Plasma modelling in RF simulations Alexej Grudiev 3/06/2013 HG2013, ICTP Trieste, Italy

Ion source for CERN LINAC4

2 MHz Antenna + “plasma”

Plasma model: dielectric or conductor For a fixed frequency the second term describing the losses in dielectrics equivalent to the one describing the losses in conductors. We will take a conductor to represent plasma with certain number of free electrons per volume: N, then the conductivity is expressed (Jeckson: Ch.7.5) It is also convenient for comparison to introduce plasma frequency Where γ0 is so-called damping term describing energy loss of free electrons moving between ions and neutrals. it is proportional to the collision frequency

Range of interest for conductivity In solid Copper: N = 8x1028/m3; σ = 58 MS/m => γ0 = 4x1013/s >> ω in microwave range ωp = 6x1016/s >> γ0 Let’s also can assume that in our case ωp >> γ0 >> ω = 4πx106/s The lowest value for conductivity: σ/ω ≈ ε0 => ωp ≈ ω ≈ γ0 : σ > 10-4 [S/m] For lower conductivity the lossy part: σ/ω is much smaller than ε0 The highest value of interest for conductivity is set by skin-effect: δ = (2/σωµ0)1/2 > 1mm => σ < 105 [S/m] For higher conductivity EM fields cannot be solved in “plasma”

E-field: antenna + “plasma” (σ = 0 S/m)

E-field: antenna + “plasma” (σ = 1e-4 S/m)

E-field: antenna + “plasma” (σ = 1e-3 S/m)

E-field: antenna + “plasma” (σ = 1e-2 S/m)

E-field: antenna + “plasma” (σ = 1e-1 S/m)

E-field: antenna + “plasma” (σ = 1e-0 S/m)

E-field: antenna + “plasma” (σ = 1e+1 S/m)

E-field: antenna + “plasma” (σ = 1e+2 S/m)

E-field: antenna + “plasma” (σ = 1e+3 S/m)

E-field: antenna + “plasma” (σ = 2e+3 S/m)

E-field: antenna + “plasma” (σ = 3e+3 S/m)

E-field: antenna + “plasma” (σ = 1e+4 S/m)

E-field: antenna + “plasma” Along the coil axis Across the coil axis Higher field in the “extraction” region

H-field: antenna + “plasma” For high sigma when skin depth is very small there are some numerical issues for magnetic field calculation

RF circuit (M.M. Paoluzzi, et. al., NIBS2010)

Measured parameters of the RF system. (M.Paoluzzi et.al. NIBS 2010) IS-01* (simulated by AG) 68/60/26 6 3.2 (3.0) 0.4 (0.26) 2.3 0.14 1.5 nF 5.5 nF ~100=2pi*2*3.2/0.4 ~9=100*0.4/(0.4+4.3) 50Ω ? * Private communication, Mauro Paoluzzi, (2013).

Input impedance of antenna + “plasma” Imaginary part almost constant It get reduced for high σ due to skin-effect in “plasma” Real part stay constant at the level of Rant for low σ < 1 S/m It is significantly high for σ > 10 S/m It peaks at σ = 1000 S/m when skin depth is about the “plasma” radius

RLC parameters of antenna + “plasma” Model used by Mauro RANT LANT RPlasma LPlasma ZPlasma = ZAnt+Plasma(σ>0) - ZAnt(σ=0) ZAnt = RAnt +jω0LAnt ZPlasma = RPlasma +jω0LPlasma IS-01 coil + “plasma” (simulated) RAnt = 0.124 Ω LAnt = 2.65 µH H- source (measured) SPL PG (measured) Mauro Paoluzzi Plasma resistance and inductance calculated following Mauro’s equivalent circuit for a given plasma shape in bare antenna agree rather well with the measurement results for H- source.

Input impedance for different “plasma” shapes Plasma R20mm Plasma R24mm The same “plasma” radius – the same Re{Zin} (red curves) Larger “plasma” radius – larger Re{Zin}, larger Im{Zin} reduction Plasma R28mm

RLC parameters for different “plasma” shapes IS-01 coil + “plasma” (simulated) RAnt = 0.124 Ω LAnt = 2.65 µH H- source (measured) SPL PG (measured) Mauro Paoluzzi Plasma R20mm Plasma R24mm Plasma R24mm Plasma R28mm Plasma R20mm Plasma R28mm The larger is “plasma” radius the higher is Rplasma and –Lplasma

Input impedance for full 3D model Plasma R24mm Plasma R24mm RAnt = 0.124 Ω LAnt = 2.65 µH Plasma Full SC RAnt = 0.263 Ω LAnt = 3.02 µH For IS-01, Mauro measured: RAnt = 0.4 Ω LAnt = 3.2 µH Rplasma = 4.3 Ω Lplasma = -0.05 µH plasma Plasma Full SC Inverted polarity RAnt = 0.263 Ω LAnt = 3.03 µH

E-field for full 3D model plasma conductivity: ~316 S/m frame # σ [S/m] plasma

Possible applications to breakdown studies Plasma ignited by a discharge Antenna couples RF power to plasma and is used to measure plasma impedance Additional passive or/and active diagnostics via damping waveguides RF output RF input Higher frequency signal for break down plasma diagnostics plasma Plasma ignited by the breakdown

Back up slides

Power loss in antenna and “plasma” Power loss on the antenna surface: Pant shows very small variation with respect to the “plasma” conductivity Power loss in the “plasma”: Ppl linearly rising from ~0 level at σ=1e-1 S/m up to maximum at σ=1e+3 S/m. Reason: E-field distribution is constant and Ppl ~ ∫dV σE2. Equivalent circuit: Rp~1/σ => at constant “plasma” voltage: Vpl, Ppl~V2σ For σ>1e+3 S/m, Skin-depth becomes comparable to the “plasma” radius what reduce E-field in the “plasma” volume Vpl goes dawn Ppl goes down ~σ-1/3.5

Equivalent circuit of a transformer Volume current distribution in plasma IAnt RAnt LP 36RPl LM Ipl VAnt Vpl RP = RAnt ; RC = ∞ ; XP+XM = ωLAnt NP:NS = 6:1 XS = 0; RS = Rpl – plasma resistance

Input impedance of antenna + “plasma” ω0LP = Im{Zin}= 25.66 Ω LP= 2.04 µH RAnt = 0.124 Ω LAnt = 2.65 µH Re{Zin}= 0.121 Ω σ=∞

Plasma impedance: different view System behaviour versus plasma conductivity: σ<1e-4: no effect of the plasma 1e-4<σ<1e-1: plasma act as a dielectric pushing Er,Ez outside 1e-1<σ<1e3: plasma heats up due to Eθ. Power ~ ∫dV σEθ2. 1e3<σ: plasma heating is gradually reduced down to 0 due to skin effect. Coupling inductance LM is gradually reduced from LM=LM0 -> LM=0