A simple formulation of the CH 2 O photolysis quantum A simple formulation of the CH 2 O photolysis quantum yields

New expressions for the various wavelength – dependent photolysis quantum yields of CH 2 O, Φ j , are presented. They are based on combinations of functions of the type The parameters A i , b i , and λ 0 i which have a physical meaning are obtained by ﬁts to the measured data of the Φ j i available from the liter- 5 ature. The altitude dependence of the photolysis frequencies resulting from the new quantum yield expressions are compared to those derived from the Φ j recommended by JPL and IUPAC.


Introduction
Formaldehyde, CH 2 O, is an important trace gas in the atmosphere. It is formed as 10 an intermediate in the oxidation of methane and non-methane hydrocarbons, and destroyed by the reaction with OH and by photolysis in the near ultraviolet. The photolysis involves several channels. Following the excitation (Reaction R1), CH 2 O * can decay into purely molecular products (Reaction R2), or into products that in the atmosphere lead to the eventual formation of hydroperoxy radicals, HO 2 , (Reactions R3 and R4).

15
The quenching Reaction (R5) and fluorescence (Reaction R6) can influence the quantum yields of the product channels. The measured wavelength dependences of the quantum yields are usually given in tabular form (see e.g. Atkinson et al., 2006;IUPAC, 2013) or as a fit by a fourth order polynomial (see Sander et al., 2011). To provide a more handy tool for atmospheric modeling we propose to use combinations of energy dependent functions of the type (2) 5 to fit Φ mol and Φ rad . These functions are well-suited to map smooth transitions. They allow to include pressure and temperature dependences. And the resulting parameters are few and have a physical meaning: in particular 1/λ 0 corresponds to the threshold energy of the respective reaction; b describes the width of the transitions. Moreover, the formalism should also provide a useful template for the formulation of the analogous 10 Φ i for the isotopologues of formaldehyde. Our analysis of the quantum yields will be based on the data filed by JPL (Sander et al., 2011) andIUPAC (2006) omitting all measurements with an obvious bias. Likewise, only publications of independent measurements were taken into account, i.e. if measured data appear in several publications by the same authors, only the latest data 15 were considered.
First, in Sects. 2 to 4, we will fit the measured wavelength dependences of the various Φ separately and compare them to those reported in the literature. In a second step, after having convinced ourselves that the parameters from the separate fits that should correspond to each other are indeed similar in value, we attempt a simultaneous fit of 20 all Φ in Sect. 5.

The quantum yield of the radical channel
Most publications on the formaldehyde photolysis deal with the radical channel (Reaction R3) -notably: Horowitz and Calvert (1978), Moortgat et al. (1983) (2002), Gorrotxategi et al. (2008), andTatum Ernest et al. (2012). Nearly all of these measurements were made at room temperature, and experiments and theory indicate that there is no pressure dependence of Φ rad . We, therefore, assume all these data to be comparable and their variance attributable to experimental error. Thus all these data are combined in Fig. 1 without any weighing. Smith et al. (2002) attributed 5 some of the variance in their data to a line structure in Φ rad . The possibility of a line structure appears corroborated by the data of Tatum Ernest et al. (2012), which show a strong feature in Φ rad at 321 nm. For comparison the data of Tatum Ernest et al. are also shown in Fig. 1, but they are not used for the fit.
To fit the experimentally observed wavelength dependence of Φ rad we use a combi-10 nation of two functions of the type mentioned above, one for the decay of Φ rad to longer wavelengths at about 328 nm, the other for the decay towards shorter wavelenghts at 277 nm. To obtain the fit parameters and their errors a simplex algorithm (Nelder and Mead, 1965) is used in combination with a bootstrapping method with 2000 arbitrary removals of 20 % of the data. The result is given by Eq.
(3), with λ in nm: It is also shown in Fig. 1. Equation (3) holds primarily for room temperature. The respective parameters will be labelled by the subscripts l, s. The λ 0 mark the inflection points in the decays: λ 0,l = 328.0 nm; λ 0,s = 278.4 nm. The corresponding b define the wavelength interval 20 within which the decrease takes place. Owing to the scatter in the measured Φ rad data all these parameters exhibit an uncertainty range. The estimated 1σ errors are listed in Table 1 along with the values of the parameters. We note that λ 0,l closely corresponds to the dissociation energy of the H-CHO bond namely 30328.5 cm −1 or 329.7 nm (Terentis et al., 1998) and that λ 0,s approximately corresponds to the heat of formation of Reaction (R4) namely 423 kJ mol −1 or 283 nm (Sander et al., 2011).  Moortgat et al. (1983) have also measured the wavelength dependence of Φ rad at 220 K. Given the experimental variance in those admittedly sparse data, Eq. (3) also fits the measured Φ rad at 220 K quite well (not shown here). Thus, as far as the experimental data on Φ rad are concerned, Eq. (3) covers the temperature range of 220 to 300 K relevant for atmospheric modeling and there is no immediate need to introduce 5 a temperature dependence. On the other hand, theoretical considerations suggest the inclusion of the internal energy of the CH 2 O molecule, and this can be easily done: following Troe (2007) one can add a term 3kT (appropriately scaled) to 1/λ in the left hand term of Eq. (3). In Sect. 5, Discussion, we will investigate the impact of this T dependence (see Eq. 12) on the altitude profile of the respective photolysis frequency. 10 In principle, another weak T dependence can arise through the parameter b. That dependence could be easily accommodated by replacing b by (b 0 + b 1 T ) should future Φ rad measurements provide enough information to warrant such a step.
The present formulation of Eq. (3) with constant parameters b -i.e. b independent of λ -forces the decrease to be nearly symmetrical around the respective λ 0 . This is 15 not necessarily realistic. Again, if future measurements or theoretical considerations should prove the need, an asymmetry could be easily accommodated by allowing b to depend on λ.
Finally, we note, that a line structure could be superimposed on Eq.
(3) without difficulty. For the moment we refrain from doing so for two reasons. (1) As Tatum Ernest 20 et al. (2012) showed even the strong feature in Φ rad at 321 nm would change the photolysis frequency in the atmosphere, j rad , by only −4 %, because it coincides with a strong minimum in the absorption coefficient of CH 2 O. Thus the error possibly introduced by its neglect is comparatively small (see discussion below). (2) The measurements of Φ rad by Smith et al. (2002), andGorrotxategi et al. (2008) contain data points close 25 to 321 nm which fall right on the average Φ rad given by Eq. (3). They were made with sufficient resolution to resolve the feature at 321 nm and are therefore somewhat at variance with the finding of Tatum Ernest et al. (2012).  Figure 1 also contains the recommended wavelength dependences of Φ rad given in the evaluations by JPL (Sander et al., 2011) andIUPAC (2006). The reason for the choice of IUPAC (2006) over IUPAC (2013) is that the former data, which were first published in 2002 and remained in the internet until 2012, had many users in the past and possibly still has at present. Further included is the theory-based dependence derived 5 by Troe (2007); it covers only the restricted wavelength range from 310 to 350 nm. As a quantitative measure of the quality of these fits we here add the coefficient of determination c. In the present case this is identical to the correlation coefficient between fitted and measured data. These correlation coefficients are: c = 0.821 (IUPAC, 2006); c = 0.840 (Troe, 2007); c = 0.898 (JPL, 2011); and c = 0.905 (this work); that is the 10 quality of these various fits does not differ drastically.

The total quantum yield
There are more direct measurements for Φ tot and its dependence on λ than for Φ mol . To obtain higher accuracy we, therefore, first obtain a fit for Φ tot (λ) and then use Eq. (1), i.e. Φ mol = Φ tot -Φ rad for a fit of Φ mol (λ). That fit is later compared to the measured 15 dependence of Φ mol on λ.
The available measurements of Φ tot (λ) at 300 K temperature and 1013 hPa pressure are reproduced in Fig. 2. The values of Φ tot at 355 and 353 nm were obtained by interpolating the respective Stern-Volmer plots given by Moortgat et al. (1979Moortgat et al. ( , 1983 to the pressure of 1 atm. The Φ tot values at λ < 340 nm are pressure independent. The 20 measured Φ tot (λ) exhibits three regions: a plateau between 290 and 330 nm, a steep decrease to zero at longer wavelengths, and a weak decrease to Φ tot ∼ 0.8 at shorter wavelengths. The average measured Φ tot in the plateau is 1.06 ± 0.09 -not significantly different from 1 -the maximum possible value. Therefore, in the fit we will fix this value to unity. The separation of the two decreases by a plateau with Φ tot = 1 also The measurements in Fig. 1 indicate that Φ rad vanishes at λ > 340 nm; at those wavelengths Φ tot becomes identical to Φ mol . Moreover, tunneling processes extend the photolysis of CH 2 O to H 2 and CO well beyond the threshold energy of about 350 nm (Troe, 2007). In this energy regime the rate of decay into the molecular channel decreases to values where collisional quenching of the excited formaldehyde molecule 5 (Reaction R5) begins to compete. Consequently, Φ mol and Φ tot become pressure dependent. Based on theoretical modeling and comparison with the data of Moortgat et al. (1978Moortgat et al. ( , 1983, Troe (2007) proposed a Stern-Volmer formulation for Φ mol for λ > 340 nm: 10 with λ 0 = 349 nm; c = 0.225 nm −1 for λ > λ 0 and c = 0.205 nm −1 for λ < λ 0 and M the number density of the bath gas. M 0 = 2.46×10 19 cm −3 , the number density at 1013 hPa pressure and 300 K temperature. Troe (2007) also pointed out that on theoretical grounds the temperature dependence of Φ mol should be small compared to the experimental uncertainties and thus negligible at this stage. This is somewhat at variance 15 to the measurements by Moortgat et al. (1983) which seem to indicate such a dependency, albeit with large uncertainties. Since Φ tot equals Φ mol for λ > 340 nm where nearly all of the change in Φ tot with wavelength is located, and since Eq. (4) approaches unity for λ < 330 nm, Eq. (4) should also provide a good approximation for Φ tot (λ). In fact we could use it with its 20 current parameters as our intended fit (see Fig. 2).
However, we prefer to formulate our fit in terms of energy, i.e. 1/λ. Moreover, a direct fit to the data in Fig. 2 will merge the pre-exponential factor in Eq. (4) with λ 0 . So, instead of using Eq. (4) we will fit Eq. (5) to the data at λ > 310 nm in Fig. 2: Our fit yields the parameters λ 0,l and b l of Table 2. In this case λ 0,l has a somewhat different meaning than before. Here, λ 0,l not only depends on the threshold energy of the reaction involved, but also on the quenching efficiency with which energy is drained from the excited CH 2 O molecule. But as before λ 0,l represents the inflection point in the decrease of Φ, at least for M = M 0 .

5
The fit of Φ tot for the short wave decrease relies on our model Eq.
(2) and yields the parameters listed in Table 2.
The equation for Φ tot (λ) over the full wavelength range therefore is: with λ given in nm. 10 We have not been able to find a ready explanation for the experimentally observed weak decrease of Φ tot at shorter wavelengths in the literature. We note, however, that λ 0,s = 284.3 corresponds closely to the heat of formation for reaction Eq. (4) (see Sect. 2).
Following the arguments by Troe (2007) we assume the temperature dependence of 15 Φ tot (λ) to be negligible. But here again, our fitting functions could readily be modified to include a T dependence. Φ tot (λ) from Eq. (6) is also shown in Fig. 2. It compares favorably to the measured data of Φ tot . For additional comparison Fig. 2 also contains the recommended wavelength dependences of Φ tot given in the evaluations by JPL (Sander et al., 2011) and

The quantum yield of the molecular channel
Since Φ mol is given by Φ tot -Φ rad , it could be simply obtained from the difference of Eqs. (6) and (3). Explicitly: On the other hand Φ mol can be obtained by a direct fit to the measured data. This requires a combination of only three functions of the Eq.
(2) type and the fit results in: Equation (8) makes the implicit assumption that the short wave decreases in Φ tot and Φ rad (second and fourth term in Eq. 7) have the same λ 0,s and b s . The estimated 1σ errors along with the fit parameters are listed in Table 3. In Fig. 3 Φ mol (λ) from Eq. (8) is compared to the measured data on Φ mol (λ). The 15 latter consist of direct measurements of Φ mol by Moortgat et al. (1979Moortgat et al. ( , 1983, and data 7248 Introduction

Tables Figures
Back Close

Full Screen / Esc
Printer-friendly Version

Interactive Discussion
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | based on measured Φ tot and Φ rad by Horowitz and Calvert (1978). The agreement of Eq. (8) with the measurements is quite reasonable. For further comparison Fig. 3 also includes the recommendations by JPL (Sander et al., 2011) and IUPAC (2006)

Simultaneous fit of Φ rad , Φ mol , and Φ tot
A comparison of the parameters and their errors obtained from the individual fits of the various Φ suggests that the λ 0,s , λ 0,m , λ 0,l and b s , b m , b l in a given fit equation do not differ significantly from the corresponding parameters in the others. We, therefore, 10 felt justified to attempt a simultaneous fit of all Φ. In this attempt we assume that the corresponding λ 0 and b parameters in the various equations for Φ are indeed identical. We further assume that Φ tot reaches a maximum value of 1 and that Eq. (1) holds. With these assumptions the total number of fit parameters for all three Φ together reduces to 9. The simultaneous calculation of the 9 unknown parameters results in the equations 15 for the Φ i listed in Table 4, The coefficients of determination together with the function parameters and their estimated 1σ errors are tabulated in Table 5. The functions of Table 4 differ somewhat, but hardly significantly, from those given by Eqs. (3), (6), and (8)  concerns the temperature dependence of Φ. Given the experimental uncertainties we have refrained from providing T dependences for the Φ's. But there are temperature dependences in the literature, which could be incorporated in our formulation (Atkinson et al., 2006;Troe, 2007;Sander et al., 2011). Below we will incorporate such a temperature dependence in Φ rad to test the sensitivity of the corresponding photolysis 5 frequencies of CH 2 O to the vertical temperature profile.
In addition the question of line structure in Φ rad needs eventually to be resolved.
Of major interest to the atmospheric chemist is the impact of this new formulation of Φ on the atmospheric photolysis frequencies of CH 2 O. That photolysis frequency j is given by: (9) i.e. it also depends on the absorption cross-section, σ(λ), of CH 2 O, and the local actinic photon flux density F λ (λ). For our calculations of j we will use the absorption spectrum measured by Gratien et al. (2007). It is, by the way, also slightly temperature dependent; the respective function can be found in Röth et al. (1997). The atmospheric 15 actinic photon flux density consists of down-welling and up-welling contributions, and depends of course on the solar zenith angle and altitude. It was calculated by the radiative transfer program ART (Röth, 2002) using the extraterrestrial solar flux from WMO (1985). All three factors under the integral strongly vary with wavelength, λ. (To various degrees they also vary with altitude.) As an example Fig. 4 shows σ(λ), F λ (λ), and 20 Φ mol (λ) at 30 km altitude and 33 • solar zenith angle. We particularly notice the sharp cutoff in F λ (λ) around λ = 320 nm caused by the absorption of solar UV in the ozone layer at lower wavelengths. This means that below 30 km altitude the exact form of the Φ i at λ < 300 nm has little influence on the various photolysis frequencies. Figure 4 further indicates how much the long-wave decrease of Φ mol is shifted towards longer 25 wavelengths at the air density at 30 km altitude. In fact, this shift is so large that the 7250 Introduction long-wave cutoff of the integrand in Eq. (9) is no longer determined by Φ mol , as it is at low altitudes, but rather by the absorption spectrum of CH 2 O. Hence, at altitudes above 30 km the exact form of the decrease in Φ mol and Φ tot at the longer wavelengths has no influence on the respective photolysis frequencies.
Given the Φ i from Eqs. (11) to (13), σ(λ) from Gratien et al. (2007) along with vertical 5 temperature and density profiles of the US standard atmosphere (NOAA, 1976) we can calculate the vertical profiles of the photolysis rates. They are shown in Fig. 5. The shaded areas mark the 1σ error bounds of the j i profiles based on the errors of the fitting parameters for Φ i given in Sect. 5. As to be expected, all j i increase with altitude. In the case of j rad that increase is essentially due to the vertical change in F λ (λ), since our Φ rad is neither temperature nor pressure dependent and thus independent of altitude, and the slight temperature dependence of σ(λ) makes a minor contribution only. j tot and j mol , however, are significantly modified by the density dependence in Φ mol . In Fig. 5 we also demonstrate the impact of a possible temperature dependence in 15 Φ rad . The temperature dependence is introduced by adding the term (300−T ) (3k/hc) in the appropriate dimensional units to 1/λ in the first term of Eq.
That means: only the long-wave decay in Φ rad is considered to be temperature depen-20 dent. Here k is the Boltzmann constant, h the Planck constant, and c the speed of light. As Fig. 5 shows a temperature dependence of this size clearly has a significant impact on j rad and by virtue of Φ mol = Φ tot − Φ rad also on j mol . The effect is largest at around 15 km, the height of the temperature minimum, and about −9 % for j rad , respectively ca. +6 % for j mol . The temperature at 15 km is 220 K, i.e. the temperature shifts in j rad and 25 j mol correspond to a temperature difference of 80 K. Apparently a correct formulation 7251 Introduction

Tables Figures
Back Close

Full Screen / Esc
Printer-friendly Version

Interactive Discussion
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | of the T dependence of Φ rad could lead to a significant improvement in the predicted vertical profiles of j rad and j mol . j tot remains unaffected by the proposed temperature dependency. In fact, even assuming a temperature dependence of the kind above for the long-wave decay of Φ tot would have comparatively little impact on the j tot profile. It would be masked by the air 5 density dependence of Φ tot : just as at lower densities, the exact form of the long-wave decay in Φ tot no longer influences j tot , so can its temperature dependence no longer influence j tot .
Finally, in Fig. 6, we compare the photolysis frequencies based on this work's quantum yields to those calculated with the quantum yields recommended by IUPAC (2006) 10 and JPL (Sander et al., 2011). The JPL recommendation includes an explicit temperature dependence for Φ rad . In addition, both, JPL and IUPAC (2006), treat the density dependence of Φ mol in terms of atmospheric pressure, which introduces a further temperature dependence. Both temperature effects are included in the calculation of the respective j i profiles. The comparison demonstrates that even at present -without 15 a representation of the temperature dependence -our Φ i provide vertical profiles of the photolysis frequency which agree well with those based on Φ i from the JPL recommendation -for all j i and both solar zenith angles considered. The comparison with the data from Atkinson et al. (2006) is less favorable, especially for j mol . This reflects the differences between Φ mol (λ) given here and that recommended by JPL on the one 20 hand to that recommended by Atkinson et al. (2006) on the other, which were already apparent in Figs. 2 and 3. The new quantum yields recommended by IUPAC in 2013 give photolysis rates which lie slightly above our curves for j mol , just outside the error bounds.
Although the derived j i profiles as well as the fits to the measured Φ i (Figs. 1-3) 25 based on the JPL recommendation and on the present work appear reasonably equivalent, we feel our formalism to be advantageous: since it consistently formulates the wavelength dependence of Φ i in terms of 1/λ, its fitting parameters are in units of energy, and represent, or are close to, molecular parameters, notably threshold     Table 4, along with the coefficients of determination c for the quantum yield functions. These parameters result from a global fit of all data, as described in Sect. 5.