Norio Kato

Pendellösung Fringes in Distorted Crystals III. Application to homogeneously bent crystals

The dynamical diffraction theory developed in Part I and II of this paper (J. Phys. Soc. Japan 18 (1963) 1785; 19 (1964) 67) is applied to homogeneously bent crystals. The Laue cases of X-ray diffraction are treated specifically. For a small bending Pendellosung fringes are still observable but fringe distances decrease with increasing the curvature. The fringes take the form of hyperbola in the same way as the perfect crystal. For a large bending and in the parts of the crystal far from the entrance surface the fringes fade out. Connecting with these aspects the trajectories, the phase changes and the intensity distributions of crystal waves are discussed in details both for non-absorbing and absorbing crystals. The formula of integrated intensity of diffracted beam is given in non-absorbing cases. The Borrmann absorption in distorted crystals is reformulated in a neat form for general cases of lattice distortions.

Pendellösung Fringes in Distorted Crystals I. Fermat's Principle for Bloch Waves

A wave-optical foundation is given for a geometrical optics for electrons and X-rays in distorted crystalline media. A “modified Bloch-wave” and the associated Eikonal function are introduced for this purpose. The modified Bloch-wave behaves like the usual Bloch-wave within local small regions. The trajectory of the wave can be given by a variational principle as to the phase integral along a path with fixed end points A and B , namely \begin{aligned} \delta\int_{A}^{B}(\textbf{\itshape k}_{0}{\cdot}d\textbf{\itshape r}){=}0 \end{aligned} k 0 being the wave-vector associated with the modified Bloch-wave. This is nothing more than the Fermats principle. The Eulers equation of the variational equation gives the change in k 0 along the trajectory.

Pendellösung Fringes in Distorted Crystals II. Application to Two-Beam Cases

Wave-optical principles of electron and X-ray diffraction in distorted crystals developed in Part I are applied to Laue cases where only one diffracted wave is excited appreciably. Using the two-beam approximations, it can be shown that the trajectory of a “modified Bloch-wave” is determined by a differential equation which has an analogous form to a special-relativistic equation of motion for a charged particle in an electric field. The apparent force which changes the direction of motion is caused by lattice distortions. The phase change along the trajectory is obtainable by the phase integral given in Part I. The amplitudes of the direct and the diffracted waves are given by considering the compatibility of crystal and vacuum waves as to the energy flows on the crystal boundaries. The boundary conditions are discussed for electron and X-ray cases separately. Pendellosung fringes are expected for both cases from an interference between two modified Bloch-waves corresponding to the different branches of ...

Dynamical Theory of Electron Diffraction for a Finite Polyhedral Crystal I. Extension of Bethe's Theory

Bethes dynamical theory of election diffraction has been extended by an approximate treatment first to the case of a wedge-shaped crystal, next to the case of a wedge-shaped crystal covered by opaque screens with an aperture and finally to the most general case of a polyhedral crystal of finite extension. If the incident wave, \(\varPhi_{\text{I}}=\varPsi_{\text{I}} \exp j(\textbf{K}_{\text{I}}\cdot\textbf{\itshape r})\), impinges on the entrance surface, Sc , and the transmitted and diffracted (interfracted) wave leave the erystal from the exit surface, Sa , their wave functions are given as a superposition of plane waves;

Contraction of Pendellösung Fringes in Distorted Crystals

It is theoretically shown that the spacing of X-ray Pendellosung fringes should contract in an arbitrarily strained crystal compared with that in the Perfect crystal, so far as the displacement of lattice points is given by a quadratic form of position. The result is an extension of the prediction previously obtained for the case of homogeneous bending. In the light of this theory several experimental results, particularly Ando and Katos experiment which is to appear in Acta cryst. (1966), are discussed qualitatively. The term “the fringes of equal strain-gradient” is proposed for the Pendellosung fringes observed in X-ray traverse topographs of strained crystals having the form of parallel slab.

Integrated Intensities of the Diffracted and Transmitted X-rays due to ideally Perfect Crystal (Laue Case)

The analytical expressions of the integrated reflecting power and transmitting power of X-rays for absorbing perfect crystals are obtained as follows: The diffracted wave: \begin{aligned} {R_{H}}^{y}{=}e^{-\mu_{0}t}{\cdot}\frac{\pi}{2}\left[2\sum\limits_{n{=}0}^{\infty}J_{2n+1}(2A)+I_{0}(h)-1\right]. \end{aligned} The transmitted wave: \begin{aligned} {R_{T}}^{y}{=}e^{-\mu_{0}t}{\cdot}(-2\pi)\left[\sum\limits_{m{=}1}^{\infty}m\cos m\beta I_{m}(h)\right]-{R_{H}}^{y}. \end{aligned} In these formulae \begin{aligned} h{=}2A\sqrt{k^{2}+g^{2}}\quad\text{and}\quad \beta{=}\tan^{-1}k/g. \end{aligned} J m is the first kind Bessel function of m -th order and I m is the modified Bessel function of m -th order, and the other notations are the same as those of Zachariasens text-book on X-ray diffraction (1944). A practical method of obtaining the numerical values of R H y and R T y , and calculated results are shown.

Absolute Positions of Pendellösung Fringes in X-Ray Cases

The positions of Pendellosung fringes with respect to the entrance surface were studied experimentally. In section-type experiments, the following results were obtained. (1) At the entrance surface the intensity of the wave field takes the maximum. (2) The distance between the first and second fringes is larger than the averaged one. (3) The positions of fringes in a deeper region are shifted by a quarter of the averaged fringe distance towards the entrance surface, compared with the fringe positions expected. from the conventional plane-wave theory of dynamical crystal diffraction. The traverse-type experiments were also carried out. These results are predictable by a spherical wave theory (N. Kato: Acta cryst. 14 (1961) 526, 627). The fact (3) is due to the phase jumps of the crystal waves at a kind of caustic region near the entrance surface. The phenomenon is similar to the phase anomaly near the focal point of lenses.

Dynamical Theory of Electron Diffraction for a Finite Polyhedral Crystal II. Fraunhofer Formula

The general formula derived in the previous paper 1) has been simplified by introducing an assumption that the wave-length of electrons compared with crystal size is very short. The simplified formula is more practical and directly applicable to the interpretation of the simple- and double-refraction as well as the Fraunhofer diffraction effects of Laue spot. A comparison of the dynamical and corresponding Kinematical formulae shows that the concept of “ Intensitatsbereich ” as advocated by von Laue 2) in his kinematical theory may be used in dynamical theory. Finally some numerical examples and comparison with theory and experiment are given.

Dynamical Theory of Electron Diffraction for Finite Polyhedral Crystal III. Fresnel Diffraction Formula

Following the first paper of the same title (J. Phys. Soc. Jap. 7 1952) 397), in which Bethes dynamical theory of electron diffraction (Ann. d. Phys. Lpz. 87 (1928) 55) was extended to the case of finite polyhedral crystal and the general expressions of wave functions for transmitted and interfracted (=reflected by crystal net plane) wave were obtained, they have been rewritten in a form of the so-called Fresnel diffraction with the purpose that we might have better understanding of electron microscope images of crystallites. From these formulae, it is concluded that the striations due to “ Pendellosung ” are observed in the just-focused images of crystallites in which Bragg reflection occurs while in defocused image they are disturbed in the neighbourhood of a crystal edge by the Fresnel diffraction effect. In a special case where the parameter indicating the deviation from the Bragg condition is very large, the present formulation applies to the phase contrast fringes due to the edge of transparent wed...

An Experimental Study on the Form of X-Ray Dispersion Surface by Means of Pendellösung Fringes

It was experimentally verified that the Pendellosung fringes in the X-ray diffraction pattern of section type are of hyperbolic form in a wedge-shaped perfect crystal as predicted theoretically (N. Kato: Acta cryst. 14 (1961) 526, 627). This amounts to a direct proof to the fact that the dispersion surface of crystal waves is also of hyperbolic form. Based on this fact, a method of obtaining the structure factors by means of Pendellosung fringes is proposed. The atomic scattering factors were redetermined by this method for (511) and (444) reflections of Si single crystal: they were found to be 5.85 and 4.20 respectively. The method is useful in the cases of higher order reflections. The agreement between the present results and the previous ones (H. Hattori, et al. : J. Phys. Soc. Japan 20 (1965) 988) is satisfactory.