Spherical distribution

We can see, in Figure 51, that the spectra form a spherical cloud in this 3-dimensional subset of the absorbance data space- In other words, this data is isotropic. No matter in which direction we look, we will see no significant (in the statistical sense of the word) difference in the distribution of the data points. If we were able to show the plot for all 10 dimensions, we would see a 10-dimensional hyperspherical cloud that is isotropic within the spherical distribution of points. 

Provided that the bulk concentrations remain constant, the flux J of Eq. (9-3) is also constant. Consider a spherical distribution of potential reactants B around a particular molecule of A. The surface area of a sphere at a distance r from A is 4irr2. Thus, the expression for the flow of B toward A is... 

Ligand field theory mainly considers the last contribution. For this contribution the geometric distribution of the ligands is irrelevant as long as the electrons of the central atom have a spherical distribution the repulsion energy is always the same in this case. All half and fully occupied electron shells of an atom are spherical, namely d5 high-spin and dw (and naturally d°). This is not so for other d electron configurations. 

Figure 5.10 Representation of the formation of the lone pair in the PF3 molecule, (a) An isolated P3 + ion consisting of a P5+ core surrounded by two nonbonding electrons in a spherical distribution, (b) Three approaching F ions distort the distribution of the two valence shell electrons pushing them to one side of the P5+ core, (c) When the F ligands reach their equilibrium positions, the two nonbonding electrons are localized into a lone pair, which acts as a pseudo-ligand giving the PF3 molecule its pyramidal geometry.

Unlike the lanthanides, the actinides U, Np, Pu, and Am have a tendency to form linear actinyl dioxo cations with formula MeO and/or Me02. All these ions are paramagnetic except UO and they all have a non-spherical distribution of their unpaired electronic spins. Hence their electronic relaxation rates are expected to be very fast and their relaxivities, quite low. However, two ions, namely NpO and PuOl", stand out because of their unusual relaxation properties. This chapter will be essentially devoted to these ions that are both 5/. Some comments will be included later about UOi (5/°) and NpOi (5/ ). One should note here that there is some confusion in the literature about the nomenclature of the actinyl cations. The yl ending of plutonyl is often used indiscriminately for PuO and PuOl and the name neptunyl is applied to both NpO and NpOi. For instance, SciFinder Scholar" makes no difference between yl compounds in different oxidation states. Here, the names neptunyl and plutonyl designate two ions of the same 5f electronic structure but of different electric charge and... 

Nuclei with spin I > are not, as a rule, perfectly spherical distributions of charge, as may be shown by quite general quantum mechanical symmetry considerations (89). All nuclei possess the spin axis as a sym-... 

Lang s result from the numerical theory is in excellent agreement with the general theory in the previous section. If both tip state and sample state are. 9-wave states, the tunneling conductance should have a spherical distribution, and the apparent radius should equal the nominal distance between the two nuclei. The radius obtained from the figure fits well with this expectation. 

The first term is the potential caused by the spherically distributed charge Q, the second term is the potential caused by redistribution of charge Q in response to the nonhomogeneous field of point charges —Q/n, and the third is the ordinary coulombic potential caused by the charges —Q/n. The potential Vl(r,0,[Pg.205]

The pictorial representation of radial and spherical distribution functions for values of n= 1, 2, 3 are shown in Figure 2.4. 

Radial and spherical distribution functions for atomic orbitals. 

For example, in the hydrogen molecule-ion in its equilibrium configuration we may say that the electron is distributed in a way equivalent to having 3/7 of the electron spherically distributed about each nucleus and 1/7 at the midpoint between the two nuclei, giving a force of attraction by the electron for each nucleus that balances the force of repulsion by the other nucleus. 

All systems studied in this way until now have a spherical distribution of the non-bonding electrons, so no account need be taken of the symmetry relationships between the ground state and the possible transition states. Consequently, the size of the reaction centre, the coordination number and the effective size and packability of the ligands determine whether the mode of activation is associative or dissociative. There is, as yet, no suitable evidence to decide whether the nucleophilicity of the entering group plays a significant part in deciding the mode of activation. 

If one considers an elementary model of a metal consisting of a latlice of fixed positive ions immersed in a sea of conduction electrons that are free to move through the lattice, every direction of electron motion will be equally probable. Since the electrons fill the available quantized energy states staring with the lowest, a three-dimensional picture in momentum coordinates will show a spherical distribution of electron momenta and, hence, will yield a spherical Fermi surface. In this model, no account has been taken of the interaction between Ihe fixed posilive ions and the electrons. The only restriction on the movement or "freedom" of the electrons is the physical confines of the metal itself. 

In the absence of an external magnetic field the Zeeman Hamiltonian provides zero energy and all the 11, Mi) levels (termed as / manifold) have the same energy. However, this may not be true for nuclei with / > V2. In this case, the non-spherical distribution of the charge causes the presence of a quadrupole moment. Whereas a dipole can be described by a vector with two polarities, a quadrupole can be visualized by two dipoles as in Fig. 1.11. 

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