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The radial electric field Er in a magnetic confined machine, such as the compact fusion reactor (CFR), the field-reserved configuration (FRC), and the tokamak, plays an essential role in affecting the confinement properties of the high energy particles. The parallel velocities of the high energy particles will be accelerated or decelerated by applying a radial electric field, which could change the loss rate of the high energy particles in the magnetic confined machines. Unlike the fourth-order Runge-Kutta method RK4, the recently-developed Boris method can strictly preserve energy conservation of the high energy particles in the case without radial electric field. The orbit of high energy α particle in compact fusion reactor (CFR) is simulated by solving the equations of motion numerically with the Boris Algorithm. The effect of radial electric field on the orbit of the high energy α particle is investigated and the confinement of plasma in different radial electric fields in the CFR machine is studied in the present paper. By changing the strength of the radical electric field and the particles' radical locations in the middle plane of the CFR configuration, the confinement property of the high energy α particle is studied. The numerical results indicate that both the positive radial electric field and negative electric field can significantly affect the confinement of the high energy α particle. When the radial electric field is increased to a threshold, the high energy α particle could be confined in the central region of the CFR machine for a long enough time. The threshold of the radial electric field depends on the initial parameters of the confined particle. Systematic investigations of the radical electronic field effect will conduce to greatly improving the performance of the designed CFR machines.
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Keywords:
- compact fusion reactor /
- radial electric field /
- particle loss /
- high energy particle
[1] Dolan T J, Brotankova J, Cadwallader L C, Costley A E, Ivanov D P, Manheimer W, Merola M, Moir R W, Neumann M J, Parrish A, Waganer L M 2013 Magnetic Fusion Technology (New York: Springer) pp23–68
[2] Baylor L R, Combs S K, Foust C R, Jernigan T C, Meitner S J, Parks P B, Caughman J B, Fehling D T, Maruyama S, Qualls A L, Rasmussen D A, Thomas C E 2009 Nucl. Fusion 49 085013
Google Scholar
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Google Scholar
Sun X, Liu M, Xie J L, Yu Y, Lin M N, Zhang Q 2014 J. Univ. Sci. Technol. China 44 374
Google Scholar
[4] Bodin H A B, Newton A A 2011 Nucl. Fusion 20 1255
[5] Steinhauer L C 2011 Phys. Plasmas 18 070501
Google Scholar
[6] Tuszewski M, Smirnov A, Thompson M C, Korepanov S, Akhmetov T, Ivanov A, Voskoboynikov R, Schmitz L, Barnes D, Binderbauer M W, Brown R, Bui D Q, Clary R, Conroy K D, Deng B H, Dettrick S A, Douglass J D, Garate E, Glass F J, Gota H, Guo H Y, Gupta D, Gupta S, Kinley J S, Knapp K, Longman A, Hollins M, Li X L, Luo Y, Mendoza R, Mok Y, Necas A, Primavera S, Ruskov E, Schroeder J H, Sevier L, Sibley A, Song Y, Sun X, Trask E, Van Drie A D, Walters J K, Wyman M D, Team T A E 2012 Phys. Rev. Lett. 108 255008
Google Scholar
[7] Binderbauer M W, Tajima T, Steinhauer L C, Garate E, Tuszewski M, Schmitz L, Guo H Y, Smirnov A, Gota H, Barnes D, Deng B H, Thompson M C, Trask E, Yang X, Putvinski S, Rostoker N, Andow R, Aefsky S, Bolte N, Bui D Q, Ceccherini F, Clary R, Cheung A H, Conroy K D, Dettrick S A, Douglass J D, Feng P, Galeotti L, Giammanco F, Granstedt E, Gupta D, Gupta S, Ivanov A A, Kinley J S, Knapp K, Korepanov S, Hollins M, Magee R, Mendoza R, Mok Y, Necas A, Primavera S, Onofri M, Osin D, Rath N, Roche T, Romero J, Schroeder J H, Sevier L, Sibley A, Song Y, Van Drie A D, Walters J K, Waggoner W, Yushmanov P, Zhai K 2015 Phys. Plasmas 22 056110
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Google Scholar
[11] Cornish S, Gummersall D, Carr M, Khachan J 2014 Phys. Plasmas 21 092502
Google Scholar
[12] Miley G H, Murali S K 2014 Inertial Electrostatic Confinement (IEC) Fusion (New York: Springer) pp1-400
[13] Hoffman A L, Guo H Y, Miller K E, Milroy R D 2005 Nucl. Fusion 45 176
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[20] Van Oost G 2006 Fusion Sci. Technol. 49 327
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Google Scholar
[22] Gorman J G 1966 Phys. Fluids 9 2504
Google Scholar
[23] Itoh K, Itoh S I 1996 Plasma Phys. Controlled Fusion 38 1
Google Scholar
[24] Silva C, Figueiredo H, Cabral J A C, GonÁalves B, Nedzelsky I, Varandas C A F 2004 Plasma Phys. Controlled Fusion 46 163
Google Scholar
[25] Sun Y, Chen Z P, Zhu T Z, et al. 2014 Plasma Phys. Controlled Fusion 56 015001
Google Scholar
[26] Zhang Q, Shi P Y, Liu M, Lin M N, Sun X 2015 Fusion Sci. Technol. 68 50
Google Scholar
[27] 张杰, 罗家融, 王少杰 2006 物理学报 55 1077
Google Scholar
Zhang J, Luo J R, Wang S J 2006 Acta Phys. Sin. 55 1077
Google Scholar
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Google Scholar
Xu X L, Zhao X M, Wang Z T, Tang C J 2012 Acta Phys. Sin. 61 185201
Google Scholar
[29] 牟茂淋, 刘宇, 王中天, 陈少永, 唐昌建 2014 物理学报 63 165201
Google Scholar
Mou M L, Liu Y, Wang Z T, Chen S Y, Tang C J 2014 Acta Phys. Sin. 63 165201
Google Scholar
[30] 张良 2009 博士学位论文 (北京:清华大学)
Zhang L 2009 Ph. D. Dissertation (Beijing: Tsinghua University) (in Chinese)
[31] Qin H, Zhang S X, Xiao J Y, Liu J, Sun Y J, Tang W M 2013 Phys. Plasmas 20 084503
Google Scholar
[32] Delzanno G L, Camporeale E 2013 J. Comput. Phys. 253 259
Google Scholar
[33] Kuley A, Wang Z X, Lin Z, Wessel F 2013 Phys. Plasmas 20 102515
Google Scholar
[34] Wei X S, Xiao Y, Kuley A, Lin Z 2015 Phys. Plasmas 22 092502
Google Scholar
[35] Winkel M, Speck R, Ruprecht D 2015 J. Comput. Phys. 295 456
Google Scholar
[36] He Y, Sun Y, Liu J, Qin H 2015 J. Comput. Phys. 281 135
Google Scholar
[37] Freidberg J P 2007 Plasma Physics and Fusion Energy (Cambridge: Cambridge University Press) pp149–160
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图 1 Lockheed Martin紧凑型聚变反应磁镜装置 (a)构造; (b)等离子体分布
Figure 1. (a) The structure of CFR machine; (b) plasma distribution in CFR (https://en.wikipedia.org/wiki/Lockheed_Martin_Compact_Fusion_Reactor)
图 2 CFR装置中R-Z平面上的磁场位形
Figure 2. The magnetic flux of the R-Z plane in the CFR machine
图 3 Boris 算法和RK4算法模拟下第一个漂移周期和最后一个漂移周期的轨道(模拟时间1200 μs) (a) Boris 算法; (b) RK4算法
Figure 3. Orbits of the first and the last drift motion periods by the Boris algorithm and the RK4 algorithm (simulation time is 1200 μs): (a) The Boris algorithm; (b) the RK4 algorithm
图 4 Boris算法和RK4算法数相对能量误差随时间变化对比
Figure 4. Comparison of the relative energy errors between Boris method and RK4 method
图 5 (a) CFR粒子轨道; (b)粒子轨道在X-Y截面的投影; (c)粒子轨道在R-Z截面的投影
Figure 5. (a) The orbit of particles in CFR; (b) the projection of the particle orbital in the X-Y plane; (c) the projection of the particle orbital in the R-Z plane
图 6 CFR中α粒子损失轨道
Figure 6. The lost orbits of theαparticles in CFR
图 7 粒子约束时间随着正向电场强度的变化
Figure 7. The particle confinement time changes with the intensity of the positive electric field
图 8 初始位置
${r_0}/a = 0.3$ 处α粒子在不同正向径向电场作用下的运动轨迹Figure 8. The orbits of α particle under the different positive radial electric fields at
${r_0}/a = 0.3$ 
图 9 粒子约束时间随着负向径向电场强度的变化
Figure 9. The variations of particle confinement time with the intensity of the negative radial electric field
图 10 初始位置
${r_0}/a = 0.3$ 处α粒子在不同负向径向电场作用下的轨道Figure 10. The orbits of α particle under the different negative radial electric fields at
${r_0}/a = 0.3$ 
图 11 不同径向电场作用下的α粒子损失率
Figure 11. The loss rates of the α particles with different radial electric fields
表 1 CFR装置线圈参数
Table 1. Main parameters of coils in CFR.
参数 中心线圈 内部线圈 封装线圈1 封装线圈2 磁镜线圈 轴向位置Z/m 0 ± 0.4 ± 0.7 ± 1 ± 1.25 半径R/m 0.7 0.25 0.7 0.5 0.3 电流I/MA –4.3 7 –1 –1 –10 
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[1] Dolan T J, Brotankova J, Cadwallader L C, Costley A E, Ivanov D P, Manheimer W, Merola M, Moir R W, Neumann M J, Parrish A, Waganer L M 2013 Magnetic Fusion Technology (New York: Springer) pp23–68
[2] Baylor L R, Combs S K, Foust C R, Jernigan T C, Meitner S J, Parks P B, Caughman J B, Fehling D T, Maruyama S, Qualls A L, Rasmussen D A, Thomas C E 2009 Nucl. Fusion 49 085013
Google Scholar
[3] 孙玄, 刘明, 谢锦林, 余羿, 林木楠, 张情 2014 中国科学技术大学学报 44 374
Google Scholar
Sun X, Liu M, Xie J L, Yu Y, Lin M N, Zhang Q 2014 J. Univ. Sci. Technol. China 44 374
Google Scholar
[4] Bodin H A B, Newton A A 2011 Nucl. Fusion 20 1255
[5] Steinhauer L C 2011 Phys. Plasmas 18 070501
Google Scholar
[6] Tuszewski M, Smirnov A, Thompson M C, Korepanov S, Akhmetov T, Ivanov A, Voskoboynikov R, Schmitz L, Barnes D, Binderbauer M W, Brown R, Bui D Q, Clary R, Conroy K D, Deng B H, Dettrick S A, Douglass J D, Garate E, Glass F J, Gota H, Guo H Y, Gupta D, Gupta S, Kinley J S, Knapp K, Longman A, Hollins M, Li X L, Luo Y, Mendoza R, Mok Y, Necas A, Primavera S, Ruskov E, Schroeder J H, Sevier L, Sibley A, Song Y, Sun X, Trask E, Van Drie A D, Walters J K, Wyman M D, Team T A E 2012 Phys. Rev. Lett. 108 255008
Google Scholar
[7] Binderbauer M W, Tajima T, Steinhauer L C, Garate E, Tuszewski M, Schmitz L, Guo H Y, Smirnov A, Gota H, Barnes D, Deng B H, Thompson M C, Trask E, Yang X, Putvinski S, Rostoker N, Andow R, Aefsky S, Bolte N, Bui D Q, Ceccherini F, Clary R, Cheung A H, Conroy K D, Dettrick S A, Douglass J D, Feng P, Galeotti L, Giammanco F, Granstedt E, Gupta D, Gupta S, Ivanov A A, Kinley J S, Knapp K, Korepanov S, Hollins M, Magee R, Mendoza R, Mok Y, Necas A, Primavera S, Onofri M, Osin D, Rath N, Roche T, Romero J, Schroeder J H, Sevier L, Sibley A, Song Y, Van Drie A D, Walters J K, Waggoner W, Yushmanov P, Zhai K 2015 Phys. Plasmas 22 056110
Google Scholar
[8] Forsen H K 1988 J. Fusion Energy 7 269
[9] Park J, Krall N A, Sieck P E, Offermann D T, Skillicorn M, Sanchez A, Davis K, Alderson E, Lapenta G 2015 Phys. Rev. X 5 021024
[10] Carr M, Khachan J 2010 Phys. Plasmas 17 052510
Google Scholar
[11] Cornish S, Gummersall D, Carr M, Khachan J 2014 Phys. Plasmas 21 092502
Google Scholar
[12] Miley G H, Murali S K 2014 Inertial Electrostatic Confinement (IEC) Fusion (New York: Springer) pp1-400
[13] Hoffman A L, Guo H Y, Miller K E, Milroy R D 2005 Nucl. Fusion 45 176
Google Scholar
[14] McGuire T J 2014 US Patent 201414242999
[15] Lockheed Martin Compact Fusion Reactor Concept, Confinement Model and T4B Experiment (PDF). Lockheed Martin Corporation. 2016. Archived from the original (PDF) on December 25, 2017. Retrieved 25 December 2017(https://en.wikipedia.org/wiki/Lockheed_Martin_Compact_Fusion_Reactor)
[16] Zhu L M, Liu H F, Wang X Q 2016 Phys. Scr. 91 095604
Google Scholar
[17] Wagner F, Becker G, Behringer K, Campbell D, Eberhagen A, Engelhardt W, Fussmann G, Gehre O, Gernhardt J, Gierke G v, Haas G, Huang M, Karger F, Keilhacker M, Klüber O, Kornherr M, Lackner K, Lisitano G, Lister G G, Mayer H M, Meisel D, Müller E R, Murmann H, Niedermeyer H, Poschenrieder W, Rapp H, Röhr H, Schneider F, Siller G, Speth E, Stäbler A, Steuer K H, Venus G, Vollmer O, Yü Z 1982 Phys. Rev. Lett. 49 1408
Google Scholar
[18] Taylor R J, Brown M L, Fried B D, Grote H, Liberati J R, Morales G J, Pribyl P, Darrow D, Ono M 1989 Phys. Rev. Lett. 63 2365
Google Scholar
[19] Shaing K C, Crume Jr E C, 1989 Phys. Rev. Lett. 63 2369
Google Scholar
[20] Van Oost G 2006 Fusion Sci. Technol. 49 327
[21] Groebner R J, Burrell K H, Seraydarian 1990 Phys. Rev. Lett. 64 3015
Google Scholar
[22] Gorman J G 1966 Phys. Fluids 9 2504
Google Scholar
[23] Itoh K, Itoh S I 1996 Plasma Phys. Controlled Fusion 38 1
Google Scholar
[24] Silva C, Figueiredo H, Cabral J A C, GonÁalves B, Nedzelsky I, Varandas C A F 2004 Plasma Phys. Controlled Fusion 46 163
Google Scholar
[25] Sun Y, Chen Z P, Zhu T Z, et al. 2014 Plasma Phys. Controlled Fusion 56 015001
Google Scholar
[26] Zhang Q, Shi P Y, Liu M, Lin M N, Sun X 2015 Fusion Sci. Technol. 68 50
Google Scholar
[27] 张杰, 罗家融, 王少杰 2006 物理学报 55 1077
Google Scholar
Zhang J, Luo J R, Wang S J 2006 Acta Phys. Sin. 55 1077
Google Scholar
[28] 徐欣亮, 赵小明, 王中天, 唐昌建 2012 物理学报 61 185201
Google Scholar
Xu X L, Zhao X M, Wang Z T, Tang C J 2012 Acta Phys. Sin. 61 185201
Google Scholar
[29] 牟茂淋, 刘宇, 王中天, 陈少永, 唐昌建 2014 物理学报 63 165201
Google Scholar
Mou M L, Liu Y, Wang Z T, Chen S Y, Tang C J 2014 Acta Phys. Sin. 63 165201
Google Scholar
[30] 张良 2009 博士学位论文 (北京:清华大学)
Zhang L 2009 Ph. D. Dissertation (Beijing: Tsinghua University) (in Chinese)
[31] Qin H, Zhang S X, Xiao J Y, Liu J, Sun Y J, Tang W M 2013 Phys. Plasmas 20 084503
Google Scholar
[32] Delzanno G L, Camporeale E 2013 J. Comput. Phys. 253 259
Google Scholar
[33] Kuley A, Wang Z X, Lin Z, Wessel F 2013 Phys. Plasmas 20 102515
Google Scholar
[34] Wei X S, Xiao Y, Kuley A, Lin Z 2015 Phys. Plasmas 22 092502
Google Scholar
[35] Winkel M, Speck R, Ruprecht D 2015 J. Comput. Phys. 295 456
Google Scholar
[36] He Y, Sun Y, Liu J, Qin H 2015 J. Comput. Phys. 281 135
Google Scholar
[37] Freidberg J P 2007 Plasma Physics and Fusion Energy (Cambridge: Cambridge University Press) pp149–160
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