Material Science & Engineering : Crystal Structure

The Space Lattices

• A space lattice provides the framework with reference to which a crystal structure can be described.
• A space lattice is defined as an infinite array of points in three dimensions in which every point has surroundings identical to that of every other point in the array.
• As an example, for ease of representation on paper, consider a two-dimensional square array of points shown in the figure below. By repeated translation of the two vectors a and b on the plane of paper.

 A two-dimensional square array of points gives a square lattice.

Two ways of choosing a unit cell are illustrated:

• The magnitudes of a and b are equal and can be taken to be unity. The angle between them is 90°; a and b are called the fundamental translation vectors that generate the square array.

   Basis

• The space lattice has been defined as an array of imaginary points which are so arranged in space that each point has identical surroundings.
• It should be noted that the crystal structure is always described in terms of atoms rather than points. Thus, in order to obtain a crystal structure, an atom or a group of atoms must be placed on each lattice point in a regular fashion. Such an atom or a group of atoms is called the basis, and this acts as a building unit or a structural unit for the complete crystal structure.

A space lattice is combined with a basis to generate a crystal structure.

Space lattice + basis → Crystal structure.

Generation of crystal structure from lattice and a basis

• Thus, whereas a lattice is a mathematical concept, the crystal structure is a physical concept.
• The generation of a crystal structure from a two-dimensional lattice is illustrated in Fig.2. The basis consists of two atoms, represented by o and ● having orientation as shown in the above figure. The crystal structure is obtained by placing the basis on each lattice point such that the centre of the basis coincides with the lattice point.
• It must be noted that the number of atoms in a basis may vary from one to several thousands, whereas the number of space lattices possible is only fourteen as described in a later section.

   UNIT CELL

• It is discussed earlier that the atomic order in crystalline solids indicates that the smallest groups of atoms form a repetitive pattern. Thus, in describing crystal structures, it is often convenient to subdivide the structure into repetitive small repeat entities called unit cells, i.e. in every crystal some fundamental grouping of particles is repeated.
• Obviously, a unit cell is the smallest component of the space lattice and thus, the unit cell is the basic structural unit or building block of the crystal structure by virtue of its geometry and atomic positions within.
• The space lattices of various substances differ in the size and shape of their unit cells. Figure below shows a unit cell of a three-dimensional crystal lattice.

 Lattice parameters of a unit cell

• The distance from one atom to another atom measured along one of the axes is called the space constant. The unit cell is formed by primitives or intercepts a, b and c along X, Y and Z axes respectively.
• A unit cell can be completely described by the three vectors

when the length of the vectors and the angles between them (α, β, γ) are specified.

• The three angles α, β and γ are called interfacial angles.
• Taking any lattice point as the origin, all other points on the lattice, can be obtained by a repeated of the lattice vectors . These lattice vectors and interfacial angles constitute the lattice parameter of a unit cell.
• Obviously, if the values of these intercepts and interfacial angles are known, one can easily determine the form and actual size of the unit cell.

2.4.   PRIMITIVE CELL 

• This may be defined as a geometrical shape which, when repeated indefinitely in 3-dimensions, will fill all space and is equivalent of one lattice point i.e. the unit cell that contains one lattice point only at the corners as shown in Fig.1:(A).
• In some cases, the unit cell may coincide with the primitive cell, but in general the primitive unit cell differs from the latter in that it is not restricted to being the equivalent of one lattice point. The unit cells, which contain more than one lattice point are called non-primitive cells.
• The unit cells may be primitive cells, but all the primitive cells need not to be unit cells.

CRYSTAL FAMILIES AND CRYSTAL SYSTEMS

• If all the atoms at the lattice points are identical, the lattice is said to be Bravais lattice.
• There are four systems and five possible Bravais lattices in two dimensions. The four crystal systems of two-dimensional space are oblique, rectangular, square and hexagonal.
• The rectangular crystal system has two Bravais lattices, namely, rectangular primitive and rectangular centred.

The effective number of lattice points in the unit cell of the three cubic space lattices:

CALCULATION OF NUMBER OF LATTICE POINTS, ATOMIC RADIUS AND ATOMIC PACKING FRACTION

The number of lattice points in unit cell can be calculated by appreciating the following:

(a). 1 Contribution of lattice point at the corner = of the point

(b). 1 Contribution of the lattice point at the face = of the point

(c). Contribution of the lattice point at the centre = 1 of the point

Every type of unit cell is characterized by the number of lattice points (not the atoms) in it.

For example: the number of lattice points per unit cell for simple cubic (SC), body centered cubic (BCC) and face centered cubic (FCC) lattices are 1, 2 and 4, respectively.

Atomic Packing Factor (APF):

• This is defined as the ratio of total volume of atoms in a unit cell to the total volume of the unit cell.

·              This is also called relative density of packing (RDP).

CO-ORDINATION NUMBER

• This is defined as the number of nearest atoms directly surrounding the given atom.
• The value of co-ordination number is 6 for simple cubic, 8 for BCC and 12 for FCC. In closely packed structures this number is 12.
• The coordination number of carbons is 4, i.e. number of nearest neighbours of carbon atom is 4. This low coordination results in a relatively inefficient packing of the carbon atoms in the crystal.

DEFECTS OR IMPERFECTIONS IN CRYSTALS

• Up to now, we have described perfectly regular crystal structures, called ideal crystals, and obtained by combining a basis with an infinite space lattice.
• In ideal crystals atoms were arranged in a regular way. However, the structure of real crystals differs from that of ideal ones. Real crystals always have certain defects or imperfections, and therefore, the arrangement of atoms in the volume of a crystal is far from being perfectly regular.
• Natural crystals always contain defects, often in abundance, due to the uncontrolled conditions under which they were formed. The presence of defects which affect the colour can make these crystals valuable as gems, as in ruby (chromium replacing a small fraction of the aluminium in aluminium oxide: Al₂O₃). Crystal prepared in laboratory will also always contain defects, although considerable control may be exercised over their type, concentration and distribution.
• The importance of defects depends upon the material, type of defect, and properties which are being considered. Some properties, such as density and elastic constants, are proportional to the concentration of defects, and so a small defect concentration will have a very small effect on these.
• Other properties, e.g. the colour of an insulating crystal or the conductivity of a semiconductor crystal, may be much more sensitive to the presence of small number of defects.

Classification of Crystal defects:

Crystalline defects can be classified on the basis of their geometry as follows:

(i). Point imperfections

(ii). Line imperfections

(iii). Surface and grain boundary imperfections

(iv). Volume imperfections

The dimensions of a point defect are close to those of an interatomic space. With linear defects, their length is several orders of magnitude greater than the width. Surface defects have a small depth, while their width and length may be several orders larger. Volume defects (pores and cracks) may have substantial dimensions in all measurements, i.e. at least a few tens of Å.

Point imperfections:

• The point imperfections, which are lattice errors at isolated lattice points, take place due to imperfect packing of atoms during crystallization.
• The point imperfections also take place due to vibrations of atoms at high temperatures.

Vacancies:

• The simplest point defect is a vacancy.
• This refers to an empty (unoccupied) site of a crystal lattice i.e. a missing atom or vacant atomic site. Such defects may arise either from imperfect packing during original crystallization or from thermal vibrations of the atoms at higher temperature.

Interstitial Imperfections:

• In a closed packed structure of atoms in a crystal if the atomic packing factor is low, an extra atom may be lodged within the crystal structure. This is known as interstitial position i.e. voids.

Frenkel defect:

• Whenever a missing atom, which is responsible for vacancy occupies an interstitial site (responsible for interstitial defect) , the defect caused is known as Frenkel defect.
• Obviously, Frenkel defect is a combination of vacancy and interstitial defects. These defects are less in number because energy is required to force an ion into new position.

Schottky Defect:

• These imperfections are similar to vacancies.
• This defect is caused, whenever a pair of positive and negative ions is missing from a crystal 

·              This type of imperfection maintains a charge neutrality

Substitutional Defect:

• Whenever a foreign atom replaces the parent atom of the lattice and thus occupies the position of parent atom. The defect caused is called substitutional defect.
• In this type of defect, the atom which replaces the parent atom may be of same size or slightly smaller or greater than that of parent atom.

Line defects or dislocations:

• Line imperfections are called dislocations.
• A linear disturbance, i.e. one-dimensional imperfections in the geometrical sense of the atomic arrangement, which can very easily occur on the slip plane through the crystal, is known as dislocation.

Edge Dislocation:

• This type of dislocation is formed by adding an extra partial plane of atoms to the crystal. An edge dislocation in its cross-section is essentially the edge of an ‘extra’ half-plane in the crystal lattice. The lattice around dislocation is elastically distorted.

Surface and grain boundary defects

• Surface and gain boundary imperfections of a structural nature arise from a change in the stacking of atomic planes on or across a boundary and are two-dimensional. The change may be one of the orientations or of the stacking sequence of the planes.

Grain Boundaries

• Engineering materials may be either polycrystalline or single crystal type.
• A polycrystalline alloy contains an enormous quantity of fine grains Grain boundary imperfections are those surface imperfections which separate crystals or grains of different orientation in a polycrystalline aggregation during nucleation or crystallization. The shape of a grain is usually influenced by the presence of surrounding grains.

Twin Boundaries

• This is another planar surface imperfection.
• The atomic arrangement on one side of a twin boundary is a mirror reflection of the arrangement on the other side.
• Twinning may result during crystal growth or deformation of materials. Twin boundaries occur in pairs, such that the orientation change introduced by one boundary is restored by the other. The region between the pair of boundaries is termed as the twinned region.

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