Intensity in absolute units

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). 

The determination of the interface surface area S, using equation (8-11), requires the measurement of the scattering intensity in absolute units. However, applying equations (8-11) and (8-12), another expression for the surface area per unit volume can be derived ... 

Thus, if the scattering intensity is only known in relative scale and provided the volume fractions of both phases are known, equation (8-13) can be applied to determine the interface surface area per unit volume. Equation (8-13) is particularly useful for powdered samples, for which the precise measurement ofthe scattering intensity in absolute units is often not possible. 

The experimental determination of the emission intensity in absolute units is quite complex as it commonly happens in luminescence measurements, /(A) and A/(A) are often measured in arbitrary units, which are dependent on the equipment and the experimental conditions adopted. The dissymmetry factor gem(- ) is a significant quantity because it is a ratio of emission intensities and is therefore unaffected by the instrumental and experimental parameters. Its value gives an absolute quantification of the chirality of the emitting excited state. 

In Eq. (7.21) the normalization to the scattering cross-section r2 leads to the definition of absolute intensity in electron units which is common in materials science. If omitted [90,91], the fundamental definition based on scattering length density is obtained (cf. Sect. 7.10.1). 

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). 

In terms of contrast, Pluronic micelles have a much more higher contrast using SANS (with D20 as a solvent) than with SAXS. It is illustrated in figure 5, where the SANS and SAXS curves in absolute units are compared for the same sample, micelles of P123 in pure D20 at 40°C and at a volume fraction of 2.6 %. The absolute intensity of the SANS curve is about 103 times greater than the SAXS ones. The maximal flux on the D22 experiment is about 10s neutrons/s. Then, in order to perform kinetics experiments with SAXS, with a temporal resolution equal or better than 30 s, the high flux of a synchrotron source (10nphotons/s or more) is needed. 

One can measure the site concentrations in absolute units to 25% by measuring the absorption coefficient and radiative transition probability (which in turn comes from the level lifetime, radiative quantum efficiency and radiative branching ratios) or to 15% by nonlinear regression fitting of relative intensities to total dopant concentration over a range of site distributions. 

Figure 14.5 Experimental Cu KL23L23 Auger spectra photoinduced from the polycrystalline Cu metal sample, presented as a function of the excess photon energy (AEph) above the K-absorption threshold. The spectra are normalized to absolute intensity (in arb. units) [17].

Figure 152 EL intensity vs. current density for DL LEDs based on the TPD/Alq3 junction (a) ITO/TPD(60nm)/ALq3(60nm)/Mg/ Ag structure (left hand scale in absolute units) (see Ref. 21) (b) ITO/TPD(20nm)/Alq3(40nm)/Mg Ag (see Ref. 356).

For this evaluation, an intensity calibration in absolute units (here crn g srad) ) is necessary. 

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. 

Electromagnetic radiation, such as a bea m of x-rays, carries energy, and the rate of flow of this energy through unit ansa perpendicular to the direction of motion of the wave is called the intensity /. proportional to the square of the amplitude In absolute units intensity is measured in joules/m /sec, but this measurement is a difficult one and is seldom carried out most x ray intensity measurements are made on a relative basis in arbitrary units, sudi as the degree of blackening of a photographic film exposed to the x-ray beam. 

Here p is the electron density of the sample, 7(s)/V is the absolute scattering intensity in electron units, Og (s) = (Yb) is the Fourier transform of Yb, the shape function of the test volume, and Vb = fffyB (r) d r is the size of the test volume. 

Normalization. The practice of subtracting the theoretical Compton-modified intensity works only if the measured intensity is available on an absolute scale. Similarly, before the intensity function I(q) can be converted into the interference function i(q) by means of Equation (4.13), I(q) must be on an absolute scale. In principle, the intensity can of course be measured from the beginning in absolute units by means of an instrument calibrated for absolute intensity, as discussed in Section 2.7. The more usual practice, however, is to measure the intensity first in relative units and then to scale it by the normalization constant determined according to the following criterion. The normalization condition is satisfied when the following is obeyed in the limit of large q... 

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... 

By comparing the experimentally obtained intensity function I(q) against the Porod law (5.71), one can determine the total interfacial area S in the sample. To accomplish this the intensity must be available in absolute units. If the intensity is determined only in relative units, there is still a way out. When the invariant Q is evaluated according to (1.86) and divided into I q), the relative unit scale used to express the intensity is canceled out from the ratio, and we obtain, in place of the Porod law (5.71),... 

SAXS profiles for two (o-fimctionalized IS diblock copdymers at T=303 K Data have been corrected fi>r the density fluctuations, and the intensities are given in absolute units. (Reproduced by permission fimn reference 33)... 

FIGURE 8.3 Time-resolved power spectra (TRPS) estimated by Fourier transforms (FTs) of the time-segmental velocity autocorrelation functions (TSVAFs) of the active center. The spectra indicate the intensities of the vibrational modes of the active center in the globin. The intensities over 3500 cm" are zero (not shown). The photodissociation is achieved at t = Ops, after the previous equilibrium MD stimulation is denoted by negative time duration. The intensities of power spectra are depicted as functions of the vibrational frequency. The ordinate axes indicate the absolute FT intensity in arbitrary units. The spectrum (-5 < t < Ops) is that of MbCO, while the other spectra are those of Mb -i- CO. (From Okazaki, I. et al., Chem. Phys. Lett., 337,151, 2001. With permission.)... 

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... 

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