Applied Reactor Physics

Alain Hébert

Second Edition

[cursus

Applied-Reactor-Physics-Copyright-2016.indd 1 2016-05-27 10:09:22

Applied Reactor Physics, Second EditionAlain Hébert

Cover page: Cyclone DesignEditing, proofreading: Judy Yelon

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Reprint, June 2016.

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Legal deposit: 1st Quarter 2016 ISBN 978-2-553-01698-1 (Printed version) Bibliothèque et Archives nationales du Québec ISBN 978-2-553-01709-4 (PDF version)Library and Archives Canada Printed in Canada

To my sons, Antoine and Guillaume.

Foreword

This text was prepared to provide teaching support to the students of graduate courses ENE6101and ENE6103 at Ecole Polytechnique de Montreal. A first version was written in French aslecture notes in May 1983 and has been used until 2009. This text was translated, correctedand completed, taking the opportunity to add more exercises. The first edition of the textbookwas released in 2009 and reprinted in 2010 with minor error corrections. The 2010 versionwas used for more than 6 years in many universities and research institutes across the world.The second edition was released in 2016 with important improvements in the text, new end-of-chapter exercises, extended bibliography, a Moodle support website1 and a new cover. I takethe opportunity to thank Prof. Ben Forget of the Massachusetts Institute of Technology forpointing out many errors in the first edition.

The first motivation for this text is to provide the fundamental knowledge required tounderstand and apply the reactor physics tools available today. Most existing textbooks are stillrelated to the first generation of tools based on the four-factor formula. The current generationof tools is heavily based on ENDF information and on multigroup lattice calculations. Wetried to cover a sufficient amount of fundamental information to help the reader understand themodern approach to reactor physics. This text is also a good starting point for understandingthe physics behind DRAGON and DONJON codes.

Numerous Matlab scripts are proposed to readers. Complete solutions of the exercises aremade available to teachers. Any suggestions to improve this material are welcome and may bedirected to me at the following address: alain.hebert@polymtl.ca.

1see https://moodle.polymtl.ca/course/view.php?id=1233

Contents

1 Introduction 1

2 Cross sections and nuclear data 52.1 Solid angles and spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Dealing with distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Dynamics of a scattering reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Collision of a neutron with a nucleus initially at rest . . . . . . . . . . . . 162.4 Definition of a cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Formation of a compound nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.1 The single level Breit-Wigner formulas . . . . . . . . . . . . . . . . . . . . 272.5.2 Low-energy variation of cross sections . . . . . . . . . . . . . . . . . . . . 32

2.6 Thermal agitation of nuclides and binding effects . . . . . . . . . . . . . . . . . . 332.6.1 Numerical convolution of cross sections . . . . . . . . . . . . . . . . . . . 342.6.2 Convolution of Breit-Wigner cross sections . . . . . . . . . . . . . . . . . 352.6.3 Convolution of a constant cross section . . . . . . . . . . . . . . . . . . . . 382.6.4 Convolution of the differential scattering cross section . . . . . . . . . . . 402.6.5 Effects of molecular or metallic binding . . . . . . . . . . . . . . . . . . . 46

2.7 Expansion of the differential cross sections . . . . . . . . . . . . . . . . . . . . . . 502.8 Calculation of the probability tables . . . . . . . . . . . . . . . . . . . . . . . . . 512.9 Production of an isotopic cross-section library . . . . . . . . . . . . . . . . . . . . 54

2.9.1 Photo-atomic interaction data . . . . . . . . . . . . . . . . . . . . . . . . . 582.9.2 Delayed neutron data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.9.3 An overview of DRAGR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3 The transport equation 693.1 The particle flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2 Derivation of the transport equation . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2.1 The characteristic form of the transport equation . . . . . . . . . . . . . . 743.2.2 The integral form of the transport equation . . . . . . . . . . . . . . . . . 743.2.3 Boundary and continuity conditions . . . . . . . . . . . . . . . . . . . . . 75

3.3 Source density in reactor physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.3.1 The steady-state source density . . . . . . . . . . . . . . . . . . . . . . . . 773.3.2 The transient source density . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.4 The transport correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.5 Multigroup discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.5.1 Multigroup steady-state transport equation . . . . . . . . . . . . . . . . . 853.5.2 Multigroup transient transport equation . . . . . . . . . . . . . . . . . . . 86

3.6 The first-order streaming operator . . . . . . . . . . . . . . . . . . . . . . . . . . 883.6.1 Cartesian coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . 88

viii Contents

3.6.2 Cylindrical coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . 903.6.3 Spherical coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.7 The spherical harmonics method . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.7.1 The Pn method in 1D slab geometry . . . . . . . . . . . . . . . . . . . . . 943.7.2 The Pn method in 1D cylindrical geometry . . . . . . . . . . . . . . . . . 983.7.3 The Pn method in 1D spherical geometry . . . . . . . . . . . . . . . . . . 1033.7.4 The simplified Pn method in 2D Cartesian geometry . . . . . . . . . . . . 106

3.8 The collision probability method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.8.1 The interface current method . . . . . . . . . . . . . . . . . . . . . . . . . 1103.8.2 Scattering-reduced matrices and power iteration . . . . . . . . . . . . . . 1123.8.3 Slab geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.8.4 Cylindrical 1D geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.8.5 Spherical 1D geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.8.6 Unstructured 2D finite geometry . . . . . . . . . . . . . . . . . . . . . . . 125

3.9 The discrete ordinates method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.9.1 Quadrature sets in the method of discrete ordinates . . . . . . . . . . . . 1323.9.2 The difference relations in 1D slab geometry . . . . . . . . . . . . . . . . 1373.9.3 The difference relations in 1D cylindrical geometry . . . . . . . . . . . . . 1393.9.4 The difference relations in 1D spherical geometry . . . . . . . . . . . . . . 1433.9.5 The difference relations in 2D Cartesian geometry . . . . . . . . . . . . . 1453.9.6 Synthetic acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.10 The method of characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.10.1 The MOC integration strategy . . . . . . . . . . . . . . . . . . . . . . . . 1503.10.2 Unstructured 2D finite geometry . . . . . . . . . . . . . . . . . . . . . . . 1563.10.3 The algebraic collapsing acceleration . . . . . . . . . . . . . . . . . . . . . 161

3.11 The multigroup Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . 1653.11.1 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1663.11.2 Rejection techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713.11.3 The random walk of a neutron . . . . . . . . . . . . . . . . . . . . . . . . 1733.11.4 Criticality calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1823.11.5 Monte Carlo reaction estimators . . . . . . . . . . . . . . . . . . . . . . . 186

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

4 Elements of lattice calculation 1934.1 A historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1944.2 Neutron slowing-down and resonance self-shielding . . . . . . . . . . . . . . . . . 197

4.2.1 Elastic slowing down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1994.2.2 A review of resonance self-shielding approaches . . . . . . . . . . . . . . . 2024.2.3 The Livolant-Jeanpierre approximations . . . . . . . . . . . . . . . . . . . 2034.2.4 The physical probability tables . . . . . . . . . . . . . . . . . . . . . . . . 2064.2.5 The statistical subgroup equations . . . . . . . . . . . . . . . . . . . . . . 2104.2.6 The multigroup equivalence procedure . . . . . . . . . . . . . . . . . . . . 213

4.3 The neutron leakage model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2154.3.1 The Bn leakage calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 2164.3.2 The homogeneous fundamental mode . . . . . . . . . . . . . . . . . . . . . 2174.3.3 The heterogeneous fundamental mode . . . . . . . . . . . . . . . . . . . . 2224.3.4 Introduction of leakage rates in a lattice calculation . . . . . . . . . . . . 2254.3.5 Introduction of leakage rates with collision probabilities . . . . . . . . . . 2274.3.6 Full-core calculations in diffusion theory . . . . . . . . . . . . . . . . . . . 2294.3.7 Full-core calculations in transport theory . . . . . . . . . . . . . . . . . . 230

4.4 The SPH equivalence technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2324.4.1 Definition of the macro balance relations . . . . . . . . . . . . . . . . . . . 233

Contents ix

4.4.2 Definition of the SPH factors . . . . . . . . . . . . . . . . . . . . . . . . . 234

4.4.3 Iterative calculation of the SPH factors . . . . . . . . . . . . . . . . . . . 237

4.5 Isotopic depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

4.5.1 The power normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

4.5.2 The saturation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

4.5.3 The integration factor method . . . . . . . . . . . . . . . . . . . . . . . . 244

4.5.4 Depletion of heavy isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . 246

4.6 Creation of the reactor database . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

4.6.1 Selected information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

4.6.2 Database information structure . . . . . . . . . . . . . . . . . . . . . . . . 251

4.7 A presentation of DRAGON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

4.7.1 A DRAGON tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

5 Full-core calculations 267

5.1 The steady-state diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . 270

5.1.1 The Fick law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

5.1.2 Continuity and boundary conditions . . . . . . . . . . . . . . . . . . . . . 272

5.1.3 The finite homogeneous reactor . . . . . . . . . . . . . . . . . . . . . . . . 274

5.1.4 The heterogeneous 1D slab reactor . . . . . . . . . . . . . . . . . . . . . . 276

5.2 Discretization of the neutron diffusion equation . . . . . . . . . . . . . . . . . . . 280

5.2.1 Mesh-corner finite differences . . . . . . . . . . . . . . . . . . . . . . . . . 281

5.2.2 Mesh-centered finite differences . . . . . . . . . . . . . . . . . . . . . . . . 284

5.2.3 A primal variational formulation . . . . . . . . . . . . . . . . . . . . . . . 286

5.2.4 The Lagrangian finite-element method . . . . . . . . . . . . . . . . . . . . 288

5.2.5 The analytic nodal method in 2D Cartesian geometry . . . . . . . . . . . 292

5.3 Generalized perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

5.3.1 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

5.3.2 State variables and reactor characteristics . . . . . . . . . . . . . . . . . . 302

5.3.3 Computing the Jacobian using the implicit approach . . . . . . . . . . . . 304

5.3.4 Computing the Jacobian using the explicit approach . . . . . . . . . . . . 305

5.4 Space-time kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

5.4.1 Point-kinetics equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

5.4.2 The implicit temporal scheme . . . . . . . . . . . . . . . . . . . . . . . . . 312

5.4.3 The space-time implicit scheme . . . . . . . . . . . . . . . . . . . . . . . . 314

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

Answers to Problems 323

A Tracking of 1D and 2D geometries 331

A.1 Tracking of 1D slab geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

A.2 Tracking of 1D cylindrical and spherical geometries . . . . . . . . . . . . . . . . . 333

A.3 The theory behind sybt1d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

A.4 Tracking of 2D square pincell geometries . . . . . . . . . . . . . . . . . . . . . . . 337

A.5 The theory behind sybt2d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

B Special functions with Matlab 345

B.1 Function taben . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

B.2 Function akin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

x Contents

C Numerical methods 349C.1 Solution of a linear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

C.1.1 Gauss elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349C.1.2 Cholesky factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351C.1.3 Iterative approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

C.2 Solution of an eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 362C.2.1 The inverse power method . . . . . . . . . . . . . . . . . . . . . . . . . . . 362C.2.2 The preconditioned power method . . . . . . . . . . . . . . . . . . . . . . 366C.2.3 The Hotelling deflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368C.2.4 The multigroup partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . 370C.2.5 Convergence acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

C.3 Solution of a fixed source eigenvalue problem . . . . . . . . . . . . . . . . . . . . 375C.3.1 The inverse power method . . . . . . . . . . . . . . . . . . . . . . . . . . . 375C.3.2 The preconditioned power method with variational acceleration . . . . . . 378

Bibliography 381

Index 391

Chapter 1

Introduction

Reactor physics is the discipline devoted to the study of interactions between neutrons andmatter in a nuclear reactor. Such an interaction is produced when a neutron collides withthe nucleus of a specific nuclide (or isotope). Interactions between neutrons and nuclei aredescribed by nuclear physics models as a function of neutron energy and nuclide characteristics.A statistical mechanics approach is also used to describe the distribution of neutrons in phasespace (position and velocity vectors) as a function of time. The neutronic number density (orneutron distribution) can be obtained as the solution of a transport equation similar to theequation used to describe photon populations. This statistical model is greatly simplified bysome characteristics of neutron-nuclide interactions:

• Relativistic effects can be neglected.

• Neutron-neutron interactions can be neglected. Consequently, the collision term of thecorresponding transport equation is linear.

• Neutrons are neutral particles; neutron mean free paths are straight lines.

• The materials are isotropic in space.

• The nuclides are in thermal equilibrium with their neighbors.

Two phenomena are responsible for the complexity of reactor physics. A nuclear reactor isgenerally a complicated three-dimensional assembly of different geometrical components madeof a variety of materials. Secondly, many materials have interaction characteristics that varystrongly with neutron energy. Neutron distribution in the reactor can be predicted using thethree main steps depicted in Fig. 1.1. The basic principle is to start the calculation with a veryfine representation in neutron energy (with a coarse representation in space) and to terminatewith a very fine representation in space (with a coarse representation in energy).

Neutron cross sections are used to characterize the neutron-nuclide interaction at differentneutron energies. Such an interaction occurs only when a neutron collides with a nucleus. Atneutron energies less than 1 electron volt (eV), it will be necessary to take into account thenuclide temperature and the effects of molecular binding for nuclides such as heavy water, lightwater and graphite. At energy greater than 1 eV, the cross sections of heavy isotopes areresonant functions of the neutron energy. In this case, we will have to compute averaged cross-section values, also called self-shielded cross sections, over energy sub-domains that may containmany resonances. These self-shielded values are also a function of the nuclide temperature dueto the Doppler effect. These concepts will be covered in Chapter 2.

Reactor physics is closely related to the solution of the transport equation for neutral par-ticles. We have chosen to present five among the most widely used techniques for solving the

2 Introduction

Neutron-nuclidecross-section

calculation

Latticecalculation

Depletioncalculation

Reactorcalculation

Space-timekinetics

calculation

Fuel management,design and

operation simulation

Depletioncalculation

Step 1 Step 2 Step 3

Figure 1.1: Global computational scheme.

transport equation in multigroup approximation. The reviewed techniques are the method ofspherical harmonics (Pn), the collision probability method (CP), the discrete ordinates method(SN ), the method of characteristics (MOC) and the multigroup Monte Carlo method. Eachof these legacy techniques is first introduced from scratch, together with elementary numericaltechniques adapted to their solution. This introduction of transport theory for neutral particlesis presented in Chapter 3.

A lattice calculation is a collection of numerical algorithms and models which are capable ofrepresenting the neutronic behaviour of a unit cell or assembly in a nuclear reactor. A unit cellor assembly is a part of the reactor that is similar to its neighboring parts. It generally consistsof a fuel channel surrounded by a moderator or a fuel assembly in a light-water reactor. Onecomponent of a lattice code permits the calculation of the neutron flux, a quantity related tothe neutron distribution, as a function of the cross sections recovered from the preceding step.The neutron flux is the solution of a transport equation defined over the unit cell or assembly.Many numerical solution techniques are available to solve this equation and will be reviewedin Chapter 3. The input cross sections are next weighted with the neutron flux to produceenergy- and region-averaged cross-section and diffusion coefficient values that will be used inthe complete reactor, or full-core, calculation. The lattice calculation is coupled to a depletioncalculation in order to represent the variation of nuclide concentrations with burnup. Othermodels are also required to represent neutron leakage or non-linear homogenization effects. Allthese models will also be reviewed in Chapter 4.

The complete reactor, or full-core, calculation permits us to obtain the primary powerdistribution in the reactor. This calculation is based on a simplified representation of the crosssections as a function of energy. Moreover, each unit cell or assembly will be replaced by a setof homogeneous parallelepipeds or hexagons, using energy- and region-averaged cross-sectionand diffusion coefficient values produced by the lattice calculation. On the scale of the completereactor, the transport equation is generally replaced by a simplified equation called the diffusionequation or simplified Pn equation. Once the neutron distribution over the complete reactor isobtained, the primary power distribution can be obtained from knowing the energy productioncross sections.

The complete reactor calculation can be performed in a steady-state or static condition,assuming that the neutron distribution is constant in time. A first-order perturbation of thesteady-state equations leads to a fixed source eigenvalue problem and to the generalized per-turbation theory (GPT). It is also possible to represent transient effects resulting from the

Introduction 3

reactor operation or from accidental events. In this case, we need to solve a space-time kineticsequation over the complete reactor domain. Calculations related to fuel management, designand operation simulation must take into account the effects of depletion for many importantnuclides with time. Such a capability is also available in the reactor calculation step. All theaspects related to the reactor calculation will be presented in Chapter 5.

The computational examples in this text were done with Matlab as this environment is anexcellent one for acquiring experience with the algorithms, for doing the exercises, and for rapidprototype development.1 These codes are for pedagogical and academic use only. They werewritten to remain simple, so they lack certain safeguards that production code would possess.They have not been optimized for speed, nor are they guaranteed to be error free.

1Matlab is a registered trademark of The MathWorks, Inc.

Chapter 2

Cross sections and nuclear data

A collision between a neutron and a nucleus is likely to produce nuclear reactions of differenttypes. In the simplest case, the neutron is simply scattered by the nucleus without penetratingit. This interaction, similar to a billiard-ball collision, is the potential scattering reaction. Thepotential scattering reaction is elastic, as it conserves both the momentum and kinetic energyof the neutron-nucleus pair.

However, a neutron-nucleus collision is likely to produce a compound nucleus where theneutron actually penetrates the nucleus and mixes with other nucleons. In this case, theincident neutron and binding energy is transmitted to the compound nucleus, making it highlyunstable. After a half-life of between 10−22 and 10−14 second, the compound nucleus losesexcitation energy by emitting particles and/or electromagnetic rays. For heavy nuclides, themost probable decay mode of the compound nucleus is fission. If the compound nucleus doesnot undergo fission, the majority of the excitation energy is removed by gamma ray or particleemission (protons, neutrons, alpha particles). The mode of decay determines the type of nuclearreaction. In a resonant scattering reaction, for example, a single neutron is emitted with orwithout emission of gamma rays. If no gamma rays are emitted, the scattering reaction is saidto be elastic; otherwise, it is inelastic. A radiative capture reaction is another type of nuclearreaction where the compound nucleus decays by emitting only gamma rays.

Various nuclear reactions will be studied with the help of two concepts:

1. the cross sections are related to the probability associated with each nuclear reaction andcan be used to compute the corresponding reaction rate, the number of nuclear reactionsof this type per unit of time.

2. the collision law describes the dynamics of a collision. The collision law is used to computethe velocity and direction characteristics of the emitted (or secondary) particles as afunction of the nuclide temperature and of the characteristics of the incident (or primary)neutron. The secondary particles are those resulting from the decay of the compoundnucleus or correspond to the scattered neutron from a potential reaction.

We shall first introduce some mathematical elements required to deal with neutron directionand distributions; then, we study the dynamics of a scattering reaction, either potential orresonant.

2.1 Solid angles and spherical harmonics

The direction of a moving particle in a three-dimensional (3D) domain is represented by itssolid angle, a unit vector pointing in the direction of the particle, as shown in Fig. 2.1. The

6 Chapter 2

velocity vector V n of the particle is written in terms of its solid angle as

V n = Vn Ω (2.1)

whereVn = |V n| and |Ω| = 1. (2.2)

μ

ξ

η

φ

dφ

d2ΩΩ

X

Y

Z

dψ

ψ

Figure 2.1: Definition of the solid angle.

The solid angle Ω is defined in terms of its three direction cosines μ, η and ξ, using

Ω = μ i + η j + ξ k (2.3)

with the constraintμ2 + η2 + ξ2 = 1 (2.4)

where the symbols i, j and k are used to denote the unit vectors in the x-, y- and z-direction,respectively.

Here, we have used the x-axis as the principal axis to define the colatitude or polar angleψ = cos−1 μ and the azimuth φ. The definition domain is 0 ≤ ψ ≤ π for the colatitude and0 ≤ φ ≤ 2π for the azimuth. On a case-by-case basis, any other axis can be used. The last twodirection cosines are written in terms of the azimuthal angle φ using

η =√

1− μ2 cosφ and ξ =√

1− μ2 sinφ. (2.5)

An increase in ψ by dψ and in φ by dφ sweeps out the area d2Ω on a unit sphere. The solidangle encompassed by a range of directions is defined as the area swept out on the surface ofa sphere divided by the square of the radius of the sphere. Thus, the differential solid angleassociated with solid angle Ω is

d2Ω = sinψ dψ dφ. (2.6)

The solid angle is a dimensionless quantity. Nevertheless, to avoid confusion when referringto a directional distribution function, units of steradians, abbreviated sr, are attributed to thesolid angle.

Cross sections and nuclear data 7

In reactor physics, many quantities are continuous and bounded distributions of the particledirection cosine μ or solid angle Ω. In the first case, such a quantity is written f(μ) and canbe approximated in terms of an L-order Legendre polynomial expansion using

f(μ) =L∑

�=0

2� + 1

2f� P�(μ). (2.7)

Legendre polynomials obey the following orthonormal relations:∫ 1

−1

dμ P�(μ)P�′(μ) =2

2� + 1δ�,�′ (2.8)

so that the �-th order coefficient is obtained using

f� =

∫ 1

−1

dμP�(μ) f(μ). (2.9)

The Legendre polynomials are defined by the relations

P0(μ) = 1, P1(μ) = μ

and

P�+1(μ) =1

� + 1[(2� + 1)μP�(μ)− � P�−1(μ)] if � ≥ 1. (2.10)

In the more general case, a bounded distribution of the particle solid angle is written f(Ω)and can be approximated in terms of an L-order real spherical harmonics expansion using

f(Ω) =L∑

�=0

2� + 1

4π

�∑m=−�

fm� Rm

� (Ω) (2.11)

where Rm� (Ω) is a real spherical harmonics component, a distribution of the solid angle Ω.

The first component μ of the solid angle is the cosine of the polar angle and φ represents theazimuthal angle. These components are expressed in terms of the associated Legendre functions

P|m|� (μ) using

Rm� (Ω) =

√(2− δm,0)

(�− |m|)!(� + |m|)! P

|m|� (μ) Tm(φ) (2.12)

where Pm� (μ) is defined in terms of the �-th order Legendre polynomial P�(μ) as

Pm� (μ) = (1− μ2)m/2 dm

dμmP�(μ), m ≥ 0 (2.13)

and where

Tm(φ) =

{cosmφ, if m ≥ 0;sin |m|φ, otherwise.

(2.14)

Note that we have used the Ferrer definition of the associated Legendre functions Pm� (μ),

in which the factor (−1)m is absent. They can be obtained using the following Matlab script:

function f=plgndr(l,m,x)

% return the Ferrer definition of the associated Legendre function.

% function f=plgndr(l,m,x)

% (c) 2008 Alain Hebert, Ecole Polytechnique de Montreal

if m < 0

8 Chapter 2

error(’bad arguments in plgndr 1’)

elseif m > l

error(’bad arguments in plgndr 2’)

elseif abs(x) > 1.

error(’bad arguments in plgndr 3’)

end

pmm=1. ;

if m > 0, pmm=prod(1:2:2*m)*sqrt((1.-x).*(1.+x)).^m ; end

if l == m

f=pmm ;

else

pmmp1=(2*m+1)*x.*pmm ;

if l == m+1

f=pmmp1 ;

else

for ll=m+2:l

pll=((2*ll-1)*x.*pmmp1-(ll+m-1)*pmm)/(ll-m) ;

pmm=pmmp1 ; pmmp1=pll ;

end

f=pll ;

end

end

The Ferrer definition helps to simplify low-order angular expansions since

Ω =

⎛⎝ μ√1− μ2 cosφ√1− μ2 sinφ

⎞⎠ =

⎛⎝ R01(Ω)

R11(Ω)

R−11 (Ω)

⎞⎠ . (2.15)

Real spherical harmonics are to be preferred to classical ones because they permit us to elim-inate imaginary components in 3D problems. The trigonometric functions, associated Legendrefunctions and real spherical harmonics obey the following orthonormal relations:∫ π

−π

dφ Tm(φ) Tm′(φ) = π (1 + δm,0) δm,m′ , (2.16)

∫ 1

−1

dμ Pm� (μ)Pm

�′ (μ) =2(� + m)!

(2� + 1)(�−m)!δ�,�′ (2.17)

so that ∫4π

d2Ω Rm� (Ω)Rm′

�′ (Ω) =4π

2� + 1δ�,�′ δm,m′ (2.18)

with d2Ω = dμ dφ, so that the components of the distribution in Eq. (2.11) are written

fm� =

∫4π

d2Ω Rm� (Ω) f(Ω). (2.19)

In the previous relations, we have introduced the integral over 4π to represent an integrationover all possible directions. This integral is defined as∫

4π

d2Ω f(Ω) ≡∫ 1

−1

dμ

∫ 2π

0

dφ f(μ, φ). (2.20)

The real spherical harmonics satisfy the addition theorem which can be written in terms oftwo different solid angles Ω and Ω′, as