Modelling and simulation methods of particle transport

Henrik SjöstrandHenrik Sjöstrand henrik.sjostrand@phycics.uu.se

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Why this course?Why this course?

kill i i h fi ld• Skills important in many research fields – Detector development, space science, accelerator physics, nuclear physics, medicine, particle physics

• Skills important outside the academic worldp– i.e., nuclear industry or medical technology

• Provide introduction to a number of differentProvide introduction to a number of different tools

Ability to choose independent of supervisors– Ability to choose independent of supervisors• Better value if attending specialized courses

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The course focus on the modelling of physical systems and the simulation of flux and interactions of neutrons protonsthe simulation of flux and interactions of neutrons, protons, (ions) and gammas. However other particles can also be investigated Questions answered• How is flux distributed (particle‐ and space physics)?• What is the reaction rate (detector development, accelerator target design)

Where is the energy deposited (cancer therapy dosimetry)– Where is the energy deposited (cancer therapy, dosimetry)• What is the attenuation of a material (shielding, imaging)Questions not fully answeredQuestions not fully answered• What is the value of keff (nuclear criticality safety, Fission and fusion reactor design)g )

• What is the isotopic content after irradiation(radio nuclide production)

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Detector developmentDetector development

dInstruments measure one quantity and users try to estimate another. E.g. a photon detector is giving the pulse height spectrum PHS The detector users arespectrum, PHS. The detector users are generally interested in estimating the original h t t Ephoton energy spectrum, Eγ.

Let us follow the energy spectrum “through the gy p gdetector” to the pulse height spectrum.

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Let us assume a mono energetic γ source which PH

P)

energetic γ‐source which will result in a detector (i.e. NaI) response ba

bility (P

( ) pdepending on the energy, Eγ.

eight P

rob

Pulse Height

Pulse

He

V

To read more, i.e., Radiation

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Detection and Measurements, F, Knoll, chapter 10.

Many mono energetic γ‐source results in a Pulse Height Probability (PHP) distributionPulse Height Probability (PHP) distribution.

The PHP‐ distribution is the detectors responsefunction DRF(x,q), which is

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the probability of the measured parameter q given the parameter x as input . In this case q = pulse height and x = gamma energy.

NaI(Tl) detector measuring the emitted spectrum from a 24Na What does folding with DRF mean?emitted spectrum from a Na source.

The gamma lines of 24Na are 1.369 Source Detector Response

gand 2.754 MeVrespectively.

DRF * =

I( ) E

MeasuredPulse Height distribution g(q)

I(x)= EγConclusionThe DRF has to be known to reconstruct the I(x). h h b d l d

,I x DRF x q x g q

The DRF can either be measured or simulated. In most cases it is impossible to measure the DRF for all x and q with sufficient statistics.→ ,q g q

→

Modelling and simulations are necessaryExample in Knoll (3rd): Chapter 18 (MPH) Spectrum analysis (p.704)

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Modelling and simulations are necessary.

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Concluding:Concluding:

Modelling and simulations are necessary• to determine the DRF and reconstruct theto determine the DRF and reconstruct the sought physical quantityi d i d i i i f d• in design and optimization of your detector system

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Cancer therapyCancer therapyX‐rays Energy deposition Protons

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ICARUS Neutrino Oscillation

BB – Beta‐beam Task

ICARUS Neutrino Oscillation Experiment

LHC – The Large n_TOF – Neutrons Time of Flight

gHadron Collider

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Building of actinides in a reactor

, ,

Bateman equation

, ,d tr d trii j i j i j i j i jj i j j j i

dN E t E dE N E t E dE Ndt

dt

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Course goalsCourse goals

• model and simulate experimental systems with different tools for particle transport p p

• choose the correct tools for a given situation, motivate the choice and identify the pros andmotivate the choice and identify the pros and cons

• evaluate the result from the simulation and describe methods to verify the tooldescribe methods to verify the tool

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Course structureCourse structureI t d t l t 1 E h t l i i t d d• Introductory lectures – me– Infrastructure, general theory

on simulations

1. Each tool is introduced with lectures and tutorials

2. An assignment is handed

The different simulations tools • PENELOPE –Richard Brenner

gout

– The assignments have a common theme (detector PENELOPE Richard Brenner

• FLUKA – Mathias Lantz • GEANT4 – Anders

Hj l

development)3. Possibility to work with

the assignment supervised Hjalmarsson– Student instructions

• MCNP/MCNPx –Sean

g pin computer labs (4102 and 6K1101)

4 You hand in theMCNP/MCNPx Sean Conroy

4. You hand in the assignment, which then is marked

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Course infrastructureCourse infrastructure

S d l• Studentportalen– Schedule– Contact to the other students– User manuals

• Computer cluster:– Login: lillekis.tsl.uu.se– User manual at studentportalen

• Bookoo– Exploring Monte Carlo Methods, e‐Book in the UU library, link at studentportaleny p

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Probabilistic and deterministic calculation schemescalculation schemes

Question: describe a particle populations.

The probabilistic way The deterministic way

Question: describe a particle populations.

The probabilistic way• Through a Monte Carlo

method the particle track

The deterministic way• The Boltzmann balance

equation is solved using a pand interactions are simulated. From sampling

q gset of algebraic equations.

• The particle flux is the average behavior is deduced Th ti l fl i

determined for the entire space

• The particle flux is determined for a limited space

• Normally simplifications necessary → systematic errorsspace

• Statistical errorserrors

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Kinetic theory particle densityKinetic theory – particle density

• The density depends on the spatial location, the velocity and the timey

• Therefore to accurately describe a population

( , , )n r v t

the particle density depends on 7 distinct i blvariables

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The phase spaceThe phase spaceM t i l ft i t i• Materials are often isotropic– The cross sections are only dependent on the energy of the

incident neutron and not on their direction– The velocity and the direction are differentiated:

vv

• This is the so called particle phase and given in spherical‐d

v

coordinates• The angular density is noted as :

18( , , , )n r v t

Particle fluxParticle flux

• The scalar angular flux is defined as the particle density times the velocity:p y y

r v t v n r v t , , , , , ,r v t v n r v t

• We can define the integrated flux:

2, , , , ,r v t r v t d 19

Reaction rateReaction rate

• Reaction rate is the number of particles undergoing a reaction [s‐1m‐3]g g [ ]

, , , , , , , , ,R r v t r v t r v t , , , , , , , , ,R r v t r v t r v t

• When the medium is isotropic:

, , , , , ,R r v t r v t r v t

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Neutral transport equationNeutral transport equationChapter 9 on a slightly different

ˆ1 ( , , , ) ˆ( , , ) ( , , , )ˆ ˆ[ · ( , , , )] tE t

v tdiv E t E t E t

r Ω r r ΩΩ Ωr

form

4 0ˆ ˆ ˆ( , , , ) ( , , , ˆ( , , ,) )s s E td dE E E t E t

r Ω Ω r ΩΩ r

- dissaperance due tch oa nge re i al ctions n the flux = e k ge a a+ inscattering from other directions and ene + sourcergies term

•Solving the equation gives the particle flux for all r and Ω.

h d l h b l d d21

•For charged particles a Force term has to be included

Deterministic = Solve the transport equation using algebraic equations

M t l ti l ti F i l tiMost cases no analytic solution. For numeric solution variables must be discretized.• Spatial meshing (FEM, FD) – 109p g ( , )• Angular discretization 102 ‐ 103• The energy (e.g the cross‐section) are discretized ‐105• Total 1017 equations for each time step ‐ impossible →

Simplifications necessarySimplifications necessary• Homogenization, 1D‐2D, physical modelling, coarser grid,

multigroup approximation

→ BIAS /systematic error22

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Probabilistic approach – focus on neutral particles

• The particle track and interactions are simulated. From sampling the average p g gbehavior is deduced.

• Exact 3DParticle born

• Exact 3D• Continuous energy

(n,n´)Tally – Energy

(n,np)(n,n´)

Energydeposited

• Correct physics (n,→γ)

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Monte Carlo methodsMonte Carlo methods

• By random sampling determine a physical quantityq y

• Theory developed by Jaco Bernouille in 1689M h d fi d f i l L• Method first used for particle transport at Los Alamos– The name is a reference to the Monte Carlo Casino in Monaco

• Early example: Buffons needle (1777)

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MC ‐mathematics

We all know that the concept of limitWe all know that the concept of limit, except in the mind of a dimwit, is used to make senseof things small or immenseof things small or immense,taming zero and the infinite.

(Chapter 2 with a slightly different approach)(Chapter 2, with a slightly different approach)

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Pseudorandom number generatorsg

Monte Carlo needs numbers quite randomBut practicality has mostly banned themBut practicality has mostly banned themSo in order to scoreWe need do no more Than generate them pseudo randomThan generate them pseudo random

(Chapter 3)

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Modelling the problemModelling the problem

fi h• Define the source• Define the

Particle born

(n,n´)

geometry– Including

( )

(n np)Including materials and cross‐sections

(n,np)(n,n´)

T ll• Define the physics to be used

(n,→γ)Tally‐Neutron flux through surfacaephysics to be used

• Define the Tally

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MC‐simulations of neutral particle Particle history1 Source

Chapter 10

1. Source – energy, position, direction

form PDF2 Distance to next

Particle born

(n,n´)2. Distance to next interaction is sampled

3. Type of interaction is l d

( )

(n np)sampled 4. Secondary particles are

banked

(n,np)(n,n´)

T ll– Goto (1) when first orginal

particle is terminated5 All paricles are terminated

(n,→γ)Tally‐Neutron flux through surfacae

5. All paricles are terminated6. The contribution zi to the

Tally of interest is calc lated Analogue simulation e g wecalculated

7. Goto (1) new history 29

Analogue simulation, e.g. we simulate the exact physical process

Tally = Estimatorsy

Th lli i ti t• The collision estimator– Flux= interactions/ (Volume*total macroscopic cross‐section)section)

• Track length estimator– Flux= tracklength / volumeFlux= tracklength / volume

• Surface estimators• Flux point estimator• Flux point estimator

– next event estimator• Energy deposition estimator• Energy deposition estimator• What ever estimator

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The Monte Carlo particle transport algorithm Emission:

Position, energy direction

Track lengthlength estimator

Can be avoided

CollisionIsotope

interactionBoundary

Interface of media

(Estimator alt 1)interaction

Reflection, symmetry

(Estimator alt 1)

ScatteringAngle, energy

symmetry. Be careful! Leakage Absorption

Can be avoided

31End of history (estimator alt 2)

Variance reduction • Chapter 10.8 and 10.12 and

(Chapter 5) but also (eq. • Truncation methods

– I.e. energy and space cutoff2.54)

• Deep penetration problemsA l t h i i ht

gy p– Physics simplification

• Modified sampling method• Analogue techniques might

not be sufficient • Use non analogue

– Implicit capture– Interaction forcing

• Population controlUse non analogue techniques

• Particle weights (4.3 in

Population control– Splitting or Russian Roulette

• Partially deterministic methodsMCNP primer) – Circumvent part of the random

walk– Point TallysPoint Tallys

• Source Biasing– Normally done by the user. Easy to

get wrong32

Partially deterministic method:Point estimatorPoint estimator

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Variance reductionVariance reductionG l C tiGoals• Maximize the FOM• Keep the number of tracks

Cautions• Do not get to elaborate to

quicklyKeep the number of tracks constant from the source region to the estimator

i

quickly• use new techniques first on

simple problemsregion• Number of tracks should

decrease gracefully in

s p e p ob e s• Study the manuals• Balance user time withdecrease gracefully in

unimportant region. (Factor of 2‐4 between neighboring ll )

Balance user time with computer time

• Study the outputcells)

• Achieve adequate sampling of all cells

y p• Importance should not

change with more than a of all cellsfactor 8 between neighboring cells 34

Accuracy vs precisionAccuracy vs precision

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Accuracy Precision• Code errors

– Wrong physical models– Poor random number

• Number of histories contributing to the Tally

generators– Coding errors– Benchmarks against other

d

• (Round of errors)

codes or measurements• Physical uncertainties

– i.e. cross‐sections– Error propagation– Benchmarks against

measurements• User errors

– Bad geometry– Misunderstanding of the

h i i l dphysics implemented– Variance reduction techniques

used in a bad wayReview your input understand– Review your input, understand the option you use and validate against analogue simulations 36

FinallyFinally

With the development of inexpensive computers and computer clusters and with the p pavailability of powerful general‐purpose MC‐codes MC has become the method of choice forcodes, MC has become the method of choice for high accuracy and reference transport analysis.

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VVQVVQ

• Verification• Validation• Qualification

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Different MC CodesNeutron and gamma (some limit in energy)• USA: MCNP5, MORSE,COG, KENO, TART, MERCURY, SCALE, , , , , ,• FRANCE: TRIPOLI ‐4• UK: MACBEND, MONKUK: MACBEND, MONKBatesman equation• FISPACT SCALE SERPENT FLUKAFISPACT, SCALE, SERPENT, FLUKANeutral and charged particles• EU: FLUKA GEANT• EU: FLUKA GEANT• HETC, LAHET, MCNPXEl t ti dElectromagnetic cascade• USA: EGS 4, SRIM,ETRAN, MCNPXPENELOPE• PENELOPE

• TRIPOLI 4 39

Applications deterministicApplications ‐ deterministic

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• FLUKA can simulate with high accuracy the interaction and propagation in matter of p p gabout 60 different particles, including photons and electrons from 1 keV to thousands of TeVand electrons from 1 keV to thousands of TeV, neutrinos, muons of any energy, hadrons of energies up to 20 TeV (up to 10 PeV by linkingenergies up to 20 TeV (up to 10 PeV by linking FLUKA with the DPMJET code) and all the corresponding antiparticles, neutrons down to thermal energies and heavy ions.

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DeterministicDeterministic

Some special cases• No external forces: neutral particlesNo external forces: neutral particles• Thin medias ‐ no collisions • Isotropic medias • I e in plasma physics the Fokker PlancksI.e in plasma physics the Fokker Plancksequation

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