A perfect (ideal) crystal is a perfectly ordered arrangement of atoms at the points of infinite space lattice. Any departure from this idealized arrangement makes the crystal imperfect. A perfect single crystal is rarely available.
Crystalline materials have real crystals. They are of finite dimensions. They have broken bonds at the boundaries where bonding forces remain unbalanced. Thus the boundary of a crystal is a defect in itself.
These imperfections lead to several deficiencies in the solids. Solids are generally imperfect. Polycrystalline solids are, inevitably, imperfect. We shall deal with all such imperfections (defects) of crystalline solids in this article. They decrease the mechanical strength of materials. A material does not attain its theoretical strength due to presence of imperfections.
Imperfections affect structure-sensitive properties of crystals. Their presence is advantageous in certain applications. For example, parts per million (ppm) doping of phosphorus in silicon changes the behavior of intrinsic semiconductor and makes it suitable for various applications.
Types of Imperfections in Crystalline Solids
Imperfections in crystalline solids are classified as follows:
- Point imperfections (zero-dimensional defects).
- Line imperfections (1-dimensional defects) Surface or planer imperfections (2-dimensional defects).
- Volume imperfections (3-dimensional defects).
Point imperfections, line and surface imperfections may occur together in crystals. These defects are not visible to the naked eye. They can be visualized by using X-rays diffraction techniques and microscopes. Imperfections are also classified on the basis of their dimensions as under:
- Nano-level (10-9 m) imperfections.
- Angstrom level (10 -10 m) imperfections.
- Micro-level (10-6 m) imperfections.
We shall now deal with each type of imperfection one by one.
As the name suggests, they are imperfect point-like regions in the crystal. These defects are of one or two atomic diameters only. Hence these are known as zero-dimensional defects. Various types of point imperfections are:
Vacancy refers to a vacant atomic site in a crystal. At these sites the atoms are missing. One or more atoms may remain absent from their respective locations. The missing of atoms is random and not according to any rule. The atoms A and B are missing from a FCC unit cell as shown in Figure 1(a) below.
It is not necessary that atoms from A and B sites only will abstain when the unit cell repeats to form a crystal. Figure 1(b) shows a simple cubic (SC) crystal. Sites marked V are the vacant atomic sites. In above figures, the atoms are shown separated for clarity.
This defect refers to a foreign atom that substitutes a parent atom at its site in the crystal. Atoms marked A in Figures 2(a) and (b) are the foreign atoms.
The substituting foreign atoms are called solute and the substituted (or dislodged) parent atoms are known as the solvent. Solute and solvent of comparable sizes mix randomly to form an alloy.
For example copper and zinc mix together to form alpha-brass. Boron or antimony doped in germanium is another example of substituted impurities in a crystal.
When a small sized foreign atom occupies a void space in the parent crystal (or its unit cell), the defect is known as interstitial impurity. Atoms marked A in Figures 3(a) and (b) are the interstitial atoms.
Dislodging of parent atoms from their sites does not occur in this case. However, they can squeeze due to forced entry of a foreign atom or even the parent atom. An atom can enter into the interstitial void when it is quite smaller in size than the parent atom.
The largest atom that can fit into tetrahedral and octahedral voids has radius of 0.225 r and 0.414 r respectively. Carbon, an interstitial solute in FCC iron (between 910°C and 1410°C), is an example of this kind.
An ion displaced from a regular location to an interstitial location, in an ionic solid is called Frenkel’s defect it shown in Figure 4(b).
The ions of two different kinds are known as cations and anions. Cations are the smaller ions while anions are the larger ones. Cations may easily get displaced into the void. Anions, on account of their larger size, do not displace in small sized voids.
The presence of this defect does not change the overall electrical neutrality of the crystal. Imperfections in CaF2 and silver halides are the examples of this kind. The number of Frenkel defects per cubic metre may be calculated from the formula given below:
nF = √(NL Ni)e(-Ef/kT) ……….(equation 1)
Where, NL = number of lattice sites occupied by ions per m3,
Ni = number of interstitial sites,
Ef = energy of formation of one Frenkel defect,
k = Boltzmann’s constant,
T = absolute temperature.
The number of lattice sites is found from the formula given below:
NL = NAρ/Mw
Where, NA is Avogadro’s number, p is density, and Mw is molecular weight. As the number of tetrahedral voids is double of the number of anions in ionic solids, therefore
Ni = 2NL
The number of anion vacancies is twice the number of cation vacancies for such equilibrium defects, hence effective value of Ef in Equation 1 is taken as half of it (i.e. Ef/2).
When a pair of one cation and one anion are absent from an ionic crystal, the defect is called Schottky’s defect. The valancy of missing pair of ions maintains electrical neutrality in the crystal. Such imperfections are dominant in alkali halides such as LiC1, LiBr, LiI etc.
Effects of Point Imperfections
Presence of point imperfections induces distortions in their surroundings. Consequently, various point imperfections have the following effects on crystals and their properties:
- When the imperfection is a vacancy, the bonds with its neighboring atoms do not exist.
- In case of substitutional impurity, elastic strains develop in the surrounding region due to size difference of parent and foreign atoms. A larger foreign atom induces compressive stress and strain while a smaller atom produces stress and strain field of tensile nature.
- An interstitial atom creates strains around its surrounding.
- Point imperfections of different types interact with each other and in doing so lower the total energy. Consequently, the stability of crystals is affected.
- These are thermodynamically stable.
Origin of Point Imperfections
From the above discussion, we know that vacancies and impurities exist in a crystal at thermal equilibrium. Question arises as to from where these point imperfections originate? Following are the possible sources that create point defects:
- Thermal fluctuations
- Thermal shock
- Mechanical deformations
- High energy particles bombardment
Materials in engineering applications are invariably exposed to temperature variation. Such temperature variation is often a functional need.
Thermal shock is given to the metals during hardening process by sudden or rapid quenching. At some stage of the process, the point defects crop-in. It is worth mentioning to recall the importance of Arrhenius equation and rate of reaction in this regard.
During fabrication and manufacturing, the materials are subjected to mechanical deformations of different kinds. In the processes like casting, forging, rolling and extrusion etc., the defects are introduced.
Bombardment of metals by high energy particles is a common feature in nuclear and X-ray industries. It may be the beam of particles from cyclotron, neutron bombardment on nuclear materials, or impingement of particles from cathode on target material in X-ray tube. During this action, the particles collide with the lattice atoms and displace them. This causes formation of point defects.
Enthalpy of Formation of Point Imperfections
To create a point imperfection, some work is required to be done. This work is called enthalpy or potential energy of formation Hf and is expressed in either kJ/mol or eV/point defect. The equilibrium concentration of vacancies n/NA in a crystal may be computed by
n = NAe-Hf/RT
Where n is number of vacancies per mole of a crystal, NA is Avogadro’s number, R is gas constant, T is absolute temperature in kelvin, and Hf is enthalpy of vacancy formation. It should be noted that the number of vacancies will be zero at T= 0K as e-∞= 0 in above equation.