In this article, we will learn about the definition of the unit cell, some introduction to the unit cell, and discuss the number of atoms per unit cell in a simple cubic structure, BCC, FCC, and HCP. In this article, we’ll explore the significance of the number of atoms per unit cell and how it can help us understand the behavior of materials at the atomic level. So let’s start with the introduction from the unit cell.
Introduction
When it comes to understanding the properties and behavior of materials, one of the most important concepts to grasp is the structure of their crystals. A crystal structure refers to the arrangement of atoms, ions, or molecules in a repeating pattern, which is responsible for the characteristic properties of a material. One crucial aspect of crystal structures is the number of atoms per unit cell, which refers to the smallest repeating unit of a crystal.
What is a unit cell?
Let’s define a unit cell first before we go into the number of atoms per unit cell. A crystal lattice is a three-dimensional arrangement of atoms, ions, or molecules.
A unit cell is the smallest repeating unit in a crystal lattice.
Unit cells are distinguished by their symmetry, which is characterized by the crystal’s space group. Translations, rotations, and reflections are examples of space group operations that leave the crystal structure unaltered.
Why is the number of atoms per unit cell important?
The number of atoms per unit cell is a crucial factor in determining the properties and behavior of a material. For example, the density of a material is directly related to the number of atoms per unit cell. Materials with a higher number of atoms per unit cell tend to have a higher density, as there are more atoms packed into a given volume.
The number of atoms per unit cell also affects the electronic properties of a material, such as its conductivity and band gap. In materials with a higher number of atoms per unit cell, the electrons are more tightly bound to the atoms, making them less mobile and less likely to conduct electricity. On the other hand, materials with a lower number of atoms per unit cell tend to have a wider band gap, which means that they require more energy to excite electrons and conduct electricity.
The number of atoms per unit cell in Crytal structure
The number of atoms per unit cell depends on the crystal structure of the material. There are several types of crystal structures, such as simple cubic, body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close-packed (HCP). Each crystal structure has a unique arrangement of atoms, which determines the number of atoms per unit cell.
1. Number of Atoms per unit cell in a Simple Cubic Structure
The below figure shows the unit cell of a simple cubic structure. In this case, there are only 8 atoms, one at each corner of the cube or the unit cell.
In actual crystals, each and every corner atom is shared by 8 adjacent unit cells. Therefore, each and every corner atom contributed \frac{1}{8} of its part to one unit cell.
Total number of corner atoms = 8
Therefore, the total share of all the corner atoms per unit cell = 8 \times \frac{1}{8} = 1.
Therefore the number of atoms per unit cell in a simple cubic structure = 1.
In other words, the effective number of lattice points in a simple cubic structure is one. Thus, a simple cubic is a primitive cell.
2. Number of Atoms per unit cell in BCC
In the BCC or Body-centered Cubic structure case, there are eight atoms one at each corner of the unit cell plus one atom at the center of the unit cell present as shown in the figure below:
Each and every corner atom is shared by eight adjacent unit cells.
Therefore, the total share of all the corner atoms per unit cell is 8 \times \frac{1}{8} = 1.
The atom at the body center is not shared by any other unit cell.
Therefore, the number of unshared atoms per unit cell = 1
Hence, the Total number of atoms per unit cell in BCC = (8 \times \frac{1}{8}) + 1 = 2
3. Number of Atoms per unit cell in FCC
In the FCC or Face-centered Cubic structure case, there are eight atoms, one at each corner of the unit cell and six atoms at the center of six faces of the unit cell as shown in the figure below:
The total share of all the corner atoms per unit cell is 8 \times \frac{1}{8} = 1.
Now, consider the toms at the face center of the unit cell. Each such atom is shared by only two unit cells, which lie on either side of the plane in which the atom is located.
Therefore, the total share of all the face-centered atoms per unit cells = 6 \times \frac{1}{2} = 3.
Hence, the total number of atoms per unit cell in FCC = (8 \times \frac{1}{8}) + (6 \times \frac{1}{2}) = 1 + 3 = 4
4. Number of Atoms per unit cell in HCP
In the below figure of HCP or Hexagonal closed packed structure, there are six corner atoms at each top D-layer and bottom A-layer. One face center atom at each top F-layer and bottom A-layer and there are three full atoms in the middle of the B-layer of the structure.
Each corner atom is shared by six surrounding hexagon cells.
Therefore, the number of corner atoms in the top layer is 6 \times \frac{1}{6} = 1.
Similarly, the number of atoms in the bottom layer is 6 \times \frac{1}{6} = 1.
Each central atom at the top and bottom layer is shared by two unit cells.
Therefore, the number of center atoms in the top layer is 1 \times \frac{1}{2} = \frac{1}{2}.
Similarly, the number of center atoms in the bottom layer is 1 \times \frac{1}{2} = \frac{1}{2}.
But the three atoms within the body of the cell are fully contributing to the cell, i.e., they are not shredded by other unit cells.
| Number of the unit cell in HCP = | Number of corner atoms per unit cell (Top and bottom layer) | Number of center atoms per unit cell (Top and bottom layer) | Number of middle layer atoms per unit cell |
| = | (1 + 1) | (\frac{1}{2} + \frac{1}{2}) | 3 |
| = | 2 | 1 | 3 |
Therefore, the total number of atoms per unit cell in HCP = 6
Conclusion
In summary, the number of atoms per unit cell is a fundamental concept in understanding the properties and behavior of materials at the atomic level. The crystal structure of a material determines the number of atoms per unit cell, which in turn affects its density and electronic properties. By understanding the number of atoms per unit cell, we can predict the behavior of materials in various applications, such as semiconductors, metals, and ceramics.
Frequently Asked Questions – FAQs
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What is unit cell and its types?
The smallest replicating portion of a crystal lattice is a unit cell. Unit cells exist in many types. The cubic crystal structure, for example, consists of three distinct unit cell types : (1) plain cubic, (2) face-centered cubic, and (3) body-centered cubic.
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What is meant by a unit cell of crystal?
The unit cell is defined as the smallest repeated unit with full crystal structure symmetry. The unit cell geometry is known as a parallelepiped, providing six lattice parameters taken as the lengths of the edges of the cells (a, b , c) and the angles between them (α, β, ÿ).
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How many kinds of primitive unit cells are possible?
Seven simple crystal structures exist; cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic, and rhombohedral. They differ in the way their crystallographic axes and angles are arranged. Bravis defined 14 possible crystal systems according to the above seven.
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What do you mean by primitive unit cell?
A primitive cell (also known as a primitive unit cell) is a minimum-volume unit cell in mathematics, biology, mineralogy (especially crystallography), and solid-state physics, referring to a single lattice point of a structure with discrete translation symmetry. The main cell is simple.
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What is an hcp unit cell?
In a unit cell, the number of coordinates of an atom is the number of atoms that it touches. The closest hexagonal packed (hcp) has a coordinating number of 12 and contains six atoms per unit cell. The face-centered cubic (fcc) has a total of 12 coordinates and contains 4 atoms per unit cell.
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What is the difference between a crystal lattice and a unit cell?
A crystal lattice is a three-dimensional arrangement of atoms, ions, or molecules, while a unit cell is the smallest repeating unit of a crystal lattice. The crystal lattice can be described by the space group, which is a set of mathematical operations that leave the crystal lattice unchanged.
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How do you determine the number of atoms per unit cell?
The number of atoms per unit cell depends on the crystal structure of the material. Each crystal structure has a unique arrangement of atoms, which determines the number of atoms per unit cell.
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What is the significance of the number of atoms per unit cell?
The number of atoms per unit cell is a crucial factor in determining the properties and behavior of a material, such as its density and electronic properties.
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Can the number of atoms per unit cell change?
The number of atoms per unit cell is determined by the crystal structure of the material and cannot change unless the crystal structure changes.
