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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 $T_{\rm{e}} < T_{\rm{i}}$. 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.
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Keywords:
- energy relaxation /
- warm and hot dense matter /
- plasma screening /
- dynamical response of plasmas
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Google Scholar
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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
Figure 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
Figure 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)
Figure 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
Figure 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)).
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[1] Lindl J 1995 Phys. Plasmas 2 3933
Google 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 106005
Google Scholar
[4] Haines B 2024 Phys. Plasmas 31 050501
Google Scholar
[5] 赵英奎, 欧阳碧耀, 文武, 王敏 2015 物理学报 64 045205
Google Scholar
Zhao Y K, Ouyang B Y, Wen W, Wang M 2015 Acta Phys. Sin. 64 045205
Google 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 035204
Google 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 035204
Google Scholar
[7] Mahieu B, Jourdain N, Ta Phuoc K, et al. 2018 Nat. Commun. 9 3276
Google Scholar
[8] Fletcher L B, Vorberger J, Schumaker W, et al. 2022 Front. Phys. 10 838524
Google Scholar
[9] Chen W T, Witte C, Roberts J L 2017 Phys. Rev. E 96 013203
Google Scholar
[10] Sprenkle R T, Silvestri L G, Murillo M S, Bergeson S D 2022 Nat. Commun. 13 15
Google Scholar
[11] Vanthieghem A, Tsiolis V, Spitkovsky A, Todo Y, Sekiguchi K, Fiuza F 2024 Phys. Rev. Lett. 132 265201
Google 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 036418
Google Scholar
[15] Brown L S, Singleton R L 2009 Phys. Rev. E 79 066407
Google Scholar
[16] Hazak G, Zinamon Z, Rosenfeld Y, Dharma-wardana M W C 2001 Phys. Rev. E 64 066411
Google Scholar
[17] Daligault J, Dimonte G 2009 Phys. Rev. E 79 056403
Google Scholar
[18] Chapman D A, Vorberger J, Gericke D O 2013 Phys. Rev. E 88 013102
Google Scholar
[19] Scullard C R, Serna S, Benedict L X, Leland Ellison C, Graziani F R 2018 Phys. Rev. E 97 013205
Google Scholar
[20] Simoni J, Daligault J 2020 Phys. Rev. E 101 013205
Google Scholar
[21] Rightley S, Baalrud S D 2021 Phys. Rev. E 103 063206
Google Scholar
[22] Glosli J N, Graziani F R, More R M, et al. 2008 Phys. Rev. E 78 025401
Google Scholar
[23] Jeon B, Foster M, Colgan J, Csanak G, Kress J D, Collins L A, Gronbech-Jensen N 2008 Phys. Rev. E 78 036403
Google Scholar
[24] Murillo M S, Dharma-wardana M W C 2008 Phys. Rev. Lett. 100 205005
Google Scholar
[25] Benedict L X, Surh M P, Stanton L G, et al. 2017 Phys. Rev. E 95 043202
Google Scholar
[26] Ma Q, Dai J Y, Kang D D, Murillo M S, Hou Y, Zhao Z X, Yuan J M 2019 Phys. Rev. Lett. 122 015001
Google Scholar
[27] Nanbu K 1997 Phys. Rev. E 55 4642
Google Scholar
[28] Zhao Y J 2018 Phys. Plasmas 25 032707
Google Scholar
[29] Gericke D O 2005 J. Phys. Conf. Ser. 11 111
Google Scholar
[30] Hansen J P, McDonald I R 2006 Theory of Simple Liquids (New York: Academic Press) p294
[31] Arista N R, Brandt W 1984 Phys. Rev. A 29 1471
Google Scholar
[32] Kremp D, Schlanges, Kraft W D 2005 Quantum Statistics of Nonideal Plasmas (Berlin: Springer-Verlag Berlin Heidelberg) Chapter 4
[33] Chapman D A, Vorberger J, Fletcher L B, et al. 2015 Nat. Commun. 6 6839
Google Scholar
[34] Vorberger J, Gericke D O 2009 Phys. Plasma 16 082702
Google Scholar
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