 Solution

For indexing a diamond-cubic powder pattern, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ and Bragg's law gives ${\sin }^2\theta ={\lambda }^2(h^2+k^2+l^2)/(4a^2)$. Using half of each printed scattering angle, the ratios to the first peak are approximatelyMultiplying by3 gives the allowed sequence $3,8,11,16,19$. ThusThe numerical calculation, expressed without assuming a historical Data Book wavelength, isTherefore $a\simeq 3.51537\lambda$. As a separately identified reference convention, choosing Cu K${\alpha }_1$ with $\lambda \simeq 0.15406\mathrm{nm}$ givesThat wavelength follows from the copper $K{L}_3$ transition near $8047.8227\mathrm{eV}$ in the NIST transition database, using $\lambda =hc/E$.^[ref]

The paper says Cu K$\alpha$ but does not give the line-selection convention or an explicit wavelength. A rounded unresolved-doublet wavelength, for example $0.15418\mathrm{nm}$, would instead give about $0.54200\mathrm{nm}$. These are conditional wavelength evaluations, not claims about the unavailable historical Data Book. The angles are internally consistent with one cubic parameter; they should not be replaced by literature peak positions simply to obtain a familiar silicon value.