The Solid State

General Characteristics Of Solid State

The solid state is one of the fundamental states of matter, characterised by its definite shape and definite volume. This distinct behaviour arises from the strong intermolecular forces that hold the constituent particles (atoms, molecules, or ions) closely together in relatively fixed positions.

The key general characteristics of the solid state are:

1. Definite Mass, Volume, and Shape: Solids have a fixed mass, occupy a definite volume, and maintain a distinct shape. They do not flow or spread out like liquids and gases.
2. Short Intermolecular Distances: The particles in a solid are packed very closely together, resulting in very short distances between them.
3. Strong Intermolecular Forces: The forces of attraction between the particles are very strong, much stronger than in liquids or gases. These forces keep the particles tightly bound.
4. Fixed Positions of Particles: The particles in a solid are not free to move from one position to another. They are confined to fixed equilibrium positions.
5. Vibratory Motion Only: The particles in a solid can only vibrate about their mean or equilibrium positions. They do not have translational or rotational motion (except possibly in some highly specific cases or at extremely high temperatures).
6. Incompressibility and Rigidity: Due to the close packing and strong forces, solids are generally incompressible and rigid. Their volume does not significantly change with pressure, and they resist deformation.
7. Slow Diffusion: Diffusion (mixing of particles) is extremely slow in solids compared to liquids and gases, as particles cannot move freely.

These characteristics are explained by the dominance of intermolecular forces over thermal energy in the solid state.

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Amorphous And Crystalline Solids

Solids can be classified into two main types based on the arrangement of their constituent particles: Crystalline solids and Amorphous solids.

Crystalline Solids:

These solids have a regular, ordered, repeating three-dimensional arrangement of constituent particles throughout the bulk of the solid. This ordered arrangement is known as a crystal lattice.

Examples: Sodium chloride (NaCl), Quartz, Diamond, Sugar, Metals (Iron, Copper, Gold).

Properties of Crystalline Solids:

1. Long-range Order: Particles are arranged in a definite, repeating pattern extending over the entire crystal.
2. Sharp Melting Point: Crystalline solids melt sharply at a specific, characteristic temperature. At the melting point, the thermal energy is sufficient to overcome the forces holding the lattice together, and the solid transitions entirely to liquid at that single temperature.
3. Anisotropy: Many physical properties (like electrical conductivity, thermal conductivity, refractive index, mechanical strength) can be different when measured in different directions within the same crystal. This is because the arrangement of particles is different along different directions.
4. Clean Cleavage: When cut with a sharp-edged tool, they break along specific planes, producing flat, smooth surfaces.
5. True Solids: Considered true solids.
6. Definite Heat of Fusion: Have a characteristic and definite heat of fusion.

Amorphous Solids:

These solids lack a long-range ordered arrangement of constituent particles. The particles are arranged randomly, similar to the arrangement in a liquid, but are held in fixed positions.

Examples: Glass, Rubber, Plastics, Tar, Fused Silica.

Properties of Amorphous Solids:

1. Short-range Order: Particles may have some local order over a short distance, but there is no repeating pattern throughout the solid.
2. Melt over a Range of Temperature: Amorphous solids do not melt at a sharp, specific temperature. Instead, they gradually soften over a range of temperatures as they are heated. They become less viscous and eventually flow.
3. Isotropy: Physical properties are the same in all directions, similar to liquids and gases. This is because the random arrangement of particles is the same throughout the material.
4. Irregular Cleavage: When cut with a sharp-edged tool, they break into irregular pieces with uneven surfaces.
5. Pseudo Solids or Supercooled Liquids: Amorphous solids are sometimes called pseudo solids or supercooled liquids because their structure is liquid-like and they have a tendency to flow very slowly over time (e.g., old window panes of historical buildings might be slightly thicker at the bottom due to slow flow of glass).
6. No Definite Heat of Fusion: They do not have a definite heat of fusion.

Glass is a common example of an amorphous solid. It is made by cooling molten silica (SiO$_2$) rapidly, preventing the formation of a crystalline structure.

Some crystalline solids can become amorphous on melting and rapid cooling, and some amorphous solids can be converted into crystalline solids by heating and slow cooling (annealing).

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Classification Of Crystalline Solids

Crystalline solids can be classified into four main types based on the nature of their constituent particles and the type of intermolecular forces or bonds holding them together. This classification helps in understanding their properties.

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Molecular Solids

Constituent Particles: Molecules (either non-polar or polar).

Bonding/Forces: Intermolecular forces (Dispersion forces, Dipole-dipole forces, Hydrogen bonds).

Types:

1. Non-polar Molecular Solids:
   • Particles: Non-polar molecules (e.g., H$_2$, Cl$_2$, I$_2$, CH$_4$, CO$_2$) or inert atoms (e.g., Ar, Ne).
   • Forces: Weak dispersion (London) forces.
   • Properties: Soft, low melting points, poor electrical conductors. Exist as liquids or gases at room temperature and pressure.
   • Examples: Solid Hydrogen, Solid Carbon Dioxide (Dry Ice), Argon.
2. Polar Molecular Solids:
   • Particles: Polar molecules (e.g., HCl, SO$_2$, Solid NH$_3$).
   • Forces: Stronger dipole-dipole forces than dispersion forces (although dispersion forces are also present).
   • Properties: Relatively soft, higher melting points than non-polar molecular solids (but still low compared to other types), poor electrical conductors. Exist as liquids or gases at room temperature and pressure.
   • Examples: Solid HCl, Solid SO$_2$.
3. Hydrogen Bonded Molecular Solids:
   • Particles: Molecules with polar covalent bonds between H and highly electronegative atoms like O, N, or F (e.g., H$_2$O (ice), Solid NH$_3$, Solid HF).
   • Forces: Strong hydrogen bonds.
   • Properties: Relatively soft, melting points are higher than other molecular solids but still below 0$^\circ$C or slightly above, poor electrical conductors.
   • Example: Ice (solid H$_2$O).

Overall, molecular solids are generally soft, have low melting points, and do not conduct electricity because the molecules are held by weak intermolecular forces, and electrons are tightly bound within the molecules or atoms.

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Ionic Solids

Constituent Particles: Cations (positive ions) and Anions (negative ions).

Bonding/Forces: Strong electrostatic forces of attraction (ionic bond) between oppositely charged ions.

Properties:

• Hard and brittle.
• High melting and boiling points (due to strong ionic bonds).
• Poor electrical conductors in the solid state (ions are fixed in the lattice).
• Good electrical conductors in the molten (liquid) state or when dissolved in water (ions become mobile).

Examples: Sodium Chloride (NaCl), Potassium Chloride (KCl), Magnesium Oxide (MgO), Calcium Fluoride (CaF$_2$).

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Metallic Solids

Constituent Particles: Positive metal ions (cations) immersed in a 'sea' of mobile valence electrons.

Bonding/Forces: Metallic bond – the attraction between the positive metal ions and the delocalised electrons.

Properties:

• Hard (though hardness varies greatly, e.g., Sodium is soft, Iron is hard).
• Malleable (can be hammered into sheets).
• Ductile (can be drawn into wires).
• Excellent electrical and thermal conductors (due to mobile electrons).
• Have metallic lustre (shiny appearance).
• Generally have high melting and boiling points (though again, varies, e.g., Mercury is liquid at room temperature).

Examples: Iron (Fe), Copper (Cu), Gold (Au), Silver (Ag), Aluminium (Al), Zinc (Zn).

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Covalent Or Network Solids

Constituent Particles: Atoms (usually non-metals).

Bonding/Forces: Covalent bonds forming a continuous, three-dimensional network throughout the crystal.

Properties:

• Very hard and brittle.
• Extremely high melting points (often decompose before melting) due to the need to break numerous strong covalent bonds throughout the network.
• Poor electrical conductors (electrons are localised in covalent bonds), except for Graphite (due to delocalised electrons in layers).

Examples: Diamond (Carbon atoms covalently bonded in a tetrahedral network), Quartz (SiO$_2$, Silicon and Oxygen atoms forming a network), Silicon Carbide (SiC).

Graphite is a special case of a covalent solid. Carbon atoms are arranged in layers held by strong covalent bonds within the layer, but the layers are held together by weak dispersion forces. This layered structure makes Graphite soft and a good lubricant, and the delocalised electrons within the layers make it a good electrical conductor.

Summary table for classification of crystalline solids:

Type of Solid	Constituent Particles	Interparticle Forces/Bonding	Examples	Physical Nature	Melting Point	Electrical Conductivity
Molecular (Non-polar)	Molecules (non-polar) or Atoms	Dispersion forces	Ar, CCl$_4$, H$_2$, I$_2$, CO$_2$ (solid)	Soft	Very Low	Insulator
Molecular (Polar)	Molecules (polar)	Dipole-dipole forces	HCl (solid), SO$_2$ (solid)	Soft	Low	Insulator
Molecular (Hydrogen bonded)	Molecules (with H-bond)	Hydrogen bonding	H$_2$O (ice), NH$_3$ (solid), HF (solid)	Soft	Low	Insulator
Ionic	Ions	Electrostatic forces	NaCl, MgO, ZnS, CaF$_2$	Hard but Brittle	High	Insulator (Solid), Conductor (Molten/Aqueous)
Metallic	Positive ions in a sea of delocalised electrons	Metallic bonding	Fe, Cu, Ag, Au, Al, Zn	Hard but Malleable & Ductile	Fairly High	Conductor (Solid and Molten)
Covalent or Network	Atoms	Covalent bonding	Diamond, Quartz (SiO$_2$), SiC	Very Hard	Very High	Insulator (except Graphite)

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Crystal Lattices And Unit Cells

The characteristic feature of a crystalline solid is the regular and repeating arrangement of its constituent particles (atoms, molecules, or ions) in three dimensions. This ordered arrangement can be represented by a crystal lattice.

Crystal Lattice: A crystal lattice is a regular three-dimensional arrangement of points in space that represents the positions of the constituent particles of a crystalline solid. Each point in a crystal lattice represents a constituent particle (atom, molecule, or ion) and is called a lattice point or lattice site. When the lattice points are connected by straight lines, the geometric shape of the crystal lattice is obtained.

Unit Cell: A unit cell is the smallest repeating structural unit of a crystal lattice. When unit cells are repeated over and over again in three dimensions, they generate the entire crystal lattice. The unit cell is essentially the building block of the crystal structure.

A unit cell is characterised by six parameters:

• The dimensions of the three edges (or axes) of the unit cell: $a, b, c$. These edges may or may not be mutually perpendicular.
• The angles between the edges: $\alpha$ (between $b$ and $c$), $\beta$ (between $a$ and $c$), and $\gamma$ (between $a$ and $b$).

The shape of the unit cell is determined by these six parameters ($a, b, c, \alpha, \beta, \gamma$). Based on the possible values of these parameters, there are seven possible types of crystal systems (Cubic, Tetragonal, Orthorhombic, Monoclinic, Triclinic, Hexagonal, Rhombohedral).

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Primitive And Centred Unit Cells

Unit cells can be classified into two categories based on the location of lattice points:

1. Primitive Unit Cell (Simple Unit Cell): A primitive unit cell has lattice points only at its corners. There is one lattice point at each of the eight corners of the unit cell.
2. Centred Unit Cell: A centred unit cell has lattice points at places other than corners, in addition to the corners. Centred unit cells are of three types:
   • Body-Centred Unit Cell (BCC): Has lattice points at all eight corners and one lattice point at the centre of the body of the unit cell.
   • Face-Centred Unit Cell (FCC): Has lattice points at all eight corners and one lattice point at the centre of each of the six faces of the unit cell.
   • End-Centred Unit Cell: Has lattice points at all eight corners and one lattice point at the centre of any two opposite faces (usually the top and bottom faces). This type is less common in basic examples than BCC and FCC.

Combining the seven crystal systems with the possible lattice point arrangements (Primitive, BCC, FCC, End-centred) leads to 14 possible 3D lattices, known as Bravais Lattices. For the cubic crystal system, the three possible Bravais lattices are: Primitive cubic, Body-centred cubic, and Face-centred cubic.

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Number Of Atoms In A Unit Cell

While a unit cell has lattice points at various positions, each point might be shared by multiple adjacent unit cells. To find the effective number of constituent particles (atoms, molecules, or ions) *belonging* to a single unit cell, we need to consider the fraction of each particle that lies within the boundaries of that specific unit cell.

Contributions of particles at different positions in a cubic unit cell:

• A particle at a corner is shared by 8 unit cells. Contribution to one unit cell = $1/8$.
• A particle at a face centre is shared by 2 unit cells. Contribution to one unit cell = $1/2$.
• A particle at a body centre belongs entirely to that unit cell. Contribution to one unit cell = 1.
• A particle at an edge centre is shared by 4 unit cells. Contribution to one unit cell = $1/4$. (Edge centres are not present in basic cubic unit cells, but in other crystal systems).

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Primitive Cubic Unit Cell

A primitive cubic unit cell has particles only at the 8 corners.

Number of atoms at corners = 8

Contribution of each corner atom = $1/8$

Total number of atoms in a primitive cubic unit cell = Number of corners $\times$ Contribution per corner

$ Z_{\text{primitive}} = 8 \times \frac{1}{8} = 1 \text{ atom per unit cell} $

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Body-Centred Cubic Unit Cell

A body-centred cubic unit cell has particles at the 8 corners and 1 particle at the body centre.

Number of atoms at corners = 8

Contribution from corners = $8 \times \frac{1}{8} = 1$

Number of atoms at body centre = 1

Contribution from body centre = $1 \times 1 = 1$

Total number of atoms in a body-centred cubic unit cell = Contribution from corners + Contribution from body centre

$ Z_{\text{BCC}} = (8 \times \frac{1}{8}) + (1 \times 1) = 1 + 1 = 2 \text{ atoms per unit cell} $

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Face-Centred Cubic Unit Cell

A face-centred cubic unit cell has particles at the 8 corners and 1 particle at the centre of each of the 6 faces.

Number of atoms at corners = 8

Contribution from corners = $8 \times \frac{1}{8} = 1$

Number of atoms at face centres = 6

Contribution from face centres = $6 \times \frac{1}{2} = 3$

Total number of atoms in a face-centred cubic unit cell = Contribution from corners + Contribution from face centres

$ Z_{\text{FCC}} = (8 \times \frac{1}{8}) + (6 \times \frac{1}{2}) = 1 + 3 = 4 \text{ atoms per unit cell} $

Note: Face-Centred Cubic (FCC) arrangement is equivalent to Cubic Close Packing (CCP), which will be discussed later.

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Close Packed Structures

In crystalline solids, the constituent particles (atoms, ions, or molecules) are packed together as closely as possible to maximise attractive forces and minimise empty space. This close packing arrangement dictates many physical properties of the solid.

We can understand close packing by considering the particles as identical hard spheres and arranging them in different dimensions.

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Close Packing In One Dimension

In one dimension, particles are arranged in a single straight line, touching each other. There is only one way to arrange spheres in a line.

In this arrangement, each sphere is in contact with two neighbouring spheres (one on each side). The number of nearest neighbours of a particle in a crystal structure is called its coordination number. So, in 1-D close packing, the coordination number is 2.

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Close Packing In Two Dimensions

In two dimensions, we can arrange layers of 1-D close-packed rows. There are two main ways to do this:

1. Square Close Packing:

   If the second row is placed directly above the first row such that the spheres of the two rows are vertically and horizontally aligned, we get a square close-packed structure. If we call the first row 'A' type, the second row is identical and placed directly above it, so it is also 'A' type. This arrangement is called AAAA type packing.

   In this arrangement, each sphere is in contact with four other spheres (two in its own row, one above, and one below). Connecting the centres of these four nearest neighbours forms a square. The coordination number is 4.

2. Hexagonal Close Packing:

   If the second row is placed in the depressions (dents) between the spheres of the first row, a more efficient packing is achieved. If the first row is 'A' type, placing the second row in the depressions creates a new arrangement, call it 'B' type. The third row can be placed in the depressions of the second row. If the third row's spheres align directly with the spheres of the first row (type A), the pattern repeats as ABAB type packing.

   In this arrangement, each sphere is in contact with six other spheres (two in its own row, two in the row above, and two in the row below). Connecting the centres of these six nearest neighbours forms a hexagon. The coordination number is 6.

   This hexagonal close packing in 2D is more efficient (fills more space) than square close packing.

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Close Packing In Three Dimensions

Three-dimensional close-packed structures are formed by stacking 2-D layers. The most efficient 3D packing arrangements arise from stacking 2-D hexagonal close-packed layers.

There are two ways to stack 2-D hexagonal layers to create close-packed structures in 3D:

1. Placing the second layer (B) over the first layer (A): The spheres of the second layer are placed in the depressions (voids) of the first layer. Two types of voids are created in the first layer: those covered by spheres of the second layer and those not covered. The voids in the first layer are triangular. When the second layer is placed, two types of voids are formed in the second layer:
   • Tetrahedral voids: Formed when a void in the second layer is directly above a void in the first layer, or vice versa. They are called tetrahedral because they are surrounded by four spheres whose centres form a tetrahedron.
   • Octahedral voids: Formed by a void in the second layer being directly above a sphere in the first layer (or vice versa). They are called octahedral because they are surrounded by six spheres whose centres form an octahedron.

   If the number of spheres in the close-packed structure is 'n', then the number of octahedral voids is 'n', and the number of tetrahedral voids is '2n'.

2. Placing the third layer: The third layer can be placed in two ways relative to the second layer (B):
   • Covering the tetrahedral voids: If the spheres of the third layer are placed in the tetrahedral voids of the second layer, the spheres of the third layer will align directly above the spheres of the first layer (type A). The stacking pattern is ABABAB... type. This arrangement is called Hexagonal Close Packing (HCP). The unit cell for HCP is hexagonal.
   • Covering the octahedral voids: If the spheres of the third layer are placed in the octahedral voids of the second layer, the spheres of the third layer will not align with either the first (A) or second (B) layer spheres. This forms a third type of layer (type C). The fourth layer will then align with the first layer (type A). The stacking pattern is ABCABCABC... type. This arrangement is called Cubic Close Packing (CCP). The unit cell for CCP is face-centred cubic (FCC).

Both HCP and CCP (FCC) are highly efficient close-packed structures. In both arrangements, each sphere is surrounded by 12 nearest neighbours (6 in its own layer, 3 in the layer above, and 3 in the layer below). Thus, the coordination number in both HCP and CCP structures is 12.

Examples:

• Metals crystallising in HCP structure: Mg, Zn, Ti, Cd, Co.
• Metals crystallising in CCP (FCC) structure: Cu, Ag, Au, Al, Ni, Pt.
• Metals crystallising in BCC structure (which is NOT a close-packed structure, coordination number 8): Li, Na, K, Rb, Cs, Fe, Cr, W.

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Formula Of A Compound And Number Of Voids Filled

In ionic compounds, larger ions (usually anions) often form a close-packed structure (HCP or CCP), and the smaller ions (usually cations) occupy the voids (tetrahedral or octahedral). The formula of the compound depends on the type of void occupied and the fraction of voids that are filled.

If anions (say, $B^-$) form the close-packed structure, and their number is $n$ per unit cell (4 in CCP/FCC, 6 in HCP unit cell, but let's consider 'n' as the number of atoms forming the lattice structure for simplicity of void calculation, i.e., in a lattice with N particles, there are N octahedral voids and 2N tetrahedral voids), then:

• Number of octahedral voids = $n$
• Number of tetrahedral voids = $2n$

If cations (say, $A^+$) occupy these voids, the formula of the compound is determined by the ratio of $A^+$ ions to $B^-$ ions.

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Answer:

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Locating Tetrahedral And Octahedral Voids

In the CCP (FCC) unit cell:

• Tetrahedral Voids: There are 8 tetrahedral voids located at the body diagonals, one-fourth of the way from each corner. Since there are 4 formula units (4 atoms effectively) per FCC unit cell, and there are 8 tetrahedral voids, it means there are $2 \times Z_{FCC}$ tetrahedral voids, consistent with the $2n$ rule.
• Octahedral Voids: There is one octahedral void at the body centre of the unit cell, and 12 octahedral voids at the centre of each edge. The edge-centre voids are shared by 4 unit cells ($12 \times 1/4 = 3$ voids), plus the one at the body centre ($1 \times 1 = 1$ void), giving a total of 4 octahedral voids per FCC unit cell. This is consistent with the $n$ rule, as $Z_{FCC} = 4$.

In the HCP unit cell:

There are 6 atoms effectively per HCP unit cell. Therefore, there are 6 octahedral voids and 12 tetrahedral voids within the hexagonal unit cell volume.

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Packing Efficiency

Packing efficiency is the percentage of the total volume of a unit cell that is occupied by the constituent particles (assumed to be spheres). The remaining space is empty space or voids.

Packing efficiency (%) = $\frac{\text{Volume occupied by spheres in the unit cell}}{\text{Total volume of the unit cell}} \times 100$

The volume occupied by spheres is calculated from the number of spheres per unit cell ($Z$) and the volume of a single sphere (assuming radius $r$): Volume $= Z \times (\frac{4}{3}\pi r^3)$. The total volume of the unit cell is calculated from its edge length(s).

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Packing Efficiency In Hcp And Ccp Structures

Both HCP and CCP (FCC) structures are close-packed and have the highest packing efficiency among simple crystal structures involving single spheres.

Consider the FCC unit cell (equivalent to CCP):

• Number of atoms per unit cell (Z) = 4
• Let the edge length of the cubic unit cell be 'a' and the radius of the sphere be 'r'.
• In FCC, particles touch along the face diagonal. The face diagonal ($d_f$) is related to the edge length by $d_f = \sqrt{a^2 + a^2} = \sqrt{2}a$.
• Along the face diagonal, three spheres touch: a corner sphere, a face-centred sphere, and another corner sphere. So, the face diagonal length is equal to $r + 2r + r = 4r$.
• Therefore, $\sqrt{2}a = 4r$, which gives $a = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r$. Or, $r = \frac{a}{2\sqrt{2}}$.
• Volume of the unit cell = $a^3 = (2\sqrt{2}r)^3 = (2\sqrt{2})^3 r^3 = 16\sqrt{2} r^3$.
• Volume occupied by spheres = $Z \times (\frac{4}{3}\pi r^3) = 4 \times (\frac{4}{3}\pi r^3) = \frac{16}{3}\pi r^3$.
• Packing efficiency = $\frac{\frac{16}{3}\pi r^3}{16\sqrt{2} r^3} \times 100 = \frac{\frac{16}{3}\pi}{16\sqrt{2}} \times 100 = \frac{\pi}{3\sqrt{2}} \times 100$.
• Using $\pi \approx 3.14159$ and $\sqrt{2} \approx 1.41421$: Packing efficiency $\approx \frac{3.14159}{3 \times 1.41421} \times 100 = \frac{3.14159}{4.24263} \times 100 \approx 0.74048 \times 100$.

$ \text{Packing efficiency (HCP/CCP)} = \frac{\pi}{3\sqrt{2}} \times 100 \approx 74.04 \% \approx 74 \% $

Both HCP and CCP (FCC) structures have a packing efficiency of approximately 74%. The remaining 26% is empty space (voids).

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Efficiency Of Packing In Body-Centred Cubic Structures

Consider the BCC unit cell:

• Number of atoms per unit cell (Z) = 2
• Let the edge length of the cubic unit cell be 'a' and the radius of the sphere be 'r'.
• In BCC, particles touch along the body diagonal. The body diagonal ($d_b$) is related to the edge length by $d_b = \sqrt{a^2 + a^2 + a^2} = \sqrt{3}a$.
• Along the body diagonal, three spheres touch: a corner sphere, the body-centred sphere, and another corner sphere. So, the body diagonal length is equal to $r + 2r + r = 4r$.
• Therefore, $\sqrt{3}a = 4r$, which gives $a = \frac{4r}{\sqrt{3}}$. Or, $r = \frac{\sqrt{3}}{4}a$.
• Volume of the unit cell = $a^3 = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}}$.
• Volume occupied by spheres = $Z \times (\frac{4}{3}\pi r^3) = 2 \times (\frac{4}{3}\pi r^3) = \frac{8}{3}\pi r^3$.
• Packing efficiency = $\frac{\frac{8}{3}\pi r^3}{\frac{64}{3\sqrt{3}} r^3} \times 100 = \frac{\frac{8}{3}\pi}{\frac{64}{3\sqrt{3}}} \times 100 = \frac{8\pi}{64/\sqrt{3}} \times 100 = \frac{8\pi\sqrt{3}}{64} \times 100 = \frac{\pi\sqrt{3}}{8} \times 100$.
• Using $\pi \approx 3.14159$ and $\sqrt{3} \approx 1.73205$: Packing efficiency $\approx \frac{3.14159 \times 1.73205}{8} \times 100 \approx \frac{5.4413}{8} \times 100 \approx 0.68016 \times 100$.

$ \text{Packing efficiency (BCC)} = \frac{\pi\sqrt{3}}{8} \times 100 \approx 68 \% $

The packing efficiency of the BCC structure is 68%. The empty space is 32%.

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Packing Efficiency In Simple Cubic Lattice

Consider the primitive cubic unit cell (simple cubic):

• Number of atoms per unit cell (Z) = 1
• Let the edge length of the cubic unit cell be 'a' and the radius of the sphere be 'r'.
• In a simple cubic lattice, the particles touch along the edges. So, the edge length 'a' is equal to the diameter of the sphere, $2r$.
• $a = 2r$, which means $r = a/2$.
• Volume of the unit cell = $a^3 = (2r)^3 = 8r^3$.
• Volume occupied by spheres = $Z \times (\frac{4}{3}\pi r^3) = 1 \times (\frac{4}{3}\pi r^3) = \frac{4}{3}\pi r^3$.
• Packing efficiency = $\frac{\frac{4}{3}\pi r^3}{8 r^3} \times 100 = \frac{4\pi/3}{8} \times 100 = \frac{\pi}{6} \times 100$.
• Using $\pi \approx 3.14159$: Packing efficiency $\approx \frac{3.14159}{6} \times 100 \approx 0.5236 \times 100$.

$ \text{Packing efficiency (Simple Cubic)} = \frac{\pi}{6} \times 100 \approx 52.4 \% $

The packing efficiency of the simple cubic lattice is 52.4%. This is the least efficient packing among the three common cubic lattices. The empty space is 47.6%.

Order of packing efficiency: HCP = CCP (FCC) (74%) > BCC (68%) > Simple Cubic (52.4%).

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Calculations Involving Unit Cell Dimensions

The knowledge of unit cell dimensions (edge length 'a' for cubic systems) and the type of unit cell (Primitive, BCC, FCC) allows us to calculate various properties of a crystalline solid, such as its density, atomic radius, or molar mass.

The most common calculation is determining the density of the solid.

Density of the Unit Cell: The density of the crystal is the same as the density of its unit cell.

Density ($\rho$) = $\frac{\text{Mass of the unit cell}}{\text{Volume of the unit cell}}$

Let:

• 'a' be the edge length of the cubic unit cell (in cm).
• 'V' be the volume of the unit cell = $a^3$ (in cm$^3$).
• 'M' be the molar mass of the element/compound (in g/mol).
• 'Z' be the number of atoms/formula units per unit cell (1 for primitive, 2 for BCC, 4 for FCC).
• '$N_A$' be Avogadro's number ($6.022 \times 10^{23}$ mol$^{-1}$).

The mass of one atom or formula unit is $M/N_A$.

The mass of the unit cell = Number of atoms/formula units in the unit cell $\times$ Mass of each atom/formula unit

Mass of unit cell = $Z \times \frac{M}{N_A}$

Volume of unit cell = $a^3$

So, the density ($\rho$) is:

$ \rho = \frac{Z \times M}{a^3 \times N_A} $

Where 'a' must be in cm for density to be in g/cm$^3$ when M is in g/mol and $N_A$ is in mol$^{-1}$. If 'a' is given in picometres (pm), remember $1 \text{ pm} = 10^{-12} \text{ m} = 10^{-10} \text{ cm}$. So, $a \text{ (in cm)} = a \text{ (in pm)} \times 10^{-10}$. The volume $a^3 \text{ (in cm}^3\text{)} = [a \text{ (in pm)} \times 10^{-10}]^3 = [a \text{ (in pm)}]^3 \times 10^{-30}$.

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Answer:

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This formula can also be rearranged to calculate the molar mass (M), number of atoms per unit cell (Z), or edge length (a) if other parameters are known.

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Imperfections In Solids

Crystalline solids are expected to have a perfectly ordered arrangement of constituent particles. However, in reality, crystals are not perfect. They contain defects or imperfections in the arrangement of particles. These defects affect the properties of the solid.

Crystal defects are broadly classified as point defects, line defects, and surface defects. We will focus on point defects.

Point Defects: These are deviations from the ideal arrangement around a point or an atom in a crystalline substance. Point defects are often caused during crystallisation or due to external factors like heating or irradiation.

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Types Of Point Defects

Point defects are primarily of two types:

1. Stoichiometric Defects: These are point defects that do not disturb the stoichiometry (the ratio of positive and negative ions) of the solid. The general formula of the compound remains unchanged. Stoichiometric defects are also called intrinsic or thermodynamic defects.
   They can be further classified as:
   • Vacancy Defect: Occurs when some lattice sites are vacant. This happens in non-ionic solids. It leads to a decrease in the density of the substance.
   • Interstitial Defect: Occurs when some constituent particles occupy an interstitial site (space between lattice points) in the crystal structure. This happens in non-ionic solids. It increases the density of the substance.
   • Schottky Defect: A type of vacancy defect found in ionic solids. It occurs when equal numbers of cations and anions are missing from their lattice sites. This maintains electrical neutrality. This defect is common in ionic compounds with similar sized cations and anions, and high coordination number. It leads to a decrease in density. Example: NaCl, KCl, CsCl, AgBr.
   • Frenkel Defect: A type of interstitial defect found in ionic solids. It occurs when an ion (usually the smaller cation) leaves its lattice site and occupies an interstitial site. This creates a vacancy at the original site and an interstitial defect. Electrical neutrality is maintained. This defect is common in ionic compounds with large difference in size between cations and anions, and low coordination number. It does not change the density significantly. Example: ZnS, AgCl, AgBr, AgI. Note: AgBr shows both Schottky and Frenkel defects.

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2. Non-stoichiometric Defects: These are point defects that disturb the stoichiometry of the solid. The ratio of positive and negative ions becomes different from the ideal chemical formula. This often occurs due to the presence of excess metal ions or non-metal ions, or due to impurities.
   They can be further classified as:
   • Metal Excess Defects:
     • Metal excess due to anion vacancies: Anions are missing from their lattice sites, and the electrical neutrality is maintained by the presence of electrons in the anionic vacancies (called F-centres). F-centres are responsible for the colour of alkali halide crystals (e.g., NaCl becomes yellow when heated in sodium vapour, LiCl becomes pink in Li vapour, KCl becomes violet in K vapour).
     • Metal excess due to presence of extra cations at interstitial sites: Extra cations occupy interstitial sites, and electrical neutrality is maintained by the presence of electrons at other interstitial sites. Example: Zinc oxide (ZnO) is white at room temperature but turns yellow on heating because it loses oxygen and excess Zn$^{2+}$ ions and electrons occupy interstitial sites.
   • Metal Deficiency Defects: Occurs when some metal ions are missing from their lattice sites, and the charge neutrality is maintained by the presence of metal ions with higher oxidation states. This defect is common in compounds of transition metals which can show variable valency. Example: FeO (Ferrous oxide) often exists as Fe$_{0.95}$O, indicating missing Fe$^{2+}$ ions compensated by some Fe$^{3+}$ ions.

The presence of defects influences the electrical, magnetic, and optical properties of crystalline solids and is crucial in understanding semiconductor behaviour and other material properties.

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