1. 1 Stratified sampling This method involves reducing variance by forcing more order onto the random number stream used as input As the simplest example, it can be implemented by FORCING the random number stream to produce EXACTLY half the numbers below 0.5 and half above 0.5 In effect, you are dividing the problem into two subproblems with ½ the width of the domain each This reduces the discrepancy (proportional to the STANDARD DEVIATION) of each to ½ its previous value, which cuts the variance (per history) in half. If time permits, we will work an example in class like the one in the course notes

2. 2 “Other”: Two alternate Choose x from Choose x from using: using: Choose x from Choose x from (Gaussian/normal) using: (Gaussian/normal) using: (Why 12?)

3. 3 Homework from text

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6. 6 Lesson 4: Application to transport distributions Choosing from multidimensional distributions Choosing from multidimensional distributions Transport distributions Transport distributions Initial particle position Initial particle position Initial particle direction Initial particle direction Initial particle energy Initial particle energy Distance to next collision Distance to next collision Type of collision Type of collision Outcome of scattering event (new energy and direction) Outcome of scattering event (new energy and direction) Flux estimation Flux estimation Collision based (2 flavors) Collision based (2 flavors) Tracklength based Tracklength based

7. 7 Choosing from a multi-D pdf Starting with Starting with Normalize to get: Normalize to get: Integrate y out to get pdf in x only: Integrate y out to get pdf in x only: Choose x from Choose x from Choose y from Choose y from

8. 8 Examples of interest to transport To keep the material real, here are some details about how the decisions are made for outcomes of neutral particle tranport “events” To keep the material real, here are some details about how the decisions are made for outcomes of neutral particle tranport “events” As you will see, all of the tools are used: discrete, direct, rejection, probability mixing As you will see, all of the tools are used: discrete, direct, rejection, probability mixing We will go over these in class as time permits, but you should study them (i.e., likely examples that will show up on the test!) We will go over these in class as time permits, but you should study them (i.e., likely examples that will show up on the test!)

9. 9 Examples from transport “events” The lifecycle decisions that we will look at are: The lifecycle decisions that we will look at are: 1. Particle initial position 2. Particle initial direction 3. Particle initial energy 4. Distance to next collision 5. Type of collision 6. Outcome of a scattering event

10. 10 Decision 1: Particle initial position Decisions about the initial position of a particle is usually a multidimensional parameter determination based on a given position distribution over volume. Decisions about the initial position of a particle is usually a multidimensional parameter determination based on a given position distribution over volume. The mathematical approach to this is to define this function in terms of an appropriate coordinate system and then independently choose random numbers in each of the dimensions according to that dimension's "part" of the total distribution. The mathematical approach to this is to define this function in terms of an appropriate coordinate system and then independently choose random numbers in each of the dimensions according to that dimension's "part" of the total distribution. We will look at: Cartesian, cylindrical, and spherical. We will look at: Cartesian, cylindrical, and spherical.

11. 11 Cartesian coordinate system The classic shape in Cartesian coordinate system is a right parallelpiped: The classic shape in Cartesian coordinate system is a right parallelpiped: A differential volume element is defined by: A differential volume element is defined by:

12. 12 Cartesian coordinate system (2) If we want to pick a point with a flat distribution (i.e., each volume element equally likely), then the total distribution would be: If we want to pick a point with a flat distribution (i.e., each volume element equally likely), then the total distribution would be:

13. 13 Cylindrical coordinate system The classic shape in Cylindrical coordinate system is a right cylinder with z axis: The classic shape in Cylindrical coordinate system is a right cylinder with z axis: A differential volume element is defined by: A differential volume element is defined by:

14. 14 Cylindrical coordinate system (2) If we want to pick a point with a flat distribution (i.e., each volume element equally likely), then the total distribution would be: If we want to pick a point with a flat distribution (i.e., each volume element equally likely), then the total distribution would be:

15. 15 Translation to Cartesian In the Cartesian coordinate system (that most Monte Carlo codes run in) these would be translated into: In the Cartesian coordinate system (that most Monte Carlo codes run in) these would be translated into:

16. 16 Spherical coordinate system The classic shape in spherical coordinate system: The classic shape in spherical coordinate system: A differential volume element is defined by: A differential volume element is defined by:

17. 17 Spherical coordinate system (2) If we want to pick a point with a flat distribution (i.e., each volume element equally likely), then the total distribution would be: If we want to pick a point with a flat distribution (i.e., each volume element equally likely), then the total distribution would be:

18. 18 Translation to Cartesian In the Cartesian coordinate system (that most Monte Carlo codes run in) these would be translated into: In the Cartesian coordinate system (that most Monte Carlo codes run in) these would be translated into:

19. 19 Other shapes In this development, I have followed the convention from MCNP, that allows the user to specify a volume source directly only if it is one of the 3 shapes discussed: Cartesian, Cylinder, Sphere. In this development, I have followed the convention from MCNP, that allows the user to specify a volume source directly only if it is one of the 3 shapes discussed: Cartesian, Cylinder, Sphere. For a situation in which the source has another shape, MCNP uses a rejection technique that requires the user to: For a situation in which the source has another shape, MCNP uses a rejection technique that requires the user to: Build a 3D region (“cell” in MCNP terminology) that is shaped the way that is desired Build a 3D region (“cell” in MCNP terminology) that is shaped the way that is desired Specify the number of the cell on the source definition input line Specify the number of the cell on the source definition input line Also create a shape of one of the three preferred shapes that completely encloses the properly-shaped cell Also create a shape of one of the three preferred shapes that completely encloses the properly-shaped cell MCNP will implement a rejection technique: Pick a point in the enclosing cell but only use it if it is also in the properly shaped cell MCNP will implement a rejection technique: Pick a point in the enclosing cell but only use it if it is also in the properly shaped cell

20. 20 Choosing from multiple sources For a situation in which source particles are chosen from multiple source (possibly of various shapes, sizes, and source rate density), the user should apply a probability mixing strategy whereby: For a situation in which source particles are chosen from multiple source (possibly of various shapes, sizes, and source rate density), the user should apply a probability mixing strategy whereby: 1. A source is chosen from the multiple sources using the total source rates in each source (in units of particles/sec) to choose among the sources. 2. The point within the chosen source is picked using the appropriate shape's equations from above.

21. 21 Non-uniform spatial distributions One additional consideration is what should be done if the spatial source distribution is not uniform. In this case, the PDFs for the individual dimensions would be multiplied by the non-uniform distribution. One additional consideration is what should be done if the spatial source distribution is not uniform. In this case, the PDFs for the individual dimensions would be multiplied by the non-uniform distribution. Example: How would you choose a point inside a spherical source if the source is distributed in volume according to Example: How would you choose a point inside a spherical source if the source is distributed in volume according to

22. 22 Decision 2: Particle initial direction The choice of direction is based on probabilities on, which is a differential element of solid angle on the surface of a unit sphere: The choice of direction is based on probabilities on, which is a differential element of solid angle on the surface of a unit sphere:

23. 23 Particle initial direction (2) Note that the specification of the polar axis to be the z axis in this figure is completely arbitrary. The polar axis can be oriented in any direction that the analyst desires. Note that the specification of the polar axis to be the z axis in this figure is completely arbitrary. The polar axis can be oriented in any direction that the analyst desires. If we define, the solid angle becomes: If we define, the solid angle becomes: where the minus sign is present because decreases as increases. where the minus sign is present because decreases as increases.

24. 24 Particle initial direction (3) This gives us a dimensional PDFs of: This gives us a dimensional PDFs of: Generally, Monte Carlo methods require directions in the form of direction cosines, which would be: Generally, Monte Carlo methods require directions in the form of direction cosines, which would be:

25. 25 Decision 3: Particle initial energy Generally, choice of the initial particle energy is based on either a continuous, discrete, or multigroup source spectrum. Generally, choice of the initial particle energy is based on either a continuous, discrete, or multigroup source spectrum. Continuous: Particular distribution must be dealt with in the usual ways – direct or rejection Continuous: Particular distribution must be dealt with in the usual ways – direct or rejection Discrete: (common for  ) Particular particle energies coupled with the yields as Discrete: (common for  ) Particular particle energies coupled with the yields as Multigroup: Group source is the integrated source over the group. Therefore, the individual group source values are exactly analogous to discrete yields, so would be used as the probabilities in a discrete distribution. (Then the starting energy is chosen uniformly within the chosen group.) Multigroup: Group source is the integrated source over the group. Therefore, the individual group source values are exactly analogous to discrete yields, so would be used as the probabilities in a discrete distribution. (Then the starting energy is chosen uniformly within the chosen group.)

26. 26 Decision 4: Distance to next collision For infinite material with, the probability distribution for collision dx is: Therefore the PDF is: which is already normalized over the range

27. 27 Expected distance to collision (2) The associated CDF is: which inverts to give us the formula: In terms of the optical path length, we can use:

28. 28 Expected distance to collision (3) Translating mean free path to actual distance is just governed by the relationship: Applied by translating the distance to the closest boundary in to a number of mean free paths that is “spent” to get there (if there is enough) or else you “buy” a fraction of the distance to the boundary

29. 29 Expected distance to collision (4)

30. 30 Decision 5: Type of collision Once a collision is known to have occurred, the choice of reaction type is based on the reaction macroscopic cross sections: This gives us probabilities of: We make the choice between reaction types by using these probabilities as a discrete distribution.

31. 31 Decision 6: Outcome of Scattering Event The outcome of a scattering event by a particle with initial energy E is given by the multi-dimensional distribution: where M = material and the primed variables are associated with the particle after the collision. Sample using:

32. 32 Outcome of Scattering Event (2) For some elastic scattering events (and inelastic scattering from known nuclear levels) there is a unique relationship between the scattering deflection angle and fractional energy loss. This would reduce this last problem to just a problem of finding new energy OR deflection angle. For multigroup, the angular dependence of the group-to-group scattering is represented by a Legendre expansion in deflection angle OR by equal- probability ranges

33. 33 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 caused that contribution 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 caused that contribution 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

34. 34 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”

35. 35 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:

36. 36 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

37. 37 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)

38. 38 Homework

39. 39 Homework