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Physical Review

Width and Shift of Spectral Lines Due to Dissipative Interaction

R. Brout

Published Aug 1, 1957• 25 citations
Publisher

Abstract

The theory of line width due to dissipative interactions, such as collisions, is examined for the case of weakly coupled systems (weak collisions). It is shown that the apparent paradox that the width of the energy of the total system is of the order of Avogadro's number is resolved by a substraction procedure. The width of a spectral line is the difference in widths of the two macroscopic states before and after the transition. This should be contrasted to the case of natural line width where the width of the line is the sum of the widths of the two states involved. The physical reason for the difference in sign is discussed.

DOI
10.1103/physrev.107.664

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The theory of line width due to dissipative interactions, such as collisions, is examined for the case of weakly coupled systems (weak collisions). It is shown that the apparent paradox that the width of the energy of the total system is of the order of Avogadro's number is resolved by a substraction procedure. The width of a spectral line is the difference in widths of the two macroscopic states before and after the transition. This should be contrasted to the case of natural line width where the width of the line is the sum of the widths of the two states involved. The physical reason for the difference in sign is discussed.
These results suggest that the natural width of a spectral line may be less than the value to be expected from the classical theory of damping by radiation.The subject of width of spectral line is one of considerable importance.If the lines are really much narrower than 0.12 X-Unit the radiation can- not come from a damped oscillating electron.The mechanism must be such as to maintain a pure harmonic oscillation of constant amplitude until the quantum of energy is completely emitted.Such a train of waves would need to have a great number of elements and so have considerable length.An alternate hypothesis would be that a quantum is an entity (the word "pulse" is avoided) that may be resolved into a train of waves by the crystal grating.In this case the width of a spectral line would de- pend on the degree of perfection of the crystal.The quantum theory of crystal grating action advanced by Duane8 might also give a narrow spec- tral line, whose width would be a property of the crystal grating and not of the radiation.
It has been postulated that the state function can be expressed as a linear superposition of eigenstates of Hermitian operators. When a system is subjected to an external impulse in the form of radiation, the probability of finding the system in different eigenstates of the state function changes. The total probability is always unity, which means the change in probabilities for different states occurs at the expense of each other. This change in probability of finding the system in different eigenstates brought about by system-radiation interaction is called the spectroscopic transition. Here, we use a two-level system to develop the details of a general spectroscopic transition, and work further to examine the effects of interruption in the coherent evolution of the state function. The physical meaning of absorption coefficient is discussed in detail. We derive the general forms of relaxation of the excited state, spectral line shape, and line broadening. The origin and the meaning of homogeneous and inhomogeneous line broadening are discussed. We describe the Doppler effect and its use in the interpretation of inhomogeneous line shape.
A general method for analysing line broadening caused by Doppler, phaseshift, inelastic and velocity-changing collisional effects is proposed for transitions at optical wavelengths. In a unified approach Poisson statistics are used to describe collision probabilities randomly distributed in a time interval which can terminate the optical coherence in a combination of the above effects; markoffian independence is assumed for each collision. Then the dipole moment correlation function is obtained in a form which is virtually exact in accounting for all effects of isotropic forces, and which is correct to second-order terms in the anisotropic interaction potentials. Phenomena as diverse as standard optical pressure broadening, broadening due to velocity-changing collisions in nonlinear spectroscopies, and Dicke narrowing of the Doppler contour are described. The implications of the theory are discussed in detail and applied to the case of the NaD lines perturbed by He buffer gas. Numerical calculations reveal that the substantially different lineshift and broadening parameters in nonlinear and in standard optical spectroscopies can be reasonably well understood within the limitations posed by present uncertainties in the Na 3 2 P-He interaction potentials. However, the large differences in shift and width between the D 2 and D 1 lines in saturation spectroscopy cannot be explained within the present theory.
Resonant scattering of intense light is analyzed in the limit in which the separation of the spectral lines is large compared to their widths. The effect on the line intensities and widths of elastic and inelastic collisions in the impact approximation and of radiative damping is found. The limit of well-separated spectral lines allows simple solutions in a wide class of cases not easily treated outside this limit, in particular the case in which the upper laser-coupled state decays through a complicated cascade sequence to the lower laser-coupled ground state, or suffers collisional reorientation (depolarization) during the emission process.
The quantum-mechanical Baranger-Kolb-Griem theory of pressure broadening is reexamined in order to include both the effects due to overlapping lines and finite duration of collision. A general line-shape formula is derived which for near wings can be presented as a linear combination of the Lorentzian and dispersion profiles. The asymmetry parameter is expressed by means of evolution operators for the initial and final states. It is shown that in the weak-coupling case only the resultant asymmetry parameter can be approximately given as the sum of parameters describing the asymmetry due to line mixing and finite collision time.