Miller Indices

Miller indices are sets of three integers, conventionally written $(hkl)$, used in crystallography to describe the orientation of parallel planes in a crystal lattice. They are defined relative to the axes of a chosen unit cell, rather than an external coordinate system. The same integers also label reciprocal-space positions and diffraction reflections, although reflection indices need not be reduced to their smallest integer ratio. (dictionary.iucr.org)

Geometrical definition

Choose a crystallographic basis consisting of vectors $\mathbf a,\mathbf b,\mathbf c$. Express a plane’s intercepts with these axes as multiples of the corresponding cell vectors. Its Miller indices are proportional to the reciprocals of those multiples. Thus, intercepts $2a,3b,c$ give reciprocal values $1/2,1/3,1$, which become $(326)$ after multiplication by six. For an orientation indexed using a primitive lattice basis, the integers are normally reduced so that they have no common factor. Centred conventional cells require additional care: valid lattice-plane indices need not be relatively prime. (dictionary.iucr.org)

An axis parallel to the plane has an infinite intercept and therefore contributes a zero index. A negative intercept produces a negative index, usually indicated by an overbar: $(\bar{1}10)$ means $(-1,1,0)$. If the plane passes through the origin, a parallel plane can be used to determine its orientation without taking reciprocals of zero intercepts. The indices specify orientation, not the absolute position of one particular plane. (doitpoms.ac.uk)

For example, $(100)$ describes planes parallel to the $\mathbf b$ and $\mathbf c$ axes; a representative $(110)$ plane intercepts the first two axes equally in cell units and is parallel to the third. A representative $(111)$ plane intercepts all three axes at one cell length. These constructions remain meaningful when the axes are unequal or oblique. (iucr.org)

Planes, directions, and symmetry

Bracket type distinguishes several related objects:

• $(hkl)$: a plane orientation or set of parallel planes.
• $\{hkl\}$: a family of planes related by crystal symmetry.
• $[uvw]$: a crystallographic direction.
• $\langle uvw\rangle$: a family of symmetry-equivalent directions.

For a cubic crystal, $\{100\}$ includes planes with normals along each positive and negative cell axis. Equivalence depends on the actual crystal symmetry; arbitrary permutations of indices do not always describe equivalent planes. (doitpoms.ac.uk)

Direction indices specify a direct-space vector proportional to $u\mathbf a+v\mathbf b+w\mathbf c$. They should not be confused with plane indices. In cubic axes, $[hkl]$ is perpendicular to $(hkl)$, but this is generally false for noncubic cells. A direction $[uvw]$ lies parallel to a plane $(hkl)$ precisely when the Weiss zone law holds:

This condition applies to every crystal system, including those with oblique axes. (doitpoms.ac.uk)

Reciprocal-space interpretation

The reciprocal lattice provides a coordinate-independent explanation of plane indices. In the crystallographic convention, its basis vectors satisfy

with corresponding relations for $\mathbf b^*$ and $\mathbf c^*$. The dots denote the Euclidean inner product. The vector

is normal to the indexed planes. For fractional coordinates $x,y,z$, their equation is

Changing $C$ selects parallel planes without changing their orientation. (dictionary.iucr.org)

The indexed spacing is

where the denominator is the vector’s norm. For a cubic cell of edge length $a$, this becomes

For an orthorhombic cell,

Oblique cells require cross terms determined by the reciprocal metric tensor. These formulas use reciprocal vectors without a $2\pi$ factor; with the alternative physics convention, the spacing is $2\pi/\|\mathbf G_{hkl}\|$. (materials.duke.edu)

Hexagonal indexing

For hexagonal axes, Miller–Bravais indices use four integers, $(hkil)$. Three equivalent basal axes are separated by $120^\circ$, while the fourth axis is perpendicular to the basal plane. Their redundancy imposes

A plane written $(hkl)$ using two basal axes and the perpendicular axis therefore becomes $(hk\bar{h+k}l)$ in four-index notation. The basal plane is $(0001)$, and a common prismatic plane is $(10\bar{1}0)$. Four-index notation makes basal-plane symmetry more explicit. Direction indices have a separate conversion rule and cannot be obtained simply by inserting $i=-(h+k)$. (dictionary.iucr.org)

Diffraction and reflection indices

In X-ray crystallography, Miller indices identify diffraction reflections through reciprocal-lattice coordinates. Bragg’s law, $2d\sin\theta=n\lambda$, relates plane spacing to wavelength and diffraction angle. A higher-order reflection from one orientation can instead be indexed using multiplied integers: the second order associated with $(100)$ is labelled $200$, with indexed spacing $d_{200}=d_{100}/2$. Reducing reflection indices would therefore discard essential information. (iucr.org)

Indices describe reflection geometry, not whether a reflection has measurable intensity. Contributions from the atoms in the cell combine in the structure factor, and destructive interference can produce systematic absences. For a face-centred lattice, the centring condition permits reflections only when $h,k,l$ are all even or all odd; additional structural conditions can remove further reflections. Assigning indices to measured peaks or spots—indexing—is a central step in interpreting both single-crystal patterns and powder diffraction. (circle-test.iucr.org)