Behavior at Large

At large distances from the nucleus, the potential energy term as well as the angular momentum term of the radial Dirac equation becomes small, and the equation approaches the limiting form  

This pair of equations is easily seen to be satisfied by exponential solutions, and in fact at large r the behavior of the solution is dominated by the exponential term. The behavior of the solutions—bound or continuum—depends on X from (7.32)  

It is interesting to note that X is related to the product of the large and small component relative normahzation factors, 

The two points at which A. = 0 correspond to = 0, where the small component vanishes, and E = -2mc, where the large component vanishes. 

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In an attempt to describe the behavior at large chain deformations, de Gennes [7] incorporated into the dumbbell model the FENE spring law along with a variable bead friction coefficient which increases linearly with the interbead distance ... 

GL 1] [R 1] [R 3] [P le] The falling film micro reactor has a better selectivity-conversion performance than the two micro bubble columns tested (Figure 5.18) (3, 38]. The micro bubble column with narrow channels has a better behavior at large conversion than the version with wide channels. The behavior of the falling film micro reactor and the micro bubble column with narrow channels is characterized by a nearly constant selectivity with increasing conversion, while the bubble column with wide channels shows notably decreasing selectivity with conversion (similar to the laboratory bubble column). 

The two-electron integrals (Equation 6.32) are determined from atomic experimental data in the one-center case, and are evaluated from a semiempirical multipole model in the two-center case that ensures correct classical behavior at large distances and convergence to the correct one-center limit. Interestingly, this parameterization results in damped effective electron-electron interactions at small and intermediate distances, which reflects a (however less regular) implicit partial inclusion of electron correlation (Thiel, 1998). In this respect, semiempirical methods go beyond the HF level, and may accordingly be superior to HF ab initio treatments for certain properties that have a direct or indirect connection to the parameterization procedure. 

Figure 7 Ground-state crossover, U(AC,N), of the modified Hubbard model (36), with periodic and antiperiodic boundary conditions for N = An and An + 2, respectively. The dashed lines are extrapolations to the infinite chain and rigorously passes through the origin. The inset shows the behavior at large site-energy A the limit A —> oo corresponds to the restricted basis in which D+2 and A-2 sites are excluded[97].

Figure 3. Interaction free energy in the discrete approach as a function of the separation distance between surfaces, calculated in the discrete model for various values of g (a, top) linear scale (which shows better the oscillatory behavior near the surface (b, bottom) logarithmic scale (which better reveals the behavior at large distances).

According to Holtzer [16], the Rayleigh ratio of rigid, rod-like scattering particles reaches a linear asymptotic behavior at large values of qL, given by ... 

For burning in a reactive atmosphere, conditions may be encountered for which the analysis of the diffusion-flame regime is relevant. Under suitable conditions there are two thin reaction sheets, an inner one at which premixed decomposition of the monopropellant occurs and an outer one having a diffusion-flame character. Categorizations are available for potential limiting behaviors at large Damkohler numbers in reactive environments [206]. There are many flame-structure possibilities, not all of which have been analyzed thoroughly. 

Valuable information on crazes can be derived from their scattering behavior at large values of s. Following a derivation given by Porod it may be shown that a two-phase system with sharp phase boundaries and a cylindrically symmetric correlation function obeys the asymptotic law... 

Figure 3.2 Trouton ratio, Tr, of uniaxial extensional viscosity to zero-shear viscosity jq after start-up of steady uniaxial extension at a rate of 1 sec i for a Boger fluid consisting of a 0.185 wt% solution of flexible polyisobutylene (Mu, = 2.11 x 10 ) in a solvent composed mostly of viscous polybutene with some added kerosene (solid line). The dashed line is a fit of a multimode FENE dumbbell model, where each mode is represented by a FENE dumbbell model, with a spring law given by Eq. (3-56), without preaveraging, as described in Section 3.6.2.2.I. The relaxation times were obtained by fitting the linear viscoelastic data, G (co) and G"(cu). The slowest mode, with ri = 5 sec, dominates the behavior at large strains the best fit is obtained by choosing for it an extensibility parameter of = 40,000. The value of S — = 3(0.82) n/C(x, predicted from the...

We can formulate the radiation conditions for elastic waves as certain requirements for the field behavior at large distances from the source region CV. These requirements should ensure that the Kirchhoff integral over the sphere Or in formula (13.207) goes to zero as r —> cx). 

It is also important to reproduce the correct behavior at large distances where a description in terms of collective coordinates is appropriate. The... 

Here we obtain A/U > 0 whenever + sBB > 2sAB and Aab < 0 whenever saa + sbb < 2sab. It is tempting to conclude that in these two cases we shall obtain stable and unstable mixtures, respectively. However, we must remember that in the limit of 0, Pt = PP and Pt -ab = P2PVint[eAA + sBB — 2 /U must be small. Therefore, one cannot predict the behavior at large values of Pt ab Thus, from the above discussion one can predict the occurrence of positive or negative deviations from SI solutions. But since in this limit P — 0, and Pt 0 we also expect Pt -ab to be small, therefore we must have miscibility in the entire region of compositions. 

Figure 2. Surface pressure-area isotherm for fatty acids and phospholipids. The inset shows the behavior at large molecular areas. Four one-phase regions are identified gaseous, liquid expanded, liquid condensed, and solid. This isotherm is typical of pentadecanoic acid at room temperature. For a higher-molecular weight acid, such as octadecanoic acid, there is no LE region at room temperature the isotherm does not rise steeply until an area of about 25 A. ...

To be fair, there are several well-known exact conditions that the LSD approximation does not get right it is not self-interaction free[37], v (r) does not have the correct — 1/r behavior at large r for finite systems[38], it does not contain the integer discontinuity[39—41], etc. These shortcomings may be overcome by other improvements[42], but not by the gradient corrections discussed in this article. 

When R is sufficiently large, the interaction potential can be expanded in terms of Since spin multiplicities of the two approaching atoms are different, the asymptotic behavior at large R is exponential, i.e.,... 

In the limit of large t both Gs and Gd become independent of r, while the behavior at large r is the same as that at large t. It follows that... 

Rubber becomes harder to deform at large strains, probably because the long flexible molecular strands that comprise the material cannot be stretched indefinitely. The strain energy functions considered up to now do not possess this feature and therefore fail to describe behavior at large strains. Strain-hardening can be introduced by a simple modification to the first term in Eq. (1.18), incorporating a maximum possible value for the strain measure J, denoted Jm (Gent, 1996) ... 

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