# Shower and Slow Particle Productions in Nucleus-Nucleus Collisions at High Energy

Authors: S. Fakhraddin; Magda A. Rahim; Hamdy M. M. Zayd
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

The multiplicity distributions of shower, grey, and black particles produced in interactions of4He, 12C, 16O, 22Ne, and 28Si with emulsion (Em) at 4.1–4.5 A GeV/c beam energies, and their dependence on target groups (H, CNO, and AgBr) is presented and has been reproduced by multisource thermal model. The multiplicity and the angular distributions of the three types of particles have been investigated. The experimental results are compared with the corresponding ones from the model. We found that the experimental data agrees with theoretical calculations using multisource thermal model.


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## Body

## 1. Introduction

Study of the secondary charged particles produced in heavy ion collisions is attracting a great deal of attention during the recent ten years. Since the first run of the Dubna Synchrophasotron, in 1980, a lot of data for nuclear fragmentation in light- and heavy-ion collisions at high energy have been collected [1–5]. The measurements show that the average multiplicity of shower, grey particles increases with increase in projectile mass, but the average multiplicity of black particles is approximately constant. These observations have generated a flurry of theoretical activities [6–11].Many models have been introduced to describe the multiparticle production in the nucleus-nucleus (N-N) interactions; some of them concern the dynamical evolution of interacting systems [12–18]. Others concern the thermal characteristics of final-state particles and fragments. One of these thermal models is the multisource thermal model, proposed to explain the multiplicity and angular distributions, based on the assumption that many emission sources are assumed to be formed in the interactions [19–30]. The aim of the present research is to check the model validity for describing the basic characteristics of particle production in the interactions of nuclei with emulsion at 4.1–4.5 A GeV/c, mainly beams of 4.1 A GeV/c Ne22 and 4.5 A GeV/c (He4, C12, O16, and Si28) from Dubna Synchrophasotron.


## 2. Multisource Thermal Model

### 2.1. Multiplicity Distribution

The physics picture of the following discussions is based on the multisource thermal model [31–34], which is mainly used in the descriptions of particle (fragment) emission angles, azimuthal angles, and transverse flows in nucleus-nucleus (NN) collisions. In the model, many emission sources of particles and fragments are assumed to be formed in high energy collisions. According to the interaction mechanisms or event sample, the sources are divided into l groups (subsamples). The source number in the jth group is assumed to be mj. Each source contributes multiplicity distribution to be an exponential distribution.The multiplicity (nij) distribution contributed by the ith source in the jth group is an exponential function:
(1)Pij(nij)=1〈nij〉exp⁡(-nij〈nij〉),
where
(2)〈nij〉=∫nijpij(nij)dnij
is the mean multiplicity contributed by the ith source in the jth group. As in [31], we assume that
(3)〈n1j〉=〈n2j〉=〈n3j〉=⋯〈nmjj〉.
The multiplicity (nch) distribution contributed by the jth group is the fold of mj exponential functions; that is,
(4)Pj(nch)=nchmj-1(mj-1)!〈nij〉mjexp⁡(-nch〈nij〉).
It is an Erlang distribution. The total multiplicity distribution contributed by the l groups can be written as
(5)Pj(nch)=1NdNdNch=∑j=1lkjPj(nch),
where kj is the relative weight contributed by the jth group.In the Monte Carlo calculation, letRij denote random variable in [0,1]. For the jth group, we have
(6)nij=-〈nij〉ln⁡Rij,nch=-∑i=1mj〈nij〉ln⁡⁡Rij.
The multiplicity distribution is obtained by a statistical method. Meanwhile, the mean multiplicities contributed by the jth group and the l groups are given by
(7)〈nch〉=mj〈nij〉,(8)〈nch〉=∑j=1lkjmj〈nij〉,
respectively.Generally speaking [35], j=1 for e-e+, pp(pp-), and e-p collisions at not too high energies (less than a few hundred GeV). The parameters for the collisions are 〈nij〉 and mj. For the mentioned collisions at very high energies (greater than a few hundred GeV), j=2 or 3 due to the different interaction mechanisms existing in the event samples. For pA and N-N collisions at a fixed impact parameter, mj can be regarded as the number of participant nucleons. The weight kj in (5) is obtained by the geometrical weight of the impact parameter. This formula was first proposed by Liu et al. to describe the multiplicity distributions of final-state particles produced in “elementary” particle interactions and heavy ion collisions at high energies. The basis of the formula is a multisource model and each source contributes multiplicity distribution to be an exponential form. The model treats uniformly the final-state particles and nuclear fragments by the same formula. It is shown that the model is successful in the descriptions of multiplicity distributions of different types of particles and projectile fragments.


### 2.2. Emission Angle of Particles

According to the multisource thermal model suggested by Liu et al. [31–36], many emission sources are assumed to be formed in the interactions. Let the beam direction of the incoming projectile be theoz-axis, and let the reaction plane be thexoz plane. Each source is assumed to emit particles isotropically in the source rest frame. As the first approximation, the three components px,y,z∖ of the particle momentum in the source rest frame are assumed to obey Gaussian distributions with the same deviation width σ [37]. Considering the motion of the emission source and the interactions among emission sources, the particle momentum components px, py, and pz in the final state in the laboratory reference frame are different from those in the rest frame of the emission source. The simplest relations between px and px∖, py and py∖, and pz and pz∖ are linear:
(9)px=axpx∖+Bx=axpx∖+axσi,py=aypy∖+By=aypy∖+ayσi,pz=azpz∖+Bz=azpz∖+azσi,
where Bx, By, and Bz are free parameters and σi is the parameter that characterizes the width of the momentum distribution in the source reference frame. ax, bx, ay, by, az, and bz are free parameters. Let R1, R2, R3, R4, R5, and R6 denote random variables distributed in [0,1]; we have
(10)px∖=-2ln⁡R1cos⁡(2πR2)σi,py∖=-2ln⁡R3cos⁡(2πR4)σi,pz∖=-2ln⁡R5cos⁡(2πR6)σi
because px∖, py∖, and pz∖ obey a Gaussian distribution law.The emission angleθ of a target fragment in the laboratory reference frame is given by
(11)θ=arctanpx2+py2pz.
Considering (9)–(11), we have(12)θ=arctan⁡[ax-2ln⁡R1cos⁡(2πR2)+bx]2+[ay-2ln⁡R3cos⁡(2πR4)+by]2az-2ln⁡R5cos⁡(2πR6)+bz.


## 2.1. Multiplicity Distribution

The physics picture of the following discussions is based on the multisource thermal model [31–34], which is mainly used in the descriptions of particle (fragment) emission angles, azimuthal angles, and transverse flows in nucleus-nucleus (NN) collisions. In the model, many emission sources of particles and fragments are assumed to be formed in high energy collisions. According to the interaction mechanisms or event sample, the sources are divided into l groups (subsamples). The source number in the jth group is assumed to be mj. Each source contributes multiplicity distribution to be an exponential distribution.The multiplicity (nij) distribution contributed by the ith source in the jth group is an exponential function:
(1)Pij(nij)=1〈nij〉exp⁡(-nij〈nij〉),
where
(2)〈nij〉=∫nijpij(nij)dnij
is the mean multiplicity contributed by the ith source in the jth group. As in [31], we assume that
(3)〈n1j〉=〈n2j〉=〈n3j〉=⋯〈nmjj〉.
The multiplicity (nch) distribution contributed by the jth group is the fold of mj exponential functions; that is,
(4)Pj(nch)=nchmj-1(mj-1)!〈nij〉mjexp⁡(-nch〈nij〉).
It is an Erlang distribution. The total multiplicity distribution contributed by the l groups can be written as
(5)Pj(nch)=1NdNdNch=∑j=1lkjPj(nch),
where kj is the relative weight contributed by the jth group.In the Monte Carlo calculation, letRij denote random variable in [0,1]. For the jth group, we have
(6)nij=-〈nij〉ln⁡Rij,nch=-∑i=1mj〈nij〉ln⁡⁡Rij.
The multiplicity distribution is obtained by a statistical method. Meanwhile, the mean multiplicities contributed by the jth group and the l groups are given by
(7)〈nch〉=mj〈nij〉,(8)〈nch〉=∑j=1lkjmj〈nij〉,
respectively.Generally speaking [35], j=1 for e-e+, pp(pp-), and e-p collisions at not too high energies (less than a few hundred GeV). The parameters for the collisions are 〈nij〉 and mj. For the mentioned collisions at very high energies (greater than a few hundred GeV), j=2 or 3 due to the different interaction mechanisms existing in the event samples. For pA and N-N collisions at a fixed impact parameter, mj can be regarded as the number of participant nucleons. The weight kj in (5) is obtained by the geometrical weight of the impact parameter. This formula was first proposed by Liu et al. to describe the multiplicity distributions of final-state particles produced in “elementary” particle interactions and heavy ion collisions at high energies. The basis of the formula is a multisource model and each source contributes multiplicity distribution to be an exponential form. The model treats uniformly the final-state particles and nuclear fragments by the same formula. It is shown that the model is successful in the descriptions of multiplicity distributions of different types of particles and projectile fragments.


## 2.2. Emission Angle of Particles

According to the multisource thermal model suggested by Liu et al. [31–36], many emission sources are assumed to be formed in the interactions. Let the beam direction of the incoming projectile be theoz-axis, and let the reaction plane be thexoz plane. Each source is assumed to emit particles isotropically in the source rest frame. As the first approximation, the three components px,y,z∖ of the particle momentum in the source rest frame are assumed to obey Gaussian distributions with the same deviation width σ [37]. Considering the motion of the emission source and the interactions among emission sources, the particle momentum components px, py, and pz in the final state in the laboratory reference frame are different from those in the rest frame of the emission source. The simplest relations between px and px∖, py and py∖, and pz and pz∖ are linear:
(9)px=axpx∖+Bx=axpx∖+axσi,py=aypy∖+By=aypy∖+ayσi,pz=azpz∖+Bz=azpz∖+azσi,
where Bx, By, and Bz are free parameters and σi is the parameter that characterizes the width of the momentum distribution in the source reference frame. ax, bx, ay, by, az, and bz are free parameters. Let R1, R2, R3, R4, R5, and R6 denote random variables distributed in [0,1]; we have
(10)px∖=-2ln⁡R1cos⁡(2πR2)σi,py∖=-2ln⁡R3cos⁡(2πR4)σi,pz∖=-2ln⁡R5cos⁡(2πR6)σi
because px∖, py∖, and pz∖ obey a Gaussian distribution law.The emission angleθ of a target fragment in the laboratory reference frame is given by
(11)θ=arctanpx2+py2pz.
Considering (9)–(11), we have(12)θ=arctan⁡[ax-2ln⁡R1cos⁡(2πR2)+bx]2+[ay-2ln⁡R3cos⁡(2πR4)+by]2az-2ln⁡R5cos⁡(2πR6)+bz.


## 3. Results and Discussions

### 3.1. Multiplicity Characteristics

To study the multiplicity behavior of the target fragmentation as function of mass number of the target nucleusAT, we classify the emulsion nuclei based on nh (nh=ng+nb) into three groups [38]: two types of light nuclei (H and CNO) and one type of heavy nuclei (AgBr). Collisions with H target nuclei are events with nh=0,1, collisions with only one bound nucleon in CNO or AgBr target nuclei these events having 2≤nh≤7 are mostly interactions with CNO targets with some admixture of peripheral AgBr interactions. All events with nh≥8 are only due to AgBr interactions. It should be noted that the classification of events in emulsion is not unique; however, there is no perfect method for classifying events due to the limitations of the emulsion technique [39].Table1 illustrates the average values of shower, grey, black, and heavy particles produced in interactions of different projectiles with Em at momentum 4.1–4.5 A GeV/c. The experimental data has been taken from [40] and the data available from the High Energy Physics Group at Sana’a University.Table 1
The average multiplicities of the charged secondary particles emitted in4He-, 12C-, 16O-, 22Ne-, and 28Si-Em interactions and those emitted in the interactions with the different groups of target nuclei (H, CNO, Em, and AgBr) compared with those obtained by the model.
Projectile
〈
n
i
〉
H
CNO
Em
AgBr
4He
MSTM
〈
n
s
〉
1.623
3.3263
3.903
5.78
EXP
〈
n
s
〉
1.619 ± 0.1
3.112 ± 0.133
3.757 ± 0.096
5.77 ± 0.151
MSTM
〈
n
g
〉
0.288
2.35
5.372
9.474
EXP
〈
n
g
〉
0.288 ± 0.03
2 ± 0.066
4.3899 ± 0.17
9.38 ± 0.3
MSTM
〈
n
b
〉
0.163
2.5
4.86
9.519
EXP
〈
n
b
〉
0.172 ± 0.024
2.25 ± 0.07
4.395 ± 0.149
9.223 ± 0.21
12C
MSTM
〈
n
s
〉
2.014
6.22
8.897
13.975
EXP
〈
n
s
〉
2.062 ± 0.186
6 ± 0.25
8.0 ± 0.24
13.975 ± 0.4
MSTM
〈
n
g
〉
0.288
2.35
5.274
10.882
EXP
〈
n
g
〉
0.293 ± 0.03
2.04 ± 0.1
5.081 ± 0.22
10.753 ± 0.4
MSTM
〈
n
b
〉
0.205
2.5
4.6346
9.519
EXP
〈
n
b
〉
0.204 ± 0.027
2.28 ± 0.075
4.57 ± 0.16
9.293 ± 0.21
16O
MSTM
〈
n
s
〉
2.196
6.621
9.741
16.664
EXP
〈
n
s
〉
2.454 ± 0.145
6.679 ± 0.197
9.417 ± 0.206
16.426 ± 0.349
MSTM
〈
n
g
〉
0.268
2.35
6.284
12.6
EXP
〈
n
g
〉
0.264 ± 0.02
2.154 ± 0.06
6.353 ± 0.17
12.454 ± 0.29
MSTM
〈
n
b
〉
0.243
2.5
4.315434
9.217
EXP
〈
n
b
〉
0.24 ± 0.02
2.17 ± 0.05
4.421 ± 0.097
8.58 ± 0.13
22Ne
MSTM
〈
n
s
〉
2.11
6.605
10.769
18.919
EXP
〈
n
s
〉
2.24 ± 0.0898
6.66 ± 0.151
10.254 ± 0.113
18.482 ± 0.29
MSTM
〈
n
g
〉
0.249
2.35
6.752
13.547
EXP
〈
n
g
〉
0.252 ± 0.013
2.2 ± 0.04
6.188 ± 0.295
13.495 ± 0.214
MSTM
〈
n
b
〉
0.243
2.301
4.288
8.978
EXP
〈
n
b
〉
0.238 ± 0.013
2.05 ± 0.034
4.151 ± 0.103
8.482 ± 0.09
28Si
MSTM
〈
n
s
〉
2.461
7.836
11.776
21.007
EXP
〈
n
s
〉
2.887 ± 0.22
7.51 ± 0.297
11.784 ± 0.343
20.163 ± 0.602
MSTM
〈
n
g
〉
0.249
2.39
6.071
13.156
EXP
〈
n
g
〉
0.2474 ± 0.03
2.266 ± 0.07
6.405 ± 0.228
13.224 ± 0.383
MSTM
〈
n
b
〉
0.163
2.5
4.678
9.504
EXP
〈
n
b
〉
0.165 ± 0.022
2.2 ± 0.064
4.449 ± 0.1244
8.651 ± 0.17The comparison between the average values of the multiplicities〈ns〉, 〈ng〉, and 〈nb〉 obtained experimentally and those obtained by multisource thermal model shows a fair agreement between the model and the experiment for a wide range of projectiles.It can be noticed from this table that the values of〈nb〉 are nearly independent of the nature of incident projectiles, while 〈ns〉, 〈ng〉, and 〈nh〉 show their dependence on the projectile mass number Ap. This fact indicates that the target evaporation fragments do not seem to depend on the mass of the projectile.Figure1 presents the multiplicity distributions of shower particles in interactions of Si28, Ne22, O16, C12, and He4 with (a) Em, (b) H, (c) CNO, and (d) AgBr. For comparison, distributions obtained by the multisource thermal model calculations are also shown as curves. All the distributions are normalized to one. 4He is represented by solid histogram, 12C by dashed one, 16O by dotted one, 22Ne by dash-dotted one, and 28Si by beaded one. It can be noticed from this figure that the model is in good agreement with the experimental data for the projectiles. The height of the multiplicity distribution of ns decreases with increase in projectile mass, while the position of the peak moves to higher multiplicities with increasing AT. The distributions get wider with increasing AT where the distributions have larger tails. This may reflect the effect of the target mass number on the number of collisions of Si28, Ne22, O16, C12, and He4 with the target nuclei.The normalized multiplicity distributions for shower particles produced in the interactions of projectiles with (a) Em, (b) H, (c) CNO, and (d) AgBr, respectively. The histograms represent the experimental data, while the curves are the corresponding ones calculated according to the multisource thermal model.
(a)
(b)
(c)
(d)Similarly the multiplicity distributions of grey particles are illustrated in Figure2. It can be seen that the dependence of the height of the distributions on Ap gets weaker than that in case of shower particles. Also one can notice that the distributions for Si28, Ne22, O16, C12, and He4 with AgBr interactions are broader than those for Si28, Ne22, O16, C12, and He4 with Em, H, and CNO.The normalized multiplicity distributions for the grey particles produced in the interactions of projectiles with (a) Em, (b) H, (c) CNO, and (d) AgBr, respectively. The histograms represent the experimental data, while the curves are the corresponding ones calculated according to the multisource thermal model.
(a)
(b)
(c)
(d)The same features can be observed for black particles (nb) distributions given in Figure 3 as for ng. Figure 3(a) shows a tow-peak structure. The two peaks are around 0 and 10; this could be due to interactions with light- and heavy-target nuclei, respectively.The normalized multiplicity distributions for black particles produced in the interactions of projectiles with (a) Em, (b) H, (c) CNO, and (d) AgBr, respectively. The histograms represent the experimental data, while the curves are the corresponding ones calculated according to the multisource thermal model.
(a)
(b)
(c)
(d)Finally, in Figure4, we investigate the heavy particles (nh=ng+nb) distribution. Two obvious peaks are observed here, one around multiplicity 0 and the other one at 5. The first peak could be due to interactions with H and the second one with CNO. There is one around 20, but it is flattened, and this could represent the collision with AgBr. In all the above distributions, the model can reproduce the multiplicity characteristics for the different particles, and the experimental distributions are in agreement (within errors) with the theoretical ones.Figure 4
Normalized multiplicity distributions for heavy particles produced in the interactions of projectiles with Em at (4.1–4.5) A GeV/c. The histograms represent the experimental data, while the curves are the corresponding ones calculated according to the multisource thermal model.The parameter values of〈ni1〉, 〈ni2〉, and 〈ni3〉, m1, m2, and m3, and k1, k2, and k3 for multiplicity distributions for shower, grey, black, and heavy particles along with χ2/dof are illustrated in Tables 2–5. Table 2 shows these parameter values for Em, while the parameter values for H, CNO, and AgBr are given in Tables 3, 4, and 5, respectively.Table 2
Parameter values for multiplicity distributions for shower, grey, black, and heavy particles. The values of〈ni1〉, 〈ni2〉, and 〈ni3〉 and m1, m2, and m3, and k1, k2, and k3 for 4He-, 12C-, 16O-, and 28Si-Em at 4.5 A GeV/c and 22Ne-Em at 4.1 A GeV/c.
Projectile
〈
n
i
1
〉
〈
n
i
2
〉
〈
n
i
3
〉
m
1
m
2
m
3
k
1
k
2
k
3
χ
2/dof
4He
n
s
2.2
—
—
2
—
—
1
0
0
1.026
n
g
2.5
9.2
—
1
1
—
0.5
0.5
0
1.379
n
b
0.87
0.96
1.16
1
4
10
0.33
0.4
0.27
1.711
n
h
0.74
0.59
5
2
10
4
0.32
0.32
0.36
1.663
12C
n
s
4.6
7.1
—
1
2
—
0.5
0.5
0
1.143
n
g
2.3
9.2
—
1
1
—
0.5
0.5
0
1.232
n
b
0.89
0.96
1.12
1
4
10
0.32
0.38
0.30
0.658
n
h
0.75
0.63
5.5
2
9
4
0.31
0.38
0.31
1.416
16O
n
s
4.9
7.8
—
1
2
—
0.5
0.5
0
0.642
n
g
2.45
5.38
3.1
1
1
7
0.42
0.42
0.26
0.887
n
b
0.93
0.88
1.05
1
4
10
0.32
0.38
0.30
0.692
n
h
0.74
0.63
7.75
2
9
3
0.31
0.31
0.38
0.611
22Ne
n
s
4.2
9.2
—
1
2
—
0.5
0.5
0
0.476
n
g
2.2
14.2
—
1
1
—
0.5
0.5
0
0.688
n
b
0.93
0.75
1.24
1
4
8
0.32
0.33
0.35
0.584
n
h
0.74
0.81
4.83
2
7
5
0.32
0.32
0.36
0.43
28Si
n
s
1.8
9.3
—
2
2
—
0.42
0.58
0
0.917
n
g
2.6
11
—
1
1
—
0.5
0.5
0
1.161
n
b
0.8
0.7
1.35
1
5
8
0.31
0.35
0.34
1.84
n
h
0.64
0.82
6
2
7
4
0.28
0.35
0.37
0.843Table 3
Parameter values for multiplicity distributions for shower, grey, and black particles. The values of〈ni1〉, 〈ni2〉, m1, m2, and k1, k2 for 4He-, 12C-, 16O-, and 28Si-H at 4.5 A GeV/c and 22Ne-H at 4.1 A GeV/c.
Projectile
〈
n
i
1
〉
〈
n
i
2
〉
m
1
m
2
k
1
k
2
χ
2/dof
4He
n
s
1.16
—
2
—
1
0
1.248
n
g
0.14
—
6
—
1
0
0.02
n
b
0.12
—
6
—
1
0
0.049
12C
n
s
0.85
1.36
1
3
0.5
0.5
1.153
n
g
0.14
—
6
—
1
0
0.039
n
b
0.127
—
6
—
1
0
0.006
16O
n
s
0.9
1.1
1
4
0.5
0.5
1.622
n
g
0.137
—
6
—
1
0
0.003
n
b
0.133
—
6
—
1
0
0.004
22Ne
n
s
0.89
0.06
1
4
0.5
0.5
1.074
n
g
0.134
—
6
—
1
0
0.02
n
b
0.133
—
6
—
1
0
0.008
28Si
n
s
1.41
0.89
1
5
0.5
0.5
0.912
n
g
0.134
—
6
—
1
0
0.006
n
b
0.12
—
6
—
1
0
0.009Table 4
Parameter values for multiplicity distributions for shower, grey, and  black particles. The values of〈ni1〉, 〈ni2〉, m1, m2, and k1, k2 for 4He-, 12C-, 16O-, and 28Si-CNO at 4.5 A GeV/c and 22Ne-CNO at 4.1 A GeV/c.
Projectile
〈
n
i
1
〉
〈
n
i
2
〉
m
1
m
2
k
1
k
2
χ
2/dof
4He
n
s
0.9
1.62
3
3
0.5
0.5
1.783
n
g
0.95
—
3
—
1
0
1.29
n
b
0.75
—
4
—
1
0
1.41
12C
n
s
1.72
2.35
2
4
0.45
0.55
1.293
n
g
0.95
—
3
—
1
0
1.027
n
b
0.75
—
4
—
1
0
1.408
16O
n
s
1.75
2.52
2
4
0.45
0.55
1.223
n
g
0.95
—
3
—
1
0
1.475
n
b
0.75
—
4
—
1
0
1.853
22Ne
n
s
1.78
3
2
4
0.58
0.42
0.736
n
g
0.95
—
3
—
1
0
1.29
n
b
0.75
—
4
—
1
0
1.776
28Si
n
s
2.35
2.1
2
8
0.70
0.30
1.831
n
g
0.96
—
3
—
1
0
0.136
n
b
0.75
—
4
—
1
0
1.964Table 5
Parameter values for multiplicity distributions for shower, grey, and  black particles. The values of〈ni1〉, 〈ni2〉, m1, m2, and k1, k2 for 4He-, 12C-, 16O-, and 28Si-AgBr at 4.5 A GeV/c and 22Ne-AgBr  at 4.1 A GeV/c.
Projectile
〈
n
i
1
〉
〈
n
i
2
〉
m
1
m
2
k
1
k
2
χ
2/dof
4He
n
s
2.45
0.9
2
8
0.4
0.6
0.72
n
g
1.23
2.3
5
6
0.5
0.5
0.878
n
b
1.67
—
6
—
1
0
1.109
12C
n
s
2.6
2.4
3
8
0.42
0.58
0.702
n
g
1.35
3.2
5
5
0.5
0.5
1.217
n
b
1.67
—
6
—
1
0
0.744
16O
n
s
3.55
2.96
3
8
0.5
0.5
1.883
n
g
1.28
2.5
6
8
0.56
0.44
1.983
n
b
1.62
—
6
—
1
0
1.296
22Ne
n
s
6.3
5.25
2
5
0.5
0.5
1.102
n
g
1.6
3.35
5
6
0.5
0.5
1.018
n
b
1.58
—
6
—
1
0
1.083
28Si
n
s
5.8
6.3
2
5
0.5
0.5
1.146
n
g
1.46
4
5
5
0.5
0.5
1.69
n
b
2
—
5
—
1
0
1.127


### 3.2. Angular Distributions

The angular distributions for the different secondary charged particlesθs, θg, and θb emitted in Si28, Ne22, O16, C12, and He4 interactions with Em, together with their corresponding distributions obtained by the multisource thermal model, are given in Figures 5–7. It is evident from these figures that the values of θs, θg, and θb are nearly independent of the nature of incident projectiles. Figure 5 illustrated the angular distributions of shower particles. The curves are the distributions obtained by the model calculations. From Figure 5, it can be noticed that the peak increases with increase in projectile mass and the model is in agreement with experimental data in describing angular distributions of the shower particles.Figure 5
Angular distributions of shower particle produced inHe4- (solid triangle), C12- (open square), O16- (solid square), Ne22- (open circles), and Si28- (solid circles) Em collisions at (4.1–4.5) A GeV/c. The symbols represent the experimental data. Capone the curves are our calculated results by multisource thermal model.Figure 6
Angular distributions of grey particle produced inHe4- (solid triangle),C12- (open square), O16- (solid square), Ne22- (open circles), and Si28- (solid circles) Em collisions at (4.1–4.5) A GeV/c. The symbols represent the experimental data. Capone the curves are our calculated results by multisource thermal model.Figure 7
Angular distributions of black particle produced inHe4- (solid triangle), C12- (open square), O16- (solid square), Ne22- (open circles), and Si28- (solid circles) Em collisions at (4.1–4.5) A GeV/c. The symbols represent the experimental data. Capone the curves are our calculated results by multisource thermal model.The angular distributions of grey particles,θg, are illustrated in Figure 6. For comparison, distributions obtained by the model calculations are also shown. The model agrees with experimental data in describing angular distributions for the grey particles. Also, it is notable that the angular distributions of grey particles become wider than those of shower particles with displacement of the peak position to higher values of θ.Figure7 illustrated the angular distributions of black particles, θb. For comparison, distributions obtained by the multisource thermal model calculations are also shown. The model agrees with experimental data. It can be noticed that the angular distributions for black particles are nearly symmetrical around the peak position.From Figure5 to Figure 7, we observe that the angular distributions of shower, grey, and black particles produced are independent of projectile mass. The peak position shifts towards higher values of θ with increase in product mass, that is, θsp<θgp<θbp, which is clear in Table 6. The parameter values for angular distributions for shower, grey, and black particles ax, ay, and az and bx, by, and bz are given in Table 7.Table 6
The peak position of angular distributions for shower, grey, and black particles.
Projectile
θ
s
p
θ
g
p
θ
b
p
4He
20
60
90
12C
20
60
90
16O
20
60
90
22Ne
20
60
90
28Si
20
60
90Table 7
Parameter values for angular distributions for shower, grey, and black particles. The values ofax, ay, and az and bx, by, and bz for 4He-, 12C-, 16O-, and 28Si-Em at 4.5 A GeV/c and 22Ne-Em at 4.1 A GeV/c.
Projectile
a
x
a
y
a
z
b
x
b
y
b
z
4He
n
s
1
1
1.3
0
0
1.7
n
g
1.25
1
1
0
0
0.47
n
b
1.2
1
1
0
0
−0.15
12C
n
s
1
1
1.3
0
0
1.7
n
g
1.25
1
1
0
0
0.47
n
b
1.2
1
1
0
0
−0.15
16O
n
s
1
1
1.3
0
0
1.85
n
g
1.25
1
1
0
0
0.47
n
b
1.2
1
1
0
0
−0.15
22Ne
n
s
1
1
1.3
0
0
1.9
n
g
1.25
1
1
0
0
0.47
n
b
1.2
1
1
0
0
−0.15
28Si
n
s
1
1
1.3
0
0
2
n
g
1.25
1
1
0
0
0.47
n
b
1.2
1
1
0
0
−0.15


## 3.1. Multiplicity Characteristics

To study the multiplicity behavior of the target fragmentation as function of mass number of the target nucleusAT, we classify the emulsion nuclei based on nh (nh=ng+nb) into three groups [38]: two types of light nuclei (H and CNO) and one type of heavy nuclei (AgBr). Collisions with H target nuclei are events with nh=0,1, collisions with only one bound nucleon in CNO or AgBr target nuclei these events having 2≤nh≤7 are mostly interactions with CNO targets with some admixture of peripheral AgBr interactions. All events with nh≥8 are only due to AgBr interactions. It should be noted that the classification of events in emulsion is not unique; however, there is no perfect method for classifying events due to the limitations of the emulsion technique [39].Table1 illustrates the average values of shower, grey, black, and heavy particles produced in interactions of different projectiles with Em at momentum 4.1–4.5 A GeV/c. The experimental data has been taken from [40] and the data available from the High Energy Physics Group at Sana’a University.Table 1
The average multiplicities of the charged secondary particles emitted in4He-, 12C-, 16O-, 22Ne-, and 28Si-Em interactions and those emitted in the interactions with the different groups of target nuclei (H, CNO, Em, and AgBr) compared with those obtained by the model.
Projectile
〈
n
i
〉
H
CNO
Em
AgBr
4He
MSTM
〈
n
s
〉
1.623
3.3263
3.903
5.78
EXP
〈
n
s
〉
1.619 ± 0.1
3.112 ± 0.133
3.757 ± 0.096
5.77 ± 0.151
MSTM
〈
n
g
〉
0.288
2.35
5.372
9.474
EXP
〈
n
g
〉
0.288 ± 0.03
2 ± 0.066
4.3899 ± 0.17
9.38 ± 0.3
MSTM
〈
n
b
〉
0.163
2.5
4.86
9.519
EXP
〈
n
b
〉
0.172 ± 0.024
2.25 ± 0.07
4.395 ± 0.149
9.223 ± 0.21
12C
MSTM
〈
n
s
〉
2.014
6.22
8.897
13.975
EXP
〈
n
s
〉
2.062 ± 0.186
6 ± 0.25
8.0 ± 0.24
13.975 ± 0.4
MSTM
〈
n
g
〉
0.288
2.35
5.274
10.882
EXP
〈
n
g
〉
0.293 ± 0.03
2.04 ± 0.1
5.081 ± 0.22
10.753 ± 0.4
MSTM
〈
n
b
〉
0.205
2.5
4.6346
9.519
EXP
〈
n
b
〉
0.204 ± 0.027
2.28 ± 0.075
4.57 ± 0.16
9.293 ± 0.21
16O
MSTM
〈
n
s
〉
2.196
6.621
9.741
16.664
EXP
〈
n
s
〉
2.454 ± 0.145
6.679 ± 0.197
9.417 ± 0.206
16.426 ± 0.349
MSTM
〈
n
g
〉
0.268
2.35
6.284
12.6
EXP
〈
n
g
〉
0.264 ± 0.02
2.154 ± 0.06
6.353 ± 0.17
12.454 ± 0.29
MSTM
〈
n
b
〉
0.243
2.5
4.315434
9.217
EXP
〈
n
b
〉
0.24 ± 0.02
2.17 ± 0.05
4.421 ± 0.097
8.58 ± 0.13
22Ne
MSTM
〈
n
s
〉
2.11
6.605
10.769
18.919
EXP
〈
n
s
〉
2.24 ± 0.0898
6.66 ± 0.151
10.254 ± 0.113
18.482 ± 0.29
MSTM
〈
n
g
〉
0.249
2.35
6.752
13.547
EXP
〈
n
g
〉
0.252 ± 0.013
2.2 ± 0.04
6.188 ± 0.295
13.495 ± 0.214
MSTM
〈
n
b
〉
0.243
2.301
4.288
8.978
EXP
〈
n
b
〉
0.238 ± 0.013
2.05 ± 0.034
4.151 ± 0.103
8.482 ± 0.09
28Si
MSTM
〈
n
s
〉
2.461
7.836
11.776
21.007
EXP
〈
n
s
〉
2.887 ± 0.22
7.51 ± 0.297
11.784 ± 0.343
20.163 ± 0.602
MSTM
〈
n
g
〉
0.249
2.39
6.071
13.156
EXP
〈
n
g
〉
0.2474 ± 0.03
2.266 ± 0.07
6.405 ± 0.228
13.224 ± 0.383
MSTM
〈
n
b
〉
0.163
2.5
4.678
9.504
EXP
〈
n
b
〉
0.165 ± 0.022
2.2 ± 0.064
4.449 ± 0.1244
8.651 ± 0.17The comparison between the average values of the multiplicities〈ns〉, 〈ng〉, and 〈nb〉 obtained experimentally and those obtained by multisource thermal model shows a fair agreement between the model and the experiment for a wide range of projectiles.It can be noticed from this table that the values of〈nb〉 are nearly independent of the nature of incident projectiles, while 〈ns〉, 〈ng〉, and 〈nh〉 show their dependence on the projectile mass number Ap. This fact indicates that the target evaporation fragments do not seem to depend on the mass of the projectile.Figure1 presents the multiplicity distributions of shower particles in interactions of Si28, Ne22, O16, C12, and He4 with (a) Em, (b) H, (c) CNO, and (d) AgBr. For comparison, distributions obtained by the multisource thermal model calculations are also shown as curves. All the distributions are normalized to one. 4He is represented by solid histogram, 12C by dashed one, 16O by dotted one, 22Ne by dash-dotted one, and 28Si by beaded one. It can be noticed from this figure that the model is in good agreement with the experimental data for the projectiles. The height of the multiplicity distribution of ns decreases with increase in projectile mass, while the position of the peak moves to higher multiplicities with increasing AT. The distributions get wider with increasing AT where the distributions have larger tails. This may reflect the effect of the target mass number on the number of collisions of Si28, Ne22, O16, C12, and He4 with the target nuclei.The normalized multiplicity distributions for shower particles produced in the interactions of projectiles with (a) Em, (b) H, (c) CNO, and (d) AgBr, respectively. The histograms represent the experimental data, while the curves are the corresponding ones calculated according to the multisource thermal model.
(a)
(b)
(c)
(d)Similarly the multiplicity distributions of grey particles are illustrated in Figure2. It can be seen that the dependence of the height of the distributions on Ap gets weaker than that in case of shower particles. Also one can notice that the distributions for Si28, Ne22, O16, C12, and He4 with AgBr interactions are broader than those for Si28, Ne22, O16, C12, and He4 with Em, H, and CNO.The normalized multiplicity distributions for the grey particles produced in the interactions of projectiles with (a) Em, (b) H, (c) CNO, and (d) AgBr, respectively. The histograms represent the experimental data, while the curves are the corresponding ones calculated according to the multisource thermal model.
(a)
(b)
(c)
(d)The same features can be observed for black particles (nb) distributions given in Figure 3 as for ng. Figure 3(a) shows a tow-peak structure. The two peaks are around 0 and 10; this could be due to interactions with light- and heavy-target nuclei, respectively.The normalized multiplicity distributions for black particles produced in the interactions of projectiles with (a) Em, (b) H, (c) CNO, and (d) AgBr, respectively. The histograms represent the experimental data, while the curves are the corresponding ones calculated according to the multisource thermal model.
(a)
(b)
(c)
(d)Finally, in Figure4, we investigate the heavy particles (nh=ng+nb) distribution. Two obvious peaks are observed here, one around multiplicity 0 and the other one at 5. The first peak could be due to interactions with H and the second one with CNO. There is one around 20, but it is flattened, and this could represent the collision with AgBr. In all the above distributions, the model can reproduce the multiplicity characteristics for the different particles, and the experimental distributions are in agreement (within errors) with the theoretical ones.Figure 4
Normalized multiplicity distributions for heavy particles produced in the interactions of projectiles with Em at (4.1–4.5) A GeV/c. The histograms represent the experimental data, while the curves are the corresponding ones calculated according to the multisource thermal model.The parameter values of〈ni1〉, 〈ni2〉, and 〈ni3〉, m1, m2, and m3, and k1, k2, and k3 for multiplicity distributions for shower, grey, black, and heavy particles along with χ2/dof are illustrated in Tables 2–5. Table 2 shows these parameter values for Em, while the parameter values for H, CNO, and AgBr are given in Tables 3, 4, and 5, respectively.Table 2
Parameter values for multiplicity distributions for shower, grey, black, and heavy particles. The values of〈ni1〉, 〈ni2〉, and 〈ni3〉 and m1, m2, and m3, and k1, k2, and k3 for 4He-, 12C-, 16O-, and 28Si-Em at 4.5 A GeV/c and 22Ne-Em at 4.1 A GeV/c.
Projectile
〈
n
i
1
〉
〈
n
i
2
〉
〈
n
i
3
〉
m
1
m
2
m
3
k
1
k
2
k
3
χ
2/dof
4He
n
s
2.2
—
—
2
—
—
1
0
0
1.026
n
g
2.5
9.2
—
1
1
—
0.5
0.5
0
1.379
n
b
0.87
0.96
1.16
1
4
10
0.33
0.4
0.27
1.711
n
h
0.74
0.59
5
2
10
4
0.32
0.32
0.36
1.663
12C
n
s
4.6
7.1
—
1
2
—
0.5
0.5
0
1.143
n
g
2.3
9.2
—
1
1
—
0.5
0.5
0
1.232
n
b
0.89
0.96
1.12
1
4
10
0.32
0.38
0.30
0.658
n
h
0.75
0.63
5.5
2
9
4
0.31
0.38
0.31
1.416
16O
n
s
4.9
7.8
—
1
2
—
0.5
0.5
0
0.642
n
g
2.45
5.38
3.1
1
1
7
0.42
0.42
0.26
0.887
n
b
0.93
0.88
1.05
1
4
10
0.32
0.38
0.30
0.692
n
h
0.74
0.63
7.75
2
9
3
0.31
0.31
0.38
0.611
22Ne
n
s
4.2
9.2
—
1
2
—
0.5
0.5
0
0.476
n
g
2.2
14.2
—
1
1
—
0.5
0.5
0
0.688
n
b
0.93
0.75
1.24
1
4
8
0.32
0.33
0.35
0.584
n
h
0.74
0.81
4.83
2
7
5
0.32
0.32
0.36
0.43
28Si
n
s
1.8
9.3
—
2
2
—
0.42
0.58
0
0.917
n
g
2.6
11
—
1
1
—
0.5
0.5
0
1.161
n
b
0.8
0.7
1.35
1
5
8
0.31
0.35
0.34
1.84
n
h
0.64
0.82
6
2
7
4
0.28
0.35
0.37
0.843Table 3
Parameter values for multiplicity distributions for shower, grey, and black particles. The values of〈ni1〉, 〈ni2〉, m1, m2, and k1, k2 for 4He-, 12C-, 16O-, and 28Si-H at 4.5 A GeV/c and 22Ne-H at 4.1 A GeV/c.
Projectile
〈
n
i
1
〉
〈
n
i
2
〉
m
1
m
2
k
1
k
2
χ
2/dof
4He
n
s
1.16
—
2
—
1
0
1.248
n
g
0.14
—
6
—
1
0
0.02
n
b
0.12
—
6
—
1
0
0.049
12C
n
s
0.85
1.36
1
3
0.5
0.5
1.153
n
g
0.14
—
6
—
1
0
0.039
n
b
0.127
—
6
—
1
0
0.006
16O
n
s
0.9
1.1
1
4
0.5
0.5
1.622
n
g
0.137
—
6
—
1
0
0.003
n
b
0.133
—
6
—
1
0
0.004
22Ne
n
s
0.89
0.06
1
4
0.5
0.5
1.074
n
g
0.134
—
6
—
1
0
0.02
n
b
0.133
—
6
—
1
0
0.008
28Si
n
s
1.41
0.89
1
5
0.5
0.5
0.912
n
g
0.134
—
6
—
1
0
0.006
n
b
0.12
—
6
—
1
0
0.009Table 4
Parameter values for multiplicity distributions for shower, grey, and  black particles. The values of〈ni1〉, 〈ni2〉, m1, m2, and k1, k2 for 4He-, 12C-, 16O-, and 28Si-CNO at 4.5 A GeV/c and 22Ne-CNO at 4.1 A GeV/c.
Projectile
〈
n
i
1
〉
〈
n
i
2
〉
m
1
m
2
k
1
k
2
χ
2/dof
4He
n
s
0.9
1.62
3
3
0.5
0.5
1.783
n
g
0.95
—
3
—
1
0
1.29
n
b
0.75
—
4
—
1
0
1.41
12C
n
s
1.72
2.35
2
4
0.45
0.55
1.293
n
g
0.95
—
3
—
1
0
1.027
n
b
0.75
—
4
—
1
0
1.408
16O
n
s
1.75
2.52
2
4
0.45
0.55
1.223
n
g
0.95
—
3
—
1
0
1.475
n
b
0.75
—
4
—
1
0
1.853
22Ne
n
s
1.78
3
2
4
0.58
0.42
0.736
n
g
0.95
—
3
—
1
0
1.29
n
b
0.75
—
4
—
1
0
1.776
28Si
n
s
2.35
2.1
2
8
0.70
0.30
1.831
n
g
0.96
—
3
—
1
0
0.136
n
b
0.75
—
4
—
1
0
1.964Table 5
Parameter values for multiplicity distributions for shower, grey, and  black particles. The values of〈ni1〉, 〈ni2〉, m1, m2, and k1, k2 for 4He-, 12C-, 16O-, and 28Si-AgBr at 4.5 A GeV/c and 22Ne-AgBr  at 4.1 A GeV/c.
Projectile
〈
n
i
1
〉
〈
n
i
2
〉
m
1
m
2
k
1
k
2
χ
2/dof
4He
n
s
2.45
0.9
2
8
0.4
0.6
0.72
n
g
1.23
2.3
5
6
0.5
0.5
0.878
n
b
1.67
—
6
—
1
0
1.109
12C
n
s
2.6
2.4
3
8
0.42
0.58
0.702
n
g
1.35
3.2
5
5
0.5
0.5
1.217
n
b
1.67
—
6
—
1
0
0.744
16O
n
s
3.55
2.96
3
8
0.5
0.5
1.883
n
g
1.28
2.5
6
8
0.56
0.44
1.983
n
b
1.62
—
6
—
1
0
1.296
22Ne
n
s
6.3
5.25
2
5
0.5
0.5
1.102
n
g
1.6
3.35
5
6
0.5
0.5
1.018
n
b
1.58
—
6
—
1
0
1.083
28Si
n
s
5.8
6.3
2
5
0.5
0.5
1.146
n
g
1.46
4
5
5
0.5
0.5
1.69
n
b
2
—
5
—
1
0
1.127


## 3.2. Angular Distributions

The angular distributions for the different secondary charged particlesθs, θg, and θb emitted in Si28, Ne22, O16, C12, and He4 interactions with Em, together with their corresponding distributions obtained by the multisource thermal model, are given in Figures 5–7. It is evident from these figures that the values of θs, θg, and θb are nearly independent of the nature of incident projectiles. Figure 5 illustrated the angular distributions of shower particles. The curves are the distributions obtained by the model calculations. From Figure 5, it can be noticed that the peak increases with increase in projectile mass and the model is in agreement with experimental data in describing angular distributions of the shower particles.Figure 5
Angular distributions of shower particle produced inHe4- (solid triangle), C12- (open square), O16- (solid square), Ne22- (open circles), and Si28- (solid circles) Em collisions at (4.1–4.5) A GeV/c. The symbols represent the experimental data. Capone the curves are our calculated results by multisource thermal model.Figure 6
Angular distributions of grey particle produced inHe4- (solid triangle),C12- (open square), O16- (solid square), Ne22- (open circles), and Si28- (solid circles) Em collisions at (4.1–4.5) A GeV/c. The symbols represent the experimental data. Capone the curves are our calculated results by multisource thermal model.Figure 7
Angular distributions of black particle produced inHe4- (solid triangle), C12- (open square), O16- (solid square), Ne22- (open circles), and Si28- (solid circles) Em collisions at (4.1–4.5) A GeV/c. The symbols represent the experimental data. Capone the curves are our calculated results by multisource thermal model.The angular distributions of grey particles,θg, are illustrated in Figure 6. For comparison, distributions obtained by the model calculations are also shown. The model agrees with experimental data in describing angular distributions for the grey particles. Also, it is notable that the angular distributions of grey particles become wider than those of shower particles with displacement of the peak position to higher values of θ.Figure7 illustrated the angular distributions of black particles, θb. For comparison, distributions obtained by the multisource thermal model calculations are also shown. The model agrees with experimental data. It can be noticed that the angular distributions for black particles are nearly symmetrical around the peak position.From Figure5 to Figure 7, we observe that the angular distributions of shower, grey, and black particles produced are independent of projectile mass. The peak position shifts towards higher values of θ with increase in product mass, that is, θsp<θgp<θbp, which is clear in Table 6. The parameter values for angular distributions for shower, grey, and black particles ax, ay, and az and bx, by, and bz are given in Table 7.Table 6
The peak position of angular distributions for shower, grey, and black particles.
Projectile
θ
s
p
θ
g
p
θ
b
p
4He
20
60
90
12C
20
60
90
16O
20
60
90
22Ne
20
60
90
28Si
20
60
90Table 7
Parameter values for angular distributions for shower, grey, and black particles. The values ofax, ay, and az and bx, by, and bz for 4He-, 12C-, 16O-, and 28Si-Em at 4.5 A GeV/c and 22Ne-Em at 4.1 A GeV/c.
Projectile
a
x
a
y
a
z
b
x
b
y
b
z
4He
n
s
1
1
1.3
0
0
1.7
n
g
1.25
1
1
0
0
0.47
n
b
1.2
1
1
0
0
−0.15
12C
n
s
1
1
1.3
0
0
1.7
n
g
1.25
1
1
0
0
0.47
n
b
1.2
1
1
0
0
−0.15
16O
n
s
1
1
1.3
0
0
1.85
n
g
1.25
1
1
0
0
0.47
n
b
1.2
1
1
0
0
−0.15
22Ne
n
s
1
1
1.3
0
0
1.9
n
g
1.25
1
1
0
0
0.47
n
b
1.2
1
1
0
0
−0.15
28Si
n
s
1
1
1.3
0
0
2
n
g
1.25
1
1
0
0
0.47
n
b
1.2
1
1
0
0
−0.15


## 4. Conclusion

We conclude that multisource thermal model gives uniform description of the target fragmentation in interaction ofHe4, C12, O16, Ne22, and Si28 with emulsion at 4.1–4.5 A GeV/c. This model has succeeded in reproduction of the general characteristics of interactions of nuclei with emulsion such as average multiplicities, multiplicity distributions, and angular distributions of particles produced in N-N collisions.



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*Source: 403504-2014-04-07.xml*