Lesson 6: Computer method
overview
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Neutron transport overviews
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Comparison of deterministic vs. Monte Carlo
User-level knowledge of Monte Carlo
Example calc: Approach to critical
Neutron transport overview
Neutron balance equation
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Scalar flux/current balance
  
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  J r , E    a r , E r , E    f r , E r , E   S r , E 
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Used in shielding, reactor theory, crit.
safety, kinetics
Problem: No source=No solution !
K-effective eigenvalue
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Changes , the number of neutrons per fission
  
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
  J r , E    a r , E  r , E    f r , E  r , E 

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Advantages:
 Everybody uses it
 Guaranteed real solution
 Fairly intuitive (if you don’t take it too seriously)
 Good measure of distance from criticality for reactors
Disadvantages:
 No physical basis
 Not a good measure of distance from criticality for CS
Neutron transport
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Scalar vs. angular flux
Boltzmann transport equation
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Deterministic
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Neutron accounting balance
General terms of a balance in energy, angle, space
Subdivide energy, angle, space and solve equation
Get k-effective and flux
Monte Carlo
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Numerical simulation of transport
Get particular flux-related answers, not flux everywhere
Deterministic grid solution
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Subdivide everything: Energy, Space, Angle
“Eulerian” grid: Fixed in space
Use balance condition to figure out each piece
Source
Spatial and angular flux for Group 1
(10 MeV-20 MeV)
Stochastic solution
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Continuous in Energy, Space, Angle
“Lagrangian” grid: Follows the particle
Sample (poll) by following “typical” neutrons
Absorbed
Fissile
Fission
Particle track of two 10 MeV fission
neutrons
Advantages and disadvantages
of each
Advantages
Discrete
Ordinates
(deterministic)
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Monte Carlo
(stochastic)
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Disadvantages
Fast (1D, 2D)
Accurate for simple geometries
Delivers answer everywhere:
= Complete spatial, energy,
angular map of the flux
1/N (or better) error convergence
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“Exact” geometry
Continuous energy possible
Estimate of accuracy given
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Slow (3D)
Multigroup energy required
Geometry must be approximated
Large computer memory
requirements
User must determine accuracy by
repeated calculations
Slow (1,2,3D)
Large computer time
requirements
1/N1/2 error convergence
Monte Carlo overview
Monte Carlo
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
We will cover the mathematical details in
3 steps
1. General overview of MC approach
2. Example walkthrough
3. Special considerations for criticality
calculations
Goal: Give you just enough details for you
to be an intelligent user
General Overview of MC

Monte Carlo: Stochastic approach
 Statistical simulation of individual particle
histories
 Keep score of quantities you care about
 Most of mathematically interesting features
come from “variance reduction methods”
 Gives results PLUS standard deviation =
statistical measure of how reliable the answer is
Mathematical basis
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Statistical simulation driven by random number generator:
0<x<1, with a uniform distribution:
p(xdxprob. of picking x in (x, x+dx)=dx

Score keeping driven by statistical formula:
N
x  xˆ 
x
n 1
n
N
N
 2  xˆ  
 x
n 1
 xˆ 
2
n
N  N  1
Simple Walkthrough
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Six types of decisions to be made:
1.
2.
3.
4.
5.
6.
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Where the particle is born
Initial particle energy
Initial particle direction
Distance to next collision
Type of collision
Outcome of scattering collision (E, direction)
How are these decisions made?
Decision 1: Where particle is born
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3 choices:
1.
2.
Set of fixed points (first “generation” only)
Uniformly distributed (first generation only)
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KENO picks point in geometry
xmin  x  xmax , uniformly distributed
x  xmin  x ( xmax  xmin )
y  ymin  x ( ymax  ymin )
z  zmin  x ( zmax  zmin )
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Rejected if not in fuel
After first generation, previous generation’s
sites used to start new fissions
Decision 2: Initial particle
energy
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From a fission neutron energy spectrum
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Complicated algorithms based on advanced
mathematical treatments
Rejection method based on “bounding box”
idea
Decision 3: Initial particle
direction
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Easy one for us because all fission is isotropic
Must choose the “longitude” and “latitude” that
the particle would cross a unit sphere centered on
original location
Mathematical results:
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m=cosine of angle from polar axis
 1  m  1, uniformly distribute d
m  2x - 1
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F=azimuthal angle (longitude)
0    2, uniformly distribute d
  2x
Decision 4: Distance to next
collision
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Let s=distance traveled in medium with t
Prob. of colliding in ds at distance s
=(Prob. of surviving to s)x(Prob. of colliding in
ds|survived to s)
=(e- t s)x(ts)

Using methods from NE582, this results in:
s
 ln x
t
Decision 5: Type of collision
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Most straight-forward of all because straight from
cross sections:
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s / t =probability of scatter
a / t =probability of absorption
Therefore, if x< s / t , it is a scatter
Otherwise particle is absorbed (lost)
“Weighting in lieu of absorption”=fancy term for
following the non-absorbing fraction of the
particle
Decision 6: Outcome of
scattering collision
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In simplest case (isotropic scattering), a
combination of Decisions 3 and 5:
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“Decision 5-like” choice of new energy group
 sg  g 
 probabilit y of scatter from group g to group g'
 sg
Decision 3 gets new direction
In more complicated (normal) case, must deal with
the energy/direction couplings from elastic and
inelastic scattering physics
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Special considerations for
criticality safety
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Must deal w/ “generations”=outer iteration
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Fix a fission source spatial shape
Find new fission source shape and eigenvalue
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User must specify # of generations AND # of
histories per generation AND # of generation to
“skip”
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Skipped generations allow the original
lousy spatial fission distribution to improve
before we really start keeping “score”
KENO defaults: 203 generations of 1000
histories per generation, skipping first 3
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Approach to critical
Approach to critical
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This is a brief overview of the basic
techniques of running a critical experiment
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The very same techniques are used to bring a
reactor from shutdown state to an initial zeropower critical condition
The basic idea is to use the equations of
subcritical multiplication to sneak up on the
critical value of some parameter (i.e., one of
the nine MAGICMERV parameters)
Approach to critical (cont’d)
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We begin with the basic definition of keffective (which we will call “k” in our
equations)
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The ratio of fission neutrons produced in any
“generation” to those in the previous generation
Actually, since the flux increases linearly with
fission rate (if you are careful) any fluxdependent parameter can be used in place of the
fission neutron production
•
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We will use the response of an external detector
In both practice and theory, there must be an
external neutron source to give the system
something to multiply
Approach to critical (cont’d)
If we assume an external source rate of S
neutrons/generation, we can see that:
1. In the first generation, S neutrons will be
released. Since no fission has taken place
yet, these will be the only neutrons in the
system.
2. In the second generation, S new neutrons
will be released and the previous
generation’s S neutrons will cause kS(<S)
new neutrons to give us a population of:

S (1  k )
Approach to critical (cont’d)
3.
In the next generation, the same thing
happens: S new neutrons will be released
and the previous population is multiplied
by k, giving us:
S  k  S (1  k )  S (1  k  k 2 )
4.
Following this through to an infinite series:
S
S (1  k  k  k  ...) 
1 k
2
3
Approach to critical (cont’d)
5.
6.
The 1/(1-k) term is a MULTIPLIER of the
source. It appears from first glance that this
equation can be used to find k, but it really
can’t, because we never really know what the
original k is.
But it CAN tell you is how much (relatively)
closer your most recent change got you to k=1.

For example, if you make a change and your
detector reading goes from 100 counts/sec to 200
counts/sec, you DO know that you went
HALFWAY to k=1. But you can’t tell whether you
went from k=.8 to .9 or (maybe) 0.96 to 0.98
Approach to critical (cont’d)
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7.
This is useful because it tells you that ANOTHER
change of (about) the same magnitude would get
you to critical.
Since criticality corresponds to the M=1/(1-k)
term (the MULTIPLIER of the source) going
infinite, we have found it is more useful to
instead watch its inverse.

Criticality corresponds to 1/M going to 0.
(Inclass example as time allows)