1
Cherenkov Radiation energy loss of runaway
electrons
Chang Liu, Dylan Brennan, Amitava Bhattacharjee
Princeton University, PPPL
Allen Boozer
Columbia University
Runaway Electron Meeting 2016
Pertuis, France
Jun 7 2016
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
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
4
Introduction: Cherenkov radiation
•  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.
5
Introduction: Lenard-Balescu collision operator and
Cherenkov radiation in plasmas
C[ f ] =
∂ ⎡⎛ q ⎞ p
∂f ⎤
⋅ ⎢⎜ ⎟ E f − D ⋅ ⎥
∂v ⎣⎝ m ⎠
∂v ⎦
Ep: Polarization drag electric field
D: Diffusion from electric field fluctuation
1⎛ q ⎞
D= ⎜ ⎟
2 ⎝ m⎠
2
Test particle
superposition principle
∫ dk δ Eδ E (k,ω = k ⋅ v)
•  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.
6
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
7
Electron waves in unmagnetized plasma
•  In unmagnetized plasma, the electron waves have two branches
(ignoring ion effects and thermal correction)
4
Langmuir wave: ω 2 = ω 2p
n
• 
3
• 
Electronmagnetic wave: ω 2 = k 2 c 2 + ω 2p
2
1
0
0.1
0.5
1
ω/ωpe
•  For Cherenkov radiation, only Langmuir wave is possible
(n>1).
5
10
8
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.
2
Ampere's law ⎫
4π iω
⎪
⎛ω ⎞
→
k
×
k
×
E
+
ε
⋅
E
=
−
j
⎬
⎜⎝ ⎟⎠
2
Faraday's law ⎪
c
c
⎭
•  The current from a single moving electron
Cherenkov resonance
j(x,t) = −evδ (x − x 0 − vt) → j(k, ω ) = −evδ (ω − k ⋅ v)exp(ik ⋅ x 0 )
•  The time-averaged electric field felt by the moving electron
Ep =
3
d
∫ k ∫ dω E(k,ω )exp [ −ik(x 0 + vt) + iω t ]
•  The energy loss is then Ep·j.
t
9
Cherenkov radiation energy loss in unmagnetized
plasma
•  For unmagnetized plasma
⎛ ω 2p ⎞
ε = ⎜ 1− 2 ⎟ 1 = ε1
⎝ ω ⎠
•  Assume electron is moving along z direction,
⎛
k!2 c 2 ⎞
4π i ev c ⎜ ε − 2 ⎟
ω ⎠
⎝
2
Ezp = ∫ d 3k dω
⎛
k 2c2 ⎞
ωε ⎜ ε − 2 ⎟
ω ⎠
⎝
δ (ω − k ⋅ v), k! =
k⋅v
v
•  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.
10
Correction to Coulomb logarithm due to Cherenkov
radiation loss
•  Using the residue theorem to calculate the integral
2
e
ω
sin θ dθ
Ezp = 2 p ∫
σ (θ ),
v
cosθ
eω 2p kmax v
= 2 ln
v
ωp
⎧ 1, if cosθ > ω p / kmax v
⎪
σ (θ ) = ⎨
0, if cosθ > ω p / kmax v
⎪
⎩
θ Is the wave emittance angle
•  Note that we use the cold plasma approximation, so we can choose
kmax=1/λD to ensure thermal effect is not important.
11
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
Edrag
eω 2p
b
= 2 ln Λ, Λ = max
v
bmin
⎧
! ⎫
⎪ e e
⎪
bmax = λ D , bmin = max ⎨ α β 2 ,
⎬
m
u
2m
u
⎪
αβ ⎪
⎩ αβ
⎭
•  Cherenkov radiation energy loss gives a correction to lnΛ
⎛ v⎞
ln Λ → ln Λ + ln ⎜ ⎟
⎝ vth ⎠
•  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.
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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
13
Cherenkov radiation in magnetized plasma
•  The dielectric tensor and the dispersion relation in magnetized plasma
(ignoring ions and thermal effects)
θ=5°
7
2
ω ce ω pe
−i
ω ω 2 − ω ce 2
0
ω 2pe
1− 2
ω − ω ce 2
0
0
ω 2pe
1− 2
ω
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟⎠
6
5
4
n
⎛
ω 2pe
⎜ 1− 2
ω − ω ce 2
⎜
⎜ ω
ω 2pe
ce
ε =⎜ i
2
2
⎜ ω ω − ω ce
⎜
⎜
0
⎜⎝
3
2
n=1/cosθ
1
0
0.1
0.5
1
ωce 5
ω/ωpe
•  The allowed wave branches for Cherenkov radiation are: Langmuir wave,
extraordinary-electron (EXEL) wave, and upper-hybrid wave (electrostatic
Bernstein wave).
G.I. Pokol, A. Kómár, A. Budai, A. Stahl, and T. Fülöp, Phys. Plasmas 21, 102503 (2014).
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Cherenkov radiation in magnetized plasma
•  The dielectric tensor and the dispersion relation in magnetized plasma
(ignoring ions and thermal effects)
θ=80°
7
ω
ω
−i ce 2
ω ω − ω ce 2
2
pe
0
ω 2pe
1− 2
ω − ω ce 2
0
0
ω 2pe
1− 2
ω
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟⎠
n=1/cosθ
6
5
4
n
⎛
ω
1−
⎜
ω 2 − ω ce 2
⎜
⎜ ω
ω 2pe
ce
ε =⎜ i
2
2
⎜ ω ω − ω ce
⎜
⎜
0
⎜⎝
2
pe
3
2
1
0
0.1
0.5
1
ωUH 5
ω/ωpe
•  The allowed wave branches for Cherenkov radiation are: Langmuir wave,
extraordinary-electron (EXEL) wave, and upper-hybrid wave (electrostatic
Bernstein wave).
G.I. Pokol, A. Kómár, A. Budai, A. Stahl, and T. Fülöp, Phys. Plasmas 21, 102503 (2014).
10
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Wave frequency for Cherenkov radiation
•  For a given emittance angle of k, one can calculate the wave
frequency that satisfies the Cherenkov radiation condition.
3.5
4
θ=0.5
ω2=ωUH
3.0
3
n
ω /ωpe
2.5
2.0
2
n=1/cosθ
1.5
1
ω1=ωpe
ωce=3ωpe
1.0
0.2
0.4
0.6
0.8
1.0
θ
1.2
1.4
0
0.1
0.5
1
5
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ω/ωpe
•  For ωce~ωpe, one emittance angle gives one solution of ω, which
shifts from ωpe to ωUH.
16
Wave frequency for Cherenkov radiation (cont’d)
•  For a given emittance angle of k, one can calculate the wave
frequency that satisfies the Cherenkov radiation condition.
10
4
ω2=ωUH
3
6
n
ω /ωpe
8
θ=0.2
2
n=1/cosθ
4
1
2
ωce=10ωpe
ω1=ωpe
0.2
0.4
0.6
0.8
θ
1.0
0
0.1
0.5
1
5
10
ω/ωpe
•  For ωce≫ωpe, there are 3 branches of ω roots, which all coexist in a
range of emittance angle.
17
Wave frequency for Cherenkov radiation (cont’d)
•  For a given emittance angle of k, one can calculate the wave
frequency that satisfies the Cherenkov radiation condition.
10
1.10
ω2=ωUH
1.08
1.06
1.04
6
n
ω /ωpe
8
θ=0.2
4
1.02
n=1/cosθ
1.00
2
ω1=ωpe
0.2
0.98
ωce=10ωpe
0.4
0.6
0.8
θ
0.96
1.0
0.5
1
5
10
ω/ωpe
•  For ωce≫ωpe, there are 3 branches of ω roots, which all coexist in a
range of emittance angle.
18
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⊥)
20
•  Although the radiation loss from
15
A(θ)
2
e
ω
sin θ dθ
Ezp = 2pe ∫
A(θ )
v
cosθ
ωce/ωpe=10
10
5
ωce/ωpe=3
ωce/ωpe=0.2
small emittance angle (Langmuir) and
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
large angle (Upper-hybrid) stays
θ
the same, intermediate angle (EXEL wave) gives additional energy loss.
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Effects of 3 modes on Cerenkov radiation energy
loss
ωce/ωpe=10
Total
•  All the 3 modes contributed to the Cherenkov radiation energy loss
•  For ωce≫ωpe, the radiation emittance is strongly peaked at certain
angle.
20
Cherenkov radiation energy loss in magnetized
plasma (cont’d)
•  The Coulomb logarithm in the drag force has a correction
⎛ v⎞
ln Λ → ln Λ + ln ⎜ ⎟ + Δ 2
⎝ vth ⎠
2.0
x(a1 x + a0 )
,
(x 2 + b1 x + b0 )
x=
ω ce
ω pe
1.5
Δ2
Δ2 =
1.0
a1 = 2.05,a0 = 9.51
b1 = 1.63,b0 = 26.23
0.5
0.0
0
2
4
6
ωce /ωpe
8
10
21
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
Magnetic pumping: FTTMP = − µ∇Bz
•  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.
22
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
FTTMP
⎛ v⊥ ⎞
≈
γ
⎜⎝ ⎟⎠
eE p
c
2
1.5
Δ3
FTTMP
2.0
2
⎤
µω ce ω pe ⎡ ⎛ v ⎞ 1 ⎛ vth ⎞
= 2
ln
−
1−
+
Δ
⎢
⎜
⎟
3⎥
c ω pe v 2 ⎣ ⎜⎝ vth ⎟⎠ 2 ⎝
v⎠
⎦
1.0
0.5
0.0
0
2
4
6
8
ωce /ωpe
•  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.
10
23
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
24
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
25
Questions for discussion
1. 
Can we detect the Cherenkov radiation from runaway electrons?
B. J. Tobias et al., Rev. Sci.
Instrum. 83, 10E329 (2012)
2. 
What is the correct physics interpretation of Cherenkov resonance
condition when collisional damping is considered?
ω = k ⋅v
exp(ikx − iω t)
⎧⎪ Im ω < 0,Im k < 0 Wave is damping in time, but growing in space?
⎨
⎪⎩ Im ω > 0,Im k > 0 Wave is damping in space, but growing in time?