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Cherenkov Radiation energy loss of runaway electronsChang Liu, Dylan Brennan, Amitava BhattacharjeePrinceton University, PPPLAllen BoozerColumbia University
1
Runaway Electron Meeting 2016Pertuis, France
Jun 7 2016
Outline• Introduction to Cherenkov radiation in plasma
• Lenard-Balescu collision operator
• Cherenkov radiation in unmagnetized plasma• Radiative energy loss• Correction to Coulomb logarithm
• Cherenkov radiation in magnetized plasma• Transit-time magnetic pumping (TTMP) momentum loss
• Summary
2
Outline• Introduction to Cherenkov radiation in plasma
• Lenard-Balescu collision operator
• Cherenkov radiation in unmagnetized plasma• Radiative energy loss• Correction to Coulomb logarithm
• Cherenkov radiation in magnetized plasma• Transit-time magnetic pumping (TTMP) momentum loss
• Summary
3
• Cherenkov radiation is emitted when a charged particle passed through a dielectric medium at a speed greater than wave phase velocity in that medium.
• The condition for Cherenkov radiation is n>1/β (n=kc/ω, β=v/c).• For plasma waves and runaway electrons (β~1), this condition is easy to
satisfy.
Introduction: Cherenkov radiation
4
Introduction: Lenard-Balescu collision operator and Cherenkov radiation in plasmas
• The polarization drag originates from the polarization of the plasma medium due to the electric field of a test particle, which is related to the Cherenkov radiation energy loss.• For ω/k~vth, the excited mode is strongly Landau damped and the energy
is absorbed by bulk electrons.• For ω/k≫vth, the mode is weakly damped (mainly through collisions), and
energy gets radiated away.
5
C[ f ]= ∂
∂v⋅ q
m⎛⎝⎜
⎞⎠⎟ E
p f −D ⋅ ∂ f∂v
⎡⎣⎢
⎤⎦⎥
Ep: Polarization drag electric fieldD: Diffusion from electric field fluctuation
D = 1
2qm
⎛⎝⎜
⎞⎠⎟2
dk δEδE (k,ω = k ⋅v)∫
Test particle superposition principle
Outline• Introduction to Cherenkov radiation in plasma
• Lenard-Balescu collision operator
• Cherenkov radiation in unmagnetized plasma• Radiative energy loss• Correction to Coulomb logarithm
• Cherenkov radiation in magnetized plasma• Transit-time magnetic pumping (TTMP) momentum loss
• Summary
6
Electron waves in unmagnetized plasma
• In unmagnetized plasma, the electron waves have two branches (ignoring ion effects and thermal correction)
•
•
• For Cherenkov radiation, only Langmuir wave is possible (n>1).
7
Langmuir wave: ω 2 =ω p2
Electronmagnetic wave: ω 2 = k2c2 +ω p2
0.1 0.5 1 5 100
1
2
3
4
ω/ωpen
The energy loss from Cherenkov radiation
• To solve the energy loss due to Cherenkov radiation, one can calculate the excited electric field from a single moving electron in the medium.
• The current from a single moving electron
• The time-averaged electric field felt by the moving electron
• The energy loss is then Ep·j.
8
Ampere's lawFaraday's law
⎫⎬⎪
⎭⎪→ k × k ×E+ ω
c⎛⎝⎜
⎞⎠⎟
2
ε ⋅E = − 4πiωc2
j
j(x,t) = −evδ (x − x0 − vt)→ j(k,ω ) = −evδ (ω − k ⋅v)exp(ik ⋅x0 )
Ep = d 3k dω E(k,ω )exp −ik(x0 + vt)+ iωt[ ]∫∫ t
Cherenkov resonance
Cherenkov radiation energy loss in unmagnetized plasma• For unmagnetized plasma
• Assume electron is moving along z direction,
• The principal value of the integral is imaginary, which will not contribute to the energy loss. • However, the residues (which correspond to the modes that satisfy the
Cherenkov radiation condition) will give real contribution and energy loss.
9
ε = 1−
ω p2
ω 2⎛
⎝⎜⎞
⎠⎟1 = ε1
Ezp = d 3kdω
4πi evc2 ε −k!2c2
ω 2⎛⎝⎜
⎞⎠⎟
ωε ε − k2c2
ω 2⎛⎝⎜
⎞⎠⎟
∫ δ (ω − k ⋅v), k! =k ⋅vv
Correction to Coulomb logarithm due to Cherenkov radiation loss• Using the residue theorem to calculate the integral
• Note that we use the cold plasma approximation, so we can choose kmax=1/λD to ensure thermal effect is not important.
10
Ezp =
eω p2
v2sinθdθcosθ
σ (θ ), σ (θ ) =1, if cosθ >ω p / kmaxv
0, if cosθ >ω p / kmaxv
⎧⎨⎪
⎩⎪∫
=eω p
2
v2ln kmaxv
ω pθ Is the wave emittance angle
Correction to lnΛ for the drag force
• Recall that the drag force in the Landau collision operator due to binary collisions can be written as
• Cherenkov radiation energy loss gives a correction to lnΛ
• In DIII-D QRE experiments, for highly relativistic runaway electrons, this correction is about 20%.
• For electron starting to run away (v≪c), this correction is small.
11
Edrag =eω p
2
v2lnΛ, Λ = bmax
bmin
bmax = λD , bmin = maxeαeβmαβu
2 ,!
2mαβu⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
lnΛ→ lnΛ + ln vvth
⎛⎝⎜
⎞⎠⎟
Outline• Introduction to Cherenkov radiation in plasma
• Lenard-Balescu collision operator
• Cherenkov radiation in unmagnetized plasma• Radiative energy loss• Correction to Coulomb logarithm
• Cherenkov radiation in magnetized plasma• Transit-time magnetic pumping (TTMP) momentum loss
• Summary
12
Cherenkov radiation in magnetized plasma
• The dielectric tensor and the dispersion relation in magnetized plasma (ignoring ions and thermal effects)
• The allowed wave branches for Cherenkov radiation are: Langmuir wave, extraordinary-electron (EXEL) wave, and upper-hybrid wave (electrostatic Bernstein wave).
13
0.1 0.5 1 5 100
1
2
3
4
5
6
7
ω/ωpen
ωce
θ=5°
n=1/cosθ
ε =
1−ω pe
2
ω 2 −ω ce2 −i
ω ceω
ω pe2
ω 2 −ω ce2 0
iω ceω
ω pe2
ω 2 −ω ce2 1−
ω pe2
ω 2 −ω ce2 0
0 0 1−ω pe
2
ω 2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
G.I. Pokol, A. Kómár, A. Budai, A. Stahl, and T. Fülöp, Phys. Plasmas 21, 102503 (2014).
0.1 0.5 1 5 100
1
2
3
4
5
6
7
ω/ωpen
Cherenkov radiation in magnetized plasma
• The dielectric tensor and the dispersion relation in magnetized plasma (ignoring ions and thermal effects)
• The allowed wave branches for Cherenkov radiation are: Langmuir wave, extraordinary-electron (EXEL) wave, and upper-hybrid wave (electrostatic Bernstein wave).
14
ωUH
θ=80°
n=1/cosθ
ε =
1−ω pe
2
ω 2 −ω ce2 −i
ω ceω
ω pe2
ω 2 −ω ce2 0
iω ceω
ω pe2
ω 2 −ω ce2 1−
ω pe2
ω 2 −ω ce2 0
0 0 1−ω pe
2
ω 2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
G.I. Pokol, A. Kómár, A. Budai, A. Stahl, and T. Fülöp, Phys. Plasmas 21, 102503 (2014).
Wave frequency for Cherenkov radiation
• For a given emittance angle of k, one can calculate the wave frequency that satisfies the Cherenkov radiation condition.
• For ωce~ωpe, one emittance angle gives one solution of ω, which shifts from ωpe to ωUH.
15
0.2 0.4 0.6 0.8 1.0 1.2 1.4
1.0
1.5
2.0
2.5
3.0
3.5
θ
ω/ω
pe
ω1=ωpe
ω2=ωUH
ωce=3ωpe
0.1 0.5 1 5 100
1
2
3
4
ω/ωpe
n
θ=0.5
n=1/cosθ
• For a given emittance angle of k, one can calculate the wave frequency that satisfies the Cherenkov radiation condition.
• For ωce≫ωpe, there are 3 branches of ω roots, which all coexist in a range of emittance angle.
Wave frequency for Cherenkov radiation (cont’d)
16
0.2 0.4 0.6 0.8 1.0
2
4
6
8
10
θ
ω/ω
pe
0.1 0.5 1 5 100
1
2
3
4
ω/ωpe
nω1=ωpe
ω2=ωUH
ωce=10ωpe
θ=0.2
n=1/cosθ
• For a given emittance angle of k, one can calculate the wave frequency that satisfies the Cherenkov radiation condition.
• For ωce≫ωpe, there are 3 branches of ω roots, which all coexist in a range of emittance angle.
0.5 1 5 10
0.96
0.98
1.00
1.02
1.04
1.06
1.08
1.10
ω/ωpe
n
Wave frequency for Cherenkov radiation (cont’d)
17
0.2 0.4 0.6 0.8 1.0
2
4
6
8
10
θ
ω/ω
pe
ω1=ωpe
ω2=ωUH
ωce=10ωpe
θ=0.2
n=1/cosθ
Cherenkov radiation energy loss in magnetized plasma• We can use the same method to calculate the Cherenkov radiation energy
loss.• Assume that electron is moving perfectly along B field (no v⊥)
• Although the radiation loss from small emittance angle (Langmuir) and large angle (Upper-hybrid) stays the same, intermediate angle (EXEL wave) gives additional energy loss.
18
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40
5
10
15
20
θ
A(θ)
ωce/ωpe=0.2
ωce/ωpe=3
ωce/ωpe=10 Ezp =
eω pe2
v2sinθdθcosθ
A(θ )∫
Effects of 3 modes on Cerenkov radiation energy loss
• All the 3 modes contributed to the Cherenkov radiation energy loss• For ωce≫ωpe, the radiation emittance is strongly peaked at certain
angle.
19
Total
ωce/ωpe=10
Cherenkov radiation energy loss in magnetized plasma (cont’d)• The Coulomb logarithm in the drag force has a correction
20
Δ2 =x(a1x + a0 )(x2 + b1x + b0 )
, x = ω ceω pe
a1 = 2.05,a0 = 9.51b1 = 1.63,b0 = 26.23
0 2 4 6 8 100.0
0.5
1.0
1.5
2.0
ωce/ωpe
Δ 2
lnΛ→ lnΛ + ln vvth
⎛⎝⎜
⎞⎠⎟+ Δ2
Transit-time magnetic pumping (TTMP) associated with Cherenkov radiation• Now we consider runaway electrons with finite v⊥.• In addition to electric field, gradient of magnetic field can also cause
momentum loss in parallel direction
• Note that for electron moving near the speed of light, the effect from the electric force (E) and Lorentz force (v×B/c) can be of similar order.
21
Magnetic pumping: FTTMP = −µ∇Bz
TTMP momentum loss• From Faraday’s law, Bz=kx cEy/ω. So the TTMP force can be calculated similarly from Ey.
• The ratio of TTMP over polarization electric force
• For high energy regime where synchrotron radiation energy loss dominates, v⊥/c~1/γ, the effect of TTMP is small.• For high energy runaway electrons, synchrotron and Bremsstrahlung radiation dominate
• With fast pitch angle scattering mechanism (different from collisional scattering) such as whistler wave, the effect can be more important.
22
FTTMP =µω cec2ω pe
ω pe2
v2ln v
vth
⎛⎝⎜
⎞⎠⎟− 121− vth
v⎛⎝⎜
⎞⎠⎟ + Δ3
⎡
⎣⎢
⎤
⎦⎥
0 2 4 6 8 100.0
0.5
1.0
1.5
2.0
ωce/ωpe
Δ 3
FTTMPeE p
≈ γ v⊥c
⎛⎝⎜
⎞⎠⎟2
Outline• Introduction to Cherenkov radiation in plasma
• Lenard-Balescu collision operator
• Cherenkov radiation in unmagnetized plasma• Radiative energy loss• Correction to Coulomb logarithm
• Cherenkov radiation in magnetized plasma• Transit-time magnetic pumping (TTMP) momentum loss
• Summary
23
Summary• Cherenkov radiation causes runaway electrons to lose energy, which can be
described by adding a correction to the Coulomb logarithm in the drag force.• The correction is about 20% compared to collisional force in DIII-D QRE
experiment
• Magnetic field enhances Cherenkov radiation and energy loss.• For runaway electrons with finite v⊥, TTMP will cause momentum loss on
parallel direction.
• Future work:• Including thermal effect in the plasma wave description• Study the spiral orbit particle rather than straight orbit• Study the electric field fluctuation given by Cherenkov radiation
24
Questions for discussion1. Can we detect the Cherenkov radiation from runaway electrons?
2. What is the correct physics interpretation of Cherenkov resonance condition when collisional damping is considered?
25
ω = k ⋅vexp(ikx − iωt)
Imω < 0,Im k < 0 Wave is damping in time, but growing in space?Imω > 0,Im k > 0 Wave is damping in space, but growing in time?
⎧⎨⎪
⎩⎪
B. J. Tobias et al., Rev. Sci. Instrum. 83, 10E329 (2012)