Time-dependent electric fields

Fig. 4.6. Piezoelectric pulse diagrams can be used to obtain explicit representations of the time dependent electric fields in piezoelectric substances. The magnitudes and orientations of these electric fields are critical to development of shock-induced conduction. As an example, the diagram on the left shows the polarization and displacement relations for a location at the input electrode. The same functions for a location within the crystal is shown on the right (after Davison and Graham [79D01]).

A simple time-dependent electric field is given by... 

Equation (4.a) states that the wave function must obey the time-dependent Schrodinger equation with initial condition /(t = 0) = < ),. Equation (4.b) states that the undetermined Lagrange multiplier, x t), must obey the time-dependent Schrodinger equation with the boundary condition that x(T) = ( /(T))<1> at the end of the pulse, that is at f = T. As this boundary condition is given at the end of the pulse, we must integrate the Schrodinger equation backward in time to find X(f). The final of the three equations, Eq. (4.c), is really an equation for the time-dependent electric field, e(f). 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

Suppose now that two different two-step fields are applied (1) at t0 the electric field 2EX is turned on and at tx the field suddenly changes to Ex and (2) the field at tx is the same as in (1) but between t0 and tx it is Ex/2. It is clear from Fig. 9.14 that the polarization in these two cases is different, even for t > tx when the applied field is the same that is, the polarization at time t depends on the history of the applied field and not merely its instantaneous value. This is a specific example of a general conclusion that we made about the response of a linear medium to a time-dependent electric field (see Section 2.3). The polarization at all times greater than tx can be obtained by following reasoning similar to that which led to (9.37) we write P(t) in the form (9.36) and require that lim,A, P(f) = Pd(tx) + e0x0vEx the result is... 

The basic aspects of the theory of the behaviour of dielectrics in time dependent electric fields have been known for a long time. We recall some elements useful for our discussion. 

An issue that has been explored is how the relative distribution of charge and mass affect the viscosity of an ionic liquid. Kobrak and Sandalow [183] pointed out that ionic dynamics are sensitive to the distance between the centers of charge and mass. Where these centers are separated, ionic rotation is coupled to Coulomb interactions with neighboring ions where the centers of charge and mass are the same, rotational motion is, in the lowest order description, decoupled from an applied electric field. This is significant, because the Kerr effect experiments and simulation studies noted in Section III. A imply a separation of time scales for ionic libration (fast) and translation (slow) in ILs. Ions in which charge and mass centers are displaced can respond rapidly to an applied electric field via libration. Time-dependent electric fields are generated by the motion of ions in the liquid... 

The dipole interaction of this molecule with an incident time-dependent electric- field E(f) is described by the total Hamiltonian ... 

In the RF trap the ion is exposed to an inhomogeneous and time dependent electric field. At each point in the trap the time average of the field vanishes, but the time averaged square of the field is proportional to the squared distance from the trap center. For an ion oscillating around the trap center the... 

Our interest in quantum dot-sensitized solar cells (QDSSC) is motivated by recent experiments in the Parkinson group (UW), where a two-electron transfer from excitonic states of a QD to a semiconductor was observed [32]. The main goal of this section is to understand a fundamental mechanism of electron transfer in solar cells. An electron transfer scheme in a QDSSC is illustrated in Figure 5.22. As discussed in introduction, quantum correlations play a crucial role in electron transfer. Thus, we briefly describe the theory [99] in which different correlation mechanisms such as e-ph and e-e interactions in a QD and e-ph interactions in a SM are considered. A time-dependent electric field of an arbitrary shape interacting with QD electrons is described in a dipole approximation. The interaction between a SM and a QD is presented in terms of the tunneling Hamiltonian, that is, in... 

B. Dielectric Polarization in Time-Dependent Electric Fields... 

In dielectric spectroscopy the polarization response P(t) of a dipolar material is monitored, which is subject to a time-dependent electric field (Maxwell field), E t). For a linear and isotropic dielectric one can write (e.g., Ref. 34) ... 

Absorption of light by molecules, resulting in electronic excitations, is caused by the interaction of the bound molecular electrons with the electric field of the radiation. In the dipolar approximation, the interaction of the dipole operator of the solute mo with the time-dependent electric field E(t)... 

In the following, we consider two examples where the Melnikov integral is explicitly calculable. The first example serves as a standard case for the calculation of the integral. The results will be used in the later sections where we study the Arnold model and tangency. The second one serves as a prototype for molecular systems under time-dependent electric fields. 

Saville, G.F. and Goodkind, J.M. (1994). Computation of tmmeling rates in time-dependent electric fields Electrons on the surface of liquid helimn, a one-dimensional hydrogen atom, Phys. Rev. A50, 2059-2067. 

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