Nuclear mass formula

Mamdouh, A., Pearson, J.M., Rayet, M., Tondeur, F. Fission barriers of neutron-rich and superheavy nuclei calculated with the ETFSI method. Nucl. Phys. A679, 337—358 (2001) Pearson, J.M., Nayak, R.C., Goriely, S. Nuclear mass formula with Bogolyubov-enhanced shell-quenching application to r-process. Phys. Lett. B387, 455-459 (1996)... 

It is mentioned in passing that the proper masses mA and mB to be used in Equation 3.3 are the atomic masses (nucleus + electrons) rather than the respective nuclear masses as might be expected from a strict Born-Oppenheimer approximation. For further discussion of this point, reference should be made to the reading lists at the end of this chapter and of Chapter 2. The combination of Equations 3.1 and 3.2 corresponds to a classical harmonic oscillator with force constant f and mass p. The harmonic oscillator frequency v is given by the well-known formula... 

The effective hamiltonian in formula 29 incorporates approximations that we here consider. Apart from a term V"(R) that originates in nonadiabatic effects [67] beyond those taken into account through the rotational and vibrational g factors, other contributions arise that become amalgamated into that term. Replacement of nuclear masses by atomic masses within factors in terms for kinetic energy for motion both along and perpendicular to the internuclear axis yields a term of this form for the atomic reduced mass. 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

The nuclei up to A 60 are produced in these equilibrium processes. In such equilibrium processes, the final yields of various nuclei are directly related to their nuclear stability with the more stable nuclei having higher yields. So one observes greater yields of even-even nuclei than odd A nuclei (due to the pairing term in the mass formula) and even N isotopes are more abundant than odd A isotopes of an element. 

The mechanisms and data of the fission process have been reviewed recently by Leachman (70). Several different approaches have been used in an effort to explain the asymmetry of the fission process as well as other fission parameters. These approaches include developments of the liquid drop model (50, 51,102), calculations based on dependence of fission barrier penetration on asymmetry (34), the effect of nuclear shells (52, 79, 81), the determinations of the fission mode by level population of the fragments (18, 33, 84), and finally the consideration of quantum states of the fission nucleus at the saddle point (15, 108). All these approaches require a mass formula whereby the masses of the fission fragments far removed from stability may be determined. The lack of an adequate mass formula has hindered the development of a satisfactory theory of fission. The fission process is highly complex and it is not surprising that the present theories fall short of a full explanation. 

One finds that the LDA overestimates the calculated binding energy. At the BHF minimum the difference is about 85 MeV for the total nucleus, which is close to the surface correction contribution in the Bethe-Weizsacker mass formula for (as 118MeV). This suggests that the LDA approximation misses the major contribution to the surface tension and also the major contribution of the surface effects to the compressibility of a finite nucleus. This is supported by the observation that the LDA result for as a function of density is only slightly above the energy versus density curve of nuclear matter. 

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. 

We note that the notation of the dispersion formula (21) does not agree with the usual one in the case of infrared vibrations. Here we have to take into account the fact that the slow nuclear masses have a much weaker vibration than the electrons, and correspondingly the large masses of the atomic nuclei are taken before the partial summations which contains the ultraviolet dispersion, in order to obtain for... 

Most of the mass formulas mentioned include the effect of nuclear deformation explicitly. Accordingly, values for the equilibrium deformation in the ground state, radial proton and/or neutron distribution as well as quadrupole moments were deduced from the mass calculations. The values agree in general quite well with experimental results [32, 33, 35]. 

Herewith, in the formula for the determination of R, instead of the nuclear mass, nif, we should substitute the macroscopic crystal mass then the recoil energy reduces practically to zero, and the y-quantum energy becomes almost precisely equal to the energy difference El-Eq. The Doppler broadening from the thermal motion also vanishes. As a result, the emitting and absorbing lines narrow down to the natural width and coincide with each other. The resonance requirement becomes satisfied. 

Among the modem procedures utilized to estabUsh the chemical stmcture of a molecule, nuclear magnetic resonance (nmr) is the most widely used technique. Mass spectrometry is distinguished by its abiUty to determine molecular formulas on minute amounts, but provides no information on stereochemistry. The third most important technique is x-ray diffraction crystallography, used to estabUsh the relative and absolute configuration of any molecule that forms suitable crystals. Other physical techniques, although useful, provide less information on stmctural problems. 

We saw in Chapter 12 that mass spectrometry gives a molecule s formula and infrared spectroscopy identifies a molecule s functional groups. Nuclear magnetic resonance spectroscopy does not replace either of these techniques rather, it complements them by "mapping" a molecule s carbon-hydrogen framework. Taken together, mass spectrometry, JR, and NMR make it possible to determine the structures of even very complex molecules. 

Nuclear binding energies are determined by applying Einstein s formula to the mass difference between the nucleus and its components. Iron and nickel have the highest binding energy per nucleon. 

A chemical equation describes a chemical reaction in many ways as an empirical formula describes a chemical compound. The equation describes not only which substances react, but the relative number of moles of each undergoing reaction and the relative number of moles of each product formed. Note especially that it is the mole ratios in which the substances react, not how much is present, that the equation describes. In order to show the quantitative relationships, the equation must be balanced. That is, it must have the same number of atoms of each element used up and produced (except for special equations that describe nuclear reactions). The law of conservation of mass is thus obeyed, and also the "law of conservation of atoms. Coefficients are used before the formulas for elements and compounds to tell how many formula units of that substance are involved in the reaction. A coefficient does not imply any chemical bonding between units of the substance it is placed before. The number of atoms involved in each formula unit is multiplied by the coefficient to get the total number of atoms of each element involved. Later, when equations with individual ions are written (Chap. 9), the net charge on each side of the equation, as well as the numbers of atoms of each element, must be the same to have a balanced equation. The absence of a coefficient in a balanced equation implies a coefficient of 1. 

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