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

Abstract

In a recent evaluation of the atomic constants, the value of Avogadro's number is $N=6.02252\ifmmode\times\else\texttimes\fi{}{10}^{23}{(\mathrm{g}\mathrm{mole})}^{\ensuremath{-}1}\ifmmode\pm\else\textpm\fi{}11$ ppm (probable error). Measurements on the atomic weight of silicon give $A=28.0857\ifmmode\pm\else\textpm\fi{}10$ ppm. Precision measurements of the density of silicon combined with the above values in the Bragg equation $a={(\frac{\mathrm{fA}}{\ensuremath{\rho}N})}^{\frac{1}{3}}$ 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 \ifmmode\pm\else\textpm\fi{}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 \ifmmode\pm\else\textpm\fi{}3.1 ppm. The measured densities of two of the 17 crystals differed from the average by more than 3 $\ensuremath{\sigma}$, 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 $K{\ensuremath{\alpha}}_{1}$ and $K{\ensuremath{\alpha}}_{2}$ lines. The wavelengths in angstroms were evaluated from the Bragg law for each of the 17 crystals and for the ${\ensuremath{\alpha}}_{1}$ and ${\ensuremath{\alpha}}_{2}$ lines. The average wavelengths were Cu $K{\ensuremath{\alpha}}_{1}=1.540563 \AA{}\ifmmode\pm\else\textpm\fi{}5$ ppm, and Cu $K{\ensuremath{\alpha}}_{2}=1.544390 \AA{}\ifmmode\pm\else\textpm\fi{}5$ ppm. Taking the peak wavelength values of 1537.400 xu\ifmmode\pm\else\textpm\fi{}1 ppm for the Cu $K{\ensuremath{\alpha}}_{1}$ and 1541.219 xu\ifmmode\pm\else\textpm\fi{}6.5 ppm for the Cu $K{\ensuremath{\alpha}}_{2}$ lines yields a wavelength conversion factor from angstrom to thousand x units of $\ensuremath{\Lambda}=1.002057 \AA{}/\mathrm{kxu}\ifmmode\pm\else\textpm\fi{}5$ 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 $\ensuremath{\Lambda}=1.00203\ifmmode\pm\else\textpm\fi{}30$ ppm, which is lower than the above values, but the probable errors overlap.