Question

3. Body-Centered Cubic (let r = 1.00A    Show all work

A) calculation of the length of the until cell edge(a):

B) calculation of the length of the face diagonal (F):

C) Calculation of the % void space:

Answer 1

A) r = 1.00 A = 1.00 x 10-8 m

The relation between radius and the edge lenght can be found by considering the pythogorean theorem

the principla diagonal distance, face diagonal distance and the edge lenght which gives

(4r)² = 3a²

a= 4/sqrt(3) x r = 12.309 x 1.0 A = 2.309 A

B) The length of face diagonal is sqrt(2) x edge length which can be obtained by considering one face and applying pythagoras theorem

F = sqrt(2) x a = 1.414 x 2.309 A = 3.264 A

C) Void space can be found by finding the packing fraction which is the ratio of V of sphere / volume of unit cell

packing fraction = 2 x 4/3 x pi x r³ /(4/Sqrt(3) x r)³ (2 is the no of atoms in the unit cell which is one in the center and 1 shared by the 8 corners of the unit cell)

Packing fraction = sqrt(3) x pi / 8 = 0.6802

Void fraction = 1 - packing fraction = 1 - 0.6802 = 0.3198