## What are space lattices and unit cells?

Every matter consists of atoms, molecules, and ions. If these atoms, molecules, and ions (micro-particles) are represented in a three-dimensional space-occupying certain specific points in space, then these arrangements are termed as space lattices and the specific points which are being occupied by atoms, molecules and ions are termed as lattice points. The atoms, molecules, and ions at the lattice points are generally grouped in a regular fashion and repeat periodically at the three-dimensional space.

These repeated arrangements of the micro-particles form a crystal lattice. For instance, consider a square having side as ABC with each atom present at the corners of the square. To form a crystal lattice, each of the atom present at the corners must join a similar square of side ABC having atoms at each corner. This orderly arrangement of ABCABCABC, and so on forms a crystal lattice.

## Unit Cells

The crystal lattice can be represented by considering a small three-dimensional portion such that the space of the three-dimensional portion can represent the whole crystal structure; this small three-dimensional portion is known as the unit cell. The entire crystal lattice can be formed by continuously repeating the unit cell. Based on the geometry of the unit cell, it can be a parallelogram, a parallelepiped or a rhombus. In chemistry, analysis of crystal arrangements is carried out by a diffractometer.

## Types of unit cell

### Simple cubic unit cell

The simple cubic unit cell is regarded as the simplest unit cell. This kind of unit cell has atoms in every corners of the cube. Atoms are absent at the center and at the center of the faces. Each unit cell can be joined together at the lattice points repeatedly to form the crystal lattice. The simple cubic unit cell does not occur in the nature due to its low atomic packing, almost 52%. The simple cubic unit cell has a single interstitial site at the center, the only element that exists as a simple cubic unit cell at normal temperature and pressure is polonium.

### Face-centered cubic unit cell

In the face-centered cubic unit cell, there is a presence of one-eighth of atoms at the corners, with presence of a single atom at each faces of the unit cell. The atoms in the face-centered cubic unit cell remains closely packed than the other unit cell arrangements. Due to this close packing, the face-centered cubic unit cell arrangement makes the metal softer and ductile. The atomic packing factor of face-centered cubic unit cell arrangement is 74%.

### Body-centered cubic unit cell

In body-centered cubic unit cell, there are one-eighth of atoms present at the eight corners of the unit cell along with a single atom present at the center of the unit cell. There are no atoms at the faces of the unit cell. All the corner atoms of the unit cell form a bond with the atom present at the center. The atoms of the body-centered cubic unit cell forms the closest-packed array of atoms. The body-centered cubic unit cell forms a very strong bond among the atoms, which makes the metal harder to deform. Metals with body-centered cubic unit cell are harder than simple cubic unit cell and face-centered cubic unit cell. For this kind of arrangement, the atomic factor is 68%. The volume of the face-centered cube in terms of ionic radii can be given as $V={\left(\frac{4}{\sqrt{3}·r}\right)}^{3}$, where $r$ = ionic radii

### Hexagonal cubic unit cell

This kind of atomic arrangement is also known as hexagonal closed packing, this kind of arrangement is commonly exhibited by atoms for most of the metals. The atoms can be found at each of the corners of the three-dimensional hexagon with three atoms inside the hexagon. This kind of unit cell arrangement is regarded as the most stable arrangement.

## Octahedral and tetrahedral holes of space lattices

An ionic compound is generally accompanied by two holes, namely, octahedral holes and tetrahedral holes. An ionic compound is a compound made up of cations and anions where the anions form the major part. An example of an ionic compound is zinc sulphate, ZnS. In a face-centered kind of arrangement for the ZnS, the smaller regions are occupied by the whereas the larger regions are occupied by the . The face-centered unit cell forms four octahedral holes. Another example of compounds forming an octahedral hole is the sodium chloride, NaCl. Sodium chloride is a compound that dissociates into cations and anions, the sodium ions form the cations, while the chloride ions form the anions. In the octahedral hole, the sodium ions occupy the smaller regions of the arrangement with the chloride ions in the larger regions. In each unit cell, there are four chloride ions.

Methane is a compound that forms a tetrahedral hole. Tetrahedral holes are generally formed when a center atom is bonded by four equally distant atoms.

## Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

• Bachelor in Science (Physics)
• Bachelor in Science (Chemistry)
• Bachelor in Technology (Mechanical Engineering)

## Practice Problems

1. Which of the following is true for space lattice?

1. In space lattice, the atoms, molecules and ions occupy certain specific points in space
2. In space lattice, only atoms occupy certain specific points in space
3. In space lattice, only the molecules occupy certain points in space
4. None of these

Correct option- a

Explanation: In a space lattice, the atoms, molecules and ions occupy certain specific points in space called lattice points.

2. Which of the following is true for lattice points?

1. Lattice points are those points in space that represent the density of atoms
2. Lattice points are those points that represent the net ion density of the matter
3. Lattice points are those points that represent the specific points of location of atoms, molecules, and ions
4. None of these

Correct option- c

Explanation: Lattice points are the points that represent the specific locus of the location of atoms, molecules, and ions.

3. Which of the following represents a unit cell?

1. Unit cell is the smallest two-dimensional portion of a crystal lattice
2. Unit cell is the smallest two-dimension portion of a space lattice
3. Unit cell is the smallest three-dimensional portion of space lattice
4. Unit cell is the smallest three-dimensional portion of a crystal lattice

Correct option- d

Explanation: Unit cells are the three-dimensional volumes inside a crystal lattice. These are the smallest possible enclosed volumes.

4. Which of the following unit cell has atoms at each face of the unit cell?

1. Simple cubic unit cell
2. Face-centered cubic unit cell
3. Body-centered cubic unit cell
4. Hexagonal cubic unit cell

Correct option- b

Explanation: A face-centered unit cell has atoms at each face of the unit cell.

5. Which of the unit cell arrangement forms the closest-packed array of atoms?

1. Face-centered cubic unit cell
2. Body-centered cubic unit cell
3. Simple cubic unit cell
4. None of these

Correct option- a

Explanation: A face-centered unit cell forms the closest-packed array of atoms.

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