scholarly journals An analytical solution of Bohr's collective hamiltonian

1983 ◽  
Vol 6 (4) ◽  
pp. 803-809
Author(s):  
M. K. El-Adawi ◽  
H. A. Ismail ◽  
S. A. Shalaby ◽  
E. M. Sayed
Keyword(s):  
Analytical Solution ◽  
Energy Levels ◽  
Analytic Form ◽  
Collective Hamiltonian ◽  
Simple Analytic Form ◽  
Collective States

The collective states of126Xe,128Xe,130Xe, and130Be,52<z,N≤116are studied using Bohr's Collective Hamiltonian and a simple analytic form for the potential. The energy levels are calculated for this model and the results are compared with previously published experimental and calculated values.

Analytical solution of the Shredinger equation for the linear combination of the Manning-Rosen and the class of Yukawa potential

2021 ◽  
pp. 157-169
Author(s):  
G.A. Bayramova ◽  
Keyword(s):  
Analytical Solution ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Yukawa Potential ◽  
Energy Levels ◽  
Energy Eigenvalue ◽  
Radial Wave ◽  
Rosen Potential ◽  
Analytical Expressions ◽  
Potential Parameters

In the present work, an analytical solution for bound states of the modified Schrödinger equation is found for the new supposed combined Manning-Rosen potential plus the Yukawa class. To overcome the difficulties arising in the case l ≠ 0 in the centrifugal part of the Manning-Rosen potential plus the Yukawa class for bound states, we applied the developed approximation. Analytical expressions for the energy eigenvalue and the corresponding radial wave functions for an arbitrary value l ≠ 0 of the orbital quantum number are obtained. And also obtained eigenfunctions expressed in terms of hypergeometric functions. It is shown that energy levels and eigenfunctions are very sensitive to the choice of potential parameters.

scholarly journals The Uehling correction to the energy levels in a pionic atom

2006 ◽  
Vol 84 (2) ◽  
pp. 107-113 ◽  
Author(s):  
S G Karshenboim ◽  
E Yu. Korzinin ◽  
V G Ivanov
Keyword(s):  
Free Electron ◽  
Vacuum Polarization ◽  
Energy Levels ◽  
Analytic Form ◽  
Electron Mass ◽  
Mass Ratio ◽  
Pionic Atom ◽  
Closed Analytic Form

We consider a correction to energy levels in a pionic atom induced by the Uehling potential, i.e., by a free electron vacuum-polarization loop. The calculation is performed for circular states (l = n–1). The result is obtained in a closed analytic form as a function of Zα and the pion-to-electron mass ratio. Certain asymptotics of the result are also presented.PACS Nos.: 12.20.Ds, 36.10.Gv

Theory of the stabilization of colloids by adsorbed polymer

1974 ◽  
Vol 337 (1611) ◽  
pp. 509-516 ◽  
Keyword(s):  
Plane Surface ◽  
Analytic Form ◽  
Polymer Chains ◽  
Contour Length ◽  
Link Length ◽  
Excluded Volume Effects ◽  
Adsorbed Polymer ◽  
Random Flight ◽  
The Stability ◽  
Simple Analytic Form

The configurational free energy of random flight polymer chains adsorbed by one end onto a plane surface as a function of the distance from a parallel plane surface is expressed to a good approximation in simple analytic form. The result is used to discuss the stabilization of a colloid suspension by adsorbed polymer. According to this theory two types of aggregation of colloid particles may occur. If LI < AS/2π 3 NkT , where l is the link length and L the contour length of a polymer chain, A is the Hamaker constant, N /S is the number of adsorbed polymer chains per unit area and kT is the Boltzman constant multiplied by temperature, the particles adhere closely, but if AS/2π 3 kT < IL < AS/nkT lg 2N a looser association is formed. It is expected that the presence of excluded volume effects would greatly increase the stability against the looser association.

Why Is Climate Sensitivity So Unpredictable?

Science ◽  
2007 ◽  
Vol 318 (5850) ◽  
pp. 629-632 ◽  
Author(s):  
Gerard H. Roe ◽  
Marcia B. Baker
Keyword(s):  
Climate Change ◽  
Atmospheric Carbon Dioxide ◽  
Probability Distributions ◽  
Analytic Form ◽  
Future Climate Change ◽  
Large Temperature ◽  
Global Mean Temperature ◽  
Simple Analytic Form ◽  
Global Mean

Uncertainties in projections of future climate change have not lessened substantially in past decades. Both models and observations yield broad probability distributions for long-term increases in global mean temperature expected from the doubling of atmospheric carbon dioxide, with small but finite probabilities of very large increases. We show that the shape of these probability distributions is an inevitable and general consequence of the nature of the climate system, and we derive a simple analytic form for the shape that fits recent published distributions very well. We show that the breadth of the distribution and, in particular, the probability of large temperature increases are relatively insensitive to decreases in uncertainties associated with the underlying climate processes.

Boundary-layer-induced potential flow on an elliptic cylinder

1974 ◽  
Vol 66 (1) ◽  
pp. 145-157 ◽  
Author(s):  
Stanley G. Rubin ◽  
Frank J. Mummolo
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Elliptic Cylinder ◽  
Analytic Form ◽  
Slender Body ◽  
Dimensional Solution ◽  
Slender Body Theory ◽  
Simple Analytic Form ◽  
Body Theory

The application of slender-body theory to the evaluation of the three-dimensional surface velocities induced by a boundary layer on an elliptic cylinder is considered. The method is applicable when the Reynolds number is sufficiently large so that the thin-boundary-layer approximation is valid. The resulting potential problem is reduced to a two-dimensional consideration of the flow over an expanding cylinder with porous boundary conditions. The limiting solutions for a flat plate of finite span and a nearly circular cross-section are obtained in a simple analytic form. In the former case, within the limitations of slender-body theory, the results are in exact agreement with the complete three-dimensional solution for this geometry.

An analytical equation derived from a perturbation theory to calculate Henry's constants for square well mixtures

1980 ◽  
Vol 45 (4) ◽  
pp. 1036-1046 ◽  
Author(s):  
M. I. Guerrero ◽  
L. Ponce ◽  
J. P. Monfort
Keyword(s):  
Perturbation Theory ◽  
Analytical Solution ◽  
Pair Potential ◽  
Analytic Form ◽  
Analytical Equation ◽  
Henry's Constant ◽  
Henry's Constants ◽  
Square Well

An analytic form for Henry's constant is derived and applied to several systems. The derivation is based on the use of Leonard-Henderson-Barker perturbation theory for a square well pair potential assuming the Ponce-Renon analytical solution of the square well fluid. Computed values of Henry's constants for CH4-Ar, CH4-N2, CH4-He, CH4-H2, C2H6-N2 and C2H6-CH4 mixtures are compared with experiment. The agreement is quite satisfactory, with mean relative deviations between 2.5 and 8 per cent. Heats of solutions are also computed and compared with experiment.

scholarly journals Scale-invariant singularity of the surface quasigeostrophic patch

2019 ◽  
Vol 863 ◽  
Author(s):  
R. K. Scott ◽  
D. G. Dritschel
Keyword(s):  
Numerical Simulations ◽  
Finite Time ◽  
Initial Conditions ◽  
Logarithmic Spiral ◽  
Analytic Form ◽  
Scale Invariant ◽  
Boundary Curvature ◽  
Simple Analytic Form

Numerical simulations of the surface quasigeostrophic patch indicate the development of a scale-invariant singularity of the boundary curvature in finite time, with some evidence of universality across a variety of initial conditions. At the time of singularity, boundary segments are shown to possess an exact and simple analytic form, described by branches of a logarithmic spiral centred on the point of singularity. The angles between the branches depend non-trivially on the shape of the smooth connecting boundary as the singularity is approached, but are independent of the global boundary.

Periodic features of the statistical atom with exchange and correlation

1968 ◽  
Vol 46 (1) ◽  
pp. 1-13 ◽  
Author(s):  
F. C. Auluck ◽  
Ashok Jain ◽  
Satish Kumar
Keyword(s):  
Angular Momentum ◽  
Mass Velocity ◽  
Energy Levels ◽  
Analytic Form ◽  
Particle Energy ◽  
Relativistic Mass ◽  
Angular Momentum Quantum Number ◽  
Velocity Correction ◽  
Present Formalism ◽  
Exchange And Correlation

The periodic features of atoms are studied using a statistical model embellished with exchange and correlation. It is shown that the earlier results, obtained on the basis of the simple TF model, can be computed in an analytic form by using the Tietz approximation. Single-particle energy levels of neutral atoms are calculated following the approach outlined by us in an earlier publication (Jain and Kumar 1967), and are compared with those obtained by other methods. A new angular momentum assignment is proposed and is shown to give values of v(l, Z) (the number of electrons with orbital angular momentum quantum number l, in an atom of atomic number Z) which are in very close agreement with the empirical values. The relativistic mass–velocity correction is incorporated in the calculation of angular-momentum groups. Finally, some suggestions are made for improving the present formalism.

scholarly journals Multiscale Electrolyte Transport Simulations for Lithium Ion Batteries

2019 ◽  
Author(s):  
Felix Hanke ◽  
Nils Modrow ◽  
Reinier Akkermans ◽  
Ivan Korotkin ◽  
Felix Mocanu ◽  
...  
Keyword(s):  
Transport Properties ◽  
Lithium Ion Batteries ◽  
Full Range ◽  
Lithium Ion ◽  
Analytic Form ◽  
Quantitative Agreement ◽  
Discharge Voltage ◽  
Battery Cell ◽  
Computational Optimization ◽  
Simple Analytic Form

Establishing a link between atomistic processes and battery cell behavior is a major challenge for lithium ion batteries. Focusing on liquid electrolytes, we describe parameter-free molecular dynamics predictions of their mass and charge transport properties. The simulations agree quantitatively with experiments across the full range of relevant ion concentrations and for different electrolyte compositions. We introduce a simple analytic form to describe the transport properties. Our results are used in an extended Newman electrochemical model, including a cell temperature prediction. This multi-scale approach provides quantitative agreement between calculated and measured discharge voltage of a battery and enables the computational optimization of the electrolyte formulation.

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