fluka fluka advanced biasing fluka users meeting cern 31/05/2007

FLUKAFLUKAAdvanced Biasing

FLUKA Users MeetingCERN 31/05/2007

CERN FLUKA Users Meeting 2

Outline Sampling

Random numbers Sampling from discrete & generic distributions Sampling from biased distributions

FLUKA User biasing routines USIMBS UDCDRL SOURCE

Application: n_TOF biasing example


Pseudo-Random number All computers provide a pseudo random number

generator. The FLUKA PRNG is provided by the function

FLRNDM(DUMMY). A 64-bit random number generator returning a double precision number.

All such generators return a sequence of numbers uniformly distributed between [0,1). To avoid overflows sometimes, you need to change to (0,1] then use 1.0-FLRNDM(x)

Uniform and Random are independent concepts. For MC sometimes uniformity is more important than randomness, but not if correlations are important

It is important also that sampled histories are uncorrelated with each other


FLUKA Random number generators Function FLRNDM(DUMMY)

The main FLUKA PRNG. Returns a double precision random number uniformly distributed in the [0,1) interval

Subroutine SFECFE(SFE, CFE)returns a random azimuthal sine and cosine

Subroutine RACO(TXX, TYY, TZZ)returns the cosines of a randomly oriented vector

Subroutine SFLOOD(XXX, YYY, ZZZ, UXX, VXX, WXX)returns a random position and direction on the surface of a sphere of unit radius (generating a uniform and isotropic field inside the sphere)

Subroutine FLNRRN(RGAUSS)returns a random number normally distributed with unit variance


Sampling from a discrete distribution Suppose to have a discrete random variable x, that can

assume values x1,x2,…,xn with probability p1,p2,…,pn

Assume ipi = 1, or normalize it Divide the interval [0,1) in n subintervals with limits

y0=0, y1=p1, y2=p1+p2, …, yn=ipi

Generate a uniform random number Find in which of the i y-intervals it is

WARNING: This can be a time consuming process: Use optimized search techniques like Binary search (look

n_TOF example) Use of equal-probability bins (not recommended when tails

are present) Select the corresponding xi as sampled value Since is uniformly random, we have

P(xi) = P(yi-1 < < yi) = yi – yi-1 = pi


Sampling from a Generic Distribution

Using one random numberUsing one random number Integrate the distribution function, analytically or

numerically, and normalize to 1 to obtain the normalized cumulative distribution

Sample a uniform pseudo-random number Get the desired result by finding the inverse value x=F-

1(), analytically or most often by interpolation (table lookup)

Since is uniformly random, we have

max

min

min

)(

)()( x

x

x

x

dxxf

dxxfxF

b

adxxfaFbFbFaFPbxaP )()()()]()([)(


Example

Practical rule: a distribution can be sampled directly if and Practical rule: a distribution can be sampled directly if and only if its pdf can be integrated and the integral invertedonly if its pdf can be integrated and the integral inverted

t = - λ log (1 – ξ)


Sampling from a Generic DistributionUsing two random numbers (Rejection technique)Using two random numbers (Rejection technique) Choose a constant value C > f(x) for any x Sample two random numbers 1 and 2

1, C2) represents a point on the 2D space

If 2<f(1)/C then X=1

otherwise re-sample 1, 2

The probability that a 1 value is

accepted is f(1)/C for any 1

P(x)dx = P(1=x)dx f(1=x)/C

Since P(1)dx = const,P(x) = const f(x)


Sampling from a biased distribution


Example


Uniform sampling from a steep distribution


User written biasingFLUKA offers the following routines for user-written biasing ubsset.f: User BiaSing SETting

called after reading in the input file and before the first event allows to alter almost any biasing weight on a region-dependent

basis usimbs.f: USer defined IMportance BiaSing

if activated, called at every particle step allows to implement any importance biasing scheme based on

region number and/or phase space coordinates udcdrl.f: User defined DeCay DiRection biasing and Lambda

only for neutrinos emitted in decays: bias the direction of emitted neutrino

Not biasing by itself, but it could be used for generating biased runs

source.f: User written source to sample primary particle properties from distribution in space,

energy, time…


User Defined Importance Biasing Typical problem: Spend a lot of time to write the problem

input, geometry and on the first runs you realize that statistics are not good

First method (and safest) is to introduce region importance biasing. In FLUKA you can introduce it with two ways:

1st Manually slice the geometry and increase the number of regions. Modifying an existing geometry to introduce biasing can be a very cumbersome process

2nd Introduce the “importance biasing” information with a user fortran routine independent of the regions defined in the geometry

Routine: usimbs.f

USer defined IMportance BiaSing

Allows to implement any importance biasing scheme based on region number and/or phase space coordinates


User Defined Importance Biasing Enable the call to USIMBS routine with the BIASING card:

WHAT(1) Particles to be biased WHAT(2) and WHAT(3) ≠ 1.0 (Any value) WHAT(4) Lower bound of region WHAT(5) Upper bound of region WHAT(6) Step SDUM = USER


usimbs.f – RoutineThe routine is called on every

particle step!WARNING: can slow down theexecution speed! Use with caution.

Input: Region information at the

beginning and end of the step X,Y,Z coordinates through the

TRACKR common.Beginning: X, Y, Ztrack(0)End: X, Y, Ztrack(Ntrack)

Particle type and Energy could be used for even more advanced biasing schemes

Output:The routine must return the importance ratio to the new position (end/beginning) in the variable FIMP

SUBROUTINE USIMBS ( MREG, NEWREG, FIMP )

INCLUDE '(DBLPRC)' INCLUDE '(DIMPAR)' INCLUDE '(IOUNIT)'**----------------------------------------------------------------------** ** USer defined IMportance BiaSing: ** ** Created on 02 july 2001 by Alfredo Ferrari & Paola Sala ** Infn - Milan ** ** Last change on 09-jul-01 by Alfredo Ferrari ** ** Input variables: ** Mreg = region at the beginning of the step ** Newreg = region at the end of the step ** (thru common TRACKR): ** Jtrack = particle id. (Paprop numbering) ** Etrack = particle total energy (GeV) ** X,Y,Ztrack(0) = position at the beginning of the step ** X,Y,Ztrack(Ntrack) = position at the end of the step ** ** Output variable: ** Fimp = importance ratio (new position/original one) ** **----------------------------------------------------------------------*e INCLUDE '(TRACKR)'* FIMP = ONEONE RETURN END


usimbs.f – Important Notes The routine has only a relative effect on the weight of the

particle Beam particles can have any weight.

Importance ratio will be limited by WW-THRESh card The Russian Roulette / Splitting will take place at the end

of the step, and not on a fixed region boundary.The biasing position will be a little fuzzy depending on the particle step. This has a visible effect when it is applied to low density materials (i.e. air)

Results will similar but not be the same as with the manual region biasing.

Is a great time saver for complex geometries, as well different biasing schemes

Combined with the particle type and energy could become very powerful


usimbs.f: Simple Example Biasing a factor of 2 every 50 cm on the Z direction from

100cm to 500cm

ZSTART = ZTRACK(0)IF (ZSTART .LT. 100.0D0) THEN

FSTART = ONEONEELSE IF (ZSTART .GT. 500.0D0) THEN

FSTART = TWOTWO ** NINT((500.0D0-100.0D0)/50.0D0)ELSE

FSTART = TWOTWO ** NINT((ZSTART-100.0D0)/50.0D0)ENDIFZEND = ZTRACK(NTRACK)

* Similarly calculate the FEND from ZEND…FIMP = FEND / FSTART

Initial position

Final position

Importance Ratio


usimbs.f: Function example Introduce an importance biasing assuming an exponential

law of attenuation in the R direction exp(- R), for R>1cm

RSTART = SQRT(XTRACK(0)**2 + YTRACK(0)**2 + ZTRACK(0)**2)REND = SQRT(XTRACK(NTRACK)**2 + YTRACK(NTRACK)**2 +

ZTRACK(NTRACK)**2)IF (RSTART .LT. ONEONE) THEN

FSTART = ONEONEELSE

FSTART = EXP(-ALAMBDA * RSTART)ENDIFIF (REND .LT. ONEONE) THEN

FEND = ONEONEELSE

FEND = EXP(-ALAMBDA * REND)ENDIFFIMP = FEND / FSTART


Neutrino Decay Biasing There is a special routine udcdrl.f where one can bias the

direction of the emitted neutrino in decays.

DOUBLE PRECISION FUNCTION UDCDRL( IJ, KPB, NDCY, UDCDRB, VDCDRB, WDCDRB)

Input variables:IJ decaying particleKPB outgoing neutrino

Output variables:U,V,W DCDRB preferential outgoing direction for the neutrinoUDCDRL Lambda for direction biasing (1-cos())

The biasing expression is of the form:

e-(1-cos/)

Useful for neutrino applications like CNGS, Beta Beams… For a fixed direction the LAM-BIAS card with SDUM=DCY-DIRE

could be used instead


Direction Biasing Example (n_TOF) Bias the direction of the neutrino in the pion decay so the

daughter muon to be directed to the exp. area @(-120,0,18500). The direction is given in the lab-frame, and in this example the energies we are dealing are small, so safely we can assume that also in the lab-frame, the neutrino and muon go in opposite directions. The lambda is ¼ wide enough to cover the whole exp area.

Direction in the lab frame

Width [1-cos()]


Source routine The user can prepare a biased source.f routine, when needed

to sample from biased distributions Most often encountered to sample from a biased source energy Common use also to sample from a biased spatial and/or angular

distribution Using a two step approach reading particle histories from

another program or another simulation Of course in such cases it is the user’s responsibility to

ensure that weights are adjusted in statistically correct way.

Tip:Tip: The user should assign the correct weight to the primary particles generated by the source routine WTFLK, but it is recommended that the WEIPRI variable which contains the total weight of the primary particles to be increased by one (instead of the weight of the particle WTFLK). At the end of the run the user should renormalize everything to the correct total weight of the primary particles


Source: Spatial distribution sampling In this example we want to create a biased-uniform

distribution on a disc surface giving emphasis to the center Sampling from 1/sqrt(r)

= 2 FLRNDM(x)SFE=sin()CFE=cos()


Source: Biased energy sampling In this example the energy is given by the tabulated

distribution SPPROT(E). The sampling is divided into two parts below the one using the functions 1/E- and the other as 1/E


Sampling from an Existing Event History Applications:

Linking different codes (e.g., tracking and cascade codes) Sampling a particle distribution given as external data file Two-step approach in order to better sample a certain part

of the geometry (e.g., detectors)

It is important to carefully adjust the source particle weight corresponding to the way how the original source was calculated

1) If the first step of a calculation included biasing the weights have to be adjusted accordingly Weight = original_Weight

2) Source particle corresponds to a single original event Weight = 1.0 (or Weight = original_Weight)

3) Selected source particle is only part of the original event Weight = 1.0 / #Particles_per_Event (or Weight = original_Weight / #Particles_per_Event)


Important for the Final NormalizationTracking Code Example: Ingredients:

Number of particles tracked (and hypothetically absorbed in all machine components) in the tracking simulations: all_Tracked

Number of particles intercepted by the element(s) of interest: sel_Events

Ratio of particles on the considered machine element(s) sel_Events / all_Tracked

This ratio has then to be used to rescale the respective normalization valid for the required application e.g., particle loss rate (LossRate): Nf =

ppb: particles per bunch #b: number of circulating bunches t: beam lifetime

Final normalization:Nf = LossRate x sel_Events/all_Tracked


Sampling from a File - Part 1 using e.g., coordinates generated by tracking codes; 1st loading the

file

* | | Load file given on SDUM of the SOURCE card -> FileName FILEN

WRITE (LUNOUT,*) 'Read source particles from '// FILEN

CALL OAUXFI(FILEN,LUNRDB,'OLD',IERR)

IF ( IERR .GT. 0 )

+ CALL FLABRT('SOURCE','Error opening file '//FILEN)

NIN = 0

10 CONTINUE

READ( LUNRDB, '(A)', ERR=9999, END=20 ) LINE

IF ( LINE(1:1) .EQ. '#' ) GO TO 10

NIN = NIN + 1

IF (NIN.GT.NINMAX) CALL FLABRT('SOURCE','Increase NINMAX')

READ (LINE,*) XIN(NIN), YIN(NIN), ZIN(NIN),

+ UIN(NIN), VIN(NIN)

GOTO 10

20 CONTINUE

WRITE (LUNOUT,*) NIN,' particles loaded'

CLOSE(UNIT=LUNRDB)

Read all coordinates to be sampled from!


Sampling from a File - Part 2 2nd sampling it* | Choose a random particle

RNDSIG = FLRNDM (RNDSIG)

N = NINT(NIN*RNDSIG)+1

IF (N.GT.NIN) N=NIN

* TEST BEAM: uncomment the following line

IJIJ = IJBEAM

WEE = BEAWEI

* Cosines (tx,ty,tz)

TXI = UIN(N)

TYI = VIN(N)

TZI = SQRT ( ONEONE - TXI**2 - TYI**2 )

IF (WBEAM.LT.ZERZER) TZI = -TZI

* Particle coordinates

XXX = XIN(N)

YYY = YIN(N)

ZZZ = ZIN(N)

* Energy and Momentum

POO = PBEAM

EKE = SQRT ( PBEAM**2 + AM (IJBEAM)**2 ) - AM (IJBEAM)

ELKE = LOG (EKE)

WTFLK (NPFLKA) = WEE

Select a random particle

Set the direction cosines

The coordinates

+ the energy and momentum

and the weight


Bibliography

A Third Monte Carlo Sampler( A revision and Extension of Samplers I and II)C.J. Everett and E.D.CashwellLA-9721-MS, UC-32, March 1983

Application: n_TOF Background

FLUKA Users MeetingCERN 31/05/2007


n_TOF Facility

High flux spallation neutron source ~200 m time of flight

Measurements of neutron induced cross-sections ADS applications Astrophysics

Proton Beam CERN-PS: 20 GeV/c, Ip=4 bunches 7 1012 pr / 14.4 s

Neutron Source Features Wide energy range: thermal up to GeV High flux: ~ 6105 n/cm2/7 1012pr @ 200 m Resolution: E/E 210-4 @ 1 keV, 210-3 @ 1 MeV Low background 10-6

FOR MORE INFO...

http://cern.ch/ntof, CERN/LHC/98-02 (EET), CERN/INTC/2000-004, CERN/SPSC 99-8


n_TOF Virtual Tour

Lead Container

Sweeping Magnet

2nd Collimator

Exp. Area


Background Problem June 2001

Background Features 50 larger than simulations Three time components

400 ns “flash” (after the -flash)

20 s – fast neutrons > 16 ms – thermal neutrons

Position depended Strong Left-Right asymmetry Strong ionization signal Not sample related


Background Possible Sources

There were plenty of theories trying to describe the origin: Elements in the neutron Tube

Collimators Escape line

Insufficient concrete shielding in the exp. area Charged Particles deflected from the magnet High energy neutron leaking from the target area …

Challenges: Tiny solid angle Huge attenuation factors


Topview of the Simulated Geometry


1st Attempt: Elements on the neutron tube

Using a two (three) step approach:1. Simulating only the spallation target. Record particle histories

with a modified FLUSCW.F routine when particles enter in the neutron time of flight tube.

2. Filter particles using a condition of vz/v > cos10o, cos5o, cos1o

bias slightly the direction with preference the element of interest, e.g. 2nd collimator, escape line.

3. Full n_TOF simulation, using as source particles from step 2 Extra Biasing used:

Importance biasing with USIMBS.F subroutine. Increasing a factor of x5 for every 50cm of concrete on the shielding upstream of the experimental area and decreasing downstream


USIMBSUsing the USRICALL to set dynamically the filename from the

inputusrini.f

tt2a.inpUSRICALL 1.0 usrbias

The usrbias.dat file defines the Z position of each change in importance, and the importance ratio with the previous Z-section

170.0 5.0175.0 5.0180.0 5.0..200.0 0.2250.0 0.2


USIMBS: Loading the file


USIMBS: Importance ratio


USIMBS: Binary Search


1st) Background from NEL & Collimator

Neutron Background from:Neutron Escape Line Collimator


2nd Attempt: Fast neutron streaming from the spallation area

There are visible paths where the shielding is not enough

Find weak points in geometry with the use of the RAY particle


2nd) Minimum mass calculation Special “simulation” using the RAY particle of FLUKA. RAY is a straight-line trajectory through the geometry. The

program tracks all the objects lying in the given direction calculating a number of various quantities like distance traversed in each material, mass, number of interaction lengths etc…

Using a modified source.f routine a scan was performed on all the possible directions starting from a vertical plane located at Z=+7m (X [-120, 280 cm], Y [-120, 230 cm]), to a vertical plane at Z=+185 m (X [-250, 200 cm], Y [-200, 210 cm]) from the center of the lead target.

500,000 rays were generated scanning the geometry starting from a grid of 1010 on the former plane to a grid of 5050 to the later plane and vice versa.

Recording on a 2D histogram the position for the trajectory were the surface density is minimized.


2nd) Minimum Mass Calculation

Revealed paths with minimum mass of 1.4 kg/cm2

Equivalent to 2 GeV/c energy losses for Minimum ionizing particle Minimum ionizing particles like muons could easily penetrate the

experimental area High energy protons could also generate a background


3rd) Simulation for High Energy neutrons Starting directly from the protons on the lead target Enabling Leading particle biasing and Weight windows

LAM-BIAS, WW-FACTOr, WW-THRESh Using the USIMBS biasing:

Factor of 2 every 100cm around the spallation area Factor of 2 every 10m in the tunnel (and neutron tube) Factor of 5 every 50cm in the concrete shielding

With LOW-BIAS stop neutrons with E<1MeV in the spallation area

No EMF calculation


3rd) Neutron Fluence (per fast neutron)


3rd) Muon Fluence

Simulated muon fluence at the entrance of the exp. area per proton pulse of 7 1012 protons


1st,2nd,3rd) Results

Bad newsBad newsAll studied possible background sources: Fast neutrons, Neutron Escape line, collimator, gave consistent result of 0.01n/cm2/pulse in the experimental area. 50-100 times lower than what was measured

Good NewsGood NewsStrong muon component, that can explain the ionization signal measured with the TLD’s and the strong asymmetry left/right in the tunnel (fast-flash signal)

Last hopeLast hopeFollowing the disappointing results from all previous efforts, last resource was to leave running the simulation as long as possible, searching blindly using brute force. 8 CPU’s


Nice Surprise After ~few weeks x 8 CPU’s of simulation one event

appeared with ~100 times higher importance than the rest.

To identify the origin of the event we needed to record the history of all particles reaching the exp. area. This includes tracking information (which regions traverse), last interaction point, last decay point (if any), parent id, energy etc.


Recording particle history (stuprf.f)Extra USER informationfor each particle ISPARK: array of integersSPAREK: array of floating point


Write to file history info (fluscw.f)Note the change of nameSPAREK SPAUSRISPARK ISPUSR


Identification of the event The event was identified as a ~1 MeV neutron generated in the

concrete surrounding the experimental area, originated from a negative muon being stopped at the experimental area, generated by a pion decay

Neutron generation by muons is a known process, important for underground sites or atmospheric showers at ground level where muons:

Photo-nuclear interactions via virtual photons Negative muon capture with - brought to rest through the weak process

The former process involves muons of (relatively) high energy and has a mean free path of a few hundred meters in earth (not important for n_TOF).

Negative muon capture occurs when - brought to rest is captured forming muonic atom. The capture usually occurs in a relatively loosely bound state and then the muon jumps into progressively deeper levels emitting hard x-rays. Due to the mass being much larger than the electron one, the muon wave functions are significantly overlapping the nucleus, and nuclear capture becomes competitive with decay.

The assumptions used in FLUKA simulation models predict that ~50% of - stopping in concrete undergo nuclear capture. Experimental spectra of neutrons emitted following - captures show an evaporation peak, plus a low intensity tail extending up to several tens of MeV.


Muons arriving at Exp. Area

Blue: Decay Position

Red: Inelastic reaction where the parent of muon was created


Biasing to enhance the - captureKnowing the origin of the problem a more sophisticated biasing

wasintroduced to enhance the muon fluence at the experimental

area.

Keep all previous settings of the high energy neutron biasing Forcing the decay of the parent pions every one meter, to

increase the number of muons arriving at the experimental area

Reducing (Increasing) the mean life of the muons by a factor of 0.01, to enhance the capture probability versus the muon decay.


Muon direction (udcdrl.f)Biasing the direction of the neutrino from the pion decay such as the daughter muon to go in the direction of the experimental area with the routine UDCDRL

Direction in the lab frame

Width [1-cos()]


Neutron Fluence at Exp. Area

Computed neutron fluence (n/cm2) inside the experimentalarea (topview) for a standard pulse of 71012 protons


Background neutron sources


Neutron Time Distribution


Possible Remedies

Stop the parent pions Iron & Concrete shielding near the target area

Stop the muons Place iron shielding near the target area Iron shielding after the sweeping magnet

Absorb the neutron in the experimental area Use borated polyethylene lining


The 3m “muon” wall The simulation results have clearly demonstrated the

ineffectiveness of a possible wall close to the target area: 50% of parent pions are still in the tube at the exit of the

target shielding 10% of muons/parent pions are still in the tube as far as

60 m from the target Therefore a suitable shielding should be located where

the fraction of muons/pions in the pipe is minimal or just after the sweeping magnet

3 m of Iron will lower the muon energy of 3.5 GeV


Muon wall effectiveness


Final Reduction

fluka fluka advanced biasing fluka users meeting cern 31/05/2007

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