Physical Quanta in Quasicrystal Convergent Beam Electron Diffraction

This paper serves two purposes. Firstly, we show how the classical mechanics of Newton—where momentum energy is less than rest mass energy, $pc？m_o{c}^2=E_o$ —transitions with increasing momentum, to the relativistic mechanics of Einstein and Klein-Gordon, where $pc？m_o{c}^2$ . To describe the transition, we replace the classical rest mass energy constant E_o by Einstein’s varying relativistic energy $E=m^{\prime }{c}^2=\sqrt{p^2{c}^2+m_o^2{c}^4}$ , and apply it to dispersion dynamics in the electron microscope. Secondly, we show how diffractive wave mechanics enables unconventional logarithmic order in hierarchic quasicrystals. This order acts simultaneously to produce conventional dynamical interference in linear order. Each of these remarkable physical effects is explained with neither abstruse mathematics nor arbitrary inventions of dimensionality; but in a realistic framework of conventional and verified hypotheses. The results describe momentum quanta in scattering by free electrons that resonate with dual diffraction in quasicrystalline convergent beam electron diffraction. Here, irrational indices in the quasicrystals separate into real and imaginary parts, where the latter part shifts the phase of the free-particle wave-function. The shift provides a new constant, a log-lin metric function, in the physics of condensed matter. The phase shift also demonstrates superposition of scattered rays simultaneously into both logarithmic space and linear space. The dual diffraction occurs, in dynamical diffraction by virtue of a