1-1 Lesson 1 Objectives Objectives of Course Objectives of Course Go over syllabus Go over syllabus Go over course Go over course Overview of Course Overview of Course The Transport Equation The Transport Equation Assumptions Assumptions Definition of basic elements Definition of basic elements Scattering cross sections Scattering cross sections Use of Legendre expansions of angular distribution Use of Legendre expansions of angular distribution Fission neutron distribution Fission neutron distribution

1-2 Objectives of Course User setup should be: Materials and geometry Materials and geometry 1. Material makeup (isotopics) 2. Material energy interactions with particles 3. Material spatial distribution Source description Source description 4. Source particles 5. Source energy distribution 6. Source spatial distribution “Detector” response “Detector” response 7. Detector particle sensitivity 8. Detector energy sensitivity (response function) 9. Detector spatial location Why are any other questions asked? Why are any other questions asked?

1-3 Objectives of Course (2) Answer: Boltzmann gave us an exact equation, but we cannot solve it. We must simplify the equation: Space: Replace continuous space with homogeneous blocks (“cells”) of material Space: Replace continuous space with homogeneous blocks (“cells”) of material Energy: Replace continuous energy with energy “groups” Energy: Replace continuous energy with energy “groups” Direction: Constrain particles to only travel in certain directions Direction: Constrain particles to only travel in certain directions Result: The deterministic discrete ordinates equation, discretized for computer solution.

1-4 Objectives of Course (3) “Deterministic codes give you exact solutions to approximate models. Monte Carlo codes give you approximate solutions to exact models.” This situation puts an extra burden on you, the user. This situation puts an extra burden on you, the user. You are required to supply computer code input that is NOT related to the description of your problem, but is related to this simpler model (which is the only one the computer can solve). You are required to supply computer code input that is NOT related to the description of your problem, but is related to this simpler model (which is the only one the computer can solve). My goal: Help you understand what is being asked of you My goal: Help you understand what is being asked of you

1-5 Overview of Course First four chapters of: First four chapters of: Lewis, E. E., and Miller, W. F., Jr.; Computational Methods of Neutron Transport, American Nuclear Society, La Grange Park, IL, Lewis, E. E., and Miller, W. F., Jr.; Computational Methods of Neutron Transport, American Nuclear Society, La Grange Park, IL, General flow of the course will be: General flow of the course will be: Derivation of the continuous-energy Boltzmann Equation (L&M, 1) Derivation of the continuous-energy Boltzmann Equation (L&M, 1) Derivation of the forward equation Derivation of the forward equation Differences in approach for source vs. eigenvalue problems Differences in approach for source vs. eigenvalue problems Derivation and use of adjoint form of equation Derivation and use of adjoint form of equation

1-6 Overview of Course (2) General flow of the course (cont’d): General flow of the course (cont’d): Energy and time discretization (L&M, 2) Energy and time discretization (L&M, 2) Multigroup approximation in energy Multigroup approximation in energy Fixed source solution strategies in energy Fixed source solution strategies in energy Eigenvalue problem solution strategies in energy Eigenvalue problem solution strategies in energy Time-dependent considerations Time-dependent considerations 1D discrete ordinates methods (L&M, 3) 1D discrete ordinates methods (L&M, 3) Angular approximation Angular approximation Spatial differencing Spatial differencing Curvilinear coordinates Curvilinear coordinates Acceleration techniques Acceleration techniques

1-7 Overview of Course (3) General flow of the course (cont’d): General flow of the course (cont’d): 2D and 3D discrete ordinates (L&M, 4) 2D and 3D discrete ordinates (L&M, 4) Angular quadrature Angular quadrature Cartesian treatments Cartesian treatments Curvilinear treatments Curvilinear treatments Ray effects Ray effects Integral transport theory (L&M, 5) Integral transport theory (L&M, 5) If time permits If time permits

1-8 The Transport Equation Introduction Introduction Particle Interaction Particle Interaction Particle Streaming Particle Streaming Transport with Secondary Particles Transport with Secondary Particles The Time-Independent Transport equation The Time-Independent Transport equation The Adjoint Transport Equation The Adjoint Transport Equation

1-9 The basic physical assumptions 1. Particles are points 2. Particles travel in straight lines, unaccelerated until they interact 3. Particles don’t hit other particles 4. Collisions are resolved instantaneously 5. Material properties are isotropic in direction 6. Composition, configuration, and material properties are known and constant 7. Only the expected (mean) values of reaction rates are needed You will think about these more deeply in HW problem 1-1.

1-10 Definition of basic elements Material cross sections: Particle/matter interaction probabilities Material cross sections: Particle/matter interaction probabilities We will use small sigma, , for for microscopic AND macroscopic cross sections: We will use small sigma, , for for microscopic AND macroscopic cross sections: =Probability of an interaction of type x per unit path length

1-11 Definition of basic elements (2) where x= where x= ‘c’ for capture=particle loss ‘f’ for fission ‘a’ for absorption=fission + capture ‘s’ for scattering=particle change of energy and direction For neutrons, the primary scattering mechanisms are elastic scattering, inelastic scattering, and (n,2n) For neutrons, the primary scattering mechanisms are elastic scattering, inelastic scattering, and (n,2n) For photons, the scattering mechanisms are Compton scattering and pair production For photons, the scattering mechanisms are Compton scattering and pair production For coupled neutron/gamma problems, neutron reactions that produce gammas are “scatter” For coupled neutron/gamma problems, neutron reactions that produce gammas are “scatter” Unit of macroscopic cross section is cm -1 Unit of macroscopic cross section is cm -1

1-12 Definition of basic elements (3) Denoting the intensity of the flow of a beam of particles as I (x), we have: Denoting the intensity of the flow of a beam of particles as I (x), we have: This is the familiar exponential attenuation This is the familiar exponential attenuation In the book, the total cross section is sometimes denoted by  (no subscript) In the book, the total cross section is sometimes denoted by  (no subscript)

1-13 Definition of basic elements (4) Background: A “weighted average” of a function of x is defined as: Background: A “weighted average” of a function of x is defined as: The most common variations we see in NE are: The most common variations we see in NE are: Unweighted average: w(x)=1 (over finite domain of x) Unweighted average: w(x)=1 (over finite domain of x) Mean (or expected) value of x: w(x)=Pr(x) (probability of x being chosen) Mean (or expected) value of x: w(x)=Pr(x) (probability of x being chosen) Nth moment of x: w(x)=x n, 0<x<infinity Nth moment of x: w(x)=x n, 0<x<infinity Nth Legendre moment: w(x)=1/2 P n (x), -1<x<1 Nth Legendre moment: w(x)=1/2 P n (x), -1<x<1

1-14 Definition of basic elements (4a) The mean free path,, is defined as the average distance traveled before a collision: The mean free path,, is defined as the average distance traveled before a collision: Work this out (Prob. 1-2) Work this out (Prob. 1-2) For reaction rate x, we have: For reaction rate x, we have:

1-15 Scattering cross sections For scattering reactions, we must consider the post-collision properties as well as the probability of interaction: For scattering reactions, we must consider the post-collision properties as well as the probability of interaction: where:

1-16 Scattering cross sections (2) Based on Assumption #5, the angular dependence is dependent on the deflection angle between the two directions: Based on Assumption #5, the angular dependence is dependent on the deflection angle between the two directions: where: Note that there is no azimuthal angular dependence Note that there is no azimuthal angular dependence

1-17 Scattering cross sections (3) The distribution function is normalized to integrate to the number of particles that are emitted by the reaction. For example, elastic scattering has: The distribution function is normalized to integrate to the number of particles that are emitted by the reaction. For example, elastic scattering has: whereas for (n,2n) we have: whereas for (n,2n) we have:

1-18 Scattering cross sections (4) Note that since we combine cross sections linearly, the relationship between the macroscopic and microscopic distribution functions is given by: Note that since we combine cross sections linearly, the relationship between the macroscopic and microscopic distribution functions is given by:

1-19 Scattering cross sections (5) The most familiar distribution is the elastic scattering distribution (from kinematics): The most familiar distribution is the elastic scattering distribution (from kinematics):

1-20 Functional expansions The general field of functional expansion involves the approximation of a continuous function as a linear combination of a basis function set: The general field of functional expansion involves the approximation of a continuous function as a linear combination of a basis function set: Use of a “complete” basis functions set means that as L goes to infinity, the approximation approaches the function Use of a “complete” basis functions set means that as L goes to infinity, the approximation approaches the function The trick is finding the coefficients that “best” fit the function The trick is finding the coefficients that “best” fit the function

1-21 Functional expansions (2) Determining the “best” comes down to finding the approximation (for a given L) that is “closest” to the function according to some “norm” Determining the “best” comes down to finding the approximation (for a given L) that is “closest” to the function according to some “norm” A “norm” is simply a measure of difference between two values (or functions) that satisfies two simple criteria: A “norm” is simply a measure of difference between two values (or functions) that satisfies two simple criteria: 1. The value of the norm is always non-negative 2. The value is only zero if the two are identical Example: Distance norms for points Example: Distance norms for points

1-22 Functional expansions (3) The norm that we will use is the least-square norm,, which is a member of the series defined by: The norm that we will use is the least-square norm,, which is a member of the series defined by: The “distance” from a function and its approximate expansion is then: The “distance” from a function and its approximate expansion is then:

1-23 Functional expansions (4) We find the optimum coefficients by setting the partial derivatives to zero: We find the optimum coefficients by setting the partial derivatives to zero: This gives us a set of linear (matrix) equations: This gives us a set of linear (matrix) equations: Thus minimizing the least squares norm comes down to preserving the moments of the expansion functions themselves (Galerkin method). Thus minimizing the least squares norm comes down to preserving the moments of the expansion functions themselves (Galerkin method).

1-24 Functional expansions (5) Solving for the coefficients becomes much easier if the basis functions are orthogonal (which means that the integral of the product of any two different basis functions is zero): Solving for the coefficients becomes much easier if the basis functions are orthogonal (which means that the integral of the product of any two different basis functions is zero): This turns the previous equation into: This turns the previous equation into:

1-25 Use of Legendre expansions Using the cosine of the deflection angle, we can represent the angular dependence of the distribution in a Legendre expansion: Using the cosine of the deflection angle, we can represent the angular dependence of the distribution in a Legendre expansion: This allows us to represent the scattering distribution by determining the Legendre coefficients: This allows us to represent the scattering distribution by determining the Legendre coefficients:

1-26 Use of Legendre expansions (2) Using the orthogonality of the Legendre polynomials: Using the orthogonality of the Legendre polynomials: we can operate on both sides of the expansion (1 st eqn. on previous slide) with:

1-27 Use of Legendre expansions (3) To get: To get: Work this out for yourself (Prob. 1-3) Work this out for yourself (Prob. 1-3)

1-28 Fission neutron distribution Two data variables you need to know are: Two data variables you need to know are: The first is a function; the second is a distribution The first is a function; the second is a distribution

1-29 Homework 1-1 For each of the assumptions listed on slide 1-9, give a physical situation (if you can think of one) for which the assumption may not be a good one. For each of the assumptions listed on slide 1-9, give a physical situation (if you can think of one) for which the assumption may not be a good one.

1-30 Homework 1-2 Use integration by parts and l’Hopital’s rule to show that: Use integration by parts and l’Hopital’s rule to show that:

1-31 Homework 1-3 a. Show that if we expand: a. Show that if we expand: that the coefficients can be found from: b. Use this fact to expand Expand it to enough terms so that the error is less than 1% at all values of x. Expand it to enough terms so that the error is less than 1% at all values of x.