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 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( 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 %Backscatter CPU Time

Solid .115842 603

01/01/01/01 .115842 632

01/40/01/01 .1146 244

01/40/01/40 .1135 91

01/40/40/40 .1306 80

40/40/01/40 .1269 38

40/40/40/40 .1529 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 Z (Z + 1) / A• Interpolate distance in terms of (t • 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 <r> emerging from end• Shorter volumes permit more histories• 1% convergence in <r> clearly more strict

criteria than 1% convergence in <t>• 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 <r> 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

Reflected E Transmitted E

EGS4 default 1.3% 49.2%

EGS4 1% ESTEPE 6.4% 61.3%

EGS5 30% EFRACH 8.1% 66.5%

EGS5 1% EFRACH 7.3% 64.4%

EGS5 2 mm charD 7.4% 64.8%

Energy Hinge

E_0

Energy hinge

ht

E_1

Mono-energetic transport between energy hingesHinges needed only for accuracy of f(E_0) variables

Energy Hinge

EGS5 integrals: f(E_0) h + f(E_1) (t – h)

h uniformly distributed in E: E / SP(E_0)

All Monte Carlo programs must deal with energydependence over steps. EGS5 relies on average values to be correct.

E (f(E_0) + f(E_1)) / 2Can show EGS5

Energy Hinge

• PEGS5 compute ESTEPE(E) such that trapezoid rule accurate to within some 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%

First Step Artifacts

EGS4

EGS5

Gamma angle correlated to electron angle after scatter

Gamma angle correlated to electron angle before scatter

EGS5 First Step for Primary Electrons

Interface

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)

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