From Newton's dynamics to quasicrystal CBED in resonant response

Classically, kinetic energy E^k=p²/2m and momentum p=mv, mass•velocity, are combined with wave optics v=fl=w/k, frequency•wavelength and angular frequency/wave-vector. The momentum is inversely proportional to wavelength because quantized waves contain two velocities, v_p and v_g, the phase and group velocities, where p=h/l, as de Broglie hypothesized. Relativistically, E²=p²c²+m_o²c⁴/h²=m’c², with m_o=m the rest mass constant and m’ a varying relativistic mass. The dynamics of this equation approximate Newton’s when p<<m_oc It turns out that w²=k²c²+m_o²c⁴/h, and that massive particles have v_p.v_g=c², where v_g=dw/dk and v_p>c._. Meanwhile the massless photon in free space has v_p=v_g=c. The quasicrystal is a relatively new solid with hierarchic structure that diffracts into geometric space by operation of a special metric that is a new constant in condensed matter [1]. The principal axes and diffraction planes are described and illustrated completely in 3-dimensions. The most remarkable properties occur in CBED (convergent beam electron diffraction). Novel dual diffraction occurs instantaneously in both geometric and linear spaces; in both quasi-Bragg and dynamical diffraction; at mutually normal angles in resonant response [2]. The properties of the momentum quantum apply generically. Von Neumann's mathematics play games with real physics. [1] Quantum mechanics:collapse, Bourdillon A.J., UHRL (2203). [2] Journal of Modern Physics (2025) 16 911-932.