Perturbation Formulas Research Articles

Theoretical Studies on the EPR Parameters and Local Structures for the Orthorhombic and Tetragonal Cu2+ Centers in Cu1-xHxZr2(PO4)3.

The defect structures of the orthorhombical and tetragonal Cu2+ centers in Cu1-xHxZr2(PO4)3 are theoretically studied by analyzing their experimental electron paramagnetic resonance (EPR) parameters, based on the perturbation formulas of these parameters for a 3d9 ion in orthorhombically and tetragonally elongated octahedra, respectively. The above centers are attributed to the Cu2+ ions locating at M(1) site, and the crystal field parameters (CFPs) are quantitatively determined from the superposition model and the local structures of the Cu2+ sites. Based on the calculation, the parallel Cu-O bonds may undergo the relative elongations ΔZO (≈ 0.113 Å) and ΔZT (≈ 0.102 Å) for the orthorhombical and tetragonal Cu2+ centers in Cu1-xHxZr2(PO4)3 along z-axis, respectively. Meanwhile, the planar Cu-O bonds are found to experience the relative variation δr (≈ 0.056 Å) along the x- and y-axes for the orthorhombical Cu2+ center because of the Jahn-Teller (JT) effect. The theoretical EPR parameters based on the above local structures agree well with the observed values.

An asymptotic formula for perturbations in an electromagnetic eigenvalue problem due to the presence of small deformable dielectric inclusion

AbstractWe consider a homogeneous electromagnetic material enclosing a bounded ‐domain with a small dielectric inclusion. We derive a rigorous asymptotic expansion of the eigenvalues of the Maxwell operator as the diameter of the inclusion goes to zero.

Primordial Perturbations Including Second-Order Derivatives of the Inflationary Potential

In inflationary cosmology, the form of the potential is still an open problem. In this work, second-order effects of the inflationary potential are evaluated and related to the known formula for the primordial perturbations at a wide range of scales. We found effects that may help to unravel the unknown inflationary potential form and impose new constraints on the parameters that define this potential. In particular, we demonstrate that even slight deviations in the inflationary potential can lead to significant differences in the calculated spectra if inflation persists sufficiently long and the normal modes of perturbations are affected by these variations.

Large deformation problem of hyperbolic shallow shells with bimodular effect: A perturbation‐variation method

AbstractAs an important analytical method, the perturbation‐variation method combines the advantages of perturbation method and variational method, and is widely used for the solution of nonlinear plate and shell problems. In this study, the perturbation‐variation method is used to solve the large deformation problem of a flexible hyperbolic thin shallow shells with different moduli in tension and compression (bimodular effect). First, we establish the governing equations expressed in terms of three displacement components, in which the bimodular effect of materials is considered in physical equations and the large deformation of shell is considered in geometrical equations. Next, we use the perturbation‐variation method to solve the governing equations established, obtaining the perturbation‐variation solution up to third‐order. During the application of the method, the displacements are expressed into perturbation formulas of all levels first, and the unknown constants and functions in the perturbation formulas are determined by the extreme value conditions of the functional established (variational principle), hence the perturbation‐variation method gets its name from this practice. The numerical results from FEM basically agree with the perturbation‐variation solution obtained. The method proposed may be extended into the solution of similar partial differential equations established in applied fields.

Inverse conductivity problem with one measurement: uniqueness of multi-layer structures

In this paper, we study the recovery of multi-layer structures in inverse conductivity problem by using one measurement. First, we define the concept of Generalized Polarization Tensors (GPTs) for multi-layered medium and show some important properties of the proposed GPTs. With the help of GPTs, we present the perturbation formula for general multi-layered medium. Then we derive the perturbed electric potential for multi-layer concentric disks structure in terms of the so-called generalized polarization matrix, whose dimension is the same as the number of the layers. By delicate analysis, we derive an algebraic identity involving the geometric and material configurations of multi-layer concentric disks. This enables us to reconstruct the multi-layer structures by using only one partial-order measurement.

Investigations on the EPR g factors and local structures for the orthorhombic and tetragonal Cu2+ centers in Pb[Zr0.54Ti0.46]O3.

The defect models of the orthorhombical and tetragonal Cu2+ centers in Pb[Zr0.54Ti0.46]O3 are attributed to Cu2+ ions occupying the sixfold coordinated octahedral Ti4+ site with and without charge compensation, respectively. The electron paramagnetic resonance (EPR) g factors gi (i = x, y, z) of the Cu2+ centers in Pb[Zr0.54Ti0.46]O3 are theoretically studied by using the perturbation formulas of a 3d9 ion under orthorhombically and tetragonally elongated octahedra. Based on the calculation, the impurity off-center displacements are about 0.253 and 0.162 Å for the orthorhombical and tetragonal Cu2+ centers, respectively. Meanwhile, the planar Cu2+-O2- bonds are found to experience the relative variation ΔR (≈0.102 Å) along the a- and b-axes for the orthorhombical Cu2+ center due to the Jahn-Teller (JT) effect. The theoretical EPR g factors based on the above local structures agree well with the observed values.

Perturbation formulae for the generalized core–EP inverse

Perturbation formulae for the generalized core–EP inverse

Excitonic instability towards a Potts-nematic quantum paramagnet

Magnetic frustration can lead to peculiar magnetic orderings that break a discrete symmetry of the lattice in addition to the fundamental magnetic symmetries (i.e., spin rotation invariance and time-reversal symmetry). In this work, we focus on frustrated quantum magnets and study the nature of the quantum phase transition between a paramagnet and a magnetically ordered state with broken threefold (Z3) crystal rotation symmetry. We show that the transition can occur in two stages, giving rise to an intermediate nematic phase in which rotation symmetry is broken but the system remains magnetically disordered. Since the nematic transition is described by the three-state Potts model, the intermediate phase is a Z3 Potts-nematic phase. Our prediction of the existence of a Potts-nematic phase is based on an analysis of bound states formed from two-magnon excitations in the paramagnet, which become gapless while single-magnon excitations remain gapped. By considering three different lattice models, we demonstrate a generic instability towards two-magnon bound state formation in the Potts-nematic channel. We present both numerical results and a general analytical perturbative formula for the bound state binding energy similar to BCS theory. We further discuss a number of different materials that realize key features of the model considered, and thus provide promising venues for possible experimental observation. Published by the American Physical Society 2024

Second‐order trace formulas

AbstractKoplienko [Sib. Mat. Zh. 25 (1984), 62–71; English transl. in Siberian Math. J. 25 (1984), 735–743] found a trace formula for perturbations of self‐adjoint operators by operators of Hilbert–Schmidt class . Later, Neidhardt introduced a similar formula in the case of pairs of unitaries via multiplicative path in [Math. Nachr. 138 (1988), 7–25]. In 2012, Potapov and Sukochev [Comm. Math. Phys. 309 (2012), no. 3, 693–702] obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon [Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107, 202; Open Question 11.2]. In this paper, we supply a new proof of the Koplienko trace formula in the case of pairs of contractions , where the initial operator is normal, via linear path by reducing the problem to a finite‐dimensional one as in the proof of Krein's trace formula by Voiculescu [Oper. Theory Adv. Appl. 24 (1987) 329–332] and Sinha and Mohapatra [Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 4, 819–853] and [Integral Equations Operator Theory 24 (1996), no. 3, 285–297]. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self‐adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko–Neidhardt trace formula for a class of pairs of contractions via multiplicative path using the finite‐dimensional approximation method.

Investigations on the EPR parameters and local structures for the substitutional Ti3+ and W5+ centers in stishovite

Abstract Local structures and electron paramagnetic resonance (EPR) parameters (g factors g x , g y , and g z ) for the substitutional Ti3+ and W5+ centers in stishovite are theoretically investigated by using the high-order perturbation formulas of these parameters for a d1 ion in rhombically compressed octahedra. In the calculation formulas, the related molecular orbital coefficients are obtained from the cluster approach, and the relevant crystal-field (CF) parameters are determined from the superposition model, which enables to connect these CF parameters and, hence, the studied g factors with the local structures of the Ti3+ and W5+ centers in stishovite. Based on the calculations, the impurity–ligand bond lengths parallel and perpendicular to the C 2-axis are found to be R′|| (≈1.751 and 1.717 Å) and R′⊥ (≈1.788 and 1.806 Å) with the planar bond angles θ′ (≈89.0° and 88.2°) for the studied [TiO6]9− and [WO6]7− clusters, respectively. The calculated results are in good agreement with the experimental data, and the validity of the results is discussed.

Theoretical studies on the local structure and spin Hamiltonian parameters for Cu2+ ions in LiTaO3 crystal.

The local structure and spin Hamiltonian parameters (SHPs) g factors (gx , gy , gz ) and the hyperfine structure constants (Ax , Ay , Az ) for Cu2+ doped in the LiTaO3 crystal are theoretically investigated by the perturbation formulas for a 3d9 ion under rhombically elongated octahedral based on the cluster approach. The impurity Cu2+ was assumed to occupy the host trigonally-distorted octahedral Li+ site and experience the Jahn-Teller (JT) distortion from the host trigonal octahedral [TaO6 ]10- to the impurity rhombically elongated octahedral [CuO6 ]10- . Based on the calculations, the impurity-ligand bond lengths parallel and perpendicular to the C2 -axis are found to be R|| (≈ 2.305 Å) and R⊥ (≈ 2.112 Å) for the studied [CuO6 ]10- cluster, with the planar bond angle θ (≈ 78.2°). Meanwhile, the ground-state wave function for Cu2+ center in LiTaO3 was also obtained. The calculated SHPs based on the above local lattice distortions agree well with the experimental data, and the results are discussed.

Q-Factor Optimization of Modes in Ordered and Disordered Photonic Systems Using Non-Hermitian Perturbation Theory.

The quality factor, Q, of photonic resonators permeates most figures of merit in applications that rely on cavity-enhanced light-matter interaction such as all-optical information processing, high-resolution sensing, or ultralow-threshold lasing. As a consequence, large-scale efforts have been devoted to understanding and efficiently computing and optimizing the Q of optical resonators in the design stage. This has generated large know-how on the relation between physical quantities of the cavity, e.g., Q, and controllable parameters, e.g., hole positions, for engineered cavities in gaped photonic crystals. However, such a correspondence is much less intuitive in the case of modes in disordered photonic media, e.g., Anderson-localized modes. Here, we demonstrate that the theoretical framework of quasinormal modes (QNMs), a non-Hermitian perturbation theory for shifting material boundaries, and a finite-element complex eigensolver provide an ideal toolbox for the automated shape optimization of Q of a single photonic mode in both ordered and disordered environments. We benchmark the non-Hermitian perturbation formula and employ it to optimize the Q-factor of a photonic mode relative to the position of vertically etched holes in a dielectric slab for two different settings: first, for the fundamental mode of L3 cavities with various footprints, demonstrating that the approach simultaneously takes in-plane and out-of-plane losses into account and leads to minor modal structure modifications; and second, for an Anderson-localized mode with an initial Q of 200, which evolves into a completely different mode, displaying a threefold reduction in the mode volume, a different overall spatial location, and, notably, a 3 order of magnitude increase in Q.

Theoretical studies on the g-factors and the local structure of W5+ ions in tungsten phosphate glasses

In this work, we adopt the three-order perturbation formulae for g-factors (g//, g^) of d1 ions in the octahedral environment to calculate the g-factors of W5+ ions in tungsten phosphate glasses containing lithium (P2O5-Li2WO4-Li2O). In the light of the high valence state of the studied W5+ centers and hence the strong covalency of the studied octahedral [WO6]7- cluster, we consider the contributions to g-factors from the ligand orbital and spin-orbit (SO) coupling interactions based on the cluster approach. The required tetragonal crystal-field parameters are calculated from the local structure of W5+ ions based on the superposition model. According to the theoretical calculations, we find that the octahedral [WO6]7- clusters possess the tetragonally compressed distortion with a shorter W-O bond length (≈1.54 Å) and a longer one (≈2.26 Å) along C4 axis and four normal W-O bond length (≈1.94 Å) in the perpendicular plane, which infers that the W5+ ions are in the form of tungstyl ions (i.e., WO3-). Based on the local structural data, the theoretical values of g// and g^ agree well with the experimental values.

Dispersive estimates for the Schrödinger equation with finite rank perturbations

In this paper, we investigate dispersive estimates for the time evolution of HamiltoniansH=−Δ+∑j=1N〈⋅,φj〉φjinRd,d≥1, where each φj satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant C=C(N,d,φ1,…,φN)>0 such that‖e−itH‖L1−L∞≤Ct−d2,fort>0. As far as we are aware, this seems to provide the first study of L1−L∞ estimates for finite rank perturbations of the Laplacian in any dimension.We first deal with rank one perturbations (N=1). Then we turn to the general case. By using an Aronszajn-Krein type formula for finite rank perturbations, we reduce the problem to the rank one case and solve it in a unified manner. Moreover, we show that in some specific situations, the constant C(N,d,φ1,…,φN) grows polynomially in N. Finally, as an application, we are able to extend the results to N=∞ and deal with some trace class perturbations.

A comparative study on the local structures of the V4+ and Cu2+ centers in WO3

Abstract The local structures and electron paramagnetic resonance (EPR) parameters (the g factors g i and the hyperfine structure constants A i , with i = x, y, z) of the V4+ and Cu2+ centers in WO3 were theoretically investigated based on the perturbation formulas of the EPR parameters for 3d1 and 3d9 ions under rhombically compressed and elongated octahedra, respectively. In these formulas, the adopted crystal-field parameters (CFPs) are obtained from the superposition model which enables connection of the CFPs and hence the EPR parameters to the rhombic distortion. Based on the calculations, the impurity-ligand distances of the studied [VO6]8− (or [CuO6]10−) cluster are found to suffer the axial compression (or elongation) δz of about 0.098 Å (or 0.132 Å) along the z-axis, and the additional planar bond length variation δr (≈0.052 or 0.047 Å) along x- and y-axis, respectively, due to the Jahn–Teller (JT) effect.

General Trace Formula for Perturbations from the Gilbert–Shmidt Class

General Trace Formula for Perturbations from the Gilbert–Shmidt Class

Investigations of the Spin-Hamiltonian Parameters and Defect Structure for the (TiO4)5− Tetrahedral Clusters in ZrSiO4: Ti3+ Crystal

The spin-Hamiltonian parameters (g factors g//, g⊥ and hyperfine structure constants 47Ai, 49Ai, where i = // or ⊥, 47Ai and 49Ai denote the values of isotopes 47Ti3+ and 49Ti3+) of Ti3+ ions at the tetragonal tetrahedral Si4+ sites in ZrSiO4 crystal are computed from the high-order perturbation formulas based on the cluster approach where the contributions due to both the d orbitals of central dn ion and p orbitals of ligand ion are considered. The calculated results are reasonably coincident with the experimental values. The defect structure of tetragonal (TiO4)5− impurity cluster which differs from the replaced host (SiO4)4− cluster in ZrSiO4 is evaluated from the calculation. The results are discussed.

A theoretical study of structure properties and stability in different ligands coordinated Cu-MOFs

A comprehensively theoretical study of structural properties and thermostability is performed to investigate the role of ligands in Metal Organic Frameworks (MOFs). By fitting the experimental EPR spectra, the high order perturbation formulae of a 3 d9 ion in four Cu-MOFs with ligands btm, bte, btp and btb are adopted for orthorhombically and tetragonally elongated octahedral [CuO2N4]12− groups so as to reveal the local environment around central Cu2+, including the detail structure information and local electromagnetic properties. Since the application of MOFs are normally limited by the unsatisfied thermostability, the molecule dynamic (MD) calculation based on density functional theory (DFT) is performed to clarify the influence on the stability of MOFs from the different ligands. Based on the above investigations, the mechanism of ligands substitution in MOFs can be comprehensively understood, which would be useful for designing and synthesizing novel MOFs.

The role of different alkaline earth oxide composition in copper borate glasses: Structure and optical properties

According to the experimental EPR, optical absorption and FTIR spectra of 10MO-(30‒x) ZnO-60B2O3-xCuO (M=Mg, Ca, and Sr) glasses, the local structure of Cu2+ center and the role of different alkaline earth oxides (MgO, CaO, SrO) are obtained from high order perturbation formulae. Based on the well fitted spin-Hamiltonian parameters, the clusters [CuO6]10‒ in different alkaline earth oxides glasses all are found to suffer the tetragonally elongated distortion (D4h) and the different composite ingredients of alkaline earth (M=Mg, Ca, and Sr) can strengthen the distortion. In addition, based on the reasonably calculated spin-Hamiltonian parameters, the paramagnetic susceptibility (χ) of MZBC, CZBC, and SZBC glasses is also calculated. As a remarkable feature of glass, the local optical basicity Λloc is obtained from the reasonably obtained local environment (e.g., occupation, coordination number, local symmetry, the strength of crystal fields) of 10MO-(30‒x) ZnO-60B2O3-xCuO glasses.

Complex Permittivity Measurement Utilizing Multiple Modes of a Rectangular Cavity

A novel method for measuring the dielectric constant and loss tangent of dielectric substrates is presented in this article. Four resonant modes of the rectangular resonant cavity, TE101, TE105, TE109, and TE1015, are utilized to realize the multiband measurement from 0.9 to 10 GHz. The introduction of additional slots cuts down the quality factor of some miscellaneous modes, making the working modes more obvious and convenient for extraction. Based on the traditional perturbation formulas, considering the sinusoidal field distribution in the sample, modified formulas are deduced to obtain more accurate results. An experimental resonant cavity is fabricated and measured to demonstrate the effectivity of the proposed method.