What Are the Key Features of the Trigonal Crystal System?
The trigonal system is also called the rhombohedral crystal system. It is one of the finest structural categories that can be assigned to a crystalline solid.
The trigonal system is considered the subdivision of the hexagonal system. We commonly call the rhombohedral crystal system the synonym of the trigonal system, however, they both vary.
This page will clear all your doubts regarding the difference between the trigonal crystal system and the rhombohedral crystal system. We will also discuss the trigonal system definition with the trigonal system examples in detail.
Crystal System Definition
Do you know what the crystal system is?
Well! A crystal system is a system with a set of point groups in which each point group itself and its corresponding space groups is assigned the lattice system.
Here,
A lattice system is defined as the class of lattices with an equal set of lattice point groups, which are subgroups of the arithmetic crystal classes.
The 14 Bravais lattices are grouped into seven lattice systems are as follows:
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Rhombohedral crystal system
Hexagonal
Cubic
Point to Note:
The hexagonal crystal family comprises two crystal systems that are trigonal and hexagonal.
The trigonal crystal system consists of the 5 point groups that have one three-fold rotation axis, which incorporates space groups 143 to 167.
Crystal Family
A crystal family is determined by lattices and point groups and it's formed by combining crystal systems that have space groups allocated to a standard lattice system.
In 3-d, the crystal families and systems are identical, except the crystal systems, hexagonal and trigonal are combined into one hexagonal crystal family.
The six crystal families are:
Triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic.
Do You Know?
The aforementioned 5 point groups (of the trigonal crystal system) have seven analogous space groups, denoted by R, which is assigned to the rhombohedral lattice system, whereas 18 comparable space groups, denoted by P, are assigned to the hexagonal lattice system.
Hence, the trigonal crystal system is that the only crystal system whose point groups have quite one lattice system related to their space groups.
The hexagonal crystal system comprises the seven-point groups that have one six-fold rotational axis.
These seven-point groups have 27 space groups from 168 to 194, and the hexagonal lattice system is allocated to all of these.
Now, let’s understand the Hexagonal Crystal System in detail:
Hexagonal Crystal System
Since the trigonal system is a part of the hexagonal crystal system, let’s start our page with an understanding of this system.
Regarding crystallography, the hexagonal crystal family is one of the six crystal families that involve two crystal systems, called hexagonal and trigonal, and two lattice systems, hexagonal and rhombohedral).
The hexagonal crystal family comprises 12 point groups, in which a minimum of their space groups has the hexagonal lattice as cardinal lattice and that is the unison of the hexagonal crystalline system, and henceforth, the trigonal crystal system.
Hexagonal Crystal Family
In a hexagonal crystal family, 52 space groups are related to it, whose space lattice is either hexagonal or rhombohedral.
The below diagram shows the hexagonal family arrangement of crystals with their Bravais lattice, Pearson symbol, and the unit cell diagrams of both hexagonal and trigonal system:
Trigonal Structure
A Trigonal Crystal System looks like the following:
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Trigonal System Definition
The unique geometry of the trigonal system says that the system is absolute with a 3-fold axis of rotational symmetry or a 3-fold axis of rotatory inversion as its optic axis.
A 6 - fold inversion axis within the hexagonal system is mistaken for the trigonal 3-fold axis. So too, an isometric 3-fold axis, when only the corner of a cube is visible, is often misinterpreted as trigonal when it's a part of isometric symmetry.
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Trigonal Crystal System
Now, let us understand the meaning of the trigonal crystal system:
The 5 point groups (discussed above, i.e., from 143 - 167) during this crystal system are listed below, with their international number and notation, their space groups in name only, and example crystals.
Trigonal Crystal System Example
Components of crystals within the trigonal crystal system are like the hexagonal system. They are located by regard to four axes - three of equal length with 120° intersections and one perpendicular to the plane of the opposite three.
The trigonal unit is distinguished by the presence of one line called an axis of three-fold symmetry about which the cell is often rotated by 120° to supply a face indistinguishable from the face presented within the starting position.
For instance, Selenium and quark may crystallize in trigonal form.
FAQs on Trigonal System in Physics Explained
1. What is the trigonal crystal system?
The trigonal crystal system, also known as the rhombohedral system, is one of the seven fundamental crystal systems used to classify crystalline solids. It is defined by the presence of a single, unique 3-fold axis of rotational symmetry. This means the crystal's unit cell can be rotated by 120° around this axis and appear unchanged.
2. What are the lattice parameters that define a trigonal system?
The trigonal system is unique because it can be described using two different lattice settings:
- Rhombohedral lattice: In this setting, the primitive unit cell has parameters where all sides are equal, and all angles are equal but not 90°. The conditions are: a = b = c and α = β = γ ≠ 90°.
- Hexagonal lattice: For convenience, a larger, non-primitive hexagonal cell is often used to describe the trigonal system. The conditions for this setting are: a = b ≠ c and α = β = 90°, γ = 120°.
3. Can you give some common examples of materials with a trigonal crystal structure?
Many common minerals and materials crystallise in the trigonal system. Some well-known examples include:
- Quartz (SiO₂)
- Calcite (CaCO₃)
- Dolomite (CaMg(CO₃)₂)
- Tourmaline
- Cinnabar (HgS)
- Corundum (including ruby and sapphire)
4. How does the trigonal system fit within the seven main crystal systems?
The trigonal system is one of the seven primary crystal systems that categorise all known crystals based on their symmetry. These seven systems are:
- Cubic
- Tetragonal
- Orthorhombic
- Hexagonal
- Trigonal (Rhombohedral)
- Monoclinic
- Triclinic
Each system is distinguished by its characteristic lattice parameters and minimum symmetry elements.
5. What is the main difference between the trigonal and hexagonal crystal systems?
The fundamental difference lies in their principal axis of symmetry. The trigonal system is defined by having a single 3-fold axis of rotation (rotation by 120°). In contrast, the hexagonal system is defined by a single 6-fold axis of rotation (rotation by 60°). While they share a similar hexagonal lattice setting, their inherent rotational symmetries are distinct.
6. Why is the trigonal system often considered a subdivision of the hexagonal crystal family?
The trigonal and hexagonal systems are grouped into the larger hexagonal crystal family because they are the only two systems whose point groups can be associated with a hexagonal lattice. While the hexagonal system *only* uses a hexagonal lattice, the trigonal system is more versatile and can be described by either a rhombohedral lattice or a hexagonal lattice. This shared lattice compatibility is the basis for their grouping.
7. What is a rhombohedral lattice, and how does it relate to the trigonal crystal system?
A rhombohedral lattice is one of the 14 Bravais lattices, characterised by a unit cell that can be visualised as a cube stretched or compressed along one of its body diagonals. All its edge lengths are equal, and all its axial angles are equal but not 90°. The trigonal crystal system is uniquely linked to this lattice because some of its space groups are exclusively based on the rhombohedral lattice, solidifying the connection and leading to the interchangeable name 'rhombohedral system'.
8. How can you identify a trigonal crystal based on its symmetry elements?
The definitive way to identify a trigonal crystal is by finding its unique, principal axis of symmetry. A crystal belongs to the trigonal system if and only if it possesses exactly one 3-fold rotation axis. It is crucial not to confuse this with the cubic system, which has four 3-fold axes, or the hexagonal system, which has one 6-fold axis that also includes 3-fold symmetry.
