MCRT: L0

•  Some background, what previous courses students should look over, gentle introduction/recap of probabilities

•  Get an idea of computer programming experience of the class

•  Overview of course structure and assessment

Monte Carlo Radiation Transport Kenny Wood

kw25@st-andrews.ac.uk •  A practical approach to the numerical simulation of

radiation transport •  Develop programs for the random walks of photons and

neutrons using Monte Carlo techniques •  Will refer to previous courses on optics, radiation,

neutrons, atomic physics, biological tissue structure, some hydrodynamics

•  Guest lectures in week 3 from Prof Steve Jacques (Oregon) on Monte Carlo radiation transfer in biological tissue and medical physics

Astrophysics Interpreting data via 3D radiation transfer modeling Heating of dust and gas, thermal pressure, radiation pressure Radiation-magneto-hydrodynamics

Dusty Ultra Compact H II Regions

Indebetouw, Whitney, Johnson, & Wood (2006)

ONE 3D model can explain ALL UCHII SEDs!!

Hierarchical Density Structure

Elmegreen (1997)

Fractal generating algorithm reproduces observed structure and fractal dimension, D, of clouds in interstellar medium 2D: P ~ AD/2 ; Circle: D =1 ISM clouds: D ~ 1.3 Radiation transfer in clouds…

Mathis, Whitney, & Wood (2002)

3D density: viewing angle effects

NGC 7023 Reflection Nebula Monte Carlo scattered light

Simulations of fractal clouds

Atmospheric Physics Clouds important for photon transport and temperature structure of atmosphere

Medical Physics Light activated treatments such as photodynamic therapy: how deep does the radiation penetrate into skin and tissue? Imaging using x-ray, ultraviolet, optical, infrared, & polarised light Optical tweezers, photo-acoustic imaging, nuclear medicine, etc, etc

Monte Carlo simulations of computed topography (CT) x-ray imaging doses Rensselaer Polytechnic Institute

Nuclear Physics & Neutron Transport Compute controlled criticality assemblies & geometries for nuclear fission reactors Nuclear safety – radioactive shielding calculations Uncontrolled reactions – critical masses for bombs

Chain reaction in 235U Chicago Pile 1, December 1942 World’s first artificial nuclear reactor

Course Structure •  Lectures on MCRT techniques and outline of

FORTRAN programs •  Computer lab sessions for what you’ll need in

FORTRAN and to develop a code for photon random walks in a uniform density sphere

•  Tutorial-style problems to do in groups •  No final exam, 100% continuous assessment:

–  40% for two homework sheets of written problems and short programs & subroutines

–  40% for one large project –  20% for one short project

Remind yourself of… •  Refractive index, Snell’s Law, Fresnel reflection &

refraction (PH1011 Waves & Optics) •  Polarization (PH3007 Electromagnetism & PH4035

Principles of Optics) •  Ionization potentials, atomic term diagrams PH4037

(Physics of Atoms) •  Equations of hydrodynamics, pressure & forces (PH4031

Fluids)

•  Probability theory: probability distribution function (PDF) and cumulative distribution function (CDF)

•  Numerical integration, Simpsons rule, quadrature

•  Intensity, luminosity, flux, radiation pressure (many astronomy courses)

•  Biological tissue optics, skin structure, light-tissue interactions (PH5016 Biophotonics)

•  Neutron cross sections: scattering, absorption, fission •  Fission products, slow/fast & prompt/delayed neutrons •  Chain reactions, critical mass, moderators •  PH4022 or PH4040 (Nuclear & Particle Physics)

Programming in FORTRAN77

•  You’ll need: text editor; FORTRAN compiler (gfortran); graphics package for plotting lines, contours, 3D visualisation

•  By Monday create file called hello.f •  Compile it: gfortran hello.f •  Run the executable: ./a.out

program hello implicit none print *, ‘Hello’ stop end

Note six spaces before text starts on each line!

Buffon’s needles

What is the probability that a needle will cross a line?

Georges-Louis Leclerc Comte de Buffon

1707-1788

Needles of length l Line separation s x = distance from needle centre to closest line Needle touches/crosses line if Probability density function: function of a variable that gives

probability for variable to take a given value Exponential distribution: p(x) = e-x , for x in range 0 to infinity Uniform distribution: p(x) = 1/L , for x in range 0 to L Normalised over all x:

€

x ≤l2sinθ

€

p(x)dx0

∞

∫ =1

s l

x θ

Probability x lies in range a < x < b is ratio of “areas under the curve” x is distributed uniformly between (0, s/2), θ in range (0, π/2)

p(x) = 2/s, p(θ) = 2/π Variables x and θ independent, so joint probability is

p(x, θ) = 4/(s π)

€

P =p(x)dx

a

b∫p(x)dx

0

∞

∫

Probability of a needle touching a line (l < s) is Drop lots of needles. Probability of needle crossing line is Can estimate π :

€

P =4sπ0

l / 2 sinθ∫0

π / 2∫ dx dθ =

2 lsπ

€

P =Number of needles crossing linesTotal number of needles dropped

€

π =2 lsP