On the deviation from the Bragg law and the widths of diffraction patterns in perfect crystals

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Abstract
According to the classical treatment of the dynamical theory of X-ray or neutron diffraction, the angular deviation Δ⊘B from the Bragg law diverges when, in the asymmetrical Bragg case, the angle of incidence α tends to zero. We have derived a more precise expression for Δ⊘B by assuming that the asymptotic forms of the dispersion surface are circles instead of straight linos as in conventional treatment. According to the new expression, when α tends to zero, Δ⊘B tends to the critical angle ⊘c of total reflection by an amorphous specimen of the same material as the crystal. A physical explanation of this fact is given. The classical expressions of dynamical theory predict that the width w of the diffraction pattern (Darwin curve), and therefore the integrated reflecting power R H ⊘, diverge when α tends to zero. We have derived a more precise expression for w by again using the assumption that the asymptotic forms of the dispersion surface are circles. According to the new expression, when α tends to zero, w and R H ⊘ tend to zero passing through a maximum, instead of diverging. A physical explanation of this fact is also given.