# The Solid State

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## General Characteristics of Solid State

As we know matter exists in mainly three states – solid, liquid and gas. Here we are discussing the solid state of matter in detail.

### Characteristics of Solid-State are Given Below

• In solids particles are tightly packed.

• Solids have definite shape and mass.

• In solids, intermolecular distances are short and intermolecular forces are strong.

• Solids have distinct boundaries. Their constituent particles have fixed positions and oscillate about their mean positions.

• Solids have fixed volumes.

• They are rigid and have negligible compressibility.

### Dielectric Properties of Solids

The study of dielectric properties concerns storage and dissipation of electric and magnetic energy in materials. Some dielectric properties of solids are as follows –

• Piezoelectricity

• Pyroelectricity

• Ferroelectricity

• Anti – ferroelectricity

Piezoelectricity – The solids in which individual dipoles are formed and align itself in an ordered manner in such a way so that a net dipole moment of the solid (crystal) shows piezoelectricity. When pressure is applied in such solids, their atoms or ions are displaced and produce electricity. Piezoelectricity is electric charge which accumulates in some crystals due to mechanical stress. It means piezoelectricity is electricity resulting from pressure and latent heat. The word piezoelectricity is derived from the Greek word piezein which means ‘to squeeze or press’ and elecktron, which means ‘amber’(an ancient source of electric charge). Piezoelectricity was discovered by French Physicists Jacques and Pierre Curie in 1880.

This dielectric property of solids is used in the medical field, automotive industry, information technology and telecommunications.

Pyroelectricity – The word ‘Pyroelectricity’ is derived from the two Greek words pyr which means ‘fire’ and elecktron which means ‘amber’(an ancient source of electric charge) or ‘electricity’.  Pyroelectricity is the ability of certain crystals to produce a temporary voltage when they are heated or cooled. Some piezoelectric crystals produce electricity on heating, thus produced electricity is called pyroelectricity and this phenomenon is called pyroelectric effect. Pyroelectric crystals are generally naturally electrically polarized and as a result contain large electric fields. Due to change in temperature positions of the atoms change within a crystal structure. Now due to change in crystal structure, polarization of the crystal changes which causes rise to a voltage across the crystal. Now if the temperature remains constant at its new value, the pyroelectric voltage disappears due to leakage current.

They are used in heat sensors, power generation and nuclear fusion. They can be used in PIR (passive infrared) sensors, infrared non – contact thermometers and motion detector thermal sensors. Motion detectors thermal sensors are used to detect the movement of human beings, animals, and objects etc.

Ferroelectricity – In some crystals the dipoles are permanently aligned even in absence of electric field. They possess spontaneous electric polarization. On application of external electric fields on such crystals their electrical polarization gets reversed. It was discovered by Valasek in Rochelle salt in 1920. The word ferroelectricity is made up of two words ferro which means iron and electricity. All ferroelectric materials are pyroelectric as well.

It is used in ferroelectric capacitors, ferroelectric RAM, high quality infrared cameras, fire sensors, sonar, vibration sensors and fuel injectors on diesel engines. It is also used in ferroelectric tunnel junction (FTJ). Ferroelectrics show catalytic properties. So, they can be used for catalysis as well. They can also act as energy harvesters. Materials which possess both ferroelectric and ferromagnetic properties are called multiferroics. Many researches are going on in multiferroics.

Anti – Ferroelectricity – As the name suggests it is opposite to ferroelectricity. The relation between anti - ferroelectricity and ferroelectricity is analogous to the relation of ferromagnetism and anti – ferromagnetism. Crystals which possess anti – ferromagnetism property consist of an ordered array of electric dipoles but with adjacent dipoles oriented in opposite (antiparallel) directions. This results in a net zero dipole moment. They possess zero spontaneous electric polarization since the adjacent dipoles cancel each other. This property of crystal can appear or disappear depending on temperature, pressure, growth method and external electric field etc. The temperature at which anti – ferroelectricity disappears is called Neel point or Curie point.

Dipole arrangement in anti – ferroelectric material on application of electric field

It is used in supercapacitors, integration with ferromagnetic materials, high energy storage devices etc.

### Amorphous and Crystalline Solids

Solids can be classified into two types on the basis of arrangement of their constituent particles –

• Crystalline solids

• Amorphous solids

Crystalline Solids – In crystalline solids, the arrangement of constituent particles is found in an ordered manner. It means in crystalline solids a regular pattern of the constituent particles is found which repeats itself periodically over the entire crystal.

Amorphous Solids – In amorphous solids constituent particles are found in irregular shape. In these solids regular arrangement of constituent particles is found over short distances only. These patterns are found scattered in the solid and in between the arrangement of the particles is found disordered.

### Classification of Crystalline Solids

Crystalline solids can be classified into following four types on the basis of different types of intermolecular forces between them –

Molecular Solids – Those crystalline solids whose constituent particles are molecules are called molecular solids. They can be divided into following three types –

Non - Polar Molecular Solids – Those molecular solids which have non - polar covalent bonds between the atoms are called non - polar molecular solids. Example, H2, Br2 etc.

Polar Molecular Solids - Those molecular solids which have polar covalent bonds between the atoms are called polar molecular solids. In such molecular solids, molecules are held together by dipole – dipole interactions. Example, HCl, SO2 etc.

Hydrogen bonded molecular solids - Those molecular solids which have polar covalent bonds between the atoms and the molecules are held together by hydrogen bonding are called hydrogen bonded molecular solids. Example – water molecules are held together by hydrogen bonding.

Ionic Solids - Those crystalline solids whose constituent particles are ions are called ionic solids. In these solids cations and anions are bonded together by strong electrostatic forces. Examples – NaCl, ZnS etc.

Metallic Solids - Those crystalline solids whose constituent particles are positive metal ions and electrons, are called metallic solids. In such solids a collection of orderly arranged positive metal ions are surrounded by or held together by free electrons. Example – Fe, Au, Ag etc.

Covalent Solids – In these crystalline solids, atoms of non – metals are joined to their adjacent atoms by a covalent bond. For example, in diamonds and graphite etc. As covalent bonds are strong, so they held all the atoms in their position firmly. That’s why these solids are hard and brittle. These solids are also known as network solids or giant molecules.

### Crystal Lattice and Unit Cell

Crystal Lattice - A lattice is a geometric arrangement of the points in space at which the atoms, molecules, ions or constituent particles of a crystal occur. It describes the arrangement of particles in the crystal.

Unit Cell - Unit cell of a crystal is defined by lattice. Lattice point is the point or position in the unit cell or on the lattice in a crystal where the probability of finding an atom or ion is the highest.

Bravais Lattice - Bravais Lattice is an infinite array of discrete points in three - dimensional space generated by a set of discrete translation operations. It is named after French physicist Auguste Bravais. He is known for his work in crystallography. He gave the concept of Bravais lattice and formulated Bravais Law.

In three – dimensional space, 14 Bravais lattices are there into which constituent particles of the crystal can be arranged. These 14 Bravais lattices are obtained by combining lattice systems with centering types.

A Lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. 14 Bravais lattices can be divided into 7 lattice systems –

• Cubic

• Tetragonal

• Orthorhombic

• Hexagonal

• Rhombohedral

• Monoclinic

• Triclinic

Centering types identify the locations of the lattice points in the unit cell.

• Primitive Unit Cell (P) – In this lattice points are found on the cell corners only. It is also sometimes called a simple unit cell. In these constituent particles are found at the corners of the lattice in the unit cell, no particles are located at any other position in the cell. Thus, a primitive cell has only one lattice point.

• Non – Primitive Unit Cells – In these unit cells particles are found in other positions of the lattices as well with corners. These can be divided into following types –

• Body Centered (I) - In this lattice points are found on the cell corners with one additional lattice point at the center of the cell. Thus, it has particles at the corners and center of the body or cell.

• Face Centered (F) - In this lattice points are found on the cell corners with one additional lattice point at the center of each face of the cell. Thus, it has particles at the corners and center of each face.

• Base Centered (C) - In this lattice points are found on the cell corners with one additional lattice point at the center of each face of one pair of parallel faces of the cell. It is also called end centered. Thus, it has particles at the corners and one particle at the center of each opposite face.

Not all combinations of lattice systems and centering types give rise to new possible lattices. After combination of them, several lattices we get are equivalent to each other.

### 14 - Types of Bravais Lattice

All 14 Bravais Lattices show few similar characteristics which are listed below-

• Each lattice point represents one particle of the crystal.

• This constituent particle of the crystal can be atom, ion, or molecule.

• Lattice points of the crystal are joined by straight lines.

• By joining the lattice point of the crystal, we get the geometrical shape of the crystal.

• Each one of the 14 Bravais lattices possess unique geometry. After combining lattice system and centering types, those lattices were equivalent and have been already excluded.

### List of 14 – Types of Bravais Lattices

For description, we are using a, b and c to denote the dimensions of the unit cells and α, β, γ to denote corresponding angles in the unit cells.

• Cubic – Cubic system shows three types of Bravais lattices – Primitive, base centered and face centered. For cubic systems –

a = b = c

α = β = γ = 90°

• Tetragonal – Tetragonal system shows two types of Bravais lattices – Primitive, body centered. For tetragonal systems –

a = b ≠ c

α = β = γ = 90°

• Orthorhombic – Orthorhombic system shows four types of Bravais lattices – Primitive, body centered, base centered and face centered. For orthorhombic systems –

a ≠ b ≠ c

α= β=γ=90°

• Hexagonal - Hexagonal system shows one type of Bravais lattice which is Primitive. For hexagonal systems –

a = b ≠ c

α =120°, β = γ = 90°

• Rhombohedral - Rhombohedral system shows one type of Bravais lattice which is Primitive. For rhombohedral systems –

a = b = c

α = β = γ ≠  90°

• Monoclinic - Monoclinic system shows two types of Bravais lattices – Primitive, base centered. For Monoclinic systems –

a = b ≠ c

α ≠ 90°, β = γ = 90°

• Triclinic – Triclinic system shows one type of Bravais lattice which is Primitive. For triclinic systems

a ≠ b ≠ c

α ≠ β ≠ γ ≠ 90°

Thus, from the cubic system – two, from tetragonal – two, from orthorhombic – four, from hexagonal – one, from rhombohedral – one, from monoclinic two and from triclinic one Bravais lattices are found. If you add all these Bravais lattices, you get a total 14 Bravais lattices.

## Number of Atoms in a Unit Cell

 Type of Unit Cell Number of Atoms in a Unit Cell Simple or Primitive Cubic Unit Cell Number of atoms in a simple or primitive unit cell = 8 corner atoms x ⅛ atom per unit cell = 1 Body Centered Cubic Unit Cell Number of atoms in a body centered unit cell = (8 corner atoms x ⅛  atom per unit cell) + (1 atom present at the center of the cube) = 1 + 1 = 2 Face Centered Cubic Unit Cell Number of atoms in a face centered unit cell = (8 corner atoms x ⅛ atom per unit cell) + (6 face atoms x ½  atom per unit cell) = 1 +3 = 4

### Close – Packed Structures

Solids possess closed packed structures. It means their constituent particles are found closely packed with each other, leaving very less or minimum vacant space between them. They have three dimensional closely packed structures.

• Close Packing in One Dimension – Consider the constituent particles as hard spheres. Their close packed structure in one dimension is given below –

• Close Packing in Two Dimensions – Consider the constituent particles as hard spheres. Their close packed structure in two dimensions is given below –

• Close Packing in Three Dimensions – Consider the constituent particles as hard spheres. Their close packed structure in three dimensions is given below

### Packing Efficiency

No matter in which way particles are arranged in the solids, there will always be some vacant space between them which is called a void. Quantitative aspect of the solid crystal structure can be understood with the help of packing efficiency.

Packing efficiency is the percentage of total space filled by the particles. Mathematically, it can be expressed by following formula –

Packing efficiency = $\frac{Volume \;Occupied \;By \;All\; Constituent\; Particles\; in \;Unit\; Cell}{Volume \;of \;The\; Unit \;Cell}$ x 100

 Type of Unit Cell Number of atoms in unit cell Coordination number Packing efficiency Void space Simple or Primitive Cubic Unit Cell 1 6 52.4% 47.6% Body Centered Cubic Unit Cell 2 8 68% 32% Face Centered Cubic Unit Cell 4 12 74% 26% Here, you can note that as the number of atoms in per unit cell increases, packing efficiency also increases while void space decreases.

### Following Factors Affect The Packing Efficiency of The Crystalline Solids

• The volume of the unit cell

• Number of atoms in the unit cell

• Volume of atoms present in the unit cell

### Importance of Packing Efficiency

• It helps to know the exact structure of the solid.

• By packing efficiency, we can know the density and consistency of the crystalline solid.

• Various attributes of the crystalline solid can be determined by the packing efficiency.

### Calculations Involving Unit Cell Dimensions

Density of the unit cell = $\frac{Mass \; of \; Unit \; Cell }{ Volume \; of \; Unit \; Cell}$---------------(1)

Mass of unit cell = mass of each constituent particle x number of particles in the unit cell -------------------(2)

Mass of a constituent particle = Molar mass of the solid / Avogadro’s number

Mass of a constituent particle = M/NA--------(3)

Number of particles in a unit cell will depend on the type of unit cell. Let us suppose the number of particles in the unit cell are n. Now, let’s put the values in the equation (2) from (3) –

Mass of unit cell = (M/NA) n -------------------(4)

Put the values of equation (4) in equation (1), we get –

Density of the unit cell =  (nM/NA) / a3

Density of the unit cell =  $\frac{nM}{N_{A}a^{3}}$

Relation between radius of an atom and edge length of the unit cell –

For simple cubic unit cell (scc) –  a = 2r

For body centered cubic cell (bcc) – a = (4/√3)r

For face centered cubic cell (fcc) – a = 2√2r

### Imperfections in Solids

Meaning of defect is shortcomings or imperfections. Solid crystals also show some kinds of defects which we study in solid state chemistry. Some units of the crystals may have one or more atoms lesser than other ideal units of crystals. These imperfections of crystals are called defects in crystals. In other words, the interruptions in regular patterns in crystalline solids is called crystallographic defects. There are many types of crystallographic defects such as point defects, line defects, planar defects etc. Frenkel defects are point defects. These defects in the crystals make them interesting to study. As said by Colin Humphreys –

“Crystals are like people, it is the defects in them which tend to make them interesting!”

### What are Point Defects?

Those defects in the crystals which occur around an atom or particle are called point defects. These defects occur only at or around a single lattice point. They do not extend in space in any dimension. That’s why they are also called zero dimensional (0-D) defects. These are the smallest possible defects in any crystalline solid material. Point defects occur when –

• One or more atoms of the crystal are missing from their corresponding lattice site.

• Atom/s is shifted from its corresponding lattice site to interstitial position in the crystal.

• Foreign atom/s occupy the interstitial position in the crystal lattice.

• Original atom of the crystal is replaced by foreign atom.

### Types of Point Defects

Point defects can be further divided into following types –

Stoichiometric Defects – The compounds which obey the law of definite proportions, the law of constant composition and the law of conservation of mass are called stoichiometric compounds. The defects in crystals which do not disturb the stoichiometry of the compound or crystal are called stoichiometric defects.

### Stoichiometric Defects Can Be Divided Into Following Types

• Vacancy Defect – The point defect which is produced when an atom goes missing from its original lattice site is called vacancy defect. It creates vacancy in the lattice site as shown in the diagram below –

It results in a decrease in density of the substance. Number of vacancy defects depends on the temperature of the crystal. It occurred due to imperfect packing during crystallization.

• Interstitial Defect – The point defect which occurs when an atom takes the interstitial position of the lattice structure is called interstitial defect. The atom can be of the same crystal or foreign crystal/material. If the atom is of the same crystal, then the defect is called self-interstitial defect. It is shown below through the diagram –

### This Defect Increases The Density of The Crystal. It Causes Atomic Distortion.

• Schottky Defect - Schottky defect is also a point defect which is observed in various crystals. Named after a German physicist, Walter H. Schottky, this defect occurs commonly in ionic crystals where the size of cation and anion is similar. The point defect which occurs when cation and anion leave their corresponding lattice sites and create a pair of vacancy defects is called Schottky defect. Take for example, KCl. Potassium (K) has an atomic number of 17 and Chlorine (Cl) has an atomic number of 19. Both the ions are of similar size, and hence it is a good candidate for showing Schottky defects.

### Characteristics of Schottky Defects

Schottky defects usually occur when heat is applied to the ionic compound crystal. Heat raises the temperature, and hence the thermal vibration within the crystal. This creates gaps in the crystal pattern. The gaps are created in stoichiometric ratio, i.e. as per the availability of ions in chemical compounds. For example, in a generic ionic compound with the formula XnYm, ‘n’ ions of X and ‘m’ ions of Y will leave to create vacancies. A group of such vacancies can also be referred to as a Schottky cluster.

Schottky defect reduces the density of ionic compounds because a fraction of ions leaves the crystal, hence reducing the overall mass at the same crystal volume.

### Concentration of Defects

As explained previously, Schottky defects are formed by applying heat. At any given temperature, there is a concentration of defects (i.e. Schottky defects per unit volume) given by the following formula:

$n_{s}\approx Ne^{\frac{-\Delta H_{s}}{2kT}}$

Where,

ns = number of Schottky defects per unit volume at temperature T (in Kelvins) in a crystal with N anion and N cations per unit volume, and ∆Hs is the enthalpy for creating one defect.

• Frenkel Defect - When an atom or smaller ion (generally a cation) leaves its place in the lattice, creating a vacancy and becomes an interstitial by lodging in a nearby location. Thus, a vacancy is created in the lattice. This type of defect is called Frenkel defect. It was discovered by soviet physicist Yakov Frenkel. It is a type of point defect which is also known as dislocation defect.

Frenkel Defect in NaCl Crystal

### Frenkel Defect Examples

Following are examples of Frenkel Defect –

• NaCl (Sodium Chloride) (Frenkel Defect is shown in NaCl in the image above)

• Zinc sulphide

• Silver (I) Chloride

• Silver (I) Bromide

• Silver (I) Iodide

Frenkel Defect in Silver Chloride

### Reason of Frenkel Defect

When the ionic compound size of anions is much larger than cations then Frenkel Defect occurs. As due to size difference in ions, ion occupies interstitial position in lattice. Ionic crystals having Frenkel defect also remain neutral in nature. As the number of cations and anions remain equal. Thus, it can be said that Frenkel defects are shown by those ionic solids which have large size differences between the cation and anion.

Frenkel defect doesn’t have any impact on density, mass or volume of the crystal as in this defect migration of ions takes place within the crystal. So, the density, mass and volume of the ionic crystal remain the same even after having a Frenkel defect.

### Calculation of Number of Frenkel Defects

Number of Frenkel Defects can be calculated by following formula –

$\sqrt{NN^{'}}e^{-\frac{\Delta H}{2kT}}$

Where N = normally occupied positions

N’ = Number of available positions

∆H = Enthalpy released by 1 Frenkel defect

R= Gas constant

T = Temperature

### Difference Between Schottky and Frenkel Defect

Like the Frenkel defect, Schottky defect is also a point defect in crystalline solids. Even a few crystalline ionic compounds such as silver bromide exhibit both Schottky and Frenkel defects as well.

 Schottky Defect Frenkel Defect In these defects ions leave from a crystal lattice in stoichiometric units. In these defects ions leave its place in lattice but remain in the interstitial space of crystal lattice. It reduces the density of the lattice. It doesn’t affect the density of the atom or lattice. Due to this effect mass of the lattice is reduced. Due to this effect mass of the lattice or atom remain unaffected. In these defects ions leave crystal lattice. In these defects ions just leave their position in the lattice. Schottky defect occurs in those crystals in which sizes of ions are almost similar. Frenkel defect occurs in those ionic crystals in which sizes of ions (anions and cations) show large differences. Compounds such as KCl, KBr, CsCl etc. show Schottky defect. Compounds such as NaCl, ZnS, AgI etc. show Frenkel defect. It is also known as valency defect. It is also known as a dislocation defect.

• Non- Stoichiometric Defects – The defects in crystals which disturb the stoichiometry of the compound or crystal are called stoichiometric defects. Non – Stoichiometric defects can be divided into following two types –

• Metal Excess Defects – As the name suggests in this defect metal ions occur in excess in the lattice of the crystal. It can take place by following two ways –

1. Anionic Vacancy – Anion goes missing from its corresponding lattice site and creates a vacancy. This vacancy is occupied by an electron to maintain the overall electric charge zero or neutral. It is called F - center.

Actually, this F-center electron gives colour to the compound.

1. Extra Cations – Sometimes in some crystals extra cation fit into the interstitial site on heating the crystal. Equal number of electrons do the same to maintain electrical neutrality of the crystal.

• Metal Deficiency Defects – In some compounds there is a deficiency of metal than their ideal stoichiometric proportions. It is normally found in transition elements as they possess multiple valencies.

### Electrical Properties

The crystalline solids can be classified into following three types on the basis of their electrical properties –

• Conductors – Conductors are those solids which conduct electricity efficiently. Such as metals like silver, aluminium, copper are good conductors of electricity. In these solids, the valence band and conduction band overlaps each other.

• Insulators –Insulators are those solids which do not conduct electricity. Such as rubber, wood and plastics are insulators of electricity. In these solids valance band and conduction band have a large energy gap which is called forbidden gap. Due to large energy gap, electrons cannot jump from valence band to conduction band.

• Semiconductors – Semiconductors are those solids which conduct electricity less than conductors but more than insulators. Their conductivity can be increased by adding a small amount of suitable impurity. Metalloids like silicon, germanium are semiconductors of electricity. In these solids valence band and conduction band have a small energy gap.