A view looking down at two layers of a hexagonal close packed structure. The uni

Question

A view looking down at two layers of a hexagonal close packed structure. The unit call perimeter is outlined. The unit cell is a hexagonal-based solid whose corners are the centers of the outside spheres in the (a) layers shown in Figure 6. A top view of the outline of the unit cell is sketched in Figure 6 What fraction of the volume of (1) the corner spheres. (2) those in the center of the hexagonal faces, and (3) those in the (b) layer is inside the unit cell? (Look carefully at Figure 5 and 6 before answering this question) 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 within the unit cell? In terms of the sphere radius r what is the side length s and the height h of the unit cell? To calculate h, note that the center of a sphere in the (h) layer and the centers of the three spheres in an (a) layer in contact with it are at the corners of a regular tetrahedron The height 1 of a regular tetrahedron of edge length a is given by What is the cell volume in terms of the sphere radius r? The volume V of a hexagonal-based solid of height h and side length a is given by: What is the fraction of space occupied by the spheres in s hexagonal closest packed unit cell?

Explanation / Answer

D.1. (1). We know that the vertex angel in hexagon is 120o. Also the complete angel in the sphere is 360o. So out of 360 only 120 part of sphere is inside the hexagonal unit cell.

That is = 120 / 360

= 1/3

Also, the face of unit cell passes from the center of the sphere, So fraction of sphere inside the unit cell

= (1/3) / 2

= 1/6

(2). We know that the hexagonal face of the unit cell passes from the center of the sphere. So only half of the portion of sphere is inside the unit cell.

Hence fraction of sphere inside the unit cell = 1/2

(3). From the figure we can see that there is no such sphere from the layer b that is inside the unit cell. Hence fraction of shpere = 0

D.2. We know that, the rank of Hexagonal close packed structure is 6.

So number of atoms within the unit cell is 6.

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