Crystal Structure in Materials

Ahmed Samy

1 Nov, 2023

The fundamental characteristics of solid materials used in engineering are primarily determined by the organization of the atoms, ions, or molecules composing the material, as well as the bonding forces holding them together. This physical structure of materials can be categorized into two main types: crystal structure and amorphous structure. Researchers gain in-depth knowledge about atoms, crystal structures, and amorphous structures through the utilization of X-ray technology.

Crystalline and amorphous solids difference

When the atoms or ions of a solid are arranged in a repeating pattern which known as a unit cell, that extends uniformly throughout three dimensions, they create a crystalline solid exhibiting a property called long-range order (LRO). Long-range order in a crystalline solid refers to the highly organized and consistent arrangement of atoms or ions over vast distances within the material. This means that if you were to examine the structure of a crystalline material at different points throughout its volume, you would find the same repeating pattern, which is characteristic of its crystalline nature. In essence, long-range order implies that the arrangement of particles within the crystal remains consistent and predictable over large spatial extents, setting it apart from materials that lack this level of organization, such as amorphous solids.

In contrast to crystalline solids with long-range order, amorphous solids exhibit a markedly different atomic arrangement characterized by short-range order (SRO). In amorphous solids, the constituent particles, whether atoms or molecules, lack a regular and repeating pattern over large spatial extents. Instead, their arrangement appears disordered and lacks the well-defined structure seen in crystalline materials. This short-range order means that while neighboring particles may have some degree of local organization, this order dissipates quickly as you move away from a given point within the material (The order only exists in the immediate neighborhood of an atom or molecule). As a result, the atomic or molecular structure in amorphous solids lacks the predictability and uniformity found in crystalline solids, making them distinct from their highly organized crystalline counterparts. Amorphous solids often possess unique properties and are frequently encountered in materials like glasses, certain polymers, and some liquids that have undergone rapid cooling or solidification processes.

Note that :

All metals, alloys and some ceramics are in crystal structure.
Ceramics can be exist in crystal or amorphous or mixture between it (both together in same material).

Crystalline Materials

Atomic arrangements in crystalline solids can be described by referring the atoms to the points of intersection of a network of lines in three dimensions. Such a network is called a space lattice and it can be described as an infinite three-dimensional array of points. Each point in the space lattice has identical surroundings (in ideal crystal).

Each space lattice can thus be described by specifying the atom positions in a repeating unit cell.

Note that :

Space Lattice : network results from intersection of a network of lines.
Crystal Structure = Basis + Lattice.
Basis or Motif : group of atoms associated with lattice points.
Lattice : a collection of points that divide space lattice in small equally segments.
The atoms not necessarily coincide with lattice points.

Unit cell in crystal structure

Unit cell is the smallest part that repeat itself in 3-dimensions to form the material, or it is the smallest subdivision of the lattice that maintains the characteristics of overall crystal.

Unit cells have size and shape determined by 3 axial lengths (a , b , c) and 3 interaxial angles. (Axial lengths and interaxial angles are lattice constant for each unit cell)

Crystal systems and Bravais lattices

By assigning specific values for axial lengths and interaxial angles, unit cells of different types can be constructed. Crystallographers have shown that only seven different types of unit cells are necessary to create all space lattices.

The 7 types of crystal systems are :

Cubic.
Tetragonal.
Rhombohedral.
Hexagonal.
Orthorhombic.
Monoclinic.
Triclinic.

August Bravais is a French crystallographer who derived the 14 possible arrangements of points in space.

The 14 Bravais conventional unit cells grouped according to crystal system

The most important type of unit cell is Cubic system. In cubic systems the axial lengths are equal ( a = b = c ) and the interaxial angles are equal 90.

There are 3 types of the cubic system :

Simple cubic system (SC).
Body centered cubic system (BCC).
Face centered cubic system (FCC).

90% of metals are in 3 densely packed crystal structure BCC, FCC, and Hexagonal Close-Packed (HCP), because energy is released as the atoms come closer together that means achieving a state of stability.

Principal metal crystal structure and unit cells: (a) body-centered cubic, (b) face-centered cubic, (c) hexagonal close-packed crystal structure

The Atomic Packing Factor (APF)

The Atomic Packing Factor (APF) is a measure used in materials science and solid-state physics to quantify the degree of space-filling or atomic packing within a crystal lattice. It represents the fraction of space within a crystal structure that is occupied by atoms.

In mathematical terms, the Atomic Packing Factor is calculated as:

APF = (Volume occupied by atoms) / (Total volume of the unit cell)

Where :

"Volume occupied by atoms" is the sum of the volumes of all atoms within the unit cell.

"Total volume of the unit cell" is the volume of the unit cell itself.

The APF typically ranges from 0 to 1, where 0 indicates that the unit cell is empty (no atoms), and 1 indicates that the unit cell is completely filled with atoms with no void spaces.

The APF is an important parameter in materials science because it provides insights into the efficiency of atom packing within a crystal structure. Materials with higher APFs are often denser and have more efficient atomic packing, while materials with lower APFs may have more open or less efficient packing arrangements. The APF is a key factor in determining the physical and mechanical properties of materials.

Simple-Cubic (SC) crystal structure

The Simple Cubic (SC) crystal structure is one of the most basic and straightforward arrangements of atoms in a crystalline lattice. It is the simplest of the three primary cubic crystal structures, the other two being the Body-Centered Cubic (BCC) and Face-Centered Cubic (FCC) structures. Here are some key features and properties of the SC crystal structure:

Lattice Structure: In an SC structure, each atom is positioned at the corners of a cube. There is one atom at each corner of the cube, and there are no atoms within the body of the cube. When a unit cell exists within a material Surrounded by many other unit cells, the unit cell shares the atom located at each of the four corners with three neighboring unit cells in the surrounding space. As a result, each unit cell's share amounts to only 1/8 of an atom from the atoms positioned at the corners. So, the number of atoms in SC crystal structure is 2 atoms.

Coordination Number: The coordination number of atoms in an SC structure is 6. This means that each atom in the lattice is in direct contact with six neighboring atoms.

Atomic Packing Factor (APF): The SC structure has a relatively low APF of about 0.52. This means that only about 52% of the available space within the crystal structure is occupied by atoms, making it less efficient in terms of atomic packing compared to other structures like FCC and HCP.

Closest Packed Directions: In the SC structure, there are no closest-packed directions since each atom is only in contact with six neighbors. The packing is less dense compared to other crystal structures.

Closest Packed Planes: Similar to directions, there are no closest-packed planes in the SC structure since the atoms are positioned only at the corners of the cube.

Examples of Materials with SC Structure: The SC structure is relatively rare in nature, and it is not commonly found in pure metals. However, certain materials at high temperatures or under specific conditions may temporarily adopt an SC structure. In practice, the SC structure is often used as a simple model to introduce the concept of crystal structures in educational settings.

Density: Materials with an SC structure tend to have lower densities compared to other crystal structures because of their relatively open and simple packing of atoms.

Mechanical Properties: The mechanical properties of materials with an SC structure are generally less desirable compared to materials with more densely packed crystal structures. SC metals are typically less dense and less strong.

Body-Centered cubic (BCC) crystal structure

The Body-Centered Cubic (BCC) crystal structure is one of the three primary cubic crystal structures, along with the Simple Cubic (SC) and Face-Centered Cubic (FCC) structures. The BCC structure is characterized by its unique arrangement of atoms within a cubic lattice. Here are some key features and properties of the BCC crystal structure:

Lattice Structure: In a BCC structure, each unit cell contains atoms at the corners of the cube and one atom at the center of the cube. This central atom is located within the body of the cube, giving the structure its name. The atoms at the corners are counted in the same way of SC, and by adding the atom at the center, the actual total number of atoms becomes 2 atoms.

Coordination Number: The coordination number of atoms in a BCC structure is 8. This means that each atom in the lattice is in direct contact with 8 neighboring atoms.

Atomic Packing Factor (APF): The BCC structure has a moderate APF of approximately 0.68. This means that about 68% of the available space within the crystal structure is occupied by atoms. To determine the APF we need to get relation between the radius (R) of the atom and the parameter of the cube.

Closest Packed Directions: The closest-packed directions in a BCC structure are along the body diagonals of the cube. These directions are characterized by a stacking sequence of ABABAB..., where A and B represent two different atomic layers.

Closest Packed Planes: The closest-packed planes in BCC are the {110} planes, which are not parallel to the faces of the cube but bisect the edges of the cube. These planes are closely packed with atoms.

Examples of Materials with BCC Structure: Some common metallic elements and alloys that have a BCC crystal structure include pure iron (Fe) at low temperatures, chromium (Cr), molybdenum (Mo), and some types of steel.

Density: Materials with a BCC structure tend to have moderate densities due to their relatively efficient packing of atoms.

Mechanical Properties: BCC metals often exhibit good strength and toughness. They may also have magnetic properties, depending on the specific material.

Polymorphism: Like other crystal structures, some materials can exist in multiple forms, and the choice between BCC, FCC, or other structures can depend on factors such as temperature and pressure. For example, iron can undergo a phase transition from BCC to FCC at higher temperatures.

Face-Centered cubic (FCC) crystal structure

The Face-Centered Cubic (FCC) crystal structure, also known as the cubic close-packed (CCP) structure, is one of the three primary cubic crystal structures, along with the Simple Cubic (SC) and Body-Centered Cubic (BCC) structures. The FCC structure is characterized by its densely packed arrangement of atoms. Here are some key features and properties of the FCC crystal structure:

Lattice Structure: In an FCC structure, atoms are arranged in a cubic lattice with atoms at each corner of the cube and additional atoms at the centers of each face of the cube. This arrangement results in a repeating ABCABC... sequence of atom layers. The atoms at the corners are counted in the same way of SC, and each atom in the center of each face is shared between two unit cells, so each unit cell's share of the atoms located in the center of each face is 1/2 of an atom. In this type, there is one atom on each of the cube's faces, which means there are a total of 6 atoms. This implies that the actual number of atoms is 1/2 * 6 + 1 atom from corners which means there are 4 atoms.

Coordination Number: The coordination number of atoms in an FCC structure is 12. This means that each atom in the lattice is in direct contact with 12 neighboring atoms.

Atomic Packing Factor (APF): The FCC structure has a high APF of approximately 0.74. This means that about 74% of the available space within the crystal structure is occupied by atoms, making it highly efficient in terms of atomic packing. The relation between the radius (R) of the atom and the parameter of the cube.

Closest Packed Directions: The closest-packed directions in an FCC structure are along the body diagonals of the cube. These directions are characterized by a stacking sequence of ABCABC..., where A, B, and C represent three different atomic layers.

Closest Packed Planes: The closest-packed planes in FCC are the {111} planes, which are parallel to the faces of the cube. These planes are closely packed with atoms.

Examples of Materials with FCC Structure: Many common metallic elements and alloys have an FCC crystal structure. Some examples include aluminum (Al), copper (Cu), gold (Au), silver (Ag), and lead (Pb).

Density: Materials with an FCC structure tend to have relatively high densities due to their efficient packing of atoms.

Mechanical Properties: FCC metals often exhibit good ductility and formability due to their closely packed structure. They can also have excellent electrical and thermal conductivity.

Polymorphism: Some materials can exist in multiple crystal structures, and the choice between FCC, BCC, or other structures can depend on factors such as temperature and pressure. For example, iron (Fe) can adopt either an FCC or BCC structure depending on its temperature.

Hexagonal Close-Packed (HCP) crystal structure

Hexagonal Close-Packed (HCP) is one of the three main crystal structures found in metallic materials, alongside Body-Centered Cubic (BCC) and Face-Centered Cubic (FCC) structures. HCP is characterized by its tightly packed arrangement of atoms in a hexagonal lattice. Here are some key features and properties of the HCP crystal structure:

Lattice Structure: In an HCP structure, the unit cell consists of two stacked layers of atoms forming a hexagonal lattice. Each layer contains a set of closely packed atoms, and the second layer is positioned above the spaces created by the first layer. This stacking pattern results in a repeating ABABAB... sequence.

Number of atoms: There are a total of 6 atoms contained within the unit cell.

Coordination Number: The coordination number of atoms in an HCP structure is 12. This means that each atom is in direct contact with 12 neighboring atoms in the same layer.

Atomic Packing Factor (APF): The HCP structure has an APF of approximately 0.74. This means that about 74% of the available space within the crystal structure is occupied by atoms, making it relatively efficient in terms of atomic packing.

Closest Packed Directions: The closest-packed directions in an HCP structure are along the hexagonal axis. These directions are characterized by a stacking sequence of ABABAB..., where A and B represent two different atomic layers.

Closest Packed Planes: The closest-packed planes in HCP are the hexagonal planes. These planes are parallel to the base of the hexagonal unit cell and are closely packed with atoms.

Examples of Materials with HCP Structure: Some common materials that exhibit the HCP crystal structure include magnesium (Mg), zinc (Zn), cadmium (Cd), and certain types of titanium (Ti) and beryllium (Be) alloys.

Mechanical Properties: HCP metals often have unique mechanical properties due to the anisotropic nature of the crystal structure. Anisotropy means that the properties of the material vary with direction. For example, HCP metals may exhibit different mechanical behavior along different crystallographic directions.

Density: Materials with an HCP structure tend to have relatively high densities compared to other crystal structures because of their efficient packing of atoms.

Polymorphism: Some materials can exist in multiple crystal structures, and the choice between HCP, FCC, or BCC can depend on factors such as temperature and pressure. For example, at certain temperatures and pressures, titanium can adopt either an HCP or an FCC structure.

Note that:

Metals in FCC more ductile than in BCC.

BCC need more energy to be deformed.

FCC more easier to be plastic deformed.

FCC more close-packed than BCC.

BCC is not close-packed.

FCC has 12 slip planes.

BCC has 48 slip planes.

slip planes are the close-packed planes.

Although the FCC is more close-packed, it needs less energy than the BCC to be deformed.

The plastic deformation occurs easier on the slip planes so, it is normal that the BCC which has more slip planes is easier and need lower energy than FCC to be deformed... then why the opposite occurs ?!

- Because the number of active slip plan(the deformation occurs on it) in the FCC is more than BCC.