Centred lattices

The higher solubility of carbon in y-iron than in a-iroii is because the face-ceiiued lattice can accommodate carbon atoms in slightly expanded octahedral holes, but the body-centred lattice can only accommodate a much smaller carbon concentration in specially located, distorted tetrahedral holes. It follows that the formation of fenite together with cementite by eutectoid composition of austenite, leads to an increase in volume of the metal with accompanying compressive stresses at die interface between these two phases. 

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... 

The order-disorder transition of a binary alloy (e.g. CuZn) provides another instructive example. The body-centred lattice of this material may be described as two interpenetrating lattices, A and B. In the disordered high-temperature phase each of the sub-lattices is equally populated by Zn and Cu atoms, in that each lattice point is equally likely to be occupied by either a Zn or a Cu atom. At zero temperature each of the sub-lattices is entirely occupied by either Zn or Cu atoms. In terms of fractional occupation numbers for A sites, an appropriate order parameter may be defined as... 

Fig. 2.4 (a) Body centred lattice of I ions in a-Agl. (b) Sites available to Ag ions in the conduction pathway. 

Fig. 8a-c. r rotation photographs of H. marismortui SOS crystals at 0 "C and at cryotemperature (obtained at XI1/EMBL/DESY and at SSRL/Stanford U.) a The hkO-orientation of a nearly perfectly aligned (although split) crystal reveals the mirror symmetry of the C-centred lattice plane. The severe overlap problem in this orientation caused by the large mosaic spread is obvious from this picture, b The Okl-orientation shows the extinctions of the twofold screw axis, c The best crystals have a Bragg resolution limit of about 6 A, which decreases to about 9 A in the course of a hundred exposures... 

The true unit cell is not necessarily the smallest unit that will account for all the reciprocal lattice points it is also necessary that the cell chosen should conform to the crystal symmetry. The reflections of crystals with face-centred or body-centred lattices can be accounted for by unit cells which have only a fraction of the volume of the true unit cell, but the smallest unit cells for such crystals are rejected in favour of the smallest that conforms to the crystal symmetry. The... 

The recognition of simple or body-centred or face-centred lattices is... 

Fig. 128. To a primitive tetragonal lattice ABCDEFGH add extra lattice points at the face-centres. The new lattice is equivalent to the body-centred lattice BJCIFLGK.

Shape-symmetry may tell us that a particular crystal has a fourfold axis, but it cannot tell us whether this axis is a simple rotation axis or a screw axis. Nor is it possible by examining the shape of a crystal to distinguish between a reflection plane and a glide plane. But X-ray diffraction patterns do make such distinctions, and in a very straightforward manner just as it is possible to detect compound ( centred ) lattices by noticing the absence of certain types of reflections (p. 233), so also it is possible to detect screw axes and glide planes, for the presence of atoms or groups of atoms related by translations which... 

There are no further systematic absences the absences of odd orders of A00, 070, and 00Z are included in the general statement that reflections having h+k- -l odd are absent. This means that, for a body-centred lattice, we cannot tell (from the systematic absences) whether twofold screw axes are present or not. The possible space-groups are... 

The relationship between cubic close-packed (ccp) structures and ionic compounds of type B1 is obvious. Interstitial sites with respect to metal positions are at fractional coordinates of the type 00 and equivalent to the ionic sites in Bl. The Madelung constant of Al type metals with interstitially localized free electrons is therefore the same as that of rocksalt structures. It is noted that the interstitial sites define the same face-centred lattice as the metal ions. 

The situation for body-centred cubic metals (A2) is more complicated, but related to the ccp arrangement. As shown in Figure 5.14 a tetragonal face-centred unit cell can be constructed around the central axis of four contiguous body-centred cells. The interstitial points in the transformed unit cell define an equivalent face-centred cell, as before, and the same sites also define a body-centred lattice (shown in stipled outline) that interpenetrates the original A2 lattice. Each metal site is surrounded by six fee interstices at an average distance d6 - four of them at a distance a/s/2 and two more at a/2. 

With two electrons per state twice as many electrons, that is 5.44/fl3, can therefore be placed. With a face-centred lattice, as in copper, we have four atoms per elementary cell of volume a3 and there are also four electrons with a monovalent metal. 

A compound lattice that contains body-, face-, or end-centred points can always be regarded as a simple lattice with a smaller unit cell. Thus, the body-centred tetragonal is a special case of the triclinic (Fig. 83). The centred lattices describe most readily the higher symmetry. 

An early X-ray study of Prussian blue and some related compounds showed that in ferric ferricyanide (Berlin green), FeFe(CN)6, Prussian blue, KFeFe(CN)6. and the white insoluble K2FeFe(CN)6. there is the same arrangement of Fe atoms on a cubic face-centred lattice. In Fig. 22.5 ferrous atoms are distinguished as shaded and ferric as open circles. In (a) all the iron atoms are in the ferric state in (b) one-half the atoms are Fe and the others Fe, and alkali atoms maintain electrical neutrality. These are at the centres of alternate small cubes, and it was supposed that in hydrated compounds water molecules could also be accommodated in the interstices of the main framework. Lithium and caesium, forming... 

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