The Chemical Science
This chapter includes knowledge about the study of various types of solids and different types of forces responsible to bind the particles together, arrangement of unit cells in lattice and packing of lattice points, and Calculation of density of unit cell and unit cell dimensions.
Calculation of packing efficiency of solids, Types of voids, their locations and number of voids in different types of arrangements, Imperfections or common defects in solid-state, and Electrical and magnetic properties of solids are also included in this chapter.
A solid-state is characterised by its lowest compressibility and highest density as compared to a liquid and gaseous state. Constituent particles are packed most closely. This state is distinguished from liquids and gases by their definite size, shape and volume. Solids have considerable mechanical strength, rigidity, very strong inter-particle force of attraction and minimum distance among them.
Due to the strong forces of attraction particles in solid-state do not possess any translatory motion but can only have a vibrational motion about their mean position. Solids have negligible fluidity as compared to liquids and gases. The diffusion of particles of a solid-state is negligible.
Solids are classified on the basis of two different parameters i.e.,
Packing of constituent particles
Forces of attraction among constituent particles.
1. Crystalline Solids
These are the solids in which the constituent particles (atoms, ions or molecules) are arranged in a regular, three-dimensional orderly arrangement, which gets repeated throughout. Hence, these are also called long-range order solids. Quartz, diamond, Boron Nitride, NaCl, ZnS, CsCl etc.
Properties
Crystalline solids have a sharp melting point, which means they change abruptly into a liquid state at a fixed temperature.
Their nature is Anisotropic which means a few of their physical properties such as mechanical strength, refractive index, electrical and thermal conductivity etc. have different values when measured from different directions, in the same crystal.
On cutting with a knife they give sharp and smooth edges.
They are considered true solids with characteristic enthalpy of fusion.
2. Amorphous Solids
Constituent particles are randomly scattered with, no regular arrangement of particles. Therefore they do not have any definite shape or form. Rubber, glass, plastic etc. are commonly known as amorphous solids.
Properties
If constituent particles are arranged in some orderly manner, then this arrangement does not continue for long distances, hence they are short-range order solids.
They have diffused melting points. It means they soften over a wide range of temperatures instead of melting at a particular temperature only.
They are isotropic by nature. It means due to the random arrangement of particles throughout the values of physical properties such as refractive index, electrical and thermal conductivity etc. will have the same value in all directions.
They form diffused and irregular edges when cut with a sharp knife.
They are called pseudo solids or super cooled liquids, as they have a tendency to flow. They have no definite value of enthalpy of fusion.
Classification on the basis of Nature of Intermolecular Forces
(i) Ionic Solids
There is a regular arrangement of positively and negatively charged ions throughout the solid where ions are held together by strong coulombic or electrostatic forces. These solids are very hard and brittle and have very high melting points. In solid-state, ions are not free to move, hence they are insulators but in a molten state or in the aqueous state, their ions become free to move and become a conductor. Ionic solids have high enthalpies of vaporisation. Ex- LiF, NaCl, KNO3, MgO, etc.
(ii) Metallic Solids
Metal cores and a sea of mobile electrons are the constituents of metallic solids. Each metal atom contributes one or more electrons to the sea of electrons. These electrons are evenly spread out throughout the crystals and weak forces of attraction or metallic bond binds together kernels and a sea of electrons.
Metallic crystals may be hard as well as soft having moderate enthalpies of fusion. A Mobile sea of electrons is responsible for many properties of metals such as malleability (can be beaten into thin sheets), ductility (can be drawn into wires), metallic lustre, thermal conductivity and electrical conductivity etc. Ex- Copper, Iron, Nickel etc.
(iii) Covalent Solids (Network Solid)
These atoms are bonded together by covalent bond formation throughout the crystal. It means there is a continuous network of covalent bonds forming a giant three-dimensional structure or giant molecule. Covalent bonds are strong and directional in nature. These solids are very hard, brittle and very high melting point. Due to the absence of any free electrons or ions, they are insulators. Their enthalpies of fusion are very high. Ex- Diamond, Graphite, Boron Nitride, Silicon Carbide (SiC) etc.
(iv) Molecular Solids
Their molecules are held together by dispersion forces, London forces, dipole-dipole forces or hydrogen bonds. Either atoms or molecules are bonded together by weak dispersion forces or London forces. These are non-conductor soft solids with low m.p. and low enthalpies of vaporisation. They are volatile in nature hence, at room temperature and pressure they are available in a liquid or gaseous state. Ex- Iodine, Solid H2 and CO2 (dry ice). Naphthalene, Camphor etc.
In a crystalline solid, constituent particles are arranged in a definite, three-dimensional regular geometrical order along all the three axes, in which each particle is depicted as a lattice point. A three-dimensional, regular arrangement of lattice points in space or in a crystal is called a crystal lattice or space lattice.
Crystal lattices have the following characteristics
Each point in the lattice is called a lattice site or lattice point.
Each lattice point represents one constituent particle i.e. atom, ion or molecule.
We join lattice points with straight lines to show the geometry of the lattice.
The smallest repeating unit in space lattice when repeated over and over again in different directions produces a complete crystal lattice. There are two important parameters of a unit cell.
Edge length or Axial Distance: Lengths or dimensions along the three edges a, b and c which may or may not be mutually perpendicular.
Interaxial Angle: Angles α, β and γ between pairs of edges are interaxial angles.
α: between axis B and C
β: between axis A and C
γ: between axis A and B
Unit cells can be broadly divided into two categories, primitive and centred unit cells.
1. Primitive Unit Cells
When constituent particles are present only in the corner positions of a unit cell, it is called the primitive unit cell.
2. Centred Unit Cells
When a unit cell contains one or more constituent particles present at positions other than corners in addition to those at corners, it is called a centred unit cell. Centred unit cells are of three types:
(i) Body-Centred Unit Cells: Such a unit cell contains one constituent particle (atom, molecule or ion) at its body centre beside the ones that are at its corners.
(ii) Face-Centred Unit Cells: Such a unit cell contains one constituent particle present at the centre of each face, besides the ones that are at its corners.
(iii) End-Centred Unit Cells: In such a unit cell, one constituent particle is present at the centre of any two opposite faces besides the ones present at its corners.
A crystal system is made up of a very large number of unit cells and every lattice point is occupied by one constituent particle i.e., atom, ion or molecule. These particles may be at corners, face-centres and body centres in different unit cells. Each unit cell is adjacent to another unit cell. Most of the atoms are shared by neighbouring unit cells.
In order to calculate the number of points in a unit cell following generalisations can be used.
A point present at the corner of a unit cell is shared by eight other unit cells, therefore its contribution to each unit cell is 1/8th.
A face is common to two unit cells therefore any point present at the face centre has a contribution of 1/2 to each unit cell.
A point present on each edge-centre is shared among four unit cells therefore its contribution to each unit cell is 1/4.
A point present at the centre of the body of the unit cell has a complete share of one, as it is not shared with any other unit cell.
By using this information, let us calculate the number of constituents or atoms per unit cell.
In this cell, particles are present only at 8 corners and an atom from each corner contributes 1/8th
∴ Total number of atoms per unit cell = (1/8)x8 = 1 atom
It has 8 points at the 8 corners and one point at the body–centre of the cube.
∴ Total number of atoms per unit cell = 1 + 1 = 2
There are 8 points at the corners and 6 points at 6 face centres of each unit cell.
Therefore, the Number of atoms from 8 corners of the unit cell = (1/8)x8 = 1
Number of atoms from 6 face centres = (1/2)×6 = 3
∴ Total number of atoms per unit cell = 1 + 3 = 4.
Let us consider all the constituents are identical to solid spheres. In a two-dimensional arrangement, spheres are arranged in rows to form a layer and packed in different layers to form a three-dimensional arrangement which is known as crystal or space lattice.
(a) Close packing in one Dimension: Arrangement of different atoms in a row touching each other forms one dimension or edge. The coordination number is 2.
(b) Close packing in two-Dimensions: The rows of particles can be stacked in two ways:
Square Close Packing: Spheres are packed in such a way that they align together vertically as well as horizontally and the centre of all spheres is in a straight line. Here each sphere is in contact with four other spheres in the same plane. This is termed as square close packing and the coordination number is 4.
2. Hexagonal close Packing: When the second row is arranged in the depressions of the first row and all atoms align diagonally to each other. Each atom is in contact with 6 other spheres in the same plane. It is known as hexagonal close packing. It is a more efficient mode of packing than the square close packing in a layer in two dimensions. Here the co-ordination number is 6. In this layer triangular voids are formed.
(c) Close Packing in three Dimensions: When layers are arranged over each other they form three-dimensional packing. It is layer packing in which the second layer is placed over the first layer in such a way that all the spheres are exactly above each other and all the spheres align horizontally as well as vertically. This arrangement forms AAA ........ type of lattice. It forms a simple cubic lattice and its unit cell is a primitive cubic unit cell.
Tetrahedral voids are holes or voids surrounded by four spheres Present at the corner of a tetrahedron. The coordination number of a tetrahedral void is 4.
Octahedral voids are holes surrounded by six spheres located on a regular tetrahedron. The coordination number of the octahedral void is 6.
In ccp or hcp the number of octahedral voids present in the lattice is equal to the number of lattice points and the number of tetrahedral voids is twice the number of lattice points.
In all the types of packings, there is always some free space in the form of voids or vacant spaces. Packing efficiency is the percentage of total space filled by the particles.
Packing efficiency = Volume of total lattice points / Total volume of unit cell
suppose edge length of unit cell = a
And radius of sphere = r
In ΔABC, AC2 = BC2 + AB2
AC2 = a2 + a2
AC2 = 2a2
Suppose the edge length = a
Radius of each sphere = r
Number of lattice points per unit cell = 2
In ΔABC, AC2 = BC2 + AB2 = a2 + a2 = 2a2
Similarly in ΔACD,
AD2 = AC2 + CD2 = 2a2 + a2 = 3a2
Let the edge length of the unit cell = a
The radius of sphere = r
As two spheres touch each other at an edge
Let the edge length of a cubic crystal of an element or compound = a cm
The volume of unit cell = a3 cm3
The density of unit cell = Mass of unit cell / Volume of the unit cell
Mass of unit cell = Z × m.
Here Z is the number of atoms present in one unit cell and m is the mass of a single atom.
m = Molar mass / Avogadro's number
Irregularities in the arrangement of constituent particles in the crystalline state are called imperfections or defects. Generally, defects arise during crystallisation processes at a fast or moderate rate.
Point defects are the irregularities or deviations from ideal arrangement around a point or an atom in a crystalline substance, whereas line defects are the irregularities or deviations from the ideal arrangement in entire rows of lattice points. These irregularities are called crystal defects.
(i). Vacancy defect
When some of the lattice sites are vacant, the crystal is said to have vacancy defects, which results in a decrease in the density of the substance. This defect can also develop when a substance is heated.
(ii) Interstitial defect
When some constituent particles (atoms or molecules) occupy an interstitial site, the crystal is said to have an interstitial defect. This defect increases the density of the substance. Such types of defects can be shown by non-ionic solids. Ionic solids must always maintain electrical neutrality, to show Frenkel defects and Schottky defects.
(iii) Schottky Defect
In ionic solids, some cations and a proportionate number of anions are missing from the lattice. While doing so electrical neutrality of the solid is not disturbed. It is termed a Schottky defect. The solids having comparable cationic and anionic radii where the Coordination number is high, generally show this defect. Highly ionic substances such as NaCl, KCl, CsCl, CsBr etc. show this defect. Due to this defect density and stability of the solid decrease while its electrical conductivity increases.
(iv) Frenkel Defect or Dislocation Defect
In some ionic solids smaller ions usually, cations are dislocated from their normal site to some interstitial site. It is termed a Frenkel defect. Ionic substances having large size variation between cation and anion and low Co-ordination numbers generally show this defect e.g., ZnS, AgCl, AgBr, AgI etc.
When there is an excess of metal ions in non-stoichiometric compounds, the crystal lattice has vacant anion sites. These sites are occupied by electrons. These anion sites occupied by electrons are called F-centres.
On the basis of conductivity, solids are classified as conductors, semiconductors and insulators.
The atomic orbitals of metal atoms form molecular orbitals which together form a band. The outermost filled energy band is the valence band and the next empty band in which electrons can move is called the conduction band.
In metals, the conduction band is close to the valence band and therefore electrons can easily move to the conduction band. Therefore they are good conductors.
If the gap between valence and conduction band is very large, electrons can't jump hence they behave as insulators.
In semiconductors, there is a small energy gap between valence and conduction band therefore some electrons can jump to the conduction band and show some conduction. On heating electrons can jump from the valence band hence these types of conductors are called intrinsic semiconductors.
The conductivity of Intrinsic semiconductors is very low which can be increased by adding the appropriate amount of suitable impurity to it. This is called doping. It means when some impurity is added to Si or Ge they can be converted to semiconductors. This doping may be of two types.
When an element of group-14 i.e. Si or Ge having 4 valence electrons are mixed with some quantity of group 15 element such a 'P' or 'As' having 5 valence electrons. They together form 4 covalent bonds and the 5th valence electron on 'P' or 'As' remains unbonded and delocalised. These delocalised electrons increase the conductivity of doped Si or Ge. As an increase in conductivity is due to the negatively charged electrons, hence these types of semiconductors are called n-type semiconductors.
When elements of group-14 i.e. Si or Ge having 4 valence electrons are doped with the elements of the group–13 elements i.e. B or Al having 3 valence electrons, 3 covalent bonds are formed. The place where the fourth valence electron is missing is called electron-hole or electron vacancy. An electron from a neighbouring atom can come and fill the electron-hole but it leaves a hole at its original position. In this way, electrons and holes seem to be moving in opposite directions and cause the conductivity of substances. These are called p-type semiconductors
Magnetic properties of materials are due to magnetic moments associated with the individual electron. The magnetic moment of each electron may originate from any of the two types of motion.
Orbital motion around the nucleus
Spin of an electron around its own axis
A moving electron is considered as a small current loop which generates a small magnetic moment along its axis of rotation as a magnetic moment which originates from electron spin and is directed along the spin axis. Each electron in an atom is considered a small magnet having a permanent orbital magnetic moment and spin magnetic moment. The fundamental magnetic moment is Bohr Magneton μB which is equal to 9.27 × 10–24 Am2. They are classified into 5 types.
Diamagnetic Material: The solids having all the paired up electronic spins are diamagnetic. They are weakly repelled by the magnetic field e.g. NaCl, C6H6, H2O etc.
Paramagnetic Substances: They have permanent magnetic dipoles due to a few unpaired electronic spins i.e. 1, 2, or 3 and they are weakly attracted by the applied magnetic field but cannot be magnetised permanently e.g. Cu+2, Fe+3, O2, CuO, TiO, VO2 etc.
Ferromagnetic Substances: These substances cannot only be attracted by magnets strongly, rather they can be permanently magnetised as well. Metal ions of ferromagnetic substances are in solid-state grouped together into small regions called domains. Each domain acts as a tiny magnet. In the external magnetic field, these domains get oriented in the direction of the magnetic field. On the alignment of magnetic moments of domains in substance, we observe that
When there is the spontaneous alignment of magnetic moments of domains in the same direction, they become ferromagnetic substances e.g. Fe, Co, Ni, CrO2 .
If magnetic moments are aligned in a compensatory way so that they all cancel out each other and the net magnetic moment becomes zero, these are antiferromagnetic substances e.g. MnO.
When magnetic moments of domains are aligned in parallel and antiparallel directions in unequal numbers, it results in some net magnetic moment. These are ferrimagnetic substances. These are weakly attracted by magnetic fields e.g. Fe3O4, MgFe2O4, ZnFe2O4 etc.
Crystalline solids: Substances having a definite and orderly arrangement of constituent particles through the whole lattice are crystalline solids e.g. diamond, ZnS etc.
Amorphous solids: Substances not having any geometrical arrangement of constituents but rather a scattered arrangement of particles are amorphous by nature. e.g. Rubber, glass etc.
Molecular solids: Made up of molecules which may interact with each other by van der Waals forces, hydrogen bonding, dipole-dipole interactions etc e.g. solid CO2, wax, iodine etc.
Ionic solids: Having cations and anions as constituent particles and coulombic forces as forces of interaction e.g., NaCl, CsCl, ZnS etc.
Lattice point: Smallest atoms or ions present in a solid which constitute the solid are lattice points.
Crystal lattice: It is a regular arrangement of constituent particles of a crystal in three-dimensional space.
Unit Cell: The smallest three-dimensional portion of the space lattice which forms a whole crystal lattice when repeatedly arranged in three dimensions.
Simple or primitive cubic unit cell: Having atoms or ions only at the corners of the unit cell.
Body-Centred cubic unit cell: It has lattice points at 8 corners and one at the body centre of the unit cell.
Face-Centred cubic unit cell: It has 8 lattice points at 8 corners and 6 lattice points at 6 face centres.
End-Centred: In addition to the 8 lattice points at 8 corners, there is one particle each at the centre of two opposite faces.
Coordination number: The coordination Number of any particle is equal to the number of its closest neighbours.
Hexagonal close packing (hcp): It is three dimensions of layerwise close packing of constituent particles called ABAB ..... type packing in which every 3rd layer is the same as the first layer.
Cubic close packing (ccp): It is three-dimensional layerwise close packing of constituent particles called ABCABC ... type packing in which every fourth layer is the same as the Ist layer.
Voids or holes: Some vacant or unoccupied spaces available in ccp or hcp are called voids or holes.
Tetrahedral Voids (T.V.): A triangular void, surrounded by four spheres arranged tetrahedrally around it is Tetrahedral voids. Its Coordination number is 4.
Octahedral Void (O.V.): A double triangular void surrounded by total of six spheres making it like an octahedron. Its Co-ordination Number is 6.
Schottky defect: Equal or proportionate number of cationic and anionic vacancies, causing a decrease in density of solid.
Frenkel defect: Dislocation of certain cations to some interstitial voids is called Frenkel defect. The density of solid remains unchanged.
F-centres: Sites from where anions are missing from lattice and the vacant holes are occupied by free electrons. These are responsible for the conductance and colour of solids.
Doping: The addition of a small amount of foreign impurity to a crystal is called doping. It helps in increasing the conductivity of solids.
Diamagnetic substances: Substances containing no unpaired electronic spins and which are weakly repelled by a magnetic field are diamagnetic e.g. Ne, Ar, Ca+2, Na+ etc.
Paramagnetic substances: Which are weakly attracted by the magnetic field and have 1, 2 or 3 unpaired electronic spins e.g. O2 etc.
Ferromagnetic substances: These substances have permanent magnetism even in the absence of a magnetic field. They have 3, 4, or more unpaired electronic spins. e.g. Fe, Ni, Co etc.
Antiferromagnetic substances: Ferromagnetic substances in which magnetic moment becomes zero due to alignment of unpaired electronic spins under the applied magnetic field e.g. FeO, MnO etc.
Ferrimagnetic substances: Ferromagnetic substances show small paramagnetic behaviour due to unequal alignment of unpaired electronic spins in opposite directions e.g. Fe3O4 etc.
Solids are substances that are characterized by a definite volume, shape, and high density. In the solid-state, the constituting particles are arranged in a variety of manners. Solid-state, in a simple manner, means "no moving parts." Thus solid-state electronic devices are the ones made up of solid components that don’t change their position. Solid is a state of matter where the constituting particles are arranged close to each other. The constituent particles can be either atoms, molecules, or ions.
Solids have fixed mass, volume, and shape.
They are compressible and rigid.
Intermolecular distances are very short, thus they are stronger.
Their constituent particles are placed at the same position and can only oscillate in their mean positions.
Types of Solids - Amorphous and Crystalline Solids
Solids can be divided into two types as crystalline or amorphous based on the nature of the order that is present in the arrangement of their constituent particles.
Amorphous solids behave the same as super cool liquids because constituent particles are arranged in short-range order. They are isotropic and have no sharp melting point.
Main features of amorphous solids:
Amorphous solids don’t have a regular shape.
They have short-range orders.
They gradually soften over arrange of temperature.
They are isotropic and their physical characteristics remain the same in all directions.
When they are cut with a sharp-edged tool, they are divided into smaller pieces having irregular surfaces.
They don’t have any fixed heat of fusion.
They are also being referred to as pseudo solids or super cooled liquids as they have flow very slowly.
Crystalline solids have a fixed shape and in it, the constituent particles are arranged in a long-range order.
Some key features of crystalline solids are-
Crystalline solids have fixed geometrical shapes.
They have a long-range order.
They have a sharp melting point.
They are anisotropic. Their physical properties represent different values on getting measured with different directions in the same crystal.
They have a fixed heat of fusion.
Crystalline has true solids.
When crystalline solids are cut with a sharp-edged tool, they get divided into two pieces. The newly created surfaces are plain and smooth.
Polymorphic forms or polymorphs
The different crystalline forms of a substance are referred to as polymorphic forms or polymorphs. Some of its examples include graphite and diamond.
Classification of Crystalline Solids
The classification of crystalline solids is done as per their property. The crystalline property of solid is dependent upon the nature of interactions between the constituent particles, and thus these solids are segmented into four different categories comprising:
Ionic solids
Covalent or Network solids
Molecular solids
Metallic solids
(1) Ionic Solids
Constituent Particles are Ions.
They are hard and brittle.
They have high melting points.
In a solid-state, they don’t conduct electricity, while in a molten state they become good conductors
They have low volatility.
They are mainly soluble in polar solvents such as water.
(2) Covalent or Network Solids
Constituent Particles are Atoms
They are hard but malleable and ductile.
They have high melting points.
They are poor in conducting electricity or heat.
A few examples include SiO2, (quartz), SiC, C (diamond), C(graphite)
(3)Molecular Solids
Constituent Particles are Molecules of a substance.
They are soft.
They have low to moderate-high melting points.
They are poor conductors of electricity or heat
(4)Metallic Solids
Constituent Particles are metal atoms.
They are ductile and malleable.
They have high boiling and melting points.
They are good conductors of electricity and heat.
The smallest atom group that can be used to form the entire lattice is known as a unit cell.
Types of unit cells:
Primitive or simple unit cells showcase constituent particles but at the corners only.
Centered unit cells are being referred to as those unit cells where more than one constituent particles are present at positions in addition to those present at the corners.
Types of centered unit cells:
Face-centered unit cell: These cells consist of one constituent particle that is present at the center of each face along with the ones that are present at the corners.
Body-centered unit cell: These cells consist of one constituent particle that is present at its body center and others are present at the corners.
End-centered unit cell: They have 1 constituent particle located at the center of any two opposite faces and the extremes.
Number of particles at different lattice positions:
Face center: In case an atom is present at the center of the face, it is shared by two unit cells. So, half of the atom belongs to the unit cell.
Body center: In case an atom is present at the body center, it is not shared by some other unit cell. So, that kind of atom completely belongs to the same unit cell.
End center: In case an atom is present at the edge center, it is shared by four unit cells. So, in this case, one-fourth of an atom belongs to the unit cell.
Primitive unit cells have 1 atom.
Face-centered unit cells have 3 atoms.
Body-centered unit cells have 2atoms.
Coordination number: The coordination number is referred to as the number of nearest particles.
Close packed structures
Close packing in two dimensions: It is done by stacking the rows of close-packed spheres in two ways including Square close packing and hexagonal close packing.
Close packing in three dimensions: They can be obtained by loading two-dimensional layers one above each other. It can be obtained in two ways, either by Square close-packed layers and hexagonal-close-packed layers.
Square close packing: The spheres of the second row are placed exactly above the first row. This way the spheres are aligned horizontally and vertically and their arrangement is AAA type. The coordination number is 4.
Hexagonal close-packing: These spheres are placed above the first one in such a manner that their spheres fit in the depression of the first row.
A unit cell can be defined as the smallest group of atoms, the repetition of which forms the entire crystal lattice. It is hence the building block of the crystal. It is a small portion of the crystal with its full overall symmetry. The crystal can be thought of as the repetition of unit cells in three-dimensional space. Each corner of a unit cell is a lattice point.
Understanding Lattice
Lattice can be defined as the three-dimensional arrangement of points in space. This means that when the constituent particles of a crystal are arranged in a three-dimensional fashion with each particle being represented as a point, the arrangement formed is known as the crystal lattice.
Arrangement of particles
Characteristics of a Unit Cell
The smallest portion of a crystal lattice is known as the unit cell. These cells represent the pattern of the crystal which has three dimensions. Given below are the characteristics of unit cells
The dimensions along three edges are termed as a, b, and c.
The mentioned edges are not always necessarily perpendicular to each other.
Angles between these edges are termed as:
α: Angle between b and c
β: Angle between a and c
γ: Angle between b and c
Types of Unit Cells
There are two categories under which unit cells can be classified which includes the following:
Primitive Unit Cell:
When the constituent particles are present only in the corners of the unit cell, it is known as the Primitive Unit Cell.
Centered Unit Cell:
In addition to corners, when constituent particles are present elsewhere in the unit cell, it is known as the Centered Unit Cell.
It is further classified into three categories. They are:
Body-Centered:
There exists one constituent particle in the body center apart from the ones in the corner.
Face-Centered:
There exists one constituent particle at the center of each face apart from the ones in the corner.
End-Centered:
There exists one constituent particle at the center of any two opposite faces apart from the ones in the corner.
Simple Cubic Unit Cell
Consider a unit cell with an edge ‘a’. The number of atoms in the unit cell is denoted by ‘z’. Mass of an atom is denoted by ‘m’. Its molar mass is denoted by M. Avogadro’s number is denoted by NA.
Its volume will be a3.
The density of a unit cell can be calculated with the ratio of mass and volume.
Density of unit cell = Mass of unit cell / Volume of unit cell
Mass of unit cell = (Z*M) / Na
Volume of unit cell = a³
Density of unit cell = (Z*M) / (Na*a³)
Density of unit cell = [(Z*M)/Na]/a³
The mass of a unit cell will be equal to the number of atoms in that unit cell times the mass of each atom.
i.e., Mass of the unit cell = number of atoms in that unit cell x mass of each atom = z x m
Mass of an atom can be calculated by the formula
m = M / NA
Density = Mass / Volume = m / V
Density of unit cell = Mass of unit cell / Volume of unit cell
= (Z*M) / (Na*a³)
Mass of unit cell = (Z*M) / Na
Volume of unit cell = a³
Density of unit cell = [(Z*M)/Na]/a³
Things to Remember
The density of the unit cell is equal to the substance density.
Irregularities in the normal arrangement of atoms are called Crystal Defects. They can be point defects or line defects.
Crystalline solids exhibit properties in accordance with the nature of interactions between their constituent particles.
Packing Efficiency is the percentage of total space in a unit cell that is filled by the constituent particles, such as atoms, ions, or molecules, packed within the lattice. It is the total amount of space occupied by these particles in three-dimensional space. Simply, it can be understood as the specified percentage of the total volume of a solid which is occupied by spherical atoms.
Packing Efficiency can be evaluated through three different structures of geometry which are:
Cubic Close Packing (CCP) and Hexagonal Close Packing (HCP).
Body-Centred Cubic Structures (BCC)
Simple Lattice Structures of Cubic
Factors that determine Packing Efficiency
The factors that determine the packing efficiency of a unit cell are:
The number of atoms in a lattice structure
The volume of a unit cell
The volume of atoms
Structures of Packing Efficiency
The packing efficiency of different lattices is as given below.
Hexagonal Close Packing (HCP) and Cubic Close Packing (CCP) are equally efficient. They have the same packing efficiency.
In Hexagonal Close Packing (HCP), the alternating layers cover each other’s gap.
The spheres in one layer line up with the gap of the previous layer.
In Cubic Close Packing (CCP), the layers are exactly placed above each other in symmetry.
The layers when placed form a cube.
The packing efficiency in a Cubic Close Packing (CCP) structure can be demonstrated as follows –
Cubic Close Packing (CCP) structure
In the above diagram, let ‘A’ be the edge length of the unit cell and AC, which is also equal to b, be the face diagonal.
While looking at the face ABCD of the cube, we can see a triangle is formed. Let r be the radius of each sphere. We proceed by correlating the radius and the edge of the cube.
In triangle ABC,
AC2 = BC2 + AB2
Since AC= b and BC = AB= a,
We get
b2 = a2 + a2 = 2a2
b = √2 a .....(i)
As the radius of each sphere is r, we can rewrite the equation as
b = 4r …..(ii)
We can write from (i) and (ii)
a = 2√2 r or 4r = √2a
Since, the volume of one sphere = 4/3 πr3
And we know that there are four spheres in a Cubic Close Packing (CCP) structure.
The total volume of four spheres is therefore equivalent to 4 × 4/3 πr3
Total Volume of a cube is (edge length)3 i.e, ( a3) or in terms of r it is (2√2 r)3
Crystalline solids are not always perfect; there are defects in them too. Irrespective of the rate of crystallization, crystals are not free of imperfections. Here, by defects, we mean irregularities in the arrangement of constituent particles. There are two types of defects in Solids – Point defects and line defects. Point defects are the ones that arise with respect to a particular point or atom. Line defects, on the other hand, are observed in an entire row or line of atoms in the solid. In this article, we will learn about point defects specifically.
Point defects
As discussed above, point defects are the irregularities in an ideal arrangement due to a point or an atom and the areas surrounding it, in a crystalline substance. It is classified into three types –
Stoichiometric defects
Non – stoichiometric defects
Impurity defects
Stoichiometric defects
Stoichiometric compounds maintain their stoichiometry, which means the ratios of cations and anions remain the same as represented by their chemical formula. Such defects or imperfections that do not disturb the ratio of cations and anions are known as stoichiometric defects. They are further divided into-
Vacancy defect
Interstitial defect
Schottky defect
Frenkel defect
Vacancy defect
This kind of stoichiometric defect exists in crystals where some lattice sites are vacant. In this case, a particle will be missing in the solid. This implies that the density of the solid will decrease. Even if the surrounding particles try to fill the gap, the vacant place will only shift its position. The solid structure of the crystal does not allow the particles around the vacant spot to collapse.
When eight spheres of a unit cell meet at the center of a solid structure it tends to leave a little space which is called the interstitial site. It is known as an interstitial defect when another particle occupies this space or the interstitial site. As a matter of fact, the density of the solid increases herewith.
Unlike vacancy and interstitial defects, Schottky defect occurs when more than one particle is missing in the solid. One important thing that must be noted is that the number of missing cations is equal to that of missing anions. So, there is no change in the overall electrical composition of the solid. However, the mass of the object will decrease. The volume remains unchanged but the density will reduce. In case of too many particles being missing, the lattice structure may be disrupted, which in turn, might affect the stability of the solid. Such a thing can be observed in NaCl, KCl and the like.
A cation when missing from its original position and instead, occupying an interstitial site gives rise to Frenkel defect. It generally happens when the cation is smaller in size and can easily fit into the interstitial site. Electrical neutrality is not hampered in this type of defect. No atom is missing as such; there is just a change in the position of the particle. This phenomenon is usually observed in compounds in which there is a noticeable size difference between cations and anions, for example, AgCl.
Non – stoichiometric defects
let’s look into the concept of the non – stoichiometric defects, the ones that appear without hampering the stoichiometry of the crystalline substance. These defects can change the ratio of cations and anions in the solid. This imperfection is found in a large number of inorganic compounds. There are two kinds of non – stoichiometric defects.
Metal excess defects
Metal deficiency defects
Metal excess defect
The substance may have a surplus of metal ions in its space lattice which is eventually occupied by electrons to maintain neutrality. This generally occurs in two ways,
(a) Metal excess defect due to anionic vacancy
In a compound with excess metal ions, a vacancy is formed when an anion is missing from its position. An electron occupies this space to maintain electrical neutrality. The space occupied by the electron is called the F – center. This F – center is the reason behind the coloration of the compound. For instance, NaCl becomes yellow when heated.
(b) Metal excess defects due to extra cations
Many compounds with non – stoichiometric defects often release extra cations when heated. These cations can fit into interstitial sites. To balance out, an equal number of electrons do the same. The ultimate result is the abundance of metal in the solid. This defect is mostly found in transition elements. For example, Zinc oxide on being heated losses oxygen.
Metal deficiency defects
Some compounds may have lesser metal than their ideal stoichiometric proportions, making it difficult to prepare. In such compounds, sometimes a cation goes missing from the structure. However, the neutrality of the solid is not compromised as a neighboring ion takes up two charges. This defect is also found in transition elements that can have multiple valencies.
Impurity defect
When foreign atoms invade the solid structure by replacing some of the constituent atoms it is known as an impurity defect. The foreign atoms occupy the interstitial site of the parent atom.
There are three types of defects - Point defects, Non-Stoichiometric defects and Impurity defect.
Vacancy defect, Interstitial defect, Schottky defect, and Frenkel defect are some of the stoichiometric defects.
Metal excess defect occurs due to anionic vacancy and metal deficiency defect occurs due to cationic vacancy.
Impurity defect occurs when any foreign element replaces a constituent element.
Each and every substance that we find in our surroundings has some magnetic properties in it. Differing kinds of materials show totally different properties within the presence of a magnetic field.
The magnetic properties of a substance originate from the electrons present within the Atoms or molecules. Each electron in an atom behaves sort of like a little magnet. Electrons may be referred to as little loops of current that retain their magnetic moment.
The magnetic properties of a matter can be concluded by testing the arrangement of its electrons. When at least one electron is irregular in their orbit then the matter is paramagnetic in nature. Whereas when two electrons are attached together then the matter is called to be diamagnetic.
In case of unpaired electrons which give a similar direction is known as Ferromagnetic. When dipoles are presented in a balanced manner it is called Antiferromagnetic and in the case of Ferrimagnetic property there is a presence of an uneven number of unique and antiparallel presentations of magnetic moments. These magnetic moments come from 2 forms of motion of electrons:
The movement of an atom around the nucleus in its orbit.
When the electron spins around its own axis.
Magnetic Properties of Solids
The magnetic properties of a solid is derived from the magnetic property of the ions or atoms present in those solids. Additionally, specifically the magnetism and magnetization of a solid can rely on the movement of electrons in an atom. It will therefore be aforesaid that every electron of an atom behaves sort of like a magnet, lending the entire solid its magnetic property.
This magnetic behavior of the electrons of an atom is because of the movement patterns. They have specifically 2 forms of movement,
Electrons makes a rotation around the nucleus of the atom
Electrons also spin on their own axis, spins on opposite sides are labelled with + and – signs.
These 2 motions of the electrons offer the atom and the substance their magnetic power. These constant motions build an electrical field around the electrons, almost like a loop of current that lends it its magnetic property. On the basis of their magnetic properties, solids are often classified into 5 classes. Let’s get into more detail below-.
Paramagnetic
These substances are feeble magnetised in an external magnetic field. The direction is the same direction of the magnetic field. So that they gain a net magnetization once we take away the paramagnetic substance from the field, the alignment of electrons is interrupted and therefore the substance will lose its magnetic property. Therefore paramagnetic substances don't seem to be permanent magnets.
Paramagnetism is because of at least one pair of unmatched electrons in its orbit shell that get magnetized within the magnetic field. Some common examples are O2, Cu2 etc. These paramagnetic substances notice a variety of applications in natural philosophy.
Diamagnetic
Just like paramagnetism, in diamagnetism too the substances are magnetised in an external magnetic field. However diamagnetic solids are repelled within the field. The magnetic property settled within them is in the opposite direction of the magnetic fields and therefore they have.
Diamagnetism
In diamagnetic substances, all electrons in their last shell are paired, there aren't any valence electrons. This can be the reason that the magnetic moment of their atoms is almost zero. Examples are substances like common salt, benzene etc. Being such bad conductors, we have a tendency to use them as insulators.
Ferromagnetic
Now, these solids are powerfully magnetised once we place them in an external magnetic field. Besides the terribly sturdy attraction forces, these solids will actually be magnetised permanently. This implies that even when the external magnetic fields are removed the solids can retain their magnetic properties.
Ferromagnetism
It is a very popular theory that the ferromagnetic composition has unique characteristics. They have what we call ‘domains’ that is a unique assembly of metal ions. Every domain is analogous to a small magnet. In an electromagnetic field, these domains set up themselves and align themselves with the magnetic field. In an exceedingly non-magnetized metal, these domains are randomly organized and it cancels out their magnetic properties.
The examples of ferromagnetic solids include cobalt, Nickel, and chromium compounds etc. and they have widespread industrial and everyday uses.
Antiferromagnetic
In antiferromagnetic, the domain structures of the solid are terribly the same as those of ferromagnetic solids. However here the domains are oppositely oriented. This implies they get rid of each other’s magnetism.
Antiferromagnetism Ordering
Ferrimagnetic
These substances occur when magnetic moments are aligned in both directions (parallel also as anti-parallel) however in unequal numbers. These are infirm attracted to magnetic fields. Additionally on heating, these substances can lose their ferromagnetism altogether. Examples are iron ore and ferrites of zinc and Mg.
Ferrimagnetic Ordering
Magnetic properties occur due to the spin and orbital angular momentum of the electrons inside a compound.
Compounds are diamagnetic when they contain no unpaired electrons.
Molecular compounds containing one or more unpaired electrons are paramagnetic.
Magnetic properties of solids are: Diamagnetic, Paramagnetic, Ferromagnetic.
An interesting characteristic of transition metals is their ability to form magnets. Metal complexes that have unpaired electrons are magnetic.
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