Characteristic width

Since the spatial locations r,- of active droplets are not correlated, we can replace the summation over the droplets by a continuous integral, assuming at the same time that the ripplon frequency corresponding to co/ varies from droplet to droplet within a (normalized) distribution Vi (oi) centered around co/ and having a characteristic width 8co/, whose value will be discussed shortly. There is no reason to believe that the frequency and location of the tunneling centers are correlated therefore one obtains... 

Fig. 3. Roll for mass fabrication of microstructured steel plates. The endless sheet can be moved at least at a velocity of 1 m/min, i.e., for a plate length of 10 cm and rolling four microstructured units in parallel, 2,400 plates/h can be manufactured. Structural dimensions of 500 pm have so far been realized the technique is expected to yield microstructures at alOO-pm characteristic width...

Physically, Eq. (7.63) specifies that for a spark to lead to ignition of an exoergic system, the corresponding equivalent heat input radius must be several times the characteristic width of the laminar flame zone. Under this condition, the nearby layers of the initially ignited combustible material will further ignite before the volume heated by the spark cools. 

Of more recent discovery are wide and shallow extinction bands with characteristic widths of about 500-1000 A and extending from about 3400 to 11,000 A (for a brief survey, see Huffman, 1977). This very broad structure (VBS) is too broad and weak to be seen in Fig. 14.4. Lack of correlation between the diffuse bands and the VBS suggests a different origin for the two. 

For a sufficiently long time, the function Siip, t) becomes narrower than the respective characteristic width ( -i- cof) of G( ) around -I- i, and one can set... 

The space charge density is g = (F/Vm)-(Nd+ -Nd-), and their characteristic width (Le., the Debye-HGckel length, which is an equilibrium property and independent of the d-mobilities) is obtained as... 

At these temperatures the distribution of occupied levels in the conduction bands ( the Fermi distributions ) in the two metal electrodes ( Fig.l ) are quite sharp, with a boundary between filled and empty states ( the Fermi level ) of characteristic width k T ( k =0.08617 meV/K=0.69503 cm Vk ). An applied bias voltage V between the two electrodes separates the Fermi levels by an energy eV. If the barrier oxide is sufficiently thin electrons can tunnel from one electrode to the other. This process is called tunneling since the electrons go through a potential barrier, rather than being excited over it. The barrier must be thin for an appreciable barrier to flow. For a typical 2 eV barrier the junction resistance is proportional to, where s is the barrier width in Angstroms (17). The... 

The function Dk -jtfj, t2) under the integral (6) remarkably differs from zero only for It] — t21 ((k — l)/)-1 t, where T is the characteristic width of the phonon spectrum. This allows one to replace the upper integration limit t + r by t. Then the residual r dependence of the integral in equation (6) is given by the factor cos( >,(t] — ti — t)), i.e., the dependence of the phonon correlation function on r indeed has the form (3). The factor of the ocexp(io),T) term equals... 

Equation (7.22) is at the heart of spectroscopy. The positions of the absorption lines reflect the energy levels of the excited complex and the widths provide information about the lifetime and therefore about the coupling to the continuum states. The latter requires, however, that the measured widths are the true homogeneous line widths, i.e., unadulterated by poor resolution and/or thermal broadening, for example. Each resonance has a characteristic width. In Chapters 9 and 10 we will discuss how the final fragment distributions reflect the initial state in the complex and details of the fragmentation mechanism. 

Here ac is the characteristic width defined by [Pg.769]

Both Baer and co-workers (34) and Fadley and Shirley (35) have studied a number of the transition metals. The Uppsala results are plotted in Fig. 11. Doublets are clearly observed for Ir, Pt, and Au the other spectra consist of single peaks, each with its characteristic width and hints of structure. In XPS, as a rule, intensity at the bottom of d bands and the indications of structure there are suppressed, apparently because of transition probability decrease. Hybridization of d with non-ci character, which is generally strong toward the bottom of the d bands and weak at the top, tends to lessen transition intensity. It is suggestive... 

If the inhomogeneous distribution in Raman transition frequency is assumed to be a Gaussian with a characteristic width (i.e. standard deviation of the transition frequencies), 5, the intensity of the Raman echo signal I(Ati,At2) is given by the following equation, ... 

The results can be used as a guide for the future experimental work. In order to determine the twin domain wall width W in the bulk, one only needs to determine the characteristic width IFs of the surface structure of the domain wall. Previously, these features of the twinning materials were investigated using mainly X-ray techniques. In fact, the theoretical work leads to the conclusion that the only necessary information for the determination of the twin domain wall width W are the real space positions of the particles in the surface layer. 

Here x is the location of the back-ing/catalyst layer interface, A is the characteristic width of the interface smearing, which must be much smaller than the catalyst layer thickness for this interpolation to... 

For calculating the dimension of the set of spatial locations which at each time are occupied by trajectories with non-typical values of the asymptotic Lyapunov exponent we cover these locations with objects of size which can be estimated from the dynamics Since the proportion of fluid elements experiencing a finite-time average stretching A / A°° decays as exp(—G(A)f), and in an incompressible flow the area of fluid element remains constant (we refer to the two-dimensional situation for simplicity), the total area covered by these fluid elements decreases also as A (t) exp(—G(X)t). This area is stretched by the chaotic dynamics locally characterized by A, so that its characteristic width shrinks as w (t) exp(—At). The number of boxes of size l = W needed to cover such set of fluid elements can be estimated as... 

It is again a Gaussian with time-dependent height but characteristic width Id = y/D/X. At variance with the KiSS approach, the lateral scale of the filament is not controlled by some externally imposed size of the suitable water, but by the competition between diffusion and advection. It is somewhat surprising that the biological growth rate does not affect the filament width in this model. As we will see in Chapter 7 this feature is not shared by more realistic models. 

In eq 29, 1/k is the characteristic decay length for the pure water. As reference 22 pointed out, Sii(R) displays periodic oscillations with some wavenumber q, modulated by a decaying envelope of some characteristic width . Then Sti(()) has a maximum centered at Q = q% with a width -1. In this situation when 2 >> / (k2 + ql)1 2 (22), the main contribution to the integral is given by Q — q, so that we obtain... 

A possible absorption spectrum for a molecule near its unimolecular dissociation threshold is shown in figure 8.1. Below the absorption lines for the molecular eigenstates are very narrow and are only broadened by interaction of the excited molecule with the radiation field. However, above the excited states leak toward product space, which gives rise to characteristic widths for the resonances in the spectrum. Since the line widths do not overlap, the resonances are isolated. Each... 

The second scan, depicted in the lower half of Fig. 27, shows a Qj scan with Q l = 1.0. The solid line is a fit of the data to a Gaussian line shape of characteristic width equal to 0.033 A Both the Gaussian line shape and its width were found to be in agreement with the Renn and Lubensky model of the TGBA phase. 

Another interesting comparison is with the optical absorption tail. In principle, the optical absorption coefiicient is a convolution of the valence-band density of states with the conduction-band density of states multiplied by a matrix element. However, if the band tails are exponential and one band tail is broader than the other, an elementary mathematical analysis shows that the optical absorption tail has the same energy dependence as the broader band tail, with the energy dependence of the matrix element neglected. In our picture of the electronic structure of a-Si H, the valence-band tail is broader, and hence the characteristic width of the absorption tail should be compared with the width of the valence-band tail ( 42 meV). The optical absorption tail for material prepared under conditions similar to the... 

Fic. 18. Composite optical absorption spectrum for a-Si H determined from optical transmission, pbotoacoustic deflection, and photoconductivity measurements. The linear fits to the data indicate exponential absorption lges with characteristic widths of 48 and 60 meV. [Reprinted with permission from Solid State Communications, C. B. Roxlo, B. Abeles, C. R. Wronski, G. D. Cody, and T. Tiedje, Comment on the optical absorption edge in a-Si H, Copyright 1983, Pergamon Press, Ltd.]... 

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