Advances in High Energy Physics

Volume 2017, Article ID 7907858, 16 pages

https://doi.org/10.1155/2017/7907858

## Analysis of Various Projectile Interactions with Nuclear Emulsion Detector Nuclei at ~1 GeV per Nucleon Using Coulomb Modified Glauber Model

^{1}Department of Physics, Institute of Science, Banaras Hindu University, Varanasi 221005, India^{2}Post Graduate and Research Department of Physics, The American College, Madurai 625002, India

Correspondence should be addressed to V. Singh; moc.oohay@zaknev and S. S. R. Inbanathan; moc.liamg@nahtanabninehpets

Received 29 June 2017; Accepted 25 September 2017; Published 31 October 2017

Academic Editor: Anna Cimmino

Copyright © 2017 N. Marimuthu 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 SCOAP^{3}.

#### Abstract

The total nuclear reaction cross section is calculated considering the cases with and without medium effect by employing Coulomb modified Glauber model (CMGM) for interactions of projectiles ^{56}Fe_{26}, ^{84}Kr_{36}, ^{132}Xe_{54}, ^{197}Au_{79}, and ^{238}U_{92} with nuclear emulsion detector (NED) nuclei at around 1 GeV per nucleon incident kinetic energy. These calculated nuclear reaction cross sections are correlated with the different target groups of the NED nuclei. The average value of various parameters is also calculated and compared with the corresponding experimental results. The number of shower particles emitted in an interaction is also calculated and showed good agreement with the experimental result. We observed that the total nuclear reaction cross section increases with increasing the target mass number in case of all the considered projectiles. In addition, it is shown that the average value of reaction cross section with nuclear medium effect is in good agreement with the experimental results for projectiles ^{56}Fe, ^{84}Kr, and ^{132}Xe, although results of projectiles ^{197}Au and ^{238}U are not in agreement with the experimental observations. This study sheds some light on the energy dependence of the nuclear reaction cross section.

#### 1. Introduction

The relativistic heavy-ion collision in the intermediate and high-energy domains has been extensively studied both theoretically and experimentally for a long time [1–6]; this study is highly interesting because of their important application and new research opportunities. In these regions, heavy-ion collision provides us with information to understand the mechanism of nuclear fragmentation, space-time development of hadronic interactions under extreme condition, and formation of exotic nuclei [1, 2]. The photographic nuclear emulsion detector is one of the excellent tools to understand the high-energy interactions because it provides excellent spatial resolution and very high efficiency of charge particle detection over complete solid angle [7–10]. In the heavy-ion collision, the projectile nuclei or hadrons interact with a target nucleus to produce the multiparticles and these particle productions should be considered as two different steps [1]. In the initial step, the interacting projectile nuclei mainly interacts with the primary reaction and completely overlap with the target nucleus and then leave the projectile-target participant region without any further interaction. This process associated with the production of singly charged relativistic particles, that is, shower particles (), which is mainly pions and small mixture of* k*-mesons, having velocity greater than 0.7 c. In the next step, the produced shower particles may be involved in the rescattering process with target nucleus and knock out the nucleons (proton) from the target nucleus. These particles are called grey particles (). The relative velocity (*v/c*) of grey particle belongs in between 0.3 c and 0.7 c, that is, (0.3 c < *β* < 0.7 c) and their kinetic energy ranges from 30 MeV to 400 MeV, that is, (30 <* E* < 400 MeV). Due to the consequence of these two different steps, the hadron production is not an instantaneous process and it will take certain time, which is said to be creation time [1]. The excited target residuals nucleus comes back to the initial state by losing their energy and attains thermal equilibrium by emitting the nuclear material in the form of fragments. These target fragments are known as black particles () [11, 12]. These black particles have relative velocities (*ν*/c) and kinetic energies less than 0.3 c and less than 26 MeV, respectively.

The total nuclear reaction cross section () is one of the most important physical quantities in the heavy-ion collision. From this reaction cross section, one can extract the fundamental information about the nuclear size and density distribution of protons and neutrons inside the nucleus [18]. Based on the nuclear reaction cross section, one can describe the strong interaction of hadron-nucleus (*h-A*) and nucleus-nucleus (*A-A*) interactions. It has application in various research fields, including shielding against heavy-ions coming from the space radiations or accelerators, cosmic ray propagation, and radio-biological effects resulting from clinical exposures [2].

The Glauber Multiple (GM) scattering theory is commonly used to describe the nuclear reaction cross section at high energies. In the high-energy collisions, the GM has been applied successfully and the total nuclear reaction cross section has been extracted [13, 19, 20]. This model has been extended for the study of total nuclear reaction cross section and differential elastic scattering cross section in the lower energy domain.

In GM model, scattering amplitude is defined as the phase shift function and is extended in the series, where it describes the different multiple scattering processes. The GM model is a semiclassical model picturing the nuclei moving along in the collision direction and it gives a nucleus-nucleus collision in terms of nucleon-nucleon (*NN*) interaction with the given density distribution. At high energies, this model provides good approximation and, in low energies, the nucleus deflected from the straight-line path due to the Coulomb repulsion. This approach is called the Coulomb modified Glauber model (CMGM) [2, 3]. Many workers have applied CMGM in theory and experiment and successfully calculated the total nuclear reaction cross sections. These calculated values are found to be in good agreement with the experimental results [1, 21].

In the present work, we have calculated the total nuclear reaction cross section for the collision of various projectiles, by using Coulomb modified Glauber model such as ^{56}Fe_{26}, ^{84}Kr_{36}, ^{132}Xe_{54}, ^{197}Au_{79}, and ^{238}U_{92} with different composition elements of the nuclear emulsion nuclei at incident energies ~ 1 GeV/n. In this model, for the reaction calculation, we consider the nuclear medium effect, because, in the medium, nucleon-nucleon (*NN*) interactions are in some cases different from the free space nucleon-nucleon interactions due to the effects of Pauli blacking and finite nuclear matter density [18]. The calculated total nuclear reaction cross section values are compared with the corresponding projectiles experimental values. From the elements of the nuclear emulsion, we consider the two different chemical compositions according to the emulsion plates company NIKFI (BR-2) and ILFORD (G5) types.

Since, according to the simple geometrical consideration, the total number of projectile participants (), target participants (), and binary collisions () is calculated [22]. The participant’s nucleons and binary collision involved in the collision lead to the calculation of nuclear matter effects. According to the Adamovich [23–27] empirical formula, one can easily calculate the average number of shower particles () value using the total number of participants and binary collision. Here we have calculated the average number of shower particles value and compared them with corresponding experimental results. We also studied and described the mean free path and nuclear reaction cross section with the projectile mass number.

#### 2. Coulomb Modified Glauber Model

According to the optical limit of the Glauber theory the total nuclear reaction cross section for nucleus-nucleus collision can be written as [2]where is the transparency function defined as the probability that a high-energy projectile with the impact parameter passes through the target without any interaction. The transparency function is calculated from the projectile and target overlap region, where interactions assumed to be single nucleon-nucleon interaction [2] and it is given bywhere the imaginary part of the thickness function or nuclear phase shift function , in the case of nucleus-nucleus (*A-A*) interaction, is given by [2, 3]

And in the case of nucleon-nucleus (*h-A*) interaction is written aswhere is the average energy dependent free space nucleon-nucleon (*NN*) cross section and it is taken from the average of and . and are defined as the nuclear density of the projectile and target nuclei. The function is the finite range of the nucleon-nucleon interaction [3]. The nucleon-nucleon (*NN*) interaction cross section at intermediate and low energies is modified with medium effect due to the Pauli blocking. The effect of Pauli blocking came from the exclusion principle and it is very essential for the internal region of internuclear distances owing to the high-density overlap region on the colliding nuclei. Therefore, the medium nucleon-nucleon cross section is different from the free space nucleon-nucleon interaction cross section [18]. In the present work, calculation is also carried out without medium effect:where , , , , , and are respective projectile and target mass, charge, and neutron numbers. The nucleon-nucleon interaction cross section is from [3, 18]where , , and is the proton-proton, neutron-neutron, and neutron-proton interaction cross section and it is expressed in millibarn (mb), , is the incident kinetic energy of the nucleon in MeV in the laboratory frame of reference, and is the nuclear matter density in unit of fm^{−3}. In (6) and (7), the first part describes the free space nucleon interaction and the second part describes the nuclear matter effects in the medium nucleon-nucleon interaction cross section. The parameter *β* is given as [18]

Expression (7) is used for the energy > 10 MeV and for energy, < 10 MeV, we have to use another expression as given in the following [3, 28]:

The projectile and target nuclear matter density distribution is assumed Gaussian in shape as given by [29, 30]where ; and are the diffuseness and central nuclear density, respectively. Both of these are related to the root-mean-square radius , through the following expressions [29, 30]:where ,* T* indicate projectile () and target (). We used the Gaussian form function for the nucleon-nucleon range function [2]:

Here parameter is related to the slope of the nucleon-nucleon differential scattering cross section. Integrate (3) with respect to , , , and ; the phase shift function , for nucleon-nucleus (*h-A*) interaction is given as [1–3, 31]

While the nucleus-nucleus (*A-A*) interactions will be obtained bywhere

According to the Coulomb modified Glauber model (CMGM), introducing the effect of Coulomb field between the projectile and target, there is a deviation in the original trajectory of the scattered particle. Therefore, the impact parameter is replaced by , which relate the closest approach distance between the interacting particles [3]:where is a wave number and is the dimensionless Sommerfeld parameter defined aswhere , are the total charge of the projectile and target nucleus, respectively, and is the velocity of the projectile in unit of c. It should be mentioned that, in all our calculations, the overlap integral of (3) and (4) is evaluated in terms of . On substituting (14) into (2), one can calculate the total nuclear reaction cross section () for the proton and for the different projectiles interactions with different targets, that is, constituents of nuclear emulsion detector. These calculated nuclear reaction cross sections are used in the calculation of average number of projectile participants [], target participants , and binary collision through the following simple geometrical consideration [22]:

In the above equations, is the total nuclear reaction cross section of the proton with target, that is, each target belongs to the nuclear emulsion detector constituent; consider, as a target, and are the total nuclear reaction cross section of the proton with projectile and proton-proton cross section. In addition, is the total nuclear reaction cross section of the projectile. The average numbers of projectile participants , target participants , and binary collision are used in the shower particle multiplicity calculation.

#### 3. Results and Discussions

We have used the approach discussed in Section 2 for the calculation of total nuclear reaction cross section for proton-emulsion (*h-A*), ^{56}Fe-Em, ^{84}Kr-Em, ^{132}Xe-Em, ^{197}Au-Em, and ^{238}U-Em at incident engines ~1 GeV per nucleon. These calculations have been performed in the Coulomb modified Glauber model (CMGM) environment using parameters related to the free space nucleon-nucleon interaction and medium nucleon-nucleon interaction . Taken into consideration the calculation of total nuclear reaction cross section () is = 0, in the case without nuclear medium effect. In the case with nuclear medium effect, we used *ρ* = 0.15 fm^{−3}, *ρ* = 0.17 fm^{−3}, and *ρ* = 0.19 fm^{−3}, for the calculation of total nuclear reaction cross section. Here, *ρ* is the saturation density of the normal nuclear matter, which ranges from 0.15 to 0.19 fm^{−3} [18]. The calculated nuclear reaction cross section with medium effects is represented as , , and . This* NN* interaction used in this calculation is defined as the medium nucleon-nucleon interaction and generally most of the previous calculations [21, 32] considered nuclear matter density = 0.17 fm^{−3} only, in their calculation. It is worth mentioning here that we have performed all theoretical calculation of nuclear reaction cross section in accordance with the zero-range approach. These nuclear reaction cross section values are plotted with respect to the mass number of the different target of the nuclear emulsion detector nuclei for different projectiles at incident kinetic energy around 1 GeV per nucleon in Figures 1 and 2.