Fundamental Concepts in Gaseous Electronics

Abstract

It is beyond the scope of this book to even attempt a comprehensive treatment of fundamental concepts of gaseous electronics, which experienced immense growth during and after the Second World War. In this chapter the fundamentals behind the generation, loss, and motion of charge carriers are discussed. This is followed by a review of thermal excitation and ionization, definition of the plasma state, quasi-neutrality, and plasma sheaths. For a comprehensive treatment of the subject, the reader is referred to a number of books (Capitelli et al. 2012; Finkelnburg and Maecker 1956; Griem 1964; Gupta 2007; Huddlestone and Leonard 1965; Lee et al. 1973; Lochte-Holtgreven 1995; Massey et al. 1969; Mitchner and Kruger 1973; Müller and Weiss 2005; Reif 2009) that may be considered classics in this field.

Nomenclature and Greek Symbols

$\overrightarrow{\mathrm{B}}$

Magnetic induction (V.s/m²)

B ⁰_υ

Intensity of blackbody radiation (J/ster.m²)

c

Light velocity (c = 3xl0⁸ m/s)

D_a

Ambipolar diffusion coefficient (m²/s)

D_e

Electron diffusion coefficient (m²/s)

D_i

Ion diffusion coefficient (m²/s)

D_k

Diffusion of plasma charge carriers (m²/s)

e

Electron charge (e = 1.6 × l0⁻¹⁹ A.s)

e_fast

Fast electrons

e_slow

Slow electrons

E_x

Electric field responsible for ambipolar diffusion (V/m)

E_r+1

Energy required for transforming an r-times ionized atom into an (r + 1)-times ionized atom

$\overrightarrow{\mathrm{E}}$

Electric field (V/m)

${\overrightarrow{\mathrm{E}}}_{\mathrm{r}}$

Electric field in radial direction (V/m)

${\mathrm{E}}_{\mathrm{r}+1}^{*}$

Reduced ionization energy ${\mathrm{E}}_{\mathrm{r}+1}^{*}$= ${\mathrm{E}}_{\mathrm{r}+1}-\updelta {\mathrm{E}}_{\mathrm{r}+1}\left(\mathrm{eV}\right)$

E_ion

Ionization energy (eV)

${\mathrm{E}}_{{\mathrm{H}}^{+}}$

Hydrogen atom ionization energy (E_H+=13.6 eV)

E_r,k

Energy of chemical species r in the excited state k (cm⁻¹)

eV

Electron volt (1 eV = 1.6*10⁻¹⁹ J)

f(v)

Maxwellian distribution function

$\overrightarrow{\mathrm{F}}$

Force vector (N)

${\overrightarrow{\mathrm{F}}}_{\mathrm{r}}$

Force in the radial direction (N)

g

Number of compartments (h³) in the phase space volume dx.dy.dz.dp_x.dp_y.dp_z

g_k

Statistical weight of excited state k

g_r,k

Statistical weight of chemical species r in excited state k

h

Planck’s constant (h = 6.626 × 10⁻³⁴ W.s²)

H

Elementary volume in phase space: dx.dy.dz.dp_x.dp_y.dp_z

I

Arc current (A)

${\overrightarrow{\mathrm{I}}}_{ \mathrm{e}}$

Electrons flux (s⁻¹·m⁻²)

${\overrightarrow{\mathrm{I}}}_{ \mathrm{k}}$

Species k flux (s⁻¹·m⁻²)

j

Electric current density (A/m²)

j_e

Electron current density (A/m²)

j_i

Ion current density (A/m²)

${\overrightarrow{\mathrm{J}}}_{ \mathrm{k}}$

Flux of charged particles (m⁻²·s⁻¹)

k

Boltzmann constant (k = 1.38 × 10⁻²³ J/K)

$\overrightarrow{\mathrm{J}}$

Current density vector (A·m⁻²)

ℓ _e

Electron mean free path (m)

ℓ _i

Ion mean free path (m)

L

length (m)

m_e

Electron mass (m_e = 9.11 × 10⁻³¹ kg)

M

Ion mass (kg)

n_e

electrons density (m⁻²)

n_i

electrons density (m⁻²)

n_r,k

Particle number density of chemical species r in excited state k (m⁻³)

N_k

Number of particles in excited state k

${\mathrm{N}}_{\mathrm{k}}^{\mathrm{o}}$

equilibrium distribution of phase points in phase space (Maxwell-Boltzmann distribution)

p

Total pressure (Pa)

p_e

Partial pressure of an electron gas (Pa)

p_x

Component of the momentum (p_x = m.v_x) (kg.m/s)

p_y

Component of the momentum (p_y = m.v_y) (kg.m/s)

p_z

Component of the momentum (p_z = m.v_z) (kg.m/s)

q

Electrical charge (A.s)

Q_r

Partition function of chemical species r

r_L

Larmor radius for the circular motion (r_L = m.v/q.B (m))

r_min

Landau parameter (m)

R

Arc radius (m)

${\overrightarrow{\mathrm{s}}}_{\mathrm{e}}$

Distance that an electron travels during time interval τ _e (m)

$\overline{{\overrightarrow{\mathrm{s}}}_{\mathrm{e}}}$

Mean distance travelled by the electron (m)

S_k

Source term (see Eq. 35) (m⁻³.s⁻¹)

t

Time (s)

T

Temperature (K)

T_e

Electron temperature (K)

T_h

Heavy species temperature (K)

T_k

Maximum kinetic energy acquired by an electron between two collisions (J)

${\overrightarrow{\mathrm{u}}}_{\mathrm{e}}$

Electron drift velocity (m/s)

${\overline{\mathrm{u}}}_{\mathrm{e}}$

Mean electron drift velocity (m/s)

${\overrightarrow{\mathrm{u}}}_{\mathrm{eo}}$

Initial velocity of the electron when the electric field is applied (m/s)

$\overline{{\overrightarrow{\mathrm{u}}}_{\mathrm{e}}}$

Mean drift electron velocity (m/s)

$\overline{{\overrightarrow{\mathrm{u}}}_{\mathrm{e}}}$

Mean drift ion velocity (m/s)

${\overrightarrow{\mathrm{u}}}_{\mathrm{i}}$

Ion drift velocity (m/s)

${\overline{\overrightarrow{\mathrm{u}}}}_{\mathrm{i}}$

Mean ion drift velocity (m/s)

$\overrightarrow{\mathrm{v}}$

Particle velocity (m/s)

${\overrightarrow{\mathrm{v}}}_{\mathrm{e}}$

Electron velocity (m/s)

${\overline{\mathrm{v}}}_{\mathrm{e}}$

Mean electron velocity (m/s)

${\overline{\mathrm{v}}}_{\mathrm{i}}$

Mean ion velocity (m/s)

${\overrightarrow{\mathrm{v}}}_{\mathrm{d}}^{\mathrm{e}}$

Mean electron drift velocity (m/s)

${\overrightarrow{\mathrm{v}}}_{\mathrm{D}}$

Electron drift velocity in magnetic and electric fields (m/s)

$\mathrm{v}\parallel$

Charged particle velocity component parallel to the magnetic field (m/s)

v_⊥

Charged particle velocity component perpendicular to the magnetic field (m/s)

V

Electrical potential (V)

W

Thermodynamic probability

x

Position coordinate (m)

X

Chemical species

X⁺

Singly ionized chemical species

y

Position coordinate (m)

z

Position coordinate (m)

β

Lagrange multiplier [β = 1/(k.T)]

δE _i

Ionization potential lowering (eV)

δN

Particle number variation

Δp(r)

Pressure variation along the plasma radius (Pa)

ε ₀

Dielectric constant $\left({\upvarepsilon}_0=8.86\times {10}^{-12} \mathrm{A}.\mathrm{s}/\mathrm{V}.\mathrm{m}\right)$

γ_e

Gvosdover parameter

λ_D

Debye length (m)

Λ

Diffusion length (m)

μ_e

Electron mobility (m²/V.s)

μ_i

Ion mobility (m²/V.s)

μ₀

Magnetic permeability constant $\left({\upmu}_0=1.26\times {10}^{-6} \mathrm{H}\mathrm{y}/\mathrm{m}\right)$

υ_i

Net ionization coefficient

ξ

Fraction of ionized atoms in the gas

ρ

Specific mass (kg/m³)

ρ_el

Electric space charge (C/m³)

σ_e

Electrical conductivity (ohm⁻¹.m⁻¹)

τ_e

Mean free flight time for electrons (s)

ω_e

Larmor frequency (s⁻¹)

$\nabla$

Laplace operator