Introduction to the Diagnosis of Magnetically Confined Thermonuclear - PowerPoint PPT Presentation

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

The plasma Sheath Langmuir probes Self-check LIF Outline 1 The plasma Sheath Debye Shielding Plasma-wall transition: the plasma Sheath Langmuir probes 2 Modes of operation Mach probes EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 2 / 28

The plasma Sheath Langmuir probes Self-check LIF Outline 1 The plasma Sheath Debye Shielding Plasma-wall transition: the plasma Sheath Langmuir probes 2 Modes of operation Mach probes EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 3 / 28

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

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, n s = n 0 e − q s ϕ/ T where ϕ ( x ) is the electrostatic potential. Quasineutrality n e = n i 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 = δρ ext + e ( δ n i − δ n e ) = δρ ext − 2 e 2 n 0 δϕ/ T , EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 5 / 28

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, n s = n 0 e − q s ϕ/ T where ϕ ( x ) is the electrostatic potential. Quasineutrality n e = n i 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 = δρ ext + e ( δ n i − δ n e ) = δρ ext − 2 e 2 n 0 δϕ/ T , EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 5 / 28

The plasma Sheath Langmuir probes Self-check LIF Debye Shielding: Derivation (II) Poisson’s equation for the perturbed potential gives � � δϕ = − δρ ext ∇ 2 − 2 , λ 2 ǫ 0 D � ǫ 0 T / ne 2 being the Debye length , which is a with λ D = characteristic plasma length scale for electrostatic phenomena (typicaly λ D ∼ 10 µ m, in the edge). Its solution for δρ ext = Q δ ( x ) is √ e − 2 r /λ D Q δϕ ( r ) = . 2 πǫ 0 r • The no-plasma solution is recovered in the limit n → 0 at Q constant T so that λ D → ∞ , i.e. δϕ no − plasma ( r ) = 2 πǫ 0 r ). EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 6 / 28

The plasma Sheath Langmuir probes Self-check LIF Debye Shielding: Derivation (II) Poisson’s equation for the perturbed potential gives � � δϕ = − δρ ext ∇ 2 − 2 , λ 2 ǫ 0 D � ǫ 0 T / ne 2 being the Debye length , which is a with λ D = characteristic plasma length scale for electrostatic phenomena (typicaly λ D ∼ 10 µ m, in the edge). Its solution for δρ ext = Q δ ( x ) is √ e − 2 r /λ D Q δϕ ( r ) = . 2 πǫ 0 r • The no-plasma solution is recovered in the limit n → 0 at Q constant T so that λ D → ∞ , i.e. δϕ no − plasma ( r ) = 2 πǫ 0 r ). EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 6 / 28

The plasma Sheath Langmuir probes Self-check LIF Debye Shielding: Derivation (II) Poisson’s equation for the perturbed potential gives � � δϕ = − δρ ext ∇ 2 − 2 , λ 2 ǫ 0 D � ǫ 0 T / ne 2 being the Debye length , which is a with λ D = characteristic plasma length scale for electrostatic phenomena (typicaly λ D ∼ 10 µ m, in the edge). Its solution for δρ ext = Q δ ( x ) is √ e − 2 r /λ D Q δϕ ( r ) = . 2 πǫ 0 r • The no-plasma solution is recovered in the limit n → 0 at Q constant T so that λ D → ∞ , i.e. δϕ no − plasma ( r ) = 2 πǫ 0 r ). EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 6 / 28

The plasma Sheath Langmuir probes Self-check LIF Debye Shielding 5 4 3 2 1 0 0 1 2 3 4 5 EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 7 / 28

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

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

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

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

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 � e ( φ ( x ) − φ p ) � � e ϕ ( x ) � n e ( x ) = n p exp ≡ n p exp , T e T e The ion density and velocity are determined by the conservation of energy 1 i + e ϕ ( x ) = constant = 1 2 m i V 2 2 m i V 2 p and particle number d ( n i V i ) = 0 ⇒ n i ( x ) V i ( x ) = constant = n p V p , dx EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 9 / 28

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 � e ( φ ( x ) − φ p ) � � e ϕ ( x ) � n e ( x ) = n p exp ≡ n p exp , T e T e The ion density and velocity are determined by the conservation of energy 1 i + e ϕ ( x ) = constant = 1 2 m i V 2 2 m i V 2 p and particle number d ( n i V i ) = 0 ⇒ n i ( x ) V i ( x ) = constant = n p V p , dx EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 9 / 28

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 d 2 ϕ ( x ) = − e ( n i − n e ) dx 2 ǫ 0  � � � − 1 / 2 � e ϕ ( x )   1 − 2 c 2 e ϕ ( x ) = − en p s − exp  , ǫ 0  V 2 T e T e p which doesn’t have general analytic solutions. Instead we approximate the equation for the quasi(quasineutral) region where e ϕ ( x ) ≪ T e EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 10 / 28

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 d 2 ϕ ( x ) = − e ( n i − n e ) dx 2 ǫ 0  � � � − 1 / 2 � e ϕ ( x )   1 − 2 c 2 e ϕ ( x ) = − en p s − exp  , ǫ 0  V 2 T e T e p which doesn’t have general analytic solutions. Instead we approximate the equation for the quasi(quasineutral) region where e ϕ ( x ) ≪ T e EDGE-SOL I: Lagngmuir Probes, A. Alonso, copyleft 2010 10 / 28

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 e ϕ ( x ) / T e one gets � � d 2 ϕ ( x ) 1 − c 2 = 1 s ϕ ( x ) , x ≫ x S . dx 2 λ 2 V 2 p D Negative values of h = ( 1 − c 2 s / V 2 p ) /λ 2 D gives oscillatory solutions (unphysical). This leads to the Bohm condition for proper sheath formation | V p | ≥ c s , 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

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 e ϕ ( x ) / T e one gets � � d 2 ϕ ( x ) 1 − c 2 = 1 s ϕ ( x ) , x ≫ x S . dx 2 λ 2 V 2 p D Negative values of h = ( 1 − c 2 s / V 2 p ) /λ 2 D gives oscillatory solutions (unphysical). This leads to the Bohm condition for proper sheath formation | V p | ≥ c s , 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