Line broadening instrumental profile

The effect of instrumental broadening can be eliminated by deconvolution (see p. 38) of the instrumental profile from the measured spectrum. If deconvolution shall be avoided one can make assumptions on the type19 of both the instrumental profile and of the remnant line profile. In this case the deconvolution can be carried out analytically, and the result is an algebraic relation between the integral breadths of instrumental and ideal peak profile. From such a relation a linearizing plot can be found (e.g., measured peak breadths vs. peak position ) in which the instrumental breadth effect can be eliminated (Sect. 8.2.5.8). 

In addition, diffraction line breadth contains information on lattice strain, lattice defects, and thermal vibrations of the crystal structure. The chief problem to determine crystallite size from line breadth is the determination of /3(20) from the diffraction profile, because broadening can also be caused by the instrument. To correct for the instrumental broadening on the pattern of the sample, it is convenient to run a standard peak from a sample in which the crystallite size is large enough to eliminate all crystallite size broadening. By use of a convolu-... 

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... 

To conclude this overview on the most common sources of line broadening it is worth considering the instrumental profile. As discussed in Chapters 5 and 6, wavelength dispersion, sample absorption and instrument optics generally produce a finite width IP that is regarded as an extrinsic profile, even if absorption is actually a sample related property. The IP is always present in a PD pattern, combined with the intrinsic profile produced by microstructural features and lattice defects present in the studied sample. 

ELIAS II (LTB Lasertechnik Berlin GmbH, Berlin, Germany), having an instrument profile width of 0.13 pm (FWHM), which is negligible compared to the line width. The measured profiles were used to determine the line positions and intensities of the Cu doublet. For these values Gauss profiles with 1.4 pm FWHM were calculated, representing the Doppler broadening for Cu at 2600 K. 

The dispersion of the distribution, or diffraction line broadening, is measured by the full width at half the maximum intensity (FWHM) or by the integral breadth ifi) defined as the integrated intensity (J) of the diffraction profile divided by the peak height (/3 = ///q). Line broadening arises from the convolution of the spectral distribution with the functions of instrumental aberrations and sample-dependent effects (crystallite size and structure imperfections). 

Microstructural imperfections (lattice distortions, stacking faults) and the small size of crystallites (i.e. domains over which diffraction is coherent) are usually extracted from the integral breadth or a Fourier analysis of individual diffraction line profiles. Lattice distortion (microstrain) represents departure of atom position from an ideal structure. Crystallite sizes covered in line-broadening analysis are in the approximate range 20-1000 A. Stacking faults may occur in close-packed or layer structures, e.g. hexagonal Co and ZnO. The effect on line breadths is similar to that due to crystallite size, but there is usually a marked / fe/-dependence. Fourier coefficients for a reflection of order /, C( ,/), corrected from the instrumental contribution, are expressed as the product of real, order-independent, size coefficients A n) and complex, order-dependent, distortion coefficients C (n,l) [=A n,l)+iB n,l)]. Considering only the cosine coefficients A(n,l) [=A ( ).AD( ,/)] and a series expansion oiAP(n,l), A (n) and the microstrain e (n)) can be readily separated, if at least two orders of a reflection are available, e.g. from the equation... 

Instrumentation for studies of this nature are usually variations of the normal powder X-ray diffractometer. Except for faulting and strain in single crystals, which are better treated as defects, the very nature of the material limits studies to powders or aggregates. X-ray powder patterns of simple metals can be analyzed to yield information on particle size, deformation fault probability, mean-square strain, and twinning. The theory and techniques used to study diffraction line broadening, peak shifts, and line profile asymmetry have been derived and applied by Warren, " and Warren and Averbach. To assess faulting probability, certain drastic assumptions are necessary, reducing the detectability limit to approximately one faulted layer in 200. 

In the limit that the instrumental profile is very much narrower than the emission and absorption profiles, the measured absorbance, ln I (0)/I (L), would approach the value given by equation (10.26) with g(wg) replaced by g(o)). This situation is difficult to achieve in practice owing to the narrow widths of the Doppler-broadened lines. Thus equations (10.27) and (10.28) must be evaluated using the known. instrumental function T(o)-a) ) and assumed values of the lamp profile and the absorption coefficient until there is agreement with the observed emission and absorption spectra. This procedure determines the absorption coefficient K and hence the atomic density, w... 

Finally, instmmental broadening results from resolution limitations of the equipment. Resolution is often expressed as resolving power, v/Av, where Av is the probe linewidth or instmmental bandpass at frequency V. Unless Av is significantly smaller than the spectral width of the transition, the observed line is broadened, and its shape is the convolution of the instrumental line shape (apparatus function) and the tme transition profile. 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

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 ... 

Calculation of X-ray profiles was performed in steps of 0.04° throughout the 6-50° 20 angular region by applying the procedure used in the structure determination of H-BOR-D (6). In this procedure the instrumental broadening was simulated by convoluting the sample profiles with two Lorentzian line functions, with a 2 1 intensity ratio and a full width at half maximum of 0.1° 20, representing the contribution of Ka. and Ka lines, respectively. 

The crudest way of estimating the particle size (D) as an average number from the breadth of a diffraction line is the widely used Scherrer approximation. It may be applied when the instrumental broadening is much smaller than the line profile (20 > 0.5°) and when a monomodal size distribution results in a homogeneous line profile. An explicit version of the equation determined by using the breadth of the diffraction line at half height (FWHM, pi 72) is given as follows ... 

The line shapes are described by Voigt functions, which reflect the Lorentzian line profiles due to natural line width and Gaussian profiles due to Doppler broadening. The instrumental broadening by the rocking curve of the crystal, de-focusing and the finite resolution of the detector is described well by a Voigt profile shape too [3[. 

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