Diffraction directions

This equation predicts, for a particular incident wavelength X and a particular cubic crystal of unit cell size a, all the possible Bragg angles at which diffraction can occur from the planes ifikl). For (110) planes, for example, Eq. (3-10) becomes 

If the crystal is tetragonal, with axes a and c, then the corresponding general equation is 

These examples show that the directions in which a beam of given wavelength is diffracted by a given set of lattice planes are determined by the crystal system to which the crystal belongs and its lattice parameters. In short, diffraction directions are determined solely by the shape and size of the unit cell. This is an important 

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Cristobalite in respirable airborne dust Lab method using X-ray diffraction (direct method) 76... 

So far, only defocus and spherical aberration have been considered as aberrations affecting the image contrast. Both depend only on the magnitude of the spatial frequency g[ but not on the diffraction direction, thus resulting in rotationally symmetric phase shifts (Equation 11). However, the objective lens may exhibit further aberrations resulting in additional phase shifts, which are not necessarily rotational symmetric. The most important of these additional aberrations are astigmatism and coma. 

Thus the diffraction directions also of X-rays give direct information on the energy zones, the intensity of the reflections... 

This formulation cleverly expands the value and scope of the expression describing the Fourier transform because s is dependent on both ko and k, and defines the relationship between them. Since altering ko alters F just as choosing a different k does, using s as the variable (the difference in direction between k and ko) encompasses all possibilities of incident and diffraction directions. Thus the expression above is the comprehensive Fourier transform of the distribution of points for all ko and k. The diffraction vector s we will see re-appear in a variety of manifestations as we proceed. 

Ordinary optical, organic chemical and physical chemical approaches also offer information which is indispensable for the present purposes. These commonly suffer from the difficulty that they may include extraneous, noncollagenous tissue elements in the analysis, but they furnish useful auxiliary information about molecular composition, size, shape and orientation. Some of these molecular characteristics are particularly difficult to derive from the methods of electron microscopy and X-ray diffraction directly. 

Equation (97) relates the diffraction direction to the orientation of the crystal plane. One of its consequences is that for a given crystal plane normal, the angle a... 

Diffraction Directional change of a wave group after it encounters an obstacle or passes through an aperture. 

Therefore, the dynamical diffraction has the specificity of the coupling of the two wave fields, transmitted and diffracted, in a reciprocal interaction also with the crystal. The purpose of this chapter is the assessment of the general form of the fields in the crystal based on the shape of the external fields as well as the intensities related to the transmitted and diffracted directions. 

In this framework the decomposition and re-composition in the dynamical propagation of X-ray in the crystals is to be emphasized. In this case, the decomposition on the alpha and beta branches, on the transmitted and diffracted directions, have been noted then followed by the re-compositions on common direction from different branches (Pendellbsung), on common branches from different directions (Standing Waves), with the possibility of the effect of the direct anomalous absorption (i.e., the alpha branch free of absorption, with the beta branch anomalous absorbing) and respectively the reverse anomaly (with the alpha branch anomalously absorbing, while... 

These quantities are connected through the Eq. (5.88) - also called as the dispersion equation just for the correlation that it includes, between the refraction indices associated to the transmitted and diffracted directions and along the structure factor of the crystal and the incident wave vector. The last correlation is even better emphasized if the relations (5.1), (5.68), and (5.85) are considered wherefrom another connection between the quantities 6 and 8, can be expressed namely ... 

Remarkable, if the relation (5.91) is inversed, the approximation of the parameter b in relation with the driving cosines of the incident-transmitted and diffracted directions can be considered ... 

From the relation (5.90), with the respective notations, the dependence of the refraction indices associated to the transmitted and diffracted directions of the incident wave vector on the crystal is emphasized, a fact that justifies the given name as dispersion equation for the relation (5.88). 

FIGURE 5.27 The reflection power in the diffracted direction for the Bragg s case for the thick crystal without absorption after Zachariasen (1946). 

Because there is about the energetic propagation, the form and the behavior of the total Poynting vector associated to the DS branches of the diffracted directions is natural to be analyzed. To this aim, the general form of the solutions of the waves propagated on the difiiacted directions associated to the DS branches will be firstly expressed in a form derived from the Laue s (5.104) and Bragg s (5.111) solution in the plane wave approximation the present discussion follows Birau and Putz (2000) and (Putz Lacr a, 2005) ... 

However, the punctual expression of this vector is of less use of interest since the temporal and volume averages are imder focus to be correlated with the energetic interchange between the two diffracted directions, as a function of penetration depth in the crystal. 

In order to simplify the situation in which the energy propagated on the diffracted direction is pendulant and transferred to the transmitted diiee-tion through the presence of the coupling term from Eqs. (5.183) and (5.184), a non-absorbent crystal is considered (for simplification), so that,... 

This observation corresponds to the theoretical explanation for the experimental observations for which the thickness of the crystal can influence the diffraction direction where the energy corresponding to the diffraction solution is carried. Moreover, even inside the crystal, based on its penetration, the total energy is pendulant between the directions of diffraction as based on the value for the trigonometric factor of coupling. 

From the analysis of Eq. (5.191) relation there noted that all the components of the electric induction are transversal respecting the wave vectors corresponding to the diffracted directions of propagation. 

With this conclusion, the phenomenological picture of the propagation of the energy with oscillation between the Po5mting vectors associated to the DS branches is completely described and generalized, allowing an even deeper imderstanding of the interpretation and observation of the intensities associated to the diffraction directions. 

From now on, all the relations regarding the intensities, the ratio of intensities, the powers of refleetion, and the powers of integrated reflections, calculated irr previous Section can be understood in the light of the inter-fererrce of the energies assoeiated to the DS branches for the diffraction directions over the Pendellostmg period. 

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