Crystal Lattices and Unit Cells

Edited By Team Careers360 | Updated on Jul 02, 2025 04:50 PM IST

The crystal lattice and the unit cell are the three-dimensional, repeating patterns of the atomic, ionic, or molecular positions of atoms in a crystal. Such a structured pattern extends and gives an orderly structure to the material. The Unit Cell is the smallest building block of the crystal lattice that can be repeated in space to form the entire lattice.

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In the article, we cover the topic of crystal lattice which is the sub-topic of the chapter on Solid states. it is important for board exams JEE Mains Exam, NEET Exam, and other entrance exams.

A portion of the three-dimensional crystal lattice and its unit cell as shown in Fig below:

3D Cube

In the three-dimensional crystal structure, a unit cell is characterized by:
(i) its dimensions along the three edges a, b, and c. These edges may or may not be mutually perpendicular.
(ii) angles between the edges, α (between b and c), β (between a and c), and γ (between a and b). Thus, a unit cell is characterized by six parameters a, b, c, α, β, and γ.
These parameters of a typical unit cell are shown in Fig given below:

Angles in 3D cube

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Primitive and Centred Unit Cells

Primitive Unit Cells
When constituent particles are present only on the corner positions of a unit Cell, it is called as primitive unit cell.

Centered Unit Cells
When a unit cell contains one or more constituent particles present at positions other than corners in addition to those at corners, it is called a centered unit cell. Centered unit cells are of three types:

Body-centered Unit Cells: Such a unit cell contains one constituent particle (atom, molecule, or ion) at its body center besides the ones that are at its corners.
Face-centered Unit Cells: Such a unit cell contains one constituent particle present at the center of each face, besides the ones that are at its corners.
End-centered Unit Cells: In such a unit cell, one constituent particle is present at the center of any two opposite faces besides the ones present at its corners.

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Inspection of a wide variety of crystals leads to the conclusion that all can be regarded as conforming to one of the seven regular figures. These basic regular figures are called seven-crystal systems.

Seven Primitive Unit Cells and Their Possible Variations as Centred Unit Cells

Crystal system	Bravias lattices	Intercepts	Interfacial angle	Examples
Cubic	Primitive, face-centered, body-centered = 3	a = b = c	⍺ = β = ? = 90^o	Ag, Au, Hg, Pb, diamond, NaCl, ZnS
Orthorhombic	Primitive, face-centered, body-centered, end centered = 4	a ≠ b ≠ c	⍺ = β = ? = 90^o	K₂SO₄,KNO₂, BaSO₄, Rhombic Sulphur
Tetragonal	Primitive, body-centred = 2	a = b ≠ c	⍺ = β = ? = 90^o	TiO₂, SnO₂, CaSO₄, White Tin
Monoclinic	Primitive, end centered = 2	a ≠ b ≠ c	⍺ = ? = 90 β ≠ 90^o	CaSO₄.2H₂O
Triclinic	Primitive = 1	a ≠ b ≠ c	⍺ ≠ β ≠ ? ≠ 90^o	CuSO₄.5H₂O, K₂Cr₂O₇, H₃BO₃
Hexagonal	Primitive = 1	a = b ≠ c	⍺ = β = 90^o ? = 120^o	Zn, Mg, Cd, SiO₂, Graphite, ZnO
Rhombohedral	Primitive = 1	a = b = c	⍺ = β = ? ≠ 90^o	Bi, As, Sb, CaCO₃, HgS
Total = 14

Unit Cells of 14 Types of Bravais Lattices

Any crystal lattice is made up of a very large number of unit cells and every lattice point is occupied by one constituent particle (atom, molecule or ion).

Primitive Cubic Unit Cell

The primitive cubic unit cell has atoms only at its corner. Each atom at a corner is shared between eight adjacent unit cells as shown in Fig. given below:

3D Cube
four unit cells in the same layer and four unit cells in the upper (or lower) layer. Therefore, only 1/8th of an atom (or molecule or ion) actually belongs to a particular unit cell. In Fig. given below, a primitive cubic unit cell has been depicted in three different ways. Each small sphere in this figure represents only the center of the particle occupying that position and not its actual size. Such structures are called open structures.
Lattice points
The arrangement of particles is easier to follow in open structures as shown in the figure given below depicts a space-filling representation of the unit cell with actual particle size
Voids
The figure given below shows the actual portions of different atoms present in a cubic unit cell. In all, since each cubic unit cell has 8 atoms on its corners, the total number of atoms in one unit cell 8x(1/8) = 1 atom.

Body Centred Cubic Unit Cell

A body-centred cubic (bcc) unit cell has an atom at each of its corners and also one atom at its body center. The figure given below depicts (a) open structure (b) space-filling model and (c) the unit cell with portions of atoms actually belonging to it. It can be seen that the atom at the body centre wholly belongs to the unit cell in which it is present.
Number of atoms in unit cell

Thus in a body-centered cubic (bcc) unit cell:

8 corners x 1/8 per corner atom = 8 x 1/8 = 1 atom
1 body centre atom = 1 x 1 = 1 atom
Thus, total number of atoms per unit cell = 2 atoms

Face Centred Cubic Unit Cell

A face-centered cubic (fcc) unit cell contains atoms at all the corners and at the center of all the faces of the cube. It can be seen in the figure given below, that each atom located at the face-centre is shared between two adjacent unit cells and only ½ of each atom belongs to a unit cell.

The fig. given below depicts (a) an open structure (b) a space-filling model and (c) the unit cell with portions of atoms actually belonging to it.
Types of unit cell
Thus, in a face-centred cubic (fcc) unit cell:

8 corners atoms x 1/8 atom per unit cell = 8 x 1/8 = 1 atom
6 face-centred atoms x 1/2 atom per unit cell = 6 x 1/2 = 3 atoms
Thus, total number of atoms per unit cell = 4 atoms

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Some Solved Examples

Example 1
Question: The smallest repeating pattern which when repeated in 3-D results in the crystal of substance is called:
1) Space lattice
2) Crystal lattice
3) (correct) Unit cell
4) Bravais lattice

Solution: The unit cell is the smallest repeating unit in the crystal which has all the properties of a crystal. Hence, the answer is the option (3).

Example 2
Question: The most unsymmetrical crystal system is:
1) Cubic
2) Hexagonal
3) (correct) Triclinic
4) Orthorhombic

Solution: The triclinic crystal system has the parameters a n_eqb n_eq c and alpha n_eq beta n_eqgamma n_eq90⁰ . Hence, the answer is the option (3).

Example 3
Question: The crystal system of a compound with unit cell dimensions a=0.387,nm,b=0.387,nm,c=0.504,nmandgamma=120⁰ is:
1) (correct) Hexagonal
2) Cubic
3) Rhombohedral
4) Orthorhombic

Solution: For hexagonal systems, the conditions are a = b neq c and alpha=beta=90⁰,gamma=120⁰ . Thus, the answer is the option (1).

Example 4
Question: For which of the given crystal families does the following relation hold? a not equal b not equal c and alpha=gamma=90⁰,betaneq90⁰
1) (correct) Monoclinic
2) Triclinic
3) Orthorhombic
4) Hexagonal

Solution: The given relation corresponds to the monoclinic crystal system. Hence, the answer is the option (1).

Example 5
Question: The crystal system characterized by a = b = c and alpha=beta=gamma=90⁰ is:
1) Cubic
2) Tetragonal
3) (correct) Orthorhombic
4) Rhombohedral

Solution: The cubic crystal system has equal edge lengths and all angles equal to 90⁰. Therefore, the answer is the option (1).

NCERT Chemistry Notes:

Frequently Asked Questions (FAQs)

1. What is lattice constant/ what is lattice parameter?

The lattice parameters or lattice constant can be defined as the quantities which specify a unit cell. These parameters (constants) are of six types. The dimensions along the edges of a unit cell are represented by a, b and c along x, y and z planes respectively and angle between b and c is represented by α, angle between a and c by β and angle between a and b by γ.

2. What are the types of crystal lattice?

crystal lattice can be of seven types: triclinic, monoclinic, orthorhombic, hexagonal, rhombohedral, tetragonal, and cubic. These collections of crystal lattice are called the Bravais lattices.

3. Define amorphous solid? Give example. (or) non-Crystalline materials.

It is a type of solid, in which the atoms (or) molecules are not arranged in an orderly manner that is, the same atomic groups are arranged more randomly.

Example: Plastic, rubber

4. What is meant by Crystallography?

The study of the geometric form and other physical properties of crystalline solids, using x-rays, or electron beam, or neuron beam etc is termed as the science of crystallography.

5. What is a single crystal?

A crystal in which solid contains only one crystal. These single crystals are produced artificially from their vapor (or) Liquid State.

6. define poly crystal? Give example.

A crystal structure in which has an aggregate of many small crystals or, grains separated by well-defined grain boundaries. These crystals will have a sharp melting point. Examples: Diamond, Copper, Platinum, Silver, Polonium, Gold, Aluminium, Nickel, Cadmium, Iron etc.

7. What are the types of unit cells?

There are several types of unit cells, commonly classified based on their geometry. The main types include:

Cubic: All sides are equal and angles are all 90 degrees (e.g., simple cubic, body-centered cubic, face-centered cubic).
Tetragonal: Two sides are equal, and angles are all 90 degrees.
Orthorhombic: All sides are of different lengths and angles are all 90 degrees.
Hexagonal: Contains two sides of equal length, with angles of 120 degrees.
Rhombohedral: All sides are equal, but angles are acute or obtuse (not 90 degrees).
Monoclinic: All sides are of different lengths, with one angle not equal to 90 degrees.
Triclinic: All sides are of different lengths and all angles are different from 90 degrees.

8. What are the 7 types of crystal lattice?

The seven types of crystal lattice are Triclinic, Monoclinic, Orthorhombic, Tetragonal, Trigonal, Hexagonal, and Cubic.

9. How does the arrangement of atoms in a crystal lattice affect its properties?

The arrangement of atoms in a crystal lattice affects numerous physical and chemical properties of materials, such as:

Mechanical strength: The type of lattice can influence how a material deforms under stress.
Electrical conductivity: The ability of atoms to move freely impacts electrical properties.
Optical properties: The symmetry and arrangement can affect how light interacts with the material.
Thermal properties: The lattice structure can affect how heat is conducted through a material.

10. What is a crystal lattice?

A crystal lattice is a three-dimensional, repeating pattern of atoms, ions, or molecules in a crystalline solid. It represents the orderly arrangement of particles in space, forming the structure of the crystal.

11. How does a unit cell differ from a crystal lattice?

A unit cell is the smallest repeating unit of a crystal lattice that still maintains the overall symmetry and properties of the entire crystal structure. The crystal lattice is made up of many unit cells repeated in three dimensions.

12. What are the key parameters of a unit cell?

The key parameters of a unit cell are:

13. How many types of unit cells exist in three-dimensional crystal systems?

There are seven types of unit cells in three-dimensional crystal systems: cubic, tetragonal, orthorhombic, rhombohedral (trigonal), hexagonal, monoclinic, and triclinic.

14. What is the significance of coordination number in crystal lattices?

The coordination number represents the number of nearest neighbors surrounding a central atom or ion in a crystal structure. It helps determine the geometry of the crystal and influences properties such as density, hardness, and melting point.

15. What is the significance of Brillouin zones in crystal lattice theory?

Brillouin zones are important constructs in reciprocal space that help describe the behavior of waves (such as electrons or phonons) in a periodic structure. They are crucial for understanding electronic band structures, phonon dispersion, and various solid-state phenomena.

16. What is the role of crystal field theory in understanding the properties of transition metal complexes in solid state?

Crystal field theory explains how the surrounding crystal lattice affects the electronic structure of transition metal ions. It helps predict color, magnetic properties, and reactivity of transition metal compounds based on the symmetry and strength of the crystal field.

17. What is the significance of the Debye model in understanding the thermal properties of crystal lattices?

The Debye model provides a quantum mechanical description of vibrations in a crystal lattice. It helps explain the temperature dependence of specific heat capacity in solids and provides insights into phenomena like thermal conductivity and sound propagation in crystals.

18. How does the atomic packing factor relate to the efficiency of space utilization in a crystal?

The atomic packing factor (APF) is the fraction of volume in a unit cell actually occupied by atoms or ions. A higher APF indicates more efficient space utilization and generally results in higher density and hardness of the crystal.

19. What is the difference between primitive and non-primitive unit cells?

A primitive unit cell contains only one lattice point, typically at the corners of the cell. A non-primitive unit cell contains additional lattice points, either on the faces or within the cell, and can be divided into multiple primitive unit cells.

20. How do crystal systems affect the properties of materials?

Crystal systems determine the symmetry and arrangement of atoms in a material, which directly influences its physical and chemical properties, such as cleavage planes, optical properties, thermal conductivity, and mechanical strength.

21. What is Bravais lattice, and how many types are there?

A Bravais lattice is a mathematical description of the geometric arrangement of lattice points in a crystal. There are 14 types of Bravais lattices, which are derived from the seven crystal systems and account for different lattice point positions.

22. How does polymorphism relate to crystal lattices?

Polymorphism occurs when a substance can crystallize in more than one crystal structure. Different polymorphs have the same chemical composition but different crystal lattices, leading to variations in physical properties such as melting point, solubility, and hardness.

23. What is the relationship between unit cell volume and density of a crystal?

The density of a crystal is inversely proportional to its unit cell volume. As the unit cell volume increases, the density decreases, assuming the mass of the atoms or molecules remains constant. This relationship is expressed as: density = (mass of atoms in unit cell) / (volume of unit cell).

24. How do interatomic forces influence the formation of crystal lattices?

Interatomic forces, such as ionic, covalent, metallic, and van der Waals interactions, determine how atoms or ions arrange themselves in a crystal lattice. These forces influence the stability, bond lengths, and overall structure of the crystal.

25. What is the significance of Miller indices in describing crystal planes?

Miller indices (h, k, l) are a set of three integers that describe the orientation of a plane or set of parallel planes in a crystal lattice. They are used to identify specific crystal faces, determine interplanar spacing, and analyze X-ray diffraction patterns.

26. How does the concept of close-packing apply to crystal structures?

Close-packing refers to the most efficient arrangement of spheres (atoms or ions) in a crystal structure. It maximizes the packing efficiency and minimizes empty space. Common close-packed structures include hexagonal close-packed (HCP) and face-centered cubic (FCC) arrangements.

27. What is the difference between a perfect crystal and a real crystal?

A perfect crystal is an idealized model with a completely regular arrangement of atoms or ions throughout its structure. A real crystal contains various defects and imperfections, such as vacancies, interstitials, and dislocations, which affect its properties.

28. How do crystal lattice vibrations relate to the thermal properties of solids?

Crystal lattice vibrations, also known as phonons, are responsible for heat conduction in solids. The way atoms vibrate in the crystal lattice affects properties such as thermal conductivity, specific heat capacity, and thermal expansion of the material.

29. What is the significance of symmetry in crystal structures?

Symmetry in crystal structures determines many of the material's properties, including its optical, electrical, and mechanical characteristics. It also influences the crystal's growth habits, cleavage planes, and how it interacts with electromagnetic radiation.

30. How do impurities affect the crystal lattice and its properties?

Impurities can distort the crystal lattice, creating local strain and defects. This can significantly alter the material's properties, such as electrical conductivity, optical characteristics, and mechanical strength. In some cases, impurities are deliberately introduced to modify material properties.

31. What is the relationship between crystal structure and mechanical properties?

The crystal structure directly influences mechanical properties such as hardness, ductility, and strength. For example, materials with strong directional bonds (like diamond) tend to be harder, while those with more isotropic bonding (like metals) are often more ductile.

32. How does anisotropy manifest in crystal lattices?

Anisotropy in crystal lattices refers to the variation of physical properties depending on the direction within the crystal. This can affect properties such as thermal conductivity, electrical resistivity, and optical refraction, which may differ along different crystallographic axes.

33. What is the significance of Wyckoff positions in crystal structures?

Wyckoff positions describe the symmetry-equivalent positions of atoms within a unit cell. They are crucial for understanding the crystal's symmetry, determining the number of atoms in a unit cell, and interpreting X-ray diffraction data.

34. How do crystal lattices influence the band structure of materials?

The periodic arrangement of atoms in a crystal lattice creates a periodic potential that electrons experience. This leads to the formation of energy bands and band gaps, which determine the electronic and optical properties of the material, such as whether it's a conductor, semiconductor, or insulator.

35. What is the role of point groups in describing crystal symmetry?

Point groups describe the symmetry operations (rotations, reflections, inversions) that leave at least one point of the crystal unchanged. They are essential for classifying crystals, predicting their physical properties, and understanding their behavior under various conditions.

36. How does pressure affect crystal lattices?

Increased pressure typically causes crystal lattices to contract, reducing the unit cell volume. This can lead to phase transitions, changes in crystal structure, and alterations in physical properties such as electrical conductivity and optical characteristics.

37. How do crystal lattices influence the diffusion of atoms or ions in solids?

The arrangement of atoms in a crystal lattice affects the energy barriers for atomic or ionic movement. Diffusion often occurs through vacancy or interstitial mechanisms, with the crystal structure determining the available pathways and activation energies for diffusion.

38. What is the relationship between crystal structure and magnetic properties?

The crystal structure influences the alignment and interaction of magnetic moments in materials. It can determine whether a material exhibits ferromagnetism, antiferromagnetism, or other magnetic orderings, and affects properties like magnetic anisotropy and Curie temperature.

39. How do crystal lattices affect the optical properties of materials?

Crystal lattices influence optical properties through their symmetry and arrangement of atoms. This affects how light interacts with the material, leading to phenomena such as birefringence, optical activity, and color center formation.

40. What is the significance of Frenkel and Schottky defects in crystal lattices?

Frenkel defects (interstitial-vacancy pairs) and Schottky defects (paired vacancies) are important point defects in crystal lattices. They affect properties like ionic conductivity, diffusion rates, and can influence the material's reactivity and mechanical behavior.

41. How do crystal lattices relate to the concept of crystallographic shear planes?

Crystallographic shear planes are structural features where one part of the crystal is shifted relative to another along a specific plane. They can occur due to non-stoichiometry or as a mechanism to accommodate defects, and they significantly affect the material's properties and reactivity.

42. How do quasicrystals differ from conventional crystal lattices?

Quasicrystals possess long-range order but lack periodic translational symmetry, unlike conventional crystals. They often exhibit forbidden symmetries (like 5-fold rotational symmetry) and have unique physical properties that challenge traditional crystallography concepts.

43. What is the significance of Goldschmidt's tolerance factor in predicting crystal structures?

Goldschmidt's tolerance factor is used to predict the stability and structure of ionic compounds, particularly perovskites. It relates the ionic radii of the constituent ions and helps determine whether a particular combination of ions will form a stable perovskite structure.

44. How do crystal lattices influence the formation and propagation of dislocations?

The crystal lattice structure determines the possible slip systems and Burgers vectors for dislocations. This influences how dislocations form, move, and interact, which in turn affects the material's plastic deformation behavior and mechanical properties.

45. What is the relationship between crystal structure and ferroelectric properties?

Ferroelectric properties arise from specific crystal structures that allow for a spontaneous electric polarization that can be reversed by an applied electric field. The crystal symmetry must be non-centrosymmetric and typically belongs to certain point groups to exhibit ferroelectricity.

46. How do crystal lattices affect the formation and properties of solid solutions?

Crystal lattices influence solid solution formation by determining the size and electronic compatibility of solute atoms or ions. Factors like atomic size mismatch, electronegativity differences, and crystal structure similarity between solvent and solute affect solubility limits and resulting properties.

47. What is the significance of Vegard's law in understanding solid solutions?

Vegard's law states that there is often a linear relationship between the crystal lattice parameter of an alloy and the concentration of its constituent elements. This helps in predicting and analyzing the composition of solid solutions based on X-ray diffraction data.

48. How do crystal lattices influence the behavior of materials under irradiation?

Crystal lattices affect how materials respond to radiation damage. The lattice structure influences the formation and migration of point defects, dislocation loops, and other radiation-induced defects, which in turn affect the material's dimensional stability and mechanical properties under irradiation.

49. What is the role of crystal lattices in determining the cleavage planes of minerals?

Cleavage planes in minerals are determined by the weakest bonds in the crystal lattice. The arrangement of atoms and the nature of bonding in different crystallographic directions influence where the crystal is most likely to split when stress is applied.

50. How do crystal lattices affect the thermal expansion behavior of materials?

The crystal lattice structure influences thermal expansion through factors like bond strength, atomic packing, and anharmonicity of lattice vibrations. Anisotropic crystal structures often exhibit different thermal expansion coefficients along different crystallographic directions.

51. How do crystal lattices influence the formation and stability of grain boundaries?

Crystal lattices determine the possible orientations of grain boundaries and their energies. The mismatch between adjacent crystal orientations affects grain boundary structure, mobility, and properties, influencing phenomena like grain growth, recrystallization, and material strength.

52. What is the relationship between crystal structure and superconductivity?

Crystal structure plays a crucial role in superconductivity by influencing electron-phonon interactions and the formation of Cooper pairs. Certain crystal structures, like layered perovskites, are particularly conducive to high-temperature superconductivity.

53. How do crystal lattices affect the formation and properties of intermetallic compounds?

Crystal lattices in intermetallic compounds often have specific stoichiometries and ordered arrangements of different metal atoms. These structures can lead to unique properties like high hardness, thermal stability, and interesting magnetic or electronic characteristics.

54. What is the significance of the concept of reciprocal lattice in crystallography?

The reciprocal lattice is a mathematical construct that represents the Fourier transform of the real crystal lattice. It is crucial for understanding X-ray diffraction patterns, electron and neutron scattering, and for describing the electronic band structure of solids.

55. How do crystal lattices influence the formation and properties of semiconductor devices?

Crystal lattices in semiconductors determine the band structure, carrier mobility, and dopant incorporation. The lattice structure influences how impurities are incorporated, how charge carriers move, and how various semiconductor junctions and interfaces behave.

56. What is the role of crystal lattices in determining the piezoelectric properties of materials?

Piezoelectric properties arise from non-centrosymmetric crystal structures that allow for the generation of an electric field under mechanical stress (and vice versa). The specific arrangement of atoms in the lattice determines the magnitude and direction of the piezoelectric effect.

57. How do crystal lattices affect the catalytic properties of solid materials?

Crystal lattices influence catalytic properties by determining the surface structure, active site distribution, and electronic properties of the catalyst. Factors like crystal face exposure, defect concentration, and atomic coordination at the surface play crucial roles in catalytic activity and selectivity.

58. What is the significance of the concept of Wigner-Seitz cells in understanding crystal structures?

Wigner-Seitz cells are the smallest volume of space that, when translated through all the vectors of a Bravais lattice, fills all of space without overlap or gaps. They provide a useful way to visualize the symmetry of a crystal structure and are important in understanding electronic and vibrational properties of solids.

59. How do crystal lattices influence the behavior of materials at extreme conditions (high pressure, temperature)?

Under extreme conditions, crystal lattices can undergo phase transitions, structural transformations, or even amorphization. The initial crystal structure influences how the material responds to these conditions, affecting properties like compressibility, melting behavior, and phase stability at high pressures or temperatures.