Sketch an sc unit cell with lattice constant 𝐚=4 A^∘, whose diatomic basis of atom A is located

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Sketch an sc unit cell with lattice constant 𝐚=4 A^∘, whose...

Sketch an sc unit cell with lattice constant $\mathbf{a}=4 \ A^{\circ}$, whose diatomic basis of atom $\mathrm{A}$ is located at the lattice sites, and with atom B displaced by $(\mathbf{a} / 2,0,0)$. Assume that both atoms have the same size and we have a close-packed structure (i.e., nearest neighbor atoms touch each other). Calculate (i) the packing fraction (i.e., fraction of the total volume occupied by atoms), (ii) the number of $\mathrm{B}$ atoms per unit volume, (iii) the number of $\mathrm{A}$ atoms per unit area on (100) planes.

Sketch an sc unit cell with lattice constant $\mathbf{a}=4 \ A^{\circ}$, whose diatomic basis of atom $\mathrm{A}$ is located at the lattice sites, and with atom B displaced by $(\mathbf{a} / 2,0,0)$. Assume that both atoms have the same size and we have a close-packed structure (i.e., nearest neighbor atoms touch each other). Calculate
(i) the packing fraction (i.e., fraction of the total volume occupied by atoms),
(ii) the number of $\mathrm{B}$ atoms per unit volume,
(iii) the number of $\mathrm{A}$ atoms per unit area on (100) planes.

Key Concepts

Unit Cell

The unit cell is the fundamental repeating unit of a crystal lattice that, when translated through space, recreates the entire crystal. It is defined by its lattice constants, in this case a = 4 Å, and contains all the essential information about the crystal structure, including the positions of the atoms inside it.

Crystal Lattice and Basis

A crystal lattice is an infinite array of points in space arranged in a regular pattern, and a basis is a group of one or more atoms associated with each lattice point. In this problem, a simple cubic (sc) lattice is used with a diatomic basis: one atom (A) located at the lattice points, and a second atom (B) displaced by a/2 along one axis. This combination of lattice and basis describes the complete three?dimensional arrangement of atoms in the crystal.

Simple Cubic Structure

A simple cubic (sc) structure is characterized by lattice points at the corners of a cube. In such a structure, the geometry and distances between atoms are directly determined by the lattice constant. The simple cubic arrangement is one of the most basic crystal structures and serves as a starting point for analyzing more complex configurations when additional atomic bases are introduced.

Close-Packed Structure

A close-packed structure is one in which atoms are arranged as tightly as possible, so that each atom is in contact with its nearest neighbors. Although the simple cubic lattice is not the densest packing for a single element, when a diatomic basis is introduced and atoms are sized appropriately so that they touch each other, the effective arrangement can mimic close packing locally. The idea of close packing is crucial when calculating the packing fraction since it determines the maximum volume fraction that the atoms can occupy.

Packing Fraction

The packing fraction is the fraction of the total volume of the unit cell that is occupied by the atoms, considered as hard spheres. It is calculated by summing the volumes of all the atoms (taking into account the fractions of atoms that lie within the unit cell) and dividing by the volume of the cell. This concept is essential for understanding how efficiently space is used in a crystal structure.

Atomic Number Density

Atomic number density refers to the number of a particular type of atoms per unit volume or per unit area. For example, calculating the number of B atoms per unit volume involves determining how many B atoms are present in one unit cell and then relating that to the volume of the unit cell, while the number of A atoms per unit area on a specific crystallographic plane requires analyzing the distribution of A atoms on that plane.

Crystallographic Planes

Crystallographic planes, denoted by Miller indices such as (100), are planes that cut through the unit cell and reveal the arrangement of atoms on a particular surface. These planes are important for understanding surface properties and for calculating quantities like the number of atoms per unit area, which is crucial in the analysis of surface phenomena and material properties.

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