Scattering of Radio Waves
by High Atomic Rydberg States

by
Thomas Smid

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

Copyright American Geophysical Union

A summary can be found under
http://www.plasmaphysics.org.uk/research/#A3

Abstract.........................................................................................................................3
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
Appendices.......................................................................................................................55
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

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.

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

N_n = q_n/n_n^loss .       (1)

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

q_n =q_n^Rec + q_n^{casc .         (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

                                              ¥
q_n^Rec,0   =  h_n^{Rec .}N_p  ^._ò de N_e(e) ^. s_n^Rec(e'(e,e_I)) ^. v'(e,e_I) ,       (3)
                                            0

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), N_p the total plasma (ion) density  which is related to the energy specific electron density distribution function N_e(e) by the condition of quasi neutrality of the plasma

               
                    ¥
N_p  =  _ò de N_e(e) .       (4)
                  0

Furthermore,  v'(e,e_I)  is the relative velocity between the recombining electron and an ion of average energy e_I in the laboratory system, s_n^Rec(e'(e,e_I)) the recombination cross section into level n for the corresponding relative kinetic energy  e' (see Eqs.(11) and (12))  and  h_n^Rec 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))

A_n^Rec  = 7 ^.10^{4 .}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  A_n^Rec 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

N_n  =  q_n^Rec,0 / (A_n^Rec+n_n^c) .       (6)

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

n_n^c  = 1.2 ^. 10^{-8 .} (N_p/e_p) ^. 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

q_n^Rec  =N_n  ^.A_n^Rec  =
          =  a_n^{Rec,c  .}q_n^Rec,0 ,       (8)

where

a_n^Rec,c  :=  A_n^Rec/ (A_n^Rec+n_n^c)       (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

v_I/v_e  =  Ö ( m/M  ^. e_I/e ) ,     (10)

and because v_I and e_I 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,e_I)  =  { ^e    for  e>eI.m/M ,     (11)
                              ^eI.m/M    for  e≤eI.m/M

to which  v' is related by

v'(e,e_I) =  v₀ ^. Ö e'(e,e_I) ,     (12)

where

v₀  :=  v_e(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

                                                              ¥
q_n^Rec  =  a_n^{Rec,c .}h_n^{Rec .}N_p ^. v₀  ^._ò de N_e(e) ^. s_n^Rec(e'(e,e_I)) ^. Öe'(e,e_I) .    (14)
                                                            0

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

                                      ¥
q_n^Rec  =  a_n^{Rec,c   .}åN_e,k ^. n_n,k^Rec,0 ,     ⁽¹⁵⁾
                                      k=1

with

N_e,k  =  (e_k-e_k-1) ^. N_e(e_k)     (16)

and

n_n,k^Rec,0   =   N_p ^. v₀ ^. s_n^Rec(e'_k) ^.h_n^{Rec .} 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 N_e(e_k) which is determined in the separate Section 2.2 (although actually a set of coupled equations describing the densities in the discrete (N_n) 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  A_n and the collisional (n_n^c) and photoionization (n_n*) frequencies out of this state, that is

n_n^loss  = A_n + n_n^c + n_n* ,     (18)

where

                n-1
A_n  =  åA_m,n          ;  (n³2) .     (19)
                  m=1

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

A_m,n  » 1.3 ^.10^{9  .} 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 ^.10^{9  .} (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

                              ¥
n_n*  =  h_n^{Ion .} _ò df  F^ph(f) ^. s_n^Ion(f) ,     (22)
                            f_n

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

s_n^Ion(f)  =  s_n^Rec(e(f)) ,     (23)

with

e(f)  =  f/f₀ - e_n                    ;  f³f_n     (24)

where 

e_n  = f_n /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, e_n  is then given by the Rydberg formula

e_n^H  = 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).

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

                              ¥
n_n^*  =  h_n^{Ion .} å(e_k- e_k-1) ^.F^ph(e_k) ^. s_n^Ion(e_k)  ,     (27)
                            k=1

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

F^ph(e)  =  F^ph(f)  ^. f_{0 ,}     (28)

as it follows from Eq.(24) (if f₀ = 3.2898 ^.10¹⁵ Hz,  F^ph(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),  N_n 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

                        ¥
q_n0^casc  = å  N_m ^. A_n0,m ,     (29)
                          m=n₀+1

where A_n0,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

                        ¥ 
q_n^casc  = å  N_m ^. A_n,m .     (30)
                          ^m=n+1

It should already be noted that, because of the very low densities N_m and the small cascading probabilities implied by Eq.(20) for high quantum numbers, q_n^casc  tends to be much smaller than the primary production rate  q_n^Rec .  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 s_n^c 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  A_n^Rec (Eq.(5))  and the elastic collision frequency n₀^c (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  n_n^c  or n_n*  may lead to a considerable decrease in the level population N_n during a time interval small compared to this. A quantitative determination of these short term variations of N_n  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 (e»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 N_e(e) (Smid,1987).
For the energies considered in Smid(1987) (e³0.01;  corresponding to n≤10 (see Eq.(26))), the individual recombination process is fast enough (A₁₀^Rec» 28 sec^-1 ; see Eq.(5))  to neglect interfering elastic collisions  (n_n^c »2.6^.10^-2 sec^-1 (see Eq.(7) for n=10, e_p=0.5 and a plasma density of N_p=10⁵    cm^-3)). For much smaller energies, however, elastic collisions will eventually limit the actual recombination rate because of the rapidly decreasing value of a_n^Rec,c (~ n^-4.4 ) if  n_n^c>A_n^Rec  (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 ^.10^{12 .}e_p/N_p)^0.23 ,     (31)

where the plasma density N_p has to be taken in [cm^-3] and the average plasma energy e_p in units of Rydberg ( for N_p =10⁵ cm^-3 and  e_p=0.5,  n*=51) (the numerical results will show that the excited atomic densities N_n 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 N_e(e), whereas for higher energies the latter reflects basically the energy dependence of the total recombination frequency n_T^Rec,0(e) ; see Appendix A.2, Eq.(A.2.17) and Sect.3.2).

A further possible mechanism affecting  N_e(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 .

                    M                          ¥                        ¥
p_1,k    + å N_e,i ^.n_i,k^c    + å N_n ^.n_n,k^c    + å N_n ^. (n_n,k^c +n_n,k*)    =
                    i=1                           n=2                      n=2
                    ¹k

                        M                  ¥
= N_e,k ^. ( å n_k,i^c   + å n_n,k^Rec,0 )              ;  k=1,M ,     (32)
                        i=1                n=1
                        ¹k

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    e_k= 0....e_M.
The various collision frequencies are (see Fig.1) :
n_i,k^c  is the frequency for elastic free-free (Coulomb-) scattering of electrons from the energy range  e_i-1, e_i  into e_k-1, e_k    ( n_k,i^c  indicates the inverse process), assuming that there is no scattering within the interval e_k-1, e_k  itself    (i.e.  i¹k) (see Appendix B , Eq.(B.17)),
n_n,k^c  is the frequency for elastic collisions from the bound level n (electron impact ionization) into the continuum energy range e_k-1, e_k  , n_n,k^c  the  (numerically identical)  frequency for the 'pre-bound' level  n, (Appendix B , Eq.(B.19))  and

n_n,k*  =  h_n^{Ion .} F^ph(e_k)^. s_n^Ion(e_k)     (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  n_n,k^Rec,0  into the pre-bound level  n  is given by Eq.(17)  and  p_1,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

p_1,k  =  p₀ ^. (e_k - e_k-1) ,     (34)

that is, the production function  p₁(e_k)  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 e®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

        M   
å p_1,k  = p₀ ^. e_M  =  q₁     (35)
    k=1

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

        M
å N_e,k  =  N_p .     (36)
    k=1

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

f_e(e_k)  :=  N_e(e_k)/ N_p  =
            =  N_e,k /(e_k-e_k-1) / N_{p ,}                     (37)

i.e.  Eq.(32) becomes

                              M                               ¥                      ¥
q₁^.De_k/e_M  + N_p^.å De_i ^.f_e(e_i)^.n_i,k^c  +å N_n ^.n_n,k^c  +å N_n ^. (n_n,k^c +n_n,k*)= 
                            i=1                             n=2                     n=2
                              ¹k

=  N_p^.De_k ^.f_e(e_k) ^. [ n_k,L^c   + n_T,k^Rec,0 ]              ;  k=1,M  ,                (38)

where

                      M               
n_k,L^c  = å n_k,i^c                             (39)
                      i=1          
                      ¹k
(see Eq.(B.18)),
                                ¥
n_T,k^Rec,0   =   å n_n,k^Rec,0                           (40)
                              n=1

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

and

De_k  = e_k - e_k-1 ,
De_i  = e_i - e_i-1 .     (41)

Eq.(38) is a system of M equations for the M unknowns f_e(e_k) . Together with Eq.(1),(2),(6) and (15) , which determine the recombination rate q₁ and the atomic level densities N_n and N_n, 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  A_m,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,S^Static =  ±  e ^. |E_||| ^. 2^._‹r_›n  =                                         =  ±  e ^. |E_||| ^. r₀ ^.n² ,     (42)

with e  the elementary charge ,  E_|| the average electric field amplitude in the orbital plane of the electron, _‹r_›n 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

cosa_p  =  (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,S^Static  =  ±  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 Dt_f (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|} ^.Dt_w  ^. {|dr/dt|} ^.Dt_w ,     (46)

where

Dt_w = Dt_f^.T_n / (Dt_f+T_n )     (47)

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

T_n  =  1/w_n  =
          =  Öm /e  ^._‹r_›n^3/2  »
      » 8.6 ^.10^{-18  .}n³                [sec] ,     (48)

as one obtains classically by considering the circular frequency of an electron with orbital radius _‹r_›n 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_|||}/Dt_f .     (49)

Furthermore , the projected velocity of the electron is
|dr/dt|  =  2^._‹r_›n / T_n  = 
                =  r₀ ^.n² / T_n ,     (50)

so that the energetical broadening of level n becomes

(dW)_n,B  =  ±  e ^. 2^.{|E_|||}^.r₀ ^.n^{2 .} z_n ,     (51)

with

z_n = Dt_f^.T_n / (Dt_f+T_n )² .     (52)

Note that z_n has a maximum for Dt_f=T_n  and decreases linearly with the ratio of the time constants in both limits Dt_f>>T_n  and Dt_f<<T_n .  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 T_n , and that in the limit of  Dt_f<<T_n the reduction is determined by the ratio of the squares of the fluctuation field gradient {|E_|||}/Dt_f to the characteristic gradient {|E_|||} /T_n (thus relating it to the associated (dynamical) field energies rather than the field values),  one can adopt the relative splitting (or shift) factor

m_n  =  1/ ( 1+(T_n/Dt_f)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.

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

r_p  = ( 4p/3 ^.N_p)^-1/3     (55)

of two particles in the plasma, that is

DE_p  =  e/r_p²   =
        =  e ^.  ( 4p/3 ^.N_p)^2/3    =
        =  1.25 ^.10^{-9 .} N_p^2/3       [statvolt/cm  =3^.10⁴  V/m]     (56)

if the plasma density  N_p  is in [cm^-3] .

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

Dt_p,i = 2 r_p/v_i ,     (57)

where v_i 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 r_p 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

j_n,i(df)  =   exp [ - ( df/ (Df)_n,B,i )²  ] ,     (59)

where 

(Df)_n,B,i  = 2/Öp  ^. (Df)₀^. n^{2 .} z_n,i            ,                 (60)

with z_n,i  given by Eq.(52) for Dt_f = Dt_p,i  and

(Df)₀  =  0.49 ^. N_p^2/3                  [Hz]     (61)

if N_p 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.

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

                    =  exp [ - ( df/ (Df)_n,B)²  ] ,     (62)

where

(Df)_n,B  =  2/Öp ^.(Df)₀^. n^{2 .} Ö(z_n,i²  + z_n,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 j_n(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  N_p),  the field fluctuation period Eq.(57)  attains the numerical values (respectively)

        Dt_e =  2r_p/v_e(e_p)  =
      =  5.6 ^.10^-9  _/N_p^1/3  /Öe_p           [sec]                  (64)

and

Dt_I = 2r_p/v_I  =
            =  9.4 ^.10^-5  /N_p^1/3  /Ö (T/A)          [sec] ,     (65)

with N_p the plasma density in [cm^-3] ,  e_p 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^. n^{2 .} Ö(z_n,e²  + z_n,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  f_m,n' _, the atomic line absorption profile in dependence of the given frequency f and a fixed level shift can therefore be written as

                                          ¥
j_m,n(f)  = 1/Öp  /(Df)*_B  ^.òdf' j_m(f') ^. j_n(f'-f +f_m,n' ) ,                (67)
                                            - ¥

with

(Df)*_B    =  (Df)_m,B ^.(Df)_n,B  / Ö((Df)_m,B² + (Df)_n,B²) .     (68)

Evaluation of the integral yields

j_m,n(f)  =  exp [ - (f -f_m,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 j_m,n(f)  is normalized to the line center (f =f_m,n')  rather than the integral over frequency because of the normalization of    òdf' j_m(f') ^.j_n(f'). 

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

Dt_e,I =  Dt_e^.Dt_I /(Dt_e +Dt_I) .     (71)

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

(df)_n,S  = ±(Df)₀ ^. n^{2 .} m_n  ^. E/DE_p ,     (72)

with  (Df)₀  (the normal frequency shift for the ground state in a field DE_p) 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

f_m,n'  =  f_m,n ± (Df)₀ ^.  g_m,n ^.E/DE_p ,                 (73)

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

  g_m,n =  n^{2 .} m_n  - m^{2 .} m_m .     (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.
                                                                     
                                                                          ¥
f_m,n(f)  = <j_m,n(f) >  = 1/Öp /DE_p ^. c_m,n ^.òdE  e^{-(E/DEp)2 .} j_m,n(f)    ,  (75)
                                                                        - ¥
where c_m,n is a normalization constant which makes the integral of f_m,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

f_m,n(f)  =  1/Öp /(Df)_S ^. exp[ - (f -f_m,n)² / ( (Df)_B² + (Df)_S² )  ] ,    (76)

where now the abbreviation

(Df)_S  =  (Df)₀ ^. g_m,n     (77)

has been introduced for the static line splitting.

f_m,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  f_m,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 f_m,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 f_m,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=DE_p)     (78)

becomes comparable and greater than the threshold ionization frequency f_n  (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

h_n^Rec  =    exp[ - f_n  /(Df)_n  ] ,                 (79)

with

(Df)_n  =  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

h_m,n^sc =  1 - exp[  - f_m,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 ele