Temperature Dependences of the Quantum-Mechanical and Semi-Classical Spectral-Line Widths and the Separation 𝑛₀ of the Impact and Non-Impact Regions for an Ar-Perturbed/K-Radiator System

Abstract

Quantum-mechanical and semi-classical spectral-line shapes are computed at 𝑇=400, 800, and 1000 K for the line core of the 5802 Å line of the Ar-Perturbed/K-Radiator system. HWHMs (𝑤's) are measured from computed full spectral-line shapes. The final-state pseudopotential is for the 7𝑠²𝑆_1/2 state, and the initial-state potential is for the 4𝑝²𝑃_3/2,3/2 state. Three high-pressure (P) log(𝑤)—versus—log(𝑃) curves, corresponding to the non-impact region, intersect a similar set of low-P, impact-region curves at intersections, 𝑃₀'s. Similarly, for two sets of log(𝑤)—versus—log(𝑛) curves, which yield intersections, 𝑛₀'s, where 𝑛 is the perturber density. These 𝑛₀'s and 𝑝₀'s separate the two regions and represent the upper limits of the impact regions. A specific validity condition for the impact region is given by the equation 𝑛≤𝑛₀. From an earlier spectroscopic, Fabry-Perot paper, 𝑤_expt=0.021 cm⁻¹ at 𝑇=400 K and 𝑃=10 torr. Two theoretical values, 𝑤_theor=0.025 and 0.062 cm⁻¹ corresponding to two different pseudo-potentials, are reported. Two 𝑇-dependent figures are given, in which the first shows an increase in the impact region with 𝑇, based on 𝑃 as the basic parameter, and the second which shows a decrease in the impact region with 𝑇, based on 𝑛 as the basic parameter.

1. Introduction

Spectral-line-shape and line-broadening studies have been performed for many years, as evidenced by the 30-year, 15-plus volumes of the AIP “Spectral Line Shapes”, typically, Back [1]. However, there are four areas that still have to be addressed in detail:(1)the effects of temperature (𝑇) upon the HWHM (𝑤) of the spectral line and upon the perturber-density separation 𝑛₀ between the impact and non-impact regions;(2)the acquisition of specific limits of the validity conditions for the impact approximation—Baranger [2] reported that it is “a subject which has been very neglected;”(3)the variation of the extent (size) of the impact (approximation) region with 𝑇—a controversy exists as to whether the region increases or decreases with 𝑇;(4)a partial testing of the validity of the present theoretical development and programming—this will be achieved by comparing present computed values of 𝑤 with previous Fabry-Perot spectroscopic measurements; see the work of Kreye in [3].

This study is a continuation of the work by Kreye [4] on the 5802 Å spectral line of the K/Ar system at 400 K. Quantum-mechanical (QM) and semi-classical (SC) theories were used to compute full spectral-line shapes, from which the values of 𝑤 and the shift (𝑑) were measured; a realistic pseudo-potential for the initial state, 4𝑝²𝑃_3/2,1/2, was used as the basis for his computations; A measured 𝑃₀, was employed as the separation of the impact region from the non-impact region and linear log(𝑤)—versus—log(𝑃) curves were introduced to show the dependence of 𝑤 upon 𝑃.

This paper computes the effects of 𝑇 upon 𝑤 and 𝑃₀, introduces the the perturber density 𝑛, and uses the pseudo-potential, 7𝑠²𝑆_1/2, to represent the final state and a pseudo-potential to represent the initial state. (This system was chosen because of its familiarity to the author, but the theory and programs are sufficiently general for other rare-gas/alkali systems.) The temperatures studied are 400, 800, and 1000 K, and the pressures range from 10³ to 10⁶ torr.

The following papers are pertinent to the present study: the review paper by Allard and Kielkopf [5], which contains the SC theory used in this study and describes the pseudo-potentials; the paper of Baranger [2] in which he derives the QM impact approximation theory, the non-impact-region theory and considers in detail transitions between two states; the previously mentioned paper by Kreye [4] in which the theoretical expressions of Baranger [2] are expanded into computable form; the paper of Szudy and Baylis [6] in which the unified theory of line broadening is developed; the experimental/theoretical works of Kreye and Kielkopf [7–9]; the 𝑇-dependent measurements by Vaughan [10] of the Kr/rare-gas systems at 89 and 295 K; the work of Baylis [11] who expands the pseudopotential theory.

2. Theory

Much of the theory is presented in [4], and a brief summary is included here. As in [4], the basic computations are performed with the autocorrelation function Φ(𝑠) rather than with the line-shape expression 𝐹(𝜔). In the above, Φ(𝑠)=𝜙(𝑠)^𝑁, where 𝑁 is the number of perturbers and where 𝜙(𝑠) is the auto-correlation function for a single perturber. The general 𝑔(𝑠)-type function is shown to enter in asΦ(𝑠)=exp−𝑛𝑔(𝑠), (1) where 𝑛 is the perturber density. The various QM and SC forms of 𝑔(𝑠) are presented below. The expression for the line shape 𝐹(𝜔) is taken from [2, equation (6)], and it is renumbered here as 𝐹(𝜔)=12𝜋⋅2⋅^𝑠max₀𝑔(𝑠𝑔(𝑠.𝑑𝑠⋅exp−𝑛⋅ℜ)×cos−𝑛⋅ℑ)+𝜔𝑠 (2) The SC expression for 𝑔_NI(𝑠) and 𝑔_I(𝑠) are given in [5]: equation (55) is for the non-impact (NI) region, and equations (56, 57) are for the impact (I) region. The corresponding expressions for the ℜ[𝑔_NI(𝑠)], ℑ[𝑔_NI(𝑠)], and the impact terms can be put into (2) for 𝐹(𝜔).

The QM expression for 𝑔₁(𝑠) in the impact region is obtained from Baranger's [2, equation (31)]; it becomes, after rearranging and modifying,𝑔₁(𝑠)=𝑖𝑠𝜓^∗_𝑓𝑘|▵𝑉|𝑘. (3) The QM expression for 𝑔(𝑠) in the non-impact region is given by Baranger's equation (29), and after rearranging and modifying, it becomes𝑔(𝑠)=𝑖𝑠𝜓^∗_𝑓𝑘||𝑘+𝑑▵𝑉₃𝑘′8𝜋³𝜓^∗_𝑓𝑘′|▵𝑉|𝑘²1−𝑒^{−𝑖(𝜖−𝜖′)𝑠},𝜖−𝜖′𝜖−𝜖′−𝑖𝜂 (4) where 𝑘=exp(−𝑖𝑘⋅𝑟′) and ▵𝑉=7𝑠²𝑆_1/2−4𝑝²𝑃_3/2,3/2.

In (4), 𝜓^∗_𝑓𝑘′ is the final-state wave function for the introduced set of 𝑘′ states, and 𝜂 is an infinitesimal. This equation holds for the complete range of 𝑠 and consists of two parts: the first term on the right is identical with (3) in the present paper, and it corresponds to the impact region with “large” 𝑠 and small 𝑃. The second term corresponds to the remaining “small” 𝑠 and large 𝑃; therefore it holds for the non-impact region. This second term is defined as 𝑔₂(𝑠) and is based on [4, equation (11)]. Our treating (4) as a sum of a small-𝑠 part and a large-𝑠 part is reminiscent of the 1956 unpublished work of Anderson and Talman [5, page 1134], who separated the auto-correlation function into two parts: one for small 𝑠, one for large 𝑠.

The terms in (3), (4): 𝜓^∗_𝑓𝑘, 𝜓^∗_𝑓𝑘′, and 𝑘 are expanded as sums of partial waves. 𝑘 is given by [4, equation (13)]. Also,𝜓^∗_𝑓𝑘′=^𝑙=𝑙max_𝑙=0(2𝑙+1)−𝑖^𝑙exp−𝑖𝛿_𝑓,𝑙𝑅_𝑓,𝑙𝑘′𝜖′𝑟′𝑃_𝑙(cosΘ). (5) The expression for 𝜓^∗_𝑓𝑘 can be obtained from 𝜓^∗_𝑓𝑘′. In these equations, 𝑙 is the angular momentum quantum number, 𝑃_𝑙 is the Legendre polynomial, 𝑅_𝑓,𝑙 is the radial wave function, and 𝛿_𝑓,𝑙 is the phase shift for the final state. The parameters are defined as: 𝜖=3𝑘_𝐵𝑇/2ℏ, 𝑘′(𝜖′)=(2𝜇𝜖′/ℏ), 𝑘=(3𝜇𝑘_𝐵𝑇/ℏ²), and 𝑣=(3𝑘_𝐵𝑇/𝜇). 𝜖′ is the independent variable. The above 𝑣 is the root mean square velocity which replaces the need for Maxwell/Boltzmann distribution of velocities.

In order to compute the phase shift 𝛿_𝑓,𝑙 and the radial wave function 𝑅_𝑓,𝑙[𝑘′(𝜖′)(𝑟′)], the Schroedinger equation is expressed in terms of the dimensionless variable, 𝜌=𝑘𝑟′ instead of 𝑟′. Therefore, we obtain the following form of the Schroedinger equation for the final state:𝑑²𝑅_𝑓,𝑙𝑑𝜌²+2𝑑𝑅_𝑓,𝑙+𝑉𝜌𝑑𝜌1−4𝜇𝜋^𝑓𝜌𝑐ℏ𝑘²−𝑙(𝑙+1)𝜌²𝑅_𝑓,𝑙=0, (6) where 𝜇 is the reduced mass.

In order to integrate (6) from the initial point 𝑎₁=5.0 Å to the final point 𝑎₂=20.0 Å, the initial conditions (IC's) at 𝑎₁ are needed. The approximate IC's for equation (31) are𝑅_𝑓,𝑙𝑘𝑎₁=𝑗_𝑙𝑘𝑎₁,𝑅′_𝑓,𝑙𝑘𝑎₁=𝑗^′_𝑙𝑘𝑎₁, (7) where 𝑗_𝑙(𝑘𝑎₁) is the spherical Bessel function. The expression for the phase shift 𝛿_𝑓,𝑙 in terms of 𝑅_𝑓,𝑙 and 𝑅^′_𝑓,𝑙 is based on, for example, Schiff [12, equation (19.14)], and the phase shift is computed at 𝑟′=20.0 Å. The equations for ℜ[𝑔₁(𝑠)] and ℜ[𝑔₂(𝑠)] are found in [4, equations (17, 18)]. Similarly for the imaginary terms. All these terms can be substituted into (2) to give the QM 𝐹(𝜔)'s.

3. Computational and Analytic Details

Several of the analytic details in the present study are the same as those in [4], such as: the use of linear log(𝑤)—versus—log(𝑃) curves, the numerical computation of the three-angle integration term in the expanded (4), the integration over 𝜖′, and the integration over 𝑠.

Additional analytic details are:

(1) the pseudo-potential was expanded into 27500 elements using a spline technique;

(2) the integration of (6) used a second-degree Runge-Kutta method;

(3) in order to determine 𝑙_max, we found for the present paper that it was impossible to use the method of [4], which consisted of letting 𝑙 increase until it leveled off at 𝑙_max. The reason is that 𝑙 does not level off to an asymptotic value, as can be shown. Therefore, the following methods were used to determine 𝑙_max: in the impact region, 𝑙_max was set equal to 101, 108, and 121 for the corresponding values of 𝑇=400, 800, and 1000 K. These were chosen so that 𝑤_QM≈𝑤_SC. In the non-impact region, 𝑙_max was set equal to 107, 153, and 171. These values were chosen to yield 𝑤_QM≈𝑤_SC. Moreover, it can be shown that these last choices have a theoretical basis which is based on Schiff's [12] equation, 𝑙_max=𝑘𝑎, where 𝑎 is defined as an “assumed radius beyond which the potential 𝑈(𝑟) is negligible”. From Schiff's theoretical equation, 𝑙_max=100, 140, and 157. These theoretical “Schiff” values are within about 8% of our chosen values, 107, 153, and 171.

4. Results

4.1. Discussion of 𝑉_𝑓—Figure 1 Depicts 𝑉_𝑓-versus-𝑟′

4.2. Variation of log(𝑤) with log(𝑃) and log(𝑤) with log(𝑛)

Figure 2 shows three lower log(𝑤)—versus—log(𝑃) curves in the impact (low-P) region for 400, 800, and 1000 K (reading down). The curves are linear with slopes, 𝑑log(𝑤)/𝑑log(𝑃), of 1.0, and there is good agreement between the SC and the QM curves. Figure 2 also shows three similar curves in the non-impact (high-P) region. The curves are linear with an average slope of 0.506±0.001.

Figure 1 

A plot of 𝑉𝑓—versus—𝑟′, where 𝑉𝑓 correponds to the electronic state, 7𝑠2𝑆1/2. The pseudopotential was formulated by Allard and Kielkopf in [5].

Figure 2 

Plots of log(𝑤)—versus—log(𝑃) in the two pressure regions: The high-P, non-impact region (104.5 to 106 torr); and the low-P, impact region (103 to 105 torr). Three sets of curves are presented at (counting down) 400, 800, and 1000 K. The QM curves are solid lines, and the SC curves are dashed lines in both regions. The dotted lines are approximate extrapolations from the QM non-impact curves. The intersection of an extrapolation with the corresponding QM impact-region curve, represented by an ∗, gives rise to 𝑃0, the approximate separation between the impact and the non-impact regions.

In Figure 2, the dotted lines are extrapolations from the non-impact-region QM curves, and they intersect the corresponding impact-region QM curves at points with ∗ markings. These intersections are defined as 𝑃₀'s, and they represent, as introduced in [4], the approximate upper limits to the impact regions and the lower limits to the non-impact regions. Rigorously, the connections between the two regions would be represented by smooth, continuous-derivative curves. There is, at present, no quantitative theory for representing such smoothed curves.

Figure 3 shows a similar set of log(𝑤)—versus—log(𝑛) curves. Only the QM curves are shown. As in Figure 2, the impact-region curves have a slope of 1.0 and the non-impact-region curves have a slope of 0.507. (An interesting comment is found in [5, page 1148], which confirms the above slope, because equation (296) in that reference predicts that “the width (HWHM) of the line grows only as 𝑛^1/2’’). The extrapolations of the non-impact curves with the impact curves intersect at ∗'s, and these points are designated as 𝑛₀'s. These 𝑛₀'s are defined as the 𝑃₀'s are defined.

Figure 3 

This plot shows a set of three QM log(𝑤)—versus—log(𝑛) curves in the non-impact region (large 𝑛) at temperatures: 𝑇=400 K (empty circles), 𝑇=800 K (empty squares), and 𝑇=1000 K (empty triangles). A similar set appears in the impact region (small 𝑛): 𝑇=400 K (filled circles), 𝑇=800 K (filled squares), and 𝑇=1000 K (filled triangles). The dotted lines are extrapolations from the non-impact-region QM curves; and they intersect the corresponding impact-region curves at 𝑛0's.

4.3. Temperature Dependences of 𝑛₀ and 𝑃₀

Figure 4 shows the curves for 𝑛₀—versus—𝑇. The solid curve corresponds to 𝑛₀ data obtained directly from the intersections of the curves in Figure 3. The 𝑛₀ data for the dotted curve in Figure 4 are obtained from the 𝑃₀ intersections in Figure 2, where 𝑛₀=𝑃₀/𝑅𝑇. Both show an initial decrease of 𝑛₀ with increasing 𝑇, followed by a gradual leveling off. Thus, the extent or size of the impact region, 𝑛₀, decreases with increasing 𝑇, when the perturber density 𝑛 is the basic parameter.

Figure 4 

Two plots of 𝑛0—versus—𝑇 are shown in this plot. The data for the solid curve are taken from the intersections 𝑃0's in Figure 2 (log(𝑤)—versus—log(𝑃)), and 𝑛0 is calculated from 𝑛0=𝑃0/𝑅𝑇. The data for the dotted curve is taken directly from the 𝑛0 intersections in Figure 3. The curves do not exactly overlap because of the uncertainties in the two sets of the extrapolations in Figures 2 and 3. These curves establish that the impact region, defined by 𝑛0, decreases with 𝑇 when 𝑛 is the basic parameter.

On the other hand, Figure 5, which exhibits the variation of 𝑃₀ with 𝑇, indicates that 𝑃₀  increases with increasing 𝑇, when 𝑃 is the basic parameter. In other words, the extent of the impact region increases with 𝑇. Although 𝑛 is the more significant parameter in line-broadening studies, (e.g., in (2), 𝑛 appears in the exponent and in the cos factors.), 𝑃 is a more important parameter from an experimental point of view. An advantage of this increase in the impact region at higher temperatures is that one can use the simpler impact-region theory to extract such results as a predicted 𝑇 from a measured 𝑤 and 𝑃.

Figure 5 

This plot shows the 𝑃0—versus—𝑇 curve, where the 𝑃0 intersections are taken from Figure 2. This curve establishes that the impact region, defined by 𝑃0, increases with increasing 𝑇 when 𝑃 is the basic parameter.

The controversy mentioned in part (3) in the Introduction as to whether the impact region increases or decreases with 𝑇 is herein clarified: the extent of the impact region both increases and decreases with 𝑇, depending upon whether 𝑃 or 𝑛 is the basic parameter.

4.4. Validity Conditions for the Impact (Approximation) Region

Baranger [2, page 482], states that the validity condition for the impact approximation is “a subject which has been very neglected.” We tackle these validity conditions with two approaches:(1)in the the first approach, we use the theoretical treatment of Baranger [2, pages 489–492], and we present it here extremely briefly: in his first method, he introduces the collision time 𝜏; and the qualitative validity condition is that the following relation must be satisfied: 𝑠≪𝜏. In his second method, he requires that the difference between the real part of the exact 𝑔(𝑠), in his equation (29), and the impact approximation 𝑔(𝑠), in his equation (31), must be smaller than either of them in order for the impact approximation to be valid. This condition requires that the following relation must hold: 4∫𝑑³𝑘′/(8𝜋³)[|⟨𝑘′|𝑉|𝜓⟩|−|⟨𝑘^′′|𝑉|′𝜓⟩|/(𝜖−𝜖′)]²≪𝑛⁻¹, where 𝑛⁻¹ is the volume of a perturber. In his last method, he introduces a “collision volume", 𝑈, which is defined as the volume in which the wave function propagates differently from the wave function with energy 𝜖. The corresponding validity condition is that 𝑈 must satisfy the relation: 𝑈≪𝑛⁻¹;(2) in the second approach, we use the results of the present study. We obtain a more specific validity condition than Baranger's validity conditions, all of which involve the less-than, less-than ≪ relation. Our study yields a specific, limit-type validity condition, namely, numbers, 𝑛₀ and 𝑃₀, which are defined as the upper limits to the impact region when 𝑇 and the particular system are given. Thus, the validity conditions for the impact region are𝑛≤𝑛₀,𝑃≤𝑃₀. (8)

To summarize, there are two types of validity conditions: the first is based on a ≪ condition and the second on a ≤ condition. We believe that the second ≤ condition is of greater practicable value since the question would arise, with the ≪ condition, whether a given 𝑛 is small enough to satisfy the ≪ condition. The question does not arise for the ≤ condition.

4.5. Comparison of Experimental with Theoretical Results

The value of an experimental slope, slope_expt≡𝑑log(𝑤)/𝑑log(𝑃), is obtained from Figure 3 in the Fabry-Perot paper [3]. The curve is in the impact region. Between ≈3.2 and ≈11.3 torr, the value of slope_expt can be calculated as 0.94±10%. A theoretical slope is obtained from Figure 2 in the present paper. In the impact region, slope_theor=1.00. Thus, there is agreement between the two slopes.

An experimental, 𝑤_expt, is obtained from [3], Figure 3. That paper shows that the instrumentally corrected 𝑤_expt=0.021 cm⁻¹, at 𝑇=400 K and 𝑃=10 torr. One theoretical value, 𝑤_theor, is taken from [4]: from Figure 3 in that reference, 𝑤_theor=0.025 cm⁻¹. A second 𝑤_theor is from the present study. Under the same conditions, 𝑤_theor=0.062 cm⁻¹.

The near agreement between the above 𝑤_expt=0.021 cm⁻¹ and 𝑤_theor=0.025 cm⁻¹ partially establishes the validity of the theory and programming in the present study.

5. Conclusions

(1)The separation between the impact and non-impact regions can be represented by the intersection, 𝑛₀ or 𝑃₀, of a non-impact-region (extended) curve, log(𝑤)—versus—log(𝑛), or a similar log(𝑤)—versus—log(𝑃) curve, with the corresponding impact-region curve.(2)An 𝑛₀—versus—𝑇 curve shows that the upper limit, 𝑛₀, of the impact region decreases with 𝑇 whereas in a 𝑃₀—versus—𝑇 curve, the upper limit, 𝑃₀, increases with 𝑇.(3)A specific limit to the validity condition for the impact region has been defined as, 𝑛≤𝑛₀, in contrast to Baranger's conditions which use the general ≪ term.(4)The near agreement between 𝑤_expt=0.021 cm⁻¹ and 𝑤_theor=0.025 cm⁻¹ partially establishes the validity of the theory and the programming in the present study. Thus, this program can be used for other rare-gas/alkali studies.

Acknowledgments

W. C. Kreye acknowledges the continued support of Professor John Kielkopf of the University of Louisville who also freely made available to me his excellent pseudo-potential programs. The author also acknowledges the support of the director and staff of CaTS at Wright State University and especially the manuscript preparatory help of Jeffery Jones, John Meyers, and Steve Wynne.