Closest-packing

Virtually all we know about crystal structure has been learned from X-ray diffraction studies. X-ray diffraction is the scattering of X rays by the units of a crystalline solid. The scattering, or diffraction patterns, produced are used to deduce the arrangement of particles in the solid lattice. 

Max Theodor Felix von Lane (1879-1960). German physicist Von Lane received the Nobel Prize in Physics in 1914 for his discover) of X-iay diffraction. 

X rays of wavelength 0.154 nm strike an aluminum ciystal the rays are reflected at an angle of 19.3°. Assuming that n = 1, calculate the spacing between the planes of aluminum atoms (in pm) that is responsible for this angle of reflection. 

Think About It The distance between layers of atoms in a crystal should be similar in magnitude to the wavelength of the X rays diffracted by the crystal (compare 0.154 mn with 0.233 nm). 

Many metals form crystals with hep or ccp structures. For example, magnesium, titanium, and zinc crystallize with their atoms in an hep array, while aluminum, nickel, and silver crystallize in the ccp arrangement. A substance will crystalhze with the arrangement that maximizes the stabihty of the solid. 

CHAPTER 12 Intennolecular Forces and the Physical Properties of Liquids and Solids 

---

In any crystal structure, the close-packed or closest-packed planes are the lowest energy planes. On all other planes, the density of atoms is lower, and the interatomic distance and the energy of the plane are greater. Contrary to intuitive expectations, the diameter of the largest holes or interstices between atoms in the close-packed f.c.c. structure is considerably greater than the diameter of the largest interstices between atoms in the non-close-packed b.c.c. structure. 

Carbon materials which have the closest-packed hexagonal structures are used as the negative electrode for lithium-ion batteries carbon atoms on the (0 0 2) plane are linked by conjugated bonds, and these planes (graphite planes) are layered. The layer interdistance is more than 3.35 A and lithium ions can be intercalated and dein-tercalated. As the potential of carbon materials with intercalated lithium ions is low,... 

Coke materials are generally made by heat-treatment of petroleum pitch or coal-tar pitch in an N2 atmosphere. Coke made from petroleum is called "petroleum coke" and that from coal is called "pitch coke". These materials have the closest-packed hexagonal structures. The crystallinity of coke materials is not so high as that of graphite. The crystallite size of coke along the c-axis (Lc) is small (about 10-20 A) and the interlayer distance (d value about 3.38-3.80 A) is large. 

The C, values for Sb faces are noticeably lower than those for Bi. Just as for Bi, the closest-packed faces show the lowest values of C, [except Bi(lll) and Sb(lll)].28,152,153 This result is in good agreement with the theory428,429 based on the jellium model for the metal and the simple hard sphere model for the electrolyte solution. The adsorption of organic compounds at Sb and Bi single-crystal face electrodes28,152,726 shows that the surface activity of Bi(lll) and Sb(lll) is lower than for the other planes. Thus the anomalous position of Sb(lll) as well as Bi(lll) is probably caused by a more pronounced influence of the capacitance of the metal phase compared with other Sb and Bi faces28... 

Back reflection of translational and rotational velocity is rather reasonable, but the extremum in the free-path time distribution was never found when collisional statistics were checked by computer simulation. Even in the hard-sphere solid the statistics only deviate slightly from Pois-sonian at the highest free-paths [74] in contrast to the prediction of free volume theories. The collisional statistics have recently been investigated by MD simulation of 108 hard spheres at reduced density n/ o = 0.65 (where no is the density of closest packing) [75], The obtained ratio t2/l2 = 2.07 was very close to 2, which is indirect evidence for uniform... 

The symbols Al, A2, and A3 represent the three simple metal structures cubic closest packed, body centered, and hexagonal closest packed, respectively. 

Alloys of lead and thallium have a structure based upon cubic closest packing from 0 to about 87-5 atomic percent thallium. The variation of the lattice constant with composition gives strong indication that ordered structures PbTl, and PbTl, exist. In the intermediate ranges, solid solutions of the types Pb(Pb,Tl)a and Pb(Pb,Tl)TlB exist. Interpretation of interatomic distances indicates that thallium atoms present in low concentration in lead assume the same valence as lead, about 2-14, and that the valence of thallium increases with increase in the mole fraction of thallium present, having the same value, about 2-50, in PbTls and PbTl, as in pure thallium. A theory of the structure of the alloys is presented which explains the observed phase diagram,... 

I have found that the assumption that in atomic nuclei the nucleons are in large part aggregated into clusters arranged in closest packing leads to simple explanations of many properties of nuclei. Some aspects of the closest-packing theory of nuclear structure are presented in the following paragraphs.1... 

The Distribution of Spherons in Layers.—Several theoretical and empirical arguments indicate that the nature of spheron-spheron interactions is not such as to limit the ligancy of a spheron to a fixed value, but that, instead, maximum stability is achieved when each spheron ligates about itself the maximum number of neighbors aggregates of spherons, like aggregates of argonon (noble-gas) atoms or metal atoms, assume a closest-packed structure. 

Fig. 1.—The arrangement of 45 spheres in icosahedral closest packing. At the left there is shown a single sphere, which constitutes the inner core. Next there is shown the layer of 12 spheres, at the corners of a regular icosahedron. The third model shows the core of 13 spheres with 20 added in the outer layer, each in a triangular pocket corresponding to a face of the icosahedron these 20 spheres lie at the corners of a pentagonal dodecahedron. The third layer is completed, as shown in the model at the right, by adding 12 spheres at corners of a large icosahedron the 32 spheres of the third layer lie at the corners of a rhombic triaconta-hedron. The fourth layer (not shown) contains 72 spheres.

We may use this example of triangular closest packing to derive an expression for the distribution of spheres in successive layers. The form of the expression (number of spheres proportional to the cube of a length, the radius) reflects the assumption of constancy of effective volume per sphere. The expression is... 

The general geometrical problem of the packing of spheres has not been solved. An example of closest packing of atoms with some variation in effective radius is the icosahedral packing found (13) in the intermetallic compound Mg3B(Al,Zn) (Fig. 1). The successive layers in this structure contain 1, 12, 32, and 117 spheres. These numbers are reproduced (to within 1) by the empirical equation (12)... 

Comparative study of LB films of cytochrome P450 wild type and recombinant revealed similar surface-active properties of the samples. CD spectra have shown that the secondary structure of these proteins is practically identical. Improved thermal stability is also similar for LB films built up from these proteins. Marked differences for LB films of wild type and recombinant protein were observed in surface density and the thickness of the deposited layer. These differences can be explained by improved purity of the recombinant sample. In fact, impurity can disturb layer formation, preventing closest packing and diminishing the surface density and the average monolayer thickness. Decreased purity of... 

An important parameter for surface reactivity is the density of atoms in the surface. The general rule of thumb is that the more open the surface, the more reactive it is. We return to this effect in much more detail in Chapter 6. Note that (110) is the most open basal plane of an fee crystal, whereas (111) exhibits the closest packing. For bcc crystals the order is the opposite, i.e. (Ill) is the most open and (110) the most packed. 

Figure 5 also shows the effect of the ionophore concentration of the Langmuir type binding isotherm. The slope of the isotherm fora membrane with 10 mM of ionophore 1 was roughly three times larger than that with 30 mM of the same ionophore. The binding constant, K, which is inversely proportional to the slope [Eq. (3)], was estimated to be 4.2 and 11.5M for the membranes with 10 mM and 30 mM ionophore 1, respectively. This result supports the validity of the present Langmuir analysis because the binding constant, K, should reflect the availability of the surface sites, the number of which should be proportional to the ionophore concentration, if the ionophore is not surface active itself In addition, the intercept of the isotherm for a membrane with 10 mM of ionophore 1 was nearly equal to that of a membrane with 30 mM ionophore 1 (see Fig. 5). This suggests the formation of a closest-packed surface molecular layer of the SHG active Li -ionophore 1 cation complex, whose surface concentration is nearly equal at both ionophore concentrations. On the other hand, a totally different intercept and very small slope of the isotherm was obtained for a membrane containing only 3 mM of ionophore 1. This indicates an incomplete formation of the closest-packed surface layer of the cation complexes due to a lack of free ionophores at the membrane surface, leading to a kinetic limitation. In this case, the potentiometric response of the membrane toward Li+ was also found to be very weak vide infra).

Tc or c cubic closest-packing of spheres Th or h hexagonal closest-packing of spheres Ts stacking sequence AA... of hexagonal layers Qs stacking sequence AA... of square layers... 

The structure of iodine at four different pressures. The outlined face-centered unit cell in the 30-Gpa figure corresponds to that of a (distorted) cubic closest-packing of spheres. At 24.6 GPa four unit cells of the face-centered approximant structure are shown the structure is incommensurately modulated, the atomic positions follow a sine wave with a wave length of 3.89 x c. The amplitude of the wave is exaggerated by a factor of two. Lower left Dependence of the twelve interatomic contact distances on pressure... 

In crystalline C60 the molecules have a face-centered cubic arrangement, i.e. they are packed as in a cubic closest-packing of spheres as they are nearly spherical, the molecules spin in the crystal. The crystals are as soft as graphite. Similar to the intercalation com-... 

In a-B12 the icosahedra are arranged as in a cubic closest-packing of spheres (Fig. 11.16). In one layer of icosahedra every icosahedron is surrounded by six other icosahedra that are linked by three-center two-electron bonds. Every boron atom involved contributes an average of electrons to these bonds, which amounts to -6 = 4 electrons per icosahedron. Every icosahedron is surrounded additionally by six icosahedra of the two adjacent layers, to which it is bonded by normal B-B bonds this requires 6 electrons per icosahedron. In total, this adds up exactly to the above-mentioned 10 electrons for the inter-icosahedron bonds. 

---