Question

Of the seven three-dimensional primitive lattices, which ones have a unit cell where all three lattice vectors are of the same length?

Short answer

Among the seven three-dimensional primitive lattices, only the cubic (P) and rhombohedral (R) lattices have a unit cell where all three lattice vectors are of the same length.

Step-by-step solution

Analyze the Cubic Lattice

In a cubic lattice, all three lattice vectors (a, b, c) have the same length and the angles between them are 90 degrees. Thus, a cubic lattice has lattice vectors with equal lengths.

Analyze the Tetragonal Lattice

In a tetragonal lattice, two of the three lattice vectors (a, b) are equal in length, while the third vector (c) is different. The angles between them are 90 degrees. Thus, a tetragonal lattice does not have lattice vectors with equal lengths.

Analyze the Orthorhombic Lattice

In an orthorhombic lattice, all three lattice vectors (a, b, c) are of different lengths and the angles between them are 90 degrees. Hence, an orthorhombic lattice does not have lattice vectors with equal lengths.

Analyze the Rhombohedral Lattice

In a rhombohedral lattice, all three lattice vectors (a, b, c) have the same length but the angles between them are not 90 degrees. Thus, a rhombohedral lattice has lattice vectors with equal lengths.

Analyze the Monoclinic Lattice

In a monoclinic lattice, all three lattice vectors (a, b, c) are of different lengths, with one angle not equal to 90 degrees. Thus, a monoclinic lattice does not have lattice vectors with equal lengths.

Analyze the Triclinic Lattice

In a triclinic lattice, all three lattice vectors (a, b, c) are of different lengths and none of the angles are equal to 90 degrees. Thus, a triclinic lattice does not have lattice vectors with equal lengths.

Analyze the Hexagonal Lattice

In a hexagonal lattice, two of the three lattice vectors (a, b) are equal in length, while the third vector (c) is different and the angles between them are 120 degrees. Thus, a hexagonal lattice does not have lattice vectors with equal lengths.

Conclusion

Among the seven three-dimensional primitive lattices, only the cubic (P) and rhombohedral (R) lattices have a unit cell where all three lattice vectors are of the same length.

Key concepts

Cubic Lattice

The cubic lattice is one of the simplest and most symmetrical types of crystal lattices. In a cubic lattice, the unit cell is characterized by three lattice vectors of equal length. This means that the sides of the cube are all the same—they are isotropic.
Additionally, each of the angles between these vectors is set at 90 degrees, making the geometry of the cubic lattice straightforward.
One key feature of cubic lattices is their full symmetry, which means they have the same properties in all directions. This makes them ideal for constructing simple crystal structures like those seen in metals like iron and gold.
They are further categorized into three types:

• Simple Cubic (SC): Here, the lattice points are located only at the corners of the cube.
• Body-Centered Cubic (BCC): In addition to the corners, there is a single lattice point in the center of the cube.
• Face-Centered Cubic (FCC): Lattice points occur at each corner and at the center of each face of the cube.

Rhombohedral Lattice

The rhombohedral lattice is less intuitive than the cubic lattice but shares one important trait: all three lattice vectors are of equal length. This can sometimes be confusing because we naturally associate equal-length vectors with a perfect cube. But, in a rhombohedral lattice, the difference lies in the angles between these vectors.
Instead of being 90 degrees, like in the cubic structure, the angles in a rhombohedral lattice are identical but can differ from a right angle. This form takes a shape similar to a distorted cube, where all sides are equal, yet it is stretched or compressed in such a way that the angles deviate from 90 degrees.
Rhombohedral lattices are seen in crystals like calcite and some forms of ice, where the uniform length of lattice vectors aids in forming strong crystal bonds despite the angular variation.

Primitive Lattice

A primitive lattice is a foundational concept in crystallography focusing on the most basic of crystal lattices. In its simplest definition, a primitive lattice has only one lattice point per unit cell.
This means that, at minimum, each repeating unit contains just one point, unlike more complex structures like body-centered or face-centered lattices. Primitive lattices are pivotal in describing the underlying structure of more complicated lattices.
Each lattice point represents an identical environment within the crystal, contributing to the periodic arrangement responsible for the crystal's overall symmetry. Found in diverse lattice systems like cubic, rhombohedral, and more, primitive lattices serve as the fundamental building blocks of crystalline forms. Interestingly, simplicity does not equal insignificance; the primitive lattice's efficient use of space is vital in forming tightly packed crystal structures.