Close-Packed Structures

The name “close

order to obtain the highest possible structures it is worthwhile to analyze the same radius, in order

possible. For this purpose one on the top of the other

within a layer is in contact with six others and a layer represents a dimensional close-packed hexagonal structure.

is shown in Fig. 75. We will differentiate the holes existing between spheres of a layer as of type (a) or type (b) (see Fig. 75).

Figure 76 shows the plane defined by centers o

and the projection of centers of spheres of the second layer. The centers of the second layer spheres are above the centers of the holes of type (a) specified in Fig. 75. The spheres of the second layer just rest

type (a).

The centers of the holes of type (a) coincide with the geometric centers of the equilateral triangles shown in

Figure 75 A close-packed layer of spheres that is a two

structure.

Packed Structures

ame “close-packed” refers to the way of packing the atoms in order to obtain the highest possible filling factor. To consider close

structures it is worthwhile to analyze first the manner of placing spheres of the same radius, in order for the interstitial volume to be as small as possible. For this purpose, the spheres are arranged in layers that are placed one on the top of the other in the way we will explain below. Each sphere within a layer is in contact with six others and a layer represents a

packed hexagonal structure. The cross section of a layer is shown in Fig. 75. We will differentiate the holes existing between spheres of a layer as of type (a) or type (b) (see Fig. 75).

76 shows the plane defined by centers of spheres of the first layer of centers of spheres of the second layer. The centers of the second layer spheres are above the centers of the holes of type (a) specified in Fig. 75. The spheres of the second layer just rest in the holes o

The centers of the holes of type (a) coincide with the geometric centers of the equilateral triangles shown in Fig. 76 (of course the same occurs in packed layer of spheres that is a two-dimensional close-packed hexagonal packed” refers to the way of packing the atoms in factor. To consider close-packed the manner of placing spheres of be as small as the spheres are arranged in layers that are placed in the way we will explain below. Each sphere within a layer is in contact with six others and a layer represents a two-

The cross section of a layer is shown in Fig. 75. We will differentiate the holes existing between spheres

first layer of centers of spheres of the second layer. The centers of the second layer spheres are above the centers of the holes of type (a) the holes of The centers of the holes of type (a) coincide with the geometric centers Fig. 76 (of course the same occurs in packed hexagonal

the case of holes of type (b), see Fig. 77a). Therefore, each sphere of the second layer is in contact with three spheres of the layer below it.

The third layer can be placed in two ways as depicted in Fig. 77. In the case shown in Fig. 77a the centers of spheres of the third layer are above the centers of the holes of type (b) of the first layer, specified in Fig. 75, whereas in the case shown in Fig. 77b the spheres of the third layer

lie directly above the spheres of the first layer.

We will show now that the close-packed arrangement displayed in Fig. 77a corresponds to the fcc structure. A part of Fig. 77a, with the cubic

cell of the fcc structure, is drawn in Fig. 78. We can see in this figure that the

fcc structure is of type ABCABC…, where A, B, and C denote three two- dimensional close-packed layers shifted horizontally one with respect to the other. The layer planes are orthogonal to a body diagonal of the cubic unit cell of this structure. The second layer, B, is shifted with respect to the first one, A, by vector t
, defined in Fig. 76. In this way, the spheres of the B layer are placed in holes of type (a), shown in Fig. 75, of layer A. The spheres of C layer are placed over the holes in the A layer not occupied by the spheres from B layer, it means, of type (b) in Fig. 75. The C layer is shifted with respect to the A layer by vector 2 t
, and with respect to the B layer by vector t
, so each sphere of the C layer is in contact with 3 spheres

Figure 76 The centers of spheres of the first layer and the projection of the centers of spheres

of the B layer. The spheres of the fourth layer lie directly above the spheres of the first one.

To conclude, we can say that in the a cubic close-packed

fcc one. This is an

the 12 NNs of an atom in the

which is placed the atom in consideration, while half of the other 6 belong to the layer below and the other half to the layer above.

In the case shown in Fig. 77b we have a structure of an ABAB…

with the hexagonal prism. We can see in Fig. 79 that the represents a hexagonal

Figure 77 (a) and (b)

described in (a) differs from that one in (b) in the respect to the spheres of the first and second layer

layer. The spheres of the fourth layer lie directly above the spheres To conclude, we can say that in the case shown in Fig. 77a we have

packed (ccp) structure that was already introduced as the

one. This is an ABCABC… type structure. Now it is easy to visualize s of an atom in the fcc structure; 6 of them belong to the layer in which is placed the atom in consideration, while half of the other 6 belong to the layer below and the other half to the layer above.

In the case shown in Fig. 77b we have a hexagonal close-packed

ABAB… type. Figure 79 shows a part of Fig. 77b together

with the hexagonal prism. We can see in Fig. 79 that the hcp structure hexagonal Bravais lattice with two-atom basis. Each atom in show two close-packed arrangements of equal spheres. The case described in (a) differs from that one in (b) in the positions of spheres of the third

of the first and second layers.

layer. The spheres of the fourth layer lie directly above the spheres

case shown in Fig. 77a we have that was already introduced as the

visualize belong to the layer in which is placed the atom in consideration, while half of the other 6 belong to

packed (hcp)

79 shows a part of Fig. 77b together structure atom basis. Each atom in packed arrangements of equal spheres. The case layer with

this structure has 12

them belong to the layer in which is placed the atom in consideration and the other 6 belong to the adjacent layers. The difference between

structures consists in the location of the adjacent layers. In the

holes and the other three (b) holes (specified in Fig. 75) to which belongs the atom in consideration. I

these 6 NNs occupy holes of type (a)

side of the layer. Twelve is the maximum number of spheres that can be arranged to touch a given sphere. The

details later.

There is an infinite number of possible ways of close spheres, since any sequence of

alike, represents a po

Therefore, a close-packed structure can be obtained only if two consecutive layers are of a different type. In this case, each sphere touches 12 other spheres and this characteristic of all close

Figure 78 The fcc structure viewed as a close

consecutive layers of this structure are marked as

this structure has 12 NNs (as it is also the case for the fcc structure); 6 of them belong to the layer in which is placed the atom in consideration and the other 6 belong to the adjacent layers. The difference between ccp

structures consists in the location of the NN atoms that belong to the adjacent layers. In the case of the ccp structure three of them occupy (a) holes and the other three (b) holes (specified in Fig. 75), present in the layer to which belongs the atom in consideration. In the case of the hcp structure s occupy holes of type (a): 3 from the top and 3 from the bottom . Twelve is the maximum number of spheres that can be arranged to touch a given sphere. The hcp structure will be discussed in There is an infinite number of possible ways of close-packing equal spheres, since any sequence of A, B, C layers, with no two successive layers alike, represents a possible close-packing arrangement of equal spheres. packed structure can be obtained only if two consecutive layers are of a different type. In this case, each sphere touches 12 other spheres and this characteristic of all close-packed structures could be seen structure viewed as a close-packed structure (cubic close-packed). Three consecutive layers of this structure are marked as A, B, and C.

re); 6 of them belong to the layer in which is placed the atom in consideration and the

ccp and hcp atoms that belong to the structure three of them occupy (a) , present in the layer structure he top and 3 from the bottom . Twelve is the maximum number of spheres that can be structure will be discussed in packing equal layers, with no two successive layers packing arrangement of equal spheres. packed structure can be obtained only if two consecutive layers are of a different type. In this case, each sphere touches 12 other ructures could be seen packed). Three

already in the case of

packed structure that represents a Bravais lattice with one

fcc structure.

Below we will give an example of a close from the fcc and

ABACABAC…. This structure is called a

(dhcp) structure.

8. Double Hexagonal Close