Ewald theory

Immediately after the introduction of a constant refractive index Bethe developed a dispersion theory of electron diffraction which is very closely related to the Darwin-Ewald theory. In this theory the propagation of de Broglie waves through a crystal is investigated, the potential being expanded in a triple Fourier series in terms of the contributions of the individual lattice planes hkl. Thus Vq in Schrodinger s equation is replaced by a triple Fourier series with the coefficients In accordance with this assumption, the solution... 

These examples show the great practical utility of the Ewald sphere construction. We did once hear Paul Ewald say, some sixty years after he laid the basis of X-ray scattering theory, that he wished people had named something else after him, as it was such a trivial idea ... 

There are also more limited treatments of scattering. McCartney (1976, Chaps. 4-6) confines his attention to scattering by atmospheric particles. This is also discussed by Twomey (1977, Chaps. 9-10) in his treatise on atmospheric aerosols. In Goody (1964, Chap. 7) there are discussions of absorption by gases and, in less detail, extinction by molecules and by droplets. Parts of books on electromagnetic theory or optics include the theory of scattering by a sphere, most notably Stratton (1941, pp. 563-573) and Born and Wolf (1965, pp. 633-664). The latter also derive the Ewald-Oseen extinction theorem and apply it to reflection and refraction at a plane interface (pp. 98-104). 

Uf all the different types of atomic aggregates, ionic crystals have been found to be most suited to simple theoretical treatment. The theory of the structure of ionic crystals described briefly in the following sections was developed about 40 years ago by Born, Haber, Land6, Madelung, Ewald, Fajans, and other investigators. The simplicity of the theory is due in part to the importance in the interionic interactions of the well-understood Coulomb terms and in part to the spherical symmetry of the electron distributions of the ions with noble-gas configurations. 

The first section recalls the Frenkel-Davydov model in terms of a set of electromagnetically coupled point dipoles. A compact version of Tyablikov s quantum-mechanical solution is displayed and found equivalent to the usual semiclassical theory. The general solution is then applied to a 3D lattice. Ewald summation and nonanalyticity at the zone center are discussed.14 Separating short and long-range terms in the equations allows us to introduce Coulomb (dipolar) excitons and polaritons.15,16 Lastly, the finite extent of actual molecules is considered, and consequent modifications of the above theory qualitatively discussed.14-22... 

In the first part of this introductory section, we summarize the main collective phenomena acquired by the dipolar exciton from the lattice-symmetry collectivization of molecular properties. The crystal is considered as an assembly of electrically neutral systems, the molecules, physically separated from each other and in electromagnetic interaction. This /V-body problem will be treated quantum-mechanically in the limit of low exciton densities. We redemonstrate the complete equivalence of this treatment with the theories of Lorentz and Ewald, as well as with the semiclassical approximation. In Section I.A, in a more compact but still gradual way, we establish the model of the rigid lattice of dipoles and the general theory of low-exciton-density systems in interaction with the radiation field. Coulombic excitons, photons,... 

Nam K, JL Gao, DM York (2005) An efficient linear-scaling Ewald method for long-range electrostatic interactions in combined QM/MM calculations. J. Chem. Theory Comput. 1 (1) 2-13... 

Ewald, P. P. Das reziproke Gitter in der Strukturtheorie. Teil I Das Reziproke eines einfachen Gitters, [The reciprocal lattice in the theory of structure. Part I. The reciprocal of a primitive lattice.] Z. Krist. 56, 129-156 (1921). 

Ewald, P. P. Introduction to the dynamical theory of X-ray diffraction. Acta... 

Colella, R. Multiple diffraction of X-rays and the phase problem. In P. P. Ewatd and his Dynamical Theory of X-ray Diffraction. A memorial Volume for Paul P. Ewald. S3 January 1888 — SS August 1985. (Eds., Cruickshank, D. W.. 1 Juretschke, H. J., and Kato, N.) International Union of Crystallography/OxforJ University Press Oxford (1992). 

Dunitz wrote of these equations Debye s paper, published only a few months after the discovery of X-ray diffraction by crystals, is remarkable for the physical intuition it showed at a time when almost nothing was known about the structure of solids at the atomic level. Ewald described how The temperature displacements of the atoms in a lattice are of the order of magnitude of the atomic distances The result is a factor of exponential form whose exponent contains besides the temperature the order of interference only [h,k,l, hence sin 9/M]. The importance of Debye s work, as stressed by Ewald,was in paving the way for the first immediate experimental proof of the existence of zero-point energy, and therewith of the quantum statistical foundation of Planck s theory of black-body radiation. ... 

In connexion with the two interesting experimental papers by Rupp and Wierl, I should like to attempt to give a concise outline of the way in which the theoretical treatment of interference phenomena with X-rays and electrons has been developed hitherto. I particularly wish to emphasize those points which eventually led to the adoption of the Darwin-Ewald dispersion theory of X-rays, in order to draw comparisons with the present position in the realm of electrons. This will perhaps make it clear to what extent we are meantime obliged in the case of electrons to accept Bethe s dispersion theory, and which experiments seem best adapted for testing the special results of this theory. 

Even before the experiments of Siegbahn and Stenstrom had necessitated an extension of the kinematic theory, the intensity problems referred to above had led Darwin, and subsequently Ewald, to calculate out the actual mutual effects of the dipoles existing and capable of vibration within a crystal. Starting with... 

On the dispersion theory, the intensity of the interference bands is obtained from the breadth of the region of total reflection according to Ewald, the theory yields the result that the intensity is proportional to the structure factor (the number of atoms causing the diffraction) and not, as in the kinematic theory, proportional to its square. The experimental verification of this result would require that ideal crystals should be available, whereas it is well known from experiments that most natural and artificial crystals consist of many small lattice blocks slightly displaced with respect to one another, so that the dynamical argument can only be applied to one of these blocks by itself. The whole crystal must be treated as a mosaic of such blocks, and the calculation of the intensity, which was carried out first by Darwin and subsequently in more detail by Waller, represents a combination of the dispersion theory and the kinematic theory. 

In order to test the dispersion theory per se the intensities of numerous reflections were measured by W. H. Bragg, Ehrenberg, Ewald, and others, for as ideal diamonds as could be got and for zinc blende. It turned out that the relationships actually are better reproduced if the intensity is taken as proportional to the structure factor S than if it is taken as proportional to the square of the structure factor. A proposal by W. L. Bragg that in mosaic crystals the reflected intensity p should be taken as proportional to... 

The Darwin-Ewald dispersion theory involves several idealizing assumptions. It does not take account of... 

The application of the Lorentz-Lorenz equation gives a convincing demonstration of the general similarity of the linear response in gas and liquid but its application in the liquid introduces an approximation which has not yet been quantified. A more precise objective for the theory would be to calculate the frequency dependent susceptibility or refractive index directly. For a continuum model this may lead to a polarizability rigorously defined through the Lorentz-Lorenz equation as shown in treatments of the Ewald-Oseen theorem (see, for example Born and Wolf, plOO),59 but the polarizability defined in this way need not refer to one molecule and would not be precisely related to the gas parameters. 

Harvey, M.J., De Fabritiis, G. An implementation of the smooth particle mesh Ewald method on GPU hardware. J. Chem. Theory Comput. 2009, 5, 2371-7. 

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