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 Effective cross-section: dσ dΩ = 2πb|db | 2π| sin (θ) dθ | = b sin (θ)

• db

dθ

• Rutherford scattering:

b = bc cot (θ/2) bc = Ze2 4πε0mv2 dσ dΩ = b2

c

4 sin (θ/2)4

b db dθ θ θ ❜

• A. Flacco/ENSTA - PA201: Introduction

Page 3 of 10

Collisions

Small and large angle collisions Impact parameter for θ = π/2: bc = Ze2 4πε0mv2 Maximumu impact parameter: bmax = λD For thermal velocity mv2

0 = 3kBT:

bc bmax = Z 12π 1 nλ3

D

∝ 1 N 3

D

bc bmax = λD

• A. Flacco/ENSTA - PA201: Introduction

Page 4 of 10

Collisions

Collision frequency Collision frequency determines the thermalization time.

Single collision at small θ:

• small energy exchange
• wide cross section

λc: Mean free path before a total deviation

• f π/2.

Single collision at large θ:

• large energy exchange
• small cross section

λπ/2: Mean free path before a deviation of 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 ∆v2

x = N(∆vx)2 i

(∆vx)2

i ≃ v2 sin2 (θ)

(∆vx)2

i

=

bmax

bc

2πb (∆vx)i db

bmax

bc

2πbdb ≃ 8πb2

cv2Λ

bmax

bc

2πbdb Coulombian Logarithm: Λ = ln (λD/bc) N =

• n

bmax

bc

2πbdb

• λc

Small deflection mean free path: λc = 1 8πnb2

cΛ

where a for a deflection of π/2 it has been considered ∆vx = v.

• A. Flacco/ENSTA - PA201: Introduction

Page 6 of 10

Collisions

π/2 mean free path Nπ/2 = 1 = nλπ/2

bc

2πbdb = nλπ/2πb2

c

π/2 deflection mean free path: : λπ/2 = 8Λλc In conclusion, λπ/2 ≪ λc 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 λc ∼ 1/nb2

0 is included in the b0 factor.

• For thermal particles it holds:

b0 ∼

• Z

4πε0

e2

µv2

t

The three possible collisions are considered (restrictions apply): electron-ion µ ≃ me v ≃ ve = vte vte = (3kBTe/me)1/2 ion-ion µ = mi/2 v = vti vti = (3kBTi/mi)1/2 electron- electron µ = me/2 v = vte (Not different from electron-ion)

• A. Flacco/ENSTA - PA201: Introduction

Page 8 of 10

Collisions

Collision frequency and thermalization Electron-Ion τei = λc vte = 1 νei =

• niZ2e4Λ

4πε2

0m1/2 e

(kBTe)3/2

−1

Ion-Ion τii = λc vti = 1 νei =

• niZ4e4Λ

4πε2

0m1/2 i

(kBTi)3/2

−1

Electron-Electron τee = λc vte = 1 νee =

• nee4Λ

4πε2

0m1/2 e

(kBTe)3/2

−1

τei τii ≃

Te

Ti

3/2 me

mi

1/2

(Note: according to previous definition, τei = τie!)

• A. Flacco/ENSTA - PA201: Introduction

Page 9 of 10

On Thermalization time

t τee τii τE mi/me (mi/me)1/2 (mi/me)1/2

• Due to e-e collisions (τ ≈ τee), electron population reaches maxwellian

distribution in short timescale;

• same applies for ions, but on a longer time scale, due to slower collision rate

(τ ≈ τii);

• Crossed specie collision happens on fast time (τee ≈ τei). However (mi/me)

collisions are needed to exchange of an amount of energy in the order of the average: ∆E E ≈ me mi

• A. Flacco/ENSTA - PA201: Introduction

Page 10 of 10