Solids closest packing

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... 

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,... 

The solid noble gases also adopt closest-packings of spheres at low temperatures Ne... Xe c helium becomes solid only under pressure (depending on pressure, c, h or i)... 

Two metals that are chemically related and that have atoms of nearly the same size form disordered alloys with each other. Silver and gold, both crystallizing with cubic closest-packing, have atoms of nearly equal size (radii 144.4 and 144.2 pm). They form solid solutions (mixed crystals) of arbitrary composition in which the silver and the gold atoms randomly occupy the positions of the sphere packing. Related metals, especially from the same group of the periodic table, generally form solid solutions which have any composition if their atomic radii do not differ by more than approximately 15% for example Mo +W, K + Rb, K + Cs, but not Na + Cs. If the elements are less similar, there may be a limited miscibility as in the case of, for example, Zn in Cu (amount-of-substance fraction of Zn maximally 38.4%) and Cu in Zn (maximally 2.3% Cu) copper and zinc additionally form intermetallic compounds (cf. Section 15.4). 

The right hand side of Fig. A.4.6 is contained in Fig. 3.3. Capacity measurements can readily be made at solid electrodes to study adsorption behavior. For a review see Parsons (1987). As Fig. A.4.7 illustrates, capacity potential curves of three low-index phases of silver, in contact with a dilute aqueous solution of NaF, show different minimum capacities (corresponding to the condition o = 0) and therefore remarkably different potentials of pzc. The closest packed surface (111) has the highest pzc and the least close-packed (110) has the lowest pcz these values differ by 300 mV. Such complications observed with single crystal electrodes, seem likely to have their parallel at other solid surfaces. For example, it is to be expected that a crystalline oxide will have different pzc values at its various types of exposed faces. 

Consider the molecule CIFjOj (with chlorine the central atom). How many isomers are possible Which is the most stable Assign point group designations to each of the isomers. 6.id The Structure for AliBr (Fig. 6. Ih) is assumed by both Al2Br6and ALCUin the gas phase. In the solid, however, the structures can best be described as closest packed arrays of halogen atoms (or ions) with aluminum atoms (or ions) in tetrahedral or octahedral holes. In solid aluminum bromide the aluminum atoms arc found in pairs in adjacent tetrahedral holes. In solid aluminum chloride, atoms are found in one-lhird of the octahedral holes... 

As mentioned above, the binder/filler ratio determines the fluidity and hence the process technology for a given syntactic composition. The solid line in Fig. 1 shows how the apparent density depends on the relative microsphere concentration in syntactic compounds, provided there are no air inclusions. The lower limit of apparent density is at the concentration at which the filler is most densily packed at this point the material has its highest specific strength (strength per unit apparent density). Experience has shown that the closest packing of the spheres within the binder is obtained when the material is mixed and cast or molded under vacuum 8 10). 

Molecular dynamics calculations of Hoover and Ree (25) have indicated that a fluid-solid transition occurs in a system of hard spheres even in the absence of attractive forces. The fluid exists for particle volume fractions up to a value rj = 0.49 and at this point, a solid phase with ij = 0.55 is predicted to coexist in equilibrium with the fluid phase. When the particle volume fraction lies in the range 0.55 < jj < 0.74, the solid phase is stable. The upper limit for ij corresponds to the density at closest packing for a face-centered-cubic (fee) arrangement of the particles. 

In order to determine the thermodynamic properties by means of the perturbation theory, the thermodynamic properties of the reference system are needed. Here, the expressions for the equation of state and the radial distribution function of a system of hard spheres are included for both the fluid and solid reference states. A face-centred-eubic arrangement of the particles at closest packing is assumed for the solid phase. 

Translation of ions within crystals is less frequently observed than is rotation. Perhaps one of the most interesting cases is that of silver iodide which may actually be said to melt in halves. When this solid is heated to 145.8° C, the crystal structure then changes and the ionic conductivity increases tremendously the iodide ions are hexagonally closest-packed below the transition temperature but at this temperature they rearrange to form a more open structure, and the silver atoms are allowed to move within the lattice. At 555° C, the network of iodide ions collapses, and the compound becomes a liquid. The solids Cul and Ag2Se show similar behavior. 

The lattice energy based on the Born model of a crystal is still frequently used in simulations [14]. Applications include defect formation and migration in ionic solids [44,45],phase transitions [46,47] and, in particular, crystal structure prediction whether in a systematic way [38] or from a SA or GA approach [ 1,48]. For modelling closest-packed ionic structures with interatomic force fields, typically only the total lattice energy (per unit cell) created by the two body potential,... 

From equation (8) it can be seen that sohds and liquids will form equilibrium shapes in an effort to minimize their surface area and thus the free energy of the system. Indeed, crystal faces with the closest packing of surface atoms have the lowest surface area and tend to be the most stable. When one considers a two-component system, with one material on top of the other, the interaction between the two will be defined by the surface tensions. The surface tensions of some selected solids and hquids are listed in Table 2. From these values, it can be easily predicted which materials will be capable of wetting another. In general, most liquids have lower surface tensions than clean solids and will therefore spread to cover them. 

In Example 16.1 we will see how the density of a closest packed solid can be calculated. 

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