How many atoms are in the following unit cells? Body centered cubic, face center

Question

How many atoms are in the following unit cells? Body centered cubic, face centered cubic (FCC), a hypothetical body centered/face centered cubic crystal, and a hypothetical diamond cubic structure with superimposed face centered cubic and body centered cubic atoms. Calculate the ratio of the packing factors for the following cases: simple cubic to face centered cubic. simple cubic to hypothetical face centered body centered cubic crystal (i.e. a face centered cubic with a similar atom placed in the center simple cubic to a simple hexagonal unit cell 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.

Explanation / Answer

For the unit cells

a) Number of atoms in the unit cell of,

body centered cubic = 2

face centered cubic = 4

face centered/body centered unit cell = 6

diamond cubic structure = 8

b) atomic packing factor for

1) simple cubic to face centered = 0.52/0.74 = 0.703

2) simple cubic to hypothetical face centered body centered = 0.52/0.67 = 0.776

3) simple cubic to hexagonal = 0.52/0.74 = 0.703

4) simple cubic central impurity to one with central simple cubic = 0.52/0.45 = 1.15

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