Body-centered cubic symmetry

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. 

BP I has body-centered cubic symmetry, BP II has simple cubic symmetry, and BP III has isotropic symmetry. 

Fig. 2. Structures for the solid (a) fee Cco, (b) fee MCco, (c) fee M2C60 (d) fee MsCeo, (e) hypothetical bee Ceo, (0 bet M4C60, and two structures for MeCeo (g) bee MeCeo for (M= K, Rb, Cs), and (h) fee MeCeo which is appropriate for M = Na, using the notation of Ref [42]. The notation fee, bee, and bet refer, respectively, to face centered cubic, body centered cubic, and body centered tetragonal structures. The large spheres denote Ceo molecules and the small spheres denote alkali metal ions. For fee M3C60, which has four Ceo molecules per cubic unit cell, the M atoms can either be on octahedral or tetrahedral symmetry sites. Undoped solid Ceo also exhibits the fee crystal structure, but in this case all tetrahedral and octahedral sites are unoccupied. For (g) bcc MeCeo all the M atoms are on distorted tetrahedral sites. For (f) bet M4Ceo, the dopant is also found on distorted tetrahedral sites. For (c) pertaining to small alkali metal ions such as Na, only the tetrahedral sites are occupied. For (h) we see that four Na ions can occupy an octahedral site of this fee lattice.

The term crystal structure in essence covers all of the descriptive information, such as the crystal system, the space lattice, the symmetry class, the space group and the lattice parameters pertaining to the crystal under reference. Most metals are found to have relatively simple crystal structures body centered cubic (bcc), face centered cubic (fee) and hexagonal close packed (eph) structures. The majority of the metals exhibit one of these three crystal structures at room temperature. However, some metals do exhibit more complex crystal structures. 

Caesium chloride is not body-centered cubic, but cubic primitive. A structure is body centered only if for every atom in the position x, y, z there is another symmetry-equivalent atom in the position x+ j,y+ j,z+ j in the unit cell. The atoms therefore must be of the same kind. It is unfortunate to call a cluster with an interstitial atom a centered cluster because this causes a confusion of the well-defined term centered with a rather blurred term. Do not say, the 04 tetrahedron of the sulfate ion is centered by the sulfur atom. 

Any study of colloidal crystals requires the preparation of monodisperse colloidal particles that are uniform in size, shape, composition, and surface properties. Monodisperse spherical colloids of various sizes, composition, and surface properties have been prepared via numerous synthetic strategies [67]. However, the direct preparation of crystal phases from spherical particles usually leads to a rather limited set of close-packed structures (hexagonal close packed, face-centered cubic, or body-centered cubic structures). Relatively few studies exist on the preparation of monodisperse nonspherical colloids. In general, direct synthetic methods are restricted to particles with simple shapes such as rods, spheroids, or plates [68]. An alternative route for the preparation of uniform particles with a more complex structure might consist of the formation of discrete uniform aggregates of self-organized spherical particles. The use of colloidal clusters with a given number of particles, with controlled shape and dimension, could lead to colloidal crystals with unusual symmetries [69]. 

Body-centered cubic (bcc) is the lattice symmetry of Fe, for instance (Fig. 16.2c). Bcc here refers to a crystal arrangement of atoms at the corners of a cube and one atom in the center of the cube equidistant from each face. 

The chemistry of Scheme 2 produces a cubic pore structure with long-range periodicity and unit cell parameter (Ko) of 8.4 nm. The material show a relatively large number of Bragg peaks in the X-ray diffraction (XRD) pattern, which can be indexed as (211), (220), (321), (400), (420), (332), (422), (431), (611), and (543) Bragg diffraction peaks of the body-centered cubic Ia-3d symmetry (Fig. 1). 

The metal substrates used in the LEED experiments have either face centered cubic (fee), body centered cubic (bcc) or hexagonal closed packed (hep) crystal structures. For the cubic metals the (111), (100) and (110) planes are the low Miller index surfaces and they have threefold, fourfold and twofold rotational symmetry, respectively. 

The SMA effect can be traced to properties of two crystalline phases, called martensite and austenite, that undergo facile solid-solid phase transition at temperature Tm (dependent on P and x). The low-temperature martensite form is of body-centered cubic crystalline symmetry, soft and easily deformable, whereas the high-temperature austenite form is of face-centered cubic symmetry, hard and immalleable. Despite their dissimilar mechanical properties, the two crystalline forms are of nearly equal density, so that passage from austenite to a twinned form of martensite occurs without perceptible change of shape or size in the macroscopic object. 

Figure 16.2. Conventional (non-primitive) unit cells of (a) the face-centered cubic and (b) the body-centered cubic lattices, showing the fundamental vectors a1 a2, and a3 of the primitive unit cells. (A conventional unit cell is one that displays the macroscopic symmetry of the crystal.)...

Calculations have thus far been performed for the three standard cubic arrays, namely simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fee). As a result of this geometric symmetry, the couple N and particle stress dyadic A are given by the configuration-specific relations... 

One of the least stable of the metal hexafluorides, RhF6, is a black volatile solid (P2q--c = 49.5 Torr). The gas phase is deep red brown and monomeric.1275 At room temperature it has a body-centered cubic lattice, and, like numerous other hexafluorides, it undergoes a phase change at reduced temperature. The low temperature phase has not been well defined, but is of lower symmetry, and is probably orthorhombic.1278 For RhF6 the phase change occurs at — 23 °C, which is comparable to the transition temperatures (°C) for other metal hexafluorides ... 

Energy bands for the transition-metal series Ti. V. Cr, (Mn, with a complex structure, is omitted). Fe. Co, Ni, and Cu. as a function of wave number along a symmetry line in the appropriate Brillouin Zone. For the face-centered cubic (fee) and body-centered cubic (bcc) structures, the symmetry line is in a [100] direction for the hexagonal close-packed (hep) structure, it is parallel to a nearest-neighbor distance in the basal plane. Pashed lines indicate estimated bands. [After Mattheiss. 1964.]... 

An LCAO description of the electronic structure requires at least the minimal basis set (all orbitals that may be occupied in the ground state of the atom) of five d states per atom and the s state. Consideration of the bands from F ig. 20-1 indicates that in fact the highest-cnergy slates shown (for examples, H,s and X4) have p-like symmetry, and we shall not reproduce this with our minimal set, but the bands at this energy arc unoccupied in any case and it will be of little consequence. For constructing bands in solids, the angular forms for the d states in terms of cartesian coordinates, shown in Eq. (1-21), arc most convenient. Here we shall carry out the calculation explicitly for chromium, in the body-centered cubic structure it is carried out for the face-centered cubic structure in Problem... 

For the eight nearest neiglibors in the body-centered cubic structure (Fig. 20-2), the direction cosines /, m, and n entering Table 20-1 are all 3 all combinations of plus and minus being u.sed 3 (11 1 ) 3 (111), 3- / (lll), 3- (lll), 3 (iTl), 3 (111), 3" (fl 1), 3 (111). Those with positive direction cosines n (in the positive z direction) have phase factors those with negative n have pliase factors e Let us then make the evaluation explicitly for states of symmetry zx. Tlie interatomic matrix element docs not appear in Table... 

ReFv is the only thermally stable binary heptahalide, and crystallographic studies indicate a high-temperature, body-centered cubic (bcc) lattice which transforms to a lattice with lower symmetry at lower temperatures. Spectroscopic evidence suggests pentagonal bipyramidal geometry. Only within ternary systems, hepta-anions are more often found. 

The cubic packing symmetry of the spherical domains is often body-centered cubic (BCC) (Thomas et al, 1987 Almdal et al. 1993 Okamoto et al. 1994a Chu et al. 1995 Adams et al. 1996) and, when solvent is present, sometimes face-centered cubic (FCC) (McConnell et al. 1993). The BCC packing is characteristic of spheres that interact via... 

Documentation exists in the literature as to the observation of anomalies in the temperature dependence of some physical properties of vanadium in the range 175-325 K. Although the anomaly was attributed by different workers to an antiferromagnetic transition, a small distortion of the body-centered cubic crystal structure, and impurities, Finkel et al. ( ) recently ascribed the anomaly to a second order phase transition at 230 K. Using low temperature x-ray diffraction techniques in the study of a single crystal of vanadium, Finkel et al. observed a decrease in crystal lattice symmetry form body-centered cubic (T > 230... 

Figure 1. Phase diagram for a structurally symmetric coil—coil block copolymer (Lam = lamellae, Hex = hexago-najly packed cylinders, Q/a3d = bicontinuous cubic with laid symmetry, Q/m3m = body-centered cubic, CPS = close packed sphere).

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