Gaussian cgs-units are used unless otherwise stated; for conversion into Practical- (SI-) units see the Conversion Table.

.

β(E_{w})= E_{w}^{2}/(E_{w}^{2} +Δ{E}_{p}^{2}) ,

where E_{w} is the electric field strength of the radiation with frequency *ν* (as given by the intensity I= E_{w}^{2}/8π) and Δ{E}_{p}=ΔE_{p}^{.}√(ζ_{ν,e}^{2}+ζ_{ν,I}^{2}) the effective plasma fluctuation field (see Plasma Field Fluctuations and
Stark Broadening with T_{n} replaced by 1/(2π *ν*)).

This leads to the circumstance that the Optical Depth with regard to photoionization is not proportional to the column density any more but has to be determined through the integral

τ(s)= σ_{Ion}(0) ^{.}_{0}∫^{s}ds' N(s')^{.}β(E_{w}(s')) .

Because generally the intensity I (i.e. E_{w}^{2}) is subject to the **Exponential Absorption Law**

I(s)= I_{0}^{.}e^{-τ(s)} ,

the optical depth is determined by an integral equation which can be solved numerically (see /research/nlabsorb.htm).

A_{i,k}= 16π^{.}e^{4}/(3c^{3}h)^{.}*ν*_{i,k}^{3}^{.}<r>_{i,k}^{2} .

(one should note that this expression is smaller by a factor 1/4 compared to the usual value quoted in the literature which is however derived inconsistently from statistical equilibrium considerations).

In general, <r>_{i,k} can only be evaluated numerically, but for large values of the principal effective quantum numbers m and n it can be approximated by a power law in terms of these parameters, with the result

A_{m,n}= 1.3^{.}10^{9}^{.}m^{-1.8}^{.}(n-1)^{-3.2} [sec^{-1}] (m,n >>1)

A numerical summation over all lower levels m reveals furthermore that the
**Total Average Atomic Decay Coefficient** (i.e. the inverse average lifetime) of level n can be approximated by

A_{n}= 1.1^{.}10^{9}^{.}(n-1)^{-3.6} [sec^{-1}] (n >>1)

(neglection of the angular momentum quantum number *l* in these approximations may lead to an error up to a factor 2 in the absolute values for A_{m,n} and A_{n}, the relative values for fixed *l* should be accurate to within a few percent however).

This process should be strongly temperature dependent, having its highest efficiency if the corresponding velocity of the approaching atoms is equal to the velocity of the bound electrons (i.e. about 10

The proposed process could be the explanation for the relatively high plasma density of the nighttime F- region of the earth's ionosphere. This would lead to an effective cross section of 10

In general this mechanism should result in a significant degree of ionization even in the absence of any UV- radiation sources, which should be highly relevant for some astrophysical problems like star formation (see /research/#A8).

ε_{rad}=(1+Δ*ν*_{m,n}^{d}/A_{m,n})^{.}h^{.}*ν* ,

where Δ*ν*_{m,n}^{d} is the dynamical Stark Broadening due to the plasma field fluctuations and A_{m,n} the Atomic Decay Probability (natural broadening). This could for instance resolve the discrepancy if one wants to explain the radiative energy output of the present day sun solely through the gravitational contraction of an initial gas cloud (see /research/#A5).

*f*(ε)= exp(-ε/ε_{0}) .

The energy distribution of electrons within an atom is generally assumed to behave in this way.

However, in most practical cases collisions are quite insignificant compared to radiative processes which are determined by the lifetime of the individual atomic levels. As a consequence, the distribution function has very little to do with a Boltzmann- distribution (see for instance /research/levpop.htm).

(see also
Maxwell Distribution,
Saha- Equation,
LTE).

∂/∂t(n(**r**,**v**,t)) +**v**^{.}grad_{r}(n(**r**,**v**,t)) +**F**/M^{.}grad_{v}(n(**r**,**v**,t)) = q^{Ion}(**r**,**v**,t) -*l*^{Rec}(**r**,**v**,t) +C(**r**,**v**,t) .

The steady-state (time independent) equation is obtained by setting ∂/∂t(n(**r**,**v**,t))=0. In this case, the production and loss rates due to convection (transport) in geometrical and velocity space (where **F** contains all external forces on the particle with mass M, i.e. electric, magnetic and gravitational forces), are exactly balanced by the local production and loss rates due to ionization (q^{Ion}), recombination (*l*^{Rec}) and velocity changing collisions (C) (in general, these last three terms do also depend on n(**r**,**v**,t) which has not been written here).

The ionization and recombination terms are usually neglected in standard treatments. However, they are vitally important as they are responsible for the inhomogeneities of the plasma density and affect therefore the velocity distribution function through the convection terms in the equation (see link below).

Also, one should note that the usual formulation in terms of the normalized distribution function *f*(**r**,**v**,t)= n(**r**,**v**,t)/N(**r**,t) (with N(**r**,t) = ∫d^{3}v n(**r**,**v**,t) ) is in general not sufficient because
of the dependence of N(**r**,t) on **r**.

For the one-dimensional case, the Boltzmann equation can be written as a first order linear differential equation in either the spatial or velocity variable. Formal solution yields a non-linear integral equation which can then be solved numerically (see /research/#A6 for an application to ion diffusion in the earth's ionosphere).

*ν*_{c}= N^{.}σ_{c}(v)^{.}v ,

where N is the volume density of the background medium and σ_{c}(v)
the cross section of the particle with velocity v for the type of collision being considered (e.g. Coulomb collisions, radiative recombination, collisional excitation).

Despite the random nature of collisions, *ν*_{c} can be considered as an exact quantity because the large number of particles usually assures that the average is very sharply defined within relatively short time scales and small volumes. This allows therefore an exact assessment of the importance of the individual collisional processes and also a comparison with the physical time scales like the Atomic Decay Probability or Plasma Frequency.

(see also
Mean Free Path,
Level Population,
Plasma).

There is also evidence that this aspect is of relevance for collisional excitation of atomic states by electrons or ions as the relevant cross section is apparently enhanced if the medium becomes continuous with regard to the wavelength of the equivalent radiative transition (see /research/striapot.htm).

The only true continuum is produced by the recombination of electrons with ions, which results in a continuum according to the energy characteristics of the free electron spectrum and the recombination cross section (synchrotron radiation could well be interpreted in this sense).

However, the discrete atomic spectrum may form a quasi- continuum if the lines are sufficiently broadened. This happens in particular for high plasma densities and/or highly excited atomic states . There is theoretical and observational evidence that under these conditions the 'continuum' of blended lines is many orders of magnitude more intense than the actual recombination continuum (see for instance /research/#A5). (for the latter aspect see also Bohr-Einstein Radiation Formula).

σ_{c}(E)= 5/16^{.}Z^{2}^{.}e^{4}/ε^{2} ,

with ε the energy of the scattered particle in the center of mass system, e its charge and Z the charge number of the target particle.

This form is different from the usual result quoted in the literature which contains the additional Coulomb- Logarithm factor. The latter can however be shown to be due to an incorrect Energy Loss- weighting function in the integration of the Rutherford formula (more).

V(r)= Q/r^{.}exp(-r/λ_{D}),

with

λ_{D}= √(kT_{e}/4πe^{2}N_{p})

the **Debye Length** for a plasma with density N_{p} and electron temperature T_{e}.

This result is merely academic because the assumption of a Boltzmann energy distribution in the Debye-Hückel theory implies a collisionally dominated isothermal situation where the pressure gradient exactly cancels the force due to the electric field. This non-vanishing potential is therefore the consequence of the implicit assumption of collisions in Thermodynamic Equilibrium preventing the purely electrostatic screening which would hold in a collisionless plasma. However, collisions (and the related pressure forces) should only be relevant in a plasma if the collision frequency is higher than the plasma frequency (which determines the timescale for the electrostatic re-arrangement of charges). Unless one is dealing with a very low degree of ionization, this condition is only satisfied for extremely high plasma densities as encountered in solids, fluids or the interior of the sun.

It is clear that in almost all cases of practical interest, a force free steady-state situation can only exist if the electric field is exactly zero within the whole plasma. This is obviously only possible if the test charge is directly neutralized at its surface by charges that have been attracted from the plasma. Charge neutrality within the volume is hereby conserved by the electrons slightly contracting towards the center, which leaves therefore the positive charge excess at the surface of the plasma volume (as one would expect for a conducting medium).

In addition, one should note that for near collisionless plasmas not only will the assumption of TE be invalid (as indicated above), but also the approximation of a **Local** Thermodynamic Equilibrium (LTE), i.e. the velocity distribution function may become non-Maxwellian due to diffusion effects in the presence of spatial inhomogeneities. This in turn will produce self-consistent electric fields which serve to adjust the electron flux balance as to maintain local charge neutrality. (see /research/#A6).
These plasma polarization fields are obviously not being screened by the plasma, as they are themselves the result of the dynamical imbalance between electrons and ions. In general, a consideration of the force balance is therefore not appropriate, but one has to consider the flux balance of particles (this is how one treats for instance the well known problem of spacecraft charging).

(see also
LTE,
Maxwell Distribution).

One should note that a detailed balance equation does, in contrast to LTE, in general not describe a closed system but assumes certain given input and output rates. For correct results it is obviously important to make sure that these are true sources and sinks for the particle or radiation densities, i.e. that these processes do in reality not significantly couple back into the system.

Elastic collisions of plasma electrons with bound atomic electrons can lead to collisional ionization and can therefore be an important factor for the atomic Level Population. It can also interfere with Radiative Recombination.

All type of collisions tend to inhibit collective processes in plasmas (see under Plasma).

Δ(Θ)= 4m_{1}m_{2}/( m_{1}+m_{2})^{2}^{.}sin^{2}(Θ/2) .

This form is normally used as a weighting function when integrating the Rutherford formula to obtain the total cross section for Coulomb scattering. However, this procedure neglects a further factor sin(Θ/2) which describes the density of particles hitting the surface of a spherical target and provides the geometrical connection between the mono-directional incident particle beam and the spherical scattering surface (this connection is ignored in the literature throughout, which invalidates in these cases the interpretation of the scattering angle as an independent variable; see (Coulomb Collision Cross Section). With the additional factor, the total cross section for Coulomb Scattering is finite and the **Total Relative Energy Loss** becomes

Δ= 8/(5π)^{.}m_{1}^{.}m_{2}/(m_{1}+m_{2})^{2} =

= 8/(5π)^{.}m_{1}/m_{2} if m_{1}<<m_{2}

The existence of an induced absorption process is therefore implausible, as the same physical cause (i.e. the external radiation field) can not result in two different effects. By means of symmetry arguments, this questions also the reality of the induced emission process.

(see also Scattering of Radiation).

Collisional excitation of hydrogen by protons is likely to be the decisive cooling process needed for star- and solar system formation and also for the low temperature of the photosphere of the present day sun (see /research/#A8 and /research/#A9 respectively).

ω_{B}= qB/mc ,

and the associated **Larmor Radius** is determined by

r_{B}= v/ω_{B} ,

where v is the velocity of the particle perpendicular to the magnetic field.

N_{n}= q_{n}/*ν*_{n}^{loss} ,

where q_{n} consists of the primary production rate due to Radiative Recombination and a secondary rate due to cascading from higher levels, i.e.

q_{n}= q_{n}^{Rec}+ q_{n}^{casc} .

q_{n}^{Rec} is determined by the plasma density and the radiative recombination cross section while q_{n}^{casc} is given by the population of levels higher than n and their Atomic Decay Probability.

The loss frequency on the other hand is given by

*ν*_{n}^{loss}= A_{n} +*ν*_{n}^{c} + *ν*_{n}^{*} ,

where A_{n} is the total decay probability to lower levels, *ν*_{n}^{c} the Collision Frequency for collisions with plasma electrons of sufficiently high energy to enable ionization from level n, and *ν*_{n}^{*} the photoionization frequency (if applicable).

(see /research/levschem.htm for a schematic illustration of these processes and /research/levpop.htm for numerical results applicable to the earth's ionosphere).

For anything but the highest gas densities, atomic processes (e.g. radiative transitions) and/or dynamical effects can be much more important than elastic collisions however and the assumption of LTE is not justified anymore within the whole energy range.

(see also Saha Equation, Planck Radiation Formula).

*f*(v)= 1/√π/v_{0}^{.}exp[-(v/v_{0})^{2}]

(which corresponds to the Boltzmann Distribution exp(-ε/ε_{0}) if formulated in terms of the energy).

For most practical applications this form is being taken for granted without further justification. However, in many cases the condition of elastic collisions dominating all other processes (LTE) is not even approximately fulfilled. This holds for instance for the physics of the ionosphere and space plasmas where recombination and collisional excitation (i.e. radiative processes) are of far greater importance in particular for the electrons. Not only would the assumption of a Maxwell distribution yield here quantitatively wrong results, but even prevent a correct qualitative understanding of the physics involved.

Even in the presence of elastic collisions only, the resultant equilibrium distribution can be different from a Maxwell distribution, namely if the situation is not isotropic.

For more see the page Collisional Relaxation of Gases and Maxwell Velocity Distribution.

L_{c}= 1/(N^{.}σ_{c})

A mean free path consideration shows for instance that the striations in glow discharges can not be explained by generally accepted values for the collisional excitation cross section (see /research/striatn.htm).

φ_{L}(*ν*)= A_{i,k}/4π^{2}/[(*ν*-*ν*_{i,k})^{2}+(A_{i,k}/4π)^{2}]

A_{i,k} is called the natural or damping width of the corresponding spectral line (this applies both for emission and absorption). The *ν*^{-2} decrease of the line intensity with increasing distance from the resonance frequency *ν*_{i,k} is characteristic of any exponentially damped oscillator.

The natural broadening may be masked both by Doppler Broadening and Stark Broadening (see also Spectral Line Shape).

τ(s)= σ^{.}_{0}∫^{s}ds'N(s') ,

where N is the density of the medium and σ the cross section for the corresponding process.

This definition assumes that σ is independent of the intensity of the radiation and therefore of the variable s (as the intensity in general will be a function of s). However, in the case of Photoionization, the disturbing influence of Plasma Field Fluctuations can reduce the absorption efficiency, resulting therefore in a more complicated behaviour (see Absorption).

In terms of τ, the intensity reduction is however always given by the **Exponential Absorption Law**

I(s)= I_{0}^{.}e^{-τ(s)} .

f_{i,k}:= A_{i,k}/Γ ,

where A_{i,k} is the quantum mechanical Atomic Decay Probability and

Γ= 2ω_{i,k}^{2}e^{2}/(3mc^{3})

is the classical damping constant for an oscillator with angular frequency ω_{i,k} (=2π*ν*_{i,k}).

Evaluation yields

f_{i,k}= 2π^{2}^{.}m/h^{.}*ν*_{i,k}^{.}<r>_{i,k}^{2} ,

where <r>_{i,k} is the quantum mechanical Overlap Integral for the states i and k in question.

One should note that this value is a factor 3/4 smaller than the usual expression for the oscillator strength quoted in the literature. However, this here should be the correct value as it has been derived strictly without any statistical assumptions. Established theory claims furthermore that the oscillator strength is subject to a normalization called the f-sum rule (i.e. Σ_{k}f_{i,k}=1). This is not correct as all transition are statistically independent possibilities for the electron and therefore all add up numerically. The probability for each transition is uniquely specified by the decay probability A_{i,k} and not the oscillator strength which, as indicated, is only a proportionality factor to derive the quantum mechanical resonance scattering cross section from the classical oscillator model.

<r>_{i,k}:= r_{0}^{.}_{0}∫^{∞}dρ Ψ_{i}(ρ)^{.}ρ^{.} Ψ_{k}(ρ) ,

where ρ is a dimensionless distance variable normalized to the Bohr radius

r_{0}= h^{2}/(4π^{2}me^{2})=

=5.3^{.}10^{-9} [cm]

with h the Planck constant, m the electron mass and e the elementary charge.

Ψ_{i} and Ψ_{k} designate the normalized radial wave functions of the energetically lower and upper state respectively. For bound-bound transitions, state k is characterized by the pair of principal quantum number and angular momentum (n,*l*) and i by (m,*l*±1), whereas for bound-free transitions (Photoionization, Radiative Recombination) k is characterized by the continuum electron energy ε (in this case, the corresponding wave function Ψ_{ε}(ρ) (the regular Coulomb wave function) can not be normalized separately and the absolute value of <r>_{i,k} has to be fixed by experimental measurements of the Photoionization cross section which is proportional to <r>_{i,k}^{2}).

In general, the overlap integral can only be calculated exactly by numerical integration. For principal quantum numbers m,n>>1 however, it can, both for bound-bound and bound-free transitions, be approximated analytically which enables a simple representation of both the cross section and decay constant in terms of these quantum numbers or the continuum energy (see under Resonance Scattering, Atomic Decay Probability, Photoionization, Radiative Recombination).

Note: the square of the overlap integral is sometimes also called 'Line Strength'. This name is only justified in absorption where the Resonance Scattering- cross section is directly determined by this quantity. In emission, the intensity of spectral lines depends (for a given Level Population) on the Atomic Decay Probability which is proportional to *ν*_{i,k}^{3}^{.}<r>_{i,k}^{2}).

For high effective quantum numbers n (for lower states it is still a good estimate), the cross section can be approximated by

σ_{n}^{Ion}(ε)= 3.7^{.}10^{-17}^{.}√(A/T)^{.}n^{2.4}^{.}h(ε) [cm^{2}] (n>>1) ,

where (roughly)

h(ε)= 1 for ε≤2ε_{n} and

h(ε)= (ε/2ε_{n})^{-2.9} for ε>2ε_{n}

and A the atomic mass number and T the neutral (ion) temperature in ^{o}K.

The power law dependence on the quantum number and energy has been derived 'empirically' from explicit numerical calculations involving exact wave functions for hydrogen-like atoms for electron energies ε=10^{-8}...4 Rydberg (1 Rydberg=13.6 eV) and quantum numbers n=1...1000 averaged over the angular momentum quantum number *l*. Over this range, the resultant absolute value for the cross section should be accurate to within a factor 2.

For sufficiently small radiation intensities, the photoionization cross section will be reduced due to the plasma field fluctuations (see Absorption) and the increase of the ionization time in comparison to the coherence time of the radiation (see Photons).

(see also the page Photoionization Theory for Coherent and Incoherent Light).

The notion of a photon still makes some sense though in as far as one is dealing with individual wave trains emitted in the course of the atomic transitions. In general there is no unique relationship however between the number of these wavetrains and the number of released photoelectrons, as the latter depends on certain factors like coherency (i.e. effective length) and amplitude of the wavetrains as well as disturbances of the Photoionization process by collisions.

It should be emphasized however that in most cases of practical interest the assumption of LTE is not appropriate and the spectrum will therefore differ from a Planck Function anyway, i.e. it becomes a function of several physical parameters instead of only the temperature.

(see also Continuum Radiation, LTE, Maxwell Distribution, Boltzmann Distribution, Saha Equation).

Generally, one has to distinguish between the microscopic and the collective properties of a plasma. The former are individual particle processes like Coulomb Scattering, Radiative Recombination or Inelastic Collisions, whereas the latter are for instance given by Plasma Polarization Fields, Plasma Oscillations or Debye Shielding. Collective processes can occur only if the Plasma Frequency is higher than the Collision Frequency. Apart from very high volume densities like those encountered in fluids, solids or the interior of stars, this is however usually fulfilled. For the latter example there is the additional property that, due to the high temperature in combination with the high density, no bound electronic states can exist and therefore no radiative processes either.

ΔE_{p}= 1.25^{.}10^{-9}^{.}N_{P}^{2/3} [statvolt/cm =3^{.}10^{4} V/m] ,

and

Δt_{f}= 2^{.}r_{p}/v .

For electrons with an energy of ε_{e} [eV], the fluctuation period becomes (N_{P} in [cm^{-3}])

Δt_{e}= 2^{.}10^{-8}/N_{P}^{1/3}/√ ε_{e} [sec] ,

whereas for ions with atomic mass number A and temperature T

Δt_{I}= 9.4^{.}10^{-5}/N_{P}^{1/3}/√(T/A) [sec] .

These plasma field fluctuations are important in several respects: they can broaden spectral lines as well as enhance their intensity and scattering cross section (see Stark Broadening, Emission Rate, Resonance Scattering). They can also interfere with other fields like that of electromagnetic waves and affect therefore for instance the Photoionization process.

E= 4π^{.}N_{p}^{.}e^{.}d ,

where e is the elementary charge.

In a collisionless plasma, this is equivalent of having an oscillator with frequency

ω_{p}= √(4π^{.}N_{p}^{.}e^{2}/m) ,

with m the electron mass (in principle the reduced mass M^{.}m/(M+m) for the corresponding ion and electron mass should be used here, but as m<<M this is practically identical with m; in any case, the usually assumed ion plasma frequency (where M replaces m in the equation above) does not occur here unless one is dealing with an ion-ion plasma).

For natural conditions, ω_{p} should be merely considered as the time scale for restoring charge neutrality rather than the frequency for a regular free plasma oscillation as the latter will not be stable in a chaotic medium. However, if the plasma is subjected to a well defined external perturbation like for instance electromagnetic waves, a precisely defined driven oscillation can be maintained in principle indefinitely (see Plasma Oscillations).

Ω^{-}= √(ω_{p}^{2}+ω_{B}^{2}) -ω_{B} and

Ω^{+}= √(ω_{p}^{2}+ω_{B}^{2}) +ω_{B} .

The two- dimensionality of the problem in the presence of a magnetic field causes a non-linearity of the displacement force which results in a modulation of any driven oscillation and therefore limits its maximum amplitude (see /research/#A2). This circumstance could be of crucial importance for the heating of plasmas by means of radio waves.

One has to be aware however that any type of sufficiently frequent collisions will prevent these systematic oscillations (see under Plasma).

E= √(4π^{.}N_{p}^{.}ε_{e}) .

However, it is obvious that classically a steady-state field can not be maintained as it would accelerate the ions and consequently make the plasma volume unstable (further processes can of course re-stabilize the situation; see below).

The only two solutions classically possible are therefore that either the electrons lose most of their kinetic energy to the ions (so that both diffuse with the same velocity), or that the electrons oscillate with regard to the ion background (see Plasma Oscillations) (a further possibility is of course that the charges can not be displaced in the first place because they are already confined locally by a magnetic field for instance).

In planetary and stellar atmospheres a steady-state plasma polarisation field does exist however because the upwards accelerated ions are lost at greater heights through recombination (see /research/#A6).

The quantum mechanical argument that radiation pressure is a necessary consequence of momentum conservation is also invalid as photons (i.e. electromagnetic wavetrains) are massless and in fact have no momentum (see the page regarding the Photoelectric Effect on my Physicsmyths website). Even if one assumes a momentum, a radiation pressure force could only be caused by a momentum change dp/dt, but this is not possible because the speed of light c has to be constant (the usual definition of the photon momentum p=E/c implies that momentum change is always associated with a given energy change, however for a particle with mass M, E=p

A true radiation pressure effect could only occur in the case of resonant scattering or absorption by bound atomic electrons (i.e. in spectral lines or for photoionization) as here the velocity of the oscillating electrons is always in phase with the driving field. For solid state materials, discrete resonances may in fact be broadened to such an extent as to result in a radiation pressure effect throughout the spectrum (see also Scattering of Radiation). The problem is however that one would have refer the velocity

It is therefore much more likely that in a given case the apparent 'radiation pressure' is caused either by thermal surface effects or electrons which are released from the surface by the radiation.

σ_{n}^{Rec}(ε)= 3.7^{.}10^{-17}^{.}√(A/T)^{.}n^{2.4}^{.}h(ε) [cm^{2}] (n>>1) ,

where (roughly)

h(ε)= 1 for ε≤2ε_{n} and

h(ε)= (ε/2ε_{n})^{-2.9} for ε>2ε_{n} ,

with ε the energy of the recombining electron, ε_{n} the ionization energy for level n, A the atomic mass number and T the neutral (ion) temperature in ^{o}K (see Photoionization for further explanations).

It should be noted that this result is very much different from the usual formulation found in the literature which is however inconsistently based on a statistical approach and does not conform with experimental and observational data (see /research/recrsect.htm).

For the calculation of the density of atomic levels populated by recombination, it is important that Radiative Recombination has to be described as a two-step process (see /research/levschem.htm), i.e. the electron recombines (more or less instantaneously) into a 'pre-bound' level *n* and from there into the actual level n with the **Recombination Decay Constant**

A_{n}^{Rec}= 7^{.}10^{4}^{.}n^{-3.4} [sec^{-1}]

This result is independent of the continuum energy ε as the energy dependence of the Overlap Integral cancels almost exactly the frequency dependence in the basic formula for the Atomic Decay Probability.

Also, it should be mentioned that, unlike the Atomic Decay Probability between bound levels , the recombination probability depends quite significantly on the angular momentum quantum numberIn certain cases, one wants a direct inversion of the radiative transfer equation, in the sense that the source function is directly determined by the measured intensities (rather than through trial and error fits of corresponding 'forward' model calculations). This poses in general much greater problems mathematically and numerically (see A Direct Numerical Solution to the Inverse Radiative Transfer Problem) .

The cross section for this process can be derived by considering the power radiated by a damped oscillator with damping constant A

For combined natural and Doppler broadening it is given by

σ_{i,k}(w)= 2π^{5/2}^{.}e^{2}/h^{.}*ν*_{i,k}/c/(Δ*ν*)_{D}^{.}
<r>_{i,k}^{2}^{.}H(a,w) ,

where

w=(*ν*-*ν*_{i,k})/(Δ*ν*)_{D}

is a dimensionless variable normalized to the Doppler width (Δ*ν*)_{D}, H(a,w) the Voigt- function for the corresponding natural and Doppler broadenings (see Spectral Line Shape), <r>_{i,k} the quantum mechanical Overlap Integral for the transition, and furthermore e the elementary charge, h the Planck constant and c the velocity of light.

For principal effective quantum numbers m,n>>1 the overlap integral <r>_{i,k} depends primarily on these parameters and can be approximated analytically, which leads to

σ_{m,n}(w)= 5.1^{.}10^{-12}^{.}[1/m^{2}-1/n^{2}]^{-3}^{.}m^{-1.8}^{.}(n-1)^{-3.2}^{.}√(A/T)
^{.}H(a,w) [cm^{2}] (m,n>>1)

where A is the atomic mass number and T the temperature in ^{o}K.

Usually, the angular distribution of the scattered radiation exhibits the typical characteristics for dipole scattering, i.e. the Rayleigh Scattering phase function 3/4^{.}(1+cos^{2}θ) for unpolarized incident radiation (θ=scattering angle).

The combination of the frequency coherence of the scattering in the atom's frame and the frequency due to the Doppler effect leads in general to complicated Partial Frequency Redistribution functions. In many cases the natural broadening is however negligible and the scattered line can be assumed to have a Doppler profile independent of the spectral shape of the incident radiation (Complete Frequency Redistribution). In this case the cross section above would of course also follow a Doppler profile, i.e. H(a,w) = exp(-w^{2}).

(see also LTE, Boltzmann Distribution, Maxwell Distribution).

It is furthermore not recognized in standard treatments that the usual effects of scattering due to the redistribution of radiation disappear in case of a continuous medium , that is if the wavelength exceeds the average distance of scatterers (see https://www.physicsmyths.org.uk/#refraction).

(see also Radiation Pressure).

H(a,w)= a/π^{.}_{-∞}∫^{∞}du exp(-u^{2})/[(w-u)^{2}+a^{2}] ,

where

w=(*ν*-*ν*_{i,k})/(Δ*ν*)_{D} and

a=A_{i,k}/4π/(Δ*ν*)_{D} ,

with (Δ*ν*)_{D} the Doppler broadening and A_{i,k} the natural damping constant.

H(a,w) combines the properties of the Maxwell velocity distribution function and Lorentz- damping profile and for a<<1 (which is usually the case) can be roughly approximated by

H(a,w)= exp(-w^{2})/(1+a) + a/√π/(1+w^{2}) .

For a more exact representation of H(a,w), one can numerically evaluate the above integral expression, but a more efficient way is to calculate it over the real part of the complex error function which is usually available in computer libraries of mathematical subroutines.

The Stark broadening due to plasma field fluctuations, which is dominant for sufficiently high plasma densities and/or quantum states, is determined by the
**Holtsmark Distribution** which can not be expressed as an elementary function but has to be approximated. In first order, the line wing intensity decreases like *ν*^{-5/2}, so it might be a sufficient approximation to replace the bracket under the integral for H(a,w) by [(w-u)^{2.5}+a_{S}^{2.5}] with a_{S} given by

a_{S}=(Δ*ν*)_{S}/4π/(Δ*ν*)_{D} ,

where (Δ*ν*)_{S} is the normal line width due to Stark Broadening.

In this case the integral would probably have to be evaluated explicitly by numerical integration (with an according re-normalization).

δε_{n}^{S} = ±e^{.}E_{S}^{.}r_{0}^{.}n^{2} ,

with e the elementary charge and r_{0} the Bohr radius.

This corresponds to the energy difference which the electron experiences in the field E_{S} on its path with radius r_{0}^{.}n^{2}.

However, the plasma microfield is not static but varies randomly with an average period Δt_{f} given by the plasma density N_{p} and the average particle velocity (see Plasma Field Fluctuations).

If Δt_{f} is much smaller than the orbital period

T_{n}= 8.6^{.}10^{-18}^{.}n^{3} [sec] ,

of the atomic electron, the latter will not experience any field at all any more. One can therefore assume that the energy splitting becomes correspondingly reduced by a factor

μ_{n}= 1/[1+(T_{n}/Δt_{e,I})^{2}] ,

where Δt_{e,I}=Δt_{e}+Δt_{I} is the average fluctuation period for the electrons and ions (because of the much smaller velocity, only the ions will usually be relevant here).

The normal level splitting in a plasma becomes therefore

Δε_{n}^{S}= ±2^{.}10^{-15}^{.}μ_{n}^{.}n^{2}^{.}N_{p}^{2/3} [eV] ,

where the value for the constants and the normal electric field strength in a plasma of density N_{p} (in [cm^{-3}]) have been inserted.

Because a spectral line frequency is given by the energy difference between states belonging to different principal quantum numbers m and n, the frequency broadening of a line is given by

Δ*ν*_{m,n}^{S}= ±0.49^{.}[μ_{n}^{.}n^{2}-μ_{m}^{.}m^{2}]^{.}N_{p}^{2/3} [Hz].

This statistical broadening is only the consequence of the variability of the energy levels due to the plasma field fluctuations and does not change the overall cross section or radiative emission rates because it is subject to the usual normalization of the particle density.

However, one can assume on the basis of observational evidence (see /research/#A3) that there is also a true broadening of each level due to the change dE_{p}/dt of the plasma field. This dynamical broadening can be taken to be of the form

Δ*ν*_{m,n}^{d}= ±0.49^{.}√(ζ_{n,e}^{2}+ζ_{n,I}^{2}) ^{.}n^{2}^{.}N_{p}^{2/3} [Hz],

where

ζ_{n,e}= Δt_{e}^{.}T_{n}/(Δt_{e}+T_{n})^{2} and

ζ_{n,I}= Δt_{I}^{.}T_{n}/(Δt_{I}+T_{n})^{2} .

Due to the functions ζ_{n,e} and ζ_{n,I}, the broadening has its maximum if either Δt_{e}= T_{n} or Δt_{I}= T_{n} (due to the small values of T_{n}, only the electrons will usually contribute here apart from sufficiently high quantum numbers).

This dynamical broadening is dominant for sufficiently high quantum numbers (e.g. n>60 for N_{p}=10^{5} cm^{-3}) and can vastly increase the scattering cross section and/or the radiative emission intensities because it is not subject to a normalization with regard to the particle density.

In both cases, the profile of the broadened line is given by the **Holtsmark Profile** which can not be expressed analytically in closed form (in first order, the line wing intensity decreases proportional to *ν*^{-5/2}, but even with this approximation it is not impossible to include the Stark Broadening into a general analytical function for the Spectral Line Shape).

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)

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