Module 3.0: Nuclear Theory
MODULE 3.0: NUCLEAR THEORYIntroductionWelcome to Module 3.0 of the Nuclear Criticality Safety Directed SelfStudy Course! This is the third of five modules available in thisdirected self-study course. The purpose of this module is to assist youin describing the fission process and basic nuclear theory concepts.This directed self-study module is designed to assist you inaccomplishing the learning objectives listed at the beginning of themodule. There are four sections in this module. The module hasself-check questions and activities to help you assess yourunderstanding of the concepts presented in the module. Please notethere are also optional activities that are not required for completionof this module. These are noted at the beginning of the activityBefore You BeginIt is recommended that you have access to the following material:Trainee GuideComplete the following prerequisite:There are no prerequisites to this module.How to Completethis Module1. Review the learning objectives.2. Read each section within the module in sequential order.3. Complete the self-check questions and activities within thismodule.4. Check off the tracking form as you complete the self-checkquestions and/or activities within the module.5. Contact your administrator as prompted for a progress reviewmeeting.6. Contact your administrator as prompted for any additionalmaterials and/or specific assignments.7. Complete all assignments related to this module.8. Ensure that you and your administrator have dated and initialedyour progress on the tracking form.
Module 3.0: Nuclear Theory9. Go to the next assigned module.TABLE OF CONTENTSLearning Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Self-Check Questions 3-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fission Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Neutron Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Nuclear Fission Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fissioning Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .What Fissionable Material Is Present at NRC Licensee Facilities? . . . . . . . . . . . . . . . .Fission Energy Release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Overall Heat Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .What Are the Consequences of Fission Energy Release to a Criticality Accident? . . . .Fission Fragments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Consequences of Fission Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fission Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Neutron Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Neutron Energy Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Moderation of Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Moderators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Effects of Moderators and Non-Moderators on Critical Systems . . . . . . . . . . . . . . . . .Neutron Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Energy Dependence of Microscopic Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . .Energy-Dependent Cross Sections for Uranium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Infinite System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Infinite Multiplication Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Four-Factor Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Heterogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Finite System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Neutron Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Reactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Nuclear Criticality Safety Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Neutron-Balance Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Activity 1 - Neutron Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Consequences/Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Activity 2 - Operating Limit Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Self-Check Questions 3-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Diffusion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Calculations Used with Single Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Geometric Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Buckling Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .USNRC Technical Training CenterNuclear Criticality 283-283-283-293-293-300905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Activity 3 - Buckling Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Six-Factor Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Age-Diffusion/ Migration-Area Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Activity 4 - Migration-Area Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fraction Critical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Activity 5 - Fraction Critical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Calculations Used with Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Surface Density Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Activity 6 - Surface Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Solid Angle Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Activity 7 - Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Activity 8 - Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Self-Check Questions 3-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Computer Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Validation, Bias, and Bias Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Computational Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Finite Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .KENO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .MCNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .MONK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Self-Check Questions 3-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Progress Review Meeting Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Module Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-623-633-653-673-70LIST OF TABLESTable 3-1.Table 3-2.Table 3-3.Table 3-4.Table 3-5.Table 3-6.Table 3-7.Table 3-8.Advantages and Disadvantages of a Fission Reaction . . . . . . . . . . . . . . . . . . .Categories of Materials That Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fission Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Neutron Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Four Terms of the Infinite Multiplication Factor . . . . . . . . . . . . . . . . . . . . . . . .keff and as Relates to Criticality State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Geometric Buckling Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Adjusted Geometric Buckling Configuration . . . . . . . . . . . . . . . . . . . . . . . . . .3-73-73-83-123-153-183-303-31LIST OF FIGURESFigure 3-1.Figure 3-2.Figure 3-3.Figure 3-4.Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6Fission Reaction of Uranium-235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7Double-humped (Bi-Model) Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9235U Fission Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14USNRC Technical Training CenterNuclear Criticality Safetyii0905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryFigure 3-5.Figure 3-6.Figure 3-7.Figure 3-8.Figure 3-9.Figure 3-10.Figure 3-11.Figure 3-12.Figure 3-13.Figure 3-14.Figure 3-15.Critical Mass of LEU as a Function of Uranium Enrichment . . . . . . . . . . . . .Critical Mass Versus Concentration Data for 235U . . . . . . . . . . . . . . . . . . . .Calculations for Critical Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Migration Area Approximation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . .Age-Diffusion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Surface Density Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Example of Calculation Using the Surface Density Method . . . . . . . . . . . . .Solid Angle Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Solid Angle Method Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Point-to-Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Monte Carlo Model KENO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .USNRC Technical Training CenterNuclear Criticality -600905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryLearning Objectives3.1Upon completion of this module, you will be able todescribe the fission process and basic nuclear theoryconcepts.3.1.1Define the terms:fissile isotopesfissile materialfissile systemfission, nuclearfissionable isotopes3.1.2Describe the fission chain reaction in terms of thefollowing:energy releaseneutron production- number- energy- timing (prompt and delayed)absorption, scattering, and leakageradiation types and time history3.1.3Define keff and qualitatively state changes whenparameters are changed.3.1.4Identify the purpose and basic features of each of thefollowing hand calculation methods:buckling conversionage-diffusion migration area approximationfraction criticalsolid anglesurface density3.1.5Identify the purpose and basic features of the Monte Carlocomputer code KENO.USNRC Technical Training CenterNuclear Criticality Safety3-10905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryLearning ObjectiveWhen you finish this section, you will be able to:3.1.1Define the terms:fissile isotopesfissile materialfissile systemfission, nuclearfissionable isotopesDEFINITIONSThese are terms that you will need to be familiar with while reviewingthis module:Fissile IsotopesFissile isotopes are the subset of the fissionable isotopes that willsustain a fission-chain reaction using thermal neutrons. The importantisotopes 233U, 235U, 239Pu, and 241Pu are fissile isotopes. Fissileisotopes in solution or in special mixtures can absorb thermalneutrons and sustain chain reactions with much smaller masses thanthose usually required for sustained fission; therefore, fissile isotopespose a greater risk of an accidental criticality.Fissile MaterialA material, other than natural uranium, that is capable of sustaining aneutron chain reaction. Source: ANSI/ANS 8.7, 1975.Fissile SystemA system containing 235U, 239Pu, or 233U nuclei and capable ofsignificant neutron multiplication.Fission, Nuclear1. Nuclear reaction in which a neutron and nucleus interact with theresult that the nucleus splits into two or more parts. Fission reactionsof interest here release net energy and produce neutrons that canparticipate in a sustained chain reaction. 2. Disintegration of anucleus (usually Th, U, Pu, or heavier) into two (rarely more) massesof similar order of magnitude, accompanied by a large release ofenergy and the emission of neutrons. Although some fissions takeplace spontaneously, neutron-induced fissions are designated f, andis the number of neutrons emitted per fission.Fissionable IsotopeA fissionable isotope is any isotope capable of sustaining a neutroninduced fission-chain reaction, regardless of the neutron energy orUSNRC Technical Training CenterNuclear Criticality Safety3-20905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear Theoryspeed necessary to induce and sustain the reaction. For practicalnuclear criticality safety purposes, these isotopes are limited to 233U,235U, 241Am, 242mAm, 243Am, 243Cm, 244Cm, 245Cm, 247Cm, 249Cf, 251Cf,237Np, 238Pu, 239Pu, 240Pu, 241Pu, and 242Pu. Although other isotopes ofsome of the above elements would sustain fission, the quantityrequired for a self-sustaining chain reaction is so great and/or themass available is so small as to make a nuclear-criticality accidentincredible.USNRC Technical Training CenterNuclear Criticality Safety3-30905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheorySelf-Check Questions 3-1Fill in the missing words in each statement. Answers are located in the answer key section ofthe Trainee Guide. Choose from the following tion235Uare the subset of the fissionable isotopes that will sustain a1. Fissile, 239Pu,fission-chain reaction using thermal neutrons. Examples of these are 233U,241and Pu.uranium that is capable of2. Fissile material is a material other thansustaining a neutron chain reaction.3. A fissile system is a system containing 235U, 239Pu, or 233U nuclei and capable of significant.neutron4. Nuclear fission is a nuclearin which a neutron and nucleus interact withthe result that the nucleus splits into two or more parts.isotope is capable of sustaining a neutron-induced fission-chain5. Areaction, regardless of the neutron energy or speed necessary to induce and sustain thereaction.You have completed this section.Please check off your progress on the tracking form.Go to the next section.USNRC Technical Training CenterNuclear Criticality Safety3-40905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryLearning ObjectivesWhen you finish this section, you will be able to:3.1.2Describe the fission chain reaction in terms of thefollowing:energy releaseneutron production- number- energy- timing (prompt and delayed)absorption, scattering, and leakageradiation types and time history3.1.3Define keff and qualitatively state changes when parameters are changed.NUCLEARREACTIONSFission ProcessThe fission process and its associated neutron chain reactionproduces energy, which is used by licensees in:Generation of electricityPropulsionProduction of radioisotopesResearchNuclear ReactionsThe physics of criticality and nuclear criticality safety depends first onidentification of nuclear reactions.The basic mechanism for a nuclear reaction involves a PROJECTILEparticle (x) and a TARGET nucleus (X), combining to form anunstable COMPOUND NUCLEUS (C*), which decomposes into twoor more PRODUCTS (Y and y) plus excess energy in the form ofradiation. See Figure 3-1.USNRC Technical Training CenterNuclear Criticality Safety3-50905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryFigure 3-1. Nuclear ReactionsThe process can be written as equations:X x C*X(x,y)Y, orX(x,y)Y y, orMass, energy, and other considerations determine which reactionproducts may result; each possible reaction has a PROBABILITY ofoccurrence.Neutron ReactionsNeutron reactions are the major contributors to fission chainreactions.Scattering reactions start and end with a neutron but do notalways reduce neutron energy (and, as shown later, changethe probability of subsequent reactions).Absorption reactions—capture and fission—each result in theloss of the projectile neutron. In capture, the loss ispermanent, while, in fission, two or more neutrons areproduced.Must account for scattering, capture, and fission to describe afission chain reaction.Nuclear FissionReactionThe fission reaction of uranium-235 is shown in Figure 3-2.USNRC Technical Training CenterNuclear Criticality Safety3-60905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryFigure 3-2. Fission Reaction of Uranium-235γ βFISSIONγγγβ90 Brn1.6s235 Unγ β90 Kr90 Rb32.5s2.7min.β90 Sr29yβ90 Zr90 Y64hstableFISSION FRAGMENTSGAMMARAYSγγγ βnnγ β γ β γ β143Xen Neutron0.69s143Cs1.7s143Ba12s143La14min.γ ββ143Ce143Pr33h13.6d143NdstableThe reaction has advantages and disadvantages. See Table 3-1.Table 3-1. Advantages and Disadvantages of a Fission ReactionAdvantagesDisadvantagesHeat - Provides compact energysourceNeutrons - Allow self-sustainingchain reactionFissioning MaterialsRadiation - Requires shielding,containment, and heat removalThe two basic categories of materials that fission are described inTable 3-2.Table 3-2. Categories of Materials That FissionCategoryDefinitionSpeciesFissileIsotopes that can befissioned by neutrons of anyenergy235FissionableIsotopes that can befissioned by HIGH energyneutrons (but not by lowenergy neutrons)238USNRC Technical Training CenterNuclear Criticality Safety2393-7ResultU, 233U,Pu, 241PuCan sustain a fissionchain reactionU, 232Th, 240PuCan participate in afission chain reaction0905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryWhat FissionableMaterial Is Presentat NRC LicenseeFacilities?The dominant fissionable material in NRC licensee fuel cyclefacilities is uranium (fissile 235U and non-fissile 238U).Plutonium is present at reactor sites and storage facilities (e.g.,GE-Morris) in spent fuel and will be a concern for wastemanagement.233U is present only in research-related activities (e.g., hightemperature gas-cooled reactor programs).Fission EnergyReleaseFission energy release is divided among the various fission productsas shown in Table 3-3.Table 3-3. Fission ProductsFission ProductsFission Fragments (usually two)168% of EnergyRelease84Neutrons52.5Prompt Gamma Rays73.5DelayedRadiationBeta Particles84Gamma Rays73.5Radiative Capture Gammas52.5TotalOverall Heat EnergyEquationMeV200100Overall heat energy from fission is produced at a rate given by thefollowing equation:3.1 x 1010 fissions1 watt-s(This expression can be used to convert total fissions or fission rate inan excursion to an equivalent energy production.)What Are theConsequences ofFission EnergyRelease to aCriticality Accident?Criticality accidents have:USNRC Technical Training CenterNuclear Criticality SafetyLarge radiation sources at the time of fissionDelayed beta and gamma radiation, which gives rise to longterm radiation sources3-80905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryFission FragmentsWhen fissionable nuclei split into two FISSION FRAGMENTS, thefragments are seldom of equal mass (e.g., only 0.01% in 235U).Instead, there is a range of products, as shown in the "doublehumped" (bi-modal) spectrum. See Figure 3-3. (Here for 235U,species in the ranges A 90 - 100 and A 135 - 145 may each haveoccurred in as many as 7% of fissions. Overall, there are 200species of fission products.Figure 3-3. Double-humped (Bi-Modal) SpectrumConsequences ofFission ProductsConsequences of fission products:Some fission products are gaseous (iodine) and others arevolatile (strontium and cesium).Gaseous and volatile fission products spread readily ascontamination, following an accident.Radioactive wastes from an accident or from spent fuel aremulti-species and, therefore, complex.USNRC Technical Training CenterNuclear Criticality Safety3-90905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryFission NeutronsNeutrons emitted from fission are characterized by:Numberthat generally is greater than two ( 2.5 for 235U)Timing that is either:PROMPT - at time of fission ( 99%) orDELAYED - following fission fragment decay(0.2%-0.7%; 0.1 seconds to 1 minute afterfission)Energy spectrum (E):Most probable energyAverage energyBasic energy rangeNeutronCharacteristics0.7 MeV2.0 MeV0.1 MeV to 10 MeVNeutron characteristics:Sufficient neutrons are needed to support a self-sustainingchain reaction.Presence of delayed neutrons allows for control of a neutronchain reaction in reactors and the possibility of a "delayedcritical" accident scenario (e.g., as described later and as mayaffect the design of criticality alarm systems).Neutron EnergyConsiderationsBecause neutrons are born fast ( 0.1 MeV) but are best at causingfission when moderated to thermal energies (on the order of less than1 eV), moderating materials like water are important to both reactordesign and nuclear criticality safety.Moderation ofNeutronsModeration of neutrons occurs when neutrons undergo scatteringreactions. Elastic scattering of neutrons with low-Z materials resultsin relatively large energy changes per collision. (For a neutron andhydrogen atom, the analogy is a collision of two billiard balls). Withhigh-Z materials, small energy changes per collision occur (analogousto a billiard ball striking a bowling ball).ModeratorsModerators include:WATER, which, due to its hydrogen content, can be veryefficient in a relatively small volume; however, it also hassignificant absorption.DEUTERIUM (as in heavy water), BERYLLIUM, andUSNRC Technical Training CenterNuclear Criticality Safety3-100905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryGRAPHITE are less efficient moderators (in terms of energychange per collision) and, thus, require successively largervolumes; however, each has much lower neutron absorptionthan water.Any of the above moderators, in the proper quantity anddistribution, can reduce substantially the minimum amount offissile material required to sustain a chain reaction.Elements in the range of 11 Z 83 are considered "NONMODERATORS," which separate fissile nuclides, thereby,reducing the probability of fission.Heavy metals, including the fissionable materials, cause verysmall energy change per collision from elastic scattering.Inelastic scattering from some nuclides also contributes toneutron energy loss but generally in a manner of consequenceonly for detailed calculations.Effects ofModerators andNon-Moderators onCritical SystemsEffects of moderators and non-moderators on critical systems:Hydrogen, deuterium, beryllium, and carbon are moderatingmaterials that, when mixed, can reduce substantially the fissilematerial mass required to support a neutron chain reaction.Under usual conditions, only hydrogen (e.g., in water, plastic,or oil) or carbon (e.g., in organic liquids or other materials)would be expected to provide a source of accidentalmoderation.Accidental mixing of a non-moderator (e.g., aluminum powderor silicon from sand) can be expected to separate the fissilenuclides, thereby reducing the probability of fission.Metallic and other solid forms of fissile material do not exhibitsignificant moderation effects.Neutron BalanceEquationA neutron balance equation for the fission chain reaction is as follows:Rate of IncreaseRate ofRate ofin Number of Production — AbsorptionNeutronsof Neutronsof NeutronsRate of— Leakageof NeutronsAccumulation Production — Absorption — LeakageSee Table 3-4.USNRC Technical Training CenterNuclear Criticality Safety3-110905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryTable 3-4. Neutron BalanceIfaccumulationis:CROSS SECTIONThesystem is:The neutronpopulationsis:The systembehavior is: 0CriticalSteady StateStatic 0SupercriticalIncreasingKinetic/Dynamic 0SubcriticalDecreasingKinetic/DynamicThe concept of a CROSS SECTION was developed from theperception that:Interaction Probability nwhere ndx density of atoms (at/cm3) cross-sectional area [of the target nucleus] (cm2/at)dx distance traveled by neutron (cm)However, with the determination that the interaction probability for agiven material changed substantially with neutron energy, andbecause it was not easy to view the nucleus as changing size, thefollowing term was defined without specific connection to physicaldimensions:The unit of measurement was selected as:1 barn [b] 10-24 cm2The name was derived from the intended-to-be-humorousobservation that such a target was "as big as the side of a barn."Energy Dependenceof MicroscopicCross SectionsENERGY DEPENDENCE of microscopic cross sections is verycomplex. A few of the key features include:POTENTIAL SCATTERING cross section is approximately the size ofnucleus.USNRC Technical Training CenterNuclear Criticality Safety3-120905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryAbsorption cross section over a wide energy range varies roughly as"ONE-OVER-v," which in equation form is:a(E) 0(v0 / v) 0(E0 / E)1/2Here the interaction probability is generally proportional to the "timespent near the nucleus' force field."In the RESONANCE region, specific "energy levels" in the nucleusgive rise to extremely high cross sections at rather precise neutronenergies. Absorption, scattering, and fission cross sections canexhibit this trait.Energy-DependentCross Sections forUraniumThe energy-dependent cross-section plots in Figure 3-4 show that:235U has thermal and fast fission and is therefore fissile238U has fast fission only and is therefore fissionable, butnon-fissile235U has a fission cross section several orders of magnitudegreater for thermal than for fast neutrons; thus, neutronmoderation can considerably increase the reaction probability.USNRC Technical Training CenterNuclear Criticality Safety3-130905 (Rev 3)Directed Self-Study
Module 3.0: Nuclear TheoryINFINITE SYSTEMAn initial look at developing a model for the fission chain reactionbegins with consideration of an infinite system. Here, by definition,there is no leakage (the term which turns out to be very difficult toquantify), so only neutron production and absorption need beconsidered.InfiniteMultiplicationFactorFISSION CROSS SECTION, barnesFigure 3-4.235U
MODULE 3.0: NUCLEAR THEORY Introduction Welcome to Module 3.0 of the Nuclear Criticality Safety Directed Self-Study Course! This is the third of five modules available in this directed self-study course. The purpose of this module is to assist you in describing the fission process and basic nuclear theory concepts.
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Any nuclear reactor or radiological accidents involving equipment used in connection with naval nuclear reactors or other naval nuclear energy devices while such equipment is under the custody of the Navy. DoD's Definition of Nuclear Weapon Accident An unexpected event involving nuclear weapons or nuclear
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