Cube corner

In the face-centred cubic structure tirere are four atoms per unit cell, 8x1/8 cube corners and 6x1/2 face centres. There are also four octahedral holes, one body centre and 12 x 1 /4 on each cube edge. When all of the holes are filled the overall composition is thus 1 1, metal to interstitial. In the same metal structure there are eight cube corners where tetrahedral sites occur at the 1/4, 1/4, 1/4 positions. When these are all filled there is a 1 2 metal to interstititial ratio. The transition metals can therefore form monocarbides, niU ides and oxides with the octahedrally coordinated interstitial atoms, and dihydrides with the tetrahedral coordination of the hydrogen atoms. 

The process requires the interchange of atoms on the atomic lattice from a state where all sites of one type, e.g. the face centres, are occupied by one species, and the cube corner sites by the other species in a face-centred lattice. Since atomic re-aiTangement cannot occur by dhect place-exchange, vacant sites must play a role in the re-distribution, and die rate of the process is controlled by the self-diffusion coefficients. Experimental measurements of the... 

In lead zh conate, PbZrOs, the larger lead ions are displaced alternately from the cube corner sites to produce an antifeiToelectric. This can readily be converted to a feiToelectric by dre substitution of Ti" + ions for some of the Zr + ions, the maximum value of permittivity occumirg at about the 50 50 mixture of PbZrOs and PbTiOs. The resulting PZT ceramics are used in a number of capacitance and electro-optic applicahons. The major problem in dre preparation of these solid soluhons is the volatility of PbO. This is overcome by... 

The structure can be instructively compared with that of fluorite, CaF2. In fluorite the calcium ions are arranged at face-centered lattice points, and each is surrounded by eight fluorine ions at cube corners. 

Figure 8.9 Relationship between the Ll2 (Ni3Al) and the A15 (Cr3Si) crystal structures. In both cases the cube corners are occupied by the non-transition elements (A1 and Si), but the face-centers are occupied differently by one transition metal atom in the Ll2 case, and by a pair of transition metal atoms in the A15 case. An additional difference is that the cube center is unoccupied in Ni3Al, but is occupied by a Cr atom in Cr3Si.

Figure 10.8 Alternative drawing of the crystal structure of Lanthanum Hexaboride with the metal atoms occupying the cube corners.

This interferometric dilatometer consists of a rather simple and small Michelson interferometer, in which the two arms are parallel, and of a 4He cryostat, in which the sample to be measured is hold. The sample is cooled to 4 K, and data are taken during the warm up of the cryostat. The optical path difference between the two arms depends on the sample length hence a variation of the sample length determines an interference signal. The Michelson interferometer consists of a He-Ne stabilized laser (A = 0.6328 xm), two cube corner prisms, a beam splitter, three mirrors and a silicon photodiode detector placed in the focal plane of a 25 mm focal length biconvex lens (see Fig. 13.1). 

The beams are backreflected by the cube corner prisms which are fixed, respectively, on the sample and on the sample holder. Since the cube corner prisms are able to make reflected beam exactly parallel to incident beam, this interferometer is tilt independent. The reflected beams get back to the beam splitter through the same path, but shifted by about 2 mm in the vertical direction. The beam splitter lets a part of the two beams go towards the photodiode sensor and lets the other part of beams reach the laser source (off axis, therefore giving no feedback effect). 

The distance along any side of the body-centred cubic lattice as shown in Figure 7.1(a) is equal to twice the metallic radius of the atom, 2rM. The distance between the centre of the atom in the centre of the unit cell and the centre of any atom at the cube corner is 31/3/ m. The distance between the centres of two atoms at the centres of adjacent cubes is 2rM. This means that any atom in the body- centred cubic arrangement is coordinated by eight atoms at the 1 cube corners with a distance 3l/3rM, and six more atoms in the centres of the six adjacent cubes with a distance 2rM. The extra six atoms contributing to the coordination number of body-centred atoms are 100 x (2rM - 3,/3rM)/(3I/3rM) = 15.5% further away from the central atom than the eight nearest neighbours. 

The data for ammonium chloride and bromide, treated similarly, lead to values of about 6 to 8 kcal/mole for the extra energy of interaction of the ammonium ion with the surrounding chloride and bromide ions. In these crystals each ammonium ion is surrounded by eight halide ions at cube corners. It can form hydrogen bonds with four, at tetrahedron corners, at a time. There... 

There are of course other causes for exit beam misalignment other than simply the process of optical retardation by mirror scanning. The overall alignment integrity of the cube-corner/beamsplitter/cube-corner module as well as its optical relationship to the collimated input beam from the NIR source, and the collinear HeNe laser are all critical. The thermal and mechanical instability of this whole assembly is roughly a function of its volume. Small compact FTIR modulators will perform better in process analytical... 

Figure 3.25 Cube-corner retroreflector interferometers with rotational scanning.

LaCrC>3 is one of the family of lanthanide perovskites RTO3, where R is a lanthanide and T is a period 4 transition element. In the cubic unit cell R occupies the cube corners, T the cube centre and O the face-centre positions. The coordination numbers of T and R are 6 and 8 respectively. LaCrC>3 loses chromium at high temperatures, leaving an excess of O2- ions. The excess charge is neutralized by the formation of Cr4+ which results in p-type semiconductivity with hole hopping via the localized 3d states of the Cr3+ and Cr4+ ions. The concentration of Cr4+ can be enhanced by the substitution of strontium for lanthanum. A 1 mol.% addition of SrO causes the conductivity to increase by a factor of approximately 10 (see Section 2.6.2). 

Fig. 13.2. Cube representing an electroactive polymer with two oxidation states (R and O), each having two solvation and two configuration states. Cube corners represent the eight possible species. Left/right, front/back and top/bottom planes, respectively, differ with respect to oxidation state, solvation state and configuration state. Insert shows 3D axes for...

The Michelson interferometer is used in most FTIR spectrometers. However, some FTIRs use a cube-corner interferometer. 

The simplest model for a tungsten bronze, M WOg, is to regard it as a host W03 structure in which M atoms have been introduced interstitially. For the simplest case, which leads to a perovskite-like structure, the unit cell can be represented as in Figure 1, with a tungsten atom at the center, six oxygen atoms at face centers, and eight interstitial sites at the cube corners more or less occupied by alkali atoms. When these sites are completely empty, x in M WOg is zero and the structure resembles that of WOa except that the tungsten atoms... 

A schematic sketch of the selenium. structure showing how the zig-zag chain can be coiled up along a [111] direction in a simple cubic lattice. Every third atom along the coil is translationally equivalent one such set is shaded. Other chains may be added such that the isolated shaded atoms shown are translationally equivalent to the shaded atoms in the chain and there is an atom site at every cube corner. The structure can then be distorted to reduce intrachain atom distances, increase interchain distances, and increase the bond angles to 105°. 

A fishnet plot of the electrostatic rigidity constant a for perovskite structures as a function of the charge Z on the body-centered ion and the charge Q on the edge-centered ion the cube-corner ion would have charge 6 - Z. Three compounds being discussed here are indicated, as well as the simple cubic and body-centered cubic structure, which can also be described this way. 

Therefore, we have to analyse the variation of the rate of permeation according to the temperature (zj), the trans-membrane pressure difference (Z2) and the gas molecular weight (Z3). Then, we have 3 factors each of which has two levels. Thus the number of experiments needed for the process investigation is N = 2 = 8. Table 5.13 gives the concrete plan of the experiments. The last column contains the output y values of the process (flow rates of permeation). Figure 5.8 shows a geometric interpretation for a 2 experimental plan where each cube corner defines an experiment with the specified dimensionless values of the factors. So as to process these statistical data with the procedures that use matrix calculations, we have to introduce here a fictive variable Xq, which has a permanent +1 value (see also Section 5.4.4). 

Close packing At cube corners Octahedral Tetrahedral... 

Barium titanate, which has many novel properties, is a mixed oxide ceramic. It has the same structure as the mineral perovskite, CaTiOs (Fig. 22.13), except, of course, that Ba replaces Ca. Perovskites typically have two metal atoms for every three O atoms, giving them the general formula ABO3, where A stands for a metal atom at the center of the unit cube and B stands for an atom of a different metal at the cube corners. 

Ferroelectric materials in this family have the perovskite crystal structure, where Ti+4(Zr+4) is in the B site at the center of the unit cell within an octrahedral coordination of O-2, and Pb+2 (La+3) occupies the A site at cube corners. Considerable discussion and experimentation have occurred to decide what charge-compensating defects are created by the... 

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