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Extra resources for An Introduction to Variational Inequals and Their Applns

Example text

With the help of the lattice planes it is possible to obtain an intuitively clear interpretation of the scattering conditions. We take the modulus of the equation G = K: Fig. 4 a, b. Sets of crystal lattice planes. The planes illustrated here have the values m = 1, n = 2, o = 2. The corresponding Miller indices are derived for the triplet (11m, 1In, 110) by mUltiplying this by a whole number p = 2 to give (hkl) = (211). Between the planes with indices m, n, 0 there lie additional planes (dotted lines).

We can see this immediately from the Ewald construction (Fig. 3). e. the magnitude of k o), or for particular angles of incidence, will a point of the reciprocal lattice fall on the Ewald sphere. The various procedures for determining structures differ in the method by which this condition is obtained. One can, for example, simply turn the crystal (preferably about a principal axis oriented perpendicular to the incident beam). Since the reciprocal lattice is associated in a fixed manner to the real crystal lattice, a rotation of the crystal corresponds to a rotation of the reciprocal lattice through the Ewald sphere (whose position in space is defined by the incident beam and is therefore fixed).

7 Supposing the atoms to be rigid spheres, what fraction of space is filled by atoms in the primitive cubic, fcc, hcp, bcc, and diamond lattices? 8 Give a two-dimensional matrix representation of a 2-, 3-, 4-, and 6-fold rotation. Which representation is reducible? 3. Diffraction from Periodic Structures A direct imaging of atomic structures is nowadays possible using the high-resolution electron microscope, the field ion microscope, or the tunnel microscope. Nonetheless, when one wishes to determine an unknown structure, or make exact measurements of structural parameters, it is necessary to rely on diffraction experiments.

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An Introduction to Variational Inequals and Their Applns by D. Kinderlehrer, et al.,


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