The imperfection in the solid due to the deviation from the periodicity of the crystal is called crystal defect.

1. Point defect/ Lattice vacancy: The defect in the crystal that arises from displacement of atoms from their normal positions is called point defect or lattice vacancy.

Crystal defects are imperfections in the regular structure of a crystal. These imperfections can arise due to various reasons, such as errors during crystal growth, thermal stresses, and radiation damage. There are several types of crystal defects, including point defects, line defects, and surface defects.

Crystal defects can have a significant impact on the properties of a crystal and are an important area of study in materials science and engineering.

Types of defect

a. Schottky defect: The defect in the crystal arising from the migration of atoms from their normal sites to the sites in the surface is called Schottky defect. This defect lowers the density of the crystal.

A Schottky defect is a type of point defect that occurs in ionic crystals. It involves the creation of a pair of vacancies, one for a cation and one for an anion, within the crystal lattice. The vacancies are located adjacent to one another and can move freely within the lattice. This type of defect is named after the German physicist Walter Schottky, who first described it in 1932.

Let Ev be the average energy required to create a single Schottky defect, N be the total number of atomic sites and n be the total number of Schottky defect at temperature T. Then according to thermodynamics, free energy of the crystal can be written as

F = U - TS

or, F = nEv - TS ...........1

where U = nEv is internal energy and S is entropy.

We have from Boltzmann entropy relation.

S = KB ln Ω ......2

Here, KB = Boltzmann constant

Ω = Probability of formation of n Schottky defects

Then Ω = N!/ (N-n)!n!

ln Ω = ln N! - ln (N-n)! - ln n!

Using stirling approximation: ln x! = x ln x - x

ln Ω = N ln N - N [ (N -n) ln (N -n) - N +n + n ln n -n ]

ln Ω = Nln N - (N-n) ln (N-n) - n ln n .....3

Substituting equation 2 and 3 in equation 1

F = n Ev - KBT [ N ln N - (N-n)ln (N-n) - n ln n] .........4

The graph of F as a function of 'n' is shown in figure. From the figure, it can be seen that at T> 0K, free energy of the crystal is minimum not at n=0 nut n=no. So, defect in the crystal is thermodynamically necessity.

At equilibrium,

N>>no

N - no ≈ N

This expression gives number of Schottky defects at absolute temperature. At T = 0 K, no =0 and no increases with increase in temperature. And this expression is varied only for mono atomic crystal. In case of ionic crystal an equal number of positive negative ion creates vacancy to make the crystal electrostatically neutral . Let Ep be the average energy required for a pair migration. Then free energy of the crystal is

This expression gives number of Schottky defects at absolute temperature Tk. At T = 0k, no = 0 and no increases with increase in temperature. And this expression is varied only for mono atomic crystal. In case of ionic crystal an equal number of positive negative ion creates vacancy to make the crystal electrostatically neutral. Let Ep be the average energy required for a pair migration. Then free energy of the crystal is

So ln Ω = 2 [ N ln N - (N -n) ln [ (N-n) - n ln n] .........7

Substituting equation 7 in equation 6 and using

Schottky defects in a crystal can provide important information about its thermodynamic properties and the conditions under which it was formed.

b. Frankel defect:

A Frankel defect is a type of point defect that occurs in ionic crystals. It involves the movement of an ion from its normal lattice position to an interstitial site, creating both a vacancy and an interstitial defect. The defect is named after the British metallurgist Edward Frankel, who first described it in 1949.

Frankel defects can be created when an ionic crystal is exposed to radiation, such as in a nuclear reactor. The radiation can cause ions to become displaced from their normal positions, leading to the formation of interstitial defects. At the same time, the displaced ions create vacancies in the lattice, which can be filled by nearby ions.

Frankel defects can have important implications for the behaviour of materials in radiation environments. In particular, they can lead to the build up of defects and the degradation of material properties over time. However, they can also have beneficial effects in some cases, such as in the creation of radiation-resistant materials.

Let us consider a crystal containing N atomic sites and N' interstitial temperature T. If Ev be the average energy required to create a energy of the crystal (F) = U -TS

F= nEv - TS .....1

where U = nEv is internal energy and S is entropy.

We have, Boltzmann entropy relation,

S = KB ln Ω .....2

where Ω is the probability of formation of n Frankel defects

ln Ω = ln N! + ln N' ! - ln (N-n)! - 2ln n!

Using Stirling approximation ln x! = x ln x - x

ln Ω = N ln N - N + N! ln N! - N! - [ (N-n) ln (N-n) - N + n + (N' -n) ln (N'-n) - N' +n + 2n ln n -2n]

or, ln Ω = N ln N + N'ln N' - (N-n)ln (N-n) - (N' -n) ln (N-n) - 2n ln n .......3

Substituting eq 3 and eq 2 in eq 1

F = n Ev - KBT [ N ln N + N'ln N' - (N-n) ln (N -n) - (N'-n) ln (N' -n) - 2n ln n] .......4

At equilibrium,

From this equation, it is seen that no = 0 at T = 0 K and no increases with increase in temperature.

Frankel defects are one of several types of point defects that can occur in ionic crystals, along with Schottky defects. Understanding the formation and behaviour of these defects is an important area of study in materials science and engineering.

2. Color defect: The defect in the crystal that arises due to the absorption of visible light is called color defect. In general a pure alkali halide crystal are transparent in the visible region of spectrum. The color defect may be due to presence of chemical impurities, introduction of excess metal ions to the crystal, bombardment of crystal with X ray, gamma ray, neutron, electron etc, electrolysis of crystal.

Color centre

a. F - centre

F center, also known as a color center, is a type of crystal defect in which an anion vacancy is created in the crystal lattice of an ionic solid, such as alkali halides. The vacancy creates a negatively charged site in the crystal lattice, which can attract free electrons.

When a free electron is trapped at the vacancy site, it creates an F center. The electron is held at the site by the electrostatic attraction between the negatively charged site and the positively charged ions surrounding it. The trapped electron absorbs light in the visible range, which gives the F center a characteristic color.

In alkali halide crystals, F centers typically appear as a yellow color, but they can also appear as blue or green, depending on the crystal structure and the type of alkali halide. F centers are often used in color center lasers and other optoelectronic devices.

b. FA centre

A color centre formed by replacing one of the nearest neighbours of F centre by different alkali atom is called as FA centre.

c. M -centre

When two F centers are located near each other in a crystal lattice, they can interact with each other to form an M center. The electrons in the two F centers can exchange positions, resulting in the formation of a new electronic state with a lower energy level than the individual F centers. This new state absorbs light in the visible range, giving the M center a characteristic color.

The color of the M center depends on the distance between the two F centers and the crystal structure. For example, in sodium chloride crystals, two adjacent F centers can form an M center that appears yellow, while in potassium chloride crystals, an M center formed by two adjacent F centers appears blue.

M centers are often used in color center lasers and other optoelectronic devices, as well as in studies of solid-state physics and crystallography.

d. R centre

When three F centers are located near each other in a crystal lattice, they can interact with each other to form an R center. The electrons in the F centers can exchange positions, resulting in the formation of a new electronic state with a lower energy level than the individual F centers. This new state absorbs light in the visible range, giving the R center a characteristic color.

The color of the R center depends on the distance between the F centers and the crystal structure. For example, in sodium chloride crystals, an R center formed by three adjacent F centers can appear red, while in potassium chloride crystals, an R center can appear yellow.

R centers have a variety of applications, including in optoelectronic devices, color displays, and as colorants in ceramics and glass.

3. Slip

In crystal, plastic flows results from sliding of one side of crystal relative to the other. This sliding process is called slip. The plane and the direction in which slip occurs are respectively called slip plane and slip direction.

4. Dislocation

The defect in the crystal that arises from missing of line segment of atoms is called dislocation. Dislocations are a type of crystal defect that can affect the color of a material. In particular, dislocations can scatter light and cause coloration due to the interference and diffraction of light waves as they pass through the crystal lattice.

Types of dislocation

a. Edge dislocation: The dislocation for which burger vector b is everywhere perpendicular to the dislocation line is called edge dislocation. If the dislocation occurs at the lower half of extra atomic plane and can be thought of as caused by inserting an extra plane of atoms as the upper half of the crystal, then the edge dislocation is positive edge dislocation otherwise negative dislocation.

b. Screw dislocation

Screw dislocations can be created in a crystal by a variety of processes, including plastic deformation, thermal cycling, or crystal growth. While cutting a crystal with a knife could potentially introduce dislocations into the material, the formation of screw dislocations is typically more complex than simply cutting the crystal.

Plastic deformation, for example, is a common mechanism for creating screw dislocations. When a crystal is subjected to stress or strain, the crystal lattice can deform and create dislocations. In the case of a screw dislocation, the deformation causes one plane of atoms to shift relative to the adjacent plane, which results in the spiral-shaped defect.

Thermal cycling, which involves heating and cooling the crystal, can also lead to the formation of screw dislocations. When the crystal is heated, the atoms in the lattice can become mobile and move around, which can create defects in the crystal structure. As the crystal cools, these defects can become locked in place, resulting in dislocations.

Energy formula for screw dislocation

Let us consider a cylindrical shell of material consisting axial screw dislocation as shown in figure. Let r be the radius and dr be the thickness of the shell. The circumference 2πr has been sheared by an amount b

Shear strain (e) = b/2πr ......1

We know, shear modulus (G) = shear stress (σ)/ shear strain (e)

=> σ = Ge = Gb/ 2πr ........2

Elastic energy density (dEs) = 1/2 stress x strain

Total elastic energy per unit length

Here R and ro are approximate upper and lower limit of variable rj, this ro is comparable to burger vector 'b' and R cannot exceed dimension of crystal.

This note is a part of the Physics Repository.

Explore

posts on Naya and its networks

Collections