Wigner-Seitz Research Articles

Neutron superfluidity and the structure of the neutron star inner crust

A self-consistent quantum approach to describe the inner crust structure of neutron stars is developed within the Wigner-Seitz (WS) approximation, based on the generalized energy functional method involving explicitly neutron and proton pairing correlations. The energy functional is constructed by matching the realistic phenomenological nuclear functional by Fayans et al. for describing the pseudonucleus in the center of the WS cell with the one calculated microscopically for neutron matter within the Brueckner approach with the Argonne v18 force. The microscopic description of the neutron superfluidity is based on the BCS approach with the same v18 force. Many-body theory corrections to the BCS method (the correlation and self-energy ones) are included in the calculation scheme in an approximate way. A wide region of average densities was investigated corresponding to the Fermi momentum values kF = 0.6–1.2 fm−1

Variational theory of average-atom and superconfigurations in quantum plasmas

Models of screened ions in equilibrium plasmas with all quantum electrons are important in opacity and equation of state calculations. Although such models have to be derived from variational principles, up to now existing models have not been fully variational. In this paper a fully variational theory respecting virial theorem is proposed-all variables are variational except the parameters defining the equilibrium, i.e., the temperature T, the ion density ni and the atomic number Z. The theory is applied to the quasiclassical Thomas-Fermi (TF) atom, the quantum average atom (QAA), and the superconfigurations (SC) in plasmas. Both the self-consistent-field (SCF) equations for the electronic structure and the condition for the mean ionization Z* are found from minimization of a thermodynamic potential. This potential is constructed using the cluster expansion of the plasma free energy from which the zero and the first-order terms are retained. In the zero order the free energy per ion is that of the quantum homogeneous plasma of an unknown free-electron density n0 = Z* ni occupying the volume 1/ni. In the first order, ions submerged in this plasma are considered and local neutrality is assumed. These ions are considered in the infinite space without imposing the neutrality of the Wigner-Seitz (WS) cell. As in the Inferno model, a central cavity of a radius R is introduced, however, the value of R is unknown a priori. The charge density due to noncentral ions is zero inside the cavity and equals en0 outside. The first-order contribution to free energy per ion is the difference between the free energy of the system "central ion+infinite plasma" and the free energy of the system "infinite plasma." An important part of the approach is an "ionization model" (IM), which is a relation between the mean ionization charge Z* and the first-order structure variables. Both the IM and the local neutrality are respected in the minimization procedure. The correct IM in the TF case is found to be Z-Z*= integral d3 r[n(r)-n0], where n(r) is the first-order electron density. It is shown that in the QAA case the same IM has to be used and that other IMs lead to unphysical solutions. With this IM R becomes in both cases (TF and QAA) equal to the WS radius and the variational calculation leads to SCF equations in an infinite plasma while n0 (or equivalently Z*) is to be found from the condition integral d3r theta(r-R)Vel(r)=0, where theta denotes Heaviside function and Vel(r) is the SCF electrostatic potential. In the SC case results are similar except that averages over all superconfigurations appear. In the TF case the condition for n0 gives the neutrality of the WS sphere and one gets the classical TF ion-in-cell average atom. The situation is different in the QAA and in the SC cases in which the cavity is not neutral and the SCF potential Vel(r) is not zero outside the cavity. Due to the fully variational character of our approach the expression for the thermodynamic pressure in all cases does not require any numerical differentiation and is consistent with the virial theorem.

A realistic model of superfluidity in the neutron star inner crust

A semi-microscopic self-consistent quantum approach developed recently to describe the inner crust structure of neutron stars within the Wigner-Seitz (WS) method with the explicit inclusion of neutron and proton pairing correlations is further developed. In this approach, the generalized energy functional is used which contains the anomalous term describing the pairing. It is constructed by matching the realistic phenomenological functional by Fayans et al. for describing the nuclear-type cluster in the center of the WS cell with the one calculated microscopically for neutron matter. Previously the anomalous part of the latter was calculated within the BCS approximation. In this work corrections to the BCS theory which are known from the many-body theory of pairing in neutron matter are included into the energy functional in an approximate way. These modifications have a sizable influence on the equilibrium configuration of the inner crust, i.e. on the proton charge Z and the radius R_c of the WS cell. The effects are quite significant in the region where the neutron pairing gap is larger.

The role of the boundary conditions in the Wigner–Seitz approximation applied to the neutron star inner crust

The influence of the boundary conditions used in the Wigner–Seitz (WS) approximation applied to the neutron star inner crust is examined. The generalized energy functional method which includes the neutron and proton pairing correlations is used. Predictions of two versions of the boundary conditions are compared with each other. The uncertainties in the equilibrium configuration ( Z , R c ) of the crust, where Z is the proton charge and R c , the radius of the WS cell, correspond to variation of Z by 2–6 units and of R c , by 1–2 fm. The effect of the boundary conditions is enhanced at increasing density. These uncertainties are smaller than the variation of Z and R c coming from the pairing effects. In the case of high densities, k F ≳ 1 fm −1 , the most important uncertainty occurs in the value of the neutron gap Δ n . In the WS approximation, it originates from the shell effect in the neutron single-particle spectrum which is rather pronounced in the case of larger k F and, correspondingly, small R c values, but it becomes negligible at lower density near the drip point. An approximate recipe to avoid this uncertainty is suggested.

A Static Analysis of Snap Back in the Cantilever with Mems Adhesion

The “Snap back” problem of the micro‐cantilever remains one of the dominant failure mechanisms in the Micro Electro‐mechanical System (MEMS). By analyzing the Hamaker micro continuum medium and solid physics principle, the consistency model of Wigner‐Seitz (W‐S) continuum medium is presented. The gap revision coefficients of the body with the face‐centered cubic structure are derived, which include the attractive force and the repulsive one. The adhesion model of the 500 µ m X 1 µ m silicon micro‐cantilever coated by Au is established. The micro‐cantilever static relationship between the elastic force and the adhesion force is discussed. The reason of the microcantilever “snap back” problem, an instable balanced point, is discovered. Increasing the rigidity of the micro‐cantilever, a method to avoid the micro‐cantilever “snap back” to happen, is put forward, which improves MEMS structure design and enhances MEMS reliability.

Frank–Kasper polyhedra, disclination nets and basic magnetism inNd3(Fe,Ti)29 andNd3(Fe,Ti)29Nx intermetallics

High-precision crystal structure parameters have been determined fromRietveld refinement of high-resolution neutron powder diffraction data inNd3(Fe,Ti)29 andNd3(Fe,Ti)29Nx intermetallics. The construction of the corresponding Wigner–Seitz (WS) polyhedra hasrevealed that some 12-fold (group-A) environments and the 14-fold (group-D, dumbbell)environments remain intact upon nitrogen uptake whereas some other 12-fold (group-B, C)environments transform according to the Frank–Kasper sequence . For comparison, WS cells were calculated for the 12-, 13-and 14-fold Frank–Kasper-likepolyhedra of 1:12 and 2:17 types of structure. A global character of WS volume (WSV)relations has been revealed, where the corresponding WSVs follow the same trend for the threetypes of structure. Disclination nets have been constructed for the three types of structure,revealing that the easy-magnetization direction (EMD) of the iron sublattice is always alongthe [–Nd–Fe–] type of disclination line. A comparative analysis of the site hyperfine fieldsBhfi in the three kinds of compound led to successful estimations of saturation magnetization(Ms(Fe) = 42.6 μB/fu) from Fe-site magnetic moments for and its nitride (Ms(Fe) = 52.8 μB/fu) at room temperature. The successful prediction of Fe-site magnetic moments, of EMD and ofMs from the comparative study of Frank–Kasper-like environments shows that this approachcan provide adequate guidance in the search for better rare-earth–transition-metal(TM = Fe,Co)permanent magnets.

Self-consistent approach for the thermodynamics of ions in dense plasmas in the superconfiguration approximation

We propose a new thermodynamic approach to ions in dense plasmas taking into account the screening by free electrons. The ions in fundamental and excited states are represented by a group of superconfigurations. Each superconfiguration containing an integer number of bound electrons is totally screened by free electrons in a Wigner–Seitz (WS) sphere. The minimisation of the plasma free energy with respect to the WS radii of the ions, under the condition that the specific mass of the plasma is preserved, leads to the equality of the electronic pressure for all ions. This pressure, as well as the WS radii and the distribution of ionic probabilities, are calculated by iteration. In this way, our approach not only gives the charge neutrality of the plasma, but also assures that the plasma environment is the same for each ion. In principle, the proposed approach allows one to apply the superconfiguration method to calculate radiative properties of plasmas for higher densities than those encountered in transmission experiments. Numerical examples for nickel, aluminium and samarium plasmas will be given. Comparisons with results from the previous statistical approach using the same WS radius for each ion charge state will also be discussed.

Wigner-Seitz model of charged lamellar colloidal dispersions

A concentrated suspension of lamellar colloidal particles (e. g. clay) is modelled by considering a single, uniformly charged, finite platelet confined with co- and counterions to a Wigner-Seitz (WS) cell. The system is treated within Poisson-Boltzmann theory, with appropriate boundary conditions on the surface of the WS cell, supposed to account for the confinement effect of neighbouring platelets. Expressions are obtained for the free energy, osmotic and disjoining pressures and the capacitance in terms of the local electrostatic potential and the co- and counterion density profiles. Explicit solutions of the linearized Poisson-Boltzmann (LPB) equation are obtained for circular and square platelets placed at the centre of a cylindrical or parallelepipedic cell. The resulting free energy is found to go through a minimum as a function of the aspect ratio of the cell, for any given volume (determined by the macroscopic concentration of platelets), platelet surface charge and salt concentration. The optimum aspect ratio is found to be nearly independent of the two latter physical parameters. The osmotic and disjoining pressures are found to coincide at the free energy minimum, while the total quadrupole moment of the electric double-layer formed by the platelet and the surrounding co- and counterions vanishes simultaneously. The osmotic equation-of-state is calculated for a variety of physical conditions. The limit of vanishing platelet concentration is considered in some detail, and the force acting between two coaxial platelets is calculated in that limit as a function of their separation.

The Thermal expansion of a two-phase composite with body-centred cubic symmetry

A theory is developed using symmetry adapted displacement vectors in Kelvin's method for the solution of problems in elasticity where there is spherical symmetry. For a particle-filled composite with b.c.c. symmetry the filler is assumed to be spherical and the matrix to fill the Wigner-Seitz (W-S) unit cell. Suitable displacement vectors are chosen from the Kubic harmonics of Von der Lage and Bethe and the first two odd-ordered harmonics used to derive an explicit expression for the composite expansion coefficient. The predictions of the present theory are compared with those from the spherical shell theory and found to be in better agreement with published data for silica particles in epoxy resin. Calculated stresses show a pronounced variation with angle and position in the W-S cell.

Magnetic-moment distribution inNi0.97Ru0.03alloy

Using polarized neutrons, the full three-dimensional magnetic-structure amplitudes in Ni 0.97 Ru 0.03 single crystals were measured up to (sinθ)/λ=0.9 A -1 . The form-factor analysis yields a local moment on the Ni atom of (0.582±0.005)μ B and one on the Ru atom of (-3.08±0.33)μ B . Also, the fractional population of the moment in the T 2g orbital is found to be 0.84±0.01, indicating that the moment is more asymmetric than in pure Ni, in contrast to what others have observed in nickel-based alloys with 3d impurities. Moment density maps in various portions of the Wigner-Seitz (WS) cell were obtained. Unlike the case of Ni, we find on the surface of the WS cell small islands of negative density around 1/20 0, for instance, and fairly large islands of positive density around 1/4 1/4 0, for example. A net small positive diffuse moment density of 0.0015μ B /A 3 is obtained by averaging over all the faces of the WS cell. The observed asymmetry in these density maps confirms the conclusion drawn from the form-factor analysis. The sign and magnitude of the Ru moment, together with the extensive perturbations introduced by it, both on the host local moment and the diffuse moment, seem to point to the existence of a virtual bound state very near the Fermi energy and support a similar conclusion drawn from resistivity and thermoelectric-power data.