Construction of a body-centered cubic unit cell. What fraction of the volume of

Question

Construction of a body-centered cubic unit cell. What fraction of the volume of the center sphere is inside the unit cell, if the corners of the unit cell lie at the centers of the corner spheres? What is the total number of spheres within the unit cell? In terms of the radius r of one of the spheres, what is the total volume of the spheres inside the unit cell? In terms of the sphere radius r what is the side length a of the unit cell? What is the cell volume? To answer this question, look at your model and note the direction in the cell along which the spheres are actually in contact. The following geometrical relationships for a cube will be useful. side length = a; face diagonal = f = a Squareroot 2: body diagonal = b = a Squareroot 3 What is the fraction of space occupied by the spheres in a body-centered cubic unit cell? What is the coordination number of a particle in a body-centered cubic structure? Construct from equal sized spheres two of the layers shown in Figure (a) and of the layers shown in Figure 4(b). Place layer (b) on one layer (a) in the same relative orientation as shown in Figure 4, so that the spheres in layer (b) rest in the spaces between the spheres in

Explanation / Answer

Consider a Body - centered Cubic unit cell

If corners of the cube lie at the center of corner sphere then r=a/2

There are 8 such corners in the cube and corresponding 8 corner of spheres

Each corner constitutes 1/8 the volume of the sphere

B1) From the above geometry when r=a/2 whole fraction of volume of center sphere lies within the cube

B2) The whole unit cell constitutes of 2 sphere , 1 is the body centered and other constitutes from the 8 corners of the sphere.

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