Miller index

Miller indices are used in crystallography the unique designation of crystal faces or planes in the crystal lattice. The notation ( hkl) was proposed in 1839 by William Hallowes Miller ( 1801-1880 ). In the same paper Miller also introduced the notations in use today [ uvw ] for directions ( direction indicators) and { hkl } for crystal forms, ie the set of all symmetrically equivalent areas, a.

  • 3.1 lattice plane
  • 3.2 lattice vector


In mineralogy, they are used for example to describe crystal faces clearly. Also to specify the cleavage or Verzwillingungen they are needed.

In diffraction methods, such as X-ray diffraction or electron diffraction, they describe a network -level crowd. Here are also higher indices - for example, 222 - used to specify the higher order diffraction. These indices are referred to as Laue indices, Laue symbol or (in the English-speaking ) as a Bragg indices. They are usually written to distinguish it from the prime by definition between miller indices without parentheses. Laue indices are used for example in the indication of systematic absences and go into the formula of the structure factor.

In materials science, both lattice planes and lattice vectors are needed to characterize lattice defects such as dislocations. Also slip systems, textures, or the crystal orientation of single crystals can be described by Miller indices.


Lattice plane ( Miller indices )

Three integer indexes, and the number of triplet form, these are the Miller indices. Negative indices are marked with a letter written on the number beams, ie, for example. This triplet designates a specific level.

If instead of a specific power level, all symmetrically equivalent levels meant, so the notation will be used. For example, designated in the cubic crystal system, the equivalent due to the cubic planes of symmetry, and which corresponds to the six surfaces of a cube.

Each network -level band in the direct lattice corresponds to a point or position vector in the reciprocal lattice of the crystal. This vector has the coordinates in reciprocal space; it is always perpendicular to the same power level and has a length of the reciprocal value of the lattice plane spacing.

Lattice vectors ( direction indicators)

Also vectors within the grid can be designated by indices. The notation is used to refer to a specific vector. The notation refers to the vector of all symmetrically equivalent directions.

Examples: In the case of a cubic crystal (that is, a cube ) is a direction parallel to a cube edge, the direction of a diagonal of the surface and the direction of a spatial diagonal.


Depending on its crystal system is assigned to each crystal a coordinate system. The three vectors, and may form the basis of this grid coordinate system (not to be confused with the primitive translations of the grid ). The base of the corresponding reciprocal lattice is by the vectors, and, where.

Lattice plane

Then there are two equivalent ways to define a lattice plane.

Looking at a level with the tracking points, and so the intercept form is given by:

Here, a normal vector of the plane. Now we form a multiple of this normal vector, so that all entries of this multiple of the normal vector are all the prime numbers. This is for example in the following by the integer ensured ( possible because, as the edit points are located on the crystal lattice ), then

The tuple (hkl ) is now called the Miller indices. Negative numbers are identified by a bar over the corresponding index instead of the minus sign. An index of zero denotes an intersection point at infinity ( as seen from the shaft portion provides for the form ), i.e., the corresponding base vector is parallel to the plane.

The other possibility is, the reciprocal lattice vector

To call. This vector is perpendicular to the corresponding lattice planes.

Here, those integers, and used, which have no common factor greater. This corresponds to the shortest reciprocal lattice vector that is perpendicular to the plane.

Lattice vector

According to the notation describes a vector in the real lattice (lattice vector)

This vector is not generally perpendicular to the plane. This is the case only in the cubic lattice.

Quad notation

In the trigonal crystal system and in the hexagonal crystal system, the notation is with four indices, commonly used. This modified between miller indices are called Bravais indices ( also Bravais - Miller indices or Miller - Bravais indices). The indices, and are consistent with the usual Miller indices, always arises as.

Also for the direction indices, there is a four - notation. In crystallography and mineralogy usually the normal direction indices [ uv.w ] or [ uv * w] be used where indicated by a placeholder for that the trigonal or hexagonal crystal system is meant. t is always zero. In materials science, a different notation is preferred, the so-called Weber- Weber indices or symbols (English ). The conversion of the three - notation here is different from the conversion of the planes indices:

The advantage of this notation is that the vector similar to the cubic crystal system is perpendicular to the plane. In the triple notation, this is in these crystal systems not generally the case.