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动态响应和屏蔽效应对稠密等离子体中电子离子能量弛豫的影响

林成亮 何斌 吴勇 王建国

Meng Zeng-Ming, Huang Liang-Hui, Peng Peng, Chen Liang-Chao, Fan Hao, Wang Peng-Jun, Zhang Jing. Raman coupling in atomic Bose-Einstein condensed with phase-locked laser system. Acta Phys. Sin., 2015, 64(24): 243202. doi: 10.7498/aps.64.243202
Citation: Meng Zeng-Ming, Huang Liang-Hui, Peng Peng, Chen Liang-Chao, Fan Hao, Wang Peng-Jun, Zhang Jing. Raman coupling in atomic Bose-Einstein condensed with phase-locked laser system. Acta Phys. Sin., 2015, 64(24): 243202. doi: 10.7498/aps.64.243202

动态响应和屏蔽效应对稠密等离子体中电子离子能量弛豫的影响

林成亮, 何斌, 吴勇, 王建国
cstr: 32037.14.aps.74.20241588

Analysis of dynamic response and screening effects on electron-ion energy relaxation in dense plasma

LIN Chengliang, HE Bin, WU Yong, WANG Jianguo
cstr: 32037.14.aps.74.20241588
Article Text (iFLYTEK Translation)
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  • 非平衡稠密等离子体中电子离子能量弛豫对理解惯性约束聚变、实验室等离子体和天体物理中的非平衡演化以及宏观热力学和输运性质至关重要. 受密度及温度等环境效应的影响, 等离子体中多种物理效应之间的竞合作用共同主导电子离子能量弛豫过程. 本文从量子Lenard-Balescu动理学方程出发, 建立了考虑电子和离子集体激发及其耦合效应的能量弛豫模型, 并在此基础上采用电子离子解耦、静态极限和长波近似构建了不同的简化模型, 系统研究了静态屏蔽、动态屏蔽、电子和离子等离激元激发及其耦合等效应对电子离子能量弛豫的影响机制. 通过不同模型之间的对比, 发现电子离子集体激发之间的耦合效应以及中等波长和短波区间的屏蔽效应对温热稠密等离子体中电子离子能量弛豫有着显著的影响. 这一结论表明, 准确描述等离子体中的动态响应和屏蔽效应将制约着相关物理体系中非平衡演化建模的精确性和有效性.
    Accurate knowledge of electron-ion energy relaxation plays a vital role in non-equilibrium dense plasmas with widespread applications such as in inertial confinement fusion, in laboratory plasmas, and in astrophysics. We present a theoretical model for the energy transfer rate of electron-ion energy relaxation in dense plasmas, where the electron-ion coupled mode effect is taken into account. Based on the proposed model, other simplified models are also derived in the approximations of decoupling between electrons and ions, static limit, and long-wavelength limit. The influences of dynamic response and screening effects on electron-ion energy relaxation are analyzed in detail. Based on the models developed in the present work, the energy transfer rates are calculated under different plasma conditions and compared with each other. It is found that the behavior of electron screening in the random phase approximation is significantly different from the one in the long-wave approximation. This difference results in an important influence on the electron-ion energy relaxation and temperature equilibration in plasmas with temperature Te<Ti. The comparison of different models shows that the effects of dynamic response, such as dynamic screening and coupled-mode effect, have stronger influence on the electron-ion energy relaxation and temperature equilibration. In the case of strong degeneracy, the influence of dynamic response will result in an order of magnitude difference in the electron-ion energy transfer rate. In conclusion, it is crucial to properly consider the finite-wavelength screening of electrons and the coupling between electron and ion plasmonic excitations in order to determine the energy transfer rate of electron-ion energy relaxation in dense plasma.
      PACS:
      32.80.Qk(Coherent control of atomic interactions with photons)
      85.30.De(Semiconductor-device characterization, design, and modeling)
      85.90.+h(Other topics in electronic and magnetic devices and microelectronics)
      通信作者: 何斌, hebin-rc@163.com ; 吴勇, wu_yong@iapcm.ac.cn
    • 基金项目: 国家重点研发计划(批准号: 2022YFA1602500)和国家自然科学基金(批准号: 12474277, 12274039, U2430208)资助的课题.
      Corresponding author: HE Bin, hebin-rc@163.com ; WU Yong, wu_yong@iapcm.ac.cn
    • Funds: Project supported by the National Key Research and Development Program of China (Grant No. 2022YFA1602500) and the National Natural Science Foundation of China (Grant Nos. 12474277, 12274039, U2430208).

    Two-temperature nonequilibrium dense plasmas widely exist in inertial confinement fusion[1-6], laser-driven plasma[7,8], ultracold plasma[9,10] and celestial environment[11].In ICF simulations, the energy transfer rate, which describes the electron-ion temperature equilibrium, is one of the core input parameters for simulating the propagation of shock waves through the shell and the formation of a central hot spot.In a two-temperature nonequilibrium dense plasma, electrons and ions transfer energy through elastic and inelastic collision processes and coupling interactions between collective excitations, and relax toward a complete thermodynamic equilibrium state to reach the same thermodynamic temperature.Therefore, an accurate understanding of the energy transfer rate between different species of particles in a plasma is essential for determining the temperature, equilibrium time, thermodynamic and transport properties of a nonequilibrium system.

    The theory of relaxation between ions and electrons in a two-temperature nonequilibrium plasma was originally developed by Landau and Spitzer[12,13].In their theory, the Coulomb logarithmln(bmax/bmin) is introduced to describe the two-body collision process of electrons and ions, wherebmax andbmin characterize the long-range static screening effect and the strong collision or quantum diffraction effect in the short-range interaction, respectively.In recent years, different theoretical models[3,14-21] have been developed to obtain more reasonable and accurate parametersbmax andbmin, or to describe the electron-ion energy transport process in more detail.At the same time, thanks to the development of modern computers, numerical methods such as molecular dynamics simulation[22-26] and Monte Carlo simulation[27,28] have also been widely used in the study of electron-ion energy relaxation and temperature equilibrium in plasmas.However, due to the competition between the effects of partial electronic degeneracy and strong ionic coupling in warm and dense systems, it is necessary to further understand the influence of different physical effects in order to establish physical models with high fidelity and easy application.For the energy relaxation and temperature balance of electrons and ions in the dense two-temperature plasma, it is necessary to clarify: 1) how the static screening effect, the dynamic screening effect, the plasmon excitation of electrons and ions and their coupling, the electron degeneracy, and the strong coupling of ions affect the electron-ion energy relaxation process; 2) what or which physical effects dominate the electron-ion energy relaxation process under the warm and dense conditions. Therefore, it is necessary to further study and clarify the warm and dense two-temperature plasma.

    Based on the quantum Lenard-Balescu kinetic framework, the effects of dynamic response and screening on the electron-ion energy relaxation process in a two-temperature dense plasma are systematically studied.Section2 elaborates the energy relaxation model considering the collective excitation of electrons and ions and their coupling effects, and how to introduce different theoretical approximations to separate out different physical effects; Section3 shows the calculation results and discusses the physical mechanism of the influence of different physical effects on the energy relaxation of electrons and ions; Section4 makes a brief summary and outlook of the research work in this paper.

    The plasma studied in this paper is composed of an electron system with densityne and temperatureTe, and an ion system with valencezi, densityni and temperatureTi.For the convenience of subsequent expression, the Brueckner parameterrS and the degeneracy parameterθe of electrons are introduced here:

    rS=1aB(34πne)1/3,θe=2mekBTe2k2F, (1)

    WherekB is the Boltzmann constant,aB is the Bohr radius, and the Fermi wave vector is defined askF=(3π2ne)1/3.

    In this two-temperature two-component plasma, the equilibrium relaxation process between the electron system and the ion system is described by the energy transfer rate[3]:

    dEidt=Rie(TiTe), (2)

    WhereEi is the energy of the ion system. According to the total energy conservation of the plasma system, the energy evolution of the electron system is determined by the relationdEe/dt=dEi/dt.Rie is the electron-ion coupling parameter, which can be expressed as

    Rie=42πz2ie4nenikB(4πε0)2mime(kBTeme+kBTimi)3/2Lei, (3)

    Whereme andmi are the masses of the electron and the ion, respectively;ε0 is the vacuum dielectric coefficient; andLei is a factor describing the collision strength between the electron and the ion, commonly referred to as the Coulomb logarithm.Considering the static screening effect of electrons and the quantum statistical properties of electrons, the Coulomb logarithm[17,29] can be calculated by the following formula without introducing additional truncation parameters:

    Lei0dkkfe(k/2)(1+κ2e/k2)2, (4)

    Where fe(y)=[exp(y2/θeηe)+1]1 is the Fermi-Dirac distribution function of electrons; κe is the shielding coefficient of electrons, which is given by the formula

    κ2e=2kFθ1/2eπaB0dxx1/2exηe+1 (5)

    The calculation yields where ηe=μe/(kBTe) is the reduced chemical potential of the electron and is determined by

    23θ3/2e=0dxx1/2exηe+1. (6)

    在量子Lenard-Balescu动理学框架中, 电子体系和离子体系之间的动量(k)和能量(ω)的交换由等离子体的介电响应描述. 其中为约化普朗克常数, kω分别为波矢和频率. 在之前的研究中, 我们发现双组分等离子体中离子体系介电响应的贡献主要来源于离子声波所描述的集体响应模式[3]. 在此基础上通过引入离子介电响应的单级近似来描述离子声波响应模式, 可以得到下述广义库仑对数[3]:

    LIADei=L00dkkWIAD(k), (7)
    WIAD(k)=Nei(ωiad)Dee(k,ωiad)Cei(k,ωiad), (8)

    其中L0=3π4θ3/2e(1+meTimiTe)3/2. 此处ωiad表征等离子体中的离子声波的色散关系, 它与等离子体中离子的结构因子Sii(k)紧密地联系在一起, 可以根据关系式ω2iad=kBTik2/[miSii(k)]计算[30]. 函数Nei(ω)描述能量交换过程中电子和离子集体激发模式之间的差异, 见下式:

    Nei(ω)=ω[nB(ωkBTi)nB(ωkBTe)]kB(TiTe), (9)

    其中nB(x)=1/(ex1)为玻色函数. 函数Dee(k,ω)表征电子体系的介电响应:

    Dee(k,ω)=4πε03k32e2m2eωImε1ee(k,ω). (10)

    函数Cei(k,ω)描述电子体系和离子体系之间的等离激元多模耦合效应(电子和离子的集体激发之间的耦合效应)带来的影响, 其形式如下:

    Cei(k,ω)=Imεii(k,ω)Imεii(k,ω)+Imεee(k,ω). (11)

    本文中采用随机相位近似(random phase approximation)计算电子的介电函数εee(k,ω), 其虚部和实部可以分别表示为[31]

    Imεee(k,ω)=α0rSθe8q3ln{1+eηe(uq)2/θe1+eηe(u+q)2/θe}, (12)
    Reεee(k,ω)=1+α0rS4πq3[g(u+q)g(uq)], (13)

    其中q=k/(2kF),u=meω/(kFk). g(x)的表达式为

    g(x)=0dyyfe(y)ln|x+yxy|. (14)

    随机相位近似的介电函数能很好地描述电子的简并效应. 相较于电子, 离子由于质量比较大, 可以经典地处理[32]:

    Imεii(k,ω)=πκ2ik2ses2, (15)

    其中s=miω2/(2k2kBTi), κ2i=z2ie2ni/(ε0kBTi)为离子的屏蔽系数.

    在上述广义库仑对数的模型计算中, 还需要双组分等离子体中离子的结构因子Sii(k)来确定离子声波的色散频率ωiad. 为了在相同的近似层次下与其他数值结果[18]对比, 进而明晰不同物理效应的影响机制, 本文采用Debye-Hückel结构因子来近似描述稠密等离子体中离子间的关联, 如下式[32]:

    Sii(k)=k2+κ2ek2+κ2e+κ2i. (16)

    下面讨论广义库仑对数(7)的4种简化模型, 以便研究不同物理效应对能量交换过程的影响. 如果忽略电子体系和离子体系之间等离激元多模耦合效应, 即采用Cei(k,ωiad)=1, 可以得到如下式所示的约化库仑对数:

    LNCMei=L00dkkDee(k,ωiad)Nei(ωiad).  (17)

    因为(17)式中仍采用离子声波的色散频率ωiad, 这一库仑对数中部分包含了电子和离子之间的耦合. 如果要完全解耦电子体系和离子体系, 可以进一步地采用单组分等离子体的结构因子SOCPii(k)=k2/(k2+κ2i)来计算. 这一近似对应于采用等离激元极近似(plasmon-pole approximation)描述等离子体的介电响应[32], 其对应的库仑对数可以表达为

    LIPMei=L00dkkDee(k,ωipm)Nei(ωipm). (18)

    此处ω2ipm=kBTik2/[miSOCPii(k)]. 这一库仑对数可视为费米黄金定则近似下能量转移率的简化模型[3]. 在电子离子能量弛豫和温度平衡过程中, Hazak等[16]发现动态介电响应的主要贡献来源于低频范围, 因而在他们的理论计算中采用了静态极限(ω=0). 这一近似在后续的研究中也有广泛的讨论[17,18]. 在(17)式或(18)式中进一步采用静态极限近似, 则库仑对数可以约化为下述表达式:

    LSMei=L00dkkDee(k,0)=L00dkkfe(k/2)ε2ee(k,0). (19)

    在静态极限下, 电子介电函数εee(k,0)为实数, 可以根据表达式(13)计算. 静态极限近似下的电子介电函数包含了静态屏蔽效应. 这一事实可以通过进一步采用长波近似来获得, 即εee(k0,ω=0)=1+κ2e/k2. 根据长波近似下的介电函数可以得到描述静态屏蔽效应的Debye势能. 长波近似下, (19)式有如下形式:

    LSLMei=L00dkkfe(k/2)(1+κ2e/k2)2. (20)

    这一形式与(4)式是一致的. 费米-狄拉克分布函数fe(k/2)随着k的增大呈现出指数衰减, 为k的积分提供一个量级约2kF1+θe的数值截断, (20)式给出形如Born近似的库仑对数.

    在温热稠密条件下, 多种等离子体环境效应之间的竞合关系使得对电子离子能量交换过程的描述十分复杂. 下面将重点讨论动态响应和屏蔽效应对电子离子能量弛豫的影响.

    首先讨论(19)式和(20)式中随机相位近似下和长波近似下静态屏蔽效应行为的差异, 结果如图1所示. 在kaB1的区间, 两种近似下的介电响应特性出现较大差异. 与长波极限εee(k0,ω=0)=1+κ2e/k2相比, 随机相位近似下的介电函数变化得更快. 此外, 不同于长波近似, 不同简并参数下的随机相位近似介电函数在短波极限下(即k1时)接近相同的渐近值. 在短波极限下, 通过研究电子介电函数(13)的渐近行为发现: 1)在k1时, 电子介电函数(13)依据k4呈现出幂次衰减. 这种渐近行为在相关实验和理论分析中也报道过[33]; 2)短波极限下幂次衰减的系数只 依赖于电子密度, 而不依赖于电子温度. 因此, 在给定密度下, 不同简并参数下的介电函数的短波 渐近行为一致. 在我们之前的研究中发现, 量子Lenard-Balescu动理学框架下对电子离子动量交换的主要贡献来自中间波长区间[3]. 因此, 准确了解中间波长区间的屏蔽特性对于确定电子离子能量转移速率至关重要.

    图 1 在密度$ n_{\rm{e}} = 10^{22} \, {\rm{cm}}^{-3} $, 简并参数$ \theta_{\rm{e}} = 0.1, 1, 10 $时, 随机相位近似和长波近似下静态屏蔽效应的行为. 实线为随机相位近似的结果, 点虚线为长波近似下结果\r\nFig. 1. Electronic static screening in the long-wavelength limit (dot-dashed lines) versus the full static screening in random phase approximation (RPA) (solid lines) for the electron number density $ n_{\rm{e}} = 10^{22} \, {\rm{cm}}^{-3} $ at three different degeneracy parameters $ \theta_{\rm{e}} = 0.1, 1, 10 $.
    图 1  在密度ne=1022cm3, 简并参数θe=0.1,1,10时, 随机相位近似和长波近似下静态屏蔽效应的行为. 实线为随机相位近似的结果, 点虚线为长波近似下结果
    Fig. 1.  Electronic static screening in the long-wavelength limit (dot-dashed lines) versus the full static screening in random phase approximation (RPA) (solid lines) for the electron number density ne=1022cm3 at three different degeneracy parameters θe=0.1,1,10.

    (19)式和(20)式中采用了静态极限近似, 忽略了双温双组分等离子体中动态响应效应对能量转移率的影响. 如果电子和离子体系之间的能量交换相较于两个子体系的平均热动能都很小, 则可以在静态极限下计算能量交换过程中电子和离子集体激发模式占有数的差Nei(ω), 此时近似结果为Nei(ω0)=1. 然而, 如果条件meTi/(miTe)1Ti/Te<1不再成立时, 静态极限近似开始变得不合理. 图2给出了密度ne=1025cm3和离子温度Ti=104K的全电离氢等离子体中, Nei(ω)在不同电子温度下随频率变化的规律. 从图2可以看出, 当满足条件Ti/Te<1时, 在离子响应主要贡献的低频区域, 即ω, 静态极限 \mathcal{N}_{\rm{ei}}(\omega \rightarrow 0) = 1 被证明是一个相当好的近似. 随着电子温度的降低(对应于 \alpha_1 = T_{\rm{i}} / T_{\rm{e}} 的增长), 静态极限近似变得越来越不适用. 使用 \mathcal{N}_{\rm{ei}}(\omega) 的完整表达式(9)对于描述 T_{\rm{i}} > T_{\rm{e}} 的等离子体中的能量弛豫至关重要. 此外, 采用静态极限近似也会影响电子介电响应的行为, 会进一步加剧静态极限近似的不适用性.

    图 2 密度$ n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} $、离子温度$ T_{\rm{i}} = 10^4 \, {\rm{K}} $的全电离氢等离子体中, 不同离子电子温度比$ \alpha_1 = T_{\rm{i}} / T_{\rm{e}} $下电子和离子集体激发模式的差异$ \mathcal{N}_{\rm{ei}}(\omega) $随约化频率$ \tilde{\omega} = \hbar \omega / (k_{\rm{B}} T_{\rm{i}}) $的变化. 灰色竖线对应该等离子体条件下离子的等离子体频率\r\nFig. 2. Occupation number difference $ \mathcal{N}_{\rm{ei}}(\omega) $ for reduced frequency $ \tilde{\omega} = \hbar \omega / (k_{\rm{B}} T_{\rm{i}}) $ and different temperature ratio $ \alpha_1 = T_{\rm{i}} / T_{\rm{e}} $ in fully ionized hydrogen plasmas with number density $ n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} $ and ion temperature $ T_{\rm{i}} = $$10^4 \, {\rm{K}} $. The gray vertical line marks the reduced ionic plasma frequency.
    图 2  密度 n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} 、离子温度 T_{\rm{i}} = 10^4 \, {\rm{K}} 的全电离氢等离子体中, 不同离子电子温度比 \alpha_1 = T_{\rm{i}} / T_{\rm{e}} 下电子和离子集体激发模式的差异 \mathcal{N}_{\rm{ei}}(\omega) 随约化频率 \tilde{\omega} = \hbar \omega / (k_{\rm{B}} T_{\rm{i}}) 的变化. 灰色竖线对应该等离子体条件下离子的等离子体频率
    Fig. 2.  Occupation number difference \mathcal{N}_{\rm{ei}}(\omega) for reduced frequency \tilde{\omega} = \hbar \omega / (k_{\rm{B}} T_{\rm{i}}) and different temperature ratio \alpha_1 = T_{\rm{i}} / T_{\rm{e}} in fully ionized hydrogen plasmas with number density n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} and ion temperature T_{\rm{i}} = 10^4 \, {\rm{K}} . The gray vertical line marks the reduced ionic plasma frequency.

    下面进一步考察电子和离子的集体激发之间的耦合效应对能量交换过程的影响, 即是否考虑函数 \mathcal{C}_{\rm{ei}}(k, \omega) 带来的影响. 这一影响反映在(7)式和(17)式的差异中, 即是否包含(11)式. 在电子温度 T_{\rm{e}} = 10^7 \, {\rm{K}} 、电子数密度 n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} 的全电离氢等离子体中, 根据(11)式研究了两个不同离子温度下的电子离子等离激元多模耦合效应, 如图3所示. 从图3可以发现, 等离激元多模耦合效应会显著改变(7)式和(17)式低k端的积分行为, 从而影响能量交换过程. 此外, 可以发现离子电子温度的比值 T_{\rm{i}} / T_{\rm{e}} 越小, 等离激元多模耦合效应的影响越大. 在以往的讨论中认为, 当离子的温度满足条件 T_{\rm{i}} < 0.28 z_{\rm{i}} T_{\rm{eff, e}} [19,34], 等离激元多模耦合效应将变得重要, 这里 T_{\rm{eff, e}} 为考虑简并效应后电子的有效温度. 然而, 在我们的研究中发现, 这一条件并不严格成立, 当离子温度高于电子温度时, 等离激元多模耦合效应也会影响能量转移率. 这一问题仍需进一步深入研究.

    图 3 密度$ n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} $、电子温度$ T_{\rm{e}} = 10^7 \, {\rm{K}} $的全电离氢等离子体中, 不同离子温度下等离激元多模耦合效应$ \mathcal{C}_{\rm{ei}}(k) = \mathcal{C}_{\rm{ei}}(k, \omega_{\rm{iad}}) $(根据(11)式计算)\r\nFig. 3. Coupled mode effects determined from the function $ \mathcal{C}_{\rm{ei}}(k) = \mathcal{C}_{\rm{ei}}(k, \omega_{\rm{iad}}) $, i.e. Eq. (11), for two-temperature hydrogen plasmas with density $ n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} $ and electron temperature $ T_{\rm{e}} = 10^7 \, {\rm{K}} $.
    图 3  密度 n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} 、电子温度 T_{\rm{e}} = 10^7 \, {\rm{K}} 的全电离氢等离子体中, 不同离子温度下等离激元多模耦合效应 \mathcal{C}_{\rm{ei}}(k) = \mathcal{C}_{\rm{ei}}(k, \omega_{\rm{iad}}) (根据(11)式计算)
    Fig. 3.  Coupled mode effects determined from the function \mathcal{C}_{\rm{ei}}(k) = \mathcal{C}_{\rm{ei}}(k, \omega_{\rm{iad}}) , i.e. Eq. (11), for two-temperature hydrogen plasmas with density n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} and electron temperature T_{\rm{e}} = 10^7 \, {\rm{K}} .

    为了更直观地反映动态响应和屏蔽效应对能量弛豫过程的影响, 以及与其他理论计算结果的比较, 图4给出了密度 n_{\rm{i}} = 10^{25} \, {\rm{cm}}^{-3} 、离子温度 T_{\rm{i}} = 10^4 \, {\rm{K}} 的全电离氢等离子体中, 不同电子温度下能量转移率. 本文不讨论电子温度改变对电离平衡过程和平均电离度的影响. 等离激元多模耦合(CM)和费米黄金定则(FGR)近似下的结果为Chapman等[18]的计算结果. 在他们的计算中, 不采用任何近似, 直接数值计算量子Lenard-Balescu动理学方程和费米黄金定则下的动理学方程得到对应的能量转移率. 从图4可以看到, 等离激元多模耦合(CM)近似下的数值结果与基于库仑对数(7)式的结果在全温度区间呈现出优异的一致性, 而费米黄金定则(FGR)近似下的数值结果则与基于库仑对数(18)式的结果符合得很好. 这一方面验证了我们理论模型的正确性和有效性, 另一方面也凸显了等离激元多模耦合效应对能量转移率的影响, 即等离激元多模耦合效应会减缓电子和离子体系之间的能量交换过程.

    图 4 密度$ n_{\rm{i}} = 10^{25} \, {\rm{cm}}^{-3} $、离子温度$ T_{\rm{i}} = 10^4 \, {\rm{K}} $的全电离氢等离子体中, 不同电子温度下的能量转移率. CM (三角符号)和FGR (五角星)的数据引自文献[18]. IAD (蓝色实线)和IPM (红色实线)对应考虑((7)式)和不考虑((18)式)电子离子耦合的能量弛豫率. 绿色点虚线和棕色虚线给出考虑((20)式)和不考虑((19)式)长波近似的静态极限弛豫率. 紫色点线(NCM)给出不考虑等离激元多模耦合((17)式)的结果\r\nFig. 4. Numerical results for energy transfer rate in fully ionized hydrogen plasmas for $ n_{\rm{i}} = 10^{25} \, {\rm{cm}}^{-3}, T_{\rm{i}} = 10^5 \, {\rm{K}} $ with different electron temperatures. The CM (orange triangles) and FGR (green stars) results for energy transfer rate are taken from Ref. [18]. The solid blue and red lines represent the results evaluated with Eq. (7) and Eq. (18), respectively. The green dot-dashed line and brown dashed line display the results in static limit with (Eq. (20)) and without (Eq. (19)) long-wavelength approximation, respectively. Predictions marked by NCM (violet dotted curve) give the results calculated from the expression (Eq. (17)).
    图 4  密度 n_{\rm{i}} = 10^{25} \, {\rm{cm}}^{-3} 、离子温度 T_{\rm{i}} = 10^4 \, {\rm{K}} 的全电离氢等离子体中, 不同电子温度下的能量转移率. CM (三角符号)和FGR (五角星)的数据引自文献[18]. IAD (蓝色实线)和IPM (红色实线)对应考虑((7)式)和不考虑((18)式)电子离子耦合的能量弛豫率. 绿色点虚线和棕色虚线给出考虑((20)式)和不考虑((19)式)长波近似的静态极限弛豫率. 紫色点线(NCM)给出不考虑等离激元多模耦合((17)式)的结果
    Fig. 4.  Numerical results for energy transfer rate in fully ionized hydrogen plasmas for n_{\rm{i}} = 10^{25} \, {\rm{cm}}^{-3}, T_{\rm{i}} = 10^5 \, {\rm{K}} with different electron temperatures. The CM (orange triangles) and FGR (green stars) results for energy transfer rate are taken from Ref. [18]. The solid blue and red lines represent the results evaluated with Eq. (7) and Eq. (18), respectively. The green dot-dashed line and brown dashed line display the results in static limit with (Eq. (20)) and without (Eq. (19)) long-wavelength approximation, respectively. Predictions marked by NCM (violet dotted curve) give the results calculated from the expression (Eq. (17)).

    更进一步的对比可以发现, 当 T_{\rm{e}} > T_{\rm{i}} 时, 图4(a)中(17)—(20)式预测的结果虽然有细微的差异, 但整体上与不考虑等离激元多模耦合效应的FGR能量转移率相一致. 然而在 T_{\rm{e}} < T_{\rm{i}} 时, 上述不同近似下的结果有明显差异. 从图4(b)可以发现, 长波近似下(SLM)的能量转移率比更完善的静态屏蔽模型(SM)的结果略小. 随着电子温度的降低, 考虑了动态响应的模型(IAD, IPM, NCM)都趋近于某些常数. 这种趋势的物理原因可以直接解释如下. 在低温强简并时 (\theta_{\rm{e}} \ll 1) , 电子介电函数 \varepsilon_{\rm{ee}}(k, \omega) 对电子温度的依赖性极弱. 此外, 由于较小的温度比 T_{\rm{e}} / T_{\rm{i}} \ll 1 , 电子温度下电子和离子集体激发模式的差异 \mathcal{N}_{\rm{ei}}(\omega) 以离子为主导, 对电子温度的依赖性也很弱. 因此, 在量子Lenard-Balescu动力学框架内, 能量转移率在低温时趋于一个常数值. 上述分析表明, 动态响应和屏蔽效应对能量弛豫过程有显著的影响, 需要正确合理地纳入理论建模中.

    本文基于量子Lenard-Balescu动理学方程研究了动态响应和屏蔽效应对双温双组分稠密等离子体中电子离子能量弛豫的影响. 通过与已有文献中的数值计算结果[18]进行对比, 我们的理论模型IAD ((7)式)与不采用近似的Lenard-Balescu动理学方程给出的理论预测符合得很好, 而理论模型IPM ((18)式)的结果则与费米黄金定则框架下的数值结果相一致. 这表明本文所采用近似的合理性和理论模型的正确性. 在屏蔽效应方面, 发现相较于常用的长波近似屏蔽模型, 基于随机相位近似的电子屏蔽效应明显不同于长波近似下的等离子体屏蔽行为. 这种差异会明显影响 T_{\rm{e}} < T_{\rm{i}} 的等离子体体系中电子离子能量弛豫和温度平衡过程. 通过不同模型之间的对比, 发现动态屏蔽和电子离子等离激元多模耦合等动态响应效应对电子离子能量弛豫和温度平衡过程的影响更加显著. 是否考虑动态响应的影响, 在强简并时会对电子离子能量转移率带来量级的差异, 在非简并区间也会对能量转移率造成倍数的差异. 为了与其他理论计算对比, 本文的研究中采用弱耦合的离子结构因子模型来计算离子声波的色散关系, 这一处理中忽略了离子强耦合效应带来的影响. 后续将进一步开展离子强耦合效应对双温稠密等离子体中电子离子能量弛豫和温度平衡的影响研究.

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    其他类型引用(5)

  • 图 1  在密度 n_{\rm{e}} = 10^{22} \, {\rm{cm}}^{-3} , 简并参数 \theta_{\rm{e}} = 0.1, 1, 10 时, 随机相位近似和长波近似下静态屏蔽效应的行为. 实线为随机相位近似的结果, 点虚线为长波近似下结果

    Fig. 1.  Electronic static screening in the long-wavelength limit (dot-dashed lines) versus the full static screening in random phase approximation (RPA) (solid lines) for the electron number density n_{\rm{e}} = 10^{22} \, {\rm{cm}}^{-3} at three different degeneracy parameters \theta_{\rm{e}} = 0.1, 1, 10 .

    图 2  密度 n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} 、离子温度 T_{\rm{i}} = 10^4 \, {\rm{K}} 的全电离氢等离子体中, 不同离子电子温度比 \alpha_1 = T_{\rm{i}} / T_{\rm{e}} 下电子和离子集体激发模式的差异 \mathcal{N}_{\rm{ei}}(\omega) 随约化频率 \tilde{\omega} = \hbar \omega / (k_{\rm{B}} T_{\rm{i}}) 的变化. 灰色竖线对应该等离子体条件下离子的等离子体频率

    Fig. 2.  Occupation number difference \mathcal{N}_{\rm{ei}}(\omega) for reduced frequency \tilde{\omega} = \hbar \omega / (k_{\rm{B}} T_{\rm{i}}) and different temperature ratio \alpha_1 = T_{\rm{i}} / T_{\rm{e}} in fully ionized hydrogen plasmas with number density n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} and ion temperature T_{\rm{i}} = 10^4 \, {\rm{K}} . The gray vertical line marks the reduced ionic plasma frequency.

    图 3  密度 n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} 、电子温度 T_{\rm{e}} = 10^7 \, {\rm{K}} 的全电离氢等离子体中, 不同离子温度下等离激元多模耦合效应 \mathcal{C}_{\rm{ei}}(k) = \mathcal{C}_{\rm{ei}}(k, \omega_{\rm{iad}}) (根据(11)式计算)

    Fig. 3.  Coupled mode effects determined from the function \mathcal{C}_{\rm{ei}}(k) = \mathcal{C}_{\rm{ei}}(k, \omega_{\rm{iad}}) , i.e. Eq. (11), for two-temperature hydrogen plasmas with density n_{\rm{e}} = 10^{25} \, {\rm{cm}}^{-3} and electron temperature T_{\rm{e}} = 10^7 \, {\rm{K}} .

    图 4  密度 n_{\rm{i}} = 10^{25} \, {\rm{cm}}^{-3} 、离子温度 T_{\rm{i}} = 10^4 \, {\rm{K}} 的全电离氢等离子体中, 不同电子温度下的能量转移率. CM (三角符号)和FGR (五角星)的数据引自文献[18]. IAD (蓝色实线)和IPM (红色实线)对应考虑((7)式)和不考虑((18)式)电子离子耦合的能量弛豫率. 绿色点虚线和棕色虚线给出考虑((20)式)和不考虑((19)式)长波近似的静态极限弛豫率. 紫色点线(NCM)给出不考虑等离激元多模耦合((17)式)的结果

    Fig. 4.  Numerical results for energy transfer rate in fully ionized hydrogen plasmas for n_{\rm{i}} = 10^{25} \, {\rm{cm}}^{-3}, T_{\rm{i}} = 10^5 \, {\rm{K}} with different electron temperatures. The CM (orange triangles) and FGR (green stars) results for energy transfer rate are taken from Ref. [18]. The solid blue and red lines represent the results evaluated with Eq. (7) and Eq. (18), respectively. The green dot-dashed line and brown dashed line display the results in static limit with (Eq. (20)) and without (Eq. (19)) long-wavelength approximation, respectively. Predictions marked by NCM (violet dotted curve) give the results calculated from the expression (Eq. (17)).

  • [1]

    Lindl J 1995 Phys. Plasmas 2 3933Google Scholar

    [2]

    Drake R 2018 High-Energy-Density Physics: Foundation of Inertial Fusion and Experimental Astrophysics (Berlin: Springer International Publishing AG) p367

    [3]

    Lin C L, He B, Wu Y, Wang J G 2023 Nucl. Fusion 63 106005Google Scholar

    [4]

    Haines B 2024 Phys. Plasmas 31 050501Google Scholar

    [5]

    赵英奎, 欧阳碧耀, 文武, 王敏 2015 物理学报 64 045205Google Scholar

    Zhao Y K, Ouyang B Y, Wen W, Wang M 2015 Acta Phys. Sin. 64 045205Google Scholar

    [6]

    张恩浩, 蔡洪波, 杜报, 田建民, 张文帅, 康洞国, 朱少平 2020 物理学报 69 035204Google Scholar

    Zhang E H, Cai H B, Du B, Tian J M, Zhang W S, Kang D G, Zhu S P 2020 Acta Phys. Sin. 69 035204Google Scholar

    [7]

    Mahieu B, Jourdain N, Ta Phuoc K, et al. 2018 Nat. Commun. 9 3276Google Scholar

    [8]

    Fletcher L B, Vorberger J, Schumaker W, et al. 2022 Front. Phys. 10 838524Google Scholar

    [9]

    Chen W T, Witte C, Roberts J L 2017 Phys. Rev. E 96 013203Google Scholar

    [10]

    Sprenkle R T, Silvestri L G, Murillo M S, Bergeson S D 2022 Nat. Commun. 13 15Google Scholar

    [11]

    Vanthieghem A, Tsiolis V, Spitkovsky A, Todo Y, Sekiguchi K, Fiuza F 2024 Phys. Rev. Lett. 132 265201Google Scholar

    [12]

    Spitzer L 1962 Physics of Fully Ionized Gases (John Wiley & Sons Inc.

    [13]

    Landau L D 1965 Collected Papers of L.D. Landau (Pergamon Press) p163

    [14]

    Gericke D O, Murillo M S, Schlanges M 2002 Phys. Rev. E 65 036418Google Scholar

    [15]

    Brown L S, Singleton R L 2009 Phys. Rev. E 79 066407Google Scholar

    [16]

    Hazak G, Zinamon Z, Rosenfeld Y, Dharma-wardana M W C 2001 Phys. Rev. E 64 066411Google Scholar

    [17]

    Daligault J, Dimonte G 2009 Phys. Rev. E 79 056403Google Scholar

    [18]

    Chapman D A, Vorberger J, Gericke D O 2013 Phys. Rev. E 88 013102Google Scholar

    [19]

    Scullard C R, Serna S, Benedict L X, Leland Ellison C, Graziani F R 2018 Phys. Rev. E 97 013205Google Scholar

    [20]

    Simoni J, Daligault J 2020 Phys. Rev. E 101 013205Google Scholar

    [21]

    Rightley S, Baalrud S D 2021 Phys. Rev. E 103 063206Google Scholar

    [22]

    Glosli J N, Graziani F R, More R M, et al. 2008 Phys. Rev. E 78 025401Google Scholar

    [23]

    Jeon B, Foster M, Colgan J, Csanak G, Kress J D, Collins L A, Gronbech-Jensen N 2008 Phys. Rev. E 78 036403Google Scholar

    [24]

    Murillo M S, Dharma-wardana M W C 2008 Phys. Rev. Lett. 100 205005Google Scholar

    [25]

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出版历程
  • 收稿日期:  2024-11-01
  • 修回日期:  2024-12-04
  • 上网日期:  2024-12-27
  • 刊出日期:  2025-02-05

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