Diatomic Line Strengths for Fitting Selected Molecular Transitions of AlO, C₂, CN, OH, ${\mathbf{N}}_{\mathbf{2}}^{\mathbf{+}}$, NO, and TiO, Spectra

Abstract

This work communicates line-strength data and associated scripts for the computation and spectroscopic fitting of selected transitions of diatomic molecules. The scripts for data analysis are designed for inclusion in various software packages or program languages. Selected results demonstrate the applicability of the program for data analysis in laser-induced optical breakdown spectroscopy primarily at the University of Tennessee Space Institute, Center for Laser Applications. Representative spectra are calculated and referenced to measured data records. Comparisons of experiment data with predictions from other tabulated diatomic molecular databases confirm the accuracy of the communicated line-strength data.

1. Introduction

Atomic, molecular, optical (AMO) spectroscopy furnishes fundamental insight by decoding light emanating from targets of interest [1,2,3,4,5,6,7,8]. Analytical studies of elements may be straightforward, especially for elements that appear in the first three rows of the period table. Balmer-series hydrogen lines or sodium D-lines usually are well separated from spectral interference for low (∼1 eV)-temperature plasma containing sodium as long as reasonable resolving power is available. For example, for the sodium D-lines, a resolving power, R, of R≃1000 is needed to distinguish the two components D1 and D2, separated by ∼0.06 nm. Resolving individual lines of molecular spectra may require $R>$ 10,000, or at least of the order of one magnitude better resolution than needed for atoms, of course depending on temperature. In molecular spectroscopy, one tends to focus on molecular bands describing electronic transitions. The study of individual atomic or molecular resonances with continuous-wave radiation typically requires GHz scans with nominal MHz or better laser bandwidths. In this work, the focus is on optical spectrometers that measure near-UV to near-IR molecular bands with a spectral resolution, $\mathsf{\delta }\mathsf{\lambda }$, of the order of $\mathsf{\delta }\mathsf{\lambda }\sim 0.1\mathrm{nm}$.

A collection of molecular diatomic spectroscopy data and expansive literature review and guidance [9] reveals a volley of recent and updated records in the UV to IR wavelength range. However, this work’s focus is the visible and near-IR, specific sets of electronic transition data that have been tested in the analysis of experimental records. The mentioned databases [9] predict OH spectra among many others for diatomic molecules, e.g., ExoMol [10] and HITEMP [11] that can be visualized using for example PGOPHER [12]—PGOPHER also allows one to model transitions for prediction and comparative analysis. Selected diatomic molecular spectra of AlO, C${}_2$, CN, OH, N${}_2^{+}$, NO, and TiO, transitions are of interest as these can be observed in laser-induced breakdown spectroscopy (LIBS) [13,14,15] at standard ambient temperature and pressure (SATP). Diatomic AlO and TiO spectra usually occur following the creation of micro-plasma near or at aluminum and titanium surfaces, respectively. In several cases, molecular spectra may not be of primary interest in elemental analysis with LIBS using nanosecond laser pulses, but molecular spectra are readily observed with femtosecond laser-plasma excitation, or after some time delay (of the order of larger than 100 ns for occurrence of CN in CO${}_2$:N${}_2$ gas mixtures) from optical breakdown when using nanosecond laser pulses. Just as for atomic spectra, reasonably accurate molecular spectra are required for analysis [16,17,18,19,20]. The construction of a molecular spectrum relies on: (i) accurate line positions, and (ii) reasonably accurate transition strengths [21,22,23,24]. For the former, numerical singular value decomposition is employed for upper and lower states of a particular transition. For the latter, Frank–Condon factors and r-centroids are computed, and then combined with the rotational factors that usually decouple from the overall molecular line strength due to the symmetry of diatomic molecules.

This work communicates data files and associated scripts for the computation of diatomic molecular spectra, and equally, for the fitting of measured data using a nonlinear fitting algorithm. Calculated spectra are presented and references to recorded datasets are provided. Applications comprise fields of chemistry, materials science, astronomy, and finally physics including astrophysics, e.g., decoding of light from white dwarf stars such as Procyon B. The data are provided as a set of wave numbers, upper-level term value, and line strength. Originally, FORTRAN/Windows 7 programs computed diatomic molecular spectra [21], but the scripts for the generation of molecular spectra are redesigned for use with MATLAB [25]. Moreover, this work communicates MATLAB-optimized line-strength files (LSFs) containing three columns, namely wave numbers, upper term values, and line strengths. The codes are operating system-independent as long as MATLAB or similar programs are available that allow scripts similar to the ones available in MATLAB. Supplementary data contain programs and nine selected diatomic molecular transitions of AlO, C${}_2$ Swan, CN red, CN violet, OH ultraviolet, N${}_2^{+}$, NO gamma, TiO $\gamma$, and TiO ${\gamma }^{\prime }$.

2. Materials and Methods

The computation of diatomic molecular spectra uses established line-strength data. Programs in FORTRAN accomplish the generation of spectra, coupled with a separate plotting program for visualization, including convenient implementation using a Microsoft-Windows 7 operating system. This work communicates equivalent MATLAB scripts that appear popular with various research groups. First, the Boltzmann equilibrium spectral program (BESP) generates a theoretical spectrum, and second, the Nelder–Mead temperature (NMT) program accomplishes fitting of experimental and theoretical spectra. In principle, BESP can be used to generate maps as a function of temperature and linewidth with subsequent determination of the optimum solution with minimal errors in the least-square sense. In turn, NMT uses nonlinear optimization using geometric constructs, viz. simplicia. The accumulation of experimental spectra in this work is in accord with laser-induced breakdown spectroscopy, or in general, laser spectroscopy [26].

2.1. MATLAB Scripts

The parameter list includes wavelength minimum, maximum, temperature, number of points, normalization factor, and file name. For the BESP.m and NMT.m scripts, the outputs are generated in graphical form. Table 1 lists constants that could be used (comment line in the scripts) for the determination of the variation of the refractive index, n, of air with wavelength [27],

$${10}^6(n-1)=a_0+\frac{a_1}{{\mathsf{\lambda }}_N^2}+\frac{a_2}{{\mathsf{\lambda }}_N^4},$$

(1)

where ${\mathsf{\lambda }}_N$ is the wavelength in normal air at 15 ${}^{\circ }$C and 101,325 Pa (760 mm Hg), expressed in terms of micrometer (range 0.2218–0.9000 $\mathsf{\mu }$m).

Table 1. Constants for variation of refractive index, n; see Equation (1).

Parameter	Value
$a_0$	272.643
$a_1$	1.2288 ($\mathsf{\mu }$m${}^2$)
$a_2$	0.03555 ($\mathsf{\mu }$m${}^4$)

Table 2 lists constants that are used to account for the variation of the refractive index, $r_i$, of air at 15 ${}^{\circ }$C, 101,325 Pa, and 0% humidity, with wavenumber [28],

$${10}^8(r_i-1)=\frac{k_1}{(k_0-{\mathsf{\sigma }}^2)}+\frac{k_3}{(k_2-{\mathsf{\sigma }}^2)},$$

(2)

where $\mathsf{\sigma }$ is the wavenumber in units of $\mathsf{\mu }$m${}^{-1}$.

Table 2. Constants for variation of refractive index; see Equation (2).

Parameter	Value ($\mathsf{\mu }$m${}^{-2}$)
$k_0$ (k0)	238.0185
$k_1$ (k1)	5,792,105
$k_2$ (k2)	57.362
$k_3$ (k3)	167,917

Table 3 and Table 4 summarize script constants and input variables that are important for spectra computations, respectively. However, redesign of BESP.m and NMT.m from the FORTRAN/Windows 7 version [21] was accomplished with extensive discussions [24]. Edited versions of BESP.m and NMT.m are communicated in this work along with nine separate data files.

Table 3. Constants in BESP.m and NMT.m.

Constant	Value
Planck constant (h)	6.62606957 × ${10}^{-34}$ (J s)
speed of light (c)	2.99792458 × ${10}^8$ (m s${}^{-1}$)
Boltzmann constant (kb)	1.3806488 × ${10}^{-23}$ (J K${}^{-1}$)

Table 4. Parameters and variables in BESP.m and NMT.m.

Description	Variable
wavelength minimum	wl_min (cm${}^{-1}$)
wavelength maximum	wl_max (cm${}^{-1}$)
temperature	T (kK)
full-width at half maximum	FWHM, $\mathsf{\delta }\mathsf{\lambda }$ (nm)
number of points	N
normalization	norm
file name	x

2.1.1. BESP.m

The script BESP.m is designed following the FORTRAN/Windows 7 version [21]. The individual diatomic molecular data files for selected transitions are concatenated to only show wavenumbers, upper-term values, and line strengths; see Table 5. Adjustments of input parameters for MATLAB [25] are rather straightforward, equally, for generalizing the script for automatic input by converting the script to a function. Individual lines are computed using Gaussian profiles [21], and for the generation of a spectrum, only one temperature is needed for equilibrium computation. Conversely, as one infers temperature from a measured spectrum, a modified Boltzmann plot [22] is constructed for the determination of the equilibrium temperature. A Gaussian line shape is selected to model the spectrometer/intensifier transfer function profile. However, one usually considers a natural linewidth for electronic state-to-state transitions, and a Gaussian line shape for Doppler broadening [29], viz.

$$\Delta \mathsf{\lambda }=7.16\times {10}^{-7}\mathsf{\lambda }\sqrt{\frac{\mathrm{T}}{\mathrm{M}}},$$

(3)

leading to Voigt line shapes. Here, $\Delta \mathsf{\lambda }$ is the full-width half-maximum, $\mathsf{\lambda }$ the wavelength, T the temperature, and M the molecular weight. For example, with $\mathsf{\lambda }=306\mathrm{nm}$, $\mathrm{T}=3.5\mathrm{kK}$, and $\mathrm{M}=17$ (OH), $\Delta \mathsf{\lambda }=0.0031\mathrm{nm}$. The spectral resolution, $\mathsf{\delta }\mathsf{\lambda }$, for the OH emission spectra-fitting, discussed in the Appendix, amounts to $\mathsf{\delta }\mathsf{\lambda }=0.33\mathrm{nm}$. Consequently, a Gaussian line shape is considered instead of a Voigt line shape for fitting of the OH data in the appendix, but the communicated MATLAB scripts can be adjusted for Voigt profiles, important for cases when individual electronic state-to-state molecular transitions/resonances are investigated. Equally, when investigating individual transitions/resonances, asymmetric molecular line shapes can be implemented in the scripts. There is usually a volley of lines for electronic transitions of a diatomic molecules, e.g., OH [30] in excess of 3 kK, within a wavelength bin and for an experimental spectral resolution of the order of 0.33 nm.

Table 5. Line-strength data contents: Vacuum wave numbers and upper term values, line strengths.

Description	Variable	Coulumn
wave number	WN (cm${}^{-1}$)	1
upper term value	Tu (cm${}^{-1}$)	2
line strength	S (stC${}^2$ cm${}^2$) ${}^a$	3

^a 1 stC = 3.356 10${}^{-10}$ C.

The program BESP.m receives input from the LSFs that contain relative line strengths. The output is generated in graphical format, and the program is slightly adjusted for the generation of the spectra illustrated in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. However, Figure 6 is generated with the BESP.m script given below.

2.1.2. NMT.m

The NMT script details are deferred to Appendix A. The adaptation of a previous FORTRAN code with Windows 7 libraries for a Microsoft platform is no longer viable due to support discontinuation of the Windows 7 operating system. However, the NMT.m script delivers spectra-fitting results identical to those obtained with the FORTRAN/Windows 7 implementation.

2.1.3. Data Files

This section explains the line-strength data communicated in this work. The line-strength files (LSFs) contain wave numbers, upper term values, and the line strengths. Table 5 summarizes contents of line-strength data. The air wavelength in the program, WL, is in units of nm. The two programs BESP and NMT convert the vacuum wave numbers to air wavelengths for the analysis of measured data; see Equation (1). Table 6 associates the diatomic molecules and their line-strength data, including the wavelength range.

Table 6. Diatomic molecules, line-strength data files, wavelength range, and number of spectral lines.

Diatomic Molecule	Line-Strength Data File	Wavelength Range (nm)	Number of Spectral Lines
aluminum monoxide (AlO)	AlO-BX-LSF.txt	430.72–997.66	33,484
carbon Swan spectra (C${}_2$)	C2-Swan-LSF.txt	410.93–678.58	29,004
cyanide red (CNr) system	CNr-LSF.txt	499.89–4997.56	40,728
cyanide violet (CNv) system	CNv-LSF.txt	372.88–425.22	7960
hydroxyl (OH) violet system	OH-LSF.txt	278.65–379.72	1683
nitrogen monoxide (NO) gamma system	NO-GAMMA-LSF.txt	200.41–285.95	13,000
singly ionized nitrogen (N${}_2^{+}$)	N2p-LSF.txt	319.04–501.46	7302
titanium monoxide (TiO) $\gamma$ band	TiO-AX-LSF.txt	599.58–945.44	66,962
titanium monoxide (TiO) ${\gamma }^{\prime }$ band	TiO-BX-LSF.txt	582.73–679.12	34,648

The LSFs contain significantly more data than illustrated in this communication. Applications of the LSFs include data analysis of laser-induced fluorescence and computation of absorption spectra. Some of these applications are elaborated on in the discussion of C${}_2$ Swan spectra [23].

3. Results

This section summarizes the communicated line-strength data. Table 7 associates the diatomic molecules and their line-strength files (LSF). The LSFs contain wave numbers, upper term values and the line strength. The two programs BESP and NMT convert the vacuum wave numbers to air wavelengths for the analysis of measured data. Table 7 displays spectral resolution, temperature, and Table 7 also communicates but one reference each for measurement and fitting selected molecular transitions of AlO, C${}_2$, CN, OH, N${}_2^{+}$, NO, and TiO. Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 illustrate computed spectra that refer to measured ones in the references.

Table 7. Diatomic molecules, spectral resolution, temperature, and one typical reference each that uses the data.

Diatomic Molecule	Line-Strength Data	Spectral Resolution (nm)	Temperature (kK)	Reference	Figure
aluminum monoxide (AlO)	AlO-BX-LSF.txt	1.0	3.33	[31]	Figure 1
carbon Swan spectra (C${}_2$)	C2-Swan-LSF.txt	0.39	6.75	[32]	Figure 2
cyanide red (CNr) system	CNr-LSF.txt	0.38	7.5	[33] ${}^a$	Figure 3
cyanide violet (CNv) system	CNv-LSF.txt	0.030	7.94	[34]	Figure 4
singly ionized nitrogen (N${}_2^{+}$)	N2p-LSF.txt	0.035	5.1	[35]	Figure 5
hydroxyl (OH) ultraviolet system	OH-LSF.txt	0.35	3.39	[30]	Figure 6
nitrogen monoxide (NO) gamma system	NO-GAMMA-LSF.txt	0.056	6.8	[36]	Figure 7
titanium monoxide (TiO) $\gamma$ band	TiO-AX-LSF.txt	0.10	3.03	[37] ${}^b$	Figure 8
titanium monoxide (TiO) ${\gamma }^{\prime }$ band	TiO-BX-LSF.txt	0.40	3.6	[38]	Figure 9

^a Experiments at Johannes Kepler University, Linz, Austria. ^b Experiments in part at Chemical Research Center of the Hungarian Academy of Science, Budapest, Hungary.

Figure 1. Computed AlO spectrum, $\Delta \mathrm{v}=0,\pm 1,\pm 2,+3$, $\mathsf{\delta }\lambda =1.0\mathrm{nm}$, T = 3.33 kK.

Figure 1. Computed AlO spectrum, $\Delta \mathrm{v}=0,\pm 1,\pm 2,+3$, $\mathsf{\delta }\lambda =1.0\mathrm{nm}$, T = 3.33 kK.

Figure 2. Computed C${}_2$ Swan spectrum, $\Delta \mathrm{v}=-1$, $\mathsf{\delta }\lambda =0.39\mathrm{nm}$, T = 6.75 kK.

Figure 2. Computed C${}_2$ Swan spectrum, $\Delta \mathrm{v}=-1$, $\mathsf{\delta }\lambda =0.39\mathrm{nm}$, T = 6.75 kK.

Figure 3. Computed CN red spectrum, $\Delta \mathrm{v}=+1$, $\mathsf{\delta }\lambda =0.38\mathrm{nm}$, T = 7.5 kK.

Figure 3. Computed CN red spectrum, $\Delta \mathrm{v}=+1$, $\mathsf{\delta }\lambda =0.38\mathrm{nm}$, T = 7.5 kK.

Figure 4. Computed CN violet spectrum, $\Delta \mathrm{v}=0$, $\mathsf{\delta }\lambda =0.030\mathrm{nm}$, T = 7.94 kK.

Figure 4. Computed CN violet spectrum, $\Delta \mathrm{v}=0$, $\mathsf{\delta }\lambda =0.030\mathrm{nm}$, T = 7.94 kK.

Figure 5. Computed N${}_2^{+}$ spectrum, $\Delta \mathrm{v}=0$, $\mathsf{\delta }\lambda =0.035\mathrm{nm}$, T = 5.1 kK.

Figure 5. Computed N${}_2^{+}$ spectrum, $\Delta \mathrm{v}=0$, $\mathsf{\delta }\lambda =0.035\mathrm{nm}$, T = 5.1 kK.

Figure 6. Computed OH spectrum, $\Delta \mathrm{v}=0$, $\mathsf{\delta }\lambda =0.35\mathrm{nm}$, T = 3.39 kK, see BESP.m script and Ref. [30].

Figure 6. Computed OH spectrum, $\Delta \mathrm{v}=0$, $\mathsf{\delta }\lambda =0.35\mathrm{nm}$, T = 3.39 kK, see BESP.m script and Ref. [30].

Figure 7. Computed NO gamma spectrum, $\Delta \mathrm{v}=-1$, $\mathsf{\delta }\lambda =0.056\mathrm{nm}$, T = 6.80 kK.

Figure 7. Computed NO gamma spectrum, $\Delta \mathrm{v}=-1$, $\mathsf{\delta }\lambda =0.056\mathrm{nm}$, T = 6.80 kK.

Figure 8. Computed TiO $\gamma$ spectrum, $\Delta \mathrm{v}=0$, $\mathsf{\delta }\lambda =0.10\mathrm{nm}$, T = 3.03 kK.

Figure 8. Computed TiO $\gamma$ spectrum, $\Delta \mathrm{v}=0$, $\mathsf{\delta }\lambda =0.10\mathrm{nm}$, T = 3.03 kK.

Figure 9. Computed TiO ${\gamma }^{\prime }$ spectrum, $\Delta \mathrm{v}=0$, $\mathsf{\delta }\lambda =0.40\mathrm{nm}$, T = 3.6 kK.

Figure 9. Computed TiO ${\gamma }^{\prime }$ spectrum, $\Delta \mathrm{v}=0$, $\mathsf{\delta }\lambda =0.40\mathrm{nm}$, T = 3.6 kK.

4. Discussion

The accurate prediction of line positions of diatomic molecules is important for the identification and of course for the fitting of measured data. The line positions are usually more accurate than the intensity values. The selected transitions for most of the communicated diatomic molecules, especially AlO, C${}_2$ Swan, CN, and OH, have been extensively tested in the study of laser-induced optical breakdown. Comparisons of analysis of an experimental OH ultraviolet data record using the communicated OH table and the recent and updated ExoMol diatomic molecular databases reveal agreements of most wavelength positions, of the order of 10% variations of the line strengths, but better than 3% agreement in fitted temperature, spectral resolution and background. This bodes well for applications of expansive databases such as ExoMol in analytical laser-plasma research for the other diatomic molecules communicated in this work.