Intensity finite-angle scattering

Small-angle scattering intensity in the high angle (q) region can be analysed to provide information on interface thickness (e.g. the lamellar interface thickness block in copolymer melts or core-corona interface widths in micelles). For a perfectly sharp interface, the scattered intensity in the POrod regime falls as q A (Porod 1951). For an interface of finite width this is modified to... 

The intensity of scattering at finite q, on the other hand, reflects the concentration fluctuations that exist on a more local scale. In the case of a single-component system, as discussed in Section 4.1, the finite-angle intensity data can be converted, through an inverse Fourier transform, to a radial distribution function g(r). With a two-component system a comparable general procedure is not available, and information on the structure is derived usually by comparing the observed intensity data, on the q plane, with expressions derived from theoretical models. 

Figure 23.7 shows SANS data from a 4.6 vol% latexes with a fully deuterated PMMA-D shell, (thickness 30 A) polymerized on a PMMA-H core (radius, a = 498 A), after desmearing corrections for the finite instrumental resolution [37,42]. The absolute intensity at zero scattering angle is given by Eq. (23.15) with P(0) = 1 and Vp equal to the volume of the D-labeled polymer in the shell with SLD", p p = 6.97 x 10 cm-2. qiie SLD of the solvent is close to that of the PMMA-H core (p j = 1.06 x 10 cm2) and thus... 

Gitxnier s expression for the intensity of the scattered radiation of particles" vrith finite dimensions furnishes a similar result as that given here for an atom. We can now consider each atom of the particle as a scattering centre. One can obtain an idea of the connection between the particle size and the size of the diffuse central spot in the following way. In Fig. 33 the incident ray and a ray diffracted at an angle s are... 

In an ideal scattering experiment the collisions are assumed to occur at a fixed point in space. In practice the collision volume is finite and the part viewed by the detecting system generally depends on the scattering angle. Care must therefore be taken in relating the scattered particle intensity to the cross section. 

Table I. Errors in 90° Scattering Intensity Due to Finite Detector Acceptance Angle...

Lorentz factor A factor that is used to correct diffraction intensities that allows for the varying time that different Bragg reflections (as reciprocal lattice points with finite sizes) take to pass through the sphere of reflection (Ewald sphere). The value of this correction factor depends on the scattering angle and the geometry of the measurement of the Bragg reflection. 

Here N/V is the number of the dissolved polyelectrolyte molecules per volume whereas /o (r/) denotes the scattering intensity of an isolated macromolecule. S(q) is the effective structure factor that takes into account the effect of finite concentrations. As shown further below its influence can be disregarded for higher scattering angles. 

The simple model implies an infinite perfect crystal. The crystal specimen studied is necessarily finite (rarely more than 0.5 mm in size) and, if perfect by conventional criteria, would be utterly unsuitable for collection of intensity data of the kind required for ordinary structure determination and refinement. What is needed is an ideally imperfect crystal shot through with dislocations and intergrain boundaries so that it behaves, so far as diffraction is concerned, like a mosaic of perfect crystal blocks of the order of micrometers or tenths of micrometers in size, tilted with respect to one another by angles of the order of a few seconds of arc and scattering independently (incoherently) with respect to one another. The assumption of an ideally imperfect ... 

The result can be generalized for copolymers made of monodisperse unequal subchains. As can be seen in Fig. 7.8, the scattered intensity vanishes in the zero angle limit. This result is in agreement with (7.3.59). For q / 0, the scattered intensity is finite (non-zero) and this effect is related to the existence of fluctuations of intramolecular composition. 

X is the wavelength in vacuo, N is Avogadro s number, while A and A are the second and third virial coefficients. The term P(9 is the form factor which is a function of the size and shape of the macromolecule in solution and represents the modulation of the intensity of scattered radiation due to the finite size of the molecule and to its deviation from sphericity. The term dn/dc is the specific refractive index increment and represents the change in solution refractive index as a function of solute concentration. If experiments are conducted in the limit of zero scattering angle where P(9) = 1 as well as at sufficiently low concentrations where only the second virial coefficient need be considered, then eq. (1) reduces to... 

An ordered spherulite is spherically symmetric. Consequently, the scattering in the interior of an Hv sample (see also Figure 5-25) should be zero. A finite scattering intensity in the center thus indicates disorder. The size of the spherulite can be determined from the angle at which maximum scattering occurs. 

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