A systematic survey of cubic crystal structures

Loeb, Arthur L.

doi:10.1016/0022-4596(70)90021-6

Cited by 38 publications

(7 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1 For practical applications of lattice complexes in 2 and 3 dimensions, see 1969); Fischer (1968) ;Smirnova (1966); Smirnova and Poteshnova (1966);Hellner (1965); Loeb (1967 and1970).…”

mentioning

confidence: 99%

Space groups and lattice complexes

Fisher¹,

Burzlaff²,

Hellner³

et al. 1973

View full text Add to dashboard Cite

The lattice complex is to the space group what the site set is to the point group -an assemblage of symmetry-related equivalent points.The symbolism introduced by Carl Hermann has been revised and extended, A total of 402 lattice complexes are derived from 67 Weissenberg com plexes. The Tables list site sets and lattice complexes in standard and alternate representations. They answer the following questions:What are the co-ordinates of the points in a given lattice complex?In which space groups can a given lattice complex occur? What are the lattice complexes that can occur in a given space group? The higher the symmetry of the crystal structures is, the more useful the latticecomplex approach should be on the road to the ultimate goal of their classification.

show abstract

“…1 For practical applications of lattice complexes in 2 and 3 dimensions, see 1969); Fischer (1968) ;Smirnova (1966); Smirnova and Poteshnova (1966);Hellner (1965); Loeb (1967 and1970).…”

mentioning

confidence: 99%

Space groups and lattice complexes

Fisher¹,

Burzlaff²,

Hellner³

et al. 1973

View full text Add to dashboard Cite

show abstract

“…The matrix −I leaves all lattices invariant. More information on these things can be found in Birman [40], Hilbert and Cohn-Vossen [297], Janssen [326], Loeb [424], Lomont [427], Nussbaum [497], and Schwarzenberger [568].…”

Section: Crystallographymentioning

confidence: 99%

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Terras

2013

View full text Add to dashboard Cite

Whenever there is a large earthquake the Earth vibrates for days afterwards. The vibrations consist of the superposition of the elastic-gravitational normal modes of the Earth that are excited by the earthquake.-From F. Gilbert [212, p. 107].A (surface or Laplace) spherical harmonic is an eigenfunction of the Laplacian on the sphere. These are the analogues of exponentials for Fourier analysis on the sphere. Laplace and Legendre introduced these functions in order to study gravitational theory in the 1780s. Spherical harmonics are necessary for the analysis of any phenomena with spherical symmetry; e.g., earthquakes, the hydrogen atom, and the solar corona. Some of these topics will be discussed later in this section.Since our treatment of harmonic analysis on the sphere is rather condensed, the reader may want to consult some of the following references for more information: Before discussing spherical harmonics, we need to understand something about the geometry of the sphere. This symmetric space is closely related to the orthogonal group O(n) of real n × n matrices U such that t UU = I, where t U denotes the transpose of U and I denotes the n × n identity matrix. The special orthogonal SO(n), is the subgroup of matrices U in O(n) such that the determinant of U is one. You can regard U in SO(n − 1) as an element of SO(n) by formingThe group O(n) is a compact group and the sphere is a compact symmetric space, making some of the analysis on it somewhat easier.Exercise 2.1.1 (The Sphere as a Quotient or Homogeneous Space). Consider the sphere S n−1 = {x ∈ R n | x = 1}. Show that S n−1 can be identified with the quotient space SO(n)/SO(n − 1), using the preceding identification of SO(n − 1) as a subgroup of SO(n). Hint.Map the coset gSO(n − 1) to the vector ge n for g in SO(n − 1) and e n = t (0,... ,0, 1).You may also want to read the discussion of the topology of spheres and orthogonal groups in Chevalley [85,. In particular, the fundamental (or Poincaré) group of SO(2) is isomorphic to Z, while that of SO(n), n ≥ 3, has order 2.From now on we shall consider only the sphere S 2 . See the references for the general case. Now the sphere S 2 is a differentiable manifold. This means that locally it looks like two-dimensional Euclidean space. To make this precise, we use the usual angular coordinates (θ , ϕ), 0 < ϕ < 2π, 0 < θ < π, pictured in Fig. 2.1, to parameterize S 2 except for the semi-circle C through the poles and (1, 0, 0). A similar coordinate patch can be constructed to cover the rest of the sphere. The equations for the rectangular coordinates (x, y, z) of a point on S 2 in terms of the angular coordinates are x = sin θ cos ϕ y = sin θ sin ϕ z = cos θ ⎫ ⎬ ⎭ (2.1)

show abstract

“…The boundary is the radical plane of the spheres, defined by equal tangential distances T,P = T 2 P space without voids. Dirichlet domains of such homogeneous sets of points have been calculated on a larger scale by several authors (Nowacki, 1935(Nowacki, , 1976Hellner and Löckenhoff, 1970;Loeb, 1970Loeb, , 1976Löckenhoff and Hellner, 1971;Koch, 1972Koch, , 1973Koch and Fischer, 1972).…”

Section: Dirichlet Domains Of Points Spheres and Atomsmentioning

confidence: 99%

Geometrical packing analysis of molecular compounds

Fischer¹,

Koch²

1979

Zeitschrift Für Kristallographie - Crystalline Materials

View full text Add to dashboard Cite

From a geometrical point of view the packing of molecules in crystal structures has been studied for 192 organic compounds. For this, Dirichlet domains of the atoms have been calculated by means of radical planes. These domains have been joined to packing polyhedra of the molecules. Two molecules are called adjacent if their packing polyhedra share faces. Utilizing this concept, coordination numbers have been calculated and a classification of structures into packing types has been performed with the aid of packing graphs. As a result, coordination number 14 was calculated more frequently (94 structures) than 12 (46 structures). In total, 51 packing types have been found, 9 of which stand out because of their frequency. It should be noted that two packing types with coordination number 14 show frequencies (18 and 17) comparable to those of the types ccp, bcc, and hep (32, 30, and 11, respectively).

show abstract

A systematic survey of cubic crystal structures

Cited by 38 publications

References 14 publications

Space groups and lattice complexes

Space groups and lattice complexes

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Geometrical packing analysis of molecular compounds

Contact Info

Product

Resources

About