A Simplification of Prins' Formula for Diffraction of X-Rays by a Perfect Crystal

Abstract

The intensity formula of Prins for diffraction of x-rays by a perfect crystal has been simplified so that $F\left(l\right)\mathrm{}$ is a real, single-valued, algebraic function of $l$, the deviation of the glancing angle from the corrected Bragg angle. By neglecting absorption in the crystal, Darwin's formula is obtained in a new form. By differentiation the maximum ordinate of the diffraction pattern is obtained. To calculate percent reflection (i.e., maximum ordinate of the rocking curve of a double crystal spectrometer in the 1, -1 position) $\int {-\infty }^{+\infty }F\left(l\right)\mathrm{dl}$ and $\int {-\infty }^{+\infty }{F}^2\mathrm{}\left(l\right)\mathrm{dl}$ are needed, and these integrals have been evaluated analytically for Darwin's case of no absorption, leading to a value of 4/5 for $P\left(0\right)\mathrm{}$. To include absorption $F\left(l\right)\mathrm{}$ has been expanded into a series in powers of $\frac{B}{D}$ and an approximate formula obtained for $P\left(0\right)\mathrm{}$ in terms of the constants of the crystal. This formula agrees with the graphically determined values of $P\left(0\right)\mathrm{}$ to within a few percent.