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The travelling wave group is a solution to the wave equation. With a Gaussian envelope, this stable wave does not spread as it propagates. The group is derived for electromagnetic waves and converted with Planck’s law to quantized photons. The resulting wave is a probability amplitude, and this is adapted to particles subject to special relativity. By including mass and by inverting the wave group, a description for antiparticles is derived. The consequent explanation is consistent with Dirac’s relativistic equation and with his theory of the electron; while being more specific than his idea of the wave packet, and more stable. The travelling wave group is extended to describe the positron, either free or in an external field.
structural study of quasicrystals is based on extremely dense icosahedral unit
cells that are systematically and consistently measured for the first time. The
structure and pattern indexation are 3-dimensional. A formula is given for
scattering from atoms in hierarchic arrangement and geometric series. The
Quasi-Bragg law is a new law in physics, with possible applications beyond
crystallography. The structure is compared with previous, unsuccessful, and
contradictory, attempts at analysis.
metric, that enables measurement of structural data from diffraction in
quasicrystals, is analyzed. A modified compromise spacing effect is the
consequence of scattering of periodic electromagnetic or electron waves by
atoms arranged on a geometric grid in an ideal hierarchic structure. This
structure is infinitely extensive, uniquely aligned and uniquely icosahedral.
The approximate analytic factor that converts the geometric terms base τ, into periodic terms modulo 2π, is . It matches the
simulated metric cs=0.947, consistently used in second (Bragg) order, over a
wide scale from atomic dimensions to sixth order superclusters.