ESR STUDY OF SPIN-HAMILTONIAN PARAMETERS AND CRYSTAL FIELD ENERGY LEVELS FOR THE LOW C3 SYMMETRY Fe3+ CENTER IN GREEN SAPPHIRE CRYSTALS AND POWDER

2007 ◽  
Vol 21 (04) ◽  
pp. 225-236 ◽  
Author(s):  
P. LIMSUWAN ◽  
N. UDOMKAN ◽  
P. WINOTAI
Keyword(s):  
Rotation Angle ◽  
Energy Levels ◽  
Heat Treating ◽  
Esr Spectra ◽  
Field Energy ◽  
Hamiltonian Matrix ◽  
Magnetic Field Direction ◽  
Spin Hamiltonian ◽  
Corundum Structure ◽  
Hamiltonian Parameters

In this report, Fe 3+ impurity ions present in green sapphire ( Al 2 O 3) were studied experimentally, by heating a light green sapphire in flowing oxygen atmosphere for 12 h from 1200, 1300, 1400, 1500 and 1600°C, respectively. Electron spin resonance (ESR) spectra in X-band (~9.45 GHz ) were recorded by mounting the crystal with the c-axis perpendicular (θ = 90°) to the magnetic field direction. The spectra were recorded and simulated by a numerical diagonalization of spin Hamiltonian matrix in the range from 0 to 180 degrees for every 15 degrees of rotation angle (φ). In our case, only the last two sets of peaks strongly depend on the rotation angle (φ), and each exhibits C 3 symmetry due to two magnetically inequivalent Fe 3+ sites in the corundum structure. For polycrystalline ESR spectra, seven main Fe 3+ ESR absorption peaks occur at the resonance magnetic fields of 100.20, 310.24, 486.80, 525.00, 550.60, 761.00 and 777.00 mT respectively. Specifically, ESR signals show that the number of paramagnetic Fe 3+ ions increase roughly linearly with the heat treating temperature, having the [Formula: see text] ratio ~1.41 at 1600°C.

scholarly journals On the relations between some well-known methods and the projective Riccati equations

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 613-618
Author(s):  
Şamil Akçağıl
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Riccati Equations ◽  
Nonlinear Evolution Equations ◽  
Alternative Methods ◽  
Computational Techniques ◽  
Partial Differential ◽  
Traditional Methods

AbstractSolving nonlinear evolution equations is an important issue in the mathematical and physical sciences. Therefore, traditional methods, such as the method of characteristics, are used to solve nonlinear partial differential equations. A general method for determining analytical solutions for partial differential equations has not been found among traditional methods. Due to the development of symbolic computational techniques many alternative methods, such as hyperbolic tangent function methods, have been introduced in the last 50 years. Although all of them were introduced as a new method, some of them are similar to each other. In this study, we examine the following four important methods intensively used in the literature: the tanh–coth method, the modified Kudryashov method, the F-expansion method and the generalized Riccati equation mapping method. The similarities of these methods attracted our attention, and we give a link between the methods and a system of projective Riccati equations. It is possible to derive new solution methods for nonlinear evolution equations by using this connection.

scholarly journals Reduction of the molecular hamiltonian matrix using quantum community detection

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Susan M. Mniszewski ◽  
Pavel A. Dub ◽  
Sergei Tretiak ◽  
Petr M. Anisimov ◽  
Yu Zhang ◽  
...  
Keyword(s):  
Community Detection ◽  
Slater Determinant ◽  
Hamiltonian Matrix ◽  
Approximate Methods ◽  
Molecular Systems ◽  
Hartree Fock ◽  
Molecular Hamiltonian ◽  
Structure Problem ◽  
Modularity Maximization ◽  
Chemical Knowledge

AbstractQuantum chemistry is interested in calculating ground and excited states of molecular systems by solving the electronic Schrödinger equation. The exact numerical solution of this equation, frequently represented as an eigenvalue problem, remains unfeasible for most molecules and requires approximate methods. In this paper we introduce the use of Quantum Community Detection performed using the D-Wave quantum annealer to reduce the molecular Hamiltonian matrix in Slater determinant basis without chemical knowledge. Given a molecule represented by a matrix of Slater determinants, the connectivity between Slater determinants (as off-diagonal elements) is viewed as a graph adjacency matrix for determining multiple communities based on modularity maximization. A gauge metric based on perturbation theory is used to determine the lowest energy cluster. This cluster or sub-matrix of Slater determinants is used to calculate approximate ground state and excited state energies within chemical accuracy. The details of this method are described along with demonstrating its performance across multiple molecules of interest and bond dissociation cases. These examples provide proof-of-principle results for approximate solution of the electronic structure problem using quantum computing. This approach is general and shows potential to reduce the computational complexity of post-Hartree–Fock methods as future advances in quantum hardware become available.

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