1. 1 Lesson 5: Flux, etc. Flux determination Flux determination Cell Cell Surface Surface Flux integral tallies (reaction rates) Flux integral tallies (reaction rates) K-effective calculations K-effective calculations Sample problem Sample problem

2. 2 Cell flux estimation Basic question: Why do we want to know the group flux in a cell? Basic question: Why do we want to know the group flux in a cell? Only reason: So that we can later turn it into some measurable (reaction rate, power distribution, dose) Only reason: So that we can later turn it into some measurable (reaction rate, power distribution, dose) Monte Carlo (rather perversely) is rather better at getting the reactions rates THEMSELVES Monte Carlo (rather perversely) is rather better at getting the reactions rates THEMSELVES Two ways to get it: Two ways to get it: After Monte Carlo gives you an incremental contribution to a reaction rate, back out the incremental flux that would have cause it and add it to a running total After Monte Carlo gives you an incremental contribution to a reaction rate, back out the incremental flux that would have cause it and add it to a running total Use an alternative flux definition to get flux directly Use an alternative flux definition to get flux directly

3. 3 Cell flux estimation (2) The first way to score flux is to add an incremental contribution every time there IS a collision in cell in by energy group (g): The first way to score flux is to add an incremental contribution every time there IS a collision in cell in by energy group (g): then a collision contributes an “incremental” RR addition of 1 and an incremental flux addition of: then a collision contributes an “incremental” RR addition of 1 and an incremental flux addition of: This is referred to as a “collision estimator” This is referred to as a “collision estimator”

4. 4 Cell flux estimation (3) Variation on this them is to score on particular TYPES of reactions and then score an amount depending on that REACTION’s cross section Variation on this them is to score on particular TYPES of reactions and then score an amount depending on that REACTION’s cross section Most common is an ABSORPTION estimator, which on each absorption event scores: Most common is an ABSORPTION estimator, which on each absorption event scores: Another way to score flux is to go back to the basic definition of total macroscopic cross section: Another way to score flux is to go back to the basic definition of total macroscopic cross section:

5. 5 Cell flux estimation (4) Substituting this into the reaction rate equation gives us: Substituting this into the reaction rate equation gives us: This is a “track length estimator” This is a “track length estimator” Notice that the number of reactions has CANCELLED. Notice that the number of reactions has CANCELLED. This estimator not only does NOT depend on an actual reaction occurring, but can even be used in a VACUUM This estimator not only does NOT depend on an actual reaction occurring, but can even be used in a VACUUM

6. 6 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 regions Track length estimator in thin regions Collision estimator in high collision regions (especially scattering) regions Collision estimator in high collision regions (especially scattering) regions Absorption estimator in high absorption regions Absorption 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)

7. 7 Surface flux estimation A surface flux estimation (useful when you want to know the MAXIMUM dose in a room with an obvious highest-dose surface) is just a degenerate case of a track length estimator, for a cell with epsilon thickness: A surface flux estimation (useful when you want to know the MAXIMUM dose in a room with an obvious highest-dose surface) is just a degenerate case of a track length estimator, for a cell with epsilon thickness:

8. 8 Surface flux estimation The equation breaks down to: The equation breaks down to:

9. 9 Reaction rate estimation At first glance, this seems like a ridiculous question: At first glance, this seems like a ridiculous question: Reactions are basic events in a Monte Carlo simulation. Reactions are basic events in a Monte Carlo 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 to. Answer: Yes you can. But you might not want to. The basic equation is, of course: The basic equation is, of course:

10. 10 Reaction rate estimation (2) Substituting our previous relation for flux, we get either: Substituting our previous relation for flux, we get either: Or Or

11. 11 K-Effective Calculations Fission in a subcritical situation—with a source—can (theoretically) can be handled as a multiple-particle-producing scattering reaction Fission in a subcritical situation—with a source—can (theoretically) can be handled as a multiple-particle-producing scattering reaction The calculation of k-effective, however, is handled in a special way in Monte Carlo codes The calculation of k-effective, however, is handled in a special way in Monte Carlo codes

12. 12 K-Effective Calculations (2) Source-less k-effective problems are solved by treating the particles coming out of fission as an external source Source-less k-effective problems are solved by treating the particles coming out of fission as an external source Problem: We know the particles’ energy and directional distributions but NOT their spatial distribution. Problem: We know the particles’ energy and directional distributions but NOT their spatial distribution. Solution: Instead of ONE problem, the calculation is handled as a SERIES of Monte Carlo problems, each of which uses the PREVIOUS problem’s fission sites as an external source of neutrons (and photons, if desired) Solution: Instead of ONE problem, the calculation is handled as a SERIES of Monte Carlo problems, each of which uses the PREVIOUS problem’s fission sites as an external source of neutrons (and photons, if desired) There are two difficulties: There are two difficulties: 1. How do we start the FIRST calculation 2. How do we deal with the fact that the fission spatial distribution is going to be TERRIBLE for a few rounds

13. 13 K-Effective Calculations (3) Procedure: Procedure: 1. Make an initial guess of SPATIAL distribution of fission (Why not energy and angle?) 2. Use this guess as a source in a typical MC calculation (tallying new fission neutron production). 3. Estimate the eigenvalue the old fashioned way (fission neutron production in present cycle/previous cycle) 4. Use solution’s fission locations as next cycle’s source spatial information

14. 14 K-effective Calculations (4) Effects on calculation flow: Effects on calculation flow: 1. Problem is subdivided into (user-specified) number of cycles with a given number of source histories per cycle. 2. Problem delivers one eigenvalue guess per cycle. 3. As a practical matter, one discards the first few eigenvalue guesses until the fission spatial distribution “settles down” 4. Theoretically not satisfying since the cycles are obviously not independent. Their dependence is smaller the larger the number of histories sampled in each cycle Their dependence is smaller the larger the number of histories sampled in each cycle

15. 15 Sample problem

16. 16 Sample problem

17. 17 Homework from text