Target size dependence of relativistic hadron emission from ³²S nuclear collisions at 3.7 and 200A GeV

The mass dependence of multifragmentation in low-energy heavy-ion collisions was investigated recently by Vermani and Puri [17]. The central collisions of different heavy ions were simulated by a microscopic quantum molecular dynamics model, over a wide range of projectile or target mass numbers (A = 20 to 197) and beam energies (E_lab = 10 to 130A MeV). This energy is the onset at which the multifragmentation exists. The multifragmentation at this energy range exhibits a complex picture which is quite sensitive to impact parameter, beam energy, as well as the total mass of the target and projectile nuclei (A_tot). The fragment product almost saturates around the time evolution of 200 fm/c. In other words, the time span of 200 fm/c is large enough to pin down the fragment structure. At the region beyond this time of 200 fm/c, the average multiplicity of the intermediate mass fragments, 〈N_imf〉, reaches a limited value irrespective of A_tot. This may encourage us to expect that the time value of 200 fm/c may be equivalent to the onset at which the limiting fragmentation begins to exist. 〈N_imf〉 could be reproduced with a power law, cA^τ_tot, with exponent close to unity [17], i.e. the dependence is reduced to linearity. This observation was checked over a wide range of fragments consisting of free nucleons and fragments having mass numbers (A = 2 to 44).

In experiments [18, 19] the central collisions of ¹²C, ¹⁶O, ³⁶Ar, and ⁸⁴Kr with emulsion nuclei were studied at 50 to 220A MeV. At this range of intermediate energies the chance of pion creation is low. The projectiles and target fragments were similar in the 4π space and limited in the shape of nuclear clusters. The fragmentation was directly proportional to the incident energy, i.e. the target and projectile fragments' (PFs) multiplicity increases with the incident energy and the system size. Hence, in this energy domain, the limiting fragmentation cannot be reached. Consequently, the selectivity of centrality degrees in such interactions was not enough on the basis of projectile or target fragments' multiplicity.

In this experiment, at relativistic and ultrarelativistic energies, the multiplicity of the produced particle is not limited to fragmented particles only, but the created particles are also enclosed. This permits a more precise tool than at low and intermediate energies. Although the energy transferred to the system by the participant projectile nucleons is the main parameter responsible for particle creation, the creation system deals with the size of the entrance reaction channel. Consequently, the fragmentation system is not in isolation from the particle creation system. The relativistic energy (Dubna energy or Bevalac energy) is a special energy at which the nuclear limiting fragmentation applies initially [20]. The limiting fragmentation hypothesis implies that both target and projectile are fragmented independently of each other [21]. The projectile angular distribution showed that the limiting fragmentation hypothesis is valid for peripheral as well as quasi-peripheral collisions. The domain of validity of the limiting fragmentation hypothesis extends as the energy of the projectile nucleus increases. The energy independence of fragmentation cross-sections can be held at energies beyond the 1 GeV region [22–24]. At relativistic energies, the separation in rapidity between projectile and target fragments is large, ≥1 unit of rapidity. At such high energies, the target and projectile fragmentation regions are well separated in the rapidity. The physics of the two regions are believed to be similar. In the context of many experiments at high energy, the distribution of target residues becomes approximately energy independent.

In free nucleon–nucleon collisions, the hadron emission in BHS is kinematically restricted in the center of mass system. The study of hadron emission beyond the kinematic limits in nucleus–nucleus collisions reveals signatures for a collective mechanism recognizing such emission.

At first, the collective mechanism was introduced in an experiment [25] held at Lawrence Berkeley National Laboratory (LBNL). This experiment used C, Al, Cu, Sn, and Pb targets collided by proton beams of 0.8, 1.05, 2.1, 3.5, and 4.89 GeV to study the pion production at 180°. The energy dependence of the slope parameter for charged pion production was discussed first. It was found that the pion spectra fall off exponentially, and the Lorentz-invariant pion cross-sections had been parameterized by the form $E\frac{{{\rm d}\sigma }}{{{\rm d}P^3 }} = C{\rm exp(} - T/T_0 {\rm )}$, where T is the pion laboratory kinetic energy and T₀ is the slope parameter. The trend in the data was similar for both positive and negative pions. Using a combination of data for various backward pion production angles at Dubna, Baldin et al [26] reported a similar trend and suggested that it was related to the onset of limiting target fragmentation. The slope parameter (T₀ ∼ 60 MeV) is found to be independent of the bombarding energy [26]. At LBNL, the experimental dependence of T₀ [9, 25] was compared with the predictions of the effective target model. In this model, the incident proton is assumed to interact in a collective fashion mechanism with the row of nucleons along its path. During the collision, this row of effective target nucleons is excited and then de-excites. Therefore, pions will be emitted in BHS in a fashion analogous to bremsstrahlung. In the BNL bubble chamber, from 28.5 GeV proton interactions, backward pions were produced from a tantalum plate [27]. The slope of the energy spectrum of those pions is consistent with the result of [26]. Schroeder et al [28] provides a definite test of a hard-scattering model [29] which was successful in explaining the scaling observed in forward pion production from nuclei at energies as low as 1 GeV [27]. That model predicts that the 180° pion spectra should be independent of energy, depending only on a scaling parameter of [26]. Using a 0.6 GeV proton, Perdrisat et al [30] observed pions at 155°. They [30] came to a similar conclusion as experiments [31, 32], in which the dominant mechanism is a single scattering. In this scattering, the incident proton interacted with a target nucleon producing the observed pion via the reaction NN → NNπ.

Our group⁴ carried out a series of experiments [10, 12–16, 33–37] looking at backward relativistic hadron production. The results show that those hadrons are not created particles. Their production system is a decay system characterized by a decay constant λ ∼ 1.3 for emulsion target nuclei. The temperature of this system is T ∼ 27 MeV. They depend mainly on the target size at the region of limiting fragmentation. Our observations give evidence confirming the above considerations of pion production beyond the kinematic limits.

In hadron–hadron, hadron–nucleus, and nucleus–nucleus inelastic interactions at high energy, it is preferable to analyze these interactions by means of the Lund Monte-Carlo program code-events generate FRITIOF [38–40].

One of the main ingredients of the FRITIOF model is the inelastic hadron–hadron (hh) collisions. At low energies particle production proceeds through states of resonance created in nucleon–nucleon (NN) reaction. At high energy, it proceeds through continuum spectra. The FRITIOF model assumes that all hh interactions are binary reactions, h₁ + h₂ → h*₁ + h*₂, where h*₁ and h*₂ are the excited states of the hadrons with continuous mass spectra. If one of the hadrons is in the ground state (h₁ + h₂ → h₁ + h*₂) the reaction is called 'single diffraction dissociation'. In the other case it is non-diffractive interaction. The excited hadrons are considered as QCD-strings. The key ingredient of the FRITIOF model is a sampling of the string masses. In principle, the mass sampling threshold can be below hadron mass. In the model it equals the ground state masses. The kinematic energy–momentum conservation law is applied through the excitation system in hh collision.

In the course of hA interactions, the FRITIOF model assumes that the string that originated from the projectile can interact with various intra-nuclear nucleons. Then it goes into a highly excited state. In this case, the same kinematics of hh collisions are applied for the first collision of the projectile with one of the target nucleons. For the second collision analogous kinematics are used, considering the change in the mass of the hadrons and in the longitudinal momenta. As a result, consequent collisions will involve a systematically increasing mass of the hadron, if the transverse momentum transfers are small. A similar approach in kinematic conservation laws is also applied to simulate AA interactions. Accordingly the mentioned kinematics in the longitudinal momentum conservation law are changed.

At relatively lower energies of the order of 3 to 10A GeV, FRITIOF falls to take into account hadron de-excitation. This overestimates the multiplicity of the produced particles in hA and AA interactions. A cascade of secondary particles was neglected as a rule. Due to these, the original FRITIOF model fails to describe a nuclear destruction and slow particle spectra.

As known, the Glauber-like approximation used in the FRITIOF model does not provide enough intra-nuclear collisions for a correct description of a nuclear destruction. The difficulties of the model are overcome by the Reggeon theory inspired model of nuclear destruction [41, 42]. This is considered as another ingredient added to improve the FRITIOF model. To overcome the problem it is necessary to take into account the so-called enhanced diagrams of the Reggeon theory as in [41, 42]. Accordingly, nucleons participating in the interactions predicted by the approximation are considered as primary ones. Other spectator nucleons are involved in the process. This process takes place in the two-dimensional space of the impact parameter only. Momenta of the nucleons ejected from a nucleus are sampled according to a Fermi motion algorithm. A simple model for estimating Reggeon cascading in hA and AA interactions was proposed by EMU01 collaboration ([40] and references therein). When considering a Fermi motion of nucleons it was found that the two FRITIOF codes (with and without de-excitation) give similar results for p, ⁴He, ¹²C, ¹⁶O, ²²Ne, and ³²S induced interactions with emulsion nuclei at 3.1 to 3.5A GeV [40]. For high energies (14.5 to 200A GeV), the experimental evidence prefers the simulation without de-excitation of resonances. In the original FRITIOF model the colliding nucleons become excited strings. If the mass of the excited string is below some critical value (e.g. 1.2 GeV for nucleons), the string is considered a nucleon. Mesons are produced after a certain time when strings and resonances decay. Particles taking part in the primary collisions are allowed to rescatter with other particles by using Reggeon interactions [41–43], which basically amounts to a cascade in the two-dimensional impact parameter space. This is the main difference between Reggeon cascading and the usual cascading in the three-dimensional space of the nucleus. It is reasonable to assume that at low energies the length is lower than the average distance between nucleons in a nucleus (∼2 fm). Thus, the usual cascading takes place. At higher energies the length will be greater, and two or more nucleons will participate in the interaction region. Only these nucleons will participate in the interactions and are described by the string model. In hA interactions for low target size the cascading of secondary particles is assumed not to play much role in the interactions. For overestimation at greater target size, a correction of the inelastic interaction number in a nuclear medium is considered [44]. This approach gives the opportunity to consider the excitation process with increasing mass and the de-excitation process with decreasing mass. This leads to improvement in the results at incident energies 3.1–3.5A GeV over those of the original model [40]. Thus the binary model gives additional cascading of low-energy secondaries in nuclei.

In the compound system, taking into account the energy–momentum conservation law, all the kinematical characteristics of the particles in the finite sets are obtained. The space angle of particle emission can be predicted [40]. Therefore, the FRITIOF model is useful in classifying particle emission into FHS and BHS. The description at large angles of the emitted pions can be improved when the FRITIOF model is coupled with the binary cascade model [44].

The modified FRITIOF code used in this work is based on version 1.6 (10 June 1986) of authors B Nilsson-Almquist and Evert Stenlund, University of Lund, Lund, Sweden [38, 39]. The modification was carried out by V V Uzhinskii, LIT, JINR, Dubna, Russia, in 1995.

The NIKFI–BR2 nuclear emulsion stack measured in this experiment was exposed by ³²S beams at the Synchrophasotron of JINR in Dubna, Russia. The beam energy is 3.7A GeV. Each emulsion pellicle size is 20 cm × 10 cm × 600 µm.

Table 1 shows the chemical composition of this emulsion type within round brackets.

The FUJI emulsion pellicles are coated on both sides by polystyrene films. They were horizontally exposed to 200A GeV ³²S-ion beams at SPS (CERN, EMU03 experiment). The dimensions of these emulsion pellicles are 12 cm × 4 cm × 700 µm. The thickness of each layer is 350 µm while that of the polystyrene layer is 70 µm. The chemical composition of this emulsion type is displayed in table 1 within square brackets.

The scanning of the emulsion pellicles was carried out using 850 050 STEINDORFF German microscopes. This has a stage of 18 × 16 cm² with an opening of 7.0 × 2.5 cm². Stage adjustment in the X-direction is possible over a total length of 7.8 cm with reading accuracy of the order of 0.1 mm. An oil immersable objective lens with 100 × magnification was used for scanning the emulsion plates.

Starting close to the entrance of the beam double, scanning was carried out going 0.5 cm into the emulsion ('fast' in the forward direction and 'slow' in the backward direction). Each primary track was picked up at the penetrating edge of the pellicle and was followed forward optically until it either interacted or escaped from the pellicles. Details of the microscopic scanning procedure are the same as those in [45].

Given in table 2 are the total scanned lengths (L) of primary beam tracks, the resulting number of inelastic interactions (N), and the corresponding average values of the experimental mean free path (λ).

Glauber's approach [46] is also used to calculate reaction mean free paths in emulsion. Running the encoded simulation [46] on the present ³²S interactions in NIKFI–BR2 and FUJI emulsion, the predicted mean free path values are found to be 8.39 and 8.60 cm, respectively. These values agree with the tabulated experimental ones.

Depending on ionization, all tracks emitted from the interaction vertices were classified according to the commonly accepted emulsion experiment terminology [47, 48] as follows:

• –Shower particle-tracks with g ≤ 1.5g_p; these are mainly pions having kinetic energy above 70 MeV with small admixture of singly charged particles with kinetic energies above 400 MeV. They have relative velocity β ≥ 0.7. Their multiplicity is denoted as n_s. The multiplicity of shower particles produced in FHS is denoted as n^f_s. The multiplicity of shower particles flying into BHS is denoted as n^b_s. This multiplicity of these kinds of particles is the focus of this paper. Here, g is the measured grain density and g_p corresponds to the grain density of a minimum ionizing track.
• –Gray particle-tracks with a range >3 mm and 1.5g_p < g <4.5g_p; these mainly consist of protons knocked out from the target nucleus during the collision with kinetic energy ranging from 26 up to 400 MeV. They have a few per cent admixture of π-mesons. Their multiplicity is denoted as N_g.
• –Black particle-tracks are those having short range in emulsion ≤3 mm with g > 4.5g_p, mostly evaporated target protons with kinetic energy <26 MeV. Their multiplicity is denoted as N_b.
• –Gray and black tracks are the group of heavily ionizing tracks: N_h = N_g + N_b.
• –Fragments of projectile nucleus with Z ≥ 1; the PFs essentially travel with the same speed as that of the parent beam nucleus, so the energy of the produced PFs is high enough to distinguish them easily from the target fragments. All PFs are emitted in a very narrow forward direction (θ_lab ≤ 3°) within an angle given by the Fermi momentum. PFs with Z = 1 and 2 are identified by the visual inspection of tracks where their ionizations are similar to those of shower and gray particles, respectively. The δ-ray method was used in the case of the identification of Z ≥ 3 fragments. El-Nagdy et al [49] explained widely the charge identification methods obeyed by δ-ray density measurement. More details on identification can be obtained from [50].

Nuclear emulsion is a composite medium composed of H, CNO, and AgBr. It is a difficult task to separate interactions on different classes of targets. Although there are many correlations between the measured parameters that give information regarding the target nuclei, it is impossible to find certain separation criteria that give no admixture between those classes.

Depending upon the target break-up, one uses the heavily ionizing particle multiplicity N_h as a parameter representing the impact parameter in the event. Hence, the N_h-integral distribution method (known as Florian's method) described explicitly in [51, 52] is used, on experimental bases, to select from the inelastic interaction samples with hydrogen H, light CNO, and heavy AgBr targets. According to this method all events with N_h > 8 are considered to be due to interactions with an AgBr group. The events with N_h ≤ 8 are attributed to interactions with H, CNO, or peripheral collisions with AgBr.

On the other hand, the target separation can be executed theoretically on the basis of Glauber's multiple scattering theory, using the simulation code of [46], as is done in the present experiment. Therefore, the statistical samples of events and their corresponding percentage probabilities, P%, for the interactions of ³²S with each target in nuclear emulsion at 3.7 and 200A GeV are given in table 3, with round and square brackets, respectively.

Figure 1 presents the backward shower particle multiplicity distributions in the interactions of 3.7 and 200A GeV ³²S with CNO, Em, and AgBr. The data belonging to the interactions with the H target have low statistics, so these target data are not presented in figure 1.

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For all targets, the characteristic feature of the distribution is the exponential decay shape. The multiplicity range (decay tail) increases with the target size. The data are reproduced qualitatively by the FRITIOF model. The characteristic exponential behavior can be approximated by equation (1). In figure 1, an exponential fit of the experimental data and their theoretical predictions are presented by solid and dashed curves, respectively:

$$\begin{equation} P\big(n_s^b \big)\% = P_0\, {\rm e}^{ - \lambda n_s^b } . \end{equation} \tag{ 1 }$$

The fit parameters P₀ and λ are given in table 4.

The fit parameters are nearly equal at the two incident energies for low target size (CNO). For heavier target size (Em and AgBr), the parameters are affected weakly by energy. The small energy effect consists of a longer decay tail of the distributions at 200A GeV than at 3.7A GeV. However, this effect of energy is not comparable with the great difference between the values of 3.7 and 200A GeV. This effect is attributed to more excitation in the target nucleus at SPS energies than at Dubna energies. Consequently, the produced compound nucleus will de-excite and decay by emitting this excess number of backward hadrons. Such a mechanism of compound target nucleus was discussed in experiments [15, 16]. Therefore, regarding the limiting fragmentation beyond 1 GeV, the projectile energy cannot be considered an effective parameter in the backward production and consequently does not mean that this system of particle production is a creation system.

The values of the FRITIOF model in table 4 often underestimate the data.

The backward emitted shower particle multiplicity at the two incident energies can be determined as a function of the effective target mass A_T, as shown in figure 2. In figure 2, the fit parameters of table 4 are correlated with A_T. The linear fitting is presented by solid and dashed straight lines according to 3.7 and 200A GeV data, respectively. The fit parameters of equations (2) and (3) are shown in table 5.

$$\begin{equation} \lambda = a_\lambda + b_\lambda \;A_T , \end{equation} \tag{ 2 }$$

$$\begin{equation} P_0 = a_p + b_p \;A_T . \end{equation} \tag{ 3 }$$

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The slope and intercept parameters decrease linearly with the target mass number up to the available value of A_T ∼ 100. Therefore, the backward emission of relativistic hadrons strongly depends on the target size.

In figure 3, the percentage probability of backward shower particle production, P(n^b_s > 0)%, is evaluated as a function of a target mass number. This probability is defined as the number of events having n^b_s > 0 normalized to the total sample of events. Presented in figure 3 are the data of 3.7 and 200A GeV ³²S interactions with emulsion nuclei. From figure 3, one observes the strong dependence of backward relativistic hadron production on the target size. This strong dependence is evaluated linearly using equation (4) and is presented in figure 3 by the straight lines. Irrespective of the projectile size (A_Proj = 1 to 32) and energy (2.1 to 200A GeV), the backward relativistic hadrons are produced with probability values of ∼20% to 30% for interactions with the Em target [36]. The theoretical predictions of the FRITIOF model slightly overestimate the data.

$$\begin{equation} P(n_s^b > 0)\% = \mu + \nu A_T . \end{equation} \tag{ 4 }$$

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The fit parameters µ and ν are given in table 6.

The average multiplicities of the backward emitted shower particle in the interactions of 3.7 and 200A GeV ³²S with emulsion nuclei at different target sizes are displayed in table 7.

Despite the wide range of energies, the values of 〈n^b_s〉 are nearly the same at the two energies in table 7. The FRITIOF model slightly overestimates the results.

The average multiplicity of backward shower particles, emitted in 3.7 and 200A GeV ³²S interactions with emulsion nuclei, is correlated with the target size in figure 4. The correlation reveals a strong linear dependence presented by the straight lines in figure 4. The linear relation of equation (5) approximates the values of fit parameters, a^b_s and b^b_s, to be displayed in table 8. The predictions of the FRITIOF model overestimate the data in figure 4.

$$\begin{equation} \left\langle {n_s^b } \right\rangle = a_s^b + b_s^b \;A_T . \end{equation} \tag{ 5 }$$

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The slope parameters are ∼0.01 and the intercept parameters are ∼0, irrespective of the wide energy difference.

In experiment [10], using a wide range of projectile sizes (A_Proj = 1 to 32) at Dubna energy, the values of 〈n^b_s〉 are found to increase with projectile size for A_Proj < 6. At A_Proj ≥ 6, they began to saturate and had a constant value of 〈n^b_s〉 ∼ 0.4. In this experiment the results imply that the energy is not an effective parameter in backward shower emission. Therefore, one can conclude that while the dependence of the average shower particle multiplicity, emitted in BHS, on the target size is strong, it depends neither on the projectile size nor energy. This confirms our expectation that the backward relativistic hadrons do not come from the fireball nuclear matter or hadronic matter. They are target source particles.

Figure 5 displays the forward shower particle multiplicity in the interactions of 3.7A GeV ³²S with (H, CNO, Em, and AgBr) different emulsion nuclei. The extension tails in the multiplicity ranges (up to values >14 and >26 for H and CNO targets, respectively) are the result of statistical reasons obeyed in the event separation method, and do not indicate the existence of particles beyond those limits. In figure 5, the characteristic feature is the peaking shape distributions, irrespective of the target size. The multiplicity range as well as the broadening of the distributions increase with the target size. The geometrical model considering the overlap size between target and projectile seems to be effective in drawing the characteristic features of the distributions. In this species, the low size target, H, seems to exhibit a complete dive in the overlap region, where the peaking shape distribution indicates a forward sideward suppression characterizing a violent collision distribution shape. For the heaviest target, AgBr, the distribution tends to have a shape like a hill extending over the multiplicity range with an often equal probability. The FRITIOF model fails to reproduce forward shower particle multiplicity with the H target. The model succeeds with the CNO target. Apart from the low multiplicity region with Em and AgBr targets, the model can reproduce the distributions with slight underestimations.

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From the observed behavior in figure 5, it is reasonable to say that the mechanism in this system of particle production in FHS is completely different from that in BHS.

In experiment [10], the forward shower particle multiplicity is examined for different projectile sizes (A_Proj = 1 to 32) at Dubna energy. A similar picture was observed [10] as with the target size (A_T ∼ 1 to 100) in the present experiment. Exactly, the dependence of this system of particle production on the target size does not mean that it is due to the target fragmentation system, which is the case for backward shower particles.

This similar picture of dependence on projectile or target sizes is interpreted briefly from the concepts of the fireball model summarized by Westfall et al [53]. In this model the nucleons are multiply swept out from the target and projectile. These nucleons form a hot quasi-equilibrated fireball which decays as an ideal gas. This model uses the geometrical concepts of the abrasion model [54]. In the geometrical concepts of such a model, it is assumed that the target and projectile are spheres with radii equal to 1.2A^1/3. The target and projectile make clean cylindrical cuts through each other, leaving the spectator piece of the target. If the impact parameter is sufficiently large, the spectator piece of the projectile will also be left. The free expansion of an ideal gas is used where the incident energy is extended to higher values. It is assumed that the available energy ε in the center of mass heats up the swept-out nucleons leading to a quasi-equilibrated nuclear 'fireball'. The fireball is treated relativistically as an ideal gas whose temperature, τ, is determined by ε [55]. At high energy such as relativistic energy (Bevalac or Dubna), ε is much bigger, beyond 350A MeV. This energy may go into internal excitation of the nucleons to baryon resonances as well as randomized kinetic energy of the nucleons. However, for small impact parameters, one can imagine that the projectile never penetrates through the target and thus the available energy is shared among all the nucleons in the target and projectile. The large number of swept-out nucleons combined with an anticipated, fairly large number of interactions per particle is presumably responsible for a quasi-equilibrated system (the fireball) which can then be described in terms of mean values and statistical distributions. The fireball model also enables the prediction of nucleon multiplicities and, with some modification, pion inclusive spectra and multiplicities. Thus, this picture seems to be suitable to describe the geometrical effect of the target and projectile on the particle creation at Dubna energy.

Figure 6 displays the forward shower particle multiplicity in the interactions of 200A GeV ³²S with (H, CNO, Em, and AgBr) different emulsion nuclei. The extension tails in the multiplicity ranges (up to values >60 and >120 for H and CNO targets, respectively) are the results of the statistical reasons obeyed in the event separation method. They do not present the existence of particles beyond those multiplicity values.

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The effect of energy on the forward shower particle, as a creation system of particles, is reflected in the long multiplicity range in figure 6. For H and CNO the multiplicity ranges reach nearly 60 and 120 hadrons per event, respectively. For emulsion and AgBr, the multiplicity ranges reach up to more than 300 hadrons per event. Figure 6 shows that the characteristic features of the multiplicity distributions at SPS energy are different from those at Dubna energy shown in figure 5. The peaking shape may only distinguish the distributions at low target sizes (H and CNO). For Em targets as a whole, the distribution loses its peak portion to have a decay shape; however it does not imply an exact decay system of particle production. For the heaviest target (AgBr), the distribution tends to fluctuate in nearly equally probable partitions, especially at n^f_s > 150.

From figure 6, the dependence of the hadron creation system on the target size at the SPS energy is similar to that at Dubna energy. Of course, this is also attributed to the geometrical system of the colliding nuclei, in the concept of the fireball model [53]. However, one cannot ignore the different shapes of the distributions at SPS energy from those at Dubna, especially for heavier targets (Em, AgBr). We can expect that the energy transfer at the SPS domain is too large to enable particle creation in the higher multiplicity region with equally probable partitions, as nearly equal as those at the lower multiplicity region, whatever the number of particles per event is. At ultrarelativistic energy (200A GeV) the temperature is expected to be higher for the system of forward relativistic hadrons. Apart from reaching the most central region, where the fireball is extended in size, the temperature will be enough to allow the major energy content of the system to be spent in the hadronization process, regardless of the fraction of energy loss in binary collisions of projectile nucleons with target, even at centralities lower than the violent region. Thus in the hadronization process at SPS energy, particle creation tends to occur in equally probable multiplicity irrespective of the centrality of the system, regarding the higher target size, as Em and AgBr.

At SPS energy in figure 6 the FRITIOF model fails to describe the particle creation system for the H target. The model described the multiplicity distribution successfully for higher targets (CNO, Em, and AgBr), as at Dubna energy.

The dependence of the average forward shower particle multiplicity on the target size at the two incident energies is examined in table 9. The dispersion values of the distributions of figures 5 and 6 are also found in table 9, where it is determined from equation (6):

$$\begin{equation} D = \sqrt {\big\langle {\big(n_s^f \big)^2 } \big\rangle - \big\langle {n_s^f } \big\rangle ^2 } . \end{equation} \tag{ 6 }$$

From table 9 one can observe the strong dependence of 〈n^f_s〉 on the energy. The values of 〈n^f_s〉 also increase with the target size at the two incident energies. The dispersion values follow the same trend as the average multiplicity. The predictions of the FRITIOF model always underestimate the data. The usual behavior reveals that the ratio of (D/〈n^f_s〉) is approximately 0.8, whatever the incident energy or target size is. This behavior is considered either experimentally or theoretically from the model. In an experiment [10] in which the projectile size was examined on the pion creation system using A_Proj = 1 to 32 at Dubna energy, the same trend was also observed, where (D/〈n^f_s〉) ∼ 0.8. Although the constancy in the ratio (D/〈n^f_s〉) cannot be considered to be a limiting behavior for a particle creation system, it can imply that the expansion or compression of this system may have standardization in the interaction mechanism under temperature and geometrical conditions.

Now the dependence of 〈n^f_s〉 on A_T through the interactions of 3.7 and 200A GeV ³²S with emulsion nuclei is determined in figure 7. The linear fitting presented by the straight lines is approximated by equation (7):

$$\begin{equation} \left\langle {n_s^f } \right\rangle = a_s^f + b_s^f \;A_T . \end{equation} \tag{ 7 }$$

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The fit parameters a^f_s and b^f_s are displayed in table 10. The theoretical predictions underestimate the data in figure 7.

The effect of energy on this particle creation system is clearer by diagrammatic presentation in figure 8. In figure 8, the ratio between 〈n^f_s〉 at 200A GeV and 〈n^f_s〉 at 3.7A GeV versus A_T is displayed. From figure 8, the mentioned ratio is a linear function of A_T given by equation (8), while the FRITIOF model nearly reveals a constant value which fluctuates about $\big( {\frac{{\langle {n_s^f } \rangle _{200A\ {\rm GeV}} }}{{\langle {n_s^f } \rangle _{3.7A\ {\rm GeV}} }}\sim 5} \big)$.

$$\begin{equation} \frac{{\big\langle {n_s^f } \big\rangle _{200A\;{\rm GeV}} }}{{\big\langle {n_s^f } \big\rangle _{3.7A\;{\rm GeV}} }} = \alpha + \beta A_T . \end{equation} \tag{ 8 }$$

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The fit parameters α and β are found to be 2.44 ± 0.19 and 0.04 ± 0, respectively.

From the geometrical concept, the effect of projectile size on the pionization system was examined, previously at Dubna energy [10], to find that 〈n^f_s〉 = aA_Proj^b, where a = 1.98 ± 0.01 and b = 0.56 ± 0.01. Therefore, the hadron creation system is dependent on the participating energy, which works according to the geometry of the colliding nuclei as observed from equation (8).

The system of relativistic hadron emission in FHS may be correlated with that in BHS. Such a correlation is displayed in figure 9. The dependence of the average forward shower particle multiplicity on the backward shower particle multiplicity is illustrated in figure 9. The data of figure 9 are attributed to the interactions of ³²S with emulsion nuclei at 3.7 and 200A GeV, considering the different target sizes (CNO, Em, and AgBr). Regardless of the low statistics belonging to the interactions with the H target, such target data are not considered in the correlation. From figure 9 one can observe the strong dependence of average forward shower particle multiplicity 〈n^f_s〉 on backward shower particle multiplicity. The heavier the target size, the stronger the dependence. The forward and backward multiplicities are correlated in a linear function relation with fit parameters given in table 11. The linear correlation function is presented in figure 9 by the straight lines.

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The intercept ratios at the two energies with each target is found to be I_{200A GeV}/I_{3.7A GeV} ∼ 3, 5.52, and 6.28 with CNO, Em, and AgBr, respectively. The ratios between the corresponding average multiplicities at the two energies give nearly the same values as the above intercept ratios, 〈n^f_s〉_{200A GeV}/〈n^f_s〉_{3.7A GeV} ∼ 2.97, 5.23, and 5.72, with CNO, Em, and AgBr, respectively. Therefore, the interception values in the correlation reflect the effect of energy on the hadron creation system. On the other hand, the slope parameters measure the violence degree of the dependence. The slope values are nearly the same for CNO target data at the two incident energies. For the Em and AgBr targets, the slope values are different at the two energies, as a result of the different values of the intercept at each energy value. However, this does not mean that the slope is different at the two energies. To elucidate this, one knows that the slope of the straight line = ΔY/ΔX. Therefore, the inclination angle of that line on the X-axis is tan^–1(slope). The values of the slope parameters belonging to (Em and AgBr) data at the two incident energies correspond to lines having nearly the same angle of inclination ∼(81°–88°). Therefore, the dependence behavior of the forward relativistic hadron production on the target size is the same at the two energies. On the other hand, these angles seem to be near to the right angle value. This encourages us to say that the forward emitted relativistic hadron is strongly dependent on the multiplicity of backward ones, along with the target size. The strong dependence, here, does not mean that the source responsible for forward relativistic hadron origination is the same as that of backward ones. This may be illuminated more from experiments [12–16, 34, 35, 45]. In experiment [34], Abdelsalam et al found that the events of central collision are accompanied by emission of at least one hadron to the backward direction. El-Nadi et al [12, 13] also showed that the dependence on backward hadron emission is nearly equivalent to the dependence on the Q = 0 criterion, where Q is the total charge of the outgoing PFs and it is the strongest and most commonly used factor to indicate the impact parameter as in experiments [56–58]. Badawy [35] used the n^b_s > 0 criterion as a successful indication to the degree of violence of collision at high energy. Therefore, one can say that the strong dependence of forward relativistic hadron multiplicity on backward ones, in figure 9, is fundamentally due to the strong correlation of n^b_s with the impact parameter. Moreover, from a geometrical point of view (at the same impact parameter) the system of hadron creation may be viewed as if it works by the same mechanism at the two incident energies, indicated by the similarity in the slope behavior. Although a similarity is found in the slope behavior for the straight lines of figure 9, the intercept values in the correlation reflect the effect of energy on the hadron creation system, regarding their excess at higher energy.

On the basis of the limiting behavior of backward shower particle production, and irrespective of the incident energy value or the projectile size, the value of 〈n^b_s〉 can be used as a good parameter to scale the system of hadron creation in FHS, using the (F/B)_s ratio, where (F/B)_s = 〈n^f_s〉/〈n^b_s〉. The asymmetry parameter A_s is used to compare the hadronization system in FHS with that in BHS. The asymmetry parameter is defined as

$$\begin{equation} A_s = \frac{{\big\langle {n_s^f } \big\rangle - \big\langle {n_s^b } \big\rangle }}{{\big\langle {n_s^f } \big\rangle + \big\langle {n_s^b } \big\rangle }}. \end{equation} \tag{ 9 }$$

Those parameters are displayed in table 12 for the interactions of 3.7 and 200A GeV ³²S with emulsion nuclei at different target sizes.

From table 12, the values of (F/B)_s increase with energy and decrease with the target size. This implies that (F/B)_s may be considered a parameter representing the expansion of the relativistic hadron creation system with the target size at a specific high energy value. Although the ranges of the energy and target size are wide, the asymmetry shows no dependence on energy or target size for this system of particle production where A_s is always ∼1. This means that the asymmetry of FHS with respect to BHS in the hadronization system has a limiting behavior over the different target sizes and energies. Moreover, the FHS is intimately different from BHS. While the FRITIOF model reproduces the asymmetry values (A_s ∼ 1), it underestimates the values of (F/B)_s.

On the other hand, the parameter (F/B)_s, characterizing the system of relativistic hadron creation in the interactions of 3.7 and 200A GeV ³²S with emulsion nuclei, can be correlated with the target mass numbers in figure 10.

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In figure 10, (F/B)_s shows a strong dependence on A_T. Such dependence can be approximated by linear fit of the data, to be presented by straight lines reproduced by equation (10). The theoretical predictions of the FRITIOF model underestimate the data.

$$\begin{equation} (F/B)_s = \alpha + \beta A_T . \end{equation} \tag{ 10 }$$

The fitting parameters are displayed in table 13.

• –As shown in the present experiment 〈n^f_s〉, 〈n^b_s〉, P(n^b_s > 0), and (F/B)_s increase linearly with A_T at both incident energies. The same behavior is predicted by the FRITIOF model. As said, in our previous work [10], the dependence of 〈n^f_s〉 reveals a power law relation with the exponent value of ∼0.6 at Dubna energy. This means that 〈n^f_s〉 is directly proportional to R²_Proj, where R_Proj is the projectile nuclear radius. Although the fragmentation mechanism in the present high-energy nuclear collision is different from that at low energy [17], as explained in the demonstrative review, a similarity is observed in the dependence trend on the size of the collision system. In the geometrical concepts of the fireball model [53], it is mentioned that the target and projectile are spheres with radii equal to 1.2A^1/3, basing on the abrasion model [54]. This may indicate the side of similarity between the fragmentation system at low energy and that at high energy. Hence, to draw of a complete picture of the geometrical mechanism of the collision system, it may be preferable to consider the total size of the colliding system (A_tot = A_Proj + A_T) and the fragmentation system along with the particle creation system.
• –The overall trend of the FRITIOF model predictions suggests satisfactory agreement with experimental data. The agreement in FHS data is a consequence of the maximum string excitation which is achieved by increasing the number of collisions until the two colliding string objects come to rest in their center-of-mass frame. As seen, this effect of the process appears for heavier targets (CNO, Em, or AgBr). Additional hadron production in this domain of targets could be achieved by creation of extra strings between quarks of the colliding hadrons. The de-excitation processes result in an excess of pions in BHS. This backward production may be expected in a simulation of binary reactions such as (NN → NΔ). Δ-isobars are created due to these in pp interactions at 3 to 15 GeV/c [44]. In the mentioned interactions the isobars can be scattered elastically after the quark exchange in order to generate a transferred momentum. Then Δ-isobar can reach the ground state (pp → ppπ°, pp → npπ⁺). In the other case, Δ-isobar is assumed to be in the excited states suffering a single diffraction dissociation. Then it is suppressed to low energy by the quark exchange (pp → ppπ⁺π⁻, pp → np2π⁺π⁻, pp → ppπ⁺π⁻π°). We expect that the latter case can draw the picture of backward hadron production, if the two produced pions (π⁺π⁻) are emitted back to back. Therefore, it will be reasonable to expect a similar number of the forward emitted hadrons to originate from the same source as the backward ones. On the other hand, the observed overestimations or underestimations may be due to calculations of the probabilities of interactions with different components of emulsion according to Glauber's approach [46]. Hence, the difference in model predictions is governed by different mechanisms of the multiparticle production process. However, a modern approach [44], in describing nuclear cascading, may lead to better results.