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Radiation Shielding and Reactor Criticality Fall 2013 By Yaohang Li, Ph.D. Review • 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 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 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 cs 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 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 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 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? 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 (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 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 » w1+…+wk=w Special Methods for the Shielding Problem • Outstanding feature of the shielding problem – The proportion of photons that penetrate the shield is very small, say one in 106. – To estimate an accuracy of 10% require the number of 108 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 The Semi-Analytic Method (I) •A random history – for a particle which undergoes a suitably large number n of scatterings in the medium E , E ,..., En h hn 0 1 0 ,1 ,..., n •The semi-analytic method – Pi() • 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) • ci=cosi; i=c(Ei); i=[1-(Ei)]c(Ei) – P0()=exp(- i/c0) • the probability that the particle passes through z= before suffering any scatterings – Pi+1() • A particle crosses z= between its (i+1)th and (i+2)th scatterings The Semi-Analytic Method (II) •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)=Pi(’) • ii: the particle suffered the (i+1)th scattering between the planes z= ’ and z= ’+d’ – P(ii)= i d’/|ci| • iii: after scattering, the particle now travels with energy Ei+1 in direction i+1 – P(iii)= exp(- i+1(- ’) /ci+1) – Then Pi 1 ( ) Pi ( ' ) exp{ i 1 ( ' ) / ci 1} 0 i d ' | ci | – The probability of penetrating the shield is E ( Pi (t )) i 0 Probability of Penetration • Replace with • Approximate unbiased estimator of penetration probability • N = 25, 12, 9, 6 is efficient for shields of water, iron, tin, and lead E ( Pi (t )) i 0 N E ( Pi (t )) i 0 Neutron Transport • 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 pc – Proportional to capture cross section • Scatter with probability ps – Proportional to scattering cross section Scattering •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 X Z v v' Solid Angle • 2D – measured by unit angles (radians) – full circle subtends 2 • 3D – measured by unit solid angles (steradians) – full sphere subtends 4 Probability of Scattering •Scattering equally in all directions – probability p(,)dd=d/4 •Definition of the Solid Angle sin dd S – then d = sindd – we can get p(,) = sin/4 •Probability density for and p( ) 2 1 p ( , ) d sin 0 2 p( ) p( , )d 0 1 2 Non-uniform Random Sample Generation Revisit •Probability Density p(x) p( x)dx 1 •Then x P( x) p( x' )dx' r – r is a uniform random number •Inverse Function Method – use r to represent x Randomizing the Angles • – =2r – is uniformly distributed between 0 and 2 • 1 r sin xdx 20 – Then we can get cos = 1-2r – cos is uniformly distributed between -1 and +1 Path Length •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 Neutron Transport Algorithm (1) • Input parameters – – – – thickness of the plate t capture probability pc scattering probability ps mean free path • Initial value z=0 Neutron Transport Algorithm (II) 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 lcos 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 An Improved Method • Instead of Considering A Neutron – Consider a set of neutrons – ps portion of neutrons are scattered • All scattered neutrons will move to a new direction – pc portion of neutrons are captured • A better convergence rate Properties of Polymers • 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 Solvent • 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 Abstraction of Polymer • 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 Radius • Average size of a polymer containing n monomers • Radius of gyration – average distance of a monomer from the polymer’s center of mass – <Rn2> ~ Anv • 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 Early Solution • Goal – Estimate <Rn2> • Method – Generate unrestricted random walks – Accept if no interception – Not accept if interception • Problem – Not efficient Self-avoiding Random Walk • 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 A Picture is Worth a Thousand Words 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 ); Self-avoiding Random Walk Algorithm // 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; } Output of Self-avoiding Random Walk Biased Random Walk • 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 Biased Random Walk Algorithm • 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) Output of Biased Random Walk 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 – Neutron Transport Problem – Nonuniform Distribution Samples Summary • Long Polymer Molecule • Self-avoiding Random Walk • Biased Random Walk What I want you to do? • Review Slides • Review basic probability/statistics concepts • Work on your Assignment 5