Question

Consider the body-centered cubic (BCC) structure in Figure 3.

 a. What is the length of the line (labeled c) that runs from one corner of the cube diagonally through the center of the cube to the other corner in terms of r (the atomic radius)?

 b. Use the Pythagorean theorem to derive the expression for the length of the line (labeled b) that runs diagonally across one the faces of the cube in terms of the edge length (l).

 c. Use the answer to part (a) and (b) along with the Pythagorean theorem to derive the expression for the edge length (l) in terms of r.

 d. Spheres of radius r occupy 68.0% of the available volume in this BCC arrangement. Use the fraction of occupied volume to calculate the value of l, the length of the edge of the cube, in terms of r.

 (Hint: Volume of a sphere = (4π/3) r³)

 

 

 

Answer 1

(a). Length of line c, = r + 2r + r

= 4r

(b). Length of line b:

Using pythagoras theorem;

b² = l² + l²

b² = 2l²

b =

b =

(c). From the figure:

Using pythagoras theorem;

c² = l² + b²

Using results from part a and b:

(4r)² = l² + 2l²

(4r)² = 3l²

l² = (4r)² / 3

l = 4/ r

(d). Edge length of cube = l

Volume of cube = side³ = l³  

Radius of sphere = r

Volume of sphere = 4/3 r³  

In BCC, total number of spheres = 2

So total volume of spheres = 2 * 4/3 r³   

= 8/3 r³   

Since, spheres occupy 68% of total volume:

8/3 r³    = 0.68 * l³  

l³   = 12.32 r³   

l = 2.31 r