Cubic lattice, types

Computer simulations of a range of properties of block copolymer micelles have been performed by Mattice and co-workers.These simulations have been based on bead models for copolymer chains on a cubic lattice. Types of allowed moves for bead chains are illustrated in Fig. 3.27. The formation of micelles by diblock copolymers under weak segregation conditions was simulated with pairwise interactions between A and B beads and between the A bead and vacant sites occupied by solvent, S (Wang et al. 19936). This leads to the formation of micelles with a B core. The cmc was found to depend strongly on fVB and % = x.w = %AS. In the range 3 < (xlz)N < 6, where z is the lattice constant, the cmc was found to be exponentially dependent onIt was found than in the micelles the insoluble block is slightly collapsed, and that the soluble block becomes stretched as Na increases, with [Pg.178]

When the cubic lattice type has been deduced, the unit-cell dimension Oq can be calculated. From the unit-cell volume, the measured crystal density, and the formula weight, the number of formulas in one unit cell can be calculated. 

The proper set of integers s is not hard to find because there are only a few possible sets. Each of the four common cubic lattice types has a characteristic sequence of diffraction lines, described by their sequential s values ... 

For specifying directions and planes in a crystal, the origin of the crystallographic coordinate system is positioned in a lattice point, and the axes are scaled so that the length of every edge of the unit cell is one. Thus, for non-cubic lattice types, this coordinate system is non-Cartesian. 

The summation is over the different types of ion in the unit cell. The summation ca written as an analytical expression, depending upon the lattice structure (the orij Mott-Littleton paper considered the alkali halides, which form simple cubic lattices) evaluated in a manner similar to the Ewald summation this typically involves a summc over the complete lattice from which the explicit sum for the inner region is subtractec... 

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. 

The initial configuration is set up by building the field 0(r) for a unit cell first on a small cubic lattice, A = 3 or 5, analogously to a two-component, AB, molecular crystal. The value of the field 0(r) = at the point r = (f, 7, k)h on the lattice is set to 1 if, in the molecular crystal, an atom A is in this place if there is an atom B, 0, is set to —1 if there is an empty place, j is set to 0. Fig. 2 shows the initial configuration used to build the field 0(r) for the simple cubic-phase unit cell. Filled black circles represent atoms of type A and hollow circles represent atoms of type B. In this case all sites are occupied by atoms A or B. 

The compounds NaNbF6 and NaTaF6 [158, 163] have similar crystal structure belonging to a NaSbF6 type structure [164]. The compounds form a cubic lattice that is packed with Na+ and MeF6 ions, similar to the structure of NaCl. 

Magnesia forms solid solutions with NiO. Both MgO and NiO have face-centered cubic lattices with NaCl-type structures. The similarity between the ionic radii of the metals (Ni2+ = 0.69 A, Mg2+ = 0.65 A) allows interchangeability in a crystal lattice, and thus the formation of solid solutions with any proportion of the two oxides is possible. Such solid solutions are more difficult to reduce than NiO alone. Thus Takemura et al. (I) demonstrated that NiO reduced completely at 230°-400°C (446°-752°F) whereas a 10% NiO-90% MgO solid solu-... 

Buckminsterfullerene is an allotrope of carbon in which the carbon atoms form spheres of 60 atoms each (see Section 14.16). In the pure compound the spheres pack in a cubic close-packed array, (a) The length of a side of the face-centered cubic cell formed by buckminsterfullerene is 142 pm. Use this information to calculate the radius of the buckminsterfullerene molecule treated as a hard sphere, (b) The compound K3C60 is a superconductor at low temperatures. In this compound the K+ ions lie in holes in the C60 face-centered cubic lattice. Considering the radius of the K+ ion and assuming that the radius of Q,0 is the same as for the Cft0 molecule, predict in what type of holes the K ions lie (tetrahedral, octahedral, or both) and indicate what percentage of those holes are filled. 

Table 1. Lattice Constant Data for the Cubic MB -Type Phases (B / M Indicates the Atomic Ratio Used in Preparation)...

In the first Factor, if the cell lengths are not equal, i.e. a b c, we no longer have a cubic lattice, but some other type. And if the angles cure no longer 90 i.e.- a b g, then we have a change in the lattice type... 

Here, the axis of slip is shown in the hexagonal lattice as being on, or near the surface of the array of atoms. In the cubic lattice, a slip plane is depicted where a line of atoms is missing and the lattice has moved to accommodate this type of lattice defect. 

All applications of the lattice-gas model to liquid-liquid interfaces have been based upon a three-dimensional, typically simple cubic lattice. Each lattice site is occupied by one of a variety of particles. In the simplest case the system contains two kinds of solvent molecules, and the interactions are restricted to nearest neighbors. If we label the two types of solvents molecules S and Sj, the interaction is specified by a symmetrical 2x2 matrix w, where each element specifies the interaction between two neighboring molecules of type 5, and Sj. Whether the system separates into two phases or forms a homogeneous mixture, depends on the relative strength of the cross-interaction W]2 with respect to the self-inter-action terms w, and W22, which can be expressed through the combination ... 

The fourth and final crystal structure type common in binary semiconductors is the rock salt structure, named after NaCl but occurring in many divalent metal oxides, sulfides, selenides, and tellurides. It consists of two atom types forming separate face-centered cubic lattices. The trend from WZ or ZB structures to the rock salt structure takes place as covalent bonds become increasingly ionic [24]. 

We consider a solution composed of two types of molecules, labelled 1 and 2, which are distributed on a cubic lattice such that each lattice... 

The total surface areas determined by the N2 BET method for the calcined, supported catalysts are listed in Table II. The X-ray diffraction (XRD) results showed diffraction peaks from a cubic lattice with a unit cell distance of 6.1 A were present on all of the calcined catalysts. Both C03O4 and C0AI2O4 have structures consistent with that lattice spacing, making assignment of the type of crystalline cobalt species present on the alumina supports difficult. 

Figure 12.2 The three types of unit cell of the cubic lattice.

Figure 5.8. Lanthanide Ln203 oxides (cubic cI80-Mn2O3 type, on the left side) and Pb alloys (LnPb3, cubic cP4-type, on the right). The trends of the lattice parameter and of the heat of formation are shown (see the text and notice the characteristic behaviour of Eu and Yb). A schematic representation of the energy difference between the divalent and trivalent states of an ytterbium compound is shown. Apromff represents the promotion energy from di- to trivalent Yb metal, A,//11, and Ar/Ynl are the formation enthalpies of a compound in the two cases in which there is no valence change on passing from the metal to the compound the same valence state (II or III) is maintained.

To determine the phase properties of the calcined bimetallic nanoparticles, a detailed x-ray diffraction (XRD) study was carried out. The XRD data of AuPt/C showed that the diffraction patterns for the carbon-supported nanoparticles show a series of broad Bragg peaks, a picture typical for materials of limited structural coherence. Nevertheless, the peaks are defined well enough to allow a definitive phase identification and structural characterization. The diffraction patterns of Au/C and Pt/C could be unambiguously indexed into an fcc-type cubic lattice occurring with bulk gold and platinum. We estimated the corresponding lattice parameters by carefully determining... 

XRD measurements showed that the compound has a cubic lattice with the lattice parameter a = 0.6443 nm and metal atom arrangement of the fluorite type (space group Fm3m). Theoretical studies of its X-ray absorption spectra were conducted soon after its discovery by Orgaz and Gupta [35]. 

Figure 16.2. Unit cells of the three most important lattice types (a) face-centered cubic (b) hexagonal close-packed (c) body-centered cubic. (From Ref 1, with permission from Noyes.)...

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