6-1

Lesson 6 ObjectivesLesson 6 ObjectivesLesson 6 ObjectivesLesson 6 Objectives

• Beginning Chapter 2: EnergyBeginning Chapter 2: Energy• Derivation of Multigroup Energy treatmentDerivation of Multigroup Energy treatment• Finding approximate spectra to make Finding approximate spectra to make

multigroup cross sectionsmultigroup cross sections• Assumed (Fission-1/E-Maxwellian)Assumed (Fission-1/E-Maxwellian)• CalculatedCalculated

• Resonance treatmentsResonance treatments• Fine-group to Multi-group collapseFine-group to Multi-group collapse

• Spatial collapseSpatial collapse

6-2

Energy treatmentEnergy treatmentEnergy treatmentEnergy treatment

• Beginning the actual solution of the B.E. with the Beginning the actual solution of the B.E. with the ENERGY variable.ENERGY variable.

• The idea is to convert the continuous The idea is to convert the continuous dimensions to discretized form.dimensions to discretized form.• Infinitely dense variables => Few hundred variablesInfinitely dense variables => Few hundred variables• Calculus => AlgebraCalculus => Algebra

• Steps we will follow:Steps we will follow:• Derivation of multigroup formDerivation of multigroup form• Reduction of group coupling to outer iteration in Reduction of group coupling to outer iteration in

matrix formmatrix form• Analysis of one-group equation as Neumann iterationAnalysis of one-group equation as Neumann iteration• Typical acceleration strategies for iterative solutionTypical acceleration strategies for iterative solution

6-3

Definition of MultigroupDefinition of MultigroupDefinition of MultigroupDefinition of Multigroup

• All of the deterministic methods (and many All of the deterministic methods (and many Monte Carlo) represent energy variable using Monte Carlo) represent energy variable using multigroupmultigroup formalism formalism

• Basic idea is that the energy variable is divided Basic idea is that the energy variable is divided into contiguous regions (called into contiguous regions (called groupsgroups):):

• Note that it is traditional to number groups from Note that it is traditional to number groups from highhigh energy to energy to lowlow..

EE00EE11EE22EE33EE44EE55EE66EE77

Group 1Group 1Group 7Group 7

EnergyEnergy

6-4

Multigroup flux definitionMultigroup flux definitionMultigroup flux definitionMultigroup flux definition

• We first define the We first define the groupgroup flux as the integral of flux as the integral of the flux over the domain of a single group, g:the flux over the domain of a single group, g:

• Then we Then we assumeassume (hopefully from a physical (hopefully from a physical basis) a flux basis) a flux shapeshape, ,within the group, , ,within the group, where this shape is normalized to integrate to 1 where this shape is normalized to integrate to 1 over the group.over the group.

• The result is equivalent to the separation:The result is equivalent to the separation:

),ˆ,()ˆ,(1

ErdErg

g

E

E

g

1

1ˆ ˆ( , , ) ( , ) ; 1 ( , )

g

g

E

g g g g g

E

r E r f E dE f E E E E

gf E

6-5

Multigroup constantsMultigroup constantsMultigroup constantsMultigroup constants

• We insert this into the continuous energy B.E. We insert this into the continuous energy B.E. and integrate over the energy group: (Why?)and integrate over the energy group: (Why?)

• where we have used:where we have used:

1 1

1 1

1 1

1 4

1 4

ˆ ˆ ˆ( , ) ( ) , ( , ) ( )

ˆ ˆ ˆ( , , ) ( , ) ( )

ˆ( , ) ( , ) ( )

g g

g g

g g

g g

g g

g g

g

g

E E

g g t g g

E E

E EG

s g ggE E

E EG

f g ggE E

E

E

dE r f E dE r E r f E

dE dE d r E E r f E

dE E dE r E d r f E

dE

1

ˆ( , , )q r E

10

10

g

g

EE G

g E

dE dE

6-6

Multigroup constants (2)Multigroup constants (2)Multigroup constants (2)Multigroup constants (2)

• Pulling the fluxes out of the integrals gives:Pulling the fluxes out of the integrals gives:

• wherewhere

1 1

1 1

1 1

1

1 4

1

ˆ ˆ ˆ( , ) ( ) ( , ) , ( )

ˆ ˆ ˆ( , ) ( , , ) ( )

( ) ( , ) ( )

ˆ( , ,

g g

g g

g g

g g

g g

g g

g

g

E E

g g g t g

E E

E EG

g s gg E E

E EG

g f ggE E

E

E

r dE f E r dE r E f E

r dE dE d r E E f E

dE E r dE r E f E

dE q r

)E

4

ˆ( ) ( , )g gr d r

6-7

Multigroup constants (3)Multigroup constants (3)Multigroup constants (3)Multigroup constants (3)

• This simplifies to:This simplifies to:

• if we define:if we define:

),()()(

)ˆ,()ˆˆ,(

)ˆ,()ˆ,(ˆ

1

1 4

rqrr

rrd

rrr

g

G

gggfg

G

gggg

gtgg

6-8

Multigroup constants (4)Multigroup constants (4)Multigroup constants (4)Multigroup constants (4)

1

11

1 1

11

),ˆ,()()(

)( ),ˆ,(),(

),()()( ),()(

),ˆˆ,()()ˆˆ,(

g

g

g

g

g

g

g

g

g

g

g

g

g

g

E

E

fgfg

E

E

g

E

E

g

E

E

t

E

E

gtgg

sg

E

E

E

E

gg

ErEfdEr

EdEErqdErq

ErEfdErErdEr

EErEfEddEr

6-9

Multigroup constants (5)Multigroup constants (5)Multigroup constants (5)Multigroup constants (5)

• For Legendre scattering treatment, the group For Legendre scattering treatment, the group Legendre cross sections formally found by:Legendre cross sections formally found by:

• From Lec. 4:From Lec. 4:

)(

)()(

),()()(

1

11

EEd

EEf

EErEfEddEr

g

g

g

g

g

g

E

E

g

sg

E

E

E

E

gg

),,(2

1),( 00

1

1

0 EErPdEEr ss

6-10

• The assumed shapes The assumed shapes ffgg(E)(E) take the take the mathematical role of weight functions in mathematical role of weight functions in formation of group cross sectionsformation of group cross sections

• We do not have to predict a spectral shape We do not have to predict a spectral shape ffgg(E)(E) that is good for ALL energies, but just that is good for ALL energies, but just accurate over the limited range of each group.accurate over the limited range of each group.• Therefore, as groups get smaller, the Therefore, as groups get smaller, the

selection of an accurate selection of an accurate ffgg(E)(E) gets less and gets less and importantimportant

Important points to makeImportant points to makeImportant points to makeImportant points to make

6-11

• There are two common ways to find the There are two common ways to find the ffgg(E)(E) for neutrons: for neutrons: • Assuming a shape: Use general physical Assuming a shape: Use general physical

understanding to deduce the expected understanding to deduce the expected SCALAR flux spectral shapes [fission, 1/E, SCALAR flux spectral shapes [fission, 1/E, Maxwellian]Maxwellian]

• Calculating a shape: Use a simplified Calculating a shape: Use a simplified problem that can be approximately solved to problem that can be approximately solved to get a shape [resonance processing get a shape [resonance processing techniques, finegroup to multigroup]techniques, finegroup to multigroup]

Finding the group spectraFinding the group spectraFinding the group spectraFinding the group spectra

6-12

• From infinite homogeneous medium equation From infinite homogeneous medium equation with single fission neutron source:with single fission neutron source:

• we get three (very roughly defined) generic we get three (very roughly defined) generic energy ranges:energy ranges:• FissionFission• Slowing-downSlowing-down• ThermalThermal

Assumed group spectraAssumed group spectraAssumed group spectraAssumed group spectra

EEEEEdEE st

)()()(0

6-13

• Fission source. No appreciable down-Fission source. No appreciable down-scattering:scattering:

• Since cross sections tend to be fairly constant Since cross sections tend to be fairly constant at high energies:at high energies:

Fast energy range (>~2 MeV)Fast energy range (>~2 MeV)Fast energy range (>~2 MeV)Fast energy range (>~2 MeV)

EE

Ef

EEEr

t

t

~)(

)(,

EEf fast ~)(

6-14

• No fission. Primary source is elastic down-No fission. Primary source is elastic down-scatter:scatter:

• Assuming constant cross sections and little Assuming constant cross sections and little absorption:absorption:

• (I love to make you prove this on a test!)(I love to make you prove this on a test!)

Intermediate range (~1 eV to ~2 MeV)Intermediate range (~1 eV to ~2 MeV)Intermediate range (~1 eV to ~2 MeV)Intermediate range (~1 eV to ~2 MeV)

)()1(

)(~)(

/

EE

EEdNEE

i

si

E

Eiit

i

EEfE int

1~)(~)(

6-15

• If a fixed number of neutrons are in a pure-scattering If a fixed number of neutrons are in a pure-scattering equilibrium with the atoms of the material, the result is equilibrium with the atoms of the material, the result is a Maxwellian distribution: a Maxwellian distribution:

• In our situation, however, we have a dynamic In our situation, however, we have a dynamic equilibrium:equilibrium:

1.1. Neutrons are continuously arriving from higher energies by Neutrons are continuously arriving from higher energies by slowing down; andslowing down; and

2.2. An equal number of neutrons are being absorbed in 1/v An equal number of neutrons are being absorbed in 1/v absorption absorption

• As a result, the spectrum is slightly hardened (i.e., As a result, the spectrum is slightly hardened (i.e., higher at higher energies) which is often approximated higher at higher energies) which is often approximated as a Maxwellian at a slightly higher temperature ergies as a Maxwellian at a slightly higher temperature ergies (“neutron temperature”)(“neutron temperature”)

Thermal range (<~1 eV)Thermal range (<~1 eV)Thermal range (<~1 eV)Thermal range (<~1 eV)

( ) ~EkTE Ee

6-16

• Mostly narrow absorption bands in the Mostly narrow absorption bands in the intermediate range:intermediate range:

• Assuming constant microscopic scatter and Assuming constant microscopic scatter and that flux is 1/E above the resonance (narrow that flux is 1/E above the resonance (narrow resonance approx):resonance approx):

Resonance treatmentsResonance treatmentsResonance treatmentsResonance treatments

)()1(

)(~)(

/

EE

EEdNEE

i

si

E

Eiit

i

EEEfE

EEE

t

sres

st

~)(~)(

)(

6-17

• Reactor analysis methods have greatly Reactor analysis methods have greatly extended resonance treatments:extended resonance treatments:• Extension to other energy scattering Extension to other energy scattering

situations (Wide Resonance and situations (Wide Resonance and Equivalence methods)Equivalence methods)

• Extension of energy methods to include Extension of energy methods to include simple spatial relationshipssimple spatial relationships

• Statistical methods that can deal with Statistical methods that can deal with unresolved resonance region (where unresolved resonance region (where resonance cannot be resolved resonance cannot be resolved experimentally although we know they are experimentally although we know they are there)there)

Resonance treatments (2)Resonance treatments (2)Resonance treatments (2)Resonance treatments (2)

6-18

• ““Bootstrap” technique whereby Bootstrap” technique whereby • Assumed spectrum shapes are used to form Assumed spectrum shapes are used to form

finegroup cross sections (G>~200)finegroup cross sections (G>~200)• Simplified-geometry calculations are done Simplified-geometry calculations are done

with these large datasets.with these large datasets.• The resulting finegroup spectra are used to The resulting finegroup spectra are used to

collapse fine-group XSs to multigroup:collapse fine-group XSs to multigroup:

Finegroup to multigroupFinegroup to multigroupFinegroup to multigroupFinegroup to multigroup

EE2020EE2121EE2222EE2323EE2424EE2525EE2626EE2727

EnergyEnergy

Fine-group structureFine-group structure

EE22EE33Multi-group structure (Group 3)Multi-group structure (Group 3)

6-19

• Energy collapsing equation:Energy collapsing equation:• Using the calculated finegroup fluxes, Using the calculated finegroup fluxes,

we conserve reaction rates to get new cross we conserve reaction rates to get new cross sectionssections

• Assumes multigroup flux will be:Assumes multigroup flux will be:

• The resulting multigroup versions are shown The resulting multigroup versions are shown on the next page. (I will leave the Legendre on the next page. (I will leave the Legendre scattering coefficients for another day.)scattering coefficients for another day.)

Finegroup to multigroup (2)Finegroup to multigroup (2)Finegroup to multigroup (2)Finegroup to multigroup (2)

ˆˆg

ˆˆ

ˆg gg g

6-20

Finegroup to multigroup (3)Finegroup to multigroup (3)Finegroup to multigroup (3)Finegroup to multigroup (3)

ˆ ˆ ˆˆ ˆ

ˆˆ

ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆˆ ˆ

ˆ ˆˆ ˆ

ˆˆ

ˆ

ˆ ˆ ˆˆ ˆ

ˆ ˆ

ˆˆ

gg gg g g g

ggg

g g

tg g fg gg g g g

tg fgg g

g g g g

g g g gg g g g

q q

6-21

• We often “smear” heterogeneous regions into We often “smear” heterogeneous regions into a homogeneous region:a homogeneous region:

• Volume AND flux weighted, conserving Volume AND flux weighted, conserving reaction ratereaction rate

Related idea: Spatial collapseRelated idea: Spatial collapseRelated idea: Spatial collapseRelated idea: Spatial collapse

V1V1V2V2 V=V1+V2V=V1+V2

gggg

ggggtggt

celltg

VV

VV

22

11

222

111

ˆˆ

ˆˆˆˆ

6-22

Homework 6-1Homework 6-1Homework 6-1Homework 6-1

For a total cross section given by the equation:For a total cross section given by the equation:

find the total group cross section for a group that spans find the total group cross section for a group that spans from 2 keV to 3 keV. Assume flux is 1/E.from 2 keV to 3 keV. Assume flux is 1/E.

2( ) 5 0.5 0.1 ; in keVt E E E E

6-23

Homework 6-2Homework 6-2Homework 6-2Homework 6-2

Find the isotropic elastic scatter cross section for Carbon-Find the isotropic elastic scatter cross section for Carbon-12 (A=12) from an energy group that spans from 0.6 to 0.7 12 (A=12) from an energy group that spans from 0.6 to 0.7 keV to a group that spans 0.4 keV to 0.5 keV. Assume the keV to a group that spans 0.4 keV to 0.5 keV. Assume the flux spectrum is 1/E and that the scattering cross section is flux spectrum is 1/E and that the scattering cross section is a constant 5 barns. a constant 5 barns.

[Hint: The distribution of post-collision energies for this [Hint: The distribution of post-collision energies for this case is uniform from the pre-collision neutron energy down case is uniform from the pre-collision neutron energy down to the minimum possible post-collision energy of to the minimum possible post-collision energy of E.] E.]