$\ell$-TYPE DOUBLING IN ENERGY LEVELS OF CARBON DIOXIDE COUPLED BY FREMI RESONANCE
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Date
1951
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Ohio State University
Abstract
In this paper we discuss the $\ell$-type doubling in a triad of energy levels of carbon dioxide, which are coupled by Fermi resonance. In principle, it becomes necessary to consider simultaneously the perturbations related to both anharmonic resonance interaction and $\ell$-type doubling and to solve a sixth-order secular determinant concerned. From the nature of this determinant it is not at all obvious that the $\ell$-type doubling in each of the three levels shall be determined by a formula of the well-known type: $\Delta \nu = qJ(J + 1)$ (1) where q is a constant, independent of the quantum number, J. A close inspection of the magnitude of all quantities involved reveals, however, that for values of J which are not too large we are justified in carrying out an approximate solution. The essence of this approximation is that we first consider the twofold degenerate resonance interaction levels as if $\ell$-type doubling did not occur. Subsequently the energy levels associated with $+\ell$ and $-\ell$ are found as a result of small perturbations on these degenerate levels. It is shown that the validity of an equation of the type (1) is inherent to this approximation. Consequently we can calculate a value for the $\ell$-type doubling constants q for each of the three levels associated with the Fermi resonance triad $\left\{\begin{array}{ccc}0 & 5_{1} & 1\\1 & 3_{1} & 1\\ 2& 1_{1} & 1\end{array}\right\}$ Experimental values for two out of three of these constants have recently been obtained by Goldberg et al. by analyzing two bands arising from a transition from a (0 $1_{1}$ 0) ground level to the upper triad under consideration. Taking into account the approximate character of the theoretical approach and the probable error in the experimental values, the agreement between the two sets of values is excellent.
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Author Institution: Department of Physics and Astronomy, The Ohio State University