Magnetic Linear Birefringence in Rare Earth Systems. II. Third-Order Approach

2004 ◽  
Vol 69 (1) ◽  
pp. 34-46
Author(s):  
Lidia Smentek
Keyword(s):  
Rare Earth ◽  
Field Potential ◽  
Second Order ◽  
Rare Earth Ions ◽  
Lanthanide Ions ◽  
Linear Birefringence ◽  
Third Order ◽  
The Third ◽  
The Impact ◽  
Effective Operators

The theory of linear magnetic birefringence of rare earth ions in crystals is extended here by the contributions that represent a direct perturbing influence of the crystal field potential surrounding the central ion. The basic assumptions of the theoretical model are the same as in the previous analysis of second-order terms. The third-order contributions introduced here break the free ionic system approximation, and they represent the impact due to configuration interaction. The effective operators include the perturbing influence of the excitations from the 4f shell to one-electron states of the same parity (as previously at the second order), and in addition, the excitations to states of opposite parity. All contributing terms are expressed by the effective operators that are defined within the perturbed function approach. The tensorial structure of these operators is discussed, and special attention is directed to newly defined radial integrals. The values of all radial integrals that are necessary for the third-order numerical analysis are presented in the case of all lanthanide ions.

Borrowing Intensity in Rare Earth Doped Materials; Magnetic Dipole Transitions

2005 ◽  
Vol 70 (7) ◽  
pp. 905-922 ◽  
Author(s):  
Brian G. Wybourne ◽  
Lidia Smentek ◽  
Andrzej Kędziorski
Keyword(s):  
Magnetic Dipole ◽  
Fourth Order ◽  
Field Potential ◽  
Transition Amplitude ◽  
Lanthanide Ions ◽  
Basis Sets ◽  
Interaction Operator ◽  
Magnetic Dipole Transitions ◽  
The Impact ◽  
Effective Operators

A theoretical model of magnetic dipole transitions in crystals doped by the lanthanide ions is based on the fourth-order perturbation theory applied for the Hamiltonian in which two perturbing mechanisms are taken into account. Namely, the impact due to the crystal field potential and also spin-orbit interaction operator is included in the transition amplitude. The analysis is performed in the language of Racah algebra applied for the formulation of the effective operators expressed in terms of unit double tensor operators. The radial integrals of the terms contributing at the fourth order are defined within the perturbed function approach, due to which their values are evaluated for the complete radial basis sets of one-electron states of given symmetry. The numerical analysis performed for the Eu3+ ion provides the information on the relative importance of various effective operators and allows one to establish a hierarchy of important terms contributing to the transition amplitude. The results of the analysis lead also to the verification of possible importance of magnetic dipole transitions in the description of 0 ↔ 0 and 0 ↔ 1 electric dipole transitions via the so-called borrowing mechanism introduced by Wybourne.

scholarly journals Changes in Incisor Third-Order Inclination Resulting from Vertical Variation in Lingual Bracket Placement

2009 ◽  
Vol 79 (4) ◽  
pp. 747-754 ◽  
Author(s):  
Michael Knösel ◽  
Klaus Jung ◽  
Liliam Gripp-Rudolph ◽  
Thomas Attin ◽  
Rengin Attin ◽  
...  
Keyword(s):  
Null Hypothesis ◽  
Interindividual Variation ◽  
Occlusal Plane ◽  
Third Order ◽  
Vertical Variation ◽  
Independent Variables ◽  
The Third ◽  
Dependent Variables ◽  
Incisor Inclination ◽  
The Impact

Abstract Objective: To test the null hypothesis that third-order measurements are not correlated to lingual incisor features seen on radiographs. Material and Methods: The lateral headfilms of 38 untreated, norm-occlusion subjects without incisor abrasions or restorations were used for third-order measurements of upper and lower central incisors and assessment of the inclination of four sites suitable for lingual bracket placement with reference to the occlusal plane perpendicular. Lingual sections were determined by the tangents at the incisal fossa (S1), at the transition plateau between incisal fossa and the cingulum (S2), by a constructed line reaching from the incisal tip to the cingulum (S3), and by a tangent at the cingulum convexity (S4). Third-order angles were also assessed on corresponding dental casts using an incisor inclination gauge. Regression analysis was performed using the third-order measurements of both methods as the dependent variables and the inclination of the lingual enamel sections (S1, S2, S3, S4) as the independent variables. Results: The null hypothesis was rejected. For the most common bracket application sites located on the lingual shovel (S1 and S2), third-order inclination changes of 0.4–0.7 degrees are expected for each degree of change in the inclination of the lingual surface. The impact of bracket placement errors on third-order angulation is similar between sections S1 and S2 and the cingulum convexity (S4). Section S3 proved to be least affected by interindividual variation. Conclusion: The third-order measurements are correlated to lingual incisor features. Accordingly, third-order changes resulting from variation in lingual bracket placement can be individually predicted from radiographic assessments.

scholarly journals B-splines Algorithms for Solving Fredholm Linear Integro-Differential Equations

2004 ◽  
Vol 1 (2) ◽  
pp. 340-346
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Spline Function ◽  
Cubic Spline ◽  
Second Order ◽  
Third Order ◽  
B Spline ◽  
Numerical Examples ◽  
B Splines ◽  
The Third ◽  
New Procedures

Algorithms using the second order of B -splines [B (x)] and the third order of B -splines [B,3(x)] are derived to solve 1' , 2nd and 3rd linear Fredholm integro-differential equations (F1DEs). These new procedures have all the useful properties of B -spline function and can be used comparatively greater computational ease and efficiency.The results of these algorithms are compared with the cubic spline function.Two numerical examples are given for conciliated the results of this method.

scholarly journals Lattice Location of Rare Earth Ions in Semiconductors: Interpretation and Limitations of Using g Values

2005 ◽  
Vol 866 ◽  
Author(s):  
David Carey
Keyword(s):  
Rare Earth ◽  
Crystal Field ◽  
Field Potential ◽  
Rare Earth Ions ◽  
Lattice Location ◽  
Paramagnetic Resonance ◽  
Cubic Symmetry ◽  
Crystal Fields ◽  
Valid Comparison ◽  
G Value

AbstractThe g values of rare earth ions obtained from either paramagnetic resonance or Zeeman measurements are often used to interpret the location and/or environment surrounding rare earth ions. In the case of centres with cubic symmetry the g value can be used to distinguish between substitutional and interstitial sites. For centres with less than cubic symmetry the average g value, taken as 1/3 trace of the g tensor, is often used as an indication of the lattice location and/or a measure of the strength of the local crystal field. This approach is widely used but is based on the assumption that the non-cubic terms in the total crystal field potential are small compared with the cubic crystal field. In this paper we have explored this assumption by calculating the principal g values in axial crystal fields for the Er3+ ion. We examine the limits over which the average g value approach is valid. Comparison is made with published results.

A Two-Dimensional Potential Flow Model of the Wave Field Generated by a Semisubmerged Body in Heaving Motion

1988 ◽  
Vol 32 (02) ◽  
pp. 83-91
Author(s):  
X. M. Wang ◽  
M. L. Spaulding
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Potential Flow ◽  
Flow Model ◽  
Second Order ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
Potential Flow Model ◽  
Good Agreement

A two-dimensional potential flow model is formulated to predict the wave field and forces generated by a sere!submerged body in forced heaving motion. The potential flow problem is solved on a boundary fitted coordinate system that deforms in response to the motion of the free surface and the heaving body. The full nonlinear kinematic and dynamic boundary conditions are used at the free surface. The governing equations and associated boundary conditions are solved by a second-order finite-difference technique based on the modified Euler method for the time domain and a successive overrelaxation (SOR) procedure for the spatial domain. A series of sensitivity studies of grid size and resolution, time step, free surface and body grid redistribution schemes, convergence criteria, and free surface body boundary condition specification was performed to investigate the computational characteristics of the model. The model was applied to predict the forces generated by the forced oscillation of a U-shaped cylinder. Numerical model predictions are generally in good agreement with the available second-order theories for the first-order pressure and force coefficients, but clearly show that the third-order terms are larger than the second-order terms when nonlinearity becomes important in the dimensionless frequency range 1≤ Fr≤ 2. The model results are in good agreement with the available experimental data and confirm the importance of the third order terms.

WENO-TVD schemes for hyperbolic conservation laws

Analysis ◽  
2007 ◽  
Vol 27 (1) ◽  
Author(s):  
Yousef Hashem Zahran
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Building Blocks ◽  
Second Order ◽  
Third Order ◽  
Weno Schemes ◽  
Hyperbolic Conservation ◽  
Systematic Assessment ◽  
The Third ◽  
Seventh Order

The purpose of this paper is twofold. Firstly we carry out a modification of the finite volume WENO (weighted essentially non-oscillatory) scheme of Titarev and Toro [14] and [15].This modification is done by using two fluxes as building blocks in spatially fifth order WENO schemes instead of the second order TVD flux proposed by Titarev and Toro [14] and [15]. These fluxes are the second order TVD flux [19] and the third order TVD flux [20].Secondly, we propose to use these fluxes as a building block in spatially seventh order WENO schemes. The numerical solution is advanced in time by the third order TVD Runge–Kutta method. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws, in one and two dimension is presented. Systematic assessment of the proposed schemes shows substantial gains in accuracy and better resolution of discontinuities, particularly for problems involving long time evolution containing both smooth and non-smooth features.

Contigious hyperquadrics of coequipped hyperbands sHm

2019 ◽  
pp. 126-132
Author(s):  
Yu. Popov
Keyword(s):  
Projective Space ◽  
Second Order ◽  
Third Order ◽  
Base Surface ◽  
The Third ◽  
Two Parameter ◽  
Order Contact

We consider hyperquadrics that are internally connected to coequipped hyperbands in the projective space. Specifically, a hyperquadric Qn1 tangent to a hyperplane at the point is called a contiguous hyper quadric of a hyperband if it has a second-order contact with the base surface of the hyperband. In a the third order differential neighborhood of the forming element of the hyperband, two two-parameter bundles of fields of adjoining hyperquadrics are internally invariantly joined, their equations are given in a dot frame. The set of hyperquadrics such that the plane and the plane of Cartan are conjugate with respect to hyperquadric Qn1 is considered. The condition is shown under which the normal of the 2nd kind and the Cartan plane are conjugate with respect to the hyperquadric Qn1 . In addition, the following theorem is proved: normalization of a coequipped regular hyperband has a semi-internal equipment if and only if its normals of the first and second kind are polarly conjugate with respect to the hyperquadric.

Perturbation treatment of the Hartree–Fock equations for the ground states of the boron-like ions

1969 ◽  
Vol 47 (7) ◽  
pp. 699-705 ◽  
Author(s):  
C. S. Sharma ◽  
R. G. Wilson
Keyword(s):  
Ground State ◽  
Atomic System ◽  
Ground States ◽  
Great Accuracy ◽  
Second Order ◽  
Expectation Values ◽  
Third Order ◽  
First Order ◽  
Hartree Fock ◽  
The Third

The first-order Hartree–Fock and unrestricted Hartree–Fock equations for the ground state of a five electron atomic system are solved exactly. The solutions are used to evaluate the corresponding second-order energies exactly and the third-order energies with great accuracy. The first-order terms in the expectation values of 1/r, r, r2, and δ(r) are also calculated.

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