Wavevector scattering

Consider an incident laser beam with wavelength A illuminating a polymer sample, represented by the large circle in Fig. 2.19, along the direction with 

The incident beam is coherent, meaning that all photons are in-phase. When the incident beam enters the sample, monomers absorb the radiation and re-emit it in all directions. The difference in optical paths between the light scattered by different monomers makes the scattered beam incoherent, meaning that the scattered photons are no longer in-phase. In the example sketched in Fig. 2.19, the difference in optical paths of the radiation scattered by the monomer j at position Rj (at point C) and by the monomer at the origin O is easily calculated  

The section AC is the projection of the vector Rj onto the incident direction 

This difference in optical paths results in the phase difference j, which is ItthIX times the optical path difference [see Eg. (1.77) with A replaced by the 

The scattering wavevector q is defined as the difference of the incident and scattered wavevectors  

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As discussed in Section II, measured excess scattering intensity, after a melt is cooled below its melting point, increases exponentially with time at all scattering wavevectors and the inverse of niax (at which intensity is a maximum) diverges as These observations are similar to those... 

Fig. 3.19 Plots of frequency shift f versus surface concentration T for six polymer monolayers at a scattering wavevector k = 323 cm-1. (From ref. [101])...

Frequency-domain BSS. In this mode, the spectrum of the diffracted probe light is obtained by using a Fabry-Perot interferometer. Light is diffracted by incoherent thermal phonons and the scattering wavevector is determined by the detection angle, which can be accurately fixed by limiting the collection aperture. 

The significance of the vector g can be understood with the aid of Figure 3.2. OP represents the incident wavevector Kq and OP the scattered wavevector Kg. P is a scattering point. The direction PP is given by the vector... 

If 0 is the glancing angle of incidence, then the angle between the incident and the scattered wavevectors is 20, which is called the scattering angle. 

Figure 9.5 Schematic illustration of the phase-separation process after a temperature quench into the spinodal region of the phase diagram. The time dependence of the temperature quench from the spinodal temperature to some final temperature Tfinai is shown in the top diagram. This quench time can be made arbitrarily fast, in which case it has no effect on the time period over which the linear or other regimes persist. The bottom diagram shows the maximum-scattering wavevector qm of the spinodal pattern as a function of time t, with qm oc r . At first, in the linear regime, qm is constant, so that a = 0 but as the pattern coarsens, qm decreases, initially as qm oc due to diffusive Ostwald ripening. Later, when the interfaces are well defined, if the morphology is bicontinuous, there is a crossover to a fast hydrodynamic regime with q , oct. (From Tanaka 1995, reprinted with permission from the American Physical Society.)...

Assuming that the number of crystallites approaches infinity (the randomness of their orientations has been postulated in the previous paragraph), the density of the scattered wavevectors, kj, becomes constant on the surface of the cone. The diffiacted intensity will therefore, be constant around the circumference of the cone base or, when measured by a planar area detector as shown in Figure 2.31, around the ring, which the cone base forms with the plane of the detector. Similar rings but with different intensities and diameters will be formed by other independent reciprocal lattice vectors, and these are commonly known as the Debye rings. ... 

From their definitions in Eqs (2.124) and (2.125), the magnitudes of the incident and scattered wavevectors are the same ... 

The isosceles triangle of wavevectors in Fig. 2.19 shows that half of the magnitude of the scattering wavevector is equal to the magnitude of wavevectors q or q times the sine of half the angle 6 between them ... 

The dependence of the scattered intensity on the size and the shape of the polymer is usually described by the form factor defined as the ratio of intensity scattered at angle 9 (scattering wavevector to that extrapolated to zero angle ( —>0) and therefore, zero scattering wavevector (1 1 - 0) ... 

All optical paths are the same at zero scattering angle (q = 0) and there is no phase shift (ipj = 0 for all j) because the scattering wavevector q = 0 [Eq. (2.131)]. The intensity of light scattered by the molecule at zero angle. 

The form factor in Eq. (2.139) is defined for a specific orientation of the molecule with respect to the scattering wavevector q. Often (but not always ), the system is isotropic with equal probabilities of all molecular... 

This form factor of an ideal linear polymer is called the Debye function and can be rewritten in terms of the product of the square of scattering wavevector q and the mean-square radius of gyration of the chain RI) ... 

Fig. 6 Computer simulations results of the scattering wavevector dependence of the form factors P( ) of a star with / = 24 arms (solid line) and a (compact lattice) hard sphere with nearly the same number of beads (dotted line). Taken from [41]... 

Because ki = kout, the scattering angle 0 and the modulus of the scattered wavevector are related to each other by... 

Figure 7. Top to bottom Fresnel reflectivity, penetration length, evanescent wave intensity and reflected beam plane calculations for a model system. The parameter bu = 2/uk i Q], where is the scattering wavevector, Qc is the critical angle wavevector, and p is the linear absorption coefficient. [From Elements of Modem X-ray Physics by Als-Neilson and DesMorrow, with permission from the editors at John Wiley and Sons.]...

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