Line Mixing and Broadening in the RamanQBranch of HD at 304.6 K

JOURNAL OF MOLECULAR SPECTROSCOPY ARTICLE NO.

192, 406 – 416 (1998)

MS987696

Line Mixing and Broadening in the Raman Q Branch of HD at 304.6 K G. D. Sheldon, P. M. Sinclair, M. P. Le Flohic, J. R. Drummond, and A. D. May Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 1A7 Received May 14, 1998; in revised form July 29, 1998

The Q-branch lines of pure HD were measured at densities ranging from 1 to 7 Amagat at 304.6 K. Each profile was fitted to the well-known Rosenkranz expression to extract the size of the asymmetry due to line mixing as well as to the linewidth. Line mixing and broadening coefficients were obtained by fitting the asymmetries and widths to a straight line as a function of gas density. Apart from a single existing measurement for the Q(0) line, our mixing coefficients are the first direct measurements of the asymmetry due to line mixing in HD. Our broadening coefficients are consistent with the best earlier measurements but are an order of magnitude more precise. Agreement is found with some existing semiclassical calculations of broadening. We have fitted our HD broadening coefficients to a variety of empirical energy gap laws. Our conclusions are that none of the exponential gap law (EGL), the modified exponential gap (MEG) law, and the statistical power gap (SPG) law successfully models our broadening coefficients. We present a modified version of the EGL and the MEG laws, which are successful in reproducing the experimental results. Using the fitted parameters of the new gap law, we have calculated the relaxation matrix of HD at room temperature. With this relaxation matrix, we have simulated the Q-branch spectrum at a number of densities between 49.1 and 490 Amagat and compared the results with previous high-density measurements. At all densities and frequencies, the simulated spectral intensity was found to agree with the measured strength within about 5% of the peak of the spectrum. In addition, the comparison provides evidence of a nonlinear vibrational dephasing shift in HD. © 1998 Academic Press INTRODUCTION

Over the past few decades, the quantification of spectral lineshape parameters in gases has consisted almost entirely of measurements of linewidths, linestrengths, and to a lesser extent, lineshifts. An accurate determination of the width of a line requires an instrumental linewidth that is small compared to the width of the line being measured and a high signal-tonoise ratio. Increasing the resolution of conventional instruments, such as grating spectrometers or interferometers, became increasingly counterproductive since the increased resolution was accompanied by a reduction in the signal. Measurements of narrow spectral lines, such as infrared or Raman lines in gases at low pressures, became increasingly limited either in resolution or signal-to-noise ratio. It is not surprising that for these applications, spectrometers based on tunable lasers, with their high intensity per unit bandwidth, have become the instruments of choice. Two notable examples are stimulated Raman gain spectroscopy (1) and difference– frequency infrared spectroscopy (2). The use of laser spectrometers has not only allowed accurate width and shift measurements to be made, but has also led to measurements of more subtle features of spectral profiles in dilute gases. For isolated spectral lines, these include Dicke narrowing (3) and speed-dependent effects (4, 5). For bands of closely spaced lines, line mixing has also been examined (6 –18). In this paper we are concerned with line mixing in the Raman Q branch of HD. If collisions cause transitions between the states associated with

two different lines, the two lines may interfere such that the intensity at any frequency is not the sum of the individual line intensities. The intensity in the region between the two lines increases, and it decreases in the outer wings of the two lines. Thus, the lines appear to “pull together.” At high enough density they collapse into a single line. This effect was first observed by Purcell et al. in NMR spectra (6). The formal theoretical treatment of this effect, within the binary collision impact approximation, was first given by Baranger (7). Some early observations of line mixing were made in vibrational spectra by May and coworkers (8–12). At low densities, line mixing makes lines slightly asymmetric (13). Direct observations of such asymmetries were previously made in our laboratory in the Raman Q branch of D2 (14, 15). A single earlier measurement was made by Rosasco et al. (16) in low-density HD. This paper is composed of three parts. In the first part, we present precise new measurements of line mixing and broadening in HD at low densities. The broadening data are compared with earlier measurements and with existing theoretical calculations. In the second part of the paper, the broadening data are fit to empirical energy gap laws. The third part of the paper is concerned with extrapolating our precise low-density measurements to very high densities and comparing the results to earlier measurements at those densities (10, 11). PART I: HD AT LOW DENSITIES

The spectrometer, which is shot noise limited and has a resolution of 2 MHz (;1024 cm21), has been described by

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LINE MIXING AND BROADENING IN HD

Line Broadening Figure 2 shows a plot of measured width versus density for each of the Raman Q-branch lines Q(0)–Q(4). The Q(2) widths have been shifted upward by 20 3 1023 cm21. If the residual width arising from the translational motion of the molecules (Dicke width) is taken into account, G (in Hz) may be written as (3) G j 5 n jr 1

FIG. 1. The spectral profile of the Q(1) line of HD at a density of 5.96 Amagat and 304.6 K. (a) Experimental data (solid circles) and fitted curve (solid line). (b) Residual to the fit (obs. 2 calc.).

Forsman et al. (19). Measurements were made of the Raman Q-branch lines of pure HD over a density range of 1–7 Amagat at 304.6 K. If we ignore the effects of the translational motion (collision-broadened regime), then each line may be written as the sum of a Lorentzian profile of width1 G and strength S and an asymmetric or dispersion shape of relative strength Y. The width G, the line mixing parameter Y, and the strength S are all proportional to the density. We write a line profile as (13) I~v! 5

S~G 1 YD n ! , G2 1 Dn2

[1]

where Dn 5 n 2 n0 is the frequency from linecenter. Each of the measured profiles was fit to Eq. [1] to determine linestrengths, widths, and mixing parameters. The linestrength is not an important parameter here, as only relative intensities were measured. The fitting routine also allowed for a small instrumental background. The results of a typical fit are shown graphically in Fig. 1. The solid circles represent the data, and the line through the points is the fit. The residual to the fit is shown in the middle panel. The flatness of the residual indicates that the data are well represented within the noise by Eq. [1]. If the root-mean-square deviation of the residuals is taken as the noise and the intensity at the peak as the signal, then the typical signal-to-noise ratios for the Q(0), Q(1), Q(2), Q(3), and Q(4) lines were 140, 260, 270, 160, and 40, respectively, independent of the density, for densities above 1 Amagat. The signal-to-noise ratio for the Q(4) line was too low to detect any asymmetry in the measured profiles. 1

All linewidths in this paper are half-widths at half-maximum (HWHM).

2 pn 2D 0 , r

[2]

where G j is the broadening coefficient (in Hz/Amagat), n is the Q-branch Raman frequency (in cm21), and D 0 is the diffusion constant (in cm2/s) at 1 Amagat unit of density. The widths of the Q(0)–Q(3) lines were fit to Eq. [2] to obtain line broadening coefficients g j and the parameter B 5 2 pn 2 D 0 . The fits are shown by solid lines in Fig. 2. These appear to be straight lines because the Dicke contribution is very small even at a density of 1 Amagat. The value of B determined from our fits was the same within experimental error for all lines of the Q branch and was equal to 7.8(6) 3 107 Hz z Amagat, which yields a diffusion constant at 1 Amagat of D 0 5 0.95(7) cm2/s. This value of D 0 is consistent with the value 0.978(29) cm2/s from the measurement of Rosasco et al. (16) and is slightly lower than the value 1.109(2) cm2/s deduced from the H2 measurements of Harteck and Schmidt (20).2 Since the Q(4) line was measured at only one density, it was not possible to fit the data to Eq. [2]. Instead, using the value of B from the other lines, the Dicke width was subtracted from the measured width, which was then divided by the density to yield the line broadening coefficient. This is the first measurement of the broadening coefficient of the Q(4) line. The broadening coefficients are summarized in Table 1 along with the measurements of other workers. They are presented in terms of the wavenumber or cm21 frequency unit. This frequency unit is the most commonly used in modern literature and will be used for the remainder of this paper. The precise measurements of Rosasco et al. (16) were made at 298 K. Taking the calculated temperature dependence of the rotational rate constants of HD–He (21) as being indicative of the temperature dependence of the broadening coefficients in pure HD, we expect a difference of about 4% between the two sets of measurements. This is consistent with the observed systematic difference of about 3%. Also listed in the table are the calculated broadening coefficients of Bonamy et al. (22) and Marsault-Herail et al. (23). The calculations of Bonamy et al. (22) are in reasonable agreement with our measurements, while those of Marsault-Herail et al. (23) are systematically higher. Both of these calculations are semiclassical; that is, they assume that the HD molecules in a gas move along classical trajectories. 2 This value was obtained by multiplying the H2 measurement (20) by ( =T/M) HD/( =T/M) H2, where M is the molecular weight and T is the temperature.

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SHELDON ET AL.

single-pass cell and is thus relatively insensitive to strain in the cell windows. There are, to date, no published calculations of line mixing coefficients of HD. However, Pine (25) has provided a sum rule that we can use to demonstrate that the observed line asymmetries are consistent with line mixing. The sum rule is written in our notation as

O ry 50 j j

[3]

j

FIG. 2. Linewidth G as a function of density for the Q-branch lines of HD at 304.6 K. The symbols represent the experimental data. The Q(2) data has been offset by 2 3 1023 cm21. A solid line represents the fit to the data.

Line Mixing Figure 3 shows plots of the line mixing parameter versus density for each of the lines measured. For clarity, the Q(3) data have been shifted by 3 3 1022 along the vertical axis. The error bars represent the statistical error reported by the fitting program. The solid lines in Fig. 3 are the “best fit” straight lines through each of the data sets. The zero-density offsets are due to a well understood instrumental effect related to two-color z-focusing (19, 24) and may be ignored. The slopes of the lines are the line mixing coefficients y j , where Y j 5 y j r . These are summarized in Table 2. These are the first measurements of line mixing coefficients for the Q(1), Q(2), and Q(3) lines of HD. The line mixing coefficient of the Q(0) line measured by Rosasco et al. (16) agrees with ours, within experimental error. This is perhaps fortuitous since those workers were forced to correct their results due to the presence in their signal of what appeared to be a Raman-induced Kerr effect signal (RIKES) contribution. The source of this contribution was attributed to depolarization of the laser beams by multiple passes through the slightly strained windows of the cell. Our system uses a

and simply states that the sum of the mixing coefficients, each weighted by the fractional population r j of the state (v 5 0, J 5 j), will be zero. Neglecting the j $ 4 states, which contain less than 3% of the population at 304.6 K, and using the experimental mixing coefficients from Table 2, the weighted sum is 0.18(25). This result shows that the measured asymmetries are consistent with line mixing. This completes the presentation of the new experimental results. In summary, we have reported precise broadening coefficients of the Q(0)–Q(4) lines in pure HD at 304.6 K and line mixing parameters for all but the Q(4) line. In the second part of this paper we use our results to examine a number of “fitting laws.” PART II: STATISTICALLY-BASED FITTING LAWS

Both line mixing and line broadening of the Raman Qbranch lines are the result of rotational relaxation, or rotationally inelastic collisions. Starting from state-to-state relaxation rates, both the line mixing parameter Y i and the inelastic contribution to the collisional width G i can be calculated. In the literature, the problem of characterizing state-to-state relaxation rates has often been approached using a variety of energy gap laws (see (26) and (27) for a review of the laws used in this paper). We use a similar approach here, but instead of dealing with widths, mixing parameters, and relaxation rates, all of which are proportional to density, we will deal with broadening coefficients, mixing coefficients, and relaxation coefficients,

TABLE 1 Line Broadening Coefficients for Pure HD at 304.6 K (HWHM in 1023 cm21/Amagat)

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LINE MIXING AND BROADENING IN HD

FIG. 3. Line mixing parameter, Y, as a function of density for the Q-branch lines of HD at 304.6 K. The symbols represent the experimental data. The Q(3) data have been offset by 3 3 1022. A solid line represents the fit to the data.

which are density independent. The broadening and mixing coefficients were defined above; the definition of the relaxation coefficients will now be discussed. The most fundamental description of rotational relaxation would be in terms of the state-to-state rotational relaxation rates R ji . For each pair of states i and j, there is a rate at which collisions induce transitions from state i to state j. These rates will, in general, depend on the vibrational level. However, the formalism used by most authors is given in terms of linecoupling rates. The line-coupling rates (denoted W ji ) refer to the rate at which coherence is transferred from a spectral line i to another spectral line j. It is important to be clear that the indices no longer refer to rotational states, i.e., to values of the rotational quantum number J. Under certain conditions, however, the line coupling rates can be identified with state-to-state relaxation rates. If the lines of interest are Q-branch lines

(DJ 5 0), then they can be labeled by a single value of the rotational quantum number. Furthermore, if it can be assumed that the rotational relaxation rates are independent of the vibrational level, then the line coupling rates can be identified with the state-to-state rotational relaxation rates. Thus, W ji 5 2R ji . A negative sign appears in this identification since by convention, the line coupling rates are defined to be negative. The relaxation coefficients w ji are defined by the equation W ji 5 r w ji . The relaxation matrix w is composed of the relaxation coefficients (negative values) w ji as off-diagonal elements and the broadening coefficients (positive values) as the diagonal elements. Before fitting our broadening coefficients to any energy gap law, we must subtract the vibrational dephasing component of the broadening. This has been calculated by Bonamy and Robert (28) to be a small constant value of 0.61 3 1023 cm21/Amagat for the Q(0)–Q(3) lines. The accuracy of this value will be taken to be 65% since that is the overall level of agreement of their calculated total broadening coefficients (22, 28) with our measurements. Table 3 shows the measured broadening coefficients minus the vibrational dephasing component of the broadening. The given errors incorporate the experimental error and the error in the calculated vibrational dephasing broadening. In the following analysis, we will still refer to these adjusted broadening coefficients as simply the broadening coefficients or “observed” broadening coefficients. The fitting routine consists of adjusting the gap law parameters until the predicted broadening coefficients best match the observed values. Each gap law specifies the state-to-state relaxation coefficients w GL ji . The broadening coefficients are related to the relaxation coefficients according to

g GL i 5 2

Ow

GL ji

.

[4]

jÞi

Clearly, a test of the gap law is to determine how well the reproduce the observed values. calculated values of g GL i A second test is to determine how well values of the mixing

TABLE 2 Line Mixing Coefficients for Pure HD at 304.6 K (in 1023 Amagat21)

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TABLE 3 Measured Line Broadening Coefficients Minus Calculated (28) Vibrational Dephasing Broadening for Pure HD at 304.6 K (HWHM in 1023 cm21/ Amagat)

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coefficients calculated using the fitted gap law reproduce the experimental values reported in Table 2. The well-known Rosenkranz (13) form for the mixing coefficients is y GL i 5 2

O n w2 n . GL ji

jÞi

j

[5]

i

We are now in a position to compare our measurements to several gap laws: the exponential gap law (EGL) (29, 30), the modified exponential gap law (MEG) (31), and the statistical power gap law (SPG) (32). Equation [6] gives the relaxation coefficient for the EGL for transitions upward in the rotational quantum number J as

F

2w ji 5 a exp 2

G

b|DE| . kT

[6]

Here DE 5 E j 2 E i is the energy gap, or the difference in energy between the states (v 5 0, J 5 j) and (v 5 0, J 5 i), and a and b are constants. Equation [6] has been successfully fit to rotational relaxation measurements in HD by Chandler and Farrow (33). The values of the EGL parameters obtained from the fit to our broadening coefficients are a 5 25.56(7) 3 10 23 cm21/ Amagat and b 5 1.745(3). Chandler and Farrow (33) measured rotational state-to-state rates using a pump–probe method. They did not report the measured rates, but they did report the parameters resulting from a fit of the rates to the EGL. Their values of a 5 27(7) 3 10 23 cm21/Amagat and b 5 1.75(10) agree with ours within experimental error.3 In the present work, the fit to the EGL successfully modeled the mixing coefficients, but failed to model the broadening coefficients within the noise of the measurements. For the Q(1) line, for example, gGL 2 g1 5 20.6 3 1023 cm21/Amagat, 1 which is well outside the uncertainty of 0.05 3 1023 cm21/ Amagat in the observed broadening coefficient. The MEG law is given by 2w ji 5 a

F

G F

G

1 1 AE~i!/kTd 2 b|DE| exp 2 . 1 1 AE~i!/kT kT

[7]

Here E(i) is the rotational energy of the ( J 5 i, v 5 0) state minus that of the ( J 5 0, v 5 0) state. The value of A is fixed and is given by A 5 (l 2c m s /r 2 m m ), where m s and m m are the reduced masses for the collision system and the molecule, respectively; r is the equilibrium internuclear distance; and l c is the distance of closest approach for a collision. The value of l c is calculated assuming a Lennard–Jones isotropic intermolecular potential (35). The resulting value of A for HD at 304.6 Kelvin is 4.71, which is very close to the value of 4.72 used by 3 To arrive at this value of a, their reported value of 0.41(1) must be multiplied by the hard-sphere collision rate of 1.52 3 107 s21 Torr21 and then converted to units of cm21/Amagat (34).

Rosasco et al. (16) for HD at 298 K. The rotational energies E(i) were determined from the calculated energy eigenvalues of Schwartz and Le Roy (36). Thus the only fitting parameters in Eq. [7] are a, b, and d. The parameters resulting from the fit to our broadening coefficients are a 5 26.63(11) 3 10 23 cm21/Amagat, b 5 1.826(7), and d 5 0.9579(31). These parameters are similar to those found by Rosasco et al. (16). Based on a x2 analysis, the quality of the fit for the MEG law is marginally better than that for the EGL. However, the MEG law also failed to reproduce the present experimental broadening coefficients. It is the higher precision of our broadening coefficients, and therefore tighter constraint of the fit, compared to those of Rosasco et al. (16) that reveals the failure of the MEG law to model the broadening. The measured mixing coefficients are less precise, and as with the EGL, are successfully modeled by a MEG law fitted to the broadening coefficients. Implicit in the EGL and MEG laws as given by Eqs. [6] and [7], is the conservation of magnetic quantum number Dm 5 0. This is equivalent to saying that inelastic collisions do not reorient the molecules. McCaffery et al. (37) present a classical picture of rotational relaxation as a transfer of linear momentum to angular momentum. They showed that the inelastic collisions which are most efficient are those which conserve m in the collision frame. They warned that the conservation of m in the collision frame does not rigorously imply conservation in the laboratory frame, but rather that a propensity for m-conservation exists. Brunner et al. (38) relaxed the m-conservation constraint and derived a generalization of the EGL analogous to Eq. [6]. Their result takes the form

F

2w ji 5 aN lexp 2

G

b|DE| , kT

[8]

where the factor N l is a statistical weighting term. It depends in a rather complex way on the initial and final j and on the number l, an integer greater than or equal to zero. The changes in the magnetic quantum number due to inelastic collisions are constrained to |Dm| # l . In the m-conserving case ( l 5 0), N l is equal to 1. The EGL, as originally presented by Procaccia and Levine (30), allows for transitions to any m state with equal probability. This case corresponds to N l 5 (2j 1 1). Fitting our observed broadening coefficients to the EGL with various values of the parameter l, we found for l 5 1, that the quality of the fits was unchanged from the original l 5 0 fits, and for higher values of l, that the quality of the fits was worse. In spite of the fact that the original EGL does not fit our measurements, we conclude that our broadening coefficients favor an m-conserving model. The results of Chandler and Farrow (33) also supported an m-conserving model for collisions in pure HD. One can also insert a factor N l into Eq. [7] for the MEG law. The results of the fits showed the same trend as the EGL fits.

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LINE MIXING AND BROADENING IN HD

That is, the broadening data favor an m-conserving model, although the MEG law still fails to model the measured broadening coefficients. We also compared the statistical power gap law (SPG) to our measurements. The drastic failure of this gap law to model broadening coefficients led us to abandon it without further consideration. It did, however, successfully model our mixing coefficients. Based on the work of Procaccia and Levine (30), a second round of gap law fits was attempted. Their Eqs. [2] and [6] can be used to write the state-to-state rates (in our notation) as

F

2wji 5 a~2j 1 1!exp~D! z D z K1~D!exp 2

G

b|DE| , kT

[9]

where D 5 DE/ 2kT, and we have collapsed all of the purely temperature-dependent terms and constants into a. Here K 1 is the modified Bessel function of the first order. Note that this is very similar to Eqs. [6] and [8]. The (2j 1 1) factor is simply the special case of N l that corresponds to m-scrambling collisions. The extra factors involving D are easily evaluated. The omission of these factors (i.e., equating them to unity) is equivalent to making the assumption that translational energy of the molecules is large compared to the rotational energy gaps, i.e., it is a high-temperature limit. In the case of HD at room temperature, it is questionable whether such a limit is valid. Using Eq. [9], the gap law fit was repeated. As before, the fit successfully modeled our measured mixing coefficients but failed to model the measured broadening coefficients. Variations of Eq. [9] were also tried, including the use of the more general factor N l in place of the factor (2j 1 1) and including the [. . .]2 term from the MEG law (Eq. [7]). None of these fits was successful. Having failed to fit our broadening coefficients to one of the standard energy gap laws, the question arose as to what empirical modification to the laws might improve the fit to our data. The gap laws assume that the rates of rotational transfer are simple functions of the energy gaps between the rotational states. However, it is well known that in homonuclear diatomics (such as D2 and H2), a Dj 5 even selection rule applies to rotational relaxation. Clearly in these systems the state-to-state rates are affected by more than just the energy gaps between rotational levels. Gap law fits for homonuclear diatomics are done by treating the ortho- and para-states independently. The interaction potential between two HD molecules is dominated by two terms (11, 39). One of these terms arises from the fact that the center of mass and the center of charge of HD do not coincide. This term is responsible for allowing odd DJ transitions in HD. It is reasonable to expect that although all DJ values are allowed, the cross section for odd transitions will be different than for even transitions of the same energy gap. In HD–He, according to Green (21), the calculated ratio of the cross sections for the J 5 2 3 3 transition to the J 5 0 3 2 transition is approximately 3:2,

FIG. 4. Results of the fit to our gap law nEGL as a function of the rotational quantum number J. (a) The experimental broadening coefficients g are shown as solid circles. The solid line joins the computed points. (b) The residual or difference between the measured and fitted values of the broadening coefficients. The error bars represent the experimental error. (c) The experimental line mixing coefficients y are shown as solid circles with error bars representing the experimental error. The computed mixing coefficients have been joined by a solid line.

even though the energy gaps are essentially equal for these two transitions. The calculated ratio for the same transitions in HD–H2 (40) is nearly 3:1. No alternation of transition probabilities is observed in Table 1 of Chandler and Farrow (33) since they have made an assumption4 that precludes such an observation. Based on these observations, a modification was made to each of the gap laws wherein the scaling parameter a was replaced with the product aF. The factor F is defined by F 5 1 6 b,

[10]

where b is a new fit parameter and the upper (lower) sign corresponds to odd (even) DJ transitions. In a global sense, the new factor F effectively allows rates of even and odd DJ transitions to scale independently. The newly modified gap laws will be referred to as nEGL and nMEG. The results of fitting with the new gap laws were significantly better as compared to the results using the original laws. Figures 4 and 5 compare the experimental data to the nEGL and the nMEG laws, respectively. The upper panel in each figure shows the experimental broadening coefficients (solid circles) and those calculated from the gap law (solid line). The middle panel in each of the figures shows 4

Those authors assume that the EGL (Eq. [6]) correctly models rotational energy transfer probabilities.

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SHELDON ET AL.

PART III: CALCULATION OF HIGH-DENSITY SPECTRA

FIG. 5. Results of the fit to our gap law nMEG as a function of the rotational quantum number J. (a) The experimental broadening coefficients g are shown as solid circles. The solid line joins the computed points. (b) The residual or difference between the measured and fitted values of the broadening coefficients. The error bars represent the experimental error. (c) The experimental line mixing coefficients y are shown as solid circles with error bars representing the experimental error. The computed mixing coefficients have been joined by a solid line.

the difference between the experimental and the fitted broadening coefficients. The error bars represent the experimental error. The lower panel compares the measured mixing coefficients (solid circles) to those determined from the gap law (solid line). We see from the figures that both the nEGL and the nMEG law successfully model the broadening and the mixing coefficients. The fact that the nEGL law successfully fits the data and the old MEG law fails (with the same number of parameters) shows that the F factor yields a significant improvement. The parameters resulting from the nMEG law fit were a 5 20.63(28) 3 1023 cm 21 /Amagat, b 5 1.698(9), d 5 1.016(4), and b 5 0.302(18). The parameters from the nEGL law fit were a 5 21.48(18) 3 1023 cm21/Amagat, b 5 1.730(3), and b 5 0.259(13). These values of b correspond to ratios of odd to even DJ cross sections of 1.87(7):1 and 1.70(5):1 for the nMEG and nEGL fits, respectively. These values are quite close to the ratio of 3:2 calculated by Green (21) for HD– He, and are lower than the ratio calculated by Chu (40) for HD–H2. Since both the nEGL and the nMEG laws successfully modeled our broadening and mixing coefficients, the one with the fewer fit parameters, namely the nEGL, will be accepted as the most successful and will be used in the third part of this paper.

As mentioned above, there have been two measurements (10, 11) of line mixing in HD, both made at high densities where the lines severely overlap and the band shows considerable narrowing. These high-density results have already been compared with a theoretical calculation by Bonamy et al. (22). The agreement between theory and experiment is impressive at densities up to about 100 Amagat, but large departures are seen at higher densities. Clearly the calculation of the spectrum by Bonamy et al. (22) required the calculation of the relaxation matrix, and it would have been of great interest to compare that calculated relaxation matrix with the matrix obtained from our gap law fit. However, the relaxation matrix was not published. While we cannot compare our results with theory, we can (given our relaxation matrix) use the impact theory formalism of Baranger (7) to calculate the spectrum exactly at any density and thus affect a comparison with the earlier experiments. This represents an enormous extrapolation from the density regime of our measurements (1–7 Amagat) to that of the earlier measurements (49 – 490 Amagat). The impact theory (7) gives the spectrum as I~v! 5

O

1 Im@ r kX kX l^^l|~ n 2 n o 2 i r w! 21|k&&#. p k,l

[11]

For the Raman Q branch, n 0 is a diagonal matrix of the free-molecule Raman frequencies, r k is the relative population of the state (v 5 0, J 5 k), w is the relaxation matrix, r is the gas density, and X is a column vector of line amplitudes. The line amplitudes for the isotropic Raman Q branch are the matrix elements of the isotropic polarizability ^v 5 1, J| a# |v 5 0, J&. This formalism is exact within the impact and binary collision approximations provided one neglects translational effects. Equation [11] can be used to calculate the spectrum at any density given the relaxation matrix W and the properties of the free molecule. At first sight it appears as if the evaluation of the spectrum from Eq. [11] would require a very intensive matrix inversion at each frequency. However, as pointed out by Baranger (7) and exploited by Gordon and MacGinnis (41), there is a mathematical manipulation that simplifies the expression immensely. The technique begins with the diagonalization of the matrix ( n 0 1 iW). The diagonalization of this matrix yields eigenvalues L k and a matrix A with columns equal to the eigenvectors. This diagonalization allows the spectrum to be expressed as I~v! 5

1 p

O Im L Re~n 2V Re1LIm! V1 ~Imn 2LRe L ! , i

ii

ii

2

i

i

i

2 i

[12]

where V 5 A 21 r XX t A. The reader may recognize the form of this equation as being the same as that of Rosenkranz (13) for

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LINE MIXING AND BROADENING IN HD

FIG. 6. Comparison of experimental and calculated profile of the Q branch of HD at 49.1 Amagat. (a) Experimental data (solid circles) and calculated profile (solid line). (b) Difference between the experimental data and the profile calculated from our low-density broadening coefficients and the fitted nEGL.

the spectrum at low density. The only difference is that the lineshape parameters (linewidth, mixing, strength, and shift) are now complicated functions of density instead of constant or proportional to the density. Specifically, for line k, the line mixing parameter is given by Y k 5 Im(V kk )/Re(V kk ), the linewidth is given by G k 5 Im(L k ), the strength is S k 5 Re(V kk ), while the Raman frequency is given by n k 5 Re(L k ). At low densities these expressions reduce to the forms implicit in Eq. [1]. It is now clear that we can use our measurements of broadening at low density to construct spectra at any density. The first step in the process was determining the relaxation matrix w from our experimental broadening coefficients. This was completed in Part II. Since we have determined the parameters of the nEGL, the components of w may be calculated. Using the relaxation matrix and known properties of the free molecule, the diagonalization can be carried out to yield eigenvalues L and the matrix A of eigenvectors. The spectrum is then calculated using Eq. [12]. Raman line frequencies were obtained from the measurements of Stoicheff (42) and from the HD energy eigenvalues of Schwartz and LeRoy (36). The populations were calculated according to a Maxwell–Boltzmann distribution using the rotational energies of Schwartz and LeRoy (36). The polarizability matrix elements were also taken from Schwartz and LeRoy (36). With the relaxation matrix obtained from the nEGL, the spectrum was calculated at a number of high densities. It was, of course, necessary to truncate the relaxation matrix to a finite size in order to perform this calculation. This does not pose a problem, however, because for high J the Q-branch lines are widely spaced and the rotational states associated with them are essentially unpopulated. They have, therefore, negligible contribution to the spectrum and can be ignored with impunity.

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Truncating the relaxation matrix to seven rows and columns includes the lines from Q(0) to Q(6). Including more lines made no significant difference to the calculated spectrum. The dephasing contributes a diagonal element to w and poses no problem. The J-independent part of the frequency shift may also be treated similarly. The shift has been calculated by Bonamy et al. (22) and measured by Rosasco et al. (16) at low densities. However, as the shifts in the hydrogens are known to have components nonlinear in density (43), the overall frequency was allowed to float in the fitting routine. The highdensity experimental points were obtained by digitizing the figures of Witkowicz (44) and Dion (45). The second and only other adjustable parameter, used in the comparison of the high-density data and the spectrum constructed from our lowdensity results, was the overall intensity of the spectrum. The results of the fits are shown in Figs. 6 –9 for 49.1, 103, 250, and 490 Amagat. The upper panel in each figure shows the calculated spectrum (solid line) and the data; the lower panel shows the residual spectrum. The computed and measured profiles were normalized to equal integrated strengths. The level of agreement between the measured profile at high density and that extrapolated from the low-density data is remarkable. Our predicted spectra agree with the high-density measurements significantly better than the semiclassical calculations of Bonamy et al. (22), which fail to describe the spectra above 100 Amagat. To be convincing, the shift of the spectrum should be consistent with our expectations of vibrational dephasing. Figure 10 shows a plot of the shift as a function of density. The 60.08 cm21 error shown for the shifts is determined from the sampling frequency 0.16 cm21, used to record the high-density spectra (44, 45). It does not include any errors in the original data or any errors introduced by our retrieval

FIG. 7. Comparison of experimental and calculated profile of the Q branch of HD at 103 Amagat. (a) Experimental data (solid circles) and calculated profile (solid line). (b) Difference between the experimental data and the profile calculated from our low-density broadening coefficients and the fitted nEGL.

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FIG. 8. Comparison of experimental and calculated profile of the Q branch of HD at 250 Amagat. (a) Experimental data (solid circles) and calculated profile (solid line). (b) Difference between the experimental data and the profile calculated from our low-density broadening coefficients and the fitted nEGL.

of the data from the published figures (44, 45). The solid line represents the calculated vibrational shift of Bonamy et al. (22), and the dashed line is an extrapolation of the average shifts of the Q(0)–Q(3) lines measured by Rosasco et al. (16) at low density (less than 10 Amagat). The systematic trend of the shift with density shows that the observed shift is most likely due to vibrational dephasing. The systematic departure from the precise low-density shift measurement of Rosasco et al. (16) might be explained by a quadratic component of the vibrational shift that becomes

FIG. 9. Comparison of experimental and calculated profile of the Q branch of HD at 490 Amagat. (a) Experimental data (solid circles) and calculated profile (solid line). (b) Difference between the experimental data and the profile calculated from our low-density broadening coefficients and the fitted nEGL.

FIG. 10. The vibrational shift in HD as a function of density. The solid circles are derived in this paper from the high-density data in Refs. (10, 11, 43, 44). The solid line represents the calculated linear shift (22), the dashed line represents the experimental linear shift at low density (16), and the dotted line represents the calculated linear shift (22) plus the measured quadratic shift of H2 (42).

important only at high densities. Such a quadratic vibrational shift has been observed in H2 by May et al. (43). Our determination of a vibrational dephasing shift is not precise enough to establish a quadratic dependence. However, Fig. 10 shows (dotted line) the quadratic shift of H2 (43) added to the linear shift of HD (22). Clearly the departure from the extrapolated, low-density, linear shift could be due to a vibrational shift in HD that is quadratic in density. Some attempt was made to fit with J-dependent shifts; however, the quality of the old high-density data is not high enough to constrain a fit with so many free parameters. In our comparison, we have neglected the fact that the line mixing and broadening measurements of this work were made at 304.6 K, while the high-density measurements were made at 293 K. To test if our calculated spectra are sensitive to this small difference, we applied a scaling to the relaxation matrix obtained from the nEGL fit. As above, where we compared broadening coefficients from experiments performed at slightly different temperatures, we assumed that the temperature dependence of the relaxation coefficients is that of the calculated rotational relaxation cross sections for HD–He (21). From the results of that calculation the temperature-scaling factor was estimated to be W (T 5 293 K) 5 0.94 z W (T 5 304.6 K). The high-density spectrum was recalculated and showed differences of less than 0.5% from the spectra calculated using the original relaxation matrix. It is clear that the temperature difference is not the primary cause of the slight deviation of the data from our calculation. This is not surprising. The temperature correction involved a uniform scaling of the relaxation matrix. However, another test of sensitivity of the high-density spectrum to errors in the relaxation matrix is to compare spectra calculated using relaxation matrices obtained from different gap laws. The two relaxation matri-

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ces used were those obtained from the nEGL fit and the MEG law fit. Recall that the nEGL successfully modeled our low-density broadening measurements, but the MEG law did not. Spectra were calculated at four densities using each of these relaxation matrices. At all four densities, the difference between the spectra was less than 1% of the Qbranch peak. The high-density data (10, 11) are not sufficiently precise to extract details at this level about the relaxation matrix. It is possible that small discrepancies between the experimental curves and our simulated high-density spectra would occur due to our neglect of the anisotropic component of the Q branch. However, since the agreement only begins to break down at the highest density, it is more likely that the reason for the discrepancy lies in the theory itself. The impact theory of Baranger (7) relies on the binary collision, impact approximation. Under this set of approximations it is assumed that all molecular collisions involve only two molecules and are of short duration relative to the time between collisions. It is most likely that at the densities studied by Dion and May (10) and Witkowicz and May (11), these approximations begin to fail. Regardless of the small disagreement between the simulated and measured spectra, the degree to which they agree is truly remarkable given that the simulation is based solely on measurements made at low density. SUMMARY AND CONCLUSIONS

relaxation in HD. This experimental ratio was found to be consistent with the ratio of calculated cross sections for HD–He (21). With the relaxation matrix, we have simulated the Q branch at a number of densities between 49.1 and 490 Amagat. These simulated spectra were compared to early high-density measurements (10, 11). Thus, indirectly we have compared our low-density measurements of the HD Q branch with the highdensity measurements. The high-density simulations and measurements were found to agree within about 5% of the peak of the spectra at all densities. This level of agreement is superior to that of the semiclassical calculation of Bonamy et al. (22). In the process of comparing the calculation to experiment, we have observed evidence of a nonlinear vibrational dephasing shift in HD. ACKNOWLEDGMENTS This work was supported by the Natural Sciences and Engineering Research Council of Canada, the Ontario Government through Photonics Research Ontario, and Atmospheric Environment Services. G.D.S. gratefully acknowledges the financial support of the Walter C. Sumner Memorial Fund.

REFERENCES 1. 2. 3. 4. 5.

We have studied the Raman Q-branch lines of pure HD at densities ranging from 1 to 7 Amagat at 304.6 K. Both the line mixing parameters and the linewidths were extracted from the measured profiles. Line mixing and line broadening coefficients were determined by fitting the mixing parameters and widths as a function of gas density. The broadening coefficients are consistent with existing measurements but are an order of magnitude more precise. There is reasonable agreement with the semiclassical calculation of Bonamy et al. (22), but not with the calculation of Marsault-Herail et al. (23). A single previous measurement (16) agrees with our mixing coefficients, and we have extended the existing data to three new lines. There are no published calculations of the mixing coefficients for HD with which to compare our results. We have fit our HD broadening coefficients to a variety of empirical energy gap laws. The quality of each of these fits has provided a critical test of the applicability of the gap laws for HD. Our findings are that neither the EGL, the MEG, nor the SPG laws successfully model our broadening coefficients. We have devised a modification to these gap laws, which allows for different relaxation rates for even and odd DJ transitions in HD. Our new gap laws successfully model our broadening coefficients as well as our mixing coefficients. Furthermore, the extra parameter included with our modified laws allowed us to determine the ratio of the cross sections for even and odd DJ

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

A. Owyoung, IEEE J. Quantum Electron. 14, 192–203 (1978). A. S. Pine, J. Opt. Soc. Am. 64, 1683–1690 (1974). R. H. Dicke, Phys. Rev. 89, 472– 473 (1953). R. L. Farrow, L. A. Rahn, G. O. Sitz, and G. J. Rosasco, Phys. Rev. Lett. 63, 746 –749 (1989). P. Duggan, P. M. Sinclair, M. P. Le Flohic, J. W. Forsman, R. Berman, A. D. May, and J. R. Drummond, Phys. Rev. A 48, 2077–2083 (1993). E. M. Purcell, R. V. Pound, and N. Bloembergen, Phys. Rev. 70, 986 –987 (1946). M. Baranger, Phys. Rev. 111, 494 –504 (1958). A. D. May, Ph.D. thesis, University of Toronto, 1959. A. D. May, J. C. Stryland, and G. Varghese, Can. J. Phys. 48, 2331–2335 (1970). P. Dion and A. D. May, Can. J. Phys. 51, 36 –39 (1973). T. Witkowicz and A. D. May, Can. J. Phys. 54, 575–583 (1976). R. C. H. Tam and A. D. May, Can. J. Phys. 61, 1558 –1566 (1983). P. W. Rosenkranz, IEEE Trans. Antennas Propag. 23, 498 –506 (1975). P. M. Sinclair, J. W. Forsman, J. R. Drummond, and A. D. May, Phys. Rev. A 48, 3030 –3035 (1993). P. M. Sinclair, Ph.D. thesis, University of Toronto, 1994. G. J. Rosasco, A. D. May, W. S. Hurst, L. B. Petway, and K. C. Smyth, J. Chem. Phys. 90, 2115–2124 (1989). R. Berman, P. Duggan, P. M. Sinclair, A. D. May, and J. R. Drummond, J. Mol. Spectrosc. 182, 350 –363 (1997). G. D. Sheldon, Ph.D. thesis, University of Toronto, 1998. J. W. Forsman, P. M. Sinclair, P. Duggan, J. R. Drummond, and A. D. May, Can. J. Phys. 69, 558 –563 (1991). P. Harteck and H. W. Schmidt, Z. Phys. Chem. B 21, 447– 458 (1933). S. Green, Physica 76, 609 – 615 (1974). J. Bonamy, L. Bonamy, and D. Robert, J. Chem. Phys. 67, 4441– 4453 (1977). F. Marsault-Herail, M. Echargui, G. Levi, J. P. Marsault, and J. Bonamy, J. Chem. Phys. 77, 2715–2727 (1982). M. Sheik-Bahae, J. Wang, R. DeSalvo, D. J. Hagan, and E. W. Van Stryland, Opp. Lett. 17, 258 –260 (1992). A. S. Pine, J. Quant. Spectrosc. Radiat. Transfer 57, 145–155 (1997). T. A. Brunner and D. Pritchard, Adv. Chem. Phys. 50, 589 – 641 (1982).

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416

SHELDON ET AL.

27. A. Le´vy, N. Lacome, C. Chackerian Jr., in “Spectroscopy of the Earth’s Atmosphere and Interstellar Medium” (K. N. Rao and A. Weber, Eds.), p. 261. Academic Press, New York, 1992. 28. J. Bonamy and D. Robert, Chem. Phys. Lett. 57, 22–28 (1978). 29. J. C. Polanyi and K. B. Woodall, J. Chem. Phys. 56, 1563–1572 (1972). 30. I. Procaccia and R. D. Levine, Physica A 82, 623– 630 (1976). 31. M. L. Koszykowski, L. A. Rahn, R. E. Palmer, and M. E. Coltrin, J. Phys. Chem. 91, 41– 46 (1987). 32. T. A. Brunner, R. D. Driver, N. Smith, and D. E. Pritchard, Phys. Rev. Lett. 41, 856 – 859 (1978). 33. D. W. Chandler and R. L. Farrow, J. Chem. Phys. 85, 810 – 816 (1986). 34. D. W. Chandler, private communication, 1997. 35. J. Van Kranendonk, Can. J. Phys. 41, 433– 449 (1963). 36. C. Schwartz and R. J. Le Roy, J. Mol. Spectrosc. 121, 420 – 439 (1987);

37. 38. 39. 40. 41. 42. 43. 44. 45.

R. J. Le Roy and C. Schwartz, Report CP-301R, Guelph Waterloo Centre for Graduate Work in Chemistry, Waterloo, Canada, 1987. A. J. McCaffery, Z. T. Alwahabi, M. A. Osborne, and C. J. Williams, J. Chem. Phys. 98, 4586 – 4602 (1993). T. A. Brunner, N. Smith, A. W. Karp, and D. E. Pritchard, J. Chem. Phys. 74, 3324 (1981). R. M. Herman, Phys. Rev. 132, 262–275 (1963). S.-I Chu, J. Chem. Phys. 62, 4089 – 4100 (1975). R. G. Gordon and R. P. McGinnis, J. Chem. Phys. 49, 2455 (1968). B. P. Stoicheff, Can. J. Phys. 35, 730 –741 (1957). A. D. May, G. Varghese, J. C. Stryland, and H. L. Welsh, Can. J. Phys. 42, 1058 –1069 (1964). T. Witkowicz, M.Sc. thesis, University of Toronto, 1975. P. Dion, M.Sc. thesis, University of Toronto, 1972.

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