An Alternative Formation Theory of Beat. (II) Revelations of Recursion Formulas of the Reflected X-rays and the Anomalous Transmission and Absorption by the Binomial Theorem

Abstract

The recursion formulas for the photon paths in the Borrmann triangle, which satisfy a new modified Pascal triangle can be derived from the binomial theorem by regarding the permutation of the stochastic variables of the diffracted and transmitted X-ray photons. The Borrmann triangle for the n-multiple X-ray reflections expanded by the n-degree binomial distribution consists of the two sub-triangles given by the (n−1)-degree binomial distribution of the diffracted and transmitted photons. The former sub-triangle shows perfectly flawless symmetry but the latter one shows inevitable asymmetry. A reasonable understanding of both the high intense and very weak photon flows in the Borrmann triangle, which are popularly known as the anomalous transmission and absorption, respectively, are derived from the binomial theorem. Incident photons irradiated at a point O that forms the vertex of the Borrmann triangle propagate through the bypasses parallel to only the complementary half of the integral whole median with the high probabilities from the binomial theorem and emanate them from a short width slit of \(\overline{O'O''}\) on the base of the high intense photon flow Borrmann triangle ▵OO′O″, which can be defined by the standard deviation of the normal distribution. The parallel paths to the whole median also pass the very weak photon flows from the high power exponent of d multinomials through the triangle ▵OO′O″. Both the above contrastive photon flows could coexist in ▵OO′O″ based upon the complementary rivalry duality from the binomial theorem of (d+t)n=1, including the very weak photon flows from the high power exponent of t multinomials near both sides of the Borrmann triangle.