Lattice body-centred

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 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. 

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... 

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.

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 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. 

By examining Figures 3.7 and 3.32, we note that the caesium cations sit on a primitive cubic unit cell (lattice type P) with chloride anion occupying the cubic hole in the body centre. Alternatively, one can view the structure as P-type lattice of chloride anions with caesium cation in cubic hole. Keep in mind that caesium chloride does not have a body centred cubic lattice although it might appear so at a first glance. The body centred lattice has all points identical, whereas in CsCl lattice the ion at fte body centre is different from those at the comers. 

From these results and Example 3.3, it becomes clear that closed-packed cubic lattice has the best space economy (best packing, least empty space), followed by the body-centred lattice, whereas the simple cubic packing has the lowest space economy with the highest fraction of unoccupied space. 

For the remaining lattice types, as well as the translational symmetry from these corner points there is additional symmetry within the unit cell. The simplest of these is the body-centred example seen in Figure 1.4, where there is also an atom at the centre of the unit cell (with fractional coordinates ( /2, Vi, 72). For a lattice to be body-centred therefore, if an atom or ion is placed on x, y, z there must be an identical one placed at X+V2, y+ /2, z+ i. A body centred lattice is given the symbol I. 

Figure 3.12 Diffraction from the 100 plane in a body-centred lattice...

The scattering from A and B (the atoms at the corners of the cell) will always be in phase, but there is also an atom at the centre of the cell. Diffraction from this atom will be exactly half a wavelength out of phase with the diffracted beam from the atoms at the corners. As we know from atom counting in Chapter 1, these sets of atoms occur in pairs throughout the lattice i.e. Z = 2). This means that there will be total destructive interference for the 100 reflection and no intensity will be seen. For any body-centred (I) lattice this condition will hold. Extension of this principle to other reflections generates the following rule. For a reflection to be seen for a body-centred lattice, then the sum of the Miller indices must be even ... 

This can lead to problems in the indexing procedure, as for a body-centred lattice the first reflection is not the 100 but the 110. 

The other crystal lattices can be generated by adding to some of the above-defined cells extra high-symmetry points by the so-called centering method. TableB.2 shows the new systems added to the simple crystal lattices (noted s, or P, for primitive) and the numbers of lattice points in each conventional unit cell. The body-centred lattices are noted be or I (for German Innenzentrierte), the face-centred, fc or F, and the side-centred or base-centred lattices are noted C (an extra atom at the Centre of the base). These 14 lattice systems are known as the Bravais lattices (noted here BLs). A representation of their unit cells can be found in the textbook by Kittel [7]. 

The most important consequence the symmetry elements present in a crystal is that some (hkl) planes have F(hkl) = zero, and so will never give rise to a diffracted beam, irrespective of the atoms present. Such missing diffracted beams are called systematic absences. This can most easily be understood in terms of the vector representation described above. Suppose that a crystal is derived from a body centred lattice. In the simplest case, the motif is one atom per lattice point, and the unit cell contains atoms at 000 and Zz V2 Zz, (Figure 6.13a). [Note the cell can have any symmetry]. The structure factor for each (hkl) set is given by ... 

All crystallographic unit cells derived from a body-centred lattice give rise to the same systematic absences. Similar considerations apply to the other Bravais lattices. The conditions that apply for diffraction to occur from Qikt) planes in the Bravais lattices, called reflection conditions, are listed in Table 6.4. 

The second structure common to a number of T1-B1 systems is that of sodium thallide, sometimes called the Zintlphase. This structure (fig. 13.12) is closely related to that of caesium chloride in that the pattern of sites occupied forms a cubic body-centred lattice. The distribution of the atoms, however, is such that each atom has four neighbours of each kind, and the true cell is therefore the larger unit shown, containing sixteen instead of only two atoms. Some phases in which the sodium thallide structure occurs are LiZn, LiCd, LiAl, LiGa, Liln, Naln and NaTl. It is a characteristic feature of all of these phases that in them the alkali metal atom appears to have a radius considerably smaller than in the structure of the element (even when allowance is made for the change in co-ordination number), suggesting that this atom is present in a partially ionized condition and that forces other than purely metallic bonds are operative in the structure. 

There are three fundamental IPMS structures with cubic symmetry. One is the body-centred lattice shown in Fig. 8.11 and another is the primitive diamond lattice proposed for mono-olein at high water content (Longley and McIntosh, 1983). The third is the so-called gyroid (Shoen, 1970) which is also body-centred with space-group number 214. This IPMS separates two helical labyrinths with... 

The body-centred lattice of the cubic phase of monoglycerides at low water content is consistent with the gyroid lattice, as well as with the structure shown in Fig. 8.11. Unfortunately the X-ray powder data available cannot unambiguously exclude one of these two alternatives. 

Oil) planes of the body-centred lattice. A more detailed account of the process is given by Pujita [52], but to understand the principle of the process, these fine details are irrelevant. 

In addition, the carbon further strengthens the alloy considerably. The solubility of carbon is much smaller in the body-centred than in the face-centred crystal structure because the interstitial spaces are smaller. If 7 iron with a sufficient amount of dissolved carbon (> 0.008 wt-%) is quenched, the carbon remains dissolved in the body-centred lattice. The carbon atoms strongly distort the body-centred cubic cell to a tetragonal one (figure 6.54). This distorted lattice structure has an extreme strength when highly oversaturated because the stress field cannot be passed by dislocations. As a rule of thumb. 

Of course, the silver content of the system can be increased beyond the 21 mol% solubility limit found for the f.c.c. structure. This leads to a two-phase region where the f.c.c. structure coexists with a body-centred phase. The latter cannot be formed for silver contents less than ca 35mol%. As for the face-centred phase, increasing the silver content depresses the structural transition temperature and so the body-centred lattice can be stabilised at temperatures as low as 420 °C. This phase has been widely studied at the composition LiAgS04 and shows a conductivity of 1 S cm at 530 Structural studies of this phase show the... 

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