Unit 4: Crystal Structure: Fundamentals and X-ray Diffraction

Amorphous and Crystalline Materials

Materials can broadly be classified based on the arrangement of their constituent atoms or molecules into two main categories: amorphous and crystalline. The key difference lies in the orderliness of their internal atomic arrangement.

Amorphous Materials: Materials that lack a long-range, ordered atomic arrangement. Their atoms are randomly arranged, similar to liquids, but are frozen in place, giving them a rigid form. These materials do not have a definite melting point; they gradually soften upon heating.
Examples: Glass, plastics, rubber.

Crystalline Materials: Materials characterized by a highly ordered, repeating atomic arrangement extending over long distances. This periodic arrangement forms a crystal lattice, resulting in a distinct, well-defined external shape and a sharp melting point.
Examples: Most metals (e.g., copper, iron), ceramics, semiconductors (e.g., silicon), salts (e.g., NaCl).

Fundamentals of Crystal Lattices

The study of crystal structure begins with understanding the basic building blocks and concepts that describe the periodic arrangement of atoms in a solid.

Lattice, Translation Vectors, Unit Cell, Primitive Cell, Basis

Crystal Lattice: An imaginary infinite array of points in space, where each point (a lattice point) has identical surroundings. It represents the periodic arrangement of atoms in a crystal, serving as a framework upon which the actual crystal structure is built.

Lattice Translation Vector (T): A vector that connects any two equivalent lattice points in a crystal lattice. It describes the periodicity of the lattice. In three dimensions, a general lattice translation vector T can be expressed as T = n₁a + n₂b + n₃c, where a, b, c are fundamental or primitive translation vectors (also known as basis vectors) and n₁, n₂, n₃ are integers.

Unit Cell: The smallest repeating unit of a crystal lattice that, when translated (repeated) in three dimensions along the lattice translation vectors, generates the entire crystal structure. It must possess the full symmetry of the crystal. The volume of the unit cell is given by V = |a ⋅ (b × c)|.

Primitive Cell: A specific type of unit cell that contains exactly one lattice point. It is the smallest possible unit cell by volume for a given lattice. All primitive cells are unit cells, but not all unit cells are primitive (e.g., a conventional BCC or FCC unit cell contains more than one lattice point).

Basis (Motif): The group of atoms, molecules, or ions associated with each lattice point. The crystal structure is formed by associating an identical basis with each lattice point. Therefore, Crystal Structure = Lattice + Basis. The basis determines the arrangement of atoms within each unit cell.

Miller Indices and Crystal Planes

Miller Indices and Representation of Crystal Planes

Miller indices are a notation system used to describe the orientation of planes and directions within a crystal lattice. They are crucial for understanding crystal symmetry, diffraction phenomena, and material properties.

To determine the Miller indices (hkl) of a crystal plane:

1. Determine the intercepts of the plane with the crystallographic axes (x, y, z) in terms of lattice constants. If a plane is parallel to an axis, its intercept is considered to be at infinity (∞).
2. Take the reciprocals of these intercepts.
3. Clear fractions by multiplying by the smallest common denominator to obtain a set of integers.
4. Enclose the resulting integers in parentheses (hkl). For negative intercepts, a bar is placed above the number (e.g., (h k l̄)).

Notation Summary:

• (hkl): Represents a specific crystal plane.
• {hkl}: Represents a family of crystallographically equivalent planes (e.g., {100} includes (100), (010), (001), (1̄00), etc., for a cubic system).
• [hkl]: Represents a specific crystal direction.
• 〈hkl〉: Represents a family of crystallographically equivalent directions (e.g., 〈100〉 includes [100], [010], [001], etc., for a cubic system).

Interplanar Spacing

The interplanar spacing is the perpendicular distance between adjacent parallel planes with the same Miller indices (hkl).

Interplanar Spacing (d_hkl): The perpendicular distance between two consecutive, parallel planes identified by Miller indices (hkl). This value is crucial for X-ray diffraction analysis. For a cubic crystal system with lattice parameter 'a', the interplanar spacing is given by the formula:
d_hkl = a / √(h² + k² + l²)

Symmetry in Crystals

Symmetry is a fundamental property of crystalline materials, defining how a crystal can be transformed (rotated, reflected, inverted, translated) while remaining indistinguishable from its original state.

Symmetry Consideration

Crystals possess various symmetry elements, which are geometric operations that leave the crystal unchanged. Common symmetry elements include:

• Rotation Axes: Rotation of 360°/n (where n = 1, 2, 3, 4, 6) around an axis.
• Reflection Planes (Mirror Planes): Reflection across a plane.
• Inversion Center: Inversion through a central point.
• Screw Axes: A combination of rotation and translation parallel to the axis.
• Glide Planes: A combination of reflection and translation parallel to the plane.

Symmetry Group

Symmetry Group (Point Group): A collection of symmetry operations (rotations, reflections, inversions) that leave at least one point fixed in space and transform the crystal into itself. These operations describe the external macroscopic symmetry of a crystal. There are 32 unique crystallographic point groups possible in three dimensions.

Space Group

Space Group: A complete description of the symmetry of a crystal, encompassing both point group operations and translational symmetry operations (pure translations, screw axes, and glide planes). Space groups describe the internal microscopic symmetry of the crystal structure. There are 230 unique space groups that describe all possible crystal structures.

Types and Classification of Crystal Structures

Different Types of Crystal Structures

Common crystal structures encountered in materials science include:

• Simple Cubic (SC): Has one lattice point per unit cell. Atoms are located at the corners. Coordination number is 6. This structure is rare in nature.
• Body-Centered Cubic (BCC): Has two lattice points per unit cell (one at each corner and one at the body center). Coordination number is 8. Examples: α-Iron, Chromium, Tungsten.
• Face-Centered Cubic (FCC): Has four lattice points per unit cell (one at each corner and one at the center of each face). Coordination number is 12. This is a close-packed structure. Examples: Copper, Aluminum, Gold, Silver.
• Hexagonal Close-Packed (HCP): Has six lattice points per unit cell. Coordination number is 12. This is also a close-packed structure. Examples: Magnesium, Zinc, Titanium.

Classification of Crystals Based on Nature of Structures

Crystals can be classified into four main types based on the primary bonding forces holding their constituent particles together: