Close packing Crystal structure

the hcp lattice (left) , fcc lattice (right)

the principles involved can understood considering efficient way of packing equal-sized spheres , stacking close-packed atomic planes in 3 dimensions. example, if plane lies beneath plane b, there 2 possible ways of placing additional atom on top of layer b. if additional layer placed directly on plane a, give rise following series:

...abababab...

this arrangement of atoms in crystal structure known hexagonal close packing (hcp).

if, however, 3 planes staggered relative each other , not until fourth layer positioned directly on plane sequence repeated, following sequence arises:

...abcabcabc...

this type of structural arrangement known cubic close packing (ccp).

the unit cell of ccp arrangement of atoms face-centered cubic (fcc) unit cell. not obvious closely packed layers parallel {111} planes of fcc unit cell. there 4 different orientations of close-packed layers.

the packing efficiency can worked out calculating total volume of spheres , dividing volume of cell follows:

4
×


4
3


π

r

3




16


2



r

3





=


π

3


2





=
0.7405...


{\displaystyle {\frac {4\times {\frac {4}{3}}\pi r^{3}}{16{\sqrt {2}}r^{3}}}={\frac {\pi }{3{\sqrt {2}}}}=0.7405...}

the 74% packing efficiency maximum density possible in unit cells constructed of spheres of 1 size. crystalline forms of metallic elements hcp, fcc, or bcc (body-centered cubic). coordination number of atoms in hcp , fcc structures 12 , atomic packing factor (apf) number mentioned above, 0.74. can compared apf of bcc structure, 0.68.