Scattering vector, measurement

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

Figure 7.5. Relationship between symmetrical (

reflection geometry. Bold bars symbolize the sample in symmetrical (dashed) and asymmetrical (solid) geometry. Incident and scattered beam are shown by dashed-dotted arrows, the incident angle is a = 0 + scattering vector s. For the tilted sample the sample-fixed scattering vector S3 is indicated (after [84])... 

This quantity has units of 1/length and is the magnitude of a vector known as the scattering vector. For our purpose here, it is sufficient to note that s is the yardstick used for measuring the distance between the scattering centers when the intensity of the scattered radiation of wavelength X is recorded at an angle 0 from the direction of the incident radiation. 

Fig. 23 Angular dependence of Rayleigh ratio (Rw(q)) of segmented PNIPAM-seg-St copolymer chains in water measured from static LLS, where K is a constant, q is the scattering vector and polymer concentration (C) was 7.2 x 10-7 g/mL. The inset shows the temperature dependence of the hydrodynamic radius distribution /(Rh) determined from dynamic LLS [94]...

In choosing beam optics to measure xrd-rsm, one must consider resolution function in the reciprocal space. The resolution function is defined by the incident beam divergence and the acceptance window of the diffracted beam side optic. Figure 6.3 schematically shows the definition of the resolution function in the reciprocal space. The X-ray detector is located at the tip of the scattering vector H in the reciprocal space. The incident beam divergence 5u> and the acceptance window of the diffracted beam optic 520 define the resolution function (gray area in Figure 6.3) in the reciprocal space. The form of the obtained diffracted intensity distribution of the crystal by xrd-rsm is defined by the convolution of the resolution function and the reciprocal lattice point of the crystal. Therefore, a resolution function smaller than... 

The other geometric consideration is the relative orientation of the scattering vector, q, and an applied field, such as a flow or electric field. For example, as seen in Chapter 6, the use of homodyne dynamic light scattering to measure velocity gradients requires that q be oriented parallel to the component of the velocity of interest. This is simply accomplished by having that velocity component directed parallel to the vector bisecting k(. and k [44],... 

A typical diffraction curve is shown in Fig. 2. It gives the measured intensity from a solution as a function of the scattering vector... 

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