The only element that crystallizes in a simple cubic unit cell is polonium. Simple cubic unit cells are, however, common among binary ionic compounds, where each cation is surrounded by six anions and vice versa.

The arrangement of atoms in a simple cubic unit cell. Each atom in the lattice has six nearest neighbors in an octahedral arrangement. The body-centered cubic unit cell is a more efficient way to pack spheres together and is much more common among pure elements. As shown in part b in Figure The third layer of spheres occupies the square holes formed by the second layer, so that each lies directly above a sphere in the first layer, and so forth. All the alkali metals, barium, radium, and several of the transition metals have body-centered cubic structures.

## Molar volume

The most efficient way to pack spheres is the close-packed arrangement, which has two variants. A single layer of close-packed spheres is shown in part a in Figure Each sphere is surrounded by six others in the same plane to produce a hexagonal arrangement. Above any set of seven spheres are six depressions arranged in a hexagon. In principle, all six sites are the same, and any one of them could be occupied by an atom in the next layer.

Actually, however, these six sites can be divided into two sets, labeled B and C in part a in Figure Sites B and C differ because as soon as we place a sphere at a B position, we can no longer place a sphere in any of the three C positions adjacent to A and vice versa. Placing the third-layer atoms over the C positions gives the cubic close-packed structure. If we place the second layer of spheres at the B positions in part a in Figure There are now two alternatives for placing the first atom of the third layer: we can place it directly over one of the atoms in the first layer an A position or at one of the C positions, corresponding to the positions that we did not use for the atoms in the first or second layers part c in Figure If we choose the first arrangement and repeat the pattern in succeeding layers, the positions of the atoms alternate from layer to layer in the pattern ABABABâ€¦, resulting in a hexagonal close-packed hcp structure part a in Figure If we choose the second arrangement and repeat the pattern indefinitely, the positions of the atoms alternate as ABCABCâ€¦, giving a cubic close-packed ccp structure part b in Figure Because the ccp structure contains hexagonally packed layers, it does not look particularly cubic.

The hcp and ccp structures differ only in the way their layers are stacked. The illustrations in a show an exploded view, a side view, and a top view of the hcp structure. The simple hexagonal unit cell is outlined in the side and top views. Note the similarity to the hexagonal unit cell shown in Figure The ccp structure in b is shown in an exploded view, a side view, and a rotated view.

The rotated view emphasizes the fcc nature of the unit cell outlined. The line that connects the atoms in the first and fourth layers of the ccp structure is the body diagonal of the cube. Table Most metals have hcp, ccp, or bcc structures, although several metals exhibit both hcp and ccp structures, depending on temperature and pressure. The smallest repeating unit of a crystal lattice is the unit cell. The simple cubic unit cell contains only eight atoms, molecules, or ions at the corners of a cube.

A body-centered cubic bcc unit cell contains one additional component in the center of the cube. A face-centered cubic fcc unit cell contains a component in the center of each face in addition to those at the corners of the cube. The simple cubic and bcc lattices have coordination numbers of 6 and 8, respectively.

A crystalline solid can be represented by its unit cell, which is the smallest identical unit that when stacked together produces the characteristic three-dimensional structure. Why is it valid to represent the structure of a crystalline solid by the structure of its unit cell? What are the most important constraints in selecting a unit cell?

All unit cell structures have six sides. Can crystals of a solid have more than six sides? Explain your answer. Explain how the intensive properties of a material are reflected in the unit cell. Are all the properties of a bulk material the same as those of its unit cell?

## DoITPoMS - TLP Library Crystallography - Close packing and packing efficiency

The experimentally measured density of a bulk material is slightly higher than expected based on the structure of the pure material. Propose two explanations for this observation. The experimentally determined density of a material is lower than expected based on the arrangement of the atoms in the unit cell, the formula mass, and the size of the atoms. What conclusion s can you draw about the material?

Only one element polonium crystallizes with a simple cubic unit cell. Why is polonium the only example of an element with this structure?

What is meant by the term coordination number in the structure of a solid? How does the coordination number depend on the structure of the metal? Arrange the three types of cubic unit cells in order of increasing packing efficiency.

What is the difference in packing efficiency between the hcp structure and the ccp structure? The structures of many metals depend on pressure and temperature. Connecting the centers of eight of the spheres, a cube emerges Steinhaus , pp. Connecting the centers of these 14 spheres gives a stella octangula , illustrated above. Consider the cube defined by 14 spheres in face-centered cubic packing.

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This "unit cell," one face of which is illustrated above in schematic form, contains eight -spheres one at each polygon vertex and six hemispheres. The total volume of spheres in the unit cell is therefore. The diagonal of a face of the unit cell is , so each side is of length. The volume of the unit cell is therefore. In face-centered cubic packing, each sphere is surrounded by 12 other spheres.

Taking a collection of 13 such spheres gives the cluster illustrated above. Connecting the centers of the external 12 spheres gives a cuboctahedron Steinhaus , pp. In cubic body-centered packing, each sphere is surrounded by eight other spheres. The figure above shoes the unit cell in body-centered cubic packing. In this configuration, a single full sphere occupies the center and is surrounded by eight -spheres.

The space diagonal of the unit cell is , so each side is of length. If spheres packed in a cubic lattice, face-centered cubic lattice, and hexagonal lattice are allowed to expand uniformly until running into each other, they form cubes, hexagonal prisms, and rhombic dodecahedra, respectively.

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## Engineering Physics by S. Mani Naidu

We already know from the crystal structure of gold that there are 4 atoms per unit cell. Home About Us Podcast! A simple cubic crystal built from 27 unit cells. These are shown in green on this drawing: A face-centered cubic unit cell. Now we are ready to think about nanoparticles! Teaching crystal structures using a transparent box with tennis balls.

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