Instrumental profiles

The observed profile P(20) of a Bragg peak at do = dhu is a convolution product between the instrumental profile r(20) and the physical profile p(20) ... 

An alternative approach, giving better results if the peak width is very close to the instrumental one, is to take into account the instrumental contribution (by a convolution of the refined peak due to the sample and the instrumental profile) during the profile fitting procedure [42]. 

J. Toft and O.M. Kvalheim, Eigenstructure tracking analysis for revealing noise patterns and local rank in instrumental profiles application to transmittance and absorbance IR spectroscopy. Chemom. Intell. Lab. Syst., 19 (1993) 65-73. 

Consideration of instrumental broadening is a merely technical issue. The instrumental profile Hj (s) must be measured. It is the shape of any peak18 of a single crystal of infinite size and perfection. For application in the field of polymers, many inorganic crystals, e.g., the common standard LaB6, are very good approximations to the ideal case. 

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). 

Convolve the synthetic spectrum with the surface velocity field (rotation, granulation) and with the instrumental profile. 

In their simplest form, radiometers monitor irradiance (in W/cm ) and radiant energy density (in J/cm ) for the bandwidth of the instrument. Profiling radiometers can in addition to that also provide irradiance profiles as a function of time. The results from the monitoring of a process can be effectively used to correlate exposure conditions to the physical properties of the cured product. If needed, they can also become the specifications of exposure in the design of production systems. Usually, radiometers are placed in the same position as the material that is being cured. 

In many cases, the profile a spectroscopist sees is just the instrumental profile, but not the profile emitted by the source. In the simplest case (geometric optics, matched slits), this is a triangular slit function, but diffraction effects by beam limiting apertures, lens (or mirror) aberrations, poor alignment of the spectroscopic apparatus, etc., do often significantly modify the triangular function, especially if high resolution is employed. 

Fig. 6.5. Computed structures due to the hydrogen dimer, in the quadrupole-induced (0223,2023) components near the So(0) line center at 120 K (the temperature of Jupiter s upper atmosphere). Superimposed with the smooth free — free continuum (dashes) are structures arising from bound — free (below 354 cm-1) and free - bound (above 354 cm-1) transitions of the hydrogen pair (dotted). The convolution of the spectrum with a 4.3 cm-1 slit function (approximating the instrumental profile of the Voyager infrared spectrometer) is also shown (heavy line) [282].

A structural model is required, and the parameters in the model are adjusted programmatically, by computer, to give the best least squares fit of the whole calculated powder diffraction pattern to the whole observed pattern. Many things besides crystal structure can be and often need be considered in the calculations, e.g., instrumental profiles, preferred orientation, and contributions from background and from other phases. It is possible to refine simultaneously the structures, or at least amounts present, of two or more phases present (7J. 

Absorption lines are defined relative to the continuum. In the case of a resolved line, one may describe the line in terms of its depth relative to the local continuum across the line, say R(AA) = F[(A )/FC where AA is the wavelength measured from line centre, Fc is the flux in the continuum, and F) is the flux at a wavelength within the line. The total absorption by the line obtained by integrating R(AA) over the line profile is known as the equivalent width W. When a line profile is not resolved yet unaffected by blending from neighbouring lines, the equivalent width is independent of the resolution even though the line profile is set by the instrumental profile and not by the intrinsic stellar profile. 

There are two different approaches for calculation of the instrumental function. The first is the convolution approach. Proposed more than 50 years ago, initially to describe the observed profile as a convolution of the instrumental and physical profiles, it was extended for the description of the instrumental profile by itself According to this approach the total instrumental profile is assumed to be the convolution of the specific instrumental functions. Representation of the total instrumental function as a convolution is based on the supposition that specific instrumental functions are completely independent. The specific instrumental functions for equatorial aberrations (caused by finite width of the source, sample, deviation of the sample surface from the focusing circle, deviation of the sample surface from its ideal position), axial aberration (finite length of the source, sample, receiving slit, and restriction on the axial divergence due to the Soller slits), and absorption were introduced. For the main contributors to the asymmetry - axial aberration and effect of the sample transparency - the derived (half)-analytical functions for corresponding specific functions are based on approximations. These aberrations are being studied intensively (see reviews refs. 46 and 47). 

In the field of powder diffraction related to the retrieval of the defects and microstructural information the requirements for using physically meaningful instrumental profile are greater than in the field related to the crystal structural analysis. Thus the following main methods are used. (1) The use of a special high-resolution diffractometer. (2) Experimental determination of the instrumental function for the same material but without defects. (3) Numerical calculations of the instrumental function with ray-tracing simulations. 

First, we compare these instrumental profiles calculated with the proposed method and with the convolution approach. Secondly, a fine difference between an exact solution and a solution based on the convolution approach is demonstrated. 

Figure 6.15 shows the instrumental profile caused by axial shift of the points Ai and A2 i.e. arbitrary axial divergence). 

Figure 6.15 Instrumental profile caused by an axial aberration (see explanations for angles

Precision and Calculation Time. To calculate one point on the instrumental profile we need to calculate a multidimensional integral. For the case where absorption can be neglected this integral reduces to a four-dimensional integral. To estimate how the number of points on the calculation grid affects the precision and calculation time, the calculations of the total instrumental profile were performed for four cases. They are given in Table 6.3. 

The number of calculation points on the instrumental profile for the given cases is 100. 

Figure 6.20 compares two total instrumental profiles calculated with 50 X 50 X 50 X 50 and 5 x 5 x 5 x 5 points. As is easy to see, it is enough to take only 5 calculation points in each direction to reach a precision of 1 %. The calculation time for this case is about 0.05 s. The calculation time can be decreased still further by taking an unequal number of calculation points in each direction from the line position. 

The inverse proportionality between peak width and mean size stated by the Scherrer equation places practical limits to the range of domain sizes that produce measurable effects in a powder pattern. While the lower bound [a few ( 2)nm, depending on the specific phase] is related to the approximations used, the upper bound depends on the instrumental resolution, i.e. on the width of the instrumental profile. Traditional laboratory powder diffractometers, using standard commercial optics, typically allow the detection of domain sizes up to 200 run. Above this value, domain size effects can hardly be distinguished from the instrumental broadening. This limit, however, can considerably be extended by using suitable high resolution optics, as is the case of many diffractometers in use with synchrotron radiation. In this case the practical limit can reach several micrometres. 

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. 

APPENDIX FOURIER TRANSFORMS OF PROFILE COMPONENTS Instrumental Profile (IP)... 

Suppose that in second-order calibration a standard Xi contains one analyte (hence, there is one chemical source of variation) and this standard is measured in such a way that the pseudo-rank of Xi equals one. The mixture X2, measured under the same experimental circumstances, contains the analyte and one unknown interferent. If the instrumental profiles (e.g. spectra and chromatograms) of the analyte and interferent are different, then the three-way array X having Xi and X2 as its two individual slices has two chemical sources of variation. This equals the number of PARAFAC components needed to model the systematic part of the data, which is the three-way rank of X, the systematic part of X. 

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