Profile distortion broadening

Fourier transform method. The method used most widely for the separation of size and distortion in peak profiles from metals and inorganic materials is the Fourier analysis method introduced by Warren and Averbach (21). The peak profile is considered as a convolution of the size-broadening profile fg and the distortion broadening profile fj), so that the resolved and corrected profile f(x) is given by... 

The methods in general use for separating the size and distortion broadening components of the resolved and corrected peak profiles can be separated into two groups, non-transform and transform methods. The non-transfona methods are essentially similar to the Jones method, being approximations to a convolution. The transform method discussed here makes use of the Fourier coefficients found after the Stokes correction. 

The most recent calculations, however, of the photoemission final state multiplet intensity for the 5 f initial state show also an intensity distribution different from the measured one. This may be partially corrected by accounting for the spectrometer transmission and the varying energy resolution of 0.12, 0.17, 0.17 and 1,3 eV for 21.2, 40.8, 48.4, and 1253.6 eV excitation. However, the UPS spectra are additionally distorted by a much stronger contribution of secondary electrons and the 5 f emission is superimposed upon the (6d7s) conduction electron density of states, background intensity of which was not considered in the calculated spectrum In the calculations, furthermore, in order to account for the excitation of electron-hole pairs, and in order to simulate instrumental resolution, the multiplet lines were broadened by a convolution with Doniach-Sunjic line shapes (for the first effect) and Gaussian profiles (for the second effect). The same parameters as in the case of the calculations for lanthanide metals were used for the asymmetry and the halfwidths ... 

X-ray diffraction patterns from fibres generally contain a few closely overlapping peaks, each broadened by the contributions of crystallite size, crystallite-size distribution, and lattice distortion. In order to achieve complete characterisation of a fibre by X-ray methods, it is first necessary to separate the individual peaks, and then to separate the various profile-broadening contributions. Subsequently, we can obtain measures of crystallite size, lattice distortion and peak area crystallinity, to add to estimates of other characteristics obtained in complementary experiments. 

Four major computational steps are necessary to separate the individual peaks and the different profile-broadening components (i) correction and normalisation of the diffraction data, (ii) resolution of the total peak scattering from the so-called background scatter, and resolution of crystallographic, para-crystalline, and amorphous peaks from each other, (iii) correction of the resolved profiles for instrumental broadening, (iv) separation of the corrected profiles into size and distortion components. In this paper we will discuss these steps in turn, but most attention will be paid to the hitherto largely neglected step of profile resolution. 

Distortions along low-frequency modes and small frequency changes between the neutral and cation states, if present, contribute to the width of the vibrational lines in the photoelectron spectra after taking into account instrumental line broadening. Such band profiles can be treated semiclassi-cally using equation (8), where IE is the ionization energy and D is related to the transition moment. The Gaussian functions used to fit the experimental spectra can be described... 

In such circumstances, the resulting spectral profile may produce a composite profile containing both isotropic and anisotropic signals. It is rare that the spectra will be completely averaged, and frequently one may only observe distortions to the anisotropic signal (i.e. a broadening of the lines). It is therefore important to consider such effects in some detail. 

Most real crystals contain imperfections producing local distortions of the lattice, resulting in a non-homogeneous strain field. The effect on position, shape and extension of reciprocal space points, and consequently on PD peak profiles, is usually more complex than that of the domain size. A formal treatment of the strain broadening is beyond the scope of the present book interested readers can refer to the cited literature. In the following a simplified, heuristic approach is proposed. 

This factorization would be strict only, if (Fourier) convolution were the mathematical operation that describes the effect of both the lattice distortions of the first and the second kind on hie profile. In fact, strain broadening is not described by Fourier convolution but by MelUn convolution, instead. 

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