A.Yu. Chirkov1), S.V. Ryzhkov1), P.A. Bagryansky2), A.V. Anikeev2) PLASMA KINETICS MODELS FOR FUSION SYSTEMS BASED ON THE AXIALLY-SYMMETRIC MIRROR DEVICES A.Yu. Chirkov1), S.V. Ryzhkov1), P.A. Bagryansky2), A.V. Anikeev2) 1) Bauman Moscow State Technical University, Moscow, Russia 2) Budker Institute of Nuclear Physics, Novosibirsk, Russia

Neutron generator concept: Simple mirror geometry with long central solenoid Injection of energetic neutrals Neutron generator concept: T ~ 10..20 keV, n ~ 1019 m–3, a ~ 1 m, L ~ 10 m, B ~ 1..2 T in center solenoid, ~ 20 T in mirrors, fast particle energy ~ 100..250 keV, Pn  Pinj

The power balance scheme Local balance Plasma amplification factor

Radiation losses Electron – ion bremsstrahlung mec2 = 511 keV Electron energy losses during slowing down on ions 1 10 100 103 104 105 1.1 1.2 1.3 1.4 1.5 1.6 2 3 Te, eV g ––––– fit – - – - Elwert - - - - - Gould CE = 0.5772... Pei – correction to the Born approximation – for Te ~ 1 keV [Gould] Integral Gaunt factor: Approximation taking into account Gaunt factor for low temperatures: Gaunt factors for low temperatures. Approximations of B: 1 – formula corresponds g  1 at Te  0; 2 – g  gElwert at Te  0; 3 – by Gould

Approximations of numerical results Electron – electron bremsstrahlung CF = (5/9)(44–32)  8 CE = 0.5772... Approximations of numerical results

Synchrotron radiation losses 1 Ps/Ps0 10–3 Te, keV 10–2 . –––– Trubnikov – – – Trubnikov + relativistic corr. - - - - Tamor, Te < 100 keV – - – - Tamor, Te = 100–1000 keV – - - – Kukushkin, et al. Emission in unity volume of the plasma: Losses from plasma volume (Trubnikov): Output factor: – Trubnikov Output factors at a = 2 m, Rw = 0.7, Bext = 7 T, 0 = 0.1 (upper curves) and 0 = 0.5 (down) – relativistic correction [Tamor] 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Trrel 0 2 3 4 1 – Te = Ti = 30 keV 2 – 50 keV 3 – 70 keV 4 – 90 keV a = 2 m, Rw = 0.7, Bext = 7 T Generalized Trubnikov’s formula for non-uniform plasma [Kukushkin et al., 2008]: Output factor vs 0 at a = 2 m, Rw = 0.7, Bext = 7 T, Te = Ti = 30 keV (1), 50 (2), 70 (3), and 90 keV (4)

Fast particle kinetics b Proton slow-down rate (a) and cross section (b) for interaction with electrons (- - - - -), deuterium ions (–––––) and helium-3 ions (– - – - –): 1, 2 – Coulomb collisions, 3 – nuclear elastic scattering D–T reaction and slow-down cross sections ratio for tritium ions in the deuterium plasma with Ti = Te = T

Optimal parameters: T  10 keV, Einj  100 keV, Pn  Pinj ~ 4 MW/m3 Some estimations High-energy approximation: MW/m3 m3/s keV keV keV Optimal parameters: T  10 keV, Einj  100 keV, Pn  Pinj ~ 4 MW/m3

The Fokker – Planck equation Boundary conditions: In the loss region Quasi isotropic velocity distribution function:

Numerical scheme Scales and dimensionless variables: Dimensionless equation (symbols “~” are not shown):

Finite difference equations: Numerical scheme Greed: Finite difference equations: Matrix form:

Solution:

Examples of numerical calculations Velocity distribution function of tritium ions and its contours at time moments after injection swich on t = 0.1s (а), 0.3s (b) и 10s (c). Deuterium density nD = 3.31019 м–3, energy of injected particles 250 keV, injection angle 455, injection power 2 MW/m3, Ti = Te = 20 keV,  = 10 keV, slow-down time s = 4.5 s, transversal loss time  = s

Relative pressure and density of alphas in D–T plasma (D:T = 1:1): Role of  particles in D–T fusion mirror systems 5 1 2 . 4 8 T, keV 3 n /n0 p /p0 5 1 2 . 6 4 8 3 T, keV  /s WL /W0 Relative pressure and density of alphas in D–T plasma (D:T = 1:1): –––––– isotropic plasma (no loss cone) – – – – mirror plasma with loss cone n0 = nD + nT = 2nD p0 = pD = pT Energy losses (WL) due to the scattering into the loss cone and corresponding energy loss time () of alphas in D–T mirror plasma W0 is total initial energy of alphas (3.5 MeV/particle) s is slowdown time

Parameters of mirror fusion systems: Neutron generator and reactors with D–T and D–3He fuels Parameter Neutron generator regimes Tandem mirror reactors Ver. # 1 Ver. # 2 Ver. # 3 Ver. # 4 D–T fuel D–3He fuel Plasma radius a, m 1 Plasma length L, m 10 44 Magnetic field of the central solenoid B0, T 1.5 2 3.3 5.4 Magnetic field in plugs (mirrors) Bm, T 11 14 14.8 Averaged  0.5 0.2 0.7 Deuterium density nD, 1020 m–3 0.26 0.22 0.21 0.415 0.82 1.35 Ion temperature Ti, keV 22 15 65 Electron temperature Te, keV 10.5 8.5 18 19 Ion electrostatic barrier , keV 16.5 33 60 260 Injection power Pinj, MW 74 55 – ECRH power PRH, MW Neutron power Pn, MW 24 30 43 59 Plasma amplification factor Qpl = Pfus/(Pinj + PRH) 0.38 0.9 1.34 Total neutron output N, 1018 neutrons/s 13 26.5 Neutron energy flux out of plasma Jn, MW/m2 0.4 0.04 Heat flux out of plasma JH, MW/m2 1.2 1.8 2.0 2.4 0.94

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