Summary
Highlights
Crystallography focuses on the form of matter. Matter can exist in disordered (gases, most liquids, amorphous solids) or ordered states (solids, some liquids like liquid crystals). Crystals are defined as ordered solids, while amorphous or vitreous solids are not typically studied in crystallography. Ordered states exhibit anisotropy, meaning their properties depend on the observation direction, unlike isotropic disordered states.
The order in crystals is determined by the nature of the chemical bonds. Strong bonds include covalent (e.g., diamond, involving non-metallic atoms), metallic (e.g., iron in the Eiffel Tower, involving metallic atoms), and ionic (e.g., salt, involving ions forming molecules). Weak bonds include hydrogen bonds (e.g., in solid water) and Van der Waals forces (e.g., in iodine crystals). Crystals formed by weak bonds are generally called molecular crystals and are less emphasized in this course.
To describe a crystal's form, we use a 'motif' (the repeating atom or molecule) placed within a three-dimensional lattice made of 'nodes' (points with known coordinates). The actual arrangement of motifs depends on the bonding. The lattice created by humans helps to organize and understand the arrangement of the actual motifs in space.
The smallest repetitive three-dimensional structure that, when repeated, can reconstruct the entire crystal lattice, is called the 'crystalline unit cell.' This unit cell provides a fundamental description of the crystal's shape. It has parameters such as lengths (a, b, c) along X, Y, and Z axes, and angles (alpha, beta, gamma) between these axes. For a cube, a=b=c and all angles are 90 degrees. A unit cell also typically has 8 vertices, 6 faces, and 12 edges.
Multiplicity (Z) represents the number of motifs belonging to a given unit cell. Motifs at vertices are shared by 8 unit cells (contributing 1/8 to each). Motifs on edges are shared by 4 unit cells (contributing 1/4). Motifs on faces are shared by 2 unit cells (contributing 1/2). Motifs inside the cell belong entirely to that cell (contributing 1). These factors are crucial for accurate counting.
The mass density (ρ) of a crystal is calculated as the mass of the unit cell divided by its volume. The mass of the unit cell is the number of motifs (Z) multiplied by the mass of a single motif. As chemists prefer molar masses, the mass of a motif is expressed as molar mass divided by Avogadro's number. Compactness (C) is the ratio of the volume occupied by the motifs to the total volume of the unit cell. Since motifs are often modeled as spheres, their volume can be calculated using their ionic radius. Compactness is always less than 1, indicating that there are always empty spaces within a crystal lattice.
The coordination number refers to the number of closest neighbors surrounding a particular motif within the crystal structure. This concept is important for understanding the local environment of atoms or molecules and will be further explored in practical sessions.