Equation of a plane (crystallography)

A crystal is composed of atoms, molecules or ions that are arranged in a repetitive manner, forming a three-dimensional space lattice. Each point in the space lattice represents either a particle or a collection of particles (known as a basis), and a set of lattice points can be connected by a plane. As there are many possible ways to connect lattice points with planes of different orientations, a system is needed to define these planes. We begin by deriving the vector and scalar equations of a plane.

P(x, y, z) and P₀(x₀, y₀, z₀) are two points on a plane with position vectors r and r₀respectively, which makes r – r₀the vector from P₀to P. The plane has a direction defined by the normal vector n(A, B, C), which is perpendicular to the vector r – r₀. Therefore,

$\textbf{\textit{n}}\cdot (\textbf{\textit{r}}-\textbf{\textit{r}}_0)=0\; \; \; \; \; \; \; (1)$

$\begin{pmatrix} A &B &C \end{pmatrix}\begin{pmatrix} x-x_0\\y-y_0 \\z-z_0 \end{pmatrix}=0$

$A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0$

$Ax+By+Cz=D$

where D = Ax₀ + By₀ + Cz₀.

$\frac{x}{D/A}+\frac{y}{D/B}+\frac{z}{D/C}=1\; \; \; \; \; \; \; (2)$

Eq1 is the vector equation of a plane, while eq2 is the scalar equation of a plane, where D/A, D/B and D/C are the intercepts of the plane with the x-axis, y-axis and z-axis respectively.

Equation of a plane (crystallography)

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