Vectors, scattering

Light scattering teclmiques play an important role in polymer characterization. In very dilute solution, where tire polymer chains are isolated from one anotlier, tire inverse of tire scattering function S (q) can be expressed in tire limit of vanishing scattering vector > 0 as 1121... 

Figure 3 Plot of the relative intensity versus the scattering vector at the indicated temperatures for a sample of P7MB cooled from the melt at 10°C/min, analyzed in a synchrotron source [12].

Fig. 9a,b. The intermoleailar scattering patterns calculated for a the smectic A b the smectic B phases of the mesogen GB(4.4, 20.0, 1, 1) with the scattering vector parallel to the layer normal. The scaled scattering vectors Q and Q range from -8n to 8n... 

France).The collected data, on beam D24, covered the scattering vector q range from 0.4 to 0.006 A"l, q = (47csin0/2)/X, where 0 was the scattering angle and X the wavelength. 

In many cases, including transport phenomena which differ from normal diffusion, Eq. (3.1.4) turns out to be a good approximation, in particular if only small magnitudes of the gradient intensity ybg (sometimes referred to as the generalized scattering vector of PFG NMR) are considered. Under these circumstances, the genuine diffusivity D is replaced by an effective diffusivity, Deff... 

Taking the photon scattering vector q in z-direction, the dynamical structure factor is related to the Compton profile J(pz) by... 

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 Schematic Zimm plot The determined Kc/(it values ( ), the extrapolated values for fixed concentrations to q — 0 ( ), and the extrapolated values for fixed scattering vector to infinite dilution (°) c — 0.

Experimental considerations Frequently a numerical inverse Laplace transformation according to a regularization algorithm (CONTEST) suggested by Provencher [48,49] is employed to obtain G(T). In practice the determination of the distribution function G(T) is non-trivial, especially in the case of bimodal and M-modal distributions, and needs careful consideration [50]. Figure 10 shows an autocorrelation function for an aqueous polyelectrolyte solution of a low concentration (c = 0.005 g/L) at a scattering vector of q — 8.31 x 106 m-1 [44]. 

Mw = 2.1 x 106g/mol) in water, which is denoted Cw(t) in the original work [44]. The subscript indicates that both the incoming beam and the scattered light are vertically polarized. The correlation function was recorded for a solution with a concentration of c = 0.005 g/L at a scattering vector of q = 8.31 x 106m-1. The inset shows the distribution function of the relaxation times determined by an inverse Laplace transformation. 

S =(s,, lf) Scattering vector in polar coordinates (polar angle, y/ azimuthal angle). - See the preamble to this List of Abbreviations ... 

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 + [Pg.97]

For symmetrical-reflection geometry the modulus of the true scattering vector is... 

To specify the components of the scattering vector by S 2 and s3 is only a suggestion. The specification meets the case that is of highest practical importance (anisotropy with fiber symmetry). 

The general definition of a projection has been given on p. 23 in Eq. (2.37). For the purpose of illustration let us write down an example. If s = (Si,Sj,Sk) is a representation of the scattering vector in orthogonal Cartesian coordinates, then the aforementioned ID projection is... 

It is well known in SAXS theory that isolated nuclei give a significant scattering intensity at a small scattering vector (q). The range of q between c/i and q2 will be simply denoted as c/n> i.e.,... 

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