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: 1) simple cubic to face centered cubic. 2) simple cubic to hypothetical face centered body centered cubic crystal (i.c. a face centered cubic with a similar atom placed in the center 3) 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

a) (i) body centered cubic

no of atoms from 8 corners of unit cell = (1/8) x 8 = 1

no. of atom from centere of cube = 1

total no of atoms = 1+1 = 2

(ii) face centered cubic

atoms from 8 corner = (1/8) x 8 = 1

no. of atoms from 6 face centere = (1/2) x 6 = 3

toyal no. of atoms = 4

(iii) a hypothetical body centered/face centered cubic

no. of atoms in face cetere is 4 and 1 atom at the centere of cubic crystal

total no of atoms = 5

(ix) hypothetical diamond crystal

at 8 corner = (1/8) x 8 = 1

at 6 face centere = (1/2) x 6 = 3

at center of cubic = 1

total no of atoms = 8

diamond lattice contains also 4 lattice point per unit cell but contains 8 atoms per unit cell.

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

pacing efficiency for body centered cubic

a = 4r/sqrt[3]

a = pi x sqrt[3] / 8

pacing efficiency = 0.68

for face center cubic

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

packing efficiency = 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

pacing efficiency = 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

packing efficiency = 0.91

ratio of packing factor = 0.52/0.91 = 0.57

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