Helium- and Lithium-like ionic sequences: Critical charges

In non-relativistic quantum mechanics we study the Coulomb systems of infinitely massive center of charge Z and two-three electrons: $(Z,e,e)$ and $(Z,e,e,e)$. It is shown that in both cases the total energy curve in $Z$ is smooth, without any visible irregularities. Thus, for both systems the physical integer charges $Z=1,2,...$ do not play a distinguished role as would be associated with charge quantization. By definition, a critical charge $Z_{cr}$ is a charge which separates a domain of the existence of bound states from a domain of unbound ones (continuum). For both systems the critical charges are found, $Z_{cr,2e}=0.91085$ and $Z_{cr,3e}=2.009$, respectively. Based on numerical analysis, the Puiseux expansion in fractional powers of $(Z-Z_{cr})$ is constructed for both systems. Our results indicate the existence of a square-root branch point singularity at $Z_{cr}$ with exponent 3/2. A connection between the critical charge and the radius of convergence of 1/Z-expansion is briefly discussed.Comment: 10 pages, LaTeX, typos corrected, Fig.1 added, a Note Added with calculated critical charge for $2^1S$ state for $(Z,e,e)$ system, $Z_{cr,2e}^{(2^1S)}\ =\ 1.02

Evolution of an elliptical bubble in an accelerating extensional flow

Mathematical models that describe the dynamical behavior of a thin gas bubble embedded in a glass fiber during a fiber drawing process have been discussed and analyzed. The starting point for the mathematical modeling was the equations presented in [1] for a glass fiber with a hole undergoing extensional flow. These equations were reconsidered here with the additional reduction that the hole, i.e. the gas bubble, was thin as compared to the radius of the fiber and of finite extent. The primary model considered was one in which the mass of the gas inside the bubble was fixed. This fixed-mass model involved equations for the axial velocity and fiber radius, and equations for the radius of the bubble and the gas pressure inside the bubble. The model equations assumed that the temperature of the furnace of the drawing tower was known. The governing equations of the bubble are hyperbolic and predict that the bubble cannot extend beyond the limiting characteristics specified by the ends of the initial bubble shape. An analysis of pinch-off was performed, and it was found that pinch-off can occur, depending on the parameters of the model, due to surface tension when the bubble radius is small. In order to determine the evolution of a bubble, a numerical method of solution was presented. The method was used to study the evolution of two different initial bubble shapes, one convex and the other non-convex. Both initial bubble shapes had fore-aft symmetry, and it was found that the bubbles stretched and elongated severely during the drawing process. For the convex shape, fore-aft symmetry was lost in the middle of the drawing process, but the symmetry was re-gained by the end of the drawing tower. A small amount of pinch-off was observed at each end for this case, so that the final bubble length was slightly shorter than its theoretical maximum length. For the non-convex initial shape, pinch-off occurred in the middle of the bubble resulting in two bubbles by the end of the fiber draw. The two bubbles had different final pressures and did not have fore-aft symmetry. An extension of the fixed-mass model was considered in which the gas in the bubble was allowed to diffuse into the surrounding glass. The governing equations for this leaky-mass model were developed and manipulated into a form suitable for a numerical treatment

A note about the ground state of the H3+{\rm H}_3^+ hydrogen molecular ion

Three simple $7-, (7+3)-, 10-$parametric trial functions for the ${\rm H}_3^+$ molecular ion are presented. Each of them provides subsequently the most accurate approximation for the Born-Oppenheimer ground state energy among several-parametric trial functions. These trial functions are chosen following a criterion of physical adequacy and includes the electronic correlation in the exponential form $\sim\exp{(\gamma r_{12})}$, where $\gamma$ is a variational parameter. The Born-Oppenheimer energy is found to be $E=-1.340 34, -1.340 73, -1.341 59$\,a.u., respectively, for optimal equilateral triangular configuration of protons with the equilibrium interproton distance $R=1.65$\,a.u. The variational energy agrees in three significant digits (s.d.) with most accurate results available at present as well as for major expectation values.Comment: 12 pages, 1 figure, 3 table

Free expansion of impenetrable bosons on one-dimensional optical lattices

We review recent exact results for the free expansion of impenetrable bosons on one-dimensional lattices, after switching off a confining potential. When the system is initially in a superfluid state, far from the regime in which the Mott-insulator appears in the middle of the trap, the momentum distribution of the expanding bosons rapidly approaches the momentum distribution of noninteracting fermions. Remarkably, no loss in coherence is observed in the system as reflected by a large occupation of the lowest eigenstate of the one-particle density matrix. In the opposite limit, when the initial system is a pure Mott insulator with one particle per lattice site, the expansion leads to the emergence of quasicondensates at finite momentum. In this case, one-particle correlations like the ones shown to be universal in the equilibrium case develop in the system. We show that the out-of-equilibrium behavior of the Shannon information entropy in momentum space, and its contrast with the one of noninteracting fermions, allows to differentiate the two different regimes of interest. It also helps in understanding the crossover between them.Comment: 21 pages, 14 figures, invited brief revie

Charged Hydrogenic, Helium and Helium-Hydrogenic Molecular Chains in a Strong Magnetic Field

A non-relativistic classification of charged molecular hydrogenic, helium and mixed helium-hydrogenic chains with one or two electrons which can exist in a strong magnetic field $B \lesssim 10^{16}$G is given. It is shown that for both $1e-2e$ cases at the strongest studied magnetic fields the longest hydrogenic chain contains at most five protons indicating to the existence of the $\rm{H}_5^{4+}$ and $\rm{H}_5^{3+}$ ions, respectively. In the case of the helium chains the longest chains can exist at the strongest studied magnetic fields with three and four \al-particles for $1e-2e$ cases, respectively. For mixed helium-hydrogenic chains the number of heavy centers can reach five for highest magnetic fields studied. In general, for a fixed magnetic field two-electron chains are more bound than one-electron ones.Comment: 32 pages, 2 figures, 9 table

Legumes as a Strategy for Reducing Greenhouse Gas Emissions of Forage-Livestock Systems

Incorporation of legumes into forage systems has been a widely adopted strategy to increase pasture productivity and forage nutritive value, while reducing N inputs. Considering the population growth, and the diminishing land resources for food production, the need to increase the food supply will have to be balanced with the environmental impact of these systems, particularly their carbon footprint. Enteric methane production represents the largest source of greenhouse gas emissions from livestock. Certain forage legumes have evolved plant secondary compounds, such as tannins and other polyphenols, which have been associated with reductions in enteric methane emissions. Studies were conducted at Utah State University (USU), and at the University of Florida, North Florida Research and Education Center (UF-NFREC) to assess in vivo methane emissions in grazing cattle, using the SF6 tracer technique. At USU, cattle grazing pastures of Birdsfoot trefoil (Lotus corniculatus; BFT) emitted less methane per unit of dry matter consumed when compared with cattle fed a totally mixed ration (50% barley grain, 25% alfalfa hay, and 25% corn silage) in ad libitum amounts. However, emissions in cattle grazing BFT did not differ from those grazing the legume Cicer milkvetch (Astragalus cicer), or a traditional pasture-finishing system based on Meadow brome (Bromus riparius). At UF-NFREC, three livestock-forage systems were tested during three consecutive years to determine the effects of including the legume Rhizoma peanut (Arachis glabrata Benth.; BHR) in bahiagrass pastures (Paspalum notatum Flügge) fertilized (BH) or not (BHF) with N during the warm season. No differences were observed in methane emissions (g d-1), or in methane emission intensity. From the legumes grazed in these experiments, only BFT contains significant concentrations of tannins. Thus, the potential to mitigate livestock enteric methane emissions by grazing legumes appears to be directly related to the presence of tannins