Question

In cach of the following cases indicate whether the structure is a Bravais lattice. If it is, give three primitive vectors; if it is not, describe it as a Bravais lattice with as small as possible a basis. (a) Base-centered cubic (simple cubic with additional points in the centers of the horizontal faces of the cubic cell). (b) Side-centered cubic (simple cubic with additional points in the centers of the vertical faces of the cubic cell). (c) Edge-centered cubic (simple cubic with additional points at the midpoints of the lines joining nearest neighbors).

Short answer

(a) Yes, it is a Bravais lattice with primitive vectors \textbf{a}_1 = \frac{1}{2}(\textbf{x} + \textbf{y})\, \textbf{a}_2 = \frac{1}{2}(\textbf{x} + \textbf{z})\, and \textbf{a}_3 = \frac{1}{2}(\textbf{y} + \textbf{z})\. (b) Not a Bravais lattice, but describable as a simple cubic lattice with a two-point basis. (c) Not a Bravais lattice, but describable as a simple cubic lattice with a four-point basis.

Step-by-step solution

Identifying the Structure of (a)

A base-centered cubic lattice is a simple cubic lattice with additional points at the centers of the horizontal faces (top and bottom faces). This is considered a Bravais lattice because it maintains translational symmetry through primitive vectors that translate from one lattice point to any other without changing the orientation of the lattice.

Primitive Vectors for (a)

For a base-centered cubic lattice, three primitive vectors can be chosen as follows: \( \textbf{a}_1 = \frac{1}{2}(\textbf{x} + \textbf{y}) \textbf{a}_2 = \frac{1}{2}(\textbf{x} + \textbf{z}) \), and \( \textbf{a}_3 = \frac{1}{2}(\textbf{y} + \textbf{z}) \), where \( \textbf{x}, \textbf{y}, \textbf{z} \) are the unit vectors along the x, y, and z directions respectively.

Identifying the Structure of (b)

A side-centered cubic lattice is not a Bravais lattice because the additional points in the centers of the vertical faces break the translational symmetry. However, it can be represented by a Bravais lattice with a basis consisting of more than one point.

Describing (b) as a Bravais Lattice with a Basis

To express the side-centered cubic structure as a Bravais lattice with a basis, it can be considered as a simple cubic lattice with a two-point basis: one point at a corner and the other in the center of one of the vertical faces. The primitive vectors are the same as those of a simple cubic lattice: \( \textbf{a}_1 = \textbf{x} \), \( \textbf{a}_2 = \textbf{y} \), and \( \textbf{a}_3 = \textbf{z} \).

Identifying the Structure of (c)

An edge-centered cubic lattice is not a Bravais lattice because the additional points at the midpoints of the edges do not preserve translational symmetry. Similar to part (b), it can be described by a Bravais lattice with a basis.

Describing (c) as a Bravais Lattice with a Basis

The edge-centered cubic structure can be viewed as a simple cubic lattice with a four-point basis: one point at each corner and a point at the midpoint of each edge. Thus, it can be represented by a simple cubic Bravais lattice with an extended basis. The primitive vectors here are also the same as those of a simple cubic lattice: \( \textbf{a}_1 = \textbf{x} \), \( \textbf{a}_2 = \textbf{y} \), and \( \textbf{a}_3 = \textbf{z} \).

Key concepts

Crystal Structure

Understanding crystal structure is crucial for students of materials science and physics. A crystal structure is a highly ordered arrangement of atoms that extends in all three spatial dimensions. The atoms within a crystal are arranged in a repetitive pattern called a lattice. Each point in the lattice is called a lattice point and is surrounded by atoms in the same orientation as every other lattice point.

The properties of materials, such as their electronic, optical, and mechanical attributes, are highly dependent on their crystal structure. For example, diamonds and graphite are both forms of carbon but have vastly different properties due to their differing crystal structures: diamond has a rigid structure that lends itself to hardness, whereas graphite is layered, leading to its softness and lubricating qualities.

In the case of a base-centered cubic lattice (part a of the exercise), the structure is a Bravais lattice, identified by its consistent pattern and symmetry.

Primitive Vectors

Primitive vectors are the building blocks of a crystal lattice. They are the minimum set of vectors required to recreate the entire lattice by successive translations. Essentially, you can think of them as the 'rulers' that measure the foundational repeat pattern of the lattice.

In mathematical terms, if you add integer multiples of the primitive vectors, you can reach any point within the lattice without coincidence. It is essential that these vectors are chosen as the smallest vectors that accomplish this feat. For the base-centered cubic in part (a), the vectors are combined unit vectors, indicating a different length and direction compared to a simple cubic lattice.

The appropriate selection of primitive vectors is imperative because it can simplify the study and mathematical modeling of the crystal's properties. They serve as critical tools for the understanding of crystal symmetry and geometry.

Translational Symmetry

Translational symmetry plays a defining role in recognizing Bravais lattices. This kind of symmetry means that a certain motif, when translated along certain vectors (the primitive vectors), can map the lattice onto itself without alteration. Translational symmetry ensures the uniformity of the structure over infinite space.

In context with the exercise, the side-centered cubic and edge-centered cubic lattices (parts b and c) lack translational symmetry as there is no set of primitive vectors that can be used to reach every lattice point by translation alone. Their structure requires additional points (called a basis) to describe the full crystal, which means they are not Bravais lattices. Recognizing translational symmetry guides us in determining the potential electronic or optical behaviors of the material, as well as its stability and density.

Simple Cubic Lattice

A simple cubic lattice is one of the most straightforward crystal structures. It consists of points (atoms, molecules, or ions) positioned at the corners of a cube. Each lattice point is equidistant from its neighbors, creating a regular and uncomplex pattern.

Due to its simplicity, the simple cubic lattice's primitive vectors are solely along the x, y, and z-axes, corresponding to the sides of the cube (as in parts b and c of the exercise). This lattice type, while easy to understand, is not a densely packed structure, and it does not occur commonly as a natural form for elemental metals. However, it provides a foundation for understanding more complex crystal structures, including the body-centered and face-centered cubic lattices.

As an educational platform, providing visuals like diagrams or interactive models could greatly enhance comprehension of these lattices and encourage engagement from students straining to visualize these structures.