Crystallography

History

The Crystal, initially simple object of curiosity, impassioned the collectors before intriguing the scientists who, by studying his structure, outlined the first theories on the intimate constitution of the matter. The law of the rational indices or simple truncations was defined by the Abbé Haüy in 1774. By observation of the phenomenon of cleavage of calcite, it determined the “integral molecules”, i.e. the identical parallelepipeds constituting the crystals and following that, it was deduced that each face of a crystal can be located in space by integers.

Bases

The solid matter is made up of Atome S, which one can see as elementary balls which are assembled. They can be assembled several manners: some balls are assembled to form a Molécule, it is the case of gases, the liquids, the molecular solids, the polymers (rubbers, plastics, papers, proteins…), these materials comprise billion similar molecules. The balls are arranged in an irregular way, one then has matter known as “amorphous” (or “vitreous”), such as for example glass, or they pile up in an ordered way, it is then a Cristal.

Perfect crystal

The “perfect crystal” is a model used to represent the structure of the crystalline matter. She considers that a crystal is an ordered and infinite stacking of Atome S, Ion S or Molécule S.

The crystal is a solid with structure made up of atoms ordered in a periodic and even triperiodic and symmetrical network. It has properties of symmetry with direct axes and opposite, mirrors, plans and centres of symmetry.

A crystal can be Isotrope (even index of refraction of the light in all the directions) or Anisotrope (two different indices in two perpendicular directions).

The elementary mesh is smallest crystalline volume preserving all the physical properties, chemical and geometrical of the crystal. It is defined by three vectors which generate six cell parameters thus: the three lengths of the vectors has, B, C and three angles α, β, γ.

The crystal lattice

A network is a whole of points or “nodes” in three dimensions which presents the following property: when one relocates oneself in L `space according to certain vectors, one finds the same environment exactly. There is thus a space periodicity. That makes it possible to define seven reticular systems basic: cubic, hexagonal, rhomboedric, quadratic (or tétragonal), orthorhombic, monoclinical, triclinic.

The Réseau of Faced

French Auguste Bravais defines, in 1848, starting from the various combinations of the elements of crystalline symmetry, 32 classes of symmetry, which themselves are divided into 14 types of networks (there does not exist other way of having the points in space, in order to carry out a network or a mesh, so as to not leave any free volume between the networks). The 14 networks of Faced are expansions of the 7 primitive crystal shapes.

Here two examples of the networks of Faced primitive:

Triclinic:

there is $a \ neq B \ neq c$ and none the angles is equal to 90°.

Monoclinical:

The second network of Faced is the monoclinical network. This one is composed of 2 rectangular bases and 4 sides having the shape of parallelograms. The three lengths has, B and C are not equal: $a \ neq B \ neq c$ , but two of the three angles is with 90º.

One can find it in primitive network ( P ), or in network at centered base ( C ) (a node in the middle of the face defined by the axes has and B ).

See the detailed articles:

Réseau of Faced
crystalline Famille

Indices of Miller

Haüy defined indices ( P , Q , R ) which make it possible to locate in space the faces of a crystal. Miller, to simplify, said that one did not have to use P , Q and R but their opposite (1 P , 1 Q , 1 R ) which will be noted H , K , L . They must be whole, first between them and of simple values.

See the detailed article Indices of Miller.

Specific groups of symmetry and groups of space

The specific Groupe of symmetry of a crystalline system is the group (with the mathematical direction) gathering the whole of the operations of symmetry which leave a node of the network invariant. This node is thus located at the intersection of all the operations of symmetry, of which the translation does not form part. There exist 32 specific groups of symmetry distinct.

The Groupe of space of a crystalline system gathers the whole of the operations of symmetry of the specific group, to which the operations of translation are added. Towards 1890, Fedorov and Schoenflies showed - independently one of the other - the existence of 230 groups, which represent all the possible combinations of networks and operations of symmetry.

For more information, to see the articles:

Crystalline defects

Crystallogenesis

The Cristallogénèse is the formation of a crystal, either in natural environment, or in an experimental way.

Diffraction

Principle

max von Laue had the idea to irradiate the crystals with X-rays, because he thought that the crystal lattice would make deviate the radiation in the same way that the Lumière is deviated in certain transparent minerals. The experiment that colleagues carried on a crystal of Copper sulfate made it possible him out to make the demonstration of the periodic structure of stackings of Atome S in the crystals and of the undulatory nature of the X-radiation. The mineralogical determination is generally carried out by measuring the Diffraction electromagnetic Rayonnement of the X-rays, whose wavelengths, ranging between 0,01 and 10 Nm, are about the distances which separate the plane atomic from the crystal lattices. When the crystal studied is irradiated by a fine beam of x-rays, each atom of the crystal reflects a wave of low amplitude, which is propagated in all the directions. The waves resulting from the atoms interfere, revealing on the photographic film which receives them spots which correspond to the maximum of the waves in phase; the others, in opposition of phase, were cancelled.

Reciprocal network

On the level of a screen located at a distance from the secondary diffusion centers, one will observe figures of diffraction which make it possible to visualize the disturbances created by the interferences quoted previously. The reciprocal Réseau is the image which one obtains starting from the figure of diffraction.

Equipment used in crystallography

the Microscope polarizer analyzer
the Diffractometer

Isotropic and anisotropic materials

With regard to the liquids and gases one can consider that the medium is Isotrope. Exception all the same, the physical properties of the Liquid crystals are anisotropic. This complexity comes from the three-dimensional arrangement of the molecules which compose the liquid. Indeed, the things become complicated when one speaks about crystals. Indeed, only the cubic crystals (as well as amorphous materials) are isotropic, the crystals tétragonaux, rhomboedric, orthorombic, hexagonal, monoclinical and triclinic being Anisotropes.

Applications

One uses the properties of diffraction of the crystals in physics, chemistry, biology, biochemistry, medicine and in sciences of the ground.

Their analysis gives information on organic and inorganic crystalline substances (distance between atoms, fitting space of the atoms, identification of crystalline phases, cuts crystallites).

See too

the Constant of Madelung for the ionic crystals
Andre Guinier

Bonds

Structural Characterizations of the Materials
the crystals of snow
(in) Dictionary in line of crystallography

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