Question

Of the seven three-dimensional primitive lattices, (a) which one has a unit cell where the \(a\) and \(b\) lattice vectors form a base that is an arbitrary parallelogram (like the unit cell of a two-dimensional oblique lattice), while the \(c\) lattice vector is perpendicular to the other two? (b) What is the lattice if the \(a\) and \(b\) lattice vectors form a base that corresponds to the two-dimensional hexagonal unit cell and the \(c\) lattice vector is perpendicular to the other two?

Short answer

(a) The lattice with an arbitrary parallelogram base formed by the $a$ and $b$ lattice vectors and a perpendicular $c$ lattice vector is the Simple Orthorhombic lattice (P). (b) The lattice with a 2D hexagonal base formed by the $a$ and $b$ lattice vectors and a perpendicular $c$ lattice vector is the Simple Hexagonal lattice (P).

Step-by-step solution

(a) Lattice with arbitrary parallelogram base and c vector perpendicular

To find the lattice that meets the conditions mentioned in (a), we must look for a lattice in which the angles between the a and b vectors can be arbitrary, and the c vector needs to be perpendicular to both the a and b vectors. Among the seven lattices listed above, the simple orthorhombic lattice (P) meets these conditions. In this lattice, the a and b vectors form an arbitrary parallelogram, and the c vector is perpendicular to both the a and b lattice vectors. So, the lattice mentioned in (a) is the Simple Orthorhombic lattice (P).

(b) Lattice with 2D hexagonal base and c vector perpendicular

To find the lattice that meets the conditions mentioned in (b), we must look for a lattice in which the a and b vectors form a base corresponding to the two-dimensional hexagonal unit cell, and the c vector needs to be perpendicular to both the a and b vectors. Among the seven lattices listed above, the simple hexagonal lattice (P) meets these conditions. In this lattice, the a and b vectors create a hexagonal structure (120-degree angles between vectors), and the c vector is perpendicular to both the a and b lattice vectors. So, the lattice mentioned in (b) is the Simple Hexagonal lattice (P).

Key concepts

Primitive Lattice

In crystallography, a primitive lattice is a type of crystal structure where there is exactly one lattice point per unit cell. This means it's the simplest form of a three-dimensional packing where the overall structure repeats itself at every point. The primitive unit cell is fundamental because it contains the smallest possible number of atoms and provides a clear basis for understanding more complex structures.

Each lattice point in a primitive lattice can be occupied by one atom, a molecule, or groups of atoms. They can be transformed into every other kind of lattice by the addition of points. The importance of primitive lattices lies in their ability to provide the base framework upon which atoms are arranged, serving as the "skeleton" of the crystal structure where translations of the lattice points create the periodic structure.

Primitive lattices are mathematically described by lattice vectors that define the size and shape of the unit cell. They serve as key building blocks in understanding how molecules pack in space.

Orthorhombic Lattice

An orthorhombic lattice is one of the seven crystal systems used to describe crystal symmetry in three-dimensional space. It is characterized by three mutually perpendicular axes, each differing in length. This means none of the three edges of the unit cell are equal, making them distinctly different from cubic lattices where all edges are equal.

A simple orthorhombic lattice forms a structure resembling an "oblong rectangle" in 3D. Here, the angles between axes, denoted by lattice vectors, are exactly 90 degrees. The lattice embodies a high level of symmetry yet with flexibility, allowing the creation of diverse crystal formations due to its freedom in axis length.

Orthorhombic lattices often advance to form complex structures like the face-centered or body-centered variants, which have additional lattice points either at the centers of faces or in the body of the unit cell itself. This multiplicity in form makes them essential for understanding various real-world materials such as ceramics and polymers.

Hexagonal Lattice

The hexagonal lattice is one of the geometrical arrangements of points in space and can be visualized as regularly placed hexagons, much like a honeycomb. In this crystalline system, the unit cell is described by two equal-length lattice vectors that meet at an angle of 120 degrees, forming a flat hexagonal base. The third vector, perpendicular to this base, supports the arrangement of points above the base.

This distinct arrangement creates a very efficient space-filling structure because hexagonal close-packing allows for maximum density. Two common forms include the simple hexagonal lattice and the more densely packed form, known as the hexagonally closed-packed (hcp) structure. The simplicity of the hexagonal lattice makes it a prominent choice for illustrating symmetric properties in materials.

Hexagonal lattices are significant in many natural forms, such as the crystalline structures of metals like magnesium and titanium. Their natural efficiency for packing atoms or molecules makes them integral in scientific fields related to material and mineral sciences.