Advances in High Energy Physics

Volume 2014 (2014), Article ID 403504, 12 pages

http://dx.doi.org/10.1155/2014/403504

Research Article

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

S. Fakhraddin,1,2 Magda A. Rahim,1,2 and Hamdy M. M. Zayd¹

¹Physics Department, Faculty of Science, Sana’a University, Sana’a, Yemen

²Physics Department, Faculty of Science & Arts (Girls Division), Najran University, Najran, Saudi Arabia

Received 11 December 2013; Accepted 17 February 2014; Published 7 April 2014

Academic Editor: Chen Wu

Copyright © 2014 S. Fakhraddin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

The multiplicity distributions of shower, grey, and black particles produced in interactions of ⁴He, 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.

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 and 4.5 A GeV/c (, , , and ) 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 groups (subsamples). The source number in the th group is assumed to be . Each source contributes multiplicity distribution to be an exponential distribution.

The multiplicity () distribution contributed by the th source in the th group is an exponential function:

where

is the mean multiplicity contributed by the th source in the th group. As in [31], we assume that

The multiplicity () distribution contributed by the th group is the fold of exponential functions; that is,

It is an Erlang distribution. The total multiplicity distribution contributed by the groups can be written as

where is the relative weight contributed by the th group.

In the Monte Carlo calculation, let denote random variable in . For the th group, we have

The multiplicity distribution is obtained by a statistical method. Meanwhile, the mean multiplicities contributed by the th group and the groups are given by

respectively.

Generally speaking [35], for , , and collisions at not too high energies (less than a few hundred GeV). The parameters for the collisions are and . For the mentioned collisions at very high energies (greater than a few hundred GeV), or 3 due to the different interaction mechanisms existing in the event samples. For and N-N collisions at a fixed impact parameter, can be regarded as the number of participant nucleons. The weight 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 the oz-axis, and let the reaction plane be the xoz plane. Each source is assumed to emit particles isotropically in the source rest frame. As the first approximation, the three components of the particle momentum in the source rest frame are assumed to obey Gaussian distributions with the same deviation width [37]. Considering the motion of the emission source and the interactions among emission sources, the particle momentum components , , and 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 and , and , and and are linear:

where , , and are free parameters and is the parameter that characterizes the width of the momentum distribution in the source reference frame. , , , , , and are free parameters. Let , , , , , and denote random variables distributed in ; we have

because , , and obey a Gaussian distribution law.

The emission angle of a target fragment in the laboratory reference frame is given by

Considering (9)–(11), we have

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 nucleus , we classify the emulsion nuclei based on () 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 , collisions with only one bound nucleon in CNO or AgBr target nuclei these events having are mostly interactions with CNO targets with some admixture of peripheral AgBr interactions. All events with 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].

Table 1 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 in ⁴He-, 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

H
CNO
Em
AgBr

4He
MSTM

1.623
3.3263
3.903
5.78

EXP

1.619 ± 0.1
3.112 ± 0.133
3.757 ± 0.096
5.77 ± 0.151

MSTM

0.288
2.35
5.372
9.474

EXP

0.288 ± 0.03
2 ± 0.066
4.3899 ± 0.17
9.38 ± 0.3

MSTM

0.163
2.5
4.86
9.519

EXP

0.172 ± 0.024
2.25 ± 0.07
4.395 ± 0.149
9.223 ± 0.21

12C
MSTM

2.014
6.22
8.897
13.975

EXP

2.062 ± 0.186
6 ± 0.25
8.0 ± 0.24
13.975 ± 0.4

MSTM

0.288
2.35
5.274
10.882

EXP

0.293 ± 0.03
2.04 ± 0.1
5.081 ± 0.22
10.753 ± 0.4

MSTM

0.205
2.5
4.6346
9.519

EXP

0.204 ± 0.027
2.28 ± 0.075
4.57 ± 0.16
9.293 ± 0.21

16O
MSTM

2.196
6.621
9.741
16.664

EXP

2.454 ± 0.145
6.679 ± 0.197
9.417 ± 0.206
16.426 ± 0.349

MSTM

0.268
2.35
6.284
12.6

EXP

0.264 ± 0.02
2.154 ± 0.06
6.353 ± 0.17
12.454 ± 0.29

MSTM

0.243
2.5
4.315434
9.217

EXP

0.24 ± 0.02
2.17 ± 0.05
4.421 ± 0.097
8.58 ± 0.13

22Ne
MSTM

2.11
6.605
10.769
18.919

EXP

2.24 ± 0.0898
6.66 ± 0.151
10.254 ± 0.113
18.482 ± 0.29

MSTM

0.249
2.35
6.752
13.547

EXP

0.252 ± 0.013
2.2 ± 0.04
6.188 ± 0.295
13.495 ± 0.214

MSTM

0.243
2.301
4.288
8.978

EXP

0.238 ± 0.013
2.05 ± 0.034
4.151 ± 0.103
8.482 ± 0.09

28Si
MSTM

2.461
7.836
11.776
21.007

EXP

2.887 ± 0.22
7.51 ± 0.297
11.784 ± 0.343
20.163 ± 0.602

MSTM

0.249
2.39
6.071
13.156

EXP

0.2474 ± 0.03
2.266 ± 0.07
6.405 ± 0.228
13.224 ± 0.383

MSTM

0.163
2.5
4.678
9.504

EXP

0.165 ± 0.022
2.2 ± 0.064
4.449 ± 0.1244
8.651 ± 0.17

The comparison between the average values of the multiplicities , , and 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 are nearly independent of the nature of incident projectiles, while , , and show their dependence on the projectile mass number . This fact indicates that the target evaporation fragments do not seem to depend on the mass of the projectile.

Figure 1 presents the multiplicity distributions of shower particles in interactions of , , , , and 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 decreases with increase in projectile mass, while the position of the peak moves to higher multiplicities with increasing . The distributions get wider with increasing where the distributions have larger tails. This may reflect the effect of the target mass number on the number of collisions of , , , , and with the target nuclei.

(a)

(b)

(c)

(d)

Figure 1: 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.

Similarly the multiplicity distributions of grey particles are illustrated in Figure 2. It can be seen that the dependence of the height of the distributions on gets weaker than that in case of shower particles. Also one can notice that the distributions for , , , , and with AgBr interactions are broader than those for , , , , and with Em, H, and CNO.

(a)

(b)

(c)

(d)

Figure 2: 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.

The same features can be observed for black particles () distributions given in Figure 3 as for . 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.

(a)

(b)

(c)

(d)

Figure 3: 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.

Finally, in Figure 4, we investigate the heavy particles () 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 , , and , , , and , and , , and for multiplicity distributions for shower, grey, black, and heavy particles along with /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 , , and and , , and , and , , and for ⁴He-, ¹²C-, ¹⁶O-, and ²⁸Si-Em at 4.5 A GeV/c and ²²Ne-Em at 4.1 A GeV/c.

Projectile

/dof

4He

2.2
—
—
2
—
—
1
0
0
1.026

2.5
9.2
—
1
1
—
0.5
0.5
0
1.379

0.87
0.96
1.16
1
4
10
0.33
0.4
0.27
1.711

0.74
0.59
5
2
10
4
0.32
0.32
0.36
1.663

12C

4.6
7.1
—
1
2
—
0.5
0.5
0
1.143

2.3
9.2
—
1
1
—
0.5
0.5
0
1.232

0.89
0.96
1.12
1
4
10
0.32
0.38
0.30
0.658

0.75
0.63
5.5
2
9
4
0.31
0.38
0.31
1.416

16O

4.9
7.8
—
1
2
—
0.5
0.5
0
0.642

2.45
5.38
3.1
1
1
7
0.42
0.42
0.26
0.887

0.93
0.88
1.05
1
4
10
0.32
0.38
0.30
0.692

0.74
0.63
7.75
2
9
3
0.31
0.31
0.38
0.611

22Ne

4.2
9.2
—
1
2
—
0.5
0.5
0
0.476

2.2
14.2
—
1
1
—
0.5
0.5
0
0.688

0.93
0.75
1.24
1
4
8
0.32
0.33
0.35
0.584

0.74
0.81
4.83
2
7
5
0.32
0.32
0.36
0.43

28Si

1.8
9.3
—
2
2
—
0.42
0.58
0
0.917

2.6
11
—
1
1
—
0.5
0.5
0
1.161

0.8
0.7
1.35
1
5
8
0.31
0.35
0.34
1.84

0.64
0.82
6
2
7
4
0.28
0.35
0.37
0.843

Table 3: Parameter values for multiplicity distributions for shower, grey, and black particles. The values of , , , , and , for ⁴He-, ¹²C-, ¹⁶O-, and ²⁸Si-H at 4.5 A GeV/c and ²²Ne-H at 4.1 A GeV/c.

Projectile

/dof

4He

1.16
—
2
—
1
0
1.248

0.14
—
6
—
1
0
0.02

0.12
—
6
—
1
0
0.049

12C

0.85
1.36
1
3
0.5
0.5
1.153

0.14
—
6
—
1
0
0.039

0.127
—
6
—
1
0
0.006

16O

0.9
1.1
1
4
0.5
0.5
1.622

0.137
—
6
—
1
0
0.003

0.133
—
6
—
1
0
0.004

22Ne

0.89
0.06
1
4
0.5
0.5
1.074

0.134
—
6
—
1
0
0.02

0.133
—
6
—
1
0
0.008

28Si

1.41
0.89
1
5
0.5
0.5
0.912

0.134
—
6
—
1
0
0.006

0.12
—
6
—
1
0
0.009

Table 4: Parameter values for multiplicity distributions for shower, grey, and black particles. The values of , , , , and , for ⁴He-, ¹²C-, ¹⁶O-, and ²⁸Si-CNO at 4.5 A GeV/c and ²²Ne-CNO at 4.1 A GeV/c.

Projectile

/dof

4He

0.9
1.62
3
3
0.5
0.5
1.783

0.95
—
3
—
1
0
1.29

0.75
—
4
—
1
0
1.41

12C

1.72
2.35
2
4
0.45
0.55
1.293

0.95
—
3
—
1
0
1.027

0.75
—
4
—
1
0
1.408

16O

1.75
2.52
2
4
0.45
0.55
1.223

0.95
—
3
—
1
0
1.475

0.75
—
4
—
1
0
1.853

22Ne

1.78
3
2
4
0.58
0.42
0.736

0.95
—
3
—
1
0
1.29

0.75
—
4
—
1
0
1.776

28Si

2.35
2.1
2
8
0.70
0.30
1.831

0.96
—
3
—
1
0
0.136

0.75
—
4
—
1
0
1.964

Table 5: Parameter values for multiplicity distributions for shower, grey, and black particles. The values of , , , , and , for ⁴He-, ¹²C-, ¹⁶O-, and ²⁸Si-AgBr at 4.5 A GeV/c and ²²Ne-AgBr at 4.1 A GeV/c.

Projectile

/dof

4He

2.45
0.9
2
8
0.4
0.6
0.72

1.23
2.3
5
6
0.5
0.5
0.878

1.67
—
6
—
1
0
1.109

12C

2.6
2.4
3
8
0.42
0.58
0.702

1.35
3.2
5
5
0.5
0.5
1.217

1.67
—
6
—
1
0
0.744

16O

3.55
2.96
3
8
0.5
0.5
1.883

1.28
2.5
6
8
0.56
0.44
1.983

1.62
—
6
—
1
0
1.296

22Ne

6.3
5.25
2
5
0.5
0.5
1.102

1.6
3.35
5
6
0.5
0.5
1.018

1.58
—
6
—
1
0
1.083

28Si

5.8
6.3
2
5
0.5
0.5
1.146

1.46
4
5
5
0.5
0.5
1.69

2
—
5
—
1
0
1.127

3.2. Angular Distributions

The angular distributions for the different secondary charged particles , , and emitted in , , , , and 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 , , and 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 in - (solid triangle), - (open square), - (solid square), - (open circles), and - (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 in - (solid triangle),- (open square), - (solid square), - (open circles), and - (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 in - (solid triangle), - (open square), - (solid square), - (open circles), and - (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, , 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 .

Figure 7 illustrated the angular distributions of black particles, . 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 Figure 5 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 with increase in product mass, that is, , which is clear in Table 6. The parameter values for angular distributions for shower, grey, and black particles , , and and , , and are given in Table 7.

Table 6: The peak position of angular distributions for shower, grey, and black particles.

Projectile

4He
20
60
90

12C
20
60
90

16O
20
60
90

22Ne
20
60
90

28Si
20
60
90

Table 7: Parameter values for angular distributions for shower, grey, and black particles. The values of , , and and , , and for ⁴He-, ¹²C-, ¹⁶O-, and ²⁸Si-Em at 4.5 A GeV/c and ²²Ne-Em at 4.1 A GeV/c.

Projectile

4He

1
1
1.3
0
0
1.7

1.25
1
1
0
0
0.47

1.2
1
1
0
0
−0.15

12C

1
1
1.3
0
0
1.7

1.25
1
1
0
0
0.47

1.2
1
1
0
0
−0.15

16O

1
1
1.3
0
0
1.85

1.25
1
1
0
0
0.47

1.2
1
1
0
0
−0.15

22Ne

1
1
1.3
0
0
1.9

1.25
1
1
0
0
0.47

1.2
1
1
0
0
−0.15

28Si

1
1
1.3
0
0
2

1.25
1
1
0
0
0.47

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 of , , , , and 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

J. X. Sun, “Application of the multisource thermal model in pseudorapidity distributions of charged particles produced in or pp collisions over an energy range from 0.053 to 7 TeV,” Indian Journal of Pure and Applied Physics, vol. 50, pp. 374–379, 2012.

S. Sarfaraz Ali and H. Khushnood, “Some aspects of secondary charged particles produced in 4.5 A GeV/c 28Si-nucleus interactions,” Research Journal of Recent Sciences, vol. 1, pp. 232–237, 2012.

A. Abd El-Daiem, M. Abdo Khalifa, and A. Abdel-hafiez, “Study of the central heavy-ion collision with emulsion at 4.5 A GeV/c,” Journal of Modern Physics, vol. 2, pp. 669–674, 2011.

S. Fakhraddin, M. A. Rahim, and F.-H. Liu, “Anisotropic emission of final-state products in ²²Ne-emulsion collisions at 4.1A GeV/c,” Nuclear Physics A, vol. 858, no. 1, pp. 95–105, 2011.

E. Q. Wang, F.-H. Liu, M. A. Rahim, S. Fakhraddin, and S. Jian-Xin, “Singly and doubly charged projectile fragments in nucleus-emulsion collisions at dubna energy in the framework of the multi-source model,” Chinese Physics Letters, vol. 28, no. 8, Article ID 082501, 2011.

M. A. Rahim and S. Fakhraddin, “Study of events in absence of target/projectile fragments in interactions of nuclei with emulsion at 4.5 and 4.1 A GeV/c,” Nuclear Physics A, vol. 831, no. 1-2, pp. 39–48, 2009.

M. S. Khan, P. P. Shukla, and H. Khushnood, “Nature of multiplicity correlations in ¹²C-nucleus reaction at 4.5 AGeV,” in Proceedings of the DAE Symposium on Nuclear Physics, vol. 56, 2011.

S. Dhamji, M. M. Aggarwal, and V. S. Bhatia, “Slow particle production in the interactions of 200 a gev sulphur projectile with nuclear emulsion,” Modern Physics Letters A, vol. 16, p. 1801, 2001.

M. E. Solit and A. Abd El-Daiem, “Multiplicity distributions in 24Mg-emulsion collisions at 4.5 A GeV/c,” International Journal of Physical Sciences, vol. 2, no. 12, pp. 324–330, 2007.

S. Fakhraddin, “Study of pseudorapidity distributions of shower particles emitted in interactions of nuclei with emulsion at 4.1-4.5 A GeV/c,” Turkish Journal of Physics, vol. 29, no. 2, pp. 79–84, 2005.

F. H. Liu, “Target black fragment flows in magnesium-emulsion collisions at 4.5A GeV/c,” Radiation Measurements, vol. 43, pp. S234–S238, 2008.

A. A. El-Daiem, “Target fragmentation at 4.5 a GeV/c in ²⁴Mg and ²⁸Si with emulsion interaction,” American Journal of Applied Sciences, vol. 6, no. 4, pp. 608–615, 2009.

A. Abd El-Daiem, “Characteristics of compound multiplicity in ²⁴Mg with emulsion at 4.5 A GeV/c,” Physics International, vol. 1, no. 1, pp. 31–37, 2010.

N. N. Abd-Alaah and M. Mohery, “Features of the compound multiplicity of the interactions of ²⁴Mg and ²⁸Si ions with emulsion nuclei at 4.5A GeV/c,” Turkish Journal of Physics, vol. 25, pp. 109–119, 2001.

B. K. Singh and S. K. Tuli, “Central collisions of ²⁸Si with Ag(Br) nuclei,” Nuclear Physics A, vol. 602, no. 3-4, pp. 487–501, 1996.

F. H. Liu, “Target fragmentation in oxygen-emulsion collisions at dubna and SPS energies,” Chinese Journal of Physics, vol. 40, no. 2, 2002.

F. H. Liu, “Emission of relativistic light fragments in nucleus-emulsion collisions at high energy,” Chinese Journal of Physics, vol. 38, no. 6, 2000.

H. L. Li, D. H. Zhang, and J. S. Li, “Multiplicities of forward-backward particles in ¹⁶O-emulsion interactions at 4.5 AGeV/c,” Chinese Physics C, vol. 33, no. 7, pp. 521–527, 2009.

A. Abdelsalam and B. M. Badawy, “About target fragmentation in nuclear emulsion at relativistic energy,” Journal of Nuclear and Radiation Physics, vol. 3, no. 2, pp. 109–121, 2008.

F. H. Liu, N. N. Abd Allah, B. K. Singh, and T. Ahmad, “angular distributions of target fragments in ⁸⁴kr-em and ¹⁶o-em collisions at high energies,” Acta Physica Polonica B, vol. 37, no. 9, 2006.

B. Bhattacharjee, A. Mukhopadhyay, V. Singh, S.K. Tuli, and S. Sengupta, “On the emission of fast and slow target fragments from ⁸⁴Kr—AgBr interactions at 0.95 GeV/A,” Radiation Measurements, vol. 36, pp. 291–294, 2003.

M. A. Jilany, “Nuclear fragmentation in interactions of 3.7A GeV ²⁴Mg projectiles with emulsion targets,” Physical Review C, vol. 70, no. 1, Article ID 014901, 2004.

A. Abdelsalam, M. S. El-Nagdy, N. Rashed, and B. M. Badawy, “Signatures of target fragmentation of nuclear emulsion by light nuclei,” Chinese Journal of Physics, vol. 45, no. 3, 2007.

B. M. Badawy, “Central collisions induced by light target nuclei,” Journal of Nuclear and Radiation Physics, vol. 3, no. 1, pp. 31–51, 2008.

Z. Dong-Hai, L. Fang, H. Chun-Le, et al., “Features of the total disintegration events of heavy emulsion targets caused by 4.5A GeV/c 16O,” Chinese Physics, vol. 1, no. 11, pp. 2564–2570, 2006.

A. Abdelsalam, M. S. El-Nagdy, N. Rashed, and B. M. Badawy E -El Falaky, “Further study on grey particles production in 3He and 4He interactions with emulsion at 4.5A GeV/c,” Journal of Nuclear and Radiation Physics, vol. 2, no. 1, pp. 49–60, 2007.

A. Al-Haydari and M. T. Hussein, “The multiplicity of particle production from hadronhadron and nucleus-nucleus interaction,” Natural Science, vol. 2, no. 1, pp. 388–397, 2010.

L. G. Moretto, L. Phair, and G. J. Wozniak, “Resilient reducibility in nuclear multifragmentation,” Physical Review C, vol. 60, no. 3, Article ID 031601, 1999.

C. Hartnack, L. Zhuxia, L. Neise, et al., “Quantum molecular dynamics a microscopic model from UNILAC to CERN energies,” Nuclear Physics A, vol. 495, pp. 303–319, 1989.

R. G. Nazmitdinov, E. I. Shahaliev, M. K. Suleymanov, and S. Tomsovic, “Analysis of nucleus-nucleus collisions at high energies and random matrix theory,” Physical Review C, vol. 79, Article ID 054905, 2009.

F. H. Liu, “Unified description of multiplicity distributions of final-state particles produced in collisions at high energies,” Nuclear Physics A, vol. 810, pp. 159–172, 2008.

F. H. Liu, “Longitudinal and transverse flows of protons in 2–8A GeVAu-Au collisions,” European Physical Letters, vol. 63, p. 193, 2003.

F. H. Liu, N. N. Abd-Allah, D.-H. Zhang, and M.-Y. Duan, “angular distributions of target black fragments in nucleus-nucleus collisions at high energy,” International Journal of Modern Physics E, vol. 12, p. 713, 2003.

F.-H. Liu, N. N. A. Allah, and B. K. Singh, “Dependence of black fragment azimuthal and projected angular distributions on polar angle in silicon-emulsion collisions at 4.5A GeV/c,” Physical Review C, vol. 69, no. 5, Article ID 057601, 2004.

F.-H. Liu, Q.-W. Lu, B.-C. Li, and R. Bekmirzaev, “A description of the multiplicity distributions of nuclear fragments in hA and AA collisions at intermediate and high energies,” Chinese Journal of Physics, vol. 49, pp. 601–620, 2011.

F. H. Liu, J. S. Li, and M. Y. Duan, “Light fragment emission in ⁸⁶Kr-¹²⁴Sn collisions at 25 MeV/nucleon,” Physical Review C, vol. 75, Article ID 054613, 5 pages, 2007.

Z. Sun, “Unified description of angular distributions of target particles in light and heavy ion induced interactions in nuclear emulsions at high energies,” Chinese Physics C, vol. 33, pp. 256–262, 2009.

M. A. Rahim, S. Fakhraddin, and H. Asharabi, “Systematic study of projectile fragments in nucleus-nucleus collisions at 4.1-4.5 A GeV/c and multi-source thermal model,” European Physical Journal A, vol. 48, article 115, 2012.

S. Fakhraddin and M. A. Rahim, “Multiplicity distributions of projectile fragments in interactions of nuclei with emulsion at 4.1-4.5 A GeV/c,” Physica Scripta, vol. 78, Article ID 015101, 2008.

S. Fakhraddin, Study of interactions of nuclei with emulsion at high energies [Ph.D. thesis], Cairo University, Cairo, Egypt, 2000.