Sphere packing and

We shall first say a little about the three vertical subdivisions of Table 3.1, confining our attention for the most part to systems in which p = 3 or 4. The coordination numbers from 6 to 12 are more conveniently considered under sphere packings, and the very high coordination numbers (13-16 in many transition-metal... 

Although Eqs. (33), (34), and especially (35), are useful they have a problem. They all predict that the hard sphere system is a fluid until = 1. This is beyond close packing and quite impossible. In fact, hard spheres undergo a first order phase transition to a solid phase at around pd 0.9. This has been estabhshed by simulations [3-5]. To a point, the BGY approximation has the advantage here. As is seen in Fig. 1, the BGY equation does predict that dp dp)j = 0 at high densities. However, the location of the transition is quite wrong. Another problem with the PY theory is that it can lead to negative values of g(r). This is a result of the linearization of y(r) - 1 that... 

Buckminsterfullerene is an allotrope of carbon in which the carbon atoms form spheres of 60 atoms each (see Section 14.16). In the pure compound the spheres pack in a cubic close-packed array, (a) The length of a side of the face-centered cubic cell formed by buckminsterfullerene is 142 pm. Use this information to calculate the radius of the buckminsterfullerene molecule treated as a hard sphere, (b) The compound K3C60 is a superconductor at low temperatures. In this compound the K+ ions lie in holes in the C60 face-centered cubic lattice. Considering the radius of the K+ ion and assuming that the radius of Q,0 is the same as for the Cft0 molecule, predict in what type of holes the K ions lie (tetrahedral, octahedral, or both) and indicate what percentage of those holes are filled. 

I have assumed that this equation applies to structures with two or more spheres in the central layer (as well as with one. as in icosahedral packing), and have applied it in the calculation of the ranges of values of the neutron number N in which successive subsubshells are occupied (12). (In this calculation the difference in radius of the different kinds of spherons is taken into consideration.) The assignment of quantum numbers is made with use of the following assumptions (14) ... 

Other stacking sequences than these are also possible, for example AaBpAaCy... or statistical sequences without periodic order. More than 70 stacking varieties are known for silicon carbide, and together they are called a-SiC. Structures that can be considered as stacking variants are called polytypes. We deal with them further in the context of closest-sphere packings (Chapter 14). 

The geometric principles for the packing of spheres do not only apply to pure elements. As might be expected, the sphere packings discussed in the preceding chapter are also frequently encountered when similar atoms are combined, especially among the numerous alloys and intermetallic compounds. Furthermore, the same principles also apply to many compounds consisting of elements which differ widely. 

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). 

Although the space filling of the body-centered cubic sphere packing is somewhat inferior to that of a closest-packing, the CsCl type thus turns out to be excellently suited for compounds with a 1 1 composition. Due to the occupation of the positions 0,0,0 and with different kinds of atoms, the structure is not... 

The required local charge balance between cations and anions which is expressed in Pauling s rule causes the distribution of cations and anions among the octahedral and tetrahedral interstices of the sphere packing. Other distributions of the cations are not compatible with Pauling s rule. 

F. C. Frank, J. S. Kasper, Complex alloy structures regarded as sphere packings. I Definitions and basic principles, Acta Crystallogr. 11 (1958) 184. II Analysis and classification of representative structures, Acta Crystallogr. 12 (1959) 483. 

F. C. Frank and J. S. Kasper, Complex Alloy Structures Regarded as Sphere Packings. II. Analysis and Classification of Representative Structures, Acta Cryst., 12, 483 (1959). 

C. G. Wilson and F. J. Spooner, A Sphere-Packing Model for the Prediction of Lattice Parameters and Order in o Phases, Acta Cryst., 39A, 342 (1973). 

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