Monte Carlo simulations

Transcription

1 MC simulations Monte Carlo simulations Eirik Malinen Simulations of stochastic processes Interactions are stochastic: the path of a single ioniing particle may not be predicted Interactions are quantified by probabilities (cross sections) Random numbers and cross sections may be used to simulate single events Better than analytical methods, but requires CPUtime MC simulations 2 Photons give rise to electrons and vice versa; coupled energy transport Analytic methods are suboptimal for: Modeling of scatter Generating electron- and photon spectra Modeling interface effects Calculating energy dependence of dosimeter response MC simulation - example Use random numbers to estimate π Ratio of areas: π/4 Sample points so that x 2 + y 2 Here, Ratio = 787/ =.787 π = 3.48

2 Random walk Photon MC γ Description of photons and their energy depositions:. Position 2. Pathlength 3. Interaction 4. Secondary photon? Depends on photon energy medium y =y max R 2 y max : Draw two random numbers; <R< Cross section of field: x max y max x =x max R x max Photon pathlength Photon attenuation: N N e µ = Describes the number of photons at depth is a type of frequency distribution: f () = Ce = µ f ()d = µ Expected pathlength: /µ,! f ()d = C = µ Photon interaction point At what depth does an event (interaction) take place? Need a cumulative distribution with respect to depth: F() = f (')d' = µ e F(): probability that a photon has interacted between and µ ' d' = e µ Depth 2

3 Photon interaction point 2 Draw a random number R what is the corresponding pathlenght for this photon? F( ) = R = e ln( R ) = µ µ Example: R =.6 =8.3 cm e µ = R Pathlength sampling Sampled pathlength of photons ( MeV): Number of photons Depth Depth (cm) Interaction sampling What interaction occur at given depth? Total probability: µ = τ + σr + σ + κ Probability for e.g. Compton scatter: σ p Compton = µ Draw random number: R Compton scatter R 2 Pair production Sampling of scattered photons Is the photon scattered? In what direction? Angular distribution follows Compton cross section: Compton distribution has no analytic cumulative Must draw two random numbers Probability Normalied scattering angle accepted rejected τ/µ σ R /µ σ/µ κ/µ Scattering angle 3

4 Sampling Compton scatter Frequency Scattering angle Compton Sampled, n= 3 Sampled, n= 5 n= number of samples Electron MC Simulations of electrons and positrons are more complicated A.5 MeV electron interacts ~ times when slowing down to kev in aluminium! Number of calculations Macroscopic Monte Carlo: Evaluate the electron after a given steplength several interactions included in one step (simulations of every interaction: microscopic Monte Carlo) Electron MC 2 Relative energy loss per step, η: Tk T + k T η = = Tk Tk T k : electron energy in interaction point k η is set by user may be sampled: η = ηr Tk Step length: s = η dt dx k,k+ Electron walk e - 4

5 Electron tree Electron MC, example 8 MeV electrons in water/bone Relative dose water water + bone (2-4 cm) Depth, cm EGSnrc EGSnrc is a widely used MC code for e.g. simulations of photon- and electron beams Complicated programming, but simplified, userfriendly interface available: egs_inpr EGSnrc/DOSRZ DOSRZ: MC in cylindrical geometry C C=, P= P C=2, P= C=3, P=2 5

6 DOSRZ The user sets: Phantom geometry Radiation type- and energy (or spectrum) Source (parallel beam, point source,...) Number of histories, i.e. number of particles Some MC parameters Some important parameters ECUT: lower limit for electron transport (includes rest mass of.5 MeV) PCUT: lower limit for photon transport AE: lower limit for generation of electrons AP: lower limit for generation of photons AE and AP is medium specific and must be set in PEGS (see below) Directories All codes found here Manuals Windows interfaces Medium data User area If DOSRZ is executed, data are place here User made medium files should be placed here Title Code Input file File of absorbers Electron and photon spectra DOSRZ etc. 6

7 long : most relevant iwatch on : all interactions are written to file (NB!) Number of histories Initial random numbers Maximum number of regions What should be calculated Arrangement of regions Distance from source to first slab Number of slabs and thickness Cylinders and radius Medium in each region Type of particle Beam type Beam radius (cm) Particle energy Provide spectrum (found under HENHOUSE\ spectra ) 7

8 Methods for calulating pair production and brehmsstrahlung Force photon to interact Stop electron transport if range is short PCUT and ECUT Photon interactions Output - *.egsgph (with IWATCH=graph) p q r x y E Output - *.egslst p: particle q: charge r: region x: x-coord E=energy 8

9 Output - *.egslst Output - *.egslst Cylinders Planes Plane number Cylinder number PEGS Preprocessor for EGS Medium definition is performed in PEGS Have to set AE og AP, in addition to UE og UP (upper limit for for electron- and photon energy) PEGS Give medium composition Provide name of subbstance Density correction file Lower limit for generation of secondary photons (AP) or electrons (AE) File name 9