Radiation Shielding and Reactor Criticality Spring 2005

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Radiation Shielding and Reactor
CriticalitySpring 2005

• By Yaohang Li, Ph.D.
• Department of Computer Science
• North Carolina AT State University
• yaohang_at_ncat.edu

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Review

• Last Class
• Test of Random Numbers
• Knuths test
• This Class
• Monte Carlo Application in Nuclear Physics
• Radiation Shielding
• Reactor Criticality
• Next Class
• Monte Carlo Application in Nuclear Physics
• Simulation of Collisions
• Assignment 4

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Monte Carlo Method in Nuclear Physics

• Flux of uncharged particles through a medium
• Uncharged particles
• paths between collisions are straight lines
• do not influence one another
• independence
• allow us to take the behavior of a relatively
  small sample of particles to represent the whole
• Randomness
• derive the Monte Carlo methods directly from the
  physical processes

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Problem Definition

• Particle (Photon or Neutron)
• energy E
• instantaneously at the point r
• traveling in the direction of the unit vector ?
• Traveling of the Particle
• At each point of its straight path it has a
  chance of colliding with an atom of the medium
• No collision with an atom of the medium
• continue to travel in the same direction ? with
  same energy E
• A probability of ?c?s that the particle will
  collide with an atom of the medium
• ?s an atom traverses a small length of its
  straight line
• ?c cross section
• depends on the nature of surrounding medium
• energy E

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Cross Section

• Determining ?c
• The medium remains homogeneous within each of a
  small number of distinct regions
• over each region, ?c is a constant
• ?c change abruptly on passing from one region to
  the next
• Example
• Uranium rods immersed in water
• ?c a function of E in the rods
• ?c another function of E in the water

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Collision

• Collision Probability
• cdf of the distance that the particle travels
  before collision
• Fc(s) 1 exp(- ?c s)
• Three situations of collision
• Absorption
• the particle is absorbed into the medium
• Scatter
• the particle leaves the point of collision in a
  new direction with a new energy with probability
  ?(Ei)
• fission (only arises when the original particle
  is a neutron)
• several other neutrons, known as secondary
  neutrons, leaves the point of collision with
  various energies and directions
• Probability of the three situations
• Governed by the physical law
• Known distribution from Monte Carlo point of view

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Shielding and Criticality Problems

• The Shielding Problem
• When a thick shield of absorbing material is
  exposed to ?-radiation (photons), of specified
  energy and angle of incidence, what is the
  intensity and energy-distribution of the
  radiation that penetrates the shield?
• The Criticality Problem
• When a pulse of neutron is injected into a
  reactor assembly, will it cause a multiplying
  chain reaction or will it be absorbed, and in
  particular, what is the size of the assembly at
  which the reaction is just able to sustain itself?

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Elementary Approach

• Elementary Approach
• Exact realization of the physical model
• Not very efficient
• Tracking of simulated particles from collision to
  collision
• Starting with a particle (E, ?, r)
• Generate a number s with the exponential
  distribution
• Fc(s) 1 exp(- ?c s)
• If the straight-line path from r to (rs?) does
  not intersect any boundary (between regions)
• the particle has a collision
• Otherwise
• proceed as far as the first boundary
• if this is the outer boundary, the particle
  escapes from the system
• Repeat the procedure

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Improvements of the Elementary Approach

• Problem
• There may be too many or too few particles
• Consider a reactor containing a very fissile
  component
• Every neutron entering this region may give rise
  to a very large number coming out
• Give us more tracks than we have time to follow
• Solution
• Russian Roulette
• Pick out one of the particles
• discard it with probability p
• otherwise allow this particle to continue but
  multiply its weight (initially unity) by (1-p)-1
• The number of particles is reduced to manageable
  size
• Splitting
• To increase the sample sizes
• a particle of weight w may be replaced by any
  number k of identical particles of weights w1, ,
  wk
• w1wkw

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Special Methods for the Shielding Problem

• Outstanding feature of the shielding problem
• The proportion of photons that penestrate the
  shield is very small, say one in 106.
• To estimate an accuracy of 10 require the number
  of 108 paths.
• Quite impossible
• Solution
• Semi-analytic method
• Allows the same random paths to be used for
  shields of other thickness
• Simplification
• Only think about three coordinates
• Energy E
• Angle between the direction of motion and the
  normal to the stab
• Distance z from the incident face of the slab

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The Semi-Analytic Method (I)

• A random history
• for a particle which undergoes a suitably large
  number n of scatterings in the medium
• The semi-analytic method
• Pi(?)
• The probability that a particle has a history hi
  and also crosses the plane z ? between its ith
  and (i1)th scatterings
• Abbreviation
• cicos?i ?i?c(Ei) ?i1-?(Ei)?c(Ei)
• P0(?)exp(- ?i?/c0)
• the probability that the particle passes through
  z ? before suffering any scatterings
• Pi1(?)
• A particle crosses z ? between its (i1)th and
  (i2)th scatterings

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The Semi-Analytic Method (II)

• The semi-analytic method
• the (i1)th scattering occurred on some a plane
  z ? where 0lt?lt ?.
• Compound event
• i immediately prior to the (i1)th scattering
  the particle crossed z ?
• P(i)Pi(?)
• ii the particle suffered the (i1)th scattering
  between the planes z ? and z ?d?
• P(ii) ?i d?/ci
• iii after scattering, the particle now travels
  with energy Ei1 in direction ?i1
• P(iii) exp(- ?i1(?- ?) /ci1)
• Then
• The probability of penetrating the shield is

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Summary

• Nuclear Simulation
• Radiation Shielding
• Reactor Criticality
• Particle Assumption
• Cross Section
• Collision
• Elementary Method
• Improvements for the Elementary Method
• Russian Roulette
• Splitting
• Special methods for the shielding problem
• Semi-Analytic Method

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What I want you to do?

• Review Slides
• Read the UNIX handbook if you are not familiar
  with UNIX
• Review basic probability/statistics concepts
• Work on your Assignment 3
• Prepare for your presentation topic and term paper