Skip to main content
Engineering LibreTexts

6.13: Simple cubic, face centered cubic and diamond lattices

  • Page ID
    52346
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Figure 6.14.1 shows the simplest 3-d crystal structure – the simple cubic lattice. The primitive lattice vectors are \({\bf{a_{1}}} = a_{0}{\bf{\hat{x}}},\ {\bf{a_{2}}} = a_{0}{\bf{\hat{y}}},\ {\bf{a_{3}}} = a_{0}{\bf{\hat{z}}}\), where \(a_{0}\) is the spacing between neighboring atoms. Very few materials, however, exhibit the simple cubic structure. The major semiconductors, including silicon and gallium arsenide, possess the same structure as diamond.

    As also shown in Figure 6.14.1, to describe the diamond structure, we first define the face centered cubic (FCC) lattice. Here the simple cubic structure is augmented by an atom in each of the faces of the cube. The primitive lattice vectors are:

    \[ {\bf{a_{1}}} = \frac{a_{0}}{2}{\bf{\hat{x}+\hat{z}}},\ {\bf{a_{2}}} = \frac{a_{0}}{2}{\bf{\hat{y}+\hat{z}}},\ {\bf{a_{3}}} = \frac{a_{0}}{2}{\bf{\hat{x}+\hat{y}}}, \nonumber \]

    where \(a_{0}\) is now the cube edge length.

    In the diamond lattice, each atom is \(sp_{3}\)-hybridized. Thus, every atom is at the center of a tetrahedron. We can construct the diamond lattice from a face centered cubic lattice with a two atom unit cell. For example, in Figure 6.14.1, our unit cell has one atom at (0,0,0), and another at \(a_{0}/4\).(1,1,1).

    Screenshot 2021-05-25 at 16.50.39.png
    Figure \(\PageIndex{1}\): A diamond lattice is simply a face-centered cubic lattice with a two atom unit cell (outlined in red).

    6.13: Simple cubic, face centered cubic and diamond lattices is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?