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Closest-Packed Structures

Packing Balls Together

1. Suppose you are given a large number of tennis balls and asked to pack them together as efficiently as possible. How would you do this?

The packing behavior of atoms closely resembles the packing of tennis balls, golf balls, or even bowling balls. The optimal packing produces a closest-packed structure. At Civil War battlefields cannon balls are sometimes stacked beside the cannons in a closest-packed structure. This type of packing involves hard spheres; it is assumed that the atoms are perfect spheres that are rigid and cannot be distorted.

The packing strategy is illustrated on this page in a series of virtual reality images. Each image is interactive; you may rotate the image (left mouse button), shift its position (right mouse button), and zoom in and out (center mouse button). Use the controls to examine the image from all perspectives. Frequently the reverse view is particularly informative. In these images, all balls are identical irrespective of their color. The different colors are used to distinguish balls lying in different layers of the closest-packed structure. This page requires Java3D. If an applet on this page is not visible, consult the Java3D FAQ. Drag with the left mouse button to rotate, the center button to zoom, and the right button to move the object.

When two balls are to be packed together, the obvious best strategy is to simply place the balls side by side in contact with each other. Additional balls could be added to create a line of balls.

2. When creating a second row of balls, however, where is the best position for the next ball?

The optimal position to achieve the most efficient packing is shown in Image 1. The ball in the second row fits nicely into the crevice between the two balls in the first row. This process can be repeated to obtain an entire row of balls, as shown in Image 2. In a single layer of balls, each row is shifted relative to the previous row so it can fit into the crevices created by the first row (Image 3).

Image 1 Image 2 Image 3

Stacking Layers of Balls

3. Once the first layer of balls (shown in red) is complete, balls must be placed in a new layer (shown in green) that rests directly on top of the first layer. What is the best position for the new layer?

As shown in Image 4, each ball in the second layer lies in a triangular dimple created by three balls in the first layer. Look at Image 4 from the reverse angle (use the left mouse button to rotate the image). There is a hole in the center of the three red balls, and the green ball is placed directly over this hole.

Image 5 shows four balls from the first layer, and the two vertical lines indicate the positions of two holes in the layer. Careful observation reveals that these two holes are too close together to place a ball in each position.

In Image 6 a green ball has been placed in the dimple containing the white line. Once a ball is in this position, it is not possible to place a ball in the adjacent dimple marked by the yellow line. When the entire second layer is filled, half of the dimples created by first-layer balls (shown in red) are filled by second-layer balls (shown in green and lying on the white lines) and half are empty (sites occupied by yellow lines).

Image 4 Image 5 Image 6

Creating a Third Layer of Balls

A structure containing two layers of balls is shown in Image 7. Just as before, the second layer of balls (green) create a set of dimples and there are two different types of dimples. The presence of the first layer now allows the two different types of dimples to be distinguished.

In Image 7 the white lines run through holes (dimples) in the second layer (green) that coincide with holes (dimples) in the first layer (red). Notice that the white lines run cleanly through both layers.

In contrast, the yellow lines in Image 7 run through holes in the second layer (green) and directly through balls in the first layer (red).

Rotate Image 7 to view the reverse side. You can clearly see white lines pass through holes in the first layer and the yellow lines pass directly through balls in the first layer.

When adding atoms to form the third layer, one must choose between placing balls over holes in the first layer (white lines) or balls in the first layer (yellow line). Thus two different structures are possible.

Image 8 shows two balls placed over holes in the first layer. The positions of these new balls, shown in blue, do not correspond with the positions of any balls in the first or second layer and thus the balls are given a new color.

Image 9 shows two balls placed over balls in the first layer. The positions of these new balls are identical to those of balls in the first layer, except that the new balls are in a different layer. Because the positions of the new balls correspond with those of balls in the first layer, they are given the same color as the red layer.

Image 7 Image 8 Image 9

Closest-Packed Structures

When adding additional layers to the structure, the new layers always lie directly on top of previous layers.

In the cubic closest-packed structure (ccp), there are three different types of layers and the structure has an ABCABC pattern. Layer A corresponds with the red atoms in Images 8 and 10, layer B corresponds with the green atoms, and layer C corresponds with the blue atoms. Aluminum, copper, and gold are examples of metals that pack in a cubic closest-packed structure.

In the hexagonal closest-packed structure (hcp), there are two different types of layers and the structure has an ABABAB pattern. Layer A corresponds with the red atoms in Images 9 and 11 and layer B corresponds with the green atoms. Magnesium, cadium, and zinc are examples of metals that pack in a hexagonal closest-packed structure.

Notice that there are no channels running through the cubic closest-packed structure. Carefully examine Image 10 from all angles. It is not possible to see completely through the structure.

On the other hand, carefully examine Image 11 (the hexagonal closest-packed structure) from all angles. Notice that the white lines run directly through the structure. There are holes/dimples in each layer that perfectly align.

Image 10: Cubic Closest-Packed Structure (ccp) Image 11: Hexagonal Closest-Packed Structure (hcp)



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