Question

Given edge lengths and face angles, predict the shape of each of the following crystals. $$ \begin{aligned} \text { a. } a &=3 \mathrm{nm}, b=3 \mathrm{nm}, c=3 \mathrm{nm} ; \alpha=90^{\circ}, \beta^{\circ}=90 \\ \gamma &=90^{\circ} \\\ \text { b. } a &=4 \mathrm{nm}, b=3 \mathrm{nm}, c=5 \mathrm{nm} ; \alpha=90^{\circ}, \beta^{\circ}=100 \\ \gamma &=90^{\circ} \\ \text { c. } a &=3 \mathrm{nm}, b=3 \mathrm{nm}, c=5 \mathrm{nm} ; \alpha=90^{\circ}, \beta^{\circ}=90 \\ \gamma &=90^{\circ} \\ \mathrm{d} &=3 \mathrm{nm}, b=3 \mathrm{nm}, c=5 \mathrm{nm} ; \alpha=90^{\circ}, \beta^{\circ}=90 \\ \gamma &=120^{\circ} \end{aligned} $$

Short answer

The following crystals belong to these crystal systems: a. Cubic (a = b = c, α = β = γ = 90°) b. Monoclinic (a ≠ b ≠ c, α ≠ β, γ = 90°) c. Tetragonal (a = b ≠ c, α = β = γ = 90°) d. Trigonal (or Rhombohedral) (a = b ≠ c, α = β = 90°, γ ≠ 90°)

Step-by-step solution

Crystal a: Finding the crystal system

We have the following edge lengths and face angles for crystal a: a = 3 nm, b = 3 nm, c = 3 nm; α = 90°, β = 90°, γ = 90° Since all the edge lengths are equal (a = b = c) and all the face angles are equal to 90° (α = β = γ = 90°), it belongs to the cubic crystal system.

Crystal b: Finding the crystal system

We have the following edge lengths and face angles for crystal b: a = 4 nm, b = 3 nm, c = 5 nm; α = 90°, β = 100°, γ = 90° Since the edge lengths are not equal and only two of the face angles are equal to 90° (α ≠ β, γ = 90°), it belongs to the monoclinic crystal system.

Crystal c: Finding the crystal system

We have the following edge lengths and face angles for crystal c: a = 3 nm, b = 3 nm, c = 5 nm; α = 90°, β = 90°, γ = 90° Since two of the edge lengths are equal (a = b ≠ c) and all the face angles are equal to 90° (α = β = γ = 90°), it belongs to the tetragonal crystal system.

Crystal d: Finding the crystal system

We have the following edge lengths and face angles for crystal d: a = 3 nm, b = 3 nm, c = 5 nm; α = 90°, β = 90°, γ = 120° Since two of the edge lengths are equal (a = b ≠ c) and only two face angles are equal to 90° (α = β = 90°, γ ≠ 90°), it belongs to the trigonal (or rhombohedral) crystal system.

Key concepts

Cubic Crystal System

The cubic crystal system, also known as isometric system, is one of the simplest and most symmetrical of the crystal systems. It is characterized by three axes of equal length that are perpendicular to each other. In this system, all edge lengths are equal (\(a = b = c\)) and all face angles are right angles (\(\alpha = \beta = \gamma = 90^\circ\)).

Crystals in the cubic system often display shapes such as cubes, octahedrons, and dodecahedrons. Common minerals like salt (halite) and pyrite exhibit a cubic crystal system. It is the high degree of symmetry in the cubic system that gives these minerals their characteristic shapes. Think of it like a perfectly proportioned box, with equal sides and sharp, 90-degree corners.

Monoclinic Crystal System

The monoclinic crystal system is less symmetrical than the cubic system. It consists of crystals with three unequal axes (\(a eq b eq c\)), with two angles that are 90 degrees (\(\alpha = \gamma = 90^\circ\)) and one angle that is not (\(\beta eq 90^\circ\)).

This lack of perfect symmetry results in prismatic crystals that are well formed but appear slightly slanted or skewed because of the non-right angle (\(\beta\)). Minerals such as gypsum and jadeite are examples of monoclinic crystals. The variation in angle gives each mineral a distinctive shape compared to the more symmetrical systems.

Tetragonal Crystal System

The tetragonal crystal system is similar to cubic but with a stretch. In this system, there are three axes at right angles just like in the cubic system, but only two are of equal length (\(a = b eq c\)). The face angles remain the same as in the cubic system (\(\alpha = \beta = \gamma = 90^\circ\)).

Imagine a square as the base of a crystal, and then pull one of the square's pillars up or push it down, creating a rectangle. This system results in elongated or flattened cubes, often forming prismatic or pyramidal shapes. The mineral zircon and the synthetic cubic zirconia are examples of tetragonal crystals.

Trigonal Crystal System

Lastly, the trigonal crystal system, sometimes called the rhombohedral system, is a subset of the hexagonal system and is unique due to its three-fold rotational symmetry. It features three axes of equal length (\(a = b = c\)) that are not at right angles to each other; instead, the angles between them are the same but not 90 degrees (\(\alpha = \beta = 90^\circ, \gamma eq 90^\circ\)).

This system is a bit harder to visualize than the others. Think of a cube that's been stretched along one of its body diagonals, creating an overall shape like a slanted box. Minerals like quartz, calcite, and cinnabar are found in this trigonal crystal system. The unique axis causes the faces of the crystals to form in spiral-like patterns, which is why quartz often has that twisty appearance.