= Interplanar spacing
{title2=$d$}
The <interplanar spacing> $d$ is the perpendicular <distance between parallel planes> in a specified family. Measuring along an oblique line gives a larger distance, so the direction of measurement matters.
There is a short <reciprocal lattice> derivation. With the angular convention $a_i\cdot b_j=2\pi\delta_{ij}$, put $G=hb_1+kb_2+lb_3$. Then $G\cdot r=2\pi m$ is the same <plane equation> as $hx+ky+lz=m$. Thus $G$ is a <normal vector>. Moving between consecutive sheets along this normal changes the <dot product> by $2\pi$, giving
$$
|G|d=2\pi,\qquad d=\frac{2\pi}{|G|}.
$$
For perpendicular cubic axes of length $a$, this becomes
$$
G=\frac{2\pi}{a}(h,k,l),\qquad d_{hkl}=\frac{a}{\sqrt{h^2+k^2+l^2}}.
$$
For example, choose a <primitive cubic lattice> with $a=0.600\,\mathrm{nm}$. Its $(210)$ <lattice planes> have $d_{210}=0.600/\sqrt5\simeq0.268328\,\mathrm{nm}$. This is a chosen geometry, not an experimental value. The <distance from a point to a plane> gives the same result: the planes $2X+Y=0$ and $2X+Y=a$ in ordinary Cartesian length coordinates are separated by $a/\sqrt5$.
Specify whether $d_{hkl}$ denotes spacing between successive occupied <lattice planes> or the formal spacing assigned to a reflection label. In a <primitive cubic lattice>, $(200)$ has formal spacing $a/2$ but every other sheet contains no <lattice points>; its reflection is the second <diffraction order> of $(100)$. In a <face-centered cubic lattice> with conventional cubic axes, the $(200)$ sheets at half-integer $X/a$ do contain <lattice points>. Here $a/2$ is the actual spacing of the complete family perpendicular to that cube edge. The <Primitive cubic Bragg peak positions> example explains how to avoid counting a reflection twice.
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