= Solution
Initially approximate every <tungsten> atom by the same <atomic scattering factor> $f$. Pairing the quarter-coordinate sites in the <crystal basis structure factor> gives
$$
\frac{F_{hkl}}f=1+(-1)^{h+k+l}+2\left[(-1)^k\cos\frac{\pi h}{2}+(-1)^h\cos\frac{\pi l}{2}+(-1)^l\cos\frac{\pi k}{2}\right].
$$
For cubic <powder diffraction>, increasing deflection corresponds to increasing $h^2+k^2+l^2$, since $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. The first possibilities are
|| Family | $h^2+k^2+l^2$ | $F/f$ \\
| $\{100\}$ | $1$ | $0$ \\
| $\{110\}$ | $2$ | $0$ \\
| $\{111\}$ | $3$ | $0$ \\
| $\{200\}$ | $4$ | $4$ \\
| $\{210\}$ | $5$ | $4$
Hence the first two strong rings have <Miller indices>
$$
\boxed{\{200\},\qquad\{210\}.}
$$
The <powder diffraction multiplicity> is six for $\{200\}$, from three choices of nonzero axis and two signs. For $\{210\}$ it is $3!\,2^2=24$, from six permutations and four sign choices. If the <atomic scattering factors> and angular corrections are taken as approximately equal for these nearby rings, their integrated <diffraction intensity> obey
$$
\boxed{I_{200}:I_{210}\simeq6|4f|^2:24|4f|^2=1:4.}
$$
This is a multiplicity-based estimate. An exact measured ratio also depends on the momentum-transfer dependence of the <atomic scattering factor>, the <Lorentz-polarization factor>, the <Debye–Waller factor>, absorption and instrument geometry. In particular, the source does not provide the beta-phase <lattice constant> or instrument corrections needed to calculate those angular factors. Peak heights additionally depend on peak widths; the estimate refers to integrated ring <intensities>.
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