Question

Calculate the ratio of the packing factors for the following cases: 1) simple cubic to face centered cubic. 2) simple cubic to hypothetical face centered body centered cubic crystal (i.e. a face centered cubic with a similar atom placed in the center 3) simple cubic to a simple hexagonal unit cell (hint: think about the hexagonal unit cell that can be made by deforming a simple cube) 4) a simple cubic crystal that has a central impurity atom that fills the interstitial space to a simple cubic crystal that has a central simple cubic structure in the interstitial space.

Answer 1

APF = Nparticle V partical / Vcrystal { agregate packing factor N is no. of ayom, V = volume}

a for body centered cubic

a = 4r/sqrt[3]

a = pi x sqrt[3] / 8 = 0.68

for face center cubic

a = 4 x4 x pi x r³ / 3 x 16 x sqrt[2] r³

a = 0.74

for hypothetical face centered body centered cubic (N = 5 atom per cube)

a = 5 x 4 x pi x r³ / 3 x 16 x sqrt[2] r³

a = 0.92

all as same

Common sphere packings taken on by atomic systems are listed below with their corresponding packing fraction.

Hexagonal close-packed (hcp): 0.74

for hypothetical face center body center cubic crystal = 0.9256

Face-centered cubic (fcc): 0.74 (also called cubic close-packed, ccp)

Body-centered cubic (bcc): 0.68

Simple cubic: 0.52

a)  simple cubic to face centered cubic. = 0.52/0.74 = 0.722

b) simple cubic to hypothetical face centered body centered cubic crystal = 0.52/0.92 = 0.56

c) simple cubic to a simple hexagonal unit cell = 0.52/0.74 = 0.722

d) in simple cubic crystal void redius ration is = 0.732 - 0.999

a = 0.91

ratio = 0.52/0.91 = 0.57