Calibration to Absolute Scattering Intensity

Calibration to absolute intensity means that the scattered intensity is normalized with respect to both the photon fiux in the primary beam and the irradiated volume V. Thereafter the scattering intensity is either expressed in terms of electron density or in terms of a scattering length density. Both definitions are related to each other by Compton s classical electron radius. 

Fields of Application. In SAXS a calibration to absolute intensity is required if extrapolated or integrated numerical values must be compared on an absolute scale. 

Examples are the determination of density fluctuations or the density difference between matrix and domains as a function of materials composition. 

In WAXS of soft condensed matter, studies of the intensity in absolute units are not common, unless the method for the exact determination of X-ray crystallinity according to Ruland is applied (cf. Sect. 8.2.4). 

General Routes. If a SAXS beamline in normal transmission geometry is used, calibration to absolute intensity is, in general, carried out indirectly using secondary standards. Direct methods require direct measurement of the primary beam intensity under consideration of the geometrical setup of the beamline. On a routine basis such direct calibration was commercially available for the historic Kratky camera equipped with zero-dimensional detector and moving slit device .  

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In practice, the invariant can be used for the purpose of calibration to absolute scattering intensity by means of samples for which the absolute invariant can easily be computed. For this purpose colloidal suspensions of noble metals with known volume concentration are suitable [96], All the noble metal particles must be small enough so that they really contribute to the observed particle scattering. They must not agglomerate. 

Electron Density. Continuing the preceding considerations, calibration to absolute intensity means normalization to the scattering of a single electron , Ie that can be expressed in electron units, [e.u.]. Inevitably a calibration to absolute units involves also a normalization with respect to the irradiated volume V. Thus, for the field of materials science a suitable dimension of the absolute intensity is [I/V] = e.u./nm3 - The intensity measured in the detector is originating from a material with an average electron density of 400 electrons per nanometers cubed . The electron density itself is easily computed from mass density and chemical composition of the material (cf. Sect. 2.2.1). 

In normal transmission geometry16 any mathematical treatment of calibration to absolute units [87-90] starts from the basic differential relation among the scattering intensity in the detector, the primary intensity and the structure... 

Direct calibration to absolute intensity is not a usual procedure at synchrotron beamlines. Nevertheless, the technical possibilities for realization are improving. Therefore the basic result for the total scattering intensity measured in normal transmission geometry is presented. At a synchrotron beamline point-focus can be realized in good approximation and the intensity /(s) is measured. Then integration of Eq. (7.19) results in... 

Pure liquids can be used for the purpose of calibration to absolute intensity, because their diffuse scattering Ipi (0) = limv qIFi (s) caused from density fluctuations can be computed theoretically. Some examples are in the literature [91,93-95],... 

Mathematically spoken k is the zero-dimensional projection / 0 of the scattering intensity. After calibration to absolute units 7(s) turns into 7(s) /V - its scattering power is known as Porod s invariant... 

We notice that anisotropic scattering patterns can be calibrated to absolute intensity, as well. 

A measurement of S(x) requires that the SANS spectrometer be calibrated by some absolute standard, a process which is often difficult to achieve with precision. An easier measurement in the ratio of scattering intensities of an anisotropic sample in two different directions. Figure 5 shows a graph of Sj (x)/S (x) versus x for the phantom model and the fixed junction case. 

Calibration is required to convert measurements of scattered light intensity from arbitrary to absolute values, an essential step in the calculation of molecular weight. Fortunately, because the calibration constant of most photometers remains stable for long periods of time, the calibration procedure need be carried out only infrequently. Should it need to be calibrated, the procedure described in ASTM D4001-93 or that of the instrument vendor should be followed. 

The raw scattering data were first corrected for instrumental background and dark current counts and then corrected for non-uniform detector efficiency at each detector channel element. Then the sensitivity corrected data were plotted as a two-dimensional contour plot with contour levels increasing by a factor of 2 from the outermost one. Lupolen 23/7 was used as a calibration standard to determine the absolute SAXS intensity (49.50). 

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