Amorphous Solids
Solids in which the constituent particles of matter are arranged in a random manner are known as amorphous solids. They are classified as pseudo-solids or super-cooled liquids because they do not possess a sharp melting point.
Examples: Plastics, glass, rubber, metallic glass, polymers, gels, fused silica, pitch tar, thin-film lubricants, wax, etc.
Crystalline Solids
Solids in which the constituent particles of matter are arranged in a regular, repeating specific manner are known as crystalline solids. These are true solids and have a definitive, sharp melting point.
Examples: Quartz, calcite, sugar, mica, diamonds, snowflakes, rock salt, calcium fluoride, silicon dioxide, alum, etc.
Crystal Lattice
A crystal lattice is a three-dimensional representation of constituent particles (atoms, molecules, or ions) arranged in a specific order. Alternatively, it is defined as the geometric arrangement of the constituent particles of crystalline solids mapped as points in space. There are a total of 14 possible three-dimensional spatial arrangements, widely recognized as Bravais Lattices.
Characteristics of a Crystal Lattice
☛ Each constituent particle is represented by exactly one point in a crystal lattice.
☛ These spatial coordinates are known as lattice points or lattice sites.
☛ Lattice points within a crystal lattice are interconnected by straight lines.
☛ Interconnecting the lattice points with straight lines highlights the geometric form of the crystal lattice.
Unit Cell
The smallest structural portion of a crystal lattice which, when repeated in different directions, generates the entire macro-lattice structure is called a Unit Cell.
Characteristics of a Unit Cell
☛ A unit cell is characterized by three axial edges ($a$, $b$, and $c$) and three interfacial angles ($\alpha$, $\beta$, and $\gamma$) between those respective edges.
☛ The edges $a$, $b$, and $c$ may or may not be mutually perpendicular.
☛ By standard notation, the angle between edge $b$ and $c$ is $\alpha$, between $a$ and $c$ is $\beta$, and between $a$ and $b$ is $\gamma$.
Number of Particles Per Unit Cell ($Z$)
☛ For Simple Cubic Unit Cell:
In a simple cubic unit cell, particles are present at the corners only. In a standard crystal lattice, every corner atom is shared equally among eight adjacent unit cells. Therefore, only $\frac{1}{8}$ of each corner particle belongs explicitly to an individual unit cell:
$$Z = 8 \times \frac{1}{8} = 1$$
Hence, the number of particles per simple cubic unit cell is $1$.
☛ For Body-Centered Cubic (BCC) Unit Cell:
In a Body-Centered Cubic unit cell, particles are located at all eight corners as well as one completely unshared particle at the exact geometric center of the cell body. Therefore:
$$Z = \left(8 \times \frac{1}{8}\right) + 1 = 2$$
Hence, the number of particles per BCC unit cell is $2$.
☛ For Face-Centered Cubic (FCC) Unit Cell:
In a Face-Centered Cubic unit cell, particles are positioned at the corners as well as at the center of all six faces. Each face-centered particle is shared equally between two adjacent unit cells. Therefore:
$$Z = \left(8 \times \frac{1}{8}\right) + \left(6 \times \frac{1}{2}\right) = 4$$
Hence, the number of particles per FCC unit cell is $4$.
☛ For End-Centered Cubic (ECC) Unit Cell:
In an End-Centered Cubic unit cell, particles are found at all eight corners along with two face-centered particles located at opposite ends. Therefore:
$$Z = \left(8 \times \frac{1}{8}\right) + \left(2 \times \frac{1}{2}\right) = 2$$
Hence, the number of particles per ECC unit cell is $2$.
Limiting Radius Ratio
The limiting radius ratio is defined as the minimum allowable value for the ratio of the cationic radius to the anionic radius: $$\rho = \frac{r^+}{r^-}$$ This structural parameter dictates the coordination threshold necessary for maintaining electrostatic stability, where $r^+$ represents the radius of the central cation and $r^-$ represents the ionic radius of the coordinated surrounding anions.
Bragg's Equation
When a monochromatic beam of X-rays strikes parallel crystal planes composed of regularly arranged atomic particles, diffraction occurs. For constructive interference to happen, the reflected waves must remain completely in phase, meaning the path difference traveled between the two parallel rays must equal an integer multiple of the input wavelength ($n\lambda$).
Based on geometric parameters:
$= \text{XY}\sin\theta + \text{XY}\sin\theta$
$= 2\text{XY}\sin\theta$
$= 2d\sin\theta$
Equating this to the condition for constructive interference yields: $$n\lambda = 2d\sin\theta$$
This fundamental relation is known as Bragg's equation, where:
$n = 1, 2, 3, \dots$ (order of diffraction)
$\lambda =$ wavelength of incident X-rays
$d =$ interplanar spacing distance between crystal layers
$\theta =$ glancing angle at which constructive interference occurs.
Law of Rational Indices
The law of rational indices was deduced by René Just Haüy. The law of rational indices states that the intercepts of any face of a crystal along the crystallographic axes are either equal to the unit intercepts, or are simple whole-number multiples of them.
The macroscopic faces and internal planes within crystals can be cleanly defined mathematically via spatial coordinate systems.
Consider three axes $\text{OX}$, $\text{OY}$, and $\text{OZ}$ intersected by a crystal face $\text{ABC}$ at distances $\text{OA}$, $\text{OB}$, and $\text{OC}$ from the origin point. Let $\text{OX}$, $\text{OY}$, and $\text{OZ}$ represent the primary crystallographic vectors and let $\text{ABC}$ represent a designated reference unit plane. The respective unit intercepts will then be $a$, $b$, and $c$.
According to this law, the parameters of any other face, such as $\text{KLM}$ on those identical vectors, can be represented as simple whole-number ratios or multiples of $a$, $b$, and $c$. If axes are chosen systematically, the respective intercepts bear clear integer configurations, or the face runs completely parallel to a vector plane (cutting it at infinity). This structural rule is alternatively called the law of rational intercepts.
Structure of Diamond
In a diamond network, every carbon atom is covalently bound to four other carbons, forming a continuous rigid tetrahedral structure.
- All carbon units are $sp^3$ hybridized.
- The bonding angle between intersecting carbon coordinates is $109.5^\circ$.
- All carbon-carbon bond distances are identical, measuring $154\text{ pm}$ ($1.54\ \text{Å}$).
- Diamond builds an expansive three-dimensional network linked via strong localized covalent bonds.
- It has a very high melting point of approximately $3843\text{ K}$ and a high density of $3.51\text{ g/cm}^3$.
- It serves as an electrical insulator because all valence electrons are locked within rigid $\sigma$ covalent bonds, leaving no delocalized electrons available for conducting current.
Structure of Graphite
In a graphite lattice, each carbon atom is covalently linked to three neighboring carbon atoms within a single plane.
- Each carbon center is $sp^2$ hybridized.
- The arrangement constructs a layered planar network of fused hexagonal rings. Because adjacent layers are held together only by weak van der Waals dispersion forces, they can slide over each other smoothly under shear stress.
- The intra-layer $\text{C-C}$ bond length is $141.5\text{ pm}$ ($0.14\text{ nm}$), whereas the inter-planar distance separating adjacent sheets is significantly larger, measuring $340\text{ pm}$ ($0.34\text{ nm}$).
- Every individual carbon atom contributes one unhybridized, non-bonded $p$-orbital electron to a giant delocalized network.
- Graphite functions as an excellent conductor of electricity and heat parallel to its sheets due to these highly mobile, delocalized electrons.
- It possesses a lower density ranging between $2.09\text{ g/cm}^3$ and $2.23\text{ g/cm}^3$.
- It remains completely insoluble in water and ordinary organic solvents since solvent interactions cannot break the strong intra-layer network covalent bonds.
- It has an exceptionally high melting point of $3650^\circ\text{C}$.
- Its slippery, layered nature makes it ideal for use as a solid-state dry lubricant in high-temperature machinery. It is also used as a moderator due to its ability to slow down high-speed neutrons.