First Experiments with Electronic Differentiators

In 1953, Collier and Singleton introduced electronic (analog) differentiation [45]. They developed an electronic device with wireless valves and patented their invention [46]. The 

In the same year, Martin [48] had given the theoretical basis for the assumption that higher-order derivatives n 2) have the potential for higher resolution compared to lower-order derivatives, but he himself used only a simple (passive) RC module d ) with connected valve amplification (Fig. 3-7). 

Amplifying the module s output is advisable because the signal voltage drops with differentiation. Moreover, the use of the valve has the added advantage that successive differentiating circuits are effectively isolated from one another. 

In contrast to the techniques discussed so far, analog differentiation simplified the differentiation of curves or other electric signals no intervention in the optical path of the spectrophotometer was necessary and the device could also be used for every other apparatus that converts a mechanical, optical, or other quantity into an electric signal. 

The invention of the transistor by Bardeen, Brattain, and Shockley in 1947, was of the utmost importance for electronic circuits. This active electronic structural member requires no heating current, as is the case for electronic tubes, and it is much smaller and also much cheaper. In the 1960s, the old electronic valve was also replaced in derivative devices (see Fig. 3-8). This circuit is not yet perfect, because the differentiator and the amplifier are not independent of each other. Better results can be obtained if the amplifier has a high-resistance input, as realized by a field-effect transistor (see Fig. 3-9). 

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The reaction-microscope technique is capable of measuring a fully (sixfold) differential cross section, by detecting the three-dimensional momenta of two particles of opposite charge, viz. the ion and one electron. Such a fully differential measurement was recently accomplished [7]. However, the very first experiments were content with recording the NSDI yield as a function of two momentum components of the ion, one parallel (P ) and one transverse (P ) to the linearly polarized laser field, while the second transverse component (P ,2) was integrated over. In terms of the amplitude (4.1), this corresponds to the momentum distribution... 

Such measurements were first applied with considerable success to elastic scattering. Indeed one was able to discuss experiments which would determine all the theoretically calculable amplitudes (Bederson, 1970). For inelastic processes, such measurements necessitate the simultaneous application of spin selection techniques and the alignment and orientation measurements discussed in the previous chapter. The experiments have become feasible with the advancement of experimental techniques. The first successful differential electron impact excitation study with spin-polarised electrons and alignment and orientation measurements was performed by Goeke et al. (1983) for the e—Hg case. McClelland, Kelley and Celotta (1985, 1986) carried out a systematic study for superelastic scattering of polarised electrons from polarised laser-excited Na (3 P) atoms. This system is essentially a two-electron collision system in which spin exchange is the dominant spin-dependent interaction. It thus allows one to obtain... 

For the evaluation of X-ray scattering experiments on multicomponent and multiphase systems such as polymer blends or partially crystalline polymer systems, a fourth form of scattering equations is often appropriate. We first refer to a one component system composed of particles with electrons, corresponding to a differential cross-section... 

Pople, Beveridge and Dobosh introduced the intermediate neglect of differential overlap model (INDO) in 1967. INDO is CNDO/2 with a more realistic treatment of the one-centre two-electron integrals. In the spirit of such models, the non-zero integrals were calibrated against experiment rather than being calculated fi om first principles. The authors concluded that, although INDO was a little better than... 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

This set of first-order differential equations can be solved, approximately or numerically, for a specific system. The theory has been applied to Li scattered from Cs/W, and gives more satisfactory agreement with experiment than does the one-electron approach. 

The integration of this set of coupled first-order differential equation can be done in a number of ways. Care must be taken since there are basically rather two different time scales involved, i.e. that of the nuclear dynamics and that of the normally considerably faster electron dynamics. It should be observed that this END takes place in a Cartesian laboratory reference frame, which means that the overall translation as well as overall rotation of the molecular system is included. This offers no complications since the equations of motion satisfy basic conservation laws and, thus, total momentum and angular momentum are conserved. At any time in the evolution of the molecular system can the overall translation be isolated and eliminated if so should be deemed necessary. This level of theory [16,19] is implemented in the program system ENDyne [20], and has been applied to atomic and molecular reactive collisions. Calculations of cross sections, differential as well as integral, yield results in excellent agreement with the best experiments. 

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