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反场构型平面薄膜电爆炸等离子体电流通道

刘永棠 盛亮 李阳 张金海 欧阳晓平

刘永棠, 盛亮, 李阳, 张金海, 欧阳晓平. 反场构型平面薄膜电爆炸等离子体电流通道. 物理学报, 2022, 71(3): 035205. doi: 10.7498/aps.71.20211495
引用本文: 刘永棠, 盛亮, 李阳, 张金海, 欧阳晓平. 反场构型平面薄膜电爆炸等离子体电流通道. 物理学报, 2022, 71(3): 035205. doi: 10.7498/aps.71.20211495
Liu Yong-Tang, Sheng Liang, Li Yang, Zhang Jin-Hai, Ouyang Xiao-Ping. Current channel in plasma of inverse exploding planar foils. Acta Phys. Sin., 2022, 71(3): 035205. doi: 10.7498/aps.71.20211495
Citation: Liu Yong-Tang, Sheng Liang, Li Yang, Zhang Jin-Hai, Ouyang Xiao-Ping. Current channel in plasma of inverse exploding planar foils. Acta Phys. Sin., 2022, 71(3): 035205. doi: 10.7498/aps.71.20211495

反场构型平面薄膜电爆炸等离子体电流通道

刘永棠, 盛亮, 李阳, 张金海, 欧阳晓平

Current channel in plasma of inverse exploding planar foils

Liu Yong-Tang, Sheng Liang, Li Yang, Zhang Jin-Hai, Ouyang Xiao-Ping
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  • Z箍缩负载初始化电流通道的形成建立过程、电流密度分布模式及其演化特征对研究等离子体动力学发展、分析等离子体不稳定性模式、开展磁流体数值模拟工作等具有重要意义. 平面薄膜是研究这一问题中具有连续二维结构的理想构型. 基于理想无限长平板假设下的反场薄膜-回流柱负载模型, 通过理论对电感主导模式下的电流通道建立和演化过程、薄膜平面磁场分布特征和薄膜各部分受力特征进行了计算分析. 通过开展对称型与非对称型下的平面薄膜电爆炸实验, 以可见光分幅相机记录薄膜等离子体自辐射光分布为主要诊断手段开展了实验验证. 结果显示, 早期电流的建立与分布符合电感模型预期, 薄膜等离子体表现出电流密度边缘聚集和钳状型等离子体发光形态. 中后期图像显示发光强区会由薄膜边界向中心位置转移并导致峰状凸起型等离子体发光形态的演化, 表明电流分配受等离子体发展影响, 边界融蚀等离子体向心汇聚导致电流随之转移, 最终造成电流通道的快速切换.
    In the research of Z-pinch, the initial establishment of current in load, current density distribution mode and current evolution characteristics in different stages are of great significance in studying the development of plasma dynamics, analyzing plasma instability mode, and carrying out the MHD numerical simulation. Thin planar foil is an ideal configuration with continuous two-dimensional structure to study the problem. Based on an ideal model that consists of foil-backpost configuration under the assumption of infinite length in reverse exploding condition, the establishment, distribution and evolution of the current channel, magnetic field distribution characteristics and the force state of foil plasma by the inductive mode are calculated and analyzed theoretically. Relevant experiments are carried out on the QG-1 facility with about 1.4 MA peak current and 100 ns rise time to verify the calculated results. The self-emission graphs of exploding foils are recorded by visible-light frame cameras from side-on view and end-on view. Different load configurations are utilized including symmetric and asymmetric condition both assembled with two 20-μm-thick aluminum foils. The results show that the establishment of the initial current in foil (0–70 ns) accord with the expectation of the inductance model. The inductance feature determines the distribution of the current in this stage, which causes the current to aggregate in foil edges and then the edges will suffer much higher magnetic field and J × B force than the center. This results in the formation of a clamp plasma morphology. The later images (70–120 ns) show that the intense emission region will transfer from the foil edges to the center, which proves a transfer of current channel. So the magnetic field and J × B force in the foil center increase and a peak convex plasma morphology forms then. It shows that the current distribution will be affected by the plasma evolution process. The convergence process of ablated plasma with current finally leads to rapidly switch the current channel (in 60–80 ns).
      PACS:
      52.80.Qj(Explosions; exploding wires)
      52.58.Lq(Z-pinches, plasma focus, and other pinch devices)
      52.77.Fv(High-pressure, high-current plasmas)
      通信作者: 盛亮, shengliang@tsinghua.org.cn
    • 基金项目: 国家自然科学基金(批准号: 11575147)资助的课题.
      Corresponding author: Sheng Liang, shengliang@tsinghua.org.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11575147).

    Z箍缩的研究涉及多物理过程的复合[1,2], 电流通道的建立与演化是其中基础而极为重要的课题之一[3]. 该课题对于研究能量馈入启动相变[4,5]、动力学发展过程[6,7]、不稳定结构源[8,9]以及开展Z箍缩磁流体数值模拟研究[10]等皆具有重要价值. 帝国理工大学的研究者在丝阵负载研究中指出平面丝阵构型下丝与丝之间的电感耦合决定了其电流分布, 导致在不同丝位处融蚀等离子体速率产生差异, 从而影响丝阵动力学演化过程[11,12]. 调制负载实验显示周期刻蚀结构的引入可导致趋肤电流分布特征改变, 并影响电热不稳定结构的种源特征, 从而实现对等离子体不稳定结构的定向调控[13]. 受限于Z箍缩实验的复杂电磁环境与有限诊断手段, 实验难以直接对等离子体局部电流密度进行精确测量, 但可通过对等离子体发光信息进行诊断以获取特征参数[14]. 平面薄膜作为具有连续二维结构的理想构型, 是由丝阵研究向套筒研究过渡的重要一环, 而目前国内外对该类构型的相关研究仍然较为缺乏, 对其电爆炸等离子体的电流通道建立与演化过程尚认知不足. 据此本文以反场构型平面薄膜负载为研究对象, 以基于电感主导模式的理论模型计算分析电流密度分布特征, 以不同参数的负载构型早期可见光自辐射图像为诊断手段开展实验验证, 实现对电流路径演化及其与融蚀等离子体发展的相互作用等关键问题的深入研究. 本文第2节介绍相关负载构型与诊断系统; 第3节介绍基于电感主导模型计算得到的负载电流分布特征及其对电爆炸发展现象的预测; 第4节为实验工作, 介绍实验观测到的薄膜的发光特征及等离子体形态演化特征, 并将提取的电流分布特征与理论模型进行了对比分析; 第5节为总结与讨论.

    强光一号加速器峰值电流约为1.4 MA, 上升时间约为100 ns. 实验负载采用回流柱位于轴心, 金属薄膜位于两侧的反场构型外爆式结构[13]. 在该构型下脉冲电流从阳极注入到两侧薄膜, 经阴极连接台和中心回流柱流出后返回阴极. 其电流位型导致近轴心处具有更强磁场, 形成与柱形丝阵和内爆套筒相反的磁压分布特征, 从而可驱动薄膜整体向外运动发展. 该构型具有探测区域视场大, 负载参数易调节, 利于激光探针以及分幅相机等诊断系统进行空间布局等优势. 该构型可用于类比研究大型丝阵或套筒实验中的局部等离子体行为, 同时降低了对驱动源规模的需求[12]. 同时该构型解决了等离子体向内汇聚所导致的诊断窗口受限问题[15], 且可用于研究非对称性磁场环境下界面不稳定性发展[16], 并在爆炸冲击波研究[17,18]、实验天体物理[19]和磁流体数值模拟[20]等领域内具有重要的应用价值. 本文利用该类负载构型进一步构造形成了对称双侧型和非对称双侧型外爆式负载结构, 即通过调整薄膜与回流柱间距构造对称性上的结构差异, 进而造成电流分配差异, 以便开展理论分析并利用实验现象上的差异特征进行验证. 负载安装的金属薄膜尺寸为20 μm厚, 10 mm宽, 20 mm长, 分别以符号h, b, l对应; 材料为高纯度铝金属, 以M表示; 两侧薄膜与回流柱边界的距离及回流柱宽分别以符号d1, d3, d2对应, 非对称结构下远端薄膜处的长边界距与符号d1对应, 近端薄膜短边界距与d3对应. 实验负载参数如表1所列, 负载装配结构与实验诊断系统光路如图1所示.

    表 1  反场构型平面薄膜负载参数
    Table 1.  Parameters of the planar foils.
    负载结构类型材料M膜厚h/μm膜宽b/mm膜长l/mm回流柱厚d2/mm边界距1 d1/mm边界距2 d3/mm
    双侧对称型2010201055
    双侧异距型20102010155
    下载: 导出CSV 
    | 显示表格
    图 1 诊断系统光路示意图与负载装配结构图\r\nFig. 1. Optical path of the diagnostic system and assembled load diagram.
    图 1  诊断系统光路示意图与负载装配结构图
    Fig. 1.  Optical path of the diagnostic system and assembled load diagram.

    采用激光阴影成像记录侧向等离子体的发展状态, 激光探针方向与薄膜表面平行, 图2给出了侧向激光阴影记录下的初始状态, 其图像经过了90°旋转处理, 图2(a)图2(b)分别显示了两种负载的初始间距结构. 利用可见光分幅相机对薄膜轴向和侧向可见光辐射强度进行记录, 其曝光时间为5 ns, 分幅间隔约为20 ns; 可见光图像辐射强度信息通过等离子源辐射、辐射输运、相机记录形成数字图像表征. 通过可见光分幅相机进行记录后得到的灰度表达式[21]

    图 2 不同结构负载侧向激光阴影图示 (a) 对称型薄膜负载; (b) 非对称型薄膜负载\r\nFig. 2. Side-on laser shadow graphs of the experimental load with different structures: (a) Symmetric load; (b) asymmetric load.
    图 2  不同结构负载侧向激光阴影图示 (a) 对称型薄膜负载; (b) 非对称型薄膜负载
    Fig. 2.  Side-on laser shadow graphs of the experimental load with different structures: (a) Symmetric load; (b) asymmetric load.
    G=KSΩIv. (1)

    其中, G为记录图像灰度值, K为比例系数, S为探测面积, Ω为接收立体角, Iv为单位频率辐射亮度. 每发次实验过程中诊断布局保持相同, 因此 K, S, Ω­固定, GIv. 辐射亮度与辐射区域的温度正相关. 温度方程可近似表达为

    ρcvTt=j2δm+(κT). (2)

    其中, ρ为密度, κ为热传导系数, δm为电阻率, cv为热容. 因此温度与电流密度呈正相关. (1)式和(2)式给出了图像灰度、可见光辐射强度、等离子体温度和局部电流密度之间的联系, 为实验验证电流通道模型提供了理论分析基础.

    文献[22-24]利用丝阵模型中通过所有可能导电回路下的磁通守恒条件, 解有限丝数线性方程组计算得到了电感主导模式下丝阵的电流分布特征. 文献[25-27]通过矢势法在回流柱呈现环形对称结构的等电流分配假设下, 将问题划归到二维复空间平面, 通过解有限线性方程组获得电流分配信息. 此两种方法的基本原理相同. 与有限个数的分立单丝组成的丝阵不同, 薄膜各部分相互连通构成整体, 且反场构型下的负载结构不同, 无法直接使用上述模型进行分析. 针对实验采用的外爆型薄膜负载结构, 本节建立了计算模型进行研究分析. 采用矢势法, 磁感应强度B与矢势A满足:

    B=×A. (3)

    矢势满足规范条件:

    A=0. (4)

    在真空满足拉普拉斯方程:

    2A=0. (5)

    考虑无限长元电流柱所产生的矢势. 设元电流柱半径为a, 电流密度为j, 空间某点距离圆心处的位置坐标为r, 设电流沿z轴向上, 则产生的空间磁感应强度分布为

    B=Byrˆx+Bxrˆy (6)
    B={μ0j2r    (0<r<a)μ0j2a2r    (ra) (7)

    其中B为磁感应强度大小; μ0为真空磁导率; x, y, z为空间坐标, ˆx, ˆy, ˆz为对应方向上的单位矢量. 得到矢势表达式为

    A=A(r)ˆz (8)
    A(r)={μ0j4r2+C1   (0<r<a)μ0ja22ln(C2r)   (ra) (9)

    其中A为矢势大小; C1, C2为积分常数, 设C2 = 1.

    为简化构型采用有限宽无限长平板模型开展理论分析, 如图3所示, 模型由回流柱与薄膜组成, 忽略其z向上长度的影响. 现考虑薄膜上一段宽dxδ的微元在真空区域中产生的矢势dA, 薄膜微元与电流柱微元通过以下近似关系对应:

    图 3 单侧薄膜-回流柱负载模型轴向图示. 电流方向为从薄膜底部流入顶部流出, 然后向下流进回流柱. w为回流柱宽度, b为薄膜宽度, d为两者间距. x为微元横坐标, x′为矢势研究点横坐标, R为微元与研究点间距, iα为薄膜面电流密度大小, iβ为回流柱边界处面电流密度大小. δα为薄膜载流厚度, δβ 为回流柱边界趋肤深度\r\nFig. 3. Diagram of 2-D foil-backpost geometry of axial direction view. Current flows into the foil from the bottom and flows out of the foil down to the return current post. w is the width of the return current post,  b is the width of the foil, d is the distance between foil and post. x is the coordinate of the element, x' is the coordinate of the studied point, R is the distance between the element and studied point, iα is the surface current density of the foil and iβ is that of the backpost. δα is the flowing depth of current in the foil and δβ is that of the post boundary.
    图 3  单侧薄膜-回流柱负载模型轴向图示. 电流方向为从薄膜底部流入顶部流出, 然后向下流进回流柱. w为回流柱宽度, b为薄膜宽度, d为两者间距. x为微元横坐标, x′为矢势研究点横坐标, R为微元与研究点间距, iα为薄膜面电流密度大小, iβ为回流柱边界处面电流密度大小. δα为薄膜载流厚度, δβ 为回流柱边界趋肤深度
    Fig. 3.  Diagram of 2-D foil-backpost geometry of axial direction view. Current flows into the foil from the bottom and flows out of the foil down to the return current post. w is the width of the return current post, b is the width of the foil, d is the distance between foil and post. x is the coordinate of the element, x' is the coordinate of the studied point, R is the distance between the element and studied point, iα is the surface current density of the foil and iβ is that of the backpost. δα is the flowing depth of current in the foil and δβ is that of the post boundary.
    dIjδdxjπ a2. (10)

    对应图3中单侧薄膜-回流柱的负载结构得

    dA(z,t)=μ02π ln(1R)i(x)dx (11)

    其中I为电流大小, R为微元中心到计算矢势点间距离, i(x) = j(x)·δ为压缩在二维x-z平面上的面电流密度. 以上对薄膜微元的矢势计算中采用了利用电流柱微元表达式近似替换的分析方法, 在数值计算过程中选取步长dx近似于电流载流深度δ, 图4显示计算结果具有良好的准确性.

    图 4 无限长有限宽理想平板构型在电流均匀分布假设下, 平板所在平面总磁感应强度分布. 载流厚度δ为20 μm, 平板范围为–5—5 mm, 平均电流密度大小为ja. 蓝色曲线为平板模型理论公式计算结果, 红色点集为单排电流柱微元近似薄膜微元下的数值计算结果\r\nFig. 4. Assuming equal distribution of current, calculated numerical B-field on slab by single-row model comparing to the theoretical value. Current depth is 20 μm, slab region is from –5 to 5 mm and ja is the average current density. Blue line is the numerical result and red plots are the theoretical result.
    图 4  无限长有限宽理想平板构型在电流均匀分布假设下, 平板所在平面总磁感应强度分布. 载流厚度δ为20 μm, 平板范围为–5—5 mm, 平均电流密度大小为ja. 蓝色曲线为平板模型理论公式计算结果, 红色点集为单排电流柱微元近似薄膜微元下的数值计算结果
    Fig. 4.  Assuming equal distribution of current, calculated numerical B-field on slab by single-row model comparing to the theoretical value. Current depth is 20 μm, slab region is from –5 to 5 mm and ja is the average current density. Blue line is the numerical result and red plots are the theoretical result.

    对于回流柱和薄膜面上任意两路径P1, P2构成的回路有

    ϕ=P2P1BdS=lArdr=l(A(r2)A(r1)) (12)
    ddtϕ=lddtΛ(t)=U (13)

    其中ϕ为磁通量, l为薄膜长度, U为电压, r1为回流柱表面上与z轴平行的直线P1的位置, r2为薄膜表面上与z轴平行的直线P2的空间位置(图3), Λ(t)为矢势差. 薄膜各处连接着相同的电极, 故在同一时刻下, 磁通量与选取的回路路径位置无关, 即A(r2)与A(r1)皆为与r无关的常数.

    A(r1)=At1  A(r2)=At2. (14)

    实际薄膜与回流柱结构不同, 因此计算的过程中需要分别考虑薄膜与回流柱上的电流分布. 在双侧非对称实验负载结构中, 回流柱为矩形截面柱状体, 其两侧界面为平面. 以其为轴两侧各安装有金属薄膜负载, 构成双侧负载结构. 因此需要划分为四部分区域进行考虑, 如图5所示.

    图 5 双侧薄膜结构负载模型轴向示意图. d1为远端薄膜距回流柱边界距离, d3为近端薄膜距回流柱边界距离, d2为回流柱宽度. i 为面电流密度, δ 为载流厚度, 其下标α1, β1, α2, β2分别对应薄膜1、回流柱界面1、薄膜2和回流柱界面2\r\nFig. 5. Diagram of the load structure with double foils from axial view. d1 is the distance between the further side foil α1 and post boundary β1, d3 is the distance between the other side foil α2 and post boundary β2, d2 is the width of the return current post, i is the surface current density, δ is the current depth where subscripts α1, β1, α2, β2 correspond to foil-1, post boundary-1, foil-2, post boundary-2, respectively
    图 5  双侧薄膜结构负载模型轴向示意图. d1为远端薄膜距回流柱边界距离, d3为近端薄膜距回流柱边界距离, d2为回流柱宽度. i 为面电流密度, δ 为载流厚度, 其下标α1, β1, α2, β2分别对应薄膜1、回流柱界面1、薄膜2和回流柱界面2
    Fig. 5.  Diagram of the load structure with double foils from axial view. d1 is the distance between the further side foil α1 and post boundary β1, d3 is the distance between the other side foil α2 and post boundary β2, d2 is the width of the return current post, i is the surface current density, δ is the current depth where subscripts α1, β1, α2, β2 correspond to foil-1, post boundary-1, foil-2, post boundary-2, respectively

    由(12)式计算四界面的叠加矢势, 薄膜边界α1处的矢势表达式为

    Aα1(x)=μ02π {w/2w/2ln(1(xx)2+(d1+δβ/δβ22)2)iβ1(x)dx + w/2w/2ln(1(xx)2+(d1+d2δβ/δβ22)2)iβ2(x)dxb/2b/2ln(1(xx)2+(δα/δα2)2)2)iα1(x)dxb/2b/2ln(1(xx)2+(d1+d2+d3+δα/δα22)2)iα2(x)dx}, (15)

    薄膜边界α2处矢势表达为

    Aα2(x)=μ02π {w/2w/2ln(1(xx)2+(d2+d3δβ/δβ22)2)iβ1(x)dx + w/2w/2ln(1(xx)2+(d3+δβ/δβ22)2)iβ2(x)dxb/2b/2ln(1(xx)2+(d1+d2+d3+δα/δα22)2)iα1(x)dxb/2b/2ln(1(xx)2+(δα/δα22)2)iα2(x)dx}. (16)

    回流柱边界β1处的矢势为

    Aβ1(x)=μ02π {w/2w/2ln(1(xx)2+(δβ/δβ22)2)iβ1(x)dx + w/2w/2ln(1(xx)2+(d2δβ/δβ22)2)iβ2(x)dxb/2b/2ln(1(xx)2+(d1+δα/δα22)2)iα1(x)dxb/2b/2ln(1(xx)2+(d2+d3+δα/δα22)2)iα2(x)dx}. (17)

    回流柱边界β2处的矢势为

    Aβ2(x)=μ02p{w/2w/2ln(1(xx)2+(d2δβ/δβ22)2)iβ1(x)dx + w/2w/2ln(1(xx)2+(δβ/δβ22)2)iβ2(x)dxb/2b/2ln(1(xx)2+(d1+d2+δα/δα2)2)2)iα1(x)dxb/2b/2ln(1(xx)2+(d3+δα/δα22)2)iα2(x)dx}. (18)

    由电流守恒:

    w/2w/2iα1(x)dx+b/2b/2iα2(x)dx=w/2w/2iβ1(x)dx+b/2b/2iβ2(x)dx (19)

    (15)式对应的区域矢势为常数:

    Aα1=Aα2=Constant 1,Aβ1=Aβ2=Constant 2. (20)

    将(19)式和(20)式作为给定条件求解泛函iα1, iβ1, iα2, iβ2的问题. 利用分段求和近似积分过程, 分立出线性方程组可以求解得到数值结果. 计算得到的面电流密度分布如图6图7所示, 以相对平均值ia的形式表示.

    图 6 双侧对称型结构负载电流密度分布(d1 = 5 mm, d2 = 10 mm, d3 = 5 mm, 对应于图5结构. 其中红色(与黑色重合)对应薄膜α1, α2, 蓝色(与绿色重合)对应回流柱界面β1, β2) (a) 无偏置; (b) 薄膜左偏置0.3 mm\r\nFig. 6. Distribution of surface current density of the symmetric load structure with double foils. d1 = 5 mm, d2 = 10 mm, d3 = 5 mm corresponding to the parameters in Fig. 5. Red (black) lines represent the foil α1 and foil α2, blue (green) lines represent the backpost boundary β1 and backpost boundary β2, respectively. (a) The foil is loaded in the center; (b) the foil has an initial 0.3 mm offset on the left.
    图 6  双侧对称型结构负载电流密度分布(d1 = 5 mm, d2 = 10 mm, d3 = 5 mm, 对应于图5结构. 其中红色(与黑色重合)对应薄膜α1, α2, 蓝色(与绿色重合)对应回流柱界面β1, β2) (a) 无偏置; (b) 薄膜左偏置0.3 mm
    Fig. 6.  Distribution of surface current density of the symmetric load structure with double foils. d1 = 5 mm, d2 = 10 mm, d3 = 5 mm corresponding to the parameters in Fig. 5. Red (black) lines represent the foil α1 and foil α2, blue (green) lines represent the backpost boundary β1 and backpost boundary β2, respectively. (a) The foil is loaded in the center; (b) the foil has an initial 0.3 mm offset on the left.
    图 7 双侧非对称型结构负载电流分布(d1 = 15 mm, d2 = 10 mm, d3 = 5 mm, 对应于图5结构. 其中红色对应薄膜α1, 黑色对应薄膜α2, 蓝色对应回流柱面β1, 绿色对应回流柱面β2) (a) 无偏置; (b) 左偏置0.3 mm\r\nFig. 7. Distribution of surface current density of the asymmetric load structure with double foils. d1 = 15 mm, d2 = 10 mm, d3 = 5 mm corresponding to the parameters in Fig. 5. Red, black, blue, green lines represent the foil α1,  foil α2, backpost boundary β1, backpost boundary β2, respectively. (a) The foil is loaded in the center; (b) the foil is set an initial 0.3 mm offset on the left.
    图 7  双侧非对称型结构负载电流分布(d1 = 15 mm, d2 = 10 mm, d3 = 5 mm, 对应于图5结构. 其中红色对应薄膜α1, 黑色对应薄膜α2, 蓝色对应回流柱面β1, 绿色对应回流柱面β2) (a) 无偏置; (b) 左偏置0.3 mm
    Fig. 7.  Distribution of surface current density of the asymmetric load structure with double foils. d1 = 15 mm, d2 = 10 mm, d3 = 5 mm corresponding to the parameters in Fig. 5. Red, black, blue, green lines represent the foil α1, foil α2, backpost boundary β1, backpost boundary β2, respectively. (a) The foil is loaded in the center; (b) the foil is set an initial 0.3 mm offset on the left.

    通过矢势法计算得到了薄膜-回流柱负载结构下基于电感主导模式的电流密度分布特征. 图6结果显示当两侧薄膜间距相同构成对称型负载结构时, 不同薄膜以及回流柱边界的状态一致, 其电流分布曲线重合. 此时两侧薄膜均呈现边界处电流密度最强, 中心处最弱, 由两侧向中心递减的谷状特征分布. 在边界处5%的长度范围内, 电流密度由平均值的10倍以上迅速下降到1倍左右, 形成了极为陡峭的边界密度分布特征. 回流柱边界处的电流密度同样最高, 但中心处高于其邻近两侧而形成局部凸起状结构, 但当薄膜与回流柱的间距增大时这一特征逐渐消失. 当薄膜中心存在偏置时, 近回流柱对称轴(x = 0)侧薄膜的电流密度值相对略高, 该半侧薄膜的电流分配增至总体的52%. 回流柱上靠近膜中心(x = –0.3)处的半侧电流较高, 其值增至总体的61%. 回流柱与薄膜对应半侧的电流增减形成互补.

    非对称双侧负载结构计算结果如图7所示. 结果显示近轴端薄膜α2各处的面电流密度均高于远轴端薄膜α1对应数值, 其总电流分配中近轴薄膜α2分配占66.4%, 远轴膜α1占33.6%. 近膜端回流柱界面β2上的各处面电流密度同样高于β1上对应数值, 其总电流分配中β2占67.2%, β1占32.8%. 参数的调节实现了对负载电流分配的调控. 薄膜中心存在偏置时的电流特征变化情况与对称结构结果相似.

    图8给出了对称结构中不同电流分配模式下薄膜所在平面的磁场分布和薄膜各处的受力分析. 计算结果表明在电感主导模式下(图8(a))薄膜边沿处总受力和磁感应强度远强于薄膜中心, 从而极大地加剧等离子体的融蚀过程. 边沿受力矢量的纵向分量fy同样高于薄膜中央, 因此边沿的向外扩展更为明显, 这将导致钳状特征形态的形成. 在横向分量fx的作用下, 等离子体同时存在向心汇聚并将导致其携带电流的转移. 图8(b)给出了假设电流达到均匀分布情况下的特征分析. 此时边沿受力明显减弱, 薄膜各处fy值近似相等而表现出火山状受力特征, 薄膜整体向外运动趋于一致, 但薄膜边沿处总受力仍然最强, 其等离子体的融蚀和汇聚进一步发展. 而当电流在中心进一步汇聚时(图8(c)高斯分布假设), 此时薄膜各部分受力已基本相近, 但薄膜边界受力的纵向分量fy进一步减弱, 薄膜中心fy显著增强. 薄膜中心电流密度与受力的增强, 将加剧中心处等离子体的融蚀与扩展, 从而将导致峰型等离子体形态的形成.

    图 8 不同模式下的薄膜各部分电流密度分布、薄膜总磁场强度分布、总受力分布与力矢量特征 (a) 电感主导模式; (b) 平均电流密度分布模式; (c) 中心汇聚型电流密度分布模式(高斯分布假设). 其中ja为薄膜平均电流密度, δ为薄膜载流厚度, μ0为真空磁导率, F为总受力大小, fx为局部受力的横向分量, fy为纵向分量\r\nFig. 8. Curves of current distribution, magnitude of B-field, magnitude of force and vectors of force along the foil: (a) Based on current division model that inductance dominates; (b) based on equal current division model; (c) based on current division model that its density concentrates on the center (assuming Gaussian distribution of current).
    图 8  不同模式下的薄膜各部分电流密度分布、薄膜总磁场强度分布、总受力分布与力矢量特征 (a) 电感主导模式; (b) 平均电流密度分布模式; (c) 中心汇聚型电流密度分布模式(高斯分布假设). 其中ja为薄膜平均电流密度, δ为薄膜载流厚度, μ0为真空磁导率, F为总受力大小, fx为局部受力的横向分量, fy为纵向分量
    Fig. 8.  Curves of current distribution, magnitude of B-field, magnitude of force and vectors of force along the foil: (a) Based on current division model that inductance dominates; (b) based on equal current division model; (c) based on current division model that its density concentrates on the center (assuming Gaussian distribution of current).

    在强光一号加速器平台开展了反场构型平面薄膜电爆炸实验研究, 诊断获取的可见光图像如图9图15所示, 通过薄膜侧向和轴向上的发光分布实现对等离子体电流分布的定性表征, 并与理论模型对比验证.

    图 9 双侧对称型铝膜负载实验侧向可见光辐射图像(No. 17222) (a) 初始激光阴影图; (b) 26.6 ns时刻图; (c) 60.4 ns时刻图.\r\nFig. 9. Optical side-on self-emission images for the symmetric structure load with double foils (No. 17222): (a) Laser shadow graph of load in the initial condition; (b) exploding time at 26.6 ns; (c) exploding time at 60.4 ns.
    图 9  双侧对称型铝膜负载实验侧向可见光辐射图像(No. 17222) (a) 初始激光阴影图; (b) 26.6 ns时刻图; (c) 60.4 ns时刻图.
    Fig. 9.  Optical side-on self-emission images for the symmetric structure load with double foils (No. 17222): (a) Laser shadow graph of load in the initial condition; (b) exploding time at 26.6 ns; (c) exploding time at 60.4 ns.

    图9图10展示了双侧对称结构铝膜负载电爆炸发光图像. 侧向结果显示负载表现出薄膜先于回流柱发光(图9(b))以及在回流柱边界形成发光带(图9(c))的特征现象. 轴向图像显示早期发光集中在薄膜边沿(图10(a)), 薄膜中央分布较弱. 随着电流上升, 薄膜中心处的发光特征开始增强, 逐渐形成由两边向中心的发光区域延伸过程(图10(b)). 图10(c)显示67 ns时刻下薄膜形成了两侧发光厚, 中心发光薄的等离子体形态, 与电感模式受力分析得到的钳状形态一致(图8(a)), 实验结果符合理论预期.

    图 10 双侧对称型铝膜负载实验轴向可见光图像(No. 19103) (a) 27 ns; (b) 67 ns; (c)局部薄膜发光伪色图与对应位置相对强度曲线, G为相对灰度值\r\nFig. 10. Optical end-on self-emission images for the symmetric structure load with double foils (No. 19103): (a) Shot time at 27 ns; (b) shot time at 67 ns; (c) the image in yellow box is treated with flase color. Corresponding mean column gray value curve G with position x is calculated on the top.
    图 10  双侧对称型铝膜负载实验轴向可见光图像(No. 19103) (a) 27 ns; (b) 67 ns; (c)局部薄膜发光伪色图与对应位置相对强度曲线, G为相对灰度值
    Fig. 10.  Optical end-on self-emission images for the symmetric structure load with double foils (No. 19103): (a) Shot time at 27 ns; (b) shot time at 67 ns; (c) the image in yellow box is treated with flase color. Corresponding mean column gray value curve G with position x is calculated on the top.

    图11图12展示了非对称型负载结构的实验图像, 结果表现出以下非对称型特征: 侧向图像中近轴端薄膜处较早观测到可见光辐射, 而同时刻下远轴端薄膜尚未观测到明显发光(图12(a)); 轴向图像中薄膜边沿早于中心辐射可见光, 中心区域后发光连通(图12(b)); 薄膜位置存在微小偏置下, 近中轴侧薄膜边沿发光区域更广. 结果显示, 异构型负载实验实现了电流分配的调控, 并通过差异化特征的构造验证了电感主导模式的预期现象.

    图 11 双侧非对称铝膜负载实验侧向可见光辐射图像(No. 16189) (a) 初始时刻, 薄膜1对应远端位置, 薄膜2对应近端位置; (b) 49.8 ns; (c) 70.8 ns\r\nFig. 11. Optical side-on self-emission images for the asymmetric structure load with double foils set in different distance (No. 16189): (a) Laser shadow graph of load in the initial condition, foil 1 corresponds to the far side and foil 2 corresponds to the near side; (b) shot time at 49.8 ns; (c) shot time at 70.8 ns.
    图 11  双侧非对称铝膜负载实验侧向可见光辐射图像(No. 16189) (a) 初始时刻, 薄膜1对应远端位置, 薄膜2对应近端位置; (b) 49.8 ns; (c) 70.8 ns
    Fig. 11.  Optical side-on self-emission images for the asymmetric structure load with double foils set in different distance (No. 16189): (a) Laser shadow graph of load in the initial condition, foil 1 corresponds to the far side and foil 2 corresponds to the near side; (b) shot time at 49.8 ns; (c) shot time at 70.8 ns.
    图 12 双侧非对称铝膜负载实验轴向可见光辐射图像(No. 16214) (a) 12.6 ns; (b) 33.6 ns; (c) 75.6 ns. 薄膜1对应远端位置, 薄膜2对应近端位置\r\nFig. 12. Optical end-on self-emission images for the asymmetric structure load with double foils (No. 16214): (a) Shot time at 12.6 ns; (b) shot time at 33.6 ns; (c) shot time at 75.6 ns. Foil 1 corresponds to the far side, and foil 2 corresponds to the near side.
    图 12  双侧非对称铝膜负载实验轴向可见光辐射图像(No. 16214) (a) 12.6 ns; (b) 33.6 ns; (c) 75.6 ns. 薄膜1对应远端位置, 薄膜2对应近端位置
    Fig. 12.  Optical end-on self-emission images for the asymmetric structure load with double foils (No. 16214): (a) Shot time at 12.6 ns; (b) shot time at 33.6 ns; (c) shot time at 75.6 ns. Foil 1 corresponds to the far side, and foil 2 corresponds to the near side.

    上述不同构型下薄膜负载实验结果证明, 早期阶段电流通道的建立与密度分布特征与电感主导模式相符. 这与反场构型丝阵实验得到的结论一致. 但边沿处等离子的融蚀与运动将导致模式特征发生改变. 图13给出了70 ns后薄膜轴向可见光辐射状态. 图像显示薄膜中心位置出现强烈的凸起发光特征, 这与图8(c)受力分析中得到的峰型结构相似. 图14图15展示了对称结构负载一侧薄膜的完整图像序列. 研究认为等离子体存在向心汇聚与电流通道转移, 该现象在图14序列中的67 ns时刻图像中已明显产生, 表现为图像中央区域亮度的突然增强, 而此时中心位置尚未发生明显凸起. 图15序列中的79 ns 时刻图像中同样观测到中央亮带的突然产生. 此后薄膜中心加速扩张, 峰型凸起形态由中心向外快速发展(88 ns)并向薄膜两侧延伸(108 ns). 薄膜中央亮区的形成现象类似于丝阵中轴心形成的先驱等离子体柱. 反场构型丝阵实验研究表明, 电流通道在一定时间内会集中在融蚀并分离于丝核的先驱等离子体中, 使其加速脱离丝阵平面, 然后电流又会快速返回到丝核区附近, 电流通道存在快速的切换过程[28]. 本文薄膜实验同样观测到, 边沿处融蚀形成的等离子体将携带电流向心汇聚导致中央区富集, 而由于薄膜等离子体并未明显脱离薄膜平面, 因此电流通道并未发生二次转移. 电流通道存在转移的现象同时说明了丝阵实验中利用电阻模型计算聚爆时间更为准确的原因[24].

    图 13 双侧对称型铝膜负载实验轴向可见光图像(No. 19110) (a) 79 ns; (b) 99 ns; (c) 119 ns\r\nFig. 13. Optical end-on self-emission images for the symmetric structure load with double foils (No. 19110): (a) Shot time at 79 ns; (b) shot time at 99 ns; (c) shot time at 119 ns.
    图 13  双侧对称型铝膜负载实验轴向可见光图像(No. 19110) (a) 79 ns; (b) 99 ns; (c) 119 ns
    Fig. 13.  Optical end-on self-emission images for the symmetric structure load with double foils (No. 19110): (a) Shot time at 79 ns; (b) shot time at 99 ns; (c) shot time at 119 ns.
    图 14 20 μm铝膜电爆炸过程轴向可见光自辐射诊断图像序列. 图像经过伪色处理, 相对灰度值与伪色域对应关系如右侧标尺所示\r\nFig. 14. Diagnosed images sequence of axial visible end-on self-emission in exploding experiments of 20 μm Al foils. Images are treated with false-color where color scale is shown on the right.
    图 14  20 μm铝膜电爆炸过程轴向可见光自辐射诊断图像序列. 图像经过伪色处理, 相对灰度值与伪色域对应关系如右侧标尺所示
    Fig. 14.  Diagnosed images sequence of axial visible end-on self-emission in exploding experiments of 20 μm Al foils. Images are treated with false-color where color scale is shown on the right.
    图 15 20 μm铝膜电爆炸过程截面可见光自辐射诊断图像序列. 图像经过了伪色处理, 相对灰度值与伪色域对应关系如右侧标尺所示. 相机拍摄方向与薄膜界面法向成45°\r\nFig. 15. Diagnosed images sequence of surface self-emission in exploding experiments of 20 μm Al foils. Images are treated with false-color where color scale is shown on the right. The shooting angle are 45° relative to foil surface orientation.
    图 15  20 μm铝膜电爆炸过程截面可见光自辐射诊断图像序列. 图像经过了伪色处理, 相对灰度值与伪色域对应关系如右侧标尺所示. 相机拍摄方向与薄膜界面法向成45°
    Fig. 15.  Diagnosed images sequence of surface self-emission in exploding experiments of 20 μm Al foils. Images are treated with false-color where color scale is shown on the right. The shooting angle are 45° relative to foil surface orientation.

    通过电感分析理论和反场构型平面薄膜电爆炸实验工作的开展, 系统地研究了其电爆炸等离子体的电流密度分布模式、磁场分布特征和受力分布特征. 结果显示在早期脉冲电流的主要上升阶段(0—70 ns), 电流通道的建立与分布符合电感主导模式, 薄膜边沿处电流密度最强, 表现出边沿聚集与近轴聚集等发光特征. 薄膜边沿处电流的集中与强电磁力加剧该处等离子体的融蚀与运动, 边沿处最先出现向外扩展现象, 薄膜整体形成钳状形态特征. 当脉冲电流逐渐建立并达到峰值(70—120 ns), 融蚀形成的等离子体在横向电磁力的作用下向中心快速运动, 并携带电流使电流通道逐渐向中心转移, 边界处受力逐渐减弱, 薄膜中心纵向受力逐渐增强, 加剧中心处等离子体的融蚀与向外扩展, 导致薄膜中心迅速发展形成强发光区与峰型等离子体形态特征. 电流同通道在40 ns内(图14中47—88 ns发展图像)完成快速切换, 导致初始阶段的电感主导模式失效. 理论模型与实验结果相符, 加深了对等离子体电流通道建立和转移的认知.

    理论模型分析中采用了简化无限长平板模型分析, 薄膜的长度限制以及阴阳极连接平台等其他复杂结构的影响尚未考虑. 同时薄膜电爆炸过程中存在侧向扩展与电流分层现象, 因此未来工作需要对理论模型进一步细化精确. 同时可尝试通过使用特殊构型的阴极片结构等开展负载设计, 对电流通道影响因素与薄膜动力学发展的调控等关键问题进一步探索. 如图16所示, 本文采用了常规阴极片结构(图16(a)), 在此基础上通过改造形成内凹型结构(图16(b))或外凸型阴极片结构(图16(c))等, 可探索对原轴向电流分布模式的影响, 进而实现对动力学过程的调控.

    图 16 阴极片结构设计 (a) 常规结构; (b) 内凹型结构: (c) 外凸型结构. 阴影区为与回流柱连接区, 蓝色区域为薄膜连接区\r\nFig. 16. Different designed cathode plate structure: (a) Normal structure; (b) inner fovea structure; (c) outer fovea structure. Shadow regions connect the backpost and blue regions connect metal foils.
    图 16  阴极片结构设计 (a) 常规结构; (b) 内凹型结构: (c) 外凸型结构. 阴影区为与回流柱连接区, 蓝色区域为薄膜连接区
    Fig. 16.  Different designed cathode plate structure: (a) Normal structure; (b) inner fovea structure; (c) outer fovea structure. Shadow regions connect the backpost and blue regions connect metal foils.

    另一方面本文通过可见光分幅成像诊断所得到的结论局限在早期阶段, 当薄膜完全等离子体化并进入明显动力学发展阶段时该诊断方法存在一定的局限. 本文中以电流汇聚模型(高斯分布假设)研究了在此假设上所导致的薄膜受力状态变化, 并与实验现象进行对比, 得到对后期的电流分布特征的间接分析. 后期图像特征一定程度验证了电流汇聚现象, 但仍需要进一步发展诊断手段以直接获取实验证据, 从而定量地表征后续阶段电流密度分布特征. 其中可考虑发展包括基于反射镜面介层烧蚀的轴向激光阴影成像系统显影电流路径, 局部磁探针或法拉第磁旋光法同步测量薄膜各局部区域磁场强度信息以还原电流信息, X箍缩点投影成像诊断薄膜截面上等离子体密度的空间分布以分析动力学过程和驱动电流分布等手段, 以进一步深化研究.

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  • 图 1  诊断系统光路示意图与负载装配结构图

    Fig. 1.  Optical path of the diagnostic system and assembled load diagram.

    图 2  不同结构负载侧向激光阴影图示 (a) 对称型薄膜负载; (b) 非对称型薄膜负载

    Fig. 2.  Side-on laser shadow graphs of the experimental load with different structures: (a) Symmetric load; (b) asymmetric load.

    图 3  单侧薄膜-回流柱负载模型轴向图示. 电流方向为从薄膜底部流入顶部流出, 然后向下流进回流柱. w为回流柱宽度, b为薄膜宽度, d为两者间距. x为微元横坐标, x′为矢势研究点横坐标, R为微元与研究点间距, iα为薄膜面电流密度大小, iβ为回流柱边界处面电流密度大小. δα为薄膜载流厚度, δβ 为回流柱边界趋肤深度

    Fig. 3.  Diagram of 2-D foil-backpost geometry of axial direction view. Current flows into the foil from the bottom and flows out of the foil down to the return current post. w is the width of the return current post, b is the width of the foil, d is the distance between foil and post. x is the coordinate of the element, x' is the coordinate of the studied point, R is the distance between the element and studied point, iα is the surface current density of the foil and iβ is that of the backpost. δα is the flowing depth of current in the foil and δβ is that of the post boundary.

    图 4  无限长有限宽理想平板构型在电流均匀分布假设下, 平板所在平面总磁感应强度分布. 载流厚度δ为20 μm, 平板范围为–5—5 mm, 平均电流密度大小为ja. 蓝色曲线为平板模型理论公式计算结果, 红色点集为单排电流柱微元近似薄膜微元下的数值计算结果

    Fig. 4.  Assuming equal distribution of current, calculated numerical B-field on slab by single-row model comparing to the theoretical value. Current depth is 20 μm, slab region is from –5 to 5 mm and ja is the average current density. Blue line is the numerical result and red plots are the theoretical result.

    图 5  双侧薄膜结构负载模型轴向示意图. d1为远端薄膜距回流柱边界距离, d3为近端薄膜距回流柱边界距离, d2为回流柱宽度. i 为面电流密度, δ 为载流厚度, 其下标α1, β1, α2, β2分别对应薄膜1、回流柱界面1、薄膜2和回流柱界面2

    Fig. 5.  Diagram of the load structure with double foils from axial view. d1 is the distance between the further side foil α1 and post boundary β1, d3 is the distance between the other side foil α2 and post boundary β2, d2 is the width of the return current post, i is the surface current density, δ is the current depth where subscripts α1, β1, α2, β2 correspond to foil-1, post boundary-1, foil-2, post boundary-2, respectively

    图 6  双侧对称型结构负载电流密度分布(d1 = 5 mm, d2 = 10 mm, d3 = 5 mm, 对应于图5结构. 其中红色(与黑色重合)对应薄膜α1, α2, 蓝色(与绿色重合)对应回流柱界面β1, β2) (a) 无偏置; (b) 薄膜左偏置0.3 mm

    Fig. 6.  Distribution of surface current density of the symmetric load structure with double foils. d1 = 5 mm, d2 = 10 mm, d3 = 5 mm corresponding to the parameters in Fig. 5. Red (black) lines represent the foil α1 and foil α2, blue (green) lines represent the backpost boundary β1 and backpost boundary β2, respectively. (a) The foil is loaded in the center; (b) the foil has an initial 0.3 mm offset on the left.

    图 7  双侧非对称型结构负载电流分布(d1 = 15 mm, d2 = 10 mm, d3 = 5 mm, 对应于图5结构. 其中红色对应薄膜α1, 黑色对应薄膜α2, 蓝色对应回流柱面β1, 绿色对应回流柱面β2) (a) 无偏置; (b) 左偏置0.3 mm

    Fig. 7.  Distribution of surface current density of the asymmetric load structure with double foils. d1 = 15 mm, d2 = 10 mm, d3 = 5 mm corresponding to the parameters in Fig. 5. Red, black, blue, green lines represent the foil α1, foil α2, backpost boundary β1, backpost boundary β2, respectively. (a) The foil is loaded in the center; (b) the foil is set an initial 0.3 mm offset on the left.

    图 8  不同模式下的薄膜各部分电流密度分布、薄膜总磁场强度分布、总受力分布与力矢量特征 (a) 电感主导模式; (b) 平均电流密度分布模式; (c) 中心汇聚型电流密度分布模式(高斯分布假设). 其中ja为薄膜平均电流密度, δ为薄膜载流厚度, μ0为真空磁导率, F为总受力大小, fx为局部受力的横向分量, fy为纵向分量

    Fig. 8.  Curves of current distribution, magnitude of B-field, magnitude of force and vectors of force along the foil: (a) Based on current division model that inductance dominates; (b) based on equal current division model; (c) based on current division model that its density concentrates on the center (assuming Gaussian distribution of current).

    图 9  双侧对称型铝膜负载实验侧向可见光辐射图像(No. 17222) (a) 初始激光阴影图; (b) 26.6 ns时刻图; (c) 60.4 ns时刻图.

    Fig. 9.  Optical side-on self-emission images for the symmetric structure load with double foils (No. 17222): (a) Laser shadow graph of load in the initial condition; (b) exploding time at 26.6 ns; (c) exploding time at 60.4 ns.

    图 10  双侧对称型铝膜负载实验轴向可见光图像(No. 19103) (a) 27 ns; (b) 67 ns; (c)局部薄膜发光伪色图与对应位置相对强度曲线, G为相对灰度值

    Fig. 10.  Optical end-on self-emission images for the symmetric structure load with double foils (No. 19103): (a) Shot time at 27 ns; (b) shot time at 67 ns; (c) the image in yellow box is treated with flase color. Corresponding mean column gray value curve G with position x is calculated on the top.

    图 11  双侧非对称铝膜负载实验侧向可见光辐射图像(No. 16189) (a) 初始时刻, 薄膜1对应远端位置, 薄膜2对应近端位置; (b) 49.8 ns; (c) 70.8 ns

    Fig. 11.  Optical side-on self-emission images for the asymmetric structure load with double foils set in different distance (No. 16189): (a) Laser shadow graph of load in the initial condition, foil 1 corresponds to the far side and foil 2 corresponds to the near side; (b) shot time at 49.8 ns; (c) shot time at 70.8 ns.

    图 12  双侧非对称铝膜负载实验轴向可见光辐射图像(No. 16214) (a) 12.6 ns; (b) 33.6 ns; (c) 75.6 ns. 薄膜1对应远端位置, 薄膜2对应近端位置

    Fig. 12.  Optical end-on self-emission images for the asymmetric structure load with double foils (No. 16214): (a) Shot time at 12.6 ns; (b) shot time at 33.6 ns; (c) shot time at 75.6 ns. Foil 1 corresponds to the far side, and foil 2 corresponds to the near side.

    图 13  双侧对称型铝膜负载实验轴向可见光图像(No. 19110) (a) 79 ns; (b) 99 ns; (c) 119 ns

    Fig. 13.  Optical end-on self-emission images for the symmetric structure load with double foils (No. 19110): (a) Shot time at 79 ns; (b) shot time at 99 ns; (c) shot time at 119 ns.

    图 14  20 μm铝膜电爆炸过程轴向可见光自辐射诊断图像序列. 图像经过伪色处理, 相对灰度值与伪色域对应关系如右侧标尺所示

    Fig. 14.  Diagnosed images sequence of axial visible end-on self-emission in exploding experiments of 20 μm Al foils. Images are treated with false-color where color scale is shown on the right.

    图 15  20 μm铝膜电爆炸过程截面可见光自辐射诊断图像序列. 图像经过了伪色处理, 相对灰度值与伪色域对应关系如右侧标尺所示. 相机拍摄方向与薄膜界面法向成45°

    Fig. 15.  Diagnosed images sequence of surface self-emission in exploding experiments of 20 μm Al foils. Images are treated with false-color where color scale is shown on the right. The shooting angle are 45° relative to foil surface orientation.

    图 16  阴极片结构设计 (a) 常规结构; (b) 内凹型结构: (c) 外凸型结构. 阴影区为与回流柱连接区, 蓝色区域为薄膜连接区

    Fig. 16.  Different designed cathode plate structure: (a) Normal structure; (b) inner fovea structure; (c) outer fovea structure. Shadow regions connect the backpost and blue regions connect metal foils.

    表 1  反场构型平面薄膜负载参数

    Table 1.  Parameters of the planar foils.

    负载结构类型材料M膜厚h/μm膜宽b/mm膜长l/mm回流柱厚d2/mm边界距1 d1/mm边界距2 d3/mm
    双侧对称型2010201055
    双侧异距型20102010155
    下载: 导出CSV
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    Ryutov D D, Derzon M S, Matzen M K 2000 Rev. Mod. Phys. 72 167Google Scholar

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    Haines M G 2011 Plasma Phys. Controlled Fusion 53 093001Google Scholar

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    Chittenden J P, Lebedev S V, Bland S N, Ruiz-Camacho J, Beg F N, Haines M G 2001 Laser Part. Beams 19 323Google Scholar

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    Zhigalin A S, Rousskikh A G, Oreshkin V I, Chaikovsky S A, Ratakhin N A, Kuznetsov V V 2014 International Congress On Energy Fluxes And Radiation Effects Tomsk, Russia, September 21–26, 2014 012027

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    Chittenden J P, Jennings C A 2008 Phys. Rev. Lett. 101 055005Google Scholar

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出版历程
  • 收稿日期:  2021-08-12
  • 修回日期:  2021-09-14
  • 上网日期:  2022-01-21
  • 刊出日期:  2022-02-05

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