Nuclear proton density distribution

In those cases where particular selected nuclides (with their proton numbers Z and neutron numbers N) are to be modelled, their corresponding experimental rms radii a(Z, N) can be imposed on every suitable nuclear charge density distribution model (for experimental values of rms radii see, e.g., [7,35]). If, on the other hand, one is interested in studying trends depending on the nuclear mass number A or on the atomic number Z, an expression for the rms radius a as a function of these numbers is required. 

In this last section we mention a few cases, where properties other than the energy of a system are considered, which are influenced in particular by the change from the point-like nucleus case (PNC) to the finite nucleus case (FNC) for the nuclear model. Firstly, we consider the electron-nuclear contact term (Darwin term), and turn then to higher quantum electrodynamic effects. In both cases the nuclear charge density distribution p r) is involved. The next item, parity non-conservation due to neutral weak interaction between electrons and nuclei, involves the nuclear proton and neutron density distributions, i.e., the particle density ditributions n r) and n (r). Finally, higher nuclear electric multipole moments, which involve the charge density distribution p r) again, are mentioned briefly. 

In certain regions of the density-temperature plane, a significant fraction of nuclear matter is bound into clusters. The EOS and the region of phase instability are modified. In the case of /3 equilibrium, the proton fraction and the occurrence of inhomogeneous density distribution are influenced in an essential way. Important consequences are also expected for nonequilibrium processes. 

Another question we might pose to ourselves is whether the neutron and proton distributions in nuclei are the same Modern models for the nuclear potential predict the nuclear skin region to be neutron-rich. The neutron potential is predicted to extend out to larger radii than the proton potential. Extreme examples of this behavior are the halo nuclei. A halo nucleus is a very n-rich (or p-rich) nucleus (generally with low A) where the outermost nucleons are very weakly bound. The density distribution of these weakly bound outermost nucleons extends beyond the radius expected from the R °c A1 /3 rule. Examples of these nuclei are nBe, nLi, and 19C. The most well-studied case of halo nuclei is 1 Li. Here the two outermost nucleons are so weakly bound (a few hundred keV each) as to make the size of 11 Li equal to the size of a 208Pb nucleus (see Fig. 2.12). 

As mentioned above, the function Pmic(r) in the PNC Hamiltonian is a weighted nuclear density function, with the weighting emphasizing the neutron density. Since there are no experimental values for the neutron density of Cs, we use instead an experimental proton density function. This proton density is taken to be a two-parameter Fermi distribution [45]... 

The properties of the diatomic hydrides MH can be derived from the density distribution of the united-atom approximation (see e. g. (79, 80) for the potential-energy-curves) of the neutral united atom (M-f 1). Platt (28, 98) used this approximation to calculate the frequencies via the force constants of these diatomic hydrides through purely electrostatic arguments. To make the hydride from the united atom one proton is removed from the nucleus without distortion of the charge cloud of the united atom (i. e. the removal of the proton is considered to be a small perturbation). If ro is the radius at which the effective nuclear charge is unity, the proton should be moved to ro. i- e- to the radius beyond which lie a total number of 1.00 electrons. If the proton is replaced outside ro, the excess electrons will attract it, and it will be repelled if it is situated inside the nucleus. This is just the situation for the vibrating motion of a proton attached to another atom through a stable chemical bond a small displacement (-i-iir) from ro will subject the proton to an added force... 

Results for the hole spectral function calculated for the Reid interaction [43] at normal nuclear matter density are shown in Fig. 4 [35]. The three curves correspond to k = 0.48, 0.79, and 0.93 fm" respectively. As discussed above, the expected quasiparticle features of the strength distribution become more pronounced when the sp momentum is closer to the Fermi momentum. The background strength outside the peak of these spectral functions contains about 10% of the sp strength. This number is very similar to the results obtained in finite nuclei and the experimental result for the 3si/2 proton orbital in ° Pb. 

Electrons and photons scatter through a Coulomb field interaction. Therefore, scattering experiments with these particles are studying the distribution of protons. The possibility exists that neutrons and protons do not have the same distribution in nuclei. Unfortunately, there are no convenient weak interactions of a neutron which would be useful for studying the spatial distributions of neutrons. As is discussed in the section on form factors (Sect. 7) it is not simple to interpret in terms of nuclear radii those measurements which are made with particles which interact strongly with neutrons and protons. Whether neutrons and protons have the same density distributions may be answered by theoretical developments rather than experimental advances. 

Next we consider the problem of extracting model independent nuclear structure information from analyses of medium energy proton-nucleus elastic scattering data. Spedfically, we have in mind the ground state neutron density distributions. Studies of this type are plentiful in the literature and will not be reviewed here. The reader may refer to refs. [Ba 87, Ra79, Ra 81aj. 

Recall that X rays are diffracted by the electrons that surround atoms, and that images obtained from X-ray diffraction show the surface of the electron clouds that surround molecules. Recall also that the X-ray diffracting power of elements in a sample increases with increasing atomic number. Neutrons are diffracted by nuclei, not by electrons. Thus a density map computed from neutron diffraction data is not an electon-density map, but instead a map of nuclear mass distribution, a "nucleon-density map" of the molecule (nucleons are the protons and neutrons in atomic nuclei). 

Differences resulting from nonisotropic electron distribution are significant only for H-X bond lengths X-rays see electrons rather than nuclei, and the simplest inference of a nuclear position is to place it at the center of a sphere whose surface is defined by the electron density around it. However, since a hydrogen atom has only one electron, for a bonded hydrogen there is relatively little electron density left over from covalent sharing to blanket the nucleus, and so the proton, unlike other nuclei, is not essentially at the center of an approximate sphere defined by its surrounding electron density ... 

Vanadium-51 is a spin 7/2 nucleus, and consequently it has a quadrupole moment and is frequently referred to as a quadrupolar nucleus. The nuclear quadrupole moment is moderate in size, having a value of -0.052 x 10 2S m2. Vanadium-51 is about 40% as sensitive as protons toward NMR observation, and therefore spectra are generally easily obtained. The NMR spectroscopy of vanadium is influenced strongly by the quadrupolar properties, which derive from charge separation within the nucleus. The quadrupole moment interacts with its environment by means of electric field gradients within the electron cloud surrounding the nucleus. The electric field gradients arise from a nonspherical distribution of electron density about the... 

The layer of decreasing density is about 2.5 fm, independently of the atomic number. The distribution of the neutrons is assumed to be approximately the same as that of the protons. Then the mass distribution in the nucleus is also the same as the charge distribution, and it follows from eq. (3.1) that the density of nuclear matter in the interior of the nuclei is given by... 

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