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John Lowengrub - One of the best experts on this subject based on the ideXlab platform.
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a numerical method for the quasi incompressible cahn hilliard navier stokes equations for variable density flows with a Discrete Energy law
Journal of Computational Physics, 2014Co-Authors: John LowengrubAbstract:Abstract In this paper, we investigate numerically a diffuse interface model for the Navier–Stokes equation with fluid–fluid interface when the fluids have different densities [48] . Under minor reformulation of the system, we show that there is a continuous Energy law underlying the system, assuming that all variables have reasonable regularities. It is shown in the literature that an Energy law preserving method will perform better for multiphase problems. Thus for the reformulated system, we design a C 0 finite element method and a special temporal scheme where the Energy law is preserved at the Discrete level. Such a Discrete Energy law (almost the same as the continuous Energy law) for this variable density two-phase flow model has never been established before with C 0 finite element. A Newton method is introduced to linearise the highly non-linear system of our discretization scheme. Some numerical experiments are carried out using the adaptive mesh to investigate the scenario of coalescing and rising drops with differing density ratio. The snapshots for the evolution of the interface together with the adaptive mesh at different times are presented to show that the evolution, including the break-up/pinch-off of the drop, can be handled smoothly by our numerical scheme. The Discrete Energy functional for the system is examined to show that the Energy law at the Discrete level is preserved by our scheme.
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A numerical method for the quasi-incompressible Cahn–Hilliard–Navier–Stokes equations for variable density flows with a Discrete Energy law
Journal of Computational Physics, 2014Co-Authors: Zhenlin Guo, Ping Lin, John LowengrubAbstract:Abstract In this paper, we investigate numerically a diffuse interface model for the Navier–Stokes equation with fluid–fluid interface when the fluids have different densities [48] . Under minor reformulation of the system, we show that there is a continuous Energy law underlying the system, assuming that all variables have reasonable regularities. It is shown in the literature that an Energy law preserving method will perform better for multiphase problems. Thus for the reformulated system, we design a C 0 finite element method and a special temporal scheme where the Energy law is preserved at the Discrete level. Such a Discrete Energy law (almost the same as the continuous Energy law) for this variable density two-phase flow model has never been established before with C 0 finite element. A Newton method is introduced to linearise the highly non-linear system of our discretization scheme. Some numerical experiments are carried out using the adaptive mesh to investigate the scenario of coalescing and rising drops with differing density ratio. The snapshots for the evolution of the interface together with the adaptive mesh at different times are presented to show that the evolution, including the break-up/pinch-off of the drop, can be handled smoothly by our numerical scheme. The Discrete Energy functional for the system is examined to show that the Energy law at the Discrete level is preserved by our scheme.
Patrycja Stefańska - One of the best experts on this subject based on the ideXlab platform.
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Dia- and paramagnetic contributions to magnetizabilities of relativistic hydrogenlike atoms in some low-lying Discrete Energy eigenstates
Atomic Data and Nuclear Data Tables, 2020Co-Authors: Patrycja StefańskaAbstract:Abstract In this paper we present tabulated data for relative diamagnetic and paramagneticcontributions to the magnetizability ( χ ) of the relativistic hydrogenlike atoms with a pointlike, motionless and spinless nucleus of charge Z e . Utilizing general analytical formulas for the diamagnetic ( χ d ) and paramagnetic ( χ p ) components of χ , recently derived by us (P. Stefanska, 2020) with the aid of the Gordon decomposition technique, valid for an arbitrary Discrete Energy state, we have computed the numerical values of χ d ∕ χ and χ p ∕ χ for the ground state and for the first and the second set of excited states (i.e.: 2 s 1 ∕ 2 , 2 p 1 ∕ 2 , 2 p 3 ∕ 2 , 3 s 1 ∕ 2 , 3 p 1 ∕ 2 , 3 p 3 ∕ 2 , 3 d 3 ∕ 2 , and 3 d 5 ∕ 2 ) of the hydrogen atom ( Z = 1 ) and for hydrogenic ions with 2 ⩽ Z ⩽ 137 . We compare also the numerical values of the total magnetizabilities for the ground state 1 s 1 ∕ 2 and for each state belonging to the first set of excited states of selected hydrogenlike atoms, obtained with the use of two different values of the fine-structure constant, i.e.: α − 1 = 137 . 035 999 139 (from CODATA 2014) and α − 1 = 137 . 035 999 084 (from CODATA 2018).
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Gordon decomposition of the magnetizability of the relativistic hydrogenlike atoms in an arbitrary Discrete Energy state
arXiv: Quantum Physics, 2020Co-Authors: Patrycja StefańskaAbstract:We present Gordon decomposition of the magnetizability of Dirac one-electron atom in an arbitrary Discrete Energy eigenstate, with a pointlike, spinless and motionless nucleus of charge $Ze$. The external magnetic field, by which the atomic state is perturbed, is assumed to be weak, static, and uniform. Analytical derivation of closed-form expressions for the diamagnetic ($\chi_{d}$) and paramagnetic ($\chi_{p}$) contributions to $\chi$ are performed with the use of the Sturmian series representation of the first-order Dirac--Coulomb Green function combined with the theory of special functions. The received formula for $\chi_{p}$ contains the generalized hypergeometric functions ${}_3F_2$ of the unit argument, while $\chi_{d}$ is of an elementary form. For the atomic ground state, our both general results reduce to formulas obtained earlier by other author. We present also numerical values of relative dia- and paramagnetic contributions to the magnetizability for some excited states of selected hydrogenlike ions with $1 \leqslant Z \leqslant 137$ and compare them with data available in the literature.
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Nuclear magnetic shielding constants of Dirac one-electron atoms in some low-lying Discrete Energy eigenstates
Atomic Data and Nuclear Data Tables, 2018Co-Authors: Patrycja StefańskaAbstract:We present tabulated data for the nuclear magnetic shielding constants ($\sigma$) of the Dirac one-electron atoms with a pointlike, motionless and spinless nucleus of charge $Ze$. Utilizing the exact general analytical formula for $\sigma$ derived by us \mbox{[P. Stefa{\'n}ska, Phys. Rev. A. 94 (2016) 012508/1-15],} valid for an arbitrary Discrete Energy eigenstate, we have computed the numerical values of the magnetic shielding factors for the ground state and for the first and the second set of excited states, i.e.: 2s$_{1/2}$, 2p$_{1/2}$, 2p$_{3/2}$, 3s$_{1/2}$, 3p$_{1/2}$, 3p$_{3/2}$, 3d$_{3/2}$, and 3d$_{5/2}$, of the relativistic hydrogenic ions with the nuclear charge numbers from the range $1 \leqslant Z \leqslant 137$. The comparisons of our results with the numerical values reported by other authors for some atomic states are also presented.
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Magnetic-dipole-to-electric-quadrupole cross-susceptibilities for relativistic hydrogenlike atoms in some low-lying Discrete Energy eigenstates
Atomic Data and Nuclear Data Tables, 2017Co-Authors: Patrycja StefańskaAbstract:Abstract In this paper we present tabulated data for magnetic-dipole-to-electric-quadrupole cross-susceptibilities ( χ M 1 → E 2 ) for Dirac one-electron atoms with a pointlike, spinless and motionless nucleus of charge Z e . Numerical values of this susceptibility for the hydrogen atom ( Z = 1 ) and for hydrogenic ions with 2 ⩽ Z ⩽ 137 are computed from the general analytical formula, recently derived by us (Stefanska, 2016), valid for an arbitrary Discrete Energy eigenstate. In this work we provide 30 tables with the values of χ M 1 → E 2 for the ground state, and also for the first, the second and the third set of excited states (i.e.: 2s1/2, 2p1/2, 2p3/2, 3s1/2, 3p1/2, 3p3/2, 3d3/2, 3d5/2, 4s1/2, 4p1/2, 4p3/2, 4d3/2, 4d5/2, 4f5/2 and 4f7/2) of the relativistic hydrogenlike atoms. The value of the inverse of the fine-structure constant used in the calculations is α − 1 = 137.035999139 , and was taken from CODATA 2014.
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closed form expression for the magnetic shielding constant of the relativistic hydrogenlike atom in an arbitrary Discrete Energy eigenstate application of the sturmian expansion of the generalized dirac coulomb green function
Physical Review A, 2016Co-Authors: Patrycja StefańskaAbstract:We present analytical derivation of the closed-form expression for the dipole magnetic shielding constant of a Dirac one-electron atom being in an arbitrary Discrete Energy eigenstate. The external magnetic field, by which the atomic state is perturbed, is assumed to be weak, uniform and time independent. With respect to the atomic nucleus we assume that it is pointlike, spinless, motionless and of charge Ze. Calculations are based on the Sturmian expansion of the generalized Dirac- Coulomb Green function [R. Szmytkowski, J. Phys. B 30, 825 (1997); erratum 30, 2747 (1997)], combined with the theory of hypergeometric functions. The final result is of an elementary form and agrees with corresponding formulas obtained earlier by other authors for some particular states of the atom.
Haihu Wen - One of the best experts on this subject based on the ideXlab platform.
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Discrete Energy levels of caroli de gennes matricon states in quantum limit in fete0 55se0 45
Nature Communications, 2018Co-Authors: M N Chen, Xiaoyu Chen, Huan Yang, Xiyu Zhu, Enyu Wang, Haihu WenAbstract:Caroli-de Gennes-Matricon (CdGM) states were predicted in 1964 as low-Energy excitations within vortex cores of type-II superconductors. In the quantum limit, the Energy levels of these states were predicted to be Discrete with the basic levels at ±μΔ2/EF (μ = 1/2, 3/2, 5/2, …) with Δ the superconducting Energy gap and EF the Fermi Energy. However, due to the small ratio of Δ/EF in most type-II superconductors, it is very difficult to observe the Discrete CdGM states, but rather a symmetric peak which appears at zero bias at the vortex center. Here we report the clear observation of these Discrete Energy levels of CdGM states in FeTe0.55Se0.45. The rather stable energies of these bound state peaks vs. space clearly validate our conclusion. Analysis based on the energies of these CdGM states indicates that the Fermi Energy in the present system is very small.
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Discrete Energy levels of Caroli-de Gennes-Martricon states in quantum limit due to small Fermi Energy in FeTe$_{0.55}$Se$_{0.45}$
Nature communications, 2018Co-Authors: M N Chen, Xiaoyu Chen, Huan Yang, Xiyu Zhu, Enyu Wang, Haihu WenAbstract:Caroli-de Gennes-Martricon (CdGM) states were predicted in 1964 as low Energy excitations within vortex cores of type-II superconductors. In the quantum limit, namely $T/T_\mathrm{c} \ll \Delta/E_\mathrm{F}$, the Energy levels of these states were predicted to be Discrete with the basic levels at $E_\mu = \pm \mu \Delta^2/E_\mathrm{F}$ ($\mu = 1/2$, $3/2$, $5/2$, ...). However, due to the small ratio of $\Delta/E_\mathrm{F}$ in most type-II superconductors, it is very difficult to observe the Discrete CdGM states, but rather a symmetric peak appears at zero-bias at the vortex center. Here we report the clear observation of these Discrete Energy levels of CdGM states in FeTe$_{0.55}$Se$_{0.45}$. The rather stable energies of these states versus space clearly validates our conclusion. Analysis based on the energies of these CdGM states indicates that the Fermi Energy in the present system is very small.
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Discrete Energy levels of caroli de gennes martricon states in quantum limit due to small fermi Energy in fete _ 0 55 se _ 0 45
arXiv: Superconductivity, 2017Co-Authors: M N Chen, Xiaoyu Chen, Huan Yang, Xiyu Zhu, Enyu Wang, Haihu WenAbstract:Caroli-de Gennes-Martricon (CdGM) states were predicted in 1964 as low Energy excitations within vortex cores of type-II superconductors. In the quantum limit, namely $T/T_\mathrm{c} \ll \Delta/E_\mathrm{F}$, the Energy levels of these states were predicted to be Discrete with the basic levels at $E_\mu = \pm \mu \Delta^2/E_\mathrm{F}$ ($\mu = 1/2$, $3/2$, $5/2$, ...). However, due to the small ratio of $\Delta/E_\mathrm{F}$ in most type-II superconductors, it is very difficult to observe the Discrete CdGM states, but rather a symmetric peak appears at zero-bias at the vortex center. Here we report the clear observation of these Discrete Energy levels of CdGM states in FeTe$_{0.55}$Se$_{0.45}$. The rather stable energies of these states versus space clearly validates our conclusion. Analysis based on the energies of these CdGM states indicates that the Fermi Energy in the present system is very small.
M N Chen - One of the best experts on this subject based on the ideXlab platform.
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Discrete Energy levels of caroli de gennes matricon states in quantum limit in fete0 55se0 45
Nature Communications, 2018Co-Authors: M N Chen, Xiaoyu Chen, Huan Yang, Xiyu Zhu, Enyu Wang, Haihu WenAbstract:Caroli-de Gennes-Matricon (CdGM) states were predicted in 1964 as low-Energy excitations within vortex cores of type-II superconductors. In the quantum limit, the Energy levels of these states were predicted to be Discrete with the basic levels at ±μΔ2/EF (μ = 1/2, 3/2, 5/2, …) with Δ the superconducting Energy gap and EF the Fermi Energy. However, due to the small ratio of Δ/EF in most type-II superconductors, it is very difficult to observe the Discrete CdGM states, but rather a symmetric peak which appears at zero bias at the vortex center. Here we report the clear observation of these Discrete Energy levels of CdGM states in FeTe0.55Se0.45. The rather stable energies of these bound state peaks vs. space clearly validate our conclusion. Analysis based on the energies of these CdGM states indicates that the Fermi Energy in the present system is very small.
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Discrete Energy levels of Caroli-de Gennes-Martricon states in quantum limit due to small Fermi Energy in FeTe$_{0.55}$Se$_{0.45}$
Nature communications, 2018Co-Authors: M N Chen, Xiaoyu Chen, Huan Yang, Xiyu Zhu, Enyu Wang, Haihu WenAbstract:Caroli-de Gennes-Martricon (CdGM) states were predicted in 1964 as low Energy excitations within vortex cores of type-II superconductors. In the quantum limit, namely $T/T_\mathrm{c} \ll \Delta/E_\mathrm{F}$, the Energy levels of these states were predicted to be Discrete with the basic levels at $E_\mu = \pm \mu \Delta^2/E_\mathrm{F}$ ($\mu = 1/2$, $3/2$, $5/2$, ...). However, due to the small ratio of $\Delta/E_\mathrm{F}$ in most type-II superconductors, it is very difficult to observe the Discrete CdGM states, but rather a symmetric peak appears at zero-bias at the vortex center. Here we report the clear observation of these Discrete Energy levels of CdGM states in FeTe$_{0.55}$Se$_{0.45}$. The rather stable energies of these states versus space clearly validates our conclusion. Analysis based on the energies of these CdGM states indicates that the Fermi Energy in the present system is very small.
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Discrete Energy levels of caroli de gennes martricon states in quantum limit due to small fermi Energy in fete _ 0 55 se _ 0 45
arXiv: Superconductivity, 2017Co-Authors: M N Chen, Xiaoyu Chen, Huan Yang, Xiyu Zhu, Enyu Wang, Haihu WenAbstract:Caroli-de Gennes-Martricon (CdGM) states were predicted in 1964 as low Energy excitations within vortex cores of type-II superconductors. In the quantum limit, namely $T/T_\mathrm{c} \ll \Delta/E_\mathrm{F}$, the Energy levels of these states were predicted to be Discrete with the basic levels at $E_\mu = \pm \mu \Delta^2/E_\mathrm{F}$ ($\mu = 1/2$, $3/2$, $5/2$, ...). However, due to the small ratio of $\Delta/E_\mathrm{F}$ in most type-II superconductors, it is very difficult to observe the Discrete CdGM states, but rather a symmetric peak appears at zero-bias at the vortex center. Here we report the clear observation of these Discrete Energy levels of CdGM states in FeTe$_{0.55}$Se$_{0.45}$. The rather stable energies of these states versus space clearly validates our conclusion. Analysis based on the energies of these CdGM states indicates that the Fermi Energy in the present system is very small.
Rajendra S. Dhaka - One of the best experts on this subject based on the ideXlab platform.
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Evidence of Discrete Energy states and cluster-glass behavior in Sr 2 − x La x CoNbO 6
Physical Review B, 2020Co-Authors: Ashwani Kumar, Björn Schwarz, Helmut Ehrenberg, Rajendra S. DhakaAbstract:We report the detailed analysis of specific heat $[{C}_{P}(T)]$ and ac susceptibility for magnetically frustrated ${\mathrm{Sr}}_{2\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{CoNbO}}_{6}$ $(x=0--1)$ double perovskites to understand low-temperature complex magnetic interactions and their evolution with $x$. Interestingly, the observed Schottky anomaly in the $x\ensuremath{\leqslant}0.4$ samples shifts gradually toward higher temperatures with magnetic field as well as $x$, and the analysis reveals the persistence of the Discrete Energy states in these samples resulting from the spin-orbit coupling and octahedral distortion. Moreover, the extracted values of the Land\'e $g$ factor indicate the existence of high-spin state ${\mathrm{Co}}^{3+}$ ions energetically close to a nonmagnetic low-spin state. The specific-heat data show the $\ensuremath{\lambda}$-type anomaly for the $x\ensuremath{\geqslant}0.6$ samples due to evolution of the long-range antiferromagnetic ordering. Our analysis of low-temperature ${C}_{P}(T)$ data for the $x\ensuremath{\geqslant}0.6$ samples demonstrates the 3D isotropic Heisenberg antiferromagnetic (AFM) interactions and the temperature-induced second-order AFM-paramagnetic phase transition. More interestingly, we demonstrate the presence of the free ${\mathrm{Co}}^{2+}$-like Kramers doublet ground state in the $x=1$ sample. Further, the ac susceptibility and time evolution of the magnetization data reveal the low-temperature cluster-glass-like behavior in the $x=0--0.4$ samples, where spin-spin correlation strength decreases with $x$.
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evidence of Discrete Energy states and cluster glass behavior in sr 2 x la x conbo 6
Physical Review B, 2020Co-Authors: Ashwani Kumar, Björn Schwarz, Helmut Ehrenberg, Rajendra S. DhakaAbstract:We report the detailed analysis of specific heat $[{C}_{P}(T)]$ and ac susceptibility for magnetically frustrated ${\mathrm{Sr}}_{2\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{CoNbO}}_{6}$ $(x=0--1)$ double perovskites to understand low-temperature complex magnetic interactions and their evolution with $x$. Interestingly, the observed Schottky anomaly in the $x\ensuremath{\leqslant}0.4$ samples shifts gradually toward higher temperatures with magnetic field as well as $x$, and the analysis reveals the persistence of the Discrete Energy states in these samples resulting from the spin-orbit coupling and octahedral distortion. Moreover, the extracted values of the Land\'e $g$ factor indicate the existence of high-spin state ${\mathrm{Co}}^{3+}$ ions energetically close to a nonmagnetic low-spin state. The specific-heat data show the $\ensuremath{\lambda}$-type anomaly for the $x\ensuremath{\geqslant}0.6$ samples due to evolution of the long-range antiferromagnetic ordering. Our analysis of low-temperature ${C}_{P}(T)$ data for the $x\ensuremath{\geqslant}0.6$ samples demonstrates the 3D isotropic Heisenberg antiferromagnetic (AFM) interactions and the temperature-induced second-order AFM-paramagnetic phase transition. More interestingly, we demonstrate the presence of the free ${\mathrm{Co}}^{2+}$-like Kramers doublet ground state in the $x=1$ sample. Further, the ac susceptibility and time evolution of the magnetization data reveal the low-temperature cluster-glass-like behavior in the $x=0--0.4$ samples, where spin-spin correlation strength decreases with $x$.