Fuel compression in the magnetized cylindrical implosion driven by a gold tube heated by heavy ion beams

1.   Introduction

Nuclear fusion would be a future energy source [1–3]. Its major schemes are magnetic confinement fusion (MCF) [4–6] and inertial confinement fusion (ICF) [7–12]. There are several typical schemes to achieve ICF, for example, ICF with extreme intensity laser, Z-pinch and heavy ion beam.

In laser-drive ICF, there are central ignition schemes such as indirect-drive on NIF [13–17], direct-drive on OMEGA [18–22], and non-central ones such as fast ignition [23–26] and shock ignition [27–30], etc. Z-pinch is one application of Lorentz force [31, 32].

Heavy ion beam (HIB) has many advantages for ICF [33–40]. HIB-ICF (HIF) is a novel approach to thermonuclear fusion energy with higher drive efficiency than most other approaches. HIB is generated by an accelerator [41] and it has the significant advantage that it can deposit HIB energy inside material [42–45]. The energy deposition spatial profile in materials is easy to predict. Therefore, the fusion fuel design becomes relatively simple [11, 46].

There are several international research projects on HIF. The FAIR (Facility for Antiprotons and Ion Research) project has been started at Darmstadt, Germany [47]. The HIAF (High Intensity Heavy Ion Accelerator Facility) project in China has been planned for HIF and HEDP studies [48–51]. HIAF construction started in 2018 in Huizhou City, China. Research on HIF has been conducted at many research labs and universities in Japan, Germany, China, France, U.S.A., Russia, Italy, Spain, Kazakhstan, etc [11, 41].

Nearly all of the studies on the targets for heavy ion fusion are of spherical geometry. The present investigation focused on the cylindrical implosions driven by a single ion beam incident with high symmetry relative to the target axis [52–56]. There are several advantages of cylindrical targets. For example, it is easier to make more targets, run more experiments, and acquire more data to study implosion dynamics [57–59].

In order to reach thermonuclear fusion conditions, the present paper will study the cylindrical implosions which can concentrate energy in a small amount of fuel [60–62]. It is reported that the ignition threshold (Lawson's criteria) [63–65] of magnetized fuel (for deuterium-tritium (DT)) is strongly reduced compared with that of nonmagnetized DT fuel [34].

The present paper details the study of a magnetized cylindrical target which consists of a gold tube filled with fuel plasma at low density, where an axial magnetic field is applied externally. The heavy ion beam heats the outer part of the hollow cylinder (gold) along the direction of the magnetic field and the heavy ion beam evaporates the gold. We assume that the energy deposition is uniform in the axial magnetic field direction. The gold in gas state will expand radially and drive the inner part of the tube (plasma) towards the axis. The purpose of the cylindrical implosions is to concentrate energy in a small amount of fuel in order to reach thermonuclear fusion conditions. The simulation results show that a shock wave is produced when a heavy ion beam heats the gold. The density peak of the shock wave increases as the intensity of the heavy ion beam increases. It also increases as the external magnetic field increases.

2.   Model

Figure 1 is a sketch of a magnetized cylindrical target which consists of a tube filled with DT plasma. There is an external axial magnetic field. The heavy ion beam will heat the outer part of the cylinder, then the cylinder will expand in the radial direction and push the inner part of the tube towards the axis. The size of the targets is given in figure 1.

Figure  1.  Sketch of magnetized cylindrical target.

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For this model, we set up the following column coordinate system (r, θ, z), see figure 2. The z-axis is in the direction of the external magnetic field, which is also the axis direction of the magnetized cylindrical target. R_t is the radius of the target, Z_t is the length of the target, and the metallic tube material is gold.

Figure  2.  (a) The column coordinate system is set for the system, where the z-axis is in the direction of the external magnetic field. (b) The simulation area: 0 ≤ r ≤ R_t, 0 ≤ z ≤ Z_t, where R_t is the the radius of the target, Z_t is the length of the target.

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In order to understand the fundamental physical phenomena, we use the ideal magnetohydrodynamics (MHD) equations to describe the plasma, as follows.

where ρ is the density of the plasma, I is the second order unit tensor, V is the velocity of the plasma, $E=\rho \epsilon +\frac{1}{2}\rho |\mathit{V}{|}^2+\frac{1}{2}|\mathit{B}{|}^2$ is the energy density of the plasma, P_plasma = ρε(γ - 1) is the pressure of the plasma, ε is internal energy density, γ is adiabatic exponent, P_gold is the pressure of the metallic tube in gas state, μ₀ is the permeability of vacuum and B is the magnetic field.

When the heavy ion beam injects into the metallic tube, the rapid deposition of heat from the heavy ion beam is so high that the solid state gold is vaporized suddenly. Then an enormous pressure gradient is formed which will let the gold gas expand rapidly into the plasma region. For simplicity and convenience, as well as to understand the fundamental physical phenomena, we use the ideal hydrodynamic equations to describe the metallic tube in gas state, as follows.

where $\rho \text{'}$ is the density of the metallic tube in gas state, u is the velocity of the metallic tube in gas state, $E\text{'}=\rho \text{'}\epsilon \text{'}+\frac{1}{2}\rho \text{'}|\mathit{u}{|}^2$ is the energy density of the metallic tube in gas state. $P_{\mathrm{gold}}=\rho \text{'}\epsilon \text{'}(\gamma -1)$ is the pressure of the metallic tube in gas state, $\epsilon \text{'}$ is internal energy density of the metallic tube in gas state. We neglect the term of the electron thermal conduction.

3.   Initial conditions

The initial conditions of the plasma are given as follows:

where ${\rho }_{\mathrm{Dmax}}$ is the density of plasma at equilibrium state when there is no heavy ion beam heat, and B₀ is the external magnetic field. To ensure the simulation, the plasma density is chosen in continuous tanh function. Furthermore, to ensure the plasma is mainly in the region of $0\le r\le \frac{R_{\mathrm{t}}}{2}$, we choose the large enough parameter 500 m^-1 and $\frac{R_{\mathrm{t}}}{2}$ in the expression of the tanh function in equation (8).

The initial conditions of the metallic tube in gas state are

where ${\rho }_{\mathrm{Gmax}}$ is the density of the metallic tube at the edge of the target. It is also the initial density of the metallic tube if there is no expansion. To ensure the simulation, as well as to ensure the metallic tube is mainly in the region of $\frac{R_{\mathrm{t}}}{2}\le r\le R_{\mathrm{t}}$, the metallic tube density is chosen in the continuous tanh function. The large enough parameter 500 m^-1 and $\frac{R_{\mathrm{t}}}{2}$ in the expression of the tanh function in equation (9) are also chosen. In our simulation the gold gas is warm dense matter, which has a density of the same order of magnitude as solid. The solid density of gold is 19.3 × 10³ kg m^-3, then we choose the density of gold gas as 14 × 10³ kg m^-3, which is smaller than the solid density of gold. The initial temperature is T = 1 eV, which can be estimated by using the parameters of the heavy ion beam. We consider a ²³⁸U³⁴⁺ ion beam with an energy of 800 MeV u^-1, a bunch length of 100 ns with a Gaussian intensity distribution along the radius characterized with FWHM = 1 mm. The beam intensity is N = 1.4 × 10¹¹ ppp. These parameters come from the HIAF project [51]. The beam is incident over one face of the target and the projectiles penetrate into the target. We assume that the energy is uniformly deposited along the cylinder length. It is found that the deposited energy is about ~5 kJ g^-1 in the gold. Then we can obtain the temperature of the gold gas, which is about ~1 eV. The adiabatic index is γ = 1.4.

The initial densities of the plasma and metallic tube are shown in figures 3(a) and (b). The other parameters are shown in table 1 for different runs.

Figure  3.  (a) Initial density of plasma, (b) initial density of metallic tube.

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Table  1.  Initial parameters for different runs.

Different runs		${\rho }_{\mathrm{Dmax}}(\mathrm{kg}{\mathrm{m}}^{-3})$	${\rho }_{\mathrm{Gmax}}(\mathrm{kg}{\mathrm{m}}^{-3})$	Bz	Br	Bθ	V	u
	(1)	1.37	1.4 × 104	3	0	0	0	0
	(2)	1.37	1.4 × 104	6	0	0	0	0
	(3)	1.37	1.4 × 104	9	0	0	0	0
Case I	(4)	0.97	1.4 × 104	3	0	0	0	0
	(5)	1.07	1.4 × 104	3	0	0	0	0
	(6)	1.17	1.4 × 104	3	0	0	0	0
	(7)	1.57	1.4 × 104	3	0	0	0	0
Case II	(1)	1.37	1.4 × 103	3	0	0	0	0

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4.   Results and discussions

The simulation period in our simulation is so short that the gold gas moves a short enough distance. Therefore, the Rayleigh–Taylor (RT) instability is not considered in the present paper. For this reason, as well as the symmetry of the system with respect to θ, we assume that $\frac{\partial }{\partial \theta }=0$.

We now numerically solve equations (1)–(7) by using a commercial software, Usim. Usim uses the Eulerian method. The Usim series of computational applications are powered by the Ulixes computational engine. Ulixes is a general purpose fluid plasma modeling code that supports shock capturing methods for MHD, Hall MHD, two fluid plasma, Navier–Stokes, Maxwell's equations, as well as multi-species, multi-temperature versions of the fluid systems mentioned [66]. The code we used in the study is a 2D cylindrical coordinate system. The simulation region is 0 ≤ r ≤ R_t, 0 ≤ z ≤ Z_t, where R_t = 5 mm and Z_t = 10 mm.

4.1   Shock wave excited by high density gold gas

For this case, the boundary conditions are as follows: the copy boundaries are used at the boundaries of z = 0 and z = Z_t, the fixed boundary is used at the boundary of r = R_t and the axis boundary is used at boundary r = 0.

The dependence of the plasma density on the spatial coordinates (r, z) with different times t = 20 ns, 30 ns, 38 ns is shown in figures 4(a)–(c), respectively, where ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, B₀ = 3.0 T, ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^4$ kg m^-3. Notice from figure 4 that there is obvious variation in the plasma density with respect to r, while it is nearly independent of z. Furthermore, the plasma region is of high density propagating in the -r direction.

Figure  4.  Dependence of plasma density on the spatial coordinates (r, z) with different times. (a) t = 20 ns, (b) t = 30 ns, (c) t = 38 ns, where the external magnetic field B₀ = 3 T, the plasma density in the center of target ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, and the density of gold in the edge of target ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^4$ kg m^-3.

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In order to further understand the dependence of the plasma density on the radial coordinate r, the variations of the plasma density with respect to the r with different times t = 20 ns, 30 ns, 40 ns at z = 5 mm are shown in figure 5. The maximum plasma density increases with time and it propagates in the -r direction.

Figure  5.  Dependence of plasma density on the radial coordinates r at different times t = 20 ns, 30 ns, 40 ns at the point z = 5 mm, where the other parameters are: the external magnetic field B₀ = 3 T, the plasma density in the center of target ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, and the density of gold in the edge of target ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^4$ kg m^-3.

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The dependence of the maximum value of the plasma density on time t is shown in figure 6(a). The maximum values of the plasma density increase with time exponentially. The fit result is $\rho ={\rho }_0+A_1{\mathrm{e}}^{\frac{t-t_0}{A_2}}$, where ρ₀ = 0.1318 kg m^-3, A₁ = 0.3589 kg m^-3, t₀ = 8.2185 ns and A₂ = 9.0859 ns. The dependence of the radial coordinate r of the maximum plasma density on time t is shown in figure 6(b). Notice that this high plasma density region propagates in the -r direction. The fit result of the propagation velocity satisfied $r=B_1{\mathrm{e}}^{-{\displaystyle \frac{t}{t_1}}}+r_0$, where B₁ = 7.199 mm, t₁ = 34.015 ns and r₀ = -1.641 mm. The propagation velocity of the high plasma density region is between 81.7 km s^-1 and 225.1 km s^-1.

Figure  6.  (a) Dependence of maximum values of plasma density on time t, where the dots are the simulation results, while the values in the line are the fit result. (b) Dependence of radial coordinate r of maximum plasma density on time t, where the dots are the simulation results, while the values in the line are the fit result.

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It is noticed that the propagation direction of this wave is perpendicular to the direction of the magnetic field, while the fluid velocity is parallel to the wave propagation direction. Therefore, it seems that this wave is the magnetosonic wave. It is found that the Mach number $M=\frac{\upsilon }{{\upsilon }_{\mathrm{s}}}$ is between 6.96 and 19.19, where υ is the magnetosonic wave speed, υ_s is the acoustic speed. Then we believe the wave is supermagnetosonic since the Mach number is larger than one. It is also found that the Alfvén Mach number $M_{\mathrm{A}}=\frac{\upsilon }{{\upsilon }_{\mathrm{A}}}$ is between 25.3 and 69.69, where υ_A is the Alfvén speed. The Alfvén Mach number is much larger than one, which indicates that the magnetic field does not have feedback on the flow.

The dependence of the axial magnetic field B_z on the spatial coordinates (r, z) with different times t = 8 ns, 16 ns, 24 ns is shown in figures 7(a)–(c), respectively, where ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^4$ kg m^-3, B₀ = 3.0 T. Notice from figure 7 that there is obvious variation of the axial magnetic field B_z with respect to r, while it is nearly independent on z. Furthermore, there is a high axial magnetic field region propagating in the -r direction. In order to further understand the dependence of the axial magnetic field on the radial coordinate r, the variations of the axial magnetic field with respect to r with different times t = 3 ns, 16 ns, 32 ns, 40 ns at z = 5 mm are shown in figure 8. It seems that the maximum axial magnetic field increases with time and it propagates in the -r direction.

Figure  7.  Dependence of the axial magnetic field B_z on the spatial coordinates (r, z) with different times t = 8 ns, 16 ns, 24 ns are shown in (a), (b) and (c), respectively, where ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^4$ kg m^-3, B₀ = 3.0 T.

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Figure  8.  Dependence of perturbed magnetic field on radial coordinates r at different times t = 3 ns, 16 ns, 32 ns, 40 ns at the point z = 5 mm, where the other parameters are: the external magnetic field B₀ = 3 T, the plasma density in the center of target ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, and the density of gold in the edge of target ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^4$ kg m^-3.

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It is found from both figures 4 and 5 and both figures 7 and 8 that the maximum plasma density and the maximum perturbed magnetic field, respectively, are at the same points. Furthermore, both of them propagate with the same velocity. This is caused by the magnetic freezing.

Dependence of the plasma density on the radial coordinate r at time t = 30 ns with different external magnetic fields B₀ = 3 T, 6 T, 9 T is shown in figure 9(a). Notice that the maximum plasma density increases as the external magnetic field increases. Moreover, dependence of the magnetic field in the z-axis direction on the radial coordinate r at time t = 30 ns with different external magnetic fields B₀ = 3 T, 6 T, 9 T is shown in figure 9(b). It indicates that the maximum magnetic field in the z-axis direction increases as the external magnetic field increases. It indicates from figure 9 that the external magnetic field can increase the maximum plasma density as well as the perturbed magnetic field.

Figure  9.  (a) Dependence of plasma density on the radial coordinate r at time t = 30 ns with different external magnetic fields B₀ = 3 T, 6 T, 9 T, (b) dependence of magnetic field in z-axis direction on the radial coordinate r at time t = 30 ns with different external magnetic fields B₀ = 3 T, 6 T, 9 T.

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The dependence of the maximum value of the plasma density on the time t with different parameters of ${\rho }_{\mathrm{Dmax}}=0.97$ kg m^-3, 1.07 kg m^-3, 1.17 kg m^-3, 1.37 kg m^-3, 1.57 kg m^-3 is shown in figure 10(a). Notice that the maximum plasma density increases as the time increases, while it decreases as the parameter ${\rho }_{\mathrm{Dmax}}$ increases. Moreover, dependence of the position of the maximum value of the plasma density on the time t with different parameters of ${\rho }_{\mathrm{Dmax}}=0.97$ kg m^-3, 1.07 kg m^-3, 1.17 kg m^-3, 1.37 kg m^-3, 1.57 kg m^-3 is shown in figure 10(b). Notice from figure 10(b) that the wave propagates in the -r direction and the propagation speed of the wave decreases as the parameter ${\rho }_{\mathrm{Dmax}}$ increases. It is noticed from figure 10 that the wave propagates in the -r direction and the density peak of wave increases as time increases. The reason for this result may be due to the cylindrical symmetry. As the wave propagates the volume inversely proportional to the parameter r, then the plasma density may be proportional to the r.

Figure  10.  (a) Dependence of the maximum value of the plasma density on the time t with different parameters of ${\rho }_{\mathrm{Dmax}}=0.97$ kg m^-3, 1.07 kg m^-3, 1.17 kg m^-3, 1.37 kg m^-3, 1.57 kg m^-3, respectively. (b) Dependence of the position of the maximum value of the plasma density on the time t with different parameters of ${\rho }_{\mathrm{Dmax}}=0.97$ kg m^-3, 1.07 kg m^-3, 1.17 kg m^-3, 1.37 kg m^-3, 1.57 kg m^-3, respectively.

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4.2   Shock wave excited by lower-density high-velocity gold gas

In order to further understand the physical mechanism of the present problem, we change the boundary conditions to study it. In the above investigations, as an approximation, we assume that the energy of the heavy ion beam is so high that the metallic tube is instantly vaporized into gas state initially at time t = 0. At the same time, the energy of the heavy ion beam becomes zero.

Now we consider another approximate case that the energy of the heavy ion beam is not so high, but the heavy ion beam continuously heats into the metallic tube. Then the gold in gas state is produced continuously. For this case, we assume that the total pressure is a constant, i.e., $P_{\mathrm{plasma}}+P_{\mathrm{Gold}}=\mathrm{const}$. The density of the gold in gas state is much lower initially. In this case, the boundary conditions are as follows. A constant velocity (U ~ 700 m s^-1, we assume that the gold gas expanded as thermal velocity) of the gold in gas state is given at the boundary r = R_t, which means that the gold in gas state is continuously produced by the heavy ion beam. The copy boundaries are used at the boundaries of z = 0 and z = Z_t, and the axis boundary is used at boundary r = 0.

The dependence of the plasma density on the spatial coordinates (r, z) with different times t = 80 ns, 160 ns, 240 ns is shown in figures 11(a)–(c), respectively, where ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^3$ kg m^-3, B₀ = 3.0 T. Notice from figure 11 that there is obvious variation of the plasma density with respect to r, while it is nearly independent of z except near the boundaries z = 0 and z = Z_t which may be affected by the boundary conditions. There is a high plasma density region propagating in the -r direction. Dependence of the plasma density on the radial coordinate r with different times t = 40 ns, 80 ns, 120 ns, 160 ns, 200 ns, 240 ns at z = 5 mm is shown in figure 12. Notice that the maximum plasma density increases with time t and it propagates in the -r direction. These results are similar to those in case 1. However, it seems from both figures 5 and 12 that both the density peak and the propagation speed of the shock wave are much larger in case 1 than those in case 2. The reason for this phenomenon may be the difference in initial gold densities between case 1 and case 2, in which the gold density in case 1 is much larger than that in case 2.

Figure  11.  Dependence of plasma density on the spatial coordinates (r, z) with different times t = 80 ns, 160 ns, 240 ns is shown in (a)–(c), respectively, where ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^3$ kg m^-3, B₀ = 3.0 T.

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Figure  12.  Dependence of plasma density on the radial coordinate r with different times t = 40 ns, 80 ns, 120 ns, 160 ns, 200 ns, 240 ns at z = 5 mm, where ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^3$ kg m^-3, B₀ = 3.0 T.

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The dependence of the maximum value of the plasma density on time t is shown in figure 13(a). It seems that the maximum value of the plasma density increases with time. The fit result is $\rho =A_2+\frac{A_1-A_2}{1+{\left(\frac{t}{t_0}\right)}^p}$, where A₁ = 1.0344 kg m^-3, A₂ = 12.1954 kg m^-3, t₀ = 127.4624 ns and p = 2.8688, which is different from that of case 1. The dependence of the radial coordinate r of the maximum plasma density on time t is shown in figure 13(b). Notice that the high plasma density region propagates in the -r direction with an almost constant velocity. The fit results is r = r₀ + at, where r₀ = 5.0557 mm and a = -18.2 km s^-1. It is the supermagnetosonic wave, which is similar with that of case 1. The Mach number is M ~ 1.55 for this case. The Alfvén Mach number is M_A ~ 5.63 for this case. It is found that the Mach number and the Alfvén Mach number are much smaller than those in case 1.

Figure  13.  (a) Dependence of maximum value of plasma density on time t, (b) dependence of radial coordinate r of maximum plasma density on time t.

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The dependence of the axial magnetic field B_z on the spatial coordinates (r, z) with different times t = 80 ns, 160 ns, 240 ns is shown in figures 14(a)–(c), respectively, where ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^3$ kg m^-3, B₀ = 3.0 T. Notice from figure 14 that there is obvious variation in the axial magnetic field B_z with respect to r, while it is nearly independent of z except near the boundaries. Furthermore, there is a high axial magnetic field region propagating in the -r direction. These results are similar to those of case 1. The dependence of the axial magnetic field on the radial coordinate r with different times t = 40 ns, 80 ns, 120 ns, 160 ns, 200 ns, 240 ns at z = 5 mm is shown in figure 15. It seems that the maximum axial magnetic field propagates in the -r direction, which is similar with that in case 1. However, it seems from both figures 8 and 15 that both the density peak and the propagation speed of the shock wave are much larger in case 1 than those in case 2. This is due to the difference of the initial gold densities between case 1 and case 2. The gold density in case 1 is much larger than that in case 2.

Figure  14.  Dependence of axial magnetic field B_z on spatial coordinates (r, z) with different times t = 80 ns, 160 ns, 240 ns is shown in (a)–(c), respectively, where ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^3$ kg m^-3, B₀ = 3.0 T.

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Figure  15.  Dependence of axial magnetic field on the radial coordinate r with different times t = 40 ns, 80 ns, 120 ns, 160 ns, 200 ns, 240 ns at z = 5 mm, where ${\rho }_{\mathrm{Dmax}}=1.37$ kg m^-3, ${\rho }_{\mathrm{Gmax}}=1.4\times {10}^3$ kg m^-3, B₀ = 3.0 T.

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5.   Conclusion

The numerical results are given in the present paper by using Usim. A shock wave is found when a heavy ion beam heats the gold along the direction of the magnetic field. The density peak of the shock wave increases with the increase of the time t and it propagates in the -r direction.

It is noted that this shock wave is the supermagnetosonic wave. The wave velocity can be much larger than the acoustic speed. The Mach number may decrease as the intensity of the heavy ion beam decreases. The density peak of the shock wave also increases as the external magnetic field increases.

6.   Future work

In the future, we may do the following work. (1) In the present study, we use the ideal gas state equation to describe both the plasma and the gold in the gas state. Actually both are at high temperature and in high pressure state, so we should use a more realistic equation of state. (2) There is an interface between plasma and gold in the gas state in which the mass density is different. Therefore, RT instability may occur, and we will study the RT instability on this interface when the gold in gas state expands.