Motional electric field

Secondly, in the atomic frame, it comes with a motional electric field E = V x B hydrogen atom are specially sensitive to this field. As this electric field is proportional to v, the corresponding shift of the S level due to the interaction with a near P level, is quadratic with v. Moreover, since the nearest level (P1/2) is below the S one, this motional electric field can give a positive frequency shift of the transition Av able to compensate the negative shift due to the second order Doppler effect. 

To carry out this scheme, a fast atomic beam (v/c = 0.01) is used to translate the required nanosecond time intervals into convenient laboratory distances. To avoid complications due to motional electric fields, the entire experiment is performed in zero magnetic field and the resonance is tuned through directly by changing the frequency of the applied rf field. Other rf fields are used to select one hyperfine state so as to simplify the line shape. 

Figure 8.35. Section of the electric resonance hyperfine spectrum for 7Li79Br in the v = 0, J = 1 rotational level [62]. The effective motional electric field was 2.33 V cm-1. The resonances are labelled to permit comparison with the transitions indicated in figure 8.34. Resonances f2 and f are Stark components of the same transition.

The origin of the electric dipole intensity for the AMj = 1 transitions studied merits further consideration. If the static magnetic field is 5 kG, the motional electric field has a magnitude of approximately 3 V cm-1 and is perpendicular to the applied magnetic field. This electric field mixes a state [./, Mj) with the states. J 1, Mj 1) and in order to obtain non-zero electric dipole transition moments for the transitions. /. Mj) o IJ, Mj 1), the oscillating electric field must be applied parallel to the static magnetic field. 

Fig. 1. The potential y (see Eq. (7)) along the direction of the motional electric field. Values of the pseudo-momentum and magnetic field strength are K = (0,1.4,0) and B = (0,0,10" ) in atomic units (B = la.u. corresponds to 2.35 x lO T)...

Rowell and co-workers [62-64] have developed an electrophoretic fingerprint to uniquely characterize the properties of charged colloidal particles. They present contour diagrams of the electrophoretic mobility as a function of the suspension pH and specific conductance, pX. These fingerprints illustrate anomalies and specific characteristics of the charged colloidal surface. A more sophisticated electroacoustic measurement provides the particle size distribution and potential in a polydisperse suspension. Not limited to dilute suspensions, in this experiment, one characterizes the sonic waves generated by the motion of particles in an alternating electric field. O Brien and co-workers have an excellent review of this technique [65]. 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

In order to illustrate some of the basic aspects of the nonlinear optical response of materials, we first discuss the anliannonic oscillator model. This treatment may be viewed as the extension of the classical Lorentz model of the response of an atom or molecule to include nonlinear effects. In such models, the medium is treated as a collection of electrons bound about ion cores. Under the influence of the electric field associated with an optical wave, the ion cores move in the direction of the applied field, while the electrons are displaced in the opposite direction. These motions induce an oscillating dipole moment, which then couples back to the radiation fields. Since the ions are significantly more massive than the electrons, their motion is of secondary importance for optical frequencies and is neglected. 

In electrophoresis, the motion of charged colloidal particles under the influence of an electric field is studied. For spherical particles, we can write... 

It is helpful to distinguish three different types of problem to which Newton s laws of motion may be applied. In the simplest case, no force acts on each particle between collisions. From one collision to the next, the position of the particle thus changes by v,5f, where v, is the (constant) velocity and 6t is the time between collisions. In the second situation, the particle experiences a constant force between collisions. An example of this type of motion would be that of a charged particle moving in tr uniform electric field. In the third case, the force on the particle depends on its position relative to the other particles. Here the motion is often very difficult, if not impossible, to describe analytically, due to the coupled nature of the particles motions. 

Unlike simple deflections or accelerations of ions in magnetic and electric fields (Chapter 25), the trajectory of an ion in a quadrupolar field is complex, and the equations of motion are less easy to understand. Accordingly, a simplified version of the equations is given here, with a fuller discussion in the Appendix at the end of this chapter. 

Electrophoresis (qv), ie, the migration of small particles suspended in a polar Hquid in an electric field toward an electrode, is the best known effect. If a sample of the suspension is placed in a suitably designed ceU, with a d-c potential appHed across the ceU, and the particles are observed through a microscope, they can all be seen to move in one direction, toward one of the two electrodes. AH of the particles, regardless of their size, appear to move at the same velocity, as both the electrostatic force and resistance to particle motion depend on particle surface this velocity can be easily measured. 

Assuming that the current in the gas is carried mostly by electrons, the induced electric field uB causes transverse electron motion (electron drift), which, being itself orthogonal to the magnetic field, induces an axial electric field, known as the Hall field, and an axial body force, F, given by... 

The absence of an electron from a covalent bond leaves a hole and the neighboring valence electron can vacate its covalent bond to fill the hole, thereby creating a hole in a new location. The new hole can, in turn, be filled by a valence electron from another covalent bond, and so on. Hence, a mechanism is estabUshed for electrical conduction that involves the motion of valence electrons but not free electrons. Although a hole is a conceptual artifact, it can be described as a concrete physical entity to keep track of the motion of the valence electrons. Because holes and electrons move in opposite directions under the influence of an electric field, a hole has the same magnitude of charge as an electron but is opposite in sign. 

Response to Electric and Acoustic Fields. If the stabilization of a suspension is primarily due to electrostatic repulsion, measurement of the zeta potential, can detect whether there is adequate electrostatic repulsion to overcome polarizabiUty attraction. A common guideline is that the dispersion should be stable if > 30 mV. In electrophoresis the appHed electric field is held constant and particle velocity is monitored using a microscope and video camera. In the electrosonic ampHtude technique the electric field is pulsed, and the sudden motion of the charged particles relative to their counterion atmospheres generates an acoustic pulse which can be related to the charge on the particles and the concentration of ions in solution (18). 

Charge carriers in a semiconductor are always in random thermal motion with an average thermal speed, given by the equipartion relation of classical thermodynamics as m v /2 = 3KT/2. As a result of this random thermal motion, carriers diffuse from regions of higher concentration. Applying an electric field superposes a drift of carriers on this random thermal motion. Carriers are accelerated by the electric field but lose momentum to collisions with impurities or phonons, ie, quantized lattice vibrations. This results in a drift speed, which is proportional to the electric field = p E where E is the electric field in volts per cm and is the electron s mobility in units of cm /Vs. 

At still higher fields carriers can acquke enough energy from motion in an electric field to create electron—hole paks by impact ionization. Eor siUcon the electron ioniza tion rate, which is the number of paks generated per cm of electron travel, depends exponentially on electric field. It is about 2 X 10 cm for a 50 kV/cm field at 300 K. The electric field causes electrons and holes so created to travel in opposite dkections. They may create other electron—hole paks causing positive feedback, which leads to avalanche breakdown at sufficiently high fields. 

A finite time is required to reestabUsh the ion atmosphere at any new location. Thus the ion atmosphere produces a drag on the ions in motion and restricts their freedom of movement. This is termed a relaxation effect. When a negative ion moves under the influence of an electric field, it travels against the flow of positive ions and solvent moving in the opposite direction. This is termed an electrophoretic effect. The Debye-Huckel theory combines both effects to calculate the behavior of electrolytes. The theory predicts the behavior of dilute (<0.05 molal) solutions but does not portray accurately the behavior of concentrated solutions found in practical batteries. 

Ions of an electrolyte are free to move about in solution by Brownian motion and, depending on the charge, have specific direction of motion under the influence of an external electric field. The movement of the ions under the influence of an electric field is responsible for the current flow through the electrolyte. The velocity of migration of an ion is given by ... 

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