= Bragg angle
{c}
{title2=$\theta$}
The <Bragg angle> is the glancing angle from the incident beam to the selected <lattice planes>, with $0<\theta\leq90^\circ$. If $\alpha$ is instead the angle from the beam to the plane normal, $\alpha=90^\circ-\theta$ and <Bragg's law> becomes $n\lambda=2d\cos\alpha$. Defining the angle explicitly prevents a <sine>-versus-<cosine> error as well as a factor-of-two error in the detector deflection.
Choose $\lambda=0.154\,\mathrm{nm}$ and $d=0.200\,\mathrm{nm}$ as an illustrative model, not historical measurements. For the first <diffraction order>,
$$
\sin\theta=\frac{0.154}{2(0.200)}=0.385,\qquad
\theta\simeq22.643740^\circ,\qquad
2\theta\simeq45.287481^\circ.
$$
The second <diffraction order> has $\sin\theta=0.770$ and $2\theta\simeq100.707778^\circ$; the third would require $\sin\theta=1.155$ and is impossible. These are different <diffraction orders> of the same <interplanar spacing> at the same <wavelength>.
Conversely, a first-order reflection at a measured deflection $2\theta$ gives the <interplanar spacing>
$$
d=\frac{\lambda}{2\sin\theta}.
$$
Using $2\theta\simeq45.287481^\circ$ recovers $d\simeq0.200\,\mathrm{nm}$. This is the useful reversal: a known <wavelength> turns an angular measurement into an atomic length. An unassigned <diffraction order> or an unknown <wavelength> prevents that single measurement from determining $d$ uniquely.
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