Hexagonal close packing of sphere

Show that cja for hexagonal close packing of spheres is 1.633. 

FIGURE 11.7. Although ideal hexagonal close packing of spheres predicts a maximum dispersed phase volume fraction of about 0.74 (a), the inherent polydispersity of most emulsions allows for the preparation of emulsions of much greater internal volume fraction, as smaller droplets are located in the interstitial spaces between larger drops (b). 

Side and expanded views of hexagonal and cubic close-packed crystal types. In the hexagonal close-packed structure, spheres on both sides of any plane are in the same positions, and the third layer is directly above the first. In the cubic close-packed structure, layers take up three different positions, and the fourth layer is directly above the first. 

It is natural that the face-centered and hexagonal close-packed structures, both corresponding to the close packing of spheres, should have the same value of c. In case the hexagonal lattices are not close packed, there is an additional correction factor in c, which we shall not evaluate. 

When all the rotations are possible in the solid state the symmetry increases to hexagonal. This form corresponds to the close packing of spheres or cylinders and the molecule is in a rotational crystalline state, characterized by rigorous order in the arrangement of the center (axes) of the molecules and by disordered azimuthal rotations [118]. If the chain molecules are azimuthally chaotic (they rotate freely around their axes), their average cross sections are circular and, for this reason, they choose hexagonal packing. The ease of rotation of molecules in the crystal depends merely on the molecular shape, as in molecules of an almost spherical shape like methane and ethane derivatives with small substituents, or molecules of a shape close to that of a cylinder (e.g., paraffin-like molecules). 

Figure 9-28. Close packing of spheres after Shubnikov and Koptsik [42], (a) Hexagonal closest packing (b) Cubic closest packing. Used with permission.

Figure 48. Hexagonal (a) and cubic (b) close packing of spheres...

The characteristic line sequences for cubic lattices are shown graphically in Fig. 10-2, in the form of calculated diffraction patterns. The calculations are made for Cu Kol radiation and a lattice parameter a of 3.50 A. The positions of all the diffraction lines which would be formed under these conditions are indicated as they would appear on a film or chart of the length shown. (For comparative purposes, the pattern of a hexagonal close-packed structure is also illustrated, since this structure is frequently encountered among metals and alloys. The line positions are calculated for Cu Kol radiation, a = 2.50 A, and cja = 1.633, which corresponds to the close packing of spheres.)... 

Figure 2.3. Close-packing of spheres (atoms) (a) one layer (top view), (b) two layers (top view), (c) three layers, hexagonal close packed (side view), and (d) three layers, cubic close packed (side view).

Fig. 5.01. The close packing of spheres, (a) A single close-packed layer. (b) A single close-packed layer showing the two alternative sets of dimples into which the spheres of a second close-packed layer can fit. (c) Hexagonal close packing. The layers repeat in the sequence ABABA. The plan of the unit cell is outlined by broken lines. (d) Cubic close packing. The layers repeat in the sequence A B C A B C A.

Fig. 1.9. Construction of the hexagonal close-packed and the face-centred cubic lattice by stacking close-packed layers of spheres. The structures differ in their stacking sequence In the hexagonal close-packed structure spheres in the third layer are placed perpendicularly above those in the first, in the face-centred cubic lattice the...

Figure 1.4. The two possible close packings of spheres Left the. ..ABCABC... stacking corresponding to the FCC crystal. Right the. ..ABABAB... stacking corresponding to the HCP crystal. The lattices are viewed along the direction of stacking of the hexagonal-...

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

DpopE related to SpQpp, which is in turn related to the closeness of packing of the powder. The number of particles adjacent to a given particle is represented by The maximum packing density for monosize spheres occurs at hexagonal close packing, where = 12 and = 0.2595 for... 

Fig. 5.2. Close packing of hard-sphere atoms - an alternative arrangement, giving the hexagonal close-packed (h.c.p.) structure.

---