The closest packing of equal circles on a sphere

The closest packing of x circles on the surface of a sphere is examined with the use of techniques that have been developed to determine the stereochemical arrangement of atoms packed around a central atom. The technique is based on the concept that the centres of the circles repel one another, and minimizing the total ‘repulsion energy’ leads to an approximate structure from which exact solutions can be determined. A number of improved packings have been discovered for values of x in the range 20–40. Many different types of structure are found that are of lower symmetries than those previously described. The packing density p , defined as the fraction of the spherical surface that is enclosed by the circles, is found to increase as the number of circles increases, in contrast to the conclusion from previous studies. However, this increase in p is very slight and the values remain substantially below that for an infinite number of circles, or a close-packed plane.

Defect Structure of Deformed Tungsten Carbide

In an effort to optimize both the hardness and toughness of carbides used in cemented cutting tools, a project has been undertaken to relate variations in these mechanical properties to the defect structure of deformed WC and (W,Ti)C. This paper describes the defects in stoichiometric WC as revealed by TEM studies. Possible relationships between them and the atomic configuration and bonding in the material will be discussed. WC has a hexagonal unit cell with W atoms at the 0,0,0 position and C atoms at the 1/3,2/3,1/2 position. The c/a ratio is approximately unity with a=2. 91Å and c=2.84Å. Although the basal plane is the close-packed plane, the slip plane has been determined by other workers and confirmed by this study to be {1010}.

Stacking Fault Contrast in an Ordered L12 Lattice

Stacking fault contrast arises from the phase shift α = 2π ḡ · experienced by the diffracted wave at the fault. In an ordered L12 lattice, the displacement <111> is not necessarily equivalent to a displacement of the type <112>. 1 and 2 differ by a displacement <110> which can introduce a phase shift of π when ḡ is a superlattice diffraction vector. The physical difference can be seen by examining the formation of an intrinsic fault. When the fault is produced by removal of a close packed plane, <111>, no nearest neighbor violations of order result; on the other hand,nearest neighbor order is disrupted by shear of the <112> type.Similarly, there are two types of extrinsic faults. Values of α are listed in Table I for [111] and [112] with ḡ of the form 100, 200, and 300. As expected, the two representations are equivalent when ḡ is a fundamental reflection. For g = 100, 010 the values of α for 1 and 2 differ by π because of the antiphase boundary associated with the latter displacement.