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The plasma Sheath Langmuir probes Self-check LIF Introduction to the Diagnosis of Magnetically Confined Thermonuclear Plasma EDGE-SOL I: Lagngmuir Probes J. Arturo Alonso Laboratorio Nacional de Fusin EURATOM-CIEMAT E6 P2.10

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The plasma Sheath Langmuir probes Self-check LIF

Introduction to the Diagnosis of Magnetically Confined Thermonuclear Plasma

EDGE-SOL I: Lagngmuir Probes

  • J. Arturo Alonso

Laboratorio Nacional de Fusión EURATOM-CIEMAT E6 P2.10 arturo.alonso@ciemat.es

version 0.1 (February 13, 2012)

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 1 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Outline

1

The plasma Sheath Debye Shielding Plasma-wall transition: the plasma Sheath

2

Langmuir probes Modes of operation Mach probes

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 2 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Outline

1

The plasma Sheath Debye Shielding Plasma-wall transition: the plasma Sheath

2

Langmuir probes Modes of operation Mach probes

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 3 / 28

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The plasma Sheath Langmuir probes Self-check LIF

The Debye Shielding: Definition (sort of)

Debye shielding

One of the salient properties of a plasma is its response to external electric fields. The free charges in a plasma are able to move and shield the eletric field caused by any local charge excess, creating a compensating cloud of polarization charge around the charge excess. This mechanism is known as Debye shielding.

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 4 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Debye Shielding: Derivation (I)

To illustrate this, consider a quasineutral plasma in thermal equilibrium Particle densities distribute according to the Maxwell-Boltzmann law, ns = n0e−qsϕ/T where ϕ(x) is the electrostatic potential. Quasineutrality ne = ni requires ϕ = 0. If this equilibrium is externarly perturbed by a small, localised charge, the electrostatic is perturbed from its constant value by δϕ. The total charge density becomes ρ = δρext + e(δni − δne) = δρext − 2e2n0δϕ/T ,

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 5 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Debye Shielding: Derivation (I)

To illustrate this, consider a quasineutral plasma in thermal equilibrium Particle densities distribute according to the Maxwell-Boltzmann law, ns = n0e−qsϕ/T where ϕ(x) is the electrostatic potential. Quasineutrality ne = ni requires ϕ = 0. If this equilibrium is externarly perturbed by a small, localised charge, the electrostatic is perturbed from its constant value by δϕ. The total charge density becomes ρ = δρext + e(δni − δne) = δρext − 2e2n0δϕ/T ,

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 5 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Debye Shielding: Derivation (II)

Poisson’s equation for the perturbed potential gives

  • ∇2 − 2

λ2

D

  • δϕ = −δρext

ǫ0 , with λD =

  • ǫ0T/ne2 being the Debye length, which is a

characteristic plasma length scale for electrostatic phenomena (typicaly λD ∼ 10µm, in the edge). Its solution for δρext = Qδ(x) is δϕ(r) = Q 2πǫ0 e−

√ 2r/λD

r .

  • The no-plasma solution is recovered in the limit n → 0 at

constant T so that λD → ∞, i.e. δϕno−plasma(r) =

Q 2πǫ0r).

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 6 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Debye Shielding: Derivation (II)

Poisson’s equation for the perturbed potential gives

  • ∇2 − 2

λ2

D

  • δϕ = −δρext

ǫ0 , with λD =

  • ǫ0T/ne2 being the Debye length, which is a

characteristic plasma length scale for electrostatic phenomena (typicaly λD ∼ 10µm, in the edge). Its solution for δρext = Qδ(x) is δϕ(r) = Q 2πǫ0 e−

√ 2r/λD

r .

  • The no-plasma solution is recovered in the limit n → 0 at

constant T so that λD → ∞, i.e. δϕno−plasma(r) =

Q 2πǫ0r).

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 6 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Debye Shielding: Derivation (II)

Poisson’s equation for the perturbed potential gives

  • ∇2 − 2

λ2

D

  • δϕ = −δρext

ǫ0 , with λD =

  • ǫ0T/ne2 being the Debye length, which is a

characteristic plasma length scale for electrostatic phenomena (typicaly λD ∼ 10µm, in the edge). Its solution for δρext = Qδ(x) is δϕ(r) = Q 2πǫ0 e−

√ 2r/λD

r .

  • The no-plasma solution is recovered in the limit n → 0 at

constant T so that λD → ∞, i.e. δϕno−plasma(r) =

Q 2πǫ0r).

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 6 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Debye Shielding

1 2 3 4 5 1 2 3 4 5

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 7 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Plasma in contact with a solid object

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 8 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Plasma in contact with a solid object

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 8 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Plasma in contact with a solid object

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 8 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Plasma in contact with a solid object

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 8 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Simple quantitative study of the plasma Sheath

1D, cold ion, maxwellian electron approximation

Dentity of maxwellian (thermal) electrons distributes according to Boltzmann law ne(x) = np exp e(φ(x) − φp) Te

  • ≡ np exp

eϕ(x) Te

  • ,

The ion density and velocity are determined by the conservation of energy 1 2miV2

i + eϕ(x) = constant = 1

2miV2

p

and particle number d(niVi) dx = 0 ⇒ ni(x)Vi(x) = constant = npVp ,

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 9 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Simple quantitative study of the plasma Sheath

1D, cold ion, maxwellian electron approximation

Dentity of maxwellian (thermal) electrons distributes according to Boltzmann law ne(x) = np exp e(φ(x) − φp) Te

  • ≡ np exp

eϕ(x) Te

  • ,

The ion density and velocity are determined by the conservation of energy 1 2miV2

i + eϕ(x) = constant = 1

2miV2

p

and particle number d(niVi) dx = 0 ⇒ ni(x)Vi(x) = constant = npVp ,

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 9 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Simple quantitative study of the plasma Sheath

Poisson’s equation

Inserting the charge density into Poisson’s equation yields an equation for the electic potential ϕ only d2ϕ(x) dx2 = − e ǫ0 (ni − ne) = −enp ǫ0   

  • 1 − 2c2

s

V2

p

eϕ(x) Te −1/2 − exp eϕ(x) Te    , which doesn’t have general analytic solutions. Instead we approximate the equation for the quasi(quasineutral) region where eϕ(x) ≪ Te

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 10 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Simple quantitative study of the plasma Sheath

Poisson’s equation

Inserting the charge density into Poisson’s equation yields an equation for the electic potential ϕ only d2ϕ(x) dx2 = − e ǫ0 (ni − ne) = −enp ǫ0   

  • 1 − 2c2

s

V2

p

eϕ(x) Te −1/2 − exp eϕ(x) Te    , which doesn’t have general analytic solutions. Instead we approximate the equation for the quasi(quasineutral) region where eϕ(x) ≪ Te

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 10 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Simple quantitative study of the plasma Sheath

Poisson’s equation (approximated)

Taylor expanding the charge densities in the small eϕ(x)/Te

  • ne gets

d2ϕ(x) dx2 = 1 λ2

D

  • 1 − c2

s

V2

p

  • ϕ(x) ,

x ≫ xS . Negative values of h = (1 − c2

s/V2 p)/λ2 D gives oscillatory

solutions (unphysical). This leads to the Bohm condition for proper sheath formation |Vp| ≥ cs , which gives exponetialy damped solutions ϕ(x) = C exp(−x/h) in the pre-sheath region.

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 11 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Simple quantitative study of the plasma Sheath

Poisson’s equation (approximated)

Taylor expanding the charge densities in the small eϕ(x)/Te

  • ne gets

d2ϕ(x) dx2 = 1 λ2

D

  • 1 − c2

s

V2

p

  • ϕ(x) ,

x ≫ xS . Negative values of h = (1 − c2

s/V2 p)/λ2 D gives oscillatory

solutions (unphysical). This leads to the Bohm condition for proper sheath formation |Vp| ≥ cs , which gives exponetialy damped solutions ϕ(x) = C exp(−x/h) in the pre-sheath region.

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 11 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Simple quantitative study of the plasma Sheath

Poisson’s equation (approximated)

Taylor expanding the charge densities in the small eϕ(x)/Te

  • ne gets

d2ϕ(x) dx2 = 1 λ2

D

  • 1 − c2

s

V2

p

  • ϕ(x) ,

x ≫ xS . Negative values of h = (1 − c2

s/V2 p)/λ2 D gives oscillatory

solutions (unphysical). This leads to the Bohm condition for proper sheath formation |Vp| ≥ cs , which gives exponetialy damped solutions ϕ(x) = C exp(−x/h) in the pre-sheath region.

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 11 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Electric potential drop in the Sheath (I)

In a stationary state a (unbiased) solid probe acquires an electric potential that balances the ion and electron fluxes to the wall, i.e. the floating potential. To get an estimate of the floating potential we use the ion and electron fluxes to the wall ⇒ Γe(ϕ) = Γi(ϕ). For ions we assume the Bohm condition is satisfied marginally, i.e. Vp = cs) is Γi(xS) = −npcs = Γi(0) The flux of maxwellian electrons to the wall can be

  • btained from gas-kinetic theory and is given by

Γe(0) = −1 4ne(0)v = −np exp eϕ(0) Te Te 2πme . where v =

  • 8Te/πme and ne(0)/np = exp
  • eϕ(0)

Te

  • .

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 12 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Electric potential drop in the Sheath (I)

In a stationary state a (unbiased) solid probe acquires an electric potential that balances the ion and electron fluxes to the wall, i.e. the floating potential. To get an estimate of the floating potential we use the ion and electron fluxes to the wall ⇒ Γe(ϕ) = Γi(ϕ). For ions we assume the Bohm condition is satisfied marginally, i.e. Vp = cs) is Γi(xS) = −npcs = Γi(0) The flux of maxwellian electrons to the wall can be

  • btained from gas-kinetic theory and is given by

Γe(0) = −1 4ne(0)v = −np exp eϕ(0) Te Te 2πme . where v =

  • 8Te/πme and ne(0)/np = exp
  • eϕ(0)

Te

  • .

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 12 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Electric potential drop in the Sheath (I)

In a stationary state a (unbiased) solid probe acquires an electric potential that balances the ion and electron fluxes to the wall, i.e. the floating potential. To get an estimate of the floating potential we use the ion and electron fluxes to the wall ⇒ Γe(ϕ) = Γi(ϕ). For ions we assume the Bohm condition is satisfied marginally, i.e. Vp = cs) is Γi(xS) = −npcs = Γi(0) The flux of maxwellian electrons to the wall can be

  • btained from gas-kinetic theory and is given by

Γe(0) = −1 4ne(0)v = −np exp eϕ(0) Te Te 2πme . where v =

  • 8Te/πme and ne(0)/np = exp
  • eϕ(0)

Te

  • .

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 12 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Electric potential drop in the Sheath (II)

Now, the ambipolar condition Γi = Γe npcs = np exp eϕ(0) Te Te 2πme gives the floating (ambipolar) potential at the wall ϕ(0) ϕ(0) = Te 2e ln mi 2πme ≈ 3Te e , for an hydrogenic (A = 1) plasma.

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 13 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Electric potential drop in the Sheath (II)

Now, the ambipolar condition Γi = Γe npcs = np exp eϕ(0) Te Te 2πme gives the floating (ambipolar) potential at the wall ϕ(0) ϕ(0) = Te 2e ln mi 2πme ≈ 3Te e , for an hydrogenic (A = 1) plasma.

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 13 / 28

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The plasma Sheath Langmuir probes Self-check LIF

The plasma Sheath: a summary

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 14 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Outline

1

The plasma Sheath Debye Shielding Plasma-wall transition: the plasma Sheath

2

Langmuir probes Modes of operation Mach probes

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 15 / 28

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The plasma Sheath Langmuir probes Self-check LIF

The Langmuir probe

A small conductor inserted into a plasma connected to a tunnable voltage (so that we can vary ϕ(0)) and a current monitor (to measure the current drawn by the probe).

A V

plasma conductor insulator

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 16 / 28

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The plasma Sheath Langmuir probes Self-check LIF

The I-V characteristics of a Langmuir probe

From what we learnt in the previous slides it is easy to get the current I[A] as a function of the electric potential at the probe surface V. I(V) = Ie(V) + Ii(V) = −eAΓe + eAΓi = eAnp

  • Te/2πme exp [e(V − φp)/Te] − eAnpcs

≡ ISe exp [e(V − φp)/Te] − ISi , where we defined the electron and ion saturation currents as ISe = eAnp

  • Te

2πme ISi = eAnpcs = eAnp

  • Te

mi

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 17 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Marginal note: effective collection area effects

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 18 / 28

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The plasma Sheath Langmuir probes Self-check LIF

The I-V characteristic of a Langmuir probe

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 19 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Modes of operation

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 20 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Turbulent flux probe

φf = φp + 3Te/e. Assume Te offset remains approximately constant in the timescales of interest ˜ φf ∼ ˜ φp → ˜ Eθ ≈ (φf1 − φf2)/∆ ISi ∝ nT1/2

e

so one can approximate ˜ n ∼ ISi. The particle flux can then be estimated as ΓE×B = ˜ n˜ vr ∝ 1 B ISi˜ Eθ ,

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 21 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Mach probes

The above cold-ion analitical treatment of the plasma sheath is sufficient to get estimates of some plasma parameters but it cannot be pushed too much further. Material probe measurements can give more information with more sophisticated modelling. A good example is the Hutchinson model of Mach probes [1] .

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 22 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Mach probes

The above cold-ion analitical treatment of the plasma sheath is sufficient to get estimates of some plasma parameters but it cannot be pushed too much further. Material probe measurements can give more information with more sophisticated modelling. A good example is the Hutchinson model of Mach probes [1] .

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 22 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Mach probes

Ions flow into the sheath at cs, but their densities at the sheath entrance depend on the distant plasma parallel flow vi M = vi/cs = 0.41 × ln Iup

Si

Idown

Si

,

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 23 / 28

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The plasma Sheath Langmuir probes Self-check LIF

After this lecture you must. . .

understand the basic behaviour of a plasma in contact with a wall: the formation of the plamsa sheath know the characteristic I-V curve of a Langmuir probe know the three basic modes of operation of a LP and which plasma parameters can be obtained from the measurements know what a Mach probe is

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 24 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Laser Induced Fluorescence (LIF)

LIF is a totally different (from what we saw in this lecture) diagnostic technique, but It can be used to measure ion distribution functions (of not-fully ionised atoms) so it can, in particular be used to study the ion flow towards a surface (i.e the Bohm condition we just introduced) and can also measure densities of impurity ions released from the wall (that we will be seeing in the next lecture), so it seemed appropriate to treat it here!

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 25 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Operating principles of LIF

Reference: T. Lunt, PhD Thesis

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 26 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Operating principles of LIF

The frequency bandwidth of the laser ∆fL is narrow compared with the ion’s energy thermal scatter (∆fL ≪ Ti), The light of the laser will be resonant only with those ions

  • f a certain velocity along the laser v.

fD = fL(1 − v/c) = f0 ≡ ∆E0/. The intensity of the fluorescence light will be then proportional to the local density of Ar+ ions with a velocity around v = c

  • 1 − f0

fL

  • = c
  • 1 − λL

λ0

  • .

Caveats

  • background emission in the fluorescence wavelength ≫

than the one induced by the laser → compare on and off

  • requires a very precise knowledge of the laser instant

power and wavelength

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 27 / 28

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The plasma Sheath Langmuir probes Self-check LIF

Ion PDFs measured by LIF

Reference: T. Lunt et al., Phys. Rev. Lett. 2008

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 28 / 28

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The plasma Sheath Langmuir probes Self-check LIF

K-S. Chung and I. H. Hutchinson. Kinetic theory of ion collection by probing objects in flowing strongly magnetized plasmas.

  • Phys. Rev. A, 38(9):4721–4731, Nov 1988.

EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 28 / 28