Plasma Physics Collisions A. Flacco Structure Rutherford - PowerPoint PPT Presentation

Plasma Physics Collisions A. Flacco

Structure • Rutherford scattering 3 • Collision frequency 5 • Thermalization 8 A. Flacco/ENSTA - PA201: Introduction Page 2 of 10

Collisions Rutherford scattering dθ Effective cross-section: d σ 2 π b | d b | d b b � � dΩ = 2 π | sin ( θ ) d θ | = � � sin ( θ ) d θ � � db θ b Rutherford scattering: b = b c cot ( θ/ 2) Ze 2 b c = 4 πε 0 mv 2 θ 0 b 2 d σ ❜ c = 4 sin ( θ/ 2) 4 dΩ A. Flacco/ENSTA - PA201: Introduction Page 3 of 10

Collisions Small and large angle collisions Impact parameter for θ = π/ 2: Ze 2 b c = 4 πε 0 mv 2 0 b c Maximumu impact parameter: b max = λ D For thermal velocity � mv 2 0 � = 3 k B T : b max = λ D b c Z 1 1 = ∝ n λ 3 N 3 b max 12 π D D A. Flacco/ENSTA - PA201: Introduction Page 4 of 10

Collisions Collision frequency Collision frequency determines the thermalization time. Single collision at small θ : Single collision at large θ : • small energy exchange • large energy exchange • wide cross section • small cross section λ c : Mean free path before a total deviation λ π/ 2 : Mean free path before a deviation of of π/ 2. π/ 2 in a single collision. In order to determine the thermalization time we need to: • find the most effective energy exchange mechanism; • determine the time scale for single specie thermalization; • determine the time cross-specie thermalization. A. Flacco/ENSTA - PA201: Introduction Page 5 of 10

Collisions Small angle mean free path � ∆ v 2 x � = N � (∆ v x ) 2 i � i ≃ v 2 sin 2 ( θ ) (∆ v x ) 2 � b max 2 π b (∆ v x ) i d b 8 π b 2 c v 2 Λ b c � (∆ v x ) 2 i � = ≃ � b max � b max 2 π b d b 2 π b d b b c b c Coulombian Logarithm: Λ = ln ( λ D / b c ) � b max � � N = n 2 π b d b λ c b c 1 Small deflection mean free path: λ c = 8 π nb 2 c Λ where a for a deflection of π/ 2 it has been considered ∆ v x = v . A. Flacco/ENSTA - PA201: Introduction Page 6 of 10

Collisions π/ 2 mean free path � b c N π/ 2 = 1 = n λ π/ 2 2 π b d b 0 = n λ π/ 2 π b 2 c π/ 2 deflection mean free path: : λ π/ 2 = 8Λ λ c In conclusion, λ π/ 2 ≪ λ c which indicates that the largest part of energy exchange happens due to small θ deflections . A. Flacco/ENSTA - PA201: Introduction Page 7 of 10

Collisions Collision frequency and thermalization • Effects of “large angle deflection” can be neglected (this is inherent to the Coulomb range). • Temperature dependence for λ c ∼ 1 / nb 2 0 is included in the b 0 factor. • For thermal particles it holds: � e 2 � Z b 0 ∼ µ v 2 4 πε 0 t The three possible collisions are considered (restrictions apply): electron-ion ion-ion electron- electron µ = m i / 2 µ ≃ m e µ = m e / 2 v ≃ v e = v te v = v ti v = v te v te = (3 k B T e / m e ) 1 / 2 v ti = (3 k B T i / m i ) 1 / 2 (Not different from electron-ion) A. Flacco/ENSTA - PA201: Introduction Page 8 of 10

Collisions Collision frequency and thermalization Electron-Ion � − 1 � n i Z 2 e 4 Λ τ ei = λ c 1 = = 0 m 1 / 2 v te ν ei ( k B T e ) 3 / 2 4 πε 2 e Ion-Ion � − 1 � n i Z 4 e 4 Λ τ ii = λ c 1 = = ν ei 0 m 1 / 2 ( k B T i ) 3 / 2 v ti 4 πε 2 i Electron-Electron � − 1 � n e e 4 Λ τ ee = λ c 1 = = 0 m 1 / 2 v te ν ee ( k B T e ) 3 / 2 4 πε 2 e � T e � 3 / 2 � m e � 1 / 2 τ ei ≃ τ ii T i m i (Note: according to previous definition, τ ei � = τ ie !) A. Flacco/ENSTA - PA201: Introduction Page 9 of 10

On Thermalization time m i / m e t τ ee τ ii τ E ( m i / m e ) 1 / 2 ( m i / m e ) 1 / 2 • Due to e-e collisions ( τ ≈ τ ee ), electron population reaches maxwellian distribution in short timescale; • same applies for ions, but on a longer time scale, due to slower collision rate ( τ ≈ τ ii ); • Crossed specie collision happens on fast time ( τ ee ≈ τ ei ). However ( m i / m e ) collisions are needed to exchange of an amount of energy in the order of the average: ∆ E ≈ m e E m i A. Flacco/ENSTA - PA201: Introduction Page 10 of 10