But if you think about it, a cube can also be rotated around the axes that extend between opposite corners in this case, it takes three 120° rotations to go through a complete circle, so these axes (also four in number) are three-fold or C 3 axes. We say that the cube possesses three mutually perpendicular four-fold rotational axes, abbreviated C 4 axes. For example, you can rotate a cube 90° around an axis perpendicular to any of its six faces without making any apparent change to it. This is to look at what geometric transformations (such as rotations around an axis) we can perform that leave the appearance unchanged. We usually think of a cubic shape in terms of the equality of its edge lengths and the 90° angles between its sides, but there is another way of classifying shapes that chemists find very useful. In doing so, we can develop the major concepts that are useful for understanding more complicated structures (as if there are not enough complications in cubics alone!) But in addition, it happens that cubic crystals are very commonly encountered most metallic elements have cubic structures, and so does ordinary salt, sodium chloride. In order to keep this lesson within reasonable bounds, we are limiting it mostly to crystals belonging to the so-called cubic system. We could alternatively use regular hexagons as the unit cells, but the x+ y shifts would still be required, so the simpler rhombus is usually preferred.Īs you will see in the next sections, the empty spaces within these unit cells play an important role when we move from two- to three-dimensional lattices. Notice that to generate this structure from the unit cell, we need to shift the cell in both the x- and y- directions in order to leave empty spaces at the correct spots. Nevertheless, its unit cell is also a rhombus, although one that encompass two carbon atoms. The graphite form of carbon is based on an hexagonal lattice, but the directed bonds prevent it from being close-packed. As is shown more clearly here for a two-dimensional square-packed lattice, a single unit cell can claim "ownership" of only one-quarter of each molecule, and thus "contains" 4 × ¼ = 1 molecule. This means that an atom or molecule located on this point in a real crystal lattice is shared with its neighboring cells. Iin both of these lattices, the corners of the unit cells are centered on a lattice point. Notice that we use a rhombus (rather than a hexagon) to define the hexagonal lattice because it is simpler. Shown here are unit cells for the close-packed square and hexagonal lattices we discussed near the start of this lesson. Adjacent sheets are bound by weak dispersion forces, allowing the sheets to slip over one another and giving rise to the lubricating and flaking properties of graphite. The coordination number of 3 reflects the sp 2-hybridization of carbon in graphite, resulting in plane-trigonal bonding and thus the sheet structure. The result is just the basic hexagonal structure with some atoms missing. Each carbon atom within a sheet is bonded to three other carbon atons. The version of hexagonal packing shown at the right occurs in the form of carbon known as graphite which forms 2-dimensional sheets. This will, of course, be the hexagonal arrangement.ĭirected chemical bonds between atoms have a major effect on the packing. If the atoms are identical and are bound together mainly by dispersion forces which are completely non-directional, they will favor a structure in which as many atoms can be in direct contact as possible. If we go from the world of marbles to that of atoms, which kind of packing would the atoms of a given element prefer?
It can be shown from geometry that that square packing of spheres covers 78 percent of the area, while hexagonal packing yields 91 percent coverage. It should also be apparent that the latter scheme covers a smaller area, meaning that it contains less empty space and is therefore a more efficient packing arrangement. Any marble within the interior of the square-packed array is in contact with four other marbles, while this number rises to six in the hexagonal-packed arrangement. The essential difference between cubic- and hexagonal close packing is illustrated by the number of tiny blue "x" marks in the two-dimensional views shown here.
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