Resonant Excitation of Airglow
by High Power Radio Waves

Thomas Smid

Copyright Thomas Smid

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1. Introduction..............................................................................................................4
2.  Evidence for Resonant Electron Impact Excitation
in Laboratory Glow Discharges...............................................................................6

3.  Excitation of Airglow by High Power Radio Waves..............................................11

3.1  Electron Energy Oscillations Induced During
      Resonant Radio Wave - Plasma Interaction  ...................................................11
3.2  Excitation Rate of Atomic Transitions in the
    Presence of Electron Energy Oscillations........................................................14
3.3  Resulting Increase of the Level Densities
      for the Metastable States of Oxygen................................................................19
3.4  Enhancement of Spontaneous Atomic Line Emissions
      Due to Level Broadening by Plasma Field Fluctuations.................................24
3.5  Resultant  OI- Airglow Volume Emission Rates and Intensities.....................28
3.6  Non Equilibrium Considerations and Concluding Remarks............................32



A theory is presented which explains the increase of the intensity of ionospheric airglow emissions observed in the presence of high power HF radio waves by driven oscillations of the energy of the plasma electrons in combination with a narrow band, resonant type excitation behaviour at the atomic transition energy.  Cross section and energetical width of the excitation band are assumed to be identical with the line profile for resonant scattering of radiation. Due to the narrow width of the line and the small number of electrons within, a significant excitation rate is generally only achieved if electrons from a wider energetical range are systematically driven through this band. This scenario, which hitherto has not yet been considered in the literature concerning atomic excitation by electron impact, is also confirmed by well known phenomena observed with glow discharges in the laboratory, the usual (non-resonant) cross section being for instance unable to account for the sharply defined spatial structures (striations) related to the applied electric field and the specific transition energy.
With the present airglow enhancement problem, the necessary variation of the electron energy is caused by the non-linearity of the plasma oscillation driven by the radio wave. Its amplitude has a sharp maximum at the altitude where the resonance frequency of the magneto-plasma is equal to the wave frequency. The thickness of the interaction layer is determined by the modulation period of the non-linear plasma oscillation and the gradient of the local plasma density and amounts to about 3-4 km for the OI- airglow experiment considered here. The energy oscillation induced in this layer is of the order of 0.1 eV and leads to excitation rates of 16 cm-3 .sec-1  and  26 cm-3 .sec-1 for the metastable O(1D) and O(1S) levels respectively. These values yield almost exactly the observed increase of the intensities of both the forbidden OI -lines at 6300 Å and 5577Å, if one assumes additionally.that the apparent photon emission rate is generally enhanced by the ratio of the line broadening due to plasma field fluctuations to the natural line broadening. The absence of any observable diffusion effects of the excited O(1D) atoms in the considered airglow images suggests also that the level broadening causes a reduction of the lifetime of the metastable OI- levels.

1.  Introduction

In the past two decades, the observed phenomenon of airglow enhancements during modification of the ionosphere by high power radio waves has been a source of a number of theoretical speculations concerning the production mechanism involved (see Bernhardt et al., 1989 and references therein). Various hypotheses have been proposed in order to explain the assumed increase of the photoelectron flux which is thought to be the cause of the enhanced airglow emission during modification of the ionosphere.  However, those explanations suffer from the drawback that the necessary electron acceleration is not consistently derived from the radio wave - plasma interaction and at one point or the other free parameters have to be assumed in order to get the required results.  Numerical solutions of the consistent non -linear equation for the radio wave -plasma interaction (Smid, 1992 ) show in fact that even for much higher field strengths of the radio wave than actually used, no systematic electron acceleration over the necessary energy range is achieved. Due to the non-linear nature of the forced two -dimensional plasma oscillation, the electron energy can only change periodically, the amplitude and period being determined by the field strength of the radio wave and the electron energy. For the applied transmitter powers, the amplitude of the energy oscillation turns out to be much too small to explain the increase of the collisional excitation rate on the basis of the generally accepted cross section for this process (which is shown for instance in Bernhardt et al. (1989), p. 9083). The present paper shows that a cross section of a resonance type centered at the transition frequency yields the observed excitation rate under these circumstances.  Although this may look like an arbitrary hypothetical assumption ignoring the results of explicit experimental determinations of the corresponding cross section, it is in fact consistent with other experimental evidence proving that the generally accepted cross sections can be at most a side effect occurring with the explicit laboratory measurements which fail to resolve the actual resonance region at the transition frequency. This aspect is treated in  more detail in Section 2 independently of the airglow excitation problem.  In Sect.3.1,  the numerical solution of the non- linear equation for plasma oscillations, as formulated in Smid (1992 ) , is discussed with regard to the energy oscillation induced in the interaction layer for the considered ionospheric modification experiment. The results of these chapters are then used to determine the related change in the excitation rate for discrete atomic transitions in general (Sect.3.2) and the associated increase of the level densities for the specific OI- airglow problem (Sect.3.3), both under the assumption of a detailed balance equilibrium situation.
As an additional fundamental aspect of importance for the magnitude of the emitted airglow intensity, Sect.3.4 shows the necessity to assume generally an enhancement of the line emission of the individual atom if the atomic level broadening due to plasma field fluctuations exceeds the natural line broadening which is the case for the considered transitions of OI.  This generalized concept for calculating the conversion from atomic into radiative energy is supported by an independent consideration of the radio emission of a plasma due to recombination and subsequent atomic cascading,  because here the effect is much more dramatic and its inclusion leads to an almost exact agreement with observed values (in this case the radio flux density of the sun).
In Sect.3.5 , the results of Sects.3.1-3.4 are being combined to yield the actual airglow volume emission rates and observable intensities in the local equilibrium approximation. The possible consequences of deviations from the equilibrium situation and other effects which may affect the airglow intensities and should eventually be included in a complete and self consistent theory are discussed qualitatively in Sect.4

2.  Evidence for Resonant Electron Impact Excitation
    in Laboratory Glow Discharges

It is the generally accepted view that the cross section for excitation of discrete atomic levels by free electrons has a similar dependence on energy as the impact ionization cross section, i.e. it increases from zero at the threshold energy e0 to its maximum value at a few times e0 and decreases only very gradually towards higher energies (see the schematic Fig.1, 'non-resonant' curve;  for the case of excitation of O, which is of relevance for the airglow problem considered here, the cross section is for instance reproduced in Bernhardt et al. (1989)).
Previous work (Smid, 1987) has already indicated that all cross sections for inelastic collisions have to be reduced by 3-4 orders of magnitude in order to be consistent with satellite measurements of ionospheric electron flux spectra, yielding therewith mean free path lengths of more than 105 km for those processes at a height of about 300 km and ruling out a considerable amount of excitation over a scale much shorter than this.
Furthermore,  phenomena observed for laboratory plasmas suggest that for the case of discrete transitions the cross section should be strongly enhanced  within a very narrow energy band around the transition energy:
from Franck-Hertz type experiments with electric currents through gases, it is for instance evident that at gas pressures around 0.1 Torr (133 dyne/cm2 , which corresponds to a density  N»3.1015 cm-3 at room temperature)  the current becomes significantly reduced because of inelastic collisions of the electrons over a characteristic path length of only 2 centimeters for electron energies exceeding only marginally the theoretical threshold value (see for instance Maier-Leibnitz (1935)). Taking the published excitation cross sections one would certainly have to assume sex<10-17 cm2 , which corresponds to a mean free path length

lex  =  1/( )  ≥  30 cm ,     (1)

in contradiction to the observations.
An even more illustrative example is the phenomenon of striations observed in the positive column of laboratory glow discharges (Fig.2, Francis, 1956,  Paul, 1935) . Both optically and electrically, these features appear to be very sharply defined at least towards to cathode side of the discharge (i.e. towards the electron source). The value of the potential change within a striation as well as a spectral analysis of the light emission prove collisional excitation of atomic levels of the neutral background gas to be responsible for the development of these layers. The Langmuir probe measurements of Paul (1935), who examined striations in H2, indicate that the electrons, once they have aquired the necessary energy, are stopped due to inelastic collisions within a distance of  lex≤ 10-2 cm  (see Fig.3).  Considering the H2 -pressure  of about 0.6 Torr  (800 dyne/cm2 ), which corresponds to a density N=1.8.1016 cm-3, this sets a lower limit to the excitation cross section

sex  ≥  1/ (N.lex» 6 .10-15  [cm2]  ,       (2)

which is even two orders of magnitude above the maximum cross section for electron impact excitation as given in the literature.  The actual value may be even much higher, because the spatial resolution of the measurement is obviously limited by the thickness of the Langmuir probe, which was just of the order of 10-2 cm for this experiment.
The following quantitative estimate will show that it is in fact consistent with the observations to assume the cross section for electron impact excitation to be essentially identical with the resonant scattering cross section for light for the corresponding transition (see the schematic Fig.1 ('resonant curve'). As a crude model, it is sufficient here to consider the transition between the ground state and the first excited state of a hydrogen- like system , for which the resonant scattering cross section is

s(w)  =  6.6 .10-12  /Z4. Ö(A/T)  .H(a,w)      [cm2] ,       (3)

where  Z is the nuclear charge number ( for non hydrogen- like systems  Z-1 may , in an approximate sense, be considered as the degree of ionization, i.e.  Z=1 for the case of all neutral gases ), A is the atomic mass number, T the temperature of the gas in oK,

w= (f-f0) /Df        (4)

is the frequency displacement from the transition frequency f0 in terms of the thermal Doppler width

Df  = f0/c  . Ö(2kT/M)  ,       (5)

with c the velocity of light, k the Boltzmann constant and M the atomic mass  M=A. 1.67.10-24 g .
The frequency dependence is determined by the well known Voigt function  H(a,w)  where

a = Ai,k /4p /Df    (6)

is the ratio of the natural broadening Ai,k of the transition between states i and k  to the Doppler width of the line (H(a,w) can be assumed to be identical with the Maxwell distribution e-w2 for this purpose, because in practically all cases a<<1 and the natural broadening becomes then only important for frequency displacements from the resonance frequency where the cross section is negligibly small compared to the line center (or average) value.
In the present context, it is more appropriate to formulate the cross section in dependence of the energy instead of the frequency, so that the line parameter becomes

w  =  (e - e0) /De ,   (7)


De = e0/c  . Ö(2kT/M)  .   (8)

Assuming now that Eq.(3) determines also the cross section for electron impact excitation, one obtains for A=2, Z»1(i.e. H2)  and T=300oK a value at the transition energy e0 of

sexH2 (e0) =  5.1 .10-13      [cm2]      (9)

and a width of the resonance region of

DeexH2  =  5.5 . 10-5    [eV]  (10)

(these are of course only approximate values and can not be expected to represent the actual H2- system (which is not (atomic) hydrogen- like), better than probably to within a factor 2; the decimal points are  given here only for formal reasons and in order to avoid an accumulation of errors in the following consideration or inaccuracies when transforming the values to actually (atomic) hydrogen- like systems).

One should note that Eq.(9) fulfils in fact the experimentally imposed condition Eq.(2) , the mean free path length aquiring a value

lexH2 (e0)  =  1/ [N .  sexH2 (e0) ]  =
                      =  1.1 . 10-4     [cm] , (11)

where N=1.8 .1016  cm-3 has been used (see above Eq.(2).

In the case of an applied electric field, one has to take into account that the energy of the electron does not remain constantly at a value e0 but is only within the resonance region e0 ± De  over a distance 

DlE  =  2.De /(e.E) , (12)

with E the electric field strength and e the elementary charge.
Taking E to be of the order of 10 V/cm , which is the value indicated by the potential drop and spatial distance between two striations for the considered case, one obtains

DlE,exH » 1.1 .10-5    [cm]  . (13)

A comparison with the mean free path length for a constant energy (Eq.(11)) shows that the overall probability for excitation would only be of the order of 10% , which is not consistent with the observation that the full potential drop related to the atomic transition occurs at the head of the striation.  One must therefore conclude that the excitation coefficient is even larger than given by Eq.(11). The only imaginable enhancement could arise from a 'coherence effect' due to the circumstance that the average distance of the neutral gas atoms is less than the optical wavelength associated with the transition. In fact, for a transition energy  e0= 10 eV, one has

lex  »1.2 .10-5    [cm] , (14)

whereas the average distance of atoms

d(N)  = (N. 4p/3)-1/3  (15)

attains for the density of N=1.8 .1016  cm-3  the value

dH2 »2.4 .10-6      [cm] . (16)

Assuming a quadratic dependence on the ratio

d(e0, N)  = l(e0)/d(N) (17)

(this functional behaviour is also indicated by the scattering coefficient for electromagnetic waves  (see Smid, 1993 )) , one can generally define an effective (coherent) mean free path length

Lex (e0, N)  =  1/ [N .  sex(e0) . (1+d2(e0, N)) ] . (18)

For the considered experiment, one gets therefore

LexH2 (e0)  =  4.2 . 10-6     [cm] , (19)

which is , as required, smaller than the length  DlE,exH2 , so that the majority of electrons will be stopped within the characteristic length  LexH2  once the electric field has accelerated them to the transition energy e0 , leaving only a small residual velocity identical with the velocity distribution of the gas atoms in the laboratory system , i.e.

Dvel,resid  =  DvH2  = 1.6 .105 cm/sec . (20)

The applied electric field then accelerates the electrons again to the velocity corresponding to the transition energy e0  (»1.9 .108 cm/sec  if  e0»10 eV) where another excitation and formation of a striation layer takes place.
Since the electron does work only in the thin excitation region of extension  LexH2 , the apparent plasma potential changes therefore only here (by a value according to the excitation energy) and is constant everywhere else. The circumstance that Fig.3 seems to indicate a characteristic length of the excitation region of the order of 10-2 cm rather than about 10-5 cm must , as already indicated above, be attributed to the limited spatial resolution of the Langmuir probe, but may in this case in principle also be due to a smaller cross section for the responsible H2 resonance than given by  Eq.(9).
In any case it can however be excluded that the generally accepted cross section for excitation by electron impact above threshold (which would lead to mean free path lenghts of more than 1 cm ) is capable to produce the observed features. This proves the existence of a narrow energy band around the transition energy with a strongly enhanced cross section (as schematically indicated in Fig.1) and the consideration of the airglow problem in the next sections will only confirm this finding.

3.  Excitation of Airglow by High Power Radio Waves

3.1  Electron Energy Oscillations Induced During
        Resonant Radio Wave - Plasma Interaction

In the separate publication of Smid (1992 ), the general non-linear equation for the interaction of electromagnetic waves with plasmas has been formulated and solved numerically for the non-resonant and resonant case.  In the latter, the amplitude of the forced plasma oscillations appears to be strongly enhanced, limited only by the non-linear component which becomes important if the oscillation amplitude is comparable in order of magnitude to the initial Larmor radius of the electron, and imposes a modulation of the oscillation amplitude and the energy. 
In order to enhance numerically the non-linear effect and reach a conveniently short modulation period with regard to the affordable total length of the numerical integration period, the explicit calculations in Smid (1992 ) assumed a much higher value for the field strength than usually used for high power radio waves in the ionosphere and, furthermore,  the electron energy  (i.e. the Larmor radius) was taken as much smaller.
The values appropriate for actual modification experiments imply much longer modulation periods and therefore only one cycle could be computed here without an unreasonably excessive effort. In the present case, the conditions valid for the experiments discussed in Bernhardt et al. (1991) have been taken, which implies a (peak)- field strength of  E0=0.37 V/m in the beam center at the interaction height of ~ 300 km (assuming E02 (beam center) = 1.33 .E02 (0, -3 dB) ) and a plasma density  Np= 2.2 .105 cm-3  as one obtains from the transmitter frequency  of n0 = 5.8 MHz  and the relations (see Smid (1992 ), Eqs.(2),(17))

w0 = 2p .n0  = W+  =  Ö (wp2 +wB2 )  +wB (21)

wp Ö (4p . Np.e2 /m) , (22)

where wis the Larmor frequency for a magnetic field of B=0.5 Gauss ( it can be assumed that the W+ - resonance is consistent with the frequency applied here, because for absorption at W- = W+-2w(see Smid (1992 )) a somewhat unrealistic  plasma density of  Np »6 .105 cm-3 would be necessary).

The electron energy was chosen to correspond to the 6300 Å transition of OI , i.e. e0 = 1.96 eV , which results in an average of the component transverse to the magnetic field of

e0,^    =  2/3 .e0    »1.3  [eV] , (23)

assuming an isotropic distribution of the velocity vectors.

Fig.4 shows the gain of kinetic energy of the electrons (referred to the initial energy) over a time interval of 9.10-5 sec  (the kinetic rather than the total energy is taken here since only the former can be transformed into excitation energy). It is obvious that also for these conditions the energy change is oscillatory, in this case with an amplitude  Dekin»9.4 .10-2 eV and a period DT »8.7 .10-5 sec  ( the apparent drift of the energy minima towards higher energy values is of a numerical nature and disappears if the integration stepwidth (here  10-12 sec ) is made small enough ).

Further calculations with different field strengths E0 , electron energies e0 (in the eV - range) and plasma densities Np (assuming a fixed value for the magnetic field strength of B=0.5 G) yield empirically the following approximate relationships for resonant excitation at the frequency  W+ (further test calculations indicate that these expressions are also valid for excitation at  W- )

Dekin (w0 =W+» 2.9 .10-3  .E00.6 .e00.6 .Np0.3            [eV] (24)

DT (w0 =W+)      »  5.0 .10-3  .E0-0.6 .e00.4 .Np-0.4     [sec]  , (25)

where E0 has to be taken in [V/m], e0 in [eV]  and Np in [cm-3]  and Eq.(23) assumed to hold (see Smid,1992 , Eqs.(20) and (21)). These dependences are of importance when considering the excitation rate of different atomic states as a function of the power of the radio wave interacting with the plasma (see Sects.3.3 and 3.5) (one should note that the numerical calculations assumed the radio wave to be linearly polarized perpendicular to the magnetic field which is therefore also representative for circular polarized waves propagating perpendicular to B; additional calculations indicate that Eqs.(24) and (25) can also be considered to be valid for circular polarized waves propagating along B if the polarization is such that the E vector rotates in the same sense as the gyro motion of the electrons, whereas with the opposite sense of polarization no resonant behaviour at all is observed).

It is furthermore vital for the present purpose to know the effective width of the resonance line centered at the frequency Eq.(21) (»5.8 MHz here).  Calculations for frequencies w0 displaced from  Whave been performed for the parameters selected for the resonant case in Smid (1992 ) since then the non-linear oscillation can be followed over more periods within a given time. It turns out that the width of the resonance profile is given by the modulation frequency of the oscillation, which is not an unexpected result since the non- linear component can be interpreted as a kind of damping term limiting the amplitude of the oscillation (although not in a strict steady- state- but a periodical sense).  Numerically, one finds that the amplitude of the energy oscillation decreases to less than half of its exact resonance value Dekin(w0 =W+)  for frequency displacements greater than approximately

Dw  »±  1.5 . (2p / DT (w0 =W+) )  , (26) 

i.e  (from Eq.(25) )

Dw  » ± 1.9 .103  .E00.6 .e0-0.4 .Np0.4     [rad/sec]  (27)

(»± 1.1 .10[rad/sec] for the shown case) , but one observes also somewhat higher oscillation amplitudes (up to a factor 2) for smaller frequency displacements because of the frequency shift of the resonance due to the non- linear nature of the oscillation (note that in Fig.4d in Smid (1992 ) the highest peaks of the oscillation spectrum do not occur at the resonance frequencies W+ and Wbut at the sub-lines up- and down shifted (respectively)  by twice the modulation frequency ).
The exact spectral shape of the resonance profile could only be inferred from a much more detailed analysis, but it appears to be represented by (or at least closely related to)  the envelope of the group of lines associated with the frequency  W(and W- ) in the power spectrum of the oscillation (i.e. by the curve connecting the peaks of the sub-lines originating from the non- linear nature of the oscillation ).
In the following , the resonance profile for the energy oscillation shall therefore be approximated by a rectangular function with an amplitude Dekin (w0 =W+)  and a width Dw.  While the former quantity determines the volume emission rate of the airglow, the latter defines the thickness of the interaction layer in the ionosphere, since over Eqs.(21) and (22) it corresponds to a relative variation of the plasma density of

DNp /Np  »± 2.Dw/wp . Ö (1 + wB2 / wp2) (28)

(this expression holds for small variations, i.e.  if  Dw << Ö( wp2 +wB2 ) assuming a constant gyrofrequency wB) .
Over the local height gradient of Np this yields therefore the height range of the resonant radio wave- plasma interaction (see  Sect.3.5).

For completeness it should be pointed out that collisions should be negligible for the present plasma oscillation problem, because the time scale defined by DT (»10-4  sec here)  is small compared to the collision time in the ionosphere which amounts to about 100 sec for energy changing (electron- electron)  and about 10-2 sec for momentum changing (electron- neutral) collisions at the considered height of 300 km. The equation of motion as established in Smid (1992 ) is therefore completely adequate in this case.

3.2  Excitation Rate of Atomic Transitions
      in the Presence of Electron Energy Oscillations

From the relatively small amplitude of the energy oscillation induced within the ionospheric plasma during typical modification experiments (see Sect.3.1, Eq.(24)) it is clear that only a resonant type excitation cross section (see discussion in Sect.1 and 2) can account for a significant increase in the excitation rate of atomic transitions by free electrons.  Similarly to the glow discharge phenomena discussed in Sect.2 ,  the observed effects are not consistent with the energetical characteristics of the excitation cross section as given in the literature. The various theoretical models based on the usual non-resonant excitation cross section (see Bernhardt et al. (1989) and references therein) purport to yield the observed airglow enhancements but have to assume that the flux of photoelectrons with energies greater than the excitation threshold is strongly increased by the high power radio wave due to an according acceleration of hypothetical ambient (thermal) electrons.  Apart from the fact that the popular assumption that the majority of electrons produced by photoionization thermalize at energies of about 0.1 eV can in no way be supported by numerical calculations involving the various elastic and inelastic energy transfers of relevance (Smid, 1987) , the theoretical problem of the radio wave- plasma interaction is not treated consistently in these works but the proposed acceleration process is ascribed to some speculative mechanisms which lack a theoretical proof.
The usual non- resonant excitation may of course still contribute to the airglow (explaining for instance parts of the background emission in the absence of induced plasma oscillations)  but the strong enhancements associated with the relatively small oscillation amplitude of the electron energy of the order of 0.1 eV (see Fig.4) is only compatible with a resonant type cross section concentrated within a small energy region (see the schematic Fig.1).
Taking the cross section postulated in Sect.2, which is identical with the resonant scattering cross section for dipole allowed Lya -transitions (Eq.(3)) and has already been proved to be consistent with the phenomenon observed in connection with glow discharge experiments in Sect.2, and considering furthermore the fact that for the optical transitions of interest here the cross section is about a factor 6 larger due to the larger dipole moment  (s(Ha)/s(Lya) »5.6 ), one obtains for the considered case of oxygen at 1200 oK a cross section at the transition energy of

sexO (e0» 4.3 .10-12      [cm2] (29)

and for the considered transition energy e0 = 1.96 eV (see next Sect.3.3) a Gaussian halfwidth of the cross section of (see Eq.(8))

DeexO (1.96 eV)  =  7.4 . 10-6      [eV]  . (30)

Since  DeexO is very small with regard to the energy scale of variations in the initial electron flux spectrum , one may approximate the cross section for the present purpose by a rectangular function

sexO (e0» 3.1 .10-12                  [cm2] (31)

DeexO (1.96 eV)  =  1.8 . 10-5        [eV] , (32)

where  Deexis now the full width of the resonant excitation region.
In the presence of energy oscillations of the plasma with amplitude Dekin , one has the schematic picture of electrons passing through the excitation region as shown in Fig.5: the two indicated cases with the initial energies eand e2 define the energy range of electrons which are initially outside the excitation region but pass through it in the presence of energy oscillations induced by the radio wave (note that Dekin is always positive (see Fig.4), i.e. only electrons with initial energies smaller than the resonant energy contribute).
Since  DeexO << Dekin ,  one can say that all electrons within the interval e0 -Dekin , e0 pass through the excitation region near e0 twice within the time interval DT (see Eq.(25))  (in principle both Dekin and DT are a function of the electron energy e ;  both quantities vary however only slightly within the energy interval Dekin<< e0 (see Eqs.(24) and (25)).
Designating the number density of of this group of electrons with DN  and the number density of electrons within the excitation region e0 ± DeexO /2  with Dn , one can formulate the equilibrium balance equations for the two quantities as  (for convenience Deexshall be abbreviated  as De  and Dekin as De  here and in the following)

Np .nc .De /e + 2 DN  /DT  =  2 Dn .De /De /DT  + Dn .(nex +nc +nrec +ni) (33)

Np .nc .De /e+  2 Dn .De /De /DT =  2 DN  /DT  + DN .(nc +nrec +ni)    .    (34)

The upper equation describes the production and loss of electrons in the narrow excitation band  e0 ± De/2 , whereas the lower holds for the related  'oscillation region'  e0 -Dee0 .
The first term on the left hand side of both equations represents the production of electrons within the considered energy intervals due to elastic collisions in the plasma of total density Np and the average plasma energy  ep .  This process , which occurs with the collision frequency

nc  » 1.2 . 10-8  .N/ ep            [sec-1] (35)

(where Np is in [cm-3]  and ep in units of Rydberg ; see Smid (1993 ), Appendix B, n0there),  can , for the present purpose, be considered as the primary production term, because on the average it exceeds the original ionization rate for the conditions of interest here  (this is evident from comparing nto the recombination frequency

nrec  =  1.2 .10-8  .Np . Ö(A/T)  . e0-1.2 [sec-1] (36)

(Smid, 1993 ), which is on the average related to the ionization rate over the principle of detailed balance  (the factor  Ö(A/T) reduces here nrec by almost one order of magnitude in comparison to n for  e0 = ep)).
The second term on the left hand and the first on the right hand side of Eqs.(33) and (34) are the gains and losses (respectively) because of the electrons entering and leaving the two energy regions in question . The second term on the right hand side represents the losses due to recombination, elastic (nc) and inelastic (ni) collisions and, in case of the first equation, excitation of the considered atomic transition which occurs with the frequency

nex  =  N(O) . sexO (e0) . v(e0) . (1 + d2(e0,N(O))  ) , (37)

where the 'coherence factor' d(e0,N(O)) is given by Eq.(17) .

Note that recombination as well as inelastic collisions (other than the explicitly considered transition) has been formally taken into account here as a loss process , which seems to be inconsistent with the neglection of production terms due to photoionization (see above) and inelastic collisions.  The present form of the equations is however appropriate if  the frequencies for recombination and inelastic collisions within the considered energy band (defined by e0 and De) are significantly higher than the average values for the plasma, which is true here (the numerical results will show, however, that these terms are anyway without importance for the present problem because of the high resulting values for the excitation frequency nex).

Solving Eqs.(33) and (34)  for Dn  , one obtains for the steady state number density of plasma electrons in the resonant excitation region

Dn  =  qp /(np +nex) , (38)


qp Np .nc   . [De  + De /g] /ep (39)

np nc +nrec +ni    + 2 . (1-1/g) . De /De /DT (40)


g  =  1+ DT . (nc +nrec +ni)/2 (41)

(note that  ep , De  and  De have to be taken in identical units in Eq.(39), the same holds for Eq.(43) and (44) below ) .

The related excitation rate is then

qex Dn . nex . (42)

In the plasma oscillation limit with g »1  (i.e. if the induced plasma oscillation dominates collisions in the excitation region, which is true for the problem considered here ( see Eqs.(25), (35) and (36))  , Eq.(38) reduces to

DnoscNp .nc  . [De  + De ]  / [nex +  (nc +nrec +ni) . (De /De  +1 ) ]  /ep    .    (43)

If  nex  is still large compared to the second term in the denominator despite  De /De  >>1  (as it is the case for the present consideration as far as transitions from the ground state are concerned,  see Eqs.(24) , (32)  and Eqs.(62) -(66) ) one obtains for the excitation rate the asymptotic expression

qex0Np .nc . De /ep    , (44)

i.e.  it is equal to the (collisional) production of electrons within the energy band of width De as defined by the amplitude of the energy oscillation near the resonance energy  e0.  Due to the forced non-linear plasma oscillation , the width of the resonant excitation region is therefore effectively broadened from De  to  De  and the excitation rate increases by a factor  De /De  compared to the undisturbed case  (one should be aware that this takes only the resonant excitation into account and a non-resonant component may possibly be dominant in the absence of induced plasma oscillations) (see also the remarks at the beginning of this section).
In this limit of high excitation frequencies nex , the equilibrium excitation rate is therefore only a function of the plasma properties and is completely independent of both the density of the excited gas and the corresponding excitation cross section.  The only difference in  qex0  for different transitions of the same gas arises here from the dependence of the amplitude of the energy oscillation De on the transition energy  e0 (see Eq.(24))    (note however that this result is based on the assumption of a homogeneous (frequency independent) primary (collisional) production rate  in Eqs.(33) and (34)  which may lead to considerable inaccuracies in the case that the electron production rate is strongly variable with energy due to strong spectral inhomogeneities in the actual primary electron production (e.g. photoionization) and/or if the characteristic energy of the atomic transition is not of the order of the average plasma energy  ep ).
For the considered case of the ionospheric  OI- airglow enhancement,  the asymptotic form given by Eq.(44) holds for all transitions originating from the  (3P) -ground state of oxygen (see next Sect.3.3).

3.3  Resulting Increase of the Level Densities
      for the Metastable States of Oxygen

The well known OI- airglow emissions in visible light result from dipole- forbidden transitions between the metastable  1D and  1S states and  the  3P  ground state of oxygen (see Grotrian diagram Fig.6) .  In order to determine the emission rate for each transition consistently, one has (at least formally) to solve the rate equations for the 1D and 1S level simultaneously.  Neglecting cascading from and losses to higher levels (i.e. assuming a 3-level problem) and designating  the 3P , 1D  and  1S  states with the indices  P ,D and S  respectively, the equilibrium equations for the production and loss rates of electrons in the metastable levels can be written as

qex,D  +  nS. AS,D   =  nD . ( AD,P + DnD,S . kD,S  + nq,D) (45)

qex,S  +  nD. DnD,S . kD,S     =  nS . ( AS,P + AS,D + nq,S) , (46)

where  nD and n are the unknown number densities of atoms in the corresponding state,  qex,D and qex,S  the production rate into the levels according to  Eq.(44)  (the natural production rate which leads to the always observed background emission shall not be considered here),
AS,P , AD,P and AS,D are the effective radiative decay constants for the transitions between the metastable levels, each of which assumed to be of the form

Aa,b    = å  [ Aa,bj  + (DfB)a,bj ] , (47)

where Aa,bj  is the usual intrinsic ('natural') decay constant for the individual multiplet line related to the lower sub-state bj  and  (DfB)a,bj   is the line broadening caused by plasma field fluctuations (see Sect.3.4, Eq.(75)) (note that this form for the decay constant, which results in a decrease of the life time of the level if the plasma broadening becomes comparable to or greater than the natural broadening , seems only to be valid for the present case of dipole forbidden transitions (see remarks at the end of Sect.3.4).

Only for the OI- 6300/6364 Å dublett emission is the multiplicity assumed to be of actual importance here. The numerical values for the individual transition constants are according to Wiese et al. (1966) or  Bernhardt et al., (1989)

AD,P1  (6300 Å)  =  5.1 .10-3     [sec-1] (48)
AD,P2  (6364 Å)  =  1.6 .10-3     [sec-1] (49)
AS,D    (5577 Å)  = 1.34            [sec-1] (50)
AS,P     (2972 Å)  =  6.7  .10-2     [sec-1] , (51)

whereas for all these transitions a constant value

DfB  = 6.6 .10-2      [Hz] (52)

can be adopted in an approximate sense (see Sect.3.4, Eq.(82)) . 

This leads to the effective decay constants

AD,P =  0.14    [sec-1] (53)
AS,D =  1.41     [sec-1] (54)
AS,P =  0.13       [sec-1] . (55)

In view of these values , both the additional frequencies nq,and nq,S  , which take eventual losses for the O(1D) and  O(1S) -level due to non-radiative processes ('quenching') into account, become insignificant here, because even the generally assumed quenching of  O(1D) by N2 is slow compared to this at a height of about 300 km ( nq,D = 2.3 .10-3 sec-1 for a density  N(N2) =108 cm-3 and a reaction coefficient  kq=2.3 .10-11 cm3/sec  (Bernhardt et al., 1989)  (quenching of the O(1S) level is anyway only of importance below 100 km ( Jones, 1974),  i.e.  nq,S = 0  here ).

Despite turning out to be insignificant here as well (see below Eq.(70)), collisional excitation between the metastable states is formally taken into account in Eqs.(45) and (46) through the term DnD,S . kD,S  , where DnD,S  is the number density of plasma electrons able for resonant excitation according to Eq.(38) and 

kD,S = sexO (eD,S) .vD,S 
»              2.7 .10-4                [cm3/sec ] (56)

is the related transition coefficient according to Eq.(31) and a velocity  vD,S = 8.9.107 cm/sec  corresponding to the transition energy  eD,S=2.2  eV.
Because of this term, Eqs.(45) and (46)  are formally non-linear in  nD , since  DnD,S (Eq.(37)) depends upon the excitation frequency 

(nex)D,S  =  nD. kD,S . (57)

Solving  Eq.(46) for nS  yields

nS  =  [ qex,S  +  (qp )D,S / ( 1 + (np )D,S / (nD. kD,S )  )  ] / [ AS,D +AS,P +nq,S ]
, (58)

where the index  D,S  at qp and np shall indicate that those quantities Eq.(39) and (40) which depend on the plasma electron energy (and therefore the wavelength of the particular transition)  (i.e nrec , De , De , DT ,g  )  have to be taken at the energy corresponding to the transition between the metastable states (i.e. e0 = 2.2 eV  @  l= 5577 Å ).

Inserting this expression into Eq.(45) , the resultant quadratic equation determines  nD as

nD  = - N/2  + Ö [N2/4  +  ( qex,D + qex,S.b ) . (np )D,S  /kD,S  /( AD,P +nq,D ) ] ,


N =  (np )D,S /kD,S    + [(qp )D,S . (1-b)  - qex,D  - qex,S.b ] /[ AD,P +nq,D ] (60)


b  =  AS,D / ( AS,D + AS,P + nq,S) . (61)

For the present consideration, the numerical values valid for the airglow experiment described in Bernhardt et al. (1991) have been adopted. This leads to a plasma density  Np = 2.2 .105 cm-3   (see also Sect.3.1),  a neutral oxygen density  N(O) »10cm-3 and an electron velocity  of  v(eP,D =1.96 eV)  = 8.4 .107  cm/sec in the case of the 6300 Å transition . Over Eq.(37) , this results in an excitation frequency for an electron at the corresponding energy of  (note that eP,D has been designated as e0 in Sects.3.1 and 3.2)

(nex)P,D  » 2.5 . 105      [sec-1] (62)

(note that  d(e0,N(O)) »0  for the given value of  N(O)  (see Eq.(17)) and therefore the usual expression for the excitation frequency results in Eq.(37)).
The 2972 Å transition, which populates the lowest 1S level from the 3P -ground state, is associated with a higher electron velocity (eP,S =4.16 eV) and yields the correspondingly higher excitation frequency

(nex)P,S  » 3.7 . 105      [sec-1] . (63)

One should note that the exact numerical values of (nex)P,D and (nex)P,S are here however without relevance , because the asymptotic expression Eq.(44) holds for the excitation rates qex,D  and qex,S as it follows from the elastic collision and recombination frequencies  (see Eqs.(35))  and (36) )

nc  »              5.3 .10-3        [sec-1] (64)
nrec (eP,D)    »  3.1 .10-3    [sec-1] (65)
nrec (eP,S )    »  1.3 .10-3    [sec-1] , (66)

an inelastic collision frequency ni of the same order as nc and the magnitude of  the relative resonance broadening De /De  (Eq.(24) and (32)).
(Note that in the case of Eq.(64) a value of ep = 0.5 (»7 eV) has been assumed for the average plasma energy which corresponds to dayside conditions at a height of 300 km (this can be shown by a theoretical computation of the electron flux spectrum involving the production by photoionization, loss by recombination and the various inelastic and elastic collision processes (Smid,1987)) ;  for nightside conditions, which are appropriate for the airglow experiment discussed in Bernhardt et al. (1991), the dayside model can of course not be applied due to the absence of photoionization. The fact that the nighttime plasma density in the ionospheric F-region is still of the same order of magnitude than the daytime value indicates however that the corresponding source of ionization (which still can not yet be considered to be known) possesses an energy spectrum which is not dramatically different from the photoionization case (i.e. the ionization rate varies significantly only over an energy scale of ≥1 Rydberg if one neglects the individual spectral lines).
For Eqs.(65) and (66) , the ionized oxygen temperature was taken as T=1200 oK, i.e it was assumed to be identical with the neutral temperature at the given height. )

On the basis of Eq.(44),  only the width of the energetical plasma oscillation De  determines  qex,D  and qex,S . With Eq.(24)  for a field strength of E0= 0.37 V/m (beam center) one obtains therefore

qex,D »16      [cm-3 . sec-1(67)
qex,S  »26      [cm-3 . sec-1 ] (68)

(note that it makes no difference for qex,D that the excitation of the 1D level does actually consist of 2 transitions (6300 and 6364 Å), because both are separated by much less than the energy oscillation amplitude Dekin and the relevant expression for the equilibrium excitation rate (Eq.(44)) is independent of the excitation frequency nex ).
With these values, Eqs.(48) and (49),(50) yield the density increases in the metastable levels

n»  280      [cm-3]            (69)
n»    17      [cm-3]        (70)

(it should be mentioned that excitation from the  1D  to the  1S state does not affect the level densities significantly here, i.e. the corresponding term in the level equations (involving kD,S) could have been neglected for the present purpose (the effect is of the order of 0.1% of for nS and even less for nD ).

One should note that the excitation rate as given by Eqs.(67),(68) amounts to an appreciable percentage of the total undisturbed electron production/recombination rate ( the latter is »250 cm-3 sec-1  for the given plasma density of Np= 2.2 .105 cm-3 and a recombination coefficient a »5.10-9 cm3/sec ). Since after excitation the corresponding electrons will have a very small energy where the recombination cross section is much higher, the recombination rate (in particular into highly excited states) could increase therefore significantly and lead to an according reduction of the plasma density (see also discussion in Sect.4).

3.4  Enhancement of Spontaneous Atomic Line Emissions
      Due to Level Broadening by Plasma Field Fluctuations

According to the 'classical' photon theory of atomic line emissions, each spontaneous decay from an upper energy level  en to a lower level em is related to the emission of a wavetrain of electromagnetic radiation with an associated energy

(erad)0n,m=  h .nn,m  en - em , (71)

where h is Planck's constant and nn,m the frequency of the wave. Reversely, the radiative energy emitted in one decay process is converted again into the atomic excitation energy  h.nn,m  on absorption by an atomic system.
This model for the relationship and interaction between a radiation field and a quantized atomic system is assumed to hold without restriction and to determine therefore uniquely the energy balance between matter and radiation in the universe.
Observational evidence suggests, however, that the conversion from the quantum mechanical energy difference of the atom into radiative energy is not under all circumstances provided by Eq.(71). This is most dramatically apparent when considering the radio flux emitted from a plasma due to recombination and subsequent cascading of the atom. A convenient example for radio emission from a plasma is provided by the sun. Fig.7 shows the solar radio flux measured at the earth from 10 - 104 MHz. In comparison to the quiet sun case (thick fully drawn curve), Fig.8 gives the flux in the same frequency range to be expected theoretically from recombination and cascading within (half) a spherical shell  (R= 7.105 km , DR = 105 km ) of a plasma of density  Np =5.108 cm-3 , an average electron energy ep =0.5 Ry and an ion temperature T= 1.106 oK  at a distance from the earth D= 1.5 .10km (Smid, 1992, unpublished). These parameters correspond to conditions valid for the lower solar corona and should result in a emission which matches the actually emitted solar flux at a frequency of about  200 MHz (Shklovskii, 1965;  Zirin,1988). 
It is obvious that with the transformation given by Eq.(71) (dashed curve), the theoretical flux at 200 MHz is about 11 orders of magnitude smaller than the observed one. This mismatch can in no reasonable way be compensated by taking different values for the above plasma parameters or assuming inaccuracies in the applied numerical recombination scheme, and one has therefore to conclude that Eq.(71) is in this case not the appropriate formula for calculating the conversion of atomic excitation energy into radiative energy, i.e. one has to assume that generally the transformation is of the form

(erad )n,m  = (erad)0n,m  . (hrad )n,m , (72)

where the enhancement factor (hrad)n,m accounts for the difference observed here. The only imaginable quantity determining  (hrad)n,m  is the broadening of the atomic levels due to plasma field fluctuations in relation to the natural broadening of the transition, i.e.

(hrad )n,m  =  1 +  (DfB)n,m /An,m  , (73)

where  An,m  is the intrinsic (natural) broadening of the decay from level n to m,  which for dipole allowed transitions takes on the form (see Smid, 1993 )

An,m  = 1.3 .109 . m-1.8 .(n-1)-3.2     [sec-1]      , (74)


(DfB)n,m  =  Ö ((DfB)n2 + (DfB)m2) (75)

is the broadening of the line due to plasma field fluctuations which cause a frequency dispersion of the individual atomic level by

(DfB)= 2/Öp . (Df).n2 . Ö (zn,e2  + zn,I2) (76)


(Df) =  0.49 . Np2/3          [Hz] (77)

(where Np has to be taken in  [cm-3]  ) and the function zn generally determined by the fluctuation period of the plasma field Dtf and the angular period of revolution of the atomic electron in level n

Tn  =  8.6 .10-18 .n3  [sec] (78)


zDtf .Tn / (Dtf +Tn )2  . (79)

The fluctuation period attains for the plasma electrons the numerical value

Dtf,e  =  5.6 .10-9 /Np1/3  / Öep        [sec]  (80)

and for the ions

Dtf,I  =  9.4 .10-5 /Np1/3  / Ö(T/A)        [sec] . (81)

where the plasma density Np has again to be taken in [cm-3] , the average electron energy in units of Rydberg and the temperature  T of the ions with atomic mass number A in  [oK]  (see Smid ,1993 ).

For frequencies  < 1GHz and the solar coronal plasma parameters given above, the main contribution of the emission is due to  na- transitions near n»150 because here the level broadening is of the same order as the level separation, resulting therefore with the above formulae in an enhancement factor of the order of 1011, as required by the observations.
The exact value at 200 MHz (obtained by a consistent summation of the contributions of all levels) is  hrad = 1.0 .1011 , yielding therewith an absolute solar flux density of  5. 10-22  W/m2/Hz  (5 solar flux units, thin solid curve in Fig.8 ) which is in good agreement with the oberserved value (7 s.f.u.) (it is obvious that both for frequencies significantly lower and higher than 200 MHz the theoretical flux values tend to diverge from the observed curve in the sense of an overestimation although only the production of radiation within the part of the solar atmosphere responsible for the 200 MHz emission (coronal base rather than the whole corona + chromosphere) has been considered. This circumstance could probably be due to radio wave propagation effects (i.e. interference and multiple scattering) which have not been taken into account here for the emission model).

For smaller quantum numbers, the radiative enhancement factor decreases rapidly due to the specific dependence on m and n in Eqs.(74) -(79) . In the case of transitions in the optical wavelength region (m=2, n=3) and for the plasma density considered for the airglow problem in this paper  (Np =2.2 .105 cm-3) the plasma line  broadening amounts only to

(DfB)opt (Np =2.2 .105)    »6.6 .10-2      [Hz] . (82)

(note that this value has been taken a factor 2 higher than it would follow from Eq.(76) , since the latter has been derived from an approximate expression for the atomic dipole moment valid for n>>1 which underestimates the orbital radius for small n by a factor 0.5).
(DfB)opt is here , in contrast to transitions in the radio region, many orders of magnitude smaller than the natural line broadening for dipole allowed transitions (A3,2 =  4.107 sec-1)  , i.e.  (hrad)opt »1 and the usual formula for conversion of atomic into radiative energy results in this case (Eq.(71)).
It is however obvious that for metastable atomic levels with transition constants smaller than (DfB)opt  , the enhancement effect (hrad >1 ) becomes also relevant in the optical region and should therefore significantly affect the ionospheric airglow emission examined in this paper (see next section). 
While for dipole allowed transitions the enhancement of the emitted radiation appears not to be associated with an according reduction in the life time of the upper level (this is suggested by a comparison of theoretical calculations of the coefficient for scattering of radio waves by high atomic Rydberg states with experimentally determined ionospheric backscatter coefficients (Smid,1993 )), the present examination of the OI- airglow problem indicates that this seems to be the case for metastable (dipole forbidden) states (see Sect.3.3).

3.5  Resultant  OI- Airglow Volume Emission Rates and Intensities

Taking the enhancement of the emitted atomic radiation due to level broadening by plasma field fluctuations (Eq.(73)) into account, the photon emission rate for a decay from the upper level n to the individual lower substate mj can be written as

qn,mj  =  n. An,mj. An/An[1 +  (DfB)n,mj /An,mj  ]  , (83)


An  = å å  An,mj  (84)
                      m      j

An    = å  An,m (85)

(see Eq.(47)).

It is evident that qn,mj does not depend explicitly on An,mj if    (DfB)n,mj/An,mj >>1.  In this limit, every sub- line of a given multiplet structure is emitted with the same intensity  (experimental evidence for this phenomenon is given by Fig.9, where the OI-  6300 and 6364 Å night airglow lines happen to have almost equal strength rather than showing an intensity ratio according to the transition probabilities Eqs.(48) and (49)).

With Eqs.(48) - (52) and (69),(70) , the enhancements of the volume emission rates for the different transition frequencies related to the metastable OI- levels become

qD,P (6300 Å)  » 416        [cm-3 . sec-1]                 (86)
qD,P (6364 Å)  » 396        [cm-3 . sec-1] (87)
qS,D (5577 Å)  »  26        [cm-3 . sec-1] (88)
qS,P (2972 Å)    »    2.5    [cm-3 . sec-1] .                 (89)

Because of the very small resonant scattering cross section for dipole forbidden transitions (which is reduced in proportion to the decay constant if compared to dipole allowed transitions of the same frequency), multiple scattering effects can be neglected here and the registered intensity at a given wavelength is simply obtained by integrating the volume emission rate along the line of sight, i.e.
I(l)  = òds' q(s',l) . e-tab(s',l) , (90)

where the exponential factor accounts for an eventual continuous absorption process resulting in an optical depth tab . The only line affected here by absorption is the near- UV emission at 2972 Å which can photodissociate O3 . Because of the small initial emission rate and the further reduction of the intensity in the ozone layer , the following treatment shall only deal with the optical emissions for which absorption can be neglected due to the low density of atmospheric constituents that can be ionized or dissociated by radiation of these wavelengths.
For the present purpose, the radiating volume shall be approximated by a layer of finite thickness Ds(l) with a constant emission rate  q(l), i.e.

I(l)  =  10-6 .Ds(l) . q(l)        [R] , (91)

where the factor 10-6 provides the transformation to the intensity unit of Rayleighs.

Ds(l)  is determined by the frequency width of the plasma resonance and the height gradient of the plasma density  (Eqs.(27) and (28) ) and is therefore in general only implicitly dependent on the emission wavelength l , since Dw  depends explicitly only on the excitation energy  e0 of the atomic level, which may be different from the de- excitation energy associated with the emission wavelength l . With the present example, the O(1D) state can be populated directly from the 3P ground state as well as indirectly over the 1S state. Because of the different values for the excitation energy e0 , lines originating from the 1D - level consist therefore of two contributions with different values for Dw and a weighted average has to be performed in order to obtain the effective value.
It is evident from Eqs.(48) - (52) and (67),(68) that excitation over the 1S- state contributes with a relative weight »0.6 to the density in the 1D- level. From Eq.(27) one gets therefore (with E0=0.37 V/m and Np=2.2.105 cm-3)

Dw (6300,6364 Å )  =  0.6 . Dw(e0 =4.16 eV) +  0.4 . Dw(e0 =1.96 eV)  »
                                          » ±  9.3 .104  [rad/sec] . (92)

Due to the small transition rate from the 1D-  to the 1S- level , the 5577 Å emission results only from direct excitation from the 1P ground state , i.e.

Dw (5577Å )  =  Dw(e0 =4.16 eV)  »
                                          » ±  8.1 .104  [rad/sec] . (93)

From Eq.(28)  one  obtains thus here ( wp =2.6 .107 rad/sec ,  wB =8.8 .106 rad/sec)

DNp/N(6300,6364 Å )  »  ±  0.75 %                 (94)
DNp/N(5577 Å )            »  ±  0.65 %  . (95)

It depends obviously on the details of the plasma density gradient how these variations map into a height range. For the airglow experiment described in Bernhardt et al. (1991), which is compatible with the present analysis with regard to the applied radio wave frequency and -power, a simultaneous height profile of the electron density has not been obtained. Even the profile published in Bernhardt et al. (1989) (p. 9078) in connection with a previous airglow experiment is not of a sufficient resolution to allow the determination of the spatial scale of such small density variations as given by Eqs.(94) and (95) . As a crude estimate for the large scale density gradient around the interaction height (which in the shown case accidentally coincides with the height of the F2- maximum) , this figure indicates however a value of about  10% /25 km  (undisturbed ('cold') profile). This would result in

Ds (6300,6364 Å ) »3.8 km (96)
Ds (5577 Å )          »3.3 km . (97)

These values correspond to the effective thickness ( height extension ) of the excitation region for each line. The initial slab shape of the layer, as defined by the extension of the radio wave beam in the horizontal plane (which has a (half power) radius of the order of 25 km for the experiments of Bernhardt et al. (1989,1991)),  becomes somewhat distorted for the red lines due to the diffusion of the excited oxygen atoms during the lifetime of the level O(1D) state which is here about 7 sec (see Eq.(53)). Considering the average velocity of oxygen atoms of 0.6 km/sec (corresponding to an oxygen temperature of 1200 oK), the resulting expansion (»4 km) is however small compared to the original extension of the airglow cloud perpendicular to the line of sight. Only the expansion along the line of sight (i.e. in height) causes therefore a reduction of the red line- volume emission rate q(6300,6364 Å), which, however, leaves the observed intensities unchanged since the thickness of the layer Ds(l) increases accordingly (see Eq.(91))  (this result is essentially confirmed by the airglow images reproduced in Bernhardt et al. (1989,1991) where no substantial increase of the size of the airglow cloud due to diffusion of the neutral oxygen atoms can be detected, which should on the other hand be the case with the usually assumed intrinsic lifetime of the 1D -level of about 150 sec (Eq.(48),(49)) and the quenching rate given below Eq.(55) (which would result in an overall lifetime of about 100 sec) , i.e. if the reduction of the lifetime for the metastable states due to the level broadening by plasma field fluctuations is not taken into account).

With Eqs.(86)-(89) and (91) and (96),(97) , the theoretical airglow enhancements in the beam center of the high power radio wave become therefore in case of the red and green OI- lines

I(6300 Å)  »158    [R] (98)
I(6364 Å)  »150    [R] (99)
I(5577 Å)  »  9    [R] .               (100)

The calculated 6300 Å intensity is in good agreement with the measurement of Bernhardt et al. (1991)  (the 5.8 MHz case should be the directly comparable one) considering the theoretical uncertainties entering into the problem (in particular the thickness of the interaction layer Ds ) which may easily change the resulting intensities by tens of percents or even a factor 2.  The result is also consistent in absolute magnitude with the airglow intensities given in Bernhardt et al. (1989) which amount only to about 40% of the values derived here because the effective transmitter power was smaller by a factor 0.22 for this experiment (see Eqs.(24) and (27) for the dependence on the wave field strength E0 ).  Furthermore, the observed (average)  6300/5577 Å intensity ratio is correctly reproduced by the present approach, confirming therewith in particular the proposed mechanism of enhancement of the atomic line emission due to level broadening by plasma field fluctuations which resulted here in an increase of this ratio by about a factor 12 since only the 6300 Å transition has a decay constant which is smaller than the plasma broadening DfB (note that it was not necessary here to assume a significant decrease of the electron flux from 2 eV to higher energies in order to explain this ratio, as it is in other theories where however, as already indicated in the introduction to this paper, the physical origin of this flux distribution remains unexplained (the proposed Maxwellian distribution can certainly not be established within the available time considering the very small density connected to the elevated electron flux (see Bernhardt et al. (1989), Fig.25) ).

One should note that presently accepted theories, which do not include an enhancement of the line emission due to level broadening, have to assume instead that the airglow region extends over a range to about 100 km below the interaction height, in order to compensate for the small volume excitation rates. Even if one does not care about the internal consistency of the related theoretical models, such a height distribution of the excitation region, which would lead to a considerable systematic displacement and widening of the airglow cloud relative to the HF transmitter beam (see Fig.11 in Bernhardt et al. (1989)), can certainly be ruled out from the observations which prove the airglow cloud to coincide with the location and size of the beam at the interaction altitude to within a distance of the order of 10 km in most cases.
Like the other theories, the present local equilibrium approach is of course unable to interpret and model the detailed space-time variations of the airglow emission for the individual lines which is very irregular at times (see for instance Bernhardt et al. (1989)) and appears to be crucially affected by small scale changes of the relevant physical parameters like plasma density and magnetic field strength (which are possibly even induced or at least affected by the events in the radio wave -plasma interaction region). This and further aspects related to the airglow enhancement problem in particular and the radio wave -plasma interaction in general are discussed qualitatively in the concluding section.
3.6    Non Equilibrium Considerations and Concluding Remarks

Several circumstances may be responsible for the fact that the observed airglow intensity enhancements for a given power of the radio wave often deviates substantially from the values derived in the last section (usually towards smaller values).
In the first place there is of course the trivial possibility that the parameters entering into the equilibrium equations may be different from those assumed here. In particular must the thickness and location of the radio wave- plasma interaction region be considered as highly variable since variations of the plasma density of the order of 1% are crucial (Eq.(94),(95)). Many of the variations shown in Bernhardt et al. (1989, 1991) can therefore still be explained within the framework of an equilibrium approach.

An examination of the relevant time and length scales involved shows however also that the equilibrium situation described by Eqs.(33),(34) may in some cases not be reached, at least not in a strictly local sense:
one reason is that the energy oscillation in the interaction region increases the effective width of the excitation band only at the expense of decreasing the excitation frequency for a single electron in this band to

nexosc  =  nex . De / [De  + De»
»nex . De / De ,               (101)

where nex is the excitation frequency within the actual resonance region of width De as defined by Eq.(37).
The numerical value for nex (Eqs.(62),(63)) indicates that  nexosc  »100 sec-1 , i.e.  texosc »10-2 sec (this value for nexosc takes into account that the excitation of the O(1D) - level is actually due to 2 transitions (6300 and 6364 Å ), each according to the frequency Eq.(62) ).  Since it takes also of the order of 10-2 sec for a 2 eV electron to traverse a distance of about 4 km (see Eq.(96)) along the magnetic field line, one mean free path length (with regard to excitation) may therefore not be completely contained within the interaction layer, i.e. the probability of excitation during a single traversion of the layer may be significantly smaller than 1, which reduces therewith the excitation rate compared to the equilibrium value (note hereby that nexosc is  ~ E0-0.6  according to Eq.(24) whereas the thickness of the layer is  ~E00.6 (Eqs.(27),(28)) so that the excitation probability for an individual electron is independent of the power of the radio wave) .
On the other hand, one has also to consider that, due to the small thickness of the interaction layer, electrons diffusing on the corresponding magnetic field line spend most of their lifetime within the region of undisturbed plasma where the electron flux in the relevant energy band De tends to be much higher because of the absence of excitation (which is restricted there to the much smaller interval De ) . However, the relatively long lifetime with regard to recombination and energy changing (electron-electron) collisions (»100 seconds, see Eqs.(64)-(66) ) enables an electron to cross the interaction layer many times while diffusing up and down the field lines over distances of the order of one plasma scale height (100-200 km) (the average velocity of a 2 eV electron parallel to the magnetic field is about 400 km/sec).  Within the whole region connected by means of diffusion along the magnetic field line, the electron flux within the band De should therefore be almost exclusively determined by the high loss probability in the interaction layer and thus adapt to the corresponding equilibrium value  (elastic (momentum- changing) collisions with neutral oxygen of density N(O)»109 cm-3 occur with a collision distance of about 15 km along the field line and will result in a statistical diffusion over distances greater than this;  the time needed to traverse a distance of about 100 km under these circumstances is however still short enough (a few seconds) to enable an individual electron to cross the excitation region several times;  only if the probability of excitation on a single occasion is much smaller than 1 (i.e. if the thickness of the layer Ds or  nex is much smaller than derived here) will the equilibrium excitation rate be significantly affected by the regions external to the layer).

It is obvious from the above consideration that in general a consistent calculation of the diffusion of electrons along a magnetic field line has to be performed in order to obtain the accurate excitation rates and related changes inside and outside of the interaction region. This has probably to include the enhanced recombinative loss of the low energy electrons which can not only lead to a decrease of the plasma density near the interaction region but also to an enhanced scattering coefficient for radio waves because of the enhanced population of atoms in highly excited states (see Smid, 1993 ). Apart from the self consistent large scale polarization field connected to the height gradient of the plasma density and the electron flux spectrum, it might in some cases also be necessary to include the effect of the small scale plasma field fluctuations related to the random spatial distribution of the charged particles. 
The diffusion of ions, on the other hand, should only become relevant if the increase of the total recombination rate due to the induced atomic excitations becomes significant. A high recombination probability in the interaction region could be a substantial loss process in particular for the region below the layer (the recombinative loss of upward diffusing ions is usually not important because of the long lifetime with regard to recombination  (»1000 sec) in comparison to the free fall time over twice a scale height (»200 sec) ).
A further possible effect, which may have to be taken into account in a self consistent treatment, is the change in the magnetic field near the interaction layer due to the currents related to the induced plasma oscillation. According to Eq.(21), the resonance frequency depends both on the plasma- and the Larmor frequency and since both are of the same order of magnitude, changes of B of the order of 1% (see Eqs.(94),(95)) may be sufficient to have a significant effect on height and structure of the resonance region.
The whole situation is furthermore complicated by the circumstance that a given magneto- plasma has two resonance points (W-, W+, see Eqs.(16),(17) in Smid (1992)) and a wave of a given frequency may therefore lead to resonant excitation at two different plasma densities, i.e. different heights  (it appears that for the experiment described in Bernhardt et al. (1989), the resonance was actually at W-, as the relatively low frequency of 3.2 MHz would imply an unrealistically small plasma density at the interaction height of 300 km (of the order of 104 cm-3) for a resonance at W+.  Since the latter is related to the smaller plasma density, it should always occur first (at the lower altitude) due to the ionospheric height structure, but may have been inhibited in this case by electron- neutral collisions interfering with the plasma oscillation at the corresponding altitude (see end of Sect.3.1)
All these effects may be of importance for the details of a given airglow feature induced by a radio wave of given frequency and power.


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This work was funded through grant SERC GRF 80784.

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