Universidade Estadual Paulista – UNESPFaculdade de Engenharia de Ilha Solteira - FEISDepartamento de Engenharia Elétrica - DEE

Noções básicas de cristalografia e tensores

Ilha Solteira - 2018

Introdução à cristalografia

Insulin Crystal Gallium Crystal Bismuth Crystal

Quartzo Crystal Silicon Crystal Calcite Crystal

Lattice and CrystalCrystals are composed of three-dimensional patterns. An ideal crystal is a repetition of identical structural units in three dimensional space.

The periodicity is described by a mathematical lattice (which are mathematical points at specific coordinates in space).

The identical structural units (or base of the crystal) are the atoms in some specific arrangement which are unambiguously placed at every lattice point.

Note that a lattice is not a crystal, even so the two words are often used synonymously in colloquial language, especially in the case of elemental crystals where the base consists of one atom only.

Calcite structureCalcium atoms, and their coordinations are yellow, carbon atoms and their coordinations are dark grey.

Basic definitionsCRYSTAL: Translationally periodic arrangement of ATOMS in space.

LATTICE: Translationally periodic arrangement of POINTS in space.

BASIS: An atom or a group of atoms associated with each lattice point in crystal. By replacing each group of atoms (base) by a representative point a crystal lattice is obtained.

CRYSTAL=LATTICE+BASIS

• LATTICE of a crystal tells how to repeat (the periodicity of the crystal).

• BASIS tells what to repeat according to the lattice to get the crystal.

Keep in mind; lattice sites are not necessarily associated with the position of atoms. Thus, a crystal lattice is a set of infinite, arranged points related to each other by transitional symmetry.

A lattice is an ordered array of points describing the arrangement of particles that form a crystal.

In a crystal lattice there is the parallelepiped constructed from vectors which correspond to translational periods called unit cells. The unit cell of a crystal is defined by the lattice points.

The unit cell is the smallest part of a crystal that repeated regularly through translation in three dimensions creates the whole crystal.

Example: unit cell of a primitive cubic structure.

All of the particles (yellow) are the same; in this particular case, the lattice points defining the unit cell coincide with the centers of the crystal's particles.

These parallelepiped can be chosen in different ways. Commonly, unit cells are chosen so that its vertex (or center) coincides with one of the atoms of the crystal. Then lattice sites are occupied by atoms, and of the atoms of the crystal.

The unit cell contains at least one atom of each of the types that make up the crystal. Providing that the unit cell is made up of only one type of atom, it is called monatomic, anymore than that and it is polyatomic.

Thus, think of a crystal lattice site as containing a series of points arranged in a specific pattern with high symmetry.The lattice sites are occupied by atoms, and vectors that connect the nearest equivalent atoms.All possible lattices can be described by a set of three linearly independent vectorsa, b, and c, the unit vectors of the lattice. Each lattice point than can be reached by a translation vector T of the lattice given by:

T = (ua, vb, wc)

with u, v, w = integers.

Note that these points don't tell you the position of an atom in a crystal. They are simply points 'in space' oriented in such a way to build a lattice structure.

T

- Auguste Bravais (1850)

Cristal macroscópico

Cristal macroscópico

Unit cell structure of orthorhombic potassium niobate

Noções Gerais de Tensores

, C.m

, C.m/m3 = C/m2

- Este vetor tem a mesma unidade, C/m2, que o vetor deslocamento elétrico.

campo elétrico no interior do dielétrico

campo elétrico no vácuo

(sistema de coordenadas principal)

Regras de Simetria

FIM