Lattice close packed

The holes in the close-packed structure of a metal can be filled with smaller atoms to form alloys (alloys are described in more detail in Section 5.15). If a dip between three atoms is directly covered by another atom, we obtain a tetrahedral hole, because it is formed by four atoms at the corners of a regular tetrahedron (Fig. 5.30a). There are two tetrahedral holes per atom in a close-packed lattice. When a dip in a layer coincides with a dip in the next layer, we obtain an octahedral hole, because it is formed by six atoms at the corners of a regular octahedron (Fig. 5.30b). There is one octahedral hole for each atom in the lattice. Note that, because holes are formed by two adjacent layers and because neighboring close-packed layers have identical arrangements in hep and ccp, the numbers of holes are the same for both close-packed structures. 

When the radius ratio of an ionic compound is less than about 0.4, corresponding to cations that are significantly smaller than the anion, the small tetrahedral holes may be occupied. An example is the zinc-blende structure (which is also called the sphalerite structure), named after a form of the mineral ZnS (Fig. 5.43). This structure is based on an expanded cubic close-packed lattice of the big S2 anions, with the small Zn2+ cations occupying half the tetrahedral holes. Each Zn2+ ion is surrounded by four S2 ions, and each S2" ion is surrounded by four Zn2+ ions so the zinc-blende structure has (4,4)-coordination. 

Many ionic compounds are considered to pack in such as way that the anions form a close-packed lattice in which the metal cations fill holes or interstitial sites left between the anions. These lattices, however, may not necessarily he as tightly packed as the label close-packed implies. The radius of an F ion is approximately 133 pm. The edge distances of the cubic unit cells of LiF, NaF, KF, RbF, and CsF, all of which... 

Consider a metallic element that crystallizes in a cubic close-packed lattice. The edge length of the unit cell is 408 pm. If close-packed layers are deposited on a flat surface to a depth (of metal) of 0.125 mm, how many close-packed layers are present ... 

FIGURE 14.32 These small crystals are fullerite, in which buckminsterfullerene molecules are packed together in a close-packed lattice. 

Fig. 6.4 Layered structure of LixTiSa, showing the lithium ions between the TiSa sheets. This is an anion close-packed lattice in which alternate layers between the anion sheets are occupied by a redox-active titanium atom. Lithium inserts itself into the empty remaining layers. (Adapted from [68])...

Here, we have arranged the layers on a two-dimensional structure, even though the layers are arranged in three dimensional order. Note that only two crystallographic axes are indicated. We call this the natural stacking sequence because of the nature of the hexagonal close- packed lattice. 

Although the bond-orientational metrics defined above have proven useful for identifying numerous space-filling crystalline morphologies43 like face-centered cubic, body-centered cubic, simple cubic, and hexagonally close-packed lattices, they are inadequate for detecting order in systems that organize... 

The ratio d3a of the lattice constants is a direct measure of the deviation of the lattice from a perfect cubic close-packed lattice, which is it measures the layeredness of the lattice. An ideal ccp lattice has a d3a ratio of 1.633, whereas a pure layered lattice with no transition metal in the lithium layer has a... 

The ease of oxygen removal from the close-packed lattice when lithium rich had been demonstrated by its ready reduction by ammonia gas at 200 °C in the case of the spinel Li[Lii/3Mn5/3]04. It was also shown that this oxygen could be removed by electrochemically charging above around 4.3 V the material then showed the 4 V discharge behavior typical of a spinel. These reduced materials can best be represented as Li[Lii/3Mn5/3]04 g. 

Rule 11.4 (Principle of close packing). Like ions tend to lie on close packed lattices, since this arrangement minimizes their repulsive energy when they are confined to a fixed volume. 

Compounds of stoichiometry AX2 with six-coordinated A require (according to eqn (11.1)) that X be three coordinate. Since none of the close packed lattices have cage points with three coordination, these structures are less simple. The rutile (202240) and anatase (202242) forms of Ti02 are based on FICP and FCC lattices of Ti respectively, but fitting the ions into positions of three coordination results in distortions that lower the symmetry. An alternative derivation of these structures is described in Section 11.2.2.4 below. 

Not all structures are based on close packed lattices. Ions that are large and soft often adopt structures based on a primitive or body centred cubic lattice as found in CsCl (22173) and a-AgI (200108). Others, such as perovskite, ABO3 (Fig. 10.4), are based on close packed lattices that comprise both anions and large cations. The larger and softer the ions, the more variations appear, but the lattice packing principle can still be used. Santoro et al. (1999,2000) have shown how close-packing considerations combined with the use of bond valences can give a quantitative prediction of the structure of BaRuOs (10253). 

The prefactor outside the curly brackets gives the number of bonds that are strengthened by the absence of the atom at the vacancy site. The contribution inside the curly brackets gives the change in the bond energy due to the change in coordination from to ( — 1). For close-packed lattices 1 so that using the binomial expansion... 

Figure 7.12 compares the theoretical predictions with the experimental values across the 4d series, assuming one valence s electron per atom and taking x = 12 corresponding to close-packed lattices. The experimental values of the bandwidth are taken from the first principles LDA calculations in Table 7.1. The ratio b2 a is obtained by fitting a bandwidth of 10 eV for Mo with Nd = 5, so that from eqn (7.42) b2/a = eV. The skewed parabolic behaviour of the observed equilibrium nearest-neighbour distance is found to be fitted by values of the inverse decay length that vary linearly across the series as... 

We have shown that A) interstitial hydride formation is observed only with partial occupation of the available holes, B) occupation of the interstitial position in isolated polyhedra is not observed, and C) occupation of all the holes in a close-packed lattice cancels metal-metal interactions. Therefore, it seems that interstitial hydrogen can be tolerated only in a fraction of the total number of holes, and with the weakening of metal-metal interactions. This behavior indicates strong competition between metal-metal and metal-hydrogen bonds, which is unique for hydrogen because interstitial carbon can stabilize some unusual arrangements in carbonyl carbide clusters (29, 30). 

A distinct dihydride phase is the strongest confirmation of the magnitude of the influence of hydrogen on the metal systems. As the metal is exposed to hydrogen gas, spontaneous uptake occurs within the metal lattice. This concentration is usually very small but can have strong influence on the mechanical properties, a phenomenon known as embrittlement. With further increase of hydrogen, the metal lattice transforms to a cubic close packed lattice with... 

Table I (taken from Martin, Sykes, and Hioe16) contains the most recent exact enumerations of C for the triangular and fee lattices. Similar enumerations for other lattices have been given elsewhere ° 11 numerical analysis indicates that the close packed lattices lead to most rapid convergence, and these were therefore selected for an extensive enumeration project. It should be noted that C12 for the fee lattice is of order 1.8 x 1012. Using a direct enumeration procedure on a digital computer, the machine time required would be quite prohibitive. It is only by the way of sophisticated counting theorems17 and skilled programming that these numbers could be obtained.

In Figure 1, a is plotted vs. ax for a face-centered lattice (a close-packed lattice) and for a simple cubic lattice (a loose-packed lattice). We notice that (1) the dependence of a on ax can be regarded as being practically the same for both lattices and that, (2) tx undergoes a rapid change around x = xc, which is the point at which a = 0 (Fig. 1). However, p/a does not attain the value it would have for the case of the unrestricted random walk model at x = xc, since at this point, p/a > 1 (Fig. 2), while for unrestricted chain pja = 1. Moreover, the dependence of p/a on ax is not the same for the two lattices while a as a function of ax is practically independent of the lattice. 

Calculate the correlation factor for tracer self-diffusion by the vacancy mechanism in the two-dimensional close-packed lattice illustrated in Fig. 8.22. The tracer atom at site 7 has just exchanged with the vacancy, which is now at site 6. Following Shewmon [4], let p. be the probability that the tracer will make its next jump to its kth nearest-neighbor (i.e., a 7 — k jump). 6k is the angle between the initial 6 —> 7 jump and the 7 —> k jump. The average of the cosines of the angles between successive tracer jumps is then... 

Figure 8.22 Two-dimensional close-packed lattice. The tracer atom at 7 has just...

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