From a combination of Maxwell's electromagnetism with Planck's law and the de Broglie hypothesis, we arrive at quantized photonic wave groups whose constant phase velocity is equal to the speed of light c = ω/k and to their group velocity dω/dk.When we include special relativity expressed in simplest units, we find that, for particulate matter, the square of rest mass 2 2 2 o m ω = k , i.e., angular frequency squared minus wave vector squared.This equation separates into a conservative part and a uniform responsive part.A wave function is derived in manifold rank 4, and from it are derived uncertainties and internal motion.The function solves four anomalies in quantum physics: the point particle with prescribed uncertainties; spooky action at a distance; time dependence that is consistent with the uncertainties; and resonant reduction of the wave packet by localization during measurement.A comparison between contradictory mathematical and physical theories leads to similar empirical conclusions because probability amplitudes express hidden variables.The comparison supplies orthodox postulates that are compared to physical principles that formalize the difference.The method is verified by dual harmonics found in quantized quasi-Bloch waves, where the quantum is physical; not axiomatic.

Einstein claimed Bohr’s theory is incomplete: “the wave function does not provide a complete description of the physical reality” [1]. Their views represent two physics in schism [2] [3]. Quanta are fundamental. The theory of diffraction in quasicrystals, that is summarized here, is falsifiable and verified. The quanta are not only harmonic; but harmonic in dual series: geometric and linear. Many have believed the quantum is real; rather than conceptual and axiomatic. The quasicrystal proves its reality.

Previous theories of quasicrystal diffraction have called it “Bragg diffraction in Fibonacci sequence and 6 dimensions”. This is a misnomer, because quasicrystal diffraction is not in integral linear order n where nλ = 2dsin(θ) as in all crystal diffraction; but in irrational, geometric series τm, that are now properly indexed, simulated and verified in 3 dimensions. The diffraction is due not to mathematical axiom, but to the physical property of dual harmony of the probe, scattering on the hierarchic structure in the scattering solid. By applying this property to the postulates of quantum theory, it emerges that the 3rd postulate (continuous and definite) contradicts the 4th (instantaneous and indefinite). The latter also contradicts Heisenberg’s “limit”. In fact, the implied postulates of probability amplitude describe hidden variables that are universally recognized, in all sensitive measurement, by records of error bars. The hidden variables include momentum quanta, in quasicrystal diffraction, that are continuous and definite. A revision of the 4th postulate is proposed.

Diffraction in quasicrystals is in irrational and geometric series with icosahedral point group symmetry. None of these features are allowed in Bragg diffraction, so a special theory is required. By means of a hierarchic model, the present work displays exact agreement between an analytic metric, with a numeric description of diffraction in quasicrystals—one that is founded on quasi-structure-factors that are completely indexed in 3-dimensions. At the quasi-Bragg condition, the steady state wave function of incident radiation is used to show how resonant response, in metrical space and time, enables coherent interaction between the periodic wave packet and hierarchic quasicrystal. The quasi-Bloch wave is invariant about all translations a τ m , where a is the quasi-lattice parameter. This is numerically derived, analyzed, measured, verified and complete. The hierarchic model is mapped in reverse density contrast, and matches the pattern and dimensions of phase-contrast, op-timum-defocus images. Four tiers in the hierarchy of icosahedra are confirmed, along with randomization of higher order patterns when the specimen foil is oriented only degrees off the horizontal. This explains why images have been falsely described as having “no translational symmetry”.

Diffraction in quasicrystals is in logarithmic order and icosahedral point group symmetry.Neither of these features are allowed in Bragg diffraction, so a special theory is required.The present work displays exact agreement between the analytic metric with a numeric description of diffraction in quasicrystals that is based on quasi-structure factors.So far, we treated the hierarchic structure as ideal; now, we detail the theory by including two significant features: firstly, the steady state wave function of the incident radiation demonstrates how harmonics, in metrical space and time, enable coherent interaction between the periodic wave packet and hierarchic quasicrystal; secondly, mapping of the hierarchic structure for any influence of defects will allow estimation of possible error margins in the analysis.The hierarchic structure has the required logarithmic periodicity: superclusters, containing about 10 3 atoms, convincingly map phase contrast images; while higher orders leave space for subsidiary speculation.The diffraction is completely explained for the first time.

Two problems in solid state physics and superconductivity are addressed by applications of dispersion dynamics. The first is the Hall effect. The dynamics of charges that yield positive Hall coefficients in material having no mobile positive charges have always been problematic The effect requires both electric and magnetic response, but magnetic deflection is only possible in mobile charges. In high temperature superconductors, these charges must be electrons. Contrary to Newton’s second law, their acceleration is reversed in crystal fields that dictate negative dispersion. This is evident in room temperature measurements, but a second problem arises in supercurrents at low temperatures. The charge dynamics in material having zero internal electric field because of zero resistivity; and zero magnetic field because of the Meissner-Ochsenfeld diamagnetism; while the supercurrents themselves have properties of zero net momentum; zero spin; and sometimes, zero charge; are so far from having been resolved that they may never have been addressed. Again, dispersion dynamics are developed to provide solutions given by reduction of the superconducting wave packet. The reduction is here physically analyzed, though it is usually treated as a quantized unobservable.

The initial purpose is to add two physical origins for the outstandingly clear mathematical description that Dirac has left in his Principles of Quantum Mechanics. The first is the “internal motion” in the wave function of the electron that is now expressed through dispersion dynamics; the second is the physical origin for mathematical quantization. Bohr’s model for the hydrogen atom was “the greatest single step in the development of the theory of atomic structure.” It leads to the Schrodinger equation which is non-relativistic, but which conveniently equates together momentum and electrostatic potential in a representation containing mixed powers. Firstly, we show how the equation is expansible to approximate relativistic form by applying solutions for the dilation of time in special relativity, and for the contraction of space. The adaptation is to invariant “harmonic events” that are digitally quantized. Secondly, the internal motion of the electron is described by a stable wave packet that implies wave-particle duality. The duality includes uncertainty that is precisely described with some variance from Heisenberg’s axiomatic limit. Harmonic orbital wave functions are self-constructive. This is the physical origin of quantization.

A set of discoveries are described that complete the structural model and diffraction theory for quasicrystals. The irrational diffraction indices critically oppose Bragg diffraction. We analyze them as partly rational; while the irrational part determines the metric that is necessary for measurement. The measurement is verified by consistency with the measured lattice parameter, now corrected with the metric and index. There is translational symmetry and it is hierarchic, as is demonstrated by phase-contrast, optimum-defocus imaging. In Bragg’s law, orders are integral, periodic and harmonic; we demonstrate harmonic quasi-Bloch waves despite the diffraction in irrational, geometric series. The harmonicity is both local and long range. A breakthrough in understanding came from a modified structure factor that features independence from scattering angle. Diffraction is found to occur at a given “quasi-Bragg condition” that depends on the special metric. This is now analyzed and measured and verified: the metric function is derived from the irrational part of the index in three dimensions. The inverse of the function is exactly equal to the metric that was first discovered independently by means of “quasi-structure factors”. These are consistent with all structural measurements, including diffraction by the quasicrystal, and with the measured lattice parameter.

Initially, all that was known about diffraction in quasicrystals was its point group symmetry; nothing was known about the mechanism. The structure was more evident, and was called quasiperiodic. From mapping the Mn atoms by phase-contrast, optimum-defocus, electron microscopy, the progress towards identifying unit cell, cluster, supercluster and extensive hierarchic structure is evident. The structure is ordered and uniquely icosahedral. From the known structure, we could calculate structure factors. They were all zero. The quasi structure factor is an iterative procedure on the hierarchic structure that correctly calculates diffraction beam intensities in 3-dimensional space. By a creative device, the diffraction is demonstrated to occur off the Bragg condition; the quasi-Bragg condition implies a metric that enables definition and measurement of the lattice constant. The reciprocal lattice is the 3-dimensional diffraction pattern. Typically, it builds on Euclidean axes with coordinates in geometric series, but it also transforms to Cartesian coordinates.

Thirty seven years after the discovery of quasicrystals, their diffraction is completely described by harmonization between the sine wave probe with hierarchic translational symmetry in a structure that is often called quasiperiodic. The diffraction occurs in geometric series that is a special case of the Fibonacci sequence. Its members are irrational. When substitution is made for the golden section τ by the semi-integral value 1.5, a coherent set of rational numbers maps the sequence. Then the square of corresponding ratios is a metric that harmonizes the sine wave probe with the hierarchic structure, and the quasi-Bragg angle adjusts accordingly. From this fact follows a consistent description of structure, diffraction and measurement.

The necessary derivation of negative mass in dispersion dynamics suggests cosmic applications. The method analyzes functional relationships between particle angular frequency, wave vector, rest mass and electromagnetic or nuclear potential, f(ω, k, m0, V) = 0. A summary of consequential predictions of the dynamics leads to a calculation of ways in which negative mass might influence such phenomena as the rotational velocities that are observed in spiral galaxies. The velocities are found to be not Newtonian in the simple two body approximations for our solar system; but nearly constant with increasing orbital radii. It has moreover been suggested that the motion is due to halo structures of dark matter or dark energy. However, the motion is simply described by many-body gravitation that is transmitted along elastic spiral arms. In this context, we calculate possible effects of negative mass, but without observational confirmation.

Because magnetic moment is spatial in classical magnetostatics, we progress beyond the axiomatic concept of the point particle electron in physics. Orbital magnetic moment is well grounded in spherical harmonics in a central field. There, quantum numbers are integral. The half-integral spinor moment appears to be due to cylindrical motion in an external applied magnetic field; when this is zero , the spin states are degenerate. Consider lifting the degeneracy by diamagnetism in the cylindrical magnetic field: a uniquely derived electronic magnetic radius shares the identical value to the Compton wavelength.

The outstanding difference between high temperature superconductors and low temperature superconductors is the sign of the Hall coefficient, properly understood. Since the Lorentz force acts on particles, not voids nor immobile ions, we propose that the experimental positive coefficient is due to dispersion dynamics in valence bands, i.e. on electrons with positive charge/mass ratio, but with negative charge and negative effective mass. In HiT ccompounds, anionic and cationic doping creates holes that substitute for the lattice distortions that bind Cooper pairs in metallic superconductors such as Nb. In both types of superconductor, the conventional notion of antiparallel spins S = 0, with paired wave vectors k and -k, is maintained; but in the ceramics “holes” h, produced by chemical doping and measured in the normal state, are available to bond super-conducting Boson pairs via h− or h02 excitons.

The outstanding difference between high temperature superconductors and low temperature superconductors is the sign of the Hall coefficient, properly understood. Since the Lorentz force acts on particles, not voids nor immobile ions, we propose that the experimental positive coefficient is due to dispersion dynamics in valence bands, i.e. on electrons with positive charge/mass ratio, but with negative charge and negative effective mass. In HiT ccompounds, anionic and cationic doping creates holes that substitute for the lattice distortions that bind Cooper pairs in metallic superconductors such as Nb. In both types of superconductor, the conventional notion of antiparallel spins S = 0, with paired wave vectors k and -k, is maintained; but in the ceramics “holes” h, produced by chemical doping and measured in the normal state, are available to bond super-conducting Boson pairs via h− or h02 excitons.

A quasi-structure factor method is used to show how sharp diffraction patterns are produced by aperiodic quasicrystals. Icosahedral symmetry is described for the tensor rank 3 solids with edge-sharing unit cells that are pentagonally close packed in hierarchic structures having a geometric reciprocal lattice. The hierarchic symmetry replaces translational symmetry in crystal diffraction. Details in the calculation show how the symmetry is simulated in diffraction.