Atomic nucleus density

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

A high electron density surface (also called a bond electron density surface) shows the core of electron density around each atomic nucleus and regions where neighboring atoms share electrons (covalent bonding regions). 

In the early thirties of the last century Baade and Zwicky conjectured in their studies of supernova explosions that supemovae represent a transition from ordinary stars to compact objects, whose size is an order of magnitude smaller than the size of a white dwarf. At that time it was already known that the atomic nucleus consists of neutrons and it was clear that the density of the remnant objects must be of the same order as the nuclear density. Baade and Zwicky predicted that a supernova explosions will result in objects composed of closely packed neutrons (neutron stars). Prior to the beginning of the second World War (1939) a number of theoretical works by Landau, Oppenheimer, Volkoff and Snider showed, that indeed objects could exist with sizes about 10 km and masses about a solar mass. The density in these objects is about the nuclear saturation density and they basically consist of neutrons with a small amount of protons and electrons. The studies of neutron stars were subsequently stopped most likely due to the engagement of the nuclear scientists in the development of the nuclear bomb both in the West and the East. 

During collapse, which lasts only a split second, the temperature goes up to 10 billion K, as it was in the infant Universe when it was barely 1 second old. The density exceeds that of an atomic nucleus (10 g cm ). Compressed like a spring, the matter then bounces back, for compression went just a little too far. This abrupt return to expansion gives rise to a shock wave that moves back out through the star. 

Table 1 Values of the atomic electron density at the nucleus, p(0) evaluated with the present modified TFD method compared to HF values by means of the percent deviation (%). Also, the values of 2 Z tq are displayed where tq is the switching point among the quantum mechanical and the semiclassical description (see text).

The wavefunction of an electron associated with an atomic nucleus. The orbital is typically depicted as a three-dimensional electron density cloud. If an electron s azimuthal quantum number (/) is zero, then the atomic orbital is called an s orbital and the electron density graph is spherically symmetric. If I is one, there are three spatially distinct orbitals, all referred to as p orbitals, having a dumb-bell shape with a node in the center where the probability of finding the electron is extremely small. (Note For relativistic considerations, the probability of an electron residing at the node cannot be zero.) Electrons having a quantum number I equal to two are associated with d orbitals. 

The electrostatic Hellmann-Feynman theorem states that for an exact electron wave function, and also of the Hartree-Fock wave function, the total quantum-mechanical force on an atomic nucleus is the same as that exerted classically by the electron density and the other nuclei in the system (Feynman 1939, Levine 1983). The theorem thus implies that the forces on the nuclei are fully determined once the charge distribution is known. As the forces on the nuclei must vanish for a nuclear configuration which is in equilibrium, a constraint may be introduced in the X-ray refinement procedure to ensure that the Hellmann-Feynman force balance is obeyed (Schwarzenbach and Lewis 1982). 

Though rare, there are cases in which the total density shows minor maxima at non-nuclear positions. As all (3, — 3) critical points are attractors of the gradient field, basins occur which do not contain an atomic nucleus. These non-nuclear basins (which have been found in Si—Si bonds1 in Li metal, and some other cases, distinguish the zero-flux partitioning from other space partitioning methods. 

Whenever the atoms under consideration in a given molecule are in the same valence states as in the reference molecule, the relaxation process is such that the potential created by the other atoms at the kth nucleus is the same as would be predicted by leaving the pertinent intemuclear distances and the shapes of atomic electron densities as they are in the reference molecule, with the electron populations changed as required by the new situation. 

The sensitivity of an atomic nucleus in the NMR experiment is related to its gyromagnetic ratio y. It determines the energy difference AE between the precession states in a magnetic field of flux density B0 (Figs. 2.1 and 2.35) ... 

For 19F Fermi contact shift of fluorine bound to sp2 carbon atoms, spin density on the nucleus arises from spin polarization by as for analogous CC moieties, and from spin polarization by /Op, which occurs via direct delocalization through C—F ji bonding (Fig. 2.18). The hyperfine coupling is therefore... 

Fig. 5.40 The distribution of electron density (charge density) p for an atom the nucleus is at the origin of the coordinate system, (a) Variation of p with distance from the nucleus. Moving away from the nucleus p decreases from its maximum value and fades asymptotically toward zero, (b) Variation of — p with distance from the nucleus —p becomes less negative and approaches zero as we move away from the nucleus. The —p picture is useful for molecules (Fig. 5.41) because it makes clearer analogies with a potential energy surface, (c) A 4-D picture (p vs x, y, z) of the variation of p in an atom the density of the dots (number of dots per unit volume) indicates qualitatively electron density p in various regions...

The electronic configuration of free atoms is an important factor in the interpretation of atomic spectra, but less so for the understanding of chemical behaviour. Chemistry happens in crowded environments, which means that atomic electron densities fades to zero far from infinity. SCF wave functions are therefore not appropriate for atoms in a chemical environment. More suitable wave functions are obtained by terminating the SCF calculations at some fixed distance p from the nucleus, rather than infinity. The effect of such a new boundary condition is like applying hydrostatic pressure to the atom. 

When we are dealing with simple atoms as substituents, these effects are straightforward and more or less additive. If we go on adding electronegative chlorine atoms to a carbon atom, electron density is progressively removed from it and the carbon nucleus and the hydrogen atoms attached to it are progressively deshielded. 

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