geant4 tracking test (d. lunesu)1 daniela lunesu, stefano magni dario menasce infn milano geant4...

Geant4 Tracking Test (D. Lunesu) 1

Daniela Lunesu, Stefano MagniDario Menasce

INFN Milano

GEANT4 TRACINGTESTs


The process of tracing a charged particle in a magnetic field is controlled, in Geant4, by several user-adjustable parameters.

This study aims at checking which is the best set of those parameters to provide both:

• a tracing resolution much better than the reconstruction resolution;• a reasonable computing time.

Our approach to understand these tracing precision problem is twofold:

• Test A: verify that tracing an electron (with physics processes with material points switched off) produces accurate results when compared to an analytical calculation.• Test B: check how accurately we get back to the starting point (tracing from left-to-right and then backward) as a function of the Geant4 parameters that basically control the tracing process.


Geometry used in the test:

Intersection points

The electron is traced from a

starting point up to the end of

the world box surrounding the

pixel detector. It’s trajectory is

then simply reversed and the

intersection of the forward and

backward trajectories with the

first pixel plane is computed.

We also compute the returning

point using an analytical

prediction.X

Y

Z Fig 1


• Number of pixel planes: 21

• Dimension of each plane: 1x100x100 cm3

• Distance between planes: 5 cm

• Fiducial volume surrounding the detector: 300x200x300 cm3


• Chord: max distance between the trajectory arc and the

straight line connecting the two arc ending points.

• OneStep: accuracy for the endpoint of ‘ordinary’ integration

steps, which do not intersect a volume boundary.

• Intersection: accuracy with which the intersection with a volume

is calculated.

Chord

Definition of the parameters used in test: test is made with several values of three G4 parameters that control the tracing:

Fig 2


This test is designed to verify the agreement between the analytically computed trajectory and the Geant4 traced trajectory.

• Test particle: an electron (kinetic energy = 1GeV), with physics processes switched off, only transport in magnetic field.

• Uniform Magnetic Field: 1 T

• The electron is traced from left to right and from right to left to measure

the amount of accumulation of rounding-off errors during the tracing

process. • The starting point is always in the same location for all events.• The difference between the intersection of the backward trajectory and

the corresponding analytical prediction at the first pixel plane is plotted

for different values of Chord and Intersection

(we will show later what happens under different conditions)

Test A


• The analytical track is calculated solving the differential equations:

xy

yx

c

c

-

m

eBc

• Solution:

tm

eB

eB

p

eB

pyty

tm

eB

eB

pxtx

cos)(

sin)(

0

0

field magneticB

chargeelectron e

masselectron m

momentumelectron p

position initialy ,x 00


va, vr

yayr

Starting point

x

yMagnetic field ( 1T )

In this picture we define the quantities used in the following plots:• va: the y coordinate computed analytically at the x of the first pixel plane in the forward direction• vr: the y coordinate computed analytically at the x of the first pixel plane in the backward direction (always the same as va)• ya: the y coordinated traced in the forward direction by GEANT4 (with a specified set of Chord, Intercept and Step values.• yr: the y coordinated traced in the backward direction by GEANT4, thus potentially accumulating rounding-off errors

Fig 3


We observe, as expected, that the difference va-vr (computed analytically in our program) turns out exactly zero (plot 4a)The ya coordinate, always traced from the same starting pointand same set of control parameters (see below) for all 244 events, not always turns out at the same value: we do NOTunderstand this behavior (plot 4c).

Chord = 0.099 mm Intercept = 0.0047 mm Step = 0.1 mm

N.b: units of measure is mm in this and all subsequent plots

a b

c

Starting point was y=120 mm for all 244 event in this plot

Fig 4


Increasing the accuracy of the control parameters, the ya point turns out to be always the same (118.6 mm, plot 5c)

Chord = 0.01 mm Intercept = 0.001 mm Step = 0.1 mm

a b

c

This behavior is totally unexpected, since the tracing is supposed to be a deterministic calculation: starting from the same point we should always reach the same intersection(at different values changing the control parameters, but keeping those fixed, the tracedintersection should always be the same)

Fig 5


We have generated a large sample (106 events) with a mesh of different values for the parameters that control the tracing in GEANT4. We have then measured the difference between intersection points of the trajectory with the first pixel plane for the forward track and for the backward track.

Values we used are:

Chord = 0.01, 0.012, 0.013, …, 0.099 mmIntercept = 0.0010, 0.0012, 0.0013, …, 0.0099 mmStep = 0.1, 0.2, 0.3, …, 1. mm

In the first plot of the next page, we plot the difference between an intersection point computed analytically and the same point reached after a complete tracing. This is done for the mesh of :• Chord vs Intercept• Chord vs Step • Intercept vs Step

1 2 3 ... 72


Yt

Yc

Intersec

tion

Chord

Ranges: 0.001 mm Intersection 0.01 mm in step of 0.0001 mm

0.01 mm Chord 0.1 mm in step of 0.001 mm

The vertical axis of this

scatter plot is the mean value

of the following quantity:

• Yc is the intersection

of the backward trajectory,

with the first pixel plane,

computed analytically

• Yt is this same intersection

of the backward trajectory

obtained after a complete

tracing.

Y

Y=Yc-Yt

X of first pixel plane

Fig 6


Intersec

tion

Chord

Y

What we observe:

• Y is almost independent from Intersection, except for particular values of Chord (see spikes in fig.4)

• Y smoothly depends on Chord. Chord = 0.01 mm Y ~ 0.2 m Chord = 0.1 mm Y ~ 3.0 m

• The value of Chord needed to achieve a tracing precision below 1 m is 30 m (red dotted line in figure)

• We really do not understand the weird behavior of this correlation distribution. It seems that for particular values of Chord the tracing precision degrades dramatically and a strong dependency from the Intersection parameter ensues.

Fig 7


In this picture one can notice

the discrepancy between the

forward and the backward

trajectory

Fig 8


Yf

Yb

Intersec

tion

Chord

Y

The vertical axis of this

scatter plot is the mean value

of the following quantity:

• Yf is the intersection

of the forward trajectory

with the first pixel plane

• Yb is the intersection of

the backward trajectory

computed after a complete

forward-backward tracing.

Y=Yf-Yb

X of first pixel planeRanges:

0.001 mm Intersection 0.01 mm in step of 0.0001 mm


Fig 9


StepChord

Y

Ranges: 0.1 mm Step 0.9 mm in step of 0.1 mm


Fig 10

This plot shows the tracing accuracy (as defined inprevious pages) as a function of both Step and

Chord.

As can be seen, there is no dependency on Step, the only variation being due to Chord.

Again, though, there are specific values (marked in the figure by a red arrow) where particular values of Chord give rise to ananomalously large value of the Y quantity.


Step Inter

cept

Y

Ranges:

Fig 11

This plot shows the tracing accuracy (as defined inprevious pages) as a function of both Step and

Intercept.

As expected, there is no dependency on bothStep or Intercept: this is expected since this plot is just another view of the precedingones.

The big jump corresponds to the already observed anomalous behavior of the Yquantity for specific values of one of the other parameters.

0.001 mm Intersection 0.01 mm in step of 0.0001 mm

0.1 mm Step 0.9 mm in step of 0.1 mm


CONCLUSIONS

• We do not understand why, with all parameters fixed to a particular set of values, the intersection of a track with the first pixel plane does not always give the same number (see fig. 4c, page 9).

• The tracing accuracy (defined as the average value of the difference between the intersection of a track computed in the forward direction with that computed in the backward direction) seems to depend only on the Chord parameter and NOT on Intersection or Step, except for particular values of Chord where a strong dependence on Intersection ensues, see fig. 6, page 12.

• A tracing accuracy better than 1 m is obtained with Chord <10 m regardless of Intersection or Step (fig. 7 page 13)

• Still missing in this study is the CPU time dependence for this set of parameters

• We are in close contact with the Geant4 development team to get help in resolving these issues

geant4 tracking test (d. lunesu)1 daniela lunesu, stefano magni dario menasce infn milano geant4...

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