Automated Electron Step Size Optimization in EGS5 Scott Wilderman Department of Nuclear Engineering and Radiological Sciences, University of Michigan

Multiple Scattering Step Sizes in Monte Carlo Electron Transport • Why is there a dependence? Transport mechanics • Optimal step: longest steps that get “right” answer • “Right” answer depends on: – Particular problem tallies -- “granularity” – Error tolerance • EGS5 automated method – Broomstick problem – Energy hinge – Initial step size restrictions

Condensed History Transport Mechanics

Why?

• Larsen convergence: with small enough steps, should get right answer • But speed requires long steps, and step lengths limited by accuracy of transport mechanics model • Anyone can get q, trick is f(x,y,z), and the best we can do is preserve averages (moments) • Even with perfect f(x,y,z), there will be a step-size dependence for any tally that is a function of what’s happening along the actual track

Problem Granularity Dependence of Step Size

EGS5 Step Size Parameters • Dual Hinge implies two step size controls, one for multiple scattering, and one for energy loss • EGS5( a) used fractional energy loss to set steps: – ESTEPE for energy loss hinge – EFRACH for multiple scattering hinge • But had both high E and low E values for each hinge variable – 4 different ESTEPES !

Results: Backscatter and Timing Run Solid 01/01/01/01 01/40/01/01 01/40/01/40 01/40/40/40 40/40/01/40 40/40/40/40 %Backscatter CPU Time .115842

.115842

.1146

.1135

.1306

.1269

.1529

603 632 244 91 80 38 26

Central Axis Depth Dose

How to Proceed?

• Accuracy depends on problem “granularity” – Long steps okay for “bulk” volume tallies – Short steps needed for fine mesh computations • Speed requires energy dependent step sizes: – Small fractional energy loss at high E for accuracy – Larger fractional loss at low E for speed • Base step sizes on some measure of problem geometry granularity (“characteristic dimension”) that can be energy dependent -- solve “broomstick problem”

Broomstick Problem

Broomstick Problem • Very sensitive to step size -- infinitesimally small broomstick, step must be 1 elastic mfp • Determine longest average hinge step which preserves correct average track for given diameter (characteristic dimension) • Measure tracklength as energy deposition • Measure hinge steps as scattering strength

Broomstick Methodology • Run EGS5 on broomstick problem for range of Z, E, hinge sizes vs. broomstick diameters t • Determine max hinge step (K_1) for 1% energy deposition convergence vs. Z, E, t • K_1 varies roughly as t r Z (Z + 1) / A • Interpolate distance in terms of (t r) • Interpolate materials in Z (Z + 1) / A

Broomstick Elements

Broomstick Parameters • Energy range: at .1, .2, .3, .5, ..17 in every decade from 2 keV to 1 TeV • Broomstick space: dimensions in terms of fraction of CSDA range at .1, .2, .3, .5, .7 in every decade from 1E-6 to .50 • Hinge step space: steps in terms of fractional energy loss at .1, .15, .2, .3, .5, .7 in every decade from 1E-4 to .30

Broomstick Results

Broomstick Drawbacks • Broomstick L = CSDA range, so long run times, limiting to 50k histories • Little scattering at high energies, so significant fraction of energy deposition occurs before step sizes are important • Net effect: Step size optimization criteria based on 1% converged energy deposition not stringent enough

Modified Broomstick

Modified Broomstick • Set broomstick length = diameter • Look at emerging from end • Shorter volumes permit more histories • 1% convergence in clearly more strict criteria than 1% convergence in • May be slower than necessary on some problems, but better accuracy on all problems

Modified Broomstick Results

Modified Broomstick Results • Determine maximum fractional energy loss for convergence to 1% in vs. t for all Z and E • Convert from EFRACH to K_1 • Perform linear fit of log(K_1) vs. log(t), all Z and E • New EGS5 subroutine RK1 prepares K_1(E) for all materials, given input t.

Modified Broomstick Results

Tutor4 with EGS5 • 2 MeV electrons on 2 mm of Si EGS4 default EGS4 1% ESTEPE EGS5 30% EFRACH EGS5 1% EFRACH EGS5 2 mm charD Reflected E 1.3% 6.4% 8.1% 7.3% 7.4% Transmitted E 49.2% 61.3% 66.5% 64.4% 64.8%

Energy Hinge Energy hinge t h E_0 E_1 Mono-energetic transport between energy hinges Hinges needed only for accuracy of f(E_0) variables

Energy Hinge All Monte Carlo programs must deal with energy dependence over steps. EGS5 relies on average values to be correct. EGS5 integrals: f(E_0) h + f(E_1) (t – h) h uniformly distributed in D E: z D E / SP(E_0) Can show EGS5 D E (f(E_0) + f(E_1)) / 2

Energy Hinge • PEGS5 compute ESTEPE (E) such that trapezoid rule accurate to within some e (.001 current default) • Checks stopping power (for energy loss) • Checks scattering power (for multiple scattering strength) • Checks on hard collision cross section, mean free path not yet implemented • Typical values for ESTEPE : between 2% and 8%

EGS4 First Step Artifacts g Gamma angle correlated to electron angle after scatter EGS5 g Gamma angle correlated to electron angle before scatter

EGS5 First Step for Primary Electrons usual EGS5 first step, as determined from K_1(Z,E,t) incident electron incident electron limited first step, K_f, determined from K_1(Z,E,t_min) Interface 2 K_f 4 K_f 8 K_f min(16 K_f, K_1(Z,E,t))

Summary • Optimal step selection will always depend on the problem tally granularity, and in particular, on the importance of events taking place on the first step • The new method for setting step sizes in EGS5 based on the “characteristic dimension” of the tally regions usually solves this problem for the user