Demo

Motivation
3 L a!TZ1 Z1 1 exp
Vacuum Radiation
Systematic Derivation of Radiation Transport with Source
Intensity
Normalized intensity 1.00
0.95 0.90 0.85 0.80 0.75
15
10
5
0
Harmonic number
Increasing radius
Ll
upper boundary
Self-quenching of radiation at high frequencies
Summary
Synchrotron Radiation From Plasmas with Sub-Relativistic Temperatures
Emission Coefficient
A. Necas, S. Putvinski , D. Ryutov, P. Yushmanov and the Entire TAE Team
TAE Technologies, Inc., 19631 Pauling, Foothill Ranch, CA 92610 Applications
Solving radiation transport equation for the slab geometry
Specular vs. Diffuse
Reflection
ce2 x Q= ! d! dxR
x 1 (1 a) (1 exp( L))dx
! Synchrotron radiation has been studied by [1-6]
! Vacuum radiation ~ 20 MW/m3 is too high to sustain
1)
2)
power balance
New Model
A. Radiation from hot electrons > 150 keV B. Effect of high harmonics > 100
C. Effect of relativistic Maxwellian truncation
New physics
A. 100x suppression of vacuum radiation
B. Suppression of low harmonics based on spectral
function and opacity
C. Study of radiation passing through high B field region
D. Effects of specular vs. diffuse reflection
E. Estimate of re-population of high energy Maxwellian
tail <-> Relativistic Maxwellian truncation
Normalized FRC Equilibrium
Midplane equilibrium profiles
A simple “sandwich” model for FRC geometry
⇡2c2L 1 (1 a)exp L 0 0 x
 = (!) x = cos ✓
a=1-r
1
Z1 (1 exp( L ))x dx a!3 T x
Q= ce !2d! 0 ⇡ 2 c 2 L
R1 0
0
x
Frequency Spectrum
Power transmitted through upper boundary
a=1-r r - reflection coefficient
Opacity ζ/a
spectral function “g” of radiation coming out of plasma slab
Total Intensity of radiation per Harmonic [5]
N.B.: Landau [5] uses harmonic # n for wce with relativistic correction, our ⇠ fin(⇠), is normalized to non-relativistic cyclotron frequency.
600
40000
200000
800000
Reflection coefficient -- r
Radiation power is weakly sensitive on reflection model. Power for specular case is 0.97 MW/m3 and power for diffuse case = 0.87 MW/m3 for r=0.8. No Maxwellian truncation assumed
Low B )ield
High B )ield
" Benchmark as well as convergence study is performed for vacuum radiation
" Horizontal line represents total radiated power
" Total radiated power can be recovered with 40 harmonics for Te=50keV and >140 harmonics for Te=150 keV
Where is the opacity and is
absorption coefficient. f(⇠)/⇠2 has been studied earlier. Qvac is vacuum
radiated power. Fitting function has been used for theta integration with
estimated 15% error introduction. Fitting approximation is removed for
Te=50kev
Te=150kev
Radiation from Low B passes through high B – high-beta effects
Photons produced in the bulk (low-B )ield) plasma and partially re-absorbed in upper high-)ield layer have following distribution on the upper boundary
• ne=3e14 cm-3
• Bl=6 T – High B )ield
• BL=2T–LowB)ield
• Te=150 keV
• Normalized to low B )ield
• No )itting function used as has been done in slab model
• No re2lection model
• Vacuum radiation from low-B 2ield = 21 MW/m3
• Reabsorption in low-B 2ield results in 2 MW/m3 transmitted into
high-B 2ield – worst case (no tail truncation in low-B) Dashed -- Upper integration limit = 3.6 MeV
Solid -- Upper integration limit = 0.6 MeV
B=6T B=4T
Thickness “l” of the high )ield region [cm]
Thermal electron Te
”sandwich” model. a=1-r where “r” is reflection coefficient. For transparent ⇣=0
1. Radiation from high harmonic electrons dominate
2. Equilibrium of populating and depopulating high energy
Equating electron energy half-life to e-e Coulomb scattering
characteristic times, gives
Nominal parameters give or energy = 600 keV.
Relativistic tail
0 20 40 60 80 100 120 140
plasma , we recover Q=Qvac since . For highly opaque plasma, we obtain black-body loss:
• Power radiated per electron depends on temperature and harmonic
• Maxwellian distribution tail for Te=150 keV extends beyond 1MeV
• In conclusion: high electron harmonics for Te>100 keV are important
n=1
Harmonic number
n=20
Te [mec2]
Reabsorption of Vacuum Radiation
Kinetic Equation for Photons
0
100 200 300 400
Maxwellian truncation @ 3.6 MeV
Maxwellian truncation @ 1.1 MeV
Maxwellian truncation @ 0.6 MeV
Opacity[⇣/a]
Theory for synchrotron radiation by high temperature (sub- relativistic) electrons has been developed
A combination of:
1. High beta effects – absorption
2. Reabsorption
3. Wall reflection and
4. Depletion of high energy tails
results in more than 100 times reduction of vacuum synchrotron radiation which seems acceptable for power balance in pB11 plasmas. Future work should include more detail analysis accounting for plasma profiles and plasma geometry.
Nominal parameters
ne=3e14 cm-3 B=2 T Te=150 keV L=100 cm
f(⇠)/⇠2
“Dimensionless” coefficient Te [keV] Te=150kev
Te=50kev
Te=10kev
Tamor [1]
Emission coefficient
absorption coefficient
Assumptions and Notations
1) Thermal equilibrium – Kirchhoff’s relation
2) Black-body
3) Rayleigh-Jeans law
4) Index of refraction ~ 1
5) Go to “primed” frame with v||=0
⇠
References
[1] Tamor, S. "The absorption coefficient of a magnetized plasma for cyclotron radiation." Nuclear Fusion 18.2 (1978): 229.
[2] Rosenbluth, M. N. "Synchrotron radiation in Tokamaks." Nuclear Fusion 10.3 (1970): 340.
[3] Drummond, W. E., and M. N. Rosenbluth. "Cyclotron radiation from a hot
plasma." The Physics of Fluids 6.2 (1963): 276-283.
[4] Bornatici, M., et al. "Electron cyclotron emission and absorption in fusion
plasmas." Nuclear Fusion 23.9 (1983): 1153.
[5] Trubnikov, B.A., in Reviews of Plasma Physics (Leontovich, M.A., Ed.) Vol. 7, Consultants Bureau, New York (1979) 345.
[6]Landau, L.D. and Lifshitz, E.M., Classical Theory of Fields, New York: Pergamon Press, 1971. Eq. 74.9
Opacity
600 B=6T,L=5cm
40000 Nominal,r=0
200000 Nominal, r=0.8
800000 Nominal,r=0.95
Truncation
3.6 MeV
0.41
0.09
0.042
0.019
0.6 MeV
0.28
0.046
0.018
0.007
• “Dimensionless” coefficient calculated via emission compares well to method using conductivity tensor
• Te=10 keV exhibits distinct harmonics for low n and continuous spectrum for high n
• Te=150 keV (no Tamor data ) shows continuous spectrum
• Te=50keV continuous for n>2
• Comparison with previous work [1] for low
⇠ = !/!ce
Te is in good agreement
59th Annual Meeting of the APS Division of Plasma Physics October 23 - 27, 2017 • Milwaukee, Wisconsin
# Total losses and absorption is strongly proportional to plasma thickness. # Thin high field layer is strongly absorbing.
# Passing through 10 cm high-B region reduces to 150 kW/m3
# Assuming tail truncation in low-B and 80% reflection, further
reduction by (0.046/0.09)*(0.018/0.046) = 0.2 can be expected.
Ratio
Q/Qvac
keV/s/electron
Power W/m3