Instability coming with the angle

1
Between Filamentation and Two-Stream
instabilities in e-beam/plasma interaction
A. Bret, M.-C. Firpo and C. Deutsch Laboratoire
de Physique des Gaz et des Plasmas Paris Orsay
Université
2
The problem of k-orientation
Beams
3
Formalism

• Linear Maxwell-Relativistic Vlasov system in 3D
• Collisionless Fixed plasma ions
• Analytical
• Dielectic tensor e (k,w) non-diagonal
• Det T(e, k) 0 proper frequencies
• E along eigenvectors gives angle (k,E)

4
The (k,E) angle
5
Dispersion Relation
Det T(e, k) 0 ?
(w2eyy - k2c2)
(w2exx - kz2c2)(w2ezz - kx2c2)
- (w2exz kxkzc2)2 0
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Two Branches
Branch 1
(w2eyy - k2c2) 0
Branch 2
(w2exx - kz2c2)(w2ezz - kx2c2)
- (w2exz kxkzc2)2 0
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Model 1 T 0 (Fluid)
Beam nbd(vx)d(vy)d(vz-Vb)
Plasma npd(vx)d(vy)d(vzVp)
nbVb npVp
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Model 2 Hot Plasma
Beam nbd(vx)d(vy)d(vz-Vb)
Plasma npd(vzVp)
Q(vxVt) Q(vx-Vt)
Q(vyVt) Q(vy-Vt)
nbVb npVp
2Vt
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Model 3 Hot Plasma Relativistic Beam

• Beam Relativistic energyE 2 MeV (gb 4)
• Plasma Non relativisticT 2 keV

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Branch 1 behavior
(w2eyy - k2c2) 0

• Stable at T 0
• Unstable T ? 0 for small angles
• No relativistic effects (model 3 model 2)
• Purely transverse mode in ANY model
• Original  Weibel  instability

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Branch 1 behavior
Z kVb /wp
Beam
103d
d
dRm ? Vtp/Vb
r Vtp/Vb, bVb/c
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Branch 2 behavior

• Yields unstable modes for all models and all
  angles
• Bridge between Two-Stream and Filamentation

(w2exx - kz2c2)(...
x
k
E
Beam
z
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Between two-streamand filamentation, T0
d/wp

• Beam

Z kVb /wp
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Between two-streamand filamentation, T0
d/wp

• Beam

Z kVb /wp
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Between Two-Streamand filamentation, T?0
d/wp

• Beam

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Between Two-Streamand filamentation, T?0
d/wp

• Beam

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Critical angle
f

• Angle comes when 2 dispersion function
  singularities cross
• High Z instability just is just shifted in
  another direction

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Between two-stream and filamentation,
T?0Relativistic effects
d

• Beam

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Between two-stream and filamentation,
T?0Relativistic effects
d

• Beam

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Highest Growth Rate
f
BEAM
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(k,E) angle
f
BEAM
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Growth Rates Scaling
d
b Vb/c
a nb/np
gb(1-b2)-1/2
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Transverse Beam Temp. Effects
Vtb
Beam
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Beam Temp. Effects
1nb /np
Tanf
(Vthb/gbVth) /Vb
Beam Temp. damps growth rate beyond f
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Non relativistic beam
Vtb0
VtbVb/30
VtbVb/30 k // E
26
Conclusion

• Electromagnetic formalism
• Exhaustive instabilities search
• Weibel Branch
• TSF Branch
• TSF Branch Two k-oblique effects
• Critical angle f
• Max. Growth Rate for oblique k (Vbc)
• Longi. Approx. Fails beyond f
• Beam Temp. damps instabilities beyond f, not
  bellow
• Maxwelian distribution, Collisions, Density
  gradient