Crystal Field Model Research Articles

A coupled crystal inelasticity-phase field model for crack growth in polycrystalline nitinol microstructures

Abstract This paper introduces a comprehensive computational framework, comprising a finite deformation crystal inelasticity constitutive model and phase field model, for modeling crack growth in superelastic nitinol polycrystalline microstructures. The crystal inelasticity model represents crystal stretching and lattice rotation from elastic mechanisms, as well as local inelastic deformation due to austenite-martensite phase transformation. The phase field formulation decomposes the Helmholtz free energy density into stored elastic energy, phase transformation energy, and crack surface energy components. The elastic energy accounts for tension-compression asymmetry with the formation of the crack through a spectral decomposition. Kinetic Monte Carlo simulations generate equilibrium area fractions of different surface orientations, which serve as weights for the surface energy. An adaptive wavelet-enhanced hierarchical finite element (FE) model is introduced to alleviate high computational overhead in phase field crack simulations. Simulations with the coupled inelasticity phase field model are conducted under various loading conditions including Mode-I tension, a quasi-static Kalthoff experiment, and cyclic loading of polycrystalline microstructures. Crack propagation is effectively predicted by this model, providing valuable insights into the material mechanical behavior with growing cracks.

Effect of Ligand Deprotonation on the Spin Transition in Fe(II) Complexes.

The effect of deprotonation of the ligands forming the second coordination sphere of the iron(II) ion on its spin crossover behavior is examined. Due to the pronounced changes in the observed magnetic characteristics of the iron(II) ion, the proton removal from the ligand in the second coordination sphere is modeled by the appearance of a negative charge on one of the neutral nitrogens in the nearest surrounding of the metal ion, thus significantly changing the crystal field acting on this ion and, respectively, the conditions for observation of the spin transition. Different symmetries of the nearest nitrogen surrounding of the iron(II) ion and different locations of the negative charge in this surrounding are examined in the frames of the crystal field model. It is demonstrated that in the model accepted, depending on the location of the negative charge on the nitrogens of the first coordination sphere, the spin conversion can be either suppressed or facilitated. Within the framework of the suggested model, a qualitative explanation of the effect of deprotonation on the observed magnetic properties in [Fe4(HnL4)](n)+ (n = 8, 6, 4) compounds is given.

On the origin of low-valent uranium oxidation state

The significant interest in actinide bonding has recently focused on novel compounds with exotic oxidation states. However, the difficulty in obtaining relevant high-quality experimental data, particularly for low-valent actinide compounds, prevents a deeper understanding of 5f systems. Here we show X-ray absorption near-edge structure (XANES) measurements in the high-energy resolution fluorescence detection (HERFD) mode at the uranium M4 edge for the UIII and UIV halides, namely UX3 and UX4 (X = F, Cl, Br, I). The spectral shapes of these two series exhibit clear differences, which we explain using electronic structure calculations of the 3d-4f resonant inelastic X-ray scattering (RIXS) process. To understand the changes observed, we implemented crystal field models with ab initio derived parameters and investigated the effect of reducing different contributions to the electron-electron interactions involved in the RIXS process. Our analysis shows that the electron-electron interactions weaken as the ligand changes from I to F, indicative of a decrease in ionicity both along and between the UX3 and UX4 halide series.

Auxiliary Rather Than Dominant. The Role of Direct Dy-S Coordination in Single-Molecule Magnet Unveiled via ab initio Study.

The role of Dy-S coordination in a single-molecule magnet (SMM) is investigated via an ab initio study in a group of mononuclear structures. The SMM performance of this group is well interpreted via a concise criterion consisting of long quantum tunneling of magnetization (QTM) time τQTM and high effective barrier for magnetic reversal Ueff. The best SMMs in the selected group, i.e., 1Dy (CCDC refcode: PUKFAF) and 2Dy (CCDC refcode: NIKSEJ), are just those holding the longest τQTM and the highest Ueff simultaneously. Further analysis based on the crystal field model and ab initio magneto-structural exploration indicates that the influence of Dy-S coordination on the SMM performance of 1Dy is weaker than that of axial Dy-O coordination. Thus, Dy-S coordination is more likely to play an auxiliary role rather than a dominant one. However, if placed at the suitable equatorial position, Dy-S coordination could provide important support for good SMM performance. Consequently, starting from 1Dy, we built two new structures where Dy-S coordination only exists at the equatorial position and two axial positions are occupied by strong Dy-O/Dy-F coordination. Compared to 1Dy and 2Dy, these new ones are predicted to have significantly longer τQTM and higher Ueff, as well as a nearly doubled blocking temperature TB. Thus, they are probable candidates of SMM having clearly improved performance.

Numerical modeling of shear band effect on Goss grain recrystallization in electrical steels: Crystal plasticity finite element and phase field modeling

This study investigates the effect of shear band evolution on the nucleation of Goss {110}<001> texture during the primary recrystallization of 3.24 wt% Si grain-oriented electrical steel. Nucleation at the early stage of primary recrystallization of the steel is explored both experimentally and numerically. The experimental approach involves cold rolling the steel specimens to obtain a thickness reduction ratio of 76 % and then applying heat treatment to them at 600 °C for less than 1 min. The numerical simulation employes crystal plasticity (CP) finite element model (FEM) to simulate the plastic deformation induced by the dislocation slips on predefined slip systems and non-crystallographic shear bands during cold rolling. Based on the CPFEM results, the generalized strain energy release maximization (GSERM) model is used to predict the preferential orientation probability of recrystallized nuclei for the steel by considering shear band formation. Subsequently, the microstructure evolution during the early stage of primary recrystallization of the steel is simulated using the phase field model (PFM). The developed CP model successfully predicted shear band activation and evolution in the γ-fibers centered on the {111}<112> texture component. The model also demonstrated that shear bands would be the preferred nucleation sites at the early stage of primary recrystallization because of their high stored energy. Moreover, by coupling with the GSERM model, the PFM could reproduce the nucleation of Goss grains at the beginning of primary recrystallization in shear bands.

Pressure-dependent multiplet-excitation energies of α-Al2O3:Cr3+ by the first-principles method

We apply a method [Phys. Rev. B 108, 035141 (2023)], which can treat the multiplet excitations in the first-principles method, to a typical pressure indicator α-Al2O3:Cr3+. After optimizing the crystal structure under hydrostatic pressure from 0.0 to 35.0 GPa in the usual first-prinicple calculations, we obtain the multiplet energies in the method. That is, we conduct the exact diagonalization of the crystal-field model Hamiltonian which is directly determined based on the result of the quasiparticle self-consistent GW method. Our method has no parameters by hand. The calculated pressure dependences of the excitation energies agree well with those of experiments.

Combining crystal plasticity and phase field model for predicting texture evolution and the influence of nuclei clustering on recrystallization path kinetics in Ti-alloys

A three-dimensional computational framework has been developed combining a crystal plasticity (CP) and a phase-field (PF) approach that can efficiently simulate static recrystallization (SRX) and grain growth during the hot-forming in Ti-alloys. In the framework, the CP slip system parameters have been accurately calibrated by solving an inverse optimization problem from available experimental tension and compression stress–strain data through CP simulations performed via an orientation distribution function (ODF)-based computational model. Using the CP model, the evolution of inhomogeneous local deformation, deformed texture, and grain dislocation density have been simulated in the plastically deformed polycrystalline Ti-alloys. The PF model then predicts microstructure evolution and kinetics of SRX from CP-informed dislocation density during the annealing phase. Experimental information on microstructural heterogeneity in terms of the initial arrangement of nuclei distribution has been used to guide the development of the framework that can provide deeper insights into unique morphological evolution for various types of grain impingement as well as experimental validation of SRX kinetics. Finally, when the proposed model has been quantitatively validated through experimentally measured texture evolution and SRX path kinetics, excellent agreement is achieved. The current study highlights a systematic modeling framework that is capable of predicting crystallographic texture, microstructural evolution, and kinetics in the course of SRX for a clear understanding of the relationship between the mechanical properties, various microstructural descriptors, and thermo-mechanical process in the regime of material design.

Analytical crystal-field models applied to compounds doped with Eu3+

The point charge electrostatic model (PCEM) and the simple overlap model (SOM), both purely analytical crystal field models, are being briefly revisited, in order to show the evolution of their application to Eu doped compounds. Calculations are made through the method of equivalent nearest neighbours (MENN), and applied to the Eu(dipivaloylmethanate)31,10-phenanthroline [Eu(DPM)3o-phen] complex, with the aim of discussing the Eu–O and Eu–N interactions. The charge of the Eu ion is calculated using the Batista-Longo improved model (BLIM) to give gBLIM = 3,64e around the Eu-NN midpoint. The 7F1 crystal-field levels and level splitting were satisfactorily reproduced with two charge factors, namely, g1 = 0.495 and g2 = 0.415, assuming the luminescent site in a tetragonal point symmetry slightly distorted. In particular, it is shown that the Eu3+, itself, is satisfied by attracting an amount of charge, which neutralize/stabilize its own site electrostatically, no matter what the ligating ions are. The Eu-NN overlap (ρ = <4f|2s2p>) is assumed to be 0.1, which is physically plausible both as a limit in respect to β, the SOM correction factor to the PCEM, and describes the covalent contribution to the Ln-NN interaction, the latter is the most important contribution of the SOM to the crystal-field theory of lanthanide containing compounds. An eloquent comparison with crystal-field levels of the Eu:LiYF4 and Eu(btfa)3(4,4-bipy)(EtOH) is made. Finally, this paper is, indeed, a very simple tribute to Professor Oscar L. Malta, as an acknowledgement of his deep contribution to the author's apprenticeship about crystal-field theory of systems containing lanthanides.

Defect and spectroscopy properties of Eu-doped RbBaPO4 phosphors: A computer simulation study

The effects of Eu incorporation in the RbBaPO4 structure was investigated using atomistic simulation based on lattice energy minimization combined with crystal field models. The defect study reveals that the most favorable intrinsic defect type is formed by the anti-site defect. The results revealed that Eu3+ prefers to enter the Ba site with a RbBa' anti-sites as charge compensation. Crystal field theory shows that the substitution of the Eu3+ ion for the Ba2+ ion in the anti-site defect accompanied by the local lattice deformation induces a local symmetry without the inversion center of the slightly distorted C4 type at the position of the impurity Eu3+ ion. The 7F1 and 7F2 energy sublevels as well as the 7F1 manifold were satisfactorily reproduced with a set of phenomenological charge factors and non-null crystal field parameters.

Crystal-field analysis and models of Eu-emission centers with C3v symmetry in insitu Eu- and Mg-codoping GaN layers

Gallium nitride active layers co-doped in situ with magnesium and europium, GaN(Mg):Eu, show strong red photoluminescence (PL), promising for red light emitting diode (LED) application (Eu@621 nm). Polarized PL lines excited by either above or below band-gap energy for the GaN(Mg):Eu layers are assigned to the 5D0→7FJ (J = 0, 1, 2) transitions lines from several Eu centers with C3v symmetry, which consist of single line, double lines and triple lines related to splitting of 7FJ (J = 0, 1, 2) by the C3v crystal-field, respectively. Cascade luminescence line-pattern of the 5D1→7FJ (J = 0, 1, 2) transitions is a complete replica of the PL line-pattern and gives evidence of the correct assignment of the polarized PL lines. Crystal-field splitting and Zeeman splitting of the PL lines have been analysed using the C3v crystal-field equivalent operator Hamiltonian. Structural Eu models with the C3v symmetry have been proposed on the basis of the analysis: the dominant Eu center is an Eu complex with defect-free perfect surrounding: the minor Eu centers are Eu complexes accompanied by nitrogen vacancies with different charge states on the c-axis in the closest vicinity of Eu. An important role of Mg atoms in the GaN(Mg):Eu active layers has been proposed to suppress creation of gallium vacancies, giving rise to enhancement of the PL emitted from the defect-free Eu centers.

Theory of magnetic 3d transition metal dopants in gallium nitride

Using first-principles density functional theory (DFT) methods and size-converged supercell models, we analyze the electronic and atomic structure of magnetic $3d$ transition metal dopants in cubic gallium nitride (c-GaN). All stable defect charge states for Fermi levels across the full experimental gap are computed using a method that correctly resolves the boundary condition problem (without a jellium approximation) and eliminates finite-size errors. The resulting computed defect levels are not impacted by the DFT band-gap problem, they span a width consistent with the experimental gap rather than being limited to the single-particle DFT gap. All defects with electronically degenerate (half-metal) ${T}_{d}$ ground states are found to have significant distortions, relaxing to ${D}_{2d}$ structures driven by the Jahn-Teller instability. This leads to insulating ground states for all substitutional $3d$ dopants, refuting claims in the literature that $+U$ or hybrid functional methods are required to avoid artificial half-metal results. Interpreting the ${d}^{n}$ atomic occupations within a crystal-field model and exchange splittings, we identify a systematic trend across the $3d$ transition metal series. Approaches to estimate excited-state energies as observed in photoluminescence from defect centers are assessed, ranging from a Koopmans-type single-particle energy interpretation to relaxed total energy differences in fully self-consistent DFT. The single-particle interpretations are found to be qualitatively predictive and the calculations are consistent with the limited available experimental data across the $3d$ dopant series. These results provide a baseline understanding to guide future studies and a conceptual framework within which to interpret new results.

Zeeman spectroscopy and crystal-field analysis of low symmetry centres in Nd3+ doped Y2SiO5

We report on infrared to visible Zeeman absorption spectroscopy and parameterised crystal-field modelling of Nd3+ centres in Y2SiO5 through the use of experimentally inferred crystal-field energy levels and Zeeman directional electronic g values. We demonstrate that good agreement between the calculated and experimental crystal-field energy levels as well as directional Zeeman g values along all three crystallographic axes can be obtained. Further, we demonstrate that the addition of correlation crystal field effects successfully account for discrepancies that arise between the calculated and experimental values relevant to the 2H(2) multiplet in a one-electron crystal field model.

Evaluating f-Orbital Participation in the UV═E Multiple Bonds of [U(E)(NR2)3] (E = O, NSiMe3, NAd; R = SiMe3)

The reaction of 1 equiv of 1-azidoadamantane with [UIII(NR2)3] (R = SiMe3) in Et2O results in the formation of [UV(NR2)3(NAd)] (1, Ad = 1-adamantyl) in good yields. The electronic structure of 1, as well as those of the related U(V) complexes, [UV(NR2)3(NSiMe3)] (2) and [UV(NR2)3(O)] (3), were analyzed with EPR spectroscopy, SQUID magnetometry, NIR-visible spectroscopy, and crystal field modeling. This analysis revealed that, within this series of complexes, the steric bulk of the E2- (E═O, NR) ligand is the most important factor in determining the electronic structure. In particular, the increasing steric bulk of this ligand, on moving from O2- to [NAd]2-, results in increasing U═E distances and E-U-Namide angles. These changes have two principal effects on the resulting electronic structure: (1) the increasing U═E distances decreases the energy of the fσ orbital, which is primarily σ* with respect to the U═E bond, and (2) the increasing E-U-Namide angles increases the energy of fδ, due to increasing antibonding interactions with the amide ligands. As a result of the latter change, the electronic ground state for complexes 1 and 2 is primarily fφ in character, whereas the ground state for complex 3 is primarily fδ.

Autonomous atomic Hamiltonian construction and active sampling of X-ray absorption spectroscopy by adversarial Bayesian optimization

X-ray absorption spectroscopy (XAS) is a well-established method for in-depth characterization of electronic structure. In practice hundreds of energy-points should be sampled during the measurements, and most of them are redundant. Additionally, it is also tedious to estimate reasonable parameters in the atomic Hamiltonians for mechanistic understanding. We implement an Adversarial Bayesian optimization (ABO) algorithm comprising two coupled BOs to automatically fit the many-body model Hamiltonians and to sample effectively based on active learning (AL). Taking NiO as an example, we find that less than 30 sampling points are sufficient to recover the complete XAS with the corresponding crystal field and charge transfer models, which can be selected based on intuitive hypothesis learning. Further applications on the experimental XAS spectra reveal that less than 80 sampling points give reasonable XAS and reliable atomic model parameters. Our ABO algorithm has a great potential for future applications on automated physics-driven XAS analysis and AL sampling.

Spin orbital reorientation transitions induced by magnetic field

Here we report on a new effect similar to the spin reorientation transition (SRT) that takes place at two magnetic fields of BSORT1 and BSORT2. The effect is observed in the magnetization curves of small Mn3+ magnetic clusters in the wurtzite GaN (being in a paramagnetic state) calculated using crystal field model approach. The obtained results suggest that the computed magnetic anisotropy (MA) reverses its sign on increasing B across BSORT. However, a detailed analysis shows that MA is unchanged for high magnetic fields. We show that the observed effect arises from the interplay of the crystalline environment and the spin–orbit coupling λSˆLˆ, therefore we name it spin orbital reorientation transition (SORT). The value of BSORT1 depends on the crystal field model parameters and the number of ions N in a given cluster, whereas BSORT2 is controlled mostly by the magnitude of the spin–orbit coupling λ. The explanation of SORT is given in terms of the spin MS and orbital momentum ML contributions to the total magnetization M=MS+ML. The similar effect should also be present in other materials with not completely quenched (non zero) orbital angular momentum L, a uniaxial magnetic anisotropy and the positive value of λ.

Crack tip enhanced phase-field model for crack evolution in crystalline Ti6Al from concurrent crystal plasticity FE-molecular dynamics simulations

Dislocations nucleating from a crack-tip contribute to the plasticity evolution in the vicinity of crack, as well as to the crack propagation process. This paper systematically develops a method for enhancing the defect energy, accounting for crack-tip nucleated dislocations in the Helmholtz free energy density functional, associated with the phase-field formulation of crack evolution, in a coupled crystal plasticity finite element phase field (CPFE-PF) model. A concurrent crystal plasticity FE- hyper-dynamics accelerated molecular dynamics (CPFE-MD) model is created. The model that identifies, transfers, and propagates dislocations from the atomistic to continuum domain, is used to generate a dataset of crack tip dislocation density evolution along with different state variables. The paper focuses on crack tip plasticity mechanisms for the crystalline alloy Ti6Al. Using a Bayesian inference approach the critical state variables that affect the evolution of crack tip nucleated dislocation density are inferred. A functional form of the evolution of dislocation density in terms of the state variables is derived by employing a genetic programming based symbolic regression (GPSR) approach. The contribution of nucleated dislocation densities to effective plastic strain evolution at the crack tip is validated using the CPFE-MD model simulations. A comparison of the crack path and other state variables in the vicinity of the crack with and without contributions from the nucleated dislocations shows the importance of this augmentation on crack evolution.

Simulation of dislocation slip and twin propagation in Mg through coupling crystal plasticity and phase field models

A numerical strategy to simulate plastic deformation in Mg alloys including dislocation slip and twin propagation is presented. Dislocation slip is included through a crystal plasticity model which is solved using the finite element method while twin propagation is taken into account by means of a phase field model which is solved using a fast Fourier transform algorithm. The coupled crystal plasticity and phase field equations were solved using different discretizations of the simulation domain using the same time step for both of them. The numerical strategy was used to simulate the deformation in compression of a Mg micro-pillar along the [101̄0] direction. The stress–strain curve predicted by the model as well as the dominant deformation mechanisms were in agreement with the experimental data in the literature and demonstrate the viability of the strategy to take explicitly into account twin propagation during the simulation of plastic deformation of Mg alloys. Finally, an example of slip/twin interaction in polycrystals was simulated to show the capabilities of the model.

Improved Polymer Crystal Phase Field Model and Numerical Simulation

The existing phase field model of polymer crystallization contains many parameters that lack actual physical meaning. Although the value of these parameters can be adjusted to obtain results consistent with the experiment, it cannot correspond to the experimental conditions. In this paper, a new phase field model is established. By adjusting the latent heat, various forms of isotactic polystyrene crystals, such as dendrites, spherulites, lamellas, etc., can be simulated. Latent heat refers to the heat absorbed or released by a substance from one phase to another and has important physical meaning during the solidification process. The finite difference method was used to solve the model, and then the data were used to visualize. The simulation results were consistent with the experiment. Numerical simulation results under pure diffusion conditions show that the newly established phase field model can qualitatively predict the polymer growth process and provide a theoretical basis for the preparation and optimization of high-performance polymers. In order to make the simulation result closer to the actual growth of the crystal, the flow velocity is added in the simulation to make the melt convection. Under forced convection, the simulated polymer crystal image is no longer symmetrical.

Comparative study of magnetic properties of Mn[formula omitted] magnetic clusters in GaN using classical and quantum mechanical approach

Currently, simulations of many-body quantum systems are known to be computationally too demanding to be solved on classical computers. The main problem is that the computation time and memory necessary for performing the calculations usually grow exponentially with the number of particles N. An efficient approach to simulate many-body quantum systems is the use of classical approximation. Such approach can reduce the overall computational complexity, but it usually encounters difficulties in proper reproduction of temperature-dependent properties of a investigated system due to neglect of spin quantization. Therefore, it is timely and important to assess the validity of the classical approximation. To this end, in this work, we compare the results of numerical calculations of small Mn3+ paramagnetic clusters in GaN, where the Mn spins are treated classically with those where they are treated quantum-mechanically (crystal field model). In the first case, we solve the Landau–Lifshitz–Gilbert (LLG) equation that describes the precessional dynamics of spins represented by classical vectors. On the other hand, in the crystal field model, the state of Mn3+ ion (d4 configuration with S=2, L=2) is characterized by the set of orbital and spin quantum numbers |mS,mL>. Particular attention is paid to use numerical parameters that ensure the same single ion magnetic anisotropy in both classical and quantum approximation. Finally, a detailed comparative study of magnetization M(H,T) as a function of the magnetic field H, temperature T, number of ions in a given cluster N and the strength of super-exchange interaction J, obtained from both approaches will be presented.

Excited-state exchange interaction in NiO determined by high-resolution resonant inelastic x-ray scattering at the Ni M2,3 edges

The electronic and magnetic excitations of bulk NiO have been determined using the ${}^{3}{A}_{2g}$ to ${}^{3}{T}_{2g}$ crystal-field transition at the Ni ${M}_{2,3}$ edges with resonant inelastic x-ray scattering at 66.3- and 67.9-eV photon energies and 33-meV spectral resolution. Unambiguous assignment of the high-energy side of this state to a spin-flip satellite is achieved. We extract an effective exchange field of $89\ifmmode\pm\else\textpm\fi{}4$ meV in the ${}^{3}{T}_{2g}$ excited final state from empirical two-peak spin-flip model. The experimental data is found consistent with crystal-field model calculations using exchange fields of 60--100 meV. Full agreement with crystal-field multiplet calculations is achieved for the incident photon energy dependence of line shapes. The lower exchange parameter in the excited state as compared to the ground-state value of 120 meV is discussed in terms of the modification of the orbital occupancy (electronic effects) and of the structural dynamics: (A) With pure electronic effects, the lower exchange energy is attributed to the reduction in effective hopping integral. (B) With no electronic effects, we use the $S=1$ Heisenberg model of antiferromagnetism to derive a second-nearest-neighbor exchange constant ${J}_{2}$ = $14.8\ifmmode\pm\else\textpm\fi{}0.6$ meV. Based on the linear correlation between ${J}_{2}$ and the lattice parameter from pressure-dependent experiments, an upper limit of 2% local Ni-O bond elongation during the femtosecond scattering duration is derived.