Narrowing function

In many cases, the nuclear densities n.<,(R, t) will be rather narrow functions, with a strong peak at the classical trajectory R f (t). In such a situation, integrals of the form d R n (R, f)G(R) are well represented by Taylor-expand-ing G(R) around the classical trajectory. This leads to... 

In a homogeneous stationary environment the distribution of n is close to Poisson, which is a narrow function with the average (n) and width V(n). Thus, averaging Eq. (8) over the fluctuating force, one gets the usual kinetic equation. 

For some applications, it is necessary to produce RGH lasers at a reasonable output power level with low spatial divergence and narrow bandwidth. To fulfill these requirements, use of either a master oscillator/power amplifier system or an injection-locked resonator is the most convenient method. Figure 26 shows a schematic of an injection-locked resonator. In this scheme the slave oscillator is an unstable resonator that can control the output energy and spatial mode quality. Input pulse can be made by a master oscillator typically consisting of a cavity with a line-narrowing function. An input power level on the order... 

Here, the function N( >o) describes a distribution of chromophores over their resonant ftequencies. We know from Fourier analysis that the broader function N( o) becomes the narrower function N(x — x ). The function N(tOo) describes... 

One drawback is that, as a result of the time-dependent potential due to the LHA, the energy is not conserved. Approaches to correct for this approximation, which is valid when the Gaussian wavepacket is narrow with respect to the width of the potential, include that of Coalson and Karplus [149], who use a variational principle to derive the equations of motion. This results in replacing the function values and derivatives at the central point, V, V, and V" in Eq. (41), by values averaged over the wavepacket. 

The Boltzmann constant is ks and T the absolute temperature. — is the Dirac delta function. Below we assume for convenience (equation (5)) that the delta function is narrow, but not infinitely narrow. The random force has a zero mean and no correlation in time. For simplicity we further set the friction to be a scalar which is independent of time or coordinates. 

Fig. 6.20 A switching function that applies over a narrow range near the cutoff and its effect on the Lennard-Janes potential.

The swelling of the adsorbent can be directly demonstrated as in the experiments of Fig. 4.27 where the solid was a compact made from coal powder and the adsorbate was n-butane. (Closely similar results were obtained with ethyl chloride.) Simultaneous measurements of linear expansion, amount adsorbed and electrical conductivity were made, and as is seen the three resultant isotherms are very similar the hysteresis in adsorption in Fig. 4.27(a), is associated with a corresponding hysteresis in swelling in (h) and in electrical conductivity in (c). The decrease in conductivity in (c) clearly points to an irreversible opening-up of interparticulate junctions this would produce narrow gaps which would function as constrictions in micropores and would thus lead to adsorption hysteresis (cf. Section 4.S). 

Next, by trial and error, we try to find a value for y such that sinh" y matches one of the 17/17 fractions in Table 2.2, say 17/17 = 0.80. This is easily done using either tables of sinh functions or the equation given in Table 2.1. The following results show that it is possible to place y within a range and then narrow that range without much difficulty. Remember, it is the inverse sinh values we are examining ... 

That the Poisson distribution results in a narrower distribution of molecular weights than is obtained with termination is shown by Fig. 6.11. Here N /N is plotted as a function of n for F= 50, for living polymers as given by Eq. (6.109). and for conventional free-radical polymerization as given by Eq. (6.77). This same point is made by considering the ratio M /M for the case of living polymers. This ratio may be shown to equal... 

Fig. 2. The distribution of silicon—oxygen—silicon bond angles in vitreous siUca (22,25). The function V(a) is the fraction of bonds with angles normalized to the most probable angle, 144°. This distribution gives quite a regular stmcture on the short range, with gradual distorting over a distance of 3 or 4 rings (2—3 nm). Crystalline siUca such as quartz or cristobaUte would have a narrower distribution around specific bond angles.

Resonant Sound Absorbers. Two other types of sound-absorbing treatments, resonant panel absorbers and resonant cavity absorbers (Helmholtz resonators), are used in special appHcations, usually to absorb low frequency sounds in a narrow range of frequencies. Resonant panel absorbers consist of thin plywood or other membrane-like materials installed over a sealed airspace. These absorbers are tuned to specific frequencies, which are a function of the mass of the membrane and the depth of the airspace behind it. Resonant cavity absorbers consist of a volume of air with a restricted aperture to the sound field. They are tuned to specific frequencies, which are a function of the volume of the cavity and the size and geometry of the aperture. 

Bulky, even if highly polari2able, functional groups or atoms that are attached anywhere but on the end of a rod-shaped molecule usually are less favorable for Hquid crystal formation. Enhanced intermolecular attractions are more than countered as the molecule deviates from the required linearity. For example, the inclusion of the bromine atom at position three of 4-decyloxy-3-bromoben2oic acid [5519-23-3] (9) prevents mesomorphic behavior. In other cases the Hquid crystal phases do not disappear, but their ranges are narrower. 

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