The Grating Constant of Calcite Crystals

The precise methods of measuring x-ray wave-lengths, now in use, raise two important questions concerning the crystal grating used. First, what variation in the grating constant is to be expected for crystals grown under different conditions? Second, what is the most accurate value of the grating constant? Since calcite crystals are the most used crystals in x-ray spectroscopy the present work was undertaken to answer as well as possible the above questions for this crystal. The variations in the grating constants of six calcite crystals from four sources (Iceland, Montana, Argentina, and Spain) have been determined by measuring the angle of diffraction for the molybdenum $K{\alpha }_1$ line in the fourth order. A high precision two crystal spectrometer was used for measuring the diffraction angles. The results are given in the table below. The density of these crystals was then carefully determined. Six to nine independent determinations were made at a temperature of 20.00±0.01°C. The averages of these values are given for a temperature of 20.00°C, in the fourth column of the table. The probable error determined by the method of least squares is given in the fifth column. The mass of the crystals used in determining the density is given in the sixth column. Con- sidering the density measurements of DeFoe and Compton the writer gives the density $\rho$ in g/${\mathrm{cm}}^{-3\mathrm{}}$ at 20°C as $\rho =2.71030\mathrm{}\pm 0.00003$. The crystals were then chemically analyzed. The results showed that all samples contained about 0.01 percent ferrous oxide, 0.01 percent manganous oxide, and 99.98 percent calcium carbonate. The angle between the cleavage faces of the calcite crystals was determined by three methods. X-rays were used for determining the angle in the first two methods and an optical method for the third. The results for 20°C were $\alpha ={105}^{\circ } 3^{\prime } {29}^{\prime \prime }$ or $\alpha ={101}^{\circ } {54}^{\prime } 4^{\prime \prime }$, where $\alpha$ is the interior obtuse dihedral angle, and $\beta$ the angle between the edges of the crystal. The grating constant of the crystal can be calculated from the equation $d={\left(\frac{\mathrm{nM}}{\rho N\phi }\right)}^{\frac{1}{3}}$ =3.02816A at 20°C, =3.002810A at 18°C. The values of the constants used were, $n=\frac{1}{2}$, $M=100.078\mathrm{}$, $\rho =2.71030\mathrm{}$, $N=6.0669\mathrm{}\times {10}^{23}$, $\phi =1.09594\mathrm{}$.