Silicon-Crystal Determination of the Absolute Scale of X-Ray Wavelengths

In a recent evaluation of the atomic constants, the value of Avogadro's number is $N=6.02252\mathrm{}\times {10}^{23}{\left(\mathrm{g}\mathrm{mole}\right)}^{-1\mathrm{}}\pm 11$ ppm (probable error). Measurements on the atomic weight of silicon give $A=28.0857\mathrm{}\pm 10$ ppm. Precision measurements of the density of silicon combined with the above values in the Bragg equation $a={\left(\frac{\mathrm{fA}}{\rho N}\right)}^{\frac{1}{3}}$ result in an absolute grating constant of high precision. X-ray diffraction measurements with the same crystal yield the grating constant in x units; thus the conversion factor from x units to cm can be evaluated. X-ray and density measurements have been made on 17 selected silicon crystals from four different sources. The statistical error in the measurement of the densities of the 17 crystals was ±0.4 ppm. To obtain the absolute density error, a 3 ppm probable error in the density of water must be added, giving a total error of ±3.1 ppm. The measured densities of two of the 17 crystals differed from the average by more than 3 $\sigma$, probably indicating a difference in the density of the crystals. The x-ray diffraction measurements were made with a double-crystal spectrometer using the copper $K{\alpha }_1$ and $K{\alpha }_2$ lines. The wavelengths in angstroms were evaluated from the Bragg law for each of the 17 crystals and for the ${\alpha }_1$ and ${\alpha }_2$ lines. The average wavelengths were Cu $K{\alpha }_1=1.540563\mathrm{} \AA \pm 5$ ppm, and Cu $K{\alpha }_2=1.544390\mathrm{} \AA \pm 5$ ppm. Taking the peak wavelength values of 1537.400 xu±1 ppm for the Cu $K{\alpha }_1$ and 1541.219 xu±6.5 ppm for the Cu $K{\alpha }_2$ lines yields a wavelength conversion factor from angstrom to thousand x units of $\Lambda =1.002057\mathrm{} \AA /\mathrm{kxu}\pm 5$ ppm. Recalculation of the best measurements in the literature with current values of the atomic weights gave values which agree with the present work within probable errors. Plane-ruled-grating measurements of x-ray wavelengths yield a value of $\Lambda =1.00203\mathrm{}\pm 30$ ppm, which is lower than the above values, but the probable errors overlap.