The Face-Centred Cubic Lattice

It should be emphasised that these are only guidelines and an actual determination of the extent of site mixing is needed for each specific case. The literature on this class of perovskites is considerable and the review here will be selective rather than exhaustive. 

Structure). One expects the descent in symmetry to be a function of a modified Goldschmidt tolerance factor, 

Note that Oq is larger by a factor of two for cubic Ba2MnW06 relative to monoclinic Sr2MnWOg. which reflects the more favourable 180° SSE of the former compared with angles of 162° for the latter. Nonetheless, neutron diffraction data show evidence for frustration in the form of diffuse magnetic scattering which persists well above T particularly for Ba2MnWOg.  

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We now introduce a Fourier transform procedure analogous to that employed in the solution theory, s 62 For the purposes of the present section a more detailed specification of defect positions than that so far employed must be introduced. Thus, defects i and j are in unit cells l and m respectively, the origins of the unit cells being specified by vectors R and Rm relative to the origin of the space lattice. The vectors from the origin of the unit cell to the defects i and j, which occupy positions number x and y within the cell, will be denoted X 0 and X for example, the sodium chloride lattice is built from a unit cell containing one cation site (0, 0, 0) and one anion site (a/2, 0, 0), and the translation group is that of the face-centred-cubic lattice. However, if we wish to specify the interstitial sites of the lattice, e.g. for a discussion of Frenkel disorder, then we must add two interstitial sites to the basis at (a/4, a/4, a]4) and (3a/4, a/4, a/4). (Note that there are twice as many interstitial sites as anion-cation pairs but that all interstitial sites have an identical environment.) In our present notation the distance between defects i and j is... 

The face-centred cubic lattice is very common. Many metallic elements crystallize in this form so also do many binary compounds such as alkali halides and the oxides of diva-lent metals. Thus the powder photo-... 

Endothermic occlusion takes place by diffusion of hydrogen into a metal lattice which is very little changed by the process. In exothermic occlusion by palladium, however, the face-centred cubic lattice (a phase) of palladium, with lattice constant 3.88 A, will accomodate, below 100°C, no more than about 5 at. % hydrogen, and then undergoes a transition to an expanded phase ( 3 phase), with lattice constant 4.02 A and H/Pd = 0.5—0.6. The H—Pd system thus splits into a and 3 phases in the manner familiar for two partially miscible liquids. The consolute temperature (rarely observable for solid phases) is about 310°C at H/Pd = 0.22. The phase diagram is, however, not well established because formation of the... 

Another example is that of lattice chain models. Simple square lattice models were established by Flory as a vehicle for calculating configurational entropies etc., and used later in the simulations of the qualitative behaviour, e.g. of block copolymer phase separation. More sophisticated models such as the bond fluctuation model, and the face centred cubic lattice chain modeP ... 

ElO.lO Recall from Sections 1.7 and 3.10(a) that the size of an ion is not really a fixed value rather it depends largely on the size of the hole the ion in question fills in a larger hole the ion will appear bigger than in the smaller one. We have also to recall periodic trends in cation variation. All hydrides of the Group 1 metals have the rock-salt structure (see Section 10.6(b) Saline hydrides). This means that in all Group 1 hydrides H" has a coordination number 6 and is surrounded by cations in an octahedral geometry. In other words, H occupies an octahedral hole within the face-centred cubic lattice of cations. The sizes of cations increase down the group (i.e., r L ) < KNa" ) < r(K )< tfCs"")) as a consequence both the sizes of octahedral holes and the apparent radius of H" will increase in the same direction. 

A The face-centred cubic lattice forms octahedral and tetrahedral holes. Although it is more densely packed that the body-centred cubic version, in which the coordination numbers are 12 rather than 8, there are fewer larger holes, as shown in Figure 6.16. 

The distance from the nearest iron atom to the interstitial site is 1.4 A and 1.8 A respectively for body-centred and face-centred polymorphs. While the site in the body-centred cubic lattice is too small to accommodate the carbon without significant distortion, the site in the face-centred cubic lattice is suitable, and the carbon can be accommodated. Approximately one-third of the empty octahedral sites can be occupied, giving the formula FejC. 

Many different lanthanides and early transition metals form A3C compounds by this type of reaction, as described for the face-centred cubic iron polymorph. The limiting factor appears to be the size of the radius of A. A critical radius of 1.35 A seems to be required to form the face-centred cubic lattice with twelve coordinate ions at room temperature. 

Figure 2.9 Wigner-Seitz cells (a) the body-centred cubic lattice (b) the Wigner-Seitz cell of (a) (c) the face-centred cubic lattice (d) the Wigner-Seitz cell of (c). The face-centred cubic lattice point marked forms the central lattice point in the Wigner-Seitz cell...

Fig.5 Two unit cells of the reciprocal lattice for the face-centred cubic lattice of Cs2NaYCl6, showing major symmetry points and directions (from [ 100] with permission)...

The Face-Centred Cubic Lattice. The other method of close packing of spheres, based on the alternative arrangement of the third layer, is shown ip Fig. 12 ( ). The second layer of spheres, shown by dashed circles, are placed as before, but the third... 

We must now consider those sets of crystallographic planes in actual metal lattices which do in fact pass through all the atom centres. In a face-centred cubic lattice (Fig. 17) all of the atoms are included in the (111) set of planes, among which are the planes ACB and EFD, but the face-centred atoms are not included in the (100) planes. In order that these atoms might be included we must consider the set involving the planes OBFC, HJKL and ADGE. This set of planes contains all of the atoms and has the index (200). Similarly, in order to include the face-centred atoms in the set parallel to the plane AEFB, we must specify planes with index (220) rather than those with index (110). Thus the planes with simplest indices in the face-centred cubic lattice are those with the indices (111), (200) and (220). 

Nevertheless, the calculated values of the critical exponents depend only slightly on the lattice type. The best values are obtained in two dimensions with the triangular lattice, and in three dimensions with the face-centred cubic lattice, for the following reasons. First, as each site has many neighbours, the angular fluctuations are reduced, and this is a favourable circumstance. Second, it is easy for the chain to come back close to itself and therefore, on these lattices, the chains very quickly know that they have to be self-avoiding. 

This would then explain the lack of magnetic ordering. The hydrogen occupies the octahedral sites in the face-centred cubic lattice. The Mossbauer spectrum (Fig. 12.7) of a sample of nickel doped with Co which has been... 

A regular octahedral CuL chromophore for the copper(II) ion is very uncommon (Figun 19.1) but does occur for the [Cu(N02)6] anion when stabilized by hi symmetry lattices such as the face-centred-cubic lattice of K2Pb[Cu(N02)6] (177)," at 298 K. All six Cu— distances of 2.11 A are equivalent from the copper site symmetry, with the actual symmetry o the anion lowered to T by the conformation of the nitro oxygen atoms. The six nitro ligand are also involved in bidentate nitro coordination to the Pb cations at 2.77 A, a distano consistent with intermediate Pb—O bonding, and which must be mainly responsible fo... 

If the angles a, and y are all different from 90° and from each other the crystal is said to be triclinic. The only simplification that may be possible in describing the structure is to find another set of parallelepipeds with angles closer to 90° that still describe the structure. For certain special values of a, and Y and special relationships among a, b and c it is possible to find other descriptions of the lattice that conform to the symmetry of the structure. A particular example is the face-centred cubic lattice, illustrated in fig. 2.6, in which the cube of edge a clearly exhibits the full symmetry of the structure, but the rhombohedron of edges a/ /2 does not. This parallelepiped is one of the infinite number of possible types of primitive unit cell that can be chosen so as to contain only one lattice point per cell. The cubic unit cell is not primitive there are four lattice points per cubic cell. Care must be taken when describing diffraction from non-primitive cells, which can occur both for polymeric and for non-polymeric crystals. 

Lomer WM (1951) A dislocation reaction in the face-centred cubic lattice. Phil Mag 42(334) 1327-1331... 

Bravais lattice is primitive If it contains only one atom, it is primitive if it contains more, it is not. While counting the atoms it has to be kept in mind to count only the appropriate fractions of those atoms occupying more than one cell. For instance, the conventional unit cell of the body-centred cubic lattice contains two atoms and is therefore not primitive, the conventional unit ceU of the face-centred cubic lattice contains four atoms and is thus not primitive either. 

Fig. 1.9. Construction of the hexagonal close-packed and the face-centred cubic lattice by stacking close-packed layers of spheres. The structures differ in their stacking sequence In the hexagonal close-packed structure spheres in the third layer are placed perpendicularly above those in the first, in the face-centred cubic lattice the...

As the name implies, the hexagonal close-packed crystal has the highest possible packing density. Its stacking sequence (cf. figure 1.9) differs from that of the face-centred cubic lattice. Only the 0001 -basal planes are close-packed. They contain the three (1120) close-packed directions, resulting in only three independent slip systems (figure 6.16). 

This visualisation of the diffusion-less transformation according to Bain is not totally correct. X-ray diffraction experiments have shown that, in reality, the (111) planes of the face-centred cubic lattice become the... 

It would have been much simpler to solve the exercise using the following argument The face-centred cubic and the hexagonal close-packed crystal are close packed with each atom having 12 nearest neighbours. Therefore, the relative density must be the same in both cases and the result for the face-centred cubic lattice can be used. 

The Brillouin zone for the simple (primitive) cubic lattice is a (simple ) cube. The Brillouin zone for the face-centred cubic lattice is pictured in Fig. 17.11 that for the body-centred cubic is shown in Fig. 17.16. Show that all three have Of, symmetry. 

Figure 8 is a spectrum taken at 33 K in the vCH region of CD3H adsorbed on the (100) face of the face-centred-cubic lattice of NaCl. It is seen that there are two well resolved absorption bands, one... 

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