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Article

Depolarization Ratio of the ν1 Raman Band of Pure CH4 and Perturbed by N2 and CO2

by
Aleksandr S. Tanichev
1,* and
Dmitry V. Petrov
1,2
1
Laboratory of Ecological Instrumentation, Institute of Monitoring of Climatic and Ecological Systems, Siberian Branch of the Russian Academy of Sciences, 634055 Tomsk, Russia
2
Department of Optics and Spectroscopy, Tomsk State University, 634050 Tomsk, Russia
*
Author to whom correspondence should be addressed.
Molecules 2022, 27(1), 144; https://doi.org/10.3390/molecules27010144
Submission received: 9 December 2021 / Revised: 22 December 2021 / Accepted: 24 December 2021 / Published: 27 December 2021
(This article belongs to the Section Physical Chemistry)
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Abstract

:
In this work, the effect of nitrogen and carbon dioxide on the depolarization ratio of the ν1 band of methane in the pressure range of 0.1–5 MPa is studied. A high-sensitivity single-pass Raman spectrometer was used to obtain accurate results. Moreover, we took into account the overlap of the ν1 band by the ν3 and ν2 + ν4 bands using the simulation of their spectra. The depolarization ratio of the ν1 band in pure methane is within 0–0.001, and the effect of nitrogen and carbon dioxide on this parameter is negligible in the indicated pressure range. The obtained results are useful for correct simulation of the Raman spectrum of methane at different pressures, which is necessary to improve the accuracy of gas analysis methods using Raman spectroscopy.

1. Introduction

The optical methods based on Raman spectroscopy for the analysis of multicomponent gaseous media have been rapidly developing in the last decades. These methods can simultaneously detect all molecular vibrational bands using one laser with a fixed wavelength. The recent appearance of high-sensitivity photodetectors and powerful small-size lasers provides an opportunity to amplify the useful signal and decrease the limit of detection (LOD) of the method. Other amplification methods include the multi-pass optical cells [1,2,3,4] or hollow-core fiber [5,6,7,8]. However, compression of the analyzed medium to a higher pressure is the most effective and easy to implement signal amplification approach [9,10]. Neglecting the compressibility factor, compression of the sample at ambient pressure to a pressure of 5 MPa leads to a 50-fold amplification [11]. The LOD below 1 ppm can be achieved using this approach. Such sensitivity opens up the possibility to analyze the composition of atmospheric and exhaled air using Raman spectroscopy [5,10,12,13,14,15,16]. This method is very promising due to its high measurement speed and the ability to determine a lot of compounds.
Methane (CH4) is an important greenhouse gas contained in atmospheric air. The average annual concentration of CH4 is continuously increasing due to the influence of natural and anthropogenic factors. Therefore, monitoring of atmospheric CH4 is necessary to detect leaks of greenhouse gases, as well as to improve climate prediction models. The measurement precision of concentration should be less than 100 ppb since CH4 content in atmospheric air is about 2 ppm [17]. Moreover, CH4 is included in the list of biomarkers of diseases [18,19]. The CH4 content in the exhaled air can reach 10 ppm [12]. Hence, high accuracy of CH4 measurement in a sample is necessary both for the accurate diagnosis and for the investigation of correlations. On the other hand, the region of stretching C–H vibrations (2900–3000 cm−1) is important in the air composition analysis since the most intense vibrational Raman bands of all volatile organic compounds (VOCs) are located there [20,21]. The CH4 content in the air is high compared to other VOCs. Therefore, the most intense fundamental vibrational ν1 band of CH4 (≈2917 cm−1) makes a significant contribution to the spectrum of these gaseous media in the indicated spectral range. It should be noted that pressure and different molecular environment significantly affect the spectroscopic characteristics of the ν1 peak of CH4 [22,23,24,25,26,27,28,29,30]. The higher the molar fraction of the perturbing component, the stronger its influence on the spectrum. The effect of the main components of air cannot be neglected since their concentration is orders of magnitude higher than CH4. Thus, it is necessary to account for these changes in the CH4 spectrum to decrease the LOD of CH4 and correctly derive the concentrations of other organic compounds from the Raman spectra of air. A simulation method of the CH4 spectrum is one of the most promising approaches for this purpose [26,31,32,33]. The required intensities and positions of spectral lines can be obtained using calculations based on the tensor formalism and the group theory methods [34]. However, it is necessary to know the tensor components of the total polarizability derivative of the molecule. The depolarization ratio (ρ) can be expressed in terms of these quantities [35,36,37]. Since the spectral sensitivity of a spectrometer is not the same for radiation in different polarization states, the use of the exact experimental value of the ρ can both increase the reliability of theoretical calculations and more accurately fit the simulated spectra to the experimental ones.
As noted by Wang and Ziegler [38], the depolarization ratio of the ν1 band of CH4 (ρ(ν1)) is 0.025 ± 0.005 at a pressure of less than 0.1 MPa. More precise measurements were performed using the photoacoustic Raman spectroscopy method by Yu et al. [39], where the ρ(ν1) = 0.002 ± 0.002 was obtained at a pressure of less than 15 kPa. According to the data reported in [40,41,42], the ρ(ν1) is a function of pressure. However, there is a discrepancy between the results obtained, for example, ρ(ν1) = 0.11 at 4 MPa [40], ρ(ν1) = 0.067 at 5 MPa [41], and ρ(ν1) = 0.0045 at 6 MPa [42]. Moreover, as shown in [40,41,43,44,45], the molecular environments also affect the ρ(ν1). Taking into account the composition of air, knowledge about the influence of nitrogen (N2) and carbon dioxide (CO2) on this parameter is important in the field of Raman analysis of atmospheric and exhaled air. However, to our knowledge, no studies have investigated the N2 environment effect. The ρ(ν1) perturbed by CO2 was measured with a very high error [43]. We suppose that this error is due to the low signal-to-noise ratio of the equipment used. In this work, the influence of the N2 and CO2 environments on the ρ(ν1) of CH4 in the pressure range of 0.1–5 MPa at 298 K was researched. A high-sensitivity Raman spectrometer and a simulation of the CH4 spectrum were used for this purpose.

2. Methods

Despite the recent advances in the field of amplification of the Raman signal [1,2,3,4], the experimental setup based on the single-pass excitation scheme was used to obtain reliable data in this study. Figure 1 and Table 1 present the scheme and the main characteristics of the setup, respectively. Plane-polarized radiation of the single-mode continuous-wave laser (Cnilaser, Changchun, China) with a wavelength of 532 nm was directed into the gas cell and excited spontaneous Raman scattering in the medium. The scattered radiation was collected at an angle of 90° to the direction of propagation of the laser beam through the side window of the gas cell using the system of two lenses. The notch filter and the polarizer were installed between them. Polarized radiation was focused on the entrance slit of the spectrometer based on the Čzerny–Turner configuration. The Raman spectra were recorded using the charge-coupled device (CCD) sensor S10141 (Hamamatsu Photonics K.K., Hamamatsu, Japan) with thermoelectric cooling to −10 °C. The simultaneously recorded spectral range was 2800–3040 cm−1.
Polarized and depolarized spectra of pure CH4, as well as mixtures of CH4/N2 and CH4/CO2 in molar ratios of 50/50, at pressures of 0.1, 0.5, 1, 2, 3, 4, and 5 MPa were recorded using this system. The signal-to-noise ratio in the polarized spectra of pure CH4 was 1500 (at 0.1 MPa) and 11,000 (at 5 MPa), where the peak intensity of the ν1 band (≈2917 cm−1) was the signal magnitude. The pressure measurement error was less than 1 kPa. The gas cell was thermally stabilized at 298 ± 1 K. Samples of CH4, N2, and CO2 with a purity of greater than 99.99% were used to prepare the studied mixtures in a separate mixing chamber connected to a gas cell. Pure gases were mixed in a specified ratio of partial pressures to obtain the required molar ratio. These partial pressures were calculated from the equation of state for gases taking into account the compressibility. Compressibility factors were taken from the NIST Chemistry WebBook [46]. The molar ratio measurement error in the mixture preparation procedure is estimated within 2–3%.
The wavenumber calibration of the spectrometer was performed using the spectrum of pure CH4 at a pressure of 0.1 MPa according to the procedure described by Brunsgaard Hansen [47]. However, the most intense lines of the ν3 band from data of Berger [48] were taken as reference lines, instead of the emission lines of a neon lamp. As a result, the third-degree polynomial was obtained, representing the relationship between the pixel numbers of the CCD sensor and the wavenumbers of the spectrometer. The calibration error and the spectrum drift due to ambient temperature fluctuations were estimated to be less than 0.02 cm−1.

3. Results and Discussion

3.1. Raman Spectra of Methane

Figure 2 shows the obtained Raman spectra of pure CH4 at various pressures in the spectral range of 2810–3030 cm−1. The polarized spectrum is the high-intensity peak formed by closely spaced rotational-vibrational lines of the Q branch of the ν1 band. This peak is overlapped by the O, P, and Q branches of the ν3 band and the Q branch of the ν2 + ν4 band. The contribution of other overtones and hot transitions can be neglected in this range. The vibrations ν1 and ν2 + ν4 are characterized by extremely weak anisotropic polarizability properties. Hereby, the ν2 + ν4 band is not observed in the depolarized spectra, and the ν1 band is a low-intensity peak. An increase in medium pressure leads to the broadening of all lines due to the collisional broadening effect. Therefore, the ν3 band is an almost continuous spectrum at a pressure of 5 MPa. However, this effect is not so pronounced for the ν1 band, since the processes of collisional line mixing dominate here [26]. The ν1 peak shifts to the region of low wavenumbers as the pressure increases, which corresponds to the data of [22,27,28,49,50]. The effect of the N2 and CO2 environments leads to different broadening and shifts of the CH4 lines. Nevertheless, the spectrum of the mixture is similar to that of pure CH4 at a different pressure. This difference is more pronounced as the pressure increases. As shown in Figure 3, the presence of N2 in the mixture leads to a narrowing of the ν1 peak, while the presence of CO2 leads to a broadening. It is also worth noting that the N2 environment leads to a smaller shift of the ν1 peak to the region of low wavenumbers than CH4 or CO2. These observations are in agreement with results presented in [22,27,51]. The contribution of the N2 and CO2 bands is negligible within the spectral range under investigation in comparison with the ν3 and ν2 + ν4 lines.

3.2. Measurement Procedure

The observed depolarization ratio of an arbitrary vibrational band can be defined by Equation (1),
ρ = ∫ ω E ⊥ ω d ω ∫ ω E ∥ ω d ω ,
where E ⊥ ω and E ∥ ω are the intensities of the experimental Raman spectra at the wavenumber ω, when the polarization planes of the scattered and exciting radiation are parallel (polarized spectrum) and perpendicular (depolarized spectrum), respectively. Here, it is necessary to take into account the overlap of the ν3 and ν2 + ν4 bands at different pressures and environments to correctly measure the integrated intensity of the ν1 band. The method of simulating the Raman spectrum as a sum of the profiles of each rotational-vibrational line was used for this purpose. A detailed description of this approach can be found in our previous work [33]. The positions and intensities of the ν3 and ν2 + ν4 lines were taken from the study of Ba et al. [52], and the pressure broadening and shift coefficients were used the same as those in [33]. According to the features of the polarizability anisotropy, only the ν3 lines were used to simulate the depolarized spectra. The ν3 and ν2 + ν4 lines were used to simulate the polarized spectra. The influence of the N2 and CO2 environments on the ν3 and ν2 + ν4 bands of CH4 was imitated by simulating the spectrum at a different pressure. The integrated intensities of the depolarized and the polarized ν1 band ( E ⊥ ν 1 , E ∥ ν 1 ) were measured in the range of 2880–2950 cm−1 in each experimental spectrum after subtracting the simulated spectrum (see Figure 4).
Further, it is necessary to take into account the fluctuations of the laser power to improve the accuracy of the intensity measurement. We used the Q branch of the ν3 band of CH4 for this purpose since the ρ of this band does not depend on pressure in the range of 0–5 MPa and equals 0.75 [42,53]. Thus, the ρ(ν1) values were obtained using Equations (2) and (3):
ρ ν 1 = E ⊥ ν 1 E ∥ ν 1 k ,
k = 0.75 E ∥ ν 3 E ⊥ ν 3 ,
where E ⊥ ν 3 and E ∥ ν 3 are the integrated intensities of the Q branch of the ν3 band in the depolarized and polarized spectra, respectively. These intensities were measured in the range of 3000–3030 cm−1. The data obtained are presented in Figure 5. The values of the ρ(ν1) are in the range of 0.0009–0.001 and the influence of the molecular environment in the pressure range of 0–5 MPa is not observed. The double standard deviation of all measurements is less than 0.0001. It should be noted that much larger values of the ρ(ν1) at 5 MPa were obtained by other authors [40,41,42]. We suppose that this discrepancy is caused by the neglect or incorrect accounting of the overlap of the ν1 peak by the ν3 and ν2 + ν4 bands, in addition to the low signal-to-noise ratio. The obtained values of the ρ(ν1) of pure CH4, where the subtraction procedure of the simulated spectra was not performed, are also shown in Figure 5 for comparison. It can be seen that the pressure dependence of the ρ(ν1) is observed in this case, which corresponds to the previous results [42]. The reason for this is that the contribution of the ν3 and ν2 + ν4 lines to the intensity of the ν1 band (in the 2910–2925 cm−1 range) increases due to the collisional broadening effect. Thus, the data in Figure 5 confirm that the contribution of depolarized lines must be taken into account to obtain the most reliable values of the ρ.

3.3. Uncertainty Evaluation

According to Figure 5, the measured value of the ρ(ν1) is not equal to zero even at a pressure of 0.1 MPa, which does not agree with theoretical calculations [35,53,54]. Let us estimate the error of our measurements. The main sources of the measurement error are imperfect polarization of the laser radiation, different transmittance of the polarizer in orthogonal orientations, and polarization scrambling by the windows of the gas cell [55,56], as well as the non-zero collection angle for the scattered radiation [38,57,58,59]. The additional experiment was carried out to evaluate the influence of the first three effects. Laser radiation was directed through the cell windows and the polarizer and was guided to the photodetector at the output (see Figure 6). At the first stage, the cell was pressurized by pure CH4 at 0.1 MPa and the power of the transmitted radiation was measured in two orthogonal polarization orientations. At the second stage, the pressure of CH4 in the cell was increased to 5 MPa and similar measurements were performed. It was found that the ratio P ∥ / P ⊥ was more than 1000 in both cases, where P ⊥ and P ∥ are the measured radiation power with perpendicular and parallel orientation of the polarization plane to the polarization plane of the exciting radiation, respectively. It should be noted that the entrance window (W1) and the exit window (W2) influenced the results obtained in this experiment, but the window W1 and the side window (W3) influenced the measurements of the ρ(ν1). Since all the cell windows are identical, we can conclude that the systematic measurement error of the ρ(ν1) is less than 0.001 at a zero-collection angle, taking into account the aforementioned effects of polarization scrambling.
The approach based on the calculations presented by Schlösser et al. [57] was used to estimate the measurement error in the case of the non-zero collection angle. A detailed description of the calculations performed is provided in Appendix A of this study. As a result, the geometric effect introduces the systematic measurement error of no more than 2% of the ρ(ν1) = 0.001, without taking into account the effects of polarization scrambling. Since the non-zero angle effect has a small contribution, the estimate of the total systematic measurement error of the ρ(ν1) is less than 0.001. Therefore, we can conclude that the true depolarization ratio of the ν1 band of CH4 is within 0–0.001 in the pressure range of 0.1–5 MPa.

4. Conclusions

In this study, the depolarization ratio of the ν1 band of CH4 was measured using the Raman spectrometer that combines both high resolution and high sensitivity. It was found that the depolarization ratio of the ν1 peak of pure CH4 or perturbed by the N2/CO2 molecular environment did not exceed 0.001 in the pressure range of 0.1–5 MPa. This value is significantly less than the measurements reported in earlier studies. In our view, this discrepancy is a consequence of correctly taking into account the overlap of the ν1 band by the ν3 and ν2 + ν4 bands using the spectra simulation in this study. These results imply that the correction of the tensor components of the total polarizability derivative of CH4 due to the effect of the N2/CO2 environment, and pressure can be neglected in the pressure range of 0.1–5 MPa. Therefore, the line intensities of CH4 in vacuum calculated using the tensor formalism approach are suitable for simulating its spectra in the field of Raman gas analysis of methane-bearing media (e.g., fuel gases, atmospheric air, exhaled air, etc.).

Author Contributions

Supervision, project administration, funding acquisition, validation, and writing—review and editing, D.V.P.; formal analysis, data curation, visualization, and writing—original draft, A.S.T.; conceptualization, investigation, and methodology, D.V.P. and A.S.T. All authors have read and agreed to the published version of the manuscript.

Funding

The work was supported by the Russian Science Foundation (RSF), grant no. 19-77-10046.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflict of interest.

Sample Availability

Not applicable.

Appendix A

The integrated intensity of an arbitrary vibrational Raman band vj→k observed in the direction (φ,θ) can be expressed as follows [60]:
E o b s ( φ , θ ) ∼ E i n N j g j Φ j ( S p , I p , φ , θ ) ,
where Ein is the intensity of the incident radiation exciting the Raman scattering; Φj is the scattering strength function of the scattered radiation in the direction (φ,θ); φ and θ are the scattering angles; Nj and gj are the population and the degeneracy of the jth vibrational energy level; and Ip and Sp are the polarization states of the plane-polarized incident and scattered radiation (p = ⊥ or ∥ ), respectively (see Figure A1 for details). The combinations of the polarization states assign the four scattering strength functions:
Φ ( S ⊥ , I ⊥ , φ , θ ) = 45 α 2 cos 2 φ + γ 2 3 + cos 2 φ ,
Φ ( S ∥ , I ⊥ , φ , θ ) = 45 α 2 cos 2 θ sin 2 φ + γ 2 3 + cos 2 θ sin 2 φ ,
Φ ( S ⊥ , I ∥ , φ , θ ) = 45 α 2 sin 2 φ + γ 2 3 + sin 2 φ ,
Φ ( S ∥ , I ∥ , φ , θ ) = 45 α 2 cos 2 θ cos 2 φ + γ 2 3 + cos 2 θ cos 2 φ ,
where α and γ are the mean and the anisotropy of the total polarizability derivative, respectively, with respect to the normal coordinate of the molecule related to the vibration vj→k. For example, Φ ( S ⊥ , I ∥ , φ , θ ) is the scattering strength function of the perpendicular plane-polarized scattered radiation observed in the direction (φ,θ), where the Raman scattering is induced by parallel plane-polarized radiation.
Molecules 27 00144 g0a1 550
Figure A1. Sketch (a) is the optical collection system of the scattered radiation. The laser beam propagates along the z-axis, the scattered radiation is collected along the x-axis. Here, the path of two light rays from an arbitrary point z at an angle φ = 0 is shown. Sketch (b) is the definition of scattering angles, where I ⊥ and I ∥ denote the polarization directions of the incident radiation, S ⊥ and S ∥ denote the polarization planes of the scattered radiation, φ is the angle between the scattering plane (green) and the plane S ∥ (blue), and θ is the angle between the z-axis and the direction of scattering Φ(φ,θ) in the scattering plane.
Figure A1. Sketch (a) is the optical collection system of the scattered radiation. The laser beam propagates along the z-axis, the scattered radiation is collected along the x-axis. Here, the path of two light rays from an arbitrary point z at an angle φ = 0 is shown. Sketch (b) is the definition of scattering angles, where I ⊥ and I ∥ denote the polarization directions of the incident radiation, S ⊥ and S ∥ denote the polarization planes of the scattered radiation, φ is the angle between the scattering plane (green) and the plane S ∥ (blue), and θ is the angle between the z-axis and the direction of scattering Φ(φ,θ) in the scattering plane.
Molecules 27 00144 g0a1
The corrected depolarization ratio (ρcorr) of the band, independent of the collection angle and in the case of perfectly plane-polarized radiation, is defined as the ratio of intensities of the perpendicular and parallel polarized scattered radiation observed from a single point at a zero solid angle in the direction (φ = 0, θ = π/2). Taking into account the introduced designations, the corrected depolarization ratio is given as:
ρ c o r r = Φ ( S ∥ , I ⊥ , φ = 0 , θ = π / 2 ) Φ ( S ⊥ , I ⊥ , φ = 0 , θ = π / 2 ) = 3 γ 2 45 α 2 + 4 γ 2 .
According to theoretical calculations reported by Abbate et al. [35], the anisotropy γ of the ν1 band of CH4 is zero, and hence, ρcorr(ν1) = 0. Besides, α = 0 for the ν3 band and ρcorr(ν3) = 0.75, respectively. Let us suppose that the scattered radiation is collected over a finite solid angle Ω and from a region of active molecules characterized by a finite volume V. Then, Equation (A1) is given as:
E o b s ( Ω , V ) ∼ E i n N j g j ∬ Ω , V Φ j ( S p , I p , φ , θ ) d Ω d V .
Let us assume that the exciting radiation in the V region is not perfectly plane-polarized due to the introduced polarization distortions and the imperfect polarization of the radiation source. Then, the exciting radiation can be considered as a superposition of two waves with mutually orthogonal polarization:
E i n = E i n ⊥ + E i n ∥ ,
where E i n ⊥ and E i n ∥ are the intensities of the perpendicular and parallel plane-polarized exciting radiation, respectively. Therefore, Raman scattering can be described as a superposition of two luminous fluxes, each of which is excited by radiation E i n ⊥ and E i n ∥ . In this case, the observed depolarization ratio (ρobs) is expressed as follows:
ρ o b s = E o b s ∥ ( Ω , V ) E o b s ⊥ ( Ω , V ) = ∬ Ω , V η 1 Φ ( S ∥ , I ⊥ , φ , θ ) + Φ ( S ∥ , I ∥ , φ , θ ) d Ω d V ∬ Ω , V η 1 Φ ( S ⊥ , I ⊥ , φ , θ ) + Φ ( S ⊥ , I ∥ , φ , θ ) d Ω d V ,
where η1 is the ratio E i n ⊥ / E i n ∥ . If the collected radiation from the region V undergoes polarization distortions introduced by the collection system, the scattered intensity can also be represented as a superposition on the analogy of Equation (A8). Let us assume that the initially parallel polarized scattered radiation is split into two waves with mutually orthogonal polarization as:
E o b s ∥ → E ˜ o b s ∥ ∥ + E ˜ o b s ∥ ⊥ ,
where E ˜ o b s ∥ ∥ and E ˜ o b s ∥ ⊥ are the intensities of the parallel and the perpendicular plane-polarized observed radiation from the initially parallel polarized scattered radiation, respectively. Then, the separated wave E ˜ o b s ∥ ⊥ contributes to the intensity of the perpendicular polarized observed radiation at the end of the collection system. Conversely, the separated wave E ˜ o b s ⊥ ∥ from the initially perpendicular polarized scattered radiation contributes to the intensity of the parallel polarized observed radiation. Therefore, Equation (A9) takes the following form:
ρ o b s = E ˜ o b s ∥ ∥ ( Ω , V ) + E ˜ o b s ⊥ ∥ ( Ω , V ) E ˜ o b s ⊥ ⊥ ( Ω , V ) + E ˜ o b s ∥ ⊥ ( Ω , V ) .
If splitting into two waves occurs in the same ratio for the E o b s ∥ and E o b s ⊥ , that is:
η 2 = E ˜ o b s ∥ ∥ E ˜ o b s ∥ ⊥ = E ˜ o b s ⊥ ⊥ E ˜ o b s ⊥ ∥ ,
then Equation (A11) is expressed as follows:
ρ o b s = η 2 A + B η 2 B + A ,
where:
A = ∬ Ω , V η 1 Φ ( S ∥ , I ⊥ , φ , θ ) + Φ ( S ∥ , I ∥ , φ , θ ) d Ω d V ,
B = ∬ Ω , V η 1 Φ ( S ⊥ , I ⊥ , φ , θ ) + Φ ( S ⊥ , I ∥ , φ , θ ) d Ω d V .
The Raman spectrometer used in this work detects only those scattered rays that propagate through the collecting lens and are focused into the region of the entrance slit (4 mm × 30 μm) of the spectrometer (see Figure 1 and Figure A1). Lenses used to collect the scattered radiation magnify the image of the object twice. Hence, the effective scattering area is about 2 mm × 15 µm (cross-section of the region V in the focal plane of the collecting lens). Therefore, the intensity distribution of the laser beam cross-section and the width (15 µm) of the effective area can be neglected. Hereby, the integration over the solid angle Ω and the region V can be replaced by the integration over φ, θ, and z in Equations (A14) and (A15). Substituting Equation (A6) in Equations (A2)–(A5), the simplified Equation (A13) as a function of the corrected ρ can be derived:
A = η 1 C S + ρ c o r r ( U − C S ) + C C + ρ c o r r ( U − C C ) ,
B = η 1 C + ρ c o r r ( U − C ) + S + ρ c o r r ( U − S ) ,
where C, S, CC, CS, and U are triple integrals over the variables θ, φ, and z:
S = 4 ∫ 0 φ max ∫ 0 z max ∫ θ min θ max ( sin 2 φ ) sin θ d θ d z d φ ,
C = 4 ∫ 0 φ max ∫ 0 z max ∫ θ min θ max ( cos 2 φ ) sin θ d θ d z d φ ,
C C = 4 ∫ 0 φ max ∫ 0 z max ∫ θ min θ max ( cos 2 θ cos 2 φ ) sin θ d θ d z d φ ,
C S = 4 ∫ 0 φ max ∫ 0 z max ∫ θ min θ max ( cos 2 θ sin 2 φ ) sin θ d θ d z d φ ,
U = 4 ∫ 0 φ max ∫ 0 z max ∫ θ min θ max sin θ d θ d z d φ ,
θ min = arctan F 1 cos φ 0.25 D 1 2 − F 1 2 tan 2 φ + z ,
θ max = π 2 + arccot F 1 cos φ 0.25 D 1 2 − F 1 2 tan 2 φ − z ,
z max = h 2 F 1 2 F 2 cos φ ,   φ max = arctan D 1 2 F 1 .
These expressions are obtained with the proviso that the aperture of the focusing lens does not limit the collection of the scattered radiation from the region of z ≠ 0 (D2 > D1). Moreover, the aperture and position of the exit window of the gas cell are neglected for the same reason.
As discussed in Section 3.3., the polarization scrambling effects of scattered radiation propagating through the path of W1 → W3 → Polarizer or W1 → W2 → Polarizer are similar (see Figure 6). Thus, the approximate equality η1 ≈ η2 ≥ 1000 holds in Equations (A13), (A16) and (A17). In other words, the perpendicular plane-polarized laser radiation will contain 1 part of the parallel polarized light to more than 1000 perpendicular polarized parts, at the end of the collection system. Figure A2 shows the deviation of the ρobs from the ρcorr as a function of ρcorr, parameter η, and the full collection angle (2φmax). These values are calculated using Equations (A13) and (A16)–(A25), and the parameters from Table 1. As expected, the collection of radiation using the lens with a larger f-number leads to a greater deviation of the ρ. However, the deviation of the ρ tends to a value of 1/η in the vicinity of the zero-collection angle. This deviation is caused by the splitting of the plane-polarized scattered radiation as 1 to η due to the polarization scrambling effect, even in the case of collecting from a single point at a zero solid angle. In turn, the attenuation of the polarization scrambling effect leads to less deviation of the ρ at a fixed collection angle. The deviation of the ρ is approximately 0.001 at the η = 1000 and 2φmax = 14.25°.
Molecules 27 00144 g0a2 550
Figure A2. Deviation of the observed depolarization ratio from the corrected one as a function of the total collection angle 2φmax (a) and the parameter η (b), calculated using Equations (A13) and (A16)–(A25). The dashed line denotes the boundary values of the angle 2φmax = 14.25° (lens f/4) and the parameter η = 1000, corresponding to the experimental setup used in this work.
Figure A2. Deviation of the observed depolarization ratio from the corrected one as a function of the total collection angle 2φmax (a) and the parameter η (b), calculated using Equations (A13) and (A16)–(A25). The dashed line denotes the boundary values of the angle 2φmax = 14.25° (lens f/4) and the parameter η = 1000, corresponding to the experimental setup used in this work.
Molecules 27 00144 g0a2

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Molecules 27 00144 g001 550
Figure 1. Schematic of the experimental setup.
Figure 1. Schematic of the experimental setup.
Molecules 27 00144 g001
Molecules 27 00144 g002 550
Figure 2. Experimental depolarized (a) and polarized (b) Raman spectra of pure CH4 at pressures of 0.5, 2, and 5 MPa. The spectra are normalized by the integrated intensity of the Q branch of the ν3 band. The insets show the effect of pressure on the ν1 peak. Panel (c) shows the positions and intensities of the rotational-vibrational lines of the ν1, ν3, and ν2 + ν4 bands calculated by Ba et al. [52].
Figure 2. Experimental depolarized (a) and polarized (b) Raman spectra of pure CH4 at pressures of 0.5, 2, and 5 MPa. The spectra are normalized by the integrated intensity of the Q branch of the ν3 band. The insets show the effect of pressure on the ν1 peak. Panel (c) shows the positions and intensities of the rotational-vibrational lines of the ν1, ν3, and ν2 + ν4 bands calculated by Ba et al. [52].
Molecules 27 00144 g002
Molecules 27 00144 g003 550
Figure 3. Experimental Raman spectra of the polarized ν1 band (a) and the depolarized ν3 band (b) of pure CH4 and CH4/N2 and CH4/CO2 mixtures at a pressure of 5 MPa. All spectra were normalized by the integrated intensity.
Figure 3. Experimental Raman spectra of the polarized ν1 band (a) and the depolarized ν3 band (b) of pure CH4 and CH4/N2 and CH4/CO2 mixtures at a pressure of 5 MPa. All spectra were normalized by the integrated intensity.
Molecules 27 00144 g003
Molecules 27 00144 g004 550
Figure 4. Subtraction procedure of the simulated spectra from the depolarized (a) and polarized (b) experimental Raman spectra of pure CH4 at a pressure of 0.5 MPa. The spectra are normalized by the integrated intensity of the Q branch of the ν3 band. Panel (c) shows the obtained differences, where the depolarized spectrum is magnified 10 times for visualization.
Figure 4. Subtraction procedure of the simulated spectra from the depolarized (a) and polarized (b) experimental Raman spectra of pure CH4 at a pressure of 0.5 MPa. The spectra are normalized by the integrated intensity of the Q branch of the ν3 band. Panel (c) shows the obtained differences, where the depolarized spectrum is magnified 10 times for visualization.
Molecules 27 00144 g004
Molecules 27 00144 g005 550
Figure 5. Depolarization ratio of the ν1 band of CH4 as a function of pressure at different molecular environments, where label a denotes the data obtained after the subtraction procedure of the simulated spectrum of the ν3 and ν2 + ν4 bands, and label b is the data obtained without the subtraction.
Figure 5. Depolarization ratio of the ν1 band of CH4 as a function of pressure at different molecular environments, where label a denotes the data obtained after the subtraction procedure of the simulated spectrum of the ν3 and ν2 + ν4 bands, and label b is the data obtained without the subtraction.
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Molecules 27 00144 g006 550
Figure 6. Schematic of the experimental setup used to evaluate the effect of polarization scrambling. Here, W1, W2, and W3 denote the windows of the gas cell.
Figure 6. Schematic of the experimental setup used to evaluate the effect of polarization scrambling. Here, W1, W2, and W3 denote the windows of the gas cell.
Molecules 27 00144 g006
Table
Table 1. Characteristics of the experimental setup.
Table 1. Characteristics of the experimental setup.
ParameterUnitQuantity
Laser output powerW5
Laser wavelengthnm532.094
Polarization ratiounitless>100:1
Collection lens diameter (D1)/focal length (F1)mm26.3/105
Focusing lens diameter (D2)/focal length (F2)mm46.7/210
Distance between lenses (d)mm250
Spectrometer f-numberunitlessf/8
Size of CCD chippixel2048 × 512
Diffraction gratingline/mm2400
Entrance slit height (h2)/widthmm4/0.03
Half-width of instrument response functioncm−10.5 (at 2917 cm−1)
Spectral dispersioncm−1/pixel0.12
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Tanichev, A.S.; Petrov, D.V. Depolarization Ratio of the ν1 Raman Band of Pure CH4 and Perturbed by N2 and CO2. Molecules 2022, 27, 144. https://doi.org/10.3390/molecules27010144

AMA Style

Tanichev AS, Petrov DV. Depolarization Ratio of the ν1 Raman Band of Pure CH4 and Perturbed by N2 and CO2. Molecules. 2022; 27(1):144. https://doi.org/10.3390/molecules27010144

Chicago/Turabian Style

Tanichev, Aleksandr S., and Dmitry V. Petrov. 2022. "Depolarization Ratio of the ν1 Raman Band of Pure CH4 and Perturbed by N2 and CO2" Molecules 27, no. 1: 144. https://doi.org/10.3390/molecules27010144

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