By Yaohang Li, Ph.D.

•
Last Class
– Test of Randomness
– Chi-Square Test
– KS Test
•
This Class
– Monte Carlo Application in Nuclear Physics
• Radiation Shielding
• Reactor Criticality
•
Simulation of Collisions
–
Self-avoiding random walk
–
Assignment #5
•
Markov Chain Monte Carlo

• 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

• 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: a particle traverses a small length of its straight line
 c
: cross section
» depends on the nature of surrounding medium
» energy E

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

• Collision Probability
– cdf of the distance that the particle travels before collision
• F c
(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

(E i
)
– 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

• 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?

•
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
–
F c
(s) = 1 – exp(-
 c s)
• If the straight-line path from r to (r+s

) 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

• 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
» w1+…+wk=w

• Outstanding feature of the shielding problem
– The proportion of photons that penetrate the shield is very small, say one in
10 6 .
– To estimate an accuracy of 10% require the number of 10 8 paths.
• Hit or miss
• Quite inefficient
•
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

• A random history
– for a particle which undergoes a suitably large number n of scatterings in the medium h
 h n



E
0

0
,
,
E
1
,...,

1
,...,
E n
 n
• The semi-analytic method
–
P i
(

)
•
The probability that a particle has a history hi and also crosses the plane z=
 between its ith and (i+1)th scatterings
– Abbreviation (α is the absorption probability)
• c i
=cos
–
P
0
(

)=exp(-
 i
;

 i
=
 c
(E i

/c
0
) i
);
 i
=[1-

(E i
)]
 c
(E i
)
• the probability that the particle passes through z=
 before suffering any scatterings
– P i+1
(

)
•
A particle crosses z=
 between its (i+1)th and (i+2)th scatterings

• The semi-analytic method
– the (i+1)th scattering occurred on some a plane z=
 ’ where 0<  ’< 
.
– Compound event
• i: immediately prior to the (i+1)th scattering the particle crossed z=
 ’
•
–
P(i)=P i
(
 ’)
• ii: the particle suffered the (i+1)th scattering between the planes z=
 ’ and z=
 ’+d  ’
–
P(ii)=
 i d
 ’/|c i
| iii: after scattering, the particle now travels with energy E
–
P(iii)= exp(-
 i+1
(

-
 ’) /c i+1
) i+1 in direction
 i+1
– Then
P i

1
(

)



0
P i
(

' ) exp{
  i

1
(
  
' ) / c i

1
}

| i d c i

|
'
–
The probability of penetrating the shield is
E (



0 i
P i
( t ) )

• Replace with i



0
E
N
(


0 i
P i
( t ) )
•
Approximate unbiased estimator of penetration probability
•
N = 25, 12, 9, 6 is efficient for shields of water, iron, tin, and lead

•
Transmission of Neutrons
–
Bulk matter
•
Plate
– thickness t
– infinite in the x and y directions
– z axis is normal to the plate
– Neutron at any point in the plate
• Capture with probability p c
– Proportional to capture cross section
• Scatter with probability p s
– Proportional to scattering cross section

•
Scattering
– polar angle

– azimuthal angle

• we are not interested in how far the neutron moves in x or y direction, the value of
 is irrelevant v
X v'

Z

• 2D
– measured by unit angles
(radians)
– full circle subtends 2

•
3D
– measured by unit solid angles
(steradians)
– full sphere subtends 4


•
Scattering equally in all directions
– probability p(

,

)d
 d

=d

/4

•
Definition of the Solid Angle
 
S
 sin
 d
 d

– then d

= sin
 d
 d

– we can get p(

,

) = sin

/4

• Probability density for
 and
 p (

)

2

0
 p (

,

) d
 
1
2 sin
 p (

)



0 p (

,

) d
 
1
2


•
Probability Density p(x)



 p ( x ) dx

1
• Then
P ( x )



 x p ( x ' ) dx '
 r
– r is a uniform random number
•
Inverse Function Method
– use r to represent x

• 
• 
– 
=2
 r
–  is uniformly distributed between 0 and 2
 r

1
2


0 sin xdx
– Then we can get cos

= 1-2r
– cos
 is uniformly distributed between -1 and +1

•
Path length
– distance traveled between subsequent scattering events
– obtained from the exponential probability density function p ( l )

( 1 /

) e
 l /

– l=-
 lnr
•  is the mean free path
• or the cross section constant
 c

• Input parameters
– thickness of the plate t
– capture probability p c
– scattering probability p s
– mean free path

•
Initial value z=0

1.
Determine if the neutron is captured or scattered. If it is captured, then add one to the number of captured neutrons, and go to step 5
2.
If the neutron is scattered, compute cos
 by cos

= 1-2r and l by l=-
 lnr. Change the z coordinate of the neutron by l cos

3.
If z<0, add one to the number of reflected neutrons. If z>t, add one to the number of transmitted neutrons. In either case, skip to step 5 below.
4.
Repeat steps 1-3 until the fate of the neutron has been determined.
5.
Repeat steps 1-4 with additional incident neutrons until sufficient data has been obtained

• Instead of Considering A Neutron
–
Consider a set of neutrons
– p s portion of neutrons are scattered
• All scattered neutrons will move to a new direction
– p c portion of neutrons are captured
•
A better convergence rate

•
Hydrophobic
– The attraction between monomers is stronger than their attraction to the molecules of the surrounding solvent, e.g., water
•
Hydrophilic
– The attraction between monomers is weaker than their attraction to the molecules of the surrounding solvent, e.g., water
•
Non Self-intersect
–
No two monomers can occupy the same place
• excluded volume

• Low Temperature (or in a poor solvent)
–
The attractive interactions between monomers pull the polymer into a dense ball-like configuration
– globule
•
High Temperature (or in good solvent)
– The interactions are mediated by the solvent molecules
–
Typical configurations are open coils
•
Phase Transition
– Coil-Globule transition

•
Real Polymer
– the monomers occupy positions in continuous space
– bonds btw. monomers are constrained to have only certain angles
• depending on the nature of the monomers
• Simplification
–
Embed the polymer into discrete space
– Require that the monomers exist at integer coordinates
• only a lattice spacing apart

•
Average size of a polymer containing n monomers
•
Radius of gyration
– average distance of a monomer from the polymer’s center of mass
–
<R n
2 > ~ An v
• v is the critical exponent
– in the swollen phase: v

0.588
– in the collapse phase: v=1/3
•
A is unknown
– use linear regression

• Goal
–
Estimate <R n
2 >
•
Method
– Generate unrestricted random walks
–
Accept if no interception
–
Not accept if interception
•
Problem
– Not efficient

•
Self-avoiding Random Walk
– Walk on 2D or 3D lattice
–
Explore the geometric properties of linear polymers in good solvent
– Constraint random walk (don’t allow to go backward)
– Introduced by Orr
• Analysis of Self-avoiding Random Walks
– At first glance, the model is far too simple
– Phenomenon of universality
• Many quantities are not dependent on the specific details of the system
•
They are determined only by its universality class
• All systems in the same universality class share the same dominant asymptotic behavior

2D Walk 3D Walk

#include <iostream.h>
#include <stdlib.h>
#include <math.h> void do_walk (int maxstep, int& nstep, double& rsquared ){ const int MAXSTEP=20; int map[ MAXSTEP*2][MAXSTEP*2]={0};
// start point int completed=0; int x = MAXSTEP; int y = MAXSTEP; int npoint = 1; map[x][y] = npoint; do { int xnew=x; int ynew=y; switch ( (int)(4 *(double)rand()/(RAND_MAX+1.0)) ) { case 0: xnew-= 1; break; case 1: xnew+= 1; break; case 2: ynew-= 1; break; case 3: ynew+= 1; break;
} if ( map[xnew][ynew] == 0 ){ npoint++; map[xnew][ynew] = npoint; x = xnew; y = ynew; if ( npoint == maxstep+1 )completed=1;
} else if ( map[xnew][ynew] != npoint-1 ) { completed=1;
}
} while ( !completed );
// Print window centred on map for ( int i=5; i<2*MAXSTEP-5; i++ ){ for ( int j=5; j < 2*MAXSTEP-5; j++ ){ cout.width(3); cout << map[i][j];
} cout << endl;
} nstep = npoint-1;
} rsquared = pow( x-MAXSTEP,2.0) + pow( y-MAXSTEP,
2.0 );
} int main(){ int maxstep=20,nstep; double rsquared; srand(987654321); for (int i=1; i<10; i++ ){ do_walk(maxstep,nstep,rsquared); cout << endl << "Nsteps: " <<nstep << " Rsquared: "
<<rsquared<<endl;
} return 0;

•
Problems of self-avoiding random walk
– Have to reject many terminated walks in order to have unbiased statistics
–
Unlikely to produce long polymer
– Inefficiency
•
Biased Random Walk
– Basic Idea
•
Instead of abandoning a walk when an illegal step is attempted, we go back and pick one of the possible legal steps
• Enable a walk to make a full distance

• Weight Factor W(N)
– Initially = 1
–
3 possibilities
• No further steps are possible, we have reached a dead end
– Abandon this walk
•
All steps, other than going directly backwards are possible
– proceed as normal, set W(N) = W(N-1)
•
Only m steps are possible
–
Randomly choose one of the possible steps
– set W(N)=m/3*W(N)

•
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
– Neutron Transport Problem
–
Nonuniform Distribution Samples

•
Long Polymer Molecule
• Self-avoiding Random Walk
•
Biased Random Walk

• Review Slides
•
Review basic probability/statistics concepts
• Work on your Assignment 5