Cadarache 2

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Monte-Carlo Primer
Motto Sir, In your otherwise beautiful poem (The
vision of Sin) there is a verse which
reads Every moment dies a man every moment
one is born Obviously, this cannot be true and I
suggest that in the next edition you have it
read Every moment dies a man every moment 1
and 1/16 is born Even this value is slightly in
error but should be sufficiently accurate for
poetry. ....Charles Babbage (in a letter to Lord
Tennyson)
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PROGRAM

• Introduction
• History of Monte Carlo
• Basics of Monte-Carlo
• Random Number Generators
• Cross sections
• Monte-Carlo vs Deterministic Methods
• Particle Transport Simulations
• Estimators and Statistical Analysis
• Non-analog Monte Carlo
• Variance Reduction Technique

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Introduction Monte Carlo
The Monte Carlo method has been used for almost
60 years to solve radiation transport problems in
high energy physics, nuclear reactor analysis,
radiation shielding, medical imaging, oil
well-logging, etc. Individual particle histories
are simulated using random numbers, highly
accurate representations of particle interaction
probabilities, and exact models of 3D problem
geometry. Monte Carlo methods are sometimes the
only viable methods for analyzing complex,
demanding particle transport problems.
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Modern applications of Monte Carlo

• Nuclear reactor design
• Quantum chromodynamics
• Radiation cancer therapy
• Traffic flow
• Stellar evolution
• Econometrics
• Dow-Jones forecasting
• Oil well exploration
• VLSI design

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History of Monte Carlo

• John Von Neumann invented scientific computing
  in 1940s
• stored programs, software
• algorithms flowcharts
• assisted with hardware design as well ordinary
  computers are called Von Neumann machines
• John Von Neumann invented Monte Carlo particle
  transport in 1940s
• Highly accurate - no essential approximations
• Expensive -typically method of last resort
• Monte Carlo codes for particle transport have
  been proven to work effectively on all computers
• Vector, parallel, supercomputers, workstations,
  clusters of workstations, . .

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Prominent Figures in Computing
If I have seen further it is by standing on the
shoulders of Giants in Isaac Newtons Letter to
Robert Hooke

• Gottfried Wilhelm Leibniz (1646 -1716)
• Charles Babbage (1791-1871)
• Augusta Ada Byron (King), countess of Lovelace,
  Lady Lovelace (1815-1852), the first computer
  programmer 1979, Programming Language ADA

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The First Digital Computer
Mid-1830s
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John von Neumann, 1903-57, American
mathematician, b. Hungary, Ph.D. Univ. of
Budapest, 1926
He came to the United States in 1930 and was
naturalized in 1937. He taught (1930-33) at
Princeton and after 1933 was associated with the
Institute for Advanced Study. In 1954 he was
appointed a member of the Atomic Energy
Commission. A founder of the mathematical theory
of games, he also made fundamental contributions
to quantum theory and to the development of the
atomic bomb (Stan Ulam was another great
mathematician in this environment). He was a
leader in the design and development of
high-speed electronic computers his development
of maniac-an acronym for mathematical analyzer,
numerical integrator, and computer-enabled the
United States to produce and test (1952) the
world's first hydrogen bomb. With Oskar
Morgenstern he wrote Theory of Games and Economic
Behavior). Von Neumann's other writings include
Mathematical Foundations of Quantum Mechanics
(1926), Computer and the Brain (1958), and Theory
of Self-reproducing Automata (1966).
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Parties and nightlife held a special appeal for
von Neumann. While teaching in Germany, von
Neumann had been a denizen of the Cabaret-era
Berlin nightlife circuit.
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Terminology
1) Deterministic
2) Probabilistic
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Basics of M-C

• Two basic ways to approach the use of Monte
  Carlo methods for solving the transport equation
• mathematical technique for numerical integration
• computer simulation of a physical process
• Each is correct
• mathematical approach useful for
• importance sampling, convergence, variance
  reduction, random sampling techniques, . ...
• simulation approach useful for
• collision physics, tracking, tallying, . . . . .
• For Monte Carlo approach, consider the integral
  form of the Boltzmann equation.
• Most theory on Monte Carlo deals with
  fixed-source problems.
• Eigenvalue problems are needed for reactor
  physics calculations

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Major Components of a Monte Carlo Algorithm

• The primary components of a Monte Carlo
  simulation method
• Probability distribution functions (pdf's) - the
  physical (or mathematical) system must be
  described by a set of pdf's.
• Random number generator - a source of random
  numbers uniformly distributed on the unit
  interval must be available.
• Sampling rule - a prescription for sampling from
  the specified pdf's, assuming the availability of
  random numbers on the unit interval, must be
  given.
• Scoring (or tallying) - the outcomes must be
  accumulated into overall tallies or scores for
  the quantities of interest.
• Error estimation - an estimate of the statistical
  error (variance) as a function of the number of
  trials and other quantities must be determined.
• Variance reduction techniques - methods for
  reducing the variance in the estimated solution
  to reduce the computational time for Monte Carlo
  simulation
• Parallelization and vectorization - algorithms to
  allow Monte Carlo methods to be implemented
  efficiently on advanced computer architectures.

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Major Shortcoming of Deterministic Appoach Too
Many Simplifications

• Physical simplifications
• Collision model
• Isotropic scattering in Lab-system
• Continuous slowing-down, etc.
• Mathematical simplifications
• Diffusion model, boundary conditions, etc.
• Multi-group model
• Discretization,
• Technological simplifications
• Geometrical simplifications
• Homogenization,

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Diffusion Model
Transport equation
Q0
Transport eq.
Free surface
x 0
Diffusion eq.
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Multi-Group Approximation
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Geometrical Simplifications
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Discretization
In space
In angle
It completely falls out
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Advantages of Monte Carlo

• Very complex 3-D configurations
• Continuous space variables
• Continuous angle variables
• Continuous energy variable

The errors in Monte Carlo calculations take the
form of stochastic uncertainties
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The Theoretical Foundation of MC
x1
x2
x3
xN
µ
z
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Elements of Random Variables
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Distribution Sampling
We want to generate a series of random numbers
that are distributed according to the number of
mean free paths
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Rejection Technique
1
0
1
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Example of Simple Simulation
x1
x2
x3
xN
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Principles of a Nuclear Reactor
Leakage
E
N2
2 MeV
N1
Fast fission
n n/fission
Energy
Slowing down
Resonance abs.
? 2.5
Non-fuel abs.
Non-fissile abs.
1 eV
Fission
200 MeV/fission
Leakage
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Flux/Current Estimators
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The Essence of Monte Carlo
Random numbers
Monte Carlo
RESULTS
Probability Distribution
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The Essence of Monte Carlo
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Simple Monte Carlo Example
Mathematical approach For k 1, . . . . N
choose randomly in (0,1) G (1-0) average
value of g(x)
Simulation approach darts game For k 1, . .
. . N choose
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Simulation approach darts game For k 1, . .
. . N choose
G area under curve
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M-C as integration tool
Monte Carlo is often the method-of-choice for
applications with integration over many
dimensions Examples high-energy physics,
particle transport (ADS)
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Probability Density Functions
Continuous Probability Density
Discrete Probability Density
x1 x2 x3 xN
x
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Basic parameters
Mean, Average, Expected Value (1st statistical
moment)
Variance (2nd statistical moment), standard
deviation
standard deviation
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Basic functions
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The key of MC methods is the notion of RANDOM
sampling
The problem can be stated this way Given a
probability density, f(x), produce a sequence of
The should be distributed in the same
manner as f(x).

• The use of random sampling distinguishes Monte
  Carlo from all other methods
• When Monte Carlo is used to solve the integral
  Boltzmann transport equation
• Random sampling models the outcome of physical
  events (e.g., neutron collisions, fission
  process, source, . . . . . )
• Computational geometry models the arrangement of
  materials

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Transport Equation

• Assumptions
• static homogeneous medium
• time-independent
• Markovian - next event depends only on current
  (r,v) not on previous events
• particles do not interact with each other
• neglect relativistic effects
• no long-range forces (particles fly in straight
  lines between events)
• material properties are not affected by particle
  reactions
• etc., etc.
• ? can use superposition principle

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Basis for Monte Carlo Solution Method
Note that collision k depends only on the results
of collision k-l, and not on any prior collisions
k-2, k-3, . . .
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Statistical approach to determining ?k
interpret terms in the following manner

• Monte Carlo method
• Randomly sample p from ?k-1(p)
• Given p, randomly sample p from R (p?p)
• If p lies within dpi at pi, tally 1 in bin i
• ? Repeat steps 1,2,3 N times,
• then

dp
p
pi
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Histories (trajectories)
After repeated substitution for ?k
A history (trajectory) is a sequence of states
(p0,p1,p2,p3...)
p1
p4
p0
p3
p3
p0
p2
p2
p1
For estimates in a given region, tally the
occurrences for each collision of each history
within a region
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Transport Equation

• Monte Carlo approach
• Generate a sequence of States, (p0, p1, . . . .
  pk), i.e., a history by
• Randomly sample from PDF for source ?0(p0)
• Randomly sample from PDF for kth transition R
  (pk-1 ? pk)
• Generate estimates of results, by averaging over
  M histories

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Simulations

• Simulation approach to particle transport
• Faithfully simulate the history of a single
  particle from birth to death
• During the particle history,
• model collisions using physics equations
  cross-section data
• model free-flight using computational geometry
• tally the occurrences of events in each region

Repeat the simulation for many histories,
accumulating the tallies
Fundamental rule Think like a neutron or other
projectile (proton)
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MC rules for simulations
Source Random sampling E, ? -analytic, discrete,
or piecewise-tabulated PDFs Computational
geometry r -sample from region in 3-D space, or
from discrete PDF Tracking Random
sampling dcollide-distance to collision, from mfp
or exponential PDF Computational geometry dgeom -
distance-to-boundary, ray-tracing, next-region,
.... Collisions Random sampling E, ? -
analytic, discrete, or piecewise-tabulated
PDFs Physics ?, f(?) - cross-section data,
angular PDFs, kinematics, ... Tallies Statistics V
ariance Reduction Random sampling
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MC rules for simulations

• Single particle
• random-walk for particle history
• simulate events, from birth to death
• tally events of interest

• Batch of histories (generation)
• random-walk for many particle histories
• tally the aggregate behavior

• Overall
• timesteps
• geometry changes
• material changes
• fuel depletion
• burnable absorbers
• control rods

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Random number generators

• Truly random - is defined as exhibiting true"
  randomness, such as the time between tics" from
  a Geiger counter exposed to a radioactive element
• Pseudorandom - is defined as having the
  appearance of randomness, but nevertheless
  exhibiting a specific, repeatable pattern.
• Quasi-random - is defined as filling the solution
  space sequentially (in fact, these sequences are
  not at all random - they are just comprehensive
  at a preset level of granularity). For example,
  consider the integer space 0, 100. One
  quasi-random sequence which fills that space is
  0, 1, 2,...,99, 100. Another is 100, 99,
  98,...,2, 1, 0. Yet a third is 23, 24, 25,...,
  99, 100, 0, 1,..., 21, 22. Pseudorandom sequences
  which would fill the space are pseudorandom
  permutations of this set (they contain the same
  numbers, but in a different, random" order).

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Random number generators

• Desirable Properties
• Random numbers are used to determine
• attributes (such as outgoing direction, energy,
  etc.) for launched particles,
• Interactions of particles with the medium.
• Physically, the following properties are
  desirable
• The attributes of particles should not be
  correlated. The attributes of each particle
  should be independent of those attributes of any
  other particle.
• The attributes of particles should be able to
  fill the entire attribute space in a manner which
  is consistent with the physics. E.g. if we are
  launching particles into a hemispherical space
  above a surface, then we should be able to
  approach completely filling the hemisphere with
  outgoing directions, as we approach an infinite
  number of particles launched. At the very least,
  holes" or sparseness in the outgoing directions
  should not affect the answers significantly.
  Also, if we are sampling from an energy
  distribution, with an increasing number of
  particles, we should be able to duplicate the
  energy distribution better and better, until our
  simulated distribution is good enough."

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Random number generators Desirable properties

• Mathematically, the sequence of random numbers
  used to effect a Monte Carlo model should possess
  the following properties
• Uncorrelated Sequences - The sequences of random
  numbers should be serially uncorrelated. Any
  subsequence of random numbers should not be
  correlated with any other subsequence of random
  numbers. Most especially, n-tuples of random
  numbers should be independent of one another. For
  example, if we are using the random number
  generator to generate outgoing directions so as
  to fill the hemispherical space above a point (or
  area), we should generate no unacceptable
  geometrical patterns in the distribution of
  outgoing directions
• Long Period - The generator should be of long
  period (ideally, the generator should not repeat
  practically, the repetition should occur only
  after the generation of a very large set of
  random numbers).

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• Uniformity - The sequence of random numbers
  should be uniform, and unbiased. That is, equal
  fractions of random numbers should fall into
  equal areas" in space. For example, if random
  numbers on 0,1) are to be generated, it would be
  poor practice were more than half to fall into
  0, 0.1), presuming the sample size is
  sufficiently large. Often, when there is a lack
  of uniformity, there are n-tuples of random
  numbers which are correlated. In this case, the
  space might be filled in a definite, easily
  observable pattern. Thus, the properties of
  uniformity and uncorrelated sequences are loosely
  related.
• Efficiency - The generator should be efficient.
  In particular, the generator used on vector
  machines should be vectorizable, with low
  overhead. On massively parallel architectures,
  the processors should not have to communicate
  among themselves, except perhaps during
  initialization. This is not generally a
  signiffcant issue. With minimal effort, random
  number generators can be implemented in a high
  level language such as C or FORTRAN, and be
  observed to consume well less than 1 of overall
  CPU time over a large suite of applications.

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Multiplicative, Linear, Congruential Generators -
the one most commonly used for generating random
integers.
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The Essence of Monte Carlo
Random numbers
Monte Carlo
RESULTS
Probability Distribution
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THE HEART OF THE MONTE CARLO sampling of mean
free path
x, x - random numbers
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Cross sections probability distribution
functions!
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DATA LIBRARIES ARE OF TWO FORMS

• Pointwise
• Cross sections are specified at a sufficient
  number of energy points to faithfully reproduce
  the evaluated cross section
• Used with
• Continuous energy Monte Carlo codes
• Multigroup
• All data are collapsed and averaged into energy
  groups
• Used with
• Diffusion codes
• Discrete ordinate codes
• Multigroup Monte Carlo codes

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CROSS-SECTION LIBRARIES

• Pointwise
• User does not have to worry about
• Group structure
• Flux spectrum
• Self-shielding
• Small amount of work in processing code
• Large amount of work in transport code
• Multigroup
• User needs to worry about everything
• Large amount of work in processing code
• Small amount of work in transport code

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238Np - fission cross section
sf (barn)
sf (barn)
E (eV)
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235U - fission cross section - pointwise JEF2
Resonance region
sf (barn)
E (eV)
E (eV)
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235U - fission and capture cross sections JEF2
multigroup
pointwise
s (barn)
E (eV)
E (eV)
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235U - fission cross sections in resonanse
region Pointwise and multigroup JEF2 data
s (barn)
E (eV)
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Neutron cross section data contain

• Cross sections for reactions
• elastic scattering, (n,2n), (n,3n), (n,n,p)
• inelastic scattering (n,n) continuum, (n,g),
  (n,p) and (n,a)
• total, absorption, proton prod. and a production
• Angular distrubutions for different reactions (45
  in MCNP)
• Energy distrubution for some reactions (4)
• Heating numbers and Q-values
• Photon production cross sections, angular
  distributions for (n,g), (n,n), (n,2n) and n,3n)

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Classes of cross-section data (MCNP)

• C - Continuous-energy neutron
• D - Discrete reaction neutron
• Y - Neutron dosimetry
• T - Neutron S(a,b) thermal
• P - Continuous-energy photon
• M- Multigroup neutron
• G - Multigroup photon
• E - Continuous-energy electron

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Major Neutron Physics Approximations

• (n,xn) sampled independently
• (n,f),(n,xn) happen instantly
• unresolved resonances treated as average cross
  section
• no delayed gamma production
• no charged particle production

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Experimental Data Theoretical Codes
ENDF/B Evaluations
T-2 Evaluations
LLNL Evaluations
NJOY/TRANSX Processing codes T-2
Multigroup or Continuous Energy File
MCPOINT CLYDE Processing Codes
Checking Codes Processsing Codes Data Analysis
Feedback to Evaluators Experimentalists, and
Processors
Multigroup or Continuous Energy File
Test Data in Transport Codes
Include Data in Libarary
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Approximations in generating data

• Evaluation Assumptions
• choice of experiments
• choice of representations, i.e., discrete lines
  for continuous distributions
• interpolation (model codes)
• bin sizes
• tresholds, Q-values, particularly for elements
• Processing Assumptions
• representation of angular distributions as
  equiprobable bins
• resonance parameter treatment

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Things to consider when choosing neutron
cross-section data

• Differences in evaluators philosophy
• Neutron energy spectrum
• Temperature at which set was processed
• Availability of photon-production data
• Sensitivity of results to different evaluations
• Use the best data that you can afford

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THERMAL EFFECTS

• Target motion (gas liquid, solid) --gt incident
  neutron sees targets with various velocities,
  i.e., many different relative velocities
• so average cross section is changed
• kinematics are different
• neutron upscattering
• Doppler broadening

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THERMAL EFFECTS (cont.)

• Lattice structure (solid)

At high energy - wavelength of neutron ltlt lattice
spacing
Lattice spacing
At low energy - wavelength of neutron _at_ lattice
spacing

• Cross sections show very jagged behaviour, each
  peak corresponds to a particular set of crystal
  planes
• Coherent scattering (interference of scattered
  waves) add constructively in some directions and
  add destructively in others
• gtAngular distribution changed
• Bragg scattering

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THERMAL EFFECTS (cont.)

• Molecular energy levels (liquid, solid)
• vibrational and rotational levels
• 0.1 eV spacing below a few eV
• Neutron loses or gains energy in discrete amounts
• gt modify DOUBLE-DIFFERENTIAL
• thermal inelastic scattering

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Deterministic Monte-Carlo

• simulates individual particles and records some
  aspects of their behaviour
• supplies information only about the specific
  sampled distribution (tally)
• No transport equation need to be written to solve
  a transport problem by Monte Carlo - but -
  equation of the probability density of particles
  integral transport equation

• solves the transport equation (integro-differentia
  l) for the average particle behaviour
• gives fairly complete information (e.g. flux)
  throughout the phase space of the problem

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Deterministic Monte-Carlo

• the discrete ordinates method visualises the
  phase space to be divided into many small boxes,
  the particles move from one box to another. Boxes
  get progressively smaller, particles moving from
  box to box take differential amount of time to
  move and differential distance in space ----gt
  approaches the integro differential tarnsport

• Monte-Carlo transports particle between events
  (e.g. collisions) that are separated in space and
  time. Neither differntial space nor time are
  inherent parameters of Monte-Carlo. The integral
  equation does not have time or space derivatives

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Phase space
mean value - Xn
simulation results
Accuracy measure of how close the mean is to the
TRUE PHYSICAL quantity (TRUE VALUE) being
estimated
TRUE VALUE

• Sometimes called the systematic error
• Monte Carlo cannot directly estimate this
• Factors related to accuracy
• Code accuracy (physics models etc.)
• Problem modelling (geometry, dource, etc
• User errors/abuse

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Phase space
mean value - Xn
simulation results
TRUE VALUE
Precision the uncertainty in Xn the statistical
fluctuations in the sampled xj

• Monte-Carlo can estimate this

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PRINCIPLES

• Central Limit Theorem
• The distribution of the sum of n independent,
  identically distrubuted random variables

Conditions Mean and Variance must exist (i.e. be
finite)
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PRINCIPLES

• Confidence Intervals

• Probability Distributions
• Discrete Probability Density Function n discrete
  events

p2
pi probability per event
p1
........
pn
n
1 2
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PRINCIPLES

• Probability Distributions
• Continuous Probability Density Function

p(x) probability per unit of x
p(x)
Normalisation
x
Cumulative Probability Function
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DEFINITIONS
standard deviation of population s
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DEFINITIONS
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DEFINITIONS
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Reminder
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Key of Symbols in Monte Carlo

• W - particle weight, Ws - source weight
• E- particle energy (MeV) ED - energy deposited
  (MeV)
• m- cosine of angle between surface normal and
  trajectory
• A- surface area (cm2) V - cell volume (cm3)
• l - track length (cm)
• p(m) - probability density function m - cosine
  of angle between particle
• trajectory and detector
• s - total mean free path to the detector
• R- distance to the detector
• sT(E)- microscopic cross section (barns), sf(E)
  - fission x-section
• H(E) - heating number (MeV/collision)
• Q - fission heating Q-value (MeV)
• ra - atom density (atoms/barn-cm), m - cell mass
  (g)

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ESTIMATORS

• Definitions
• Particle speed - v(cm/s)
• Particle density - n (particles/cm3), a function
  of position, energy, angle, and time
• Flux (particles/cm2s)
• Fluence (particles/cm2)
• Cell fluence estimator (track length)

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• Total heating
• Fission Heating
• keff

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DEFINITIONS
Precision the uncertainty in Xn caused by the
statistical fluctuations in the sampled xis

• Monte Carlo codes can estimate this (relative
  error)

• User controlled factors related to precision
• Forward versus adjoint calculation
• Estimator type (tally)
• Variance reduction technique
• Number of histories run

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DEFINITIONS
Efficiency a measure of how quickly the desired
precision is achieved

• Monte Carlo codes can estimate this (Figure of
  Merit - FOM)

FOM should be constant with N Larger FOM more
efficient

• Factors affecting efficiency
• History-scoring efficiency
• Dispersions in nonzero history scores
• Computer time per history

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VARIANCE REDUCTION MOTIVATION
ý

q fraction with xi¹0
R2eff
R2non
R2eff can be a significant part of the relative
error Variance reduction techniques should strive
to equalize the xi scores as well as reduce the
fraction of zero scores.
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VARIANCE REDUCTION TECHNIQUE
Type Time and energy cutoffs
T Weight window generator P Energy splitting
and Russian roulette P Forced
collisions M Exponential transform D
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TYPES OF VARIANCE REDUCTION
(T) Truncation method - trancates parts of phase
space that do not contribute significantly to the
solution (P) Population control method - uses
particle splitting and Russian roulette to
control the number of samples taken in various
regions of phase space (M) Modified sampling
method - alters the statistical sampling of a
problem to increase the amount of information
squeezed from each trajectory. Sampling is done
from distributions that send particles in desired
directions or into other desired regions of phase
space such as time, energy, or change the
location or type of collision (D) Partially
deterministic methods - circumvents part of the
normal random walk process (e.g. next event
estimator)
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VARIANCE REDUCTION TECHNIQUE
Implicit Capture - a splitting process where the
particle is split into absorbed weight (which is
not followed) and serviving weight. Applied after
each collision Surviving weight is given by
Generally this decreases Reff faster than
increasing T.
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VARIANCE REDUCTION TECHNIQUE
Weight cutoff - Russian roulette is played if a
particles weight drops below a user-specified
weight cutoo. The particle is either killed or
its weight is increased. Unlike other cutoffs,
weight cutoff is NOT biased. Applied when W lt Rj
WC2 and survival is given by probability
W/(WC1 Rj) weight WC1 Rj
Rj ratio of source cell/ cell j
importances
This decreases Rnon faster than increasing T
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VARIANCE REDUCTION TECHNIQUE
Geometry splitting and Russian roulette -
particles in important regions are increased in
number (or weight) for better sampling and
decreased in number in other regions.
Applied at surface crossing. If importance ratio
(previous cell to new) is an integer gt 2, the
particle is split into n particles with weight
W/n If ratio is less than one, Russian roulette
is played.
This decreases Reff faster than increasing T
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SPLITTING
Lower importance region
Higher importance region
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RUSSIAN ROULETTE
Bang!
Lower importance region
Higher importance region
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VARIANCE REDUCTION TECHNIQUE
General source biasing - allows the production of
more source particles. with suitably reduced
weights, in the more important regimes of each
variables. Most common variables Angle Ene
rgy Position This decrease Rnon and Reff
faster than increasing T.
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The END
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EXPONENTIAL TRANSFORM
Requirements for preserving expected
collided weight
uncollided weight
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VARIANCE REDUCTION CAUTIONSI could have it done
in a much more complicated way said the red
Queen, immensely proudLewis Caroll

• Be cautious
• the advantage of one variance reduction technique
  may cancel another
• balance user time with computation time
• analyse carefully the results
• make cells small enough so neighbouring
  importances do not exceed a factor of 8-10

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Monte Carlo and Transport Equation
Boltzman Transport Equation - Time-independent,
Linear A general integral form
97
Monte Carlo and Transport Equation
Source term for the Boltzmann equation
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Photon Physics

• Coherent (Thomson) Scattering Form Factors
• Inhorenet (Compton) Scattering Form Factors
• Pair Production
• Photoelectric Absorption and Fluorescence
• Thick-Target Bremsstrahlung

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Photon Physics Approximations

• Only K,L edges treated for photoelectric
  absorption
• Distance to collision sampled, not distance to
  scatter
• Thick/thin-target bremsstrahlung
• No distinction between pair and triplet
  production
• No anomalous scattering factors
• No photoneutrons

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Surface flux

• Cell flux estimator
• Surface flux estimator

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