Correlation between magnetic field and nuclear stopping in different rapidity segments during heavy ion collisions

Research in heavy ion [1] physics is mainly concentrated on exploring the various hidden phenomena of nuclear matter in extreme circumstances and its relation with the nuclear equation of state (NEOS) [2, 3]. Much success has been achieved for heavy ion collisions (HICs) by unveiling the complex dynamical properties under different conditions of temperature and pressure [4], but more effort is still required. Therefore, various reaction observables, such as anisotropic flow [5–10], nuclear stopping [11, 12] and the electromagnetic field [13–18], need further study extract knowledge about the NEOS during HICs.

HICs thermalize [19] nuclear matter, which is influenced by both Pauli-blocking and binary collisions [20, 21]. This thermalization of nuclear matter is linked to the degree of nuclear stopping, which is a good probe to provide valuable insights about the dynamics of HICs. Furthermore, non-central collisions result in the development of a pressure gradient that transforms initial spatial eccentricity into a final momentum anisotropy, represented by anisotropic flow that is expressed in terms of Fourier expansion [22, 23]. This anisotropic flow is a phenomenon observed during HICs and, moreover, azimuthal asymmetry, reflected in the second coefficient of its Fourier expansion, carries information about elliptical flow. In other words, the phenomenon that occurs in the presence of substantial elliptical anisotropy in the momentum spectra of observed particles is commonly referred to as elliptical flow. Thus, elliptical flow is one form of anisotropic flow; it enjoys a special status in the intermediate-energy region, and explains the dynamics of HICs and NEOS due to complexity of nucleonic mean field and nucleon–nucleon collisions. Elliptical flow can potentially help us comprehend the early time thermalization in HICs that is driven by the initial eccentricity of the matter generated in the overlapping zone. Furthermore, the magnitude of the observed anisotropic flow is shown to be substantially associated with the anisotropic form of the overlapping zone.

The mid-rapidity zone has a very high nucleonic density and is therefore distinguished by significant particle production as well as collective phenomena. The nucleons ejected from the mid-rapidity zone are vitally important for understanding the equation of state, collective flow and other observables that provide valuable insights into the extremely hot and dense nuclear matter produced in collisions. Also, this region can be considered as a probe for exploring the intricate dynamics of nucleons and thermalization processes. The asymmetry in the charged nucleon distribution and motion of the charged nucleons along the beam axis leads to the development of a magnetic field with varying intensity. Recently, the magnetic field generated by nucleons during HICs has emerged as a promising avenue for exploring the intricate dynamics of nucleons in the early reaction phase. Due the wide application areas in defence and energy sectors there has been a lot of research on this topic [24–31]. Therefore, comprehensive theoretical and experimental attempts have been made over the years to gain insight into the relationship between momentum anisotropy and the magnetic field generated in different rapidity bins. Significant work has been reported on intriguing phenomena such as the chiral magnetic effect [32–34], chiral vortical effect [35] and inverse magnetic catalysis [36–38]. Despite this, a thorough study of the electromagnetic field during HICs and its correlation with other observables in the intermediate-energy range is still missing. Theoretical investigations and the study of astrophysical objects with strong magnetic fields provide valuable insights into the behaviour of matter under such extreme conditions.

The scope of the present research is three-fold:

• to investigate the variation of momentum anisotropy with centrality and incident energy in different segments of the mid-rapidity zone;
• to analyse the effect of various segments of the mid-rapidity zone on the correlation of eccentricity and momentum anisotropy;
• to investigate the correlation between the magnetic field produced due to nucleons in various segment of the mid-rapidity zone and nuclear stopping.

In the current study, the isospin-dependent quantum molecular dynamics (IQMD) model [39, 40] has been employed to generate the phase space of the nucleons that is elucidated in section 2. The results and discussions are chronicle explained in section 3, followed by summary in section 4.

The IQMD [39, 40] model offers a sophisticated framework for simulating and analysing the complex dynamics of HICs. IQMD is an advanced version of the QMD model [41] and employs the isospin degree of freedom via cross-sectional Coulomb interactions and symmetry potential that enables the investigation of nuclear matter under extreme density and temperature conditions. It is a time-dependent many-body theoretical approach that treats nucleon–nucleon correlation explicitly. The IQMD model seamlessly integrates the time evolution of nuclear systems, providing insights into the formation of excited states, multi-fragmentation phenomena and the emergence of collective flow patterns [42–45]. This model involves three main steps:

• (i)  
  Initialization: the first step is initialization of the target and projectile nuclei. Each nucleon is represented by a Gaussian wave packet and occupies volume h³ which is distributed in a sphere of radius R = 1.12A^1/3. Rejection and acceptance of nucleons depends on the Fermi momentum and distance between the nucleons. The distribution of all nucleons is expressed by the Gaussian-shaped density distribution

  $$\begin{eqnarray}&&{f}_{i}(\vec{r},\vec{p},t)=\displaystyle \frac{1}{{\pi }^{3}{{\hslash }}^{3}}\,{e}^{-{\left({\bf{r}}-{{\bf{r}}}_{i}(t)\right)}^{2}\displaystyle \frac{2}{L}}\,{e}^{-{\left({\bf{p}}-{{\bf{p}}}_{i}(t)\right)}^{2}\displaystyle \frac{L}{2{{\hslash }}^{2}}}.\end{eqnarray} \tag{ 1 }$$

  The symbol $\vec{r}$ denotes a dynamic variable that characterizes the central position of the wave packet in coordinate space, whereas $\vec{p}$ refers to a variable that varies with time and describes the central position of the wave packet in momentum space. Likewise, L indicates the Gaussian width and h represents the Planck constant.
• (ii)  
  Propagation: after the initialization, the target and projectile nuclei are boosted toward each other in the nucleon–nucleon centre of mass reference frame with appropriate energy. In the IQMD model, the Hamilton equation of motion is used to depict the position and momenta of all hadrons:

  $$\begin{eqnarray}&&\displaystyle \frac{{{dr}}_{i}}{{dt}}\,=\,\displaystyle \frac{d\langle H\rangle }{{{dp}}_{i}};\,\,\displaystyle \frac{{{dp}}_{i}}{{dt}}\,=\,-\displaystyle \frac{d\langle H\rangle }{{{dr}}_{i}},\end{eqnarray} \tag{ 2 }$$

  with

  $$\begin{eqnarray}\begin{array}{rcl}\langle H\rangle & = & \langle T\rangle +\langle V\rangle \\ & = & \displaystyle \sum _{i}\displaystyle \frac{{p}_{i}^{2}}{2{m}_{i}}+\displaystyle \sum _{i}\displaystyle \sum _{j\gt i}\displaystyle \int {f}_{i}(\vec{r},\vec{p},t){V}_{{\text{}}{ij}}({\vec{r}}^{{\prime} },\vec{r})\\ & & \times {f}_{j}({\vec{r}}^{{\prime} },{\vec{p}}^{{\prime} },t)d\vec{r}d{\vec{r}}^{{\prime} }d\vec{p}d{\vec{p}}^{{\prime} }.\end{array}\end{eqnarray} \tag{ 3 }$$

  Here, V_ij , the total nucleon–nucleon potential, is the sum of the Skyrme interaction, Yukawa potential and Coulomb interaction as well as symmetry potential

  $$\begin{eqnarray*}\begin{array}{rcl}{V}_{{ij}} & = & {V}_{{ij}}^{{Skyrme}}+{V}_{{ij}}^{{Yukawa}}+{V}_{{ij}}^{{Coul}}+{V}_{{ij}}^{{Sym}}.\end{array}\end{eqnarray*}$$

  At standard nuclear matter density, the symmetry energy exhibits a magnitude of 32 MeV. In the IQMD model, symmetry energy is a function of nuclear matter density during HICs. Therefore, E_sym (ρ) transforms into

  $$\begin{eqnarray}&&{E}_{{sym}}(\rho )\,=\,32\times {\left(\displaystyle \frac{\rho }{{\rho }_{0}}\right)}^{\gamma }\,{MeV}.\end{eqnarray} \tag{ 4 }$$

  Here, ρ₀ represents the standard nuclear matter density while ρ reflects the nuclear matter density achieved during HICs. The γ term gives an estimation of the strength of the symmetry energy. In the present work we use a soft equation of state and density-dependent symmetry energy, i.e. γ = 0.66. Further details of the IQMD model can be found in previous reports [39, 41].
• (iii)  
  Nucleon–nucleon collisions: in this model it has been presumed that every nucleon occupies a volume h³ in phase space. To examine the probability of allowed nucleon–nucleon collisions, the availability of phase space around the colliding partners should be considered. The nucleons will scatter in the course of propagation when the approach distance between them becomes smaller than the distance $\sqrt{\tfrac{{\sigma }_{{tot}}}{\pi }}$. σ_tot (i.e. ${\sigma }_{{tot}}=\sigma (\sqrt{s}$, type) is the total energy-dependent cross section and is expressed as

$$\begin{eqnarray}&&{\sigma }_{{tot}}={\sigma }_{{el}}+{\sigma }_{{inel}}={\sigma }_{{el}}+\displaystyle \sum _{{channels}}{\sigma }_{i}\end{eqnarray} \tag{ 5 }$$

where $\sqrt{s}$ is the centre of mass energy of two colliding partners and 'type' represents the in-going collision partners (N–N, N–π, N–Δ, ...). σ_el and σ_inel are the elastic and inelastic cross-sections, respectively. Moreover, binary nucleon–nucleon collision is allowed with a probability 1 – P_block . P_block is defined as

$$\begin{eqnarray}&&{P}_{{block}}=1-[1-\min ({P}_{1},1)][1-\min ({P}_{2},1)].\end{eqnarray} \tag{ 6 }$$

P₁ and P₂ are the fractions of final phase space occupied by the two scattering partners.

The electromagnetic field produced due to moving charged particles can be smoothly integrated with the transport model by introducing Lineard–Wiechert potential

$$\begin{eqnarray}&&e\vec{{\boldsymbol{B}}}(\vec{{\boldsymbol{r}}},t)=\displaystyle \frac{{e}^{2}}{4\pi {\epsilon }_{0}c}\displaystyle \sum _{n}{Z}_{n}\displaystyle \frac{{c}^{2}-{v}_{n}^{2}}{{\left({{cR}}_{n}-{\vec{{\boldsymbol{R}}}}_{n}\cdot {\vec{{\boldsymbol{v}}}}_{n}\right)}^{3}}({\vec{{\boldsymbol{v}}}}_{n}\times {\vec{{\boldsymbol{R}}}}_{n})\end{eqnarray} \tag{ 7 }$$

where Z_n represents the charge number of the nth particle and α = e²/4π = 1/137 is the electromagnetic fine structure constant. ${\vec{{\boldsymbol{R}}}}_{n}=\vec{{\boldsymbol{r}}}-{\vec{{\boldsymbol{r}}}}_{n}^{{\prime} }$ indicates the relative distance of the field point $\vec{{\boldsymbol{r}}}$ from the nth proton that is at a position ${\vec{{\boldsymbol{r}}}}_{n}^{{\prime} }$ at retarded time ${t}_{{rn}}=t-\left|\vec{{\boldsymbol{r}}}-{\vec{{\boldsymbol{r}}}}_{n}^{{\prime} }\left({t}_{{rn}}\right)\right|/c$ and ${\vec{{\boldsymbol{v}}}}_{n}$ depicts the velocity of the nth proton. When ${\vec{{\boldsymbol{v}}}}_{n}$ ≪c, the above equation turns into [15, 16]

$$\begin{eqnarray}&&e\vec{{\boldsymbol{B}}}(\vec{{\boldsymbol{r}}},t)=\displaystyle \frac{{e}^{2}}{4\pi {\epsilon }_{0}{c}^{2}}\displaystyle \sum _{n}{Z}_{n}\displaystyle \frac{1}{{R}_{n}^{3}}({\vec{{\boldsymbol{v}}}}_{n}\times {\vec{{\boldsymbol{R}}}}_{n}).\end{eqnarray} \tag{ 8 }$$

In this work, simulations were carried out for the ${}_{79}^{197}\mathrm{Au}$ + ${}_{79}^{197}\mathrm{Au}$ reaction in the IQMD model framework. Correlation between momentum anisotropy and the magnetic field generated in different rapidity zones provides a unique insight into the complex dynamics of HICs. Here, momentum anisotropy is defined as the unequal distribution of particle momenta in different directions relative to the collision axis, and reveals important information about the early stage of HICs. To correlate these observables, a rapidity zone in the range of –0.6 ≤ Y_c.m./Y_beam ≤ 0.6 was chosen and segregated into three different rapidity bins, as shown in figure 1. Here, bin 1 is the most central and symmetric part of target and projectile rapidity, which is taken in the range –0.2 ≤ Y_c.m./Y_beam ≤ 0.2; this is the highly dense and chaotic part of the overlapped region. Further, bin 2 is considered as −0.4 ≤ Y_c.m./Y_beam ≤ −0.2 in the target rapidity region; keeping the symmetry in consideration, projectile rapidity 0.2 ≤ Y_c.m./Y_beam ≤ 0.4 is also taken as part of bin 2. Bin 3 is considered to be from −0.6 ≤ Y_c.m./Y_beam ≤ –0.4 and 0.4 ≤ Y_c.m./Y_beam ≤ 0.6 in the target and projectile rapidity regions, respectively.

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Furthermore, nuclear stopping is the computation of the dissipation of projectile energy into transverse energy during HICs. As documented in the literature, different parameters can be used to characterize the extent of nuclear stopping [46], and here it is investigated using the anisotropic ratio 〈R_P 〉, which is expressed as [46]

$$\begin{eqnarray}&&\langle {R}_{P}\rangle =\displaystyle \frac{2}{\pi }\displaystyle \frac{\left({\sum }_{i}{p}_{\perp }(i)\right)}{\left({\sum }_{i}{p}_{\parallel }(i)\right)}\end{eqnarray} \tag{ 9 }$$

where ${p}_{\perp }(i)=\sqrt{({p}_{x}^{2}(i)+{p}_{y}^{2}(i))}$ is the transverse momentum and p_∥(i) = p_Z (i) the longitudinal momentum of the ith nucleon. Likewise, to check the relationship between the mean field and nucleon–nucleon collisions, the elliptical flow, which is a second coefficient of Fourier expansion [22], has been scaled through eccentricity described as [47–49]

$$\begin{eqnarray}&&\langle \epsilon \rangle =\displaystyle \frac{\langle {y}^{2}-{x}^{2}\rangle }{\langle {y}^{2}+{x}^{2}\rangle }.\end{eqnarray} \tag{ 10 }$$

Furthermore, the rapidity is expressed as [41]

$$\begin{eqnarray}&&Y(i)=\displaystyle \frac{1}{2}{ln}\displaystyle \frac{E(i)+{p}_{z}(i)c}{E(i)-{p}_{z}(i)c},\end{eqnarray} \tag{ 11 }$$

where p_z (i), E(i) are the longitudinal momentum and total energy of the ith nucleon and c is the velocity of light.

Before investigating the correlation between magnetic field and nuclear stopping, we first explore a dominant parameter, i.e. nucleon–nucleon collision that influences the strength of both the magnetic field and nuclear stopping. The fluctuations in the number of collisions with time during HICs is an intricate phenomenon that is affected by various factors such as collision energy, impact parameter and rapidity. Study of this variation provides critical insights into the dynamics of nuclear matter and advances our understanding of the fundamental properties of nuclear matter under extreme conditions. Figure 2(a) demonstrates the progression in the number of collisions per nucleon with time in the determined scaled impact parameter range ($\hat{b}$ = 0.45–0.55) in different rapidity zones. The number of collisions increases very rapidly during the initial phase of collision and then the rate of increment reduces and subsequently almost saturates. This might be because the nucleons have a larger possibility of engaging with one other in the overlapping phase. Moreover, the increased density of nucleons in the overlapping zone also enhances the likelihood of nucleon–nucleon interactions. Further, the nuclei begin to split as the collision progresses, and the nucleonic density in the overlapped zone drops, resulting in a slowing down of the rate of increase in the number of collisions per nucleon. Moreover, the progression in the number of collisions per nucleon with the time in different rapidity regions during HICs is a complex phenomenon that sheds light on the dynamic nature of these interactions. During the early stages of a HIC, the number of collisions per nucleon is generally higher for bin 1 than for bins 2 and 3. This might be attributed to the denser overlapped region in bin 1, where the colliding nuclei interact strongly, resulting in numerous collisions between the nucleons. In rapidity bins 2 and 3, the number of collisions per nucleon is generally less than in bin 1, albeit with a similar pattern of time evolution of N_coll /nucleon in all the rapidity bins. When the density of nucleons falls too low for additional significant collisions to occur the saturation point occurs for all the rapidity bins.

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The dependence of collision multiplicities per nucleon on the impact parameter in different rapidity regions is also a crucial aspect that needs investigation to gain insight into the collision dynamics and the spatial distribution of nucleon–nucleon interactions. Figure 2(b) depicts variation of the number of collisions with the impact parameters, keeping all the other parameters the same in different rapidity bins. A smaller impact parameter leads to a larger overlapping zone that results in a higher number of nucleons from each nucleus participating in the interaction. Hence, the higher density of nucleons for a smaller impact parameter results in a greater number of collisions, as shown in figure 2(b). On the other hand, with a higher impact parameter the probability of nucleon–nucleon interactions decreases. This is why the number of collisions per nucleon decreases with increase in the impact parameter. Furthermore, the rapidity, which is a relativistic quantity reflecting particle velocity along the beam axis, plays a crucial role in the collision dynamics. Specifically, the bin 1 rapidity region that encompasses the most central zone of the collisions experiences a higher density of nucleons; thus the highest number of collisions per nucleon takes place compared with other rapidity bins.

Through meticulous experimental measurements and advanced theoretical models, researchers have striven to unravel the complex dynamics of nuclear stopping and provide critical insights into the fundamental processes that occur during HICs and the properties of nuclear matter under extreme conditions. Nuclear stopping alludes to the reduction in the longitudinal momentum of the colliding nucleon as a result of its interactions. Therefore, in the mid-rapidity segment (indicating an extremely dense zone during collision) an extremely high nucleonic density results in strong nucleon–nucleon interactions leading to significant nuclear stopping. Hence, the dependence of nuclear stopping on incident energy in the mid-rapidity region enables scientists to deepen their understanding of the collision dynamics and underlying mechanisms governing nuclear interactions. Figure 3(a) shows the variation of nuclear stopping with incident energy in different rapidity bins. It has been observed that as the collision energy increases, the nuclear stopping per nucleon tends to decrease in the mid-rapidity segment. This decrease might be attributed to the dominance of more violent collisions at higher energies. In such types of collisions, the nucleons have a higher probability of undergoing scattering processes rather than experiencing significant momentum transfer and stopping. Consequently, as the collision energy increases, the relative contribution of these events with less stopping becomes more pronounced, leading to a decrease in the overall nuclear stopping per nucleon in the mid-rapidity zone. Noticeably, nuclear stopping in bin 1 is much higher than in the other two bins, indicating that more squeezing of nucleons takes place in bin 1. Additionally, the rate of decrement of nuclear stopping with incident energy is higher for bin 1 than for bins 2 and 3.

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The variation of nuclear stopping per nucleon with impact parameter in different rapidity regions during HICs is a subject of great significance, as shown in figure 3(b). This figure reflects that going towards a higher impact parameter there is decrease in the nuclear stopping per nucleon. This is attributed to the increased overlapping zone between colliding nuclei at lower impact parameters, leading to a stronger interaction and a higher degree of nuclear stopping. In contrast, for collisions with larger impact parameters, the nuclear stopping per nucleon decreases, indicating a smaller overlap region and reduced interaction between the nuclei, resulting in less stopping of nucleons along the beam axis. Likewise, it is also worth mentioning that bin 1 experiences more nuclear stopping than the other rapidity bins.

To acquire insight into the collective behaviour and underlying transport features of the generated high-density nuclear matter, it is critical to explore the correlation between elliptical flow and nuclear stopping in different rapidity zones to unravel the dynamical evolution of HICs. A higher elliptical flow leads to higher initial eccentricity, and therefore eccentricity in different rapidity bins is used to scale the elliptical flow. Furthermore, to investigate this correlation, the effect of some important parameters, i.e. incident energy and the impact parameter, on the eccentricity is demonstrated in figure 4. It can be observed in figure 4(a) that the eccentricity goes on decreasing with increase in the incident energy for a certain value of impact parameter. Likewise, figure 4(b) depicts the variation of eccentricity with the scaled impact parameter at a fixed value of incident energy (400 MeV/nucleon) in different rapidity bins. It can be inferred that eccentricity goes on increasing as we go for a higher scaled impact parameter.

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Eccentricity plays a vital role in the generation of elliptical flow, and there is a correlation between elliptical flow and nuclear stopping in HICs. The degree of nuclear stopping governs the transfer of energy and momentum between colliding nuclei, which affects initial spatial anisotropy and the eventual development of elliptical flow. As a result, the anisotropic pressure gradient in the system is triggered by the initial eccentricity, which is dictated by the geometry and overlapping region of colliding nuclear particles that are emitted perpendicularly with respect to the reaction plane. Figure 5 represents the correlation between eccentricity and nuclear stopping at 400 MeV/nucleon in different rapidity bins. An inverse correlation is observed between eccentricity and nuclear stopping in all the rapidity bins. Furthermore, it can be inferred from figure 5 that the rate of change of eccentricity with respect to nuclear stopping goes on increasing as we move from bin 1 to bin 3.

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The time evolution of magnetic field during HICs is very important for unveiling the complex nature of nucleon dynamics. Figure 6 demonstrates the production of a magnetic field with time at 400 MeV/nucleon in the different rapidity segments. It indicates that the intensity of eB_y (0, 0, 0) increases rapidly to reach its maximum and then starts to decrease. The intensity of the magnetic field attains its peak value going through maximum compression of the nuclei. Noticeably, the intensity of eB_y (0, 0, 0) is highest for bin 3 compared with the other two bins. This is because bin 3 nucleons feel less nuclear stopping, in addition to the occurrence of fewer collisions compared with bins 1 and 2. Likewise, the phase space evolution of eB_y (0, 0, 0) is demonstrated in figure 7 using MATLAB software. The phase space evolution is shown for all the three rapidity bins for different time steps (5 fm/c, 30 fm/c and 60 fm/c) at 400 Mev/nucleon.

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In-depth knowledge of the intricate interplay between the dynamics of charged particles and their resulting impact on the generation of magnetic fields in different rapidity segments is crucial. Therefore, it is of the utmost importance to carry out a systematic investigation into the variation of magnetic field as a function of incident energy and impact parameter. Figure 8(a) depicts the variation of ${\left({{eB}}_{y}(0,0,0)\right)}_{\max }$ with incident energy during HICs for various rapidity regions. Increased incident energy will result in a higher magnetic field intensity, as can be clearly observed in figure 8(a). This is attributed to increase in the acceleration of charged nucleons. Moreover, the number of collisions and nuclear stopping (discussed in figures 2 and 3) in bin 3 is comparatively less compared with the other rapidity bins. Therefore, the magnetic field intensity is higher for rapidity bin 3 at each incident energy of HICs compared with rapidity bins 1 and 2. Similarly, the variation of magnetic field intensity ${\left({{eB}}_{y}(0,0,0)\right)}_{\max }$ with the impact parameter for different rapidity zones is displayed in figure 8(b). As we go toward a higher impact parameter, the participating zone decreases, resulting in a higher magnetic field intensity in all rapidity bins. The magnetic field per nucleon increases with increase in the impact parameter when we go from bin 1 to bin 3 during HICs. This is attributed to the fact that bin 3 nucleons experience fewer nucleon–nucleon collisions and reduced nuclear stopping (as discussed in figures 2 and 3). In the convoluted landscape of HICs, a meticulous study of the correlation between nuclear stopping and magnetic field in different rapidity segments help us to grasp the detailed information about the dynamics of nucleons and their contribution to the generation of magnetic field. Figure 9 demonstrates the correlation between magnetic field and nuclear stopping in various rapidity bins at 400 MeV/nucleon. In the central rapidity zone (bin 1), the intensity of the magnetic field is comparatively smaller. There are two main reasons for this: the first one is more nuclear stopping and the greater number of nucleon–nucleon collisions, resulting in a weaker magnetic field; the second reason is the maximum squeezing out of nucleons taking place in bin 1 at 400 MeV/nucleon that make an insignificant contribution to the production of a magnetic field in the y-direction. Going towards the higher rapidity bins, nuclear stopping begins weakening and nucleon–nucleon collisions start to decrease, as shown in figure 2, therefore enhancement in the intensity of ${\left({{eB}}_{y}(0,0,0)\right)}_{\max }$ can be clearly observed in figures 9(a) and (b). In addition, the squeezing out of nucleons is hampered for higher rapidity bins, and therefore the nucelons will develop in-plane momentum. All these factors contribute significantly to enhancement of the intensity of ${\left({{eB}}_{y}(0,0,0)\right)}_{\max }$ in rapidity bins 2 and 3.

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Finally, to check the robustness and reliability of the theoretical framework of the IQMD model, a comparative study with another theoretical framework was carried out. In recent times, several transport models have been developed to simulate HICs. For example, Li et al simulated the time evolution of the magnetic field in the y-direction (eB_y (0, 0, 0)) using the IBUU11 theoretical framework at 500 MeV/nucleon [16]. Figure 10(a) compares our findings with the work reported by Li and his collaborators. It is worth mentioning that the pattern followed by the time evolution of magnetic field in the y-direction is almost same but with a different intensity due to different method of calculation using same input mean field potential. This study has also been compared with the work reported by Ma and his collaborators [15]. They studied the time evolution of eB_y (0, 0, 0) using the BUU theoretical framework at 80 MeV/nucleon at b = 0.5 b_max for Pb + Pb collision (shown in figure 10(b)). Figure 10(b) shows that the trend followed is almost same but notable disparity arises in the magnitude of eB_y (0, 0, 0). In addition to this, the maximum intensity of eB_y (0, 0, 0) is obtained at about 30 fm/c which is almost same as reported in this work.

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This study contributes to enhancing our understanding of the complex interaction and dynamics of nucleons in the mid-rapidity zone. The mid-rapidity zone is segregated into three different bins to investigate their effects on various observables. It has been observed that the most central region (bin 1) of mid-rapidity zones has highest number of nucleon–nucleon collisions per nucleon; this decreases going toward the higher centrality region. Therefore, bin 1 nucleons feel more nuclear stopping compared with the other bins and their nuclear stopping decreases with increase in the incident energy and impact parameter. Furthermore, an inverse correlation has been observed between the eccentricity and nuclear stopping in all bins in mid-rapidity zones. Finally, the effect of various rapidity bins has been observed on the time and phase space evolution of magnetic field in the y-direction (eB_y ). The intensity of ${\left({{eB}}_{y}(0,0,0)\right)}_{\max }$ increases with increase in incident energy and impact parameter in all the rapidity bins. Moreover, correlation between momentum anisotropy and the magnetic field provides unique and valuable insights to understand the early stage of HICs. It can be summarized that the strong nucleon–nucleon interactions in bin 1 lead to a significant number of collisions and nuclear stopping. Due to more nucleon–nucleon collisions and nuclear stopping, bin 1 nucleons contribute less to the production of a magnetic field than bins 2 and 3.

Authors express their gratitude the Professor R K Puri for valuable discussion and providing access to different computer programs utilized in present study. One of author, Dhanpat Sharma is thankful to UGC-CSIR for providing senior research fellowship.

The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.