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Molecular Physics: An International Journal at theInterface Between Chemistry and PhysicsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tmph20

Line mixing in the ν 3 and forbidden ν 2 bands of CH4

in gaseous heliumN. N. Filippov a , I. M. Grigoriev a , N. M. Grigorovich a & M. V. Tonkov aa St. Petersburg State University , St. Petersburg, RussiaPublished online: 28 Nov 2010.

To cite this article: N. N. Filippov , I. M. Grigoriev , N. M. Grigorovich & M. V. Tonkov (2006) Line mixing in the ν 3 andforbidden ν 2 bands of CH4 in gaseous helium, Molecular Physics: An International Journal at the Interface Between Chemistryand Physics, 104:16-17, 2711-2718

To link to this article: http://dx.doi.org/10.1080/00268970600857727

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Molecular Physics, Vol. 104, Nos. 16–17, 20 August–10 September 2006, 2711–2718

Line mixing in the m3 and forbidden m2 bands ofCH4 in gaseous helium

N. N. FILIPPOV*, I. M. GRIGORIEV, N. M. GRIGOROVICH and M. V. TONKOV

St. Petersburg State University, St. Petersburg, Russia

(Received 31 January 2006; in final form 17 June 2006)

The paper reports on studies of the possibility of the transfer of the adjustable parametersof the line mixing model describing the �3 absorption band shape of CH4 in helium to the case

of the forbidden �2 band shape. This transfer provides a reasonable overall agreement with themeasured spectra, with deviations greater than the experimental uncertainties remaining insome J-manifolds. The model has been improved by taking into account the mixing of spectral

lines with the same initial level. It was found that the description of the shapes of certainmanifolds can be improved by fitting corresponding matrix elements of the optical rotationalrelaxation matrix, providing, in some particular cases, more detailed information on the

matrix structure.

1. Introduction

Methane is an important component of the Earth’satmosphere and those of the outer planets of the solarsystem, and information on its spectra is currentlyrequired for several applications [1–3]. As demonstratedin [4, 5], the vibration–rotation band shapes ofmethane are significantly influenced by the line mixingeffect, which can lead to significant deviations of themeasured profiles from those calculated as a sum ofLorentz or Voigt overlapping line shapes. In this case,we need an optical rotation relaxation matrix tomake a correct calculation of the band shapes. Thismatrix can be constructed by means of the empiricalmethod proposed in [4], which uses an ab initiocalculated state-to-state rotational relaxationmatrix for the methane molecule in the ground vibra-tional state colliding with helium atoms. This approachwas successful when applied to CH4 vibrational transi-tions of type F2 A1, namely, to the methane bands �3and �4 [4, 5]. In this work, we examine the validity ofthis approach and the transfer of the empiricalparameters of the relaxation matrix determined for the�3 band to the �2 forbidden E A1 band shapecalculation using previously obtained experimentaldata [5, 6].

The spectra of the CH4–He mixtures considered inthis work were registered using the Fourier transformspectrometer Bruker HR 120 in the regions of thebands �3 and �2, with a spectral resolution rangingfrom 0.003 to 0.008 cm�1, depending on gas pressure.The gas mixtures were inserted into a White cellwith a 2m base and an optical path of either 24m(�3) or 48m (�2). The pressures varied over theinterval 0.25–1.0 atm; the spectrum in the �2 bandregion was also registered at 5 atm. The methaneconcentration in any mixture was less than 1%. Inthese experimental conditions, the manifold structureis not washed up, and the lines of different manifoldsdo not overlap. Since it was possible to vary themethane concentration, the spectral shapes wereregistered under optimal conditions providing anaccuracy of about 0.01 absorbance units. Themeasured profiles demonstrate considerable deviationsfrom the sum of isolated line shapes in someJ-manifolds.

2. Evidence of line mixing

The �2 band (E A1) is forbidden by the selection ruleson IR absorption, and it appears in the IR spectrum ofmethane due to the Coriolis interactions, mainlybetween the states forming the �2 and �4 bands. The �2band structure significantly differs from the other*Corresponding author. Email: nfilippov@yandex.ru

Molecular PhysicsISSN 0026–8976 print/ISSN 1362–3028 online � 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/00268970600857727

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fundamental bands of methane (F2 A1): the selectionrules for rotational transitions lead to a differentrotational structure of the J-manifolds. As a result ofthe Coriolis interactions, the structure of a J-manifoldis generally wider than, for example, in the case of the�3 band.We compared the J-manifold shapes of the �2 band

for mixtures with helium at pressures from 0.38 up to5 atm [6] with the profiles calculated in the isolated lineapproximation. At pressures greater than 0.1 atm, thisapproximation gives a sum of Lorentz line shapes.At lower pressures it is necessary to account for theDoppler effect using the Voigt profile for an individualline shape. The frequencies and intensities of the lineswere taken from the HITRAN database [7], and thebroadening coefficients were determined following theresults in [6].In the chosen experimental conditions, a greater part

of the manifold shapes in the �2 band can be adequatelydescribed by a sum of Lorentz profiles. Certaindeviations of the calculated shapes from the experi-mental ones can be accounted for by inaccuracies of lineparameters, primarily of line widths. Line mixing has asmaller effect on the �2 band shape as compared to the �3band since the Coriolis perturbation of the �2 band ismuch greater, which increases the widths of themanifolds and therefore decreases the overlap of theircomponents.At the pressures involved, substantial deviations

(410%) of the experimental shapes from thosecalculated as a sum of isolated line profiles in therotation-vibration spectra of the CH4–He mixture inthe region of �2 were mainly observed for the Pbranch manifolds with greater values of J¼ 12, . . . , 15.We attributed the observed deviations to the linemixing effect since the character of these deviations isquite similar to the previously studied one for the �3band. In figure 1, there are examples from the �3(upper plot) and �2 (lower plot) bands. In both cases,we observed excessive absorption at the experimentalprofile maxima, a corresponding decrease of absorp-tion in the wings of the group of interacting lines,and a good description by the isolated lines approx-imation out of the centre of the line cluster. Forcomparison, the absorption coefficients in figure 1and throughout the article are given in arbitrary unitsso that the integrated intensities of the measured andcalculated band shapes are equal for a givenmanifold. For the sake of the further analysis,symbols (squares and triangles) designating positionsand relative intensities of the lines of a givenmanifold are used in figure 1 and thereafter.

3. Band shape calculations

To account for line mixing, the band shapes werecalculated using a conventional expression

Kð!Þ ¼4�!

3�hcð1� e��h!=kTÞ

�ReXk,k0

Pkdkdk01

ið!�L0ÞþpW

� �kk0

ð1Þ

where k and k0 are transition numbers corresponding tothe transitions {i! f } and {i0 ! f 0}, Pk is the populationof the initial state i for the k-th transition, dk is thereduced matrix element of a dipole moment for the kthtransition, L0 is the Liouvillian of an unperturbedmolecule, p is the foreign gas pressure, and W is theoptical rotational relaxation matrix. The diagonalelements of the relaxation matrix W describe the linebroadening and shifting effects, and non-diagonalelements describe the line mixing effect.

Non-diagonal elements of the relaxation matrix W

can be related to the matrix K of state-to-state collisionrotation relaxation rates, as it was done for the caseof linear molecules [8]. As far as we know, nocalculations of this type have been made for sphericaltop molecules. To describe this relation, we appliedthe scheme proposed in [4]:

Wkk0 ¼ �Akk0Kii0

2�cð2Þ

where Akk0 are the empirical factors relating thesematrices and depending on transition types. Thematrix K was calculated by Gabard [5, 6] formethane–helium collisions.

In the case of Ji 6¼ Ji 0 (inter-manifold mixing,�J¼�J0) we take the uniform value from [6]:Akk0 ¼ 0.49. However, the exact value of this parameteris not significant since the overlap of lines from differentmanifolds is negligible for the experimental data inquestion. For the same reason, the inter-branch(�J 6¼�J0) mixing is neglected.

In the case of Ji¼ Ji 0 (intra-manifold mixing), thesecoefficients can be found by fitting the calculated shapesto the measured ones. For the manifolds of the �3 band,this was done in [5, 6]. According to the proposedscheme, for the case of intra-manifold mixing, the Akk0

coefficients are taken to be equal for all the transitions inthe manifold, even through their upper states aredifferent. However, as will readily be observed, thischoice of Akk0 does not permit a quantitative descriptionof the interactions of the lines k and k0 with a common

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initial level (i¼ i0). According to equation (2), we cannotdefine the value of the W12 matrix element (see figure 2);besides, the matrix elements W13 and W23 should alwaysbe equal, which is hardly correct.To correct this drawback of the model, we modified

the structure of the relaxation matrix W.

4. Model modification

In the case of the existence of two or more lines withi¼ i0 in a manifold, we propose additional rules based onthe well known sum rules for relaxation matrices [9, 10].Considering the interaction of the lines k and k0 with the

Figure 1. Evidences of line mixing in the �3 (upper plot) and �2 (lower plot) absorption bands.

Line mixing in the �3 and �2 bands of CH4 2713

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same initial level, the corresponding matrix element was

taken as equal to the product of the weaker line

multiplied and an empirical factor:

i ¼ i0 Wkk0 ¼ �a�weak

ffiffiffiffiffiffiffiffiffiffiffiffiSweak

Sstrong

sðlike W12 in figure 2Þ,

ð3Þ

where Sk ¼ Pkjdkj2. We have found that the value

a¼ 0.1 provides a good agreement of the calculationswith the experimental data.

For the interaction of lines with different initial levelsin the manifold under consideration, we take intoaccount the relative intensities Sk00 of all the lines k00 inthe manifold starting at the initial levels i00 which are thesame for one (i00 ¼ i) or for another (i00 ¼ i0) of thecoupled lines. Then

i 6¼ i0 Wkk0 ¼ �Akk0Kii0

2�c

SkSk0Pk00:i00¼i Sk00

� � Pk00:i00¼i0 Sk00

� � :ðlike W13 and W23 in figure 2Þ ð4Þ

We used the lower pressure data from [5] to fit thevalue of Akk0 for each manifold in the frame of theproposed modifications. The analysis was performed forthe �3 band shape of CH4 mixed with He. The Akk0

values obtained at J¼ 17, . . . , 22 practically coincidewith those found in [5].

Figure 3 shows the dependence of the empiricaldimensionless factors Akk0 on the manifold number m(m¼ Jþ 1 for R-branch and m¼�J for P-branch) alongwith a second order polynomial, the coefficients ofwhich have been determined by the least-squaresmethod:

Akk0 ¼ 0:24ð4Þ � 0:0003ð15Þmþ 0:0020ð2Þm2: ð5Þ

Figure 3. Dependence of the empirical dimensionless factors Akk0 as a function of the manifold number m and its polynomialapproximation.

Figure 2. Interaction scheme for transitions with common ordifferent initial levels.

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This approximation is more appropriate than the

linear one as it does not yield senseless negative values

for Akk0 and describes the given data by a single curve.

The dependence obtained is symmetrical with respect

to m since the inaccuracy of the linear coefficient is

much greater that the value itself: 0.0003� 0.0015 in

equation (5).The interpolated parameters Akk0 provide a good

agreement of the calculated and experimental shapes

of the J-manifolds in the �3 band. Figure 4 gives an

example for the R(17) manifold of the �3 band: the

deviations of about þ25% in the profiles calculated

as a sum of Lorentz lines from the experimental ones in

the absorption peak are almost entirely compensated

for by using the proposed model.

5. Comparison of the measured and calculated profiles

For the �2 band shape calculations, we use the factors

Akk0 found according to equation (5) to determine the

elements of the relaxation matrix W with corrections

given by equations (3) and (4). It is possible to use a part

of the relaxation matrix W only if the lines on a given

interval do not overlap with the other lines; this part

describes the coupling of the lines in question. This

approach substantially accelerates the matrix inversion

procedure (see equation (1)). The most considerable

deviations from the isolated line profiles are observed

for the P branch manifolds with greater values of

J numbers from P(12) to P(15). At the pressures

under study (0.3–5 atm), no similar deviations in the

P-manifolds at lower J as well as in the Q and R

branches are observed. We have agreement between the

experimental and calculated spectra of the forbidden �2band transferring the same Akk0 coefficients obtained

for the �3 band. However, the divergence of the

experimental shapes and the calculated shapes is

slightly greater than that for the �3 band.In figure 5 we can see that there are manifolds of the

P(14) type (see the lower plot) for which the line mixing

calculations with the transferred factors Akk0 provide a

satisfactory agreement with the experiment on the level

of the experimental accuracy. However, there are other

manifolds, like P(12) (see the upper plot), for which

there are some residual deviations, or, in other words,

the modified line mixing model with transferred factors

Akk0 does not provide a quantitative description of the

Figure 4. Quality of description of the measured absorption profiles by the modified model of line mixing: the R(17) manifold ofthe �3 band at 0.258 atm. The upper plot presents the experimental profile (open circles), the isolated line approximation (dottedline), the line mixing model calculation (solid line), and the positions and relative intensities of the lines (solid squares forA-symmetry, solid triangles for E-symmetry, and open triangles for F-symmetry). The lower plot presents relative deviations of theexperimental profile from the isolated line approximation (dotted line) and from the line mixing model calculation (solid line).

Line mixing in the �3 and �2 bands of CH4 2715

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measured absorption. We attribute the observeddiscrepancies to the insufficient accuracy of the modelused to calculate the rotational relaxation matrix W.For further improvement of the model, it is necessary

to determine the spectral regions that are sensitive to thedetails of the structure of the matrix W. In general, linemixing effects can be described most directly if weconsider an interaction of two adjacent lines overlappingat rather low pressures and distant from the otherlines of the band, which leads to a weak influence of thelatter at moderate pressures. It is also preferablethat the intensities of these lines predominate over theother lines of the same manifold. Evidently, these linesmust be of the same symmetry since there is no mixing ofthe lines of different symmetries.In the considered bands, we choose two examples: the

R(18) manifold of the �3 band and the P(13) manifold of

the �2 band. For the R(18) manifold of the �3 band, as itis seen in figure 1, there are two lines 23 1 (A1)and 22 2 (A2) situated very closely at 3191.396 cm�1

with the wavenumber difference equal to 0.0022 cm�1.In addition, there is only one line of the same symmetry,namely, 24 2 (A2) at 3191.365 cm

�1. The other linesforming the manifold in this spectral region are ofdifferent symmetries (E and F). In the case of the P(13)manifold of the �2 band, the profile in the region of1418.6–1419.1 cm�1 is formed at lower pressures,primarily by three lines with comparable intensities(see figure 6): the pair of transitions 15 3 (F2) and16 4 (F1) and the ‘isolated’ line 11 2 (E), which doesnot interact with the pair.

Providing that there is no significant overlapping ofthe chosen lines with the other lines of the samesymmetry in a manifold, we can suppose that line

Figure 5. Residual deviations of the modified line mixing model calculations for the P(14) and P(12) manifolds of the �2 bandfrom the experimental profiles at 0.855 atm. The dimensionless factors Akk0 have been directly transferred from the �3 band study.The notation is the same as in figure 4, with the positions and relative intensities of the lines of the �4 band marked with asterisks.

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mixing within a given spectral region is determined bythe rotational relaxation matrix elements Wkk0 which isrelated to the involved transitions only. In thisapproximation, we can vary the matrix elements of thematrix W and minimize deviations of the calculatedprofiles from the measured ones.

For the R(18) manifold of the �3 band, weobtained a value of the interaction coefficient for thelines 23 1 (A1) and 22 2 (A2) equal to8� 10�3 cm�1/atm by fitting, while the value given bythe basic model is equal to 1.510�3 cm�1/atm. In the caseof the �2 band, the coefficient for the lines 15 3 (F2)

Figure 6. The P(13) manifold of the �2 band at different pressures of the CH4–He mixtures: (a) 0.38 atm; (b) 0.855 atm; (c)4.93 atm. In each case, the upper plot presents the experimental profile (open circles), the line mixing model calculation (dotted line),and the calculation with corrected elements of the matrix W (solid line), and the lower plot presents relative deviations of theexperimental profile from the line mixing model calculation (dotted line) and from the calculation with a corrected matrix (solid line)in the region of sufficient accuracy of measurements. The notations for the positions and relative intensities of the lines are the sameas in figure 4.

Line mixing in the �3 and �2 bands of CH4 2717

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and 16 4 (F1) in the P(13) manifold appeared to be

equal to 19� 10�3 cm�1/atm instead of 13� 10�3 cm�1/

atm. In both cases, we observe a better agreement of the

calculated profiles with the measured ones at all

pressures used in the experiment. An example is given

in figure 6: there is a residual difference of about 20%

provided by the model of line mixing at the pressures

from 0.38 to 4.93 atm. The change of a single matrix

element, which is fitted using a single spectrum,

improves the description for the total set of spectra at

various pressures.So, the consistency of measured and calculated

profiles can be substantially improved by varying certain

matrix elements Wkk0. It is worth noting that, in the case

of the P(13) manifold of the �2 band, the interaction

coefficient can be determined unambiguously since only

two matrix elements are involved in the description of

line interaction.

6. Conclusions

We studied the description of the forbidden �2 band

shape of methane mixed with helium provided by an

empirical method of the optical rotational relaxation

matrix determination based on the calculated matrix of

state-to-state collision rates. The parameters of the

calculations were obtained by fitting the spectra of the �3band and transferred directly to the studied case. An

acceptable overall agreement with the experiment proves

(a) that the proposed method can be used for a

consistent description of band shapes in the spectra of

gaseous methane and (b) that the vibrational depen-

dence of matrix elements Wkk0 is weak. However, certain

deviations of the calculated profiles require more

accurate knowledge on the structure of the rotational

relaxation matrix; its improvement provides a much

better agreement for the profiles influenced by line

mixing effects. In some cases, using a wider structure of

the J-manifolds in the �2 band, a study of spectral

profiles can yield more detailed information on the

optical rotational relaxation matrix elements.The proposed empirical model leads to a much better

agreement of the experimental band shapes and the

calculated ones than the isolated line model. However, it

does not provide a satisfactory result, taking into

account the experimental accuracy of the measured

spectra. Further improvements are to be sought by

means of non-empirical models and more rigorouscalculation schemes.

Acknowledgement

This work was supported in part by the RussianFoundation for Basic Research (Grant No. 05-03-32227).

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