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Formation of Standing Waves Within Crystal Unitcell

Bhanumoorthy Pullooru

Published May 28, 2021• 2 citations
Publisher

Abstract

Abstract We show that the historic Davisson-Germer experiment demonstrates formation of standing waves within nickel crystal unitcell. Cartesian Fourier transform cannot offer description in terms of standing waves because Cartesian Fourier theory cannot accommodate π in place of 2π. Thus, formation of standing waves within unitcell in Davisson-Germer experiment necessarily requires spherical polar coordinate description of crystal diffraction. Description in spherical polar coordinates permits to incorporate precision angles from the experiment for better convergence in structure determination calculations.

DOI
10.21203/rs.3.rs-506068/v1

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Abstract We show that the historic Davisson-Germer experiment demonstrates formation of standing waves within nickel crystal unitcell. Cartesian Fourier transform cannot offer description in terms of standing waves because Cartesian Fourier theory cannot accommodate π in place of 2π. Thus, formation of standing waves within unitcell in Davisson-Germer experiment necessarily requires spherical polar coordinate description of crystal diffraction. Description in spherical polar coordinates permits to incorporate precision angles from the experiment for better convergence in structure determination calculations.
Abstract Application of unitcell defined boundary conditions leads to the question on the formation of standing waves within unitcell. We design and propose X-ray version of Davisson-Germer experiment as the answer. In the proposed experiment, a tunable synchrotron beam replaces the variable de Broglie wavelength of incident electron beam. Among the two series of Davisson-Germer peaks from original experiments available in literature, first series shown in figure 1 demonstrates standing wave description and the second series shown in figure 2 demonstrates running wave description. Both running waves and standing waves cannot simultaneously exist within the unitcell and the proposed experiment alone can resolve between the two. The experiment can be conducted on a macromolecular crystal also.
Abstract The famed Davisson-Germer Experiments demonstrated the wave phenomenon of electrons similarly to X-Ray scattering from Sir Lawrence Bragg’s X-ray experimentations on crystals c. 1913. Their empirical deduction of electrons behaving as waves (i.e. oscillatory) ignores the possibility of an electron beam behaving harmonically upon elastic collision with a diffraction grating - represented by nickel crystal - in their experiment. However, it is well established in the electrical engineering science that crystals possess piezoelectric effects and are used ubiquitously in electronic circuit designs for causing stable harmonic oscillation responses to direct current voltages. In light of this, the current mathematical model proposes the Davisson-Germer results to be the effect of a nickel crystal oscillator circuit which amplifies a direct voltage source – the electron beam – causing the phenomenon of inductance from the resultant electrical feedback with the crystal atom’s electromagnetic field.
Abstract We identify Maxwell-Schrödinger equation as the governing partial differential equation. We name the solution as unitcell function such that Born probability rule interprets electron density as modulus square of the unitcell function. The existing traditional full-amplitude Fourier transform relation with electron density as directly the inverse Fourier transform of structure factors therefore violates Born probability rule. We derive the partial wave Parse-val relation as the probability conservation law using Rayleigh expansion. We subdivide the experimental full-amplitudes into partial wave sub-amplitudes, which become amplitudes of spherical waves. The subscript on a sub-amplitude represents localization within the unit-cell and therefore spherical waves become localized wave packets. As a probability measure, the amplitude of a localized wave packet is exact and merits over the amplitude of a non-localized plane wave. We measure angle dependence as the precision angles data. We derive the partial wave sub-amplitude Fourier transform relation to determine orbital structures of individual atoms using the precision angles data. The partial wave sub-amplitude Fourier transform relation alone obeys Born probability rule and hence becomes the only acceptable Fourier transform relation in quantum crystallography.
This article is taken from the manuscript prepared by the author for his address at the joint meeting of Section B of the American Association for the Advancement of Science and the American Physical Society on December 28, 1927, at Nashville, Tennessee. An account of this work giving fuller experimental details is given by Davisson and Germer in the December, 1927, issue of the Physical Review. These experiments are fundamental to some of the newer theories in physics. Until they were performed, it could be said that all experimental facts about the electron could be explained by regarding it as a particle of negative electricity. It now appears that in some way a “wave-length” is connected with the electron's behavior. The work thus shows an interesting contrast with the discovery of A. H. Compton that a ray of light (a light pulse) suffers a change of wave-length upon impact with an electron, the change of wave-length corresponding exactly to the momentum gained by the electron. Until Compton's work, all the known facts about light could be explained by thinking of light as a wave motion. The Compton effect seems to prove the existence of particles of light. Physics is thus faced with a double duality. Compton showed that light is in some sense both a wave motion and a stream of particles. Davisson and Germer have now shown that a beam of electrons is in some sense both a stream of particles and a wave motion. At the same time, theoretical advances have been made which seem to pave the way for an understanding of this curious situation. A general account of these new developments was given by K. K. Darrow in his series “Contemporary Advances in Physics” in the Bell System Technical Journal for October, 1927. Some remarks on the relation of the Davisson and Germer experiments to the new mechanics were given in this article, p. 692 et seq. — Editor.
Abstract Discusses the Davisson–Germer experiment in which electrons pass through a screen having one or two slits, and shows the necessity for a wave theory. The operation of an electron microscope is explained and some properties of the waves including the uncertainty between the simultaneous determination of position and momentum of the particle. Analogies with classical phenomena are discussed, namely the Fourier transform relationship between pulse length and spectral width, and the size of an aperture aerial and its relationship to the beamwidth.