Specimen intensity transform

However the recent development of a technique for calculating a quasi-continuous map of the specimen intensity transform Q) affords the possibility of realising the full potential of fiber diffraction data. 

Comparison of the observed specimen intensity transform with that calculated for a model of the structure of the specimen provides a powerful test of the correctness of the model. In the present contribution we describe some preliminary attempts to simulate fiber diffraction patterns. When the observed and simulated intensity transforms are displayed visually they provide a useful guide to the progress of a structure refinement as well... 

The specimen intensity transform X is a type of convolution product of the particle intensity transform Ip and the particle orientation density function ( 1,2). The procedure that we have used to simulate Ip involves firstly the calculation of the intensity transform for an infinite particle, with appropriate allowances for random fluctuations in atomic positions and for matrix scattering. A mapping of Xp is then carried out which includes the effects of finite particle dimensions and of intraparticle lattice disorder, if this is present. A mapping of Is is then obtained from Tp by incorporating the effects of imperfect particle orientation. 

Figure 2. Intensity transforms for points uniformly distributed on a helix with radius r = 4.5 A, unit height h = 3 A, and unit twist t = 108°. (a) Particle intensity transform for p = 500 A (b) particle intensity transform for p = 100 A (c) specimen intensity transform for p = 100A,ao = 3°.

Figure 3. Comparison of (a) simulated specimen intensity transform calculated for a model of the structure of collagen with (b) a mapping of the observed specimen intensity transform derived from a diffraction pattern obtained with the specimen tilted at 15.75° to the normal to the x-ray beam.

The parameters used in the simulation are those derived from the observed pattern (H). No allowance has been made for interparticle interference effects which are responsible for the sampling of the particle imensity transform along the equator in the observed specimen intensity transform. 

Following Fraser et al. (4), we choose to represent the scattered intensity in terms of a cylindrically symmetric "specimen intensity transform" I (D), where D is a position vector in reciprocal space. Figure 10 shows the Ewald sphere construction, the wavelength of the radiation being represented by X. The angles p and X define the direction of the diffracted beam and are related to the reciprocal-space coordinates (R, Z) and the pattern coordinates (u,v) as follows ... 

When the microdensitometer data have been corrected for oblique incidence, they are in a form suitable for substitution into equation 18 for the specimen intensity transform. In practice, the correction is made as the integration is performed (see section 10.2). 

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

The single most severe drawback to reflectivity techniques in general is that the concentration profile in a specimen is not measured directly. Reflectivity is the optical transform of the concentration profile in the specimen. Since the reflectivity measured is an intensity of reflected neutrons, phase information is lost and one encounters the e-old inverse problem. However, the use of reflectivity with other techniques that place constraints on the concentration profiles circumvents this problem. 

Regarding the spatial aspects of the enzymatic degradation of CA-g-PLLA, a surface characterization [30] was carried out for melt-molded films by atomic force microscopy (AFM) and attenuated total-reflection Fourier-transform infrared spectroscopy (ATR-FTIR) before and after the hydrolysis test with proteinase K. As exemplified in Fig. 3 for a copolymer of MS = 22, the AFM study showed that hydrolysis for a few weeks caused a transformation of the original smooth surface of the test specimen (Fig. 3a) into a more undulated surface with a number of protuberances of 50-300 nm in height and less than a few micrometers in width (Fig. 3b). The ATR-FTIR measurements proved a selective release of lactyl units in the surface region of the hydrolyzed films, and the absorption intensity data monitored as a function of time was explicable in accordance with the AFM result. 

The power spectrum of high-resolution electron microscopy (HREM) images contains usefiil diffraction information, which can be used for studying defects, small particles or other local structures that are difficult to study by diffraction alone. For thin specimens below a certain thickness, the Fourier transform of the image intensity (/ (5)) can be related to the Fourier transform of the potential (F (5), the structure factor) ... 

In the first three chapters of this book, we considered the fundamentals of crystallographic symmetry, the phenomenon of diffraction from a crystal lattice, and the basics of a powder diffraction experiment. Familiarity with these broad subjects is essential in understanding how waves are scattered by crystalline matter, how structural information is encoded into a three-dimensional distribution of discrete intensity maxima, and how it is convoluted with numerous instrumental and specimen-dependent functions when projected along one direction and measured as the scattered intensity V versus the Bragg angle 20. We already learned that this knowledge can be applied to the structural characterization of materials as it gives us the ability to decode a one-dimensional snapshot of a reciprocal lattice and therefore, to reconstruct a three-dimensional distribution of atoms in an infinite crystal lattice by means of a forward Fourier transformation. 

The latter instrument is of particular value in work of this kind because it allows continuous observation of a diffraction line. For example, the temperature below which a high-temperature phase is unstable, such as a eutectoid temperature, can be determined by setting the diffractometer counter to receive a prominent diffracted beam of the high-temperature phase, and then measuring the intensity of this beam as a function of temperature as the specimen is slowly cooled. The temperature at which the intensity falls to that of the general background is the temperature required, and any hysteresis in the transformation can be detected by a similar measurement on heating. 

Complete information about the specimen would be available only by tomographic methods with a stepwise rotation of the sample (see e.g. Schroer, 2006) or using inherent symmetry properties of the sample. Under the assumption of fibre symmetry of the stretched specimen around the tensile axis, from the slices through the squared FT-structure the three-dimensional squared FT-structure in reciprocal space can be reconstructed and hence also the projection of the squared FT-structure in reciprocal space. The Fourier back-transformation of the latter delivers slices through the autocorrelation function of the initial structure. Stribeck pointed out that the chord distribution function (CDF) as Laplace transform of the autocorrelation function can be computed from the scattering intensity l(s) simply by multiplying I(s) by the factor L(s) = prior to the Fourier back-... 

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