The 3.3 μm Infrared Emission Feature: Observational and Laboratory Constraints on Its Carrier

The gas phase laboratory measurements of Joblin et al. (1994, 1995) provide high spectral resolution (1 cm⁻¹) spectra of the 3.3 μm band for different size PAHs over a temperature range of about 400–900 K. We focus on the interpretation of their laboratory spectra for the 3.3 μm CH(S) band of naphthalene (C₁₀H₈), pyrene (C₁₆H₁₀), coronene (C₂₄H₁₂), and ovalene (C₃₂H₁₄). The temperature and size dependent widths and central wavelengths for N_C > 32 are difficult to obtain both experimentally and theoretically. Since there is general agreement that N_C > 50 for the interstellar PAHs we extrapolate the laboratory results to larger N_C . We consider in Section 3.3 two extrapolation approaches that we argue represent plausible limits to the actual behavior. The theoretical underpinnings of the extrapolation are discussed in Section 3.4.

Joblin et al. (1995) found that the widths and shifts for each PAH scaled linearly with temperature. Thus, each PAH is characterized by two temperature slopes, one for the bandwidth, and one for the band shift. Here, we carry the analysis of Joblin et al. (1995) a step further and show that each of these temperature slopes also scales linearly with N_C . Pech et al. (2002) did a similar analysis but they did not extrapolate to N_C > 50.

Our fits to the laboratory temperature-dependent slopes of the bandwidths and shifts of the center absorption frequencies, denoted as χ_w and χ_c , are presented in Figure 4. Both slopes are well represented by linear fits, suggesting that extrapolation to larger PAH molecules is meaningful. In addition to the slopes, there is an intercept specific to each molecule that corresponds to the bandwidth and peak frequency at a reference temperature, which will be discussed later. For the smallest PAHs, like naphthalene, the width of its rotational envelope is approximately equal to its vibrational bandwidth. This requires a correction to the observed width in order to compare to the observed bandwidths of the larger PAH molecules, for which the rotational envelope contribution is negligible (see Appendix A).

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Additionally, we have extended the lower size limit of Joblin et al. (1995) to benzene (Cook et al. 1998), such that the linear fits are demonstrated over a factor of approximately 5 in size (i.e., N_C = 6–32). This also suggests that a linear dependence may plausibly extend to N_C > 32. On the other hand, we recognize that a linear dependence may not be accurate for very large N_C , so we consider the possibility of an asymptotic limiting behavior for the slopes. We define a conservative limit to the quantification of the asymptotic limit (see Section 3.3 for details). Most importantly, we note that the basic conclusions of this work that the widths and central wavelengths will vary significantly with variations of excitation energy and PAH size do not depend on precise knowledge of the behavior of the slopes for larger N_C .

Cook et al. (1998) used UV laser single photon excitation of PAH molecules in the size range of N_C = 6–24 to observe the resulting spectral emission of the 3.3 μm band. The experimental conditions enabled high levels of internal excitation while maintaining low levels of rotational excitation. In this respect, these measurements are a more realistic simulation of the interstellar environment than those of Joblin et al. (1995). For benzene, we used Equations (1) and (2) presented in Section 3.3, to determine χ_w and χ_c , respectively, based on the spectrum shown by Cook et al. (1998) in their Figure 8. A comparable approach was used for naphthalene based on their Figure 7 and the parameter values reported in their Table 13. The results for χ_w represent an average based on the widths for the top three, highest vibrational energies, differenced with the lower energy width for T = 930 K.

Because benzene is a relatively small molecule it has relatively large rotational constants. Even though the rotational temperature was low, around 300 K, the breadth of the rotational envelope (≈24 cm⁻¹) is significant compared to the observed width of 73 cm⁻¹ for the 193 nm (6.4 eV) photon excitation. The low measurement spectral resolution, 18 cm⁻¹, is also a significant source of broadening. We retrieved the intrinsic vibrational broadening by deconvolving the rotational envelope and spectral resolution broadening to get ${\gamma }_{\mathrm{vib}}\,=66.5={({73}^{2}-{18}^{2}-{24}^{2})}^{1/2}$, which follows the example shown in Appendix A, Figure 7.

There are a number of factors to consider in the application of laboratory data to modeling and analysis of the observational data. The laboratory data is measured at discrete temperatures for which all the rotational and vibrational degrees of freedom are characterized by the gas kinetic temperature. For the observational data, each rotational and vibrational degree of freedom may be characterized by different effective temperatures. As discussed below, the spectral broadening due to the rotational envelope is not substantial for PAHs with N_C greater than approximately 20. Thus, for interstellar PAHs where N_C > 50, the rotational envelope broadening is negligible. However, we do need to remove the effect of rotational envelope broadening for the smaller PAHs measured in the laboratory, since we are seeking to quantify the purely vibrational contributions for the widths and shifts.

It is commonly assumed, as we do here, that for a fixed internal energy there is a single effective vibrational temperature that characterizes all the modes. An observational spectrum arises from the absorption of a UV photon followed by fast intramolecular energy transfer and then an IR emission cascade within the ground electronic state. This means that an observational spectrum is a superposition of a continuous distribution of spectra, each corresponding to a different internal energy, or equivalently, to a different internal vibrational temperature as the molecule cools.

Laboratory emission spectra at different temperatures can be summed to simulate a cascade spectrum. A significant advantage to using laboratory emission data is that it correctly represents the complexities of Fermi resonances, which are difficult to accurately model with theoretical methods, particularly for large PAHs. If only temperature-dependent absorption spectra are available, they can be converted to emission spectra by multiplying each absorption spectrum by the Planck blackbody function at the appropriate temperature. The methodology for calculating a cascade spectrum using an ensemble of temperature-dependent measured or simulated laboratory spectra is well established (i.e., Léger & Puget 1984; Joblin et al. 1995). Our goal here is to modify the laboratory widths and shifts for the effects of the IR cascade.

We are looking to associate a cascade width and shift to the UV excitation energy and PAH size. The absorption of a UV photon results in an initial, maximum internal vibrational temperature, T_max. A method for determining T_max for a spectral distribution of UV photons from a stellar source with effective temperature T_eff is presented in the following section. Due to the radiative cooling, the temperatures that characterize the width and shift are lower than T_max. We represent these temperatures using βT_max where β < 1. As described in Section 3.3, we empirically estimated β from cascade calculations presented in Joblin et al. (1995). Their cascade calculations were based on Lorentzian profiles tied to the temperature-dependent width and peak location of the experimental data.

We expect that β will be sensitive to the frequency of the vibrational fundamental. If one is considering a relatively high frequency mode, such as a CH(S) mode, then a relatively small decrease in temperature will significantly decrease the emission intensity. Hence, only a small temperature interval will contribute to the cascade spectrum. Thus, we anticipate β will be close to 1.

We expect that β will be different for the band profile width and shift. For the band center shift, the component spectral profiles shift increasingly to higher frequency as the cascade temperature decreases (Joblin et al. 1995). This means that the magnitude of the cascade shift will be less than the initial shift for T_max, hence β < 1.

The cascade width is less straightforward to understand because it arises from the interplay of several factors. Each component of the cascade spectrum shifts to higher frequency, thus extending higher frequency edge of the band, which suggests that β > 1. However, both the width of each component spectrum and its overall contribution (i.e., weighting factor) decrease with the decreasing cascade temperature. The combined effect of these factors results in a cascade width that is very close to the initial width for T_max (i.e., β ≈ 1).

Given the time-dependent cooling curve for a particular PAH, the width and peak frequency shifts in Figure 4 can be used to model the full radiative cascade spectrum as discussed by Joblin et al. (1995). By scaling the calculated line profiles in Figure 8 of Joblin et al. (1995), we derived a reasonably accurate (approximately ±10%), simple approximation to the width, Δν, of the cascade spectrum given by

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}\nu & = & {\rm{\Delta }}\nu \left(500\,{\rm{K}}\right)+{\chi }_{w}\left({\beta }_{w}{T}_{\max }-500\right)\\ & & \times \left(1-\exp \left(-{N}_{S}/{N}_{C}\right)\right)\quad \mathrm{for}\,{T}_{\max }\gt 500\,{\rm{K}},\end{array}\end{eqnarray} \tag{ 1 }$$

where 500 K is a reference temperature, χ_w is the bandwidth slope as a function of N_C (see Figure 4), T_max is the vibrational temperature corresponding to the energy of the absorbed photon, β_w accounts for the modest reduction of T_max due to the radiative cascade, and N_S is a PAH size associated with the asymptotic large molecule/solid-state limit (see below). The value of T_max is determined by integration of the molecule-specific heat capacity (Appendix B) until the integrated energy is equal to the absorbed photon energy, E_UV. Some examples of the relationship of T_max to E_UV for different size PAHs are shown in Table 2. Δν(500 K) is molecule dependent and varies from ≈15–20 cm⁻¹ Joblin et al. (1995). For simplicity we adopted a value of Δν(500 K) = 20 cm⁻¹ because the majority of molecules (3 out of 4) investigated by Joblin et al. (1995) exhibited values close to 20 cm⁻¹. A value of β_w = 0.96 was derived from the computed cascade spectra in Joblin et al. (1995) (see their Figure 8 for the 10 eV excitation spectra).

Analogous to the widths, we use the relationships and cascade calculations in Figure 8 of Joblin et al. (1995) to derive an approximate relationship for the band center frequency, ν_c , to PAH size and its associated excitation temperature. This relationship is given by

$$\begin{eqnarray}\begin{array}{rcl}{\nu }_{c} & = & {\nu }_{c}\left(0\,{\rm{K}}\right)+{\chi }_{c}\left({\beta }_{c}{T}_{\max }-200\right)\\ & & \times \left(1-\exp \left(-{N}_{S}/{N}_{C}\right)\right)\quad \mathrm{for}\,{T}_{\max }\gt 200\,{\rm{K}},\end{array}\end{eqnarray} \tag{ 2 }$$

where χ_c is the band center slope as a function of N_C (see Figure 4) and ν_c (0 K) is the fundamental frequency for a selected PAH. We note that χ_c exhibits a very tight linear relationship to N_C , suggesting that the extrapolation to larger PAH is reasonable. While the absolute band shift, ν_c , is molecule specific through ν_c (0 K) (Joblin et al. 1995), we consider the relative band shift, Δν_c = ν_c − ν_c (0 K), which depends principally on the PAH size via T_max and N_C in Equation (2). As for β_w , a value of β_c = 0.87 was determined from the laboratory data of Joblin et al. (1995).

Pech et al. (2002) suggested that for very large PAH molecules the width and shift slopes should reach asymptotic values, which they refer to as the solid-state limit. We have introduced the solid-state size parameter N_S into Equations (1) and (2) to account for the asymptotic behavior as a function of N_C . The value of this parameter is not known; however, we can estimate a lower limit by tying the quickest onset of the solid-state limit to the uncertainty in the linear fits of the width and shift with respect to N_C . A value of N_S = 103 is obtained from the exponential term in Equations (1) and (2) from which N_S = −N_C ln(0.04) = 103, where N_C = 32 and the average uncertainty of the two linear fits is 0.04.

Joblin et al. (1995) derived a theory-based expression (Equation (6) in their paper) for the temperature-dependent band shift based on the assumption that that the shifts were primarily due to the anharmonic coupling terms, χ_ij , between a C–H stretch band and the much lower frequency vibrational modes (ν_max ≪ 700 cm⁻¹). They demonstrated that for a given PAH, the band shift scales linearly with the vibrational temperature. We reexpress their formula to show that it also supports a linear scaling with PAH size. The modified expression is

$$\begin{eqnarray}&&{\rm{\Delta }}{\nu }_{c,i}=\left(3{N}_{C}-6\right)\displaystyle \frac{{kT}}{{hc}}\sum _{j=1}^{{j}_{\max }}\displaystyle \frac{{f}_{j}\langle {\chi }_{i,j}\rangle }{{\nu }_{j}},\end{eqnarray} \tag{ 3 }$$

where Δν_c,i is the band center shift for the ith C–H stretch fundamental, 3N_C − 6 is the total number of skeletal carbon vibrational modes, ν_j is the mean frequency for the low frequency modes in the jth frequency bin (i.e., we are binning the modes into frequency intervals), f_j is the fraction of the total modes in the jth bin, 〈χ_ij 〉 the average anharmonic coupling constant for all the modes in the jth bin, and j_max corresponds to the high frequency cutoff, ν_max.

Assuming the same frequency bins for different size PAHs, and the same ν_max, we empirically determine using the PAHdb that f_j for compact PAHs does not depend on PAH size. Thus, the only size dependence in the summation terms must arise from the 〈χ_ij 〉 term. The observational fact that Δν_c,i varies linearly with N_C implies that 〈χ_ij 〉 is size independent up to N_C = 32. Since there is no physical reason to expect a strong discontinuity in the scaling beyond N_C = 32 it is not unreasonable to assume that the linear scaling with N_C persists for larger N_C . It seems plausible to expect that 〈χ_ij 〉 may gradually decrease for larger N_C , which provides the motivation for introducing a solid-state limit.

The assumption that f_j is independent of PAH size for the low frequency C skeletal modes of compact PAHs is tested as follows. Consider the compact PAHs coronene, C₂₄H₁₂, ovalene, C₃₂H₁₄, circumovalene, C₆₆H₂₀, and circumcircumcoronene, C₉₆H₂₄, which have 3N_C -6 vibrational degrees of freedom of 66, 90, 192 and 282, respectively. According to Joblin et al. (1995) the anharmonicity arises primarily from modes with frequencies ≪700 cm⁻¹. We assume an upper limit frequency of 350 cm⁻¹ and find from the NASA PAH database that the number of modes below this limit are 9, 16, 29, and 43 for C₂₄H₁₂, C₃₂H₁₄, C₆₆H₂₀, and C₉₆H₂₄, respectively. The corresponding f factors are 9/66 = 0.14, 16/90 = 0.18, 29/192 = 0.15, and 43/282 = 0.15. There is good consistency of the f factors over a substantial range of PAH sizes.

We note that the five laboratory PAHs in Figure 4 are relevant exemplars of the proposed interstellar PAHs in that they are examples of highly symmetric compact PAHs. This offers some confidence that extrapolation of their properties to larger PAHs of the same symmetry type is reasonable. Ricca et al. (2012) have shown that a small number of highly symmetric, compact PAHs in the N_C = 60−100 size range may be the dominant carriers for the Class A profiles.

The preceding discussions pertains solely to the band center shifts. We do not have a comparably simple expression for the bandwidths. However, the observational fact that the bandwidths correlate tightly with the band shifts (the absolute values of the slopes are nearly identical) suggests the use of the same extrapolation assumptions as for the band center shifts.

Theoretical methods for calculating anharmonicity effects in PAH molecules are under active development (Mackie et al. 2018; Mulas et al. 2018; Peeters et al. 2021). The theoretical calculations are challenging and are not yet sufficiently accurate to substitute for experimental measurements. For example, using the band profiles calculated by Mackie et al. (2018, Supplemental Material) we estimate the vibrational width slopes, χ_w , for naphthalene and pyrene to be 0.057 and 0.045 cm⁻¹ K⁻¹, respectively. These estimates are based on the excitation temperatures and widths for the 3 and 5 eV cascade spectra. For comparison, the experimental width slopes, with rotational broadening removed, for naphthalene and pyrene are 0.026 and 0.031 cm⁻¹ K⁻¹, respectively. The theoretical predictions are high by roughly a factor of 1.5–2.0 and they decrease with increasing PAH size, N_C .

We used the scaling relationships given in Equations (1) and (2) to predict the widths and shifts in the ISM, accounting for the radiative cooling of the molecule, for a range of PAH sizes and for two UV photon excitation energies. The predictions are displayed in Figures 5 and 6 for excitation energies of 7.0 and 10.6 eV. For comparison, we indicate the observed width of 38 cm⁻¹ for the ubiquitous Class A profile in Figure 5 and the band center frequency shift of −30 cm⁻¹ in Figure 6.

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We selected E_UV = 7.0 and 10.6 eV because they are representative of low and high average UV photon absorption energies for PAHs in different environments. For simplicity, we frame our analysis around the widths and shifts based on a single profile at the average excitation energy, 〈E_UV〉. The distribution of excitation energies is broad and the average of the profiles over E_UV may not produce a width and shift identical to those for a single profile evaluated at E_UV. However, we are focused on the general trends with respect to PAH size and excitation energy, which should be insensitive to the method used to calculate the widths and shifts.

In order to associate a particular stellar UV source and its effective temperature, T_eff, with an average UV excitation energy, 〈E_UV〉, we derived an empirical relationship between 〈E_UV〉 and T_eff based on the points from Andrews et al. (2015), where (〈E_UV〉, T_eff) is (6.2 eV, 14,000 K) for NGC 1333, (7.0 eV, 17,000 K) for NGC 7023, and (8.1 eV, 23,000 K) for NGC 2023. The relationship is

$$\begin{eqnarray}&&\langle {E}_{\mathrm{UV}}\rangle =12.3\{1-\exp \left(-4.9\times {10}^{-5}\,{T}_{\mathrm{eff}}\right)\},\end{eqnarray} \tag{ 4 }$$

where 〈E_UV〉 = 12.3 eV is an asymptotic limit at high T_eff. For T_eff = 40,000 K, which is characteristic of the Orion Bar (Salgado et al. 2016), we obtain 〈E_UV〉 = 10.6 eV.

The 〈E_UV〉 derived above from T_eff represent an upper limit to the local excitation energy for a PAH carrier. This is due to the spatially variable extinction from dust, which preferentially absorbs far-UV compared to near-UV photons (Draine 2003). Consequently, as the stellar UV photons travel further from their source, 〈E_UV〉 decreases. Furthermore, assuming that the 3.3 μm IEF arises primarily from neutral PAHs, the neutrals will populate regions with relatively low values of 〈E_UV〉 where ionization does not occur. Model predictions from Montillaud et al. (2013) show (see their Figure 2) that, for NGC 7023, the neutral PAHs favor regions further from the stellar source, with higher UV extinction (i.e., lower 〈E_UV〉), than PAH cations.

These considerations may contribute to an explanation of the invariance of the 3.3 μm band. It would require that the dust extinction for neutral PAHs always results in a low and relatively constant excitation energy. In general, the UV excitation spectrum of a PAH is bimodal (for example, see Figure 2 in Pech et al. 2002). The excitation spectrum is bimodal because the PAH absorption spectrum is bimodal with centroids around 6 and 12 eV (Malloci et al. 2004) and there is a cutoff in the stellar spectrum at 13.6 eV due to the onset of H atom absorption). Thus, enhanced relative extinction of the 12 eV excitation could result in a source invariant excitation energy peak of approximately 6 eV.

We infer from Figures 5 and 6 that neutral PAHs with E_UV ≈ 6 eV could produce shifts and widths consistent with observations of the 3.3 μm IEF for a distribution of PAHs with an average size of 〈N_C 〉 ≈ 70. The preceding arguments may explain a constant excitation energy. However, it is also necessary to explain the profile invariance which implies that the relative abundances of the PAHs remain invariant among diverse sources.

The general characteristics (e.g., size, shape, chemical composition, and charge state) of the PAH carriers associated with the 3.3 μm IEF band are inferred from spectral fitting studies employing the PAHdb (i.e., Bauschlicher et al. 2008, 2009; Ricca et al. 2012; Boersma et al. 2018). The PAHdb software tool computes the band positions and takes into account the full radiative cascade to the ground state. In order to account for the vibrational excitation energy and anharmonic effects a bandwidth of 15 cm⁻¹ and a frequency shift to lower frequency of 15 cm⁻¹ is typically assumed. This constant bandwidth and frequency shift are applied irrespective of UV radiation field and PAH size. This fitting procedure is at odds with the laboratory studies, such as those of Joblin et al. (1995), which show that the shifts and widths depend on the PAH size and the excitation energy. Furthermore, the laboratory shifts and widths are different for every band.

Given that the 3.3 μm IEF is due to aromatic C–H stretching modes, then there must be accompanying IEFs due to out-of-plane (OOP) C–H bending modes. One of the main IEF bands occurs at 11.2 μm and is ascribed to the OOP mode (Peeters 2011). Like the 3.3 μm IEF, the majority of observed 11.2 μm emission profiles are also of Class A. The 3.3 and 11.2 μm bands are thought to arise primarily from neutral PAH molecules (Allamandola et al. 1989; Tielens 2008). For neutral PAHs, these two features are the dominant features. In contrast, PAH cations produce most of their emission in several features spanning the 5–10 μm region. One of the strengths of the PAH model is that it can quantitatively account for the observed ratios of the integrated fluxes for these bands for different sources as well as the spatial variability of the ratio within a source (i.e., Schutte et al. 1993; Croiset et al. 2016; Maragkoudakis et al. 2020).

Several studies have included some laboratory-based constraints in their IEF spectral models (Verstraete et al. 2001; Pech et al. 2002). While many of the details differ between the two models, they share key assumptions central to achieving a nearly invariant 3.3 μm spectral profile for very different environments. One is that the bandwidth slope is fixed at that for coronene and does not vary with PAH size. A second is a power-law size distribution, ${N}_{C}^{-3.5}$, with a sharp turn-on at a minimum size, N_Cmin. The final assumption is that N_Cmin varies with variations in 〈E_UV〉 in such a manner as to preserve a constant excitation temperature. Since the initial excitation temperature depends on the ratio 〈E_UV〉/N_C (Berné & Tielens 2012), keeping this ratio constant leads to a constant excitation temperature.

There are several problems with these assumptions. First is that the results of this study strongly suggest that the bandwidth slope varies with PAH size. Second, the values of N_Cmin determined for the sources considered in these studies vary over the range of 20−30 C atoms. This is problematic because PAHs with N_C less than ≈50 should be fully dehydrogenated (Allain et al. 1996a; Montillaud et al. 2013; Andrews et al. 2016; Joblin et al. 2020). The relatively fast size distribution falloff combined with the decrease in excitation temperature with increasing size means that only a small range of sizes, anchored by the minimum size, contribute to the modeled spectrum. For example, we estimate that relative to N_Cmin = 50 the contribution of N_C = 55 is decreased by about 50%.