Lorentz line broadening

Collisions between molecules are the greatest cause of line broadening at the pressures normally employed for MMW spectrometry. In the Lorentz theory (ref 2, p. 338) the lifetime of the rotational state involved in the transition is ended abruptly by collision with another molecule which stops the rotation. When the molecule starts to rotate again, its phase with respect to the other molecules is random. For an assembly of molecules this will give rise to an absorption line profile with a FWHM of Xjlm, where r is the mean time between collisions. This is the linear sum of two terms, one for the upper and one for the lower state, having the shape of the Lorentz function (Figure 1.4) when Av [Pg.12]

Pressure or Lorentz broadening is the second important factor for line broadening. It is a result of the interaction of the emitting species and other, non-emitting particles. Its contribution to line broadening is ... 

Both lines are broadened independently and solely by adiabatic phase shift as in Lorentz and Weisskopf theories. They are Lorentzians of width (1 — cosa) and frequency shift (sin a). In general off-diagonal elements of f are not zero though they are less than diagonal elements. Consequently, the spectrum may collapse even in the adiabatic case when A 1/tc. However, adiabatic collapse is hardly ever achieved in the gas phase where l/rc > l/t0 > jS since A > 1/tc > j8 and hence only the resolved doublet limit is available. 

Thus the random phase gives rise to a Lorentz broadening of the spectral line. 

Blickensderfer et al. (119) have recently extended the Samson theory to include various vessel geometries under conditions of pure Doppler, pure Lorentz, and Voigt (a combination of Doppler, Lorentz, and natural broadening) line profiles. The quenching cross section can be obtained also by a dynamic method in which the decay of the fluorescence intensity is measured after the exciting light is cut off. If the initial fluorescence intensity is / and the intensity after a lime f is If, we obtain... 

It has been shown [Mitchell and Zemansky (21), p. 197] that the quenching of the Hg 2537 A line by foreign gases follows the Stern-Volmcr formula only when (1) Lorentz broadening [sec Section 1-6.1] by foreign gases is... 

For Doppler lines there is little broadening with increasing self-absorption whereas for lint s with a Lorentz distribution of frequencies, self-absorption leads to a marked broadening, especially at the wings of the line. 

There are other factors affecting the intensity of the peaks on a x-ray diffraction profile of a powdered sample. We have analyzed the structure factor, the polarization factor, and the broadening of the lines because of the dimensions of the crystallites. Now, we will analyze the multiplicity factor, the Lorentz factor, the absorption factor, the temperature factor, and the texture factor [21,22,24,26],... 

The factors that are included when calculating the intensity of a powder diffraction peak in a Bragg-Brentano geometry for a pure sample, composed of three-dimensional crystallites with a parallelepiped form, are the structure factor Fhkl 2=l/ TS )l2, the multiplicity factor, mm, the Lorentz polarization factor, LP(0), the absorption factor, A, the temperature factor, D(0), and the particle-size broadening factor, Bp(0). Then, the line intensity of a powder x-ray diffraction pattern is given by [20-22,24-26]... 

This is collision between atoms of the same element in the ground state and results in an intensity distribution similar to Lorentz broadening but without line asymmetry or shift. The effect depends on concentration and half-widths which are very small and negligible when compared with other collisions. 

The perturbational effects have been variously described as interruption broadening, resonance broadening, and statistical broadening. The perturbational line shape derived by Michelson was not correct even for pure interruption broadening because he neglected to average over all times between collisions. To do so results in the simplified Lorentz model. ... 

Equations 1.16 and 1.17 are based on the assumption that all the molecules undergoing the transition m —> do so at the same frequency v. In reality they will have slightly different transition frequencies centred around the centre frequency Vq due predominately to collisional interactions between molecules. Doppler broadening also makes a small contribution giving a Gaussian shape to the line (Figure 1.4), but the overall result is a profile approximated by the Lorentz shape function 5 (v) ... 

This can be substituted into Equation 1.17 in the same way as was the Lorentz function, to give a value for the absorption coefficient of the Doppler broadened line profile. 

Equation 1.37 shows that ctmax hardly changes from the Lorentz case, as the right-hand term in the bracket is dominated by the l/v term, but the line becomes asymmetric, broadening towards the higher frequency as shown for water in Figure 1.5. This effect is of some consequence if analysis of gaseous mixtures at pressures around atmospheric is considered (Section 4.3 and Chapter 5). 

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