Electromagnetic field effects on nucleon transverse momentum and fragment distribution for heavy ion collisions around 100A MeV

Special Section on the Celebration for 55 Years’ Dedicated Research on Heavy-Ion Physics of Natowitz and his 80th Birthday

Electromagnetic field effects on nucleon transverse momentum and fragment distribution for heavy ion collisions around 100A MeV

Xian-Gai Deng，

Yu-Gang Ma

Nuclear Science and Techniques

Vol.28, No.12

Article number 182

Published in print 01 Dec 2017

Available online 22 Nov 2017

DOI：10.1007/s41365-017-0337-1

41101

With taking electromagnetic field into account for the transport model of Boltzmann-Uehling-Uhlenbeck, electromagnetic effects are studied for ²⁰⁸Pb+²⁰⁸Pb collisions around 100A MeV. Electromagnetic field evolution during the collisions was estimated. It was found that the electric field has an obvious effect on the transverse momentum (p_T) spectra of nucleons during heavy ion collisions, and leads to different minimum position for the peak of p_T spectra of nucleons versus beam energy when the electric field is switched on. For the magnetic field, it affects the z-axis direction distributions of nucleons for central heavy ion collisions at lower energy.

Heavy-ion collisionElectromagnetic filedTransverse momentum

1 Introduction

In heavy-ion collisions, strong electrical fields and magnetic fields should be considered which was firstly pointed out by the Rafelski and Müller in Ref.[1]. With our well knowledge, the Coulomb force causes the repulsive interactions for the charge particles and the Lorentz force can alter the velocities of charge particles. Many efforts have been paid for the Coulomb effects in heavy ion collisions [2-8]. The electric fields treated as static electric field (Coulomb fields) has been taken into account in most transport models for the study of nuclear matter of intermediate energy heavy ion collision. In Ref. [9], the electromagnetic fields are considered with the velocities of charge particles in heavy ion collisions at intermediate energies within the Boltzmann-Uehling-Uhlenbeck (BUU) transport model. The magnetic field created in noncentral Au+Au collision at $\sqrt{s} = 200$ GeV at the Relativistic Heavy Ion Collider (RHIC) can reach about $e B \sim m_{π}^{2} \sim 10^{18}$ Gauss (in Lorentz units, 1 MeV² ≈ 5.11 × 10¹³ Gauss) [9-13]. Experimental and theoretical indications showed that the QCD topological effects such as the Chiral Magnetic Effect (CME) in the intense electromagnetic fields may be the evidence of local parity violation in strong interactions [15-21]. The properties of strange quark matter in a strong external electric and magnetic field will be also significantly influenced [22-24]. Very recently, the strongest votical motion has been deduced from the polarization measurement of Λ hyperon in Au + Au collisions at 200 GeV/c by the STAR collaboration [25] which is consistent with the prediction of the global polarization of the quark-gluon plasma [26].

But electromagnetic field, especially for the magnetic field, are seldom investigated at low energy for heavy ion collisions where is far from pion production energies. So the aim of this paper is to study the electromagnetic effects at low energies ranging from 40 MeV/nucleon to 120 MeV/nucleon for Pb + Pb heavy ion collisions.

The organization of the paper is as follows: In Sec. 2 we give a brief introduction of the simulation model including electromagnetic field. Results of electromagnetic field and electromagnetic field effects are discussed in Sec. 3.1. Finally, a summary is given in the end.

2 Theoretical Framework

2.1 Boltzmann-Uehling-Uhlenbeck model

The BUU model is a very popular tool for describing intermediate-energy heavy ion collision [27, 28], which is a one-body mean-field theory based upon the Boltzmann equation [29]. The BUU equation reads [30]:

\begin{array}{l} \frac{\partial f}{\partial t} + υ \nabla_{r} f - \nabla_{r} U \nabla_{p} f \\ = \frac{4}{{(2 π)}^{3}} \int d^{3} p_{2} d^{3} p_{3} d Ω \frac{d σ_{NN}}{d Ω} υ_{12} \times [f_{3} f_{4} (1 - f) \\ \times (1 ​ - ​ f_{2}) ​ - ​ f f_{2} (1 ​ - ​ f_{3}) (1 ​ - ​ f_{4})] δ^{3} (p + p_{2} ​ - ​ p_{3} ​ - ​ p_{4}), \end{array}

(1)

where f = f(r,p,t) is the phase-space distribution function. One can solve this equation by the method of Bertsch and Das Gupta [31]. In Eq.(1), $\frac{d σ_{NN}}{d Ω}$ and υ₁₂ are nucleon-nucleon cross section and the relative velocity for the colliding nucleons, respectively. The nucleon-nucleon cross sections include the elastic and inelastic ones, and the well-known parametrization of Cugnon, Mizutani, and Vandermeulun [32] as a function of the available center-of-mass energy has been used. The details of the BUU model were described in Ref. [31].

And the mean-field potential U including the isospin-dependent symmetry energy term can be given:

U (ρ, τ_{z}) = a (\frac{ρ}{ρ_{0}}) + b {(\frac{ρ}{ρ_{0}})}^{κ} + C_{sym} (\frac{ρ_{n} - ρ_{p}}{ρ_{0}}) τ_{z},

(2)

where ρ₀ (ρ₀ = 0.168 fm^-3), ρ_n and ρ_p are the normal nuclear matter, neutron and proton densities; and τz equals 1 or -1 for neutrons and protons, respectively; and C_sym is the coefficient of the symmetry energy term (here C_sym = 32 MeV is used). The coefficients a, b, and κ are parameters for the nuclear equation of state (EoS). In this work, we use one set of mean-field parameters, the semi-soft EoS with the compressibility K of 235 MeV (a = -218 MeV, b = 164 MeV, κ = 4/3).

One can appropriately implement the electrical and magnetic fields into transport model according to the Liénard-Wiechert potentials at a position $\vec{r}$ and time t [9, 33]:

e \vec{E} (\vec{r}, t) = \frac{e^{2}}{4 π ϵ_{0}} \sum_{n} Z_{n} \frac{c^{2} - υ_{n}^{2}}{{(c R_{n} - {\vec{R}}_{n} \cdot {\vec{υ}}_{n})}^{3}} (c {\vec{R}}_{n} - R_{n} {\vec{υ}}_{n}),

(3)

e \vec{B} (\vec{r}, t) = \frac{e^{2}}{4 π ϵ_{0} c} \sum_{n} Z_{n} \frac{c^{2} - υ_{n}^{2}}{{(c R_{n} - {\vec{R}}_{n} \cdot {\vec{υ}}_{n})}^{3}} {\vec{υ}}_{n} \times {\vec{R}}_{n},

(4)

where in the left-hand side, an additional charge e is in order to get the electromagnetic fine structure constant α = e²/4π = 1/137 (setting ϵ₀= ħ = c = 1) in the right side of Eqs.(3) and (4). And Zn is the charge number of the nth particle; ${\vec{R}}_{n} = \vec{r} - \vec{r}'_{n}$ , where $\vec{r}'_{n}$ is position of charge particle moving with the velocity ${\vec{υ}}_{n}$ at retarded time $t_{r n} = t - | \vec{r} - \vec{r}'_{n} (t_{r n}) | / c$ . In non-relativistic approximation, when υ << c, one can obtain:

e \vec{E} (\vec{r}, t) = \frac{e^{2}}{4 π ϵ_{0}} \sum_{n} Z_{n} \frac{1}{R_{n}^{3}} {\vec{R}}_{n},

(5)

e \vec{B} (\vec{r}, t) = \frac{e^{2}}{4 π ϵ_{0} c} \sum_{n} Z_{n} \frac{1}{R_{n}^{3}} {\vec{υ}}_{n} \times {\vec{R}}_{n},

(6)

With adding the electrical and magnetic fields, the Hamilton’s equations of propagations of nucleons become:

\frac{d {\vec{p}}_{i}}{d t} = - \nabla_{r} U ({\vec{r}}_{i}) + {\vec{F}}_{C} ({\vec{r}}_{i}) + {\vec{F}}_{L} ({\vec{r}}_{i}),

(7)

\frac{d {\vec{r}}_{i}}{d t} = \frac{{\vec{p}}_{i}}{\sqrt{m^{2} + {\vec{p}}_{i}^{2}}},

(8)

where ${\vec{F}}_{C} ({\vec{r}}_{i})$ and ${\vec{F}}_{L} ({\vec{r}}_{i})$ are Coulomb and magnetic force of i-th particle, respectively. We have to mention that for dealing with the initial distance of target and projectile, we did it as the same way which was mentioned in Ref. [9].

3 Results and discussion

3.1 Space-time evolution of the electromagnetic field

First we plot the time evolutions of nucleon density (top panels), electric field in the z direction (middle panels) and magnetic field in the y direction (bottom panels) in the x-z plane at impact parameter of b = 0.5×b_max (b_max= $1.12 * (A_{P}^{1 / 3} + A_{T}^{1 / 3})$ ) fm for the 40 MeV/nucleon ²⁰⁸Pb+²⁰⁸Pb collisions in Fig. 1. The characteristics of electric fields (middle panels) in the z direction display the geometric configurations of two collision nuclei as shown in top panels of Fig. 1. There are two parts in the zone of each nucleus for the electric field eEz at the early stage of reaction, as shown in Fig. 1 (b1) and (b2). At later stage, a compound system occurs between nuclei and the shapes of nuclei influence the distribution of electric field which are formed two parts as shown in Fig. 1 (b3). As time increases, they maintain it except for the strength decreasing. The magnetic field eBy forms differently in comparison with electric field eEz. The magnetic field forms a centre and two spectator zones about 20 fm/c. When the reactions reach maximum compression, the centre zone of magnetic field becomes smaller and strength becomes larger. Compared the strength of electromagnetic field in relativistic heavy ion collisions [10], the strengths (electric field ≈ 5×10^-3 mπ and magnetic field ≈ 5×10^-3 mπ) are too small here. Our calculations for the distributions of nucleon density and electromagnetic field are similar to the calculation by Ou and Li [9].

Figure 1:

(Color online) Time evolutions of distributions of nucleons density ρ (top), x-axis of electrical field eEx (middle), and y-axis of magnetic field eBy (bottom) in the x–z plane at impact parameter of b = 0.5×b_max fm for the 40 MeV/nucleon ²⁰⁸Pb+²⁰⁸Pb collisions.

In addition, we give the evolution of magnetic field in the y direction at centre position R(0,0,0) at different impact parameters and incident energies in Fig. 2 and Fig. 3, respectively. The time evolution of average eBy(0,0,0) around zero for the central collision is shown in Fig. 2. As the impact parameters increase, the maximum values increase but for the situation of b=1.5×b_max fm. It is due to different definition of initial position of target and projectile. In Fig. 3, the maximum values of the average eBy(0,0,0) increase as the beam energies increase. One should notice that the strengths of eBy(0,0,0) are negative in both Fig. 2 and Fig. 3. This is caused by the relative position of target and projectile which is shown in the top of Fig. 1. It is opposite compared with the results in Ref. [9].

Figure 2:

(Color online) Time evolution of eBy(0,0,0) at different impact parameters for the 40 MeV/nucleon ²⁰⁸Pb+²⁰⁸Pb collisions.

Figure 3:

(Color online) Time evolution of eBy(0,0,0) at different beam energies for ²⁰⁸Pb+²⁰⁸Pb collisions at the impact parameter of b = 0.5×b_max fm.

3.2 Electric effects on the p_T spectra

One should be acquainted with Coulomb interaction for the EoS in heavy ion collisions. Coulomb effects on flows have been studied around the balance energy in many literatures, eg. Refs.[8, 34, 35]. In our work, the electric field is included in the simulation model with considering the velocities of charge particles in the reaction system. Here electric effects are explored by the p_T spectra of nucleons at beam energy of 40 MeV/nucleon and impact parameter of b=0 fm as shown in Fig. 4. It is worth to mention here that distributions of p_T (= $\sqrt{p_{x}^{2} + p_{y}^{2}}$ ) of nucleons will keep stable when the reactions reach freeze-out stage. And we extracted the distributions from the final stage of reactions in Fig. 4. One can see that the p_T peak shifts to left when including electric field (with E). It indicates that the beam energy becomes difficult to convert into the transverse energy with the Coulomb interaction is included at 40 MeV/nucleon, which also makes the system hard thermalized at lower beam energy. Two peak values of the solid line (without E) and dash line (with E) are extracted by the fitting and shown in Fig. 4. Interestingly, a minimum value emerges from the dependence of peak values on beam energy as shown in Fig. 5(a). From the square-blue line without E, the minimum value emerges around 80 MeV/nucleon. Also as the dot-red line shows, there is another minimum value when the electric field is considered. In the first case, it could be related to the competition between the attractive part and repulsive part of EoS. For the latter, the additional repulsive interaction can be generated by electric field and makes the minimum value shift to lower energy. In contrast to the lower beam energy case (eg. 40 MeV/nucleon), the peak value of p_T becomes larger when the Coulomb interaction is on when beam energy is larger (eg. 120 MeV/nucleon), which indicates the beam energy has higher efficiency to convert into transverse energy. Furthermore, differences of peak point values between the cases w/ and w/o Coulomb interaction increase as beam energy as shown in Fig. 5(b). And the difference is vanishing around 55 MeV/nucleon. Thus, it should be paid more attentions to the physics mechanism at beam energy of ranging 40 MeV/nucleon to 80 MeV/nucleon.

Figure 4:

(Color online) The distribution of p_T of all nucleons at impact parameter of b = 0 fm for the 40 MeV/nucleon ²⁰⁸Pb+²⁰⁸Pb collisions.

Figure 5:

(Color online) The peak point values of p_T spectra of nucleons (a) and the change of peak point values between the cases w/ and w/o the Coulomb interaction (b) as a function of beam energies.

To see the how electric field affects on the nucleons, the evolutions of distributions of p_T of proton, neutron and all nucleons are shown in Fig. 6. Here we take the case without electric field as a reference. At early reaction stage of 40 fm/c, the protons are directly affected by the electric field as Fig. 6(a1) shows. And the distribution of p_T is shifted to right. The shift becomes smaller as time increases. It, however, is inverse for the neutron. The difference becomes larger at the final stage. Although electric field affects on the protons directly, the neutrons are also affected during collisions. It is indicated that there is an energy transfer between the protons and neutrons.

Figure 6:

(Color online) Time evolutions of distributions of p_T of proton (top), neutron (middle), and all nucleons (bottom) at impact parameter of b = 0 fm for the 40 MeV/nucleon ²⁰⁸Pb+²⁰⁸Pb collisions with and without electric field.

3.3 Observation for magnetic effects

Pion production as an probe has been employed to investigate the magnetic effects in Ref. [9], however, pion production energy is far from the energy range we discussed in this paper. First, magnetic effects are likewise considered with the distribution of p_T as shown in Fig. 7. But here, the simulations include the electric field. One can see that the magnetic effects are not obvious from distribution of p_T. And Fig. 8 shows the ratios of neutron to proton of all and free nucleons as a function of rapidity at different impact parameters with and without magnetic field. The magnetic effects on the ratios are very tiny at impact parameters of b = 0 fm and b = 0.5×b_max fm for 40 MeV/nucleon collisions. They are well in agreement with the results in Ref. [9].

Figure 7:

(Color online) The distribution of p_T at impact parameter of b=0.5×b_max fm for 40 MeV/nucleon and 120 MeV/nucleon ²⁰⁸Pb+²⁰⁸Pb collisions.

Figure 8:

(Color online) The neutron to proton ratio of free (a) and all (b) nucleons as a function of rapidity at impact parameters of b=0 fm and b=0.5×b_max fm for the 40 MeV/nucleon ²⁰⁸Pb+²⁰⁸Pb collisions with and without magnetic field.

The observables which considered in this paper basing on the momentum space of nucleons are not sensitive to the magnetic field at low energy heavy ion collisions. For the knowledge, however, the directions of charged particles can be changed by the magnetic field and fluctuations can be arisen. Thus, we consider more observables with the z-axis direction distributions of nucleons.In Fig. 9(a), the z-axis direction distribution of nucleons are almost symmetric in impact parameter of b=0 fm. When we consider the magnetic field, less nucleons in the central region will be. As beam energy increases in Fig. 9(b), the difference nearly vanishes. It indicates that nucleons with high kinetic energies are not nearly affected by magnetic field generated in heavy ion collisions.

Figure 9:

(Color online) The z-axis direction distributions of nucleons at impact parameters of b=0 fm for 40 MeV/nucleon and 120 MeV/nucleon ²⁰⁸Pb+²⁰⁸Pb collisions with and without magnetic field.

4 Conclusion

In summary, coupling with electromagnetic filed in the transport model BUU, the evolutions of electric field in the z direction and magnetic field in the y direction in the x-z plane are given and the impact parameters and incident energies dependence of the magnetic field at the centre of mass are discussed. Further, we consider the electric effects from the p_T spectra of nucleons and two minimum values of peaks of p_T spectra were found from different situations of with and without electric field. The Coulomb interaction is very important for the study of heavy ion collisions. The magnetic effects are not obvious from the p_T spectra of nucleons and the ratio of neutron to proton of all and free nucleons as a function of rapidity. Nevertheless, magnetic filed have some effects on the z-axis direction distribution of nucleons.

References

[1]

J. Rafelski, B. Müller,

Magnetic Splitting of Quasimolecular electronic states in strong fields

. Phys. Rev. Lett. 36, 517 (1976). doi: 10.1103/PhysRevLett.36.517

Electromagnetic field effects on nucleon transverse momentum and fragment distribution for heavy ion collisions around 100A MeV

1 Introduction

2 Theoretical Framework

2.1 Boltzmann-Uehling-Uhlenbeck model

3 Results and discussion

3.1 Space-time evolution of the electromagnetic field

3.2 Electric effects on the pT spectra

3.3 Observation for magnetic effects

4 Conclusion

3.2 Electric effects on the p_T spectra