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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

Theory of line-breadth phenomena

Edmond Arnous, Walter Heinrich Heitler

Published Dec 8, 1953• 20 citations
Publisher

Abstract

Abstract A general and exact formalism is developed that permits the calculation of the line shape in all line-breadth phenomena, in principle to any order of approximation. Expansions for the line breadth and the numerator occurring in the line-shape formula are given. Corrections to the classical line shape include: (i) radiative corrections to the line breadth, (ii) a level shift additional to the usual radiative shift for levels with finite transition probability, (iii) radiative corrections to the numerator and (iv) a slight frequency dependence of the numerator. All these corrections, with the exception of (iv), are exceedingly small. The case of radio-frequency transitions is discussed.

DOI
10.1098/rspa.1953.0188

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Abstract A general and exact formalism is developed that permits the calculation of the line shape in all line-breadth phenomena, in principle to any order of approximation. Expansions for the line breadth and the numerator occurring in the line-shape formula are given. Corrections to the classical line shape include: (i) radiative corrections to the line breadth, (ii) a level shift additional to the usual radiative shift for levels with finite transition probability, (iii) radiative corrections to the numerator and (iv) a slight frequency dependence of the numerator. All these corrections, with the exception of (iv), are exceedingly small. The case of radio-frequency transitions is discussed.
We obtain expressions for the radiative level shifts of a two-level system in terms of (i) recurrence relations, (ii) ratios of determinants, (iii) continued fractions, and (iv) a Lidstone expansion. These expressions are shown to be very useful for numerical computations. It is pointed out that perturbation series in powers of the coupling constant are not the most appropriate way of representing the solutions of the problem, but if they are to be used, different series should be employed depending on the relative value of the frequency of the field to the frequency of the two-level system. The significance of these results in the general theory of perturbation is discussed.
An elementary account is given of the absorption and emission of radiation by molecules employing a particular formulation of Schrödinger theory which facilitates the treatment of radiative line widths and level shifts. The familiar Fermi golden rule for transition rates, the Rabi Bloch-Siegert absorption profile, and the Weisskopf-Wigner approximation for spontaneous decay are obtained under appropriate conditions in the context of a common perspective. Comparisons are made with other recently devised perturbative and nonperturbative formulations of electromagnetic transition phenomena which correctly include radiative line widths and level shifts, and the importance of proper treatment of the normalization and phase of the Schrödinger wave function in this connection is emphasized.
The theory of the line shape in coherent electric quadrupole absorption and emission processes involving NNN interacting molecules is developed, both quantum mechanically and with the use of a classical model. The theory is applied to small clusters and to linear chains of molecules. As in the dipole case, the effects of the interaction include in general a splitting of each molecular transition line into a number of components, a shift of each component from the molecular transition frequency, and a broadening or narrowing of each component relative to the width of the molecular line.