Beginning with a sketch of an fcc lattice, add atoms at ((1)/(4), (1)/(4), (1)/(4)) from each fc

step 1 The unit of diamond is made of of it is made up of it is made of of 8 carbon atoms, 8 carbon ato

Beginning with a sketch of an fcc lattice, add atoms at...

Beginning with a sketch of an fcc lattice, add atoms at $\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right)$ from each fcc atom to obtain the diamond lattice. Show that only the four added atoms in Fig. 1-8a appear in the diamond unit cell.

Key Concepts

Face Centered Cubic (FCC) Lattice

This is a type of crystal lattice where atoms are positioned at each of the corners and the centers of all the faces of the cube. It is highly symmetric and is a common underlying structure for many materials. In the problem, the FCC lattice serves as the base structure upon which the diamond lattice is constructed.

Lattice Plus Basis

In crystallography, a crystal structure can be described by a lattice combined with a basis, which are the positions of atoms attached to each lattice point. For the diamond lattice, additional atoms are placed at specific fractional coordinates relative to the FCC lattice points. This concept is used to generate more complex structures by supplementing a simple lattice with a nontrivial basis, leading to a crystal with lower symmetry than the parent lattice.

Crystallographic Unit Cell

The unit cell is the smallest repeating unit that defines the entire crystal structure. In this case, while the FCC lattice contributes many atoms, only a subset, specifically the four additional ones added at shifted positions, appear uniquely in the diamond unit cell. Understanding the unit cell is key to analyzing the distribution and repetition of atoms within the crystal.

Fractional Coordinates

These are values that describe atom positions within a unit cell relative to its dimensions and are expressed as fractions. The non-integer additions like (1/4, 1/4, 1/4) define the positions of the extra atoms that, when applied to every FCC lattice point, generate the diamond structure. This concept is essential for precisely characterizing and visualizing atomic arrangements in a crystal.

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