How to Calculate the Packing Fraction of a Diamond Lattice

Calculating the diamond packing fraction
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Atoms within solids are arranged in one of several periodic structures known as a lattice. Crystalline structures, as opposed to amorphous structures, show a definite repetitive pattern of atom arrangements. Most solids form a regular arrangement of atoms as a way to minimize energy in the system. The simplest repeating unit of atoms in a structure is called the unit cell. The entire solid structure consists of this unit cell repeated in three dimensions.

TL;DR (Too Long; Didn't Read)

The diamond lattice is face-centered cubic. The simplified packing fraction is 8 x (V atom) / V unit cell. After making substitutions for known volume of spheres and cubes and simplifying, the equation becomes √3 x π/16 with a solution of 0.3401.

There are 14 types of lattice systems in total, which are subdivided into seven categories. The seven types of lattices are cubic, tetragonal, monoclinic, orthorhombic, rhombohedral, hexagonal and triclinic. The cubic category includes three types of unit cells: simple cubic, body-centered cubic and face-centered cubic. The diamond lattice is face-centered cubic.

The face-centered cubic structure has eight atoms per unit cell located at each of the corners and the centers of all the cubic faces. Each of the corner atoms is the corner of another cube, so the corner atoms are shared among eight unit cells. Additionally, each of its six face centered atoms is shared with an adjacent atom. Since 12 of its atoms are shared, it has a coordination number of 12.

The ratio of the volume of atoms in a cell compared to the total volume of a cell is the packing factor or packing fraction. The packing fraction indicates how closely atoms pack in a unit cell.

You can calculate the diamond packing density of a material with some material parameters and simple mathematics.

How to Calculate the Packing Fraction of a Diamond Lattice

The equation for packing fraction is:

Packing fraction = (N atoms) x (V atom) / V unit cell

N atoms is the number of atoms in a unit cell. V atom is the volume of the atom, and V unit cell is the volume of a unit cell.

Substitute the number of atoms per unit cell into the equation. Diamond has eight atoms per unit cell, so the diamond packing fraction equation now becomes:

Packing fraction = 8 x (V atom) / V unit cell

Substitute the volume of the atom into the equation. Assuming atoms are spherical, the volume is: V = 4/3 × π × r3

The equation for packing fraction now becomes:

Packing fraction = 8 x 4/3 × π × r3/ V unit cell

Substitute the value for the unit cell volume. Since the unit cell is cubic, the volume is V unit cell = a3

The formula for packing fraction then becomes:

Packing fraction = 8 x 4/3 × π × r3/ a3

The radius of an atom r is equal to √3 x a/8

The equation is then simplified to : √3 x π/16 = 0.3401

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