H2-Induced Pressure Broadening and Pressure Shift in the P-Branch of the v3 Band of CH4 from 300 to 700 K

TL;DR

This study provides detailed measurements of pressure broadening and shifts in methane's v3 band at high temperatures, improving models of exoplanet atmospheres.

Contribution

It offers new high-temperature pressure broadening and shift data for methane's v3 band, enhancing atmospheric modeling accuracy.

Findings

Measured linewidths for 116 transitions between 300 and 700 K.

Derived rotation- and tetrahedral-symmetry-dependent broadening coefficients.

Data will improve radiative-transfer models for exoplanet atmospheres.

Abstract

For accurate modelling of observations of exoplanet atmospheres, quantification of the pressure broadening of infrared absorption lines for and by a variety of gases at elevated temperatures is needed. High-resolution high-temperature H2-pressure-broadened spectra are recorded for the CH4 v3-band P-branch. Measured linewidths for 116 transitions between 2840 and 3000 cm^{-1} with temperature and pressures ranging between 300 and 700 K, and 10 and 933 Torr, respectively, were used to find rotation- and tetrahedral-symmetry-dependent coefficients for pressure and temperature broadening and pressure-induced lineshifts. The new pressure-broadening data will be useful in radiative-transfer models for retrieving the properties of observed expolanet atmospheres.

Tables

Table 1. Table 1 : Experimental conditions and characteristics of the spectra.

Parameter	Value
Spectral coverage	2800 – 3200 cm-1
Temperature range	300 – 700 K
\ceCH4 pressure	0.8 – 7.0 Torr
\ceH2 pressure	10.0 – 933.3 Torr
Cells path length	10.0$\pm$0.2 cm
Number of averaged scans	100 – 400
Gas cell material/windows	Quartz (\ceSiO2)
Gas cell transmission range	2750 – 3250 cm-1
Light source	SiC Globar
Beam splitter	KBr
Detector	MCT LN2
Filter	fused silica and Ge
Spectral resolution (cm-1)	0.01 – 0.005
Apodization function	Box-car

Table 2. Table 2 : Summary of the experimental conditions

$\#$	Tube	$T$[K]	R†[cm-1]	Scan‡	$P_{\text{ce}CH4}$[Torr]	$P_{\text{ce}H2}$[Torr]
1	1	300	0.005	400	0.8	10.0
2	1	500	0.005	200	1.3	16.7
3	1	700	0.005	200	1.9	23.3
4	2	300	0.01	150	1.1	100.0
5	2	500	0.01	100	1.8	166.7
6	2	700	0.01	100	2.6	233.3
7	3	300	0.01	200	2.2	200.0
8	3	500	0.01	200	3.7	333.3
9	3	700	0.01	200	5.1	466.7
10	4	300	0.02	100	3.0	400.0
11	4	500	0.02	100	5.0	666.7
12	4	700	0.02	100	7.0	933.3

Table 3. Table 3 : Predicted thermodynamic \ce CH4 volume mixing ratios †

$T$(K)	\ceCH4($\%$)	\ceH2($\%$)	\ceC2H2($\%$)	\ceC2H4($\%$)
Case 1: pure (100$\%$) 0.8 Torr of \ceCH4 at 300 K ‡
300	100	0.0	0.0	0.0
500	99.9	0.01	0.0	0.0
700	98.6	0.9	0.0	0.5
900	81.2	13.1	1.7	4.0
Case 2: 0.8 (7.4$\%$) \ceCH4 in 10.0 Torr (92.6$\%$) \ceH2 at 300 K ††
300	7.4	92.6	0.0	0.0
500	7.4	92.6	0.0	0.0
700	7.4	92.6	0.0	0.0
900	7.4	92.6	0.0	0.0

Table 4. Table 4 : J 𝐽 J -dependent Lorentzian coefficients averaged over ν 3 subscript 𝜈 3 \nu_{3} P 𝑃 P branch line cluster of \ce CH4.

$J^{\prime }$	$J^{\prime \prime }$	${\gamma }_{\text{L}}$	${\gamma }_L^{min}$–${\gamma }_L^{max}{}^{\star }$	$n_{\gamma }$	$n_{\gamma }^{min}$–$n_{\gamma }^{max}{}^{\star }$	${\delta }_{\text{L}}$†	$n_{\delta }$
1	2	0.069	0.068–0.069	0.65	0.59–0.66
2	3	0.067	0.066–0.068	0.56	0.55–0.57	$-$0.0040(5)	1.4(5)
3	4	0.065	0.061–0.068	0.56	0.51–0.60	$-$0.0035(4)	1.2(3)
4	5	0.066	0.063–0.067	0.60	0.54–0.68	$-$0.0032(4)	1.0(3)
5	6	0.066	0.063–0.068	0.58	0.55–0.63	$-$0.0041(4)	1.1(3)
6	7	0.064	0.054–0.067	0.59	0.51–0.65	$-$0.0038(7)	1.5(6)
7	8	0.063	0.057–0.061	0.55	0.54–0.57	$-$0.0031(5)	1.2(4)
8	9	0.063	0.062–0.063	0.55	0.53–0.56	$-$0.0035(5)	1.5(5)
9	10	0.059	0.045–0.062	0.52	0.44–0.54	$-$0.0030(8)	1.6(9)
10	11	0.058	0.049–0.060	0.48	0.42–0.52	$-$0.0027(5)	1.6(6)
11	12	0.057	0.054–0.058	0.46	0.41–0.47	$-$0.0032(8)	1.9(9)
12	13	0.053	0.035–0.057	0.42	0.25–0.48
13	14	0.051	0.037–0.054	0.36	0.20–0.54
14	15	0.046	0.041–0.051	0.24	0.14–0.41
15	16	0.043‡		0.24‡
16	17	0.041	0.038–0.049	0.20	0.14–0.65

Table 5. Table 5 : Overview of previous measured temperature-dependence coefficients, n γ subscript 𝑛 𝛾 n_{\text{$\gamma$}} , of \ce CH4 in various broadeners

Broadener	T[K]	Band	Lines	$n_{\gamma }$†	Ref.
\ceH2	300   – 700	${\nu }_3$	116	0.2 – 0.65	PS‡
	77   – 295	6${\nu }_1$,5${\nu }_3$	2	0.45, 0.53	[64]
	130 – 295	${\nu }_4$	6	0.46 – 0.51	[65, 66]
	161 – 295	${\nu }_4$	6	0.35 – 0.52	[67]
Air	200 – 300	${\nu }_3$	3	0.62 – 1.0	[46]
	211 – 314	${\nu }_4$	148	0.50 – 0.80	[47]
	212 – 297	${\nu }_1+{\nu }_4$	130	0.50 – 0.85	[68]
	212 – 297	${\nu }_3+{\nu }_4$	406	0.50 – 0.90	[68]
	212 – 297	${\nu }_2+{\nu }_3$	71	0.40 – 0.85	[68]
\ceN2	215 – 297	${\nu }_3$	3	0.94 – 0.97	[43]
	215 – 297	${\nu }_2+{\nu }_4$	2	0.86,   0.92	[43]
	77   – 295	6${\nu }_1$,5${\nu }_3$	2	0.77,   0.97	[64]
	130 – 295	${\nu }_4$	6	0.75 – 0.83	[65, 66]
	161 – 295	${\nu }_4$	6	0.71 – 0.82	[67]
	211 – 314	${\nu }_4$	148	0.55 – 0.85	[47]
	90   – 296	${\nu }_3$	4	0.84 – 0.86	[69]
Self	77   – 295	6${\nu }_1$,5${\nu }_3$	2	0.84, 0.93	[64]
\ceHe	77 – 295	6${\nu }_1$,5${\nu }_3$	2	0.37, 0.67	[64]
	130 – 295	${\nu }_4$	6	0.28 – 0.38	[65, 66]
	161 – 295	${\nu }_4$	6	0.26 – 0.38	[67]
\ceAr	130 – 295	${\nu }_4$	2	0.80 – 0.83	[65, 66]
	161 – 295	${\nu }_4$	6	0.72 – 0.82	[67]

Table 6. Table 6 : Fitted constants for global equations †

Case	$m_1$	$m_2$	$m_3$
${\gamma }_L$ ($J^{\prime \prime }$)	0.066	0.0008	$-$0.00014
$n_{\gamma }$ ($J^{\prime \prime }$)	0.520	0.0290	$-$0.00290
${\gamma }_L$ ($J^{\prime \prime }$, $F_1/F_2$)	0.0657	0.0012	$-$0.00017
${\gamma }_L$ ($J^{\prime \prime }$, $A_1/A_2$)	0.0650	0.0007	$-$0.00011
${\gamma }_L$ ($J^{\prime \prime }$, $E$)	0.0690	$-$0.0010	$-$0.00010

Table 7. Table 7 : Broadening coefficients ⋆ averaged over N 𝑁 {\it N} .

$J^{\prime }$	$J^{\prime \prime }$	Sym	${\gamma }_L$ [cm-1/atm]	${\gamma }_L^{min}$–${\gamma }_L^{max}$	$n_{\gamma }$	$n_{\gamma }^{min}$–$n_{\gamma }^{max}$
1	2	$E$	0.069(1)		0.59(4)
1	2	$F$	0.068(1)		0.66(3)
2	3	$A$	0.066(1)		0.55(3)
2	3	$F$	0.067(1)		0.57(2)
3	4	$A$	0.064(2)		0.51(3)
3	4	$E$	0.061(2)		0.56(4)
3	4	$F$	0.068	0.068–0.069	0.60	0.60–0.62
4	5	$E$	0.063	0.063–0.073	0.60	0.60–0.95
4	5	$F$	0.067	0.065–0.072	0.60	0.54–0.68
5	6	$A$	0.065	0.063–0.067	0.55	0.53–0.58
5	6	$E$	0.063(1)		0.55(3)
5	6	$F$	0.068	0.063–0.070	0.63	0.51–0.66
6	7	$A$	0.067	0.067–0.069	0.65	0.59–0.65
6	7	$E$	0.054(1)		0.51(2)
6	7	$F$	0.066	0.064–0.068	0.60	0.57–0.62
7	8	$A$	0.061(1)		0.56(2)
7	8	$E$	0.057	0.056–0.059	0.54	0.52–0.53
7	8	$F$	0.065	0.063–0.067	0.57	0.55–0.60
8	9	$A$	0.062	0.061–0.079	0.53	0.52–0.88
8	9	$E$	0.063	0.058–0.063	0.56	0.56–0.67
8	9	$F$	0.062	0.061–0.064	0.56	0.51–0.60
9	10	$A$	0.061	0.053–0.062	0.53	0.46–0.57
9	10	$E$	0.045	0.045–0.062	0.44	0.07–0.45
9	10	$F$	0.062	0.061–0.063	0.54	0.52–0.56
10	11	$A$	0.060(2)		0.50(3)
10	11	$E$	0.049	0.045–0.057	0.42	0.41–0.43
10	11	$F$	0.060	0.055–0.059	0.50	0.32–0.59
11	12	$A$	0.058	0.052–0.059	0.47	0.46–0.50
11	12	$E$	0.054	0.049–0.054	0.41	0.40–0.62
11	12	$F$	0.056	0.055–0.058	0.45	0.41–0.77
12	13	$A$	0.057(1)		0.46(3)
12	13	$E$	0.035	0.030–0.064	0.25	0.16–0.48
12	13	$F$	0.056	0.055–0.062	0.48	0.36–0.60
13	14	$A$	0.053(9)		0.54(18)
13	14	$E$	0.037	0.031–0.048	0.19	0.19–0.29
13	14	$F$	0.054	0.052–0.063	0.35	0.15–0.57
14	15	$A$	0.051	0.049–0.055	0.35	0.22–0.50
14	15	$E$	0.048	0.043–0.052	0.41	0.31–0.40
14	15	$F$	0.041	0.027–0.059	0.14	0.15–0.53
15	16	$A$	0.048†	0.041–0.068	0.24†	0.24–0.59
15	16	$E$	0.027‡		0.24‡
15	16	$F$	0.041	0.041–0.053	0.19	0.24–0.66
16	17	$A$	0.049	0.045–0.065	0.24	0.18–0.59
16	17	$E$	0.039	0.023–0.034	0.65	0.18–0.73
16	17	$F$	0.038	0.037–0.051	0.14	0.18–0.24

Equations

$$f_{\text{G}}(\nu -{\nu }_{ij},{\Gamma }_{\text{D}})=\sqrt{\frac{\ln (2)}{\pi {\Gamma }_{\text{D}}^2}}\exp \left(-\frac{\ln (2)(\nu -{\nu }_{ij}{)}^2}{{\Gamma }_{\text{D}}^2}\right)$$

$${\Gamma }_{\text{D}}(T)=\frac{{\nu }_{ij}}{c}\sqrt{\frac{2\ln (2)N_{\text{A}}{k}_{\text{B}}T}{M}}$$

$$f_{\text{L}}(\nu ;{\nu }_{ij},{\Gamma }_{\text{L}},{\Delta }_{\text{L}})=\frac{1}{\pi }\frac{{\Gamma }_{\text{L}}(p,T)}{[{\Gamma }_{\text{L}}(p,T){]}^2+[\nu -({\nu }_{ij}+{\Delta }_{\text{L}}){]}^2}$$

$${\Gamma }_{\mathrm{L}}={\left(\frac{T}{T_0}\right)}^{-n_{\text{γ}}}{\gamma }_{\text{L}}{P}_{\text{ce}H2}$$

$${\Delta }_{\mathrm{L}}={\left(\frac{T}{T_0}\right)}^{-n_{\text{δ}}}{\delta }_{\text{L}}{P}_{\text{ce}H2}$$

$${\Gamma }_{\mathrm{L}}=\frac{n{\bar{\nu }}_{\mathrm{th}}{\sigma }_r}{2\pi }$$

$${\Gamma }_{\mathrm{L}}=\sqrt{\frac{2}{k_{\mathrm{B}}m{\pi }^3}}{P}_{\text{ce}H2}{T}^{-0.5}{\sigma }_r$$

$$X(J^{\prime \prime })=m_1+m_2{J}^{\prime \prime }+m_3{J^{\prime \prime }}^2$$

$$X(J^{\prime \prime },\mathrm{sym})=m_1+m_2{J}^{\prime \prime }+m_3{J^{\prime \prime }}^2$$

Full text

††journal: Journal of Quantitative Spectroscopy and Radiative Transfer

1 Introduction

Methane (\ceCH4) has been observed in the infrared spectra of different solar-system atmospheres including those of terrestrial planets (e.g., on the surface of Mars [1, 2]), Jovian planets (e.g., Jupiter, Saturn, Uranus [3, 4]), and Titan [5, 6]. The abundance of \ceCH4 is also important in constraint understanding the C/O ratio in the atmospheres of brown dwarfs and exoplanets, as well as understanding their formation history [7, 8]. Because the thermochemically dominant carbon-bearing molecule at T$>$1200 K is CO and at T$<$800 K is \ceCH4 [6], their mixing ratios with \ceCO2 are used as a temperature probe and to determine super-Earths/sub-Neptune metallicities [9, 10]. Moreover, \ceCH4 near-infrared (NIR) spectra are an important tool for classifying brown dwarf types (e.g., T-dwarfs [11]). Despite extensive endeavors to model the chemical composition of exoplanetary atmospheres by means of radiative transfer modeling (i.e., transmission and emission exoplanetary spectra [12, 13, 14]), a proposed detection of \ceCH4 is still under debate [15]. Additionally, high-resolution Earth-based searches of methane through the cross-correlation technique have been unsuccessful [16]. However, it has been argued thermochemically that \ceCH4 is one of the main absorbers in super-Earth to sub-Neptune atmospheres [17]. \ceH2 is the major broadening molecule (or perturber) in these exoplanetary atmospheres, and therefore, the accuracy of radiative transfer modeling, particularly for the cross-correlation technique (see section 3.5 in [18]), relies strongly upon the accuracy and completeness of \ceCH4 spectroscopic data including rovibrational transitions and pressure-broadening coefficients appropriate for high-temperature and \ceH2-dominated atmospheres [19, 20]. Accurate quantification of pressure-broadening coefficients at room- and high-temperature is fundamental because they influence the absorption cross-section data and, therefore, the modeled exoplanet atmospheric spectra [20, 21].

Methane is a tetrahedral molecule with five symmetry species: $A_{1}$, $A_{2}$, $E$, $F_{1}$, and $F_{2}$. The $\nu_{3}$ fundamental band arises from asymmetric C–H stretching (see chapter 7 in [22]). Given the relevance of \ceCH4 infrared (IR) absorption and emission in the study of brown dwarfs and planetary/exoplanetary atmospheres, many experimental and theoretical studies recorded or computed the relevant rovibrational transitions. High-resolution IR spectra of \ceCH4 have been recorded at both room [23, 24] and high temperatures [25, 26, 27, 28, 29, 30]. Additionally, several ab-initio studies have computed the \ceCH4 rovibrational transitions [25, 31, 32, 33, 34].

Since the 1980s, several laboratory measurements of the pressure-broadening of \ceCH4 by various broadeners (hereafter referred to as absorber@[broadener], e.g., \ceCH4@[\ceH2]) at room temperature have been made. Non-Voigt pressure-broadening coefficients of the \ceCH4@[\ceH2, \ceN2, \ceAr, or \ceHe] $\nu_{3}$ band $Q$ branch were analyzed using a laser spectrometer at high resolution [35, 36] and showed a strong dependency of linewidths on broadener and total angular-momentum quantum number, $J$. In addition, the measured linewidths are dependent on the tetrahedral symmetry species (i.e., $A_{1},A_{2},E,F_{1},F_{2}$). The $R$-branch of the 3$\nu_{3}$ overtone ($\sim\!9000$ cm*-1*) of \ceCH4@[\ceH2] was measured up to $J_{\rm{lower}}=J^{\prime\prime}$=6 by Fourier transform spectroscopy with 0.01 cm*-1* spectral resolution [37, 38].

Several studies have used quantum or semi-classical approaches to calculate, predict, and explain pressure broadening of \ceCH4 in different broadeners (or perturbers)[39]. Anderson theory, for instance, utilizes a perturbation approach to compute the line broadening and their temperature-dependence coefficients through electrostatic interactions [40, 41, 42]. However, it was shown later that electrostatic forces are not able to explain the broadening for some perturbers such as \ceO2 and \ceN2 [43]. In comparison, Robert-Bonamy theory [44] was used to show the atom-atom potential energy is the main cause of collisional broadening for these species [45].

To the best of our knowledge, there are no measurements of \ceCH4@[\ceH2] (or any other broadeners) at T$>$ 315 K. Measurements for temperatures between 200 and 300 K show the temperature-dependence coefficient (n_{\text{\gamma}}, see section 4.2) of the $\nu_{3}$ band of \ceCH4@[Air] and \ceCH4@[\ceN2] is 0.6 – 1.0 [46] and 0.94–0.97 [43], respectively. For the $\nu_{4}$ band of \ceCH4@[Air] and \ceCH4@[\ceN2], n{}_{\text{\gamma}} is 0.5 – 0.8 [47]. A complete list of literature regarding measurement of temperature-dependence coefficients is reported in Table 5.

For this study, we used Fourier-transform infrared spectroscopy (FTIR) to record rovibrational lines of \ceCH4@[\ceH2] in the $P$ branch of the $\nu_{3}$ band over the temperature range 300–700 K (Sec. 2). Then, using a least-squares fitting analysis, the Lorentzian linewidth ($\gamma_{\text{L}}$) and temperature-dependence coefficients (n_{\text{\gamma}}) are determined for $J^{\prime\prime}$= 2 – 17 (Sec. 3). The dependency of the Lorentzian coefficients on total quantum number $J^{\prime\prime}$ and the tetrahedral symmetry species is discussed in Sec. 4.

2 Experimental details

2.1 Instrumental setup

All spectra in this study were recorded with a Bruker 125HR infrared Fourier-transform spectrometer located at the Advanced Light Source (ALS) of the Lawrence Berkeley National Laboratory (LBNL). As shown in Fig. 1, the evacuated sample chamber in this model of spectrometer is located between the beam splitter and detector. In this case, the thermal IR emission from the heated sample gas cells does not contribute to the recorded interferogram, and no post-analysis correction for the cell emission is required in comparison with other studies in which the heated cell was placed at the entrance to the spectrometer (e.g., Ref. [26]). For measurements at high temperature, we designed a sealed monolithic gas cell. Due to its high transmittance over a spectral range of 2750–3250 cm*-1* and high melting point, the whole gas cell and spectral windows are fabricated from fused quartz.

2.2 Recorded Spectra

We recorded spectra for temperatures 300, 500 and 700 K and over a pressure range of $\sim$0.8 – 7.0 Torr for \ceCH4 gas and $\sim$10 – 933 Torr (0.013–1.2 atm) of \ceH2 broadening gas. In total, four quartz gas cells with a path length of 10$\pm$0.2 cm were used in these measurements. Different amounts of \ceCH4 and \ceH2 gases were inserted in each tube at room temperature and then the port was sealed. After sealing the gas cell at room temperature with known number densities of \ceCH4 and \ceH2, FTIR transmittance spectra of each tube were recorded at three different temperatures: 300, 500, 700 K. Table 1 reports the experimental conditions in detail.

Overall, 12 different spectra of \ceCH4 were recorded at various resolutions. Figure 2 represents an overview of spectrum $\#$4 (i.e., P${}_{\ce{CH4}}$=1.1 Torr and P${}_{\ce{H2}}$=100.0 Torr), which encompasses $P$, $Q$, and $R$ branches up to $J^{\prime\prime}$=17. In addition, each $J^{\prime\prime}$ consists of a cluster of transitions with various symmetry species and ${\it N}$ quantum index111 The \ceCH4 energy levels are labelled by different quantum numbers such as $J$ and $C$ (tetrahedral symmetry), and ${\it N}$ (quantum index) defined in Brown et al. [48].. Figure 3 illustrates the modeled spectra for $P$(7) transitions. The elevated temperature gas pressures, P${}_{\ce{CH4}}$ and P${}_{\ce{H2}}$, were then calculated using the ideal gas law. Table 2 lists the resolution, number of scans, and the P${}_{\ce{CH4}}$ and P${}_{\ce{H2}}$ values for all measurements. Spectrum $\#$1 was used to measure the unbroadened Doppler-width and intensity of each line.

The decomposition of \ceCH4 is an important issue for high-temperature measurements [49]. To decrease the potential for loss of \ceCH4, we added 10 Torr of \ceH2 into the first gas cell at room temperature. The main product of \ceCH4 + CH4 bimolecular dissociation in the absence of \ceH2 is \ceCH3, but in the presence of \ceH2 gas as a third-body component, \ceCH4 will reform. In other words, \ceH2 gas will decrease the amount of decomposition by increasing the back reaction. Additionally, 10 Torr of \ceH2 has a negligible pressure-broadening effect. The volume mixing ratios of these gases can be calculated through minimizing Gibbs free energy which is dependent on the temperature, pressure, and gas concentrations. Therefore, we used the online thermodynamical simulator222http://navier.engr.colostate.edu/code/code-4/index.html to calculate the fraction of decomposition of pure \ceCH4 at different temperatures and pressures. Table 3 (case 1) represents the thermodynamic mixing ratios of 0.8 Torr of pure \ceCH4. Note that these calculations are done up to 900 K while the maximum laboratory temperature in this work is 700 K. Thermal decomposition of pure \ceCH4 is predicted to occur for temperatures 700 K and above but is suppressed by the mixture of a small amount of \ceH2. Ultimately, no significant decrease of the \ceCH4 column density was noted even at 700 K. In this study, the line assignments and the line positions of \ceCH4 were adopted from HITRAN2016 [50, 51].

The \ceCH4 and \ceH2 gases were 99.99$\%$ and obtained from Matheson. The gas pressure while filling the sample tubes was measured using two different MKS Baratron pressure gauges (maximum range 100 and 1000 Torr). For controlling the temperature, heat tapes from BriskHeat company (Type BW0) were used. A thermocouple connected to each gas cell was used in a feedback loop with the heat-tape controller to maintain a constant temperature. Omega company states typical uncertainties as 0.1$\%$ of the displayed reading for their digital readers. The uncertainty for type K probes is estimated to be 0.75$\%$ (2.2 K at 300 K). Therefore, the overall uncertainty is due to the probe, not the reader, and T is good to within $\pm$2 K at the location of the junction. There is a possibility of temperature nonuniformity in our gas cell. We expect this effect to be small given the high heating-element coverage of the cell excluding the transmitting windows but including its support structure, the small size of the cell, and its vacuum environment. The uncertainty in the measurement of $P_{\ce{H2}}$ and $P_{\ce{CH4}}$ is less than 0.5$\%$, and is also negligible.

3 Data Analysis

Our main goal is to extract pressure-induced broadening coefficients by modeling all lines with Voigt line profiles. Lorentzian and temperature-dependence coefficients for each rovibrational line are determined from linewidths extracted from spectra $\#$1–12 using a least-squares fitting method. The signal-to-noise ratio (S/N) is insufficiently high to justify modeling the spectra with non-Voigt profiles.

The negligibly pressure-broadened sample tube $\#$ 1 was analyzed first to determine the correct \ceCH4 line assignments, wavenumber calibration, and the presence of other \ceCH4 bands and other contaminant species. Line strengths were determined separately at each measured temperature. The highly-blended pressure-broadened spectra were analyzed with line strengths fixed to their unbroadened values and line widths and positions freely modified.

3.1 Continuum / baseline fitting

All \ceCH4 spectra were converted from their interferograms with a Boxcar apodization using the OPUS software333www.bruker.com. The effect of instrumental broadening was modeled using a custom fitting code as a sinc function [52]. The background continuum is also modeled using cubic splines optimised during the least-squares fitting procedure [53]. Additionally, interference between the two cell windows that affects the recorded spectra by introducing sinusoidal behavior into the spectral continuum. We modeled this interference effect by employing two sine functions scaling the modeled spectrum.

3.2 Line position corrections

Line assignments are determined from the recent version of HITRAN [51, 50]. All corresponding line positions from HITRAN were input into the fitting code, and a global fit was made to calculate a single global shift induced by any slight miscalibration of the spectrometer. Afterward, the calculated shift was applied to our low pressure spectra (i.e., spectra $\#$1–3 in Table 2). Later, the corrected/shifted line positions from the low pressure spectra were used to fit high pressure spectra (i.e., spectra $\#$4–12 in Table 2), where pressure-induced lineshifts were also evident.

3.3 Line profiles

At very low pressure, the effect of collisions on molecular spectra is negligible. However, molecular velocities are distributed according to the Maxwell–Boltzmann statistics resulting in Doppler broadening (see chapter 1 at [54]). The Doppler half-width at half-maximum (HWHM) linewidth (${\it\Gamma}_{\text{D}}$) were individually modeled using Gaussian line profile $f_{\text{G}}$ :

[TABLE]

[TABLE]

where $M$ is the molar mass of the absorber molecule in grams, $N_{\text{A}}$ is the Avogadro constant, $k_{\text{B}}$ is the Boltzmann constant, and $\nu_{ij}$ is the line position or the energy gap between quantum levels $i$ and $j$ in any arbitrary energy unit (e.g., cm*-1*). ${\it\Gamma}_{\text{D}}$ values for our various measurements are in the range 0.004 – 0.007 cm*-1* given the dependence of ${\it\Gamma}_{\text{D}}$ on the temperature and wavenumber. The natural radiative linewidth of the \ceCH4 $\nu_{3}$ band is $\sim$ 10*-9* cm*-1* (i.e., in the range of 10–100 Hz) [55], which is fully negligible when fitting the spectra.

Since the intensity of each line is distributed as a result of pressure-broadening, we increased the column density of \ceCH4 when a high \ceH2 pressure is present in order to obtain optimal S/N ratios without saturating any lines. As a result of this change, the modeled \ceCH4 optical depth of high-pressure spectra (i.e., spectrum $\#$ 4–12) were scaled up uniformly.

The Lorentzian HWHM linewidth ${\it\Gamma}_{\text{L}}$ and lineshift ${\it\Gamma}_{\text{L}}$ were fitted individually for each line using the Lorentzian line profile $f_{\text{L}}$:

[TABLE]

[TABLE]

[TABLE]

in which $\gamma_{\text{L}}$ (cm*-1*/atm) and nT are the Lorentzian linewidth coefficient and its temperature-dependence coefficient, respectively. $\delta_{\text{L}}$ (cm*-1*/atm) and n_{\text{\delta}} are the Lorentzian lineshift coefficient and its temperature-dependence coefficient, respectively. $T_{0}$ is a reference temperature, and it is set equal to 300 K. Note, all these coefficients are dependent on the total rotational quantum number of $J$, tetrahedral (T${}_{\text{d}}$) symmetry species, and the broadeners. The code computes the Voigt profile as the Faddeeva function.444http://ab-initio.mit.edu/wiki/index.php/Faddeeva_Package

The Lorentzian coefficients ${\it\Gamma}_{\text{L}}$, extracted from the recorded spectra result from the effect of $P_{\rm{\ce{H2}}}$ collisional-induced broadening. The pressure-broadening from \ceCH4 self broadening is negligible since $P_{\rm{\ce{CH4}}}$ $\leqslant$ 1.1$\%\times P_{\rm{\ce{H2}}}$. Regarding Dicke narrowing, this effect becomes important at intermediate pressures or the Doppler–Lorentzian transition region because Doppler broadening at low pressures and Lorentzian broadening at high pressures mask the narrowing. For example, Pine [35] found the largest discrepancy between Voigt and Rautian at 50 Torr \ceH2, and a corresponding 5$\%$ difference in the derived $\gamma_{\text{L}}$ for the two cases. This difference will be reduced by about half at 100 Torr (the lowest pressure we use). Then our Lorentzian linewidths fitted at 100 Torr may be underestimated by up to 3$\%$ (in comparison with random fitting uncertainties of at 4$\%$ or more).

Other formulations for the temperature-dependence of Eq. 5 have been adopted elsewhere [47]. We use the most conventional single-parameter temperature dependence formula above given the limited temperature sampling of our data.

4 Results and Discussion

4.1 Pressure broadening coefficients: ${\gamma}_{\rm L}$ and ${n}_{\gamma}$

After fitting all 12 spectra from 300 to 700 K, the Lorentzian HWHM (i.e., ${\it\Gamma}_{\text{L}}$ in Eq. 4) is extracted for each tetrahedral rovibrational transition555 Each tetrahedral transition is labelled by total rotational quantum number $J$, symmetry species $C$, and quantum index $N$ [48].. Then, the $\gamma_{\text{L}}$ and n_{\text{\gamma}} coefficients are computed in three different ways: 1) for all lines individually including its own $J$, symmetry and $N$ numbers, 2) averaged over lines with the same $J^{\prime\prime}$ but different symmetry and $N$ index (i.e., the multiplicity index), and 3) all lines with the same $J^{\prime\prime}$ and symmetry but different $N$ index were fitted. As a sample fitting, Fig. 4 illustrates ${\it\Gamma}_{\text{L}}$ versus (T/300 K){}^{-n_{\text{\gamma}}}$$P_{\ce{H2}} for $J^{\prime\prime}$=7 and different symmetry species (n_{\text{\gamma}} is computed below). Figure 4 shows the fitted slope (i.e., $\gamma_{\text{L}}$) of transitions with $A_{1}$/$A_{2}$ and $F_{1}$/$F_{2}$ is higher than for the $E$ symmetry lines.

Figure 5(I–III) illustrates the trend of $\gamma_{\text{L}}$ and n_{\text{\gamma}} with $J^{\prime\prime}$. Figure 5(I) represents $\gamma_{\text{L}}$ and n_{\text{\gamma}} fitted to all lines individually. At each $J^{\prime\prime}$ value, there is the scatter of both $\gamma_{\text{L}}$ and n_{\text{\gamma}} coefficients which arise from the difference between T${}_{\text{d}}$ symmetries, $N$ indexes, and random fitting errors. In the first analysis step, individual lines with the same $J,symmetry,N$ from all spectra were fitted to extract the $\gamma_{\text{L}}$ and n_{\text{\gamma}} coefficients data. From this we determine the Lorentzian linewidth of each individual line as a result of \ceH2 collisional impact. Figures 4 and 5(I) as well as the supplementary Table (S1) represent these results. The error bars shown in these figures and the table uncertainties are due to the fitting uncertainties, noise, and the low signal-to-noise of some lines. These line-by-line coefficients are the main outcome of this study and they can be utilized in generating absorption cross-section (or opacity) data the standard HITRAN code666i.e. HITRAN Application Programming Interface (HAPI) [56], https://github.com/hitranonline/hapi or the NASA Ames Freedman’s code [19, 57].

In contrast, if we average the coefficients for all lines with the same $J^{\prime\prime}$ value, then $\gamma_{\text{L}}$ and n_{\text{\gamma}} coefficients fall in the range of 0.07–0.03 and 0.65–0.25, respectively (see Fig. 5(II) and Table 4). In Table 4, the scatter of these coefficients are mostly due to the scattering of lines with the same $J^{\prime\prime}$ but different symmetries and ${\it N}$ dependencies, as well as, the uncertainty in fitting the Lorentzian linewidths from the recorded spectra. Another motivation for this step is to provide data for opacity codes which input only $J$-dependent pressure-broadening values such as the current version of EXOCROSS code777https://github.com/Trovemaster/exocross[[58](#bib.bib58)]. Figure 5(II) shows that there is a clear dependency of the Lorentzian coefficient and its temperature-dependence with ($\gamma_{\text{L}}$ and n_{\text{\gamma}}) on $J$. This data are also presented in Table 4, and the range of scatter for each one is shown as a range of $\gamma_{L}^{min}$–$\gamma_{L}^{max}$ and n_{\text{\gamma}}^{min}–n_{\text{\gamma}}^{max}. According to the Anderson collisional theory [40], the n_{\text{\gamma}} coefficient is expected to be 0.5; however, our analysis shows that n_{\text{\gamma}} coefficients deviate from this value by up to 30$\%$. We also find that $\gamma_{\text{L}}$ and n_{\text{\gamma}} decrease by 25$\%$ and 80$\%$ , respectively, between $J^{\prime\prime}$=2 and 17 in agreement with the trend calculated by Neshyba et al. [45] and Gabard [44].

Next, we grouped the lines with similar symmetries, and extracted the Lorentzian coefficients from each group. Figure 5(III) shows the symmetry-dependence of $\gamma_{\text{L}}$ and n_{\text{\gamma}}. The bars shown in this figure are due to the uncertainty in fitting this data (similar to Fig. 5(III)) and also the scatter imposed by different values of the $N$ quantum index. It should be noted that only some symmetry-$J$ combinations have multiple $N$ values. Therefore, two kinds of uncertainties are shown in Table 7: statistical fitting uncertainties for singular-$N$ values, and the range of scatter for values averaged over mutiple $N$ transitions. In general, within each $J$ manifold, $E$-lines are the weakest and also have the narrowest Lorentzian linewidth ${\it\Gamma}_{\text{L}}$. In contrast, lines with ${A_{1}/A_{2}}$ and ${F_{1}/F_{2}}$ symmetries are generally the strongest, and have the broadest linewidth.

Following the complex Robert-Bonamy theory[59], Neshyba et al. [45] calculated the impact of electrostatic and atom-atom intermolecular potential on the line broadening and line shift of the \ceCH4@[\ceN2] system. They found that the atom–atom potential component is the main reason for the line broadening with a corresponding decrease with increasing total angular momentum, $J$. In addition, the broadening effect is symmetry dependent and it was shown [60, 47, 44] the total collisional cross-section for $E$ symmetry is lower than for $A_{1}/A_{2}$ and $F_{1}/F_{2}$ at low $J$, which results in smaller perturbation and collisional-broadening for the $E$-symmetry species, as we observed.).

4.2 Lorentzian temperature-dependence coefficient: n_{\text{\gamma}}

According to early Anderson collisional theory [40, 61] a broadened line has a Lorentzian profile (Eq. 3), and the broadening linewidth is proportional to T*-0.5* following Eqs. 6 and 7:

[TABLE]

where $n$ is the broadener column density (i.e., $n$=$n_{\ce{H2}}$= $P_{\ce{H2}}/k_{b}T$), $\bar{\nu}_{\rm th}$ is the mean thermal velocity from Maxwell-Boltzmann distribution (i.e., $\bar{\nu}_{\rm th}=\sqrt{8k_{b}T/\pi m}$ where $m$ is the \ceH2 mass), and $\sigma_{r}$ is the real component of the collisional cross-section (see discussion in [62]).

[TABLE]

Following Eqs. 6 $\&$ 7, the temperature-dependence coefficient, n_{\text{\gamma}} is 0.5. Note, there are different assumptions at play in Eq. 7 including the hard-sphere approximation, ideal gas law, and also a single thermal velocity $\bar{\nu}_{th}$ for all broadeners. Therefore, this n_{\text{\gamma}}=0.5 value should be considered as a gas kinetic value, and a more sophisticated picture is reviewed by Gamache and Vispoel [63]. Our results show that n_{\text{\gamma}} strongly depends on $J$, and it is in the range of $\sim$0.65–0.2 (see Fig. 5(II)). No significant dependence of n_{\text{\gamma}} on the tetrahedral symmetry species is found.

Table 5 lists most previous temperature-dependence measurements of \ceCH4 in different broadeners. In addition, the measurements are for different fundamental and combination vibrational modes providing insight into the vibrational dependency of n_{\text{\gamma}}. The n_{\text{\gamma}} of \ceCH4@[\ceN2] and \ceCH4@[Air] falls in the range of $\sim$0.55–1.0 and $\sim$0.4–0.9, respectively, which are roughly 30$\%$ larger than our results for \ceCH4@[\ceH2]. In comparison, \ceCH4@[\ceHe] is about half that of \ceCH4@[\ceH2]. Table 5 also illustrates the slight vibrational-dependency of n_{\text{\gamma}} for various broadeners.

4.3 Lorentzian line-shift coefficients: $\delta_{\text{L}}$ and n_{\text{\delta}}

The S/N ratio in the current study is insufficiently high to extract pressure shifts for all lines. Hence, a pressure-shift coefficient is calculated only for lines with $J^{\prime\prime}$=2–11, and falls in the range of $-0.004$ to $-0.002$ cm*-1*/atm. We discern no significant dependence of the Lorentzian pressure-shift coefficients on the $J$ values (Fig. 6), and the mean value of $\delta_{\text{L}}$ and n_{\text{\delta}} are $-$0.0035 cm*-1*/atm and 1.24, respectively. The reported uncertainties for Lorentzian pressure-shift coefficients ($\delta_{\text{L}}$ and n_{\text{\delta}}) are due to the scatter of the symmetry- and ${\it N}$-dependency. Note that our n_{\text{\delta}} is larger than n_{\text{\gamma}}, and this difference has been reported for water self-broadening as well [70].

The form of Eq. 5 is based on Eq. 4, which is derived from the ideal gas law and hard-sphere approximation. Some studies of other systems such as Frost [41] and Baldacchini et al. [71] have shown temperature-dependence has more complex form than our selected formula in Eq. 5. Additionally, Smith et al. [47] found both positive and negative $\delta_{\text{L}}$ and n_{\text{\delta}} values for the \ceCH4 $\nu 4$ band. However, we exclusively observed negative $\delta_{\text{L}}$ and positive n_{\text{\delta}} values.

4.4 Global equations for Lorentzian coefficients

In order to provide Lorentzian broadening coefficients ($\gamma_{\text{L}}$ and n_{\text{\gamma}}) appropriate for high-temperature \ceH2-dominanted exoatmospheres (i.e., super-Earth or warm-Neptunes with 400–900 K temperature), the dependency of these coefficients with $J^{\prime\prime}$ is presented by fitting the experimental results to a second-order polynomial $J^{\prime\prime}$-dependence (e.g., Eq. 8 see the red-dashed line in Fig. 5(II)). Additionally, due to the significant dependence of $\gamma_{\text{L}}$ on the symmetry species, the fitting coefficients are extracted from them separately (i.e., Eq. 9 dashed lines in Fig. 5(III)) conforming to:

[TABLE]

[TABLE]

where $m_{1}$, $m_{2}$, and $m_{3}$ are the fitted constants, $X$ is the Lorentzian coefficient i.e., $\gamma_{\text{L}}$, n_{\text{\gamma}}, and “sym” can be $A_{1}/A_{2}$, $F_{1}/F_{2}$, or $E$ symmetry species. All the polynomial fitted constants are presented in Table 6.

4.5 Comparison with existing data

Since \ceCH4 is an important molecule in the atmosphere of the Earth, other planets, and brown dwarfs, many experiments have been carried out for broadeners in the atmosphere of Earth (i.e., \ceN2 and \ceO2), Jupiter (i.e., \ceH2 and \ceHe), and other broadeners such as Ar- and self-broadening. In the following, we will discuss the comparison of our results with the most relevant literature data.

Figure 7 represents the comparison of our results with the literature data [72, 35, 65, 66, 73] for the \ceCH4@[\ceH2] $\nu_{3}$ band. Note that most of the previous studies have been for the ${\it Q}$ branch [72, 35] and employed Rautian line profiles; while there are a few measurements on the ${\it P}$ branch [65, 73], none employ \ceH2 as a broadener. Figure 7 (I,II) shows the comparison of all lines with their $J$-, symmetry- and ${\it N}$-dependencies. In Fig. 7 (II), the Pine [72] results are slightly lower than ours which might be due to the selection of different line profiles and branches. In Fig. 7, a comparison of lines within different symmetry classes is shown.

Figure 8 illustrates the comparison between our temperature-dependence coefficients n_{\text{\gamma}} (i.e., \ceCH4@[\ceH2] for $\nu_{3}$ $P$ branch) with both \ceCH4@[\ceN2] and \ceCH4@[Air] for $\nu_{4}$ band. Note, there are a two differences between these measurements: 1) our broadener \ceH2 is different from the previous works, 2) there might be some vibrational-dependency of n_{\text{\gamma}}. In general our n_{\text{\gamma}} coefficients (@[\ceH2]) is smaller than both @[Air]-broadening and @[\ceN2]-broadening 5.

Figure 9 represents the effect of various broadeners (i.e., self or \ceCH4, \ceN2, and \ceHe) on $\gamma_{L}$ for different $T_{d}$ symmetry species [72, 35, 65, 66, 73, 43]. In general $\gamma_{\text{L}}$(Self) $>$ $\gamma_{\text{L}}$(\ceH2) $\geqslant$ $\gamma_{\text{L}}$(\ceN2) $>$ $\gamma_{\text{L}}$(\ceHe). In earlier work [42], electrostatic forces (dipole, quadrupoles, and higher-order multipoles) were theorised to cause the differing broadening effects of various broadeners (or perturbers) on \ceCH4. However, the quadrupole moments of \ceO2 and \ceN2 could not explain their similar broadening of \ceCH4 (e.g., [43]), given that their quadrupole moments differ by a factor of 3. Later, Neshyba et al. [45] showed that in fact atom-atom interactions supplant electrostatic interactions is a minor reason, and atom–atom interaction is the major source of broadening using Robert-Bonamy theory [59] (see the theory section in Ref. [44]).

Figure 10 compares our Lorentzian pressure-shift results $\delta_{\text{L}}(\ce{CH4}@[\ce{H2}],P$ branch, see Table 4) with literature values for different broadeners and branches. The reported $\delta_{\text{L}}$ are averaged over of $J^{\prime\prime}$, and they are in the range of the Pine [35] $\nu_{3}$ $Q$-branch data. In addition, this figure shows that the collisional effect of \ceN2 and \ceAr species on pressure shift is larger than \ceH2. The largest $\delta_{\text{L}}$, however, would be due to the \ceCH4 self-broadening interactions, and it is $\sim$2$\times$ higher than our results ($\delta_{\text{L}}(\ce{CH4}@[\ce{H2}])$).

5 Summary $\&$ Conclusion

High-temperature Lorentzian broadening and shift coefficients of \ceCH4@[\ceH2] for more than 100 individual rovibrational transitions in the $\nu_{3}$ $P$ branch are obtained using high resolution (0.01–0.005 cm*-1*) FTIR spectroscopy. We find that $\gamma_{\text{L}}$ falls in the range 0.03–0.07 cm*-1*/atm, and is strongly dependent on molecular rotation and symmetry dependent. The temperature-dependence broadening coefficient, n_{\text{\gamma}} falls in the range 0.20–0.65. The averaged shift pressure and its temperature-dependence coefficient, $\delta_{\rm L}$ and n_{\text{\delta}} are $-$0.0035 cm*-1*/atm and 1.24, respectively, and these are constant with J as far as our data can determine.

All these coefficients were fitted to simple polynomial equations in terms of $J^{\prime\prime}$ and neglecting symmetry and $N$ quantum index for the benefit of the astrophysical/exoplanetary community. Table S1 lists the $\gamma_{\rm{L}}$ and n_{\text{\gamma}} for all individual lines, showing the change in these coefficients with $J$, symmetry, and ${\it N}$ numbers, and is recommended to use these data where these details are important. The detection of \ceCH4 spectral features in hot-Jupiters to super-Earths needs these pressure-broadening data because of their high-temperature and \ceH2-dominant atmospheres.

These pressure-broadening and pressure-shift coefficients can be directly incorporated into current databases, such as HITRAN/HITEMP or EXOMOL.

6 Acknowledgment

We kindly thank Glenn Stark, David Wright, Adam Schneider, and the Arizona State University exoplanet group for many useful discussions. E.G.N. especially thanks Mike Line for invaluable numerous invalvuable discussions during this work as well as Richard Freedman and Mark Marley invaluable discussions regarding the intricacies of opacity data. E.G.N. acknowledges funding from the GRSP research grant from the Arizona State University Graduate office program award XH51027. A.N.H.’s research was supported by an appointment to the NASA Postdoctoral Program at Arizona State University and the NASA Astrobiology Institute, administered by Universities Space Research Association under contract with NASA. This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility under contract no. DE-AC02-05CH11231.

7 Reference