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Lesson 1 ObjectivesLesson 1 ObjectivesLesson 1 ObjectivesLesson 1 Objectives
• Objectives of CourseObjectives of Course• Go over syllabusGo over syllabus• Go over courseGo over course• Overview of CourseOverview of Course• The Transport EquationThe Transport Equation
• AssumptionsAssumptions• Definition of basic elementsDefinition of basic elements• Scattering cross sectionsScattering cross sections• Use of Legendre expansions of angular distributionUse of Legendre expansions of angular distribution• Fission neutron distributionFission neutron distribution
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Objectives of CourseObjectives of CourseObjectives of CourseObjectives of Course
User setup User setup shouldshould be: be:• Materials and geometryMaterials and geometry
1.1. Material makeup (isotopics)Material makeup (isotopics)2.2. Material energy interactions with particlesMaterial energy interactions with particles3.3. Material spatial distributionMaterial spatial distribution
• Source descriptionSource description4.4. Source particlesSource particles5.5. Source energy distributionSource energy distribution6.6. Source spatial distributionSource spatial distribution
• ““Detector” responseDetector” response7.7. Detector particle sensitivityDetector particle sensitivity8.8. Detector energy sensitivity (response function)Detector energy sensitivity (response function)9.9. Detector spatial locationDetector spatial location
• Why are any other questions asked?Why are any other questions asked?
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Objectives of Course (2)Objectives of Course (2)Objectives of Course (2)Objectives of Course (2)
Answer: Boltzmann gave us an exact equation, but Answer: Boltzmann gave us an exact equation, but we cannot solve it.we cannot solve it.
We must simplify the equation:We must simplify the equation:• Space: Replace continuous space with homogeneous Space: Replace continuous space with homogeneous
blocks (“cells”) of materialblocks (“cells”) of material• Energy: Replace continuous energy with energy Energy: Replace continuous energy with energy
“groups”“groups”• Direction: Constrain particles to only travel in certain Direction: Constrain particles to only travel in certain
directionsdirections
Result: The deterministic discrete ordinates equation, Result: The deterministic discrete ordinates equation, discretized for computer solution.discretized for computer solution.
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Objectives of Course (3)Objectives of Course (3)Objectives of Course (3)Objectives of Course (3)
““Deterministic codes give you exact solutions to Deterministic codes give you exact solutions to approximate models. Monte Carlo codes give approximate models. Monte Carlo codes give you approximate solutions to exact models.” you approximate solutions to exact models.”
• This situation puts an extra burden on you, the This situation puts an extra burden on you, the user.user.
• You are required to supply computer code input You are required to supply computer code input that is NOT related to the description of your that is NOT related to the description of your problem, but is related to this simpler model problem, but is related to this simpler model (which is the only one the computer can solve).(which is the only one the computer can solve).
• My goal: Help you understand what is being My goal: Help you understand what is being asked of youasked of you
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Overview of CourseOverview of CourseOverview of CourseOverview of Course
• First four chapters of: First four chapters of: • Lewis, E. E., and Miller, W. F., Jr.; Lewis, E. E., and Miller, W. F., Jr.; Computational Computational
Methods of Neutron TransportMethods of Neutron Transport, American Nuclear , American Nuclear Society, La Grange Park, IL, 1993.Society, La Grange Park, IL, 1993.
• General flow of the course will be:General flow of the course will be:• Derivation of the continuous-energy Derivation of the continuous-energy
Boltzmann Equation (L&M, 1)Boltzmann Equation (L&M, 1)• Derivation of the forward equationDerivation of the forward equation• Differences in approach for source vs. eigenvalue Differences in approach for source vs. eigenvalue
problemsproblems• Derivation and use of adjoint form of equationDerivation and use of adjoint form of equation
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Overview of Course (2)Overview of Course (2)Overview of Course (2)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 energyMultigroup approximation in energy• Fixed source solution strategies in energyFixed source solution strategies in energy• Eigenvalue problem solution strategies in energyEigenvalue problem solution strategies in energy• Time-dependent considerationsTime-dependent considerations
• 1D discrete ordinates methods (L&M, 3)1D discrete ordinates methods (L&M, 3)• Angular approximationAngular approximation• Spatial differencingSpatial differencing• Curvilinear coordinatesCurvilinear coordinates• Acceleration techniquesAcceleration techniques
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Overview of Course (3)Overview of Course (3)Overview of Course (3)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 quadratureAngular quadrature• Cartesian treatmentsCartesian treatments• Curvilinear treatmentsCurvilinear treatments• Ray effectsRay effects
• Integral transport theory (L&M, 5)Integral transport theory (L&M, 5)• If time permitsIf time permits
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The Transport EquationThe Transport EquationThe Transport EquationThe Transport Equation
• IntroductionIntroduction• Particle InteractionParticle Interaction• Particle StreamingParticle Streaming• Transport with Secondary ParticlesTransport with Secondary Particles• The Time-Independent Transport The Time-Independent Transport
equationequation• The Adjoint Transport EquationThe Adjoint Transport Equation
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The basic physical assumptionsThe basic physical assumptionsThe basic physical assumptionsThe basic physical assumptions
1.1. Particles are pointsParticles are points2.2. Particles travel in straight lines, unaccelerated until they Particles travel in straight lines, unaccelerated until they
interactinteract3.3. Particles don’t hit other particlesParticles don’t hit other particles4.4. Collisions are resolved instantaneouslyCollisions are resolved instantaneously5.5. Material properties are isotropic in directionMaterial properties are isotropic in direction6.6. Composition, configuration, and material properties are Composition, configuration, and material properties are
known and constantknown and constant7.7. Only the expected (mean) values of reaction rates are Only the expected (mean) values of reaction rates are
neededneeded
You will think about these more deeply in HW problem 1-1.You will think about these more deeply in HW problem 1-1.
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Definition of basic elementsDefinition of basic elementsDefinition of basic elementsDefinition of basic elements
• Material cross sections: Particle/matter Material cross sections: Particle/matter interaction probabilitiesinteraction probabilities
• We will use small sigma, We will use small sigma, , for for , for for microscopic AND macroscopic cross microscopic AND macroscopic cross sections:sections:
ErnEr ixi
N
ix
~,1
=Probability of an interaction of type x per =Probability of an interaction of type x per unit path lengthunit path length
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Definition of basic elements (2)Definition of basic elements (2)Definition of basic elements (2)Definition of basic elements (2)• where x=where x=
‘‘c’ for capture=particle loss c’ for capture=particle loss ‘‘f’ for fissionf’ for fission‘‘a’ for absorption=fission + capturea’ for absorption=fission + capture‘‘s’ for scattering=particle change of energy and directions’ for scattering=particle change of energy and direction
• For neutrons, the primary scattering mechanisms are elastic scattering, inelastic scattering, and For neutrons, the primary scattering mechanisms are elastic scattering, inelastic scattering, and (n,2n)(n,2n)
• For photons, the scattering mechanisms are Compton scattering and pair productionFor 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 cmUnit of macroscopic cross section is cm-1-1
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Definition of basic elements (3)Definition of basic elements (3)Definition of basic elements (3)Definition of basic elements (3)
• Denoting the intensity of the flow of a beam of Denoting the intensity of the flow of a beam of particles as particles as II(x)(x), we have:, we have:
x
t
teIxI
xIdx
xdI
)0()(
)()(
• This is the familiar exponential attenuationThis is the familiar exponential attenuation• In the book, the total cross section is sometimes In the book, the total cross section is sometimes
denoted by denoted by (no subscript) (no subscript)
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Definition of basic elements (4)Definition of basic elements (4)Definition of basic elements (4)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:
( ) ( )
( )
w x f x dx
Average
w x dx
• 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 Mean (or expected) value of x: w(x)=Pr(x) (probability of x
being chosen)being chosen)• Nth moment of x: w(x)=xNth moment of x: w(x)=xnn , 0<x<infinity , 0<x<infinity• Nth Legendre moment: w(x)=1/2 PNth Legendre moment: w(x)=1/2 Pnn(x), -1<x<1(x), -1<x<1
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Definition of basic elements (4a)Definition of basic elements (4a)Definition of basic elements (4a)Definition of basic elements (4a)
• The mean free path, The mean free path, , is defined as the , is defined as the average distance traveled before a collision: average distance traveled before a collision:
t
t
t
x
x
t
t
dxe
dxxe
dxxI
dxxIx
t
t
11
1
)(
)(2
0
0
0
0
• 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)(
EE
xx
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Scattering cross sectionsScattering cross sectionsScattering cross sectionsScattering cross sections
• For scattering reactions, we must consider For scattering reactions, we must consider the post-collision properties as well as the the post-collision properties as well as the probability of interaction:probability of interaction:
)ˆˆ,()()ˆˆ,( EEfEEE ss
where:where:
scattering for section cross )(Es
particles emitted for
function onDistributi )ˆˆ,( EEf
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Scattering cross sections (2)Scattering cross sections (2)Scattering cross sections (2)Scattering cross sections (2)
• Based on Assumption #5, the angular Based on Assumption #5, the angular dependence is dependent on the dependence is dependent on the deflectiondeflection angle between the two directions:angle between the two directions:
where:where:
),()ˆˆ,()ˆˆ,( 0 EEfEEfEEf
ˆˆ0 and between angle the of cosine
• Note that there is no azimuthal angular Note that there is no azimuthal angular dependencedependence
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Scattering cross sections (3)Scattering cross sections (3)Scattering cross sections (3)Scattering cross sections (3)
• The distribution function is normalized to integrate The distribution function is normalized to integrate to the number of particles that are emitted by the to the number of particles that are emitted by the reaction. For example, elastic scattering has:reaction. For example, elastic scattering has:
• whereas for (n,2n) we have:whereas for (n,2n) we have:
1),( 0
1
1
0
0
EEfdEd es
2),( 02
1
1
0
0
EEfdEd nn
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Scattering cross sections (4)Scattering cross sections (4)Scattering cross sections (4)Scattering cross sections (4)
• Note that since we combine cross sections Note that since we combine cross sections linearly, the relationship between the macroscopic linearly, the relationship between the macroscopic and microscopic distribution functions is given by:and microscopic distribution functions is given by:
Isotopes
iisii
ss
EEfEn
EEfEEE
10
00
),()(~
),()(),(
Isotopes
isii
Isotopes
iisii
En
EEfEn
EEf
1
10
0
)(~
),()(~
),(
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Scattering cross sections (5)Scattering cross sections (5)Scattering cross sections (5)Scattering cross sections (5)
• The most familiar distribution is the elastic The most familiar distribution is the elastic scattering distribution (from kinematics):scattering distribution (from kinematics):
0
0
2
ii2
i
i i
1, for
(1 )( , )
0, otherwise
where:
A -1 , A atomic mass of isotope i (multiples of neutron mass)
A 1
1 A 1 A 1
2
ii
i
E E EEf E E
E E
E E
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Functional expansionsFunctional expansionsFunctional expansionsFunctional expansions• The general field of functional expansion involves the approximation of a continuous function as The general field of functional expansion involves the approximation of a continuous function as
a linear combination of a basis function set:a linear combination of a basis function set:
• Use of a “complete” basis functions set means that as L goes to infinity, the approximation Use of a “complete” basis functions set means that as L goes to infinity, the approximation approaches the functionapproaches the function
• The trick is finding the coefficients that “best” fit the functionThe trick is finding the coefficients that “best” fit the function
xefxfL
0
)(
f
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Functional expansions (2)Functional expansions (2)Functional expansions (2)Functional expansions (2)• Determining the “best” comes down to finding the approximation (for a given L) that is “closest” Determining the “best” comes down to finding the approximation (for a given L) that is “closest”
to the function according to some “norm”to the function according to some “norm”• A “norm” is simply a measure of difference between two values (or functions) that satisfies two A “norm” is simply a measure of difference between two values (or functions) that satisfies two
simple criteria:simple criteria:1.1. The value of the norm is always non-negativeThe value of the norm is always non-negative
2.2. The value is only zero if the two are identicalThe value is only zero if the two are identical
• Example: Distance norms for pointsExample: Distance norms for points
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Functional expansions (3)Functional expansions (3)Functional expansions (3)Functional expansions (3)• The norm that we will use is the least-square norm, , which is a member of the The norm that we will use is the least-square norm, , which is a member of the
series defined by: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:
dxxfxfLdomain
n
n 21
nL2L
dxxefxfLdomain
L
n
2
0
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Functional expansions (4)Functional expansions (4)Functional expansions (4)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). 0
2 0L
nk
k domain
Lf x f e x e x dx
f
dxxexefdxxexfdomain
k
L
k
domain
0
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Functional expansions (5)Functional expansions (5)Functional expansions (5)Functional expansions (5)• Solving for the coefficients becomes much easier if the basis functions are orthogonal (which 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):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:
kdxxexedomain
k if 0
dxxexedxxexffdomain
kkk
domain
k
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Use of Legendre expansionsUse of Legendre expansionsUse of Legendre expansionsUse of Legendre expansions
• Using the cosine of the deflection angle, we Using the cosine of the deflection angle, we can represent the angular dependence of the can represent the angular dependence of the distribution in a Legendre expansion:distribution in a Legendre expansion:
00
0 )(12),(
PEEEE ss
• This allows us to represent the scattering This allows us to represent the scattering distribution by determining the Legendre distribution by determining the Legendre coefficients:coefficients:
..., 1, 0, ),( EEs
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Use of Legendre expansions (2)Use of Legendre expansions (2)Use of Legendre expansions (2)Use of Legendre expansions (2)
• Using the orthogonality of the Legendre Using the orthogonality of the Legendre polynomials:polynomials:
1
0 0 0
1
2
2 1m md P P
we can operate on both sides of the we can operate on both sides of the expansion (1expansion (1stst eqn. on previous slide) with: eqn. on previous slide) with:
1
0 0
1
md P
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Use of Legendre expansions (3)Use of Legendre expansions (3)Use of Legendre expansions (3)Use of Legendre expansions (3)
• To get:To get:
1
0 0 0
1
1( ) ( , )
2sm m sE E P E E d
• Work this out for yourself (Prob. 1-3)Work this out for yourself (Prob. 1-3)
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Fission neutron distributionFission neutron distributionFission neutron distributionFission neutron distribution
• Two data variables you need to know are:Two data variables you need to know are:
energy) itemitted/un (#
neutrons, fission of ondistributienergy E
neutrons Eenergy by caused fission
from released neutrons of #mean
E
• The first is a function; the second is a The first is a function; the second is a distributiondistribution
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Homework 1-1Homework 1-1Homework 1-1Homework 1-1
• For each of the assumptions listed on slide 1-9, give a physical situation 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.(if you can think of one) for which the assumption may not be a good one.
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Homework 1-2Homework 1-2Homework 1-2Homework 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:
t
t
t
x
x
t
t
dxe
dxxe
dxxI
dxxIx
t
t
11
1
)(
)(2
0
0
0
0
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Homework 1-3Homework 1-3Homework 1-3Homework 1-3• a. Show that if we expand:a. Show that if we expand:
that the coefficients can be found from:that the coefficients can be found from:
b. Use this fact to expand b. Use this fact to expand Expand it to enough terms so that the error is less than 1% at all values of Expand it to enough terms so that the error is less than 1% at all values of xx. .
00
0 )(12),(
PEEEE ss
),(2
1)( 00
1
1
0
EEPdEE smsm
11, xexf x
