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Symmetry crystallographic

The clock example illustrates most principles of importance in discrete symmetry groups with translation, also known as crystallographic symmetry groups. For simplicity consider a two-dimensional unit cell with a two-fold axis T, pictured as a pointed ellipse, and two mirror planes my and mu-To construct a multiplication table any general position (not coincident with [Pg.33]

It is easy to show that the table remains the same while labelling the sites differently, starting from different sites for E. It is therefore more practical [Pg.33]

Several important concepts emerge from this simple model  [Pg.34]

Translational symmetry is assumed in both directions, X and Y. This assumption implies that if an operator moves any site into a neighbouring cell it is equivalent to that site entering the reference cell from the opposite side. The coordinates x,y are therefore fractional and symmetry translations are unity, whereby 1 + x x x — 1 — x, etc. [Pg.34]

Operations of the second kind are identified by commas, noting that a given operator that transforms an empty circle into a circled comma also does the opposite. [Pg.34]


The crystallization of glassy Pd-Ni-P and Pd-Cu-P alloys is complicated by the formation of metastable crystalline phaf s [26]. The final (stable) crystallization product consists of a mixture of a (Pd,Ni) or (Pd,Cu) fee solid solution and more than one kind of metal phosphide of low crystallographic symmetry. Donovan et al. [27] used transmission electron microscopy (TEM) and X-ray microanalysis to study the microstructure of slowly cooled crystalline Pd4oNi4oP2o- They identified the compositions of the metal phosphides to be Pd34Ni45P2j and Pdg8Ni[4Pjg. [Pg.295]

Creation operator, 505 representation of, 507 Critical value, 338 Crystallographic point groups irreducible representations, 726 Crystallographic symmetry groups construction of mixed groups, 728 Crystal, eigenstates of, 725 Crystal symmetry, changes in, 758 Crystals... [Pg.772]

Symmetry of the anion in the crystal. Data in parentheses give the approximate (non-crystallographic) symmetry Solvent molecule(s) in the crystal structure... [Pg.136]

The analysis of XRPD patterns is an important tool studying the crystallographic structure and composition of powder compounds including the possibility to study deviation from ideal crystallinity, i.e. defects. Looking at an X-ray powder diffractogram the peak position reflects the crystallographic symmetry (unit cell size and shape) while the peak intensity is related to the unit cell composition (atomic positions). The shape of diffraction lines is related to defects , i.e. deviation from the ideal crystallinity finite crystallite size and strain lead to broadening of the XRPD lines so that the analysis of diffraction line shape may supply information about sample microstructure and defects distribution at the atomic level. [Pg.130]

The Rietveld Fit of the Global Diffraction Pattern. The philosophy of the Rietveld method is to obtain the information relative to the crystalline phases by fitting the whole diffraction powder pattern with constraints imposed by crystallographic symmetry and cell composition. Differently from the non-structural least squared fitting methods, the Rietveld analysis uses the structural information and constraints to evaluate the diffraction pattern of the different phases constituting the diffraction experimental data. [Pg.135]

The oligomeric structures of chemokines vary. CXCL8 (IL-8), the prototypical CXC chemokine, is dimeric in solution (26) and in its crystalline form (due to twofold crystallographic symmetry) (27). The three-stranded P-sheet of each subunit joins to form a six-stranded P-sheet (Figure IB). The two C-terminal... [Pg.11]

The "unfolded-drum" or "ladder" compound 2 has crystallographic symmetry. This corresponds to the idealized molecular symmetry and, therefore, there are three chemically inequivalent types of Sn atoms in the molecule, although all are hexacoordinated. The oxygen atoms in the open form can be subdivided into two types, as in the case of the drum molecule tricoordinate framework oxygen atoms and the dicoordinate oxygen atoms of the bridging carboxylate ligands. [Pg.475]

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. [Pg.243]

The complex has crystallographic ///-symmetry, the mirror plane bisecting the unique benzyl group, the nitrogen atom to which it is attached, and the ethylzinc moiety. The pseudo-tetrahedral zinc atom has a short (1.930(4) A) zinc-ethyl bond, but comparatively long (2.230(2) A) nitrogen-zinc donor bonds. [Pg.341]

The structure of the anion [Be3(OH)3(malonate)3]3 (95) is illustrated in Fig. 18. The Be3(OH)3 ring has a flattened chair conformation with no crystallographic symmetry. Nevertheless, the arrangement of oxygen atoms around each beryllium is near tetrahedral. [Pg.145]

The symmetry of the chain tends to be maintained in the crystalline lattice, and the local symmetry becomes a crystallographic symmetry. [Pg.112]

The orientation of the primary alcohol group is gauche-trans. The hydrogen bonding consists of finite chains which intersect at four-coordinated water molecules on two-fold, crystallographic-symmetry axes. [Pg.436]

Quasicrystals, the diffraction patterns of which show non-crystallographic symmetry. [Pg.190]

In a famous paper by Shechtman et al. (1984) electron diffraction patterns were shown of rapidly quenched and solidified aluminium-manganese alloys. Sharp diffraction peaks, suggesting long-range translational order, were observed with the presence however of five-fold symmetry (that is of a non-crystallographic symmetry see 3.6.1.1). By different orientation of the specimen five-fold axes (in 6 directions), three-fold axes (in 10 directions) and two-fold axes (in 15 directions) were identified with the subsequent observation of the existence also of an inver-sion centre the assignment of this phase to the icosahedral point group, m36, was defined. [Pg.198]

To the extent that a crystal is a perfectly ordered structure, the specificity of a reaction therein is determined by the crystallographic symmetry. A crystal is built up by repeated translations, in three dimensions, of the contents of the unit cell. However, the space group usually contains elements additional to the pure translations, such as a center of inversion, rotation axis, and mirror plane. These elements can interrelate molecules within the unit cell. The smallest structural unit that can develop the whole crystal on repeated applications of all operations of the space group is called the asymmetric unit. This unit can consist of a fraction of a molecule, sometimes fractions of two or more molecules, a single whole molecule, or more than one molecule. If, for example, a molecule lies on a crystallographic center of inversion, the asymmetric unit will contain half... [Pg.134]

It should be remembered that conclusions based on crystallographic symmetry apply strictly only to the average geometry of the molecule at each site. Crystals often suffer various kinds of disorder, in which case individual molecules may depart considerably from the crystallographically determined average geometry. [Pg.135]

The surfaces that form subunit-subunit contacts are very much like parts of a protein interior detailed fit of generally hydrophobic side chains, occasional charge pairing, and both side chain and backbone hydrogen bonds. Twofold symmetry is the most common relationship between subunits. The 2-fold is often exact and can be part of the actual crystallographic symmetry, as for the prealbumin dimer in Fig. 62. However, in many cases (e.g., Tulinsky et al., 1973 Blundell et al., 1972) individual side chains very close to the approximate 2-fold axis must take up nonequivalent positions in order to avoid overlapping (see Fig. 63). Conformational nonequivalence can extend further away from the axis and produce such effects as different... [Pg.242]

The two most fundamental macroscopic properties of a mineral are its chemical composition and crystallographic symmetry. Together these serve as the basis for naming and classifying minerals. Other macroscopic properties (e. g., optical, electrical, magnetic, thermal, mechanical, etc.) can be considered manifestations of a mineral s average structure. [Pg.422]


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See also in sourсe #XX -- [ Pg.340 ]

See also in sourсe #XX -- [ Pg.4 ]




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