General Characteristics Of Solid State

Matter exists in three states: solid, liquid, and gas. The state of a substance at a given temperature and pressure is determined by a balance between two opposing factors:

• Intermolecular forces: Tend to hold constituent particles (atoms, molecules, ions) close together.
• Thermal energy: Causes particles to move and tends to keep them apart.

At sufficiently low temperatures, thermal energy is minimal, allowing strong intermolecular forces to hold particles in fixed positions. Although fixed, these particles can still oscillate around their mean positions, defining the solid state.

Characteristic properties of solids:

• They possess a definite mass, volume, and shape.
• The distances between constituent particles are short.
• Intermolecular forces holding the particles together are strong.
• Constituent particles occupy fixed positions and can only perform vibrational motion around their average locations.
• Solids are generally incompressible and rigid.

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Amorphous And Crystalline Solids

Solids are classified based on the arrangement pattern of their constituent particles:

• Crystalline solids: Consist of a large number of small crystals, each having a well-defined, characteristic geometrical shape. The arrangement of particles is highly ordered and repeats periodically in three dimensions throughout the entire crystal. This ordered arrangement extends over long distances, showing long-range order. Examples: Sodium chloride ($\textsf{NaCl}$), quartz, diamond, metals like iron and copper.
• Amorphous solids: Lack a definite characteristic geometrical shape. The term "amorphous" means "no form." The arrangement of constituent particles has only short-range order, meaning a regular, repeating pattern is observed only over small distances. The arrangement is disordered over longer distances. Examples: Glass, rubber, plastics, quartz glass.

Differences in particle arrangement lead to distinct properties:

• Melting Point: Crystalline solids have a sharp, definite melting point; they melt abruptly at a specific temperature. Amorphous solids soften gradually over a range of temperatures.
• Cleavage Property: When cut with a sharp tool, crystalline solids cleave along specific planes, producing pieces with smooth surfaces. Amorphous solids cut into pieces with irregular surfaces.
• Heat of Fusion: Crystalline solids have a definite, characteristic enthalpy of fusion (energy required to melt a specific amount). Amorphous solids do not have a definite heat of fusion as they melt over a range.
• Anisotropy vs Isotropicity: Crystalline solids are anisotropic. Some physical properties (like electrical resistance, refractive index) have different values when measured along different directions due to the ordered but direction-dependent arrangement of particles. Amorphous solids are isotropic. Properties are the same in all directions because the short-range order and overall disorder make the arrangement statistically equivalent in all directions.
• Nature: Crystalline solids are considered true solids. Amorphous solids are sometimes called pseudo solids or supercooled liquids due to their tendency to flow, although very slowly (e.g., ancient glass objects can appear milky due to slow crystallisation).

Summary of distinctions:

Property	Crystalline solids	Amorphous solids
Shape	Definite characteristic geometrical shape	Irregular shape
Melting point	Melt at a sharp and characteristic temperature	Gradually soften over a range of temperature
Cleavage property	Split into two pieces with plain, smooth surfaces when cut	Cut into two pieces with irregular surfaces when cut
Heat of fusion	Definite and characteristic enthalpy of fusion	Do not have definite enthalpy of fusion
Anisotropy	Anisotropic in nature (properties vary with direction)	Isotropic in nature (properties are same in all directions)
Nature	True solids	Pseudo solids or supercooled liquids
Order in arrangement	Long range order	Only short range order

Some solids appear amorphous but are actually polycrystalline, consisting of many randomly oriented tiny crystals. Metals are often polycrystalline, appearing isotropic overall despite individual anisotropic crystals.

Amorphous solids like glass, rubber, and plastics are widely used. Amorphous silicon is important in solar cells.

Classification Of Crystalline Solids

Crystalline solids can be classified based on the nature of the intermolecular forces or bonds holding their constituent particles together. This classification divides them into four categories:

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Molecular Solids

Constituent particles are molecules. These are further sub-divided based on the intermolecular forces:

1. Nonpolar Molecular Solids: Constituent particles are either atoms (like Ar, He) or nonpolar molecules ($\textsf{H}_2$, $\textsf{Cl}_2$, $\textsf{I}_2$, $\textsf{CO}_2$). Held together by weak dispersion forces (London forces). These are soft, non-conductors of electricity, have low melting points, and are usually gases or liquids at room temperature and pressure.
2. Polar Molecular Solids: Constituent particles are molecules formed by polar covalent bonds ($\textsf{HCl}$, $\textsf{SO}_2$, $\textsf{NH}_3$). Held together by stronger dipole-dipole interactions. These are soft, non-conductors of electricity, have higher melting points than nonpolar molecular solids, but are often gases or liquids at room temperature and pressure.
3. Hydrogen Bonded Molecular Solids: Molecules contain polar covalent bonds between $\textsf{H}$ and highly electronegative atoms like $\textsf{F}$, $\textsf{O}$, or $\textsf{N}$. Held together by strong hydrogen bonds. Example: Ice ($\textsf{H}_2\textsf{O}$). These are non-conductors of electricity. Generally volatile liquids or soft solids at room temperature and pressure.

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Ionic Solids

Constituent particles are ions (cations and anions). Formed by three-dimensional arrangement of ions held together by strong coulombic (electrostatic) forces. Examples: $\textsf{NaCl}$, $\textsf{MgO}$, $\textsf{ZnS}$, $\textsf{CaF}_2$. These solids are hard and brittle. They have high melting and boiling points. In the solid state, ions are fixed, so they are electrical insulators. In the molten state or when dissolved in water, ions are free to move and conduct electricity.

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Metallic Solids

Constituent particles are positive metal ions (cations) surrounded by and held together by a 'sea' of mobile or delocalised electrons (metallic bonding). Each metal atom contributes valence electrons to this mobile pool, which are spread throughout the crystal. Examples: $\textsf{Fe}$, $\textsf{Cu}$, $\textsf{Ag}$, $\textsf{Mg}$. These solids are typically hard (though variable), malleable (can be hammered into sheets), and ductile (can be drawn into wires). Mobile electrons are responsible for their high electrical and thermal conductivity in both solid and molten states, as well as their characteristic lustre and color.

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Covalent Or Network Solids

Constituent particles are atoms held together by a network of continuous covalent bonds extending throughout the crystal. These are also called giant molecules. Covalent bonds are strong and directional, holding atoms in fixed positions. Examples: Diamond ($\textsf{C}$), silicon carbide ($\textsf{SiC}$), quartz ($\textsf{SiO}_2$), aluminum nitride ($\textsf{AlN}$). These are very hard and brittle, have extremely high melting points (often decompose before melting), and are electrical insulators.

An exception is Graphite ($\textsf{C}$). It is a covalent solid but is soft and conducts electricity. Its structure consists of layers of carbon atoms, where each carbon is covalently bonded to three neighbors in the same layer. The fourth valence electron is delocalised between layers, allowing it to conduct electricity. Weak Van der Waals forces between layers allow them to slide, making graphite soft and a good lubricant.

Summary of properties of different crystalline solids:

Type of Solid	Constituent Particles	Bonding/Attractive Forces	Examples	Physical Nature	Electrical Conductivity	Melting Point
Molecular solids	Molecules	Dispersion or London forces	Ar, CCl4, H2, I2, CO2	Soft	Insulator	Very low
	Molecules	Dipole-dipole interactions	HCl, SO2	Soft	Insulator	Low
	Molecules	Hydrogen bonding	H2O (ice)	Hard	Insulator	Low
Ionic solids	Ions	Coulombic or electrostatic	NaCl, MgO, ZnS, CaF2	Hard but brittle	Insulators in solid state but conductors in molten state and in aqueous solutions	High
Metallic solids	Positive ions in a sea of delocalised electrons	Metallic bonding	Fe, Cu, Ag, Mg	Hard but malleable and ductile	Conductors in solid state as well as in molten state	Fairly high
Covalent or network solids	Atoms	Covalent bonding	SiO2 (quartz), SiC, C (diamond), AlN	Very hard	Insulators	Very high
	Atoms		C (graphite)	Soft	Conductor (exception)	Very high

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Crystal Lattices And Unit Cells

In crystalline solids, constituent particles are arranged in a highly ordered, repeating pattern. This pattern can be visualised using a space lattice or crystal lattice, which is a three-dimensional arrangement of points representing the locations of the constituent particles (motifs).

Characteristics of a crystal lattice:

• Each point in the lattice is a lattice point or lattice site.
• Each lattice point represents one constituent particle (atom, molecule, or ion).
• Lattice points are connected by straight lines to show the lattice geometry.

A crystal lattice can be completely described by its smallest repeating unit, called a unit cell. The entire crystal structure is generated by repeating the unit cell through translational displacements in three dimensions.

In three dimensions, a unit cell is characterised by six parameters:

• Three edge lengths: a, b, and c.
• Three angles between the edges: $\alpha$ (between b and c), $\beta$ (between a and c), and $\gamma$ (between a and b).

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Primitive And Centred Unit Cells

Unit cells are broadly categorised into two types:

1. Primitive Unit Cells: Constituent particles are located only at the corner positions of the unit cell.
2. Centred Unit Cells: Contain one or more constituent particles at positions other than the corners, in addition to those at the corners. There are three types of centred unit cells:
   • Body-Centred Unit Cells (bcc): Have one particle at the body centre in addition to the corner particles.
   • Face-Centred Unit Cells (fcc): Have one particle at the center of each face in addition to the corner particles.
   • End-Centred Unit Cells: Have one particle at the center of any two opposite faces in addition to the corner particles.

Based on the possible variations in edge lengths and angles, there are seven basic types of primitive unit cells, known as the seven crystal systems. A French mathematician, Bravais, showed that considering the possible centred unit cells within these crystal systems, there are only 14 unique three-dimensional lattices, called Bravais lattices.

Characteristics of the seven primitive unit cells and their possible centred variations:

Crystal system	Possible variations	Axial distances or edge lengths	Axial angles	Examples
Cubic	Primitive, Body-centred, Face-centred	a = b = c	$\alpha = \beta = \gamma = 90^\circ$	NaCl, Zinc blende, Cu
Tetragonal	Primitive, Body-centred	a = b $\neq$ c	$\alpha = \beta = \gamma = 90^\circ$	White tin, SnO2, TiO2, CaSO4
Orthorhombic	Primitive, Body-centred, Face-centred, End-centred	a $\neq$ b $\neq$ c	$\alpha = \beta = \gamma = 90^\circ$	Rhombic sulphur, KNO3, BaSO4
Hexagonal	Primitive	a = b $\neq$ c	$\alpha = \beta = 90^\circ$, $\gamma = 120^\circ$	Graphite, ZnO, CdS
Rhombohedral or Trigonal	Primitive	a = b = c	$\alpha = \beta = \gamma \neq 90^\circ$	Calcite (CaCO3), HgS (cinnabar)
Monoclinic	Primitive, End-centred	a $\neq$ b $\neq$ c	$\alpha = \gamma = 90^\circ$, $\beta \neq 90^\circ$	Monoclinic sulphur, Na2SO4.10H2O
Triclinic	Primitive	a $\neq$ b $\neq$ c	$\alpha \neq \beta \neq \gamma \neq 90^\circ$	K2Cr2O7, CuSO4. 5H2O, H3BO3

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Number Of Atoms In A Unit Cell

In a crystalline solid, constituent particles are located at the lattice points of the unit cell. We need to calculate the effective number of particles belonging to a single unit cell based on their positions (corners, faces, body centre).

Let's consider cubic unit cells (primitive, body-centred, face-centred), assuming identical atoms as constituent particles.

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Primitive Cubic Unit Cell

A primitive cubic unit cell has atoms only at its eight corners. As seen in Fig. 1.12, each corner atom is shared by eight adjacent cubic unit cells (four in the same layer, four in the layer above/below). Thus, only $1/8$th of a corner atom belongs to any one unit cell.

Effective number of atoms per primitive cubic unit cell:

• Contribution from corners = 8 corners $\times$ $1/8$ atom/corner = 1 atom

Total number of atoms per primitive cubic unit cell = 1 atom.

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Body-Centred Cubic Unit Cell

A body-centred cubic (bcc) unit cell has atoms at its eight corners and one atom at the body centre. The atom at the body centre is entirely within the unit cell and is not shared with any other unit cell.

Effective number of atoms per body-centred cubic unit cell:

• Contribution from corners = 8 corners $\times$ $1/8$ atom/corner = 1 atom
• Contribution from body centre = 1 atom at body centre $\times$ 1 = 1 atom

Total number of atoms per body-centred cubic unit cell = $1 + 1$ = 2 atoms.

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Face-Centred Cubic Unit Cell

A face-centred cubic (fcc) unit cell has atoms at its eight corners and one atom at the center of each of its six faces. Each atom at a face centre is shared between two adjacent unit cells.

Thus, $1/2$ of a face-centred atom belongs to any one unit cell.

Effective number of atoms per face-centred cubic unit cell:

• Contribution from corners = 8 corners $\times$ $1/8$ atom/corner = 1 atom
• Contribution from face centers = 6 face centers $\times$ $1/2$ atom/face = 3 atoms

Total number of atoms per face-centred cubic unit cell = $1 + 3$ = 4 atoms.

Close Packed Structures

Constituent particles in solids are often packed as closely as possible to minimise vacant space. This close packing can be built up by stacking layers of identical hard spheres.

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Close Packing In One Dimension

Spheres are arranged in a single row, touching each other.

Each sphere is in contact with two neighbors. The coordination number (number of nearest neighbors) is 2.

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Close Packing In Two Dimensions

Formed by stacking rows of one-dimensionally close-packed spheres. Two ways of stacking:

1. Square Close Packing (AAA type): Second row spheres are placed directly above the first row, aligned horizontally and vertically (A type over A type).
   Each sphere contacts four neighbors. Two-dimensional coordination number is 4. Joining centers of neighbors forms a square.
2. Hexagonal Close Packing (ABAB type): Second row spheres are placed in the depressions of the first row (staggered). Third row aligned with first (A), fourth with second (B), etc. (ABAB type).
   This arrangement has less empty space and is more efficient. Each sphere contacts six neighbors. Two-dimensional coordination number is 6. Joining centers of neighbors forms a hexagon. Triangular voids (empty spaces) are present between the spheres. These voids have two different orientations (apex pointing up or down).

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Close Packing In Three Dimensions

Three-dimensional structures are built by stacking two-dimensional layers. Two types of 2D layers are possible: square close-packed and hexagonal close-packed.

1. From 2D Square Close-Packed Layers (AAA... type): Stacking square close-packed layers directly on top of each other (A type over A type over A type...).
   This forms a simple cubic lattice, with a primitive cubic unit cell.
2. From 2D Hexagonal Close-Packed Layers: More efficient 3D packing is obtained by stacking hexagonal close-packed layers.
   1. Placing the second layer (B) over the first layer (A): Spheres of the second layer are placed in the depressions of the first. This generates two types of voids (empty spaces):
      • Tetrahedral voids (T): Formed wherever a sphere of the second layer is above a void of the first layer (or vice versa). Surrounded by four spheres, forming a tetrahedron when centers are joined.
      • Octahedral voids (O): Formed where triangular voids in the first and second layers overlap (but their apexes point in opposite directions). Surrounded by six spheres, forming an octahedron when centers are joined.
      If N is the number of close-packed spheres, then the number of octahedral voids is N, and the number of tetrahedral voids is 2N.
   2. Placing the third layer over the second layer: Two possibilities:
      • Covering Tetrahedral Voids (ABAB... type): Spheres of the third layer cover the tetrahedral voids of the second layer and are aligned with the first layer. This results in a repeating pattern ABAB... This structure is called hexagonal close packed (hcp) structure. Metals like Magnesium ($\textsf{Mg}$) and Zinc ($\textsf{Zn}$) often crystallise in hcp.
      • Covering Octahedral Voids (ABCABC... type): Spheres of the third layer cover the octahedral voids of the second layer. The third layer is not aligned with the first or second layer. The fourth layer is aligned with the first. This results in a repeating pattern ABCABC... This structure is called cubic close packed (ccp) structure or face-centred cubic (fcc) structure. Metals like Copper ($\textsf{Cu}$) and Silver ($\textsf{Ag}$) crystallise in ccp/fcc.
   Both hcp and ccp/fcc structures are highly efficient, filling 74% of the total space. In both structures, each sphere has 12 nearest neighbors, so the coordination number is 12.

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Formula Of A Compound And Number Of Voids Filled

In ionic solids, larger ions (usually anions) often form a close-packed structure (ccp or hcp), while smaller ions (usually cations) occupy some of the tetrahedral or octahedral voids. The fraction of voids occupied depends on the compound's stoichiometry.

Answer:

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Locating Tetrahedral And Octahedral Voids

Let's locate these voids within a cubic close-packed (ccp) or face-centred cubic (fcc) unit cell.

Effective number of atoms in a ccp unit cell is 4.

(a) Locating Tetrahedral Voids:

• Consider a ccp unit cell. It can be divided into eight small cubes.
• Each small cube has atoms at alternate corners. In total, a small cube has 4 atoms (if we consider the atoms on the corners of the main unit cell).
• A tetrahedral void is located at the body centre of each of these eight small cubes. It is surrounded by the 4 atoms located at alternate corners of the small cube.
• Since there are eight such small cubes in a ccp unit cell, there are eight tetrahedral voids per ccp unit cell.

This confirms that the number of tetrahedral voids (8) is twice the effective number of atoms (4) in a ccp unit cell (2N).

(b) Locating Octahedral Voids:

• Consider a ccp unit cell. There is one octahedral void located at the exact body centre of the cube. This void is surrounded by the 6 atoms at the face centers of the unit cell.
• Octahedral voids are also located at the centre of each edge of the cube. There are 12 edges. Each edge center is surrounded by 6 atoms (4 from the same unit cell's corners and face centers, 2 from adjacent unit cells).
• An octahedral void at an edge centre is shared between four adjacent unit cells. So, each unit cell gets $1/4$th contribution from each edge void.

Total number of octahedral voids in a ccp unit cell:

• Void at body centre = 1
• Voids at edge centres = 12 edge centers $\times$ $1/4$ void/edge = 3

Total number of octahedral voids per ccp unit cell = $1 + 3$ = 4.

This confirms that the number of octahedral voids (4) is equal to the effective number of atoms (4) in a ccp unit cell (N).

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Packing Efficiency

Packing efficiency is the percentage of the total space in a crystal that is filled by the constituent particles. The remaining space is in the form of voids. Packing efficiency is calculated as:

$\textsf{Packing efficiency} = \frac{\textsf{Volume occupied by particles in the unit cell}}{\textsf{Total volume of the unit cell}} \times 100\%$

Let's calculate the packing efficiency for different cubic structures, assuming particles are identical hard spheres.

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Packing Efficiency In Hcp And Ccp Structures

Both hcp and ccp (fcc) structures have the highest packing efficiency among common crystal lattices. Let's calculate the efficiency for the ccp structure.

In a ccp structure, the unit cell is a face-centred cubic (fcc) cell. Atoms are at corners and face centers. Particles touch along the face diagonal.

Let the edge length of the unit cell be 'a' and the radius of each sphere be 'r'. The face diagonal (b) is related to the edge length by the Pythagorean theorem in a face-diagonal triangle (e.g., ABC in Fig 1.24): $b^2 = a^2 + a^2 = 2a^2$, so $b = a\sqrt{2}$.

Along the face diagonal, the spheres at the corners and the face center touch each other. The length of the face diagonal (b) is equal to $4r$ (r from one corner atom, 2r from the face-centered atom, r from the other corner atom): $b = r + 2r + r = 4r$.

So, $4r = a\sqrt{2}$, which gives the relationship between edge length and radius: $a = \frac{4r}{\sqrt{2}} = 2r\sqrt{2}$.

In a ccp/fcc unit cell, the effective number of atoms is $z = 4$.

Volume occupied by spheres in the unit cell = $z \times \text{Volume of one sphere} = 4 \times (\frac{4}{3}\pi r^3) = \frac{16}{3}\pi r^3$.

Volume of the unit cell = $a^3 = (2r\sqrt{2})^3 = (2\sqrt{2})^3 r^3 = 8 \times 2\sqrt{2} r^3 = 16\sqrt{2} r^3$.

Packing efficiency = $\frac{\textsf{Volume occupied by spheres}}{\textsf{Volume of unit cell}} \times 100\% = \frac{\frac{16}{3}\pi r^3}{16\sqrt{2} r^3} \times 100\%$

Packing efficiency = $\frac{\frac{1}{3}\pi}{\sqrt{2}} \times 100\% = \frac{\pi}{3\sqrt{2}} \times 100\% \approx \frac{3.14159}{3 \times 1.41421} \times 100\% \approx \frac{3.14159}{4.24264} \times 100\% \approx 0.7404 \times 100\% \approx 74\%$.

Both hcp and ccp/fcc structures have a packing efficiency of 74%.

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Efficiency Of Packing In Body-Centred Cubic Structures

In a body-centred cubic (bcc) structure, atoms are at corners and the body centre. Particles touch each other along the body diagonal.

Let the edge length be 'a' and radius 'r'. Consider a triangle along a face diagonal (e.g., EFD in Fig 1.25) with face diagonal $b = a\sqrt{2}$. Now consider a triangle along the body diagonal (e.g., AFD) with edge 'a', face diagonal 'b', and body diagonal 'c'. $c^2 = a^2 + b^2 = a^2 + (a\sqrt{2})^2 = a^2 + 2a^2 = 3a^2$, so $c = a\sqrt{3}$.

Along the body diagonal, the corner atom, the body-centered atom, and the opposite corner atom touch each other. The length of the body diagonal (c) is equal to $4r$: $c = r + 2r + r = 4r$.

So, $4r = a\sqrt{3}$, which gives the relationship between edge length and radius: $a = \frac{4r}{\sqrt{3}}$.

In a bcc unit cell, the effective number of atoms is $z = 2$.

Volume occupied by spheres in the unit cell = $z \times \text{Volume of one sphere} = 2 \times (\frac{4}{3}\pi r^3) = \frac{8}{3}\pi r^3$.

Volume of the unit cell = $a^3 = (\frac{4r}{\sqrt{3}})^3 = \frac{64r^3}{3\sqrt{3}}$.

Packing efficiency = $\frac{\textsf{Volume occupied by spheres}}{\textsf{Volume of unit cell}} \times 100\% = \frac{\frac{8}{3}\pi r^3}{\frac{64r^3}{3\sqrt{3}}} \times 100\%$

Packing efficiency = $\frac{8\pi/3}{64/(3\sqrt{3})} \times 100\% = \frac{8\pi}{64/\sqrt{3}} \times 100\% = \frac{\pi\sqrt{3}}{8} \times 100\% \approx \frac{3.14159 \times 1.73205}{8} \times 100\% \approx \frac{5.4413}{8} \times 100\% \approx 0.6802 \times 100\% \approx 68\%$.

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Packing Efficiency In Simple Cubic Lattice

In a simple cubic lattice, atoms are only at the corners. Particles touch each other along the edges of the cube.

Let the edge length be 'a' and radius 'r'. Since particles touch along the edge, the edge length is equal to twice the radius: $a = 2r$.

In a simple cubic unit cell, the effective number of atoms is $z = 1$.

Volume occupied by spheres in the unit cell = $z \times \text{Volume of one sphere} = 1 \times (\frac{4}{3}\pi r^3) = \frac{4}{3}\pi r^3$.

Volume of the unit cell = $a^3 = (2r)^3 = 8r^3$.

Packing efficiency = $\frac{\textsf{Volume occupied by spheres}}{\textsf{Volume of unit cell}} \times 100\% = \frac{\frac{4}{3}\pi r^3}{8r^3} \times 100\%$

Packing efficiency = $\frac{4\pi/3}{8} \times 100\% = \frac{\pi}{6} \times 100\% \approx \frac{3.14159}{6} \times 100\% \approx 0.5236 \times 100\% \approx 52.4\%$.

Conclusion: Close-packed structures (hcp and ccp) have the highest packing efficiency (74%), followed by bcc (68%), and simple cubic (52.4%) has the lowest packing efficiency among these cubic structures.

Calculations Involving Unit Cell Dimensions

The dimensions of a unit cell can be used to calculate properties like density or atomic/molar mass if other parameters are known. The density of a unit cell is the same as the density of the substance it represents.

Density (d) of a unit cell is given by:

$\textsf{d} = \frac{\textsf{Mass of the unit cell}}{\textsf{Volume of the unit cell}}$

Mass of the unit cell = Number of atoms in the unit cell (z) $\times$ Mass of a single atom (m).

The mass of a single atom (m) can be calculated from the molar mass (M) and Avogadro's number ($\textsf{N}_\textsf{A}$): $\textsf{m} = \frac{\textsf{M}}{\textsf{N}_\textsf{A}}$.

Volume of a cubic unit cell with edge length 'a' is $\textsf{a}^3$.

So, for a cubic crystal, the density formula is:

$\textsf{d} = \frac{\textsf{z} \times \textsf{m}}{\textsf{a}^3} = \frac{\textsf{z} \times \frac{\textsf{M}}{\textsf{N}_\textsf{A}}}{\textsf{a}^3} = \frac{\textsf{zM}}{\textsf{a}^3 \textsf{N}_\textsf{A}}$

This equation relates density (d), number of atoms per unit cell (z), molar mass (M), edge length (a), and Avogadro's number ($\textsf{N}_\textsf{A}$). If any four of these are known, the fifth can be calculated.

Answer:

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Imperfections In Solids

Even though crystalline solids have long-range order, they are not perfectly ordered. Real crystals contain irregularities in the arrangement of constituent particles, called crystal defects or imperfections. Defects are more common when crystallisation occurs rapidly or at moderate rates. Even slow crystallisation yields crystals with some defects.

Defects are classified into two main types:

• Point defects: Irregularities or deviations from the ideal arrangement around a point (atom or lattice site) in the crystal.
• Line defects: Irregularities or deviations from the ideal arrangement along entire rows of lattice points (e.g., dislocations).

We will focus on point defects.

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Types Of Point Defects

Point defects are categorised into three types:

1. Stoichiometric defects: These point defects do not change the overall stoichiometry of the solid. They are also called intrinsic or thermodynamic defects.
2. Impurity defects: Arise due to the presence of foreign atoms or ions in the crystal lattice.
3. Non-stoichiometric defects: Disturb the stoichiometry of the solid, resulting in a non-stoichiometric composition.

(a) Stoichiometric Defects

These are defects where the ratio of positive and negative ions (or atoms) remains the same as in the ideal stoichiometric formula. Two basic types:

• Vacancy Defect: Occurs when some lattice sites that should be occupied by constituent particles are vacant.
  This defect results in a decrease in the density of the substance. It can be caused by heating the solid.
• Interstitial Defect: Occurs when some constituent particles (atoms or molecules) occupy interstitial sites (spaces between lattice points) in addition to their normal lattice sites.
  This defect results in an increase in the density of the substance.

Vacancy and interstitial defects, as described above, are typically observed in non-ionic solids. In ionic solids, defects must maintain electrical neutrality. Instead of simple vacancy or interstitial defects, ionic solids show these defects in slightly different forms: Frenkel and Schottky defects.

• Frenkel Defect: Shown by ionic solids. The smaller ion (usually the cation) is displaced from its normal lattice site to an interstitial site within the crystal.
  This creates a vacancy defect at the original site and an interstitial defect at the new location. It is also called a dislocation defect. It does not change the density of the solid (as particles are just relocated within the crystal). Frenkel defects occur in ionic compounds with a large difference in the size of cations and anions, where the cation is small enough to fit into interstitial sites. Examples: $\textsf{ZnS}$, $\textsf{AgCl}$, $\textsf{AgBr}$, $\textsf{AgI}$ (due to small $\textsf{Zn}^{2+}$ and $\textsf{Ag}^{+}$ ions).
• Schottky Defect: Shown by ionic solids. It is essentially a vacancy defect involving missing ions from the lattice. To maintain electrical neutrality, equal numbers of cations and anions are missing from their lattice sites.
  Like simple vacancy defects, Schottky defects cause a decrease in the density of the solid. They are common in ionic compounds where cations and anions are of similar size. Examples: $\textsf{NaCl}$, $\textsf{KCl}$, $\textsf{CsCl}$, $\textsf{AgBr}$. $\textsf{AgBr}$ is unique in showing both Frenkel and Schottky defects. Schottky defects can be present in significant numbers (e.g., $\sim 10^6$ Schottky pairs per $\textsf{cm}^3$ in $\textsf{NaCl}$ at room temperature).

(b) Impurity Defects

Impurity defects arise when foreign atoms or ions are present in the crystal lattice. In ionic solids, if the impurity ions have a different charge (valence) than the host ions, vacancies may be created to maintain electrical neutrality.

Example: Crystallising molten $\textsf{NaCl}$ containing a small amount of $\textsf{SrCl}_2$. $\textsf{Sr}^{2+}$ ions (from $\textsf{SrCl}_2$) have a +2 charge, while $\textsf{Na}^{+}$ ions have a +1 charge. To maintain neutrality, one $\textsf{Sr}^{2+}$ ion replaces two $\textsf{Na}^{+}$ ions. One $\textsf{Na}^{+}$ site is occupied by $\textsf{Sr}^{2+}$, and the other $\textsf{Na}^{+}$ site remains vacant. This creates cation vacancies equal in number to the $\textsf{Sr}^{2+}$ ions introduced.

Similar impurity defects are observed in solid solutions of $\textsf{CdCl}_2$ and $\textsf{AgCl}$ (where $\textsf{Cd}^{2+}$ replaces $\textsf{Ag}^{+}$ ions, creating $\textsf{Ag}^{+}$ vacancies).

(c) Non-Stoichiometric Defects

These defects disturb the ideal stoichiometric ratio of elements in a crystal. They result in solids with a non-stoichiometric composition (e.g., $\textsf{Fe}_{0.95}\textsf{O}$). Two types:

• Metal Excess Defect: The crystal contains an excess amount of the metal element.
  • Metal excess due to anionic vacancies: Alkali halides ($\textsf{NaCl}$, $\textsf{KCl}$) show this. When heated in metal vapor (e.g., $\textsf{NaCl}$ in $\textsf{Na}$ vapor), metal atoms deposit on the surface. Halide ions ($\textsf{Cl}^{-}$) diffuse to the surface to combine with metal atoms ($\textsf{Na}$), forming metal halide. Metal atoms lose electrons ($\textsf{Na} \to \textsf{Na}^{+} + \textsf{e}^{-}$). The released electrons diffuse into the crystal and occupy the anionic vacancies created by the halide ions leaving. These anionic sites occupied by unpaired electrons are called F-centres. F-centres absorb light and impart color to the crystal (e.g., yellow for $\textsf{NaCl}$ with excess $\textsf{Na}$, pink for $\textsf{LiCl}$ with excess $\textsf{Li}$, violet for $\textsf{KCl}$ with excess $\textsf{K}$).
  • Metal excess due to extra cations at interstitial sites: $\textsf{ZnO}$ is white but turns yellow on heating. It loses oxygen ($\textsf{ZnO} \to \textsf{Zn}^{2+} + 1/2\textsf{O}_2 + 2\textsf{e}^{-}$). Excess $\textsf{Zn}^{2+}$ ions move to interstitial sites, and the electrons occupy neighboring interstitial sites to maintain neutrality. The formula becomes $\textsf{Zn}_{1+\text{x}}\textsf{O}$.
• Metal Deficiency Defect: The crystal contains less amount of metal compared to the stoichiometric ratio. These solids are often difficult to prepare in exact stoichiometric composition.
  • Example: $\textsf{FeO}$ is often found as $\textsf{Fe}_{0.95}\textsf{O}$ (ranging from $\textsf{Fe}_{0.93}\textsf{O}$ to $\textsf{Fe}_{0.96}\textsf{O}$). Some $\textsf{Fe}^{2+}$ cations are missing from the lattice sites. To compensate for the loss of positive charge, some $\textsf{Fe}^{2+}$ ions are replaced by $\textsf{Fe}^{3+}$ ions in the lattice. The presence of $\textsf{Fe}^{3+}$ ions maintains electrical neutrality despite the missing $\textsf{Fe}^{2+}$ vacancies.

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Electrical Properties

Solids exhibit a vast range of electrical conductivities. Based on conductivity, solids are classified into three types:

1. Conductors: High conductivity ($10^4$ to $10^7 \text{ ohm}^{-1}\text{m}^{-1}$). Metals are good conductors ($\sim 10^7 \text{ ohm}^{-1}\text{m}^{-1}$).
2. Insulators: Very low conductivity ($10^{-20}$ to $10^{-10} \text{ ohm}^{-1}\text{m}^{-1}$).
3. Semiconductors: Intermediate conductivity ($10^{-6}$ to $10^4 \text{ ohm}^{-1}\text{m}^{-1}$).

Electrical conductivity in solids can be due to the movement of electrons (metallic conductors) or ions (electrolytes).

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Conduction Of Electricity In Metals

Metals conduct electricity in both solid and molten states. Their conductivity is explained by band theory, based on the formation of molecular orbitals from atomic orbitals of metal atoms. These molecular orbitals are very close in energy, forming continuous energy bands.

• Valence band: Contains valence electrons.
• Conduction band: A higher energy band, unoccupied or partially occupied.

In metals, the valence band is either partially filled or overlaps with the conduction band. Electrons can easily move into the empty energy levels within the partially filled band or readily jump to the overlapping conduction band when an electric field is applied. This free movement of electrons results in high conductivity.

In insulators, there is a large energy gap (called the forbidden gap) between the filled valence band and the unoccupied conduction band. Electrons require significant energy to jump across this gap, so conductivity is very low.

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Conduction Of Electricity In Semiconductors

In semiconductors, the energy gap between the valence band and the conduction band is smaller than in insulators. At room temperature, some electrons gain enough thermal energy to jump from the valence band to the conduction band, allowing for some conductivity. Conductivity increases with increasing temperature as more electrons can jump to the conduction band.

Pure semiconductors like silicon ($\textsf{Si}$) and germanium ($\textsf{Ge}$) are called intrinsic semiconductors. Their conductivity is too low for practical use.

The conductivity of intrinsic semiconductors can be significantly increased by adding a suitable impurity, a process called doping. Doping introduces electronic defects and creates extrinsic semiconductors.

Types of doping based on impurity valence:

1. Doping with electron-rich impurities (Group 15 elements): Elements from Group 14 ($\textsf{Si}$, $\textsf{Ge}$) have 4 valence electrons. When doped with a Group 15 element (P, As) which has 5 valence electrons, the impurity atom replaces a host atom in the crystal lattice. Four of the impurity's valence electrons form covalent bonds with neighbors, but the fifth electron is extra and becomes delocalised. These free electrons increase conductivity. Since conductivity is due to negatively charged electrons, the semiconductor is called an n-type semiconductor ('n' for negative).
2. Doping with electron-deficit impurities (Group 13 elements): When doped with a Group 13 element (B, Al, Ga) which has only 3 valence electrons, the impurity atom replaces a host atom. The three valence electrons form covalent bonds with neighbors, but there is a missing electron in the fourth bond, creating an electron hole or electron vacancy. An electron from a neighboring atom can fill this hole, leaving a hole at its original position. Under an electric field, electrons move towards the positive terminal by filling holes, making it appear as if the positive holes are moving towards the negative terminal. Since conductivity is due to these apparent positive holes, the semiconductor is called a p-type semiconductor ('p' for positive).

Applications of n-type and p-type semiconductors:

• Diode: Formed by joining n-type and p-type semiconductors, used as a rectifier (converts AC to DC).
• Transistors: Formed by sandwiching a layer of one type between two layers of the other (npn or pnp). Used for detecting or amplifying signals.
• Solar cells: Photo-diodes that convert light energy into electrical energy.

Other semiconductors are formed by combining elements from different groups to achieve an average valence of four, similar to Group 14 elements. Examples: Group 13-15 compounds (InSb, AlP, GaAs - Gallium arsenide is fast response), Group 12-16 compounds (ZnS, CdS, CdSe, HgTe - ionic character varies).

Transition metal oxides also show varying electrical properties (metallic or insulating) depending on the material and temperature (e.g., $\textsf{TiO}$, $\textsf{CrO}_2$, $\textsf{ReO}_3$ are metallic; $\textsf{VO}$, $\textsf{VO}_2$, $\textsf{VO}_3$, $\textsf{TiO}_3$ show temperature-dependent behavior).

Magnetic Properties

Every substance has magnetic properties originating from the electrons within its atoms. Each electron acts as a tiny magnet due to its motion:

• Orbital motion: Electron orbiting the nucleus creates a magnetic moment (like a current loop).
• Spin motion: Electron spinning on its own axis creates a magnetic moment.

Each electron has a permanent spin and orbital magnetic moment. The magnitude is measured in Bohr magneton ($\mu_\textsf{B}$), equal to $9.27 \times 10^{-24} \text{ A m}^2$. Magnetic properties of substances arise from the net magnetic moment of the atoms, which depends on the number of unpaired electrons and how their magnetic moments are oriented.

Based on magnetic behavior in an external magnetic field, substances are classified into five types:

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Paramagnetism

Paramagnetic substances are weakly attracted by a magnetic field. They become weakly magnetised in the same direction as the applied field but lose magnetism when the field is removed. This property is due to the presence of one or more unpaired electrons, whose magnetic moments align with the external field. Examples: $\textsf{O}_2$, $\textsf{Cu}^{2+}$, $\textsf{Fe}^{3+}$, $\textsf{Cr}^{3+}$.

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Diamagnetism

Diamagnetic substances are weakly repelled by a magnetic field. They become weakly magnetised in the opposite direction to the applied field. This property is shown by substances where all electrons are paired. Paired electrons have opposite spins, and their magnetic moments cancel each other out, resulting in no net magnetic moment. Examples: $\textsf{H}_2\textsf{O}$, $\textsf{NaCl}$, $\textsf{C}_6\textsf{H}_6$, $\textsf{Ti}^{4+}$ (e.g., in $\textsf{TiO}_2$).

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Ferromagnetism

Ferromagnetic substances are very strongly attracted by a magnetic field. They can become permanently magnetised even after the external field is removed. In solid state, metal ions of ferromagnetic substances are grouped into small regions called domains, each acting as a tiny magnet with aligned magnetic moments. In an unmagnetised state, these domains are randomly oriented, canceling out their magnetic moments. When a magnetic field is applied, domains align in the direction of the field, producing a strong net magnetic moment. This alignment persists after the field is removed. Examples: $\textsf{Fe}$, $\textsf{Co}$, $\textsf{Ni}$, $\textsf{Gd}$, $\textsf{CrO}_2$. Ferromagnetism is the basis for permanent magnets.

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Antiferromagnetism

Antiferromagnetic substances have a domain structure similar to ferromagnetic substances, but their domains are aligned in opposite directions in equal numbers. This results in the cancellation of their magnetic moments, giving a net magnetic moment of zero. These substances do not show magnetism in an external field. Example: $\textsf{MnO}$.

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Ferrimagnetism

Ferrimagnetic substances have domains with magnetic moments aligned in parallel and anti-parallel directions in unequal numbers. The anti-parallel moments are not equal and do not completely cancel, resulting in a net magnetic moment, but it is smaller than that of ferromagnetic substances. They are weakly attracted by a magnetic field compared to ferromagnetic substances. Examples: $\textsf{Fe}_3\textsf{O}_4$ (magnetite), ferrites like $\textsf{MgFe}_2\textsf{O}_4$, $\textsf{ZnFe}_2\textsf{O}_4$. Ferrimagnetic substances become paramagnetic on heating as the domain alignment is lost above a certain temperature.

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