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

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 ²²Ne and 4.5 A GeV/c (⁴He, ¹²C, ¹⁶O, and ²⁸Si) 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 𝑚_𝑗. Each source contributes multiplicity distribution to be an exponential distribution.

The multiplicity (𝑛_𝑖𝑗) distribution contributed by the 𝑖th source in the 𝑗th group is an exponential function: 𝑃_𝑖𝑗𝑛_𝑖𝑗=1𝑛_𝑖𝑗−𝑛exp^𝑖𝑗𝑛_𝑖𝑗, (1) where 𝑛_𝑖𝑗=𝑛_𝑖𝑗𝑝_𝑖𝑗𝑛_𝑖𝑗𝑑𝑛_𝑖𝑗 (2) is the mean multiplicity contributed by the 𝑖th source in the 𝑗th group. As in [31], we assume that 𝑛_1𝑗=𝑛_2𝑗=𝑛_3𝑗=⋯𝑛_𝑚𝑗𝑗. (3) The multiplicity (𝑛_ch) distribution contributed by the 𝑗th group is the fold of 𝑚_𝑗 exponential functions; that is, 𝑃_𝑗𝑛_ch=𝑛^𝑚𝑗−1_ch𝑚_𝑗−1!𝑛_𝑖𝑗^𝑚𝑗−𝑛exp^ch𝑛_𝑖𝑗. (4) It is an Erlang distribution. The total multiplicity distribution contributed by the 𝑙 groups can be written as 𝑃_𝑗𝑛_ch=1𝑁𝑑𝑁𝑑𝑁_ch=^𝑙_𝑗=1𝑘_𝑗𝑃_𝑗𝑛_ch, (5) where 𝑘_𝑗 is the relative weight contributed by the 𝑗th group.

In the Monte Carlo calculation, let 𝑅_𝑖𝑗 denote random variable in [0,1]. For the 𝑗th group, we have 𝑛_𝑖𝑗=−𝑛_𝑖𝑗ln𝑅_𝑖𝑗,𝑛_ch=−^𝑚𝑗_𝑖=1𝑛_𝑖𝑗ln𝑅_𝑖𝑗. (6) 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 𝑛_ch=𝑚_𝑗𝑛_𝑖𝑗,𝑛_ch=^𝑙_𝑗=1𝑘_𝑗𝑚_𝑗𝑛_𝑖𝑗, (7)(8) 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 ⟨𝑛_𝑖𝑗⟩ and 𝑚_𝑗. 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, 𝑚_𝑗 can be regarded as the number of participant nucleons. The weight 𝑘_𝑗 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 𝑝^⧵_{𝑥,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 𝑝_𝑥, 𝑝_𝑦, and 𝑝_𝑧 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 𝑝_𝑥 and 𝑝^⧵_𝑥, 𝑝_𝑦 and 𝑝^⧵_𝑦, and 𝑝_𝑧 and 𝑝^⧵_𝑧 are linear: 𝑝_𝑥=𝑎_𝑥𝑝^⧵_𝑥+𝐵_𝑥=𝑎_𝑥𝑝^⧵_𝑥+𝑎_𝑥𝜎_𝑖,𝑝_𝑦=𝑎_𝑦𝑝^⧵_𝑦+𝐵_𝑦=𝑎_𝑦𝑝^⧵_𝑦+𝑎_𝑦𝜎_𝑖,𝑝_𝑧=𝑎_𝑧𝑝^⧵_𝑧+𝐵_𝑧=𝑎_𝑧𝑝^⧵_𝑧+𝑎_𝑧𝜎_𝑖, (9) where 𝐵_𝑥, 𝐵_𝑦, and 𝐵_𝑧 are free parameters and 𝜎_𝑖 is the parameter that characterizes the width of the momentum distribution in the source reference frame. 𝑎_𝑥, 𝑏_𝑥, 𝑎_𝑦, 𝑏_𝑦, 𝑎_𝑧, and 𝑏_𝑧 are free parameters. Let 𝑅₁, 𝑅₂, 𝑅₃, 𝑅₄, 𝑅₅, and 𝑅₆ denote random variables distributed in [0,1]; we have 𝑝^⧵_𝑥=−2ln𝑅₁cos2𝜋𝑅₂𝜎_𝑖,𝑝^⧵_𝑦=−2ln𝑅₃cos2𝜋𝑅₄𝜎_𝑖,𝑝^⧵_𝑧=−2ln𝑅₅cos2𝜋𝑅₆𝜎_𝑖 (10) because 𝑝^⧵_𝑥, 𝑝^⧵_𝑦, and 𝑝^⧵_𝑧 obey a Gaussian distribution law.

The emission angle 𝜃 of a target fragment in the laboratory reference frame is given by 𝜃=arctan𝑝²_𝑥+𝑝²_𝑦𝑝_𝑧. (11) Considering (9)–(11), we have𝜃=arctan𝑎_𝑥−2ln𝑅₁cos2𝜋𝑅₂+𝑏_𝑥²+𝑎_𝑦−2ln𝑅₃cos2𝜋𝑅₄+𝑏_𝑦²𝑎_𝑧−2ln𝑅₅cos2𝜋𝑅₆+𝑏_𝑧. (12)

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 𝐴_𝑇, we classify the emulsion nuclei based on 𝑛_ℎ (𝑛_ℎ=𝑛_𝑔+𝑛_𝑏) 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 𝑛_ℎ=0,1, collisions with only one bound nucleon in CNO or AgBr target nuclei these events having 2≤𝑛_ℎ≤7 are mostly interactions with CNO targets with some admixture of peripheral AgBr interactions. All events with 𝑛_ℎ≥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].

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.

The comparison between the average values of the multiplicities ⟨𝑛_𝑠⟩, ⟨𝑛_𝑔⟩, and ⟨𝑛_𝑏⟩ 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 ⟨𝑛_𝑏⟩ are nearly independent of the nature of incident projectiles, while ⟨𝑛_𝑠⟩, ⟨𝑛_𝑔⟩, and ⟨𝑛_ℎ⟩ show their dependence on the projectile mass number 𝐴_𝑝. 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 ²⁸Si, ²²Ne, ¹⁶O, ¹²C, and ⁴He 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 𝑛_𝑠 decreases with increase in projectile mass, while the position of the peak moves to higher multiplicities with increasing 𝐴_𝑇. The distributions get wider with increasing 𝐴_𝑇 where the distributions have larger tails. This may reflect the effect of the target mass number on the number of collisions of ²⁸Si, ²²Ne, ¹⁶O, ¹²C, and ⁴He 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 𝐴_𝑝 gets weaker than that in case of shower particles. Also one can notice that the distributions for ²⁸Si, ²²Ne, ¹⁶O, ¹²C, and ⁴He with AgBr interactions are broader than those for ²⁸Si, ²²Ne, ¹⁶O, ¹²C, and ⁴He 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 (𝑛_𝑏) distributions given in Figure 3 as for 𝑛_𝑔. 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 (𝑛_ℎ=𝑛_𝑔+𝑛_𝑏) 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 ⟨𝑛_𝑖1⟩, ⟨𝑛_𝑖2⟩, and ⟨𝑛_𝑖3⟩, 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 ⟨𝑛_𝑖1⟩, ⟨𝑛_𝑖2⟩, and ⟨𝑛_𝑖3⟩ 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.

Table 3 

Parameter values for multiplicity distributions for shower, grey, and black particles. The values of ⟨𝑛_𝑖1⟩, ⟨𝑛_𝑖2⟩, 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.

Table 4 

Parameter values for multiplicity distributions for shower, grey, and black particles. The values of ⟨𝑛_𝑖1⟩, ⟨𝑛_𝑖2⟩, 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.

Table 5 

Parameter values for multiplicity distributions for shower, grey, and black particles. The values of ⟨𝑛_𝑖1⟩, ⟨𝑛_𝑖2⟩, 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.

3.2. Angular Distributions

The angular distributions for the different secondary charged particles 𝜃_𝑠, 𝜃_𝑔, and 𝜃_𝑏 emitted in ²⁸Si, ²²Ne, ¹⁶O, ¹²C, and ⁴He 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 𝜃_𝑠, 𝜃_𝑔, and 𝜃_𝑏 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 4He- (solid triangle), 12C- (open square), 16O- (solid square), 22Ne- (open circles), and 28Si- (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 4He- (solid triangle),12C- (open square), 16O- (solid square), 22Ne- (open circles), and 28Si- (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 4He- (solid triangle), 12C- (open square), 16O- (solid square), 22Ne- (open circles), and 28Si- (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, 𝜃_𝑔, 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.

Figure 7 illustrated the angular distributions of black particles, 𝜃_𝑏. 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 𝜃 with increase in product mass, that is, 𝜃_𝑠𝑝<𝜃_𝑔𝑝<𝜃_𝑏𝑝, which is clear in Table 6. The parameter values for angular distributions for shower, grey, and black particles 𝑎_𝑥, 𝑎_𝑦, and 𝑎_𝑧 and 𝑏_𝑥, 𝑏_𝑦, and 𝑏_𝑧 are given in Table 7.

Table 6 

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

Table 7 

Parameter values for angular distributions for shower, grey, and black particles. The values of 𝑎_𝑥, 𝑎_𝑦, and 𝑎_𝑧 and 𝑏_𝑥, 𝑏_𝑦, and 𝑏_𝑧 for ⁴He-, ¹²C-, ¹⁶O-, and ²⁸Si-Em at 4.5 A GeV/c and ²²Ne-Em at 4.1 A GeV/c.

4. Conclusion

We conclude that multisource thermal model gives uniform description of the target fragmentation in interaction of ⁴He, ¹²C, ¹⁶O, ²²Ne, and ²⁸Si 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.