Plasma Radiation Transport

Abstract

Radiative energy transfer is one of the principal properties of gases under plasma conditions. It is a direct consequence of the excitation to higher energy states of the elementary particles in a plasma and their return to lower energy states, or the ground state, by emission of radiation over a wide range of the spectrum. In this chapter, following a general definition of general concepts of blackbody and gaseous radiation, a review is presented of the radiation emission and absorption in plasmas. This includes line and continuum radiation, total effective radiation of plasmas, and thermal plasma radiation modeling. Examples are given of the contribution of line and continuum emission to the total volumetric emission of argon and nitrogen at atmospheric pressure as function of temperature. This is followed by an introduction to the concept of effective or net emission coefficient (NEC) as a means of taking into account self-absorption in plasmas. This is followed by a discussion of mixing rules for complex plasma gas mixtures. Examples are given of the total volumetric emission coefficients of gases such as argon, nitrogen, hydrogen, helium, air, water vapor, and their mixtures at atmospheric pressure over the temperature range from 5000 to 25,000 K. The effect of the presence of metal vapors such as copper and iron in the plasma gases is discussed. Data are provided for different metal vapor concentrations ranging from a few percentage points up to pure metal vapor plasmas. A brief discussion is presented of blackbody radiation of high temperature, of high-pressure plasmas, and of two-temperature nonequilibrium plasmas.

Nomenclature and Greek Symbols

A ⁱ_ul

Transition probability (s-1) for spontaneous emission.

B_lu

Transition probability for absorption (m³/J.s²).

B_λ

Blackbody monochromatic radiation intensity (W/m³·ster).

B_v

Blackbody monochromatic radiation intensity (J/m⁴·ster).

B_ul

Transition probability for induced emission (m³/J.s²).

b

Impact parameter.

c

Velocity of light (c = 2.998 × 10⁸ m/s).

e

Charge of the electron (e = 1.60217 × 10⁻¹⁹ A.s or C).

E

Energy (J).

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

Ionization energy of the hydrogen atom (${\mathrm{E}}_{{\mathrm{H}}^{+}}^{\mathrm{l}}=13.6\ \mathrm{eV}$).

E_i,u

Energy of the excited state u of the chemical species i (eV).

${\mathrm{E}}_{{\mathrm{X}}^{+}}^{\mathrm{l}}$

Ionization energy of the atom X (eV).

${\mathrm{E}}_{{\mathrm{X}}_{\mathrm{j}}}$

Excited state of the atom X (eV).

E_v

Monochromatic radiation energy (eV).

F

Energy of the rotational excited state in (cm⁻¹).

F_Rv

Monochromatic radiation flux.

G

Energy of the vibrational excited state expressed in (cm⁻¹).

G₁

Function accounting for the cylindrical geometry of the plasma.

G ⁿ_i,z

Gaunt factor.

g_u

Statistical weight or degeneracy.

g_x

Statistical weight of the ground state of the negative ion.

H⁺

Total radiation flux in positive direction (W/m²).

H⁻

Total radiation flux in negative direction (W/m²).

H⁰

Total flux (intensity emitted per unit surface per unit time into the half sphere) for a blackbody.

h

Planck’s constant (h = 6.6 × l0⁻³⁴ W.s²).

I_ν(θ, φ)

Monochromatic radiation intensity (refers to unit surface, unit time, and unit frequency) for the frequency v (J/m² · ster).

I_λ(θ, φ)

Monochromatic radiation intensity for the wavelength λ, see Eq.(4) (W/m³·ster).

I(θ, φ)

Total directional radiation intensity, see Eq.(2) (W/m² · ster).

I

Total radiation intensity (W/m²).

I_v

Monochromatic radiation intensity for the frequency, v (J/m² · ster).

I_λ

Monochromatic radiation intensity for the wavelength, λ (J/m² · ster).

J

Rotational quantum number.

J_R

Total radiation flux.

J_υ

Mean radiation intensity (W/m³.ster).

k

Boltzmann constant (k = 6.610⁻³⁴ J.s).

K_υ

Absorption coefficient (m⁻¹).

K ^′_v

Absorption coefficient taking into account induced emission (m⁻¹).

ℓ

Azimuthal quantum number.

M

Atomic mass (kg).

m_e

Mass of the electron (kg).

N_u(t)

Population of excited state u (m⁻³).

n

Principal quantum number.

$\overrightarrow{\mathrm{n}}$

Surface normal.

n_e

Electron density (m⁻³).

n_i,u

Density of the excited state u of the chemical species i (m⁻³).

n_r

Refractive index.

n ^n,ℓ_i,z

Density of the chemical species i;

n ^*_ℓ

Effective quantum number.

p

Pressure (Pa).

$\mathrm{P}\left(\mathrm{v}-{\mathrm{v}}_{\mathrm{o}}\right)$

Shape factor of a spectral line (s).

$\mathrm{P}\left(\uplambda -{\uplambda}_{\mathrm{o}}\right)$

Shape factor of a spectral line (m⁻¹).

Q ^el_i,z

Electronic partition function of the chemical species i with electrical charge z.e.

r₁

Bohr radius of the ground state (r₁ = 5.3 × 10⁻¹¹ m).

R

Radius of the elemental plasma control volume (m).

S_v

Source function: (${\mathrm{S}}_{\mathrm{v}}={\upvarepsilon}_{\mathrm{v}}/{\upkappa}_{\mathrm{v}}^{\prime }\ .{\mathrm{n}}_{\mathrm{r}}$) (J/m²·ster).

S

Cross section (m²).

T_e

Energy of the electronic excited state expressed in (cm⁻¹).

t

Time (s).

u

Total radiation density (J/m³).

u_v ( θ, φ)

Monochromatic radiation density (J.s/m³.ster).

u ^o_v (T)

Blackbody monochromatic radiation density (J.s/m³.ster).

v

Vibrational quantum number.

$\overline{\mathrm{v}}$

Mean velocity of an atom or an ion (m/s).

v_e

Velocity of the electron (m/s).

α

Constant of fine structure (2πe²/hc)

δ_e,i

Stark width at half the maximum intensity (nm).

${\updelta}_{{\mathrm{E}}_{{\mathrm{X}}^{+}}}$

Lowering of ionization energy of the atom X (eV).

δ_λ

Width of the spectral line (nm).

δ_D

Doppler width at half the maximum intensity (nm).

ΔJ

Difference in rotational quantum numbers related, respectively, to the upper ′ and lower ″ states $\Delta \mathrm{J}={\mathrm{J}}^{\prime }-{\mathrm{J}}^{{\prime\prime} }$.

Δv

Difference in vibrational quantum numbers related, respectively, to the upper and lower ″ states $\Delta \mathrm{v}={\mathrm{v}}^{\prime }-{\mathrm{v}}^{{\prime\prime} }$.

ε_E

Effective emission coefficient (W/m³·ster).

ε_fb(υ)

Emission coefficient for free–bound transition (J/m³·ster).

ε_ff(υ)

Emission coefficient for free–free transition (J/m³·ster).

ε ^{e. i}_ff (υ)

Emission coefficient for free–free transitions due to the field of ions (J/m³·ster).

ε_L(λ_o)

Integrated volumetric emission coefficient of a spectral line centered on the wavelength λ_o (W/m⁴·ster).

ε_L(υ_o)

Integrated volumetric emission coefficient of a spectral line centered on the frequency v_o (W/m³·ster).

ε ^e,a_ff

Emission coefficient for the free–free transitions due to elastic collisions.

${\upvarepsilon}_{\mathrm{i},\mathrm{z}+1}^{\mathrm{n},\ell }$

Emission coefficient of particles of chemical species i with electrical charge z.e; the excited state is defined by the quantum numbers n and ℓ.

ε_N

Net emission coefficient see Eq.(131) (W/m³·ster).

ε_T

Total emission coefficient (W/m³·ster).

ε_λ

Monochromatic emission coefficient (W/m⁴·ster).

ε_υ

Emission coefficient (J/m³·ster).

${\upzeta}_{\mathrm{i},\mathrm{z}}\left(\upupsilon, \mathrm{T}\right)$

Biberman factor.

θ

Angle with respect to the surface normal $\overrightarrow{\mathrm{n}}$

κ_υ

Monochromatic absorption coefficient including induced emission $\left({\upkappa}_{\uplambda}^{\prime }={\upkappa}_{\upupsilon}^{\prime}\right)$ (cm⁻¹).

κ ^′_υ

Monochromatic absorption coefficient per unit length without induced emission (cm⁻¹).

${\upkappa}_{\mathrm{i},\mathrm{z}+1}$

Absorption coefficient for a particle of species i and electrical charge z (z = 0 for an atom, z = l for its first ion) (cm⁻¹).

λ

Wavelength (nm).

λ_max

Wavelength giving the maximum value of B_v at a given temperature.

Λ

Escape factor.

υ

Radiation frequency.

σ_u,ℓ

Wave number for the transition between the state u and the state ℓ (m^-l).

σ ^n,ℓ_i,z

Cross section for photoionization (m²).

σ ^n,ℓ_i,z,class

Classical photoionization cross section (m²).

τ_s

Perturbation time resulting from the motion of a charged particle (s).

τ_u,ℓ

Lifetime of the excited state u (s).

τ_υ

Optical depth (dimensionless).

φ

Azimuthal angle with respect to the normal $\overrightarrow{\mathrm{n}}$

Ω

Solid angle.

′

Upper energy level

″

Lower energy level

ℓ

Azimuthal quantum number

n

Principal quantum number

I

Chemical species i

ℓ

Lower excited state

u

Upper excited state

z

Electrical charge of the particle