Gaussian component, line broadening

Figure 1.1a shows the Gaussian function. The Lorentzian shape is similar to the Gaussian, but falls off more slowly. The Doppler shift of radiation from an emitting molecule is proportional to its velocity component in the direction of observation. The one-dimensional distribution of speeds in a gas is a Gaussian function. (See any physical-chemistry text.) Hence when Doppler broadening is dominant, we get a Gaussian-shaped line. 

Unfortunately, few cases can be properly described under the assumptions of Equations (14) or (15) (or by other possible combinations " ) real life cases usually do not match perfectly any simple combination of Lorentzian or Gaussian profiles. Generally speaking the additivity rule for different IB components is not known a priori, so using Equations (14), (15) or other combinations of terms is somewhat arbitrary, unless specific assumptions are made on the line broadening sources." ... 

Stoichiometry fluctuations are also responsible for a line broadening effect that, with the exception of cubic phases, is also hkl) dependent. The effect on the line profile is directly related to the nature of the fluctuation if the compositional variation is described by a suitable function, e.g. a Gaussian curve, then the resulting peak profile component is also Gaussian and the FT for this effect can be written as ... 

Doppler broadening has a Gaussian lineshape, and its convolution with the Lorentzian natural lineshape yields a Voigt profile. In typical experiments, this effect can be neglected since the Doppler width is usually much smaller than the resolution of the apparatus. Collisional line broadening is also Lorentzian, and the Lorentzian component of measured lines must be carefully extrapolated to zero pressure. 

Figure 19 The VE prediction of the diffusive component of the vibrational coherence decay CpiD as a function of the solvent viscosity (ij = 1, 2, 4, 8, 16, 32 and oo cP) for typical parameters. At low viscosity, the decay is exponential, its rate is inversely proportional to the viscosity, and the corresponding Raman line is homogeneously broadened. At high viscosity, the decay becomes Gaussian, its decay time reaches a limiting value, and the Raman line is inhomogeneously broadened. (From Ref. 8.)...

The linewidth, or lineshape, is the third aspect of the ESR spectrum important in any study of radicals. The two most frequently encountered lineshapes observed are the Lorentzian and Gaussian. The Gaussian line-shape is observed when the line is the superposition of a large number of unresolved individual components, such a line being referred to as inho-mogeneously broadened. More important is the Lorentzian lineshape which is often observed for radicals in solution, when complications due to unresolved hyperfine splittings do not occur. The Lorentzian lineshape may be defined by... 

Figure 2 The bulk FID from a chocolate sample before contact with hazelnut oil (lower line), and after 54 hrs of contact time with the hazelnut oil (upper line). The short component (expanded in inset) decays as a Sine function with Gaussian broadening, while the long one decays exponentially.

More detailed consideration shows that a Doppler-broadened spectral line cannot be strictly represented by a pure Gaussian profile as has been assumed in the foregoing discussion, since not all molecules with a definite velocity component emit or absorb radiation at the same frequency co = < o(l Because of the finite lifetimes of the molecular energy levels,... 

The total peak profile is the sum of these two components, with a relative weight that depends on 0 and I. The Bragg component is mathematically a 5-function, but is in reality broadened by instrumental and sample imperfections. The shape of the diffuse component depends on the form of the correlation function. Many forms are possible, but two common ones are an exponential correlation function, leading to a Lorentzian line shape, and a Gaussian correlation function, leading to a Gaussian Hne shape. One can also use correlation functions that describe a preferred distance (e.g., island-island correlations) or with an in-plane anisotropy [42]. 

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