Natural line shape

Spectral lines are fiirther broadened by collisions. To a first approximation, collisions can be drought of as just reducing the lifetime of the excited state. For example, collisions of molecules will connnonly change the rotational state. That will reduce the lifetime of a given state. Even if die state is not changed, the collision will cause a phase shift in the light wave being absorbed or emitted and that will have a similar effect. The line shapes of collisionally broadened lines are similar to the natural line shape of equation (B1.1.20) with a lifetime related to the mean time between collisions. The details will depend on the nature of the intemrolecular forces. We will not pursue the subject fiirther here. 

In our account here we neglect a third aspect of a spectral line, specifically its shape, beyond its characteristic frequency and strength. A natural line shape is almost impracticable to observe and would yield on analysis little or no additional information about intrinsic molecular properties. Another shape merely reflects components of molecular velocities in a direction parallel to the direction of propagation. Apart from these effects, further broadening of spectral lines due to finite durations, between collisions, of molecules in particular quantum states is attributed to interactions between colliding molecules rather than directly to... 

If the spectral lines are broadened but have the same profile, the moments of the spectrum can be expressed in terms of the moments of the non-broadened spectrum and the moments of a separate line. When the moments of the separate line do not exist (for natural line shape), the global characteristics of the spectrum can be found simulating its structure by Monte-Carlo or distribution function methods and then convoluting each calculated line. 

Most lines in liquids arise from systems where there is interaction between nuclear spins, and are not Lorentzian in shape. Natural line shapes are closer to a Gaussian shape in the tails of the intensity distribution in liquids. The relationship between peak width at half height and T2 is still retained and gives values in agreement with more rigorous transient methods (Hahn, 1950). 

If we continue to assume that the average energy loss is large compared with the line width, (16) and (18) will give the mass and surface absorptions. An easy integration over the natural line shape gives for these... 

This function is the Doppler broadened line shape. It is the counterpart of the natural line shape... 

The actual line shape in a spectrum is a convolution of the natural Lorentzian shape with the Doppler shape. It must be calculated for a given case as there is no simple fomuila for it. It is quite typical in electronic... 

Wlrile tire Bms fonnula can be used to locate tire spectral position of tire excitonic state, tliere is no equivalent a priori description of the spectral widtli of tliis state. These bandwidtlis have been attributed to a combination of effects, including inlromogeneous broadening arising from size dispersion, optical dephasing from exciton-surface and exciton-phonon scattering, and fast lifetimes resulting from surface localization 1167, 168, 170, 1711. Due to tire complex nature of tliese line shapes, tliere have been few quantitative calculations of absorjDtion spectra. This situation is in contrast witli tliat of metal nanoparticles, where a more quantitative level of prediction is possible. 

Equation (2.27) illustrates what is called the natural line broadening. Since each atom or molecule behaves identically in this respect it is an example of homogeneous line broadening, which results in a characteristic lorentzian line shape. 

In principle all the X-ray emission methods can give chemical state information from small shifts and line shape changes (cf, XPS and AES in Chapter 5). Though done for molecular studies to derive electronic structure information, this type of work is rarely done for materials analysis. The reasons are the instrumental resolution of commercial systems is not adequate and the emission lines routinely used for elemental analysis are often not those most useftil for chemical shift meas-ure-ments. The latter generally involve shallower levels (narrower natural line widths), meaning longer wavelength (softer) X-ray emission. 

After the completion of this manuscript a paper concerning conformational analyses of 1,1, 3,3 -tetra-r< H-alkylmctallocene of iron and ruthenium including 6 based on thorough NMR spectroscopic measurements (line-shape analysis) has appeared in which the nature of the transition states has conclusively been discussed in detail [164]. 

In the ultraslow exchange limit, O -4 1/T, the line shape becomes insensitive to the motion because the extra broadening is much smaller than the natural line width. 

Natural Line Width and Spectral Line Shape... 

---