Displacements, radial

The radial interference, 5, necessary to achieve pressure P may be calculated from the radial displacements Uj and Ujj generated during assembly, assuming that the shrinkage is carried out without generating an axial stress in either component. 

As indicated earlier, protective oxide scales typically have a PBR greater than unity and are, therefore, less dense than the metal from which they have formed. As a result, the formation of protective oxides invariably results in a local volume increase, or a stress-free oxidation strain" . If lateral growth occurs, then compressive stresses can build up, and these are intensified at convex and reduced at concave interfaces by the radial displacement of the scale due to outward cation diffusion (Fig. 7.7) . 

Figure 4 [29] shows the (s) versus profiles for potato amylose and the amylose/amylopectin mixture from wheat starch corresponding to the concentration versus radial displacement data of Fig. 3. The s data used in the concentration dependence plot of Fig. 3 for wheat amylopectin comes from (s) vs. s analysis data of Fig. 2b and similar. The concentrations shown in the abscissa in Fig. 4 have been obtained from the total starch loading concentration normalised by the weight fraction of the amylopectin component estimated from the (s) vs. s profiles.

Fig. 4 Sedimentation velocity g (s) profiles for starch polysaccharides using DCDT+. The profiles correspond to the radial displacement plots of Fig. 2. a Potato amylose, sample concentration 8 mg/ml in 90% in dimethyl sulphoxide. Rotor speed was 50 000 rpm at a temperature of 20 °C. b Wheat starch (containing amylose, left peak and the faster moving amylopectin, right peak), (total) sample concentration 8 mg/ml in 90% dimethyl sulphoxide. Rotor speed was 35 000 rpm at a temperature of 20 °C. From [29]...

One can see the M procedure has a parallel to either g (s) vs. s or c(s) vs. s in sedimentation velocity where the data are transformed from radial displacement space [concentration, c(r) versus r] to sedimentation coefficient space [g s) or c(s) versus s]. Here we are transforming the data from concentration space [concentration relative to the meniscus j(r) versus r] to molecular weight space [M r) versus r]. 

In the REC model, the ligand is modelled through an effective point charge situated in the axis described by the lanthanide-coordinated atom axis, at a distance R, which is smaller than the real metal-ligand distance (Figure 2.6). To account for the effect of covalent electron sharing, a radial displacement vector (Dr) is defined, in which the polar coordinate R is varied. At the same time, the charge value (q) is scanned in order to achieve the minimum deviation between calculated and experimental data, whereas 9 and cp remain constant. 

It is assumed here that the axial displacements are independent of the radial position, and the stress components in the radial and circumferential directions are neglected for Eqs. (4.8) and (4.9). Also, the radial displacement gradient with respect to the axial direction is neglected compared to the axial displacement gradient with respect to the radial direction in Eq. (4.10). Combination of Eqs. (4.10) and (4.16) for the boundary condition of the axial displacement continuity at the bonded interface (i.e. i/ (a,z) = u z)) and integration gives ... 

This equation incorporates Darcy s Law for fluid flow as well as conservation of mass. The position Z is related to the original fiber position r by the radial displacement u (Eq. 13.10)... 

With an appropriate (quadratic) field profile E(r), where r is the radial displacement, it might be exploited to focus an electron beam. Quite sufficient for this purpose would be, for example, the transverse laser mode TEMoi,... 

Consider the following perfectly conventional procedure that assumes that we know what is meant by a given radial displacement, say, R. On a sufficeintly large scale (e.g., > 108 lightyears), we can reasonably assume that it is possible to write down a relationship describing the amount of mass contained within a given spherical volume, for instance... 

We calibrate the radial displacement parameter so that it coincides with the geodesic radius, and find the remarkable result that, on sufficiently large scales, the calibrated radius of a sphere centered on the chosen origin in the model universe then varies as the square root of the mass contained within the sphere. 

Using Jl = dt)M + m again, the mass distribution function can be expressed in terms of the invariant radial displacement as... 

Here, B is the magnetic field strength, mc is the critical mass (40), q is the charge of the ion in question, rm is the maximum radius required for ejection, a is the inside length of the cubic cell, r is a radial displacement, V-p is the voltage applied to the trap plates, and a = 1.3871 (49) a geometry constant. 

Charge transfer (CT) excited states are characterized by a radial displacement of charge from one group or atom to another. In arylboranes (Figure 6.18), CT excited states are responsible for the compound s luminescence.50 The reverse solvatochro-mism of the fluorescence and the CT absorption bands appear to be related to an inversion of the dipole moment in the electronic transition, S(l S1 that is, from the... 

A further property associated with the radial displacement of charge associated with CT electronic transitions is a change in the dipolar moment of the molecule. If the electronic transition causes, for example, an increase in the dipolar moment, the energy of the CT excited state will decrease (other factors aside) with the polarity of the solvent. Therefore, the CT absorption bands will experience solvatochromic shifts of tens of nanometers. Related solvatochromic effects will be detected in the emission spectrum of CT excited states. While the solvatochromism of absorption bands is a tool for the assignment of CT transitions in the absorption spectrum of complexes, the rationalization of such effects in terms of the solvent properties, for example, the dielectric constant, is not always possible. 

Redox Photochemistry of MLCT The existence of intense absorption bands in the spectra of MZL complexes where the redox potentials of the M1 /M1 couples, for example, Cu(I), Fe(II), and L/L , are not too negative was associated by spectroscopists with MLCT electronic transitions.125 128 The MLCT electronic transition causes a radial displacement of charge from the metal to the ligand. There... 

Similarly, the total radial displacement on the boundary surface is given by... 

In chemical compounds and minerals containing transition elements, Racah B parameters are decreased relative to the ffee-ion values. This implies that both the mean radial displacement of 3d electrons has increased and the effective charge experienced by the electrons has decreased when a transition metal is bonded to ligands in a coordination site. Since the Racah B parameter is always smaller than the free-ion value it is used as a qualitative measure of bond covalency. 

Figure 10.7 Snapshots from a simulation of accretion in the inner Solar System, from O Brien et al. (2006). Jupiter and Saturn are present at t = 0 on their current orbits, black particles are embryos and gray particles are planetesimals. By 30 Myr all embryos and many of the planetesimals are cleared from the Asteroid Belt region (shown as the black line). The remaining asteroids are dynamically excited and have experienced significant radial displacement from their initial locations.

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