Study of the Electronic-Vibrational and Line Structure of the Aluminum Oxidation Spectrum
Abstract Generated abstract
This study examines the spectrum produced by aluminum powder burning in oxygen in order to clarify excitation processes in AlO molecules and assess equilibrium conditions in the aluminum flame. The authors photographed the flame spectrum, measured AlO band intensities, calculated vibrational transition probabilities using harmonic-oscillator overlap integrals, and tested whether the electronic-vibrational band distribution follows the Boltzmann law. The measured band intensities were consistent with a Boltzmann distribution within experimental accuracy, giving vibrational temperatures of about 3700 to 4000 K, while temperatures derived from aluminum atomic lines were about 5000 K. The lower molecular temperature is attributed to AlO emission occurring mainly in cooler flame zones where the molecules can persist.
Full Text
INVESTIGATION OF THE ELECTRONIC-VIBRATIONAL AND LINE STRUCTURE OF THE ALUMINUM OXIDATION SPECTRUM
L. V. ZHDANOVA, V. A. SOKOLOV
The question of the nature of the unusually bright flame accompanying the oxidation of aluminum remains unresolved. Our task was to investigate the character of the excitation processes of AlO molecules and the conditions for attainment of equilibrium in an aluminum flame.
In the present work, the spectrum of aluminum oxidation was investigated; it was obtained by photographing, with a KSA-1 spectrograph, the flame of aluminum powder burning in oxygen.
A characteristic feature of the aluminum flame spectrum is intense continuous radiation. It had previously been established that it obeys Kirchhoff’s law, which indicates the thermal character of the excitation of incandescent particles of aluminum oxide formed in the combustion process and accounting to a considerable extent for the radiation of the aluminum flame [1].
At the same time, AlO molecules present in the flame in the gaseous phase also make a noticeable contribution to the flame radiation. This is indicated by the large number of bands of the electronic-vibrational structure of the AlO molecule that stand out rather sharply against the continuous background (in the region 5400–4400 Å). The bands correspond to the transition \(B^{2}\Sigma—{}^{2}\Sigma\) [2].
A necessary condition for the equilibrium character of the radiation of AlO molecules is the subordination of the intensity distribution of the bands of the electronic-vibrational system to the Boltzmann law
\[ I_{v'v''}=A^{*}_{v'v''} C e^{-\frac{E_{v'}}{kT}}, \]
where \(E_{v'}\) is the energy of the upper vibrational level.
In order to determine the applicability of the law, we measured the intensities of the bands \(I_{v'v''}\) and calculated the probabilities of vibrational transitions \(A^{*}_{v'v''}\).
In calculating \(A^{*}_{v'v''}\) we proceeded from the quantum-mechanical expression for the probability of the dipole transition \(v'\to v''\) in the first approximation
\[ A^{*}_{v'v''}=\text{const}\cdot \nu_{v'v''}^{3} \left(\int_{-\infty}^{+\infty}\psi_{v'}(r)\psi_{v''}(r)\,dr\right)^{2}. \]
The solution of the Schrödinger equation, \(c(v', v'')=\int_{-\infty}^{\infty}\psi_{v'}(r)\psi_{v''}(r)\,dr\) (the overlap integral), was based on the assumption that the harmonic-oscillator model is applicable, in first approximation, to the AlO molecule. The possibility of applying the harmonic-oscillator model in our case was demonstrated by comparing potential-energy curves calculated for the AlO molecule as a harmonic and as an anharmonic oscillator.
The solution of the Schrödinger equation describing the vibrational state of a harmonic oscillator is given in the form of a Hermite polynomial
\[ \psi_v(r)=Ne^{-\frac{1}{2}\left(\frac{r-r_e}{\rho}\right)^2}\cdot H_v\left(\frac{r-r_e}{\rho}\right), \]
where \(\rho\) is the maximum elongation of a diatomic classical harmonic oscillator.
After carrying out a series of mathematical transformations, Manneback [3] proposed the following recurrence formulas for the overlap integrals:
\[ C(v'+1,v'')= -\left[\frac{v'}{v'+1}\right]^{\frac12}k\,C(v'-1,v'') + \]
\[ +\left[\frac{v''}{v'+1}\right]^{\frac12}k^{+}C(v',v''-1) -(v'+1)^{-\frac12}a\,C(v',v''); \]
\[ C(v',v''+1)= \left[\frac{v''}{v''+1}\right]^{\frac12}k\,C(v',v''-1) + \]
\[ +\left[\frac{v'}{v''+1}\right]^{\frac12}k^{+}C(v'-1,v'') +(v''+1)^{-\frac12}b\cdot C(v',v''); \]
\[ C(0,0)=\frac{e^{-M^2}}{\sqrt{\sin 2\vartheta}}. \]
Here
\[ k=\cos 2\vartheta, \]
\[ k^{+}=\sin 2\vartheta, \]
\[ \tan\vartheta=\sqrt{\frac{\omega'}{\omega''}}, \]
\[ M=0.172205\,\mu^{\frac12}(r_e''-r_e')\cdot \sqrt{\frac{\omega'\omega''}{\omega'+\omega''}}, \]
\(\mu\) is the reduced mass of the molecule in atomic-weight units,
\[ a=\sqrt{2}\,M\cos\vartheta, \]
\[ b=\sqrt{2}\,M\sin\vartheta. \]
For the AlO molecule, the natural vibrational frequencies \(\omega\) in the upper and lower states and the equilibrium internuclear distances \(r_e\) were equal to [4]:
\[ \omega''=977\ \text{cm}^{-1}, \qquad \omega'=868.15\ \text{cm}^{-1}, \]
\[ r_e''=1.614\ \text{\AA}, \qquad r_e'=1.663\ \text{\AA}. \]
For the calculations, two electronic-vibrational systems of AlO were selected: a system with vibrational transitions \(0—1,\ 1—2,\ 2—3,\ 3—4,\)
4—5, 5—6 and a system with vibrational transitions 1—0, 2—1, 3—2, 4—3, 5—4.
The calculated values of the relative probabilities of the vibrational transitions under consideration are given in Table 1.
Table 1
| Band \(v', v''\) | \(A^{*}_{v'v''}\) | Band \(v'v''\) | \(A^{*}_{v'v''}\) |
|---|---|---|---|
| 0—1 | 0,154 | 1—0 | 0,251 |
| 1—2 | 0,194 | 2—1 | 0,380 |
| 2—3 | 0,174 | 3—2 | 0,422 |
| 3—4 | 0,160 | 4—3 | 0,406 |
| 4—5 | 0,086 | 5—4 | 0,354 |
| 5—6 | 0,047 |
The relative intensities \(J_{v'v''}\) of the spectral lines were determined by the usual method of photographic photometry.
Applying Boltzmann’s law in logarithmic form,
\[ \lg \frac{J_{v'v''}}{A^{*}_{v'v''}} = \lg C-\frac{\lg e}{kT}E_{v'} \]
and using the obtained values of the intensities \(J_{v'v''}\) and of the transition probabilities \(A^{*}_{v'v''}\), a graphical dependence of \(\lg \dfrac{J_{v'v''}}{A^{*}_{v'v''}}\) on \(E_{v'}\) was obtained for both systems of vibrational transitions (Fig. 1). The deviation
Fig. 1. Dependence of \(\lg \dfrac{J_{v'v''}}{A^{*}_{v'v''}}\) on \(E_{v'}\) for two systems of vibrational transitions of the AlO molecule.
of the experimental points from the straight line does not exceed the accuracy of measurement. This indicates that the distribution of molecules over vibrational states obeys Boltzmann’s law.
In the spectrum of aluminum burning in oxygen, a large number of lines emitted by aluminum atoms is also observed. This made it possible to compare the temperatures of different types of emitting particles (molecules and atoms) from the vibrational and line structure of the spectrum.
The temperature determined from the vibrational structure, by the formula
\[ T_{\mathrm{vib}}=\lg \frac{e}{k}\, \frac{E_{v'}-E_{v''}} {\left(\lg \frac{J_{v'v''}}{A^{*}_{v'v''}}\right)_{1} - \left(\lg \frac{J_{v'v''}}{A^{*}_{v'v''}}\right)_{2}} \]
was on the average \(3850^\circ\mathrm{K}\) (more precisely: for the first system, \(3700^\circ\); for the second, \(4000^\circ\)).
To calculate the temperature from the atomic lines, we used the most intense lines \(3944.1\ \text{\AA}\) (transition \({}^{2}P_{1/2}-{}^{2}S_{1/2}\)), \(3961.5\ \text{\AA}\) (transition \({}^{2}P_{1/2}-{}^{2}D_{1/2}\)) and \(3082.7\ \text{\AA}\) (transition \({}^{2}P_{1/2}-{}^{2}D_{3/2}\)), with known values of \(E\) [4] and \(Ag\) [5]. The temperature was determined by the formula
\[ T=\frac{\lg e}{k}\, \frac{E_{2}-E_{1}} {\left\{\lg \frac{J_{1}}{J_{2}}+\lg \frac{\lambda_{1}}{\lambda_{2}}-\lg \frac{(Ag)_{1}}{(Ag)_{2}}\right\}} . \]
The temperature \(T\) calculated from the atomic lines was \(\sim 5000^\circ\mathrm{K}\).
A comparison was also made of the relative intensities of aluminum lines in the flame and in the arc; it showed the same distribution of intensities in both cases.
The relatively low temperature obtained from the vibrational structure of AlO molecules is explained by the fact that these molecules, owing to their comparatively small dissociation energy, exist and emit only in the cooler zones of the flame.
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