I. INTRODUCTION

For the first few Balmer lines of hydrogen, recent papers

(Gerardo and Hill, 1966; Bacon and Edwards, 1968; Kepple and Griem, 1968; Birkeland, Oss and Braun, 1969) have demonstrated fairly good agreement between measurements in high electron density plasmas

(n > 10 ne

16cm-3

) and improved calculations of the so called "modified impact theory". The experimental and theoretical half-widths differ less than about 10%. However, measurements of the Lyman-a wings (Boldt and W. Cooper, 1964; Elton and Griem, 1964) and low electron

1013 cm-3

density measurements (n = 1013cm3) of the higher Balmer and

e

Paschen lines (Ferguson and Schlüter, 1963; Vidal, 1964; Vidal, 1965) have revealed parts of the hydrogen line profile, for which the modified impact theory appears to break down. For the higher series members better agreement has been obtained with quasi-static calculations (Vidal, 1965). The reason the current impact theories break down is that these theories correct the completed collision assumption by means of the Lewis cutoff (Lewis, 1961) which is only correct to second order. With this cutoff it was possible to extend

the range of validity for the impact theory beyond the plasma frequency. However, in the distant wings, where the electron broadening becomes quasistatic, the second order perturbation treatment with the Lewis cutoff breaks down because the time development operator must then be evaluated to all orders. Attempts to correct the second order theory have been made already (Griem, 1965; Shen and J. Cooper, 1969), but these theories still make the completed collision assumption by replacing the time development operator by the corresponding S-matrix, and so it has to be emphasized that in conjunction with the Lewis cutoff these theories would only be correct to second order. The impact

theory in its present form is intrinsicly not able to describe the static wing and the transition region to the line center where dynamic effects

cannot be neglected. To overcome this problem, several semiempirical procedures (Griem, 1962; Griem, 1967a; F. Edmonds, Schlüter and Wells, 1967) have been suggested to generate a smooth transition from the modified impact theory to the static wing.

Recently the classical path methods in line broadening have been reinvestigated in two review papers (E. Smith, Vidal and J. Cooper, 1969a, 1969b), which are from now on referred to as papers I and II. The purpose of I and II was to state clearly the different approximations which are required to obtain the classical path theories of line broadening and to find out where these theories are susceptible to improvements. In a manner similar to the Mozer-Baranger treatment of electric microfield distribution functions (Baranger and Mozer, 1959, 1960), it was shown that the general thermal average can be expanded in two ways, one of which leads to the familiar impact theory describing the line center (Baranger, 1958, 1962; Griem, Kolb and Shen, 1959, 1962). The other expansion represents a generalized version of the one electron theory (J. Cooper, 1966), which holds in the line wings. It is also shown that there is generally a considerable domain of overlap between the modified impact theory and the one electron theory. Based on these results, a "unified theory" was then developed (E. Smith, J. Cooper and Vidal, 1969), henceforth referred to as paper III, which presents the first line shape expression which is valid from the line center out to the static line wing including the

where d, Aw and £(w) are operators. In paper III it was shown that the familiar impact theories, which hold in the line center, may be obtained by making a Markoff approximation in the unified theory, while the one electron theory describing the line wings is just a wing expansion of the unified theory. Consequently the crucial problem for any line broadening calculation is to evaluate the matrix elements of (w), which is essentially the Fourier transform of the thermal average see Eq. (46) and (47) of paper III). This will be done in detail in this paper for the general case of upper and lower state interactions.

In the following Sec. II we start with a brief summary of the basic relations which are required for the classical path approach pursued here. We then generalize the results of the unified theory to include lower state interaction (Sec. IV) after introducing a more compact tetradic notation (Sec. III). From this general result we turn to the specific problem of hydrogen by discussing briefly the no

quenching assumption (Sec. V) and deriving the thermal average 3(1)(t)

(see Eq. (47) of paper III) for the general case of upper and lower state interaction (Sec. VI). We next investigate the multipole expansion of the classical interaction potential in the time development operator

(Sec. VII). 3 The thermal average (1)(t) is then evaluated in two steps

by first performing a spherical average (Sec. VIII) and then an average over the collision parameters: some reference time t to' impact parameter p and velocity v (Sec. IX). Appendix A gives the computer program which we used in calculating the thermal average for dipole interactions including lower state interactions. The large time limit of the thermal average, which leads to the familiar impact theories in the line center, is investigated in detail in Appendix B for different cutoff procedures and compared with the results in the literature.

In

Sec. X, a method is developed for performing the Fourier transformation of the thermal average and it leads us to the crucial function for any classical path theory of Stark broadening. This function is finally applied in Sec. XI to the one electron theory, which forms the basis for the asymptotic wing expansion, and in Sec. XII to the unified theory, which describes the whole line profile from the line center to the static wing. Numerical results are given for the hydrogen line profiles as measured by Boldt and W. Cooper, 1964; Elton and Griem, 1964, and Vidal, 1964, 1965. The computer program for the unified theory calculations and the asymptatic wing expansion is given and explained in Appendix C.

II. BASIC RELATIONS

In this section we will briefly outline the basic relations which are used in our classical path treatment of line broadening.

i

As discussed in Sec. 2 of paper I, we are considering a system containing a single radiator and a gas of electrons and ions. We will make the usual quasi-static approximation for the ions by regarding their electric field, as being constant during the times of interest 1/ Δω . This approximation is usually very good because the region where ion dynamics are important is normally well inside the half width of the line except for a few cases such as the n-a lines of hydrogen (Griem, 1967b). The complete line profile I(w) is then given by the microfield average (see Eq. (3) of paper II)

where the normalized distribution function P(e) is the low frequency component of the fluctuating electric microfields. Due to shielding effects P(e) depends on the shielding parameter r/D where r and D are the mean particle distance and the Debye length (for electrons only respectively.

With the static ion approximation we have reduced the problem to a calculation of the electron broadening of a radiator in a static

electric field

i'

The resulting line profile I(w, Ɛ;) is then simply averaged over all possible ion fields to give the complete line profile I(w). The static ion field will be used to define the z-axis for the radiator and the ion-radiator interaction will be taken to be the dipole interaction e Ze, where -e Z denotes the Z-component of the radiators dipole moment.

If the unperturbed radiator is described by a Hamiltonian H we may then define a Hamiltonian for a radiator in the static field

a'

The complete Hamiltonian for the system is then given by

H= H + V ̧(À, ✯, v, t)

(II. 2)

(II. 3)

which denote the positions and velocities of the N electrons and R denotes some internal radiator coordinates. For one-electron atoms, Ris the position of the "orbital" electron relative to the nucleus.

The interaction V will be regarded as a sum of binary interactions,

e

v (Ř, x, v, t) = Σ 1

denotes the interaction between the radiator and a single
As is well known the line shape I(w, E.) may be given by

(II. 4)

the Fourier transform of an autocorrelation function C(t) (Baranger, 1962)