Quantum theory with conjecture of fractional charge quantization, eigenfunctions for fractional charge quantization, fractional Fourier transform, Hermite function for fractional charge quantization, and eigenfunction for a twisted and twigged electron quanta is developed and applied to resistivity, dielectricity, giant magneto resistance, Hall effect and conductance. Our theoretical relationship for quantum measurements is in good conformity and in agreement with most of the experimental results. These relationships will pave a new approach to quantum physics for deciphering measurements on single quantum particles without destroying them. Our results are in agreement with 2012 Physics Nobel Prize winning Scientists, Serge Haroche and David J. Wineland.

Cobalt (Co) doped magnesium hydroxide Mg(OH)2 nanoparticles are synthesized by a surfactant-free co-participation method. Scanning electron microscopy (SEM) images show nanometer size Mg(OH)2 particles in spherically shaped particle-like morphology. Synthesis of these Mg(OH)2 nanocrystals involves the formation of monomeric MgOH[Formula: see text] ions as the precursor for the Mg(OH)2 nuclei which finally evolves in spherical particle-like morphology. X-ray diffraction (XRD) confirms the hexagonal crystal structure of the samples. With increasing Co concentration, the absorption spectra of the samples show narrowing of the bandgap from 5.47 eV (for pure Mg(OH2)) to 5.26 eV (for 10% Co-doped Mg(OH2)) effect is attributed to changes in the interaction potentials between Co and the host Mg(OH)2 lattice due to dopant-induced lattice distortion and the presence of a mixed valance Co[Formula: see text]/Co[Formula: see text] state.

We demonstrate a morphological change in [Formula: see text] ([Formula: see text] 0–5%) samples from rod-like ([Formula: see text] 2%) to flower-like ([Formula: see text] 2%) with increasing Cu concentration. This effect is attributed to Cu induced change in the ZnO growth mode from one-dimensional (forming rod-like) to two-dimensional (forming flower-like) structures. The optical properties of the samples were also found to be well correlated with the observed morphological changes in the samples. Optical bandgap decreases with increasing [Formula: see text], exhibiting a minimum (2.88[Formula: see text]eV) at [Formula: see text] 2% and afterwards it increases slightly up to 3.00[Formula: see text]eV for [Formula: see text] 5%. Our results show an intriguing possibility of bandgap tuning by the sample morphology.

In this work, Cobalt (0-10) % doped MgO nanoparticles are prepared for magnetic and optical studies. X-ray diffraction studies show the particle size increases from 3.10 nm to 9.50 nm with increasing Cobalt concentration in MgO samples. The secondary phases are observed beyond 5% Cobalt in the host MgO which indicates the solubility limit of dopant in this material. SEM analysis further reconfirms the purity and spherical morphology of nanoparticles in all samples. Its result also reveals that the smallest particles have the largest oxygen vacancy concentration as compared to other samples. The dopant and size-induced effects are observed for electronic absorption spectra and bandgap of the samples.  UV-visible absorption spectra show a quantum confinement effect as the optical band gap decreases with the increase of the particle size. The optical band gap of the nanoparticles ranges from 4.68 eV to 3.15 eV which is very less in comparison with the bulk MgO band gap up to 7.2eV.  Vibrating sample magnetometer studies at room temperature clearly show the ferromagnetic behavior for pure MgO nanoparticles in contrast with the paramagnetic behavior for other samples.  It is suggested that the presence of the ferromagnetic behavior in pure MgO nanoparticles are due to the presence of oxygen vacancies.