Calculated Equilibrium Configurations Research Articles

The impact of superconductivity and the Hall effect in models of magnetised neutron stars

AbstractEquilibrium configurations of the internal magnetic field of a pulsar play a key role in modelling astrophysical phenomena from glitches to gravitational wave emission. In this paper, we present a numerical scheme for solving the Grad–Shafranov equation and calculating equilibrium configurations of pulsars, accounting for superconductivity in the core of the neutron star, and for the Hall effect in the crust of the star. Our numerical code uses a finite difference method in which the source term appearing in the Grad–Shafranov equation, which is used to model the magnetic equilibrium is non-linear. We obtain solutions by linearising the source and applying an under-relaxation scheme at each step of computation to improve the solver’s convergence. We have developed our code in both C++ and Python, and our numerical algorithm can further be adapted to solve any non-linear PDEs appearing in other areas of computational astrophysics. We produce mixed toroidal–poloidal field configurations, and extend the portion of parameter space that can be investigated with respect to previous studies. We find that in even in the more extreme cases, the magnetic energy in the toroidal component does not exceed approximately 5% of the total. We also find that if the core of the star is superconducting, the toroidal component is entirely confined to the crust of the star, which has important implications for pulsar glitch models which rely on the presence of a strong toroidal field region in the core of the star, where superfluid vortices pin to superconducting fluxtubes.

Slowly rotating supercompact Schwarzschild stars

The Schwarzschild interior solution, or `Schwarzschild star', which describes a spherically symmetric homogeneous mass with constant energy density, shows a divergence in pressure when the radius of the star reaches the Schwarzschild-Buchdahl bound. Recently Mazur and Mottola showed that this divergence is integrable through the Komar formula, inducing non-isotropic transverse stresses on a surface of some radius $R_{0}$. When this radius approaches the Schwarzschild radius $R_{s}=2M$, the interior solution becomes one of negative pressure evoking a de Sitter spacetime. This gravitational condensate star, or gravastar, is an alternative solution to the idea of a black hole as the ultimate state of gravitational collapse. Using Hartle's model to calculate equilibrium configurations of slowly rotating masses, we report results of surface and integral properties for a Schwarzschild star in the very little studied region $R_{s}<R<(9/8)R_{s}$. We found that in the gravastar limit, the angular velocity of the fluid relative to the local inertial frame tends to zero, indicating rigid rotation. Remarkably, the normalized moment of inertia $I/MR^2$ and the mass quadrupole moment $Q$ approach to the corresponding values for the Kerr metric to second order in $\Omega$. These results provide a solution to the problem of the source of a slowly rotating Kerr black hole.

Interaction of Surfactants Containing a Sulfuric Group with a (5,5) Carbon Nanotube

Sulfuric acid is a good nanotube disperser but surfactants containing a sulfuric head group with a Na atom are better for dispersing carbon nanotubes. Nevertheless, there is no explanation for the importance of the sodium atom in the head group. We have studied the interaction of sulfuric acid, sodium bisulfate and sodium butyl sulfate molecules with a (5,5) single-walled carbon nanotube using density functional theory (DFT), calculating equilibrium configurations, binding energies, charge transfers, and densities of states. In all cases there is charge transfer from the tube to the molecules when these are adsorbed, as has been previously found for sulfuric acid adsorbed on graphene. The presence of these molecules with their hydrogen atoms pointing to the tube does not affect the density of states of the tube close to the Fermi energy, but the presence of sodium bisulfate or sodium butyl sulfate with the Na atom close to the tube enhances the density of states in the region below and near the Fermi ener...

Deformations and Stability of an Elastica Subjected to an Off-Axis Point Constraint

This paper studies, both theoretically and experimentally, the deformation, the vibration, and the stability of a buckled elastic strip (also known as an elastica) constrained by a space-fixed point in the middle. One end of the elastica is fully clamped while the other end is allowed to slide without friction and clearance inside a rigid channel. The point constraint is located at a specified height above the clamping plane. The elastic strip buckles when the pushing force reaches the conventional buckling load. At this buckling load, the elastica jumps to a symmetric configuration in contact with the point constraint. As the pushing force increases, a symmetry-breaking bifurcation occurs and the elastica evolves to one of a pair of asymmetric deformations. As the pushing force continues to increase, the asymmetric deformation experiences a second jump to a self-contact configuration. A vibration analysis based on an Eulerian description taking into account the sliding between the elastica and the point constraint is described. The natural frequencies and the stability of the calculated equilibrium configurations can then be determined. The experiment confirms the two jumps and the symmetry-breaking bifurcation predicted theoretically.

Magnetic fields in axisymmetric neutron stars

We derive general equations for axisymmetric Newtonian MHD and use these as the basis of a code for calculating equilibrium configurations of rotating magnetised neutron stars in a stationary state. We investigate the field configurations that result from our formalism, which include purely poloidal, purely toroidal and mixed fields. For the mixed-field formalism the toroidal component appears to be bounded at less than 7%. We calculate distortions induced both by magnetic fields and by rotation. From our non-linear work we are able to look at the realm of validity of perturbative work: we find for our results that perturbative-regime formulae for magnetic distortions agree to within 10% of the nonlinear results if the ellipticity is less than 0.15 or the average field strength is less than $10^{17}$ G. We also consider how magnetised equilibrium structures vary for different polytropic indices.

Numerical simulations of plasma equilibrium in a one-dimensional current sheet with a nonzero normal magnetic field component

The force balance in a thin collisionless current sheet in the Earth’s magnetotail with a given constant magnetic field component Bz across the sheet is numerically studied for the first time in a self-consistent formulation of the problem. The current sheet is produced by oppositely directed plasma flows propagating from the periphery of the sheet toward the neutral plane. A substantially improved version of a macroparticle numerical model is used that makes it possible to simulate on the order of 107 macroparticles even with a personal computer and to calculate equilibrium configurations with a sufficiently low discrete noise level in the first-and second-order moments of the distribution function, which determine the stress tensor elements. Quasisteady configurations were calculated numerically for several sets of plasma parameters in some parts of the magnetotail. The force balance in the sheet was checked by calculating the longitudinal and transverse pressures as well as the elements of the full stress tensor. The stress tensor in the current sheet is found to be nondiagonal and to differ appreciably from the gyrotropic stress tensor in the Chew-Goldberger-Low model, although the Chew-Goldberger-Low theory and numerical calculations yield close results for large distances from the region of reversed magnetic field.

On Bifurcations of Equilibria of Intrinsically Curved, Electrically Charged, Rod-like Structures that Model DNA Molecules in Solution

DNA molecules in the familiar Watson–Crick double helical B form can be treated as though they have rod-like structures obtained by stacking dominoes one on top of another with each rotated by approximately one-tenth of a full turn with respect to its immediate predecessor in the stack. These “dominoes” are called base pairs. A recently developed theory of sequence-dependent DNA elasticity (Coleman, Olson, & Swigon, J. Chem. Phys. 118:7127–7140, 2003) takes into account the observation that the step from one base pair to the next can be one of several distinct types, each having its own mechanical properties that depend on the nucleotide composition of the step. In the present paper, which is based on that theory, emphasis is placed on the fact that, as each base in a base pair is attached to the sugar-phosphate backbone chain of one of the two DNA strands that have come together to form the Watson–Crick structure, and each phosphate group in a backbone chain bears one electronic charge, two such charges are associated with each base pair, which implies that each base pair is subject to not only the elastic forces and moments exerted on it by its neighboring base pairs but also to long range electrostatic forces that, because they are only partially screened out by positively charged counter ions, can render the molecule’s equilibrium configurations sensitive to changes in the concentration c of salt in the medium. When these electrostatic forces are taken into account, the equations of mechanical equilibrium for a DNA molecule with N + 1 base pairs are a system of μN non-linear equations, where μ, the number of kinematical variables describing the relative displacement and orientation of adjacent pairs is in general 6; it reduces to 3 when base-pair steps are assumed to be inextensible and non-shearable. As a consequence of the long-range electrostatic interactions of base pairs, the μN × μN Jacobian matrix of the equations of equilibrium is full. An efficient numerically stable computational scheme is here presented for solving those equations and determining the mechanical stability of the calculated equilibrium configurations. That scheme is employed to compute and analyze bifurcation diagrams in which c is the bifurcation parameter and to show that, for an intrinsically curved molecule, small changes in c can have a strong effect on stable equilibrium configurations. Cases are presented in which several stable configurations occur at a single value of c.

Buckling instabilities in coupled nano-layers

We study the dynamic buckling of a pair of dissimilar Euler–Bernoulli beams subject to compressive edge loading whose transverse displacements are coupled through non-linear interactions, a problem motivated by the mechanics of graphene layers. The transverse coupling models van der Waals interaction and is derived from a Lennard-Jones 12–6 potential. The beams are assumed to be a fixed distance apart at their ends, although this distance is not necessarily equal to the equilibrium distance as identified from the Lennard-Jones potential. Therefore, the equilibrium configuration is not necessarily straight. Via a Galerkin method, the governing equations are reduced to a system that can be used to calculate equilibrium configurations as well as the stability of these configurations. We show that the buckling instability in this model is significantly affected by the presence of the interaction force as well as the separation of the graphene layers at the boundaries.

An extension of Hill's three‐component model to include different fibre types in finite element modelling of muscle

AbstractMost of the proposed versions of the Hill's model use a sliding‐element theory, considering a single sarcomere. However, a muscle represents a collection of different fibre types with a large range in contractile properties among them. An extension of Hill's three‐component model is proposed here to take into account different fibre types. We present a model consisting of a number of sarcomeras of different types coupled in parallel with the connective tissue. Each sarcomere is modelled by one non‐linear elastic element connected in series with one non‐linear contractile element.Using the finite element method, in an incremental‐iterative scheme of calculating equilibrium configurations of a muscle, the key step is the determination of stresses corresponding to strain increments. The stress calculation procedure for the extended Hill's model is reduced to the solution of a number of independent non‐linear equations with respect to the stretch increments of the serial elastic elements in each sarcomere.Since the distribution of the specific fibre type is non‐uniform over the muscle volume, we have material heterogeneity which we modelled by using the so‐called ‘Generalized Isoparametric Element Formulation’ for functionally graded materials (FGMs).The proposed computational scheme is built in our FE package PAK, so that muscles of complex three‐dimensional shapes can be modelled. In numerical examples, we illustrate the main characteristics of the developed numerical model and some possibilities of realistic modelling of muscle functioning. Copyright © 2006 John Wiley &amp; Sons, Ltd.

The gyromagnetic ratio of rapidly rotating compact stars in general relativity

We numerically calculate equilibrium configurations of uniformly rotating and charged neutron stars, in the case of insulating material and neglecting the electromagnetic forces acting on the equilibrium of the fluid. This allows us to study the behaviour of the gyromagnetic ratio for those objects when varying rotation rate and equation of state for the matter. Under the assumption of low charge and incompressible fluid, we find that the gyromagnetic ratio is directly proportional to the compaction parameter M/R of the star, and very little dependent on its angular velocity. Nevertheless, it seems impossible to have g = 2 for these models with low charge-to-mass ratio, where matter consists of a perfect fluid and the collapse limit is never reached.

End-Capped Associative Polymer Chains between Nanospheres: Attractions in Ideal Solutions

We calculate equilibrium configurations of telechelic associative polymer chains without excluded volume interactions between flat plates and curved surfaces, obtaining analytical expressions for flat plates and numerical results for chains between two spheres. Ideal chains with strongly but reversibly adsorbed terminal groups produce an attraction of O(kT) per chain for planar surfaces as opposed to ≪ kT per chain in marginal and good solvents at high densities. Our numerical results for two spheres yield a weaker repulsion and an attraction of shorter range than predicted by the Derjaguin approximation based on the flat plate solution. These effects are due to deflection of chains out of the gap at close separations.

Monte Carlo calculation of orientationally anisotropic pair distributions and energy transfer in a model monolayer

The NPT (number of particles, pressure, temperature) Monte Carlo method has been used to calculate the equation of state and orientation- and distance-dependent pair distributions for a model monolayer of hard «UFO»s. A UFO is defined to be the intersection volume of two identical spheres whose centers are more than one and less than two radii apart. As pressure is increased from zero, the UFOs rapidly tip perpendicular to the plane and then more gradually align. By use of the calculated equilibrium configurations, and Forster's orientation-dependent dipole-dipole energy-transfer rate constant, the bimolecular energy-transfer rate constant was calculated as a function of pressure and density. The macroscopic rate constant is sensitive to the extent of orientational ordering in the model monolayer

Calculations of [001] Tilt Grain Boundaries Energy in TiO&lt;SUB&gt;2&lt;/SUB&gt; (Rutile) Using a Polarizable Point Ion Shell Model

A series of [001] tilt grain boundaries in TiO2 (rutile) has been studied using a polarizable point ion shell model. The energy as a function of misorientation angle consists of peaks and cusps. Cusps occur at the (230), (120), (130) and (150) interfaces and others correspond to peaks. Especially the (120) and (230) boundaries have low energies. The (001) projections of calculated equilibrium configurations of these boundaries are also shown. The rutile structure makes these boundaries quite different from the interfaces in NiO which has a simple structure (rock salt) compared with rutile. The results obtained are discussed in terms of monopole fields acting on ions and also the reciprocal density of coincidence sites of ions (Σ). It is found that boundaries with Σ-values smaller than 39 correspond to cusps in the diagram of energy versus misorientation angle, while peaks comprise of boundaries which have large values over 51 for Σ. This fact suggests that the boundary energy is related directly to the reciprocal density of coincidence sites in the case of TiO2.

Estimations of Plasma Position and βp+li/2 from Magnetic Measurements under High-β Conditions in JET-2

The MHD equilibria of a high-β plasma are studied by carring out magnetic measurements in JFT-2. Equilibrium configurations agreeing well with the experimental results are obtained by numerical calculations, and βp, in the NB injection phase is estimated. Using these calculated equilibrium configurations, two methods of estimating Rp and βp+li/2 from magnetic probe signals are also discussed, and the Fourier analysis method is shown to give a more accurate value of Rp and βp+li/2 even under high-β conditions.

The study of dipoles in Cu—Al alloys using weak-beam electron microscopy

Abstract In f.o.c. metals and alloys, dipoles are the predominant dislocation structure during the early stages of plastic deformation. Single crystals of Cu-Al alloys, having low stacking fault energies, have been deformed in tension to the end of stage I of the work-hardening curve. The weak-beam method of electron microscopy has then been used to study the equilibrium configurations of dissociated dipoles having different character and glide-plane separation. By analysing the micrographs, the Burgers vectors of all four partial dislocations can be indexed, the various equilibrium positions identified, and the arrangement of the partial dislocations determined in full. The experimental results are compared with the calculated equilibrium configurations derived by Morton and Forwood (1974, private communication) using linear anisotropic elasticity theory. Various types of equilibrium positions predicted by the theory are observed experimentally and quantitative agreement is good. Deviations from the stre...

Alkali halide surfaces: Adsorbate binding energies and structures at surface defects

We describe an electrostatic method of calculating equilibrium configurations of alkali halide diatomic molecules (monomers) and tetratomic dimers adsorbed on rigid alkali halide crystal surfaces. Results are presented for adsorbate binding energies and geometries on the smooth surface, at single-step ledge defects, and at kink (corner) defects. The results are consistent with our recent experimental measurements and we explore several features of the adsorbate–substrate interaction. Cyclic dimers are more weakly bonded than monomers on the smooth surface, but binding strengths for both are nearly equivalent at ledges. Linear dimers are stabilized at the defect binding sites. For all species, the kink position is the strongest adsorption site. Adsorbate structures at the different sites are dependent on the relative magnitudes of adsorbate–adsorbate and adsorbate–substrate interactions.

Effect of quadrupolar deformability on off-centre behaviour of Ag+impurities

A molecular model allowing relaxation up to second-nearest neighbours and polarization of the negative ions is used to calculate equilibrium configurations of Ag+ substitutional defects in RbCl and RbBr. Including the quadrupolar deformability of the Ag+ ion can lead to an equilibrium configuration of the defect which is off-centre along the (110) axis, with a quadrupolar distortion of the surroundings, in agreement with experiment. Dipole moment and pressure dependence of the defect configuration also agree quite well with experiment. The high-pressure on-centre configuration is predicted to show a quadrupolar distortion.

One-Center Basis Set SCF MO's. II. NH3, NH4+, PH3, PH4+

One-center-expanded SCF MO wavefunctions are reported for the ground states of NH3 and PH3 in their planar and in their calculated equilibrium configurations and of NH4+ and PH4+ in their calculated equilibrium configurations. The calculated molecular energies and the geometrical parameters of the determined equilibrium configurations agree well with the experimental values. Good agreement is also obtained for the electric dipole moment and the proton affinity of NH3 while the calculated value of the height of the potential barrier for inversion appears to be too small. For PH3 a good value for the height of the potential barrier is obtained while the calculated electric dipole moment is too large. The calculated proton affinity of PH3 seems to be reasonable. A critical examination of the wavefunctions gives an account of the quality of the several results obtained.

One-Center Basis Set SCF MO's. III. H2O, H2S, and HCl

One-center expanded SCF MO wavefunctions are reported for H2O, H2S, and HCl in their ground states. The wavefunctions refer to the calculated equilibrium configurations which agree well with the experimental data. The corresponding computed molecular energies are —75.922436, —397.58906, and —458.83776 a.u., respectively, which are reasonably close to the experimental values. The energy obtained for H2O is found to be near the estimated Hartree—Fock value. The computed electric dipole moment of H2O agrees well with the experimental value, while for H2S and HCl larger discrepancies are found.