Question

Consider a hypothetical metal that has a density of 8.40 g/cm3, an atomic weight of 108.6 g/mol, and an atomic radius of 0.136 nm. Compute the atomic packing factor if the unit cell has tetragonal symmetry; values for the A and C lattice parameters are 0.494 and 0.352, respectively.

Answer 1

The conventional unit cell of the body centered tetragonal structure is characterized by a lattice parameter a in the basal plane and a lattice parameter c in the z-direction. The packing fraction, f, of a crystal structure is defined as the maximum fraction of the volume of the structure that can be occupied by non-overlapping hard spheres. The packing fraction is thus given by the ratio of the volume of spheres, V_s, that can be accommodated within a unit cell to the total volume of the unit cell, V_c;

f = V_s/V_c   

For the body centered tetragonal structure there are two spheres contained within the conventional unit cell and the packing fraction may be determined as a function of the ratio of the basal plane lattice parameter, a, and the z-axis lattice parameter, c, that is, c/a, from a geometrical consideration of the packing of hard spheres. Three distinct regimes need to be considered on the basis of constraints placed on the diameter of the spheres by the geometry of the crystal structure. For c/a < the distance between the spheres is limited along the z-direction. The spheres along a unit cell edge touch along the z-direction leading to the constraint that the radius of the spheres, r, is given by

(1)

Thus the total volume of two spheres within the unit cell will be

(2)

and for a unit cell volume of V_c = a ² c, the packing fraction, is found to be

(3)

For the region the distance between the spheres is limited along the body diagonal of the tetragonal cell. In this case the sphere radius is 1/4 of the body diagonal or

(4)

and the total sphere volume will be

(5)

Dividing by the cell volume and rearranging gives the packing fraction as

(6)

In the third regime c/a > and the distance between the spheres is limited in the basal plane. Here the radius will be one half of the basal plane lattice parameter or

(7)

The total sphere volume will be

(8)

and the packing fraction is found to be

(9)

Now in this case

thus,

from equation (3)

atomic packing fraction