Nuclear masses models

Nearly all kinetic isotope effects (KIE) have their origin in the difference of isotopic mass due to the explicit occurrence of nuclear mass in the Schrodinger equation. In the nonrelativistic Bom-Oppenheimer approximation, isotopic substitution affects only the nuclear part of the Hamiltonian and causes shifts in the rotational, vibrational, and translational eigenvalues and eigenfunctions. In general, reasonable predictions of the effects of these shifts on various kinetic processes can be made from fairly elementary considerations using simple dynamical models. 

There are several possible ways of introducing the Born-Oppenheimer model " and here the most descriptive way has been chosen. It is worth mentioning, however, that the justification for the validity of the Bom-Oppenheimer approximation, based on the smallness of the ratio of the electronic and nuclear masses used in its original formulation, has been found irrelevant. Actually, Essen started his analysis of the approximate separation of electronic and nuclear motions with the virial theorem for the Coulombic forces among all particles of molecules (nuclei and electrons) treated in the same quantum mechanical way. In general, quantum chemistry is dominated by the Bom-Oppenheimer model of the theoretical description of molecules. However, there is a vivid discussion in the literature which is devoted to problems characterized by, for example, Monkhorst s article of 1987, Chemical Physics without the Bom-Oppenheimer Approximation... ... 

As we learned in Chapter 2, it is necessary to include shell effects in the liquid drop model if we want to get reasonable values for nuclear masses. Similarly, we must devise a way to include these same shell effects into the liquid drop model description of the effect of deforming nuclei. Strutinsky (1967) proposed such a method to calculate these shell corrections (and also corrections for nuclear pairing) to the liquid drop model. In this method, the total energy of the nucleus is taken as the sum of a liquid drop model (LDM) energy, LDM and the shell (8S) and pairing (8P) corrections to this energy,... 

Mass models seek, by a variety of theoretical approaches, to reproduce the measured mass surface and to predict unmeasured masses beyond it Subsequent measurements of these predicted nuclear masses permit an assessment of the quality of the mass predictions from the various models Since the last comprehensive revision of the mass predictions (in the mid-to-late 1970 s) over 300 new masses have been reported Global analyses of these data have been performed by several numerical and graphical methods These have identified both the strengths and weaknesses of the models In some cases failures in individual models are distinctly apparent when the new mass data are plotted as functions of one or more selected physical parameters ... 

It is quite instructive to compare these new measurements (which lie outside the data bases available at the time the various mass models were formulated) with predictions from the models. For such comparisons it is convenient to define A = Predicted Mass - Measured Mass. A > 0 thus denotes cases where the binding energy has been predicted to be too low and conversely, A < 0 corresponds to a prediction of too much nuclear binding. Table 1 summarizes average and root-mean-square deviations for twelve models. 

Notice that one can only measure the product pao with this technique. Notice also that the event rate at energy E depends on the WIMP velocity distribution at speeds v > ME/2p . This integration limit depends on the nuclear mass, and thus detectors with different kinds of nuclei are sensitive to different regions of the WIMP velocity space. Moreover, the cross section gq scales differently for spin-dependent and spin-independent WIMP-nucleus interactions. Finally, while there is a consensus on the spin-independent nuclear form factors, spin-dependent form factors are sensitive to detailed modeling of the proton and neutron wave functions inside the nucleus , seeand references therein] Jungman 1996. 

In these expressions written with use of so-called atomic units (elementary charge, electron mass and Planck constant are all equal to unity) RQs stand as previously for the spatial coordinates of the nuclei of atoms composing the system r) s for the spatial coordinates of electrons Mas are the nuclear masses Zas are the nuclear charges in the units of elementary charge. The meaning of the different contributions is as follows Te and Tn are respectively the electronic and nuclear kinetic energy operators, Vne is the operator of the Coulomb potential energy of attraction of electrons to nuclei, Vee is that of repulsion between electrons, and Vnn that of repulsion between the nuclei. Summations over a and ft extend to all nuclei in the (model) system and those over i and j to all electrons in it. 

In 1955, J.A. Wheeler [1] concluded from a courageous extrapolation of nuclear masses and decay half-lives the existence of nuclei twice as heavy as the heaviest known nuclei he called them superheavy nuclei. Two years later, G. Scharff-Goldhaber [2] mentioned in a discussion of the nuclear shell model, that beyond the well established proton shell at Z=82, lead, the next proton shell should be completed at Z=126 in analogy to the known TV = 126 neutron shell. Together with a new A=184 shell, this shell closure should lead to local region of relative stability. These early speculations remained without impact on contemporary research, however. 

Bearing in mind the large-deviation considerations for the Curie-Weiss model, one could try to characterize molecules by some large-deviation entropy that describes how fast a nuclear molecular structure appears with increasing molecular nuclear masses. Such a large-deviation entropy would describe the decrease in fuzziness of the molecular nuclear structure when the nuclear masses increase. In the limit of infinite nuclear masses one expects a strictly classical nuclear framework, this not being fuzzy anymore at all. Such a large-deviation entropy would also nicely describe the quantum fluctuations round the strictly classical nuclear structure. 

To characterize the geometry of molecules, a model of point nuclear masses with fixed intemuclear distances has proved very useful in many instances. As long as intemuclear distances are not required with an accuracy of better than a few hundredths of an angstrom, this model is quite adequate, and it is possible to speak of the geometrical parameters of a molecule. If greater accuracy is required, however, it becomes necessary to consider the consequences of the nonrigidity of molecules. 

Fig. 36. Snapshots in the nuclidic chart of flow patterns in a ID model of a detonating He layer accreted onto a 0.8M WD. The selected times and corresponding temperatures or densities are given in different panels. The stable nuclides are indicated with open squares. The magic neutron and proton numbers are identified by vertical and horizontal double lines. The drip lines predicted by a microscopic mass model are also shown. The abundances are coded following the grey scales shown in each panel. At early times (bottom left panel), an r-process type of flow appears on the neutron-rich side of the valley of nuclear stability. At somewhat later times (top left panel), the material is pushed back to the neutron-deficient side rather close to the valley of /3-stability. As time passes (two right panels), a pn-process [87] develops...

Now due to the well-known isotope effect, the change of 7(. with the change of nuclear mass upon isotopic exchange, provides direct evidence for the phonon-mediated mechanism in the BCS superconductor. Since the discovery of high-7). materials, many effects have been shown in isotope effect studies. There exist small isotope exponents, a = — log 7)/log M, 0.12 and 0.04 for LSCO and YBCO7, respectively. The non-zero isotope exponent produces doubts about the exclusion of a phonon-mediated model although the discussion above supports strongly the spin-fluctuation-mediated model. 

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. 

A simple relation between any length parameter and the nuclear mass number A follows directly from geometrical considerations and the assumption of constant nuclear (mass) density ( liquid drop model or homogeneous model ), e.g. for the rms radius... 

The repulsive force produced between the positive nucleus and the positive alpha particles causes the deflections. Figure 4.13 illustrates how Rutherfords nuclear atomic model explained the results of the gold foil experiment. The nuclear model also explains the neutral nature of matter the positive charge of the nucleus balances the negative charge of the electrons. However, the model still could not account for all of the atoms mass. 

With the use of the Hartree-Fock method (Hartree 1928 Fock 1930), one can calculate the average potential starting from an effective nucleon-nucleon interaction. The Hartree-Fock calculations give a firm basis for the nuclear shell model (see, e.g., Heyde 1990) and allow the calculation of many observables of nuclei (nuclear binding energy, mass, charge radius, see section Charge radii ). 

On the other hand, the liquid drop model can correctly reproduce the binding energies of nuclei (see, e.g.,0 Eq. (2.3)), nuclear masses, and the threshold potential of nuclear fission, but it cannot describe the shell effects, ground state spins, and many other quantum characteristics. 

The liquid drop model can also explain many nuclear phenomena successfully. The most important ones are as follows the nuclear volume is proportional to the mass number (A) (O Eq. (2.15)) the binding energy per nucleon is approximately constant in a wide mass-number region (O Fig. 2.3) the nuclear masses can be rather well described by the Weizsacker formula (O Eq. (2.3)) nuclear fission (see Chap. 3 in this Volume) Hofstadter s electron scattering experiments show that the nuclear volume is filled up with nucleons rather uniformly. However, the liquid drop model also has its weak points, e.g., it cannot give account of the shell effects. 

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