Symmetry cubic

A similar effect occurs in highly chiral nematic Hquid crystals. In a narrow temperature range (seldom wider than 1°C) between the chiral nematic phase and the isotropic Hquid phase, up to three phases are stable in which a cubic lattice of defects (where the director is not defined) exist in a compHcated, orientationaHy ordered twisted stmcture (11). Again, the introduction of these defects allows the bulk of the Hquid crystal to adopt a chiral stmcture which is energetically more favorable than both the chiral nematic and isotropic phases. The distance between defects is hundreds of nanometers, so these phases reflect light just as crystals reflect x-rays. They are called the blue phases because the first phases of this type observed reflected light in the blue part of the spectmm. The arrangement of defects possesses body-centered cubic symmetry for one blue phase, simple cubic symmetry for another blue phase, and seems to be amorphous for a third blue phase. 

In solids of cubic symmetry or in isotropic, homogeneous polycrystalline solids, the lateral component of stress is related to the longitudinal component of stress through appropriate elastic constants. A representation of these uniaxial strain, hydrostatic (isotropic) and shear stress states is depicted in Fig. 2.4. Such relationships are thought to apply to many solids, but exceptions are certainly possible as in the case of vitreous silica [88C02]. 

Another phase which has attracted recent interest is the gyroid phase, a bicontinuous ordered phase with cubic symmetry (space group Ia3d, cf. Fig. 2 (d) [10]). It consists of two interwoven but unconnected bicontinuous networks. The amphiphile sheets have a mean curvature which is close to constant and intermediate between that of the usually neighboring lamellar and hexagonal phases. The gyroid phase was first identified in lipid/ water mixtures [11], and has been found in many related systems since then, among other, in copolymer blends [12]. 

FIGURE 6.44 Several possible symmetric arrays of identical protein snbnnits, inclnding (a) cyclic symmetry, (b) dihedral symmetry, and (c) cubic symmetry, inclnding examples of tetrahedral (T), octahedral (O), and icosahedral (I) symmetry. (Irving GAs)... 

Both of the current models for the central mode scattering contain the implicit assumption of cubic symmetry above Tm. Possibly because of the dramatic nature of the soft-mode behaviour and a ready understanding of the structural transformation in terms of it, there was a strong incentive to establish a link between it and the central mode scattering. A consistent difficulty with this approach is the failure to establish an intrinsic line-width for the central mode peak and the unspecified nature of the mechanism responsibly for a low-frequency resonance in the energy of the soft mode. ... 

The observation of the departure from cubic symmetry above Tm co-incident with the appearance of the central peak scattering serves to resolve the conflict between dynamic and lattice strain models. The departure from cubic symmetry may be attributed to a shift in the atomic equilibrium position associated with the soft-mode anharmonicity. In such a picture, the central peak then becomes the precusor to a Bragg reflection for the new structure. 

The compounds of the MMe205F type, where Me = Nb or Ta M = Rb, Cs, Tl, crystallize in cubic symmetry and correspond to a pyrochlore-type structure [235-237]. This structure can be obtained from a fluorite structure by replacing half of the calcium-containing cubic polyhedrons with oxyfluoride octahedrons. 

By contrast, lithium extraction from the tetrahedral sites in Li[Mn2]04, i.e., for 0spinel structure [105, 114, 120]. It is difficult to extract all the lithium electrochemi-cally from Li[Mn2]04, at least at practical voltages, without causing decomposi-... 

Electrodes that are prepared from acid-leached LT-LiCo, xNix02 compounds (0< x<0.2) show significantly enhanced electrochemical behavior over the parent LT-LiCo1 xNix02 structure. The improved performance has been attributed to the formation of compounds with a composition and cation arrangement close to the ideal Li[B2]04 spinel structure (B = Co, Ni) [62]. These spinel-type structures have cubic symmetry, which is maintained on lithiation the unit cells expand and contract by only 0.2 percent during lithium insertion and extraction. 

If the perturbation function shows cubic symmetry, and in certain other special cases, the first-order perturbation energy is not effective in destroying the orbital magnetic moment, for the eigenfunction px = = i py leads to the same first-order perturbation terms as pi or pv or any other combinations of them. In such cases the higher order perturbation energies are to be compared with the multiplet separation in the above criterion. 

To avoid this phase change, zirconia is stabilized in the cubic phase by the addition of a small amount of a divalent or trivalent oxide of cubic symmetry, such as MgO, CaO, or Y2O3. The additive oxide cation enters the crystal lattice and increases the ionic character of the metal-oxygen bonds. The cubic phase is not thermodynamically stable below approximately 1400°C for MgO additions, 1140°C for CaO additions, and below 750°C for Y2O3 additions. However, the diffusion rates for the cations are so low at Xhtstsubsolidus temperatures that the cubic phase can easily be quenched and retained as a metastable phase. Zirconia is commercially applied by thermal spray. It is also readily produced by CVD, mostly on an experimental basis. Its characteristics and properties are summarized in Table 11.8. 

Hexaborides of a CaBg type are formed by K, the alkaline earths, Y and the larger lanthanides, as well as Th and some actinides ". The crystal structure of these compounds with cubic symmetry (Pm3m, O, ) (see Fig. 1) is characterized by a three-dimensional skeleton of Bg boron octahedra, the interstices of which are filled by metal atoms. The connection between two octahedra is by a B—B bond of length 1.66 X 10 pm, whereas the B—B bond lengths in one octahedron are 1.76 X 10 pm. ... 

Similarly, monometallic Rh, Pd, and Au and bimetallic Pt-Rh and Pt-Pd nanowires were prepared in FSM-16 or HMM-1 by the photoreduction method [30,33,34]. The bimetallic wires gave lattice fringes in the HRTEM images, and the EDX analysis indicated the homogeneous composition of the two metals. These results show that the wires are alloys of Pt-Rh and Pt-Pd. Mesoporous silica films were also used as a template for the synthesis of uniform metal particles and wires in the channels [35,36]. Recently, highly ordered Pt nanodot arrays were synthesized in a mesoporous silica thin film with cubic symmetry by the photoreduction method [37]. The... 

Most complexes showing spin-state transitions are in fact of low symmetry. In order to describe their electronic structure it is convenient to employ term symbols appropriate to cubic symmetry and this practice will be followed below. The most common transition-metal ions for which spin-state transitions have been observed are Fe " (3d ), Fe " (3d ) and Co (3d ), a minor role being played by Co " (3d ), Mn " (3d ), as well as Cr " and Mn " (3d ). The relevant ground states for an octahedral disposition of the ligands are LS Ui,(t ,) and HS r2,(t ,e ) for iron(II), LS and HS Ai,(t, e ) for... 

Sauer et al. [185] derived a weak quadmpole interaction from the asymmetry of a poorly resolved Zeeman split spectmm of in W metal versus a Ta metal absorber. They also ascribed the unexpected weak quadmpole effect to deviations from cubic symmetry at the source or absorber atom arising from either interstitial impurities or crystal defects. 

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