Fokker-Planck Equation for Superthermal Electron Energy Deposition

doi:10.19596/j.cnki.1001-246x.8614

Abstract

Abstract:

We study the transport and energy deposition of superthermal electrons in high-density plasma, starting from the basic physics of relativistic particle collisions, considering relativistic Coulomb collisions and the collective effect, combined with the background plasma electron return and temperature equation, developed a hybrid model of the electronic relativistic Fokker-Planck equation. The finite volume algorithm of the Fokker-Planck equation in the rectangular-momentum spherical coordinate system is constructed, and the numerical algorithm and numerical simulation program are verified by calculating the energy deposition and magnetic field generation process of the monoenergetic electron beam in the high-density plasma. For evaluating of the preheat effects of superthermal electrons during the implosion process, the energy deposition processes and energy distributions in the implosion target under the single-energy and bi-Maxwellian energy spectrum are calculated.

Key words: relativistic Fokker-Planck equation, electron beam, super-thermal electrons, energy deposition

Hua ZHANG, Mingqiang LI, Li PENG, Minqing HE, Sizhong WU, Cangtao ZHOU. Fokker-Planck Equation for Superthermal Electron Energy Deposition[J]. Chinese Journal of Computational Physics, 2023, 40(2): 222-231.

Figures/Tables 8

Fig.1 At t = 1.5 ps, (a) electron density, (b) plasma temperature and (c) magnetic field contours; At t = 3.0 ps, (d) electron density, (e) plasma temperature and (f) magnetic field contours

Fig.2 Energy vs. time

Fig.3 Penetration depth vs. electron beam energy

Fig.4 Radiation vs. time, the main pulse is in the interval of 18 ns~22 ns, with peak power t = 21.6 ns

Fig.5 Electron, ion temperature and plasma density at t = 21.6 ns

Fig.6 Energy deposition in energy ranging from 50 keV to 200 keV at (a) t = 19 ns and (b) t = 21.6 ns

Fig.7 Superthermal electron spectrum, where Thot = 18 keV for E < 150 keV

Fig.8 Superthermal electron preheating under different energy spectrum (a) Thot = 60 keV; (b) Thot = 80 keV; (c) Thot = 120 keV; (d) Thot = 150 keV

References 25

1	TABAK M , HAMMER J , GLINSKY M E , et al. Ignition and high gain with ultrapowerful lasers[J]. Physics of Plasmas, 1994, 1 (5): 1626- 1634. DOI
2	LINDL J D . Development of the indirect-drive approach to inertial confinement fusion and the target physics[J]. Physics of Plasmas, 1995, 2 (11): 3933- 4024. DOI
3	SENTOKU Y , KEMP A J . Numerical methods for particle simulations at extreme densities and temperatures: Weighted particles, relativistic collisions and reduced currents[J]. Journal of Computational Physics, 2008, 227 (14): 6846- 6861. DOI
4	DAVIES J R , BELL A R , HAINES M G , et al. Short-pulse high-intensity laser-generated fast electron transport into thick solid targets[J]. Physical Review E, 1997, 56 (6): 7193- 7203. DOI
5	XU H , ZHUO H B , YANG X H , et al. Hybrid particle-in-cell/fluid model for hot electron transport in dense plasmas[J]. Chinese Journal of Computational Physics, 2017, 34, 505- 525. DOI
6	BELL A R , ROBINSON A P L , SHERLOCK M , et al. Fast electron transport in laser-produced plasmas and the KALOS code for solution of the Vlasov-Fokker-Planck equation[J]. Plasma Physics & Controlled Fusion, 2006, 48 (3): R37- R57.
7	TZOUFRAS M , BELL A R , NORREYS P A , et al. A Vlasov-Fokker-Planck code for high energy density physics[J]. Journal of Computational Physics, 2011, 230 (17): 6475- 6494. DOI
8	THOMAS A G R , TZOUFRAS M , ROBINSON A P L , et al. A review of Vlasov-Fokker-Planck numerical modeling of inertial confinement fusion plasma[J]. Journal of Computational Physics, 2012, 231 (3): 1051- 1079. DOI
9	YOKOTA T , NAKAO Y , JOHZAKI T . Two-dimensional relativistic Fokker-Planck model for core plasma heating in fast ignition targets[J]. Physics of Plasmas, 2006, 13 (2): 22702. DOI
10	WU S Z , ZHOU C T , ZHU S P , et al. Relativistic kinetic model for energy deposition of intense laser-driven electrons in fast ignition scenario[J]. Physics of Plasmas, 2011, 18 (2): 022703. DOI
11	WU S Z , ZHANG H , ZHOU C T , et al. Kinetic model for energy deposition in fast ignition[J]. European Physical Journal EDP Sciences, 2013, 59 (1): 05021.
12	吴思忠, 张华, 周沧涛, 等. 快点火中相对论电子束能量沉积的动理学研究[J]. 强激光与粒子束, 2015, 27 (3): 77- 86.
13	BRAAMS B J , KARNEY C F F . Differential form of the collision integral for a relativistic plasma[J]. Physical Review Letters, 2005, 59 (16): 1817- 1820.
14	BRAAMS B J , KARNEY C F F . Conductivity of a relativistic plasma[J]. Physics of Fluids, 1989, 1B (7): 1355- 1368.
15	ATZENI S , SCHIAVI A , DAVIES J R . Stopping and scattering of relativistic electron beams in dense plasmas and requirements for fast ignition[J]. Plasma Physics & Controlled Fusion, 2008, 51 (1): 015016.
16	DEUTSCH C , FURUKAWA H , MIMA K , et al. The interaction physics of the fast ignitor concept[J]. Laser and Particle Beams, 1997, 256 (1): 2483- 2486.
17	SOLODOV A A , ANDERSON K S , BETTI R , et al. Integrated simulations of implosion, electron transport, and heating for direct-drive fast-ignition targets[J]. Physics of Plasmas, 2009, 16 (5): 056309. DOI
18	LI C K , PETRASSO R D . Stopping, straggling, and blooming of directed energetic electrons in hydrogenic and arbitrary-z plasmas[J]. Physical Review E, 2006, 73 (1): 016402. DOI
19	LI C K , PETRASSO R D . Energy deposition of MeV electrons in compressed targets of fast-ignition inertial confinement fusion[J]. Physics of Plasmas, 2006, 13 (5): 056314. DOI
20	MORE R . Pressure ionization, resonances, and the continuity of bound and free states[J]. Advances in Atomic and Molecular Physics, 1985, 21 (8): 305- 356.
21	CAO L H , CHEN M , HE X T , et al. Relative importance of mega electronvolt-electron energy deposition by collisions and field effects in fast ignition[J]. Physics of Plasmas, 2012, 19 (4): 044503. DOI
22	DAVIES J R . Electric and magnetic eld generation and target heating by laser generated fast electrons[J]. Physical Review E, 2003, 68 (5): 056404. DOI
23	WENG S M , SHENG Z M , HE M Q , et al. Plasma currents and electron distribution functions under a dc electric field of arbitrary strength[J]. Physical Review Letters, 2008, 100 (18): 185001.
24	ZHANG H , WU S Z , ZHOU C T , et al. Numerical method for relativistic Vlasov equation in cartesian-spherical coordinate system[J]. Chinese Journal of Computational Physics, 2017, 34 (5): 12.
25	LINDL J , LANDEN O , EDWARD J , et al. Review of the national ignition campaign 2009-2012[J]. Physics of Plasmas, 2014, 21 (2): 339- 566.

[1]	Taiwu HUANG, Ke JIANG, Ran LI, Cangtao ZHOU. Propagation of Intense Laser and Transport of Relativistic Electron Beam in Inhomogeneous Plasmas [J]. Chinese Journal of Computational Physics, 2023, 40(2): 147-158.
[2]	Shouguang CHENG, Yunqian YIN, Julian ZHONG, Kunyuan XU. Goos-Hanchen Shift of Electrons Based on Wigner Equation [J]. Chinese Journal of Computational Physics, 2021, 38(4): 431-440.
[3]	ZHANG Hua, WU Sizhong, ZHOU Cangtao, HE Minqing, CAI Hongbo, CAO Lihua, ZHU Shaoping, HE Xiantu. Numerical Method of Relativistic Fokker-Planck Equation for Energy Deposition of Fast Electrons [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2017, 34(5): 555-562.
[4]	QIU Youheng, YING Yangjun, WANG Min, CHEN Xingliang. Self-adaptive Method for Energy Deposition of Photons and Electrons [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2011, 28(3): 317-322.
[5]	WAN Jun-sheng, ZHANG Ying, ZHANG Li-xing, XIA Hai-hong, DING Da-zhao. Spallation Target of ADS Neutron Source [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2003, 20(5): 408-412.
[6]	PENG Chang-xian, LIN Peng, TANG Yu-zhi. Properties of Energy Deposition and Thermal Shock Wave in Material Radiated by Pulsed Electron Beam [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2003, 20(1): 51-58.
[7]	Lei Wei. ANALYSE THE FOCUS OF ELECTRON BEAM FROM PROBABILITY FUNCTION [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 1998, 15(2): 177-183.
[8]	Tan Zhenyu, He Yancai. MONTE CARLO METHOD AND ENERGY DISSIPATION DISTRIBUTION IN LOW-ENERGY ELECTRON BEAM LITHOGRAPHY [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 1997, 14(S1): 499-501.
[9]	Wang Min, Li Jing. SOME BASIC PROPERTIEW OF VIRCATOR GENERETING MICROWAVE [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 1996, 13(1): 38-42.
[10]	Shao Qiyun, Pan Zhengying, Huo Yukun. MONTE CARLO SIMULATION OF DEPTH DISTRIBUTION OF DISPLACED ATOMS IN AMORPHOUS TARGETS [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 1992, 9(S1): 551-554.
[11]	Wang Xiaosha, Chen Dongquan, Wang Zhengyan. RESEARCH ON CHEMICAL KINETICS AND CALCULATION OF ENERGY DEPOSION IN KrF^* EXCIMER SYSTEM PUMPED BY PROTON-BEAM [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 1992, 9(3): 263-266.