Herman–Wallis factor to improve thermometric accuracy of vibrational coherent anti-Stokes Raman spectra of H2

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Proceedings of the Combustion Institute 32 (2009) 863–870

Combustion Institute www.elsevier.com/locate/proci

Herman–Wallis factor to improve thermometric accuracy of vibrational coherent anti-Stokes Raman spectra of H2 Michele Marrocco * ENEA (Ente per le Nuove tecnologie, l’Energia e l’Ambiente), via Anguillarese 301, 00123 Santa Maria di Galeria (Rome), Italy

Abstract The Herman–Wallis factor is a molecular parameter that measures the influence of centrifugal force on the intensity of spectral lines. Understandably, the effect is significant for very light molecules that necessarily have large couplings between vibrational and rotational degrees of freedom. Although known, the conceptual basis of the Herman–Wallis factor are nevertheless not clearly established in the literature. Over the years, different approaches have been proposed to explain the corrections to spectral line-strengths and, recently, an experimental study has demonstrated that Q-branch Raman transitions of H2 are highly sensitive to the theoretical model employed to determine the Herman–Wallis factor. In this paper, this fact is used to analyze the consequences on thermometry based on coherent anti-Stokes Raman scattering (CARS) designed to probe H2 molecules in combustion studies. It is found that the different Herman–Wallis factors lead to relative thermometric disagreements from several tens up to hundreds of degrees. This analysis could explain why H2 CARS thermometry has been considered less reliable than thermometric predictions based on CARS of more common molecules such as N2, O2 and others. In particular, it is remarked that unreliable expressions of Herman–Wallis factors have been used so far to interpret Q-branch H2 CARS experiments. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Coherent anti-Stokes Raman scattering (CARS) thermometry; Hydrogen flames; Non-linear diagnostics; Herman–Wallis factor

1. Introduction In combustion science, it is known that H2, when present, is the ideal molecule to extract thermometric information from spectra of coherent anti-Stokes Raman scattering (CARS) [1]. The ideality of H2 is rooted in the large values of both *

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the rotational constant and the vibration–rotation coupling. These two parameters determine the frequency spacing between Raman transitions and, in this regard, H2 spectral lines are characterized by large separations that make CARS measurements relatively easy to understand. For instance, the problem of line-mixing [2], which complicates spectra of more common molecular targets, such as N2, and whose treatment in precise elaboration of CARS measurements requires the so-called G-matrix approach [3,4], is non-existent in the

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spectral synthesis of rovibrational transitions belonging to H2. Not only is the anti-Stokes signal simpler to interpret, but also the Raman cross-section bears the additional advantage that H2 CARS measurements are practically free from non-resonant background contribution for reasonably low H2 concentrations [1]. Furthermore, other properties of the molecule could be invoked to complete the multifaceted view on the greater simplicity of H2 among several CARS molecules, but their mention is omitted here for the sake of conciseness. It is nonetheless striking that, in contrast with the promising aspects of H2 CARS thermometry and with the amount of work devoted to understand CARS spectra of molecular hydrogen [1,5–18], the technique is prone to criticisms of various authors, who reported on persistent disagreements between H2 CARS thermometric measurements and predictions based on alternative methods [7,13,16]. The research on possible flaws points to the uncertainty associated with the knowledge of Raman linewidths [10,13,15,16,19]. For instance, in the work of Hancock et al. [13], the discrepancy with more reliable N2 CARS thermometry is thought to be caused by unclear broadening mechanisms of spectral lines. In effect, the linewidths have a significant contribution to the heights of spectroscopic line intensities, especially in collision-broadened spectra [1,10,15,16,19] where the gas pressure has a fundamental role and more sophisticated lineshape models become necessary to take into account interactions between molecules leading to speeddependent effects [15,16,19]. On the other hand, the measurements shown in Ref. [13] were made at atmospheric pressure, where, as demonstrated in the work of Hussong et al. [16], the spectral lines are primarily Doppler broadened. The unresolved discrepancy appears again in the measurements of Bombach et al. [7], where CARS temperatures are compared with pyrometric temperatures and recurrent relative deviations as large as 100 K or more are observed. The perspective that Raman linewidths could be responsible for part of the troubles with H2 CARS thermometry is agreeable, but is this the only answer? Especially, at low or modest pressures, where spectral lines can be satisfactorily described by the regular Doppler broadening [1,16]. Very recently, another possible flaw in the evaluation of Q-branch H2 CARS spectra has been found in the inaccurate understanding of the coupling between vibrational and rotational degrees of freedom [20]. Here, differently from intermolecular effects apparent in changes of spectral linewidths [15,16,19], the rovibrational coupling has a mere intra-molecular genesis and has two main spectral consequences that are independent from the pressure regime. One is observed in the splitting of Q-branch lines belonging to different

rotational transitions. The second (less known) effect is on the intensity of spectral lines. In particular, the coupling introduces a correction, called Herman–Wallis (HW) factor, which quantifies the centrifugal distortion in the line-strength of small rotating Raman-active molecules [21,22]. In the past decades, different expressions of the HW factor have been proposed [23–26] and, lately, they have been scrutinized to single out the conceptual model that guarantees the most accurate interpretation of H2 CARS spectra at high temperatures [20]. The findings of this work are fundamental to thermometric uses of CARS applied to molecular hydrogen, because the temperature-dependent rotational distribution of molecular population can be significantly different depending on the HW factor adopted. Note that this does not hold true for heavier molecules, like N2, that are less subjected to centrifugal distortion and, for this reason, provide more reliable results despite the more complex elaboration of their spectra. In other words, precise understanding of H2 CARS measurements cannot overlook the troubles created by unreliable choices of the HW factor (or, even worse, no choice at all). Bearing this in mind, we could finally affirm that the above-mentioned unreliability found for H2 CARS measurements [7,13] could be helpful to underline the importance of the HW factor, which was undoubtedly underrated in the specialized literature before Ref. [20]. In an effort to develop a better understanding of CARS diagnostics in combustion, this paper is about thermometric uses of H2 CARS measurements viewed in light of all the considerations made so far. The layout is organized as follows. First of all, it will be demonstrated that relative uncertainties on the HW factor are more critical than those on Raman linewidths (Section 2). This higher sensitivity will then be aimed at the description of temperature variations expected for modifications of the HW factor (Section 3). In conclusion, to deepen the whole subject, available expressions of the Q-branch HW factor will be introduced (Section 4) for the purpose of evaluating their relative thermometric potential to account for the measured inaccuracies of H2 CARS spectroscopy (Section 5). 2. Dependence of CARS spectra on HW factor To get at the central subject of this study, it is necessary to appreciate first the influence of the HW factor on CARS spectra. This is self-explanatory once that the strict correspondence between gas temperatures and relative heights of spectral peaks is taken into consideration. However, throughout this work, we will avoid the intricacy of the physical models behind the lengthy calcula-

M. Marrocco / Proceedings of the Combustion Institute 32 (2009) 863–870

tion of HW factors. Actually, these can be found elsewhere [21–23,26–28]. We focus, instead, on the possible alterations of the CARS intensity depending on different choices of the HW factor and, in doing so, we consider the most common situation of a CARS signal I(xa) at the antiStokes frequency xa generated by two quasimonochromatic pump lasers and a broadband Stokes laser [1,5–13,15–18,20]. Under this ordinary condition, the starting point is represented by the so-called Kataoka– Teets integral [1,29,30] Z Iðxa Þ ¼ K jv12S j2 L1 ðx1 ÞL2 ðx2 ÞLS ðxS Þdðx1 þ x2  xS  xa Þdx1 dx2 dxS :

ð1Þ

where, with an obvious meaning of the notation, K is a proportionality constant, while Lp(xp), with p = 1.2, and LS(xS) indicate, respectively, the lineshapes of the pump and Stokes lasers that are spectrally convolved with the non-linear susceptibility written as v12S = vNR + [v1S(x1  xS) + v2S(x2  xS)]/2 and made of a non-resonant and featureless contribution vNR added to the two possible spectral components v1S and v2S taken, respectively, at the frequency difference x1  xS and x2  xS. The delta function in Eq. (1) ensures the energy conservation in the non-linear process that transforms the three incoming waves at frequencies xp and xS into a wave at the anti-Stokes frequency xa. The key dependence on the HW factor appears in the resonant susceptibilities v1S and v2S, which are described as a summation over the possible transitions q between the initial and final rovibrational states (v, J) and (v0 , J0 ). In the isolated lines approximation (valid for H2), we then have X aq =ðXq þ xS  xp  iCq Þ: ð2Þ vpS ðxp  xS Þ ¼ q

In Eq. (2), the Raman frequency and the linewidth (namely the half width at half height) are, respectively, indicated as Xq and Cq. The amplitude aq denotes the strength of the associated transition and takes in the dependences on the population difference and the Raman cross-section (or/oX)q. More precisely, aq is given as   N Dq or aq ¼ k 4 ð3Þ xS oX q where k is a constant, N is the total population and Dq is the fractional population difference [1]. The cross-section is the quantity that incorporates the HW factor Fq   or ¼ jx4S Uða2 ; b2 ; #ÞF q ð4Þ oX q where j is a constant, U(a2, b2, #) is a function of the mean value of the derived polarizability tensor

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a2 and the anisotropy tensor b2 [1,31]. In case of the strongest vibrational Q-branch (or Dv = 1), within which the rotational state does not change (DJ = J0  J = 0), the Raman cross-section takes the following form     or 4 2 2 ð5Þ ¼ Kx4S a0 þ S J ;J b0 F ðJ Þ oX Q 45 where the terms a0 and b0 are the isotropic and anisotropic invariants of the static dipole polarizability derivatives, and SJ,J are the Placzek–Teller coefficients [1,31]. In Eq. (5), it is made clear that the HW factor F(J) now depends on the rotational level J only. Assuming broadband Stokes profiles and degenerate pump beams with linewidths much smaller than the Raman widths CJ (monochromatic approximation), the peaks of I(xa) are found in correspondence with the maxima of the rotational transitions at xa,J 2 2 Iðxa;J Þ ¼ CLS ðxS;J ÞD2J ½a02 þ S J;J b02 a  F ðJ Þ=CJ

b0a

ð6Þ

0

In Eq. (6), C is a constant, ¼ 4b =45 and xS,J is the Stokes frequency of the Jth anti-Stokes line. This equation is useful to reach a first important conclusion: the dependence of the peak height I(xa,J) on the HW factor F(J) is characterized by a power law that is visibly stronger than the dependence on the Raman width CJ. Since much effort to explain the unreliability of H2 CARS thermometry was concentrated on the J dependence of CJ [10,15,16,19] which is, nevertheless, sufficiently understood in terms of the Doppler broadening as long as the pressure remains close to the atmospheric value [1,5–20], we are left with the question of what is the contribution of F(J) to the claimed inaccuracies of H2 CARS thermometry? To answer it, it is now compulsory to clarify the role of the HW factor relatively to the thermometric use of CARS. 3. Dependence of temperature predictions on HW factor Having shown that the HW factor can have a large influence on CARS measurements, we are ready to look more closely into the main thesis of the current paper. To that end, leaving aside the complications of collision dependent linewidths [10,15,16,19], we will consider the case of hydrogen combustion at atmospheric pressure. On the other hand, this corresponds to the perspective of those experimental works where the inaccuracy of H2 CARS thermometry was most notably addressed [7,13]. In line with this intent, we will make the same assumptions of Eq. (6) and, in this manner, the answer to our problem will be conveniently set out by means of the socalled Boltzmann plot analysis [1,7,9,11–14,18],

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according to which the energies f(J) of the rotational lines are plotted as a function of the variable b ¼ k B =ðhcÞLogfDJ =½gs ð2J þ 1Þg

ð7Þ

where gs is the nuclear spin statistical weight (gs = 1 for even J and gs = 3 for odd J), (2J + 1) is the rotational degeneracy and the quantity DJ is related to fractional population difference through Eq. (6), or DJ ¼ DJ =C ¼ ½Iðxa;J ÞCJ = LS ðxS;J Þ1=2 =½ða02 þ S J;J b02 a ÞF ðJ Þ. Under the hypothesis of thermal equilibrium, the fractional population difference obeys the proportionality relationship DJ / gs(2J + 1) exp[hcf(J)/kBT] and, when inverted, the Boltzmann plots are derived from the previous equations. This means that the experimental points should make up the linear dependence f ðJ Þ ¼ const þ T b

ð8Þ

whose angular coefficient is just equal to the gas temperature T. In Eq. (8), it is possible to note that the temperature determination depends on the possible change in the HW factor contributing to b together with other experimental parameters. In other terms, the temperature variation dT caused by the associated variation dF of the HW factor can be calculated as dT =T ¼ k B =ðhcbÞðdF =F Þ:

ð9Þ

Generally speaking, Eq. (9) suggests that the measured temperature is logarithmically related to the HW factor, or T = T0/[1 + Log(F)/(hc/kBb0)] where b0 is Eq. (7) calculated for a rigid rotor (i.e. F = 1) and T0 is its corresponding temperature. Since hc/kBb0 is positive, we find that measured temperatures are greater than T0 for F < 1 and vice versa in the opposite context of F > 1. The logarithmic behavior is only indicative and any less qualitative analysis of such problems cannot go further without the knowledge of the proposed expressions for F. This is why we now specialize the problem to realistic considerations about the J dependence of the HW factor. 4. Different expressions of HW factor The HW factors suitable for the correction of Q-branch Raman signals of H2 are at least three

(without considering the manifestly wrong conceptual option of those works where the problem of centrifugal corrections to H2 CARS intensity is not even mentioned [1,5,6,8–12,15,16,18]). These factors have been examined in Ref. [20] and, in the present Section, they are revised very briefly. Traditionally, the pioneering paper reporting on the calculation of Q-branch HW factors is attributed to James and Klemperer [23]. Their approach is based on a perturbative method and leads to the result FJK = 1  3c2J(J + 1)/2 with the parameter c measuring the rovibrational coupling, i.e. c = 2Be/xe (Be and xe are, respectively, the rotational constant and the vibrational frequency). Although the factor FJK dates back to almost 50 years ago, it is still in use for thermometric purposes [13,17] and, for this reason, cannot be disregarded. Other experimental investigators [7,14] are, instead, more familiar with the correction of Luthe et al. [24], whose result is derived from Ref. [26] and is written as FLBY = [1  3c2(a1 + 1)J(J + 1)/4]2, with a1 the first Dunham coefficient [32]. The factors FJK and FLBY, or no correction at all (i.e. F = 1) are enough to encompass the whole set of experimental works based on Q-branch CARS of molecular hydrogen. Nonetheless, Ref. [20] suggests a third HW factor as derived by Tipping and Bouanich [25], that is FTB = 1  [3(a1 + 1)/2  4p2/p1]c2 J(J + 1), with p1 and p2 the coefficients of the polarizability expansion. The numerical values of these factors are given in Table 1 for the first ten rotational levels and the immediate conclusion is that their differences are significant. To start with, FJK decreases with the rotational level J, whereas the negative value of the first Dunham coefficient is responsible for the opposite behavior of FLBY and FTB. The reason of this initial discrepancy is simple. The factor FJK is incomplete because there are missing terms of order c2 that are instead present in the expansion leading to FLBY and FTB. On the other hand, the two more complete factors do not coincide. First of all, FLBY contains a term of order c4 that is absent in FTB. Secondly, the contribution of the polarizability expansion is not contemplated in FLBY. This means that, although FLBY has higher order terms, FTB has an additional contribution of order c2. To sum up, the differences in Table 1 can be explained either totally or largely by the differ-

Table 1 Comparison of HW factors for the Q-branch of molecular hydrogen HW factors

J=1

J=2

J=3

J=4

J=5

J=6

J=7

J=8

J=9

J = 10

FJK FLBY FTB

0.998 1.001 1.003

0.993 1.004 1.009

0.986 1.008 1.018

0.977 1.014 1.030

0.966 1.021 1.046

0.952 1.030 1.064

0.936 1.040 1.085

0.917 1.051 1.109

0.897 1.064 1.137

0.874 1.078 1.167

The known parameters are cH2 ¼ 2:77102 , a1 = 1.6, while p1 and p2 have been obtained from the polarizability expansion based on Ref. [33].

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ences in the conceptual approaches to the issue, but the three factors are mutually exclusive. According to the conclusion of Ref. [20], the most reliable factor is FTB, which shows the highest numerical values (see Table 1) and has never been used in H2 CARS spectroscopy. This means that, in agreement with the qualitative logarithmic dependence found at the end of Section 3, ordinary thermometry (with HW factors smaller than FTB) is affected by a systematic error that provokes higher evaluations of rotational temperatures. Recalling that the experimental thermometric disagreements are unanimous in their observations of an excess of measured H2 temperatures [7,13], we are led to hypothesize that the deviation of FLBY and FJK from FTB can be responsible for a good part of the imprecision intrinsic to H2 CARS thermometry. To verify this conjecture, we now proceed with the comparisons of temperatures obtained alternatively with the three factors FJK, FLBY and FTB. 5. Thermometric comparisons The qualitative reasoning followed so far has pointed out that the thermometric inaccuracy of collision-free vibrational H2 CARS spectroscopy is related to the differences among FJK, FLBY and FTB. In order to take this argument to its logical conclusion, it is surely instructive to see how the accuracy of past experimental works can be improved by using the new factor FTB instead of FJK or FLBY. For this reason, we will consider the CARS measurements of Refs. [7,13], where important experimental disagreements have been more clearly demonstrated in comparison with alternative techniques. First of all, it should be remarked that these two research groups used two incompatible factors: FJK and FLBY, whose large difference has gone unnoticed for years. This fact alone suggests that something unclear as regards the issue of CARS interpretation of H2 spectra should lie behind the understanding of the vibration–rotation coupling and, in an attempt to clarify the whole matter, we could take the easy option of Refs. [7,13] summarized in the Boltzmann plot analysis of Eq. (8). Suppose then that the energies of the rotational levels are correctly described by

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T ¼ df =db ¼ ½f ðJ 2 Þ  f ðJ 1 Þ=½bðJ 2 Þ  bðJ 1 Þ ð11Þ for a given HW factor contained in the b variable. But, at the same time, we find T  ¼ df =db ¼ ½f ðJ 2 Þ  f ðJ 1 Þ=½b ðJ 2 Þ  b ðJ 1 Þ ð12Þ *

for another choice of the HW factor (with b – b) and, combining Eq. (11) with (12), we end up with the result that the two different temperatures T and T* can be simply related through their corresponding HW factors F(J) and F*(J), T  ¼ T =½1  k B =ðhcÞLogfF ðJ 2 ÞF  ðJ 1 Þ=½F ðJ 1 ÞF  ðJ 2 Þg ð13Þ

Choosing any combination of rotational levels, the plot of Eq. (13) is always the same and equal to what is reported in Fig. 1. This is arranged in such a way that F* coincides with FJK, whereas F is given by FTB. The dashed line bisecting the graph corresponds to the temperature T obtainable with the HW factor FTB. Considering the other factor FJK leads instead to a relevant increase in the temperature prediction T * (continuous line). Such increase is calculated in the inset and it is seen that around 2000 K we expect a thermometric excess on the order of 100 K. This result can be compared with the experimental disagreement found in the work of Hancock et al. [13], where atmospheric pressure hydrogen/air flame temperatures were tested against nitrogen CARS

f ðJ Þ ¼ Bv J ðJ þ 1Þ  Dv J 2 ðJ þ 1Þ2 þ H v J 3 ðJ þ 1Þ3 ð10Þ (see for example Ref. [24] for numerical values of the vibration dependent rotational constant Bv and its corrections Dv and Hv at higher orders). It is obvious that Eq. (10) is not altered by a different choice of the HW factor. This means that, assigning two rotational levels J1 and J2, the angular coefficient in Eq. (8) is

Fig. 1. Plot of Eq. (13) with T and T * determined for F = FTB and F * = FJK, respectively. The dashed line, corresponding to temperatures predicted with FTB, bisects the graph. Above, the deviation T *  T is reported.

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thermometry. In their work, detailed equilibrium code calculations of flame temperatures confirmed the experimental analysis based on N2 CARS. On the contrary, the authors of Ref. [13] declared an overestimation of real temperatures of about 70 K on average when H2 was employed. Since they used the oldest factor FJK, it is then not unexpected that measurements with molecular hydrogen were problematic (actually, this problem is again present in more recent measurements [17]). In particular, looking at Fig. 9 of Ref. [13], it is seen that, away from stoichiometry, where hydrogen concentration is sufficiently high, the mismatch is exactly on the order of 100 K as found in the plot of Fig. 1 of this work. Vice versa, the agreement between the analysis of this paper and the hydrogen CARS temperatures determined by Hancock et al. is less satisfactory near stoichiometry, where the expected temperature are above 2200 K. In this condition, however, the average measurements show significant oscillations that are suggestive of probable alterations caused by increasingly lower concentrations of hydrogen. Another application of Eq. (13) can be found in Ref. [7]. There, it is convincingly argued that an unexplainable excess of H2 CARS temperatures occurs when these are compared with pyrometric temperatures (Fig. 2 of Ref. [7]). Differently from Hancock et al. [13] or Roy et al. [17], the authors of Ref. [7] used the more modern factor FLBY and, in addition, they correctly emphasized that ‘‘the line strengths of Raman transitions depend. . . on the correction factor for the vibra-

Fig. 2. Plot of Eq. (13) with T and T * determined for F = FTB and F* = FLBY, respectively. The dashed line, corresponding to temperatures predicted with FTB, bisects the graph. Above, the deviation T *–T is reported. The layout of the main plot is made to facilitate the comparison with Fig. 2 of Ref. [7].

tion–rotation interaction” [7], but they did not realize that the HW factor is one of the critical points of H2 CARS spectroscopy at high temperatures. For this reason, they concluded, with reference to the poor thermometric comparison between pyrometry and CARS, that ‘‘the origin of this difference is unclear at present and could not be attributed unambiguously to the pyrometric or the CARS measurements” [7]. But, if Eq. (13) is adjusted to F * = FLBY, it possible to verify that the use of FLBY in place of FTB introduces an increase of temperatures similarly to what is seen in Fig. 2 of Ref. [7], where the experimental points suggest a trend lying above the ideal line. This is shown in Fig. 2 of this work. Further revisions of known experimental results could be carried out in agreement with what we learnt above. For instance, Fig. 11 of Ref. [8] shows the molar fraction of H2 as a function of the measured rotational temperature. The immediate comment on this figure is that the temperatures are surely overestimated because the authors did not consider any line-strength correction (or F = 1, as though H2 was a rigid rotor). But, taking into account the physics of the vibration–rotation interaction through the HW factor FTB, more realistic thermometric CARS measurements make the experimental points scale toward lower temperatures with the final result that the measurements approach more clearly the theoretical curve (this result is not shown here). Having analyzed the possible improvements on H2 CARS thermometry, one may wonder why the subtleties concerning the HW factor were not examined before. As a matter of fact, anti-Stokes Raman signals of molecular hydrogen have been known for decades [5–18] and the confusing uses of different HW factors (including the unitary factor for a rigid rotor) did not arouse any suspicion for many years. The reason of this lack of attention has a simple explanation. The experimental H2 CARS spectra are very regular and, in principle, they are equally reproducible by theory regardless of the HW factor adopted. The sole difference is in the temperature corresponding to the calculated spectrum fitting the measurement. This is why the problem can only be brought up in thermometric comparisons with alternative techniques (as shown in Refs. [7,13]) or in experiments like in Ref. [20]. However, to investigate the relationship between spectra obtained with different HW factors, it can be proven that simulated CARS signals IJK, ILBY and ITB (respectively, associated with FJK, FLBY and FTB) become indistinguishable for adequate temperature shifts. For example, this can be shown by synthesizing some spectra in agreement with the prescription of Ref. [34] and for physical conditions defined somewhere else [20], i.e. broadband CARS with multi-mode and degenerate pump laser probing a H2/air flame at atmospheric pressure and

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temperature close to the adiabatic limit (i.e. TA = 2380K). Keeping constant the temperature T = TA of the spectrum ITB while varying the temperature of the other two spectra IJK and ILBY, it is found that these two approximate very well the spectrum ITB for temperature shifts of about 200 and 60 K, respectively. This means that, in principle, an experimental CARS spectrum can be well represented by IJK, ILBY or ITB and, of course, this indetermination is paid in terms of consequent inaccuracies of temperature measurements. A situation of this kind is depicted in Fig. 3, where large deviations from ITB (calculated as the best fit of experimental measurements of Ref. [20]) are found when the factor FJK is used at the same temperature T = TA of ITB. However, when the temperature is chosen as a free parameter of the simulations, the deviation becomes increasingly smaller and reaches a minimum for T = 2575 K. This value is remarkably higher than the possible temperature accepted for the physical conditions of the flame and it must be discarded. Similar considerations can be made for the factor FLBY (Fig. 4). This time the deviation is less pronounced, but it is still possible to appreciate the differences between the theoretical predictions based on two different temperatures. Once again, the realistic flame temperature of 2380 K produces relevant deviations, whereas the temperature minimizing the difference is sufficiently higher than the maximum value achievable in the experiment. 6. Conclusions In summary, this paper addresses the subject of H2 CARS thermometry in reference to the known inaccuracies claimed by some authors for measurements on flames at atmospheric pressure [7,13]. Since collision broadening of spectral lines plays a marginal role at modest gas pressures, the

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Fig. 4. Relative deviation of CARS intensity ILBY from ITB. In the simulations, the physical parameters are taken from Ref. [20].

weak understanding of the vibration–rotation interaction is identified as a possible reason for the lower thermometric reliability in comparison with other experimental techniques. This idea stems from a recent work where different models of the vibration–rotation coupling were analyzed [20]. Such coupling, otherwise called Herman– Wallis (HW) factor, has a visible effect on spectroscopic line intensities and, in the present context, it is pointed out that old and contrasting expressions of the HW factor are commonly used in the interpretation of vibrational H2 CARS measurements. More importantly, the analysis based on Boltzmann plots shows that the new HW factor of Tipping and Bouanich [25] can correct the thermometric inaccuracy of H2 CARS. This explains why the spectroscopic technique, widely recognized as a precise tool when applied to heavier molecules, was found less effective when applied to hydrogen [7,13]. Finally, it must be added that the whole issue treated in this paper emphasizes a topic that is lacking in fundamental textbooks devoted to linear and non-linear Raman spectroscopy [1,31].

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Fig. 3. Relative deviation of CARS intensity IJK from ITB. In the simulations, the physical parameters are taken from Ref. [20].

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