Analytical solution of the Coulomb potential for spherical nuclei

2019 ◽  
Vol 34 (29) ◽  
pp. 1950237
Author(s):  
Hüseyin Koç ◽  
Erhan Eser ◽  
Cevad Selam

A lot of problems of atomic and nuclear physics depend on Coulomb potential with high accuracy. Therefore, it is very important to carefully and accurately calculate the Coulomb potential. In this study, a new analytical expression was obtained for calculating the Coulomb potential by choosing the Fermi distribution function, which is suitable for charge distribution in nuclei. The proposed formula guarantees an accurate and simple calculation of the Coulomb potential of nuclei. Using the analytical expression obtained, the Coulomb potentials for several spherical nuclei were calculated for all values of the parameters. It is shown that the results obtained for arbitrary values of the radius are consistent with the literature data. In this study, the accepted values in the literature of the two parameters (Coulomb radius [Formula: see text] and diffuseness parameter [Formula: see text] which are important for the Coulomb potential are also discussed.

Author(s):  
J. Bonevich ◽  
D. Capacci ◽  
G. Pozzi ◽  
K. Harada ◽  
H. Kasai ◽  
...  

The successful observation of superconducting flux lines (fluxons) in thin specimens both in conventional and high Tc superconductors by means of Lorentz and electron holography methods has presented several problems concerning the interpretation of the experimental results. The first approach has been to model the fluxon as a bundle of flux tubes perpendicular to the specimen surface (for which the electron optical phase shift has been found in analytical form) with a magnetic flux distribution given by the London model, which corresponds to a flux line having an infinitely small normal core. In addition to being described by an analytical expression, this model has the advantage that a single parameter, the London penetration depth, completely characterizes the superconducting fluxon. The obtained results have shown that the most relevant features of the experimental data are well interpreted by this model. However, Clem has proposed another more realistic model for the fluxon core that removes the unphysical limitation of the infinitely small normal core and has the advantage of being described by an analytical expression depending on two parameters (the coherence length and the London depth).


2005 ◽  
Vol 23 (6) ◽  
pp. 429-461
Author(s):  
Ian Lerche ◽  
Brett S. Mudford

This article derives an estimation procedure to evaluate how many Monte Carlo realisations need to be done in order to achieve prescribed accuracies in the estimated mean value and also in the cumulative probabilities of achieving values greater than, or less than, a particular value as the chosen particular value is allowed to vary. In addition, by inverting the argument and asking what the accuracies are that result for a prescribed number of Monte Carlo realisations, one can assess the computer time that would be involved should one choose to carry out the Monte Carlo realisations. The arguments and numerical illustrations are carried though in detail for the four distributions of lognormal, binomial, Cauchy, and exponential. The procedure is valid for any choice of distribution function. The general method given in Lerche and Mudford (2005) is not merely a coincidence owing to the nature of the Gaussian distribution but is of universal validity. This article provides (in the Appendices) the general procedure for obtaining equivalent results for any distribution and shows quantitatively how the procedure operates for the four specific distributions. The methodology is therefore available for any choice of probability distribution function. Some distributions have more than two parameters that are needed to define precisely the distribution. Estimates of mean value and standard error around the mean only allow determination of two parameters for each distribution. Thus any distribution with more than two parameters has degrees of freedom that either have to be constrained from other information or that are unknown and so can be freely specified. That fluidity in such distributions allows a similar fluidity in the estimates of the number of Monte Carlo realisations needed to achieve prescribed accuracies as well as providing fluidity in the estimates of achievable accuracy for a prescribed number of Monte Carlo realisations. Without some way to control the free parameters in such distributions one will, presumably, always have such dynamic uncertainties. Even when the free parameters are known precisely, there is still considerable uncertainty in determining the number of Monte Carlo realisations needed to achieve prescribed accuracies, and in the accuracies achievable with a prescribed number of Monte Carol realisations because of the different functional forms of probability distribution that can be invoked from which one chooses the Monte Carlo realisations. Without knowledge of the underlying distribution functions that are appropriate to use for a given problem, presumably the choices one makes for numerical implementation of the basic logic procedure will bias the estimates of achievable accuracy and estimated number of Monte Carlo realisations one should undertake. The cautionary note, which is the main point of this article, and which is exhibited sharply with numerical illustrations, is that one must clearly specify precisely what distributions one is using and precisely what free parameter values one has chosen (and why the choices were made) in assessing the accuracy achievable and the number of Monte Carlo realisations needed with such choices. Without such available information it is not a very useful exercise to undertake Monte Carlo realisations because other investigations, using other distributions and with other values of available free parameters, will arrive at very different conclusions.


2020 ◽  
Vol 495 (2) ◽  
pp. 2428-2435 ◽  
Author(s):  
Y H Chen

ABSTRACT wdec is used to evolve grids of DA-variable (DAV) star models adopting the element diffusion scheme with pure and screened Coulomb potentials. The core compositions are thermonuclear burning results derived from mesa. mesa yields composition profiles that the version of wdec used in this work could not accommodate (most notably, the presence of helium in the core of the model). According to the theory of rotational splitting, Fu and colleagues identified six triplets for the DAV star HS 0507 + 0434B based on 206 h of photometric data. The grids of DAV star models are used to fit the six reliable m = 0 modes. When adopting the screened Coulomb potential, a best-fitting model of log(MHe/M*) = −3.0, log(MH/M*) = −6.1, Teff = 11 790 K, M* = 0.625 M⊙, log g = 8.066 and σRMS = 2.08 s was obtained. Compared with adopting the pure Coulomb potential, the value of σRMS is improved by 34 per cent. This study may provide a new method for research into mode-trapping properties.


1970 ◽  
Vol 48 (3) ◽  
pp. 349-354 ◽  
Author(s):  
C. R. James ◽  
F. Vermeulen

The shielding potential for a test charge in a plasma is derived for the case in which the velocity of the charge is much less than the thermal velocity. This potential is an analytical expression valid for any angle in space. The potential contains the Debye term and also a term which is due to the particle's motion. At large distances from the test charge the latter term asymptotically becomes proportional to the inverse third power of the distance. On the other hand, as the distance to the test charge approaches zero, this term vanishes and the potential becomes the Coulomb potential. It is shown that, although the inverse third-power term dominates at very large distances, the Debye term dominates over the very important region out to at least several Debye lengths.


1979 ◽  
Vol 19 (1) ◽  
pp. 413-415 ◽  
Author(s):  
C. S. Lam ◽  
Y. P. Varshni

A statistical view of fracture at cracks is presented that is also appropriate for failures in singularity-dominated, self-similar fields other than those at crack tips. Consideration of the behaviour of the distributions of stress and strain near crack tips results in the development of a new two-parameter distribution function for the probability of failure. The two fundamental premises on which the function is based are, firstly, that the failure of any part of the material near to the crack tip leads to total failure along the whole crack front or at least represents total failure; and secondly, that the variability of strength in material is due to micro-structural inhomogeneity. The new function is tested by means of several large sets of toughness data from other workers, and is found to give with only two parameters better fits than can the three-parameter distribution function of Weibull. The Weibull function is capable of giving reasonable fits in its extremely flexible three-parameter form, but that very flexibility means that these fits may be no more than descriptions without theoretical foundation. It is found also that the new function is applicable equally to ceramics and steels. The very good fits afforded by the new function are further support for previous findings in two basic areas in the science of fracture. Firstly, previous work concerned with the distributions of stress and strain at crack tips and with crack-opening displacement upon which the new function is based, is supported, as is the idea that the crack-opening displacement is fundamental in determining the possibility and prob­ability of failure. Secondly, the present work is in agreement with widely accepted ideas concerned with stress-controlled mechanisms of failure in materials.


Author(s):  
M. S. Longuet-Higgins

ABSTRACTThe distribution of the total (or ‘second’) curvature of a stationary random Gaussian surface is derived on the assumption that the squares of the surface slopes are negligible. The distribution is found to depend on only two parameters, derivable from the fourth moments of the energy spectrum of the surface. Each distribution function satisfies a linear differential equation of the third order, and the distribution is asymmetrical with positive skewness, in general. A special case of zero skewness occurs when the surface consists of two intersecting systems of long-crested waves.


1958 ◽  
Vol 36 (7) ◽  
pp. 944-962 ◽  
Author(s):  
E. M. Pennington ◽  
M. A. Preston

A system of coupled differential equations relates the amplitudes of alpha particle waves emitted from a spheroidal nucleus described by the Bohr–Mottelson model. These equations in spheroidal coordinates have been solved for five even–even nuclides with the aid of the electronic computer FERUT. There are four possible cases for each nuclide which are consistent with the boundary conditions. The solutions of the equations are used to calculate the probability density of alpha particles on the nuclear surface for each case. For Case I the peak in the distribution function shifts from the nuclear symmetry axis to the equator with increasing mass number. The probability density for Case II is always peaked between the symmetry axis and the equator, while for Cases III and IV it is always peaked strongly at the equator. The change in the distribution function with increasing distance from the nucleus is considered for a typical case. Barrier penetration factors are calculated and found to differ from those for spherical nuclei by factors of the order of 2 or 3. Comparison with the calculations of an approximation method of Fröman is made for one nuclide.


2019 ◽  
Vol 17 (1) ◽  
pp. 1774-1793 ◽  
Author(s):  
Mario A. Sandoval-Hernandez ◽  
Hector Vazquez-Leal ◽  
Uriel Filobello-Nino ◽  
Luis Hernandez-Martinez

Abstract In this work, we propose to approximate the Gaussian integral, the error function and the cumulative distribution function by using the power series extender method (PSEM). The approximations proposed in this paper present a high accuracy for the complete domain [–∞,∞]. Furthermore, the approximations are handy and easy computable, avoiding the application of special numerical algorithms. In order to show its high accuracy, three case studies are presented with applications to science and engineering.


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