1. 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

2. Neutron generator concept:
Simple mirror geometry with long central solenoid
Injection of energetic neutrals
Neutron generator concept:
T ~ 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 ~ keV, Pn  Pinj

3. The power balance scheme
Local balance
Plasma amplification factor

4. 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 =
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

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

6. Synchrotron radiation losses1
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)

7. Fast particle kineticsb
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

8. 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

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

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

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

12. Solution:

13. 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

14. 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

15. 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

16. Thank you!