14.5: Collisional Broadening

To this point, we have described the broadening of spectral lines arising from intrinsic properties of the atom and the collective effects of the motion of these atoms. However, in all but the most extreme cases of macroscopic broadening, the most prominent source of broadening of spectral lines results from the interaction of the absorbing atom with neighboring particles of the gas. Since these particles are often charged (even the neutral atoms possess the potential field of an electric dipole), their potential will interact with that of the atomic nucleus which binds the orbiting electrons. This interaction will perturb the energy levels of the atom in a time-dependent fashion. The collective action of these perturbations on an ensemble of absorbing atoms is to broaden the spectral line. The details of this broadening depend on the nature of the atom and energy level being perturbed and the properties of the dominant perturber. All phenomena that fall into this general class of broadening mechanisms are usually gathered under the generic term collisional broadening. However, some authors refer to this concept or a subset of it as pressure broadening, on the grounds that there can be no collisions unless the gas has some pressure. The use of the different terms is usually not of fundamental importance, and the basic notion of what is behind them should always be kept in mind.

There is some confusion in the literature (and much more among students) regarding the terminology for describing these processes. Some of this results from a genuine confusion among the authors, but most derives from an unfortunate choice of terms to describe some aspects of the problem. You should keep clearly in mind what is being described during any discussion of this topic - the broadening of atomic energy levels resulting from the perturbations of neighboring particles. We adopt a variety of theoretical approaches to this problem, each of which has its own name. Care must be taken lest the name of the theoretical approach be confused with a qualitatively different type of broadening. We discuss perturbations introduced by different types of perturbers, each of which will produce a characteristic line profile for the absorbing gas. Each of these profiles has its own name so as to delineate the type of perturbation. However, they are all just perturbations of the energy levels. Each type will generally be discussed in a "vacuum", in that we assume that it is the only form of perturbation that exists, when in reality virtually all types of perturbations are present at all times and affect the energy levels. Fortunately, one of them usually does dominate the level broadening.

There are two main theoretical approaches to collisional broadening. One deals with the weak, but numerous, perturbations that cause small amounts of broadening. The other is concerned with the large, but infrequent, perturbations that determine the shape of the wings of the line. It as somewhat unfortunate that the former theoretical approach is known as impact phase-shift theory, while the latter is called the statistical or static broadening theory. The word impact conjures up visions of violence, yet the theoretical approach labeled by this word is concerned only with the weakest and least violent of the interactions. Similarly, the term static implies calm, but this approach deals with the most violent perturbations. So be it. We try to justify this apparent anomaly during the specific discussions of these approaches. In addition, we clearly label the myriad terms as they are introduced so that those which are synonymous are clearly separated from those which have unique meanings.

To estimate the perturbation to the atom that changes the energy of the transition and thereby broadens the line, we must characterize the nature of the collision. The two theoretical approaches to collisional broadening differ in this description. Both approaches are largely classical in form so that whatever is true for absorption is also true for emission. So we often deal with the effects of a collision on a radiating atom with the full intention of applying the results to absorption.

a. Impact Phase-Shift Theory

The approach of impact phase-shift theory assumes that the collision is of a very short duration compared to the span of time during which the atom is actually radiating (or absorbing) the photon. Thus, \[t_{\mathrm{col}} \ll t_{\mathrm{rad}}\label{14.4.1}\]

It is the short duration of the collision that is responsible for the name impact for the theoretical approach.

Determination of the Atomic Line Absorption Coefficient Suppose that the atom radiates in an undisturbed manner between collisions with a frequency ω₀. The electric field of the emitted photon will vary as \[E(t)=E_0 e^{-i \omega_0 t} \quad-\frac{T}{2} \leq t \leq \frac{T}{2}\label{14.4.2}\]

where \(T\) is the time between collisions. Further assume that the radiation of the photon does not continue before or after the collision, so that \[E(t)=0 \quad t>\left|\frac{T}{2}\right|\label{14.4.3}\]

\[E(\omega)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} E(t) e^{i \omega t} d t=\frac{E_0}{\sqrt{2 \pi}} \int_{-T / 2}^{+T / 2} e^{i\left(\omega-\omega_0\right) t} d t\label{14.4.4}\]

or \[E(\omega)=+\frac{E_0}{\sqrt{2 \pi}} \frac{2 \sin \left[\left(\omega-\omega_0\right) T / 2\right]}{\omega-\omega_0}\label{14.4.5}\]

Since the power spectrum of the emitted photon will depend on the square of the electric field, \[I(\omega)=\frac{A^2 E_0^2}{2 \pi} \frac{\sin ^2\left[T\left(\omega-\omega_0\right) / 2\right]}{\left[\left(\omega-\omega_0\right) / 2\right]^2}\label{14.4.6}\]

Here we have assumed that we will be dealing with emissions and absorptions that are totally uncorrelated, which for random collisions occurring in a sea of unrelated atoms is a perfectly reasonable assumption.

Now to determine the effects of multiple collisions (or numerous atoms), we must combine the effects of these collisions, which means that we must have some estimate of the time between them \(T\). Let \(P(t)\) be the probability that a collision has not occurred in a time t measured from the last collision. Now if the collisions are indeed random, the differential probability \(dp\) that a collision will occur in a time interval \(dt\) is \[d p=\frac{d t}{T_0}\label{14.4.7}\]

where \(T_0\) is the average time between collisions. Thus, the differential change in the probability \(P(t)\) is \[d[1-P(t)]=-d P(t)=P(t) d p=\frac{P(t) d t}{T_0}\label{14.4.8}\]

or \[P(t)=B e^{-t / T_0}\label{14.4.9}\]

Since a collision must occur at some time, we can determine the constant in equation \ref{14.4.9} by normalizing that expression to unity and integrating over all time. Thus we see that the collision frequency distribution is a Poisson distribution of the form \[P(t)=\frac{1}{T_0} e^{-t / T_0}\label{14.4.10}\]

and the constant of proportionality in equation \ref{14.4.6} is \(1/T_0\).

To obtain the total energy distribution or power spectrum resulting from a multitude of collisions, we must sum the power spectra of the individual collisions multiplied by the probability of their occurrence. Thus, \[\begin{aligned}
I_t(\omega) & =\frac{A^2 E_0^2}{2 \pi T_0} \int_0^{\infty} \frac{\sin ^2\left[T\left(\omega-\omega_0\right) / 2\right]}{\left[\left(\omega-\omega_0\right) / 2\right]^2} e^{-T / T_0} d T \\
& =\frac{A^2 E_0^2}{\pi T_0} \frac{1}{\left(\omega-\omega_0\right)^2+\left(1 / T_0\right)^2}
\end{aligned}\label{14.4.11}\]

We can use the same normalization process implied by equations \ref{14.1.3}, \ref{14.1.4}, and \ref{14.2.21} to write the atomic absorption coefficient as \[S_\nu(\mathrm{col})=\frac{\pi e^2 f_{i k}}{2 \pi T_0 m_e c} \frac{1}{\left(\nu-\nu_0\right)^2+\left(1 / T_0\right)^2}\label{14.4.12}\]

In going from equation \ref{14.4.11} to \ref{14.4.12}, we have changed from circular frequency w to frequency n so that the appropriate factors of \(2\pi\) must be introduced. The quantity \(2/T_0\) is usually called the collisional damping constant so that \[\Gamma_c=\frac{2}{T_0}\label{14.4.13}\]

Since the form of equation \ref{14.4.12} is identical to that of equation \ref{14.3.21}, we can immediately obtain the convolution of the collisional damping absorption coefficient with that for radiation damping by simply adding the respective damping constants: \[\Gamma=\Gamma_{i k}+\Gamma_c\label{14.4.14}\]

The combined absorption coefficient could then be convolved with that appropriate for microturbulent Doppler broadening producing a total line profile that is still a Voigt profile but with \[a=\frac{c \Gamma}{4 \pi \nu_0 v_0}\label{14.4.15}\]

where \(\Gamma\) is the combined damping constant for radiation and collisional damping. However, before this result can be of any practical use, we must have an estimate of the collisional damping constant in terms of the state variables of the atmosphere.

Determination of the Collisional Damping Constant is equivalent to determining the average time between collisions \(T_0\). To do this, it is necessary to be quite specific about exactly what constitutes a collision. We follow a method originally due to Victor Weisskopf¹⁶ and described by many authors^17_-19. Consider that the perturbation of an energy level \(∆E\) caused by a passing perturber has the distance dependence \[\Delta h \nu=\hbar \Delta \omega \simeq r^{-n}\label{14.4.16}\]

which will produce a change in the frequency of the emitted photon of \[\Delta \omega=\frac{2 \pi C_n}{r^n}\label{14.4.17}\]

The constant \(C_n\) is known as the interaction constant, and it must be determined empirically from laboratory experiments involving the kinds of particles found in the collisions. Since all these collisions are mediated by the electromagnetic force, the typical interaction can be viewed as a "long range" one so that the short collisions [see equation \ref{14.4.1}] refer to distant collisions where the colliding particle is located near its point of closest approach. This distance is commonly referred to as the impact distance, or impact parameter. Since the collision is short and the interaction weak, we can assume that the perturbing particle is largely unaffected by the encounter, and its path can be viewed as a straight line (see Figure 14.4). This assumption is usually referred to as the classical path approximation and it appears in one form or another in all theories of collisional broadening.

We wish to calculate the entire frequency shift caused by the collision because when the accumulated phase shift becomes large enough, it is reasonable to say that the wave train has been interrupted and a collision has occurred.

Figure 14.4 shows the "classical path" taken by a perturbing particle under the Weisskopf approximation. The point of closest approach \(\rho\) is called the impact parameter.

To estimate this total phase shift, it is necessary to describe the path taken by the particle, so we use of the classical path approximation. It is clear from Figure 14.4 that \[r^2=\rho^2+(v t)^2\label{14.4.18}\]

This enables the total phase shift \(\eta\) caused by the encounter to be calculated from \[\eta=\int_{-\infty}^{+\infty} \Delta \omega d t=2 \pi C_n \int_{-\infty}^{+\infty} \frac{d t}{\left[\rho^2+(v t)^2\right]^{n / 2}}=\frac{2 \pi C_n a_n}{v \rho^{n-1}}\label{14.4.19}\]

where \[a_n=\frac{\sqrt{\pi} \Gamma[(n-1) / 2]}{\Gamma(n / 2)}\label{14.4.20}\]

Here \(\Gamma(x)\) is the gamma function, and it should not be confused with the symbol for the damping constant. Before using this for the determination of the average time between collisions, we must decide what constitutes an interruption in the wave train. Weisskopf took this value of \(\eta\) to be 1 radian. The smaller the value for the phase shift, the larger the value of the impact parameter may be that will produce that phase shift. The value of the impact parameter \(\rho_0\) that produces the minimum phase shift \(\eta\) which constitutes an interruption in the wave train is known as the Weisskopf radius: \[\rho_0=\left(\frac{2 \pi C_n a_n}{\eta_0 v}\right)^{1 /(n-1)}\label{14.4.21}\]

Since the Weisskopf radius defines the distance inside of which any encounters will produce a large enough phase shift to be considered a collision, it may be used to calculate a collision cross section \(\sigma=\pi \rho_0^2\) and an average time between collisions \(T_0\). The collision frequency is \[\frac{1}{T_0}=\frac{\langle v\rangle_{\mathrm{rel}}}{l}=\pi \rho_0^2 N\langle v\rangle_{\mathrm{rel}}=\frac{\Gamma_c}{2}\label{14.4.22}\]

where N is the number density of the perturbing particles, \(l\) is the mean free path between collisions, and \(<\mathrm{v}>_{\text{rel}}\) is the relative velocity between the perturber and the perturbed atom. That relative velocity is \[\langle v\rangle_{\mathrm{rel}}=\left[\frac{8 k T}{\pi m_h}\left(\frac{1}{A_1}+\frac{1}{A_2}\right)\right]^{1 / 2}\label{14.4.23}\]

where \(A_\mathrm{i}\) is the atomic weight of the constituents of the collision in units of the mass of the hydrogen atom. Thus, we can write the collisional damping constant as \[\Gamma_c=\text { (const) }\langle v\rangle_{\mathrm{rel}}^{1-2 /(n-1)} \eta_0^{-2 /(n-1)} N\label{14.4.24}\]

where N is the number density of perturbers and \[\text { const }=(2 \pi)^{(n+1) /(n-1)}\left(a_n C_n\right)^{2 /(n-1)}\label{14.4.25}\]

All that remains is to specify the power law that describes the perturbing force and the interaction constant \(C_n\). Since the force that mediates the collision is electromagnetic, the exponent of the perturbing field is determined by the electric field of the perturber. A simple way of understanding this is to view the passage of the perturber as interposing a "screening" potential energy between an optical electron and the nucleus. The screening potential energy will depend on the locally interposed energy density of the perturber's electric field which is proportional to E² so that for a perturbing ion or electron, \(n=4\). This is called the quadratic Stark effect because it depends quadratically on the perturber's electric field. If the perturber is a neutral atom, it still possesses a dipole moment that produces a measurable field near the particle. However, this field varies as r^-3 so that the perturbing energy density varies as r^-6 and \(n = 6\). Broadening of this type is called van der Waal's broadening and it will to play a role in relatively cool gases where there are few ions.

If the atomic energy level of interest is degenerate, the interposition of an external electric field will result in the removal of the degeneracy and a splitting of the energy level into a set of discrete energy levels. The amount of the splitting is proportional to the electric field. Since a time-dependent splitting is equivalent to a broadening brought about by a shift of the energy level itself, the broadening of degenerate levels will occur, but their broadening will be directly proportional to the

electric field of the perturber rather than to its square. Thus the broadening of a degenerate level by ions or electrons will produce \(n = 2\). This form of broadening is known as the linear Stark effect. This form of broadening creates an interesting problem for the impact phase-shift theory since the integral for the minimum phase shift [equation \ref{14.4.19}] will not converge for \(n = 2\) and the theory is not applicable. Since the energy levels of hydrogen are degenerate, and the hydrogen lines are among the most prominent in stellar spectra, we are left with the somewhat embarrassing result that these lines can not be dealt with by the impact phase-shift theory and we have to resort to some other description of collisional broadening to obtain line profiles for hydrogen. The problem basically arises from the 1/r² nature of the perturbing field and is not restricted to the theory of line broadening. Since the number of perturbers increases as r2 while the perturbation from any one of them declines as r², the contribution to the total perturbation from particles at a given distance is independent of distance. Thus some cutoff of the distances to be considered must be invoked. This problem arises frequently in gravitation theory where there can be no screening of the potential field and the fundamental force is also long-range. In our case, the Heisenberg uncertainty principle sets a limit on the smallest perturbation that can matter and hence an upper limit on the volume of space to be considered.

The case of the broadening of degenerate levels by neutral atoms does present a situation that can be dealt with by the impact phase-shift theory. Here the electric field near the perturber varies as r^-3, so that the proper value of \(n\) is \(n = 3\). In the special case where the broadening is by atoms of the same species as the atom being perturbed a significant enhancement of the broadening occurs. Indeed, for astrophysical cases, the broadening of spectral lines arising from degenerate levels by neutral particles is of interest only when the broadening occurs from collisions with atoms of the same species. For that reason, this kind of broadening is known as self-broadening. These considerations are summarized in table 14.1.

An important improvement was made by Lindholm²⁰ and Foley²¹ which included the effects of multiple collisions on the line. Although the multiple collisions are weak, they are frequent. The result of their work is that the secondary collisions introduce a slight shift in the line center of the atomic absorption coefficient so that \[S_\omega(\mathrm{col})=\frac{2 \pi e^2 f_{i k}}{m c} \frac{\Gamma_c / 2}{\left(\omega-\omega_0-\beta\right)^2+\left(\Gamma_c / 2\right)^2}\label{14.4.26}\]

where \[\beta=2 \pi N\langle v\rangle_{\mathrm{rel}} \int_0^{\infty} \sin [\eta(\rho)] \rho d \rho\label{14.4.27}\]

Limits of Validity for Impact Phase-Shift Theory In developing the impact phase-shift theory, we tacitly assumed that the collisions were adiabatic. By that we mean that all the perturbing energy was contained in the perturbation and none was lost to other processes. There were no collisional transitions within the energy level or between the split levels of the degenerate cases. This will be a reasonable approximation as long as the splitting of the degenerate levels or the width of the perturbed level is greater than the uncertainty of energy of the perturber due to the Heisenberg uncertainty principle.

Table 14.1 Types of Collisional Broadening

	Type of Energy Level
Type of Perturber	Degenerate	Nondegenerate
Ion or electron	Linear Stark effect \(n=2\)	Quadratic Stark effect \(n=4\)
Neutral atom	Self-broadening \(n=3\)	van der Waal's broadening \(n=6\)

Since the duration of the collision is of the order of \(\rho/\mathrm{v}\), the uncertainty of the colliding particle's energy is of the order \[\Delta E \frac{\rho}{v} \simeq \hbar\label{14.4.28}\]

In equation \ref{14.4.16} we estimated the energy of the perturbation itself so that \[\frac{v \hbar}{\rho}<\Delta \omega \hbar=\frac{2 \pi C_n \hbar}{r^n} \simeq \frac{2 \pi C_n \hbar}{\rho^n}\label{14.4.29}\]

which requires that \[\rho<\left(\frac{2 \pi C_n}{v}\right)^{1 /(n-1)} \simeq \rho_0\label{14.4.30}\]

So it appears that only collisions that occur inside the Weisskopf radius will be adiabatic, and all the energy of the collision goes into perturbing the energy level.

However, if the impact parameter is too small, the classical path approximation will be violated and the duration of the collision will exceed the radiation time [equation \ref{14.4.1}]. The extent of this constraint can be estimated by noting that the duration of the collision is of the order of \(\rho/\mathrm{v}\). The radiation time is of the order of 1/∆ω, so that \[\frac{\rho \Delta \omega}{v} \ll 1\label{14.4.31}\]

if the classical path approximation is to be valid. However, equation \ref{14.4.29} places a constraint of the impact parameter \(\rho\) that must be met if the collisions are to be adiabatic. Using equation ref{14.4.29} to eliminate \(\rho/\mathrm{v}\) from equation \ref{14.4.31} we get \[\Delta \omega \ll \frac{v^{n /(n-1)}}{\left(2 \pi C_n\right)^{1 /(n-1)}}\label{14.4.32}\]

Obviously the impact phase-shift theory will be valid only for the inner part of the line. For the outer part we must turn to another description of collisional broadening.

b. Static (Statistical) Broadening Theory

In some real sense, the impact phase-shift theory follows the life history of a single radiating (or absorbing) atom which is subject to numerous weak collisions of short duration. The atomic absorption coefficient is then represented by the average of many atoms in various phases of that temporal history. In static broadening theory, the atomic absorption coefficient is constructed from the average of many atoms that are subject to the electric field of perturbers scattered randomly about. The opposite assumption is made concerning the duration of the collision compared to the radiation time. That is, the collision time is much longer than the radiation time, so that \[t_{\mathrm{col}} \gg t_{\mathrm{rad}}\label{14.4.33}\]

It is as if we took a picture of the perturbed atom with a shutter duration of the radiation time for the photon. In the impact phase-shift theory, we would see a blur of colliding tracks of the perturbers, while in the case of statistical broadening the picture would show individual perturbers fixed in space and some might be quite close to the atom in question. We are most interested in these near perturbers, for they are responsible for the largest perturbations to the atomic energy levels which in turn generate the broadest part of the line. This is precisely the part of the line for which the impact phase-shift theory fails.

Figure 14.5 shows a schematic view of the universe of perturbers under the assumptions of the Static Theory of broadening. The perturbers are randomly distributed in space, but only the "nearest neighbor" will be used for calculating the perturbing electric field.

Remember that the perturbation arises from the presence of an external electric field. In the static theory of line broadening, all particles are fixed in space, and the perturbing electric field is the vector sum of the electric fields of all the perturbers (see Figure 14.5). However, since we are concerned mostly about the strongest perturbations that form the wings of the line, we address only the perturbers closest to the atom in question

Nearest-Neighbor Approximation and the Distribution of Electric Fields The assumptions required for the development of the static theory of broadening are similar in form and content to those required of the impact phase-shift theory. The collision time is assumed to be much larger than the radiation time [equation \ref{14.4.33}] so that from the point of view of the radiating atom, the universe is frozen in time. Implicit in this assumption is the notion that every perturber has a well-defined position (zero) and momentum with regard to the perturbed atom. This is equivalent to the classical path approximation of the impact phase-shift theory in that the position and momentum of the perturber are specified throughout the interaction time and are such that they are unaffected by the interaction.

To these complementary assumptions we add one more. Let us assume that the perturbative electric field can be represented by the electric field of the perturber closest to the atom and by that perturber alone. This is known as the nearest-neighbor approximation. Our task, then, is to find the probability distribution function for the perturber lying within a specified distance and thereby producing a perturbing electric field of a particular strength. Consider a spherical shell of thickness \(dr\) located a distance r from the perturbed atom (see Figure 14.5), and let the probability that the nearest neighbor is located within that shell be \(P(r)dr\). Then the probability that the nearest neighbor lies within a sphere of radius \(r\) is \(\int_0^r P(r) d r\). Since the universe is not empty, there must be a nearest neighbor somewhere, so that the probability that the nearest neighbor does not lie within that sphere is \(\left(1-\int_0^r P(r) d r\right)\). Now if the region around the perturbed atom is of uniform density, the probability of finding any perturber within the spherical shell of thickness \(dr\) located at \(r\) is \(4πr^2ndr\), where \(n\) is the perturber density. Thus, the probability that the particle in that shell is the nearest neighbor is just the probability that there is a particle there multiplied by the probability that there is no particle nearer to the perturbed atom. So \[P(r) d r=\left(4 \pi r^2 n d r\right)\left[1-\int_0^r P(r) d r\right]\label{14.4.34}\]

This is an integral equation for the distribution function of nearest neighbors \(P(r)\). We can solve it most easily by differentiating with respect to \(r\) and forming a differential equation for, \(P(r)/4πr^2n\). The solution to this equation is \[P(r) d r=4 \pi r^2 n e^{-4 \pi r^3 n / 3} d r\label{14.4.35}\]

However, we need the probability distribution of perturbing electric fields, so we assume that the perturber has a field that behaves as \[E=\frac{a}{r^m}\label{14.4.36}\]

Then, by substituting this dependence of the electric field on \(r\) into equation \ref{14.4.35}, the probability that an atom will see a perturbing electric field of strength E is \[W_m(E) d E=\frac{3}{m} \frac{E_0^{3 / m}}{E^{(m+3) / m}} e^{-\left(E_0 / E\right)^{3 / m}} d E\label{14.4.37}\]

where \[E_0=a\left(\frac{4 \pi n}{3}\right)^{m / 3}\label{14.4.38}\]

and it is sometimes called the normalizing field strength. If we further define the dimensionless quantity \[\beta=\frac{E}{E_0}\label{14.4.39}\]

we can write the probability distribution for this dimensionless field strength as \[W_m(\beta) d \beta=\frac{3}{m} \beta^{-(m+3) / m} e^{-\beta-3 / m} d \beta\label{14.4.40}\]

Finally, if we consider the case for broadening by ions or electrons, then \(m = 2\) and we have \[W_2(\beta) d \beta=\frac{3}{2} \beta^{-5 / 2} e^{-\beta-3 / 2} d \beta\label{14.4.41}\]

which is usually called the Holtsmark distribution function. As can be seen from Figure 14.6, the probability of finding a weak field due to the nearest neighbor is very small simply because it is unlikely that the nearest neighbor can be so far away and still be the nearest neighbor. As the field strength rises, so does the probability of it being the perturbing field, peaking between 1 and 2 times the normalized field strength. Stronger fields become less likely because the volume of space surrounding the atom within which the perturber would have to exist becomes just too small.

Behavior of the Atomic Line Absorption Coefficient If we assume that the perturbative change in the atomic energy level is proportional to the electric field to some power, then we can write \[\Delta E=\Delta(h \nu)=h \Delta \nu \propto E^q \propto \beta^q\label{14.4.42}\]

We can then use the nearest-neighbor distribution function to generate a probability density distribution function for the absorption of photons at a particular frequency shift \(\Delta\nu\) as \[P(\Delta \nu) d \nu \propto \Delta \nu^{-[1+3 /(m q)]} e^{-\Delta \nu^{-3 /(m q)}} d \nu\label{14.4.43}\]

For large frequency or wavelength shifts, the argument of the exponential approaches zero, so the wavelength-dependent probability of absorption becomes \[P(\Delta \lambda) d \lambda \propto \Delta \lambda^{-[1+3 /(m q)]}\label{14.4.44}\]

Figure 14.6 shows the Nearest Neighbor distribution function for the perturbing electric field of the nearest neighbor assuming that it is an ion or electron as the solid line. The dashed line is for the Holtsmark distribution that includes the contribution from the rest of the gas. The parameter \(\delta\) is a measure of the screening potential of the nearest neighbor [see Mihalas¹¹ (pp. 292-295)].

This is precisely the range for which the static theory through the nearest-neighbor approximation was expected to be accurate. Since the atomic line absorption coefficient is indeed proportional to the probability of photon absorption, its behavior in the far wings of a line is \[S_{\Delta \lambda} \propto \Delta \lambda^{-[1+3 /(m q)]}\label{14.4.45}\]

Table 14.2 provides a brief summary of the asymptotic dependence of the absorption coefficient in the wings of the line for the various types of interactions discussed.

Table 14.3 Asymptotic Behavior of the Atomic Line Absorption Coefficient

	Type of Energy Level
Type of Perturber	Degenerate \(q=1\)	Nondegenerate \(q=2\)
Ion or electron \(m=2\)	Linear Stark effect \(\mathbf{S}_\lambda \sim \Delta \lambda^{-5 / 2}\)	Quadratic Stark effect \(\mathbf{S}_\lambda \sim \Delta \lambda^{-7 / 4}\)
Neutral atom \(m=3\)	Self-broadening \(\mathbf{S}_\lambda \sim \Delta \lambda^{-2}\)	van der Waal's broadening \(\mathbf{S}_\lambda \sim \Delta \lambda^{-3/2}\)

Finally, we may find the constant of proportionality for the atomic line absorption coefficient in terms of the interaction constant for the force law \(C_l\). This is analogous to the constant \(C_n\) that appears in equation \ref{14.4.17} and is usually determined empirically. In terms of this constant, the atomic line absorption coefficient becomes \[S_\nu=\frac{3 C_l}{(4 \pi N / 3)^{1 / 3} a_m} \Delta \nu^{-(1+3 / l)} e^{-\Delta \nu^{-3 / l}}\label{14.4.46}\]

where \(a_m\) is given by equation \ref{14.4.20}, and \[l \equiv m q\label{14.4.47}\]

In the broadening of degenerate levels, the splitting of the energy levels is so large that the line should be considered to consist of individual linear Stark components, each of which is quadratically Stark broadened. Under these conditions, the atomic line absorption coefficient for the combined Stark components becomes \[S_\nu=\sum_k \frac{3 C_{k l}}{(4 \pi N / 3)^{1 / 3} a_m} \Delta \nu_k^{-(1+3 / l)} e^{-\Delta \nu_k^{3 / l}}\label{14.4.48}\]

If one were to improve on the static theory, the most obvious place would be to relax the nearest-neighbor approximation. The problem of including an ensemble of perturbers, all with their electric fields adding vectorially, was considered by J.Holtsmark²² and solved by S. Chandrasekhar²³, who also provides tables of the results. As one might expect, the resultant form is similar to that of the nearest-neighbor distribution in shape but somewhat more spread out (see Figure 14.6). Unfortunately the result takes the form of an integral so a complete description must be obtained numerically. However, for β in the vicinity of 1 we get the following asymptotic formula for \(W(β)\): \[W(\beta)=1.496 \beta^{-5 / 2}\left(1+5.107 \beta^{-3 / 2}+14.43 \beta^{-3}+\cdots+\right)\label{14.4.49}\]

As we might expect, the lead term of this series is just that of the nearest-neighbor approximation [see equation \ref{14.4.41}.]

Limits of Validity and Further Improvements for the Static Theory Since the assumption relating the collision time to the radiation time led to a limit on the range of validity for the impact phase-shift theory [equation \ref{14.4.32}], we should not be surprised if the same were true for the static theory. This is indeed the case and the result is known as the Holstein relation, can be deduced from equation \ref{14.4.32} almost by inspection: \[\Delta \omega \gg \frac{v^{n /(n-1)}}{\left(2 \pi C_n\right)^{1 /(n-1)}}\label{14.4.50}\]

So, as we hoped at the outset of the development of the static theory, it will be valid for precisely those regions of the line profile for which the impact phase-shift theory fails.

Of course, any microscopic inspection of a problem usually finds phenomena that provide additional complications for the solution. For example, we have assumed that the perturbers interact with the atom in question but do not interact among themselves. In reality an ion will attract electrons so as to create a neutral plasma on as small a scale as possible. In effect, then, the plasma will try to shield the ions from even more distant perturbers. This phenomenon is known as Debye shielding and is discussed in some detail by Mihalas¹¹ (pp. 292-295). The basic effect is density dependent and tends to flatten the Holtsmark distribution still further, thereby broadening the line even more. Fortunately, for normal stellar atmospheres the densities are not large enough to make Debye shielding a major effect until one reaches optical depths in the line that are quite remote from the boundary.

The treatment of collisional line broadening described so far has been based on purely classical considerations and has now been largely replaced by quantum mechanical calculations of the atomic line absorption coefficient for the more important stellar spectral lines. However, the quantum mechanical treatment is considerably less transparent than the classical one, so we give only the basic form. The power spectrum for the line is given by \[I(\omega)=\frac{2 \omega^4}{3 \pi c^2} \operatorname{Tr}\left[\mathbf{P} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \mathbf{Q}(t) e^{-i \omega t} \mathbf{Q}\left(t^{\prime}\right) e^{i \omega t \prime} d t d t^{\prime}\right]\label{14.4.51}\]

where P is the probability density matrix for the atomic states involved in the formation of the spectral line, and Q is the matrix of dipole moments with elements \[Q_{i j}=\int \Psi_i^* q \Psi_j d V\label{14.4.52}\]

The wave functions \(\Psi_{\mathrm{i}}\) must include the effects of the perturbers as well as the atomic states of interest. A very complete discussion of the quantum theory of spectral line formation is given by Hans Griem²⁴.

For simple lines the classical theories of collisional broadening produce line profiles that agree well with observation. However, for the stronger lines of hydrogen and helium, any serious model should involve an atomic absorption coefficient based on the quantum mechanical description. While tables of these coefficients exist for many important lines (see Griem²⁴ and references there), much remains to be done to produce accurate values for many lines of astrophysical interest.