Oscillatory electric charges

Eigenstates of a crystal, 725 Eigenvalues of quantum mechanical angular momentum, 396 Electrical filter response, 180 Electrical oscillatory circuit, 380 Electric charge operator, total, 542 Electrodynamics, quantum (see Quantum electrodynamics) Electromagnetic field, quantization of, 486, 560... 

Fig. 6.3. Motions and crystalline arrangements of trapped charged particles. The left figure is a photo of a cluster of 32 charged micro-particles stored in a Paul trap. The two other figures are results from a numerical simulation of a 1000-ion Coulomb crystal confined in a linear quadrupole trap. In the left and the center panel the micro-motion of the particles in the oscillatory electric field can be seen. The amplitude increases proportionally to the distance from the center. The time-averaged positions plotted in the right part for a selected sample shows that the ions remain well-localized. This is the basis for defining an effective translational temperature of the ion cluster by subtracting the periodic oscillation from the overall motion.

For X-rays, the interaction between radiation and matter is primarily via interaction between the electrons and the oscillatory electric field of the electromagnetic beam. E = EoCOs(a>t). An electron will, in the X-ray beam, be accelerated by the oscillating electric field an accelerated charged particle, on the other hand, will... 

At lower frequencies, orientational polarization may occur if the glass contains permanent ionic or molecular dipoles, such as H2O or an Si—OH group, that can rotate or oscillate in the presence of an appHed electric field. Another source of orientational polarization at even lower frequencies is the oscillatory movement of mobile ions such as Na". The higher the amount of alkaH oxide in the glass, the higher the dielectric constant. When the movement of mobile charge carriers is obstmcted by a barrier, the accumulation of carriers at the interface leads to interfacial polarization. Interfacial polarization can occur in phase-separated glasses if the phases have different dielectric constants. 

L. Mandelstam and N. Papalexi performed an interesting experiment of this kind with an electrical oscillatory circuit. If one of the parameters (C or L) is made to oscillate with frequency 2/, the system becomes self-excited with frequency/ this is due to the fact that there are always small residual charges in the condenser, which are sufficient to produce the cumulative phenomenon of self-excitation. It was found that in the case of a linear oscillatory circuit the voltage builds up beyond any limit until the insulation is ultimately punctured if, however, the system is nonlinear, the amplitude reaches a stable stationary value and oscillation acquires a periodic character. In Section 6.23 these two cases are represented by the differential equations (6-126) and (6-127) and the explanation is given in terms of their integration by the stroboscopic method. 

The harmonic oscillator is an important system in the study of physical phenomena in both classical and quantum mechanics. Classically, the harmonic oscillator describes the mechanical behavior of a spring and, by analogy, other phenomena such as the oscillations of charge flow in an electric circuit, the vibrations of sound-wave and light-wave generators, and oscillatory chemical reactions. The quantum-mechanical treatment of the harmonic oscillator may be applied to the vibrations of molecular bonds and has many other applications in quantum physics and held theory. 

A simple model that illustrated the behavior of the polarization when the water molecules are organized in water layers between perfectly flat surfaces was previously suggested.13 That model took into account the nearest-neighbor dipole interactions, but ignored the surface charges and the electrolyte ions. The model is extended here to cases in which an electrolyte as well as surface charges are also present. It will be shown that a treatment of all electrostatic interactions, in the assumption of an icelike structuring of water near interfaces, can predict an oscillatory behavior for both the polarization and the electric potential as well as a nonproportionality between the polarization and the electric fields. 

In simulations of the full electric double layer, ionic density profiles are oscillatory in the concentration range between 1 and 3 mol/1. Surface charges are screened by free ions over a distance of several Debye lengths. 

This term is a correction to the interaction of the charge of an electron with an electric field due to the Zitterbewegung —a trembling motion (a highly oscillatory part of the motion with a frequency of the order 2m0c2/h and an amplitude of the order R/m c). 

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