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Lesson 5: Flux, etc.Lesson 5: Flux, etc.

• Flux determinationFlux determination• CellCell• SurfaceSurface• Flux integral tallies (reaction rates)Flux integral tallies (reaction rates)

• K-effective calculationsK-effective calculations• Sample problemSample problem

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Cell flux estimationCell flux estimation

• Basic question: Why do we want to know Basic question: Why do we want to know the group flux in a cell?the group flux in a cell?

• Only reason: So that we can later turn it into some Only reason: So that we can later turn it into some measurable (reaction rate, power distribution, dose)measurable (reaction rate, power distribution, dose)

• Monte Carlo (rather perversely) is rather better at getting Monte Carlo (rather perversely) is rather better at getting the reactions rates THEMSELVESthe reactions rates THEMSELVES

• Two ways to get it:Two ways to get it:• After Monte Carlo gives you an incremental contribution to After Monte Carlo gives you an incremental contribution to

a reaction rate, back out the incremental flux that would a reaction rate, back out the incremental flux that would have cause it and add it to a running totalhave cause it and add it to a running total

• Use an alternative flux definition to get flux directlyUse an alternative flux definition to get flux directly

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Cell flux estimation (2)Cell flux estimation (2)• The first way to score flux is to add an The first way to score flux is to add an

incremental contribution every time there IS a incremental contribution every time there IS a collision in cell in by energy group (g):collision in cell in by energy group (g):

• then a collision contributes an “incremental” then a collision contributes an “incremental” RR addition of 1 and an incremental flux RR addition of 1 and an incremental flux addition of:addition of:

• This is referred to as a “collision estimator”This is referred to as a “collision estimator”

, , ,g cell tg cell g cell cellRR V

,,

, ,

1g cellg cell

tg cell cell tg cell cell

RR

V V

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Cell flux estimation (3)Cell flux estimation (3)

• Variation on this them is to score on particular Variation on this them is to score on particular TYPES of reactions and then score an amount TYPES of reactions and then score an amount depending on that REACTION’s cross sectiondepending on that REACTION’s cross section

• Most common is an ABSORPTION estimator, which Most common is an ABSORPTION estimator, which on each absorption event scores:on each absorption event scores:

• Another way to score flux is to go back to the basic Another way to score flux is to go back to the basic definition of total macroscopic cross section:definition of total macroscopic cross section:

cellcellagcellcellag

cellagcellg VV

RR

,,

,,

1

particlesby travelleddistance Total

reactions ofnumber Expectedt

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Cell flux estimation (4)Cell flux estimation (4)

• Substituting this into the reaction rate equation gives Substituting this into the reaction rate equation gives us:us:

• This is a “track length estimator”This is a “track length estimator”• Notice that the number of reactions has Notice that the number of reactions has

CANCELLED.CANCELLED.• This estimator not only does NOT depend on an actual This estimator not only does NOT depend on an actual

reaction occurring, but can even be used in a VACUUMreaction occurring, but can even be used in a VACUUM

cellcellg V

cellg, groupin particlesby travelleddistance Total,

cellcellgcellg VRR ,, cellgroup,in particlesby travelleddistance Total

cellgroup,in reactions ofNumber

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Cell flux estimation (5)Cell flux estimation (5)

• When to use which? General rules of thumb: When to use which? General rules of thumb: • Track length estimator in thin regionsTrack length estimator in thin regions• Collision estimator in high collision regions (especially Collision estimator in high collision regions (especially

scattering) regionsscattering) regions• Absorption estimator in high absorption regionsAbsorption estimator in high absorption regions

• Examples. Which estimator is most efficient for a:Examples. Which estimator is most efficient for a:• Thin foils Thin foils • Thick control rod (and thermal neutrons)Thick control rod (and thermal neutrons)• Diffusive low-absorber (e.g., D2O, graphite)Diffusive low-absorber (e.g., D2O, graphite)

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Surface flux estimationSurface flux estimation• A surface flux estimation (useful when you A surface flux estimation (useful when you

want to know the MAXIMUM dose in a room want to know the MAXIMUM dose in a room with an obvious highest-dose surface) is just a with an obvious highest-dose surface) is just a degenerate case of a track length estimator, degenerate case of a track length estimator, for a cell with epsilon thickness:for a cell with epsilon thickness:

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Surface flux estimationSurface flux estimation• The equation breaks down to:The equation breaks down to:

,

Total distance travelled by particles in group g, -cell

/(cosine of angle of incidence)

1

(cosine of angle of incidence)

g surfacecell

cell

cell

V

A

A

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Reaction rate estimationReaction rate estimation

• At first glance, this seems like a ridiculous At first glance, this seems like a ridiculous question: question:

• Reactions are basic events in a Monte Carlo Reactions are basic events in a Monte Carlo simulation.simulation.

• So, can’t you just COUNT them as they occur So, can’t you just COUNT them as they occur • Answer: Yes you can. But you might not want Answer: Yes you can. But you might not want

to.to.

• The basic equation is, of course:The basic equation is, of course:

, , ,1

G

x cell xg cell g cell cellg

RR V

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Reaction rate estimation (2)Reaction rate estimation (2)

• Substituting our previous relation for flux, Substituting our previous relation for flux, we get either:we get either:

• OrOr

,, ,

1 , ,

Ggg xg cell

x cell xg cell cellg tg cell cell tg cell

RR VV

, ,1

,

Total distance in group g ,cell

Total distance in group g ,cell

G

x cell xg cell gg cellg cell

xg cell

RR VV

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K-Effective CalculationsK-Effective Calculations

• Fission in a subcritical situation—with a Fission in a subcritical situation—with a source—can (theoretically) can be handled source—can (theoretically) can be handled as a multiple-particle-producing scattering as a multiple-particle-producing scattering reactionreaction

• The calculation of k-effective, however, is The calculation of k-effective, however, is handled in a special way in Monte Carlo handled in a special way in Monte Carlo codescodes

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K-Effective Calculations (2)K-Effective Calculations (2)• Source-less k-effective problems are solved by treating the Source-less k-effective problems are solved by treating the

particles coming out of fission as an external sourceparticles coming out of fission as an external source• Problem: We know the particles’ energy and directional Problem: We know the particles’ energy and directional

distributions but NOT their spatial distribution.distributions but NOT their spatial distribution.• Solution: Instead of ONE problem, the calculation is handled Solution: Instead of ONE problem, the calculation is handled

as a SERIES of Monte Carlo problems, each of which uses as a SERIES of Monte Carlo problems, each of which uses the PREVIOUS problem’s fission sites as an external source the PREVIOUS problem’s fission sites as an external source of neutrons (and photons, if desired)of neutrons (and photons, if desired)

• There are two difficulties:There are two difficulties:1.1. How do we start the FIRST calculationHow do we start the FIRST calculation

2.2. How do we deal with the fact that the fission spatial distribution is How do we deal with the fact that the fission spatial distribution is going to be TERRIBLE for a few roundsgoing to be TERRIBLE for a few rounds

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K-Effective Calculations (3)K-Effective Calculations (3)

• Procedure:Procedure:1.1. Make an initial guess of SPATIAL distribution of Make an initial guess of SPATIAL distribution of

fission (Why not energy and angle?)fission (Why not energy and angle?)

2.2. Use this guess as a source in a typical MC Use this guess as a source in a typical MC calculation (tallying new fission neutron production).calculation (tallying new fission neutron production).

3.3. Estimate the eigenvalue the old fashioned way Estimate the eigenvalue the old fashioned way (fission neutron production in present (fission neutron production in present cycle/previous cycle)cycle/previous cycle)

4.4. Use solution’s fission locations as next cycle’s Use solution’s fission locations as next cycle’s source spatial informationsource spatial information

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K-effective Calculations (4)K-effective Calculations (4)

• Effects on calculation flow:Effects on calculation flow:1.1. Problem is subdivided into (user-specified) number of Problem is subdivided into (user-specified) number of

cycles with a given number of source histories per cycles with a given number of source histories per cycle.cycle.

2.2. Problem delivers one eigenvalue guess per cycle.Problem delivers one eigenvalue guess per cycle.

3.3. As a practical matter, one discards the first few As a practical matter, one discards the first few eigenvalue guesses until the fission spatial eigenvalue guesses until the fission spatial distribution “settles down”distribution “settles down”

4.4. Theoretically not satisfying since the cycles are Theoretically not satisfying since the cycles are obviously not independent.obviously not independent.• Their dependence is smaller the larger the number of Their dependence is smaller the larger the number of

histories sampled in each cyclehistories sampled in each cycle

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Sample problemSample problem

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Sample problemSample problem

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Homework from textHomework from text