Crystal Face

Crystal faces are not accidents of growth - they are expressions of the crystal lattice beneath them. Every face on a well-formed crystal occupies a rational orientation relative to the internal lattice, meaning it is parallel to a plane that passes through lattice nodes in a geometrically orderly way. ^[1] The most common faces tend to run parallel to the surfaces of the unit cell itself - the principal planes of the lattice - so that the external shape of the crystal directly mirrors the geometry of its internal building block. ^[1] This is why minerals with a cubic unit cell so often produce cubic crystals, and minerals built on a hexagonal unit cell so often present a hexagonal cross section in hand specimens - the outer shape is, in a real sense, the unit cell made large. ^[1] Faces can also develop along simple diagonals through the lattice, though these are generally less prominent than faces running parallel to the cell surfaces. ^[1]

The Two Laws of Crystal Faces

The tendency of faces to occupy preferred orientations was formalised into two complementary laws that together explain both which orientations can occur and which are most likely to be prominent.

The first is the Law of Haüy: crystal faces make simple rational intercepts on the crystal axes. ^[1] A rational intercept means the face crosses a given crystal axis at a distance that is a simple integer multiple of the unit cell dimension along that axis - not at some arbitrary, irrational distance. This law is why Miller indices, which express those intercepts in inverted form, always turn out to be small integers rather than arbitrary decimals. The law excludes certain orientations entirely: a face whose intercept ratios could not be expressed as simple integers simply does not form. This constraint dramatically limits the range of crystal faces that any mineral can develop.

The second law, the Law of Bravais, adds a probabilistic dimension: common crystal faces are parallel to lattice planes that have high lattice-node density. ^[1] Lattice-node density measures how many lattice nodes - the points where atoms or atom groups sit - are packed into a given area of a plane through the lattice. A plane that passes through many nodes per unit area is densely populated. A plane that passes through few nodes has low density. The Bravais law says that high-density planes are not just allowed faces - they are the faces you will actually see prominently developed on a real crystal. Low-density planes, even when they satisfy the Law of Haüy, tend to be minor or absent.

High and Low Node Density in Practice

The principal planes of the lattice - those running parallel to the faces of the unit cell - always have the highest lattice-node density, because the unit cell is defined precisely by the most densely populated planes of the lattice. These planes satisfy both laws simultaneously: they make the simplest possible intercepts (one unit cell dimension along two axes, infinity along the third), and they concentrate the most nodes per unit area. In a primitive monoclinic lattice, three such planes are all expected to appear as crystal faces, and they typically do. ^[1]

A face that cuts through the lattice on a simple diagonal - intersecting two axes at equal distances, so that a : c = 1 : 1 - still makes rational intercepts and carries a moderately high node density. Such a face is a reasonable candidate and typically does appear on crystals, though it is less prominent than the principal faces. ^[1] By contrast, a face cutting at a steeper ratio - say, two unit cells along a for every one along c (a : c = 2 : 1) - still has rational intercepts and therefore satisfies the Law of Haüy, but it threads through fewer lattice nodes per unit area. ^[1] The Law of Bravais predicts - correctly, as observed in real crystals - that this face is either minor or absent.

Taken together, the two laws explain the practical observation that crystals tend to be dominated by a small number of prominent faces, all with small Miller index integers, while faces with larger indices are either minor or entirely absent. The Laws of Haüy and Bravais are thus why Miller indices work: they are small because only simple-intercept, high-density planes grow into the faces that define the shape of the crystal.

References

1. Nesse, W. D. (2018). Introduction to Mineralogy, 3rd ed. Oxford University Press.

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