Effect of internal magnetic field on collective flow in heavy ion collisions at intermediate energies

The properties of nuclear matter under extreme conditions of high temperature, density, and isospin asymmetry have attracted wide attention in recent years. At present, heavy ion reactions in combination with corresponding model simulations are one of the most important ways to investigate this subject. It is known that a strong magnetic field can be created in heavy ion collisions. However, its effect on the motion of charged particles is usually neglected in previous transport model simulations. In this work, within the ultrarelativistic quantum molecular dynamics (UrQMD) model, the temporal evolution and spatial distribution of the internal magnetic field are calculated. The magnetic field strength is found to reach about $eB\approx 470{\mathrm{MeV}}^2$ ($B\approx 8\times {10}^{16}$ G) for Au $+$ Au collisions at $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with impact parameter of 7 fm. The magnetic field in Cu $+$ Au collisions exhibits somewhat different spatial distribution from that in Au $+$ Au collisions. The magnetic field is found to affect the directed flow of pions at forward and backward rapidities to some extent, dependent of the impact parameter and beam energy while the effect on the elliptic flow is small. In addition, the effects of the magnetic field on the ${\pi }^{-}/{\pi }^{+}$ ratio over the whole rapidity range and the elliptic flow difference $v_2^n\text{−}{v}_2^p$ between neutrons and protons at forward and backward rapidities are on the same order as those from the nuclear symmetry energy. The $v_2^n\text{−}{v}_2^p$ difference in the midrapidity region is not strongly affected by the magnetic field, and the total ${\pi }^{-}/{\pi }^{+}$ yield ratio is immune to it. It is advisable to include the magnetic field effects in future studies using pion flow, pion yield ratio, and nucleon elliptic flow difference to probe the symmetry energy at super saturation densities.

Figure 1

The nuclear symmetry energy obtained from Skz4, SV-mas08 (default), and SV-sym34 are shown as a function of density.

Figure 2

Temporal evolution of the magnetic field component $-e{B}_y(0,0,0)$ produced by spectator protons in Au $+$ Au collisions at the beam energy $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with (a) $b=7$ fm and (b) $b=10$ fm. The legend “non-rel. wave,” “non-rel. point,” and “rel. point” indicate the results obtained using Eqs. (10), (5), and (4), respectively.

Figure 3

The magnetic field strengths $-e{B}_y(0,0,0)$ produced by participants or spectators in Au $+$ Au collisions at the beam energy $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with impact parameter $b=2$, 7, and 10 fm.

Figure 4

Distributions of the magnetic field strength $-e{B}_y$ in the reaction plane at $t=0,10,20,30$ fm/$c$ in Au $+$ Au (top) and Cu $+$ Au (bottom) collisions at the beam energy $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with impact parameter $b=7$ fm.

Figure 5

Temporal evolution of the magnetic field $-e{B}_y$ at the point ($x$, 0, 0) in (a) Au $+$ Au and (b) Cu $+$ Au, and at the point (0, $y$, 0) in (c) Au $+$ Au and (d) Cu $+$ Au at the beam energy $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with impact parameter $b=7$ fm.

Figure 6

The peak value of the time evolution of $-e{B}_y(0,0,0)$ as a function of the impact parameter $b$ in Au $+$ Au/Cu $+$ Au collisions at the beam energy $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$.

Figure 7

Beam energy dependence of $-e{B}_y$(0, 0, 0) in (a) Au $+$ Au and (b) Cu $+$ Au collisions with $b=7$ fm.

Figure 8

The integral of ${\int }_0^{\infty }-e{B}_y(0,0,0)dt\approx {\int }_0^{100\text{fm/c}}-e{B}_y(0,0,0)dt$ as a function of beam energy in Au $+$ Au and Cu $+$ Au collisions with $b=7$ fm.

Figure 9

A sketch illustrating the Lorentz force on charged pions.

Figure 10

The change of protons and pions' (a) $\langle p_x\rangle$ and (b) $\langle p_y\rangle$ by the Lorentz force as a function of normalized center-of-mass rapidity in Au $+$ Au collision at $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with $b=7$ fm.

Figure 11

$\langle p_x\rangle$ of (a) protons and (b) pions as a function of normalized center-of-mass rapidity in Au $+$ Au collision at $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with $b=7$ fm.

Figure 12

$v_1$ and $v_2$ of protons as a function of normalized center-of-mass rapidity in Au $+$ Au collisions at $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with impact parameters of 1, 3, 5, and 7 fm.

Figure 13

$v_1$ and $v_2$ of charged pions as a function of normalized center-of-mass rapidity in Au $+$ Au collisions at $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with impact parameters of 1, 3, 5, and 7 fm.

Figure 14

$v_1$ and $v_2$ of charged pions as a function of normalized center-of-mass rapidity in Cu $+$ Au collisions at $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with impact parameters of 1, 3, 5, and 7 fm.

Figure 15

$v_1$ and $v_2$ of charged pions as a function of normalized center-of-mass rapidity in Au $+$ Au collisions at $E_{\text{lab}}=0.6$, 1, 1.5 GeV/nucleon with $b=7\mathrm{fm}$.

Figure 16

$v_1$ and $v_2$ of charged pions as a function of normalized center-of-mass rapidity in Cu $+$ Au collisions at $E_{\text{lab}}=0.6$, 1, and 1.5 GeV/nucleon with $b=7\mathrm{fm}$.

Figure 17

The ${\pi }^{-}/{\pi }^{+}$ yield ratio as a function of the normalized center-of-mass rapidity in Au $+$ Au collisions at (a) $E_{\text{lab}}=0.4\mathrm{GeV}/\mathrm{nucleon}$ and (b) $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with $b=7\mathrm{fm}$. Calculations with Skz4 (the corresponding slope parameter $L=5.7$ MeV) and SV-sym34 ($L=81$ MeV) interactions are shown.

Figure 18

The elliptic flow difference between neutrons and protons $v_2^n\text{−}{v}_2^p$ as a function of the normalized center-of-mass rapidity in Au $+$ Au collisions at (a) $E_{\text{lab}}=0.4$ GeV/nucleon and (b) $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with $b=7\mathrm{fm}$. Calculations with Skz4 ($L=5.7$ MeV) and SV-sym34 ($L=81$ MeV) interactions are shown.

Figure 19

The elliptic flow difference between neutrons and protons $v_2^n\text{−}{v}_2^p$ as a function of $u_{t0}$ in Au $+$ Au collisions at [(a), (b)] $E_{\text{lab}}=0.4$ GeV/nucleon and [(c), (d)] $E_{\text{lab}}=1\mathrm{GeV}/\mathrm{nucleon}$ with $b=7\mathrm{fm}$. Calculations with Skz4 ($L=5.7$ MeV) and SV-sym34 ($L=81$ MeV) interactions are shown.