Cubic And Hexagonal Close Packing


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1.9 Description of crystal structures
The most common way for describing crystal structure is to refer the structure to the unit cell. The structure is given by the size and shape of the cell and the position of atoms (i.e. atomic coordinates). However, this is insufficient to visualize the whole crystal structures in 3D. Referring to a larger part of the structure (comprising perhaps several unit cells) and considering the arrangement relative to each other, their coordination numbers, interatomic distances, types of bonding, etc are more important.
Two of the most useful ways of describing structures are based on close packing and space-filling polyhedra. They provide greater insight into the crystal chemistry than is obtained using unit cells alone.
1.10 Close packed structures – cubic and hexagonal close packing
Many crystal structures can be described using the concept of close packing. ※Structures are usually arranged to have the maximum density. ※Consider the most efficient way of packing equal-sized spheres in 3D.
Fig. 1.12 shows the most efficient way to pack spheres in 2D. Each sphere, Ⓐ is surrounded by, and is in contact with, six others; i.e. Ⓐ has six nearest neighbors. The coordination number of six is the maximum possible for a planar arrangement of contacting, equal-sized spheres.
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Regular repetition Ö Close packed layer (cp layer) The cp layer has three close packed directions: XX’, YY’, ZZ’ (spheres are in contact in the directions).
A non-cp layer is shown in the inset of Fig. 1.12; a lower coordination number is observed (4 for Ⓐ).
※The most efficient way to pack spheres in 3D is to stack cp layers on top of each other: [hexagonal close packed] and [cubic close packed].
- The most efficient way for two cp layers A and B to be in contact Each sphere of one layer to rest in a hollow between three spheres in the other layer, e.g. at P or R in Fig. 1.12
- Fig. 1.13 shows such a position for 2 layers. Atoms in second layer may occupy either P or R (Fig. 1.12), not both, nor a mixture of the two.
Hexagonal Close Packing (hcp) - Fig. 1.13, suppose that A lies underneath the B layer.
A third layer on top of B: positions S or T For S position, the third layer directly over the A layer. As subsequent layers are added, the following sequence arises:
…ABABAB… This is known as hexagonal close packing (hcp)
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Cubic Close Packing (ccp) - In Fig. 1.13, the third layer is placed at T, then all three layers are
staggered relative to each other and it is not until the fourth layer is positioned (at A) that the sequence is repeated..
The third layer is called C, and it gives …ABCABCABC…
i.e. cubic close packing (ccp) Fig. 1.14

hcp and ccp are two simplest stacking sequences and are by far the most important in structure chemistry.

Other more complex sequences with larger repeat units, e.g. ABCACB or ABAC are related to the phenomenon of polytypism (多樣型).

In 3D cp structures, each sphere is in contact with twelve others and this is the maximum coordination number possible for contacting and equal-sized spheres. (A common non-cp structure is body centered cube, with CN = 8, Fig. 1.9c for α−Fe)

6 of these neighbors are coplanar (Fig. 1.12)

Two groups of 3 spheres

One group in the plane above One group in the plane below

Fig. 1.13 and 1.14

Fig. 1.15: hcp and ccp differ only in the relative orientations of these two groups of 3 spheres.

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1.11 Relationship between cubic close packed and face centered cubic
The unit cell of a ccp arrangement is the face centered (fcc) unit cell (Fig. 1.9(a) with spheres at the corners and face centers).
- The relationship between ccp and fcc is not obvious since the faces of the fcc unit cell do not correspond to cp layers.
- The cp layers are parallel to the {111} planes of the fcc unit cell (Fig. 1.16, the spheres labeled 2−7 in (a) and (b) form part of a cp layer, revealed by removing sphere 1)
- In an fcc structure, cp layers occur in four orientations and are perpendicular to the body diagonals. (the fcc cube has four body diagonals, originating from 8 corners)
- Fig. 1.16(c), the cp layers in one orientation are seen edge-on. 從側邊看
- Fig. 1.16(d), the cp layers are seen perpendicular to the layers. (對角 corner 的 atoms 剛剛好對到, 均為 A layers)
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1.12 Hexagonal unit cell and close packing
An hcp arrangement of spheres has a hexagonal unit cell. - The basal plane of the cell coincides with a cp layer of spheres (Fig, 1.17b) - The unit cell centains two spheres One at the origin (and hence at all corners) One inside the cell at position 1 2 1 (dashed circle in b).
3, 3, 2
- cp Layers occur in only one orientation in a hcp structure (hexagonal unit cell).
- Fig. 1.17 shows that the two axes in the basal plane are of equal length (a = 2r, if the spheres of radius r touch)
- The angle γ is 120°, c-axis is a sixfold rotation axis. See further from the auxiliary materials
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1.13 Density of close packed structures

In cp structures, 74.05% of the total volume is occupied by spheres. This is the maximum density possible if structures constructed of spheres of only one size.

Ex: the fcc unit cell (4 spheres in one unit cell). One at a corner and three at face centers

The cp directions (XX’, YY’, ZZ’ in Fig. 1.12) occur parallel to the face

diagonal of the unit cell, i.e. spheres 2,5,6 in Fig. 1.16b.

The length of the face diagonal: 4r The length of the cell edge: 2 2 r

(2 2γ )2 + (2 2γ )2 = 4γ

The volume of the cell: 16 2 r3

(2 2γ )3 = 16 2γ 3

4 × 4 πγ 3 ∴ total sphere volume = 3 = 0.7405
unit cell volume 16 2γ 3
Similar results are obtained for hcp.

In non-cp structures, densities lower than 0.7405 are obtained - Body centered cubic, bcc, is 0.6802 (the cp directions (Fig. 1.12) in bcc are parallel to the body diagonals of the cube).

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1.14 Unit cell projections and atomic coordinates
To give 3D perspective to crystal structures, they are often drawn as oblique projections (Fig, 1.16a).
For accurate and unambiguous descrriptionm, it is necessary to project them down particular crystallographic directions and/or on to crystallographic faces.
Ex. Fig. 1.18b (Unit Cell Projections), a face centered cube projected down the z-axis onto the ab unit cell face.
- All sense of vertical perspective is lost - To restore vertical perspective, their vertical height in the cell is
given as a fraction of c, beside each atom. - It is not necessary to specify the x, y coordinates if the structure
is drawn to scale. - The origin is often taken as the top left hand corner. - Two atoms at each corners and two at top and bottom face
centers, with z-coordinates of 0 and 1. - The four side face centers, each with z =1/2
Fig. 1.18c (Atomic Coordinates): It is important to use Fig. 1.18b to relate the fractional atomic coordinates.
- A face centered cube contains, effectively, four positions: one corner and three face centers, i.e. 000 , 1 1 0 , 1 0 1 , 0 1 1
22 2 2 22
- Each coordinates specifies the fractional distance of the atom from the origin in the directions a, b, and c.
- The more complete structure shown in Fig. 1.18b is obtained by
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Cubic And Hexagonal Close Packing