Scattering of Radio Waves
by High Atomic Rydberg States

Thomas Smid

as published in
Radio Science 28,3,361,S93-001,1993

Copyright American Geophysical Union

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Contents                                                                                                page

1. Introduction...............................................................................................................4
2. Theory........................................................................................................................6
    2.1 Determination of the Level Population in Detailed BalanceEquilibrium............6    
    2.2  Determination of the Plasma Electron Distribution Function...........................14
    2.3  Stark Broadening of Atomic Levels by Plasma Field Fluctuations ..................19
    2.4  Effect of External Perturbations and Wave Coherence
          on the Photoionization Cross Section.................................................................28
2.5  Total Scattering Coefficient...............................................................................31
3. Numerical Results.....................................................................................................36
3.1  Details of the Computational Method.............................................................. .36
3.2  Plasma Electron Distribution Function..............................................................39
3.3  Excited Atom Densities......................................................................................40
3.4  Equilibrium Scattering Coefficient for Radio Waves.........................................43
3.4.1.  Atomic (Incoherent) Scattering Coefficient.................................................43
3.4.2.  Effective Density of Excited Atoms at a Given Frequency.........................45
3.4.3.  Effective (Coherent) Scattering Coefficient.................................................46
3.4.4.  Further Discussion of the Level Broadening................................................47
3.4.5.  Dependence of the Scattering Coefficient
              on the Ionizing Radiation Flux.....................................................................49
3.5  Non- Equilibrium Considerations........................................................................51
4. Concluding Remarks and Outlook..............................................................................53
A. Cross Sections and Decay Constants for Atomic Dipole Transitions.....................55
A.1.  Transitions Between Bound States...................................................................55
A.2.  Photoionization and Radiative Recombination................................................59
B.  Cross Sections and Collision Frequencies for Energy Transfer
  by Electron- Electron (Coulomb-) Scattering...........................................................64


The scattering coefficient of highly excited atomic levels formed by recombination from a plasma and energetically broadened by plasma field fluctuations is determined. For this purpose, the population of the levels is calculated from a detailed balance equation which is solved consistently together with the corresponding equation for the free electron spectrum. All relevant quantum mechanical cross sections and decay constants as well as the elastic electron scattering cross sections are hereby derived from first principles in a form directly suitable for numerical application.
A special emphasis in the theoretical treatment is laid on the problem of the determination of the energetical level broadening for a given plasma density and -energy , because the spectral line width of a discrete transition is the crucial quantity affecting the resultant scattering coefficient as a function of frequency, and because usual theoretical approaches concerning Stark broadening are either inconsistent or insufficient for the present purpose.  Furthermore,  as a new aspect concerning the interaction of radiation with atoms, the effect of a finite wave field strength  (compared to the plasma fluctuation field)  and the wave coherence on the photoionization cross section is considered as an important mechanism which has to be taken into account in a complete and consistent theory.
Numerical results are obtained for initial values appropriate for ionospheric conditions. The solution proves to be in no way related to LTE- situations,  therewith invalidating usual treatments of the problem, which are not consistently based on a detailed balance approach.
With the present theory, the scattering coefficient for radio waves ( in particular for frequencies  < 100 MHz) turns out to depend sensitively on the electron impact-  and (radio-)photo  -ionization frequencies since they determine the level population for high quantum numbers (this offers for instance a straightforward explanation for the so called ionospheric 'short wave fadeout' which is observed in connection with solar radio bursts).
The theoretical result for the equilibrium coefficient for resonant scattering of radio waves by Rydberg atoms is consistent with experimental data obtained from the ionosphere. This suggests that the considered mechanism is exclusively responsible for the scattering of radio waves in particular and electromagnetic radiation in general.

1.    Introduction

Though being generally of little importance for the overall energy and radiation budget in natural plasmas, atoms in highly excited states may become significant and observable if energies very small compared to the average plasma energy are being examined.
From a variety of astrophysical objects for example, such as HII-regions, planetary nebulae and galaxies, line transitions up to quantum numbers as high as  n=350 have been detected by means of radio telescopes.  These observations are being used to derive physical properties of these objects like density and temperature.
The observed line radiation related to the excited states is generally due to spontaneous decay to lower lying states, whereby the intensity of the lines is proportional to the level population (the usual designation of these lines as  'recombination lines' is therefore somewhat inaccurate, since one is not dealing with photons related to the radiative recombination process (bound-free transitions) but to bound-bound transitions following radiative capture into high levels).
On the other hand, the non-zero density of atoms in high levels also provides the chance that radiation with a frequency corresponding to the energy difference between those states may be resonantly scattered exactly analogous to transitions involving the ground state. This possibility has so far been neglected in the literature concerning the physics of highly excited atomic states.
Several aspects have to be considered for calculating the scattering efficiency of Rydberg states correctly:
a)  for the determination of the production rate for each level:  a non-LTE calculation of the energy distribution of free plasma electrons for energies as small in value as the energy of this bound state; the radiative recombination cross sections and decay constants for those continuum energies and discrete levels; the rate of elastic collisions interfering with the recombination process and cascading between the bound states ;
b) for the loss processes: knowledge of the life time of each level with regard to radiative decay to lower levels; collisional -and photo -ionization out of these states;
c) determination of the level broadening by plasma field fluctuations which enables line scattering even if the wave frequency does not match the transition frequency to within a natural or Doppler linewidth;
d) the resonance scattering cross section involving two states;
e) a weighted summation of the resultant scattering coefficient over all broadened states;
f) (in connection with b)): the dependence of the photoionization cross section upon the field strength and coherence of the wave field .
In the present literature, points a) and b) are usually treated only by methods relating to LTE-approximations.
For the case of low density plasmas and quantum states n≤10, it has however already been shown by means of a detailed balance approach (Smid, 1987) that the LTE- concept lacks completely a physical justification. The method developed there will be extended for the present purpose to the appropriate energy range and quantum numbers.
Some confusion exists presently still also about the degree of line broadening by plasma field fluctuations, which is for instance reflected in the discussion of corresponding astronomical observations. This indicates also the necessity for a different theoretical approach to this problem.
Since a consistent theoretical treatment of the relevant atomic processes (photoionization, radiative recombination, resonance scattering) is not available in the generally accessible literature, it is given here in suitable form in a separate appendix section, based on the approach used in the thesis of Smid (1987). The same holds for the problem of elastic electron- electron -(Coulomb-) scattering which is relevant for calculating the free-free and bound-free collision rates.

Numerical results are primarily obtained for physical situations corresponding to scattering of radio waves in the ionosphere and (where it is possible) compared to experimental data,  but there is also a reference to astrophysical plasmas since transitions between high Rydberg states are a more or less well known phenomenon in this area of research and numerous high resolution measurements of the radio emission of astronomical objects exist which can aid as a test of the theory.

2.  Theory

2.1    Determination of the Level Population
        in Detailed Balance Equilibrium

In detailed balance equilibrium, the density of atoms in the excited state  n (where n=1 shall designate the atoms ground state, i.e. n is the effective quantum number)  is given by the quotient of the production rate and loss (depopulation) frequency

Nn = qn/nnloss .       (1)

The production rate consists of the primary production rate due to radiative recombination  qnRec  and a secondary rate due to cascading from higher levels, that is

qn =qnRec + qncasc .         (2)

Induced excitation from lower levels by radiation or collisions is neglected here because the cross section for a resonant transition from level  n-Dn to  n (n>>1; Dn<<n) decreases like (Dn)-3 (as it follows from Eq.(A.1.15))  and limits thus eventual transitions to neighboring states for which, however, the production and loss rates will be almost the same (at least for natural conditions) so that the effect cancels out.

In the absence of collisions interfering with the recombination process, the first term in Eq.(2) is given by

qnRec,0   =  hnRec .Np  .ò de Ne(e) . snRec(e'(e,eI)) . v'(e,eI) ,       (3)

where e is a dimensionless energy variable characterizing the energy of the free  plasma electron in the laboratory system ( as defined by the center of mass of the ions), Np the total plasma (ion) density  which is related to the energy specific electron density distribution function Ne(e) by the condition of quasi neutrality of the plasma

Np  =  ò de Ne(e) .       (4)

Furthermore,  v'(e,eI)  is the relative velocity between the recombining electron and an ion of average energy eI in the laboratory system, snRec(e'(e,eI)) the recombination cross section into level n for the corresponding relative kinetic energy  e' (see Eqs.(11) and (12))  and  hnRec a reduction factor which takes the Stark broadening of the level n into account if this becomes comparable to the level energy (see Sect.2.3, Eq.(79)).

However, Eq.(3) neglects the fact that the recombination process does not proceed infinitely fast but within a finite time given by the effective quantum mechanical decay constant (see Appendix A, Eq.(A.2.19))

AnRec  = 7 .104 .n-3.4          [sec-1]       (5)

and can thus be interrupted by energy changing collisions with the other plasma electrons.
In order to describe this effect correctly, one has to make assumptions about the details of the recombination process, because it represents a fundamental problem that the decay time of the continuum electron into a bound state is many orders of magnitude larger than the time the electron spends in the vicinity of the ion (as given by the quantum mechanical wave function for the level n).
It is therefore assumed here that the electron first 'recombines' radiationless into a 'pre-bound' state n (|n| = |n|) within a sufficiently short time and then decays into the actual state n according to the constant  AnRec under emission of a photon with a frequency corresponding to the energy difference between the initial energy of the free electron and the bound level n  (it is beyond the scope of this paper and requires further basic theoretical work to answer the question, how energy is conserved during this two stage process;  it should be obvious, however, that this model is the only physically reasonable one considering the different time scales involved in the problem).
The density in the 'pre-bound' state n is then given by

Nn  =  qnRec,0 / (AnRec+nnc) .       (6)

Apart from the decay to the actual bound state n (determined by AnRec), a further loss for the 'level' n is assumed here to arise from elastic scattering by the bulk of plasma electrons (of density Np and average energy ep) back into the continuum, which (for large n) occurs with a collision frequency

nnc  = 1.2 . 10-8 . (Np/ep) . n       [sec-1] ,       (7)

if one assigns classically to the state n the energy of the related actual quantum level n (Eq.(7) corresponds thus to the frequency for collisional ionization by electrons from the atomic level  n; see Appendix B, Eq.(B.20)).
The recombinative production rate for level n is then

qnRec  =Nn  .AnRec  =
          =  anRec,c  .qnRec,0 ,       (8)


anRec,c  :=  AnRec/ (AnRec+nnc)       (9)

is the recombinative - collisional 'branching- ratio' for 'level' n  (see also Fig.1  for a schematic illustration of this 'two-step' recombination model).

The relative quantities v' and e'  in Eq.(3) can usually be identified with the velocity and energy of the electron because of the small velocity of the ions related to their higher mass , and only for electron velocities within the core of the ion velocity distribution function does the latter become relevant. Since the velocity ratio between the ion of mass M and the electron of mass m  is connected to their kinetic energy ratio through

vI/ve  =  Ö ( m/M  . eI/e ) ,     (10)

and because vI and eI shall represent here an average ion velocity and energy (as given by their thermal values for instance), one can therefore adopt in a simplifying manner                     
e'(e,eI)  =  { e    for  e>eI.m/M ,     (11)
                              eI.m/M    for  eeI.m/M

to which  v' is related by

v'(e,eI) =  v0 . Ö e'(e,eI) ,     (12)


v0  :=  ve(e=1)     (13)

is the normalizing electron velocity  (=2.2 .108 cm/sec  if  e is in units of Rydberg).

With this, the recombination rate into level n becomes

qnRec  =  anRec,c .hnRec .Np . v0  .ò de Ne(e) . snRec(e'(e,eI)) . Öe'(e,eI) .    (14)

By assuming the integrand constant within subintervals  ek-1, e( e0=0), one can finally write the discrete sum approximation

qnRec  =  anRec,c   .åNe,k . nn,kRec,0 ,     (15)


Ne,k  =  (ek-ek-1) . Ne(ek)     (16)


nn,kRec,0   =   Np . v0 . snRec(e'k) .hnRec . Öe'k ,                (17)

where the explicit argument of e' has been dropped here .

Since the recombination cross section in Eq.(17) is a known quantity (see Appendix A.2) and the associated reduction factor due to excessive level broadening is independently derived in Sect.2.3 (Eq.(79)), the only unknown in Eq.(15) is the electron spectrum Ne(ek) which is determined in the separate Section 2.2 (although actually a set of coupled equations describing the densities in the discrete (Nn) and continuous energy region is being solved).

The loss frequency for the excited state n is given by the sum of the total spontaneous decay probability to lower levels  An and the collisional (nnc) and photoionization (nn*) frequencies out of this state, that is

nnloss  = An + nnc + nn* ,     (18)


An  =  åAm,n          ;  (n³2) .     (19)

The decay constant for a dipole transition from the upper level n to the lower level m can be numerically approximated by

Am,n  » 1.3 .109  . m-1.8 .(n-1)-3.2    .Min[1, m/(n-1)0.75]          [sec-1]    ,        (20)

where the last factor is a correction for contributions of higher angular momentum values considering the initial population of the level n due to recombination (see  Appendix A.1, Eq.(A.1.14) and App.A.2, Eq.(A.2.10)).
(in the case n=2, Eq.(20) holds of course only for the 2p state, since the 2s level is metastable with regard to a decay into the (1s) ground state).
An explicit numerical evaluation of Eq.(19) shows that the total decay probability from level n can be approximated by

A n  » 1.1 .109  . (n-1)-3.6            [sec-1]                 (21)

(one should note,however, that only Eq.(19) itself gives exact consistency with the cascading scheme Eq.(30)).

The collision frequency for ionization by plasma electrons is given by Eq.(7)  (or Eq.(B.20) ) and the photoionization frequency for level n is

nn*  =  hnIon . ò df  Fph(f) . snIon(f) ,     (22)

where Fph is the ionizing radiation flux at frequency f  in photons/unit area/sec/unit frequency interval  ( fn being the minimum (threshold) frequency for ionization of level n) and  snIon  the ionization cross section for level n for which the symmetry property holds (see also Appendix A.2)

snIon(f)  =  snRec(e(f)) ,     (23)


e(f)  =  f/f0 - en                    ;  f³fn     (24)


e= fn /f0 ,     (25) 

and f0 is a unit frequency which provides the conversion to the dimensionless energy unit e . It is convenient to choose f0 to be the threshold frequency for ionization of the ground state of hydrogen (3.2898 .1015 Hz) which normalizes e to the energy of 1 Rydberg (13.6058 eV).
In the case of hydrogen, eis then given by the Rydberg formula

enH  = 1/n2 ,     (26)

which can also be considered to be a good approximation for other elements  if n>>1, since highly excited states are hydrogenlike due to the large distances of the electron orbit from the atomic nucleus and the other atomic electrons (Eq.(26) holds for the case of singly excited neutrals (recombination of a singly ionized gas), which is the assumption throughout this paper;  otherwise a factor Z2, where Z is the degree of ionization, would occur).

The factor hnIon in Eq.(22) takes into account that the ionization efficiency of a given radiation field may depend on its degree of coherence, its field strength and on disturbances by collisions with other particles during the ionization process. Because this topic has to go beyond the usual perturbation theory used for calculating the ionization cross section, it  is discussed in the separate section 2.4 where the appropriate form for hnIon  is derived (Eq.(91)).

Eq.(22) can be discretized by transforming to the excess energy unit e defined by Eq.(24) and assuming the integrand constant over subintervals  ek-1, ek , so that

nn*  =  hnIon . å(ek- ek-1) .Fph(ek) . snIon(ek)  ,     (27)

where the normalization is now such that the photon flux has to be taken
per energy interval De, i.e.

Fph(e)  =  Fph(f)  . f0 ,     (28)

as it follows from Eq.(24) (if f0 = 3.2898 .1015 Hz,  Fph(e) is thus in units of photons/Rydberg  ; if the energy flux of the radiation is given instead of the photon flux, the former has to be transformed by means of Eq.(94)).

Through Eqs.(1),(2),(15),(18),(21),(7) and (27),  Nn is therefore determined if the production due to cascading is negligible, which is true for sufficiently high quantum numbers, because the decay constant becomes rapidly smaller with increasing n and ,on the other hand, the collision frequency increases.
Defining a quantum number n0 such that the level population for n=n0+1 can be determined without considering cascading, the cascading rate into level  n0 becomes thus

qn0casc  = å  Nm . An0,m ,     (29)

where An0,m is the spontaneous decay constant for transitions from level m to n0 as given by Eq.(20) (note however the interchange of the variables designating the lower and upper state respectively, which arises from the convention used throughout this paper that the first subscript always indicates the lower state).
Repeating this procedure recursively for values  n=n0-1, n0-2 ....etc. yields then the cascading rate into each level n  via Eqs.(1) and (2)  and

qncasc  = å  Nm . An,m .     (30)

It should already be noted that, because of the very low densities Nm and the small cascading probabilities implied by Eq.(20) for high quantum numbers, qncasc  tends to be much smaller than the primary production rate  qnRec .  Only for sufficiently small values of n, for which the level population decreases strongly because of the rapidly increasing probability of decay to lower levels (Eq.(21)) and the resulting decreasing importance of collisions, does cascading become significant.

Compared to cascading, induced redistribution of the electrons (either by radiation or collisions) among the levels can be neglected, because , as already indicated below Eq.(2), transitions are effectively limited to neighboring states for which (under natural conditions) the initial production and loss rates and therewith the occupation numbers are almost the same, so that the net effect of induced transitions between those states will be zero. Only for an extremely selective, coherent and powerful excitation, which can be achieved in laser applications for instance, could this process become significant.
One should also note that the collisional ionization of each level has been assumed here to be only due to free plasma electrons, i.e. the contribution of ions and neutrals to the loss frequency (Eq.(18)) is considered as insignificant. This simplification is justified, because for all atomic levels of interest here the orbital velocity of the bound electron is much higher than the thermal velocity of ions and neutrals, which makes energy changing collisions with the latter very ineffective (see also Appendix B, Eq.(B.25)). In the case of neutrals, the ionization efficiency is especially small since the collision cross section is much smaller than for Coulomb scattering where snc increases proportional to the quantum number n due to the unscreened potential of the colliding particles (see Eq.(B.15)) .
Only if a very high neutral/plasma density ratio compensates for the small collision cross section, may the excited atom density be controlled by neutral constituents. With regard to the earths ionosphere, this may happen at D-region heights and below.

Concerning the applicability of the results obtained with a detailed equilibrium approach, one should be aware that one is essentially limited to situations where there is no variation in the physical parameters over the longest characteristic time scale entering into the problem, which is here given by the recombination constant  AnRec (Eq.(5))  and the elastic collision frequency n0c (see Eq.(B.23)).  Because this amounts up to many hours for those atomic levels and plasma densities of interest here, a relatively small increase in the loss frequencies  nnc  or nn*  may lead to a considerable decrease in the level population Nn during a time interval small compared to this. A quantitative determination of these short term variations of Nn  is therefore only possible by means of a time dependent consideration of the problem. On the other hand, the detailed balance approach may be applied in a quantitative sense even to non-stationary situations if only long term (several hours in this context) integrals or averages are being considered  (see also Sect.3.5 and the discussion of the theoretical and experimental radar scattering coefficients in Sect.3.4.3).

2.2    Determination of the Plasma Electron Distribution Function

The energy distribution functions of the plasma electrons can be calculated from a balance equation including the production and loss rates due to photoionization, radiative recombination and inelastic and elastic collision processes. For ionospheric conditions and energies  e>0.01 (e=1 @1 Ry =13.6058 eV), it is almost exclusively determined by the first three processes, elastic (electron-electron) collisions being only of a secondary nature. Above approximately 3 eV (0.2) inelastic collisions (with neutral constituents) constitute the main loss mechanism determining the electron spectrum, whereas for smaller energies the strongly increasing cross section for radiative recombination into high levels (see Appendix A.2) leads to a rapidly decreasing value of Ne(e) (Smid,1987).
For the energies considered in Smid(1987) (0.01;  corresponding to n≤10 (see Eq.(26))), the individual recombination process is fast enough (A10Rec» 28 sec-1 ; see Eq.(5))  to neglect interfering elastic collisions  (nnc »2.6.10-2 sec-1 (see Eq.(7) for n=10, ep=0.5 and a plasma density of Np=105    cm-3)). For much smaller energies, however, elastic collisions will eventually limit the actual recombination rate because of the rapidly decreasing value of anRec,c (~ n-4.4 ) if  nnc>AnRec  (Eq.(9)).  One can find the quantum state n* where the transition from the collisionally undisturbed to the disturbed recombination occurs by equating Eqs.(5) and (7), with the result

n* =  (5.4 .1012 .ep/Np)0.23 ,     (31)

where the plasma density Np has to be taken in [cm-3] and the average plasma energy ep in units of Rydberg ( for Np =105 cm-3 and  ep=0.5,  n*=51) (the numerical results will show that the excited atomic densities Nn have a maximum value just for a state n which is close to n*).
For continuum energies e<1/n*2 (in units of Rydberg), the interference of elastic collisions with the recombination process will therefore lead to a saturation of the electron spectrum Ne(e), whereas for higher energies the latter reflects basically the energy dependence of the total recombination frequency nTRec,0(e) ; see Appendix A.2, Eq.(A.2.17) and Sect.3.2).

A further possible mechanism affecting  Ne(e) even at very low energies could be inelastic scattering of photoelectrons of higher energy (impact- ionization and  -excitation of atoms or molecules) .
A detailed knowledge of the relationship between photoelectron energy and the energy loss spectrum related to the relevant inelastic transitions (including the associated cross sections) would be necessary in order to treat the problem consistently, which is beyond present theoretical understanding of inelastic collision processes. On the other hand, experimental data concerning inelastic scattering usually have a spectral resolution of not much better than 0.01 Ry, which is by far insufficient for the present consideration which deals with energies and energy differences as small as 10-6 Ry and less.
However, from the energy loss data published in Massey (1969) (Chpts.11 and 13) and Rees (1989) (pp.114 , 272-273) one can conclude that the cross section for inelastic collisions involving an energy loss comparable to the electron energy itself ('near threshold excitation') is very small, i.e. only very few electrons will end up with an energy as close to zero as indicated above. For the case of the earth's ionosphere, computations show (Smid, 1987) that the frequencies for the elastic and inelastic collisions are comparable in magnitude at energies corresponding to the maxima of the continuous cross section curves for the latter process (>1 Ry) (although elastic collisions are insignificant below a height of 500 km for the shape of the electron distribution function because the related changes in the production and loss rates partially compensate and the life time with regard to radiative recombination is not long enough for 'thermalization' to occur).
Because the 'near threshold' cross section for an inelastic collision can be expected to be several orders of magnitudes smaller than the maximum value, one can thus assume that the corresponding production rate at very small energies is smaller by this factor if compared to the production due to elastic collisions.
For the present purpose, it has therefore been decided to neglect the effect of inelastic collisions completely, because also below about 10-3 Ry there is no significant energy loss due to excitation of rotational, vibrational or fine structure transitions .

With this assumption, the balance equation for the production and loss rates determining the density  Ne,k of free electrons in the energy interval ek-1, ek (Eq.(16)) can be written as

                    M                          ¥                        ¥
p1,k    + å Ne,i .ni,kc    + å Nn .nn,kc    + å Nn . (nn,kc +nn,k*)    =
                    i=1                           n=2                      n=2

                        M                  ¥
= Ne,k . ( å nk,ic   + å nn,kRec,0 )              ;  k=1,M ,     (32)
                        i=1                n=1

where the summation over the energy index of the plasma electrons has now been limited to a maximum value M rather than infinity as in Eq.(15), therewith restricting the energy spectrum from    ek= 0....eM.
The various collision frequencies are (see Fig.1) :
ni,kc  is the frequency for elastic free-free (Coulomb-) scattering of electrons from the energy range  ei-1, ei  into ek-1, ek    ( nk,ic  indicates the inverse process), assuming that there is no scattering within the interval ek-1, e itself    (i.e.  i¹k) (see Appendix B , Eq.(B.17)),
nn,k is the frequency for elastic collisions from the bound level n (electron impact ionization) into the continuum energy range ek-1, e, nn,kthe  (numerically identical)  frequency for the 'pre-bound' level  n, (Appendix B , Eq.(B.19))  and

nn,k*  =  hnIon . Fph(ek). snIon(ek)     (33)

is the photoionization frequency from level n (see Eq.(27)) (note that it is assumed in Eq.(32) that there is no photoionization from the 'pre-bound' level n ).
The recombination frequency  nn,kRec,0  into the pre-bound level  n  is given by Eq.(17)  and  p1,k  is the primary production due to ionization of the ground state of neutrals,  which is taken here separately as an external boundary condition ( note that the ground state n=1 (and n=1) has been  excluded from the sum describing collisional and photoionization from the individual levels) and is assumed to be of the form

p1,k  =  p0 . (ek - ek-1) ,     (34)

that is, the production function  p1(ek)  is taken as const. =  p0 ,  which is a reasonable approximation in view of the fact that the photoionization cross section for the ground state decreases from its maximum value roughly as Öe for 0 (see Appendix A.2), whereas on the other hand the cross section for electron impact ionization (which contributes nearly the same amount to the total production rate as photoionization (Smid, 1987)) varies in the inverse manner (see Appendix B, Eq.(B.20)).
Rather than calculating  p0 explicitly from the local density of neutrals and the ionizing radiation flux, it is a more convenient method here to define it implicitly by noting that in equilibrium the total ionization rate from the ground state must equal the recombination rate (including cascading from higher levels)  into the ground state, that is

å p1,k  = p0 . eM  =  q1     (35)

(where q1  is given by Eq.(2),(15) and (30) ) and imposing instead the external boundary condition of a given plasma density Nover the normalizing equation  (see Eq.(4))

å Ne,k  =  Np .     (36)

In this form, Eq.(32) reduces to a system of equations determining the normalized electron distribution function

fe(ek)  :=  Ne(ek)/ N=
            =  Ne,k /(ek-ek-1) / Np ,                     (37)

i.e.  Eq.(32) becomes

                              M                               ¥                      ¥
q1.Dek/eM  + Np.å Dei .fe(ei).ni,kc  +å Nn .nn,kc  +å Nn . (nn,kc +nn,k*)= 
                            i=1                             n=2                     n=2

Np.Dek .fe(ek) . [ nk,Lc   + nT,kRec,0 ]              ;  k=1,M  ,                (38)


nk,Lc  = å nk,ic                             (39)
(see Eq.(B.18)),
T,kRec,0   =   å nn,kRec,0                           (40)

(see Eqs.(17), (A.2.17) ,


Dek  = ek - ek-1 ,
Dei  = ei - ei-1 .     (41)

Eq.(38) is a system of M equations for the M unknowns fe(ek) . Together with Eq.(1),(2),(6) and (15) , which determine the recombination rate q1 and the atomic level densities Nn and Nn, they represent a consistent set of coupled equations for the discrete and continuous electron spectrum (see also Fig.1, where the transitions between the various electronic states are illustrated through a schematic energy level diagram  (using the definitions of this chapter)).

2.3    Stark Broadening of Atomic Levels by Plasma Field Fluctuations

For atomic transitions related to small quantum numbers it is usually (under natural conditions) sufficient to calculate the corresponding line profile only from the apparent natural and Doppler broadening involved (see Appendix A1. Eq.(A.1.4) and following).  In order to obtain the correct frequency dependent scattering cross section for transitions between highly excited states, it is however essential to consider the explicit energetical shift or broadening of the atomic levels by external perturbations, because disturbing fields which are insignificant for small quantum numbers may become important for higher states.  This is due  to the strongly decreasing values for the natural broadening  Am,n (Eq.(A.1.14), Doppler broadening (if transitions between neighboring states are being considered; Eq.(A.1.6) and Eq.(A.1.10)) and the level separation (Eq.(A.1.10)  with increasing m,n.
On the other hand, the energy shift caused by external fields is at least independent of the quantum number (as for the magnetic field; Zeeman effect). In the case of an electric field (Stark effect) it increases  even for higher states, which is evident from considering the work a homogeneous static electric field does on an electron in level n during one revolution in its orbit, namely

(dW)n,SStatic =  ±  e . |E||| . 2.rn  =                                         =  ±  e . |E||| . r0 .n2 ,     (42)

with e  the elementary charge ,  E|| the average electric field amplitude in the orbital plane of the electron, rn the expectation value for the radius of the electron orbit in state n>>1 (Eq.(A.1.12)), r0 the Bohr radius and the  ± sign arises from the circumstance  that during one half of the orbit the electron gains this  energy, while on the other half it looses the same amount.
(Note added later: Eq.(42) effectively assumes a linear Stark effect. This is justified here because high Rydberg states can be considered to be hydrogen-like. Additionally, the electric microfield due to the plasma is not static but varies on a time scale which is practically for all cases much shorter than the time required to polarize the atomic charge distribution (which should be given by the linear Stark frequency). The present treatment should therefore also be applicable to low lying states of multi-electron atoms).

In the case of a static (the conditions under which a field can be considered as static will become evident below) and macroscopically directed electric field of strength |E|, the projected field is classically simply given by

|E|||  = |E| . cosa ,     (43)

where a =0...p/2  is the angle between the electric field vector and the orbital plane of the electron. If  the latter is randomly orientated in space,  a  would, for a large number of atoms, classically take on all possible continuous values.  Because of directional quantization, however,  only certain discrete values are allowed. From the observation of spectral lines, one finds for level n  the  n possibilities

cosap  =  (n-p)/n            ;    p=1...n .     (44)

This gives the splitting scheme of the atomic level n in a static electric field of strength |E|

(dW)n,SStatic  =  ±  e.|E| . r0 n .(n-p)          ;    p=1...n .     (45)

In the case of a time dependent electric field E||(t) fluctuating with (an average) period Dtf (defined by the average time between two zero crossings of E||(t)), each shifted level will additionally be broadened according to the associated change of work ({  }= time average)
(dW)n,B  =  ±  e . {|dE||(t)/dt|} .Dtw  . {|dr/dt|} .Dtw ,     (46)


Dtw = Dtf.Tn / (Dtf+Tn )     (47)

is the average period of the work done on the electron due to the field fluctuation period Dtf and the angular period of revolution of the electron in state n

Tn  =  1/wn  =
          =  Öm /e  .rn3/2  »
      » 8.6 .10-18  .n3                [sec] ,     (48)

as one obtains classically by considering the circular frequency of an electron with orbital radius rn in a neutral atom.

If the field variation is equal in magnitude to the field amplitude, then the average change of work becomes
{|dE||(t)/dt|}2.{|E|||}/Dtf .     (49)

Furthermore , the projected velocity of the electron is
|dr/dt|  =  2.rn / Tn  = 
                =  r0 .n2 / Tn ,     (50)

so that the energetical broadening of level n becomes

(dW)n,B  =  ±  e . 2.{|E|||}.r0 .n2 . zn ,     (51)


zn = Dtf.Tn / (Dtf+Tn )2 .     (52)

Note that zn has a maximum for Dtf=Tn  and decreases linearly with the ratio of the time constants in both limits Dtf>>Tn  and Dtf<<Tn .  As a consequence (see Eq.(48)),  (dW)n,B  increases  ~n5  in the first limit ('small' n) but decreases  ~n-1  in the second ('large' n) with increasing  n.

Due to the fluctuating field, the actual level shift  (dW)n,S will be reduced compared to its static limit given by Eq.(42) or (45) . Assuming that the relevant time scale which defines a static field is Tn , and that in the limit of  Dtf<<Tn the reduction is determined by the ratio of the squares of the fluctuation field gradient {|E|||}/Dtf to the characteristic gradient {|E|||} /Tn (thus relating it to the associated (dynamical) field energies rather than the field values),  one can adopt the relative splitting (or shift) factor

mn  =  1/ ( 1+(Tn/Dtf)2 ) .     (53)

For the present problem, the electric field is neither uniquely determined nor macroscopically directed, since it is produced by the superposition of the fields of all charged particles in the plasma which are distributed randomly in space and furthermore are in constant motion.

Therefore, the zero order static field (obtained by integrating over a homogeneous density distribution) vanishes at any time instant t and only statistical fluctuations around this quasi-neutrality situation occur. 
If deviations from the purely statistical field due to close encounters of charged particles with the considered atom are being neglected, the (unnormalized) distribution about this mean value is a Gaussian probability function
p( E||(t))  =  exp[-( E||(t)/DEp)2] .                                                    (54)

The dispersion value DEp is the first order electric field which a single charged particle produces at a distance corresponding to the average separation

rp  = ( 4p/3 .Np)-1/3     (55)

of two particles in the plasma, that is

DEp  =  e/rp2   =
        =  e .  ( 4p/3 .Np)2/3    =
        =  1.25 .10-9 . Np2/3       [statvolt/cm  =3.10V/m]     (56)

if the plasma density  Nis in [cm-3] .

The average field fluctuation period associated with Eq.(55)  (which replaces Dtin Eq.(52) and (53) )  is given by

Dtp,i = 2 rp/vi ,     (57)

where vi is the average velocity of the charged particle species i relative to the considered atom (the factor 2 arises from the circumstance that a particle has to travel twice the average distance rp in order for the resultant field to change sign).

The Gaussian field distribution Eq.(54) for species i transforms now over Eq.(51) and the energy-frequency relationship

(dW)n,B,i =  h . (df)n,B,i     (58)

(with h the Planck constant) into the frequency broadening function for level n

jn,i(df)  =   exp [ - ( df/ (Df)n,B,i )] ,     (59)


(Df)n,B,i  = 2/Öp  . (Df)0. n2 . zn,i            ,                 (60)

with zn,i  given by Eq.(52) for Dtf = Dtp,i  and

(Df)0  =  0.49 . Np2/3                  [Hz]     (61)

if Np is  again in  [cm-3]  (the factor  1/Öp  in Eq.(60)  arises from the field average in Eq.(51) with the distribution function Eq.(54) ).
One should be aware that Eq.(52) will lead to a quite inaccurate representation of the line profile in the wings (df>>(Df)n,B,i ) because of the neglection of close encounters in Eq.(54). The exact statistical field distribution would however make an analytical treatment impossible, and the Gaussian profile, adequate in the line core, yields still a sufficiently accurate approximation for the overall behaviour of the scattering coefficient in the region of blended lines as well as giving correct values for the equivalent widths of the individual lines.

The presence of a second species j of charged particles causes a further, independent dispersion for each of the possible values of df, so that the resulting field distribution function is obtained by folding the individual distributions, i.e.

jn(df)  ~  òdf'  jn,i(f') . jn,j(f'-df)      =         
                    - ¥

                    =  exp [ - ( df/ (Df)n,B)] ,     (62)


(Df)n,B  =  2/Öp .(Df)0. n2 . Ö(zn,i2  + zn,j2)           .     (63)

The generalization to more than two species is obviously achieved by adding the further contributions under the square root in Eq.(63).
(Note that jn(df) has been taken as an unnormalized function here because the normalizable quantity is the product of the distributions for the two levels involved in a radiative transition).

For a plasma consisting of electrons and a single species of ions (both of density  Np),  the field fluctuation period Eq.(57)  attains the numerical values (respectively)

        Dte =  2rp/ve(ep)  =
      =  5.6 .10-9  /Np1/3  /Öep           [sec]                  (64)


DtI = 2rp/vI  =
            =  9.4 .10-5  /Np1/3  /Ö (T/A)          [sec] ,     (65)

with Np the plasma density in [cm-3] ,  ep the average electron plasma energy in units of Rydberg ,  T the ion temperature in  oK  and  A the ion mass number.
With these values inserted into Eq.(52) for each species, the frequency dispersion of level n becomes

(Df)n,B  =  2/Öp .(Df)0. n2 . Ö(zn,e2  + zn,I2)         .     (66)

Assuming now a lower level m with broadening  (Df)m and an upper level n with broadening  (Df)n while the centers of the Stark shifted levels are separated by the frequency  fm,n' , the atomic line absorption profile in dependence of the given frequency f and a fixed level shift can therefore be written as

jm,n(f)  = 1/Öp  /(Df)*B  .òdf' jm(f') . jn(f'-f +fm,n' ) ,                (67)
                                            - ¥


(Df)*B    =  (Df)m,B .(Df)n,B  / Ö((Df)m,B2 + (Df)n,B2) .     (68)

Evaluation of the integral yields

jm,n(f)  =  exp [ - (f -fm,n' )2 / (Df)B2 ]  ,                (69)

where now the abbreviation

(Df)B Ö( (Df)m,B2 + (Df)n,B2)     (70)

has been introduced for the microscopic line broadening.
One should  note that the atomic absorption profile jm,n(f)  is normalized to the line center (f =fm,n')  rather than the integral over frequency because of the normalization of    òdf' jm(f') .jn(f'). 

The frequency separation of the levels fm,n' depends on degree of static splitting of each of the levels which is determined by the quantities mm and m( Eq.(53)) for the combined (reduced) fluctuation period for the fields of electrons and ions (replacing Dtf there)

Dte,IDte.DtI /(Dte +DtI) .     (71)

For a level n, the absolute frequency shift of the level centers is then

(df)n,S  = ±(Df)0 . n2 . m. E/DEp ,     (72)

with  (Df)0  (the normal frequency shift for the ground state in a field DEp) given by Eq.(61)
Assuming that only those sub-states with the same sign of displacement contribute significantly to the transition cross section, the frequency separation between the centers of two Stark shifted levels m,n can therefore be written as

fm,n'  =  fm,n ± (Df)0 gm,n .E/DEp ,                 (73)

where fm,n is the original (unshifted) transition frequency between the two levels and

  gm,n =  n2 . m- m2 . mm .     (74)

The macroscopic frequency scattering profile is now obtained by averaging Eq.(69) over the Gaussian probability distribution for the plasma fluctuation field (ensemble average) , i.e.
fm,n(f)  = <jm,n(f) >  = 1/Öp /DEp . cm,n .òdE  e-(E/DEp)2 . jm,n(f)    ,  (75)
                                                                        - ¥
where cm,n is a normalization constant which makes the integral of fm,n(f) over frequency equal to 1 in the limit of zero microscopic line broadening (Df)B (thus normalizing only the macroscopic (ensemble) part of the profile due to the line splitting (Df)S since only this is connected to the given volume density of scatterers and has therefore to be normalized).
After substitution of Eq.(73) into (69) and elementary integration one obtains the result

fm,n(f)  =  1/Öp /(Df)S . exp[ - (f -fm,n)2 / ( (Df)B2 + (Df)S2 )  ] ,    (76)

where now the abbreviation

(Df)=  (Df)0 . gm,n     (77)

has been introduced for the static line splitting.

fm,n(f)  represents the actual macroscopic absorption profile for resonance scattering of a wave of frequency f by the statistically Stark broadened and shifted states m and n under neglection of further Doppler and natural broadening (with the restriction mentioned below Eq.(61)).
One should be aware of the different physical origin of the two contributions to its width (Df)B and  (Df)S , since the continuous broadening associated with the latter is only a result of the macroscopic (volume) averaging process, whereas the former is an actual continuous broadening of the level of an individual atom (due to a time average rather than a volume average) and therefore resembles more the natural broadening of spectral lines .
For this reason, the emitted (in contrast to the absorbed) line is only represented by  fm,n(f)  in case of a spontaneous transition between levels n and m  or for scattering of a wave with a sufficiently broad frequency spectrum. The line profile of an initially monochromatic signal scattered by Stark broadened and shifted states is however different from the absorption profile fm,n(f)  because the scattering is coherent in the atoms frame and (at least if the atomic broadening (Df)B dominates) only the Doppler effect connected to the velocity distribution of the scattering atoms will affect the line width. The inclusion of the latter line broadening mechanism and the natural broadening into the absorption profile, which has been neglected in this chapter since it is not related to the Stark broadening , is performed separately in the next Section 2.4 where the actual scattering cross section is derived by folding fm,n(f) with the corresponding distributions and finally adding the contributions of all quantum states in order to get the total scattering coefficient for a wave of a given frequency.

A further consequence of Stark broadening should be a reduction of transition cross sections if the level width is not small compared to the transition frequency.
For recombination into a given level n one has to assume that it is reduced if either the average broadening (Df)n,B or the average splitting  (see Eq.(72))

(Df)n,S    =    (df)n,S (E=DEp)     (78)

becomes comparable and greater than the threshold ionization frequency f(Eq.(25),(26) and (A.1.11)) from this level, because then the discrete level becomes energetically undefined and blends into the continuum. Assuming an exponential reduction of the corresponding recombination cross section with the ratio of the total level broadening to the threshold ionization frequency, one gets the reduction factor for recombination

hnRec  =    exp[ - fn  /(Df)n  ] ,                 (79)


(Df)Ö((Df)n,B2 +(Df)n,S2) .     (80)

In the case of discrete transitions, i.e. for resonance scattering, the efficiency factor is given by the degree of (non)-overlap of the two states m and n involved, which yields the expression

hm,nsc =  1 - exp[  - fm,n2 / ( (Df)B2 + (Df)S2 )  ] .     (81)

2.4    Effect of External Perturbations and Wave Coherence
        on the Photoionization Cross Section

The cross section for photoionization (Eq (A.2.4) ) is derived from the assumption of an ideal radiative transition, i.e. an atom interacting with an ideally coherent photon (infinitely long sinusoidal wavetrain) with no time dependent perturbations by other electrostatic or electromagnetic sources.
It is however obvious that the ionization efficiency can not stay the same if either the duration (coherence length) of the wave train is shorter than the characteristic transition time for this process or if the electric field strength of the photon is not large compared to disturbing fields varying irregularly on a similar time scale, because (in contrast to a pure scattering process) actual work must be done on the atomic electron by the wave field in order to transfer it to the free electron spectrum.
The plasma fluctuation field (see Sect.2.3) will therefore affect the photoionization process if it becomes comparable in magnitude to the field strength of the incident wave. Assuming that the corresponding reduction will be determined by the field energy (i.e. the square of the field strengths) rather than the field strengths themselves, one can adopt the efficiency factor

bn(Ew)  =  Ew2 /  (Ew2 + D{E}n 2 ) ,     (82)

where Ew is the field strength of the incident wave and

D{E}n  =  DEp . Ö(zn,e2  + zn,I2)     (83)

the effective plasma fluctuation field for the level n from which the transition originates (Sect.2.3,Eqs.(56), (66) ).
It is in general (although not for the results of the present paper) important that by taking the field strength dependence of the  photoionization cross section into account, one obtains a finite optical depth over infinite path lengths for this radiative process. This is because the square of the field strength of the original wave decreases in an absorbing (scattering) medium with distance from the centre of emission as

Ew2(s)  =  Ew2(s0) .e -tn(s-s0) .(s0/s)2 .                 (84)

The last factor is the usual inverse square law behaviour in vacuum conditions and the exponential takes an absorption process related to level n into account characterized by the optical depth

tn(s-s0)  = sn0  . ò ds' Nn (s')  .bn(Ew(s') .cIon(Ew(s'))) ,     (85)

with Nn the density in level n and sn0 the cross section in the absence of plasma fluctuations ( cIon is an additional reduction factor taking coherence effects into account and is derived below (Eq.(90)).
Since bn is itself dependent on the optical depth through Eqs.(82) and (84), Eq.(85) constitutes an integral equation in tn  which can be solved numerically.
The result is shown in Fig.2 as a function of the field independent optical depth tn0  (bn=1)  for various values of the ratio

G  =  Ew/ D{E}n     (86)

(only the effect due to bn has been considered here, i.e. it was assumed that  cIon =1; it is evident from Eq.(90) below that the dependence on cIon  produces analogous effects with regard to the ratio  Twcoh/TIon(Ew(s0)) ). For simplicity, the 'geometrical factor' s0/s has hereby been assumed as constant=1, i.e. the results correspond effectively to the case of the propagation of a plane wave. Because in this case the asymptotic solution of Eq.(85) for sufficiently large distances is logarithmic (as is easily verified), the optical depth over infinite path lengths is also infinite (with the power of the wave decreasing inversely proportional with distance). For realistic geometries with a finite divergence of the plane of propagation of the signal,however, Ew2(s) decreases faster than 1/s2 and the statement above Eq.(84) holds.

The second aspect of importance for the photoionization efficiency is the coherence of the wave field.  Assuming the oscillating electron always to be in phase with the incident wave of field strength Ew , one obtains the time TIon required for photoionization by substituting the classical relationship

Dv = a(Ew) .TIon   =
      = e/m .Ew .TIon     (87)

into the (correspondence-like) energy equation

m/2 .(Dv)2  =  h .fw     (88)

where  fw is the effective frequency of the ionizing wave (see Eq.(97)).
After inserting the numerical values of the physical constants, one gets therefore

TIon =  7.2 .10-18  .Öfw  /Ew        [sec] ,     (89)

with fw in [Hz] and Ew  in cgs-units (statvolt/cm  = 3.104 V/m).

As long as TIon is small compared to the coherence time Twcoh  of the wave ( time during which a sinusoidal waveshape is maintained to within one cycle) ,  the ionization process should not depend on Twcoh and the ionization efficiency  in this case be given by Eq.(82). Otherwise, it will be further reduced by a factor which in the opposite limit should be the square of the ratio of Twcoh and TIon since the ionization cross section (Eq.(A.2.4)) depends on the square of the dipole moment which for a classical oscillator is proportional to the field strength of the incident wave. Over Eq.(89) this implies therefore also a quadratic dependence on the time scales involved and one can thus adopt the coherence efficiency factor for photoionization

cIon (fw,Ew )=1/ [ 1 + (TIon/ Twcoh)2 ] .                      (90)

Note added later: this result is only applicable if the field strength Ew of the wave is obtained from the intensity of light over the classical relationship Eq.(96); in this case, one can still make the technical assumption of the ionization time given by Eq.(89); if Ew is given explicitly however, the ionization time takes on a different dependence on field strength and frequency (see .

An additional dependence of the ionization efficiency on the corresponding quantum mechanical transition constant (which by symmetry arguments should be identical to the recombination constant AnRec ; see Appendix A.2) can not be logically excluded with certainty by means of theoretical arguing ; from experimental results it is however evident that even if  Twcoh <<1/AnRec, the photoionization cross section remains unaffected (each photon produced by a discrete atomic transition has a coherence time which is about 4 orders of magnitude smaller than the characteristic time for bound-free transitions (cf. Eq.(5) and (20)); the ionization efficiency for those 'discrete' photons is however not known to be accordingly smaller than for 'continuous' photons produced by radiative recombination (in fact, experimental photoionization cross sections are usually derived by measuring the attenuation of certain discrete spectral lines by the examined gas and they are also consistent with ionization efficiencies of continuous spectral radiation)). Although this evidence might in principle be due to the circumstance that the electric field strength of the radiation field was high enough so that the classical coherence criterion is valid (cIon » 1, which might be a sufficient but not necessary condition for ionization to occur with maximum efficiency), it is somewhat physically unreasonable to assume that the quantum mechanical transition constant is a further independent reference quantity for the coherence time of the wave  , since this would mean that even for a vanishingly small field strength Ew , ionization occurs with maximum efficiency as long as Twcoh >>1/AnRec.
Therefore, one can take Eq.(90) as the actual efficiency factor for ionization with regard to wave coherence , so that the total relative efficiency factor (considering the combined effects of plasma field fluctuations and wave coherence) is obtained as

hnIon(fw,Ew)  =  bn(Ew) . cIon(fw,Ew ) .                 (91)

(the cross section for resonance scattering between discrete states, on the other hand, does not depend on the wave coherence, since the wave is re-radiated instantaneously in a coherent way by the atomic electron and no work in a quantum mechanical sense is done on the latter).

If the radiation field does not consist of a monochromatic wave but a more or less extended energy spectrum, the effective field strength Ew and frequency fw  can be derived by considering the amount of energy flux which the atomic transition 'cuts' out of the available spectrum according to the frequency characteristics of the corresponding cross section.
For photoionization from level n, the total effective energy flux is given by

Fn  =  1/snIon . ò df  PnIon(f) ,                 (92)


PnIon(f)  =  F(f) . snIon(f)     (93)

is the power of photoionization at frequency f  . The associated energy flux F(f) relates to the photon flux used in Eq.(22)  by

F(f)  =  Fph(f) . h.f     (94)

with h designating Planck's constant.

The average effective cross section for photoionization is hereby determined through

                          ¥                                                  ¥
snIon  =      ò df  snIon(f) . PnIon(f)    /      ò df  PnIon(f)  .     (95)
                          fn                                                fn

From Eq.(92) , the effective field strength is then derived by means of the well known relationship of electrodynamics

Ew Ö (Fn .8p/c) ,     (96)

with c the velocity of light.

The average effective frequency fw of the ionizing radiation , which goes into Eq.(89) is also determined by using the ionizing power as a weighting function, that is

              ¥                                      ¥
fw  =      ò df  f . PnIon(f)    /      ò df  PnIon(f)  .     (97)
                    fn                                    fn

For the resonance scattering process, on the other hand, the field strength and frequency are assumed to be known here, since one considers the scattering of (nearly) monochromatic waves.

The only unknown left (in connection with determining the photoionization efficiency) is the effective coherence time of the radiation field Twcoh . Since generally this quantity is not provided by measurements of natural radiation sources (the solar radiation field  for instance), one has to make assumptions about the generation of the radiation with regard to its coherency, i.e. the atomic decay times for the relevant transitions and the effect of interfering collisions, and rely on the conservation of the coherence time if many sources are being superposed, even if the latter are distributed randomly in space and time (numerical calculations of the field resulting from the superposition of many overlapping wavetrains with random relative phase shifts indicate in fact that the effective coherence time of the total wave field is (approximately) identical with the coherence time of the individual waves, despite the fact that the superposition is statistical and the total amplitude increases only with the square root of the number of wavetrains).

2.5    Total Scattering Coefficient

The actual macroscopic frequency profile for resonance scattering between states m and n can be found be folding the Stark broadening function fm,n(f)  with the Doppler and natural profile  H(a,w) (Eq.(A.1.7)). As a result one obtains the function

Fm,n(f)  =  1/Ö((Df)S2 + (Df)D2) .  H(a,w(f)) ,     (98)


w(f)  =  (f- fm,n )/Df ,     (99)

      a =  Am,n/  4p /Df   (100)


Df  =  Ö((Df)B2 + (Df)S2 + (Df)D2) ,   (101)

i.e. the the widths of the individual Gaussian components simply add in the usual way.
The normalization of  Fm,n(f) takes into account that Df  consists of a microscopic (atomic, coherent) broadening (Df)and the macroscopic (ensemble, incoherent) broadenings (Df)S and (Df)D of which only the latter component has to be normalized (see below Eq.(75)).

The cross section related to Fm,n(f)  is obtained by substituting it into Eq.(A.1.4) for the function 1/(Df)D  . H(a,w) . This yields the result

sm,n(f)  =  2p5/2 .e2/h . Ö(M/2kT) . rm,n2 . gm,n . hm,nsc . H(a,w(f)) ,   (102)


gm,n  =  1/ Ö(1+ (Df)S2 /(Df)D2)   (103)

and the dimensionless efficiency factor hm,nsc (which takes the dependence of the scattering cross section on the energetical overlap of the Stark broadened states into account) given by Eq.(81).
It is important that only the Stark splitting and Doppler broadening determines the absolute (peak) magnitude of the cross section, the atomic Stark broadening (Df)B leading only to an increase of the line width by affecting the scale of  w  (note that H(a,w)  = e-w2  for a<<1, which holds in most cases of practical interest) (it should again be emphasized (see Sect.2.3 below Eq.(61) ) that a Gaussian or Voigt profile has in the first instance been chosen for the sake of convenience and does not represent the actual profile in the collisional line wings).

Inserting now the approximate expression for  rm,n for large quantum numbers (Eq.(A.1.13)), one obtains the generalized form of Eq.(A.1.15)

sm,n(f)  »5.1.10-12 . [1/m2-1/n2]-3.m-1.8 .(n-1)-3.2 .Ö(A/T) .gm,,nsc. H(a,w(f))
      [cm2]                (m,n  >>1)  ,  (104)

with A the atomic mass number and T the atom temperature in oK.
Note again that, apart from the explicitly labelled quantities, also the line profile parameters a and w depend upon the quantum numbers m and n, in particular through the line broadening Df.
The  scattering coefficient related to sm,n(f) is now given by

km,n(f)  =  N. sm,n(f) ,   (105)

and the total scattering coefficient at the given frequency f is

                      ¥          ¥ 
k0(f)  = å      å  km,n(f) .             (106)                           m=1      n= m+1

The index 0 shall indicate that the contributions of the individual atoms are assumed to superpose linearly to the total scattering coefficient ('incoherent' superposition). This assumption will however fail if the total effective density of scatterers for the wave of frequency f

Neff(f)  =  k0(f) /s(f) ,   (107)


                                  ¥        ¥
s(f)  =  1/k0(f) .å      å  sm,n(f) . km,n(f) ,   (108)
                                        m=1    n= m+1

is such that the related average distance of scatterers

deff(f) =  (4p/3 . Neff(f))-1/3   (109)

is less than one wavelength l=c/f , i.e. if the scattering is specular (or 'coherent'). In this case the scattering coefficient can be expected to increase in a non-linear way analogously to the total energy of a wave if the individual sources are being superposed in phase rather than incoherently.
Assuming a quadratic dependence on the parameter

d(f)  =  l / deff(f)    =    c/f/ deff(f) ,   (110)

the total macroscopic (effective) scattering coefficient can be written as

k(f)  = k0(f) . (1 +d2(f) )   (111)

(in the following results section, k(f) will be referred to as 'effective' or 'coherent'  scattering coefficient, whereas for k0(f)  'atomic' or 'incoherent' scattering coefficient will be used ).

One should note that apart from affecting the total (integrated) scattered power as given by k(f) , the quantity  d(f) determines also the characteristics of the scattered wave field by the well known principles of interference optics (e.g. the so called magnetic aspect sensitivity  (orthogonality effect) which is observed with the scattering of radar signals from the ionosphere ( as a necessary condition for the magnetic field to affect the scattering characteristics of electromagnetic waves, it is reasonable to assume that the local gyrofrequency exceeds the natural width of the atomic states involved in the scattering and eventually the frequency of disturbing collisions with other atoms ;  in the radio wave region and for ionospheric conditions this is certainly always the case due to the small values of Am,n and nnfor the associated quantum numbers (see Eqs.(20) and (7) ) ).

3.    Numerical Results

3.1    Details of the Computational Method

The coupled system of equations determining the free electron energy spectrum and the density of neutral atoms in excited states (see Sect.2.1 and 2.2) was solved iteratively with the numerical grid for the free electron energy scaled like the discrete atomic levels, that is, the energy values ek  in Eq.(38) were determined by

ek  =  1/ k2              ;  k  =1....kmax .               (R.1)

(in order to achieve a better resolution at high energies, additional non-integer values k=1.1 ,1.2....1.9  were added)
The maximum energy of the free electrons was therefore assumed to be 1 Rydberg and the minimum energy was given by the highest quantum number considered which was in this case  kmax=7000 , i.e.  emin =2.10-8 .
This numerical scaling symmetry between the continuous and discrete electron spectrum is justified by the functional dependence of the recombination cross section (which determines the free electron spectrum) on the bound level quantum number n and the continuous energy e (see Appendix A.2).
The circumstance that Eq.(R.1) implies only a relatively low resolution of the free electron spectrum at high energies is not important here, since one is interested in the recombination into states with n>>1 which is primarily due to low energy electrons (k>>1) although the latter constitute only a small fraction compared to the bulk of higher energy plasma electrons.
Because a matrix of  7000.7000 coefficients is inconvenient to handle numerically and the physical quantities determining them vary only relatively slowly with the bound level quantum number n if n is large (at least for natural conditions), the whole grid has been divided into four regions with different grid resolutions, namely

Dn =  1          if  1≤n<10
Dn =  2          if  10≤n<40
Dn =  20        if  40≤n<2000
Dn =  100      if  2000≤n<7000               (R.2)

and the same division was taken for the continuous energy numbers k, Dk (with the exception k<2;  see below Eq.(R.1)).
With the coefficients and densities assumed constant between two points, this defines a step function of 173 intervals for the discrete and 182 for the continuous energy spectrum, which turns out to result in a relative error for the electron density distribution function fe(e) and the excited atom densities Nn of about 20% at each point (this corresponds to the change of the densities within one unresolved interval Dn;  a higher numerical accuracy could be achieved by decreasing Dn accordingly, but this would be pointless here considering the fact that the numerical expressions for the atomic cross sections and decay constants  have been derived by means of inter- and extrapolations of only a few explicitly calculated values and are unlikely to be more accurate than this throughout (see Appendix A)).
About 10 iterations were necessary in order to make the solution  converge numerically to within an error of less than 10%.
It should be mentioned that for all calculations presented in this paper, the free electron spectrum turned out to depend only on the 'pre-bound' densities Nn but not the actual level densities Nn (with n≥2) which are too low to cause any significant production of free electrons due to ionization at any energy. In principle, the bound levels could have been therefore excluded from the iteration procedure (calculations for some extreme (although unrealistic in this context) ionizing radiation flux spectra showed however that under certain circumstances there can be some effect at certain energies of the free electron spectrum, therewith justifying the formal coupling of the whole system of equations as illustrated by Fig.1).  Due to the course grid resolution implied by Eq.(R.2) , conservation of electrons within the cascading scheme given by Eqs.(19) and (30) could not be achieved with a sufficient accuracy. The total recombination rate into the ground state q1 , which defines the ionization equilibrium through Eq.(35), was therefore simply determined by adding up the initial recombination into all levels, that is q1=SqnRec , taking advantage of the above mentioned circumstance that only a negligible amount of the total number of recombined electrons become re-ionized before having cascaded into the ground state.

The final summation of the contributions from all levels to the total atomic scattering coefficient( Eq.(106)) was performed by first adding up all transitions between states n,n+1 (n1- or na-transitions) then with n,n+2 (n2 or nb) and so on.  This method provides a better convergence, since the scattering cross section decreases rapidly with increasing m for n,m transitions (as it follows from Eq.(104) ). The actual number of transitions which has to be taken into account in order to achieve a given accuracy depends strongly on the assumed physical parameters, in particular the plasma density and the related level broadening as well as the considered wave frequency. If the latter is in the continuous region of the scattering coefficient (which is for instance the case for  f<1GHz  if  Np = 105 cm-3 ;  see Sect.3.4, Fig.12), it is sufficient to add up n1-n3 transitions in order to achieve an error of less than 10%, whereas in the line region (f>500 MHz  if Np = 104 cm-3 ; see Fig.11) n1-n10 transitions have to be taken into account ( the general tendency is therefore an increase in the number of contributing transitions with decreasing plasma density and increasing wave frequency).

In order to avoid confusion when discussing the dependence of the results on the total plasma density  Np , the values for the mass and temperature of neutrals and ions (which are assumed to be identical) have been kept constant throughout, namely  A=32  and T=300 oK. This is appropriate for the lower ionosphere from where the measurements of scattering coefficients used for comparison with the theoretical results have been obtained.  The circumstance that these values are not in any case consistent with the assumed plasma densities if referred to the ionosphere is only of minor importance because the results are only weakly dependent on A and T.  For ionospheric F2- region conditions for instance (A=16, T=1200 oK), the only effect would be to reduce the scattering coefficient by a constant factor 0.6 due to an according depletion in the excited atom densities.

3.2    Plasma Electron Distribution Function

Fig.3 shows the normalized electron energy spectrum fe(e) (as defined in Sect. 2.2) appropriate for lower ionospheric conditions (Np = 105 cm-3 , ep=0.5 Ry,  T=300oK, A=32; see also remarks at the end of Sect.3.1) (thick solid step function). The dashed step function gives the electron spectrum when neglecting all elastic collisions, i.e.  only the primary production rate q1 and the (undisturbed) recombination frequency  nT,kRec,0 (thin solid curve, left ordinate) are hypothetically assumed to determine the spectrum (see Eq.(38)). It is evident that elastic collisions are the dominant production mechanism apart from the highest energies (e>0.1) where the primary production q1 (photo- and electron impact ionization) dominates.  Within the elastic scattering domain, one can further differentiate into free-free scattering of electrons and scattering out of the 'pre-bound' state n. The former is dominant for energies greater than the characteristic energy corresponding to the quantum number n* (Eq.(31)) where the recombination constant AnRec (dashed-dotted line, left ordinate) and the elastic collision frequency  nn(thin dashed line) become equal (the upper abscissa gives the quantum number scale which is related to the energy scale through  e=1/ n2 ), whereas the latter is dominant for energies smaller than this. Only in this region does  fe(e)  depend on the total plasma density at all (i.e. is the specific electron density Ne(e) a non-linear function of the total plasma (electron) density  Np ). A comparison with Figs.4 and 5 (Np = 104 cm-3 and 106 cm-3 respectively) indicates a dependence  fe(e) ~ Np0.23 (which can be attributed to the function anRec,c , see Eq.(31))  in the energy region where recombination is not yet affected by plasma field fluctuations (see Sect.2.3)  and  ~ Np0.4 for energies even smaller than this (corresponding to the saturation region of the recombination frequency  nT,kRec,0 ).
One should furthermore note that for energies  e<<1 ,  nT,kRec,0   is (for the given ion mass and temperature; see Eq.(A.2.17)) always large compared to the elastic collision frequency  nn (which is effectively equal to the free-free loss frequency  nk,L(Eq.(B.18)) ), i.e. the loss rates at a given electron energy are almost exclusively determined by recombination. This means that the differences in  fe(e) for different plasma densities are only caused by differences in the production rates due to elastic scattering out of the 'pre-bound' state n.  Only for much higher ion temperatures and/or smaller masses does the recombination cross section become small enough so that elastic collisions dominate the loss rates also for  e<<1 with the consequence of  fe(e) being much closer to 1 in this region.

3.3    Excited Atom Densities

With knowledge of the plasma electron distribution function  fe(e) (or equivalently the specific electron density  Ne(e) ) the production rate qn and thus the excited atom density Nn  can be calculated (Sect.2.1). For the conditions valid for Fig.3 (Np = 105 cm-3 ) the result is shown in Fig.6.
The thin solid curve gives the production rate qn (left ordinate, to be multiplied by 105 )  as a function of  n  (lower abscissa, taken as a continuous variable here), with the close dashed-dotted curve showing the initial production rate qnRec (i.e., neglecting the cascading rate qncasc)
One can distinguish two regions of the production rate function :  for levels n lower than n* (see Eq.(31) and previous paragraph 3.2;  n*»50 here),  the recombination rate decreases approximately in proportion with the narrowing of the continuum energy range from which electrons can recombine into level n (De~1/n2), whereas for higher levels, where collisions become important, the recombination rate decreases even much more rapidly with increasing n due to the additional reduction factor anRec,c  ( ~n-6.4) (see Eqs.(9) and (14)).
The loss frequency nnloss is given by the thick short dashed curve (right ordinate). It is the sum of the decay constant An (long dashed) which is dominant below a quantum number of n»350 in this case, the collisional ionization frequency nnc ( short-dashed line) which dominates for n>350 and the photoionization frequency nn* (dotted curve) corresponding to an average (quiet conditions ; Fsol =1)  solar radiation flux which turns out to be unimportant for all n although at about n=300 it comes close in order of magnitude to both An and nnc .
By taking the ratio of the production rate and the loss frequency, one obtains the level population Nn  (thick solid curve). It increases from n=2  to  n=n* because the decay constant An decreases more rapidly with n than the production rate (the steep increase from n=2p to n=1 indicates only schematically the much higher density in the ground state with A1=0,  since it is not self-consistently contained in the level rate equations).  Only in this region does qncasc  become significant (the cascading rate is of the same order of magnitude here as the primary production rate qnRec ), because the rapid increase in Nn with n can compensate for the low densities and small cascading probabilities.
A comparison with the solution for other plasma densities (Figs.7 and 8, note the absolute scale) shows that  Nn~ Np for n<n*,  Nn~ Np  for  n*<n<nc  (where nis the level at which the collisional ionization frequency  nnc becomes dominant (»350  Np = 105 cm-3 ) )  and Nn independent of  Np  for n>nc
It is important that with a solar radio flux much higher than the quiet sun flux (as it occurs for instance during a solar burst), the photoionization frequency  nn* can become the main loss process for high quantum numbers (where usually collisions are dominant) and the dependence of Nn on Np will then also be linear there ( Fig.9 shows the photoionization frequency and corresponding excited atom density for the strongest observed solar bursts (FsolBurst =1, 'Type IV', see Fig.10) for a plasma density of Np = 105 cm-3 (compare to the quiet solar conditions of Fig.6)) (note that this is an equilibrium consideration and that much smaller changes in the radiation flux may be sufficient to produce short term non-equilibrium variations of the same magnitude; this argument holds also for the collisional ionization frequency nn;  see Sect.3.5 for a more detailed discussion of the non- equilibrium aspect).
In both cases (quiet and enhanced solar flux), the efficiency factor for photoionization (Eq.(90)) was calculated by assuming the coherence time Twcoh  of the solar radiation flux to be determined by elastic collisions in a plasma of a density Npsol = 108 cm-3  which is appropriate for the lower solar corona. Since the frequency fw of the emitted radiation is approximately related to the quantum number n over (twice) the threshold frequency f(Eqs.(25),(26)),  the coherence time is therefore given by the associated collision frequency nnc (Eq.(7)) for the plasma density Npsol , i.e.

Twcoh (fw)  =  1/nnc (Npsol)
                  = 7.7.107 .Ö(fw/6.6.1015) . ep/Npsol          [sec] ,              (R.3)

where fw (as defined by Eq.(97)) has to be taken in [Hz] and Npsol in    [cm-3]. The average plasma (electron) energy was chosen to be identical with the ionospheric value ep= 0.5.

For frequencies higher than about 104 MHz, the quiet sun radiation flux of Fig.10 was extrapolated  by means of the the power law  F(f) ~ f1.85 (see Eq.(94)) which yields consistency with measurements of the solar spectrum in the optical region (Unsöld, 1955, p.34). A crude exponential interpolation was then applied between the maximum of the solar spectrum and the far UV -region, where the flux for the latter was taken from Rees (1989) (p. 9). With this, the photoionization frequency  nn* (Eq.(22)) could be calculated consistently for all states n=2.
As a general aspect arising from the results presented in the last two sections, one should note that both for the continuous and discrete electron energies, the derived densities are in no way related to some kind of thermal equilibrium. In the light of the consistent detailed equilibrium theory developed in this paper, it is in fact hard to see how approaches based on Saha-Boltzmann statistics, which are widely if not exclusively used in areas as astrophysics, plasma physics etc., could be connected to it and yield any physically reasonable approximation to the true situation or only order of magnitude estimates thereof.  Such concepts should therefore be abandoned and replaced by detailed equilibrium approaches like the one presented in this paper.  In the case that one has to consider time dependent situations which vary significantly within the longest time scale entering into the problem , it becomes of course even necessary to generalize the rate equations to time dependent expressions (see end of Sect.2.1 and Sect.3.5 for a qualitative discussion of the non-equilibrium aspect).

3.4    Equilibrium Scattering Coefficient for Radio Waves

3.4.1    Atomic  (Incoherent)  Scattering Coefficient

The total atomic scattering coefficient for a radio wave of frequency f is obtained from the excited atom densities calculated in Sect.3.3 via Eqs.(105) and (106) (and preceding formulae). In Figs.11, 12, and 13 the result for k0(f) (thick short-dashed curve, left ordinate) is plotted for radar frequencies from  820 KHz to 1.9 GHz for those plasma densities discussed in Sects.3.2 and 3.3 (undisturbed conditions).
The frequency dependence of the atomic scattering coefficient is strongly determined by the line frequency broadening , which is shown here by the thin solid curve (right ordinate) for the case of na (n®n+1) -transitions as a function of n (upper abscissa, which is scaled such that the frequency of the related na -transition matches the radar frequency (lower abscissa)).
Three different domains can be distinguished (see in particular Fig.11): 
1)  an almost frequency independent region corresponding to scattering by states which are broadened in excess of the level separation (long-short-dashed line, right ordinate)  (level continuum);   
2)  a  f -2  region related to levels which are well separated but broader than the frequency difference between two neighboring  (na, (n+1)a ) -lines (long-dashed line)  (line continuum);
3)  a region where the broadening is smaller than the line separation and the individual lines become visible (discrete line region).
The functional behaviour in region 2 is explained by the increasing number of lines able to get into resonance with the given frequency f as the latter decreases and probes higher (i.e. more closely spaced) levels with an increasing collisional broadening.
Once the frequency (long-short-dashed line) gets smaller than the level broadening, however, all lines with a resonance frequency below this 'critical' frequency scatter the wave and the atomic scattering coefficient becomes constant (region 1 ;  this is effectively the limit  f<<fm,n<<Ö((Df)B2+(Df)S2)  in Eq.(76), which makes the scattering profile  f independent of f ;  it should be noted in this context that for the considered frequencies, i.e the corresponding states n, the line broadening is only determined by the component  (Df)B which is related to the actual collisional level broadening , whereas the apparent line broadening due to a further static level shift (Df)S (see Sect.2.3) is only dominant for much smaller quantum numbers (=60 for the conditions considered here) and therefore higher frequencies).
In region 3, the dominant lines are formed by the na-transitions, which have the largest scattering cross section (sn,n+Dn~Dn-3 , as it follows from Eq.(104)) and the smallest broadening (because they are associated with smaller quantum numbers than nb- and higher order transitions in the same frequency region). With increasing radar frequency, the number of orders of lines (na, nb....) which become visible increases, the rest of the lines forming a quasi-continuum with the lower order lines superposed (see Fig.14  for a frequency range of 900-1000 MHz and a plasma density of 104 cm-3  (k0(f)  and k(f) coincide here);  the small peaks at 975.6 MHz and 988 MHz are for instance due to the 237b- and 236b-  transitions respectively;  note however that the wings of the lines are quantitatively not correctly reproduced by a Gaussian profile; the actual scattering coefficient may therefore be somewhat higher between the line peaks (see Sect.2.3 below Eq.(61) )
One should note that the 'continuum' formed by the blended higher order lines shows also the ~f -2 decrease with frequency which is characteristic for region 2, whereas the completely resolved lines have approximately equal peak amplitude in this frequency region, because for quantum numbers n>n* (see Eq.(31)) the increase in the scattering cross section with increasing n (~n4 for transitions between neighboring states) is just compensated by the decrease of the level densities Nn (see Fig.7).

Concerning the dependence of the atomic (incoherent) scattering coefficient on the total plasma density, it is evident from Figs.11, 12, 13 that k0(f) shows an (approximate) proportionality to Np only for frequencies near the transition from region 1 to 2 (»50 - 200 MHz for those densities considered here).
In region 1, k0 is a weaker function of Np (~Np0.4 ) since the density Nn of the responsible (high) levels is almost invariable ( the relevant levels are within the collisionally determined regime; see Sect.3.3, Figs.6, 7, 8), but also because the increased broadening with higher Np is assumed here to lead to a reduction in the scattering cross section if the levels are blended into each other (as it is just the case in region 1).
In region 2 , on the other hand, the dependence is stronger than linear (~Np1.4 ). For frequencies located between the dominant line peaks in region 3 (that is, in its 'continuum' part) the dependence may be even stronger because the increase in the line broadening related to an increase in  Np can bring the frequency point from the line wing close to the line center of the dominant line, an effect which adds to the ~Np0.7 increase of k0  in the line center ( see  Figs.14 and 15 where k0 increases by about 2 orders of magnitude between the na- line peaks as Np changes from 104 to 105 cm-3 (as already mentioned above , the line profile differs significantly from the assumed Gaussian function in the line wings (see Sect.2.3, below Eq.(61)) and the actual relative increase may therefore be somewhat less)  ).

Fig.16 has been added here in order to justify the above argumentative use of the na-transitions to explain the behaviour of k0 with frequency. It shows that they contain in fact all the features of the total scattering coefficient and amount at least to 50% of its absolute magnitude apart from frequencies between the na- peaks in region 3 (compare to Fig.12).

3.4.2  Effective Density of Excited Atoms at a Given Frequency

The scattering coefficient k0(f), as given by Eq.(106) and discussed in the previous section,  is related to an effective density of excited atoms for the wave of frequency f through Eqs.(107) and (108).
From the results displayed in Figs.11, 12 and 13 one derives in this way the effective density  Neff(f)  as shown in Figs.17, 18 and 19 respectively  (thick solid curve, left ordinate).  It is evident that for frequencies in region 1 and 2 (that is, in the  'continuum ' part of the scattering coefficient where the line broadening (thin solid curve in Figs.11-13 ) exceeds the line separation (long-dashed line in Figs.11-13) )  the effective density at frequency f is much higher than the number density  Nn (thin solid curve in Figs.17-19) of those levels  n  associated to f by the frequency relationship for  na- transitions (upper abscissa).  In region 2, the ratio of the two curves for a given point on the abscissa can thus be interpreted as the number of levels able to scatter the wave of the given frequency (this is not possible in region 1, since scattering is here by all states higher than the state where the level broadening exceeds the level separation, and quantum number n and frequency f are therefore not directly related anymore).

The 'scattering coherence' parameter d(f) (Eq.(110)) is shown in Figs.17-19 as the dashed curve (right ordinate). Remarkably, the transition between the optically different regions of 'incoherent' (d(f) <1 ) and 'coherent' (specular) scattering (d(f) >1)  occurs just within the frequency range considered here, i.e. for frequencies usually applied for probing the ionosphere. With increasing plasma density the critical frequency shifts to higher frequencies (»250 MHz  if Np = 104 cm-3 , 700 MHz  if Np = 105 cm-3, 2 GHz if Np = 106 cm-3 ). Relatively small variations of Np may therefore bring a given frequency from one characteristic region into the other and change the backscatter efficiency considerably in a non-linear way.
One should again be aware, however, that these are considerations based on the assumption of a time independent equilibrium situation and that changes in the plasma density will not immediately be reflected in the scattering coefficient since it takes a considerable time until the additional free electrons have recombined into excited atomic states.  More important are therefore probably time dependent non-equilibrium effects due to increases in the loss frequencies  nnc  and (or) nn*  (see end of Sect.2.1 and Sect.3.5).

3.4.3    Effective (Coherent) Scattering Coefficient

The effective (coherent) scattering coefficient k(f) is obtained from the atomic (incoherent) scattering coefficient k0(f) (see Sect.3.4.1) and the parameter d(f) (see Sect.3.4.2) by means of Eq.(111).  In Figs.11, 12, 13 it is shown as the thick solid curve. It is evident that k(f) becomes strongly enhanced over k0(f) with decreasing frequency due to the increasing value of d(f) >1.  The existence of three different regions (see Sect.3.4.1) is still preserved, with the constant region 1 of k0(f) changing to a f-2 slope and the f-2  behaviour of k0(f) in region 2 to f-4.  In region 3 both curves coincide, which is due to the accidental circumstance that d(f) becomes smaller than 1 at about the same frequency where the line separation begins to exceed the line broadening.
It is furthermore evident from the figures that k(f)  depends much stronger on the plasma density than k0(f)  because of the additional dependence of d(f) on Np  (~Np1.2  in region 1 , ~Np2.2  in region 2 and between the na -lines in region 3  and ~Np0.9 in the centers of the na-lines in region 3  ).
Interestingly, k(f) yields an optical depth >1 for the ionosphere for frequencies in the HF region and lower ( f< a few MHz)  (for Np = 105 cm-3  and f=4 MHz,  tsc=1 would be reached within a distance of 10 km). This means that for those frequencies the ionosphere becomes opaque and (in the limit of large optical depths) totally reflective.

A comparison of k(f) with the experimental determination by Moorcroft (1987) (Fig.20,  the absolute values can be compared directly) shows quite a good agreement for frequencies from 30 MHz to 1.2 GHz if one drops Moorcrofts assumption that k(f) can be represented by a simple power law and includes the measured values for the lowest and highest frequencies which have been excluded from Moorcrofts least square fit.  One should hereby note that an exact agreement would be rather accidental, because the experimental determination is subject to several uncertainties (note Moorcrofts title 'Estimates of...'). In particular is the assumption of an isotropic scattering phase function not consistent with the 'orthogonality effect' (magnetic aspect angle dependence) which is always associated with radar backscatter as long as one has 'specular' scattering (d(f)>1) . This could lead here to an overestimation of the backscatter coefficient by a few (up to 4, most likely 2-3) orders of magnitude. On the other hand, destructive interference of the contributions of scattering atoms along the line of sight could reduce the received power by a similar amount which only accidentally would cancel with the first effect.

In view of the compatibility of the theoretical (as obtained with the present approach) and experimental determinations of k(f),  it seems therefore to be doubtful that any other mechanism than resonant scattering by atomic Rydberg states (i.e. the well known diverse 'coherent' and 'incoherent'  backscatter 'theories') could be relevant at all for explaining the scattering of radio waves in general.

3.4.4  Further Discussion of the Level Broadening

It is worthwhile to have a closer look at the level broadening as a function of the quantum number (thin solid curve in  Figs.11, 12, 13 ), since this is the crucial quantity determining the scattering coefficient. As indicated at the end of Sect.3.4.1, the broadening (which is only due to the component (Df)B for those n values of interest here  (see remarks in Sect.3.4 and Eq.(76) and following) ) increases strongly with n (~n5 ) if n is small enough so that Tn is small compared to the shortest plasma fluctuation time scale which is in this case  Dte (Eq.(64)). If only electrons would be present, the broadening should reach a maximum with increasing n at a quantum number where the usual ~n2 increase of the broadening is just compensated by the function zn (see Eq.(60);  this happens at about  if nis the state where  Tne= Dte ) and then decrease with a ~1/n behaviour in the limit  Tn>> Dtas it follows from Eqs.(52) and (60).  The presence of ions, however, changes this picture since an additional characteristic fluctuation period DtI  (Eq.(65)) is introduced, which leads to a superposition of the individual broadenings as given by Eq.(66) with the consequence of a strong increase in the broadening as soon as the ion component becomes dominant. If n becomes so large that Tn>DtI, also the ion broadening becomes saturated and eventually decreases again ~1/n if  Tn>>DtI    ( for the plasma densities considered here, the ion broadening exceeds the electron broadening for quantum numbers of the order of n>1000  which is here however without any significant consequence for the scattering coefficient at the corresponding na-transition frequencies, since the broadening exceeds the level separation already for smaller n (> 500-700) so that all lines corresponding to levels higher than this contribute anyway with their  (close to -) line center cross section. Apart from this, scattering by levels blended into each other because of collisional broadening is here supposed to be strongly reduced (see discussion in Sect.3.4.1 concerning the interpretation of the atomic (incoherent) scattering coefficient in region 1)  ).
It should be noted that it is of course only accidental that the electron dominated broadening reaches its maximum close to the same quantum number where it exceeds the level separation (at about n»500 for Np = 105 cm-3).  With a different bulk energy of the plasma electrons than 0.5 Rydberg for instance , the maximum  shifts to different values of n because the fluctuation period Dte (Eq.(64)) changes.

With regard to astrophysical applications, it is interesting that the observed width of emission lines in the radio spectrum of HII - regions has about the same value as one would expect from the Stark broadening due to plasma field fluctuations for those conditions. This is evident from Fig.21 which shows the H157a  - line at 1.6832  GHz  emitted from the Orion nebula (Pankonin, 1980, p. 121)  (the halfwidth corresponds to about 100 kHz if transformed into frequencies) and Fig.22 giving the theoretical scattering profile for this line as one would expect it from Stark broadening in a plasma of  Np = 2.103 cm-3 (this is thought to be the highest plasma density in the Orion nebula  (Lang, 1974, p.121)).  Note that the resultant (half) line width of about 75 kHz is solely due to the Stark broadened atomic levels , since a temperature of 10 oK  has been assumed here which yields a negligible thermal Doppler broadening compared to this.  It is therefore likely that the commonly accepted interpretation of the line broadening in HII- regions as a thermal effect (which implicates temperatures of about 104 oK ) is at least in some cases incorrect.

3.4.5  Dependence of the Scattering Coefficient
          on the Ionizing RadiationFlux

As already indicated in Sect.3.3,  the excited atom density  Nn , and therewith also the scattering coefficient k,  depends on the ionizing radiation flux if the latter leads to photoionization frequencies nn* which are greater than the decay constant An  and the collisional ionization frequency nnc  .  For the case of the earths ionosphere, the solar radiation flux corresponding to quiet (normal) conditions  is just not strong enough to affect the equilibrium densities significantly (see Fig.6) (only for plasma densities as low as Np = 104 cm-3  it might become moderately relevant; see Fig.7  for quantum numbers n=500-1000).  During times of enhanced solar activity however,  the radio flux increases by orders of magnitude in particular at low frequencies (see Fig.10) and the resultant photoionization becomes the dominant loss process for levels n≥300 , thus reducing  Nn  for those states. Fig.23 shows the result for k0(f) and k(f) for the strongest (Type IV) solar bursts observed and a plasma density of  Np = 105 cm-3  (that is, corresponding to Fig.9)) which has to be compared to the quiet conditions scattering coefficient of Fig.12.  It is evident that the scattering coefficient is significantly reduced for  f<200 MHz  (by about a factor 0.5 for f =100 MHz and a factor 0.2 if f =1 MHz). With a plasma density of  Np = 104 cm-3  (which may be even more appropriate for the lower ionosphere), the effect is even more dramatic and results in a more than 2 orders of magnitude decrease in k(f)  for  f<10 MHz  with the related optical depth changing from optically thick to thin for frequencies around 1 MHz (see Fig.24 in comparison to Fig.11 ).
Since k(f) is a direct measure of the reflectivity of the ionosphere, it is a straightforward conclusion to relate this effect to the so called short wave fadeout which is observed for the transmission of radio waves between frequencies of 500 kHz and 10 MHz  after the onset of a solar flare (burst) (Mitra, 1974; pp. 5, 27) .  The almost simultaneous fading of the signal with the increase of the solar radio flux at frequencies f≥600 MHz (Fig.25) is in fact consistent with the assumption that photoionization of highly excited states is responsible for this effect, since the photoionization frequencies are so high (see dotted curve in Fig.9) that the Rydberg atoms become ionized within fractions of a second for the strongest bursts. It is therefore not a 'curious requirement' (Mitra, 1974, p.93) that a threshold solar radio flux is needed for 'Sudden Ionospheric Disturbances' to occur, but an expression of the fact that only above a certain value of the radio flux the photoionization frequency nn* is high enough (in comparison to the collisional ionization frequency nnc ) to affect the population number of atoms in high Rydberg states significantly and cause the reflectivity of the ionosphere to change.
Obviously, the same 'fadeout' effect can be achieved artificially by illuminating the relevant region of the ionosphere with a sufficiently powerful radio signal of the appropriate frequency. In this sense, Fig.26 gives (in comparison to Fig.12)  the scattering coefficient for a plasma of density  Np = 105 cm-3  subject to an ionizing microwave radiation with a frequency of  fw* = 35 GHz  and a field strength of Ew* = 0.2 V/m at the location of scattering.  For frequencies  f<200 MHz the scattering coefficient is here reduced by about 2 orders of magnitude because of the strong additional loss of electrons in states n higher than the threshold level Ö(3.2898.1015 Hz /fw*)  (»300 here) ( Fig.27, thin solid curve in comparison to Fig.18 ; see Eq.(24) and following for the relation between threshold frequency and quantum number  (the small scale variations in the curve for n>300 are not real but of a numerical nature due to the finite grid resolution and the almost monochromatic microwave frequency spectrum).

It should again be emphasized that these are equilibrium results,  i.e. in both cases (natural or artificial) will the observed effects be even much stronger if one considers times after the increase of the radiation flux which are short compared to the equilibrium relaxation time (this is of the order of several hours for ionospheric plasma densities and the frequencies considered here;  see Sect.3.5).  For this reason, only a relatively small increase in the ionizing radiation flux is needed to produce a short term non- equilibrium reduction of the scattering coefficient comparable in magnitude to the equilibrium effects shown here.

3.5    Non- Equilibrium Considerations

All numerical results presented in chapter 3 are based on the initial assumption of a detailed balance equilibrium. As already indicated at the end of Sects.2.1 , 3.3, 3.4.2  and 3.4.5,  they are therefore only strictly valid if the considered physical conditions vary slowly compared to the longest characteristic time constant entering into the problem.  It is obvious that for instance a variation in the total density of free plasma electrons  Np will not lead to a different scattering coefficient for radio waves until it causes a corresponding change in the level densities Nn . On the other hand, a sudden removal of electrons in the excited atomic states (by means of photo- or electron impact - ionization for instance)  will be related to an instantaneous drop in the scattering coefficient which will then only gradually  return to its equilibrium value as the states are re-populated from the free electron spectrum.  For level n this takes (on the statistical average) a time  1/AnRec  (Eq.(5)) which amounts from about 1 to many hours for the states of interest here.  With variations in the bulk plasma parameters (total electron density and mean energy), there is even a further delay 1/n0(see Appendix B, Eq.(B.23) ) until the change becomes apparent (by means of elastic collisions) at electron energies which are responsible for the recombination into those levels n of interest here. For ionospheric plasma densities, the collisional time scale becomes dominant if n<150, whereas for larger n (which covers those states relevant for the numerical results for the scattering coefficient presented in this work), the recombination time scale is larger and determines therefore the time dependent behaviour (one should again realize that 1/AnRec  is the duration of the quantum mechanical recombination process (the effective life time of the 'pre-bound' level n , see Sect.2.1 and Fig.1) which has to be distinguished from the recombinative collision time for the free plasma electron which is given by Eq.(17) and is much shorter than  1/AnRec  for the levels and plasma densities of interest here and is in particular always smaller than the time for elastic collisions  1/n0c for the ion mass and temperature assumed here (see Eq.(A.2.9)).

The long time scale associated with the establishment of an equilibrium between the free and bound electron spectrum enables considerable deviations from this equilibrium situation over time scales short compared to this , as indicated above . This could be one of the reasons that one observes for instance sometimes orders of magnitude variations in radar backscatter from  the ionosphere within a few minutes  or gets an almost instantaneous fading of radio signals transmitted over the ionosphere (short wave fade out) after the onset of strong solar flares (see also Sect.3.4.5).
It is therefore evident that a time dependent approach has to be formulated if one is interested in a time resolution which is better than the characteristic time scales  1/AnRec  and 1/n0c . In the context of ionospheric physics this is certainly predominantly the case unless one is dealing with data sets integrated or averaged over several hours or more. In the latter case, the detailed equilibrium theory used in this paper should, apart from giving general insight into the relationship between the relevant physical quantities, also yield quantitatively correct numerical results (see for instance the discussion of the theoretical and experimental radar scattering coefficients in Sect.3.4.3).

4.    Concluding Remarks and Outlook

The results obtained with the present detailed equilibrium approach show that commonly used methods to calculate excited atom densities, which are always related to some kind of LTE- assumption, can not even approximately describe the actual situation which is strongly determined by the quantum mechanical cross sections and time constants of the individual levels.  Any theory which does not proceed from these atomic properties (e.g.  the usual statistical equilibrium approach) is therefore a priori invalid and must necessarily lead to wrong conclusions.  This may be the reason why the process of resonance scattering by high atomic Rydberg states has not yet been considered as a possible mechanism affecting the propagation of radio waves, since the usual Boltzmann distribution underestimates the excited level density dramatically.
The present theoretical work attempts to give a general, complete and detailed coverage of the basic concepts necessary to calculate the efficiency for the scattering of electromagnetic waves by excited atoms without referring to any inconsistent and misleading ad hoc concepts but using only elementary physical laws on a microscopic level and deriving the macroscopic quantities in a logical and deductive way.
A modification of the theory to a time dependent formalism, which is necessary for instance for ionospheric problems if one is interested in a time resolution shorter than a few hours, should in principle be straightforward to accomplish without further basic theoretical work. 
With regard to certain situations, it might also become unavoidable to consider inelastic collisions of electrons as an important mechanism affecting the scattering coefficient for radio waves due to the related production rate of electrons at very small energies exceeding the one due to elastic collisions and enhancing therefore the recombination rate into high Rydberg states accordingly.  For example, strong increases in the rate of excitation of discrete atomic transitions by electrons are observed during non-linear plasma oscillations in the ionosphere induced by means of high power radio waves  (Smid, 1992a) and may in this way explain associated changes in the characteristics of the propagation of radio waves through the affected region.
As a further aspect, future work has to include the problem of the frequency- and angular redistribution of the radiation resonantly scattered by Stark shifted and -broadened levels in the presence of a magnetic field and the task of determining the total spatial interference wave field from the contributions of the individual atoms.  With this, one would then for instance be able to calculate the exact frequency dependence of radio wave spectra backscattered from the ionosphere, if additionally the modulation of the wave by the non- linear interaction with the magnetized plasma is taken into account (Smid, 1992b).

In consideration of the addressed theoretical aspects and the numerical results of this paper, it appears to be possible that the concept of resonance scattering by atomic states can generally serve as a basis for a unified theory for the interaction and propagation of electromagnetic waves in arbitrary media, which may replace inconsistent and unphysical ad hoc theories presently applied to corresponding problems.

Appendix A

Cross Sections and Decay Constants for Atomic Dipole Transitions

A.1  Transitions Between Bound States

The physical parameters for atomic dipole transitions can generally be obtained by considering the power radiated by an oscillator whose dipole moment is given by the quantum mechanical overlap integral

ri,k  :=  r0 . ò dr  Yi(r) .r .Yk(r) (A.1.1)

where  Yi and Yk designate the normalized radial wave functions of the energetically lower and upper state respectively (where state k is characterized by the pair of principal quantum number and angular momentum  (n,l) and  i  by  (m,l±1))  and r  is a dimensionless distance parameter which is normalized to the  Bohr radius

r0  =  h2/ ( 4p2me2)
    =  5.3 .10-9          [cm] ,             (A.1.2)

where h is Planck's constant, m the electron mass and e the elementary charge.

As a result (Smid, 1987), the damping constant related to the transition with the associated frequency  fi,k  is obtained as

Ai,k  =  16p4 .e2 .fi,k3 /  3c3h . ri,k2 , (A.1.3)

where c is the velocity of light,
and the cross section for resonant absorption or scattering of radiation of frequency  f  is determined through 

si,k(w)  =  2p5/2 .e2/h  . fi,k/c/(Df)D  . ri,k2  .H(a,w) , (A.1.4)


w  =  (f-fi,k)/(Df)D (A.1.5)

being the difference between the frequency of the radiation and the resonance frequency  normalized to the Doppler width

(Df)D  =  fi,k/c .  Ö(2kT/M) , (A.1.6)

where k is the Boltzmann constant, T the temperature and M the mass of the radiating atoms (ions) which are assumed here to have a Maxwellian velocity distribution.

The frequency dependence of the cross section is determined by the well known Voigt- function

H(a,w)  =  a/p  . ò du  e-u2 / [(w-u)2+a2] ,                (A.1.7)


a  =  Ai,k/ 4p /(Df)D . (A.1.8)

In most cases of interest, a<<1,  so that  H(a,w) can be approximated by the pure Doppler line profile e-w2 .  Note however that  (A.1.7), (A.1.8)  neglects any initial level broadening of the individual atom, since it contains only the apparent ('natural') frequency broadening due to the finite duration  1/Ai,k  of the transition between the two sharp levels involved  and the Doppler broadening caused by the Maxwellian velocity distribution of the atoms.
The case of an explicit level broadening is discussed in Sects.2.3 and 2.5,  where the resonant cross section including collisional Stark broadening is derived.

Explicit numerical calculations show now that for degenerate l-states (which can be assumed in the case of highly excited atoms), the dipole integral (A.1.1) depends primarily upon the principal quantum numbers, in the approximate form

rm,n  » rm,m+1  . (fm,m+1/fm,n)1.5 . (m/(n-1))1.6 .            (A.1.9)

In the case of large m, the Rydberg formula yields the further approximation

fm,m+1  =  f0 .[1/m-1/(m+1)2 »
                  »  2f0/m3 ,             (A.1.10)


f0  =  3.2898 . 1015  Hz ,           (A.1.11)

and the dipole integral for the ma-transition can be written as

rm,m+1  » r0/2 . m2 ,           (A.1.12)

so that Eq.(A.1.9) becomes (after substituting the explicit Rydberg formula for fm,n)

rm,n  »  1.4 . r0. [ 1/m2 -1/n2 ]-1.5  . m-0.9 .(n-1)-1.6 .        (A.1.13)

With this, one obtains the numerical approximations

Am,n  » 1.3 .109  .m-1.8 .(n-1)-3.2        [sec-1]        (m,n  >>1)         (A.1.14)


sm,n(w)  » 5.1 .10-12 . [ 1/m2-1/n2 ]-3 .m-1.8 .(n-1)-3.2 .Ö(A/T).H(a,w)      [cm2]  ,
        (m,n  >>1) ,        (A.1.15)

where A is the atomic mass number (=M/MH with MH the mass of a hydrogen atom)  and  T the temperature  in oK.

The summation of Eq.(A.1.14) over m yields the total decay probability for the level n

An  =  åAm,n          ;  (n³2) .           (A.1.16)

which can be numerically approximated by

A n  » 1.1 .109  . (n-1)-3.6            [sec-1] ,           (A.1.17)

as it emerges from an explicit numerical evaluation of (A.1.16) with (A.1.14)  including a correction for the contribution of higher angular momentum values (see Eq.(20)  and Appendix A.2, Eq.(A.2.10) and following) .

It should be noted that the dependence on the angular momentum quantum number, which has been neglected from Eq.(A.1.9)  onwards, might introduce an error up to about a factor 2 for the absolute values of Am,n and sm,n as obtained by Eq.(A.1.14) and (A.1.15) (this results from numerical calculations of the dipole integral involving the actual wave functions Y(m,l±1) and Ynl ). The relative values for a fixed angular momentum, on the other hand, which determine the branching ratios into the different levels, can be considered to be accurate to within a few percent. Even here, one has nevertheless to exclude very high angular momenta, because the dipole integral turns out to 'collapse' to extremely small values if  l comes close in value to the principal quantum number m if m is large. Since those transitions contribute however only a small fraction to the sum of all possible ones (in particular in view of the fact that recombination tends to populate only the lowest angular momenta of a given level, see next Sect. A.2), the error introduced by this circumstance can be neglected for the present purpose.

Furthermore, the explicit numerical calculations show that even for the lowest quantum numbers (for which the approximations (A.1.10) and (A.1.14) are in principle not valid anymore) Eqs.(A.1.14) and (A.1.17) are reasonably good approximations to Am,n , An and  sm,n .

A.2    Photoionization and Radiative Recombination

The concept outlined in Section A.1 can easily be extended to bound-free transitions by replacing the wave function of the upper state in Eq.(A.1.1) by a continuum wave function of the energy e , i.e.

rnl(e)  =  NB,F .r0 . ò dr Ynl(r) .r . [ cl.Fl-1(e,r) +Fl+1(e,r) ]      ,        (A.2.1)

where  Fl-1(e,r)  and  Fl+1(e,r)  are the regular Coulomb wave functions for the two angular momenta allowed by the selection rules for dipole transitions ( the angular momentum quantum number l has now been explicitly included because of its greater importance for bound- free transitions (see Eq.(A.2.10)). 
Note that the dipole integral is evaluated with the sum of the two possible continuous wave functions which are related to the angular momentum l of the bound state by the l- selection rule for dipole transitions, rather than by calculating the dipole integral separately for each l value and adding the results later.  This form is assumed here to make more sense for the present case of bound- free transitions, since it is not possible to define the angular momentum of a free electron uniquely before recombination into the bound state  nl  of a particular ion has taken place.  Reversely, the notion of an angular momentum for an electron is in principle meaningless after photoionization out of the atom has occurred (it should be noted that the two indicated methods for calculating the dipole integral produce anyhow only insignificantly (for the present purpose) different results, which are within a factor of 2).

The factor

cl  =  { 0  for  l=0   (A.2.2)
                    1  for  l>0

takes the special situation for l=0 into account, where only one transition is allowed by the selection rule,
and the dimensionless normalization constant 

NB,F  =  8.3  .10-6 (A.2.3)

has to be applied here because the integral over  Fl(e,r) can not be normalized separately like the wave functions for the bound states Ynl(r). It is determined from experimental measurements of the photoionization cross section by comparing it with the unnormalized theoretical result . The latter is obtained by inserting (A.2.1) instead of (A.1.1) into Eq.(A.1.4) and integrating over the atomic line profile H(a,w) with respect to w as (Smid, 1987)

snlIon(f(e))  =  2p3 .e2/h  . Ö(M/ 2kT) . rnl2(e.NB,F , (A.2.4)

where f(e) is the frequency of the ionizing radiation which is associated with the continuum energy  e through Eq.(24), Sect.2.1.

The cross section for radiative recombination is obtained from Eq.(A.2.4) by means of symmetry arguments, i.e.

snlRec(e)  =  snlIon(f(e)) (A.2.5)

(note that here and in the following formulae, the index n rather than the (numerically identical) quantum number n is being used with the recombination cross section, indicating that it is strictly to be understood as the cross section for the (collisionfree) recombination into the 'pre-bound' state n from which then the decay into the actual level n occurs if collisions do not interfere during its lifetime 1/AnRec (see Eq.(A.2.19)  and Section 2.1, Eq.(5) and following)).

Averaging over the angular momentum yields the cross section into the level n

snRec(e)  =  1/n . å  snlRec(e)      . (A.2.6)

from which the total cross section for recombination of an electron of energy e into all bound states is obtained as

sTRec(e)  =  å  snRec(e)      . (A.2.7)

Explicit numerical calculations of the recombination cross section involving the exact hydrogenic wave functions have been performed for continuum energies  e = 10-8 to 4 Rydberg  and bound states  n=1....1000  (Smid, 1987  and unpublished work).
As a symmetry property, one observes that for any n,  snRec(e)  has its maximum value approximately at an energy

en  =  1/n2 , (A.2.8)

with en in units of  Rydberg ,  and that this peak cross section can be approximated by the power law

snRec(en» 3.7 .10-17 . Ö(A/T)  . n2.4      [cm2] , (A.2.9)

where A is the ion mass number and T the ion temperature in oK ( note that for other elements than hydrogen,  n has to be interpreted here as designating the effective quantum number  (i.e. n=1 for the lowest unoccupied state of the ion) rather than the actual quantum number).
The distribution over the different angular momenta is hereby such that the recombination cross section is only slowly decreasing with l for l=n0.75 but drops rapidly if l is greater than this,  i.e. one can approximately set  (see Eq.(A.2.6))

snlRec(en)  = { snRec(en) .n0.25  for  l≤n0.75
                                0          for  l>n0.75 ;        (A.2.10)
(note that those electrons which have recombined into the bound state  nl  are not able to decay directly into a lower lying energy level  m in the case  l>m because of the l-selection rule for dipole transitions;  this circumstance together with (A.2.10) leads directly to the correction factor for the transition matrix  Eq.(20) ,Sect.2.1).

If  e<en  ,  snRec(e)  turns out to be approximately constant with energy (only for n=1 does one observe a dependence roughly  ~Öe if e<<1),  whereas for  e>>en  the cross section decreases rapidly ~ e-2.9. The transition region is obviously difficult to approximate analytically. Since it occurs however only within a relatively narrow energy band, it is sufficient for the present purpose to approximate the energy dependence of  snRec(e)  for  fixed n by the schematic function

h(e,en)  =  { 1  for  e=2en
                            (e/ (2en))-2.9  for  e>2en ,           (A.2.11)

so that

snRec(e)  =  snRec(en) . h(e,en) .           (A.2.12)

In order to obtain the cross section for a given energy instead of a given quantum number, one transforms the function  h  by means of Eq.(A.2.8) into

h(n,e)  =  { 1  for  n≤Ö(2/e)                                                 (n/ Ö(2/e) )-5.8  for  n>Ö(2/e) ,        (A.2.13)

and gets thus the alternative relationship

snRec(e)  =  sneRec(e) . h(n,e) ,           (A.2.14)


ne =  1/Öe           (A.2.15)

is the inversion of Eq.(A.2.8)  for continuous energy values  and  sneRec(e) is the corresponding generalization of the peak cross section Eq.(A.2.7) to the continuous quantum number  ne .

(It should be noted that if  h(e,en) (or  h(n,e)) <<1 ,  the distribution of the recombination cross section over angular momentum turns out to be even more concentrated towards small l-values than given by the peak cross section scheme (A.2.10).  Since those cross sections are however much smaller, their contribution to the recombination rate into level n  is only of minor importance, so that (A.2.10) can still be assumed to determine the angular momentum correction factor for the decay probability of the bound states  (Eq.(20), Sect.2.1)).
By using Eq.(A.2.14) and evaluating Eq.(A.2.7) numerically, one obtains the total cross section for recombination of an electron of energy e into all bound states as

sTRec(e» 5.4 .10-17  . Ö(A/T)  .e-1.7 [cm2]   (e≤2)    (A.2.16)

with  e again in units of Rydberg (for energies e>2, Eq.(A.2.9) for n=1 has to be used instead). .

In the case of negligible Stark broadening of the bound levels (compared to their energy) , i.e. if  hnRec=1 (see Sect.2.3, Eq.(79)), the related total recombination frequency (see Sect.2.1, Eqs.(17) and (40)) becomes therewith numerically

nTRec,0(e)   =  Np .ve(e) . sTRec(e)  =
                        =  Np . ve(e=1) . Öe . sTRec(e»
                        »  1.2 .10-8 .Np . Ö(A/T) .e-1.2      [sec-1]  ;  (hnRec=1; e≤2))    ,    (A.2.17)

where the plasma density Np has to be taken in  [cm-3]  and the superscript 0 has been added to nTRec in order to emphasize that this quantity describes the collisionfree recombination frequency into the 'pre-bound' state  n (for energies e>2, Eq.(A.2.9) for n=1 has to be used instead for the cross section in the above equation) .

The lifetime of the the 'pre-bound' state  n with regard to recombination into the actual bound level n is obtained by generalizing  Eq.(A.1.3) to bound- free transitions by multiplying the expression with the normalizing factor NB,F  (Eq.(A.2.3)) (one has to take the same factor here as for the cross section because both quantities depend on the square of the dipole integral (from which the need of a normalization arises ;  see below Eq.(A.2.3)).
One can therefore write

AnlRec(e)  =  [ 16p4 .e2 .fnl,e3 /  3c3h ] .rnl2(e.NB,F ,        (A.2.18)

which  (for those angular momenta where the recombination cross section Eq.(A.2.10) is different form zero),  can be numerically approximated by

AnRec  = 7 .104  .n-3.4          [sec-1] .           (A.2.19)

Note that the recombinative decay constant depends only on the bound state n but not the continuous energy e, which is due to the fact that the energy dependences of rnl2(e)  and  fnl,ecancel almost exactly (see Eq.(A.2.11)).

Appendix B

Cross Sections and Collision Frequencies for Energy Transfer by Electron-Electron  (Coulomb-) Scattering

For particles of arbitrary energy, the calculation of the cross section for Coulomb scattering represents a difficult problem in the laboratory system, which cannot always be solved in closed form or at least leads to complex analytical expressions because of the complicated formula which connects the scattering angles in the laboratory and center of mass system (Landau and Lifshitz, 1982).
With the present problem, however, a great simplification is introduced by the fact that one is primarily interested in electrons which have an energy which is small compared to the energy  ep  of the bulk of the plasma electrons.  In this case, the well known Rutherford formula gives to a good approximation the differential scattering cross section, since it just holds for one particle initially resting.

With  ep  being the dimensionless plasma energy (energy of the incident electron)  normalized to  E0 = 1 Rydberg (in cgs- units) and q  the scattering angle in the center of mass system, the differential cross section for scattering with a low energy electron can thus be written as

dsc(ep,q)/dW  =  e4 / [ 16.E02 . ep2 . sin4(q/2) ] , (B.1)

where  e  is the elementary charge ( in cgs- units) and

dW  =  2p  sinq  dq . (B.2)

Even energy changes very small compared to the plasma energy  ep  are fully taken into account by Eq.(B.1), because those can be primarily attributed to small angle scattering of the bulk of higher energetic electrons rather than large angle collisions of the few low energetic electrons.
This is due to the fact that the ratio of the cross section for the latter possibility to the cross section for the former increases more slowly for De®0  ( ~ De-1.5 ) (see Eq.(B.6) ) than the number of electrons within the energy range  0...De  decreases (~ De1.7, taking the consistent electron distribution function into account  (Figs.345).

The energy change  De  is generally related to the scattering angle through the energy transfer function for particles of equal mass (Landau and Lifshitz, 1982)

De/ep  =  sin2(q/2) .   (B.3)

If one would be interested in the average energy transfer ( i.e. scattering about all angles),  the related cross section would be obtained by integrating  ( from  q =0....p) the product of the differential cross section Eq.(B.1),  the energy transfer function Eq.(B.3)  and a further geometrical weighting function

f(q)  = 1/p  . sin(q/2) ,   (B.4)

which takes care of the sphericity of the scattering potential (Smid, 1987).
In the present case, however, one is just interested in those small energy transfers which the function (B.3) suppresses, so that only  f(q) has to be multiplied to the differential cross section. By doing this, changing to the differential (see Eq.(B.3))

d(De)  =  d.ep . sin(q/2) .cos(q/2)   (B.5)

and subsequently to De  by means of Eq.(B.3), one gets for the differential cross section with regard to a given energy change

dsfc(ep,De)/d(De)  =  s0c /2  . ep-1.5  .(De)-1.5 ,   (B.6)


s0c  =  5.6  .10-17       [cm2]   (B.7)

and  ep  and  De  are in energy units of  1 Rydberg.

Assuming that the energy of an electron within the continuum energy range  ei-1, ei  can be represented by

e  =  (ei-1 + ei)/2 ,   (B.8)

the cross section for scattering from within this range into the given interval ek-1, ek  is therefore

si,kc   =  s0c /2  . ep-1.5 . ò de e -+ei)-1.5        =

          =  s0c . ep-1.5  . [ ±1/Öek-1-+ei)   -+ 1/Öek -+ei) ] ,   (B.9)

where the upper sign holds for k>i and the lower for k<i  (the exclusion of k=i means that scattering within the interval  ek-1, ek  itself is neglected).

The cross section for collisional loss out of the interval ek-1, e into the whole numerically available energy range  0....eM  is given by

sk,Lc  = å sk,ic             =          

                                                ek-1                              eM
                  =   s0c /2  . ep-1.5 . [ ò de (ek -e)-1.5    +  ò de (e -ek)-1.5 , (B.10)
                                                    0                                  ek

where, in consistency with Eq.(B.8),

ek   =  (ek-1 + ek)/2 . (B.11)

Evaluation of the integral yields

sk,Lc  =  s0c . ep-1.5  . [ 2Ö2/Ö(Dek) - 1/Öek  -1/Ö(eM -ek ) ] (B.12)

with Dek  as defined by Eq.(41).

Furthermore, the cross section for collisional ionization from the bound level n (with energy  -en ) into the continuum interval  ek-1, ek  is

    sn,kc  =   s0c /2  . ep-1.5 . òde   (e +en)-1.5               =

          =    s0c . ep-1.5  . [1/Ö(ek-1 + en)   - 1/Ö(ek + en)   ] . (B.13)

It has been assumed here that the energy  -en , which is a quantum mechanical eigenvalue, is identical with the classical energy of the electron in its orbit, which is at least approximately the case.
The cross section for collisional ionization into the whole (numerically) available continuum energy range  0...eM  is obtained from Eq.(B.13)  by setting  ek-1 = 0  and  e= eM  ,i.e.

snc  =  s0c . ep-1.5  . [1/Öe - 1/Ö(eM + en) ]  . (B.14)

In the case eM >>en ,this reduces to the simple expression

snc  =  s0c . ep-1.5  . n . (B.15)

The collision frequency for elastic scattering is generally related to the corresponding cross section by

n c  = Np .ve(ep) .s c  =
          =  Np . 2.2.108  . Öep .s c        [sec-1] , (B.16)

if  eis in units of Rydberg, Np in  [cm-3]  and s c  in  [cm2] .

From the various cross sections derived in this chapter one gets thus

ni,kc   =  n0c  . [ ±1/Öek-1-+ei)   -+ 1/Öek -+ei) . di (B.17)

nk,Lc   =  n0c  . [ 2Ö2/Ö(Dek) - 1/Öek  -1/Ö(eM -ek ) ]  . dk (B.18)

nn,kc   =  n0c  .  [1/Ö(ek-1 + en)   - 1/Ö(ek + en)   ]             (B.19)

nnc   =  n0c  . [1/Öe - 1/Ö(eM + en) ]        »
              »  n0c  .n      if  en <<eM (energy of excited state small compared to
                                                                            the maximum numerical continuum energy) (B.20)
nn,kc   =  nn,kc   (B.21)
nnc     =  nnc   , (B.22)


n0 1.2.10-8  .Np /ep        [sec-1] , (B.23)

with Np in [cm-3] and ep  in units of Rydberg,    and

dk  =  (Np  - Ne,k )  / Np (B.24)

is the fraction of electrons outside the energy interval  ek-1, ek  (see Sect.2.2, Eq.(32)) in consistency with the assumption that scattering within this interval itself is neglected (see below Eq.(B.9)) .

From experimental results, it is evident (Smid, 1987) that the cross section for electron impact ionization of neutrals in the ground state by electrons with an energy comparable to the ground state energy is about 4 orders of magnitude smaller as would follow from Eq.(B.14) (at least is this true for ionospheric conditions).
For this reason, Eqs.(B.19)-(B.22) have been additionally multiplied by the factor

hnc  =  10-4 . Min(104, ep/en ) . (B.25)

It should be noted that for the present purpose this correction is however only of minor relevance, since electron collisions are only important for large n for which hnc is close to 1 (see Sect.3.3).

With regard to the application of the expressions derived in this chapter to the balance equation in Sect.2.2 , one should again realize that in principle they are only strictly valid if the energy transfer is small compared to the bulk plasma energy ep . The numerical error introduced by this inconsistency for the highest energies of the electron spectrum is however insignificant for the present purpose, because the high energy bulk of electrons serves only as a background 'pool' for the production of electrons at smaller energies by means of elastic scattering and errors in the assumption for the form of the differential scattering cross section for those energies will more or less average out (this is also the justification for choosing the low numerical resolution of the energy grid for the electron spectrum at high energies (see Sect.3.1) ).


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pp. 465-481,  Wiley Interscience, London, New York, Sydney, 1967.

Landau, L.D. and E.M. Lifshitz,  Mechanics,  Pergamon Press, Oxford, 1976.

Lang, K.R.,  Astrophysical Formulae,  Springer-Verlag, Berlin, Heidelberg, New York,

Massey, H.S.W.,  Electronic and Ionic Impact Phenomena, Vol.2,  Oxford University
Press, 1969.

Mitra, A.P.,  Ionospheric Effects of Solar Flares,  D. Reidel Publishing Company,
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Moorcroft, D.R.,  Estimates of absolute scattering coefficients of radar aurora,
J.Geophys.Res. 92, 8723-8732, 1987.

Pankonin, V.,  The partially ionized medium adjacent to HII regions, in
RadioRecombination Lines , edited by P.A. Shaver, pp. 111-125, 
D. Reidel Publishing Company, Dordrecht, Holland, 1980.

Rees, M.H.,  Physics and Chemistry of the Upper Atmosphere, Cambridge University
Press, 1989.

Smid, T.S.,  Theoretische Aspekte der Ionosphärenphysik,  Ph.D. Thesis, Bonn
University, 1987.

Smid, T.S.,  The resonant excitation of airglow by high power radio waves, submitted to
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Smid, T.S.,  The excitation of oscillations of a magnetized collisionless plasma, submitted
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