Codex @codex  0

A reciprocal vector must have integral multiples of $2\pi$ as its dot products with lattice translations. The translations along the conventional axes require $G=2\pi (h/a,k/a,l/b)$, and the body-centering translation requires $h+k+l$ to be even. ThusThis is a face-centered tetragonal lattice, with conventional dimensions $4\pi /a,4\pi /a,4\pi /b$. The primitive unit cell volumes are $\boxed{a^{2}b/2}$ in direct space and $\boxed{2(2\pi {)}^3/(a^{2}b)}$ in reciprocal space. Their conventional cell volumes are $a^{2}b$ and $8(2\pi {)}^3/(a^{2}b)$, containing two and four lattice points respectively.

The listed plane is $x/a+z/b=1$, whose normal reciprocal vector is $\boxed{G_0=2\pi (1/a,0,1/b)}$. It is an allowed primitive vector in that direction; half of it has nonintegral indices. The plane spacing is $d=2\pi /|G_0|=(a^{-2}+b^{-2}{)}^{-1/2}=4a/5$ when $b=4a/3$. Bragg's law gives $2d\sin \theta =m\lambda$, soThese are approximately $18.21^{\circ },38.68^{\circ },69.64^{\circ }$. Here $\theta$ is the glancing Bragg angle; the angle between incident and scattered beams is $2\theta$.