Infrared Line Collisional Parameters of PH3 in Hydrogen: Measurements with Second-Order Approximation of Perturbation Theory

Room-temperature absorption by PH3–H2 mixtures in the ν2 and ν4 bands of phosphine (PH3) has been measured for low pressures. Fits of these spectra are made to determine the width of isolated lines and line mixing in a first-order Rosenkranz approximation. From the previous determinations, we deduce remarks on the lack of accuracy of predicting the collisional process. With the first-order Rosenkranz approximation, the collisional parameters are considered linear with pressure. In this work, we have considered spectra recorded for three doublets: A1 and A2 lines in the ν2 and ν4 bands of PH3 diluted with higher H2 pressure. We show that the line shifts are nonlinear with perturber pressures, which requires testing the fits of the recorded spectra with profiles developed in the secondorder approximation of the perturbation theory. Consequently, the first- and second-order mixing coefficients are determined and discussed. Throughout this study, we also show that the change in the intensity distribution is provided by the population exchange between low energy levels for the two components of doublets A1 and A2 lines and is described through the second-order mixing parameter. Thereby, we show the mixing effect on line width.

the second order in the framework of perturbation theory. His work yields a collisional profi le, considering the fi rst-and second-order parameters of line mixing effects. In this same profi le expression, the line shift versus the perturber pressure has a parabolic form.
We have used the collisional profi le expression given by Smith to analyze some spectra of three doublets A 1 and A 2 line recorded using a diode-laser spectrometer in the ν 2 and ν 4 bands of PH 3 diluted with H 2 at higher pressure and room temperature. Then, we specifi ed the collisional parameters to adjust. In addition, the reconstruction of the recorded spectra justifi es the need to consider second-order mixing and shifting parameters. Consequently, the fi rst-and second-order H 2 -line mixing coeffi cients in the ν 2 and ν 4 bands of PH 3 are presented and discussed. Through this study, we show the line mixing effect on the intensity distributions and line widths.
Experimental Analysis. Experimental conditions. The absorption spectra of the PH 3 -H 2 mixture in the ν 2 and ν 4 bands of phosphine were recorded at high resolution using a tunable diode-laser spectrometer (Laser Analytics Model LS3). The experimental techniques are detailed in [21,22]. In this work, we give the experimental conditions of the measurements, verifying our analytic procedure.
The absorption path length of the IR radiation provided by the diode-laser in the multipass white-type cell is fi xed to 20.17 m. The phosphine sample is supplied by Union Carbide with a stated purity of 99.999%, and the hydrogen sample is supplied by Air Liquide with a stated purity of 99.99%. The gas pressure is measured by two Baratron MKS gauges with full-scale measurements of 1.2 and 120 mbar, with an accuracy of 5 × 10 -4 and 2 × 10 -2 mbar. All spectra are recorded at room temperature (297.2 ± 1.5 K). Table 1 summarizes the data needed for the transitions studied in this work: wave numbers, pressure of PH 3 (P PH3 ), Doppler half-width (γ Dop ), effective Doppler half-width (γ Deff ), temperature T, and pressure of H 2 (P H2 ). Moreover, Fig. 1 shows an example of the recorded spectra of the doublet R R(4,3,A 1 ) and R R(4,3,A 2 ) lines of the ν 4 band of PH 3 diluted with pressures of H 2 , where the transmittance is plotted vs. the point numbers. The relative calibration of spectra is performed using a confocal etalon with an interfringe spacing of 0.007958 cm -1 . The etalon fringe pattern provides a check of the laser mode's quality for correcting the slightly nonlinear tuning of the diode-laser. Furthermore, it aids in linearization of the spectra with a constant step of 0.000121 cm -1 . All spectra are linearized using the cubic splines techniques [23].
Profi les and fi tting procedure. The spectra recorded using the diode-laser spectrometer allow the writing of the Beer-Lambert law: where α(σ) is the experimental absorbance per unit length at wavenumber σ in cm -1 , l is the path length, and I 0 (σ) and I t (σ) are the transmitted intensities measured with the cell under a vacuum and fi lled with the gas sample, respectively. To fi t the recorded spectra, three physical effects must be considered: weak instrumental distortion and Doppler and collisional effects. The fi rst is implicitly considered through the effective half-width γ Deff obtained by fi tting the effective Doppler line [25] ( Table 1). The Voigt profi le (VP) results from the convolution of the Doppler and collisional profi les, refl ecting the latter two effects. The expression of this profi le depends on the extension of the collisional profi le to be considered. Consider the collisional profi le proposed by Smith [19] and developed within the framework of the second-order approximation of perturbation theory: where the index k represents transitions in Liouville or "line" space, and S k , γ k , and Y k are the coupled line strength, the collisional half-width, and the fi rst-order line mixing coeffi cient, respectively. The wavenumber σ k = σ 0k -δ k , where σ 0k is the line center wavenumber and δ k is the line shift. In addition, g k is the second-order line-mixing coeffi cient, and δσ k is the second-order line-shift coeffi cient. We deduce the Voigt profi le (VPI2) corresponding to this collisional profi le as where W(x,y) is the complex probability function expressed by [26]: where x = (ln 2) 1/2 (σ -σ k + P 2 δσ k )/γ Deff and y = (ln 2) 1/2 γ k /γ Deff .
It should be noted that the parameters σ k , P 2 δσ k , γ k , PY k , and P 2 g k are related to the diagonal (W kk ) and off-diagonal (W kk‫׳‬ with k ‫׳‬ ≠ k) elements of the collisional relaxation matrix (W ) [19]. If the second-order parameters (δσ k and g k ) are equal to zero, then Eq. (3) reduces to the expression of the fi rst-order Rosenkranz approximation model (VPI1) [18,27].
To fi t the observed spectra, we set the intensity parameter to the value deduced from the absolute line intensity [18]. Figure 2 shows an example of the fi ts for the A 1 and A 2 lines of the doublet R R (4,3) in the ν 4 band of PH 3 diluted with 100.77 mbar of H 2 by the theoretical profi les VPI1 and VPI2. The (obs-calc) residuals are multiplied by 5 and displaced vertically for visibility. The (obs-calc) residuals of VPI2 show a better reproduction of the observed lines than that given by VPI1, where the second-order line mixing parameter is not considered. The better reconstruction given by VPI2 mainly refl ects the contribution of the second-order mixing parameter to the reproduction of the lines at the peak.   [18], the absorption path length l, and the constant partial pressure of PH 3 (P PH3 ) in the gas mixtures, we have deduced the intensity parameter S for each studied transition. This parameter is fi xed in the fi t profi les used for all four recorded spectra broadened by four H 2 pressures. Consequently, we can fi t the fi rst-and second-order line mixing parameters PY and P 2 g, respectively. This concept allows us to distinguish the proper line intensity from the rate of intensity transferred with the neighboring line during the overlap. Figure 3 gives qualitative examples showing the difference between the intensity distribution in two overlapping lines obtained in this work, S(1 + P 2 g); the line intensities are presented with their errors (S ± ΔS) deduced from the results of [18]. These examples are presented for the A 1 and A 2 lines of the doublets Q R (8,3) in the ν 2 band and R R (4,3) in the ν 4 band of PH 3 vs. the square of the pressure of hydrogen P 2 . We observe an almost linear variation of the intensity distribution between the two overlapping A 1 and A 2 lines vs P 2 . The slopes of the straight lines of these variations have opposite signs. Table 2 shows that the variation of the intensity distribution obtained by this work sometimes exceeds the measurement  uncertainties of the line intensities. They also differ from Brown's measurements [28], and they can reach 6.68% in the case of the line Q R (8,3,A 1 ).
Broadening coeffi cients and line mixing effects. Figure 4 shows a typical linear regression of the values of the collisional half-width measured at each of the four pressures of H 2 for the R R(4,3) doublet A 1 and A 2 lines of PH 3 . The collisional half-widths are measured using the VPI2 profi le. The slopes of the straight lines correspond to the H 2 -broadening coeffi cients γ 0 (in 10 -3 cm -1 ·atm -1 ). Here, we have systematically considered the small self-broadening contributions (represented by a point close to the origin) derived from the self-broadening coeffi cients calculated using the theoretical model detailed [6] and from the constant partial pressure of PH 3 in the gas mixtures. The measurements of γ 0 was presented in Table 3 with their errors given by the standard deviation derived from the linear least-squares fi t. The average values of the broadening coeffi cients of the A 1 and A 2 lines are in good agreement with those obtained in [18], where the secondorder mixing parameter is neglected. However, any appreciable difference between the coeffi cients of each line is shown as a percentage in Table 3. This behavior refl ects the line mixing effect on the line widths, and it is shown by considering the second-order mixing term given by Smith's development. In the same branch, the line mixing effect on the width decreases with the rotational quantum number J, i.e., when the difference wavenumber Δσ increases (Table 3). Note. Δσ = |σ(A 1 ) -σ(A 2 )|, γ 0Av = [γ 0 (A 1 ) + γ 0 (A 2 )]/2, Δγ 0 = |γ 0 (A 1 ) -γ 0 (A 2 )|.
Line shifting parameters. Figure 5 depicts a typical plot of the line shift δ derived from the VPI2 profi le versus the H 2 pressure P for the R R(4, 3, A 1 ) and R R(4, 3, A 2 ) lines in the ν 4 band of PH 3 . The point close to the origin represents the self-shifting contribution (δ self ). The measured values show a quadratic dependence on pressure, which agrees with the theoretical analyses given by the development of Smith's second-order perturbation theory [19]. Consequently, from the unconstrained second-order polynomial least-squares procedures, we deduce the fi rst-and second-order coeffi cients of the curves, which are the fi rst-order δ 0 and second-order δσ H 2 -shift coeffi cients, respectively. Indeed, the line-shift parameter for each line k is expressed in the framework of the development of the second-order perturbation theory as 2 0 k s e l f k k P P δ = δ + δ + δσ ,   (4, 3) doublet A 1 and A 2 lines in the ν 4 band of PH 3 , derived from the fi t of VPI2 (▪). The point close to the origin represents the self-shifting contribution. The best fi t curves represent the second-order polynomial functions whose fi rst-and second-order coeffi cients are, respectively, the fi rst-and second-order H 2 -shifting coeffi cients for each transition. where the fi rst-order coeffi cient δ 0k is related to the imaginary part of the diagonal (W kk ) elements of the collisional relaxation matrix (W), and the second-order coeffi cient δσ k is related to their off-diagonal elements (W kk ‫׳‬ with k ′ ≠ k) by [19]: We only present the qualitative behavior of the line-shift parameter with the perturber pressure. This behavior agrees with the hypothesis of second-order Smith's development.
First-order line mixing parameter. Figure 6 depicts two examples of the variation of the fi tted fi rst-order line mixing parameter (PY) by VPI2, with H 2 pressure P for the A 1 and A 2 lines of the R R(4, 3) doublet in the ν 4 band of PH 3 .  The fi rst-order H 2 -line mixing coeffi cients Y are deduced from the slope of the straight lines resulting from unconstrained linear least-squares procedures. These values obtained with their errors, given by the standard deviation on Y derived from the linear least-squares fi t, are presented in Table 4. For the components A 1 and A 2 of the doublet lines, the fi rst-order mixing coeffi cients are opposite. Except for the mixing coeffi cients of Q R (8,3) doublet lines, where it is underestimated, the results presented in this work satisfactorily agree with those given in [18]. In the Q R branch, the fi rst-order mixing coeffi cients show a decrease in absolute value with the rotational quantum number J, that is, when the difference wavenumber Δσ increases ( Table 4). The line mixing (off-diagonal relaxation elements) coeffi cients (W ij ) for the A 1 A 2 pairs of transitions in the phosphine pentad are given by V. Malathy Devi et al. [29] are seen and compared to ours.
Second-order line mixing parameter. Two typical examples of the variation of the second-order line mixing parameter (P 2 g) deduced by fi tting with the VPI2 profi le vs. the square of the pressure of hydrogen P 2 are shown in Fig. 7 for the R R(4, 3) doublet A 1 and A 2 lines in the ν 4 band of PH 3 . The second-order H 2 -line mixing coeffi cients g are derived from the slope of the straight lines resulting from unconstrained linear least-squares procedures. Presented in Table 4, these values are obtained with their errors, given by the standard deviation on g derived from the linear least-squares fi t. The measurement uncertainties are less than 31.8% for the studied transitions, except for the Q R(8, 3, A 1 ) and R R(4, 3, A 2 ) lines, which are in the range of 53 and 45%, respectively. For each doublet line A 1 and A 2 , these second-order mixing coeffi cients are the opposite, which refl ects the rate of intensity exchange during the overlap due to the population transfer between low energy levels of the two transitions. With the pressure range considered in this work, the second-order mixing term becomes appreciable. Thereafter, it is measurable, while at lower pressures it is indistinguishable from the measured uncertainties of line intensities, such as the case of the spectra studied in [18]. Like the observed behavior of the fi rst-order mixing parameter, the latter shows a decrease with the rotational quantum number J in the ν 2 band as the difference wavenumber Δσ increases (Table 4).
Conclusions. This work presents reasonable fi rst-and second-order mixing coeffi cients within the framework of the second-order approximation of perturbation theory for some lines in the ν 2 and ν 4 bands of PH 3 perturbed by H 2 at room temperature. To achieve these results, we used the spectra recorded at pressures ranging from 55 to 117 mbar using a diode-laser spectrometer. We also have considered the collisional profi le proposed by Smith [19] and developed within the framework of the second-order perturbation theory. This allows us to deduce the VPI2 profi le by convolution with the Doppler profi le.
The obtained shifting parameter shows a parabolic variation with the perturber pressure, which is justifi ed by the second-order approximation used. We have set the intensity parameter in the fi t profi les; this allows us to distinguish between the appropriate line intensity and the intensity rate exchanged with the neighboring line during the overlap. Consequently, we have shown that the second-order mixing parameter is appreciable and measurable. Indeed, it expresses the rate of intensity transferred between the overlapping lines A 1 and A 2 of each doublet. During this work, we have demonstrated the line mixing effect on the line widths, which allows us to better understand the collisional dynamics of the molecules.