Absolute units , scattering

Fig. 6.5. Coherent structure function S(q) in absolute units in comparison to amorphous cell simulations [194] and neutron scattering data [185]...

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... 

Thereafter the slit-smeared scattering intensity is readily expressed in absolute units [7(x)/y] =e.u./nm4. 

Guinier s law exhibits two parameters, I (0) and R2, which describe structural aspects of the sample. The experimentalist should consider their determination, if the recorded SAXS data show a monotonous decay that is indicative for the scattering from uncorrelated1 particles. Particularly useful is the evaluation of Guinier s law, if almost identical particles like proteins or latices are studied in dilute solution (cf. Pilz in [101], Chap. 8). The absolute value of 1(0) is only accessible, if the scattering intensity is calibrated in absolute units (Sect. 7.10.2). 

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... 

Figure 5 Comparison between SAXS and SANS. The two experimental scattering curves are given in absolute units (cm1) for PI23 micelles in D20 at 40°C, at a volume fraction of 2.6 %.

Further details are discussed in [2]. Figure 1 shows the PDF of SrTiOs [4]. The first negative peak at 1.95 A is due to Ti - O pairs, and is negative because the scattering length of Ti is negative. The PDF is given in the absolute units. Thus... 

If the shape of the particles or pores is known, a geometrical size can be calculated from Is or Ip. For thin rods, / or Ip are equivalent to the diameter of a cylinder [8]. The TPM model also allows the determination of the skeletal density if the scattering intensity can be obtained in absolute units ... 

The scattered intensity is nearly always measured in relative and not in absolute units, which necessarily introduces a proportionality coefficient, C. As we established before, when the phase angle is mr (n is an integer), the corresponding interference functions in Eq. 2.18 are reduced to U, U2 and Ui and they become zero otherwise. Hence, assuming that the volume of a crystalline material producing a diffraction pattern remains constant (this is always ensured in a properly arranged experiment), the proportionality coefficient C can be substituted by a scale factor K = CU U- Uz. 

Figure 7. Two-body depolarized scattering Stokes spectrum of the Vi Raman band of gaseous CF4 in absolute units at 294.5 K. Full circles ( ) indicate experimental data together with error bars.

In summary, since the early 1980s or so, collision-induced light scattering experiments on molecular fluids have demonstrated that the study of the collision-induced absolute-unit spectra (depolarized and/or isotropic ones) is an useful tool, and up to now the only one, to measure multipolar polarizabilities of molecules. 

The correlation function rn(r), according to (5.61), can be obtained by taking the inverse Fourier transform of the scattered intensity I(q), which, however, must be measured in absolute units if T (r) is to be obtained also in absolute units. When the intensity is known only in arbitrary (relative) units, y(r) can still be obtained in view of the normalization condition y (0) = 1. 

This simple relationship turns out to be useful in a surprising variety of situations, provided the intensity is determined in absolute units so that the invariant Q can also be evaluated in absolute units. If the scattering length densities of the two phases are known, for example, from knowledge of their chemical compositions, the experimental value of Q can be used to provide the relative amounts of the two phases. If, on the other hand, the relative amounts are known beforehand, for example, when the sample is prepared by mixing known quantities of two substances, the invariant can be used to determine the difference pi — P2. An instance in which such information could be useful is with a polymer blend in which a partial miscibility is suspected. When known quantities of two polymers are mixed together, any decrease in p — p2... 

Figure 2.6 (a) The X-ray scattered intensities, l q), of the 5 wt% CioCi/liquid paraffin in absolute unit at different temperatures (50, 55, and 60°C), and (b) the pair-distance distribution functions (PDDFs), p(r), extracted from these scattering curves with the GIFT method. Solid and broken lines in (a) represent GIFT fit and the calculated (total) form factor for n particles... 

Figure 2.14 (a) The X-ray scattering intensities l q) of 5 wt% CnG /octane with different concentration of added water obtained in absolute unit at 60°C, (b) the... 

Figure 2.19 (a) The scattering functions of 5 wt% C16G2 in liquid paraffin and squalene in absolute unit at 50°C, and (b) the p(r)-functions. The arrows and a broken line in panel b indicate the maximum length and cross-sectional diameter of reverse micelles. 

Figure 2.21 The X-ray scattered intensities, l q), of 5 vrt% Ci4C2/oil systems at higher temperature SO C in absolute unit and the p(r)-functions (a) the l(q) of the Ci4G2/oil systems in aromatic oils and cyclohexane, and (b) the corresponding p(r)-functions, (c) The l(q) of the Q4G2 in n-alkanes, and...

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