Spherical co-ordinates

Added February 10, 1927.—J. H. Van Vleck in Proc. Nat. Acad. America, vol. 12, p. 662 (December, 1926), has discussed the mole refraction and the diamagnetic susceptibility of hydrogen-like atoms with the use of the wave mechanics, obtaining results identical with our equations (24) and (34). He also considered the effect of the relativity corrections (which is equivalent to the effect of a central field) and concluded that equation (24), derived by the use of parabolic instead of spherical co-ordinates, is not invalidated.]... 

In Eq. (22), r is the spherical co-ordinate starting at the center of the drop. The variable represents the convective velocity due to the moving surface of the expanding drop. Within a reference framework centered on the moving drop surface, this is given by... 

Firstly, a point dipole placed at the surface of a glass substrate is considered. The orientation of the dipole is considered to be random. The angular distribution plotted in spherical co-ordinates is shown in Figure 12. The radiation propagating in the positive (down) and negative (up) z-directions corresponds to that radiated into the glass substrate and the environment (air), respectively. 

The Kf matrix is related to the conventional K matrix of MQDT by a frame transformation from parabolic to spherical co-ordinates the K matrix is then related by a further frame transformation to the quantum defects [43, 45], The first term in Eq. (4) gives a contribution to the photoionization intensity borrowed from the bound-state spectrum. The dlyom term represents direct photoionization, and the overall expression allows Fano-type interference between these terms. In Eq. (5) A is a phase shift in the parabolic rep-... 

Equation (l-3a) may be transformed into either cylindrical or spherical co ordinates by standard calculus techniques. The results are as follows ... 

Another ultrasonic absorption study of the rates of complexation-decom-plexation of 15-crown-5 and 18-crown-6 complexes for a series of bivalent cations in aqueous solution shows that non-spherically co-ordinated Hg + (similarly to the previously studied - Ag+) reacts faster with the crown ethers than spherically co-ordinated metal ions, which gives additional evidence for water loss as the rate-determining step. The more rigid crown ethers have to undergo considerable conformational strain to accommodate themselves to the linear geometry of silver(i) and mercury(ii), so the rigidity of the ligand is especially important for the rate of complexation with these metal ions. ... 

The optimum hybridization schemes for coordination compounds are summarized in Table 3. In Fig. 1 the hybridization characteristics of spherical co-ordination polyhedra are summarized together with those for polyhedra with... 

Fig. 6. The positions of the nodal lines in hybrids for spherical co-ordination polyhedra...

The occurrence of these non-bonding orbitals can be associated with either spherical or non-spherical co-ordination polyhedra. The choice of co-ordination polyhedra depends on several factors and notably ... 

The electric field intensity is connected with the flowing current density and is determined by vector Eq. (8-30). With spherical co-ordinates, the equation assumes the... 

Olbrich proposed representing the multinomials in spherical co-ordinates as points on an n-shphere using these proportions. Denote these points by respectively. Then... 

It can be more useful to use spherical co-ordinates [3] (see FIGURE 2). Any cylindrically symmetrical function with a centre of inversion can be developed into a series of even-order Legendre polynomials P2n, in a fashion similar to the way a periodic function is depicted by a Fourier series. The scattered intensity distribution over the reciprocal space can thus be represented as... 

As shown in Figure 12.13, we have performed the pressure dependence study of the Ge-Ge modes of the NCs for one pressure cycle and there is no hysteresis observed. This confirms that the pressure-induced strain in the NCs is reversible. With this elastic behavior of our Ge/Si02 NCs-matrix system, we assume both the NCs and the matrix as isotropic elastic continua [46]. We modeled the NC as a sphere of radius ri in a spherical Si02 matrix of radius t2, where r2 2> ri. The system is subjected to a hydrostatic pressure P. Using spherical co-ordinates with the origin at the center of the NC sphere, our system has a spherical symmetry where the displacement vector u is everywhere radial and can be written d u = ar + b/r. The components of the strain tensor are Urr = a — and uee = =... 

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