Interpretation of measured spectral line shapes using a square-well potential: The absorption lines of cesium broadened by rare gases

J. Quant. $l~ctrosc. Radiat. Ttans[erVoL 23, pp. 201-210 Pergamon Press Ltd., 1980. Printed in Great Britain

INTERPRETATION OF MEASURED SPECTRAL LINE SHAPES USING A SQUARE-WELL POTENTIAL: THE ABSORPTION LINES OF CESIUM BROADENED BY RARE GASES D. E. GILBERT Eastern Oregon State College, La Grande, OR 97850, U.S.A. N. F. ALLARD Observatoire de Paris, 92190 Meudon, France S. Y. CH'EN University of 'Oregon, Eugene, OR 97403, U.S.A. (Received 18 April 1979)

Abstract--A simple square-well potential is used as an aid in understanding the complicated evolution of the absorption lines and their associated satellites for cesium perturbed by the rare gases. New experimental halfwidth data for cesium perturbed by Xe, Ar, and He are presented. In the case of Xe, data for the second, third and fourth members are presented to 30, 7, 1 rd (relative density), respectively. For At, data are presented for the third through the ninth member.The pressure range for the third and fourth members is from 0 to 20 rd. For the other members, it ranges to 8 rd for the fifth and to 2 rd for the ninth. Third and fourth member helium data are presented to about 20 and 8 rd, respectively. Several theoretical computer profiles are used to illustrate the evolution of the line and satellites with foreign gas pressure and these are compared with the experimental data.

1. INTRODUCTION When an optically active atom is placed in a foreign gas, not only are the wavelengths and widths of the spectral lines modified, but asymmetries as well as satellite bands appear. Experimental data are presented by showing graphically how the various line parameters, shift, halfwidth, and asymmetry change with the relative density (rd) of the foreign gas. These curves are quite often very complicated and because of the interactions of resolved/unresolved satellites, difficult to understand, In the present work, new data are presented for cesium in the presence of various foreign gases. By using the results of a simple theoretical model, we show how these, as well as previously published parameter curves, can be understood. 2. THEORETICAL MODEL In two other papers, Allard [ and Allard and Biraud, 2 a theoretical model based upon a simple square-well potential was developed. This model represents a first step toward calculations involving more realistic potentials. Even though simple and not realistic, the model at this stage of development nevertheless provides one with a means of clearly understanding many features in the experimental parameter curves. Line profiles are calculated using the theory of Anderson and Talman 3 and represent the Fourier transform of an autocorrelation function. By using a square-well potential, an analytical calculation of this autocorrelation function is possible. The profile is defined by two parameters x and h, the number of perturbers in the well. Then, x =2aV]~, where V is the depth of the well, a is the range of the potential and = (SkT/wm)lt2; also h = ~3¢ra3n, where n is the number density. The variations of width at half intensity (halfwidth) and shift with h (Fig. 1) are obtained for four different characteristic values of x. Physically, these correspond to different line-satellite separations; Fig. l(a) represents the case when the satellite is well resolved from the line; Fig. l(c) represents the case when the satellite produces a definite shoulder in the line. Fig. l(b) represents the transition between these two cases; Fig. l(d) represents the case when the QSRT Vol. 23. No. 2--F

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satellite is completely unresolved and the line appears to have a smooth red asymmetry for all densities. These cases are more fully discussed in Allard and Biraud. 2 In order to compare with specific experimental results (Fig. 4), a and ~ are chosen to give the corresponding rd and temperature. For Figs. 1, 4, 5, and 6, t~ = 4 x l04 cm/s (T = 478°K) and a = 17.7 x 10-a cm. For this a, h = 1.25 which corresponds to 2rd. Under these conditions (x = 9), V will be 5.4 cm -I. This is not a realistic well depth and will not lead to quantitative agreement between the model and experiment. Since our purpose is the interpretation of the experimental curves, only qualitative agreement is necessary. 3. EXPERIMENTAL SYSTEM The experimental profiles presented in this paper were obtained from a digital data acquisition and computer processing systems which were added to the analog display system described in Gilbert and Ch'en. 5 The analog line-shape signal was digitized on an H-P 2116 B

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computer. Once in the computer, the line shape is displayed on a CRT. At this point, the operator may correct for any linear baseline drift, subtract a predetermined complex baseline as determined in the absence of absorption, average several profiles, or call up a "smoothing" routine. By using a movable cursor line on the CRT, the area of any segment of the line can be calculated. Likewise, shifts, halfwidths and asymmetries are automatically computed. Finally, a hard copy of the results along with a plot of the profiles is made. Other details of the experiment arrangement and conditions are described in detail in a series of papers (see Refs. 4-10). 4. RESULTS: BROADENING

AND SHIFT

(a) Xenon The relative intensities of the two fine-structure components and their spacings relative to the red satellites make it convenient to discuss first the changes for the third member (Cs(3)/Xe) of the principal series (6S,~2~SPl~.s~2) with increasing Xenon rd.+ Once the influence of the satellites on the parameter curves (shift and halfwidth vs rd) is understood for this case, the line profile evolution for the other members is clearer. Cs(3)/Xe. As can be seen in Fig. 2, at low rd, both fine-structure components are little affected by the red satellites and do not appreciably interact with each other; hence, for low rd, the halfwidth changes linearly with rd (Fig. 3) at the rate of 5.6 cm-'/rd. The same is true for the shift, previously reported in Ch'en, Gilbert, and Tan4 and reproduced for convenience here as Fig. 4(a). Referring to the theoretical profiles (Fig. 5) generated from the square-well model and the shift curve (Fig. 4(b)) as well as the halfwidth curve Fig. 6(a), both taken from these theoretical profiles, one finds remarkable qualitative agreement. Referring to Fig. 7, as the pressure is increased to about 1~16rd, the red satellite in the 2pI/2 component becomes nearly as intense as the line. For the 2p3j2 component, the effect of the satellite begins to be appreciable. This effect is dramatically illustrated by a sudden change in the slope of the halfwidth curve to about 55 cm-~/rd between about 1 and 1.6 rd: The reason is that one measures first the halfwidth of the line proper and then, as the satellite grows to more 0

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than one half the height of the line, one measures the combined width of line and satellite. At 1.94 rd, the satellite is nearly as intense as the line itself and can be seen to surpass the line in intensity at 2.73 rd. Hence, the "discontinuity" appears on the shift curve at around 2 rd, that is, when the he/ght of the satellite exceeds that of the line and one measures the shift from the satellite peak rather than the line peak. This same phenomenon is seen on the theoretical shift curve of Fig. 4(b). In the reg/on from about 2-3 rd, the slope on the halfwidth curve decreases to about 28 cm-Z/rd due to the fact that the satellite is stronger than the line. As the satellite continues to grow with foreign gas pressure, the line intensity decreases. Thus, we are now measuring the halfwidth of the satellite with minimal influence from the line. This is quite evident if one studies the intensity curves of line and satellite as predicted by the theory; see Fig. 5 in Allard. !

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D.E. GILBERTet al.

206

distinct peaks for the 2p1/2 but not for 2p3/2. Hence, the 2pi/2 has a sharper transition. At pressures lower than this transition value, the spacing of the satellite compared to the width of the line makes the influence of the red satellite unimportant; hence the curve is non-linear from about 4 rd on. The same is true on the halfwidth curve (Ch'en eta[., 4 Fig. 6). The influence of the satellite at low rd can be seen from the experimental profiles in Gilbert et al. 5 (Fig. 1) and in Allard et al. 2 (Fig. 16). The non-linear nature of both the shift and width curves in Ch'en eta[., 4 after the transition, is possibly due to the influence of an unresolved second satellite as discussed above for Cs(3)/Xe. For the '2p3/2 component; the influence of the blue satellite is appreciable after 20 rd or so. This effect will be discussed more fully in the section on argon. The parameter curves are much like the theoretical curves 1-b and 1-c for the 2p1/2 and 2/)3/2 components, respectively. Cs(2)/Xe. Shift data for Cs(2)/Xe (6S->7P1/2,3/2) are reported by Ch'en et al., 4 (Fig. 2). The halfwidth curves are shown here in Fig. 8. As can be seen, both line-structure components are about equally broadened. The halfwidth increases linearly with rd at 2.9 cm-l/rdl At about 4 rd, the slope increases and is again linear (33.Scm-I/rd) as the two components become unresolved. After about 6 rd, the curve becomes linear again (23.5 cm-I/rd), as can be seen in the high rd extension of the previous curve (Fig. 9). Even though the satellites are well separated from the line at low rd (39 cm -1 and 55 cm -1 for the 2PI/2 and 2p3/2, respectively), the satellite appears rather broad. Typical experimental profiles are shown in Allard I (Fig. I). There are smooth transitions on the parameter curves. Hence, the explanation is the same as for Cs(3)/Xe; however, the features are smoother and appear a t higher density. The parameter curves are much like the theoretical curves (Fig. l(d)). Cs(4)/Xe. Figure 3 gives the halfwidth for Cs(4)/Xe(6Slr2-,9P3t2); the shift is reported in Ch'en et a[. 4 (Fig. 3). One sees the same general shape as reported for Cs(3)/Xe. There is good qualitative agreement with the theoretical curve (Fig. 6(b)). Examination of the evolution of the line profiles reveals the same influences by the red satellites. However, corresponding features occur at lower rd due to the closer relative proximity of the first satellite to the line--about 29 cm -1 for Cs(4)/Xe. (b) Krypton The shift and halfwidth of several members of the cesium principle series were reported by Ch'en, Looi, and Garrett. 6 Although we have, at present, no new experimental data on krypton ' 120

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to present, a brief comparison of the previous work with our theoretical model will clarify the argon data which follow. Specifically, the satellite of the 2Pt12 component, for Cs(1)/Kr although closer to the line (5 cm -I compared to 17 cm -1 for ~P3/2), is rather sharp. Hence, the 2Pm parameter curves closely resemble the curves discussed above for Xe, including a rather clear "discontinuity" on the shift curve corresponding to the point where the satellite is higher than the line. This corresponds to the theoretical curve of Fig. l(b). The ZP312 component for Cs(1)/Kr, on the other hand, provides an example of an intermediate transition between the very evident satellites seen with Xe and the totally unresolved satellites associated with Ar. If one looks at the pressure evolution of the zP3/2profiles, one sees a very clear red satellite at pressure below 9 rd, where the "top" of the satellite is about one third the way up the line. By the time one reaches 12 or 13 rd, the satellite completely blends with the line. Because this satellite is rather broad, there is little effect on the shift at this point and the profile has the appearance of a reasonably sharp line with a pronounced smooth red asymmetry. The parameter curves correspond to the theoretical curve of Figs. l(c). (c) Argon The shift and broadening of the first two members of the principal series perturbed by argon are reported in Ch'en and Garrett] Shift data for Cs(3)/Ar to Cs(20)/Ar are reported by Tan and Ch'en. 8 In the present work, we report halfwidth data for Cs(3)/Ar through Cs(9)/Ar, see (Fig.

11). Even at low pressures, one does not observe the slightest hint of an unresolved satellite. The profile simply appears to have a red asymmetry (see Fig. 10). Extrapolating from the previous analysis and utilizing the theoretically generated curves for a line with a very close and completely unresolved red satellite (Fig. l(d)), one can see very close agreement between theory and experiment. Because of the close weak satellites, the parameter curves are quite

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smooth and featureless. In conclusion, one can simply say that the parameter curves are explained quite well by assuming the existence of a close, weak satellite. The actual data points are not drawn for Cs(5)/Ar through Cs(9)/Ar for the sake of clarity. Although, there is some scatter, the plots appear quite linear throughout the range. This can, again, be attributed to the existence of close, weak satellites. (d) The influence of the violet satellite There is a well resolved violet satellite present only on the 2P3/2 component of Cs(l) perturbed by each rare gas. As the mass of the perturber goes up, this satellite becomes stronger and closer to the parent line. No doubt, this satellite complicates the profiles and parameter curves as one goes up in pressure. Its effects are mixed with the effects of the red satellites. Unfortunately, the present theory is not equipped to handle this case. At best, one can say that the analysis given above is complicated by the violet satellite for this one component. (e) Helium The shift and broadening of the first two members of the principal series perturbed by helium are reported in Garrett and Ch'en. 9 Shift data for Cs(3)/He to Cs(20)/He are reported by Tan et aLe In the present work, the halfwidths of Cs(3,4)/He are presented (see Fig. 12). The aj

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Interpretation of measured spectral line shapes using a square-well potential

209

linear regions have slopes of 3.74 cm-l/rd for the 2Pro. For Cs(4)/He, the slopes.arc 2.3 cm-l/rd and 1.67 cm-l/rd in the 2P3/2 and 2P1/2 components, respectively. The lines of Cs perturbed by He show a marked violet asymmetry. The contours are smooth (Figs. 13 and 14)and featureless, with the exception of the 2P3/2 component of Cs(l)/He which, as previously discussed (Section D), has a resolved weak violet satellite at a displacement of 367 cm -1. Although the present theory does not address violet satellites, we can still draw some conclusions by reversing the role of the line and satellite in the theory. For the purpose of comparison, then, we assume that the violet asymmetry is due to an unresolved symmetric satellite. If this were true, one would expect all of the parameter curves to be smooth, as in Fig. l(d). Further, one would expect the broadening curves to be of about the same qualitative shape for He as for Ar since Ar presumably has an unresolved red satellite. For the same reasons, one would expect the shift curves to have the same quantitative shape except that Ar shifts red and He shifts blue. Examination of the corresponding curves shows this agreement to be reasonably good. It is also interesting to study the asymmetry curves for Cs(1,2)/He (Figs. 9 and 10 in Garrett et al.9). Using our present model, one can now understand the hitherto unexplained maximum that occurs at higher rd in the 2Pl~ components. For Cs(1)/He, as the pressure increases, so does the violet asymmetry due to the fact that our unresolved violet satellite grows with respect to the line. At about 30 rd, the satellite begins to be important with respect to the line and the increase in violet asymmetry slows. As the satellite, assumed to be symmetric, predominates, the asymmetry becomes more red. Finally, at about 70 rd, the red tail of the 2P3/2 component

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begins to mix with the 2P~t2profile and the trend is reversed. This reversal is complete in the case of Cs(2)/He because of the closer spacing of the components. It is interesting to note that the same effects are present with Ne (see Garrett, Ch'en, and Looi~°). 5. CONCLUSION

By understanding how corresponding features on a theoretical profile affect the various parameter curves, one has a powerful aid in interpreting complicated experimental results. Even though the model is quite simple, much qualitative understanding is achieved. Acknowledgements--This work was supported, in part, by the National Science Foundation, the Research Corporation, and NATO Research Grant No. 1198, which made collaboration between the authors possible.

REFERENCES I. N. F. AUard, .I. Phys. B. (Atom. Motec. Phys.) ll, 1383 0978). 2. N. F. Allard and Y. G. Biraud, JQSRT to be published. 3. P. W. Anderson and J. D. Talman, Proc. Con[. Broadeningof Spectral Lines, p. 29. Bell Telephone System Technical Publications, Murray Hill, New Jersey (1956). 4. S. Y. Ch'en, D. E. Gilbert, and D. K. L. Tan, Phys. Rev. 184, 51 (1969). 5. D. E. Gilbert and S. Y. Ch'en, Phys. Rev. IU, 1 0969). 6. S. Y. Ch'en, E. C. Looi, and R. O. Garrett, Phys. Rev. 155, 38 (1967). 7. S. Y. Ch'en and R. O. Garrett, Phys. Rev. 144, 59 0966). 8. D. K. L. Tan and S. Y. Ch'en, Phys. Rev. A. 2, 1 (1970). 9. R. O. Garrett and S. Y. Ch'en, Phys. Rev. 144, l (1966). 10. R. O. Garrett, S. Y. Ch'en, and E. C. Looi, Phys. Rev. 156, 1 (1967).