Examples of directions

Miller indices are a notation used to describe lattice planes and directions in a crystal. Image File history File links Size of this preview: 800 × 171 pixel Image in higher resolution (957 × 205 pixel, file size: 4 KB, MIME type: image/png) Exemple dindices de Miller de directions Examples of Miller indices for directions Auteur/author : Christophe Dang Ngoc Chan (cdang) File links The... Image File history File links Size of this preview: 800 × 171 pixel Image in higher resolution (957 × 205 pixel, file size: 4 KB, MIME type: image/png) Exemple dindices de Miller de directions Examples of Miller indices for directions Auteur/author : Christophe Dang Ngoc Chan (cdang) File links The... Quartz crystal Copper(II) sulfate and iodine crystal Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid...

In particular, a family of lattice planes is determined by three integers , , and , the Miller indices. They are written and denote planes orthogonal to a direction in the basis of the reciprocal lattice vectors. By convention, negative integers are written with a bar, as in for − 3. The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1. The integers are commonly denoted by the above symbol. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...

The precise meaning of this notation depends upon a choice of lattice vectors for the crystal, as described below. Usually, the three primitive lattice vectors are used. However, for cubic crystal systems, the cubic lattice vectors are used even when they are not primitive (e.g., as in body-centered and face-centered crystals). In geometry, solid state physics and mineralogy, particularly in describing crystal structure, a primitive cell, is a minimum cell corresponding to a single lattice point of a structure with translational symmetry in 2D, 3D, or other dimensions. ... The cubic crystal system is a crystal system where the unit cell is in the shape of a cube. ...

There are also several related notations (Ashcroft & Mermin, 1976). , with square instead of round brackets, denotes a direction in the basis of the direct lattice vectors instead of the reciprocal lattice. The notation denotes all planes that are equivalent to by the symmetry of the crystal. Similarly, the notation denotes all directions that are equivalent to by symmetry.

Definition

There are two equivalent ways to define the meaning of the Miller indices (Ashcroft & Mermin, 1976): via a point in the reciprocal lattice, or as the inverse intercepts along the lattice vectors. Both definitions are given below. In either case, one needs to choose the three lattice vectors , , and as described above. Given these, the three primitive reciprocal lattice vectors are also determined (denoted , , and ). In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. ... In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. ...

Then, given the three Miller indices , denotes planes orthogonal to:

That is, simply indicates a normal to the planes in the basis of the primitive reciprocal lattice vectors. Because the coordinates are integers, this normal is itself always a reciprocal lattice vector. The requirement of lowest terms means that it is the shortest reciprocal lattice vector in the given direction. In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

Equivalently, denotes a plane that intercepts the three points , , and , or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane, in the basis of the lattice vectors. If one of the indices is zero, it means that the planes do not intersect that axis (the intercept is "at infinity").

The related notation denotes the direction:

That is, it uses the direct lattice basis instead of the reciprocal lattice. Note that is not generally normal to the planes, except in a cubic lattice as described below.

Definition of the index for a plane

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The crystallographic planes and directions

Dense crystallographic planes

The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. The crystallographic planes are fictitious planes linking nodes. Some directions and planes have a higher density of nodes; these dense planes have an influence on the behaviour of the crystal: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ... Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle still characterizing a chemical element. ... “Multivalent” redirects here. ... In science, a molecule is a group of atoms in a definite arrangement held together by chemical bonds. ... Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ...

• optical properties: in condensed matter, the light "jumps" from one atom to the other with the Rayleigh scattering; the velocity of light thus varies according to the directions, whether the atoms are close or far; this gives the birefringence
• adsorption and reactivity: the adsorption and the chemical reactions occur on atoms or molecules, these phenomena are thus sensitive to the density of nodes;
• surface tension: the condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species; the surface tension of an interface thus varies according to the density on the surface
  • the pores and crystallites tend to have straight grain boundaries following dense planes
  • cleavage
• dislocations (plastic deformation)
  • the dislocation core tends to spread on dense planes (the elastic perturbation is "diluted"); this reduces the friction (Peierls-Nabarro force), the sliding occurs more frequently on dense planes;
  • the perturbation carried by the dislocation (Burgers vector) is along a dense direction: the shift of one node in a dense direction is a lesser distortion;
  • the dislocation line tends to follow a dense direction, the dislocation line is often a straight line, a dislocation loop is often a polygon.

For all these reasons, it is important to determine the planes and thus to have a notation system. Table of Opticks, 1728 Cyclopaedia Optics ( appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye (visible light) or, in a technical or scientific context, electromagnetic radiation of any wavelength[1]. The elementary particle that defines light is the photon. ... Rayleigh scattering causing a reddened sky at sunset Rayleigh scattering (named after Lord Rayleigh (RAY-lee)) is the scattering of light, or other electromagnetic radiation, by particles much smaller than the wavelength of the light. ... Cherenkov effect in a swimming pool nuclear reactor. ... A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... Adsorption is a process that occurs when a gas or liquid or solute (called adsorbate) accumulates on the surface of a solid or more rarely a liquid (adsorbent), forming a molecular or atomic film (adsorbate). ... Reactivity refers to the rate at which a chemical substance tends to undergo a chemical reaction in time. ... In physics, surface tension is an effect within the surface layer of a liquid that causes that layer to behave as an elastic sheet. ... A pore, in general, is some form of opening, usually very small. ... A crystallite is a domain of solid-state matter that has the same structure as a single crystal. ... Cleavage, in mineralogy, is the tendency of crystalline materials to split along definite planes, creating smooth surfaces, of which there are several named types: Basal cleavage: cleavage parallel to the base of a crystal, or to the plane of the lateral axes. ... In materials science, a dislocation is a crystallographic defect, or irregularity, within a crystal structure. ... In physics and materials science, plasticity is a property of a material to undergo a non-reversible change of shape in response to an applied force. ... Friction is the force that opposes the relative motion or tendency toward such motion of two surfaces in contact. ... In materials science, a dislocation is a linear crystallographic defect, or irregularity, within a crystal structure. ... Look up polygon in Wiktionary, the free dictionary. ...

Case of the cubic structures

For the special case of cubic crystals, the lattice vectors are orthogonal and of equal length; similarly for the reciprocal lattice. So, in this common case, the Miller indices and both simply denote normals/directions in Cartesian coordinates. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

Case of the hexagonal and rhombohedral structures

Miller-Bravais index

With hexagonal and rhombohedral crystal systems, it is possible to use the Bravais-Miller index which has 4 numbers (h k i l) Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In crystallography, the hexagonal crystal system is one of the 7 lattice point groups. ... In crystallography, the rhombohedral (or trigonal) crystal system is one of the 7 lattice point groups. ...

i = -h-k

where h, k and l are identical to the Miller index.

The (100) plane has a 3-fold symmetry, it remains unchanged by a rotation of 1/3 (2π/3 rad, 120°). The [100], [010] and the directions are really similar. If S is the intercept of the plane with the axis, then

i = 1/S

i is redundant and not necessary.

See also

• Crystal structure
• Crystal

Enargite crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... Quartz crystal Copper(II) sulfate and iodine crystal Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid...

References

• Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: New York, 1976).

External links

• http://www.ece.byu.edu/cleanroom/EW_orientation.phtml - Miller index description with diagrams