[MAT, 1, 452/ ZA, AWR, 0, LNU, 0, 0]HEAD (LNU=1)

[MAT, 1, 452/ 0.0, 0.0, 0, 0, NC, 0/ C1, C2, ...CNC]LIST

[MAT, 1,99999/ 0.0, 0.0, 0, 0, 0, 0]SEND

If LNU=2, the structure of the section is

[MAT, 1, 452/ ZA, AWR, 0, LNU, 0, 0]HEAD (LNU=2)

[MAT, 1, 452/ 0.0, 0.0, 0, 0, NR, NP/Eint /⎯ν(E)]TAB1

[MAT, 1,99999/ 0.0, 0.0, 0, 0, 0, 0]SEND

1.2.2. Procedures

If a polynomial representation (LNU=1) has been used to specify⎯ν (E), this representation is valid over any range in which the fission cross section is specified (as given in Files 2 and 3). When using a polynomial to fit⎯ν (E), the fit shall be limited to a third-degree polynomial (NC4). If such a fit does not reproduce the recommended values of⎯ν (E), a tabulated form (LNU=2) should be used.

If tabulated values of⎯ν (E) are specified (LNU=2), then pairs of energy-⎯ν values are given. Values of⎯ν (E) should be given that cover any energy range in which the fission cross section is given in File 2 and/or File 3.

The values of⎯ν (E) given in this section are for the average total number of neutrons produced per fission event. When another section (MT=455) that specifies the delayed neutrons from fission is given, the average number of delayed neutrons per fission (⎯νd) must be added to ⎯νp given in section MT=456 and included in the value of⎯ν(E) given in this section (MT=452). In this case, only LNU=2 is allowed for MT=452.

For spontaneous fission, the polynomial representation (LNU=1) is used with NC=1 and Cl = ⎯νtotal. There is no energy dependence.

1.3. Delayed Neutron Data, ⎯νd, (MT=455)

This section describes the delayed neutrons resulting from either particle induced or spontaneous fission. The average total number of delayed neutron precursors emitted per fission,⎯νd, is given, along with the decay constants, λi, for each precursor family. The fraction of⎯νd generated for each family is given in File 5 (section 5 of this report). The energy distributions of the neutrons associated with each precursor family are also given in File 5.

For particle-induced fission, the total number of delayed neutrons is given as a function of energy in tabulated form (LNU=2). The energy dependence is specified by tabulating⎯νd(E) at a series of neutron energies using the same format as for MT=452. For spontaneous fission LNU=1 is used with NC=1 and C1=⎯νd as for MT=452.

The total number of delayed neutron precursors emitted per fission event, at incident energy E, is given in this file and is defined as the sum of the number of neutrons emitted for each of the precursor families,

where NNF is the number of precursor families. The fraction of the total, Pi(E), emitted for each family is given in File 5 (see section 5) and is defined as

1.3.1. Formats

The following quantities are defined.

LNU

Test indicating which representation is used:

LNU=1 means that polynomial expansion is used;

LNU=2 means that a tabulated representation is used.

LDG

Flag indicating energy dependence of delayed-group constants:

LDG=0 means that decay constants are energy independent;

LDG=1 means that decay constants are energy independent.

NE

Number of energies at which the delayed-group constants are given.

NC

Number of terms in the polynomial expansion. (NC4).

NR

Number of interpolation ranges used. (NR20).

NP

Total number of energy points used in the tabulation of ν(E)

Eint

Interpolation scheme (See Section 0.6.2)

(E)

Total average number of delayed neutrons formed per fission event.

NNF

Number of precursor families considered.

λi (E)

Decay constant (sec-1) for the ith precursor. (May be constant)

αi (E)

Delayed-group abundances.

The structure when values of⎯νd are tabulated (LNU=2) and the delayed-group constants are energy independent (LDG=0) is:

[MAT, 1,455/ ZA, AWR, LDG, LNU, 0, 0]HEAD (LDG=0, LNU=2)

[MAT, 1,455/ 0.0, 0.0, 0, 0, NNF, 0/λ1,λ2,...λNNF]LIST

[MAT, 1,455/ 0.0, 0.0, 0, 0, NR, NP/ Eint /⎯νd(E)]TAB1

[MAT, 1, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

The structure when values of⎯νd are tabulated (LNU=2) and the delayed-group constants are energy dependent (LDG=1) is:

[MAT, 1,455/ ZA, AWR, LDG, LNU, 0, 0]HEAD (LDG=1, LNU=2)

[MAT, 1,455/ 0.0, 0.0, 0, 0, NR, NE/ Eint]TAB2

[MAT, 1,455/ 0.0, E1, 0, 0, NNF*2, 0/

λ1(E1), α1(E1), λ2(E1), α2(E1), λ3(E1), α3(E1),

...λNNF(E1), αNNF(E1)]LIST

------------------------------------

------------------------------------

[MAT, 1,455/ 0.0, ENE, 0, 0, NNF*2, 0/

λ1(ENE), α1(ENE), λ2(ENE), α2(ENE), λ3(ENE), α3(ENE),

...λNNF(ENE), αNNF(ENE)]LIST

[MAT, 1,455/ 0.0, 0.0, 0, 0, NR, NP/ Eint /⎯νd(E)]TAB1

[MAT, 1, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

If LNU=1 (spontaneous fission) and the delayed-group constants are energy independent (LDG=0), the structure of the section is:

[MAT, 1,455/ ZA, AWR, LDG, LNU, 0, 0]HEAD (LDG=0, LNU=1)

[MAT, 1,455/ 0.0, 0.0, 0, 0, NNF, 0/λ1,λ2,...λNNF]LIST

[MAT, 1,455/ 0.0, 0.0, 0, 0, 1, 0/⎯νd]LIST

[MAT, 1, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

If LNU=1 (spontaneous fission) and the delayed-group constants are energy dependent (LDG=1), the structure of the section is:

[MAT, 1,455/ ZA, AWR, LDG, LNU, 0, 0]HEAD (LDG=1, LNU=1)

[MAT, 1,455/ 0.0, 0.0, 0, 0, NR, Ne/ Eint]TAB2

[MAT, 1,455/ 0.0, E1, 0, 0, NNF*2, 0/

λ1(E1), α1(E1), λ2(E1), α2(E1), λ3(E1), α3(E1),

...λNNF(E1), αNNF(E1)]LIST

------------------------------------

------------------------------------

[MAT, 1,455/ 0.0, ENE, 0, 0, NNF*2, 0/

λ1(ENE), α1(ENE), λ2(ENE), α2(ENE), λ3(ENE), α3(ENE),

...λNNF(ENE), αNNF(ENE)]LIST

[MAT, 1,455/ 0.0, 0.0, 0, 0, 1, 0/⎯νd]LIST

[MAT, 1, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

1.3.2. Procedures

When tabulated values of⎯νd(E) are specified, as is required for particle-induced fission by Section 1.2., they should be given for the same energy range as that used to specify the fission cross section.

The probability of producing the precursors for each family and the energy distributions of neutrons produced by each precursor family are given in File 5 (section 5 of this report). It is extremely important that the same precursor families be given in File 5 as are given in File 1 (MT=455) with the same abundances, and the ordering of the families should be the same in both files. It is recommended that the families be ordered by decreasing half-lives (λ1<λ2<…<λNNF).

For spontaneous fission, the polynomial form (LNU=1) is used with only one term (NC=1, C1=⎯νd).

If MT=455 is used, then MT=456 must also be used as well as MT=452.

1.4. Number of Prompt Neutrons per Fission,⎯νp, (MT=456)

If the material fissions (LFI=1), a section specifying the average number of prompt neutrons per fission,⎯νp, (MT=456) can be given using formats identical to MT=452. For particle-induced fission,⎯νp is given as a function of energy. The prompt⎯ν for spontaneous fission can also be given using MT=456, but there is no energy dependence.

1.4.1. Formats

The following quantities are defined:

LNU

Indicates what representation of⎯νp(E) has been used:

LNU=1, polynomial representation has been used;

LNU=2, tabulated representation.

NC

Count of the number of terms used in the polynomial expansion. (NC4)

NR

Number of interpolation ranges used to tabulate values of⎯νp(E). (See 0.7.7.)

NP

Total number of energy points used to tabulate⎯νp(E).

Eint

Interpolation scheme (see section 0.6.2.)

⎯νp(E)

Average number of prompt neutrons per fission.

If LNU=2, (tabulated values of ), the structure of the section is:

[MAT, 1,456/ ZA, AWR, 0, LNU, 0, 0] HEAD (LNU=2)

[MAT, 1,456/ 0.0, 0.0, 0, 0, NR, NP/Eint/⎯νp(E)]TAB1

[MAT, 1, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

If LNU=1 (spontaneous fission) the structure of the section is:

[MAT, 1,456/ ZA, AWR, 0, LNU, 0, 0]HEAD (LNU=1)

[MAT, 1,456/ 0.0, 0.0, 0, 0, 1, 0/⎯νp]LIST

[MAT, 1, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

1.4.2. Procedures

If tabulated values of⎯νp(E) are specified (LNU=2), then pairs of energy- values are given. Values of ⎯νp(E) should be given that cover any energy range in which the fission cross section is given in File 2 and/or File 3. The values of ⎯νp(E) given in this section are for the average number of prompt neutrons produced per fission event. The energy independent⎯νp for spontaneous fission is given using LNU=1 with NC=1 and C1= as described for MT = 452.

If MT=456 is specified, then MT=455 must also be specified as well as MT=452.

1.5. Components of Energy Release Due to Fission (MT=458)

The energy released in fission is carried by fission fragments, neutrons, gammas, betas (+ and –), and neutrinos and anti-neutrinos. The term fragments includes all charged particles that are emitted promptly, since for energy-deposition calculations, all such particles have short ranges and are usually considered to lose their energy locally. Neutrons and gammas transport their energy elsewhere and need to be considered separately. In addition, some gammas and neutrons are delayed, and in a shut-down assembly, one needs to know the amount of energy tied up in these particles and the rate at which it is released from the metastable nuclides or precursors. The neutrino energy is lost completely in most applications, but is part of the Q-value. As far as the betas are concerned, prompt betas, being charged, deposit their energy locally with the fragments, and their prompt energies are correctly included with the fragment energies.

ET

Sum of all the partial energies that follow. This sum is the total energy release per fission and equals the Q value.

EFR

Kinetic energy of the fragments.

ENP

Kinetic energy of the "prompt" fission neutrons.

END

Kinetic energy of the delayed fission neutrons.

EGP

Total energy released by the emission of "prompt" γ rays.

EGD

Total energy released by the emission of delayed γ rays.

EB

Total energy released by delayed β's.

ENU

Energy carried away by the neutrinos.

ER

Total energy less the energy of the neutrinos (ET - ENU); equal to the pseudo-Q in File 3 for MT=18.

All of these energies are given for an incident energy of zero.2

(1.2)

2 Taken from R. Sher and C. Beck, Fission Energy Release for 16 Fissioning Nuclides, EPRI-NP-1771 (1981).

where Ei is any of the energy release components;

Ei(0) is the value at Einc = 0; Einc = 0 is fictitious and represents an artifice by which it is possible to recover the values at any Einc.

Ei(Einc) is the value at incident energy Einc.

The δEi's are given by the following:

1.5.1. Formats

The structure of this section always starts with a HEAD record and end with a SEND record. The section contains no subsections and only one LIST record.

The structure of a section is:

[MAT, 1,458/ ZA, AWR, 0, 0, 0, 0]HEAD

[MAT, 1,458/ 0.0, 0.0, 0, 0, 18, 9/

EFR, EFR, ENP, ENP, END, END,

EGP, EGP, EGD, EGD, EB, EB,

ENU, ENU, ER, ER, ET, ET]LIST

[MAT, 1, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

where the 's allow the error estimates on the quantities listed above.

1.5.2. Procedures

This section should be used for fertile and fissile isotopes only.

Consistency should be maintained between the Q values in File 3, the energies calculated from File 5 and 15 and the energies listed in File 1. Note that ER = the pseudo-Q for fission (MT=18) in File 3.

Other components are not so readily determined or checked. The procedure should be that File 5 and File 15 data take precedent, whenever available. That is, "prompt" fission neutron energy calculated from File 5 spectra from MT=18 should be used in File 1; the same holds true for the delayed neutron spectra given in File 5, MT=455. The "prompt" gamma energy calculated from File 15 (MT=18 for fission) should be input into File 1, that is the prompt gammas due to the fission process.

These quantities should be calculated at the lowest energy given in the Files for MT=18 except for fissile isotopes for which the thermal spectra should be used. For fertile materials, the spectrum given at threshold would be appropriate. Note that the File 5 spectra for MT=18 should be used with⎯ν prompt (not⎯ν total) for the fission neutrons. MT=455 in File 5 contains the delayed fission neutron spectra.

In many reactor applications, time dependent energy deposition rates are required rather than the components of the total energy per fission which are the values given in this MT. Time-dependent energy deposition parameters can be obtained from the six-group spectra in File 5 (MT=455) for delayed neutrons. Codes such as CINDER, RIBD, and ORIGEN must be used, however, to obtain more detailed information on the delayed neutrons and all time-dependent parameters for the betas and the gammas due to the fission process.

The time-integrated energies for delayed neutrons, delayed gammas, and delayed betas as calculated from the codes listed above may not always agree with the energy components given in File 1. The File 1 components must sum to ET (the total energy released per fission).

In heating calculations, the energy released in all nuclear reactions besides fission, principally the gamma-energy released in neutron radiative capture, enters analogously to the various fission energy components. Thus the (n,γ) energy-release would be equal to the Q-value in File 3, MT=102, of the capturing nuclide. The capture gammas can be prompt or delayed, if branching to isomeric states is involved, and this is relevant to various fission- and burnup-product calculations. The "sensible energy" in a heating calculation is the sum of ER, defined previously, and the energy released in all other reactions.

2. FILE 2. RESONANCE PARAMETERS

2.1. General Description

The primary function of File 2 is to contain data for both resolved and unresolved resonance parameters. It has only one section, with the reaction type number MT=151. A File 2 is required for incident-neutron evaluations, but it may be omitted in other cases. The use of File 2 is controlled by the parameter LRP (see section 1.1):

LRP=-1 No File 2 is given. Not allowed for incident neutrons.

LRP= 0 No resonance parameters are given except for the scattering radius AP.

AP is included for the convenience of users who need an estimate of the potential scattering cross section. It is not used to calculate a contribution to the scattering cross section, which in this case is represented entirely in File 3.

LRP= 1 Resonance contributions for the total, elastic, fission, and radiative capture cross sections are to be computed from the resonance parameters and added to the corresponding cross sections in File 31.

The File 2 resonance contributions should also be added to any lumped reactions included in File 3. For SLBW and MLBW, any other competing reactions in the resonance range must be given in their entirety in File 3 and included in the background for the total cross section. The effects of the competing reactions on the resonance reactions are included using a single competitive width, Γx. This width is given explicitly in the unresolved resonance region, and implicitly in the resolved region. In the latter region, it is permissible for the total width to exceed the sum of the neutron, radiative capture, and fission widths. The difference is interpreted as the competitive width:

For the Reich-Moore or Adler-Adler formalisms competitive reactions are not used.

LRP= 2 Resonance parameters are given in File 2 but are not to be used in calculating cross sections, which are assumed to be represented completely in File 3. Used for certain derived libraries only.

1 In the unresolved resonance region, the evaluator may, optionally, specify a different procedure, which uses the unresolved resonance parameters in File 2 solely for the purpose of computing an energy-dependent self-shielding factor. This option is governed by a flag, LSSF, defined in Section 2.3.1, and discussed in Section 2.4.21. When this option is specified, File 3 is used to specify the entire infinitely-dilute cross section, and the function of File 2 is to specify the calculation of self-shielding factors for shielded pointwise or multigroup values.

The resonance parameters for a material are obtained by specifying the parameters for each isotope in the material. The data for the various isotopes are ordered by increasing ZAI values (charge-isotopic mass number). The resonance data for each isotope may be divided into several incident neutron energy ranges, given in order of increasing energy. The energy ranges for an isotope should not overlap; each may contain a different representation of the cross sections.

In addition to these parameterized resonance ranges, the full energy range may contain two additional non-resonance ranges, also non-overlapping. Comments on these ranges follow:

1. The low energy region (LER) is one in which the cross sections are tabulated as smooth functions of energy. Doppler effects must be small enough so that the values are essentially zero degrees Kelvin. For light elements, i.e., those whose natural widths far exceed their Doppler widths and hence undergo negligible broadening, the entire energy range can often be represented in this way. For heavier materials, this region can sometimes be used below the lowest resolved resonances. With a good multilevel resonance fit, the LER can often be omitted entirely, and this is preferred. An important procedure for the LER is described in Section 2.4.6.4.

2. The resolved resonance region (RRR) is one in which resonance parameters for individual resonances are given. Usually this implies that experimental resolution is good enough to "see" the resonances, and to determine their parameters by area or shape analysis, but an evaluator may choose to supply fictitious resolved parameters if he so desires. If the evaluator does this, the resonances must have physically-allowed quantum numbers, and be in accord with the statistics of level densities (Appendix D, Section D.2.2). A File 3 background may be given. The essential point is that resonance self-shielding can be accounted for by the user for each resonance individually.

3. The unresolved resonance region (URR) is that region in which the resonances still do not actually overlap, so that self-shielding is still important, but experimental resolution is inadequate to determine the parameters of individual resonances. In this situation, self-shielding must be handled on a statistical basis. A File 3 may be given. The interpretation of this cross section depends on the flag LSSF (see Sections 2.3.1 and 2.4.21). It may be interpreted either as a partial background cross section, to be added to the File 2 contribution, as in the resolved resonance region or it may be interpreted as the entire dilute cross section, in which case File 2 is to be used solely to specify the self-shielding appropriate to this energy region. It is important to choose the boundary between the RRR and the URR so that the statistical assumptions underlying the unresolved resonance treatments are valid. This problem is discussed further in Section 2.4.

4. The high-energy region (HER) starts at still higher energies where the resonances overlap and the cross sections smooth out, subject only to Ericson fluctuations. The boundary between the URR and HER should be chosen so that self-shielding effects are small in the HER.

File 3 may contain "background cross sections" in the resonance ranges resulting from inadequacies in the resonance representation (e.g., SLBW), the effects of resonances outside the energy range, the average effects of missed resonances, or competing cross sections. If these background cross sections are nonzero, there must be double energy points in File 3 corresponding to each resonance range boundary (except 10-5eV). See Section 2.4 for a more complete discussion of backgrounds.

Several representations are allowed for specifying resolved resonance parameters. The flag, LRF, indicates the representation used for a particular energy range:

LRF=1 Single-level Breit-Wigner; (no resonance-resonance interference; one single-channel inelastic competitive reaction is allowed).

LRF=2 Multilevel Breit-Wigner (resonance-resonance interference effects are included in the elastic scattering and total cross sections; one single-channel inelastic competitive reaction is allowed).

LRF=3 Reich-Moore (multilevel multichannel R-matrix; no competitive reactions allowed).

It is possible to define partial widths and with two different values of the channel spin, as is required when both the target spin and the orbital angular momentum are greater than zero. This is accomplished by setting the resonance spin parameter AJ to a positive value for the larger channel spin (s = I + 1/2), and negative for the smaller channel spin (s = I - 1/2). (See definition of AJ in 2.2.1.) Older ENDF files have not used this feature, but instead have only positive AJ; in this case, all resonances of a given l,J are assumed to have the same channel spin.

For a given resonance, the only rigorously conserved quantities are J (total angular momentum) and π (total parity). Nevertheless, this format assumes that both l (orbital angular momentum) and s (channel spin) are also conserved quantities.

LRF=4 Adler-Adler (level-level and channel-channel interference effects are included in all cross sections via "effective" resonance parameters; usually applied to low-energy fissionable materials; no competitive reactions).

LRF=5 This option is no longer available.

LRF=6 This option is no longer available.

LRF=7 R-Matrix Limited format, which contains all the generality of LRF=3 plus unlimited numbers and types of channels.

Preferred formalisms for evaluation are discussed in Section 2.4.17. Further discussion of the above formalisms is contained in the Procedures Section 2.4.

Each resonance energy range contains a flag, LRU that indicates whether it contains resolved or unresolved resonance parameters. LRU=1 means resolved, LRU=2 means unresolved.

Only one representation is allowed for the unresolved resonance parameters, namely average single-level Breit-Wigner. However, several options are permitted, designated by the flag LRF. With the first option, LRF=1, only the average fission width is allowed to vary as a function of incident neutron energy. The second option, LRF=2, allows the following average parameters to vary: level spacing, fission width, reduced neutron width, radiation width, and a width for the sum of all competitive reactions.

The data formats for the various resonance parameter representations are given in Sections 2.2.1 (resolved) and 2.3.1 (unresolved). Formulae for calculating cross sections from the various formalisms are given in Appendix D.

The following quantities have definitions that are the same for all resonance parameter representations:

NIS

Number of isotopes in the material (NIS<10).

ZAI

(Z,A) designation for an isotope.

NER

Number of resonance energy ranges for this isotope.

ABN

Abundance of an isotope in the material. This is a number fraction, not a weight fraction, nor a percent.

LFW

Flag indicating whether average fission widths are given in the unresolved resonance region for this isotope:

LFW=0, average fission widths are not given;

LFW=1, average fission widths are given.

NER

Number of resonance energy ranges for isotope.

EL

Lower limit for an energy range2.

EH

Upper limit for an energy range3.

LRU

Flag indicating whether this energy range contains data for resolved or unresolved resonance parameters:

LRU=0, only the scattering radius is given (LRF=0, NLS=0, LFW=0 is required with this option);

LRU=1, resolved resonance parameters are given.

LRU=2, unresolved resonance parameters are given.

LRF

Flag indicating which representation has been used for the energy range. The definition of LRF depends on the value of LRU:

If LRU=1(resolved parameters), then

LRF=1, single-level Breit-Wigner (SLBW);

LRF=2, multilevel Breit-Wigner (MLBW);

LRF=3, Reich-Moore (RM);

LRF=4, Adler-Adler (AA);

LRF=5, no longer available;

LRF=6, no longer available;

LRF=7, R-Matrix Limite (RML).

If LRU=2 (unresolved parameters), then

LRF=1, only average fission widths are energy-dependent;

LRF=2, average level spacing, competitive reaction widths, reduced neutron widths, radiation widths, and fission widths are energy-dependent.

NRO

Flag designating possible energy dependence of the scattering radius:

NRO=0, radius is energy independent;

NRO=1 (not allowed in the ENDF/B-VI library)3.

NAPS

Flag controlling the use of the two radii, the channel radius a and the scattering radius AP.

For NRO=0 (AP energy-independent), if:

NAPS=0, calculate a from Equation (D.0) given in Appendix D, and read AP as a single energy-independent constant on the subsection CONT (range) record; use a in the penetrabilities and shift factors, and AP in the hard-sphere phase shifts;

NAPS=1, do not use Equation (D.0); use AP in the penetrabilities and shift factor as well as in the phase shifts.

For NRO=1 (AP energy-dependent), if:

NAPS=0, calculate a from the above equation and use it in the penetrabilities and shift factors. Read AP(E) as a TAB1 quantity in each subsection and use it in the phase shifts;

NAPS=1, read AP(E) and use it in all three places, Pl, Sl, φl ;

NAPS=2, read AP(E) and use it in the phase shifts. In addition, read the single, energy-independent quantity "AP, see following, and use it in Pl and Sl, overriding the above equation for a.

2 These energies are the limits to be used in calculating cross sections from the parameters. Some resolved resonance levels, e.g., bound levels, will have resonance energies outside the limits.

3 Formerly used for radius expressed as a table of energy, radius pairs.

File 2 contains a single section (MT=151) containing subsections for each energy range of each isotope in the material.

The structure of File 2, for the special case in which just a scattering radius is specified (no resolved or unresolved parameters are given), is as follows: (such a material is not permitted to have multiple isotopes or an energy-dependent scattering radius)

[MAT, 2,151/ ZA, AWR, 0, 0, NIS, 0] HEAD (NIS=1)

[MAT, 2,151/ ZAI, ABN, 0, LFW, NER, 0] CONT

(ZAI=ZA,ABN=1.0,LFW=0,NER=1)

[MAT, 2,151/ EL, EH, LRU, LRF, NRO, NAPS] CONT

(LRU=0,LRF=0,NRO=0,NAPS=0)

[MAT, 2,151/ SPI, AP, 0, 0, NLS, 0] CONT (NLS=0)

[MAT, 2, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND

[MAT, 0, 0/ 0.0, 0.0, 0, 0, 0, 0] FEND

If resonance parameters are given, the structure of File 2 is as follows:

[MAT, 2,151/ ZA, AWR, 0, 0, NIS, 0] HEAD

[MAT, 2,151/ ZAI, ABN, 0, LFW, NER, 0] CONT (isotope)

[MAT, 2,151/ EL, EH, LRU, LRF, NRO, NAPS] CONT (range)

<Subsection for the first energy range for the first isotope>

(depends on LRU and LRF)

[MAT, 2,151/ EL, EH, LRU, LRF, NRO, NAPS] CONT (range)

<Subsection for the second energy range for the first isotope>

-------------------------------

-------------------------------

[MAT, 2,151/ EL, EH, LRU, LRF, NRO, NAPS] CONT (range)

<Subsection for the last energy range for the last isotope for this material>

[MAT, 2, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND

The data are given for all ranges for a given isotope, and then for all isotopes. The data for each range start with a CONT (range) record; those for each isotope, with a CONT (isotope) record. The specifications for the subsections that include resonance parameters are given in Sections 2.2.1 and 2.3.1, below. A multi-isotope material is permitted to have some, but not all, isotopes specified by a scattering radius only. The structure of a subsection for such an isotope is:

[MAT, 2,151/ SPI, AP, 0, 0, NLS, 0] CONT (NLS=0)

and as above LFW=0, NER=1, LRU=0, LRF=0, NRO=0, and NAPS=0 for this isotope.

In the case that NRO0, the "range" record preceding each subsection is immediately followed by a record giving the energy dependence of the scattering radius, AP.

[MAT, 2,151/ 0.0, 0.0, 0, 0, NR, NP/ Eint / AP(E)] TAB1

If NAPS is 0 or 1 the value of AP on the next record of the subsection should be set to 0.0. If NAPS is 2, it should be set equal to the desired value of the channel radius.

2.2. Resolved Resonance Parameters (LRU=1)

2.2.1.Formats

Six different resonance formalisms are allowed to represent the resolved resonance parameters. Formulae for the various quantities, and further comments on usage, are given in Appendix D. The flag LRU=1, given in the CONT (range) record, indicates that resolved resonance parameters are given for a particular energy range. Another flag, LRF, in the same record specifies which resonance formalism has been used.

The following quantities are defined for use with all formalisms:

SPI

Spin, I, of the target nucleus.

AP

Scattering radius in units of 10-12cm. For LRF=1-4, it is assumed to be independent of the channel quantum numbers.

NLS

Number of l-values (neutron orbital angular momentum) in this energy region.

LRF=1-4, a set of resonance parameters is given for each l-value.

LRF=5 and 6, NLS is the number of l-values required to converge the calculation of the scattering cross section (see Sections 2.4.23 and 2.4.24). Another cutoff, NLSC, is provided for converging the angular distributions. Currently, NLS<4.

AWRI

Ratio of the mass of a particular isotope to that of a neutron.

QX

Q-value to be added to the incident particle's center-of-mass energy to determine the channel energy for use in the penetrability factor. The conversion to laboratory system energy depends on the reduced mass in the exit channel. For inelastic scattering to a discrete level, the Q-value is minus the level excitation energy. QX=0.0 if LRX=0.

L

Value of l.

LRX

Flag indicating whether this energy range contains a competitive width:

LRX=0, no competitive width is given, and Γ = Γn + Γγ+ Γf in the resolved resonance region, while <Γx>=0 in the unresolved resonance region; LRX must be 0 for LRF=3 or 4;

LRX=1, a competitive width is given, and is an inelastic process to the first excited state. In the resolved region, it is determined by subtraction, Γx= Γ - [Γn + Γγ + Γf ]

NRS

Number of resolved resonances for a given l-value. (NRS<600.)

ER

Resonance energy (in the laboratory system).

AJ

The absolute value of AJ is the floating-point value of J (the spin, or total angular momentum, of the resonance).

When two channel spins are possible, if the sign of AJ is negative, the lower value for the channel spin is implied; if positive, the higher value is implied. When AJ is zero, only one value of channel spin is possible so there is no ambiguity; the channel spin s is equal to the orbital angular momentum l.

GT

Resonance total width, Γ, evaluated at the resonance energy ER.

GN

Neutron width evaluated at the resonance energy ER.

GG

Radiation width, Γγ, a constant.

GF

Fission width, Γf, a constant.

GX

Competitive width, Γx, evaluated at the resonance energy ER.

It is not given explicitly for LRF=1 or 2 but is to be obtained by subtraction,

GX = GT – (GN+ GG + GF), if LRX0.

a

Channel radius, in 10-12 cm. An uppercase symbol is not defined because it is not an independent library quantity. Depending on the value of NAPS, it is either calculated from the equation given earlier (and in Appendix D), or read from the position usually assigned to the scattering radius AP.

2.2.1.1. SLBW and MLBW (LRU=1, LRF=1 or 2)

The structure of a subsection is:

[MAT, 2,151/ 0.0, 0.0, 0, 0, NR, NP/ Eint / AP(E)] TAB1

(if NRO 0)

[MAT, 2,151/ SPI, AP, 0, 0, NLS, 0] CONT

Use AP=0.0, if AP(E) is supplied and NAPS=0 or 1.

[MAT, 2,151/ AWRI, QX, L, LRX, 6*NRS, NRS/

.ER1, AJ1, GT1, GN1, GG1, GF1,

.ER, AJ, GT, GN, GG, GF2,

22222---------------------------------

ERNRS, AJNRS, GTNRS, GNNRS, GGNRS, GFNRS] LIST

The LIST record is repeated until each of the NLS l-values has been specified in order of increasing l. The values of ER for each l-value are given in increasing order.

2.2.1.2. Reich-Moore (LRU=1, LRF=3)

The following additional quantities are defined:

LAD

Flag indicating whether these parameters can be used to compute angular distributions.

LAD=0 do not use

LAD=1 can be used if desired. Do not add to file 4.

NLSC

Number of l-values which must be used to converge the calculation with respect to the incident l-value in order to obtain accurate elastic angular distributions. See Sections D.1.5.and D.1.6.5. (NLSCNLS).

APL

l-dependent scattering radius. If zero, use APL=AP.

GFA

First partial fission width, a constant.

GFB

Second partial fission width, a constant.

GFA and GFB are signed quantities, their signs being determined by the relative phase of the width amplitudes in the two fission channels. In this case, the structure of a subsection is similar to LRF=1 and 2, but the total width is eliminated in favor of an additional partial fission width. GFA and GFB can both be zero, in which case, Reich-Moore reduces to an R-function.

The structure for a subsection is:

[MAT, 2,151/ 0.0, 0.0, 0, 0, NR, NP/ Eint / AP(E)] TAB1

(if NRO0)

[MAT, 2,151/ SPI, AP, LAD, 0, NLS, NLSC] CONT

[MAT, 2,151/ AWRI, APL, L, 0, 6*NRS, NRS/

ER1, AJ1, GN1, GG1, GFA1, GFB1,

ER2, AJ2, GN2, GG2, GFA2, GFB2,

--------------------------------

ERNRS,AJNRS, GNNRS,GGNRS,GFANRS,GFBNRS] LIST

The LIST record is repeated until each of the NLS l-values has been specified in order of increasing l. The values of ER for each l-value are given in increasing order.

2.2.1.3. Adler-Adler (LRU=1,LRF=4)

For the case of (LRU=1, LRF=4) additional quantities are defined:

LI

Flag to indicate the kind of parameters given:

LI=1, total widths only

LI=2, fission widths only

LI=3, total and fission widths

LI=4, radiative capture widths only

LI=5, total and capture widths

LI=6, fission and capture widths

LI=7, total, fission, and capture widths.

NX

Number of sets of background constants given. There are six constants per set. Each set refers to a particular cross section type. The background correction for the total cross section is calculated by using the six constants in the manner following .

σT

Background =

where and k is defined in Appendix D.

The background terms for the fission and radiative capture cross sections are calculated in a similar manner.

NX=2, background constants are given for the total and capture cross sections.

NX=3, background constants are given for the total, capture, and fission cross sections.

NJS

Number of sets of resolved resonance parameters (each set having its own J-value) for a specified l.

NLJ

Number of resonances for which parameters are given, for a specified AJ and L.

AT1, AT2, AT3, AT4, BT1, BT2

Background constants for the total cross section.

AF1, AF2, AF3, AF4, BF1, BF2

Background constants for the fission cross section.

AC1, AC2, AC3, AC4, BC1, BC2

Background constants for the radiative capture cross section.

DETr

4Resonance energy, (µ), for the total cross section. Here and below, the subscript r denotes the rth resonance.

DEFr

5Resonance energy, (µ), for the fission cross section.

DECr

5Resonance energy, (µ), for the radiative capture cross section.

DWTr

5Value of Γ/2, (v), for the total cross section.

DWFr

5Value of Γ/2, (v), for the fission cross section.

DWCr

5Value of Γ/2, (v), for the radiative capture cross section.

GRTr

Symmetrical total cross section parameter, GrT.

GITr

Asymmetrical total cross section parameter, HrT.

GRFr

Symmetrical fission parameter, Grf.

GIFr

Asymmetrical fission parameter, Hrf.

GRCf

Symmetrical capture parameter, Grγ.

GICr

Asymmetrical capture parameter, Hrγ.

4 Note: DETr=DEFr=DECr and DWTr=DWFr=DWCr. The redundancy is an historical carryover.

The structure of a subsection for LRU=1 and LRF=4 depends on the value of NX (the number of sets of background constants). For the most general case (NX=3) the structure is

[MAT, 2,151/ 0.0, 0.0, 0, 0, NR, NP/ Eint / AP(E)] TAB1

optional record for energy-dependent scattering radius.

[MAT, 2,151/ SPI, AP, 0, 0, NLS, 0] CONT

[MAT, 2,151/ AWRI, 0.0, LI, 0, 6*NX, NX/

AT1, AT, AT, AT, BT, BT2,

2341 AF1, ---------------------, BF2,

AC, ---------------------, BC] LIST

12[MAT, 2,151/ 0.0, 0.0, L, 0, NJS, 0] CONT(l)

[MAT, 2,151/ AJ, 0.0, 0, 0, 12*NLJ, NLJ/

DET1, DWT1, GRT1, GIT1, DEF1, DWF1,

GRF1, GIF1, DEC, DWC, GRC, GIC,

1111 DET, DWT, -----------------------

22 ----------------------------, GIC,

2 DET,------------------------------

3 -----------------------------------

-----------------------------, GICNLJ] LIST

The last LIST record is repeated for each J-value (there will be NJS such LIST records). A new CONT (l) record will be given which NJS LIST records will follow. Note that if NX=2 then the quantities AF1, ---BF2 will not be given in the first LIST record. Also, if LI7 then certain of the parameters for each level may be set to zero, i.e., the fields for parameters not given (depending on LI) will be set to zero.

The format has no provision for giving Adler-Adler parameters for the scattering cross section. The latter is obtained by subtracting the capture and fission cross sections from the total.

Although the format allows separation of the resonance parameters into J-subsets, no use is made of J in the A-A formalism. There is no analog to the resonance-resonance interference term of the MLBW formalism. Such interference is represented implicitly by the asymmetric terms in the fission and capture cross sections.

2.2.1.4. (LRU=1, LRF=5) no longer available

2.2.1.5. (deleted)

2.2.1.6. (LRU=1, LRF=6) no longer available

2.2.1.7 R-Matrix Limited format (LRU=1,LRF=7)

In R-matrix scattering theory, a channel is defined by the two particles inhabiting that channel and by the quantum numbers for the combination. The two particles are hereafter referred to as a particle-pair (PP), and are defined by their properties: neutron (or other particle) plus target nuclide (in ground or excited state), with individual identifiers such as mass, spin, parity, and charge. The additional quantum numbers defining the channel include orbital angular momentum l, channel spin s and associated parity, and total spin and parity J π .

NOTE: This format is NOT restricted to one neutron (entrance) channel and two exit channels. There may be several entrance channels and a multitude of exit channels. Charged-particle exit channels are not excluded.

The term “spin group” may be used to define the set of resonances with the same channels and quantum numbers. For any given spin group, only total spin and parity are constant; there may be several entrance channels and/or several reaction channels (and, hence, several values of l or s, etc.) contributing to the spin group.

The “R-Matrix Limited” (RML) format was designed to accommodate the features of R-Matrix theory as implemented in analyses codes being used for current evaluations. In this format, relevant parameters appear only once. Particle-pairs are given first: the masses, spins and parities, and charges for the two particles are specified, as well as the Q-value and the MT value (which defines whether this particle-pair represents elastic scattering, fission, inelastic, capture, etc.). Two particle-pairs will always be present: gamma + compound nucleus, and neutron + target nucleus in ground state. Other particle-pairs are included as needed.

The list of resonance parameters is ordered by J π, which (as stated above) is the only conserved quantity for any spin group. For each spin group, the channels are first specified in the order in which they will occur in the list of resonances. For each channel, the particle-pair number and the values for l and s are given, along with the channel radii.

2.2.1.7.1 Formats for the basic RML subsection

Additional quantities are defined (or, in some cases, re-defined):

KRM Flag to specify which formulae for the R-matrix are to be used. KRM = 1 for single-level Breit-Wigner, KRM = 2 for multilevel Breit-Wigner, KRM = 3 for Reich Moore, KRM = 4 for full R-matrix. (Others may be added at a later date.)

KRL Flag is zero for non-relativistic kinematics, 1 for relativistic.

NJS Number of values of J π to be included.

NPP Total number of particle-pairs.

IA Spin (and parity, if non-zero) of one particle in the pair (the neutron or projectile, if this is an incident channel).

IB Spin of the other particle in the pair (target nuclide, if this is an incident channel). IB is set to zero and ignored if the first particle is a photon.

PA Parity for first particle in the pair, used only in the case where IA is zero and the parity is negative. (Value = +1.0 if positive, -1.0 if negative.)

PB Parity for second particle, used if IB= 0 and parity is negative.

MA Mass of first particle in the pair (in units of neutron mass).

MB Mass of second particle (in units of neutron mass).

ZA Charge of first particle.

ZB Charge of second particle.

QI Q-value for this particle-pair. (See Sect. 3.3.2 for details)

PNT Flag is 1 if penetrability is to be calculated, -1 if not (default depends on number; MT=108 implies PNT = -1, others are generally PNT = +1)

SHF Flag is 1 if shift factor is to be calculated, -1 if not (default = not)

MT Reaction type associated with this particle-pair; see Appendix B.

AJ Floating point value of J (spin); sign indicates parity.

PJ Parity (used only if AJ = 0.0).

NCH Number of channels for the given J π.

IPP Particle-pair number for this channel (written as floating-point number).

L Orbital angular momentum (floating-point value).

SCH Channel spin (floating-point value).

BND Boundary condition for this channel (needed when SHF=+1)

APE Effective channel radius (scattering radius), used for calculation of phase shift only. Units are 10-12 cm.

APT True channel radius (scattering radius), used for calculation of penetrability and shift factors. Units are 10-12 cm.

KBK Non-zero if background R-matrix exists; sees Subsect. 2.2.1.7.2. (Often set to 0.)

KPS Non-zero if non-hard-sphere phase shift are to be specified. (Often set to 0.)

NRS Number of resonances for the given J π.

NX Number of lines required for all resonances for the given J π, assuming each resonance starts on a new line; equal to (NCH/6+1)*NRS. If there are no resonances for a spin group, then NX = 1.

ER Resonance energy in eV.

IFG Flag is 0 if GAM is channel width in eV, 1 if reduced-width amplitude in eV1/2.

GAM Channel width in eV or reduced-width amplitude in eV1/2.

NOTE: For IFG = 0, the input quantity GAM is the width at the energy of the resonance; reduced width amplitudes are calculated from Eq. (7) of D.1.7, with E set to Eλ. (For negative-energy dummy resonances, the convention is that the input quantity is the width evaluated at the absolute value of the resonance energy.) In all cases, if the value GAM given in File 2 for the partial width is negative, the standard convention is assumed: the negative sign is to be associated with the reduced width amplitude γλc rather than with Γλc (since Γλc is always a positive quantity). More specifically, Γλc = |GAM| and , with P evaluated at the energy of the resonance.

If IFG =1, the input quantity is the reduced width amplitude .

Formats are as follows:

[MAT,2,151/ 0.0, 0.0, IFG, KRM, NJS, KRL ] CONT

(The following record provides all particle-pair descriptions. For KRM=1,2, or 3, the first particle-pare is the gamma plus compound nucleus pair.)

[MAT,2,151/ 0.0, 0.0, NPP, 0, 12*NPP, 2*NPP/

MA, MB, ZA, ZB1, IA1, IB1,

111 Q, PNT, SHF, MT1, PA1, PB1,

111 MA, MB, ZA, ZB2, IA2, IB1,

222 Q, PNT, SHF, MT, PA, PB,

222221 -------------------------------------------

MA, MBP, ZAP, ZBNPP, IANPP, IBNPP,

NPPNPNP QNPP, PNTNPP, SHFNPP, MTNPP, PANPP, PBNPP ] LIST

The following record provides the channel descriptions for one spin group.)

[Mat,2,151/ AJ, PJ, KBK, KPS, 6*NCH, NCH/

IPP1, L1, SCH1, BND1, APE1, APT1,

IPP, L, SCH, BND, APE, APT,

222222 -----------------------------------------------

IPPNCH, LNCH, SCHNCH, BNDNCH, APENCH, APTNCH] LIST

(The following record gives the values for resonance energy and widths for each resonance in this spin group.)

[Mat,2,151/ 0.0, 0.0, 0, NRS, 6*NX, NX/

ER, GAM1,1, GAM2,1 GAM3,1, GAM4,1, GAM5,1,

1 GAM, GAMNCH,

6,1,1 ER, GAM1,2, GAM2,2, GAM3,2, GAM4,2, GAM5,2,

2 GAM, …, GAM,

6,2NCH,2 -----------------------------------------------

ERS, GAMS,GAM, GAM, GAM, GAM,

NR1,NR2,NRS3,NRS4,NRS5,NRS GAM6,NRS,…,GAMNCH,NRS ] LIST

(If the number of resonances is zero for a spin group, then NRS = 0 but NX =1 in this record.)

Other records may be included here, as described below in Sect. 2.2.1.7.2. If KBK is greater than zero, a “background R-matrix” is given. If KPS is greater than zero, tabulated values exist for phase shifts. If KBK = 0 and KPS = 0, no additional records are needed.

The above records, beginning with “channel descriptions,” are repeated until each of the NJS J π spin groups has been fully specified.

2.2.1.7.2 Formats for optional extensions to the RML

The formats described in the previous section are sufficient for most evaluations currently (2003) available (using KRM = 3, KBK = 0, and KPS = 0 ). For the sake of generality, and to accommodate expected future developments in R-matrix analysis codes, additional capabilities are included in the RML format.

2.2.1.7.2.1 Different R-Matrix formulations (KRM = 1,2,4)

Equations given in Appendix D.1.7 are relevant to the Reich-Moore approximation to R-Matrix theory. The format, however, can also be used for single-level Breit-Wigner (KRM = 1), multilevel Breit-Wigner (KRM = 2), R-Matrix without approximations (KRM = 4). Equations for KRM = 1 or 2 will be written up if/when the need arises. Equations for KRM = 4 are identical to those given in Appendix D.1.7 with the elimination of the imaginary term in the denominator of Eq. (6), and the inclusion of each gamma-channel on a equal basis with all other channels.

2.2.1.7.2.2 Background R-matrix (KBK > 0):

As described in Appendix D (D.1.7.7), a background R-Matrix can be defined in a variety of different methods.

For KBK = 0, Option 0 is used everywhere (that is, for all channels for this spin group) for the background R-Matrix. No additional formats are required and no additional records need to be written; the dummy resonances are included along with the physical resonances in the list record described above.

For KBK > 0, one LIST record (and two TAB1 records, for tabulated values) is included for each channel of the current spin group, a total of NCH records. The particular option to be used for the channel is identified by parameter LBK. The formats for the four options are as follows:

Option 0. Dummy resonances (LBK = 0)

No additional information is conveyed in this record, other than LBK = 0. No terms are added to the R-matrix for this channel.

[Mat,2,151/ 0.0, 0.0, 0, 0, LBK, 1/

0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ] LIST

Option 1. Tabulated complex function of energy (LBK = 1)

Notation:

RBR Value of real part of tabulated function

RBI Value of imaginary part of tabulated function

[Mat,2,151/ 0.0, 0.0, 0, 0, LBK, 1/

0.0, 0.0, 0.0, 0.0, 0.0, 0.0] LIST

[Mat,2,151/ 0.0, 0.0, 0, 0, NR, NP/ Et / RBR(E) / TAB1

in[Mat,2,151/ 0.0, 0.0, 0, 0, NR, NP/ Eint / RBR(E) / TAB1

(Recall that NR and NP are parameters, which define the interpolation scheme for TAB1 records, as defined in Section 0.7.7. Energy values given by Eint are in units of eV.)

Option 2. SAMMY’s logarithmic parameterization (LBK = 2)

Notion (See Eq. (47) of Sect. D.1.7.7 for meanings of these quantities):

R0 S0

R1 S1

R2

EU ED

[Mat,2,151/ ED, EU, 0, 0, LBK, 1/

R0, R1, R2, S0, S1, 0.0] LIST

Option 3. Fröhner’s parameterization (LBK = 3)

Notion (See Eqs. (48 and 49) Sect. D.1.7.7 for meanings of these quantities):

R0 S0

GA

EU ED

[Mat,2,151/ ED, EU, 0, 0, LBK, 1/

R0, SO GA, 0.0, 0.0, 0.0] LIST

2.2.1.7.2.3 Tabulated phase shifts (KPS = 1)

When phase shifts are calculated externally (from optical model potentials, for example), rather than generated from the usual hard-sphere phase shift formulae, then the phase shifts must be presented in tabular form.

If parameter KPS is equal to 0, all phase shifts are calculated from the hard-sphere phase shift formulae (see Table D.1.7.1 for non-Coulomb, Sect. D.1.7.2 for Coulomb hard-sphere phase shifts).

For KPS >0, one LIST record (and two TAB1 records, for tabulated values) is included for each channel of the current spin group, a total of NCH records. The particular option to be used for the channel is identified by parameter LPS. The formats for the two options are as follows:

Option 0. Hard-sphere phase shifts (LPS = 0)

No additional information is conveyed in this record, other than LPS = 0.

[Mat,2,151/ 0.0, 0.0, 0, 0, LPS, 1/

0.0, 0.0, 0.0, 0.0, 0.0, 0.0] LIST

Option 1. Phase shift is a tabulated complex function of energy (LPS = 1)

Notation:

PSR Value of real part of tabulated phase shift

PSI Value of imaginary part of tabulated phase shift

[Mat,2,151/ 0.0, 0.0, 0, 0, LPS, 1/

0.0, 0.0, 0.0, 0.0, 0.0, 0.0] LIST

[Mat,2,151/ 0.0, 0.0, 0, 0, NR, NP/ Et / PSR (E) / TAB1

in 0.0, 0.0, 0, 0, NR, NP/ Eint / PSR (E) / TAB1

(Recall that NR and NP are parameters, which define the interpolation scheme for TAB1 records, as defined in Sect. 0.7.7. Energy values given by Eint are in units of eV.)

2.3. Unresolved Resonance Parameters (LRU=2)

2.3.1. Formats

Only the SLBW formalism for unresolved resonance parameters is allowed (see Appendix D for pertinent formulae). However, several options are available for specifying the energy-dependence of the parameters, designated by the flag LRF. Since unresolved resonance parameters are averages of resolved resonance parameters over energy, they are constant with respect to energy throughout the energy-averaging interval. However, they are allowed to vary from interval to interval, and it is this energy-dependence, which is referred to above and in the following paragraphs.

The parameters depend on both l (neutron orbital angular momentum) and J (total angular momentum). Each width is distributed according to a chi-squared distribution with a certain number of degrees of freedom. This number may be different for neutron and fission widths and for different (l,J) channels.

The following quantities are defined for use in specifying unresolved resonance parameters (LRU=2):

SPI

Spin of the target nucleus, I.

AP

Scattering radius in units of 10-12 cm. No channel quantum number dependence is permitted by the format.

LSSF

Flag governing the interpretation of the File 3 cross sections.

LSSF=0, File 3 contains partial "background" cross sections, to be added to the average unresolved cross sections calculated from the parameters in File 2.

LSSF=1, File 3 contains the entire dilute cross section for the unresolved resonance region. File 2 is to be used solely for the calculation of the self-shielding factors, as discussed in Section 2.4.21.

NE

Number of energy points at which energy-dependent widths are tabulated. (NE250).

NLS

Number of l-values (NLS3).

ESi

Energy of the ith point used to tabulate energy-dependent widths.

L

Value of l.

AWRI

Ratio of the mass of a particular isotope to that of the neutron.

NJS

Number of J-states for a particular l-state. (NJS6).

AJ

Floating-point value of J (the spin, or total angular momentum of the set of parameters).

D

Average level spacing for resonances with spin J. (D may be energy dependent if LRF=2).

AMUX

Number of degrees of freedom used in the competitive width distribution. (Assuming it is inelastic, 1.0AMUX2.0, determined by whether the spin of the first excited state is zero or not.)5

AMUN

Number of degrees of freedom in the neutron width distribution. (1.0AMUN2.0)

AMUG

Number of degrees of freedom in the radiation width distribution. (At present AMUG = 0.0. This implies a constant value of Γγ.)

AMUF

Number of degrees of freedom in the fission width distribution. (1.0AMUF4.0)

MUF

Integer value of the number of degrees of freedom for fission widths. (1MUF4)

INT

Interpolation scheme to be used for interpolating between the cross sections obtained from average resonance parameters. Parameter interpolation is discussed in the Procedures Section 2.4.2.

GN0

Average reduced neutron width. It may be energy-dependent if LRF=2.

GG

Average radiation width. It may be energy-dependent if LRF=2.

GF

Average fission width. It may be energy-dependent if LRF=1 or 2.

GX

Average competitive reaction width, given only when LRF=2, in which case it may be energy-dependent.

The structure of a subsection6 depends on whether LRF=1 or LRF=2. If LRF=1, only the fission width is given as a function of energy. If LRF=1 and the fission width is not given (indicated by LFW=0), then the simplest form of a subsection results. If LRF=2, energy-dependent values may be given for the level density, competitive width, reduced neutron width, radiation width, and fission width. Three sample formats are shown below (all LRU=2).

A. LFW=0 (fission widths not given),

LRF=1 (all parameters are energy-independent).

The structure of a subsection is:

[MAT, 2,151/ SPI, AP, LSSF, 0, NLS, 0] CONT

[MAT, 2,151/ AWRI, 0.0, L, 0, 6*NJS, NJS/

D1, AJ1, AMUN1, GN01, GG1, 0.0,

5 See Appendix D. Section D.2.2.6.

6 The structure of a section was defined previously, and covers both resolved resonance and unresolved resonance subsections.

D, AJ, AMUN, GN0, GG, 0.0,

22222--------------------------------------

DNJS, AJNJS, AMUNNJS, GN0NJS, GGNJS, 0.0] LIST

The LIST record is repeated until data for all l-values have been specified. In this example, AMUG is assumed to be zero, and there is no competitive width.

B. LFW=1 (fission widths given),

LRF=1 (only fission widths are energy-dependent; the rest are energy-independent).

The structure of a subsection is:

[MAT, 2,151/ SPI, AP, LSSF, 0, NE, NLS] CONT

ES, ES, ES, ------------

123-------------------------- ES] LIST

NE[MAT, 2,151/ AWRI, 0.0, L, 0, NJS, 0] CONT

[MAT, 2,151/ 0.0, 0.0, L, MUF, NE+6, 0/

D, AJ, AMUN, GN0, GG, 0.0,

GF, GF, GF,--------------

123---------------------------- GFNE] LIST

The last LIST record is repeated for each J-value (there will be NJS such LIST records). A new CONT(l) record will then be given which will be followed by its NJS LIST records until data for all l-values have been specified (there will be NLS sets of data).

In the above section, no provision was made for INT, and interpolation is assumed to be lin-lin. AMUG is assumed to be zero, AMUF equals MUF, and there is no competitive width.

C. LFW=0 or 1 (does not depend on LFW).

LRF=2 (all energy-dependent parameters).

The structure of a subsection is:

[MAT, 2,151/ SPI, AP, LSSF, 0, NLS, 0] CONT

[MAT, 2,151/ AWRI, 0.0, L, 0, NJS, 0] CONT

[MAT, 2,151/ AJ, 0.0, INT, 0,(6*NE)+6, NE/

0.0, 0.0, AMUX, AMUN, AMUG, AMUF,

ES1, D1, GX1, GN01, GG1, GF1,

ES, D, GX, GN0, GG, GF2,

22222 -----------------------------

ESNE, DNE, GXNE, GN0NE, GGNE, GFNE] LIST

The LIST record is repeated until all the NJS J-values have been specified for a given l-value. A new CONT(l) record is then given, and all data for each J-value for that l-value are given. The structure is repeated until all l-values have been specified. This example permits the specification of all four degrees of freedom.

2.4. Procedures for the Resolved and Unresolved Resonance Regions

CONTENTS OF THIS SECTION

2.4.1 Abbreviations

2.4.2 Interpolation in the Unresolved Resonance Region

2.4.3 Unresolved Resonances in the Resolved Resonance Range

2.4.4 Energy Range Boundary Problems

2.4.5 Numerical Integration Procedures in the Unresolved Resonance Region

2.4.6 Doppler-Broadening of File 3 Background Cross Sections

2.4.7 Assignment of Unknown J-values

2.4.8 Equivalent Single-Level Representations

2.4.9 Use of the Reich-Moore Formalism

2.4.10 Competitive Width in the Resonance Region

2.4.11 Negative Cross Sections in the Resolved Resonance Region

2.4.12 Negative Cross Sections in the Unresolved Resonance Region

2.4.13 Use of Two Nuclear Radii

2.4.14 The Multilevel Adler-Gauss Formula for MLBW

2.4.15 Notes on the Adler Formalism

2.4.16 Multilevel Versus Single-level Formalisms in the Resolved and Unresolved Resonance Regions

2.4.17 Preferred Formalisms for Evaluating Data

2.4.18 Computer Time for Generating MLBW Cross Sections

2.4.19 Amplitude-Squared Form of the MLBW Formulas

2.4.20 Degrees of Freedom for Unresolved Resonance Parameters

2.4.21 Procedures for the Unresolved Resonance Region

2.4.22 Procedures for Computing Angular Distributions in the Resolved Resonance Range

2.4.23 Completeness and Convergence of Channel Sums

2.4.24 Channel Spin and Other Considerations

2.4.1. Abbreviations

UR(R) unresolved resonance (region)

RR(R) resolved resonance (region)

RRP resolved resonance parameter(s)

URP unresolved resonance parameter(s)

SLBW single-level Breit-Wigner

MLBW multi-level Breit-Wigner

MLAG multi-level Adler-Gauss

UCS unresolved cross section(s)

2.4.2. Interpolation in the Unresolved Resonance Region (URR)

For energy-dependent formats (LRF=2, or LRF=1 with LFW=1), the recommended procedure is to interpolate on the cross sections derived from the unresolved resonance parameters (URP). This is a change from the ENDF/B-III and IV procedure, which was to interpolate on the parameters. The energy grid should be fine enough so that the cross sections at intermediate energy values can be computed with sufficient accuracy using this procedure. Normally, three to ten points per decade will be required to obtain reasonable accuracy. Some evaluations prepared for earlier versions of ENDF/B do not meet these standards. In such cases, if two adjacent grid points differ by more than a factor of three, the processing code should add additional intermediate energy points at a spacing of approximately ten-per-decade and compute the cross sections at the intermediate points using parameter interpolation. Additional cross sections can then be obtained by cross section interpolation in the normal way.

For many isotopes, there is not sufficient information for a full energy-dependent evaluation. In these cases, the evaluator may provide a single set of unresolved resonance parameters based on systematics or extrapolation from the resolved range (see LRF=1, LFW=0). Such a set implies a definite energy-dependence of the unresolved cross sections due to the slowly-varying wave number, penetrability, and phase shift factors in the SLBW formulas. It is not correct to calculate cross sections at the ends of the URR, and then to compute intermediate cross sections by cross section interpolation. Instead, the processing code should generate a set of intermediate energies using a spacing of approximately ten-per-decade and then compute the cross sections on this grid using the single set of parameters given in the file. Additional intermediate values are then obtained by linear cross section interpolation as in the energy-dependent case.

It is recommended that evaluators provide the URP's on a mesh dense enough that the difference in results of interpolating on either the parameters or the cross sections be small. A 1% maximum difference would be ideal, but 5% is probably quite acceptable.

Finally, even if the evaluator provides a dense mesh, the user may end up with different numbers than the evaluator "intended". This is particularly true when genuine structure exists in the cross section and the user chooses different multigroup breakpoints than those in the evaluation. There is no solution to this problem, but the dense mesh procedure minimizes the importance of the discrepancy.

In order to permit the user to determine what "error" he is incurring, it is recommended that evaluators state in the documentation what dilute, unbroadened average cross sections they intended to represent by the parameters in File 2. Note that the self-shielding factor option specified by the flag LSSF (Sections 2.3.1 and 2.4.21) greatly reduces the impact of this interpolation ambiguity.

2.4.3. Unresolved Resonances in the Resolved Resonance Range

As discussed in section 2.4.4, the boundary between the resolved and unresolved resonance regions should be chosen to make the statistical assumptions used in the URR valid. This creates problems in evaluating the resonance parameters for the RRR.

Problem l: At the upper end of the resolved range, the smaller resonances will begin to be missed. An equivalent contribution could be added to the background in File 3. This contribution will not be self-shielded by the processing codes, so it cannot be allowed to become "significant". A better procedure is to supply fictitious resolved parameters, based on the statistics of the measured ones, checking that the average cross section agrees with whatever poor-resolution data are available.

If both procedures are employed, care should be taken not to distort the statistics of the underlying parameter distributions.

Problem 2: Because d-wave resonances are narrower than p-waves, which are narrower than s-waves, everything else being equal, the point at which p-waves will be instrumentally unresolved can be expected to be lower in energy than for s-waves, and lower still for d-waves. Thus the unresolved region for p-waves will usually overlap the resolved region for s-waves, and similarly for d-waves. Current procedure does not permit representing this effect explicitly - one cutoff-point must serve for all l-values.

The remedies are the same as above, either putting known or estimated resonances into the background in the URR, or putting fictitious estimated resonances into the RRR. The latter is preferred because narrow resonances tend to self-shield more than broad ones, hence the error incurred by treating them as unshielded File 3 background contributions is potentially significant.

2.4.4. Energy Range Boundary Problems

There may be as many as four different kinds of boundaries under current procedures which permit multiple RRR's:

1. between a low-energy File 3 representation (range 1) and EL for the RRR (range 2),

2. between successive RR ranges,

3. between the highest RRR and the URR,

4. between EH for the URR and the high-energy File 3 representation.

Discontinuities can be expected at each boundary. At 1, a discontinuity will occur if range 1 and range 2 are not consistently Doppler-broadened. In general, only an identical kernel-broadening treatment will produce continuity, i.e., only if the range-1 cross sections are broadened from the temperature at which they were measured, and range-2 is broadened from absolute zero. A kernel treatment of range 1, or no broadening at all, will be discontinuous with a Ψ-χ treatment of range 2. This effect is not expected to be serious at normal reactor temperatures and presumably, the CTR and weapons communities are cognizant of the Doppler problem. In view of these problems, a double energy point will not usually produce exact continuity in the complete cross section, (file 2 + file 3), unless evaluator and user employ identical methods throughout.

Discontinuities will occur between successive RRR's, unless the evaluator takes pains to adjust the "outside" resonances for each RRR to produce continuity at absolute zero. If the unbroadened cross sections in two successive RRR's are broadened separately, the discontinuity will be preserved, and possibly enhanced. These discontinuities are not believed to be technologically significant.

A discontinuity at #3 is unavoidable, because the basic representation has changed. However if the RRR cross sections are group-averaged or otherwise smoothed, the discontinuity7 should be reasonably small. A discontinuity greater than 10 or 15% obtained with a suitable averaging interval indicates that the evaluator might want to reconsider his parameterization of the poor-resolution data. Some materials have large genuine fluctuations in the URR, and for these the 10-15% figure is not applicable. A double energy point will normally occur at this boundary, but will not eliminate the discontinuity.

7 This refers to the discontinuity between the average cross section in the RRR, and the dilute (unshielded) pointwise cross section in the URR, which has been generated from the URR parameters. If the self-shielding factor option has been chosen (LSSF=1, Section 2.3.1), File 3 will contain the entire dilute cross section and no File 2 unresolved region calculation will be needed to ascertain the discontinuity.

Discontinuity at #4 should be small, since both the URR and the high-energy range represent rather smooth cross sections, and the opportunity for error ought to be small. Anything over 5% or so should be viewed with suspicion.

The upper and lower energy limits of any energy range indicate the energy range of validity of the given parameters for calculating cross sections. Outside this energy range the cross sections must be obtained from the parameters given in another energy range and/or from data in file 3.

The lower energy limit of the URR should be chosen to make the statistical assumptions used in this range valid. The basic requirement is that there be "many" resonances in an energy-averaging interval, and that the energy-averaging interval be narrow with respect to slowly-varying functions of E such as wave number and penetrability. As an example, assume that the energy-averaging interval can extend 10% above and below the energy point, that the average resonance spacing is 1 eV, and that "many" is 100. Then the lowest reasonable energy for the URR would be about 500 eV, as given by 0.2 E=100×1. Some implications of this choice for the RRR-URR boundary were discussed in Section 2.4.3.

It is sometimes necessary to give parameters whose energies lie outside a specified energy range in order to compute the cross section for neutron energies that are within the energy range. For example, the inclusion of bound levels may be required to match the cross sections at low energies, and resonances will often be needed above EH to compensate the opposite, positive, bias at the high energy end.

For materials that contain more than one isotope, it is recommended that the lower energy limit of the resolved resonance region be the same for all isotopes. If resolved and/or unresolved resonance parameters are given for only some of the naturally occurring isotopes, then AP should be given for the others.

If more than one energy range is used, the ranges must be contiguous and not overlap.

Overlapping of the resolved and unresolved ranges is not allowed for any one isotope, but it can occur in an evaluation for an element or other mixture of different isotopes. In fact, it is difficult to avoid since the average resonance spacing varies widely between even-even and even-odd isotopes. Such evaluations are difficult to correctly self-shield. A kernel broadening code must first subtract the infinitely-dilute unresolved cross section, broaden the pointwise remainder, then add back the unresolved component. A multigroup averaging code that uses pointwise cross sections must first subtract the infinitely-dilute unresolved cross section to find the pointwise remainder, and then add back a self-shielded unresolved cross section computed for a background cross section which includes a contribution from the pointwise remainder.

2.4.5. Numerical Integration Procedures in the URR

The evaluation of effective cross sections in the URR can involve Doppler effects, flux-depression, and resonance-overlap as well as the statistical distributions of the underlying resonance parameters for a mixture of materials.

The previous ENDF/B recommendation for doing the complicated multi-dimensional integrations was the Greebler-Hutchins scheme, Reference 1, basically a trapezoidal integration. For essentially the same computing effort, a more sophisticated weighted-ordinate method can be used and it has been shown that the scheme in MC2-II, Reference 2, produces results differing by up to several percent from G-H. The MC2-II subroutine8, is the recommended procedure.

8 This subroutine was provided by H. Henryson, II (ANL).

The M. Beer [Ref.3], analytical method has also been suggested, and is quite elegant, but unfortunately will not treat the general heterogeneous case.

2.4.6. Doppler-Broadening of File 3 Background Cross Sections

1. In principle, the contribution to each cross section from File 3 should be Doppler-broadened, but in practice, many codes ignore it. It is therefore recommended that the evaluator keep file 3 contributions in the RRR and URR small enough and/or smooth enough so that omission of Doppler-broadening does not "significantly" alter combined File 2 plus File 3 results up to 3000 K. Unfortunately, the diversity of applications of the data in ENDF/B makes the word "significantly" impossible to define.

2. A possible source of structured File 3 data is the representation of multilevel or MLBW cross sections in the SLBW format, the difference being put into File 3. This difference is a series of residual interference blips and dips, which may affect the between-resonance valleys and possibly the transmission in thick regions or absorption rates in lumped poisons, shields, blankets, etc. Users of the SLBW formalism should consider estimating these effects for significant regions. A possible remedy is available in the Multilevel Adler-Gauss form of MLBW. (See Section 2.4.14). If the resonance-resonance interference term in MLBW is expanded in partial fractions, it becomes a single sum of symmetric and asymmetric SLBW-type terms. Two coefficients occur which require a single sum over all resonances for each resonance, but these sums are weakly energy dependent and lend themselves to approximations that could greatly facilitate the use of ψ- and χ-functions with MLBW.

3. An "in-principle" correct method for constructing resonance cross sections is:

a.) Use a Solbrig kernel [Ref.12] to broaden File 2 to the temperature of File 3, since the latter may be based on room-temperature or other non-zero 0K data.

b.) Add File 2 and File 3.

c.) Sollbrig-broaden the result to operating temperature.

Using a Gaussian kernel instead of Sollbrig incurs a small error at low energies, unless it is misused, in which case the error can be large. Using Ψ- and χ-functions introduces further errors. In fact, the Sollbrig kernel already approximates the true motion of the target molecules by a free-gas law, but anything more accurate is quite difficult to handle.

4. Some heavy element evaluations use a File 3 representation below the resolved resonance region. Often these cross sections are room-temperature values, so that if they are later broadened assuming they are zero-degrees Kelvin, they get broadened twice.

A simple way to reduce the impact of this procedure without altering the representation of the data is to calculate the cross sections from the resonance parameters, broadened to room temperature, and carry the calculation down through the low-energy region. Subtract these broadened values from the file 3 values and leave only the difference in file 3. Then extend the lower boundary of the resonance region to the bottom of the file. Now the "double-broadening" problem affects only the (small) residual file 3 and not the entire cross section.

Note that subtracting off a zero degree resonance contribution would accomplish nothing.

2.4.7. Assignment of Unknown J-values

In all multilevel resonance formalisms except Adler-Adler, the J-value determines which resonances interfere with each other. Usually, J is known only for a few resonances, and measurers report 2gΓn for the others. If this number is assumed to be Γn, one incurs an error of uncertain magnitude, depending on how different

is from l/2, how large Γn is relative to the other partial widths, and how important resonance- resonance interference is.

It is recommended that evaluators assign J-values to each resonance, in proportion to the level density factor 2J+l. To reduce the amount of interference, the J-values of strong neighboring resonances, which would produce the largest interference effects, can be chosen from different families.

In the past, some evaluations have put J=I, the target nucleus spin, for resonances with unknown J-values. This corresponds to putting g=1/2, rather than its true value. Mixing of the J=I resonances with the physically correct I±1/2 families can result in negative scattering cross sections, or distortions of the potential scattering term, depending on what formalism is used and how it is evaluated. For this reason, such J=I resonances must not be used.

In the amplitude-squared form of the MLBW scattering cross section,

(2.1)

the use of J=I resonances will destroy the equivalence between this form and the "squared" form of MLBW in Appendix D since the sum on lsJ does not go over physically-correct values.

An exception to the prohibition against J=I is the case where no J-values are known, since if all resonances are assigned J=I, the MLBW scattering cross section will be non-negative.

2.4.8. Equivalent Single-Level Representations

The single-level Breit-Wigner formalism is incorporated into the basic structure of many engineering codes used for reactor design. Its use is so widespread, that despite any shortcomings in the calculational procedures, such codes must be supplied with SLBW parameters. For ENDF/B evaluations employing other representations, one requires an "equivalent" set of SLBW parameters. This is not to minimize the importance of using improved methods, but such improved methods do not eliminate the need for SLBW parameters in reactor design. For example, the Adler formalism provides a multilevel, multichannel fission cross section in pseudo-SLBW format, permitting ψ,χ-broadening. This is very useful, but not to a code that does not recognize asymmetric fission or capture.

The following equivalences are recommended:

1. MLBW. Use the parameters "as is".

2. Reich-Moore. Use the parameters "as is", except that the absolute values of the partial fission widths are added together to form Γf. (Alternatively, convert Reich-Moore to Adler-Adler, and use the equivalence for that formalism).

3. Adler-Adler. Reasonable success in converting A-A parameters for 241Pu and 233U was obtained using a method described in Reference 4.

2.4.9. Use of the Reich-Moore Formalism

If the evaluation of fissionable, low energy, s-wave-only, materials is carried out with a Reich-Moore formalism, then the parameters may be transformed to the Adler-Adler representation. R-M has some advantages in evaluating data, mainly that it uses resonance spins, is more closely tied to familiar resonance parameters, and is more "physical", but the Adler format is more convenient for the user since it permits ψ- and χ-functions for Doppler broadening.

The computer code POLLA [Ref.5], as well as some others, will convert a set of Reich-Moore multilevel s-wave resonance parameters to Adler format. If the conversion causes differences between the Adler and R-M cross sections which exceed 0.l%, these should be put into File 3, since it is not the intent of the procedure to in any way alter the original cross sections. Such differences can possibly be reduced by feeding the POLLA output parameters to a least-squares search code based on the Adler formalism, and "fitting" the original R-M values.

According to the discussion in BNL-502969, the Reich-Moore code RAMPl, incorporated in RESEND, sets the shift factor equal to zero. This is correct for s-waves, and should pose no problem for p- and d-waves, provided that the evaluator has included this shift factor when the calculation was performed.

2.4.10. Competitive Width in the Resonance Region

2.4.10.1. Resolved Region

Procedures for the Resolved Resonance Region are contained in Section D.3.l of Appendix D.

2.4.10.2. Unresolved Region

Procedures for the Unresolved Resonance Region are contained in Section D.3.2 of Appendix D. Users are directed to the discussion of the total cross section in Appendix D, Section D.3.3, since, as pointed out by H. Henryson, II, in connection with MC2 procedures, a possibility for erroneous calculations exists.

2.4.11. Negative Cross Sections in the Resolved Resonance Region

2.4.11.1. In the SLBW Formalism

Capture and fission use the positive symmetric Breit-Wigner shape and are never negative. Scattering involves an asymmetric term which goes negative for E < ER and can cause negative cross sections. A single resonance, or a series of well-separated resonances, will usually not produce negative cross sections, but when two or more resonances "cooperate", their negative tails can combine to produce negative values. In nature, the negative tails are compensated by either the positive tails of lower-lying resonances or multilevel interference effects. However, in evaluated data files the resonances are usually given only down to "E=0", a quite arbitrary point from the standpoint of the compound nucleus, so that "negative-energy" resonances are needed to compensate the negativity bias.

9 M.R. Bhat, BNL-50296 (ENDF l48) ENDF/B Processing Codes for the Resonance Region, June, l97l,

Although the negative scattering cross sections themselves can usually be classed as an inconvenience, their effect in distorting the total cross section, which governs neutron penetration, can be more serious. Perhaps more important is the fact that even when the cross section remains positive, it is still often too low due to the same effect and this bias again affects the total cross section and calculated absorption rates. To compensate this bias, the evaluator should put in either a series of negative energy resonances with reasonable size and spacing ("picket fence", or reflect the positive-energy ones around E=0) or a few large fictitious ones ("barber poles"), or a compensating background in File 3 (e.g., see Reference 6).

To compensate interior-region negativity requires a multilevel treatment of which MLBW is the simplest. Although there is no guarantee that MLBW cross sections will be more accurate than SLBW, they are guaranteed to be non-negative (but see next section) and are generally to be preferred over SLBW.

A similar bias occurs at the upper end of the resolved resonance range, where it is less noticeable because it is a positive bias, and most calculations are not as sensitive to this region as they are to the low-energy end. The remedy is the same - extra resonances above the RRR, or compensation in file 3. The latter remedy requires a negative file 3 contribution, which is physically acceptable, but produces undesirable side-effects in some processing codes, hence the extra-resonance remedy is preferred. It is probably safe to say that there is rarely a compelling reason to use the SLBW formula for the calculation of pointwise scattering cross sections. If one is doing a calculation that is sophisticated enough to warrant the use of pointwise cross sections, then a multilevel formalism is certainly justifiable. If one is merely deriving multigroup cross sections, then the other approximations involved justify the use of any reasonable "fix" for the negative scattering, such as simply setting σS = 0 when it goes negative. Such a procedure should usually be accompanied by a corresponding increase in the total cross section.

2.4.11.2. In the MLBW Formalism

Capture and fission use the SLBW formulas and are positive. Scattering uses a formula, which can be written as an absolute square and as such is non-negative. The use of J=I resonances (Section 2.4.7) can destroy the correspondence between the absolute-square form and the expanded form given in Appendix D and result in negative scattering cross sections. Despite its non-negativity, MLBW still produces biased cross sections at both ends of the RRR unless compensating extra resonances or File 3 contributions are included above and below. The evaluator should generally correct for this effect.

2.4.11.3. In the R-matrix, Reich-Moore, and R-function Formalisms

These are again based on an absolute square and cannot be negative. However, they can be biased and extra resonances, background R-values, or File 3 contributions should be provided. If conversion of Reich-Moore to Adler format produces negative cross sections, dummy parameters should be provided to eliminate them.

2.4.11.4. In the Adler Formalism

Although the formulae are derived from an absolute square and are in principle non-negative, in practice the parameters are chosen to fit measured data, so that the physical and mathematical constraints among the parameters, which prevent negative cross sections, are lost, and any of the cross sections can be negative. If the Adler formalism is used for evaluations, negativity should be checked for. The end-effect bias exists in this formalism also and should be checked for in the scattering and total cross sections by comparing with experiment.

2.4.12. Negative Cross Sections in the Unresolved Resonance Region

R. Prael, while at ANL, reported a difficulty with SLBW resonance ladders created by VIM from the unresolved resonance parameters in Mo (MAT l287), namely that the negative File 3 capture background sometimes caused negative capture cross sections in the resonance valleys. The evaluator intended the background to compensate for an excess of capture in the average unresolved capture cross section, but did not anticipate the problem that would arise when the parameters were used in a different context. One remedy is to drop out the negative File 3 background and adjust <Γγ> on whatever energy mesh is needed to produce agreement with the dilute poor-resolution data.

The creation of SLBW ladders from average parameters can be expected to produce the same kind of end-effect bias and frequent negative scattering cross sections found in the resolved resonance region. Again, the scattering cross section per se may not be important, but the biased total cross section may adversely affect calculated reaction rates.

2.4.l3. Use of Two Nuclear Radii

The current ENDF formats defines two different nuclear radii:

a) the scattering radius, AP, and

b) the channel radius, a.

The scattering radius is also referred to as "the effective scattering radius" and "the potential scattering radius". The channel radius is also referred to as "the hard-sphere radius", or "the nuclear radius". The former is the quantity defined as AP (for a+ or â) in File 2, which must be given even if no resonance parameters are given. The nuclear radius is defined in Appendix D, Equation (D.0).

The channel radius is a basic quantity in R-matrix theory, where the internal and external wave-functions are joined and leads to the appearance of hard-sphere phase shifts defined in terms of it. The necessity to relax the definition and permit two radii can be thought of as a "distant-level effect", sometimes not explicit in R-matrix discussions.

The original ENDF/B formats made provision for an AM, or "A-minus", although it was always required that evaluators put AM=0, to signify that it was equal in value to AP. In the current formats, AM is eliminated, but one can anticipate that more sophisticated evaluation techniques may eventually force the reinstatement of not only AM, but a more general dependence of the scattering radius on the channel quantum numbers, especially as higher energies become important.

In theory, the scattering radius depends on all the channel quantum numbers, and in practice it is common to find that different optical model parameters are required for different l-values (s, p, d,...) and for different J-values (p1/2, p3/2, ,,,). This implies that one would require a different scattering radius for each of these states.

For the special case of s-waves, only two J-values are possible, namely I±1/2, commonly denoted J+ and J-. This is the origin of the terminology a+ and a-.

Up through ENDF/B-V, the recommended ENDF/B procedure was to use the above equation for the channel radius in the penetrabilities Pl(ka) and the shift factors Sl(ka), but to use the scattering radius to calculate the hard-sphere phase shifts φl(ka).

Since the phase-shifts define the potential scattering cross section, the evaluator had the freedom to fit AP to a measured cross section while still leaving undisturbed those codes that use the A1/3 formula to calculate the channel radius.

For ENDF/B-VI, new parameters NRO and NAPS are available to give the evaluator more flexibility for the SLBW, MLBW, and RM formalisms, by allowing the evaluator to use AP everywhere and to make AP energy-dependent (Section 2.1).

The full flexibility of channel-dependent radii is provided for the RML format.

2.4.14. The Multilevel Adler-Gauss Formula for MLBW

Appendix D gives (implicitly) for the MLBW formalism the equations:

(2.2)

where RRI labels the resonance-resonance-interference term for a given l-value:

(2.3)

As most users are aware, this double sum over resonances can eat prodigious amounts of computer time unless handled very tactfully. Thus, for a 200-resonance material, there are ~40,000 cross terms, of which only 20,000 need to be evaluated because the expression is symmetric in r and s.

It has been noted many times in the past that partial fractions can reduce Equation (2.3) to a form with only a single Breit-Wigner denominator. Most recently, DeSaussure, Olsen, and Perez (Reference 6) have written it compactly as

(2.4)

The authors give the special case for I=l=0, but it is valid for any set of quantum numbers. Thus an existing SLBW code can be converted to MLBW by adding GrΓr to the symmetric part of the SLBW formula, , and 2H r to the coefficient of (E-Er) in the asymmetric part, .

Since Gr and Hr are weakly energy-dependent, via the penetrabilities and shift factors, they lend themselves to approximations that can sharply reduce computing time compared to the form with the "double" Breit-Wigner denominator. In fact, if the resonances are all treated as s-wave (shifts of zero, penetrabilities of ), and the total widths are taken as constant, then Gr/k2 and Hr/k2 become independent of the neutron energy and consume a negligible amount of computing time so that MLBW and SLBW become equivalent in that respect.

The amplitude-squared form of MLBW, Section 2.4.19, also reduces computing time.

2.4.l5. Notes on the Adler Formalism

Questions concerning the ENDF/B treatment of the Adler formalism are enumerated below10, together with recommended procedures for handling them:

1. The resonance energy µ and total half-width v are the same for each reaction for a given resonance in the Adler formalism, but, for the October l970 version of ENDF-l02, the formulae on page D-7, and the format descriptions of pages 7.9 and N-l2 permit different values for the total, fission, and capture cross sections.

This is a misreading of the formalism; the remedy is to constrain the equalities DETN= DEFN= DECN and DWTN = DWFN = DWCN. The formulas for capture and fission should also have the phases eliminated in Appendix D.

2. The Adler formalism, as applied by the Adlers, breaks the resolved resonance region up into sub-regions, and each is analyzed separately. This avoids problems with contributions from distant resonances, but requires that the polynomial background be tailored to each sub-region. However, the ENDF/B formats allow only one resolved resonance energy region, so this procedure cannot be used.

If a single set of polynomial background constants is insufficient, additional background can be put into File 3, point-by-point.

3. The ENDF/B formats formerly permitted incomplete specification of the cross sections. The allowed values of LI were 5 (total and capture widths); 6 (fission and capture); and 7 (total, fission, and capture). LI=6 leaves the scattering (and total) undefined and LI=5 is deficient for fissile elements. LI=6 is now restricted to ENDF/A, and LI=5 should be used only for non-fissile elements.

4. The nomenclature for the G's and H's is not entirely consistent among different authors. The Adlers use for the total cross section the definitions:

Gt = α cos(2ka)+ β sin(2ka) ;

Ht = β cos(2ka)- α sin(2ka) ;

10 The following is a condensation and updating of the Appendix in the June, l974, Minutes of the Resonance Region Subcommittee.

and then the combination:

νGt + (µ – E) Ht .

For the reaction cross sections there are no phases, and they write

νGc + (µ – E) Hc (capture);

νGf + (µ – E) Hf (fission);

G and H are properly designated as "symmetrical" and "asymmetrical" parameters. This manual changes α to Gt and to Ht, viz:

ν [Gt cos(2ka) + Ht sin(2ka)] + (µ – E) [Ht cos(2ka) – Gt sin(2ka)]

These Gt's and Ht's are no longer symmetrical and asymmetrical, but are referred to that way. The precedent for this nomenclature is probably Reference 7.

DeSaussure and Perez, in their published tables of G and H, incorporate the Adler's constant c into their definition, but otherwise leave the formalism unchanged.

Users and evaluators should adhere to the definitions in this manual.

5. The flag NX, which tells what reactions have polynomial background coefficients given, should be tied to LI, so that the widths and backgrounds are given for the same reactions, i.e., use NX=2 with LI=5 (total and capture), and NX=3 with LI=7 (total, capture, and fission). Since no NX is defined for LI=6 (fission and capture), one is forced to use NX=3 with the background total coefficients set equal to zero, but this now occurs only in ENDF/A, if at all.

2.4.16. Multilevel Versus Single-Level Formalisms in the Resolved and Unresolved Resonance Regions

2.4.16.1. In the Resolved Resonance Region

The SLBW formalism is adequate for resonance treatments that do not require actual pointwise scattering cross sections, as, e.g., multigroup slowing-down codes. Because of the frequent occurrence of negative scattering cross sections, when two or more resonance-potential interference terms overlap, SLBW should not be used to compute pointwise scattering cross sections. Instead, the MLBW formalism should be used, although MLBW is not a true multilevel formalism, but a limit, which is valid if Γ/D is small.

The Reich-Moore reduced R-matrix formalism is a true multilevel formalism, and is recommended for low-energy fissionable s-wave evaluations. All of its cross sections are non-negative, and its only significant drawbacks, apart from the effort required for its application, are the difficulty of determining a suitable R to represent distant-level effects, and of determining the parameters of negative-energy resonances.

The Adler form of the Kapur-Peierls formalism is also a true multilevel treatment, but in actual applications the parameters are determined by fitting data and the theoretical constraints among them are lost, so that any Adler cross section can be negative.

The simplest true multilevel formalism is the reduced R-function, in which all channels except elastic scattering have been eliminated. It makes a very adequate evaluation tool for non-fissile elements up to the threshold for inelastic scattering, since below that the eliminated channels are (usually) simply radiative capture. It can be corrected for distant-level effects by substituting optical-model phase shifts for the hard-sphere ones which occur in the formalism, and by introducing an appropriate R. It can be carried above the inelastic threshold by augmenting it with the use of SLBW formulas for the reactions other than elastic scattering, since such reactions often show negligible multilevel effects. For structural and coolant materials, either Reich-Moore or R-Matrix Limited can be used. The latter provides more detail in describing competitive reactions, plus angular distributions, and allows treating resonances with both l>0 and I>0.

Multi-channel multilevel fitting is also feasible for light elements, and permits the simultaneous use of non-neutron data leading to the same compound nucleus. Due to the complexity of such calculations, they may be presented in ENDF/B as file 3 pointwise cross sections, although the R-Matrix Limited format can handle this case.

2.4.16.2. In the Unresolved Resonance Region

In principle, if the statistical distributions of the resolved resonance parameters are known, any formalism can be used to construct fictitious cross sections in the unresolved region. At the present time, only the SLBW formalism is allowed in ENDF/B, for the reason that no significant multilevel effect can be demonstrated, when SLBW is properly handled.

If resolved region statistics are used without adjustment to poor resolution data, then large multilevel/single-level differences can result, but there is no simple way to determine which is better. If both are adjusted to yield the same average cross sections, and for fissile materials, the same capture-to-fission ratio, then the remaining differences are within the statistical and measurement errors inherent in the method. The above comments on multilevel effects in the unresolved resonance region are based on the work of DeSaussure and Perez [Ref.8].

As noted in Section 2.4.l2, the use of SLBW to construct resonance profiles in the unresolved region will result in the defects associated with this formalism elsewhere, and is not recommended. This application calls for MLBW or better, and the SLBW scheme should be used only for constructing average cross sections where the negative scattering effects will combine with the other approximations and presumably be "normalized out" somewhere along the line.

2.4.17. Preferred Formalisms for Evaluating Data

Unless there is strong reason to do otherwise, the R-Matrix Limited format (LRF=7) should be used for reporting results of new evaluations, as it is the most comprehensive of the current formats.

1. Light nuclei: Use multilevel, multichannel R-matrix. Present either as pointwise cross sections in file 3, or as R-matrix parameters using LRF=7.

2. Materials with negligible or moderate multilevel effects, and no multichannel interference: Reich-Moore or MLBW. These are equivalent in computing time and all require kernel broadening, although MLBW lends itself to the ψ,χ-approximation discussed in Section 2.4.14. However, RM and RML provide the angular distribution of elastically-scattered neutrons, which MLBW does not.

3. Materials with strong multilevel effects, but no multichannel interference: Reich-Moore or R-Matrix Limited. The structural materials do not exhibit channel-channel interference, but have level-level interference that is too strong for an MLBW treatment.

4. Materials with observable channel-channel interference: Reich-Moore or R-Matrix Limited. In the past, only low-energy fissionable materials have shown channel-channel interference, and this is unlikely to change. Reich-Moore evaluations can be converted to Adler format for presentation in ENDF/B. The reason why Reich-Moore is preferred to Adler-Adler as the basic evaluation tool is that it has less flexibility and is therefore better able to distinguish between various grades of experimental data. However, it requires kernel broadening whereas Adler-Adler uses ψ and χ, making the latter more convenient to broaden. Unfortunately some of this convenience is lost in practice because there is no simple equivalence between Adler-Adler and SLBW (see Section 2.4.8). With modern computers and modern computer codes, the slight advantage offered by kernel broadening is no longer an important issue.

5. Materials with channel-channel interference and one or more competitive reactions: R-matrix, using the format LRF=7 to present the parameters.

2.4.18. Computer Time for Generating MLBW Cross Sections

Previous solutions to the problem of evaluating the double-sum form of the MLBW resonance-resonance interference term in a reasonable amount of time have been to use the amplitude-squared form from which it was derived, and kernel-broaden it, or to optimize the calculation of inner and outer loop quantities.

A third solution is to use the Multilevel Adler-Gauss formulas discussed in Section 2.4.14 and possibly approximate the energy-dependence of the Gr- and Hr-coefficients.

The amplitude-squared form of MLBW is discussed in Section 2.4.l9.

2.4.19. Amplitude-Squared Form of the MLBW Formulas

The form of the MLBW scattering cross section given in Appendix D and in Section 2.4.l4, is mathematically identical to the more fundamental "amplitude-squared" form given in Appendix D, as Equations (5) - (7) of Section D.l.2.

Those equations can be coded in complex Fortran, or broken up into their real and imaginary parts before coding. The essential point is that they sum the resonances before squaring. This avoids turning two "linear" sums into one "quadratic" one. If an isotope has 200 resonances, the above formulas have two sums with 200 terms each, whereas the ENDF form has a sum with 40000 cross terms. A discussion of points to consider in coding the above equations is given in Sections 2.4.23 and 2.4.24.

The main drawback to the above equations is that they do not admit Doppler-broadening with ψ- and χ-functions, but require kernel methods instead.

2.4.20. Degrees of Freedom for Unresolved Resonance Parameters

A resonance in the system (neutron plus a target of mass A) corresponds to a quasi-stationary state in the compound nucleus A + 1. Such a resonance can decay in one or more ways, each described as a channel. These are labeled by the identity of the emitted particle (two-body decay), the spins I and i of the residual nucleus and the emitted particle, and the orbital angular momentum l of the pair. To uniquely specify the channel, two more quantum numbers are needed, since the magnetic quantum numbers can be eliminated for unpolarized particles.

It is common to give the channel spin, s, which is the vector sum of I and i, plus , since this facilitates the isolation of the l-dependence of all channel quantities. The important point is that the same set of three ingredient angular momenta, I, i, and l, will give rise to a number of different channels, according to the rules for coupling angular momenta. The resonance will decay into each of these channels, with a probability that is governed by a real number γαIiJls, the reduced width amplitude, where α gives the identity of the emitted particle, the state of excitation of the daughter nucleus, etc. The partial width for the channel is:

ΓαIiJls = 2PαIiJls γ2αIiJls .

The penentrabilities depend only on l, and are given in Appendix D for uncharged particles. For charged particles, their Coulomb analogs can be found in texts on the subject, and for gamma rays one uses rather than γ and P.

If the collection of channel quantum numbers (αIiJls) is denoted by c, then the total width for the level is Γ = ΣcΓc. [Σc means a sum over all channels]. The argument from statistical compound nucleus theory is that the γc's are random variables, normally distributed with zero mean and equal variance. The population referred to is the set of γc's for a given channel and all the levels (or resonances). It follows that the total width is distributed as a chi-squared distribution with N degrees of freedom, since this is the statistical consequence of squaring and adding N normal variates. For N=l, this is the Porter-Thomas distribution. In determining the behavior of any quantity that is going to be averaged over resonances, it is necessary to know the way in which the widths are distributed, hence the inclusion of these degrees of freedom in ENDF/B.

1. The neutron width is governed by AMUN, which is specified for a particular l-value. Usually, only the lowest allowed l-value will be significant in any decay, although the formats would allow giving both s- and d-wave widths for the same resonance. Since there is only one J-value for a given resonance, and we label the widths by one l-value, there can be at most two channels for neutrons (i = 1/2), labeled by the channel spin values s = I±1/2. If I = 0, there is only one channel, s = i = ½; hence the restriction, 1.0AMUN 2.0. AMUN is the quantity µl,J, discussed in Section D.2.2.2.

Although there is no supporting evidence, it is assumed that the average partial widths for each channel spin are equal, and that <Γn> is the sum of two equal average partial widths. In Appendix D this factor of two is absorbed into the definition of <Γn>, through the use of a multiplicity, which is the number of channel spins, 1 or 2.

2. The competitive width is currently restricted to inelastic scattering, which has the same behavior as elastic scattering, measured from a different "zero channel energy," hence

1.0 AMUX 2.0

Note that one should not set AMUX = 0 out of ignorance of its true value, as suggested in previous versions of ENDF-l02. This implies a constant from resonance to resonance, since the chi-squared distribution approaches a delta function as N→∞. An inelastic reaction can be expected to proceed through a small number of channels and hence to fluctuate strongly from level to level.

Specifically, AMUX = , where J is the spin of the resonance, and is the orbital angular momentum of the inelastically scattered neutron. Since the daughter nucleus may have a spin different from the target spin I, may be different from l, and the number of channel spin values may be different from .

3. For the radiative capture process, AMUG should be set equal to zero. Radiative capture proceeds through many channels and it is not worthwhile deciding if AMUG is 30 or 40. (If some nucleus has selection rules that restrict radiative decay to a few channels, then a different value of AMUG might be appropriate.)

4. The fission value should be given as 1.0AMUF4.0 and the value zero would be incorrect. These small values violate the previous discussion of (Wigner-type) channels and obey instead statistics governed by fission barrier tunneling (Bohr-channels). The actual value of AMUF is determined by comparison between calculated and measured cross sections.

The degrees of freedom are constant throughout the unresolved resonance region.

2.4.21. Procedures for the Unresolved Resonance Region

Up to 250 energy points are permitted for specifying energy-dependent average parameters. This number is presumed to be sufficient to reproduce the gross structure in the unresolved cross sections. Within a given isotope the same energy grid must be used for all J- and l-values. The grids may be different for different isotopes. Unresolved resonance parameters should be provided for neutron energy regions where temperature-broadening or self-shielding effects are important. It is recommended that the unresolved resonance region extend up to at least 20 keV.

If the flag LSSF (Section 2.3.1) is set equal to one, the evaluator can specify the gross structure in the unresolved range on as fine an energy grid as he desires, subject only to the overall 10000-point limitation. Under this option, File 3 represents the entire dilute unresolved cross section, and no File 2 contribution is to be added to it. Instead, File 2 is to be used to compute a "slowly-varying" self-shielding factor that may be applied to the "rapidly-varying" File 3 values. The self-shielding factor is defined as the ratio of File 2 average shielded cross section to the average unshielded value computed from the same parameters. This ratio is to be applied as a multiplicative factor to the values in File 3.

If LSSF is set equal to zero, File 3 will be interpreted in the same way as a resolved-region File 3, i.e., it will represent a partial background cross section to be added to the average cross section, dilute or shielded that is computed from File 2.

The self-shielding-factor procedure has certain advantages over the "additive" procedure:

1. The energy-variation of the dilute cross section in the unresolved region can be more accurately specified, without the 250-point limitation imposed in File 2.

2. The energy grids in File 2 and File 3 are basically uncoupled, so that the File 2 grid can be made coarser and easier to process.

3. In principle, the results can be more accurate, since File 2 can be devoted entirely to representing changes in the average parameters that are significant for shielding. The burden of representing fluctuations in the size of the dilute cross section is taken over entirely by File 3.

4. The same representation can be used by codes requiring probability tables. For this application, the average parameters in File 2 can be used to generate random ladders of resonances, and the resulting cross sections can be used to calculate probability tables in the usual way. However, instead of using the tables directly, they are normalized by dividing the various cross section bands by the average cross section in the interval. These normalized probabilities are then converted back to cross sections by multiplying them into the File 3 values. The rationale is the same as for the shielding-factors - the dilute cross section is represented in "poor-resolution" format in File 3, while the real fine-structure is established in File 2.

The following caution should be noted by evaluators in choosing this option:

Because File 3 is energy varying, it inherently has the possibility to energy-self-shield itself. If File 2 also shields it, one may actually "double-shield". The problem will probably be most acute just above the boundary between the resolved and unresolved regions, since the experimental resolution may still be good enough to see clumps of only a few resonances.

One might consider "correcting" for this in the choice of File 2 parameters, but this would be difficult because the degree of shielding is application dependent. A better procedure would be to insure that each significant structure in File 3 actually represents a statistically meaningful number of resonances, say ten or more. If the raw data do not satisfy this criterion, then additional smoothing should be applied by the evaluator to make it a correct condition on the data. A careful treatment will require the use of statistical level theory to determine the true widths and spacings underlying the File 3 structures.

2.4.22 Procedures for Computing Angular Distributions in the Resolved Resonance Range

2.4.22.1. Background

Quantum mechanical scattering theory, which underlies all of the resonance formalisms in this chapter, describes the angular distribution of exit particles as well as the magnitudes of the various reactions. When the R-matrix formalism is used to parameterize the collision matrix, as in the Reich-Moore format (Section D.1.3) or the RML format (Section D.1.7), then the angular distributions exhibit a resonant behavior, in the sense that they may change substantially in passing through a resonance. An explicit tabulation of this detailed resonance behavior will usually imply a very large data file.

Blatt and Biedenharn [Ref. 7] simplified the general expression for the angular distribution, which is an absolute square of an angle-dependent amplitude, so that it became a single sum over Legendre polynomials. Their expression, particularized to the RML format, is given in Sections D.1.7. In the past, Reich-Moore has been a vehicle for low-energy fissionable isotope evaluations, usually s-waves only, so that the angular distribution is isotropic. If it were used for higher energies and higher angular momenta, then the angular distributions would become anisotropic. Of course, since the formulas define a center-of-mass distribution, even the isotropic case generally defines an anisotropic laboratory distribution.

In principle, similar angular distribution formulas underlie the SLBW, MLBW, and Adler-Adler formalisms, but since these are not formulated in terms of collision matrix elements (UlsJ), the Blatt and Biedenharn formulas are not immediately applicable to them.

Although the Blatt and Biedenharn formulas have been around for thirty-five years, and have been much used in the physics literature of scattering theory, they have not been widely employed in neutron cross section evaluation. ENDF/B files most often contain either experimental data or calculated data derived from an optical model. Both of these types represent a "smoothing" or "thinning" of the underlying resonant angular distributions. In the case of experiment, the smoothing is done by the resolution-broadening of the measuring apparatus, combined with the necessarily limited number of energies at which data can be taken. In the optical model case, the smoothing is done in an obscure, highly implicit manner. It seems quite clear that an explicit energy-average over resonant Blatt and Biedenharn Legendre coefficients will differ from both of the above representations.

This raises the question of whether the Blatt and Biedenharn average will be better or worse than the others. That question is dealt with in the following paragraphs, which are somewhat "theoretical", since there is not much hard experience in this area.

2.4.22.2 Further Considerations

Firstly, if in some ideal case, all the resonance spins and parities were precisely known, then the Blatt and Biedenharn values would be exact, and clearly superior to any other representation. The next step down the accuracy ladder would be a case where the major resonances, or anti-resonances ("windows") were known, but some minor, narrower ones were uncertain. For this case, one might find that errors in the "minor" resonances canceled each other, again producing a superior result, or one might find an erroneous cooperation, resulting in spurious-values.

Finally, there are evaluations that use compiled resonance parameters, with many guessed J and l-values, in which case the cancellations and/or cooperations dominate the angular distributions. In both of the two latter cases, the evaluator either will or will not have compared with experiment and made a decision on the accuracy of the Blatt and Biedenharn representation. The flag LAD allows him to inform the processing code whether or not it is "safe" to calculate from the Blatt and Biedenharn formulas. Such a flag is necessary because File 4 is limited to 1200 angular distributions, which is usually not enough to represent a fully-detailed Blatt and Biedenharn representation. The recommended ENDF/B procedure is for the evaluator to provide an under-1200-energy-point representation in File 4, and to signal the user with LAD whether he can independently generate σ(θ) on a finer energy mesh.

For the File 4 representation, the evaluation should smooth the data so as to preserve significant structure in the first Legendre coefficient, or µ. As always, the word significant is difficult to define exactly but the File 4 representation should be adequate for most ordinary reactor engineering applications.

In any case, a user who wishes to examine the implications for his own work of a finer mesh is free to use the Blatt and Biedenharn formulas. The flag LAD tells him either that the evaluator has approved this procedure (LAD=1), or that it is either of unknown quality or known to be poor (LAD=0). In the case of LAD=0, the evaluator should tell which of these is the case by putting comments into File 1 and the associated documentation.

2.4.22.3. Summary of Recommendations for Evaluation

1. Supply an under-1200-point representation of the elastic scattering angular distribution in File 4. Preserve significant structure in µ.

2. If the Blatt and Biedenharn angular distributions were not examined, or if they were examined and found to be inaccurate, supply LAD=0 in File 2. Tell which of these is the case in File 1 and in the associated documentation.

3. If the Blatt and Biedenharn angular distributions were found, or are believed, to be accurate, supply LAD=1, and describe the evaluation procedures in the documentation.

2.4.23. Completeness and Convergence of Channel Sums

Two possible errors in the calculation of cross sections from a sum over individual channels are:

1.) omission of channels because they contain no resonances (such "non-resonant" or "phase-shift-only" channels must still be included because they contribute to the potential scattering cross section), and

2.) failure to include enough non-resonant channels to insure convergence of the potential-scattering cross section with respect to l at high energy.

Avoiding the first is the responsibility of the processing codes for the SLBW, MLBW, A-A, and R-M formalisms, since the formats do not allow the evaluator to specify empty channels explicitly. For the RML formalism, where such specification is explicit, the responsibility is the evaluator's. Avoiding the second is always the evaluator's responsibility, since it would be awkward for a processing code to decide whether the omission was intentional or not.

In the channel spin representation, the incident spin, i, is coupled to the target spin, I, to form the channel spin, s, which takes on the values:

|I–i| s I+i.

The channel spin couples to the orbital angular momentum to form the total angular momentum J, with the values:

|l–s| J l+s.

If I > 0 and l > 0, the same J-value may occur for each of the two channel spins, s = I ± 1/2, and each of these J-values must be separately included. A width Γ l J is a sum of the two components, and ; and in the SLBW, MLBW, and A-A formalisms, only the sum is used. In the Reich-Moore format, the specification of J implicit (via the use of a signed AJ value). For the RML format, the evaluator should specify two separate channels (for the two values of s) within the same spin group in this situation.

There is rarely enough information on channel-spin widths to guide the evaluator in apportioning the total width between the two sub-channels, but fortunately, most neutron reactions are insensitive to the split, so that putting it all in one and none in the other, or splitting it 50/50 works equally well. Angular distributions are in principle more sensitive, but it is similarly unusual to find measured data of sufficiently high precision to show an effect.

The channel sums are infinite,

so the question of convergence arises. The simplest case is where the summand is an SLBW reaction term, as in Section D.1.6.2, in which case one sums only over channels in which there are resonances. There are then no convergence considerations.

If one is summing scattering cross section terms, as in Section D.1.6.1, there is a potential-scattering amplitude in every channel, independent of whether there are resonances or not. The l=0, or s-wave amplitudes, are finite at zero energy, but the higher l-waves only come in at higher energies. The convergence criterion is therefore that the addition of the next higher l-wave produces a negligible change in the cross section at the highest energy covered by the resonance region. In a conventional R-matrix treatment, the non-resonant channels contain hard-sphere phase shifts whose behavior has to be compared with experiment.

For the R-Moore format, NLS is defined as that value which converges the cross section calculation. This is different from the SLBW/MLBW definition, which is the number of l-channels with resonances. The latter is more liable to cause neglect of higher-l non-resonant channels. Such neglect would show up as incorrect between-resonance scattering at high energies, admittedly not the easiest defect to see.

If angular distributions are to be calculated, as in Sections D.1.5.9 and D.1.6.5, besides having more complicated sums, the range of l-values is much greater, the requirement being that the angular distributions converge at the highest energies. Because the high l-amplitudes interfere with the low ones, non-negligible cross terms occur which are absent from the cross section sums. The different convergence criteria, NLS and NLSC, are compatible because only the B0 moment contributes to the cross sections. All the higher moments integrate to zero. Computer codes which reconstruct such moments should have recursive algorithms for l-dependent quantities up to l=20.

For the R-Matrix Limited format, all terms and only those terms specified by the evaluator (i.e., included in the File 2 information) are to be included in every calculation.

2.4.24. Channel Spin and Other Considerations

For the R-Matrix Limited format, channel spin is explicit and the evaluator must in general provide partial widths that depend on s as well as l and J.

For the Adler-Adler formalism, the usual area of application is to low-energy fissile nuclides, with l = 0, so that channel spin is not mentioned in the formulae of Appendix D.

For the Reich-Moore formalism, in those cases where two channel spins are possible, the channel spin is specified by the sign of the AJ parameter. In older evaluations where the channel spin is not specified (i.e., where all AJ are positive), all resonances are assumed to have the same channel spin and the hard-sphere contribution from the second channel spin must be added separately.

For MLBW the absolute square has been expanded out and all imaginary quantities eliminated. This has several consequences.

1. Channel spin is effectively eliminated, because the partial widths occur in "summed" form.

Since only the sum is required, the evaluator is spared the necessity of specifying the separate s-values. This converts an (l,s,J) formalism into an (l,J) formalism. The same effect can be achieved by assuming that I=0, a popular assumption often made independently of the truth, as in many optical model calculations.

2. The convergence criterion is more transparent, because the potential-scattering cross section splits off from the resonance and interference terms, as

Despite the simpler nature of this term than its parent amplitudes, one must still carry enough terms to make the results physically correct, and if this cannot be done, then File 3 must be invoked to achieve that goal.

3. The resonance profiles are expressible in terms of symmetric and asymmetric Breit-Wigner shapes, and thus admit ψ,χ Doppler broadening. The price one pays for these three advantages is increased computing time, when the number of resonances is large.

Similar remarks apply to the SLBW formalism, which is MLBW without the resonance- resonance interference terms. The computing time goes way down, but the scattering cross section is very poor. SLBW has useful applications in certain analytical and semi-analytical procedures, but should never be used for the calculation of explicit pointwise scattering cross sections.

The omission of an explicit channel-spin quantum number in the SLBW formalism, while convenient in the resolved resonance region, has occasioned some difficulty in the unresolved region. Sections D.2.2 through D.2.4 attempt to clarify the situation with respect to level densities, strength functions, and spin statistics.

2.5. References for Chapter 2

1. P. Greebler and B. Hutchins. Physics of Fast and Intermediate Reactors, Vienna, 3-11 August 1961, Vol. III (International Atomic Energy Agency, 1962) p. 121

2. H. Henryson II, B. J. Toppel, and C. G. Stenberg, Argonne National Laboratory report ANL-8144 (1976)

3. M. Beer, Nuc. Sci. Eng. 50 (1973) 171

4. C.R. Lubitz, Equivalent Single-Level Breit-Wigner Resonance Parameters for Adler-Adler Evaluations, Letter to CSEWG, August 20, 1985

5. G. deSaussure and R.B. Perez, Oak Ridge National Laboratory report ORNL-TM-2599 (1969)

6. G. deSaussure, G. Olsen, and R.B. Perez, Nuc. Sci. Eng. 61 (1976) 496

7. D.B. Adler, Brookhaven National Laboratory report BNL 50045 (1967) page 7

8. G. deSaussure and R.B. Perez,, Nuc. Sci. Eng. 52 (1973) 382

9. A.M. Lane and R. G. Thomas, Rev. Mod. Phys. 30 (1958) 257

10. G.F. Auchampaugh, Los Alamos National Laboratory report LA-5473-MS (1974)

11. N.M. Larson, Oak Ridge National Laboratory report ORNL/TM-9179/R6 and ENDF-364 (2003)

12. A.W. Solbrig, Am. J. Phys. 24 (1961) 257

3. FILE 3, REACTION CROSS SECTIONS

3.1. General Description

Reaction cross sections and auxiliary quantities are given in File 3 as functions of energy E, where E is the incident energy in the laboratory system. They are given as energy-cross section (or auxiliary quantity) pairs. An interpolation scheme is given that specifies the energy variation of the data for incident energies between a given energy point and the next higher point. File 3 is divided into sections, each containing the data for a particular reaction (MT number); see Section 0.5 and Appendix B. The sections are ordered by increasing MT number. As usual, each section starts with a HEAD record and ends with a SEND record. The file ends with a FEND record.

3.2. Formats

The following quantities are defined

ZA,AWR

Standard material charge and mass parameters.

QM

Mass-difference Q value (eV): defined as the mass of the target and projectile minus the mass of the residual nucleus in the ground state and masses of all other reaction products; that is, for a+Ab+c+...+B, QM=[(ma+mA)-(mb+mc+...+mB)](9.315016x108) if the masses are in amu. (See paragraph 3.3.2).

QI

Reaction Q value for the (lowest energy) state defined by the given MT value in a simple two-body reaction or a breakup reaction. Defined as QM for the ground state of the residual nucleus (or intermediate system before breakup) minus the energy of the excited level in this system. Use QI=QM for reactions with no intermediate states in the residual nucleus and without complex breakup (LR=0). (See paragraph 3.3.2.)

LR

Complex or "breakup" reaction flag.

Indicates that additional particles not specified by the MT number will be emitted. See Sections 0.5.5 and 3.4.4.

NR,NP,Eint

Standard TAB1 parameters.

σ(E)

Cross section (barns) for a particular reaction (or the auxiliary quantity) given as a table of NP energy-cross section pairs.

The structure of a section is

[MAT, 3, MT/ ZA, AWR, 0, 0, 0, 0] HEAD

[MAT, 3, MT/ QM, QI, 0, LR, NR, NP/ Eint / σ(E)] TAB1

[MAT, 3, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND

3.3. General Procedures

3.3.1. Cross Sections, Energy Ranges, and Thresholds

For incident neutrons, the cross-section data must cover an energy range up to a common upper limit of at least 20 MeV, and the data must extend to a lower limit of the reaction threshold or 10-5 eV whichever is higher. For other reactions, the cross section should start at the reaction threshold energy (with a value of 0.0 barns) and should continue up to a common upper energy limit.

In the case where there is a change in the representation above a given energy, e.g., a change from separate reactions to MT=5 (sum of reactions not given separately in other sections), the following procedure should be used. For the cross sections in the lower energy region, there should be a duplicate point with a value of zero at the last energy for a which a non-zero cross section is given, and a point with a value of zero at the common upper energy limit. Similarly, for the cross sections in the upper energy region, there should be a duplicate point with a value of zero at the first energy for a which a non-zero cross section is given, and a zero value at the low energy limit. The evaluator should document the change in representation in the File 1 comments.

For charged-particle emission, the cross section is usually very small from the threshold (or lower limit) up to an effective threshold defined by a noticeable cross section (for example, 10-10 barns). The evaluator should tabulate a cross section of 0.0 in such a range in order to avoid interpolation problems.

Sometimes ENDF reactions have an apparent upper limit lower than the upper limit for the material due to changes in representation in different sections. For example, there might be a change from discrete levels to a continuum rule, or from separate reactions to MT=5. Such cross sections must be double valued at the highest energy for which the cross section is nonzero. The second cross section at the discontinuity must be zero, and it must be followed by another zero value at the upper limit. This will positively show that the cross section has been truncated. For such reactions, there will be another reaction with an artificial threshold at the discontinuity. The cross sections must be chosen in such a way that their sum is continuous.

The limit on the number of energy points (NP) to be used to represent a particular cross section is given in Appendix G. The evaluator should not use more points than are necessary to represent the cross section accurately. When appropriate, resonance parameters can be used to help reduce the number of points needed. The evaluator should avoid sharp features such as triangles or steps (except for the required discontinuities at the limits of the resonance ranges or where reactions change representation), because such features cannot be realistically Doppler broadened.

3.3.2. Q Values

Accurate Q values should be given for all reactions, if possible. If QI is not well defined (as for a range of levels in MT=91, 649, 699, 749, 799, or 849), use the value of QI which corresponds to the threshold of the reaction. Similarly, if the value of QM is not well defined (as in elements or for summation reactions like MT=5), use the value of QM which gives the threshold. If there is no threshold, use the most positive Q value of the component reactions. Note that these ill-defined values of QM cannot be relied on for energy-release calculations.

As an example to clarify the use of QM and QI, consider the reaction α+9Ben+X. After the neutron has been emitted, the compound system is 12C with QM=5.702 MeV and energy levels (Ex) at 0.0, 4.439, 7.654, and 9.641 MeV. The ground state is stable against particle breakup, the first level decays by photon emission, and the higher levels decay with high probability by breaking up into three alpha particles (7.275 MeV is required). This pattern can be represented as follows.

Reaction

QM

QI

EX

MT

9Be(a,n0)12C

5.702

5.702

0.000

50

9Be(a,n1)12C

5.702

1.263

4.439

51

9Be(a,n2)12C(3α)

-1.573

-1.952

7.654

52

9Be(a,n3)12C(3α)

-1.573

-3.939

9.641

53

9Be(a,nC)12C(3α)

-1.573

-1.573

91

The gamma for the second reaction is not written explicitly in this notation. The last reaction

includes the contributions of all the levels above 9.641 MeV, any missed levels, and any direct

four-body breakup; therefore, the threshold for MT=91 may be lower than implied by the fourth level of 12C. Note the value used for QI.

3.3.3. Relationship Between File 3 and File 2

If File 2 (Resonance Parameters) contains resolved and/or unresolved parameters (LRP=1), then the cross sections or self-shielding factors computed from these parameters in the resonance energy range for elastic scattering (MT=2), fission (MT=18), and radiative capture (MT=102) must be combined with the cross sections given in File 3. The resonance contributions must also be included in any summation reactions that involve the three resonance reactions (for example, MT=1, 3, or 5). The resonance energy range is defined in File 2. Double-valued energy points will normally be given in File 3 at the upper and lower limits of the unresolved and resolved resonance regions.

Some materials will not have resonance parameters but will have a File 2 (LRP=0) that contains only the effective scattering radius. This quantity is sometimes used to calculate the potential scattering cross section in self-shielding codes. For these materials, the potential scattering cross section computed from File 2 must not be added to the cross section given in File 3. The File 3 data for such materials comprise the entire scattering cross section.

In certain derived libraries, the resonance cross sections have been reconstructed and stored in File 3. Such files may have LRP=0 as described in the preceding paragraph. Alternatively, they may have LRP=2 and include a full File 2 with complete resonance parameters. In this case, resonance cross sections or self-shielding factors computed from File 2 are not to be combined with the cross sections in File 3.

3.4. Procedures for Incident Neutrons

Cross section data for non-threshold reaction types must cover the energy range from a lower limit of 10-5 eV to an upper limit of at least 20 MeV for all materials. For non-threshold reactions, a cross section value must be given at 0.0253 eV. The limit on the number of energy points (NP) to be used to represent a particular cross section is 50,000. The evaluator should not use more points than are necessary to represent the cross section accurately.

The choice of data to be included in an evaluation depends on the intended application. For neutron sublibraries, it is natural to define "transport" evaluations and "reaction" evaluations. The transport category can be further subdivided into "low-energy transport" and "high-energy transport."

A reaction evaluation will contain File 1, File 2, File 3, and sometimes File 32 and/or File 33. File 2 can contain resonance parameters. If radioactive products must be described, Files 8, 9, 10, 39, and/or 40 may be present. File 3 may tabulate one or more reaction cross sections. The total cross section is not usually well defined in reaction evaluations since they are incomplete. Examples of this class of evaluations include activation data and dosimetry data. A low-energy transport evaluation should be adequate for calculating neutron transport and simple transmutations for energies below about 6-10 MeV. Photon production and covariance data should be included when possible. Typical evaluations will include Files 1, 2, 3, 4, 5, and sometimes Files 8, 9, 10, 12, 13, 14, 15, 31, 32, 33, 34, 35, 39, and/or 40. Resonance parameters will usually be given so that self shielding can be computed. Charged-particle spectra (MT=600-849) and neutron energy-angle correlation (MF=6) will usually not be given. File 3 should include all reactions important in the target energy range, including the total (MT=1) and elastic scattering (MT=2). Other reactions commonly included are inelastic scattering (MT=4,51-91), radiative capture (MT=102), fission (MT=18,19-21,38), absorption (MT=103,104,105,...), and other neutron emitting reactions such as MT=16,17,22,28,.... Specific procedures for each reaction are given below. Examples of this class of evaluations include fission-product data and actinide data.

A high-energy transport evaluation should be adequate for calculating neutron transport, transmutation, photon production, nuclear heating, radiation damage, gas production, radioactivity, and charged-particle source terms for energies up to at least 20 MeV. In some cases, the energy limit needs to be extended to 40-100 MeV. These evaluations use Files 1, 2, 3, 4, 5, 6, 12, 13, 14, 15, and sometimes 8, 9, 10, 31, 32, 33, 34, 35, 39, and/or 40. Once again, File 3 should give cross sections for all reactions important in the target energy range, including MT=1 and 2. This will normally include many of the reactions mentioned above plus the series MT=600-849. At high energies, some reactions may be combined using the "complex reaction" identifier MT=5. File 6 will normally be needed at high energies to represent energy-angle correlation for scattered neutrons and to give particle and recoil energies for heating and damage calculations. Special attention to energy balance is required. High-energy evaluations are important for materials used in fusion reactor designs, in shielding calculations, and in medical radiation-therapy equipment (including the components of the human body).

3.4.1. Total Cross Section (MT=1)

The total is often the best-known cross section, and it is generally the most important cross section in a shielding material. Considerable care should be exercised in evaluating this cross section and in deciding how to represent it.

Cross section minima (potential windows) and cross section structure should be carefully examined. Sufficient energy points must be used in describing the structure and minima to reproduce the experimental data to the measured degree of accuracy.

The total cross section, as well as any partial cross section, must be represented by 50,000 incident-energy points or fewer. The set of points or energy mesh for the total cross section must be a union of all energy meshes used for the partial cross sections. Within the above constraints, every attempt should be made to minimize the number of points used. The total cross section must be the sum of MT=2 (elastic) and MT=3 (nonelastic). If MT=3 is not given, then the elastic cross section plus all nonelastic components must sum to the total cross section.

The fact that the total cross section is given at every energy point at which at least one partial cross section is given allows the partial cross sections to be added together and checked against the total for any possible errors. In certain cases, more points may be necessary in the total cross section over a given energy range than are required to specify the corresponding partial cross sections. For example, a constant elastic scattering cross section and a 1/v radiative capture cross section could be exactly specified over a given energy range by log-log interpolation (INT=5), but the sum of the two cross sections would not be exactly linear on a log-log scale. If a precise total cross section is required between the energy points provided, it is recommended that the total be calculated from the sum of the partials rather than interpolated directly from MT=1.

3.4.2. Elastic Scattering Cross Section (MT=2)

The elastic scattering cross section is generally not known to the same accuracy as the total cross section. Frequently, the elastic scattering cross section is obtained by subtracting the nonelastic cross section from the total cross section. This procedure can cause problems. The result is an elastic scattering cross section that contains unreal structure. There may be several causes. First, the nonelastic cross section, or any part thereof, is not generally measured with the same energy resolution as the total cross section. When the somewhat poorer resolution nonelastic data are subtracted from the total, the resolution effects appear in the elastic cross section. Second, if the evaluated structure in the nonelastic cross section is incorrect or improperly correlated with the structure in the total cross section (energy-scale errors), an unrealistic structure is generated in the elastic scattering cross section.

The experimental elastic cross section is obtained by integrating measured angular distributions. These data may not cover the entire angular range or may contain contributions from nonelastic neutrons. Such contamination is generally due to contributions from inelastic scattering to low-lying levels that were not resolved in the experiment. Care must be taken in evaluating such results to obtain integrated cross sections. Similarly, experimental angular distribution data can also cause problems when used to prepare File 4.

3.4.3. Nonelastic Cross Section (MT=3)

The nonelastic cross section is not required unless any part of the photon production multiplicities given in File 12 uses MT=3. In this case, MT=3 is required in File 3. If MT=3 is given, then the set of points used to specify this cross section must be a union of the sets used for its partials.

3.4.4. Inelastic Scattering Cross Sections (MT=4,51-91)

A total inelastic scattering cross section (MT=4) must be given if any partials are given; that is, discrete level excitation cross sections (MT=51-90), or continuum inelastic scattering (MT=91). The set of incident energy points used for the total inelastic cross section must be a union of all the sets used for the partials.

Values should be assigned to the level excitation cross sections for as many levels as possible and extended to as high an energy as possible. Any remaining inelastic scattering should be treated as continuum. In particular, low-lying levels with significant direct interaction contributions (such as deformed nuclei with 0+ ground states) should be extended to the upper limit of the file (at least 20 MeV) in competition with continuum scattering. The secondary energy distribution for such neutrons resembles elastic scattering more than an evaporation spectrum.

Level excitation cross sections must start with zero cross section at the threshold energy. If the cross section for a particular level does not extend to the upper limit for the file (e.g., 20 MeV), it must be double-valued at the highest energy point for which the cross section is non-zero. The second cross section value at the point must be zero, and it should be followed by another zero value at the upper limit. This will positively show that the cross section has been truncated.

If LR=0, a particular section (MT) represents (n,n'gamma). The angular distribution for the scattered neutron must be given in the corresponding section of File 4 or 6. The associated photons should be given in a corresponding section of File 6 or 12, if possible. If the inelastic photons cannot be assigned to particular levels, they can be represented using MT=4 in File 6, 12 or 13. When inelastic photons cannot be separated from other nonelastic photons, they can be included in MF=13, MT=3.

A LR flag greater than zero indicates inelastic scattering to levels that de-excite by breakup, particle emission, or pair production rather than by photon emission (see Section 0.6)1 . If LR=1, the identities, yields, and distributions for all particles and photons can be given in File 6. If LR>1, angular distributions for the neutron must be given in File 4, and distributions are not available for the emitted particles. In this case, photon production is handled as described above for LR=0.

If a particular level decays in more than one way, then File 6 can be used or several sections can be given in File 3 for that level. Consider the case in which an excited state sometimes decays by emitting a proton, and sometimes by emitting an alpha particle. That part of the reaction that represents (n,n'α) would use LR=22, and the other part would be given the next higher section number (MT) and would use LR=28 (n,n'p). The angular distribution for the neutron would have to be given in two different MT numbers in File 4, even though they represent the same neutron. The sections must be ordered by decreasing values of QI (increasing excitation energy).

3.4.5. Fission (MT=18,19-21,38)

The total fission cross section is given in MT=18 for fissionable materials. Every attempt should be made to break this cross section up into its various parts:first-chance fission (n,f), MT=19; second-chance fission (n,n'f), MT=20; third-chance fission (n,2nf), MT=21; and fourth-chance fission (n,3nf), MT=38. The data in MT=18 must be the sum of the data in MT=19, 20, 21, and 38. The energy grid for MT=18 must be the union of the grids for all the partials.

1 LR=31 is still allowed, however, to uniquely define the γ-decay when using MF=3, and MF=12 (or 15) and MF=4.

If resolved or unresolved resonance parameters are given in File 2, the fission cross section computed from the parameters must be included in both MT=18 and MT=19.

The Q value for MT=18, 19, 20, 21, and 38 is the energy released per fission minus the neutrino energy. It should agree with the corresponding value given in MT=458 in File 1.

Secondary neutrons from fission are usually stated to be isotropic in the laboratory system in File 4. Energy distributions are given in File 5. The complex rules associated with the partial fission reactions are described in Section 5.

3.4.6. Charged-Particle Emission to Discrete and Continuum Levels (MT=600-849)

The (n,p) reaction can be represented using a summation cross section, discrete levels, and a continuum (MT=103, 600-648, and 649) in the same way that the (n,n') reaction is represented using MT=4, 51-90, and 91 (see Section 3.4.4). Similarly, (n,d) uses MT=104 and 650-699, and so on for t, 3He, and α. Of course, MT=600, 650, 700, etc., represent the ground state and would not have corresponding sections in the photon production files, unless the flag LR>0 (such as in the 10B(n,t)8Be reaction).

3.5. Procedures for Incident Charged Particles and Photons

See Table 0.1 for sublibrary numbers for incident charged particles and photons. Procedures for incident charged particles are generally the same as for neutrons as given in section 3.4. The exceptions are noted below.

3.5.1. Total Cross Sections

The total cross section is undefined for incident charged particles. MT=1 should be used for the photonuclear total cross section, while MT=501 is used for the total atomic photon interaction cross section.

3.5.2. Elastic Scattering Cross Sections

As discussed in detail in Section 6.2.6, it is not possible to construct an integrated cross section for charged-particle elastic scattering because of the Coulomb term. Therefore, σ is either set to 1.0 or to a "nuclear plus interference" value using a cutoff angle. This value may in theory be 0.0, and, in this case, should be set to epsilon, e.g., 10-38. The first and last energy points used for MT=2 in File 3 define the range of applicability of the cross section representation given in File 6. The cross section need not cover the complete range from 10-5 eV to 20 MeV. MT=2 is used for the elastic scattering cross section for all incident particles and photons (resonance fluorescence). For photons, MT=502 and 504 are used for coherent and incoherent atomic scattering, respectively.

3.5.3. Inelastic Scattering Cross Sections

The procedure for inelastic cross section for incident charged particles and photons is the same as for neutrons. The following MT combinations should be used.

Incident Particle

MT's for Excited States

MT's for Total Inelastic σ

γ

undefined

102

n

51-91

4

p

601-649

103

d

651-699

104

t

701-749

105

3He

751-799

106

α

801-849

107

3.5.4. Stopping Power

The total charged-particle stopping power in eV×barns is given in MF=3, MT=500. This is basically an atomic property representing the shielding of the nuclear charge by the electrons, but it should be repeated for each isotope of the element. It is a "total" stopping power in that most tabulations implicitly include large-angle coulomb scattering which is also represented here in File 6. In practice, this contribution is probably small enough to keep double counting from being a problem. At low particle energies, mixture effects are sometimes noticeable. They are not accounted for by this representation.

4. FILE 4, ANGULAR DISTRIBUTIONS OF SECONDARY PARTICLES

4.1. General Description

File 4 is used to describe the angular distribution of emitted particles. It is used for reactions with incident neutrons only, and should not be used for any other incident particle. Angular distributions of emitted neutrons should be given for elastically scattered neutrons, and for the neutrons resulting from discrete level excitation due to inelastic scattering. However, angular distributions must also be given for particles resulting from (n,n continuum), (n,2n), and other neutron emitting reactions. In these cases, the angular distributions will be integrated over all final energies. File 4 may also contain angular distributions of emitted charged particles for a reaction where only a single outgoing charged particle is possible (MT=600 through 849, see section 3.4.6). Emitted photon angular distributions are given in File 14 when the particle angular distributions are given in File 4.

The use of File 6 to describe all emitted particle angular distributions is preferred when charged particles are emitted, or when the emitted particle's energy and angular distributions are strongly correlated. In these cases, Files 4 and 14 should not be used.

In some cases, it may be possible to compute the angular distributions in the resolved range from resonance parameters (see section 2.4.22 for further discussion). In such cases, the computed distributions may be preferable to the distributions from File 4 for deep penetration calculations. However, for many practical applications, the smoothed distributions in File 4 will be adequate.

Angular distributions for a specific reaction type (MT number) are given for a series of incident energies, in order of increasing energy. The energy range covered should be the same as that for the same reaction type in File 3. Angular distributions for several different reaction types (MT's) may be given in File 4 for each material, in ascending order of MT number.

The angular distributions are expressed as normalized probability distributions, i.e.,

where f(µ,E)dµ is the probability that a particle of incident energy E will be scattered into the interval dµ about an angle whose cosine is µ. The units of f(µ,E) are (unit cosine)-1. Since the angular distribution of scattered neutrons is generally assumed to have azimuthal symmetry, the distribution may be represented as a Legendre polynomial series,

where µ = cosine of the scattered angle in either the laboratory or the center-of-mass system

E = energy of the incident particle in the laboratory system

σs(E) = the scattering cross section, e.g., elastic scattering at energy E as given in File 3 for the particular reaction type (MT)

l = order of the Legendre polynomial

σ(µ,E) = differential scattering cross section in units of barns per steradian

al = the lth Legendre polynomial coefficient and it is understood that a0 = 1.0.

The angular distributions may be given by one of two methods, and in either the center-of-mass (CM) or laboratory (LAB) systems. Using the first method, the distributions are given by tabulating the normalized probability distribution, f(µ,E), as a function of incident energy. Using the second method, the Legendre polynomial expansion coefficients, al(E), are tabulated as a function of incident neutron energy.

Absolute differential cross sections are obtained by combining data from Files 3 and 4. If tabulated distributions are given, the absolute differential cross section (in barns per steradian) is obtained by

where σs(E) is given in File 3 (for the same MT number) and f(µ,E) is given in File 4. If the angular distributions are represented as Legendre polynomial coefficients, the absolute differential cross sections are obtained by

where σs(E) is given in File 3 (for the same MT number) and the coefficients al(E) are given in File 4.

Elastic transformation matrices are no longer permitted in ENDF formatted files.

4.2. Formats

File 4 is divided into sections, each containing data for a particular reaction type (MT number) and ordered by increasing MT number. Each section always starts with a HEAD record and ends with a SEND record. If the section contains a description of the angular distributions for elastic scattering, the transformation matrix is given first (if present) and this is followed by the representation of the angular distributions.

The following quantities are defined.

LTT

Flag to specify the representation used and it may have the following values:

LTT=0, all angular distributions are isotropic

LTT=l, the data are given as Legendre expansion coefficients, al(E)

LTT=2, the data are given as normalized probability distributions, f(µ,E)

LTT=3, low energy region is represented by as Legendre coefficients; higher region is represented by tabulated data.

LI

Flag to specify whether all the angular distributions are isotropic

LI=0, not all isotropic

LI=1, all isotropic

LCT

Flag to specify the frame of reference used

LCT=l, the data are given in the LAB system

LCT=2, the data are given in the CM system

NE

Number of incident energy points at which angular distributions are given (NE2000)

NL

Highest order Legendre polynomial that is given at each energy (NL64)

NM

Maximum order Legendre polynomial that will be required (NM64) to describe the angular distributions of elastic scattering in either the center-of-mass or the laboratory system. NM should be an even number.

VK

Matrix elements of the transformation matrices

VK=U-1l,m, if LCT=1 (data given in LAB system)

VK=Ul,m, if LCT=2 (data are given in CM system)

NP

Number of angular points (cosines) used to give the tabulated probability distributions for each energy (NP20l)

Other commonly used variables are given in the Glossary (Appendix A).

The structure of a section depends on the values of LTT (representation used, al(E) or f(µ,E)) but it always starts with a HEAD record of the form

[MAT, 4, MT/ ZA, AWR, 0, LTT, 0, 0]HEAD

4.2.l. Legendre Polynomial Coefficients: LTT=1 and LI=0

When LTT=1 (angular distributions given in terms of Legendre polynomial coefficients), the structure of the section is

[MAT, 4, MT/ ZA, AWR, 0, LTT, 0, 0]HEAD (LTT=1)

[MAT, 4, MT/ 0.0, AWR, LI, LCT, 0, 0] CONT (LI=0)

[MAT, 4, MT/ 0.0, 0.0, 0, 0, NR, NE/ Eint] TAB2

[MAT, 4, MT/ T, E1, LT, 0, NL, 0/al(E1)]LIST

[MAT, 4, MT/ T, E2, LT, 0, NL, 0/al(E2)]LIST

----------------------------------------------

----------------------------------------------

[MAT, 4, MT/ T, ENE, LT, 0, NL, 0/al(ENE)]LIST

[MAT, 4, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

Note that T and LT refer to temperature (in K) and a test for temperature dependence, respectively. These values are normally zero, however.

4.2.2. Tabulated Probability Distributions: LTT=2 and LI=0

If the angular distributions are given as tabulated probability distributions, LTT=2, the structure of a section is

[MAT, 4, MT/ ZA, AWR, 0, LTT, 0, 0]HEAD (LTT=2)

[MAT, 4, MT/ 0.0, AWR, LI, LCT, 0, 0]CONT (LI=0)

[MAT, 4, MT/ 0.0, 0.0, 0, 0, NR, NE/Eint] TAB2

[MAT, 4, MT/ T, E1 , LT, 0, NR, NP/µint /f(µ,E1)]TABl

[MAT, 4, MT/ T, E2 , LT, 0, NR, NP/µint /f(µ,E2)]TABl

--------------------------------------

--------------------------------------

[MAT, 4, MT/ T, ENE , LT, 0, NR, NP/µint /f(µ,ENE)]TABl

[MAT, 4, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

T and LT are normally zero.

4.2.3. All angular Distributions are Isotropic: LTT=0 and LI=1

When all angular distributions for a given MT are assumed to be isotropic then the section structure is:

[MAT, 4, MT/ ZA, AWR, 0, LTT, 0, 0]HEAD (LTT=0)

[MAT, 4, MT/ 0.0, AWR, LI, LCT, 0, 0]CONT (LI=1)

[MAT, 4, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

4.2.4. Angular Distribution over Two Energy Ranges: LTT=3 and LI=0

If LTT=3, angular distributions are given as Legendre coefficients over the lower energy range and as Probability Distributions over the higher energy range. The structure of a subsection is

[MAT, 4, MT/ ZA, AWR, 0, LTT, 0, 0]HEAD (LTT=3)

[MAT, 4, MT/ 0.0, AWR, LI, LCT, 0, NM]CONT (LI=0)

(Legendre coefficients)

[MAT, 4, MT/ 0.0, 0.0, 0, 0, NR, NE1/Eint]TAB2

[MAT, 4, MT/ T, E1, LT, 0, NL, 0/al(E1)]LIST

-----------------------------

-----------------------------

[MAT, 4, MT/ T, ENE, LT, 0, NL, 0/al(ENE1)]LIST

(Tabulated data)

[MAT, 4, MT/ 0.0, 0.0, 0, 0, NR, NE2/Eint]TAB2

[MAT, 4, MT/ T, E1, LT, 0, NR, NP/µint /f(µ,ENE1)]TABl

------------------------------

------------------------------

[MAT, 4, MT/ T, ENE, LT, 0, NR, NP/µint /f(µ,ENET)]TABl

(NET = NE1+NE2-1)

[MAT, 4, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

Note that there is a double energy point at the boundary.

4.3. Procedures

The angular distributions for two-body reactions should be given in the CM system (LCT=2). It is recommended that other reactions (such as continuum inelastic, fission, etc.) should be given in LAB system. All angular distribution data should be given at the minimum number of incident energy points that will accurately describe the energy variation of the distributions. Legendre coefficients are preferred unless they cannot give an adequate representation of the data.

When the data are represented as Legendre polynomial coefficients, certain procedures should be followed. Enough Legendre coefficients should be used to accurately represent the recommended angular distribution at a particular energy point, and to ensure that the interpolated distribution is everywhere positive. The number of coefficients (NL) may vary from energy point to energy point; in general, NL will increase with increasing incident energy. A linear-linear interpolation scheme (INT=2) must be used to obtain coefficients at intermediate energies. This is required to ensure that the interpolated distribution is positive over the cosine interval from 1.0 to +1.0; it is also required because some coefficients may be negative. In no case should NL exceed a value of 64. If more than 64 coefficients appear to be required to obtain a non-negative distribution, a constrained Legendre polynomial fit to the data should be given. NL=1 is allowed at low energies to specify an isotropic angular distribution.

When angular distributions are represented as tabular data, certain procedures should be followed. Sufficient angular points (cosine values) should be given to accurately represent the recommended distribution. The number of angular points may vary from distribution to distribution. The cosine interval must be from 1.0 to +1.0. The interpolation scheme for f(µ,E) vs. µ should be log-linear (INT=4), and that for f(µ,E) vs. E should be linear-linear (INT=2).

Accurate angular distributions for the thermal energy range can be obtained by using File 7 or a detailed free-gas calculation. File 4 contains distributions for stationary free targets only.

The formats given above do not allow an energy-dependent transformation matrix to be given, so transformation matrices may not be given for nonelastic scattering reaction types. When a processing code wishes to transfer inelastic level angular distributions expressed as Legendre polynomial coefficients from the LAB to the CM system, or CM to LAB system, a distribution should be generated and transformed point by point to the desired frame of reference. The point-wise angular distributions can then be converted to Legendre polynomial coefficients in the new frame of reference.

4.4. Procedures for Specific Reactions

4.4.l. Elastic Scattering (MT=2)

1. Legendre polynomial representations should be used for the elastic scattering angular distributions and discrete channel scattering, and must be given in the CM system. When this representation is given, the maximum order of the polynomial for each incident energy should be even and lmax must be 64.
2. Care must be exercised in selecting an incident energy mesh for certain light-to-medium mass materials. Here it is important to relate any known structure in the elastic scattering cross section to the energy dependent variations in the angular distributions. These two features of the cross sections cannot be analyzed independent of one another. Remember, processing codes operate on MT=2 data are given in Files 3 and 4. (Structure of the total cross section is not considered when generating energy transfer arrays). It is better to maintain consistency in any structure effects between File 3 and File 4 data than to introduce structure in one File and ignore it in the other.
3. Consistency must be maintained between angular distribution data given for elastic and inelastic scattering. This applies not only to structural effects, but also to how the distributions were obtained.

Frequently, the evaluated elastic scattering angular distributions are based on experimental results that, at times, contain contributions from inelastic scattering to low-lying levels (which in turn may contain direct interaction effects). If inelastic contributions have been subtracted from the experimental angular distributions, this process must be done in a consistent manner. The same contributions must be subtracted from both the integrated elastic scattering and the angular distribution. Be sure that these contributions are included in the inelastic scattering cross section (both integrated data and angular distributions). This is particularly important when the inelastic contributions are due to direct interaction, since the angular distributions are not isotropic or symmetric about 90°, but they are generally forward peaked.

4. Do not use an excessive number of incident energy points for the angular distributions. The number used should be determined by the amount of variation in the angular distributions but should not exceed 2000.
5. An incident energy point must be given at l0-5 eV. A point must be given at the highest energy point for which the angular distribution is isotropic. The highest incident energy must be the highest energy for which cross section data are given in File 3, at least 20 MeV.
6. In the case of neutrons, a relationship exists between the total cross section and the differential cross section at forward angles (Wick's limit or optical theorem).

where E0 is in eV and σT in barns; see Appendix H for value of X.

Care should be taken to observe this inequality, especially at high energies.

4.4.2. Inelastic Scattering Cross Sections

1. Do not give angular distribution data for MT = 4.
2. Always give angular distribution data for any one of the following if they are given in File 3: MT = 51 through 91.
3. Discrete channel (two body) angular distributions (e.g., MT = 2, 51 - 90, 701...) should be given as Legendre coefficients in the CM system.
4. The continuum reaction (MT = 91) should be given in the LAB system.
5. Isotropic angular distributions should be used unless the degree of the isotropy exceeds 5%. If any level excitation cross sections contain significant direct interaction contributions, angular distributions are very important.
6. Use the precautions outlined above when dealing with the level excitation cross sections that contain a large amount of structure.
7. Do not overcomplicate the data files. Restrict the number of distributions to the minimum required to accurately represent the data.

4.4.3. Other Neutron Producing Reactions

Neutron angular distribution data must be given for all other neutron producing reactions, such as fission, (n,n′α), or (n,2n) in File 4 or File 6. File 4 is only appropriate if the distributions are fairly isotropic without strong pre-equilibrium components. The LAB system should be used.

4.4.4. Charged-Particle Producing Reactions

Distributions for charged particles from two-body reactions in the 600 series can be given in File 4 using the CM system, if desired. (Continuum reactions where only one charged particle is possible (e.g., 649, 699, etc.) can also be given in File 4 using the LAB system. If angular data is needed for more complex reactions, File 6 is more appropriate.)

5. FILE 5, ENERGY DISTRIBUTION OF SECONDARY PARTICLES

5.1. General Description

File 5 is used to describe the energy distributions of secondary particles expressed as normalized probability distributions. File 5 is for incident neutron reactions and spontaneous fission only, and should not be used for any other incident particle. Data will be given in File 5 for all reaction types that produce secondary neutrons, unless the secondary neutron energy distributions can be implicitly determined from data given in File 3 and/or File 4. No data will be given in File 5 for elastic scattering (MT=2), since the secondary energy distributions can be obtained from the angular distributions in File 4. No data will be given for neutrons that result from excitation of discrete inelastic levels when data for these reactions are given in both File 3 and File 4 (MT=51, 52, ..., 90).

Data should be given in File 5 for MT=91 (inelastic scattering to a continuum of levels), MT=18 (fission), MT=16 (n,2n), MT=17 (n,3n), MT=455 (delayed neutrons from fission), and certain other nonelastic reactions that produce secondary neutrons. The energy distribution for spontaneous fission is given in File 5 (in sub-library 4).

File 5 may also contain energy distributions of secondary charged particle for continuum reactions where only a single outgoing charged particle is possible (MT=649, 699, etc.). Continuum photon distributions should be described in File 15.

The use of File 6 to describe all particle energy distributions is preferred when several charged particles are emitted or the particle energy and angular distribution are strongly correlated. In these cases Files 5 and 15 should not be used.

Each section of the file gives the data for a particular reaction type (MT number). The sections are then ordered by increasing MT number. The energy distributions p(EE), are normalized so that

(5.1)

where Emax is the maximum possible secondary particle energy and its value depends on the incoming particle energy E and the analytic representation of p(EE). The secondary particle energy E is always expressed in the laboratory system.

The differential cross section is obtained from

(5.2)

where σ(E) is the cross section as given in File 3 for the same reaction type number (MT) and m is the neutron multiplicity for this reaction (m is implicit; e.g., m=2 for n,2n reactions).

The energy distributions p(EE) can be broken down into partial energy distributions, fk(EE), where each of the partial distributions can be described by different analytic representations;

(5.3)

and at a particular incident neutron energy E,

where pk(E) is the fractional probability that the distribution fk(EE) can be used at E.

The partial energy distributions fk(EE) are represented by various analytical formulations. Each formulation is called an energy distribution law and has an identification number associated with it (LF number). The allowed energy distribution laws are given below.

Secondary Energy Distribution Laws

LF = l, Arbitrary tabulated function:

A set of incident energy points is given, E and g(EE) is tabulated as a function of E.

LF = 5, General evaporation spectrum:

θ(E) is tabulated as a function of incident neutron energy, E; g(x) is tabulated as a function of x, x = E/θ (E).

LF = 7, Simple fission spectrum (Maxwellian):

I is the normalization constant,

θ is tabulated as a function of energy, E;

U is a constant introduced to define the proper upper limit for the final particle energy such that 0 E′≤ (E – U).

LF = 9, Evaporation spectrum:

I is the normalization constant,

θ is tabulated as a function of incident neutron energy, E;

U is a constant introduced to define the proper upper limit for the final particle energy such that 0 E (E–U)

LF = 11, Energy dependent Watt spectrum:

I is the normalization constant,

a and b are energy dependent;

U is a constant introduced to define the proper upper limit for the final particle energy such that 0 E (E-U)

LF = 12, Energy dependent fission neutron spectrum (Madland and Nix):

where

EFL and EFH are constants, which represent the average kinetic energy per nucleon of the average light and heavy fragments, respectively.

TM depends upon the incident neutron energy,

E1(x) is the exponential integral,

γ(a,x) is the incomplete gamma function.

The integral of this spectrum between zero and infinity is one. The value of the integral for a finite integration range is given in Sec. 5.4.10.

The data are given in each section by specifying the number of partial energy distributions that will be used. The same energy mesh should be used for each one. The partial energy distributions may all use the same energy distribution law (LF number) or they may use different laws.

Note: Distribution laws are not presented for LF = 2, 3, 4, 6, 8, or 10. These laws are no longer used.

5.2. Formats

Each section of File 5 contains the data for a particular reaction type (MT number), starts with a HEAD record, and ends with a SEND record. Each subsection contains the data for one partial energy distribution. The structure of a subsection depends on the value of LF (the energy distribution law).

The following quantities are defined.

NK

Number of partial energy distributions. There will be one subsection for each partial distribution.

U

Constant that defines the upper energy limit for the secondary particle so that 0 E E - U (given in the LAB system).

θ

Effective temperature used to describe the secondary energy distribution for LF = 5, 7, or 9.

LF

Flag specifying the energy distribution law used for a particular subsection (partial energy distribution). (The definitions for LF are given in Section 5.l.).

pk(EN)

Fractional part of the particular cross section which can be described by the kth partial energy distribution at the Nth incident energy point.

NOTE:

fk(EE)

kth partial energy distribution. The definition depends on the value of LF.

NR

Number of interpolation ranges.

NP

Number of incident energy points at which pk(E) is given.

a,b

Parameters used in the energy dependent Watt spectrum, LF = 11.

EFL,EFH

Constants used in the energy-dependent fission neutron spectrum (Madland and Nix), LF = 12.

TM

Maximum temperature parameter, TM(E), of the energy-dependent fission neutron spectrum (Madland and Nix), LF =12.

NE

Number of incident energy points at which a tabulated distribution is given (NE200.)

NF

Number of secondary energy points in a tabulation. (NFl000.)

The structure of a section has the following form:

[MAT, 5, MT/ ZA, AWR, 0, 0, NK, 0]HEAD

-----------------------

[MAT, 5, MT/ 0.0, 0.0, 0, 0, 0, 0]SEND

The structure of a subsection depends on the value of LF. The formats for the various values of LF are given following.

LF = 1, Arbitrary tabulated function

[MAT, 5, MT/ 0.0, 0.0, 0, LF, NR, NP/ Eint/p(E)]TAB1 LF=1

[MAT, 5, MT/ 0.0, 0.0, 0, 0, NR, NE/ Eint]TAB2

[MAT, 5, MT/ 0.0, E1, 0, 0, NR, NF/ Eint /

E1, 0.0, E2,g(E1E2), E3,g(E1E3),

---------------------------------

---,ENF-1,g(E1ENF-1), ENF , 0.0]TAB1

[MAT, 5, MT/ 0.0, E2 , 0, 0, NR, NF/ Eint /

E1, 0.0, E2,g(E2E2), E3 ,g(E2E3),----

----------------------------------

---, ENF-1,g(E2ENF-1), ENF , 0.0]TAB1

[MAT, 5, MT/ 0.0, ENE, 0, 0, NR, NF/ Eint /

E1, 0.0, E2,g(ENEE2), E3,g(ENEE3)----

--------------------------------

--- ,ENF-1,g(ENEENF-1), ENF, 0.0]TAB1

Note that the incident energy mesh for pk(E) does not have to be the same as the E mesh used to specify the energy distributions. The interpolation scheme used between incident energy points, E, and between secondary energy points, E, should be linear-linear.

LF = 5, General evaporation spectrum

[MAT, 5, MT/ U, 0.0, 0, LF, NR, NP/ Eint / p(E)]TAB1 (LF=5)

[MAT, 5, MT/ 0.0, 0.0, 0, 0, NR, NE/ Eint / θ(E)]TAB1

[MAT, 5, MT/ 0.0, 0.0, 0, 0, NR, NF/ xint / g(x)]TAB1 (x=E/θ(E))

LF = 7, Simple fission spectrum (Maxwellian)

[MAT, 5, MT/ U, 0.0, 0, LF, NR, NP/ Eint / p(E)]TAB1 (LF=7)

[MAT, 5, MT/ 0.0, 0.0, 0, 0, NR, NE/ Eint / θ (E)]TAB1

LF = 9, Evaporation spectrum

[MAT, 5, MT/ U, 0.0, 0, LF, NR, NP/ Eint / p(E)]TAB1 (LF=9)

[MAT, 5, MT/ 0.0, 0.0, 0, 0, NR, NE/ Eint / θ (E)]TAB1

LF = 11, Energy-dependent Watt spectrum

[MAT, 5, MT/ U, 0.0, 0, LF, NR, NP / Eint / p(E)]TAB1 (LF=11)

[MAT, 5, MT/ 0.0, 0.0, 0, 0, NR, NE / Eint / a(E)]TAB1

[MAT, 5, MT/ 0.0, 0.0, 0, 0, NR, NE / Eint / b(E)]TAB1

LF = 12, Energy-dependent fission neutron spectrum (Madland and Nix)

[MAT, 5, MT/ 0.0, 0.0, 0, LF, NR, NP/ Eint / p(E)]TAB1 (LF=12)

[MAT, 5, MT/ EFL, EFH, 0, 0, NR, NE/ Eint /TM(E)]TAB1

5.3. Procedures

As many as three different energy meshes may be required to describe the data in a subsection (one partial distribution). These are the incident energy mesh for pk(E), the incident energy mesh at which the secondary neutrons are given, fk(EE), and the secondary energy mesh for fk(EE). It is recommended that a linear-linear or a linear-log interpolation scheme be used for the first two energy meshes, and a linear-linear interpolation for the last energy mesh.

Double energy points must be given in the incident energy mesh whenever there is a discontinuity in any of the pk(E)'s (this situation occurs fairly frequently). This energy mesh must also include threshold energy values for all reactions being described by the pk(E)'s. Zero values for pk must be given for energies below the threshold (if applicable).

Two nuclear temperatures may be given for the (n,2n) reaction. Each temperature, θ, may be given as a function of incident neutron energy. In this case p1(E) = p2(E) = 0.5. A similar procedure may be followed for the (n,3n) and other reactions.

A constant, U, is given for certain distribution laws (LF = 5, 7, 9, or 11). The constant, U, is provided to define the proper upper limit for the secondary energy distribution so that 0 E E - U. The value of U depends on how the data are represented for a particular reaction type. Consider U for inelastic scattering.

Case A: The total inelastic scattering cross section is described as a continuum. U is the threshold energy for exciting the lowest level in the residual nucleus.

Case B: For the energy range considered, the first three levels are described explicitly (either in File 3, MT = 51, 52, and 53, or in File 5), and the rest of the inelastic cross section is treated as a continuum. U is the threshold energy (known or estimated) for the fourth level in the residual nucleus.

If the reaction being described is fission, then U should be a large negative value (U = 20.0×106 eV to 30×106 eV). In this case neutrons can be born with energies much larger than the incident neutron energy. It is common practice to describe the inelastic cross section as the sum of excitation cross sections (for discrete levels) for neutron energies up to the point where level positions are no longer known. At this energy point, the total inelastic cross section is treated as a continuum. This practice can lead to erroneous secondary energy distributions for incident neutron energies just above the cutoff energy. It is recommended that the level excitation cross sections for the first several levels (e.g., 4 or 5 levels) be estimated for several MeV above the cutoff energy. The continuum portion of the inelastic cross section will be zero at the cutoff energy, and it will not become the total inelastic cross section until several MeV above the cutoff energy.

It is recommended that the cross sections for excitation of discrete inelastic levels be described in File 3 (MT = 51, 52, ..., etc.). The angular distributions for the neutrons resulting from these levels should be given in File 4 (the same MT numbers). The secondary energy distributions for these neutrons can be obtained analytically from the data in Files 3 and 4. This procedure is the only way in which the energy distributions can be given for these neutrons. For inelastic scattering, the only data required in Files 5 are for MT = 91 (continuum part).

5.4. Additional Procedures

5.4.l. General Comments

1. Do not give File 5 data for the discrete level excitation data given in File 3 as MT = 51, 52, ..., 90. If MT=91 is given in File 3, a section for MT=91 must be given in File 5 or File 6. A section must also be given in File 5 or File 6 for all other neutron-producing reactions. Continuum energy distributions for emitted protons, deuterons etc., may be given in MT=649 etc., and for photons, in File 6 or File 15. When more than one particle type is emitted, File 6 should be used to assure energy conservation.
2. Care must be used in selecting the distribution law number (LF) to represent the data. As a rule, use the simplest law that will accurately represent the data.
3. A section in File 5 must cover the same incident energy range as was used for the same MT number in File 3. The sum of the probabilities for all laws used must be equal to unity for all incident energy points.
4. If the incident neutron energy exceeds several MeV, pre-equilibrium neutron emission can be important, as illustrated from high-resolution neutron and proton spectra measurements and analysis of pulsed sphere experiments. In these cases either tabulated spectra or "mocked-up" levels can be constructed to supplement or replace simple evaporation spectra.
5. Note that prompt fission spectra are given under MT=18, 19, 20, 21, and 38. The delayed fission spectra are given under MT=455. The energy distribution for prompt spontaneous fission is given in File 5 for MT=18, but in sublibrary 4. It is used with⎯νp from File 1 (MT=456) to determine the prompt spontaneous fission spectrum. The delayed spontaneous fission spectrum is determined from⎯νd from File 1 (MT=455) and the delayed energy spectrum in File 5 MT=455. Note that for the specification of spontaneous spectra no cross sections from File 3 are required.

5.4.2. LF = 1 (Tabulated Distributions)

Use only tabulated distributions to represent complicated energy distributions. Use the minimum number of incident energy points and secondary neutron energy points to accurately represent the data. The integral over secondary neutron energies for each incident energy point must be unity to within four significant figures. All interpolation schemes must be with linear-linear or linear-log (INT=1,2, or 3) to preserve probabilities upon interpolation. All secondary energy distributions must start and end with zero values for the distribution function g(EE).

5.4.3. LF = 7 (Maxwellian Spectrum)

A linear-linear interpolation scheme is preferred for specifying the nuclear temperature as a function of energy.

5.4.4. LF = 9 (Evaporation Spectrum)

An evaporation spectrum is preferred for most reactions. Care must be taken in describing the nuclear temperature near the threshold of a reaction. Nuclear temperatures that are too large can violate conservation of energy.

5.4.5. LF = 11 (Watt Spectrum)

A linear-linear interpolation scheme is preferred for specifying the parameters a and b as a function of energy.

5.4.6. LF = 12 (Madland-Nix Spectrum)

A log-log interpolation scheme may be used for specifying the parameter TM as a function of incident neutron energy.

5.4.7. Selection of the Integration Constant, U

1. When LF = 5, 7, 9, or 11 is used, an integration constant U is required. This constant is used in defining the upper energy limit of secondary neutrons; i.e., Emax = En – U, where En is the incident neutron energy. U is a constant for the complete energy range covered by a subsection in File 5 and is given in the LAB system.
2. U is negative for fission reactions. The preferred value is -20 MeV.
3. In practice, U can be taken to be the absolute value of Q for the lowest level (known or estimated) that can be excited by the particular reaction within the incident energy range covered by the subsection. U is actually a function of the incident neutron energy, but it can be shown that it is always greater than the absolute value of Q and less than the threshold energy of the reaction. At large AWR, since Eth and |Q| are approximately equal, either could be used but the absolute value of Q is preferred. At small AWR, using |Q| for U is the best approximation and must be used.
4. The following four cases commonly occur in data files; procedures are given for obtaining U values.

Case A: The complete reaction is treated as a continuum.

U = Q, where Q is the reaction Q-value.

Case B: The reaction is described by excitation of three levels (in File 3 as MT = 51, 52, 53) and a continuum part where Q4 is the known or estimated Q-value for the fourth level.

U = Q4.

Case C: The reaction is described by excitation of three levels (in File 3 as MT=51, 52, and 53) and a continuum part which extends below the threshold for MT=51. If, for example, the reaction is a 3-body breakup reaction, use

U = Q , where Q is the energy required for 3-body breakup.

Case D: The reaction is described by excitation of the first three levels (in File 3 as MT=51, 52, 53) for neutron energies from the level thresholds up to 20 MeV, excitation of the next five levels (in File 3 as MT=54, ..., 58) from their thresholds up to 8 MeV, and by a continuum part that starts at 5 MeV.

In this case two subsections should be used, one to describe the energy range from 5 to 8 MeV and another to describe the energy region from 8 to 20 MeV. In the first subsection (5 - 8 MeV),

U = Q9,

and the second (8 - 20 MeV),

U = Q4.

5.4.8. Multiple Nuclear Temperatures

Certain reactions, such as (n,2n), may require specification of more than one nuclear temperature. θ(E) should be given for each neutron in the exit channels; this is done by using more than one subsection for a reaction. The U value is the same for all subsections. The upper energy limit is determined by the threshold energy and not by level densities in the residual nuclei.

5.4.9. Average Energy for a Distribution

The average energy of a secondary neutron distribution must be less than the available energy for the reaction:

where Eavail is greater than the neutron multiplicity times the average energy of all the emitted neutron ν⎯E, where ν is the multiplicity. The mean energy should be calculated from the distribution at each value of E. This mean is analytic in the four cases given below.

LF

7

9

11

where Er = E-U

12

U is described in Section 5.3. The analytic functions for I are given in Section 5.1 for LF = 7, 9, 11. For LF = 12, Section 5.4.10 gives the method for obtaining the integral of the distribution function.

5.4.10. Additional procedures for LF = 12, Energy-Dependent Fission Neutron Spectrum (Madland and Nix)

Integral over finite energy range [a,b].

Set:

Then, the integral is given by one of the following three expressions depending on the region of integration in which a and b lie.

Region I (a > EF, b > EF)

Region II (a < EF, b < EF)

Region III (a < EF, b > EF)

The expression for Region III would be used to calculate a normalization integral I for the finite integration constant U, if a physical basis existed by which U could be well determined.

6. FILE 6. PRODUCT ENERGY-ANGLE DISTRIBUTIONS

6.1. General Description

This file is provided to represent the distribution of reaction products (i.e., neutrons, photons, charged particles, and residual nuclei) in energy and angle. It works together with File 3, which contains the reaction cross sections, and replaces the combination of File 4 and File 5. Radioactive products are identified in File 8. The use of File 6 is recommended when the energy and angular distributions of the emitted particles must be coupled, when it is important to give a concurrent description of neutron scattering and particle emission, when so many reaction channels are open that it is difficult to provide separate reactions, or when accurate charged-particle or residual-nucleus distributions are required for particle transport, heat deposition, or radiation damage calculations.

For the purposes of this file, any reaction is defined by giving the production cross section for each reaction product in barns/steradian assuming azimuthal symmetry:

(6.1)

where i denotes one particular product,

E is the incident energy,

E is the energy of the product emitted with cosine µ,

σ(E) is the interaction cross section (File 3),

yi is the product yield or multiplicity, and

fi is the normalized distribution with units (eV-unit cosine)-1 where

(6.2)

This representation ignores most correlations between products and most sequential reactions; that is, the distributions given here are those, which would be seen by an observer outside of a "black box" looking at one particle at a time. The process being described may be a combination of several different reactions, and the product distributions may be described using several different representations.

6.2. Formats

The following quantities are defined for all representations.

ZA, AWR

Standard material charge and mass parameters.

LCT

Reference system for secondary energy and angle (incident energy is always given in the LAB system).

LCT=1, laboratory (LAB) coordinates used for both;

LCT=2, center-of-mass (CM) system used for both;

LCT=3, center-of-mass system for both angle and energy of light particles (A4), laboratory system for heavy recoils (A>4).

NK

Number of subsections in this section (MT). Each subsection describes one reaction product. There can be more than one subsection for a given particle or residual nucleus (see LIP). NK2000.

ZAP

Product identifier 1000*Z+A with Z=0 for photons and A=0 for electrons and positrons. A section with A=0 can also be used to represent the average recoil energy or spectrum for an elemental target (see text).

AWP

Product mass in neutron units.

LIP

Product modifier flag. Its main use is to identify the isomeric state of a product nucleus. In this case, LIP=0 for the ground state, LIP=1 for the first isomeric state, etc. These values should be consistent with LISO in File 8, MT=457.

In some cases, it may be useful to use LIP to, distinguish between different subsections with the same value of ZAP for light particles. For example, LIP=0 could be the first neutron out for a sequential reaction, LIP=1 could be the second neutron, and so on. Other possible uses might be to indicate which compound system emitted the particles, or to distinguish between the neutron for the (n,np) channel and that from the (n,pn) channel. The exact meaning assigned to LIP should be explained in the File 1, MT=451 comments.

LAW

Flag to distinguish between different representations of the distribution function, fi:

LAW=0, unknown distribution;

LAW=1, continuum energy-angle distribution;

LAW=2, two-body reaction angular distribution;

LAW=3, isotropic two-body distribution;

LAW=4, recoil distribution of a two-body reaction;

LAW=5, charged-particle elastic scattering;

LAW=6, n-body phase-space distribution; and

LAW=7, laboratory angle-energy law.

NR, NP, Eint

Standard TAB1 parameters.

A section of File 6 has the following form:

[MAT, 6, MT/ ZA, AWR, 0, LCT, NK, 0]HEAD

[MAT, 6, MT/ ZAP, AWP, LIP, LAW, NR, NP/Eint/yi(E)]TAB1

[LAW-dependent structure for product 1]

----------------------------

NK subsections>

----------------------------

[MAT, 6, MT/ 0.0, 0.0, 0, 0, 0, 0]SEND

File 6 should have a subsection for every product of the reaction or sum of reactions being described except for MT = 3, 4, 103-107 when they are being used to represent lumped photons. The subsections are arranged in the following order: (1) particles (n, p, d, etc.) in order of ZAP and LIP, (2) residual nuclei and isomers in order of ZAP and LIP, (3) photons, and (4) electrons. The contents of the subsection for each LAW are described below.

6.2.1. Unknown Distribution (LAW=0)

This law simply identifies a product without specifying a distribution. It can be used to give production yields for particles, isomers, radioactive nuclei, or other interesting nuclei in materials that are not important for particle transport, heating, or radiation damage calculations. No law-dependent structure is given.

6.2.2. Continuum Energy-Angle Distributions (LAW=1)

This law is used to describe particles emitted in multi-body reactions or combinations of several reactions, such as scattering through a range of levels or reactions at high energies where many channels are normally open. For isotropic reactions, it is very similar to File 5, LF=1 except for a special option to represent sharp peaks as "delta functions" and the use of LIST instead of TAB1.

The following quantities are defined for LAW=1:

LANG

Indicator which selects the angular representation to be used; if

LANG=1, Legendre coefficients are used,

LANG=2, Kalbach-Mann systematics are used,

LANG=11-15, a tabulated angular distribution is given using NA/2 cosines and the interpolation scheme specified by LANG10 (for example, LANG=12 selects linear-linear interpolation).

LEP

Interpolation scheme for secondary energy;

LEP=1 for histogram,

LEP=2 for linear-linear, etc.

NR, NE, Eint

Standard TAB2 parameters.

INT=1 is allowed (the upper limit is implied by file 3),

INT=12-15 is allowed for corresponding-point interpolation,

INT=21-25 is allowed for unit base interpolation.

NW

Total number of words in the LIST record; NW = NEP (NA+2).

NEP

Number of secondary energy points in the distribution.

ND

Number of discrete energies given.

The first ND0 entries in the list of NEP energies are discrete, and the remaining (NEP-ND)0 entries are to be used with LEP to describe a continuous distribution. Discrete primary photons should be flagged with negative energies.

NA

Number of angular parameters.

Use NA=0 for isotropic distributions (note that all options are identical if NA=0).

Use NA=1 or 2 with LANG=2 (Kalbach-Mann).

The structure of a subsection is

[MAT, 6, MT/ 0.0, 0.0, LANG, LEP, NR, NE/Eint]TAB2

[MAT, 6, MT/ 0.0, E1, ND, NA, NW, NEP/

E1, b0(E1,E1),b1(E1,E1),------ bNA(E1,E1),

E2, b0(E1,E2), ----------------------

E,b(E,E), ------------ b(E,E)]LIST

NEP01NEPNA1NEP ---------------------------------------

---------------------------------------

where the contents of the bi depend on LANG.

The angular part of fi can be represented in several different ways (denoted by LANG).

LANG=1

If LANG=1, Legendre coefficients are used as follows:

(6.3)

where NA is the number of angular parameters, and the other parameters have their previous meanings. Note that these coefficients are not normalized like those for discrete two-body scattering (LAW=2); instead, f0(E,E) gives the total probability of scattering from E to E integrated over all angles. This is just the function g(E,E) normally given in File 5. The Legendre coefficients are stored with f0 in b0, f1 in b1, etc.

LANG=2

For LANG=2, the angular distribution is represented by using the Kalbach-Mann systematics [Ref.1] in the extended form developed by Kalbach [Ref.2], hereinafter referred to as KA88. The distribution is given in terms of the parameters r and a, which are described below. If NA = 1, the parameter r is given and a is calculated. If NA = 2, then both parameters r and a are given explicitly.

This formulation addresses reactions of the form

A + a C B + b,

where: A is the target, a is the incident projectile,

C is the compound nucleus,

b is the emitted particle, B is the residual nucleus.

The following quantities are defined:

Ea

energy of the incident projectile a in the laboratory system

εa

entrance channel energy, the kinetic energy of the incident projectile a and the target particle A in the center-of-mass system

Eb

energy of the emitted particle in the laboratory system

εb

AWRemission channel energy, the kinetic energy of the emission particle b and the residual nucleus B in the center-of-mass system

µb

cosine of the scattering angle of the emitted particle b in the center-of-mass system

It is required that LCT=2 with LANG=2.

The KA88 distribution is represented by

(6.4)

where r(Ea,Eb) is the pre-compound fraction as given by the evaluator and a(Ea,Eb) is a simple parameterized function that depends mostly on the center-of-mass emission energy Eb, but also depends slightly on particle type and the incident energy at higher values of Ea.

The center-of-mass energies and angles Eb and µb are transformed into the laboratory system using the expressions

(6.5)

The pre-compound fraction r, where r goes from 0.0 to 1.0, is usually computed by a model code, although it can be chosen to fit experimental data.

The formula for calculating the slope value a(Ea,Eb)1 is:

where

ea = εa + Sa

eb = εb + Sb

R1 = minimum(ea,Et1)

R3 = minimum(ea,Et3)

X1 = R1eb/ea

X3 = R3eb/ea

The parameter values for light particle induced reactions as given in KA882 are:

C1 = 0.04/MeV

C2 = 1.8×10-6/MeV3

C3 = 6.7×10-7/MeV4

Et1 = 130 MeV

Et3 = 41 MeV

Mn = 1

Mp = 1

Md = 1

Mα = 0

mn = 1/2

mp = 1

md = 1

mt = 1

m3He = 1

mα = 2

Sa and Sb are the separation energies for the incident and emitted particles, respectively, neglecting pairing and other effects for the reaction A + a C B + b. The formulae for the separation energies in MeV3 are:

1 Equation 10 of Ref. 2.

2 Table V of ref. 2.

3 Equation 4 of Ref. 2.

and

where

subscripts A, B, and C refer to the target nucleus, the residual nucleus, and the compound nucleus, as before,

N, Z, and A are the neutron, proton, and mass numbers of the nuclei,

Ia and Ib are the energies required to separate the incident and emitted particles into their constituent nucleons (see Appendix H for values used for given particles).

The parameter f0(Ea,Eb) has the same meaning as f0 in Eq. (6.3); that is, the total emission probability for this Ea and Eb. The number of angular parameters (NA) is always 1 for LANG=2, and f0 and r are stored in the positions of b0 and b1, respectively.

This formulation uses a single-particle-emission concept; it is assumed that each and every secondary particle is emitted from the original compound nucleus C. When the incident projectile a, and the emitted particle b, are the same, Sa = Sb, regardless of the reaction. For incident projectile z, if neutrons emitted from the compound nucleus C are detected, there will be one and only one Sb appropriate for all reactions, for example, (z,nα), (z,n3α), (z,2nα), (z,np), (z,2n2α), and (z,nt2α). Furthermore, if the incident projectile is a neutron (z=n in previous examples), then Sa = Sb in all cases; even for neutrons emitted in neutron-induced reactions, Sa and Sb will be identical.

LANG = 11-15

For LANG=11-15, a tabulated function is given for f(µ) using the interpolation scheme defined by LANG minus 10. For example, if LANG=12, use linear-linear interpolation (don't use log interpolation with the cosine). The cosine grid of NA/2 values, µi, must span the entire angular range open to the particle for E and E, and the integral of f(µ) over all angles must give the total emission probability for this E and E (that is, it must equal f0, as defined above). The value of f below µNA/1 or above µNA/2 is zero.

The tabulation is stored in the angular parameters as follows:

b0 = f0,

b1 = µ1,

b2 = 0.5f11)/f0,

b3 = µ2,

...

...

bNA = 0.5fNA/2NA/2)/f0.

The preferred values for NA are 4, 10, 16, 22, etc.

In order to provide a good representation of sharp peaks, LAW=1 allows for a superposition of a continuum and a set of delta functions. These discrete lines could be used to represent particle excitations in the CM frame because the method of corresponding points can be used to supply the correct energy dependence. However, the use of LAW=2 together with MT=50-90, 600-650, etc., is preferred. This option is also useful when photon production is given in File 6.

6.2.3. Discrete Two-Body Scattering (LAW=2)

This law is used to describe the distribution in energy and angle of particles described by two-body kinematics. It is very similar to File 4, except its use in File 6 allows the concurrent description of the emission of positrons, electrons, photons, neutrons, charged particles, residual nuclei, and isomers. Since the energy of a particle emitted with a particular scattering cosine µ is determined by kinematics, it is only necessary to give

(6.6)

where the Pl are the Legendre polynomials with the maximum order NL. Note that the angular distribution pi is normalized.

The following quantities are defined for LAW=2:

LANG

flag that indicates the representation:

LANG=0, Legendre expansion;

LANG=12, tabulation with pi(µ) linear in µ;

LANG=14, tabulation with 1n pi linear in µ.

NR, NE, Eint

standard TAB2 parameters.

NL

for LANG=0, NL is the highest Legendre order used;

for LANG>0, NL is the number of cosines tabulated.

NW

number of parameters given in the LIST record:

for LANG=0, NW=NL;

for LANG>0, NW=2*NL.

Al

for LANG=0, the Legendre coefficients,

for LANG>0, the, µ,pi pairs for the tabulated angular distribution

The format for a subsection with LAW=2 is

[MAT, 6, MT/ 0.0, 0.0, 0, 0, NR, NE/ Eint ]TAB2

[MAT, 6, MT/ 0.0, E,LANG, 0, NW, NL/A(E)]LIST

1 l ---------------------------

rest of the incident energies>

---------------------------

Note that LANG=0 is very similar to File 4, LTT=1, and LVT=0. The tabulated option is similar to File 4, LTT=2, LVT=0, except that a LIST record is used instead of TAB1. The kinematical equations require AWR and AWP from File 6 and QI from File 3.

LAW=2 can be used in sections with MT=50-90, 600-648, 650-698, etc., only, and the center-of-mass system must be used (LCT=2).

6.2.4. Isotropic Discrete Emission (LAW=3)

This law serves the same purpose as LAW=2, but the angular distribution is assumed to be isotropic in the CM system for all incident energies. No LAW-dependent structure is given. This option is similar to LI=1 in File 4. The energy of the emitted particle is completely determined by AWR and AWP in this section and QI from File 3.

6.2.5. Discrete Two-Body Recoils (LAW=4)

If the recoil nucleus of a two-body reaction (e.g., nn?, pn) described using LAW=2 or 3 doesn't break up, its energy and angular distribution can be determined by kinematics. No LAW-dependent structure is given. If isomer production is possible, multiple subsections with LAW=4 can be given to define the energy-dependent branching ratio for the production of each excited nucleus. Finally, LAW=4 may be used to describe the recoil nucleus after radiative capture (MT=102), with the understanding that photon momentum at low energies must be treated approximately.

6.2.6. Charged-Particle Elastic Scattering (LAW=5)

Elastic scattering of charged particles includes components from Coulomb scattering, nuclear scattering, and the interference between them. The Coulomb scattering is represented by the Rutherford formula and electronic screening is ignored.

The following parameters are defined.

σcd(µ,E)

differential Coulomb scattering cross section (barns/sr) for distinguishable particles

σci(µ,E)

cross section for identical particles

E

energy of the incident particle in the laboratory system (eV)

µ

cosine of the scattering angle in the center-of-mass system

m1

incident particle mass (AMU)

Z1 and Z2

charge numbers of the incident particle and target, respectively

s

spin (identical particles only, s = 0, 1/2, 1, 3/2, etc.)

A

target/projectile mass ratio

k

particle wave number (barns-1/2)

η

dimensionless Coulomb parameter

The cross sections can then be written

(6.7)

and

(6.8)

where

(6.9)

(6.10)

Note that A = 1 and Z1 = Z2 for identical particles.

The net elastic scattering cross section for distinguishable particles may be written as

(6.11)

and the cross section for identical particles is

(6.12)

where the al are complex coefficients for expanding the trace of the nuclear scattering amplitude matrix and the bl are real coefficients for expanding the nuclear scattering cross section. The value of NL represents the highest partial wave contributing to nuclear scattering. Note that σei(-µ,E) = σei(µ,E).

The three terms in Equations (6.11) and (6.12) are Coulomb, interference, and nuclear scattering, respectively. Since an integrated cross section is not defined for this representation, a value of 1.0 is used in File 3.

When only experimental data are available, it is convenient to remove the infinity due to σC by subtraction and to remove the remaining infinity in the interference term by multiplication, thereby obtaining the residual cross sections

(6.13)

and

(6.14)

Then σR can be given as a Legendre polynomial expansion in the forms

(6.15)

and

(6.16)

A cross section value of 1.0 is used in File 3.

Because the interference term oscillates as µ goes to 1, the limit of the Legendre representation of the residual cross section at small angles may not be well defined. However, if the coefficients are chosen properly, the effect of this region will be small because the Coulomb term is large.

It is also possible to represent experimental data using the "nuclear plus interference" cross section and angular distribution in the CM system defined by

(6.17)

and

(6.18)

where µmin is -1 for different particles and 0 for identical particles. The maximum cosine should be as close to 1.0 as possible, especially at high energies where Coulomb scattering is less important. The Coulomb cross section σc(µ,E) is to be computed using Eqs. (6.7) or (6.8) for different or identical particles, respectively. The angular distribution pNI is given in File 6 as a tabulated function of µ, and σNI(E) in barns is given in File 3.

The following quantities are defined for LAW=5:

SPI