Higher Order Electric Fields

The ENBO method, which is a method for incorporating polarization and higher order electric field-molecule interaction terms into the theory, is discussed in Section IV. Nearly all OCT experiments actually use laser pulses that give rise to strong electric fields that are sufficiently strong to significantly... 

Higher Order Electric Fields. The electric field of equation (47) can be regarded as a field of the first order, F (10- Similarly, we can define fields of higher order thus F = VF (F), i.e. the field gradient a field of the third order F = WF (F), i.e. the gradient of a field gradient and quite generally fields of the w-th order ... 

The basis set requirements for the calculation of first hyperpolarizabilities are much the same as for the linear polarizability. However, as the first hyperpolarizabUity probes even higher-order electric-field-perturbed densities of the molecule, care should be exercised to ensure that the basis set is sufficiently saturated with respect to diffuse polarizing functions. Special basis sets have been developed for the calculation of hyperpolarizabilities by Pluta and Sadie) (1998), though the same basis sets that can be used for polarizabilities in most cases give reliable estimates also for first hyperpolarizabUities. 

Keeping only the electric field interaction in the perturbing operator, that is, neglecting magnetic and higher-order electric multipolar interactions, yields a... 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

In strong electric fields contributions to the induced dipole moment that are proportional to or E can also be important, and higher-order moments such as quadrupoles can also be induced. We will not be concerned with such contributions. 

One important sem source that is not based on thermionic emission is the field emission (fe) source. Fe-sem systems typically give images of much higher resolution than conventional sems due to the much narrower energy distribution (on the order of 0.25 eV) of the primary electron beam. A fe source is a pointed W tip from which electrons tunnel under the influence of a large electric field. This different mechanism of electron generation also results in a brightness comparable to a conventional thermionic source with much less current. 

The source requited for aes is an electron gun similar to that described above for electron microscopy. The most common electron source is thermionic in nature with a W filament which is heated to cause electrons to overcome its work function. The electron flux in these sources is generally proportional to the square of the temperature. Thermionic electron guns are routinely used, because they ate robust and tehable. An alternative choice of electron gun is the field emission source which uses a large electric field to overcome the work function barrier. Field emission sources ate typically of higher brightness than the thermionic sources, because the electron emission is concentrated to the small area of the field emission tip. Focusing in both of these sources is done by electrostatic lenses. Today s thermionic sources typically produce spot sizes on the order of 0.2—0.5 p.m with beam currents of 10 A at 10 keV. If field emission sources ate used, spot sizes down to ca 10—50 nm can be achieved. 

Electric Wind By virtue of the momentum transfer from gas ions moving in the electrical field to the surrounding gas molecules, a gas circiilation, known as the electric or ionic wind, is set up between the electrodes. For conditions encountered in electrical precipitators, the velocity of this circulation is on the order of 0.6 m/s. (2 ft/s). Also, as a result of this momentum transfer, the pressure at the collecting eleclrode is slightly higher than at the discharge electrode (White-head, op. cit., p. 167). 

For the linear response of the dipole to an electric field, this calculation is fairly straightforward. However, the dipole can also be calculated for a range of magnitudes of applied field to obtain higher order hyperpolarisabilities such as j . 

When the electric field ( ) is strong, as in a laser light source, the second and third terms cannot be neglected and N LO processes take place. A remarkable consequence of the higher-order terms in Eq. (5.4) is that the frequency of the light can change. If we consider two electric fields with frequencies coi and (O2,... 

Predictive methods that calculate u for the next time step of a MD simulation based on information from previous timesteps have been developed to minimize the computational cost. Ahlstrom et al. [13] used a first-order predictor algorithm, in which values of u from the two previous times steps are used to determine u at the next time step. A very serious drawback of this method is that it is not stable for long simulation times. However, it has been combined with iterative solutions, either by providing the initial iteration of the electric field values [163, 164], or by performing an iterative SCF step less frequently than every step [13,165], Higher-order predictor algorithms have also been described in the literature [13,163, 166],... 

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