Intensity scattered

The corresponding momentum transfer, q, is expressed in terms of the incident and scattered wave vectors, k o x i ( X i) respectively, as 

5 = ki — koand ko = ki =(6.17) The phase change in terms of momentum transfer is therefore 

Therefore, the magnitude of wave vector q is similar to s (modulus of the scattering vector), and using Eq. (6.19), is given by 

the total amplitude of scattered waves (of X-rays) from two such electrons at positions r, and rj, respectively, will simply be given by the sum of the two amplitudes 

Since experimental detectors measure the intensity I(q) and not amplitude A q). The intensity is obtained by taking a product of the amplitude A ° q) with its complex conjugate. Therefore, intensity of scattered radiation at the detector is given by  

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The physics of X-ray refraction are analogous to the well known refraction of light by optical lenses and prisms, governed by Snell s law. The special feature is the deflection at very small angles of few minutes of arc, as the refractive index of X-rays in matter is nearly one. Due to the density differences at inner surfaces most of the incident X-rays are deflected [1]. As the scattered intensity of refraction is proportional to the specific surface of a sample, a reference standard gives a quantitative measure for analytical determinations. 

We are interested in < E (0[,(t)i)E3(62,, where <> means the average over the ensemble of surfaces, the subindexes 1 and 2 refer to two different points of observation and the subindexes A and B belong to two different conditions of illumination, which for example arise from two different wavelengths, two different incident angles, etc.. If A = B and 1 = 2, the above expression gives the angular distribution of the mean scattered intensity, otherwise it turns to be the intensity correlation coefficient y from < E Eb >, assuming that we deal with fully developed speckle. 

It is also possible to deal with the mean scattered intensity by using a geometrical model [11]. Following the procedure of that reference, we finally get... 

Figure 3 The angular distribution of the mean scattered intensity for a rough wire accordingly to a) eq. (7) and b) eq. (9). The parameters in this case were a = 7t/6 and T/a = 1/10. Here X = tandcosfandy = tan9sin(j>...

Experimentally, it is these invariants (equation (B 1.3.17), equation (B 1.3.18) and equation (B 1.3.19)) that can be obtained by scattering intensity measurements, though clearly not by measuring the total cross-section only. 

RRS has also introduced the concept of a Raman excitation profile (REPy for thefth mode) [46, 4lZ, 48, 49, 50 and M]. An REP. is obtained by measuring the resonance Raman scattering strength of thefth mode as a fiinction of the excitation frequency [, 53]. Flow does the scattering intensity for a given (thefth) Raman active vibration vary with excitation frequency within an electronic absorption band In turn, this has led to transfomi theories that try to predict... 

Equation (B 1,9.11) is valid only for plane polarized light. For unpolarized incident light, the beam can be resolved into two polarized components at right angles to each other. The scattered intensity can thus be expressed as (figure Bl.9.2)... 

For random locations of scattering objects in volume V, the scattered intensity can be found by summing the scattering from each object ... 

For molecules having dimensions comparable with the wavelength, phase differences will occur between waves scattered from different regions of the molecule. These phase differences result in an angular dependence of the scattered intensity. The reduction may be expressed in temis of a particle interference factor P(2Q) such that... 

There are two ways to calculate the scattered intensity. One is to first calculate the magnitude of the structure... 

The second method to calculate the scattered intensity or R the Rayleigh ratio) is to square the sum in I... 

Equation (B 1,9.47) applies to the general scattering expression of any system. With spherical synnnetry, the scattered intensity becomes [20]... 

In the case of anisotropic systems with a cylindrical synnnetry (such as rods or fibres), the scattered intensity can be expressed as (derivation to be made later in section Bl.g.dl ... 

X-ray scattering arises from fluctuations in electron density. The general expression of the absolute scattered intensity (simplified as I(q) from now on) from the tliree-dimensional objects iimnersed in a different... 

As we will discuss in the next section, the scattered intensity I q) at very large q values will be proportional to the q tenn. This is the well known Porod approximation, which has the relationship... 

It is sometimes more convenient to nonnalize the absolute scattered intensity I(q) and use the following expression ... 

This equation is the Porod law for the large-angle tail of the scattering curve along the equatorial direction, which indicates that the equatorial scattered intensity I q is proportional to in the Porod... 

The scattered intensity measured from the isotropic three-dimensional object can be transfonned to the onedimensional mtensity fiinction/j(<3 ) by means of the Lorentz correction [15]... 

In oriented systems (fibres or stretched films), the scattered image often appears as a two-bar or a four-point pattern with the scattering maximum at or near the meridian (fibre axis). The one-dimensional scattered intensity along the meridian must be calculated by the projection method using the following fonnalism... 

Let us consider die scattered intensity from a binary incompressible mixture of two species (containing molecules of particle 1 and molecules of particle 2) as in (B 1.9.112) we can rewrite the relationship as... 

Let us take two polymers (one deuterated and one hydrogenated) and dissolve them in a solvent (or another polymer) having a scattering length b. The coherent scattered intensity can be derived from (B 1.9.117), which gives... 

If we deal with a solution at very low concentrations, we can ignore the interactions between the particles and express the scattered intensity as... 

The quantitative analysis of the scattering profile in the high q range can be made by using the approach of Debye et aJ as in equation (B 1.9.52). As we assume tiiat the correlation fiinction y(r) has a simple exponential fomi y(r) = exp(-r/a ), where is the correlation length), the scattered intensity can be expressed as... 

In polymer solutions or blends, one of the most important thennodynamic parameters that can be calculated from the (neutron) scattering data is the enthalpic interaction parameter x between the components. Based on the Flory-Huggins theory [4T, 42], the scattering intensity from a polymer in a solution can be expressed as... 

Figure Bl.23.8. Scattering intensity of 2 keV Ne versus azimuthal angle 6 scans for Pt 110] in the (1 x 2) and (1 X 3) reconstructed phases. Scattering angle 0 = 28° and incident angle a = 6°.

Figure Bl.23.9. Scattering intensity of 4 keV Ne versus azimuthal angle 8 for a Ni 110] surface in the clean (1 X 1), (1 X 2)-H missing row, and (2 x l)-0 missing row phases. The hydrogen atoms are not shown. The oxygen atoms are shown as small open circles. 0-Ni and Ni-Ni denote the directions along which O and Ni atoms, respectively, shadow the Ni scattering centre.

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