Cubic form

Abstract: A cubic fourfold is a smooth cubic hypersurface of dimension four; it is special if it contains a surface not homologous to a complete intersection. Special cubic fourfolds form a countably infinite union of irreducible families $$C_d $$ , each a divisor in the moduli space $$C $$ of cubic fourfolds. For an infinite number of these families, the Hodge structure on the nonspecial cohomology of the cubic fourfold is essentially the Hodge structure on the primitive cohomology of a K3 surface. We say that this K3 surface is associated to the special cubic fourfold. In these cases, $$C_d $$ is related to the moduli space $$N_d $$ of degree d K3 surfaces. In particular, $$C $$ contains infinitely many moduli spaces of polarized K3 surfaces as closed subvarieties. We can often construct a correspondence of rational curves on the special cubic fourfold parametrized by the K3 surface which induces the isomorphism of Hodge structures. For infinitely many values of d, the Fano variety of lines on the generic cubic fourfold of $$C_d $$ is isomorphic to the Hilbert scheme of length-two subschemes of an associated K3 surface.

Abstract: The broadening of the X-ray diffraction lines which occurs when the crystals composing the specimen are smaller than about 10-5 cm. edge length is well known. Since the discovery of this phenomenon by Scherrer in 1920 a fair amount of work has been done on the determination of the particle size from the breadth of the lines. Scherrer (1920), Bragg (1933) and Seljakow (1925) have calculated the diffraction broadening for crystals of cubic form belonging to the cubic system. For parallel monochromatic radiation and a point specimen, they agree upon a formula βx = C λ/t cos 1/2 χ , where βx = angular breadth of the line defined below, t — edge length of cubic crystal, λ= wave-length of X-radiation, 1/2 χ = θ , the Bragg angle. The values given for the constant C are 0·94, 0·89 and 0·92 respectively.

Abstract: A review is given of some conventions and definitions required for the derivation of the irreducible representations of the space groups, and of a method to obtain lattice harmonics. These are given for all the irreducible representations of the simple cubic ($\mathrm{Pm}3m$), face-centered cubic ($\mathrm{Fm}3m$) and body-centered cubic ($\mathrm{Im}3m$) space groups for $l\ensuremath{\le}12$. The expansions are given in polar coordinates and care has been taken that different bases corresponding to the same representation span identical, rather than equivalent, representations, which are given in full. Moreover, all the different expansions listed in the tables are fully orthogonal.