Miller Index

A Miller index describes the orientation of a crystal face or crystallographic plane by expressing how it intersects the unit cell. The index takes the form (hkl), where h, k, and l are integers tied to the three crystal axes a, b, and c, and the value of each integer is inversely proportional to where the face intercepts the corresponding unit cell edge. ^[1] The inverse relationship is the key to reading what an index means: a face that intercepts a given axis far from the unit cell origin produces a small index for that axis, while a face that intercepts close in produces a large one. A face running perfectly parallel to an axis never meets it - its intercept is infinity - so its index for that axis is 1/∞ = 0.

The Inverse-Intercept Rule

To find a Miller index, the unit cell intercepts of the face along each crystal axis are measured (wa along a, wb along b, wc along c) and then inverted to produce the index integers. ^[1] Consider a face with wa = 1, wb = 1, and wc = ½: the unit cell intercepts along a and b are one full cell dimension each, and the intercept along c is half a cell dimension. Inverting those three values gives 1/1 = 1, 1/1 = 1, and 1/(½) = 2, so the Miller index is (112), read “one one two.” ^[1] The fractional intercept along c becomes the largest integer precisely because of the inversion - the closer a face cuts to the origin, the bigger its index for that axis.

Miller index integers must always be co-prime - meaning they share no common factor greater than one. ^[1] If the inversion procedure produces (224), those integers are divided by their common factor of 2 to give (112). This requirement ensures that each distinct face orientation has exactly one Miller index, rather than a whole family of equivalent multiples.

Negative intercepts occur whenever a face cuts the negative end of a crystal axis. Such indices are written with a bar above the digit - so negative one is written 1̄ and read “bar one” or “negative one.” ^[1] A face with intercepts wa = 1, wb = -1, wc = ½ produces the index (11̄2̄), read “one negative-one two.” ^[1]

Notation: Parentheses, Brackets, and Braces

The type of bracket signals which geometric feature is being described. ^[1] Parentheses (hkl) identify a specific face or crystallographic plane. Square brackets [uvw] identify a crystallographic direction - a line through the lattice. Braces {hkl} identify a crystal form, meaning the complete set of equivalent faces generated by the mineral’s symmetry. The same numerals in different brackets mean completely different things: (110) is a single plane, while {110} is every plane equivalent to (110) under the mineral’s symmetry, which may be four, eight, twelve, or more planes depending on the crystal class.

Calculating a Miller Index from Measurements

When working with a real crystal, the Miller index is found by dividing the axial ratio of the mineral by the measured axial intercepts of the face, then scaling to the smallest whole numbers. ^[1] The axial ratio is calculated by dividing each unit cell edge by the b dimension, so it expresses all three edges as proportions of b. For augite, with unit cell dimensions a = 9.73 Å, b = 8.91 Å, and c = 5.25 Å, the axial ratio is 1.09 : 1 : 0.59. ^[1]

For a face on that same augite crystal whose measured axial intercepts are ia = 3.82 cm, ib = 3.5 cm, and ic = 2.07 cm, dividing each axial ratio value by the corresponding intercept gives 1.09/3.82 : 1/3.50 : 0.59/2.07 = 0.29 : 0.29 : 0.29. ^[1] Dividing through by 0.29 scales these to the smallest integers, yielding the Miller index (111). ^[1] The equal proportions here are coincidental to this particular face - in general the three ratios will differ, and the scaling step converts whatever ratios arise into co-prime integers.

Assignment by Inspection

Because crystal faces prefer simple rational orientations through the crystal lattice, prominent faces have small Miller index integers. ^[1] This makes it practical to assign Miller indices to the major faces of a well-formed crystal by inspection rather than by calculation. Faces cutting only one axis get indices like (100), (010), or (001) - the zeros showing the face runs parallel to the other two axes. Faces cutting two axes commonly get (110) or (101), and faces cutting all three commonly get (111).

The (111) face is called the unit face because it meets all three crystal axes at exactly one unit cell dimension each. ^[1] Since Miller indices are inverses of intercept distances, a face with a larger index for a given axis cuts that axis at a shorter intercept distance than (111) does. A (211) face, for example, cuts the a axis at half the distance of the unit face.

Assignment by inspection is not always reliable. ^[1] Some minerals develop unexpectedly prominent faces with higher-integer indices due to their growth chemistry. Additionally, different publications may assign different crystal axis orientations to the same mineral, leading to different Miller indices for the same physical face - a caution worth keeping in mind when comparing sources.

Miller-Bravais Indices in the Hexagonal System

The hexagonal crystal system is conventionally described with four crystal axes rather than three: three horizontal axes (a₁, a₂, a₃) placed at 120° to each other and parallel to the 2-fold rotation axes of the crystal, plus the vertical c axis. ^[1] Because this convention uses four axes, the standard three-integer Miller index is expanded to four integers written (hkil), where h, k, i, and l refer to a₁, a₂, a₃, and c respectively. These four-integer indices are called Miller-Bravais indices. ^[1]

The four integers are not independent. Because the three a axes are at 120° to each other, the third index i is always determined by the first two: h + k + i = 0 for every valid Miller-Bravais index in the hexagonal system. ^[1] The mathematical consequence is that either all three a-axis indices are zero, or at least one must be negative and one positive - the three can never all be positive or all negative at the same time. In practice this rule acts as a self-check: if h + k + i ≠ 0, the index has been calculated incorrectly.

To convert a Miller-Bravais index to a standard Miller index, simply delete the third digit (i). ^[1] For example, (101̄0) becomes (100) by dropping the i digit. ^[1] The reverse - deriving a Miller-Bravais index from a Miller index - requires using the interaxial relationship h + k + i = 0 to calculate i from h and k.

Crystallographic Planes and Directions

Internal planes - most importantly cleavage surfaces - are assigned Miller indices by exactly the same method as external crystal faces, referenced to the same unit cell. ^[1] A cleavage plane that runs parallel to the (001) crystal face gets the index (001), because its geometrical relationship to the unit cell is identical to that of the face. ^[1] The three principal crystallographic planes - parallel to the faces of the unit cell itself - are therefore (100), (010), and (001), each sharing its index with the corresponding crystal face.

Crystallographic directions use the same integers as Miller indices but placed in square brackets. ^[1] The three crystal axes are the directions [100], [010], and [001]. ^[1] A direction [111] runs through lattice nodes at coordinates 1,1,1 and 2,2,2 - diagonally through the body of the unit cell. This is conceptually distinct from the face (111): the direction [111] is a line through the crystal, while the face (111) is a plane that cuts all three axes at one unit cell dimension. Identical numbers in different brackets describe different geometrical objects.

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