UNIT 02 THE SOLID STATE
SYLLABUS
· Classification of solids
· Determination of Structure of Solids
· Lattices and unit cells
· Number of Atoms in a Unit Cell
· Packing in Metallic Crystals
· Efficiency of Packing in hcp and ccp Structures
· Calculations involving Unit Cell
· The structure of Ionic Crystals
· Imperfections in Solids
· Electrical Properties
· Magnetic Properties
· Dielectric Properties
· Amorphous solids
Solids are characterised by incompressibility, rigidity and mechanical strength. This indicates that the molecules, atoms or ions that make up the solids are closely packed , i.e., they are held together by strong forces and cannot move at random. Thus, in solids there is well ordered arrangement of molecules, atoms or ions. The properties of solids not only depend upon the number and kinds of constituents but also on their arrangements.
Classification of solids
Solids can be classified into two categories :
i) Crystalline solids
ii) Amorphous solids
1. Crystalline Solids
A substance whose constituents are arranged in an orderly manner in definite geometric form is called crystalline solid. X-ray diffraction studies of these solids reveal that the regular arrangement of constituents (molecules, atoms or ions) extends over large distances in three-dimensional network of crystals. In other words, the crystalline solids are said to exhibit long range order. Some common examples of crystalline solids are sodium chloride, sulphur , diamond, sugar etc.
2. Amorphous Solids
A substance whose constituents are not arranged in orderly manner is called amorphous solid. These substances do possess properties such as incompressibility and rigidity to certain extent but they do not have definite geometric forms. Some amorphous solids have some orderly arrangement but it does not extend to more than few Angstrom units. The amorphous solids have only short range order. Some common examples are glass, rubber, plastics etc.
Differences between crystalline and amorphous Solids
Crystalline and amorphous solids differ from one another in the following respects.
1. Characteristic geometry : A crystalline solid has a definite and regular geometry due to definite and orderly arrangement of molecules, atoms or ions in three-dimensional space. An amorphous solid , on the other hand, does not have any pattern or arrangement of molecules , atoms or ions and therefore does not have any definite geometrical shape.
2. Melting points : As a solid is heated, its molecular vibrations increases and ultimately become so great that molecules break away from their fixed positions. They now begin to move freely and have rotational motion as well . The solid now changes into liquid state. The temperature at which this occurs is known as melting point. A crystalline solid has a sharp melting point, i.e., it changes abruptly into liquid state at a fixed temperature. An amorphous solid does not have a sharp melting point. For example when glass is heated gradually, it softens and starts to flow without undergoing a definite and abrupt change into liquid state. The amorphous solids are therefore regarded as ‘liquids at all temperatures’. There is some justification for this view because it is known from X-ray examination that amorphous substances do not have well-ordered molecular or atomic or ionic arrangements. Strictly speaking, solid state refers only to crystalline state, i.e., only a crystalline material can be considered to a true solid.
3. Anisotropy and Isotropy : Amorphous substances differ from crystalline solids and resemble liquids in many respects. The properties such as electrical conductivity, thermal conductivity, mechanical strength and refractive index are the same in all directions. Amorphous substances are therefore said to be isotropic. Liquids and gases are also isotropic. Crystalline solids on the other hand , are anisotropic, i.e., their physical properties are different in different directions. For example, the velocity of light passing through a crystal may split up into two components each following a different path and traveling with a different velocity. This phenomenon is known as double refraction. Thus anisotropy itself is a strong evidence for the existence of ordered molecular arrangements in such materials. In amorphous solids as well as in liquids and gases, atoms or molecules are arranged at random and in a disorderly manner and therefore, all directions are identical and all properties are alike in all directions.
Most of the substances (elements and compounds ) form crystalline solids. Elements like iron, copper, silver , phosphorus and iodine and compounds such as sodium chloride , potassium nitrate all form crystals. Some substances adopt different structural arrangements under different conditions. Such arrangements are called polymorphous. Thus, diamond and graphite are two different polymeric forms of carbon. These different structures have different properties, such as melting point and density. Graphite is soft and good conductor of electricity whereas diamond is hard and poor conductor of electricity. The specific arrangements of atoms , ions or molecules within a crystal can be determined by X-ray diffraction studies.
Crystalline solids based on the nature of bonding are classified into four main categories viz., covalent, ionic, metallic and molecular solids. The TABLE provides examples of different types of solids. The physical properties like standard molar enthalpy of fusion, electrical and thermal conductivity provide information to which of these four categiries of the solid belongs.
TABLE Different types of solids
Type of solid | Constituent particles | Bonding / attractive forces | Binding energy KJ/mol | Examples | Physical nature | m p ( K) | Electrical conductivity |
Molecular | molecules | i) dispersion ii) dipole interaction iii) hydrogen bonding | 0.05 – 40 5 – 25 10 - 40 | Argon HCl H2O (ice) | soft soft hard | 84 158 273 | insulator insulator insulator |
Metallic | atoms | Positive ions and electrons(deloc-alized electrons) | 70 – 1000 | Ag, Cu, Mg | hard | ~800 to 1000 | Conductor |
Net work or covalent | atoms | covalent | 150 – 500 | SiO2, SiC, C(diamond) | hard | ~4000 | insulator |
Ionic | Ions | Coulombic or electrostatic | 400 – 4000 | NaCl, MgO, KCl, BaCl2 | Hard but brittle | High ~ 1500 | insulator |
DETERMINATION OF STRUCTURE OF SOLIDS
In general , the structure of solids is determined by diffraction methods . However, X-ray diffraction has been used more extensively for this structure determinations.
Structure determination by X-ray Diffraction
In a crystalline solid, the constituent particles are arranged in a regular fashion. The exact arrangement of the particles within a crystalline solid , i.e., the study of internal structure crystals has been carried out by their interaction with X-rays. The method most commonly employed is called Bragg’s method as it was introduced by W.H Bragg and W.L Bragg in 1913.
Principle : This method is based upon the principle that a crystal may be considered to be made up of parallel equidistant atomic planes, as represented by the lines XX’, YY’, ZZ’ etc. in Fig 1 .
Fig1 Reflection of X-rays by equidistant parallel atomic planes.
Suppose a beam of X-rays is incident on the crystal at the point B along the path AB . On the other hand , some rays like DE, GH etc penetrate into the crystal and are reflected by the atomic planes YY’ , ZZ’ etc. as shown in Fig 1.
Evidently, as compared to the ray AB , a ray like DE has to travel a longer distance JEK in order to emerge out of the crystal where BJ and BK are perpendiculars drawn on the lines DE and EF respectively. The reflected beams like BC , EF etc then undergo interference with each other . If these are in phase (i.e., in the reflected rays, crests fall over the crests and troughs over the troughs), they reinforce into each other and the intensity of the reflected rays is maximum. On the other hand , if the reflected rays
are out of phase, (ie crests fall over the troughs) , the intensity of the reflected beam is very low. If a photographic plate is placed to receive the reflected rays , a diffraction pattern is obtained.
In order that the reflected rays BC and EF may be in phase, the extra distance JEK traversed by the ray DE should be an integral multiple of the wave length l of the X-rays , ie.,
Distance JEK = n l ………(1)
where n is an integer i.e., 1, 2, 3, 4, etc. If d is the distance between the successive atomic planes, it is obvious from Fig1 that
JE = EK = d Sinq
so that JEK = 2 d Sinq ……….(2)
putting this value in equation (1) , we get
2 d Sinq = n l ……….(3)
The equation is called Bragg’s equation. It gives us the condition under which the reflected beam will have maximum intensity. Using monochromatic X-rays , l is constant. Also for a particular crystal when a particular face is facing X-rays , d is constant. Hence n will be integral only for certain values of the angle q. Thus by gradually increasing the value of q , a number of positions are observed corresponding to n = 1, 2, 3, 4, etc. where the reflected beam will have maximum intensity. Thus a diffraction pattern will be obtained in which the various maxima corresponding to n = 1, 2, 3, etc.
Fig 2 Signals from a crystal of sodium chloride as the angle between the direction of X-ray beam and crystal face increases.
These are respectively called diffraction maxima of the first order, second order, third order etc.
Measuring the angle q at which the first maxima occurs (i.e., first maximum intensity is observed) so that n = 1 and knowing the wave length l of the X-rays used , the value of d can be calculated using equation (3).
Apparatus : A simple set up of the X-ray diffraction apparatus is shown in Fig 3.
Fig 3 A simple set up for X-ray diffraction
It may be seen that if the incident beam makes an angle q with the face of the crystal on which it is incident , then the diffracted beam will make an angle 2q with the direction of the incident beam as shown in Fig 4.
Fig 4 Diffracted beam makes double the angle with the direction of incident beam than the incident angle.
Bragg’s method as explained above is also called ‘rotating crystal method’ . It required a single crystal of the substance. Another method introduced later by Debye and Scherrer (1916) and Hull
(a)
(b)
Fig 5 Powder method for the study of crystal structure
Problems
1. A sample of a crystalline solid scatters a beam of X-rays of wave length 70.93 pm at an angle 2 q of 14.66°. If this is a second order reflection , calculate the distance between the parallel planes of atoms from which the scattered beam appears to have been reflected.
2. X-rays of wave length 0.134 nm give a first order diffraction from the surface of a crystal when the value of q is 10.5°. Calculate the distance between the planes in the crystal parallel to the surface examined.
3. A beam of X-rays gave a first order diffraction at an angle 2q = 18.70° for a crystal. If the distance between the planes of the crystal parallel to the surface examined is 3.8 x 10-10 m, find the wae length of X-rays used.
SPACE LATTICE AND UNIT CELL
An example of array of points in a three dimensional space lattice is shown in Fig 6.
Fig 6 Space lattice and unit cell
(solid line show unit cell)
Fig 7 unit cell
Each point represents an identical atom or group of atoms. If we carefully look at a space lattice, it is observed that the entire lattice can be considered as a repetition of a small pattern . This smallest repeating pattern is called a unit cell. Thus, a unit cell is the smallest repeating unit in space lattice which when repeated over and over again results in a crystal of the given substance. The unit cell , is the smallest sample that represents the picture of the entire crystal. The crystal may be considered to consist of infinite number of unit cells. Each unit cell in a three dimensional space has, three vectors, a, b and c as shown in Fig 7.
Note that these are the points and not the lines which constitute the space lattice. The lines joining the points are drawn simply for representing three axes by means of which the relative positions of the points can be described. For example, in Fig. 7 , three imaginary axes , OX, OY and OZ , which are used to represent the unit cell , have been shown. In order to describe a unit cell, we should know : (i) the distances a , b and c which give the lengths of the edges of the unit cell and (ii) the angles a, b and g which give the angles between the three imaginary axes, OX , OY and OZ.
Types of Unit Cells
Based on the dimensions of the unit cells (i.e., lengths a, b and c and the angles a, b and g ) , there are seven types of unit cells. These are also called crystal systems or crystal habits because any crystalline solid must belong to one of these unit cells. These different types along with their characteristics and examples are listed below.
Seven crystal systems
Axial distance or edge lengths | Axial angle | Examples | |
Cubic | a = b = c | a = b = g = 90° | Cu, Zinc blende, KCl |
Tetragonal | a = b ¹ c | a = b = g = 90° | Sn(white tin), SnO2, TiO2 |
Orthorhombic | a ¹ b ¹ c | a = b = g = 90° | Rhombic sulphur, CaCO3 |
Monoclinic | a ¹ b ¹ c | a = g = 90° ; b ¹ 90° | Monoclinic sulphur, PbCrO4 |
Hexagonal | a = b ¹ c | a = b = 90° ; g = 120° | Graphite, ZnO |
Rhombohedral | a = b = c | a = b = g ¹ 90° | CaCO3 (calcite), HgS (cinnabar) |
Triclinic | a ¹ b ¹ c | a ¹ b ¹ g ¹ 90° | K2Cr2O7, CuSO4 5 H2O |
Types of Lattices
In the various types of lattices , it was assumed that particles are present only at the corners of the unit cells. Such type of unit cells in which the particles are present only at the corners are called ‘simple unit cells’ or ‘primitive unit cells’ . However, it has been observed that the particles may be present not only at the corners but also present at some other special positions within the unit cell. Such unit cells are called ‘non-primitive unit cells’. There are three types of non-primitive unit cells as follows :
(1) Face- centred : When the particles are present not only at the corners but also at the centre of each face of the unit cell.
(2) End-centred : When in addition to the particles at the corners, there are particles at the centre of the end faces.
(3) Body-centred : When in addition to the particles at the corners, there is one particle present at the centre within the body of the unit cell.
Every crystal systems does not have all the four types of unit cells , i.e., simple, face-centred, end-centred and body-centred. Hence there are only 14 types of space lattices corresponding to seven crystal systems as shown below (Fig 8):
Fig 8 Fig Seven types of unit cells or crystal systems
The fourteen lattices corresponding to seven crystal systems are known as bravais lattices.
CUBIC CRYSTAL
Cubic crystal system is one in which the axial lengths of the unit cell are equal ie., a = b = c and each of the axial angle is 90° ie., a = b = g = 90°. Further , as shown in Fig , there are three lattices corresponding to cubic system. These are simple, face-centred and body-centred. Further , in Fig 9 , the unit cells are represented by points connected by lines. The points represent the constituent particles i.e., atoms, ions or molecules whereas the lines help to visualize the symmetry of the crystal. In actual practice, the particles occupy much more space of the crystal lattice and are held together by one or the other type of force. For example, a more realistic picture of simple , face-centred and body-centred cubic showing how the particles actually pack within the solid is given in Fig 9.
Fig 9 Three different types of cubic unit cells
This type of arrangement extends in three dimensional space. In general , for any crystal system, the number of spheres which are touching a particular sphere is called its co-ordination number.
In ionic crystals , the coordination number may be defined as the number of oppositely charged ions surrounding a particular ion.
Caculation of Number of Atoms per Unit cell
1. Simple Cubic or Primitive cubic
In this structure , the structural units of the crystal (atoms, molecules or ions ) are situated at all corners of a cube. This shown in the Fig 10 (a).
Fig 10 Simple cubic lattice
It is clear from the the Fig 10 (b) that the atom present at each corner contains 1/8 to each cube because it is shared by 8 cubes at the corners and therefore,
2. Body centred cubic
It has points at all the corners as well as at the centre of the cube. It is shown in Fig 11 (a).
(a) (b)
Fig 11 Body Centred Cubic arrangement
It is clear from the figure that , there are eight atoms at the corners and each is shared by 8 unit cells, so that the contribution of each at the corners is 1/8 . In addition, there is one atom in the body of the cube as shown in Fig 11(b). Thus,
Thus , a body centred cube has two atoms per unit cell.
3. Face centred cubic
This is also called cubic close packed arrangement . It has points at all the corners as well as at the centre of each of the six faces. This is shown in Fig 12 (a).
(a) (b)
Fig 12 Face Centred Cubic arrangement
In this arrangement , there is one atom at each of the eight corners . It is clear from the Fig 12 (b) that the atom present at each corner contributes 1/8 to each cube because it is shared by 8 cubes. In addition, there are six atoms at the faces of the cube and each is shared by two unit cells. Therefore, the contribution of each atom at the face per unit cell is ½ [Fig 12 (b)]. Thus,
The total number of atoms per unit cell of different cubic lattices are given below
Unit cell | Number of atoms at corners | Number of atoms at faces | Number of atoms in centre | Total |
Simple cubic | 8 x (1/8) = 1 | 0 | 0 | 1 |
Body centred cubic | 8 x (1/8) = 1 | 0 | 1 | 2 |
Face centred cubic | 8 x (1/8) = 1 | 6 x (1/2 ) = 3 | 0 | 4 |