= Primitive cubic Bragg peak positions
{c}
{title2=$d_{hkl}=a/\sqrt{h^2+k^2+l^2}$}
Consider a chosen <primitive cubic lattice> $a\mathbb Z^3$ with $a=0.400\,\mathrm{nm}$ and one identical point scatterer per <primitive unit cell>. Retain the incident <wavelength> $\lambda=0.154\,\mathrm{nm}$. This is an ideal point model. Its <reciprocal lattice> vectors are $q=(2\pi/a)(h,k,l)$ for integer triples other than $(0,0,0)$. Write $S=h^2+k^2+l^2$. The associated <interplanar spacing> and first-order <Bragg angle> satisfy
$$
d_{hkl}=\frac{2\pi}{|q|}=\frac{a}{\sqrt S},\qquad
\sin^2\theta=\frac{\lambda^2}{4a^2}S=0.03705625S.
$$
The reflection labels $(hkl)$ use the <Miller index> convention, keeping unreduced triples to encode <diffraction order>. With that convention, set $n=1$ in <Bragg's law> for each label. The $(200)$ reflection is equivalently the second <diffraction order> of $(100)$: $d_{200}=a/2$ does not imply extra scatterers halfway between the occupied $(100)$ planes. Counting both descriptions as distinct reflections would count the same direction twice.
For $(100)$, $(110)$, $(111)$ and $(200)$, respectively, $S=1,2,3,4$ and the calculated <interplanar spacings> are $0.400000$, $0.282843$, $0.230940$ and $0.200000\,\mathrm{nm}$. Their calculated deflections $2\theta$ are $22.197435^\circ$, $31.594744^\circ$, $38.952948^\circ$ and $45.287481^\circ$. Equal steps in $S$ give equal steps in $\sin^2\theta$, not equal steps in $2\theta$. This regularity makes an angular pattern a test of a proposed <crystal structure>.
\Image[/bragg-primitive-cubic-calculated-peak-positions.png]
{title=Calculated angular positions for four reflections of an ideal primitive cubic point lattice}
{description=Separate strips show positions only, for $a=0.400\,\mathrm{nm}$ and $\lambda=0.154\,\mathrm{nm}$. Vertical row placement and marker length encode no <wave intensity>; this is not a simulated intensity spectrum or measured data. The four labels are a selected subset.}
A real <crystal structure> has a <Bravais lattice> together with an <atomic basis of a crystal>. The <crystal lattice structure factor> selects the periodic directions; the <crystal basis structure factor> combines the <scattering amplitudes> of atoms within a <unit cell> and can cancel a reflection. Consequently <Bragg's law> alone predicts neither allowed reflections nor their <wave intensities>. <Online Dictionary of Crystallography by International Union of Crystallography/Structure factor>[Online Dictionary of Crystallography by International Union of Crystallography; Definition, sum over atoms in the unit cell; Derivation, addition of companion atoms]{ref} <Systematic absences> further restrict observed families; for example, <face-centered cubic lattice> centering permits only indices all even or all odd in conventional cubic indexing. The four strips above apply to the stated <primitive cubic lattice>, not to <sodium chloride> or a <face-centered cubic lattice>. <Online Dictionary of Crystallography by International Union of Crystallography/Reflection conditions>[Online Dictionary of Crystallography by International Union of Crystallography; Definition and Table 1: primitive versus all-face-centered integral reflection conditions]{ref}
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