Simplified Calorimeter: shower moments Andrea Dotti (andrea.dotti

CERN-LCGAPP-2012-01
April 25, 2012
Version 1.0
Simplified Calorimeter:
shower moments
Andrea Dotti ([email protected])
Contents
1 Introduction
1
2 Definition of Shower Moments
2.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
4
3 Results
3.1 Energy deposit moments . . . . . . . . . . . . . . .
3.2 Moments related to the shower profile . . . . . . . .
3.3 Moments related to the core and halo of the shower
3.4 Behavior of Moments . . . . . . . . . . . . . . . . .
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4 Conclusions
22
A Appendix: Technical implementation details
24
1
Introduction
In this note we describe the use of shower moments to present in a compact
way properties of showers induced by high energy primaries in calorimeters.
This work has been developed in the context of the SimplifiedCalorimeter
testing suite developed for the Geant4 simulation toolkit. The testing suite
implements different types of calorimeter technologies in a simplified setup
(no read-out, large dimensions). The materials and segmentations used are
similar to the calorimeters of LHC as well as to some calorimeters from past
experiments (HERA) as well as future calorimeters currently in the R&D
1
phase (CALICE). The testing suite is used to validate and compare the
different models available in Geant4 (with data when available), it is also
used as a regression testing suite to validate new developments.
The typical calorimetric observables (response, resolution, shower shapes)
are measured and analyzed.
In this note we concentrate our attention on a new approach to measure
the shower dimensions: we describe the implementation of the shower moments, with details on the presented algorithm. Results will be present with
the goal to show the sensitivity of the method. We will not go in the details
of the comparison of different physics lists and different versions of Geant4
.
The appendix contains details of the developed code. It is intended to
serve as a guide to extend the application to include new shower moments.
2
Definition of Shower Moments
The detailed study of hadron interaction with calorimeters can be studied
in terms of shower shapes. Usually the longitudinal and lateral profile are
calculated as fraction of energy in different sections of the calorimeter (shower
profile).
Detailed analysis of the shower profiles has been carried out for different
physics lists in the past years and results have been presented in [1] and [2].
The general conclusions is that the effect of intra nuclear cascade models,
in particular with the Bertini model, is to make showers wider, in the right
direction with respect to test-beam data. The Fritiof model used in FTF BIC
and FTFP BERT is a promising alternative to QGSP BERT , in particular for the
longitudinal profile of protons.
The definition of shower moments has been inspired by the ATLAS Local Hadron Calibration method [3]. This method aims to calibrate, at the
hadronic scale, the clusters (groups of calorimeter cells) recognizing the fraction of electromagnetic energy (energy deposits π 0 ) in a hadron induced
shower. After reconstructing the cluster (trying to minimize the influence
of the electronic noise), it is classified as electromagnetic or hadronic. Several weights are applied depending on the magnitude of different classifiers
(for example: energy density in the cells, shower depth in the calorimeter)
to correct for non-compensation, cracks and leakages.
In the context of this work we are not interested in the calibration of
2
hNumSubSteps
Number of trasnversed pseudo-cells
h
dE/dx of steps in 5 cm voxels
Entries
Mean
RMS
Underflow
Overflow
Mean
1.069
RMS
0.2939
a.u.
Entries 4.440784e+07
79805
16.68
46.95
0
984
107
104
106
103
105
104
102
103
10
102
1
10
0
2
4
6
8
10
12
14
Nvoxels
0
50
100
150
200
250
300
350
400 450 500
dE/dx (MeV/cm)
Figure 1: Left: distribution of the number of voxels spanned by a G4Step.
Right: dE/dx deposits in the voxels.
the clusters, but we want to re-use some methodology to summarize, in few
numbers, the characteristics and shape of the showers. Since we are dealing
with a simplified calorimeter we do not have to deal with experimental issues
like the contribution of noise, cracks and leakage.
We will show that the conclusions we make using this approach is equivalent to the ones obtained with the classical approach of the shower profiles.
The moments method, has few benefits:
• It leads to a simplified classification of each shower with its shape defined by a (relatively small) set of parameters
• The shower moments parameters, calculated for each shower, allow for
a direct correlation with quantities of interest at the microscopic level
(number and type of hadronic interactions in the shower, fraction of
energy carried by the different particle species, see section 7)
• The moments are only weakly dependent on the segmentation used to
calculate moments themselves
3
2.1
Algorithm
Since no electronic noise is simulated, we are able to define the cluster as the
set of all cells with E > 0, the energy of the cluster is thus simply defined as:
X
Ecluster =
Ec
(1)
c∈{cells}
This, by definition, is the total energy deposited by the impinging hadron in
the calorimeter. We are left with the need of defining the cells. Our setup
is already equipped with a ReadOut Geometry used to calculate the shower
profile. This is defined dividing the entire calorimeter in as a series of concentric tubes. This segmentation is particular useful, for its phi-symmetry,
to calculate the shower profiles, but we prefer to add a second segmentation
made of boxes (voxels) that define a three dimensional mesh: an envelope of
the calorimeter. By default the voxels have a size of 5 × 5 × 5 cm3 .
hEdep_xz
Energy Deposit_xz
Entries
Mean x
Mean y
RMS x
RMS y
80
60
10227
-71.28
-0.8567
16.86
1000
9.823
40
800
20
Entries
Mean x
Mean y
RMS x
RMS y
80
60
11381
-66.78
1800
-1.05
18.73
1600
10.24
1400
40
1200
20
600
0
-20
400
-40
1000
0
800
-20
600
-40
200
-60
-80
hEdep_xz
Energy Deposit_xz
-100
-50
0
50
Energy Deposit_yz
100
0
hEdep_yz
Entries
Mean x
Mean y
RMS x
RMS y
80
60
10227
-71.28
-1.189
800
16.86
10.45
400
-60
-80
200
-100
-50
0
50
Energy Deposit_yz
100
0
hEdep_yz
Entries
Mean x
Mean y
RMS x
RMS y
80
60
11381
-66.78
1.733
1000
18.73
10.52
700
40
40
800
600
20
20
500
0
400
-20
300
-40
200
-60
-80
100
-100
-50
0
50
100
0
hEdep_yx
Energy Deposit_yx
Entries
Mean x
Mean y
RMS x
RMS y
80
60
10227
-0.8567
-1.189
2500
9.823
10.45
40
2000
20
600
0
-20
400
-40
200
-60
-80
-100
-50
0
50
100
0
hEdep_yx
Energy Deposit_yx
Entries
Mean x
Mean y
RMS x
RMS y
80
60
11381
-1.05
3500
1.733
10.24
10.52
3000
40
2500
20
1500
0
-20
1000
-40
2000
0
1500
-20
1000
-40
500
-60
-80
-80
-60
-40
-20
0
20
40
60
80
0
-60
500
-80
-80
-60
-40
-20
0
20
Figure 2: Examples of showers for two events . The top plots shows the
projection in the xz plane, the middle plots for the yz projection and the
bottom ones show the yx projection. Beam axis is along z axis. The colored
arrows show the principal axes of the showers.
4
40
60
80
0
For each G4Step we search which voxels are spanned and we accumulate,
for each event, the energy deposited. We note that it is possible that a
single G4Step spans more than one voxel: the left plot of Figure 1 shows the
distribution of the number of voxels crossed by each G4Steps. The energy
deposit in each voxel is then calculated as:
dE
× ∆Lv , v ∈ {voxels}
dx
dE
∆EG4Step
≈
dx
∆LG4Step
∆Ev =
Where ∆Lv is the length of the G4Step in the voxel v. The distribution of
the dE/dx is shown in the right plot of 1.
Since the voxels are quite small the hadronic shower is sampled with high
granularity, as shown in Figure 2. The figures show the projections in the
different planes of two different showers originating from impinging protons
at 20 GeV.
The shower moments are calculated starting from the energy deposits in
the mesh. A shower moment of degree n in an observable o is given by:
< on >=
1
Ecluster
×
X
Ev onv
(2)
v∈{voxels}
With Ecluster defined by equation 1.
For many of the moments we use the shower axis is needed as reference.
The axis is calculated via a principal component analysis. Defining:
Mij =
1
w
X
Ev2 (iv − < i >)(jv − < j >)
v∈{voxels}
i, j ∈ {x, y, z}
X
w =
Ev2
v∈{voxels}
The principal axis are the eigenvectors of the symmetric matrix:


Mxx Mxy Mxz
M = Mxy Myy Myz 
Mxz Myz Mzz
5
y
htemp
x
Entries
3
10
Mean
RMS
10000
htemp
10
RMS
10
10
1
1
0
2
4
<x> (cm)
-4
htemp
z
10000
-0.01132
0.8101
0.8006
102
-2
Mean
-0.004018
102
-4
Entries
3
-2
0
2
4
<y> (cm)
Shower Axis angle
Entries 10000
Mean
-25.28
RMS
23.67
3
10
102
102
10
10
1
-80 -60 -40 -20
1
0
20
40
60
80 100
<z> (cm)
0.965 0.97 0.975 0.98 0.985 0.99 0.995 1
cost(θ)
Figure 3: Distribution of the three components of the shower center ~c and of
the angle between the shower axis ~s and the gun-to-center axis.
6
Cell energy density log10
Entries
2200
Entries
ncells
QGSP_BERT
FTF_BIC
FTFP_BERT
QGSC_BERT
2000
1800
QGSP_BERT
QGSP_BERT_HP
FTFP_BERT
QGSC_BERT
LHEP
QGSP
FTF_BIC
QGSP_BIC
10-1
LHEP
QGSP
QGSP_BERT_HP
QGSP_BIC
1600
1400
10-2
1200
1000
800
10-3
600
400
200
0
0
500
1000
1500
2000
2500
3000
3500
cells
NE>0
10-4
-8
-6
-4
-2
0
2
log10(Ecell/V cell) (MeV/cm3)
Figure 4: Number of voxels with E > 0 (right) and energy density ρ (left)
for the different physics lists. Data obtained with a π + with Ekin = 20 GeV.
In Figure 2 the principal axis for the two showers are displayed with colored
arrows. It should be noted that the axis are relative to the shower center
~c = (< x >, < y >, < z >)
(3)
The shower axis ~s which is closest to the direction pointing from the particle
gun origin to the shower center (gun-to-center) is used as the shower axis.
There can be (verified to be well below 1%) events in which the shower
shape is such that the shower axis has a large angular deviation from the
gun-to-center axis1 , in this case the gun-to-center axis is used as the shower
axis.
Figure 3 shows the position of the shower center ~c and the cosine of the
angle between the shower axis ~s and the gun-to-center axis. Since the primary
particles have momentum along the z axis, the shower center is very close to
the z axis: the < x > and < y > shower moments have an average value very
close to 0 with a RMS of about 8 mm. The shower axis ~s is almost parallel
to the gun-to-center axis (and thus to the z axis).
Once the vectors ~c and ~s have been defined for each voxel two quantities
are calculated:
rv = |(~xv − ~c) × ~s|
λv = (~xv − ~c) · ~s
1
(4)
(5)
This can happen when the shower has a shape particularly spherical or that elongates
in the x or y direction.
7
respectively: the distance of the voxel v from the shower axis and its distance
from the shower center measured along the shower axis.
It is also useful to define the energy fraction in the core of the shower. To
define this quantity two additional observable have been calculated for each
shower using a sliding window algorithm approach2 :
n X
o
1
max
max
fn,m,l =
Ev
(6)
Ecluster w∈{n×m×l}
v∈{w}
nX o
1
core
max
fn,m
=
Ev
(7)
Ecluster t∈{n×m}
v∈{t}
for each shower the group of cells with size n × m × l with maximum energy
defines the core of the shower, similarly we search the group of cells in a
tower of size n × m.
For each event the following moments are calculated (in addition to the
shower center and Ecluster ):
• < ρ >: the first moment in the energy density3 ρ = E/V , with V
volume of the voxels.
• < ρ2 >: the second moment in the energy density. Together with the
previous moment these are sensitive to the electromagnetic fraction of
the shower. Showers with larger fractions of energy carried by π 0 will
have higher values of these shower moments.
• < r > and < r2 >: first and second moments on r (as defined in 4),
these moments are sensitive to the shower lateral profile.
• < λ2 >: second moment in λ (as defined in 5).
• λcenter : the distance of the shower center ~c from the calorimeter front
face measured along the shower axis. This moment is sensitive to the
depth of the shower, together with the previous moment allow for a
description of the longitudinal profile of the shower.
2
These quantities are also called moments. However they do not follow the general
definition of moment. In a more general sense a moment is a parameter describing the
shower, function of the energy deposited in each voxel.
3
In the case of our simplified calorimeter from the definition of cluster moment and from
the definition of the mesh follows that < ρ >= V −1 < E 2 > and < ρ2 >= V −1 < E 3 >
since the volume V of all voxels is a constant.
8
QGSP_BERT (µ =14202.006 σ =759.757 S=-5.300 K=70.657)
103
FTF_BIC ( µ =14744.565 σ =1058.037 S=-6.799 K=73.361)
Entries
ene
FTFP_BERT ( µ =14594.398 σ =707.617 S=-9.597 K=148.323)
QGSC_BERT ( µ =14323.784 σ =754.221 S=-7.173 K=103.005)
LHEP ( µ =12405.968 σ =921.799 S=-1.479 K=14.620)
102
QGSP ( µ =13138.141 σ =915.449 S=-1.911 K=17.985)
QGSP_BERT_HP ( µ =13656.340 σ =903.815 S=-4.652 K=51.111)
QGSP_BIC (µ =13435.060 σ =894.502 S=-2.856 K=30.222)
10
0
2000
4000
6000
1
8000 10000 12000 14000 16000 18000
Ecluster
Figure 5: Total energy deposits for the different physics lists.
max
max
max
• f2,2,2
, f3,3,3
, f5,5,5
: the fraction of energy released in a group of cells of
sizes 2 × 2 × 2, 3 × 3 × 3, 5 × 5 × 5 (see Equation 6).
core
core
core
: the fraction of energy released in the sliding tower of
, f5,5
, f3,3
• f2,2
sizes 2 × 2, 3 × 3, 5 × 5 (see Equation 7). Together with the previous set
of moments these are sensitive to the dimension of the shower shape.
3
Results
The distributions for the number of voxels with E > 0 is shown in the left
plot of Figure 4, this is a first, qualitatively, measurement of the size of
the shower, large shower will have a higher number of voxels with energy
deposits. LHEP and QGSP physics lists have the smallest shower, the inclusion
of the Binary cascade models increase the size of the shower, the Bertini
9
Ecluster
(MeV)
LHEP
QGSP
QGSP BIC
QGSP BERT
QGSP BERT HP
QGSC BERT
FTF BIC
FTFP BERT
Mean
Variance
Skewness
12406
13138
13435
14202
13656
14324
14745
14594
849798
838130
800213
577288
816962
568906
1119555
500771
-1.479
-1.9024
-2.8798
-5.2882
-4.6336
-7.1694
-6.8207
-9.5910
Excess
Kurtosis
14.5657
17.8895
30.6165
70.3345
50.7452
103.2357
73.6487
148.2123
Table 1: Mean, variance, skewness and excess kurtosis of the total energy
deposit obtained with 10000 events of π + with Ekin = 20 GeV.
intra-nuclear cascade models give the higher number of voxels interested by
the shower (almost a factor 4 with respect the LHEP physics list).
The right plot of Figure 4 shows the distributions of the energy density
for the different physics lists. The most probable value for the energy density is about 10 keV/cm3 , we can also note that the QGSP BERT HP and the
QGSC BERT physics lists have a tail to very low values of energy density. At
the opposite we can find the LHEP and QGSP physics lists have more voxels
with high energy density deposits.
3.1
Energy deposit moments
The distributions for the total energy deposit (Ecluster ) for different physics
lists are shown in Figure 5, the different physics lists are shown with different
colors and markers, and Table 1 summarizes the results on the total energy
deposits for the different physics lists.
10
The statistics4 are calculated on ten thousands events of π + of Ekin =
20 GeV. The LHEP (−5.57% with respect to QGSP ) physics lists has the smallest energy deposit, followed by the QGSP physics list. The use of the Binary
cascade increases the response (QGSP BIC +2.26%), Bertini (QGSP BERT HP
+3.94%, QGSP BERT +8.10%) increases even more the response in the calorimeter, QGSC BERT (+9.02%), FTFP BERT (11.08%) and the FTF BIC (+12.23%)
give the highest responses.
The variance of the different samples fluctuates around the QGSP physics
list of about 40%, however no clear trend is visible.
From the analysis of the skewness and kurtosis we can understand the
importance of the low energy tail. In this case the situation is similar to what
we have observed for the mean value of the distributions. The LHEP and QGSP
physics lists have the smaller tails, adding the intra nuclear cascade models
increases the size of the tail (QGSP BIC , QGSP BERT and QGSP BERT HP ), the
most important tail is obtained with the Fritiof and CHIPS models.
The first and second moment on the energy density < ρ > and < ρ2 > is
another way to study the energy deposited. Figure 6 shows the distribution
for the first (left) and second (right) energy density moments < ρ > and
< ρ2 >. These distributions show a characteristic very long tail, present
in all physics lists. The core of the distribution is at relatively low values
of the energy density, while the tail can extend to very high values. The
events in the tail represents showers in which the energy density fluctuates
to considerably high values. This is characteristic of the electromagnetic
component of the hadronic showers: the energy carried by π 0 is higher in this
showers. We can show the correlation between energy deposit and density
(left plot Figure 7), the events with higher Ecluster populate the long tail of
the energy density moment distribution. The right plot of Figure 7 shows
the correlation between the electromagnetic component of the shower (energy
deposited π 0 ) and < ρ >: the showers with higher value of < ρ > have also
4
Skewness is calculated as:
p
Pn
n(n − 1) n1 i=1 (xi − x)3
P
S=
n − 2 ( n1 ni=1 (xi − x)2 )3/2
The excess kurtosis is calculated as:
(n + 1)n
K=
(n − 1)(n − 2)(n − 3)
Pn
i=1 (xi −
V2
x)4
−3
(n − 1)2
(n − 2)(n − 3)
Where V is the unbiased estimator of the population variance.
11
rho2
QGSP_BERT (µ =25.155 σ =10.959 S=1.259 K=1.641)
FTF_BIC ( µ =26.817 σ =12.126 S=1.086 K=0.815)
400
FTFP_BERT ( µ =26.924 σ =12.155 S=1.059 K=0.705)
350
QGSC_BERT ( µ =27.061 σ =10.412 S=1.020 K=1.038)
LHEP ( µ =26.949 σ =10.627 S=1.380 K=2.100)
300
Entries
Entries
rho
QGSP_BERT (µ =1111.265 σ=734.691 S=1.120 K=0.665)
FTF_BIC (µ =1178.746 σ=787.646 S=1.004 K=0.226)
300
FTFP_BERT (µ =1187.370 σ=785.438 S=0.967 K=0.151)
QGSC_BERT ( µ =1268.465 σ=747.496 S=0.907 K=0.163)
250
LHEP (µ =1180.624 σ=702.675 S=1.122 K=0.785)
QGSP ( µ =28.351 σ =10.829 S=1.062 K=1.072)
QGSP_BERT_HP ( µ =26.004 σ =11.050 S=1.191 K=1.383)
250
QGSP (µ =1292.950 σ=744.177 S=0.917 K=0.206)
200
QGSP_BERT_HP (µ =1149.061 σ=753.337 S=1.067 K=0.449)
QGSP_BIC (µ =26.936 σ =10.739 S=1.120 K=1.255)
200
QGSP_BIC (µ =1226.751 σ=750.323 S=1.005 K=0.361)
150
150
100
100
50
50
0
0
10
20
30
40
50
60
70
3
<ρ> (MeV/cm )
0
0
500
1000
1500
2000
2500
Figure 6: First (left) and second (right) energy density moments.
higher energy released by π 0 .
Since it is difficult to appreciated the small differences between the different physics lists, it is more convenient to summarize the shape of the
distributions through few numbers. Mean, variance, skewness and excess
kurtosis have been calculated for each physics list.
For highly asymmetric distributions the characterization of the tail in
terms of kurtosis and skewness is limited: we analyze this distributions in
terms of quantiles.
The 50th (median), 84th and 98th percentile have been calculated for
each distribution, we have then defined the following ratios:
R(50) = P50 /G50 (µ, σ)
R(84) = P84 /G84 (µ, σ)
R(98) = P98 /G98 (µ, σ)
(8)
(9)
(10)
Pn is the nth percentile and Gn (µ, σ) is the nth percentile for a gaussian
distribution with mean µ and standard deviation σ obtained from the original
distribution. Being:
1
x−µ x−µ
φ( √ ) = 1 + erf ( √ )
2
2σ
2σ
the cumulative distribution function for a gaussian with mean µ and standard
deviation σ, the nth quantile is the probit function, inverse of φ. It can be
12
3000
3500
<ρ2> (MeV2/cm6)
<ρ>
Mean
(MeV/cm3 )
LHEP
28.1
QGSP
29.0
QGSP BIC
27.8
QGSP BERT
25.8
QGSP BERT HP
26.7
QGSC BERT
27.4
FTF BIC
27.6
FTFP BERT
27.8
2
<ρ >
Mean
2
6
(MeV /cm ) (×103 )
LHEP
1.71
QGSP
1.72
QGSP BIC
1.64
QGSP BERT
1.51
QGSP BERT HP
1.55
QGSC BERT
1.60
FTF BIC
1.74
FTFP BERT
1.78
Variance
189
154
152
161
161
135
191
196
Variance
(×106 )
4.58
3.39
2.96
3.14
3.00
2.41
3.90
4.11
Skewness
Excess
Kurtosis
2.33
7.87
1.99
7.91
1.76
5.20
1.98
6.41
1.80
5.07
1.66
5.24
1.63
3.70
1.66
4.19
Skewness Excess
Kurtosis
4.35
26.2
5.37
53.3
4.14
27.7
4.37
29.3
3.98
25.1
4.21
30.8
3.51
19.1
3.72
22.8
R
50
0.87
0.91
0.90
0.87
0.88
0.91
0.87
0.87
R
50
0.61
0.70
0.69
0.65
0.65
0.73
0.63
0.62
R
84
0.89
0.95
0.95
0.94
0.95
0.96
0.96
0.96
R
84
0.65
0.74
0.76
0.72
0.75
0.78
0.73
0.76
R
98
1.27
1.15
1.18
1.18
1.19
1.14
1.20
1.17
R
98
1.43
1.25
1.31
1.32
1.37
1.26
1.40
1.34
Table 2: Statistics (see text) for the first and second moments on the
energy density. Measurements obtained with 10000 events of π + with
Ekin = 20 GeV.
expressed in terms of the error function:
√
x−µ
√
= probit(p) = 2erf −1 (2p − 1) , p = n/100
2σ
And thus:
Gn = µ + σ ×
√
2probit(n/100)
(11)
For the energy density momenta, R(50) and R(84) are smaller then unity,
while R(98) > 1, the distributions present a core that is more compact than
the equivalent gaussian and have a very long (asymmetric) tail.
LHEP , QGSP and the FTF based models give higher values for the mean
value of < ρ > and < ρ2 >.
13
SimplifiedCalorimeterHV.Etot_em_pi0:rho
E (e+/e- from π0) (MeV)
Ecluster
ene:rho2
20000
15000
18000
14000
16000
13000
14000
12000
12000
10000
11000
8000
6000
10000
4000
9000
2000
8000
0
2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
<ρ2>
0
0
20
40
60
80
100
120
140
160
<ρ> (MeV/cm3)
Figure 7: Left: correlation between the energy released in the calorimeter
and the second moment in the energy density. Right: correlation between
the energy released by the electromagnetic component of the shower and the
first moment in the energy density. LHEP physics list.
For LHEP and the two Fritiof physics lists also the variance of the distribution is higher, indicating larger event-by-event fluctuations.
This is confirmed also analysing the tail of the distributions. The LHEP
and FTF BIC physics lists have a higher value of R(98).
3.2
Moments related to the shower profile
The longitudinal shower profile is characterized by the shower depth and by
the length of the shower: λcenter and the second moment in the longitudinal
length, < λ2 >.
Figure 8 shows the distributions of these moments for the different physics
lists. As in the case of the energy density moments the distributions are
characterized by a well defined core and a very long tail. The statistics
(mean, variance, skewness, excess kurtosis and R ratios) are summarized in
Table 3.
The shower center λcenter is found deeper inside the calorimeter for the
Fritiof based model. FTF BIC and FTFP BERT show the highest value for this
moment (respectively +6% and +7% with respect QGSP ). The LHEP , QGSP
and QGSP BIC have, at the opposite, early starting showers. The Bertini code
increases the shower depth and the BERT physics lists have mean value of
14
lambda_center
QGSP_BERT (µ =69.795 σ=24.881 S=1.265 K=1.803)
lambda2
QGSP_BERT (µ=895.634 σ=394.330 S=1.535 K=2.276)
FTFP_BERT (µ =70.394 σ=25.292 S=1.291 K=1.886)
QGSC_BERT ( µ =67.966 σ=25.249 S=1.357 K=2.282)
LHEP (µ =64.916 σ=23.533 S=1.361 K=2.227)
FTF_BIC (µ=890.767 σ=390.590 S=1.506 K=2.226)
Entries
Entries
FTF_BIC (µ =69.836 σ=25.213 S=1.287 K=1.873)
FTFP_BERT (µ=877.255 σ=382.433 S=1.569 K=2.464)
QGSC_BERT ( µ=884.066 σ=401.461 S=1.555 K=2.258)
LHEP (µ=778.873 σ=356.501 S=1.972 K=4.270)
QGSP (µ=762.100 σ=343.909 S=2.082 K=5.045)
QGSP (µ =66.256 σ=24.677 S=1.305 K=1.957)
10
QGSP_BIC (µ =67.234 σ=24.769 S=1.359 K=2.221)
10
1
QGSP_BERT_HP (µ=876.621 σ=393.040 S=1.579 K=2.462)
QGSP_BIC (µ=816.508 σ=368.064 S=1.786 K=3.381)
QGSP_BERT_HP (µ =68.881 σ=24.421 S=1.269 K=1.872)
2
102
10
20
40
60
80
100
120
140
160
180
200
λcenter (cm)
1
500
1000
1500
Figure 8: Shower center λcenter (left) and second moment on λ (right).
λcenter in the intermediate region. The variance of the distributions follows
the same classification: LHEP , QGSP and QGSP BIC shower depth fluctuates
less with respect the other physics lists. From the analysis of the tail of the
distributions (skewness, kurtosis and R ratios) we do not observe particular
differences between physics lists, all values are quite similar to each other.
The skewness and kurtosis are higher for the early showering models. This is
due to a more compact core of the distribution (smaller variance) and to few
events with higher values of λcenter . Kurtosis and skewness are indeed more
sensitive to outliers in the distribution, with respect to the percentile ratios.
For all physics lists the importance of the tail of the distribution is similar.
If λcenter is a measure of the shower depth, the second moment in the
shower length < λ2 > is used to characterize the longitudinal dimension of
the shower. From the right plot of Figure 8 it is possible to distinguish the
different shape of the distribution of < λ2 > for the QGSP physics list. In this
case the longitudinal profile is visibly more compact. Similarly to the case
of shower depth the LHEP and QGSP BIC physics lists have a smaller value
for the mean of < λ2 >: the showers are more compact in the longitudinal
dimension. The other physics lists have higher mean values but, differently
from what observed with λcenter , the Fritiof models do not present a particular
enhancement of the longitudinal shower profile. Also in this case LHEP , QGSP
and QGSP BIC have smaller variance of the distribution of < λ2 > and the
showers fluctuate less.
15
2000
2500
<λ2> (cm2)
λcenter
Mean Variance
(cm)
LHEP
65.8
591
QGSP
65.9
601
QGSP BIC
67.4
619
QGSP BERT
69.8
637
QGSP BERT HP 69.3
619
QGSC BERT
68.6
663
FTF BIC
69.9
633
FTFP BERT
70.7
671
2
<λ >
Mean Variance
2
(cm )
(×103 )
LHEP
826
246
QGSP
798
212
QGSP BIC
853
239
QGSP BERT
949
283
QGSP BERT HP 933
288
QGSC BERT
942
293
FTF BIC
942
276
FTFP BERT
934
277
Skewness
Excess
Kurtosis
1.39
2.44
1.40
2.49
1.35
2.11
1.29
1.97
1.26
1.80
1.32
2.04
1.29
1.92
1.32
1.91
Skewness Excess
Kurtosis
3.51
18.5
3.58
20.1
3.15
14.7
2.73
11.0
2.88
12.4
2.75
10.7
2.82
11.8
2.87
12.3
R
50
0.91
0.91
0.91
0.91
0.91
0.91
0.91
0.91
R
50
0.80
0.81
0.81
0.83
0.82
0.82
0.83
0.83
R
84
0.97
0.98
0.98
0.98
0.99
0.98
0.98
0.98
R
84
0.83
0.84
0.86
0.88
0.88
0.88
0.88
0.87
R
98
1.17
1.15
1.15
1.15
1.13
1.16
1.15
1.15
R
98
1.30
1.29
1.28
1.27
1.27
1.29
1.27
1.29
Table 3: Statistics (see text) for λcenter and the second λ moment. Measurements obtained with 10000 events of π + with Ekin = 20 GeV.
For what the longitudinal profile is concerned we can thus conclude that
the use of Bertini or Fritiof models makes showers start later in the calorimeter (this is particularly true for the FTF physics lists) but also it makes them
longer.
The lateral development of the showers is characterized by the first and
second moment in the variable r (see equation 4). In this case there are
clear differences between physics lists: Figure 9 shows the first (left) and
second (right) moments on r. The distributions, characterized by a bell shape
with a small tail towards higher values of the observable, have distinct mean
values, summarized (together with variance, skewness and excess kurtosis)
in Table 4.
We can observe that QGSP , LHEP and QGSP BIC predict again the most
compact showers . QGSC BERT , QGSP BERT HP and FTF BIC physics lists
16
r
Entries
FTF_BIC ( µ =8.042 σ =2.091 S=0.310 K=0.304)
1200
FTFP_BERT ( µ =8.314 σ =2.092 S=0.166 K=0.250)
QGSC_BERT ( µ =7.934 σ =1.865 S=0.387 K=0.547)
1000
LHEP ( µ =7.134 σ =1.776 S=0.645 K=1.369)
Entries
r2
QGSP_BERT (µ =8.480 σ =2.063 S=0.274 K=0.614)
QGSP_BERT (µ =153.346 σ =65.317 S=0.750 K=1.370)
FTF_BIC ( µ =129.760 σ =62.189 S=1.020 K=1.989)
900
FTFP_BERT ( µ =145.963 σ =63.805 S=0.687 K=1.074)
800
QGSC_BERT ( µ =138.593 σ =59.930 S=0.789 K=1.182)
LHEP ( µ =101.419 σ =47.954 S=1.331 K=3.778)
700
QGSP ( µ =6.837 σ =1.610 S=0.777 K=2.033)
QGSP_BERT_HP ( µ =7.929 σ =1.971 S=0.446 K=0.763)
800
QGSP_BIC (µ =7.458 σ =1.773 S=0.584 K=1.189)
600
QGSP ( µ =89.931 σ =42.079 S=1.549 K=5.178)
600
QGSP_BERT_HP ( µ =130.329 σ =60.103 S=0.999 K=1.987)
QGSP_BIC (µ =115.381σ =52.513 S=1.067 K=2.120)
500
400
300
400
200
200
100
0
0
2
4
6
8
10
12
14
16
18
20
<r> (cm)
0
0
50
100
150
200
250
Figure 9: First (left) and second (right) moment on r.
have shower a bit wider in the radial dimension followed by FTFP BERT and
QGSP BERT . The use of Bertini intra nuclear cascade code makes shower wider
but, differently from the case of longitudinal profile, the FTF model does not
play a particular role. This is expected since the Fritiof model is important
to describe the forward-fast moving secondaries.
3.3
Moments related to the core and halo of the shower
The shower shape is mainly characterized by the moments in the longitudinal
and lateral profile, however it is also useful to distinguish between the energy
released in the core and in the halo of the shower. It is possible to group
the cells in different ways to define those regions. For this study we decided
to use the high granularity of the voxels and define two segmentation. A
three dimensional one, f max variables and a two dimensional segmentation
(tower structure) f core variables . Figure 10 shows the distributions for three
f ax moments (top) and the three f core moments (bottom). Table 5 summarizes the mean value, variance, skewness and excess kurtosis for the different
physics lists.
The distributions are all very similar, only the LHEP , QGSP and QGSP BIC
physics lists present distributions with higher mean value and lower variance.
This is again a symptom of more compact shower shapes. A higher fraction
of the shower energies indeed is released in a relatively smaller number of
17
300
350
400
<r2> (cm2)
<r>
Mean Variance
(cm)
LHEP
7.15
3.21
QGSP
6.86
2.66
QGSP BIC
7.46
3.29
QGSP BERT
8.52
4.25
QGSP BERT HP 7.91
3.93
QGSC BERT
7.90
3.61
FTF BIC
8.04
4.42
FTFP BERT
8.31
4.45
2
<r >
Mean Variance
2
(cm )
(×103 )
LHEP
102
2.40
QGSP
91
1.82
QGSP BIC
115
2.91
QGSP BERT
154
4.24
QGSP BERT HP 130
3.61
QGSC BERT
137
3.69
FTF BIC
130
3.92
FTFP BERT
146
4.13
Skewness
Excess
Kurtosis
0.68
1.47
0.76
1.93
0.65
1.36
0.26
0.50
0.42
0.85
0.32
0.47
0.32
0.44
0.18
0.19
Skewness Excess
Kurtosis
1.40
4.05
1.67
9.27
1.24
3.09
0.67
1.02
1.04
2.36
0.82
1.69
1.01
1.94
0.65
0.79
Table 4: Statistics (see text) for the first and the second moment on the r.
Measurements obtained with 10000 events of π + with Ekin = 20 GeV.
voxels. This result is in agreement with the conclusions we obtained from
the analysis of the longitudinal and lateral profile moments.
Finally we can consider the increase of the fraction of energy collected
in larger and larger volumes, i.e. the increase in f max (f core ) moving from a
2 × 2 × 2 (2 × 2) configuration to a 5 × 5 × 5 (5 × 5). The results are shown
in Figure 11. From left plot we can see that a relatively large fraction of the
shower energy (between 60% and 70%) is already contained in a 2 × 2 towers,
extending the tower to 3×3 collects about three quarter of the shower energy
and the larger towers (5 × 5) allows us to increase the collected energy of
only an additional 10%. With f core we are integrating the shower profile
along the longitudinal direction: we can see that the shower is composed
of a compact core where the majority of the energy released and a halo
with small importance: far away from the shower center a small amount of
18
Entries
1
2
FTF_BIC (µ =0.285 RMS=0.092 γ =0.463 γ =0.421)
1
2
FTFP_BERT (µ =0.288 RMS=0.091 γ =0.296 γ =-0.398)
1
2
QGSC_BERT ( µ =0.289 RMS=0.086 γ =0.349 γ =-0.232)
1
102
ShowerMoment.fmax555 {ShowerMoment.fmax555<1.1&&ShowerMoment.fmax555>0}
Entries
ShowerMoment.fmax333 {ShowerMoment.fmax333<1.1&&ShowerMoment.fmax333>0}
QGSP_BERT (µ =0.280 RMS=0.091 γ =0.369 γ =-0.354)
Entries
ShowerMoment.fmax222 {ShowerMoment.fmax222<1.1&&ShowerMoment.fmax222>0}
102
2
102
LHEP (µ =0.307 RMS=0.089 γ =0.227 γ =-0.507)
1
2
QGSP (µ =0.317 RMS=0.090 γ =0.175 γ =-0.523)
1
2
QGSP_BERT_HP (µ =0.290 RMS=0.091 γ =0.326 γ =-0.403)
1
2
QGSP_BIC (µ =0.306 RMS=0.090 γ =0.228 γ =-0.465)
1
2
QGSP_BERT (µ =0.407 RMS=0.120 γ =0.204 γ =-0.574)
10
10
1
10
2
FTF_BIC (µ =0.417 RMS=0.118 γ =0.173 γ =-0.516)
1
QGSP_BERT (µ =0.590 RMS=0.145 γ =-0.106 γ =-0.688)
1
2
1
1
1
1
1
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1
0
0.9
1
E2,2,2/E tot
1
0.2
0.3
0.4
0.5
1
2
0.6
0.7
0.8
Entries
1
2
FTF_BIC (µ=0.743 RMS=0.103 γ =-0.028 γ =-0.401)
2
FTFP_BERT ( µ=0.740 RMS=0.104 γ =-0.046 γ =-0.379)
1
0.1
103
1
102
0.5
1
2
2
0.8
0.9
1
E5,5,5/E tot
0.8
0.9
1
E5,2,inf/E tot
2
1
2
2
QGSC_BERT ( µ =0.872 RMS=0.060 γ =-0.419 γ =0.150)
2
1
2
LHEP ( µ =0.905 RMS=0.052 γ =-1.465 γ =5.166)
2
1
102
2
2
QGSP ( µ =0.913 RMS=0.048 γ =-1.219 γ =3.465)
1
2
QGSP_BERT_HP ( µ =0.876 RMS=0.063 γ =-0.711γ =1.066)
2
1
QGSP_BIC ( µ=0.782 RMS=0.086 γ =-0.412 γ =0.561)
1
0.7
FTFP_BERT ( µ =0.860 RMS=0.067 γ =-0.404 γ =0.007)
QGSP_BERT_HP (µ=0.757 RMS=0.098 γ =-0.237 γ =0.040)
1
0.6
1
QGSP ( µ=0.804 RMS=0.082 γ =-0.619 γ =1.063)
1
0.4
FTF_BIC ( µ =0.868 RMS=0.066 γ =-0.492 γ =0.162)
LHEP (µ=0.791 RMS=0.086 γ =-0.730 γ =1.484)
1
0.3
QGSP_BERT (µ =0.854 RMS=0.066 γ =-0.557 γ =1.068)
2
QGSC_BERT (µ=0.764 RMS=0.093 γ =-0.126 γ =-0.183)
102
0.2
ShowerMoment.fcore55 {ShowerMoment.fcore55<1.1&&ShowerMoment.fcore55>0}
QGSP_BERT ( µ=0.733 RMS=0.101 γ =-0.136 γ =0.094)
1
1
0
0.9
1
E3,3,3/E tot
ShowerMoment.fcore33 {ShowerMoment.fcore33<1.1&&ShowerMoment.fcore33>0}
Entries
ShowerMoment.fcore22 {ShowerMoment.fcore22<1.1&&ShowerMoment.fcore22>0}
1
QGSP_BIC (µ =0.636 RMS=0.142 γ =-0.334 γ =-0.574)
QGSP_BIC (µ =0.442 RMS=0.117 γ =0.038 γ =-0.606)
0.1
2
QGSP_BERT_HP ( µ =0.608 RMS=0.146 γ =-0.168 γ =-0.645)
2
Entries
1
0
2
QGSP ( µ =0.655 RMS=0.141 γ =-0.375 γ =-0.596)
2
QGSP_BERT_HP (µ =0.421 RMS=0.120 γ =0.155 γ =-0.567)
1
2
LHEP ( µ =0.639 RMS=0.144 γ =-0.327 γ =-0.633)
2
QGSP (µ =0.458 RMS=0.118 γ =-0.016 γ =-0.656)
1
2
QGSC_BERT ( µ =0.606 RMS=0.143 γ =-0.190 γ =-0.594)
2
LHEP (µ =0.444 RMS=0.117 γ =0.044 γ =-0.647)
1
2
FTFP_BERT ( µ =0.606 RMS=0.142 γ =-0.199 γ =-0.576)
2
QGSC_BERT ( µ =0.418 RMS=0.115 γ =0.136 γ =-0.509)
1
2
FTF_BIC ( µ =0.607 RMS=0.142 γ =-0.189 γ =-0.658)
FTFP_BERT (µ =0.419 RMS=0.119 γ =0.123 γ =-0.550)
2
QGSP_BIC (µ =0.893 RMS=0.053 γ =-0.871 γ =1.809)
2
1
2
QGSP_BERT (µ =0.626 RMS=0.126 γ =0.135 γ =-0.223)
1
2
FTF_BIC ( µ =0.635 RMS=0.131 γ =0.194 γ =-0.465)
10
1
2
FTFP_BERT ( µ =0.636 RMS=0.132 γ =0.146 γ =-0.476)
1
10
10
2
QGSC_BERT ( µ =0.669 RMS=0.118 γ =0.021 γ =-0.269)
1
2
LHEP ( µ =0.675 RMS=0.116 γ =-0.248 γ =0.234)
1
2
QGSP ( µ =0.698 RMS=0.111γ =-0.250 γ =0.108)
1
2
QGSP_BERT_HP ( µ =0.648 RMS=0.126 γ =0.039 γ =-0.244)
1
2
QGSP_BIC (µ =0.677 RMS=0.115 γ =-0.114 γ =-0.070)
1
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
2
0.9
1
E2,2,inf/E tot
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
E3,3,inf/E tot
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
max
Figure 10: Energy fractions moments calculated with sliding window: f2,2,2
,
core
core
core
max
max
f3,3,3 , f5,5,5 (top row) and f2,2 , f3,3 , f5,5 (bottom row).
energy is released on a large volume. We can give the same conclusions if
we consider the three dimensional variable f max : more than a quarter of the
max
corresponds to
shower energy is released in a small volume (in our case f2,2,2
3
3
a cube of volume 10 cm ).
From Table 5 and Figures 10 and 11 we can observe that, with the exclusion of LHEP , QGSP and QGSP BIC all physics lists are quite similar.
3.4
Behavior of Moments
The characterization of showers via moments is only weakly dependent on
the voxel dimensions. In addition the moments show a smooth behavior as a
function of the primary energy.
Figure 12 shows the dependence of the lateral and longitudinal moments
r, r2 , λ2 , λcenter as a function of the voxel dimension. For the same simulations the shower dimension moments are calculated for different voxel sizes.
The ratio of the calculated moments with respect to a reference mesh size
(with cubic voxels of 5 cm linear dimension) are shown. Left plot shows the
longitudinal moments (λcenter in blue and < λ2 > in red). In both cases
there is a very weak dependence on the voxel dimension. The plots on the
19
Figure 11: Division of the shower energy in subsequently bigger towers (left)
and in three dimensional windows (right).
Ratio
Ratio
1.2
æ
1.6
1.1
æ
æ
æ
æ
0
æ
1
2
æ
1.4
æ
æ
æ
3
æ
3
Log10 HVL Hcm L
1.2
æ
æ
æ
æ
0.9
0
æ
æ
0.8
1
æ 2æ
3
Log10 HVL Hcm3 L
0.8
Figure 12: Relative variations of shower moments as a function of voxel
dimension with respect the reference voxel size (5 × 5 × 5 cm3 ). Left:
λcenter (blue) and < λ2 > (red). Right: < r > (blue) and < r2 > (red).
20
max
f3,3,3
Mean Variance
(×10−2 )
LHEP
0.444
1.37
QGSP
0.458
1.39
QGSP BIC
0.442
1.17
QGSP BERT
0.407
1.43
QGSP BERT HP 0.421
1.44
QGSC BERT
0.418
1.31
FTF BIC
0.417
1.40
FTFP BERT
0.419
1.41
core
f3,3
Mean Variance
(×10−2 )
LHEP
0.791
0.74
QGSP
0.804
0.66
QGSP BIC
0.782
0.75
QGSP BERT
0.733
1.01
QGSP BERT HP 0.757
0.97
QGSC BERT
0.764
0.86
FTF BIC
0.743
1.07
FTFP BERT
0.740
1.09
Skewness
Excess
Kurtosis
0.04
-0.65
-0.01
-0.66
0.04
-0.61
0.20
-0.57
0.15
-0.57
0.13
-0.51
0.20
-0.37
0.12
-0.55
Skewness Excess
Kurtosis
-0.73
1.47
-0.62
1.05
-0.41
0.56
-0.13
0.10
-0.24
0.04
-0.12
-0.18
-0.02
-0.39
-0.04
-0.38
max
core
Table 5: Statistics (see text) for the f3,3,3
and f3,3
moments. Measurements
+
obtained with 10000 events of π with Ekin = 20 GeV.
right show the dependence for the lateral moments. In this case a stronger
dependence is visible at large voxel volumes. Between < r > and < r2 > the
latter has a smaller dependence at small and moderate voxel sizes. This is
an important result: with voxels of moderate dimensions, the characteristics
of the shower do not depend on the read-out geometry used to calculate the
moments. They are sensitive to the physics of the shower and not to the
detector segmentation.
The distributions of the moments are characterized by long tails (see,
for example, Figure 8). Even if the distributions cannot be described by a
simple analytical form, they are regular as a function of the primary energy
as shown in Figure 13. The figure shows a simplified Whisker plot for one
particular moment (λcenter ) for different primary beam energies. The star is
the position of median of the distribution, while the full dot represents the
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position of the mean of the distribution. The error bars are the first and
third quartile distribution. The distribution shapes are well behaved as a
function of the beam energy and statistical error on the sample mean are
small enough with 5000 events.
Figure 13: Whisker plots for λcenter distributions as a function of beam energy. See text for details.
4
Conclusions
We have presented a method to characterize the hadronic showers. Relying
on a highly segmented calorimeter we have constructed some observables that
characterize the shower shape and its composition. From the analysis of ρ,
f core and f max we have shown that the hadronic showers are characterized by
a relatively compact core with high energy density and a low energy density
halo.
We have introduced the variables λcenter , < λ2 >, < r > and < r2 > that
can summarize the longitudinal and lateral profile of the showers.
Our method gives the same conclusions as discussed in [1] [2] and is
thus equivalent with the results obtained in the past. However the use of
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shower moments allows for a more detailed and quantitative description of
the shower shapes.
References
[1] A. Ribon et al.; Hadronic shower shapes studies in Geant4: Update; 2008
(CERN); CERN-LCGAPP-2008-01
[2] A. Ribon et al.; Hadronic Shower Shape Studies in Geant4; 2007 (CERN);
CERN-LCGAPP-2007-02
[3] T. Barillari et al.; Local Hadron Calibration; 2008 (CERN); ATL-LARGPUB-2009-001
23
A
Appendix: Technical implementation
Figure 14: Class diagram for the shower moment calculations.
One of the main components to calculate the shower moments is the
ability to accumulate, in a three-dimensional structure of voxels, the energy
deposits. This is implemented through a 3D histogram. The class Mesh
implements a proxy pattern to the underlying structure implementing the
accumulation. In this case a 3D ROOT histogram. The use of the proxy
pattern allows for a possible replacement of the TH3 object with a userdefined histogramming object.
The class SimpleMomentCalc implements the main logic of the calculation
of shower moments, via the call to the GetMoment function:
G4double SimpleMomentCalc::GetMoment( Mesh& quantity, G4int exp ) {
// implements the calculation
//<quantity^exp> = Sum_i E_i*q_i^exp / Sum_i E_i
}
This class holds the mesh of the energy deposits for each voxel, it thus allows
to calculate the totally energy of the cluster.
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The VMoment class implements the interface for a generic moment, implementing a functor pattern. It holds a reference to the SimpleMomentCal
instance associated to the current cluster. The class Moment implements the
generic moment calculation implementing the function:
G4double operator()() {
return calculator->GetMomet( *quantity , order );
}
The quantity parameter of type Mesh and the order (of type integer) are
moment specific. For example the LongMoment and RMoment classes implement the longitudinal and lateral shower moments.
The class ShowerCenter implements the principal component analysis on
the mesh of the energy deposits for each voxel. It uses ROOT classes and
holds a TVector3 representing the shower center and one TVector3 representing the shower axis. The class also implements the calculation of the R and
λ for a given point in space with respect shower center and shower moment.
The class uses an object of type SimpleMomentCalc to access the energy of
the cluster.
A typical example of the use of shower moments is shown in the following
pseudo-code example. If a given observable o can be defined for each voxel it
is enough to implement a class, inheriting from Moment that creates a Mesh
object containing the observable o. The code of RMoment is a good and
complete example. The analysis is performed in this way:
beginOfEvent() {
// Create an empty Mesh with the desired granularity
Energy = new Mesh( divisions );
aQuantity = new Mesh( divisions );
}
processStep( G4Step* aStep) {
//for each step accumulate the energy :
Energy->Add( x, y, x, aStep->GetEnergyDeposit() );
aQuantity->Add( x, y, x, somederivedquantity );
}
endOfEvent() {
SimpleMomentCalc smc( Energy );
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//If shower center and axis are needed:
ShowerCenter sc( smc , BeamOrigin );
sc.CalcShowerAxis();
MyMoment mom( aQuantity , &smc, 1);
MyMoment mom2(aQuantity, &smc , 2);
G4double firstOrderMomentOfMom = mom();
G4double secondOrderMomentOfMom = mom2();
}
The use of the moments is general enough to allow for calculating several
quantities even if these do not follow strictly the definition of moment. For
example the class RBinEsum calculates the energy in all voxels with rmin <
r < rmax . The calculation uses EConditionalSum that implements the sum
of the energies in a given set of voxels. Using RBinEsum as an example
it is possible to define the summation of energies for any pattern of voxels.
Another example of the flexibility of the use of moments is shown by Max
class: it searches for the set of nx × ny × nz adjacent voxels that contain the
maximum energy.
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Class Name
Parameters
Etot
EConditionalSum Mesh of 0 or 1
LongMoment
Max
ShowerCenter vector
nx , ny , nz
RBinEsum
ShowerCanter vector,
rmin , rmax
a Mesh, an order
ShowerCenter vector,
an order
Moment
RMoment
Function
Calculates total energy of the shower
Calculates the sum of energies
in a set of voxels
Calculates < λ2 >
Searches the set of nx × ny × nz
adjacent voxels containing
the maximum energy
Calculates the energy
in voxels with rmin < r < rmax
Calculates the general moment < mo >
Calculates < ro >
Table 6: List of moments classes used by default in the application. These
can be used as an example to extend the application with new moments.
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