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The magnetic fields generated in plasmas have extensive influences on many processes of the inertial confinement fusion and the astrophysics. Therefore, the quantitative diagnosis of the magnetic field is quite essential. Proton radiography is a widely used experimental technique to diagnose the electric field or magnetic field in high-energy-density plasma. The effective explanation of the results of proton radiography depends on the reliability and availability of the inversion method. Traditional inversion methods can only provide one- or two-dimensional structure of the self-generated magnetic field. In this study, it is found that there is an Abel transformation relationship between the deflection velocity and the magnetic field with column symmetry, which allows us to reconstruct the three-dimensional structure of the magnetic field for the first time. We theoretically deduce the process of reconstructing the cylindrical magnetic field through proton radiography with the Abel inversion algorithm. The feasibility of this method is verified by numerical simulation as well. Based on this inversion method, we reanalyze the proton radiography experimental results of Li et al. (
2016 Nat. Commun. 7 13081 ) on the self-generated magnetic field of plasma jets. The inversion results show that the maximum magnetic field intensity is about 1.9 times the traditional inversion results. We discuss a new proton radiography inversion method for the existence of magnetic fields with cylindrical symmetry in thiswork, which will contributes to an intensive understanding of the self-generated electromagnetic field and its spatiotemporal evolution related to the laser fusion and the laboratory astrophysics.-
Keywords:
- proton radiography /
- reversion /
- magnetic diagnostics /
- Abel inversion
1. 引 言
自生磁场是等离子体的基本特征之一, 普遍存在于激光聚变[1,2]、实验室天体物理[3-5]相关的高能量密度物质中. 因其可对电子热传导[6]、冲击波形成[4,7]和带电粒子输运[8,9]等物理过程产生影响而受到广泛关注, 一直是等离子体物理领域的研究热点之一.
准确认识等离子体自生磁场的时空演化行为离不开实验中的磁场诊断. 一般而言, 磁场的实验诊断主要有3种方法. 1) 磁探针法[10], 测量线圈在磁场中的感生电流来获取磁通量的平均大小. 受限于线圈的加工精度, 该方法往往无法获得较高的空间分辨能力, 且不能对等离子体内的自生磁场进行直接测量. 2) 法拉第旋转法[11,12], 通过测量偏振光在等离子体中的偏振面旋转量来推测磁场的强度. 因偏振面的旋转是磁场与等离子体密度耦合作用的结果, 该方法依赖于等离子体密度空间分布的准确诊断. 3) 质子照相法[13-15], 随着国际上靶后法向鞘场加速(target normal sheath acceleration, TNSA)[16,17]技术不断发展成熟, 该方法已经成为诊断等离子体中自生磁场的常用实验方法. 质子照相中的探针质子常通过TNSA 机制产生, 其能量一般可达到10 MeV量级[18] (对于通过D-He3内爆产生的质子, 能量则为3.03 MeV和14.7 MeV[19]). 假设探针质子束的运动速度为u0, 质子的电荷和质量分别为q和mp, 经磁场B偏转后, 质子束在离开磁场区域时将获得的偏转速度为
{{\boldsymbol u}_{\text{d}}} = q/{m_{\text{p}}}\displaystyle\int {{{\boldsymbol u}_0} \times {\boldsymbol B}{\text{d}}t} . 受此偏转速度的影响, 质子束的通量密度将得到调制并被记录在探测面上(一般为RCF堆栈片或CR-39探测器)[20]. 再从探测面上的通量密度扰动分布获得ud, 就可以反推获得磁场B的平均强度等信息.目前, 质子照相技术的应用对激光聚变和实验室天体物理相关的自生磁场研究有推动作用. 例如, Huntington等[21]和Zhou等[22]通过质子照相证实了等离子体对穿过程中存在的离子、电子Weibel不稳定性形成的丝状磁场结构; Li等[23]和 Gao等[24]利用质子照相观察了纳秒激光烧蚀CH材料时的表面磁场产生过程, 可以清楚判断出Biermann电池效应产生的环形磁场结构; Tzeferacos等[25]借助质子照相证实了磁场的压缩放大过程. 然而目前从质子照相反演自生磁场的空间结构时只能得到磁场的一维或二维结构, 沿着质子运动方向维度的分布信息往往被平均[26]. 这将不利于对磁场的强度和空间分布的认识, 以及实验诊断与数值模拟的相互校验[2].
等离子体中存在具有柱对称结构的自生磁场, 如等离子体喷流[3]或电容线圈靶[12]中的磁场. 本研究通过理论分析发现当磁场具有柱对称的结构时, 侧向照相的探针质子束偏转速度与磁场之间满足Abel变换关系. 这有别于传统的质子照相反演方法, 采用本方法能实现针对这类柱对称磁场的三维反演重建.
2. 理论分析
等离子体中自生磁场的质子照相过程如图1 所示. 不失一般性, 假设磁场Bz沿着ez方向, 场区域在ex方向的长度为Lx.
t = 0 时刻, 初始速度为ux0的质子束近似平行地进入磁场区域, 沿着ex方向穿过场区域并在t = {t_0} 时刻离开.探针质子束在穿过磁场区域时因受到洛伦兹力的作用而偏转, 离开磁场区域时, 质子将具有ey方向上的偏转速度:
{{\boldsymbol{u}}_{{\text{d}}y}}(y,z) = - \frac{q}{{{m_{\text{p}}}}}\int_0^{{t_0}} {{{\boldsymbol{u}}_x} \times {{\boldsymbol{B}}_z}(x,y,z){\text{d}}t} , (1) 其中
{\text{d}}t = {\text{d}}x/|{{\boldsymbol{u}}_x}{\text{|}} . 经过时间{t_0} 质子在ex方向的位移为Lx, 偏转速度udy简化为{{\boldsymbol{u}}_{{\text{d}}y}}(y,z) = - {{\boldsymbol{e}}_y}\frac{q}{{{m_{\text{p}}}}}\int_0^{{L_x}} {{B_z}(x,y,z){\text{d}}x} . (2) 传统的质子照相反演方法中, 可由(2)式估算沿ex方向的路径平均磁场
{{\boldsymbol B}'_z}(y, z) :{{\boldsymbol B}'_z}(y,z) = {{\boldsymbol u}_{{\text{d}}y}}(y,z)\frac{{{m_{\text{p}}}}}{{q{L_x}}}, (3) (3) 式得到的平均磁场损失了ex方向的空间信息. 相较于结构不规则、表征复杂的自生磁场, 本研究发现对具有柱对称分布的磁场进行适当的质子照相反演分析, 能获得磁场的三维结构信息.
当磁场Bz具有柱对称分布时, (2)式中
{{\boldsymbol B}_z}(x, y, z) = {{\boldsymbol B}_z}(r, z) , 其中r = \sqrt {{x^2} + {y^2}} ,x = r\cos \theta , y = r\sin \theta , 夹角\theta \in [0, 2{\text{π }}] . 众所周知, Abel变换及Abel逆变换的表达式[27]分别为I(y) = \int_{ - \infty }^\infty {g(r){\text{d}}x} ,\;g(r) = - \frac{{\text{1}}}{{\text{π }}}\int_r^\infty {\frac{{{\text{d}}I}}{{{\text{d}}y}}\frac{1}{{\sqrt {{y^2} - {r^2}} }}} {\text{d}}y. (4) (4) 式中
g(r) 为柱对称函数. 对比(2)式和(4) 式, 对于柱对称的磁场, 两个公式具有相同的形式, 仅仅是积分上下限不同. 而磁场在( - \infty , 0) 和({L_x}, + \infty ) 区域内近似为0, 则柱对称磁场Bz与偏转速度udy满足Abel变换关系. 如果udy可被反演获得, 则可通过Abel逆变换公式重建得到{{\boldsymbol B}_z}(r, z) , 即:{{\boldsymbol B}_z}(r,z) = \frac{{{m_{\text{p}}}}}{{q{\text{π }}}}\int_r^\infty {\frac{{{\text{d}}{{\boldsymbol u}_{{\text{d}}y}}(y,z)}}{{{\text{d}}y}}} \frac{1}{{\sqrt {{y^2} - {r^2}} }}{\text{d}}y. (5) 偏转速度udy的分布通常可通过密度扰动法实现重建[28,29], 有
{{\boldsymbol{u}}_{{\text{d}}y}}(y,z) = - \frac{{{u_x}M}}{{{L_{\text{D}}}}}\int {\frac{{{\text{δ}} n(My,Mz)}}{{{n_0}}}{{\rm{d}}} {\boldsymbol{l}}} , (6) 其中,
M={L_{\text{D}}}/{L_{\text{S}}} + 1 是几何放大因子, LD和LS分别为待诊断场到探测器和质子源的距离, dl是探测器上沿着偏转速度方向ey的单位长度, n和n0分别为有、无磁场时探测器上获得的质子通量密度[30]. 需要注意的是, 在利用(6)式反演udy时, 需要判断偏转速度的方向. 对于更加复杂的偏转速度分布, 则可以参考Bott等[26]介绍的蒙日-安培法来实现任意udy二维分布的重建. 此外, (6) 式还需要通量密度扰动满足\text{δ} n/{n_0} = n/{n_0} - 1 < 1 , 以表示质子束径迹未发生交叉或重叠[29]. udy也可以通过纹影法[12]获得, 即在质子源和待诊断场之间放置一个栅格, 通过读取网格的相对形变来表征探针质子在穿过磁场后的偏转速度[31]. 由于受到栅格加工工艺的限制, 纹影法的空间分辨能力具有局限性[12].上述分析表明, 当磁场具有柱对称分布时, 可以利用质子照相实现其三维结构的反演. 为了考察该方法的可行性, 本文进行了数值模拟来验证.
3. 数值模拟
在质子照相的数值模拟中, 设置待诊断的磁场B沿着ez方向, 在x-y平面内具有柱对称分布, 即:
{\boldsymbol{B}}(r,z) = {B_0}\exp \bigg[ { - \frac{{{{(r - {r_0})}^2}}}{{R_0^2}} - \frac{{{{(z - {z_0})}^2}}}{{Z_0^2}}} \bigg]{{\boldsymbol{e}}_z}. (7) 进一步设磁场的峰值强度为B0 = 5 T, R0 = 25 μm, Z0 = 100 μm, 磁场的空间范围为lx = ly = lz = 100 μm, 如图2(a) 所示. 平行的探针质子束沿着ex方向照射磁场区域, 其动能为20 MeV, 受磁场影响, 探针质子获得ey方向上的偏转速度. 探针质子的运动过程可采用七阶龙格-库塔法计算. 在穿出磁场区域后, 质子经自由飞行后被记录在约LD = 1 cm外y-z平面内的探测器上, 统计得到的通量密度扰动δn/n0如图2(b) 所示.
图 2 (a)预设磁场B在x = 50 μm平面上的分布; (b)探测面上的质子通量密度扰动Fig. 2. (a) Distributions of the preset magnetic field at x = 50 μm; (b) the flux density perturbations of the protons in the detection plane.从图2(b)可见, 质子最大通量密度扰动约为
{(\delta n/{n_0})_{\max }} = 0.2 < 1 , 说明探针质子的轨迹未发生交叉或重叠, 因此可以利用(6)式实现偏转速度udy的反演, 获得的udy如图3(a)所示. 通过正算模拟得到的偏转速度usim和反演重建得到的偏转速度urec的一维对比如图3(b)所示, 分别用红线和蓝线表示, 其最大值分别为2.07×104 m/s和2.04×104 m/s, 相差仅约1%, 表明偏转速度得到了较好的反演.
图 3 (a)质子偏转速度的反演结果; (b)质子的模拟偏转速度和反演偏转速度在z = 50 μm时的径向分布Fig. 3. (a) Reconstruction of the protons deflection velocities; (b) the radial distributions of the protons inversion deflection velocities and simulated deflection velocities at z = 50 μm.将图3(a) 中的udy代入(5)式中, 反演获得的磁场结构Brec(r, z)如图4(a) 所示, 与预设磁场, 即图2(a), 具有相近的分布. 图4(b) 给出的是z = 50 μm处的预设磁场Bset (红线), 由(5)式反演得到的磁场Brec (当
y = r\sin \theta 的夹角\theta = {{\text{π }}/2} 时)(蓝线)和由(3)式反演得到的路径平均磁场Bavg (黑线)在ey方向上的一维分布. 三者的半高全宽分别为41.6 μm, 37.9 μm和41.0 μm, 反演磁场和平均磁场相较于预设磁场相差分别为8.9%和1.4%; 磁场的峰值强度分别为5.0 T, 4.9 T, 2.1 T, 传统反演方法给出的磁场峰值强度相较于预设磁场相差58.0%, 而本文提出的方法给出的磁场峰值强度相较于预设磁场仅相差2.0%.
图 4 (a)反演磁场Brec在r-z平面的投影; (b)预设磁场Bset、反演磁场Brec及路径平均磁场Bavg的一维分布Fig. 4. (a) Projection of the inversion magnetic field Brec on the r-z plane; (b) the one-dimensional (1D) distributions of the preset magnetic field Bset, the inversion magnetic field Brec and the path average magnetic field Bavg.利用上述数值模拟, 定性及定量地从结构上和数值上证明了Abel逆变换反演方法能很好地重建磁场, 验证了该方案诊断待测量柱对称磁场的可行性.
4. 讨 论
在高能量密度物理磁场的质子照相研究中, 待诊断磁场可能具有柱对称的分布(例如电容线圈靶磁场[12]和等离子体喷流自生磁场[3]等)或局部具有柱对称结构. 本文介绍的反演方法可帮助对磁场的强度和结构进行更加精确的反演诊断. 以参考文献[3]中Li实验组有关等离子体喷流自生磁场的质子照相实验为例进行演示.
图5(a)引用自参考文献[3]中的图3(c), 是ns激光与CH靶相互作用中形成等离子体喷流的侧向质子照相结果, 其中质子探测器为RCF堆栈片, 视场大小为7 cm×7 cm, 探针质子束的能量为14.7 MeV, 实验中放大倍数M = 29. 理论和模拟结果表明, 该等离子体内喷流可以携带着冻结在其中的垂直于靶面的磁场Bz, 在x-y平面内近似具有柱对称结构. 对距离视场左边缘3.5—4 cm的质子照相通量密度扰动δn/n0的分析表明, 可以满足
\text{δ} n/{n_0} < 1 , 见图5(b). 同理于第3节中的方法, 在通过(6)式反演获得偏转速度后, 借助(5)式最终重建出磁场Brec. 当y = r\sin \theta 的夹角\theta = {{\text{π }} /2} 时, 磁场Brec在图5(a)中z = 3.75 cm处的一维分布见图5(c) (红线). 作为对比, 图5(c)还给出了由(3)式求得的路径平均磁场Bavg在z = 3.75 cm处的分布(蓝线).![图 5 (a)等离子体喷流的质子照相实验原图[3]; (b)局部的质子通量密度扰动; (c)反演磁场Brec和路径平均磁场Bavg的一维分布\r\nFig. 5. (a) Original proton radiographic image of the plasma jet; (b) the flux density perturbations of the protons at the local area; (c) the 1D distributions of the inversion magnetic field Brec and the path average magnetic field Bavg.](/fileWLXB/journal/article/wlxb/2022/24/PIC/24-20221848-5.png)
图 5 (a)等离子体喷流的质子照相实验原图[3]; (b)局部的质子通量密度扰动; (c)反演磁场Brec和路径平均磁场Bavg的一维分布Fig. 5. (a) Original proton radiographic image of the plasma jet; (b) the flux density perturbations of the protons at the local area; (c) the 1D distributions of the inversion magnetic field Brec and the path average magnetic field Bavg.从图5(c)可知, 传统方法与本文提出的方法给出的磁场峰值位置基本相同, 但前者给出的磁场最大强度为3.9 T, 半高全宽为144 μm. 而后者给出的磁场最大强度为7.7 T, 半高全宽则为136 μm. 可见, 传统方法会低估磁场, 本文所述方法给出的磁场峰值强度约为传统方法反演结果的1.9倍. 此外, 如果将重建得到的三维磁场沿着ex方向进行平均, 得到的磁场最大强度约3.7 T, 与传统方法给出的结果一致. 相比于传统方法, 本文中三维反演方法的最大优势是可以提供更细致的磁场强度空间分布.
通过对上述实验数据的处理, 直观验证了本方法实际应用于质子照相实验中的有效性, 基于Abel逆变换的反演方法可促进对等离子体喷流自生磁场的产生和电子热传导形成更加清晰的认识.
5. 结 论
基于对质子照相及其反演过程的理论分析, 本文提出了一种针对柱对称磁场三维结构的质子照相反演方法, 其核心是偏转速度和待诊断场之间存在的Abel变换关系. 通过数值模拟, 验证了该方法在重建磁场三维结构时的可行性. 此外, 本文还将该反演方法应用于等离子体喷流自生磁场的质子照相实验数据的分析处理中, 获得了磁场的空间结构, 其强度约为传统路径平均方法反演结果的1.9倍, 证明了该方法实验应用上的价值. 该方法提供了更多可待探究的磁场空间信息, 为质子照相反演方法及其应用提供了新的思路, 有助于加深对等离子体中磁场的认识.
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图 1 质子照相示意图
Figure 1. Schematic diagram of the proton radiography.
图 2 (a)预设磁场B在x = 50 μm平面上的分布; (b)探测面上的质子通量密度扰动
Figure 2. (a) Distributions of the preset magnetic field at x = 50 μm; (b) the flux density perturbations of the protons in the detection plane.
图 3 (a)质子偏转速度的反演结果; (b)质子的模拟偏转速度和反演偏转速度在z = 50 μm时的径向分布
Figure 3. (a) Reconstruction of the protons deflection velocities; (b) the radial distributions of the protons inversion deflection velocities and simulated deflection velocities at z = 50 μm.
图 4 (a)反演磁场Brec在r-z平面的投影; (b)预设磁场Bset、反演磁场Brec及路径平均磁场Bavg的一维分布
Figure 4. (a) Projection of the inversion magnetic field Brec on the r-z plane; (b) the one-dimensional (1D) distributions of the preset magnetic field Bset, the inversion magnetic field Brec and the path average magnetic field Bavg.
图 5 (a)等离子体喷流的质子照相实验原图[3]; (b)局部的质子通量密度扰动; (c)反演磁场Brec和路径平均磁场Bavg的一维分布
Figure 5. (a) Original proton radiographic image of the plasma jet; (b) the flux density perturbations of the protons at the local area; (c) the 1D distributions of the inversion magnetic field Brec and the path average magnetic field Bavg.
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