THE COMPLETE, TEMPERATURE-RESOLVED EXPERIMENTAL SPECTRUM OF ETHYL CYANIDE (CH₃CH₂CN) BETWEEN 210 AND 270 GHz

The spectrometer used in this work is based on a synthesized ×24 frequency multiplied probe, with a heterodyne receiver, coupled to a temperature-controlled 6 m long cell. The temperature of the cell (Fortman et al. 2010a) was slowly ramped from 240 K to 385 K over a period of 295 minutes. During this period, 437 spectral scans were recorded. Each scan measured the spectral intensity at 25 kHz intervals, with an integration time of ∼15 μs bin⁻¹ and a total scan time of 40 s. The temperature variation over each scan was small enough that the assumption of a constant temperature did not significantly impact the error budget of the experiment.

The microwave spectrometer is described in a recent publication (Medvedev et al. 2010). Because an analysis of the experimental spectrum, without QM assignment, is the product of this work, it is important that this system not have spurious responses. These are most likely at multiples of the 8.75–11.25 GHz drive frequency of the ×24 multiplier chain. In our earlier work in the 570–645 GHz band, spurious responses were below 5 × 10⁻⁴, even with the use of a broadband detector. However, the broader fractional bandwidth of the ×24 probe system (25% versus 12.5%) resulted in unacceptably large (∼5%) power at ×23 and ×25 of the drive frequency. The use of the frequency selective heterodyne receiver reduced these spurious responses to a level well below the weakest lines reported here.

A foundation of spectroscopy in the millimeter and submillimeter is that it is possible to use the sharpness of spectral features to separate them from wider system variations in power. However, this ordinarily sacrifices knowledge of absolute power and often modifies lineshapes as well. In our system, we directly measure the output of the intermediate frequency (IF) detector to provide a DC-coupled measure of power.

After using an automated procedure to cut each spectral line from the data, we use a spline fit to interpolate across each region that contains one or more spectral lines (Fortman et al. 2010a). The resulting line-free spectrum is subtracted from the signal channel spectrum to remove the baseline. This is done individually for each of the 437 temperature scans because the baseline is a function of temperature. Finally, to produce the absolute intensities, the spectrum is normalized by dividing by the channel that preserves the DC power levels.

As in our earlier work, we used the intensity residuals of our temperature calibration fit (see Sections 3 and 4) as a measure of the accuracy of the intensity calibration. While we expected that the more conventional electronic detector approach of the 210–270 GHz system would be more linear than the helium temperature bolometer used in our earlier work, this did not prove to be the case. However, we found that by calibrating the response of the IF amplifier–diode detector combination and by attenuating the probe power by ∼100, that excellent (∼1%) intensity calibration could be obtained. Inspection of the spectra showed that they contained about 1% vinyl cyanide. However, because of the intensity calibration and well-defined Doppler linewidths it was straightforward to subtract this spectrum to high accuracy.

In the first approach, after a peak finder locates the individual lines, they are fit to Gaussian lineshape functions, producing a line list of frequencies, line strengths, and linewidths. In congested areas multiple Gaussians are fit simultaneously. Software then collects the information for each of the 3096 lines and performs a least squares fit as a function of the 437 temperature scans to determine S_ijμ² and E_l for each of the lines, both assigned and unassigned. This produces the usual astrophysical catalog data. Because of the possibility of blends, this software also provides figures of merit to inform the user of spectral features whose origins may be from more than a single transition. As we have previously noted, these blends are a problem not only for this approach, but also for the usual QM catalog approach because a number of cataloged lines are overlapped by significantly stronger lines that are not included in the catalogs. In Section 4.3, we demonstrate a simulation strategy that minimizes the effect of these blends and overlaps.

The second approach makes no attempt to identify and characterize individual lines, but rather seeks to predict, on a point-by-point basis, the spectra as a function of temperature. Because of the exponentials in the Gaussian lineshape and in the lower state energy term, the absorbance as a function of frequency can be recast as

$$\begin{equation} A(\nu) = \frac{{8\pi ^3 }}{{3ch}}\sqrt {\frac{{\ln (2)}}{\pi }} \frac{{nL}}{Q}\frac{{\nu _0 \left({1 - e^{ - \frac{{h\nu _0 }}{{kT}}} } \right)}}{{\delta \nu _D }}S_{\it ij} \mu ^2 e^{ - \frac{{E_l }}{{kT}}} e^{ - \ln (2)\left({\frac{{\nu - \nu _0 }}{{\delta \nu _D }}} \right)^2 }, \end{equation} \tag{ 2 }$$

with the Doppler width (HWHM) given by

$$\begin{equation} \delta \nu _D = \sqrt {\frac{{2N_a k\ln (2)}}{{Mc^2 }}} \sqrt T \nu _0 = W\sqrt T \nu _0, \end{equation} \tag{ 3 }$$

where N_a is Avogodro's number and M is the molecular mass.

The absorbance normalized by the nL/Q factor then becomes

$$\begin{eqnarray} \frac{{A(\nu)}}{{nL/Q}} &=& \frac{{8\pi ^3 }}{{3ch}}\sqrt {\frac{{\ln (2)}}{\pi }} \frac{1}{W}\frac{{\left({1 - e^{ - \frac{{h\nu _0 }}{{kT}}} } \right)}}{{\sqrt T }}S_{\it ij} \mu ^2 e^{ - \frac{{E_l }}{{kT}}} e^{ - \frac{{\ln (2)}}{{W^2 T}}\left({1 - \frac{\nu }{{\nu _0 }}} \right)^2 } \nonumber\\ &=& K\frac{{\left({1 - e^{ - \frac{{h\nu _0 }}{{kT}}} } \right)}}{{\sqrt T }}\tilde S_{\it ij} \mu ^2 e^{ - \frac{{\tilde E(\nu)}}{{kT}}}, \end{eqnarray} \tag{ 4 }$$

with

$$\begin{equation} \tilde E(\nu) = E_L + k\frac{{\ln (2)}}{{W^2 }}\left({1 - \frac{\nu }{{\nu _0 }}} \right)^2. \end{equation} \tag{ 5 }$$

According to Equation (4) every frequency slice of the data (437 points, one for each temperature) can be represented by two parameters $\tilde S$_ijμ² and $\tilde E$. On line center, when ν = ν₀, $\tilde E$ equals the lower state energy, while $\tilde S$_ijμ² corresponds to the line strength. Off line center, the meanings of $\tilde S$_ijμ² and $\tilde E$ are less physical, but Equation (4) is still a valid fitting function for describing the spectral intensity for an unblended line.

Table 2 shows the results for 13 of the 3096 lines analyzed in this work in the catalog format described in Section 3.1. As described in our earlier work (Fortman et al. 2010a), this table includes a column that tags potentially unreliable results according to "W" (a line that is too wide in comparison to its expected Doppler width), "G" (a line whose Gaussian fits did not converge), and "T" (a line whose temperature dependence is unphysical). The full table is available in the electronic archives of this journal. While these tags are primarily designed to find problems associated with blends, they also serve to highlight problems in the highly automated analysis of a very large amount of spectral data.

To provide a measure of the physical basis of this approach, we compared both the lower state energies and transition strengths obtained from the experimental data with the results of the QM calculations. As expected, the accuracy is related to the experimental absorbances, both because of the experimental noise in the system and the impact of uncataloged overlapping lines on this comparison. Of the 317 lines that exist both in the catalogs and in our spectra, for the 74 strongest lines the rms difference is 8.4 cm⁻¹ in their lower state energies and for the 134 strongest lines the rms difference is 10.4 cm⁻¹ in their lower state energies. Although it is difficult to evaluate the impact of overlaps on this comparison, a reasonable elimination of lines suspected to be due to overlap results in a large majority of the differences being <50 cm⁻¹. Most of the larger of these are due to the weakest of the 317 lines.

Table 3 shows a subset of the 2.4 million pairs of $\tilde S$_ijμ² and $\tilde E$ coefficients of Section 3.2. The complete set is available in the electronic archives of this journal. In it $\tilde S$_ijμ² is reported in units of D² (Debye squared) and $\tilde E$ in units of cm⁻¹. To provide an easy way to calculate the absorbance scaled by the nL/Q parameter at an arbitrary temperature using archived $\tilde S$_ijμ² and $\tilde E$ coefficients, we reformulate Equation (4) as

$$\begin{equation} \frac{{A(\nu)}}{{nL/Q}} = {\rm }C_1 \cdot \sqrt {\rm M} \frac{{\left({1 - e^{ - C_2 \cdot \frac{\nu }{T}} } \right)}}{{\sqrt T }}\tilde S_{\it ij} \mu ^2 e^{ - {\rm C}_3 \cdot \frac{{\tilde E}}{T}}, \end{equation} \tag{ 6 }$$

with

$$\begin{eqnarray} C_1 &=& {\rm 54}{\rm.5953 }\frac{{{\rm nm}^{\rm 2} {\rm K}^{{\rm 1/2}} }}{{{\rm amu}^{{\rm 1/2}} {\rm D}^{\rm 2} }},\quad C_2 = {\rm 4}{\rm.799237} \times {\rm 10}^{ - {\rm 5}} {\rm }\frac{{\rm K}}{{{\rm MHz}}},\nonumber\\ C_3 &=& {\rm 1}{\rm.43877506 }\frac{{\rm K}}{{{\rm cm}^{ - {\rm 1}} }}, \end{eqnarray} \tag{ 7 }$$

and M the molecular weight in amu, T the temperature in K, and ν the frequency in MHz.

Table 3. Parameters of the Point-by-point Fit

$\tilde S$	$\tilde E$
(D2)	(cm−1)
206.104	962.644
280.137	982.967
316.401	967.523
388.789	971.572
447.064	962.762
458.688	934.691
549.453	941.656
610.615	936.118
605.825	908.508

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

Because this approach is based on frequency points, not spectral lines, there is no explicit line count. However, there are more than 9500 features at 300 K with signal-to-noise ratio greater than 2 and whose absorbances are greater than 0.001. This number is larger than the 3096 lines included in the line-by-line catalog in part because the fitting procedure effectively averages over the 437 spectra. This not only reduces the random noise in the spectrum, but also improves the spline baseline subtraction because much of the baseline varies as a function of temperature over the 437 spectral scans. Because of the rapid growth of the density of vibrational states with energy, these 9500 lines have their origins in approximately 40 vibrational states. Some of the lines in this analysis are also due to the ¹³C isotopic species.

Of the 3096 lines in the catalog of Table 2, 1068 are tagged, usually indicating a blend. While the point-by-point approach of Table 3 provides an alternative, it involves the use of a relatively large amount of data. It is also not in the standard astrophysical catalog data format. Accordingly, we have explored an alternative spectral simulation approach that may be of more use to the astrophysical community. It also shows the impact of overlaps and blends in both laboratory and astrophysical spectra.

The Gaussian fitting procedure used in the production of the catalog used up to 11 overlapping Gaussians to describe features in the observed spectra. Since we constrain the center frequency of these Gaussians to remain constant as the temperature is varied, the parameters of the Gaussians (their frequencies, intensities, and widths) constitute a well-defined set of parameters that describe the observed spectra, but with many fewer data points than the point-by-point files: specifically, 3096 Gaussians in comparison to 2.4 million spectral points. While, because of the strong correlations among the overlapping Gaussians, they may fail to provide meaningful characterizations of the line strengths and lower state energies of individual lines, their sum can still accurately provide the overall spectral information.

The blue trace in Figure 3 shows such a simulation based on all of the 3096 lines in the catalog, irrespective of whether or not they are tagged as a potential problem. We note the following.

• 1.  
  Both approaches include many lines that are not in the QM catalogs.
• 2.  
  The simulation based on the complete catalog is comparable to that based on the more data intensive point-by-point fit.
• 3.  
  Because we use the Doppler width in this simulation, lines that are broadened but still fit by a single Gaussian have linewidths that are too narrow. The alternative of including the measured Gaussian linewidth in the simulation would require us to produce a non-standard catalog with an additional column for the measured linewidth.
• 4.  
  The point-by-point analysis correctly includes the contributions from blends that were not separately identified in the Gaussian analysis. See for example the blends at 224.457 and 224.469 GHz (both are these tagged in the catalog).
• 5.  
  The line at 224.465 is included in the point-by-point analysis but not the simulation based on Gaussian analyses because the average over both white noise and baseline effects that are a part of the point-by-point analysis is more sensitive to weak lines than analyses that begin by identifying lines in individual temperature scans.

Thus, we conclude that while it appears that the tagging of the lines is in general accurate, if the goal is to accurately reproduce the spectrum, it is better to include them in simulations.

Figure 4 shows the differences between the observed absorbances and those calculated via the catalog method as described in Section 4.1 for the 317 lines that are both in the experimental data and the QM catalog. For the 105 reference lines, these are the same as the residuals of the temperature fit described in Section 3. Since at the line centers, all three approaches provide very similar results, this distribution is representative of the methods described in Sections 4.2 and 4.3 as well and can be considered as a measure of the uncertainty of the results reported here. Large negative differences are likely due to overlapping lines which add intensity at the location of catalog lines rather than to uncertainties associated with the analysis procedures reported here. Inspection of the spectra shows that the two largest positive differences involve blends, but the third largest appears to result from uncertainties in our analysis procedure related to the removal of the baseline.

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