Coherent diffraction

The physical stmcture of mixed-layer minerals is open to question. In the traditional view, the MacEwan crystallite is a combination of 1.0 nm (10 E) non-expandable units (iUite) that forms as an epitaxial growth on 1.7 nm expandable units (smectite) that yield a coherent diffraction pattern (37). This view is challenged by the fundamental particle hypothesis which is based on the existence of fundamental particles of different thickness (160—162). 

This result is very interesting as we increase the degree of compensation, the amount of energy in the coherent diffraction-limited core increases and the size of the residual halo decreases. This behavior can be seen in Fig. 3. 

Frishberg, C.A., Goldberg, M.J. and Massa, L.J. Quantum model of the coherent diffraction experiment recent generalizations and applications . In Ref. [23] ofthis bibliography, p. 101. 

In addition to relaxing the fabrication tolerances, the mixed Bragg order implementation induces a coherent diffraction component in the vertical direction23. Although this mechanism reduces the overall Q of the cavity, it facilitates the observation and measurement of the resonator emission. 

Besides the inelastic component, always a certain number of He atoms are elastically scattered in directions lying between the coherent diffraction peaks. We will refer to this scattering as diffuse elastic scattering. This diffuse intensity is attributed to scattering from defects and impurities. Accordingly, it provides information on the degree and nature of surface disorder. It can be used for example to study the growth of thin films or to deduce information on the size, nature and orientation of surface defects Very recently from the analysis of the diffuse elastic peak width, information on the diffusive motion of surface atoms has been obtained. ... 

Fig. 44. Thermal dependence of coherently diffracted He intensities from Cu(100) and Cu(115). (Afier Ref. 112.)...

Hematite derived from dehydroxylation of FeOOH at temperatures below 600 °C shows marked, non-uniform (differential) broadening of the XRD lines. Some authors have attributed this effect to the anisotropic shape of the coherently diffracting domains of hematite (Duvigneaud Derie, 1980), and others to the development... 

The change in the intensity with temperature is calculated with the temperature factor. This change is produced by the crystal lattice vibrations, that is, the scattering atoms or ions vibrate around their standard positions as was previously explained (see Section 1.4) consequently, as the crystal temperature increases, the intensity of the Bragg-reflected beams decreases without affecting the peak positions [25], Debye and Waller were the first to study the effect of thermal vibration on the intensities of the diffraction maxima. They showed that thermal vibrations do not break up the coherent diffraction this effect merely reduces the intensity of the peaks by an exponential correction factor, named the temperature factor, D(0) [2,26], given by... 

The determination of particle sizes using the Scherrer equation A (20) = KA/Acos0 [4] where A is the diameter of the coherently diffracting domain, K, a constant close to unit and A (20) represents the broadening of the FWHM of the Bragg reflection in 20. 

Figure 7 Experimental XRD spectra for odinite and calculated diagrams for hypothetical minerals using the NEWMOD program (Reynolds, 1985) are shown. The berthierine spectrum was simulated using a ferrous-aluminous 7 A clay structure with 3-6 coherent diffracting domain structure. The mixed layer 7 A/smectite mineral berthierine/nontronite was modeled using 60% berthierine layers and 3-6 layer coherent diffracting domains in a disordered (R = 0) structure (after Odin, 1988, p. 162).

Although the stress-induced a-j6 polymorphic transition in PBT is well documented, comparative WAXS measurements of the samples were carried out in the same deformation range at which the H measurement are performed. In addition, the size of the coherently diffracting domains (crystal size) D/,ki in the (100) and (010) and (104) ( c axis direction) during stretching was calculated from the integral breadth of the equatorial reflections according to 1 /Dhki (Baltd Calleja Vonk, 1989). 

The beam of secondary radiation issuing from the sample consists largely of fluorescent radiation, but there are some other weak components present as well. These are coherent scattered radiation, coherent diffracted radiation, and incoherent (Compton modified) radiation. These components appear as a background on which the spectral lines are superimposed. This background is normally low (see Fig. 15-3), but it may become rather high if the sample contains a large proportion of elements of low atomic number, because the sample will then emit a large amount of Compton modified radiation. 

Finally, Bertaut imagined the homogeneous domains of coherent diffraction as constituted by columns of elementary cells juxtaposed orthogonally to the diffracting planes, and he defined a size distribution P(n) as the numerical fraction of columns of length n cells. From that size distribution function, the size-only Fourier coefficients can be defined " ... 

As indicated above, physical methods provide a means by which the dispersion is derived from measurement of particle size. In actual fact, the value referred to as particle size when obtained by X-ray diffraction, should be termed crystallite size as the particle may contain several crystallites and the parameter being measured is the effective length in the direction of the diffraction vector, along which there is coherent diffraction. Additionally, for the case of X-ray diffraction, the value obtained is inherently an average value given that the sample will contain a distribution of particle sizes and the quantity to be determined from the XRD pattern is the... 

This is the intensity of the exposure radiation in the plane of the wafer. The extended source method, or Hopkins method,is often used to predict the aerial image of a partially coherent, diffraction-limited, low-numerical-aperture-aherrated projection system based on scalar diffraction theory. For very high NA, vector calculations involving the complete solution of Maxwell s equation are used. The illumination may be of a single wavelength or it may be broadband. The illumination source may be a conventional disk shape or other more complicated shapes as in off-axis illumination. ... 

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