| Citation: |
Lili DONG, Mingyuan WANG, Wenjun LIU, Yuejiang SHI, Jiaqi DONG, Dong GUO, Tiantian SUN, Xiang GU, Xianming SONG, Baoshan YUAN, Yueng-Kay Martin PENG, the ENN Team. Experimental confirmation of the linear relation between plasma current and external vertical magnetic field in EXL-50 spherical torus energetic electron plasmas[J]. Plasma Science and Technology, 2024, 26(8): 085104. DOI: 10.1088/2058-6272/ad4f23
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A three-fluid equilibrium plasma with bulk plasma and energetic electrons has been observed on the Xuanlong-50 (EXL-50) spherical torus, where the energetic electrons play a crucial role in sustaining the plasma current and pressure. In this study, the equilibrium of a multi-fluid plasma was investigated by analyzing the relationship between the external vertical magnetic field (BV), plasma current (Ip), the poloidal ratio (βp) and the Shafranov formula. Remarkably, our research demonstrates some validity of the Shafranov formula in the presence of multi-fluid plasma in EXL-50 spherical torus. This finding holds significant importance for future reactors as it allows for differentiation between alpha particles and background plasma. The study of multi-fluid plasma provides a significant reference value for the equilibrium reconstruction of burning plasma involving alpha particles.
The interaction of relativistic electron beams with plasmas has gained an intensity level of attention in various applications, such as high energy density physics [1], inertial confinement fusion [2] and plasma-based accelerators (PBAs) [3–11]. In the fast ignition, the ignition is achieved by depositing energy into the dense core with the relativistic electron beams. PBA [3] was firstly proposed by Tajima in 1979. By employing a high intensity laser or relativistic charged particle beam as the driver, PBA schemes are categorized into two kinds: laser wakefield acceleration [3, 5, 9] and plasma wakefield acceleration [4, 5, 12]. The acceleration gradients of PBA are currently able to achieve an order of several hundred GeV m-1, much larger than those produced by conventional radio frequency accelerators. With continuous progress in high power laser technology, multi-GeV energy gain of electrons is achievable over a short distance in the experiments [9]. Meanwhile, a safety design of the beam dump [13–16] is urgently acquired to decelerate particles into a safe energy region without radiation. Thus, based on the collective electromagnetic field of short particle bunches in the plasma, plasma-based beam dump has recently received tremendous interest to develop safer and greener facilities.
Many simulations and analytical works [13–18] have been carried out to investigate plasma-based beam dump. The strong collective stopping of few-fs electron beams inside mm-scale underdense plasma was firstly demonstrated in two independent experiments [19]. It is shown that the plasma beam dump can be the most straightforward application for absorbing the kinetic energy of the EuPRAXIA beam over short distance [13–16]. Generally, two types of plasma beam dump (passive beam dump and active beam dump) are considered. In the passive beam dump, relativistic electron beam travels through the undisturbed plasma and achieves the deceleration by beam self-driven wakefield. Tailored plasma-density profiles in the passive scheme are demonstrated to improve the beam-energy loss [13].
In the active scheme, relativistic electron beam propagates with the wake excited by laser pulse and then disposes large energy in the plasma due to the self-excited and laser-driven wakefields. The plasma-based beam dump can greatly improve the overall compactness of PBA and reduces high-energy radiation caused by scattering in the material beam dump.
It should be noted that in actual applications, the energetic electron beams might be defocused and have a radius much larger than plasma skin depth in a plasma-based beam dump. In this work, we consider in detail the density evolution and energy deposition of relativistic electron beams with large radius (larger than plasma skin depth) in plasmas and mainly focus on the beam phase space evolution and energy deposition. Multi-layer structures in beam phase space are clearly observed. The two-stream instability (TSI) with short electron beam is limited in the present simulations. The current filamentation instability (CFI) breaks up the relativistic electron beam into small filaments and causes large energy deposition in the plasma. The paper is organized as follows. A two-dimensional (2D) particle-in-cell (PIC) simulation model is presented in section 2, along with beam and plasma parameters. We analyze the formation of multi-layer structure in section 3. The influences of beam parameters on the multi-layer structure and beam phase space are studied in detail in section 4. Finally, conclusions are given in section 5.
A 2D3V electromagnetic PIC simulation code IBMP [20, 21] is employed to study density evolution and collective stopping of electron beams in background plasmas. A cell size of ∆x = ∆y = 2.56 × 10-9 m, time step of ∆t = 5.12 × 10-18 s and nine particles per cell per species are used here. A moving-window approach is used in these simulations to reduce the computation time. It carries out in the laboratory frame, in which the simulation window is shifted by a distance in the beam propagation direction every a few time steps (which can be set in the simulation) to ensure that the simulation window moves with the electron beam on the average. Absorbing boundary conditions are adopted in both longitudinal (along the x-axis) and transverse (along the y-axis) directions. We model a hydrogen plasma with a real ion mass mi/me = 1836 and charge Zi = e for simulations, which fills the simulation box uniformly at the initialization stage. The density and electron temperature of the plasma are set to be npe = 1027 m-3 and Tpe = 4 eV respectively. Electron beams with energy Ebe = 113 MeV and density nbe = 0.1npe are adopted in the simulations. The longitudinal spatial profile of the electron beam is assumed to be Gaussian with a width (FWHM) τbe = π/ωpe, in which ωpe is the plasma electron frequency. We keep relativistic electron beam density fixed and investigate the effects of beam parameters (beam radius and transverse density profile) in detail in the next sections.
We first show here the structure evolution of the electron beam with a Gaussian transverse density profile and radius rb = 5c/ωpe. Here, c is the speed of light and ωpe =
| Ex(eV/m)=√npenbenpe√2πkpσxe-k2pσ2x/21+1k2pσ2ysinkp(x-ct), | (1) |
where npe and nbe are the plasma and beam density in m-3 respectively, σx and σy are the rms dimensions of the beam, and kp = ωpe/c is the plasma wave number. Figure 2 clearly displays the distributions of the wakefield at ωpet = 280.5. The transverse electric field Ey at the tail of the beam is seen to strongly focus the beam electrons, which can be observed from figure 2(a). Thus, under the focusing effect of the transverse wakefield (Ey and Bz), the beam density at the tail increases significantly and reaches 1027 m-3 in figure 1(e), which is about 10 times larger than the initial value. From equation (1), the magnitude of the longitudinal electric field Ex is estimated to be 4 × 1011 V m-1 with the given plasma and beam parameters (figure 1(a)). Once focused, the beam density increases and approaches plasma density (figure 1(c)), the magnitude of Ex increases significantly and reaches 1.5 × 1012 V m-1, as indicated in figure 2(c).
By inspecting figures 1(d)–(f), we note here the formation of the multi-layer structure (figure 1(d)) and later growth of the slice numbers (figure 1(f)) at the tail of the beam. To clearly explain this, we also display the longitudinal slice distributions of the transverse wakefield at y = 0.5c/ωpe in figure 2(c). Meanwhile, the longitudinal and transverse spatial profiles of beam density nbe are also shown in figures 2(c) and (d) for illustration respectively. Under the focusing force of the transverse wakefield at the tail, the betatron frequency of the beam electron is relatively larger. Thus, the slice structure is firstly observed at the tail, as indicated in figure 1(d). The transverse wakefield Ey–cBz is essentially close to zero at the front. This implies that the slices at the beam head are also gradually presented in figure 1(f) as the travel time increases. Furthermore, the radius of the slice close to the beam front is larger than that at the beam tail, as clearly shown in figures 1(e) and (f).
Representative snapshots of the longitudinal phase space are displayed in figure 3. Prior studies [24, 25] have demonstrated that majority of the beam electrons at the middle are decelerated and a small amount of beam electrons at the tail are accelerated obviously beyond the initial energy, which can be clearly identified in figures 3(a)–(f). It can be envisioned by noting that the longitudinal wakefield Ex (figure 2(b)) is positive at the middle of the beam and negative at its higher energy tail relatively. The most attractive feature of figures 3(e) and (f) is the formation of multi-layer structures in the longitudinal phase space, indicating the beam energy modulation in addition to the collective stopping. To show this clearly, the corresponding longitudinal wakefield Ex and density of the beam electron nbe along the y-axis at the position x = 52.5c/ωpe are displayed in the figure 2(d). It is obvious from this figure that the longitudinal wakefield Ex is nonuniform in the transverse direction and decreases gradually towards beam edge. As a result, the electrons at the beam edge are expected to have higher energies than those at the beam axis. Figure 3(f) clearly illustrates that the kinetic energy of slice is nonuniform and then the layers structure of longitudinal phase space is formed consequently.
We proceed by considering the influences of different radius on the density evolution and collective stopping of electron beams with high energy traveling through background plasmas. Three cases are considered in the simulations: rb = c/ωpe, 5c/ωpe and 7.6c/ωpe. The centers of the electron beams for three cases are located at x = 23c/ωpe and y = 0 initially. The other parameters are the same as presented in section 2. We compare the longitudinal slice of the longitudinal wakefield Ex at the position y = 0 for three cases in figure 4(a). In terms of short electron beam, quantitative characteristics of Ex are beam charge dependent. The electron beam with a larger radius excites a stronger wakefield due to a higher beam charge, as indicated in figure 4(a). For the electron beam with rb = c/ωpe, the magnitude of longitudinal wakefield Ex is seen to be 300 GV m–1 from equation (1), showing agreement with the figure. Meanwhile, the magnitude further increases to 800 GV m-1 for the case rb = 7.6c/ωpe. The significant beam energy loss due to Ex can be expected in figure 4(b). Some of beam electrons are seen to have a kinetic energy of 35 MeV after a travel time of ωpet = 704 for the case of rb = 7.6c/ωpe, losing 70% of its initial energy. It should be noted here that multi-peaks in the beam energy spectrum can be identified in the figure due to the multi-layer structure.
Figure 5 presents detailed comparisons of beam density distributions nbe ((a)–(c)) and longitudinal phase space ((d)–(f)) for three radius cases. The competition between the beam focusing effect and CFI can be clearly expected in figure 5(c). Some filaments at the beam front regions can be observed from the figure. The modulation of beam density (at the beam center regions) in the transverse direction due to the CFI can also be observed, indicating the competition between the beam focusing effect and CFI. As mentioned before, the transverse electric fields at the beam center (defocusing force) increase with the beam radius. Thus, comparing figures 5(a)–(c), one can find that the beam slice radius of multi-layer structure increases with initial beam radius. From the longitudinal phase space distributions, the multi-layer structure is more significant for relativistic electron beam with radius rb ≫ c/ωpe (i.e. figures 5(e), (f)), indicating the significant nonuniformity of Ex along the transverse direction.
We also compare the density evolutions of electron beams with two transverse density profiles: Gaussian (figures 6(a), (c)) and flat-top (figures 6(b), (d)). The radius of the electron beam is selected to be rb = 7.6cωpe. For the flat-top distribution, the beam electrons are only focused at the beam edge and the CFI can be fully developed. After filaments merging, three electron filaments with small radius are formed, as indicated in figure 6(d). Figure 7 shows comparisons of the longitudinal beam phase space ((a) and (b)) and energy spectrum ((c) and (d)) with two density profile cases. The multi-layer structure of the longitudinal phase space for the Gaussian distribution is clearly indicated in figure 7(a). However, for the flat-top case (figure 7(b)), the energy spread of beam electrons is shown to be smaller, which can also be identified by comparing figures 7(c) and (d). In addition, the peak in the beam energy spectrum is seen to move to the low energy side for the flat-top case and the number of beam electrons with high energy (113 MeV) decreases.
The multi-layer structures indicate that the beam electrons with high energy are located at the beam edge (where the magnitude of longitudinal wakefield is smaller than that at the beam center), which is negative for the beam stopping. As indicated in figure 7(c), a peak can be observed at the high energy regions. This cannot be fixed with a plasma of different density. The reason is that the transverse wakefield co-exists with the longitudinal wakefield (which is for collective beam stopping) and the nonuniformity of the transverse field is mainly determined by the beam density profile. The beam duration is a critical parameter for the multi-layer structures in this work. For the beam duration larger than the plasma period, the charge neutralization can be achieved and the TSI can be excited. The coupled TSI and CFI develops and the multi-layer structure cannot be observed anymore. The magnitude of the transverse wakefield depends on the beam density, but is independent on the beam energy. Therefore, as the beam energy increases, the structure is formed on a longer time scale due to a heavier beam electron mass. Meanwhile, as the ratio of beam density to plasma density increases, a stronger wakefield can be expected and the structure is formed on a shorter time scale. These findings may help us to understand the dynamic of the beam with large radius propagating through the plasma, which should provide some references for the plasma based beam dump.
Motivated by science and commerce, investigation of beam-plasma system is a topic of significant interest. The density evolution and collective stopping of relativistic electron beams in plasmas are frequently encountered in many applications, such as the fast ignition and plasma-based beam dump. The wakefield excited by short electron beam plays a vital role in the time evolutions and energy loss. The longitudinal and transverse wakefields are responsible for the beam stopping and focusing respectively. In this work, 2D PIC simulations were used to study the density evolution and collective stopping of the short electron beam with rb ≫ c/ωpe traveling through the plasmas. Due to the longitudinal nonuniformity of the transverse wakefield, the multi-layer structure is formed in the plasmas. Furthermore, the nonuniformity of the longitudinal wakefield in the transverse direction contributes to the formation of the multi-layer structure in beam phase space. The longitudinal wakefield causes a large energy spread of beam electrons and significant beam energy extraction in dense plasmas. These dynamic evolutions are essential for the plasma-based beam dump. Dimensional effect (2D versus 3D) in PIC simulation, as a key role, can change quantitative results significantly. Especially for the nonlinear interactions between the beam and plasma, the magnitude of wakefield generated by a short beam in background plasmas will vary a lot from 2D to 3D simulations. However, the nonuniformity of the transverse wakefield (Ey and Bz) in the longitudinal direction is the essential reason for the multi-layer structure formation, which is independent on the dimension of the simulation. Therefore, we believe that the findings in this work should also be presented in the 3D simulations.
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