Transform, spherical

Figure 5.14 a) The formation of linear hydrogen bond chains transforms spherical micelles to micellar disks with hydrophobic surfaces (see text). 

When R is very small, the third term on the right side of (8.24) is dominant and X is positive. Consequently, spontaneous formation of transformed spherical particles is ruled out unless the interface energy Us is very small. 

Figure 7 The transformation spherical micelles cylindrical micelles —> linear assem-...

Consider an alchemical transformation of a particle in water, where the particle s charge is changed from 0 to i) (e.g., neon sodium q = ). Let the transformation be performed first with the particle in a spherical water droplet of radius R (formed of explicit water molecules), and let the droplet then be transferred into bulk continuum water. From dielectric continuum theory, the transfer free energy is just the Born free energy to transfer a spherical ion of charge q and radius R into a continuum with the dielectric constant e of water ... 

High-resolution transmission electron microscopy (HREM) is the technique best suited for the structural characterization of nanometer-sized graphitic particles. In-situ processing of fullerene-related structures may be performed, and it has been shown that carbonaceous materials transform themselves into quasi-spherical onion-like graphitic particles under the effect of intense electron irradiation[l 1],... 

Fig. 3. Schematic illustration of the growth process of a graphitic particle (a)-(d) polyhedral particle formed on the electric arc (d)-(h) transformation of a polyhedral particle into a quasi-spherical onion-like particle under the effect of high-energy electron irradiation in (f) the particle collapses and eliminates the inner empty space[25j. In both schemes, the formation of graphite layers begins at the surface and progresses towards the center.

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

Since the vector g is represented above in terms of the g-coordinate system (i i is) having — g as the i3 axis, it is necessary to determine the transformation to the (iI,iJ/,i2) coordinate system in which the particle velocities are written, in order to evaluate certain integrals. If we let be the spherical coordinate angles of the vector v2 — vlt in the v-coordinate system, then ... 

The thermal transformation of laminar cementite in the pearlite phase of carbon steel to spherical grains. 

Solving equation (1-8) (using Laplace transform techniques) yields the time evolution of the current of a spherical electrode ... 

According to Eq. (2.13), the spectra we are interested in are given by a Fourier transform of the orientational correlation functions of the corresponding order Similarly to what was done in Chapter 3, the correlation functions for linear and spherical molecules may be represented as a superposition of the partial (marginal) components... 

Here D(Q) = D(a,f, y), Euler angles a, (5 and y being chosen so that the first two coincide with the spherical angles determining orientation e = e(j], a). Using the theorem about transformation of irreducible tensor operators during rotation [23], we find... 

Temkin S. I., Suvernev A. A., Burshtein A. I. Pressure transformation of the Q-branch of the CARS spectrum of spherical molecules. Opt. ... 

The geometry and surface chemistry of the dendrimer assemblies can be varied through the addition of surfactants. These dendrimer/surfactant aggregates can be tuned to template the formation of the different phases of calcium carbonate [40]. In combination with hexadecyltrimethylammonium bromide (CTAB), small spherical aggregates were formed that induce the formation of vaterite. Over a period of five days, the vaterite was transformed into calcite. The use of the negatively charged surfactant, sodium dodecylsulfonate (SDS), result-... 

The UPS indicated structure change is associated with size reduction as the discontinuous gold film is transformed into rod-shape and spherical particles with size of 5-10 nm. Accordingly, with size reduction the activity displayed in CO oxidation is also altered the rate increased from 6.7 X 10 to 2 X 10 molmin cm . Consequently, not only the gold-reducible oxide interaction is responsible for the increased activity, but also size reduction. Indeed, small clusters themselves are able to activate the reaction components shown by theoretical calculations performed for 10-15-atom clusters, which can activate easily oxygen [177,200], but in real catalyst, even at the smallest active ensemble, it consists of a few hundreds atoms. 

The atomic PDF is related to the probability to find a spherical shell around a generic atom (scattering center) in the material - it is defined as G(r) = Anp[p r)-p(, where p r) and po are, respectively, the local and average atomic number densities and r the radial distance. G(r) is the Fourier transform of the total structure factor Sid). ... 

Our next objective is to find the analytical forms for these simultaneous eigenfunctions. For that purpose, it is more convenient to express the operators Lx, Ly, Zz, and P in spherical polar coordinates r, 6, q> rather than in cartesian coordinates x, y, z. The relationships between r, 6, q> and x, y, z are shown in Figure 5.1. The transformation equations are... 

Equation (6.12) cannot be solved analytically when expressed in the cartesian coordinates x, y, z, but can be solved when expressed in spherical polar coordinates r, 6, cp, by means of the transformation equations (5.29). The laplacian operator in spherical polar coordinates is given by equation (A.61) and may be obtained by substituting equations (5.30) into (6.9b) to yield... 

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