Interplanar distance (crystallography)

The interplanar distance is the perpendicular distance between two adjacent planes of a family of planes.

Consider a family of parallel planes {hkl} with the reference plane shown in the diagram below.

The interplanar distance, d_hkl, is given by

$\mathit{d_{hkl}}=OD=\frac{a}{h}cos\alpha \; \; \; \Rightarrow \; \; \; cos\alpha =\frac{h}{a}\mathit{d_{hkl}}$

Similarly, we have

$cos\beta =\frac{k}{b}d_{hkl}$

$cos\gamma =\frac{l}{c}d_{hkl}$

For orthogonal axes, cos²α + cos²β + cos²γ = 1 (see below for proof). So,

$(\frac{h}{a})^2{d_{hkl}}^{2}+(\frac{k}{b})^2{d_{hkl}}^{2}+(\frac{l}{c})^2{d_{hkl}}^{2}=1$

$\frac{1}{{d_{hkl}}^{2}}=\frac{h^2}{a^2}+\frac{k^2}{b^2}+\frac{l^2}{c^2}\; \; \; \; \; \; \; (3)$

Eq3 describes the interplanar distance for an orthorhombic unit cell. For a primitive cubic unit cell, a = b = c and eq3 becomes

$\frac{1}{{d_{hkl}}^{2}}=\frac{h^2+k^2+l^2}{a^2}\; \; \; \; \; \; \; (4)$

The diagram below shows two parallel planes with Miller indices of (110) and (220).

The interplanar distance for (110), i.e. between the orange plane and the origin, is twice of that for (220). If we rewrite (220) as (n1 n1 n0), then d₁₁₀ = nd_n1,n1,n0, where n = 2. In general, for parallel and equidistant planes,

$d_{hkl}=nd_{nh,nk.nl}\; \; \; \; \; \; \; (5)$

where n ∈ $\mathbb{Z}$ and n is called the common factor between planes.

Substituting eq5 in eq3 and eq4, we have:

$\frac{1}{{d_{nh,nk,nl}}^{2}}=\frac{(nh)^2}{a^2}+\frac{(nk)^2}{b^2}+\frac{(nl)^2}{c^2}\; \; \; \; \; \; \; (6)$

$\frac{1}{{d_{nh,nk,nl}}^{2}}=\frac{(nh)^2+(nk)^2+(nl)^2}{a^2}\; \; \; \; \; \; \; (7)$

Question

Show that cos²α + cos²β + cos²γ = 1 for orthogonal axes.

Answer

With reference to the diagram below,

$cos\alpha =\frac{a}{\left | N \right |}\; \; \;\; \; \; \; cos\beta =\frac{b}{\left | N \right |}\; \; \; \; \; \; \; cos\gamma =\frac{c}{\left | N \right |}$

$cos^2\alpha +cos^2\beta +cos^2\gamma =(\frac{a}{\left | N \right |})^2+(\frac{b}{\left | N \right |})^2+(\frac{c}{\left | N \right |})^2$

Since a² + b² + c² = INI²

$cos^2\alpha +cos^2\beta +cos^2\gamma =1$

Interplanar distance (crystallography)

Question

Answer

next article: bragg’s law

Previous article: miller indices

Content page of X-ray crystallography

Content page of advanced chemistry

Main content page