Imperfections in Crystalline Solids

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.

Point Imperfections

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

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.

Substitutional Impurity

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.

Interstitial Impurity

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.

Frenkel’s Defect

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:

n_F = √(N_L N_i)e^(-Ef^/kT)  ……….(equation 1)

Where, N_L = number of lattice sites occupied by ions per m³,

N_i = number of interstitial sites,

E_f = 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:

N_L = N_Aρ/M_w

Where, N_A is Avogadro’s number, p is density, and M_w is molecular weight. As the number of tetrahedral voids is double of the number of anions in ionic solids, therefore

N_i = 2N_L

The number of anion vacancies is twice the number of cation vacancies for such equilibrium defects, hence effective value of E_f in Equation 1 is taken as half of it (i.e. E_f/2).

Schottky’s Defect

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 H_f and is expressed in either kJ/mol or eV/point defect. The equilibrium concentration of vacancies n/N_A in a crystal may be computed by

n = N_Ae^-Hf/RT

Where n is number of vacancies per mole of a crystal, N_A is Avogadro’s number, R is gas constant, T is absolute temperature in kelvin, and H_f 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.

Surface Imperfections

Surface imperfections are observed on the surfaces of the crystals. Surface, in broad sense, means a depth of few degree angstrom (^oA) too. Due to finite size of crystals, bonds are broken on the, surfaces for want of neighboring atoms. The number of bonds broken per atom, on different planes, due to formation of one square meter (1m²) surface, are:

1/2r²    on  (100) plane,

√2/8r²  on  (110) plane, and

3√3/r²  on (111)plane.

where r is atomic radius.

Moreover during solidification, formation of new crystals in a polycrystalline material is random. Interaction of crystals between themselves inherits some imperfection on the inside of surface. We shall now deal with different types of surface imperfections in this article.

Grain Boundary

Mechanism of grain boundary defect: Polycrystalline solids consist of several crystals of different sizes oriented randomly with respect to each other. They grow during the process of re-crystallization. Growth of crystals is a direct consequence of addition of atoms.

If the number of atoms are more in a crystal, the size of crystal will be large. As shown in Figure 1, crystal marked A is smaller than crystal marked B because of lesser number of atoms joining the crystal A.

The crystals grow randomly and in doing so impinge upon each other. When adjoining crystals impinge together, some atoms marked C are caught in between them. These atoms are forced to join one or the other crystal. But they do not join any crystal due to their opposing forces. These atoms occupy positions at the junction of adjoining crystals.

Junction or the boundary region is distorted and behaves as non-crystalline material. This boundary region is the defect called grain boundary. Lines joining various atoms C are grain boundaries, and θ is the grain orientation angle.

Twin or Twinning

This defect is also called twin boundary. As the name implies, twin boundaries occur in pairs. The arrangement of atoms is such that one side of twin boundary is a mirror replica of the other side.

As shown in Figure 2, side A is a mirror image of side B. The zone CDEF is known as twinned zone. DE and FC are twin boundaries.

Annealing twin and deformation twin: Twins can form during the process of re-crystallization or during plastic deformation of materials. In the first case it is called annealing twin and in the other as deformation twin.

Occurrence of twins is common in brass and metallic sheets. Twins in crystalline solids can be visualized by an optical microscope. The twin planes and twin directions in different crystal structures are given in the following table:

Structure	Twin plane	Twin Direction	Example
BCC	(112)	[111]	Fe, W, V
FCC	(111)	[112]	Cu, Al, Ag
HOP	(101bar2)	[101bar1bar]	Mg, Ti, Zn

Low Angle Tilt Boundary

When crystals orientation is such that the orientation angle (Figure 3) is less than 10°, the structure is called low angle tilt boundary. Here the adjacent crystals are oriented at a small angle to each other.

Geometry of the tilt boundary h may be taken as a vertical distance between equally spaced edge dislocations of same kind (of inverted T) located one over the other. The schematic arrangement is shown in Figure 3.

A, B, C and D are like (positive) edge dislocations of the same kind. A relation between the geometrical dimensions b and h of edge dislocation gives

b/h = tanθ

For small angles (θ<4.5^o) , it becomes

b/h = θ (approximately)

Here b is the Burgers vector and h is the vertical distance between two adjoining edge dislocations.

Other Crystal Boundaries

Besides the low angle tilt boundary discussed above, surface boundary imperfections also occur in other forms such as:

1. high angle boundary,
2. inter-phase boundary, and
3. twist boundary.

• Orientation between two adjoining crystals having q > 10° is referred to as high angle boundary.
• Two crystals having either different crystalline configurations or different composition form an inter-phase or interface boundary.
• Boundaries, formed by screw dislocations on small orientation angle θ are called twist boundaries.

Stacking Fault

The stacking sequence in FCC crystals, of ABCABCABC …. type,  and in HCP crystals of ABABAB …. type. However if any one or more stacking plane of atoms are missing, the solid configuration becomes faulty. The defect is then called stacking fault or piling-up fault. Typical stacking fault in FCC crystals is illustrated below

ABCAB*A*CA*BC….

and in HCP crystal as

ABA*AB*BAB

The * mark in above two stackings indicate the fault.

Volume Imperfections

Volume imperfections, also known as 3-dimensional imperfections are found inside the solids. These may form due to one or more of the following reasons.

1. foreign-particle inclusions,
2. regions of non-crystallinity,
3. pores,
4. dissimilar natured regions.

The dimensions of these are of the order of tens of ^oA. The inclusions, pores etc. may be randomly located at one or many positions in the volume of the material.