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Imperfection in Solids

According to the law of nature nothing is perfect, and so crystals need not be perfect. They always found to have some defects in the arrangement of their constituent particles. These defects affect the physical and chemical properties of the solid and also play an important role in various processes. For example, a process called doping leads to a crystal imperfection and it increases the electrical conductivity of a semiconductor material such as silicon.

The ability of ferromagnetic material such as iron, nickel etc., to be magnetized and demagnetized depends on the presence of imperfections. Crystal defects are classified as follows

1. Point defects
2. Line defects
3. Interstitial defects
4. Volume defects

In this portion, we concentrate on point defects, more specifically in ionic solids. Point defects are further classified as follows.

Stoichiometric Defects in Ionic Solid:

This defect is also called intrinsic (or) thermodynamic defect. In stoichiometric ionic crystals, a vacancy of one ion must always be associated with either by the absence of another oppositely charged ion (or) the presence of same charged ion in the interstitial position so as to maintain the electrical neutrality.

Schottky Defect:

Schottky defect arises due to the missing of equal number of cations and anions from the crystal lattice. This effect does not change the stoichiometry of the crystal. Ionic solids in which the cation and anion are of almost of similar size show schottky defect. Example: NaCl.

Presence of large number of schottky defects in a crystal, lowers its density. For example, the theoretical density of vanadium monoxide (VO) calculated using the edge length of the unit cell is 6.5 g cm^-3, but the actual experimental density is 5.6 g cm^-3. It indicates that there is approximately 14% Schottky defect in VO crystal. Presence of Schottky defect in the crystal provides a simple way by which atoms or ions can move within the crystal lattice.

Frenkel Defect:

Frenkel defect arises due to the dislocation of ions from its crystal lattice. The ion which is missing from the lattice point occupies an interstitial position. This defect is shown by ionic solids in which cation and anion differ in size. Unlike Schottky defect, this defect does not affect the density of the crystal. For example AgBr, in this case, small Ag⁺ ion leaves its normal site and occupies an interstitial position as shown in the figure.

Metal Excess Defect:

Metal excess defect arises due to the presence of more number of metal ions as compared to anions. Alkali metal halides NaCl, KCl show this type of defect. The electrical neutrality of the crystal can be maintained by the presence of anionic vacancies equal to the excess metal ions (or) by the presence of extra cation and electron present in interstitial position.

For example, when NaCl crystals are heated in the presence of sodium vapour, Na⁺ ions are formed and are deposited on the surface of the crystal. Chloride ions (Cl^–) diffuse to the surface from the lattice point and combines with Na⁺ ion.

The electron lost by the sodium vapour diffuse into the crystal lattice and occupies the vacancy created by the Cl^– ions. Such anionic vacancies which are occupied by unpaired electrons are called F centers. Hence, the formula of NaCl which contains excess Na⁺ ions can be written as Na_1+xCl.

ZnO is colourless at room temperature. When it is heated, it becomes yellow in colour. On heating, it loses oxygen and thereby forming free Zn²⁺ ions. The excess Zn²⁺ ions move to interstitial sites and the electrons also occupy the interstitial positions.

Metal Deficiency Defect:

Metal deficiency defect arises due to the presence of less number of cations than the anions. This defect is observed in a crystal in which, the cations have variable oxidation states. For example, in FeO crystal, some of the Fe²⁺ ions are missing from the crystal lattice.

To maintain the electrical neutrality, twice the number of other Fe²⁺ ions in the crystal is oxidized to Fe³⁺ ions. In such cases, overall number of Fe²⁺ and Fe³⁺ ions is less than the O^2- ions. It was experimentally found that the general formula of ferrous oxide is Fe_xO, where x ranges from 0.93 to 0.98.

Impurity Defect:

A general method of introducing defects in ionic solids is by adding impurity ions. If the impurity ions are in different valance state from that of host, vacancies are created in the crystal lattice of the host. For example, addition of CdCl₂ to AgCl yields solid solutions where the divalent cation Cd²⁺ occupies the position of Ag⁺. This will disturb the electrical neutrality of the crystal. In order to maintain the same, proportional number of Ag⁺ ions leaves the lattice. This produces a cation vacancy in the lattice, such kind of crystal defects are called impurity defects.

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Packing in Crystals

Let us consider the packing of fruits for display in fruit stalls. They are in a closest packed arrangement as shown in the following fig we can extend this analogy to visualize the packing of constituents (atoms/ions/ molecules) in crystals, by treating them as hard spheres.

To maximize the attractive forces between the constituents, they generally tend to pack together as close as possible to each other. In this portion we discuss how to pack identical spheres to create cubic and hexagonal unit cell. Before moving on to these three dimensional arrangements, let us first consider the two dimensional arrangement of spheres for better understanding.

Linear Arrangement of Spheres in One Direction:

In a specific direction, there is only one possibility to arrange the spheres in one direction as shown in the fig. in this arrangement each sphere is in contact with two neighbouring spheres on either side.

Two Dimensional Close Packing:

Two dimensional planar packing can be done in the following two different ways.

(i) AAA Type:

Linear arrangement of spheres in one direction is repeated in two dimension i.e., more number of rows can be generated identical to the one dimensional arrangement such that all spheres of different rows align vertically as well as horizontally as shown in the fig.

If we denote the first row as A type arrangement, then the above mentioned packing is called AAA type, because all rows are identical as the first one. In this arrangement each sphere is in contact with four of its neighbours.

(ii) ABAB Type:

In this type, the second row spheres are arranged in such a way that they fit in the depression of the first row as shown in the figure. The second row is denoted as B type. The third row is arranged similar to the first row A, and the fourth one is arranged similar to second one. i.e., the pattern is repeated as ABAB.

In this arrangement each sphere is in contact with 6 of its neighbouring spheres. On comparing these two arrangements (AAAA…type and ABAB….type) we found that the closest arrangement is ABAB…type.

Simple Cubic Arrangement:

This type of three dimensional packing arrangements can be obtained by repeating the AAAA type two dimensional arrangements in three dimensions. i.e., spheres in one layer sitting directly on the top of those in the previous layer so that all layers are identical.

All spheres of different layers of crystal are perfectly aligned horizontally and also vertically, so that any unit cell of such arrangement as simple cubic structure as shown in fig. In simple cubic packing, each sphere is in contact with 6 neighbouring spheres Four in its own layer, one above and one below and hence the coordination number of the sphere in simple cubic arrangement is 6.

Packing Efficiency:

There is some free space between the spheres of a single layer and the spheres of successive layers. The percentage of total volume occupied by these constituent spheres gives the packing efficiency of an arrangement. Let us calculate the packing efficiency in simple cubic arrangement,

Let us consider a cube with an edge length ‘a’ as shown in fig. Volume of the cube with edge length a is = a × a × a = a³ Let ‘r’ is the radius of the sphere. From the figure, a = 2r ⇒r = \(\frac{a}{2}\)

∴ Volume of the sphere with radius ‘r’
= \(\frac{4}{3}\)πr³
= \(\frac{4}{3}\)π(\(\frac{a}{2}\))³
= \(\frac{4}{3}\)π(\(\frac{\mathrm{a}^{3}}{8}\))
= \(\frac{\pi \mathrm{a}^{3}}{6}\) ……………. (1)

In a simple cubic arrangement, number of spheres belongs to a unit cell is equal to one
∴ Total volume occupied by the spheres in sc unit cell = 1 × (\(\frac{\mathrm{a}^{3}}{6}\)) ……………… (2)
Dividing (2) by (3)
Packing Fraction = \(\frac{\left(\frac{\pi a^{3}}{6}\right)}{\left(a^{3}\right)}\) × 100 = \(\frac{100π}{6}\)
= 52.38%

i.e., only 52.38% of the available volume is occupied by the spheres in simple cubic packing, making inefficient use of available space and hence minimizing the attractive forces.

Body Centered Cubic Arrangement

In this arrangement, the spheres in the first layer (A type) are slightly separated and the second layer is formed by arranging the spheres in the depressions between the spheres in layer A as shown in figure. The third layer is a repeat of the first. This pattern ABABAB is repeated throughout the crystal. In this arrangement, each sphere has a coordination number of 8, four neighbors in the layer above and four in the layer below.

Packing Efficiency:

Here, the spheres are touching along the leading diagonal of the cube as shown in the fig.

In ∆ABC
AC² = AB² + BC²
AC = \(\sqrt{\mathrm{AB}^{2}+\mathrm{BC}^{2}}\)
AC = \(\sqrt{\mathrm{a}^{2}+\mathrm{a}^{2}}\) = \(\sqrt{2 a^{2}}\) = \(\sqrt{2}\)a

In ∆ACG

∴ Volume of the sphere with radius ‘r’

Number of spheres belong to a unit cell in bcc arrangement is equal to two and hence the total volume of all spheres

i.e., 68 % of the available volume is occupied. The available space is used more efficiently than in simple cubic packing.

The Hexagonal and Face Centered Cubic Arrangement:

Formation of First Layer:

In this arrangement, the first layer is formed by arranging the spheres as in the case of two dimensional ABAB arrangements i.e. the spheres of second row fit into the depression of first row. Now designate this first layer as ‘a’. The next layer is formed by placing the spheres in the depressions of the first layer. Let the second layer be ‘b’.

Formation of Second Layer:

In the first layer (a) there are two types of voids (or holes) and they are designated as x and y. The second layer (b) can be formed by placing the spheres either on the depression (voids/holes) x (or) on y let us consider the formation of second layer by placing the spheres on the depression (x).

Wherever a sphere of second layer (b) is above the void (x) of the first layer (a), a tetrahedral void is formed. This constitutes four spheres – three in the lower (a) and one in the upper layer (b). When the centers of these four spheres are joined, a tetrahedron is formed.

At the same time, the voids (y) in the first layer (a) are partially covered by the spheres of layer (b), now such a void in (a) is called a octahedral void. This constitutes six spheres – three in the lower layer (a) and three in the upper layer (b).

When the centers of these six spheres are joined, an octahedron is formed. Simultaneously new tetrahedral voids (or holes) are also created by three spheres in second layer (b) and one sphere of first layer (a).

Formation of Third Layer:

The third layer of spheres can be formed in two ways to acheive closest packing

• Aba arrangement – hcp structure
• Abc arrangement – ccp structure

The spheres can be arranged so as to fit into the depression in such a way that the third layer is directly over a first layer as shown in the figure. This “aba’’ arrangement is known as the hexagonal close packed (hcp) arrangement. In this arrangement, the tetrahedral voids of the second layer are covered by the spheres of the third layer.

Alternatively, the third layer may be placed over the second layer in such a way that all the spheres of the third layer fit in octahedral voids. This arrangement of the third layer is different from other two layers (a) and (b), and hence, the third layer is designated (c). If the stacking of layers is continued in abcabcabc pattern, then the arrangement is called cubic close packed (ccp) structure.

In both hcp and ccp arrangements, the coordination number of each sphere is 12 – six neighbouring spheres in its own layer, three spheres in the layer above and three sphere in the layer below. This is the most efficient packing.

The cubic close packing is based on the face centered cubic unit cell. Let us calculate the packing efficiency in fcc unit cell.

Total number of spheres belongs to a single fcc unit cell is 4

Radius Ratio:

The structure of an ionic compound depends upon the stoichiometry and the size of the ions.generally in ionic crystals the bigger anions are present in the close packed arrangements and the cations occupy the voids.

The ratio of radius of cation and anion (\(\frac{\mathrm{r}_{\mathrm{C}^{+}} {\mathrm{r}_{\mathrm{A}^{-}}}\)) plays an important role in determining the structure. The following table shows the relation between the radius ratio and the structural arrangement in ionic solids.

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Primitive and Non-Primitive Unit Cell

There are two types of unit cells:

There are two types of unit cells primitive and non-primitive. A unit cell that contains only one type of lattice point is called a primitive unit cell, which is made up from the lattice points at each of the corners. In case of non-primitive unit cells, there are additional lattice points, either on a face of the unit cell or with in the unit cell.

There are seven primitive crystal systems; cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic and rhombohedral. They differ in the arrangement of their crystallographic axes and angles. Corresponding to the above seven, Bravis defined 14 possible crystal systems as shown in the figure.

Number of atoms in a cubic cell:

Primitive (or) Simple Cubic Unit Cell(SC)

In the simple cubic unit cell, each corner is occupied by an identical atoms or ions or molecules. And they touch along the edges of the cube, do not touch diagonally. The coordination number of each atom is 6. Each atom in the corner of the cubic unit cell is shared by 8 neighboring unit cells and therefore atoms per unit cell is equal to \(\frac{\mathrm{N}_{\mathrm{c}}}{8}\), where N_c is the number of atoms at the corners.
∴ Number of atoms in a SC unit cell = (\(\frac{Nc}{8}\))
= (\(\frac{8}{8}\)) = 1

Body Centered Cubic Unit Cell(BCC)

In a body centered cubic unit cell, each corner is occupied by an identical particle and in addition to that one atom occupies the body centre. Those atoms which occupy the corners do not touch each other, however they all touch the one that occupies the body centre.

Hence, each atom is surrounded by eight nearest neighbours and coordination number is 8. An atom present at the body centre belongs to only to a particular unit cell i.e unshared by other unit cell.
∴ Number of atoms in a bcc unit cell = (\(\frac{Nc}{8}\)) + (\(\frac{\mathrm{N}_{\mathrm{b}}}{1}\))
= (\(\frac{8}{8}\) + \(\frac{1}{1}\))
= (1 + 1)
= 2

Face Centered Cubic Unit Cell(FCC)

In a face centered cubic unit cell, identical atoms lie at each corner as well as in the centre of each face. Those atoms in the corners touch those in the faces but not each other. The atoms in the face centre is being shared by two unit cells, each atom in the face centers makes (\(\frac{1}{2}\)) contribution to the unit cell.
∴ Number of atoms in a fcc unitcell = (\(\frac{\mathrm{N}_{\mathrm{c}}}{8}\)) + (\(\frac{\mathrm{N}_{\mathrm{f}}}{8}\))
= (\(\frac{8}{8}\) + \(\frac{6}{2}\))
= (1 + 3)
= 4

Drawing the crystal lattice on paper is not an easy task. The constituents in a unit cell touch each other and form a three dimensional network. This can be simplified by drawing crystal structure with the help of small circles (spheres) corresponding constituent particles and connecting neighbouring particles using a straight line as shown in the figure.

Calculations Involving Unit Cell Dimensions:

X-Ray diffraction analysis is the most powerful tool for the determination of crystal structure. The inter planar distance (d) between two successive planes of atoms can be calculated using the following equation form the X-Ray diffraction data 2dsinθ = nλ

The above equation is known as Bragg’s equation.

Where

λ is the wavelength of X-ray used for diffraction.
θ is the angle of diffraction
n is the order of diffraction

By knowing the values of θ, λ and n we can calculate the value of d.

d = \(\frac{nλ}{2sinθ}\)

Using these values the edge length of the unit cell can be calculated.

Calculation of Density:

Using the edge length of a unit cell, we can calculate the density (ρ) of the crystal by considering a cubic unit cell as follows.

Equation (6) contains four variables namely ρ, n, M and a. If any three variables are known, the fourth one can be caluculated.

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Classification of Crystalline Solids

The structural units of an ionic crystal are cations and anions. They are bound together by strong electrostatic attractive forces. To maximize the attractive force, cations are surrounded by as many anions as possible and vice versa. Ionic crystals possess definite crystal structure; many solids are cubic close packed. Example: The arrangement of Na⁺ and Cl^– ions in NaCl crystal.

Characteristics:

1. Ionic solids have high melting points.
2. These solids do not conduct electricity, because the ions are fixed in their lattice positions.
3. They do conduct electricity in molten state (or) when dissolved in water because, the ions are free to move in the molten state or solution.
4. They are hard as only strong external force can change the relative positions of ions.

Covalent Solids:

In covalent solids, the constituents (atoms) are bound together in a three dimensional network entirely by covalent bonds. Examples: Diamond, silicon carbide etc. Such covalent network crystals are very hard, and have high melting point. They are usually poor thermal and electrical conductors.

Molecular Solids:

In molecular solids, the constituents are neutral molecules. They are held together by weak vander Waals forces. Generally molecular solids are soft and they do not conduct electricity. These molecular solids are further classified into three types.

(i) Non-Polar Molecular Solids:

In non polar molecular solids constituent molecules are held together by weak dispersion forces or London forces. They have low melting points and are usually in liquids or gaseous state at room temperature. Examples: naphthalene, anthracene etc.,

(ii) Polar Molecular Solids

The constituents are molecules formed by polar covalent bonds. They are held together by relatively strong dipole-dipole interactions. They have higher melting points than the nonpolar molecular solids. Examples are solid CO₂, solid NH₃ etc.

(iii) Hydrogen Bonded Molecular Solids

The constituents are held together by hydrogen bonds. They are generally soft solids under room temperature. Examples: solid ice (H₂O), glucose, urea etc.,

Metallic Solids:

You have already studied in XI STD about the nature of metallic bonding. In metallic solids, the lattice points are occupied by positive metal ions and a cloud of electrons pervades the space. They are hard, and have high melting point. Metallic solids possess excellent electrical and thermal conductivity. They possess bright lustre. Examples: Metals and metal alloys belong to this type of solids, for example Cu, Fe, Zn, Ag, Au, CuZn etc.

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Crystal Lattice and Unit Cell

Crystalline solid is characterised by a definite orientation of atoms, ions or molecules, relative to one another in a three dimensional pattern. The regular arrangement of these species throughout the crystal is called a crystal lattice. A basic repeating structural unit of a crystalline solid is called a unit cell. The following figure illustrates the lattice point and the unit cell.

A crystal may be considered to consist of large number of unit cells, each one in direct contact with its nearer neighbour and all similarly oriented in space. The number of nearest neighbours that surrounding a particle in a crystal is called the coordination number of that particle.

A unit cell is characterised by the three edge lengths or lattice constants a, b and c and the angle between the edges α, β and γ

The crystal lattice is the arrangement of the constituent particles like atoms, molecules, or ions in a three dimensional surface. On the other hand, the unit cell is known to be the building blocks of the crystal lattice, as they get repeated in three-dimensional space to yield shape to the crystal.

A unit cell is the smallest portion of a crystal lattice that shows the three-dimensional pattern of the entire crystal. A crystal can be thought of as the same unit cell repeated over and over in three dimensions.

The total three-dimensional arrangement of particles of a crystal is called the crystal structure. The actual arrangement of particles in the crystal is a lattice. The smallest part of a crystal that has the three dimensional pattern of the whole lattice is called a unit cell.

What is Crystal Lattice? The crystal lattice is the symmetrical three-dimensional structural arrangements of atoms, ions or molecules (constituent particle) inside a crystalline solid as points. It can be defined as the geometrical arrangement of the atoms, ions or molecules of the crystalline solid as points in space.

In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.

A lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.

A lattice is a hypothetical regular and periodic arrangement of points in space. It is used to describe the structure of a crystal. A basis is a collection of atoms in particular fixed arrangement in space.

Lattice Points:

Point in a crystal with specific arrangement of atoms, reproduced many times in a macroscopic crystal. The choice of the lattice point within the unit cell is arbitrary.

Crystal Basis:

Arrangement of atoms within the unit cell.

There are four types of crystals:

1. Ionic
2. Metallic
3. Covalent network, and
4. Molecular

A lattice is an ordered array of points describing the arrangement of particles that form a crystal. The unit cell of a crystal is defined by the lattice points. For example, the image shown here is the unit cell of a primitive cubic structure. In the structure drawn, all of the particles (yellow) are the same.

The arrangement of atoms in a crystal. Each point represents one or more atoms in the actual crystal, and if the points are connected by lines, a crystal lattice is formed; the lattice is divided into a number of identical blocks, or unit cells, characteristic of the Bravais lattices.

Diamond is composed of carbon atoms stacked tightly together in a cubic crystal structure, making it a very strong material. This shows us that it is not only important to know what elements are in the mineral, but it is also very important to know how those elements are stacked together.

They are cubic, tetragonal, hexagonal (trigonal), orthorhombic, monoclinic, and triclinic. Seven-crystal system under their respective names, Bravias lattice.

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infinitum (also called a greatest lower bound or meet).

Crystals are composed of three-dimensional patterns. These patterns consist of atoms or groups of atoms in ordered and symmetrical arrangements which are repeated at regular intervals keeping the same orientation to one another.

A lattice is made by crisscrossing pieces of lath, thin strips of wood, at right angles. The small squares left open between the strips of wood create a gridlike, ornamental pattern. Panels of lattice often enclose other structures, such as a garden bench or gazebo.

The crystal structure is formed by associating every lattice point with an assembly of atoms or molecules or ions, which are identical in composition, arrangement and orientation, is called as the basis. The atomic arrangement in a crystal is called crystal structure.

A lattice point is a point at the intersection of two or more grid lines in a regularly spaced array of points, which is a point lattice. In a plane, point lattices can be constructed having unit cells in the shape of a square, rectangle, hexagon, and other shapes.

The definition of lattice is a structure made from wood or metal pieces arranged in a criss-cross or diamond pattern with spaces in between. A metal fence that is made up of pieces of metal arranged in criss-cross patterns with open air in between the pieces of metal is an example of lattice.

The most common and important are face-centred cubic (FCC) and hexagonal close-packed (HCP) structures. To get a clear picture of arrangements of atoms in these two crystal structures, it is necessary to examine the geometry of possible close-packing of atoms.

There are four types of crystals: covalent, ionic, metallic, and molecular. Each type has a different type of connection, or bond, between its atoms. The type of atoms and the arrangement of bonds dictate what type of crystal is formed.