Continuum range

In an electron-excited X-ray spectrum the discrete X-ray lines are superimposed on a continuous background this is the well-known bremsstrahlung continuum ranging from 0 to the primary energy Eq of the electrons. The reason for this continuum is that because of the fundamental laws of electrodynamics, electrons emit X-rays when they are decelerated in the Coulomb field of an atom. As a result the upper energy limit of X-ray quanta is identical with the primary electron energy. 

Cheng and coworkers have examined the photodissociation spectroscopy of MgCH4 in detail . The photofragmentation action spectrum has a broad featureless continuum ranging from 310 to 342 nm, with a maximum at 325 nm. In this region the channels observed are nonreactive (equation 36, ca 60%), H abstraction (equation 37, ca 1%) and CH3 abstraction (equation 38, ca 33%). Recent theoretical calculations on the C—H bond activation in MgCH4+ reveal that the formation of the insertion intermediate, CH3MgH, proceeds via a three-centered transition state . ... 

Liver disease is a continuum, ranging from abnormalities of liver function tests found on routine biochemical screening, with no adverse clinical consequences, to severe end-stage liver failure. 

All the situations between these two cases can be encountered with selective modification of the steric barrier at 0° compared to the one at 180° or by specific electronic stabilization of one stereoisomer. One interesting situation along the continuum ranging between the two extreme cases, obtained from suitable substitution and -contribution, can result in the definition of two identifiable barriers (a steric one and a n one) of rather similar heights (77JA4526). 

Humic substances occur in every natural water sample which has been analyzed for their presence. The amount and composition of humic substances vary considerably from soils, surface waters, and groundwaters, but their ubiquity in water is without question. Humic substances in water occur as a size continuum ranging from dissolved through colloidal, to particulate phases. The dissolved phase is a predominant phase in most streams and is the phase emphasized in this chapter. The geochemical activity and reactivity of dissolved and particulate organic phases are thought to be of sufficient difference in magnitude to merit the separation based on size at 0.45 /xm. 

It should be noted that the decay rate is proportional to Noo and not to N, as in the case of coagulation by Brownian motion in the continuum range. Integrating from the initial stale for which N = NgoiO) at f = 0, we obtain... 

For Brownian coagulation in the continuum range, the collision frequency function is given by (7.16). Substitution of the similarity form (7.69) reduces the coagulation equation for the continuous distribution (7.67) with (7.16) to the following form ... 

The rate of heterogeneous condensation depends on the exchange of matter and heat between a particle and the continuous phase. The extreme cases of a particle much larger or much smaller than the mean free path of the suspending gas are easy to analyze. In the continuum range (dp ip), diffusion theory can be used to calculate the transport rate. For a single sphere in an infinite medium, the steady-state equation of diffusion in spherical coordinates takes the form... 

As an example, this result can be substituted in (10.19) for growth by diffusion in the continuum range ... 

When growth is limited by gas-phase transport, the rate can be determined from the expressions derived in the previous section. For the continuum range, the growth law based on ( 0,I9) is... 

Examples of growth laws including those limited by chemical reaction in the aerosol phase are summarized in Table 10.3. The growth rate dvjdt is proportional to dp for diffusion in the continuum range and to dp for droplet phase chemical reaction. Different... 

In Fig. 11.5. the data of Fig. 11.3 have been replotted in the self-preserving form. As a good approximation, all the data fall on a single curve. The theory for the continuum range... 

The densities of vibrational states for the two potentials (Fig. 15) are actually very similar, exeept for a somewhat wider gap for RIMl between the two peaks at higher frequencies, corresponding to inner modes of CO3. As for the Kieffer s model, the cut off frequencies of acoustic modes were derived from elastic constants by the Voigt-Reuss-Hill approximation[38], amounting to 51, 66 and 90 cm"i. An optic continuum ranging from 113 to 287 cm was used, and four Einstein oscillators at 708,867, 1042 and 1470 cm with appropriate weights represented the internal optical modes. The... 

The spectrum of L(V) depends on the intermolecular potential of the gas molecules. For hard-sphere molecules the spectrum of L has a discrete and a continuous part, with the continuum ranging from — m to —00 with m a positive constant. If the potential is repulsive and varies with r as (r) = Kr with 5 >2, then the spectrum is discrete. For the special case s = 4, the so-called Maxwell molecules, the eigenvalues and eigenfunctions of L are completely known. However, nothing is known about the spectrum of L... 

The director deformations described by that do not lead to layer compressions, in the continuum range where the wavelengths A of the deformation are much larger than the molecular dimensions (A 10 nm) can be induced by stress K 27t/pf <10T N/m. This is usually smaller than of the layer compression modulus B l(f N/m , For this reason, deformations that do not lead to layer compression (such as splay in SmA) are usually called soft deformations, whereas those that require layer compression (such as bend and twist in SmA) are the so-called hard deformations. In SmC there will be six soft and three hard deformations, so it is basically impossible to take into account all elastic terms while keeping the transparent physics. (In the chiral smectic C materials, additional three terms are needed, as shown by de Gennes. ) Fortunately, however, the larger number of soft deformations enable for the material to avoid the hard deformations, which makes it possible to understand most of the elastic effects, even in SmC materials. 

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