Energy levels of Nd(IV) and Er(IV) in Y2O3

1969 ◽  
Vol 47 (24) ◽  
pp. 2753-2761 ◽  
Author(s):  
P. A. Narayana
Keyword(s):  
Taylor Series ◽  
Energy Levels ◽  
Ion Energy ◽  
Partial Derivatives ◽  
Free Ion ◽  
Linear And Nonlinear ◽  
Derivatives Of

The free-ion energy levels of Nd(IV) and Er(IV) in Y2O3 are expanded in a Taylor series, taking the effects of both linear and nonlinear configurational interactions into account. Numerical values of the partial derivatives of the energy levels with respect to F2, F4, F6, ξ, α, β, Y(22,1), and Y(22,3) are tabulated. The r.m.s. deviation in the case of Nd(IV) is improved from 90 cm−1 to 48 cm−1 and, in the case of Er(IV), the r.m.s. deviation is improved from 61.5 cm−1 to 43 cm−1. The improvements are the result of the inclusion of the extra parameters.

Comparative analysis of free-ion energy levels of Er3+ (4f11) in various crystal hosts

1993 ◽  
Vol 193 (1-2) ◽  
pp. 203-206 ◽  
Author(s):  
C.K. Jayasankar ◽  
A. Renuka Devi ◽  
E. Rukmini ◽  
M.F. Reid
Keyword(s):  
Comparative Analysis ◽  
Energy Levels ◽  
Ion Energy ◽  
Free Ion

Analogical Modelling and Numerical Simulation of the Adsorption Process for Poly-Phenothiazine Formaldehyde on Graphite Electrodes

2010 ◽  
Vol 7 (1) ◽  
Author(s):  
Mihaela-Ligia M. Unguresan ◽  
Delia Maria Gligor ◽  
Francisc Dulf ◽  
Tiberiu Colosi
Keyword(s):  
Numerical Simulation ◽  
Taylor Series ◽  
Adsorption Process ◽  
Phenothiazine Derivative ◽  
Graphite Electrodes ◽  
Partial Derivatives ◽  
Independent Variables ◽  
Regulation Scheme ◽  
Derivatives Of ◽  
Two Independent Variables

The paper presents the dispersion of the concentration y(t, s) on the length (s) with respect to time (t), corresponding to the adsorption process of a phenothiazine derivative on graphite electrodes. The numerical simulation by equations with partial derivatives of the second order with two independent variables (t and s) (PDE II.2), based on (Mpdx) which associates with Taylor series was performed. Also, the adsorption process defined by PDE II.2 was included in a regulation scheme of concentration y(t, s) with multiple freedom levels. It insures good performances and a remarkable flexibility for extending the method in similar categories of applications.

Full-Scale Bounds Estimation for the Nonlinear Transient Heat Transfer Problems with Interval Uncertainties

2021 ◽  
pp. 2150026
Author(s):  
Ruifei Peng ◽  
Haitian Yang ◽  
Yanni Xue
Keyword(s):  
Heat Transfer ◽  
Series Expansion ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Transient Heat Transfer ◽  
High Fidelity ◽  
Interval Scale ◽  
Nonlinear Transient ◽  
Transfer Problems ◽  
Derivatives Of

A package solution is presented for the full-scale bounds estimation of temperature in the nonlinear transient heat transfer problems with small or large uncertainties. When the interval scale is relatively small, an efficient Taylor series expansion-based bounds estimation of temperature is stressed on the acquirement of first and second-order derivatives of temperature with high fidelity. When the interval scale is relatively large, an optimization-based approach in conjunction with a dimension-adaptive sparse grid (DSG) surrogate is developed for the bounds estimation of temperature, and the heavy computational burden of repeated deterministic solutions of nonlinear transient heat transfer problems can be efficiently alleviated by the DSG surrogate. A temporally piecewise adaptive algorithm with high fidelity is employed to gain the deterministic solution of temperature, and is further developed for recursive adaptive computing of the first and second-order derivatives of temperature. Therefore, the implementation of Taylor series expansion and the construction of DSG surrogate are underpinned by a reliable numerical platform. The parallelization is utilized for the construction of DSG surrogate for further acceleration. The accuracy and efficiency of the proposed approaches are demonstrated by two numerical examples.

Hard Carbon Films with Various Microstructure Prepared by Ion Assisted Evaporation

1993 ◽  
Vol 316 ◽  
Author(s):  
J. Ullmann ◽  
A. Weber ◽  
U. Falke
Keyword(s):  
Growth Mechanism ◽  
Film Growth ◽  
Carbon Films ◽  
Carbon Surface ◽  
Vickers Indentation ◽  
Structure Modification ◽  
Hard Carbon ◽  
Ion Etching ◽  
Ion Energy ◽  
Free Ion

ABSTRACTFor a deeper understanding of the creation of carbon films the hydrogen-free ion assisted evaporation (IAE) method with neon species was used. Variation of the ion parameters energy and ion to neutral arrival ratio, delivering the necessary energy for modification of the film growth, results in different microstructures investigated with EELS, HRTEM and TED as well as different microhardnesses measured by dynamical Vickers indentation. A possible film growth mechanism is proposed based on an ion etching of mainly sp2-bonded carbon surface atoms and on defect dominated structure modification below the surface depending on the ion energy

Partial Derivatives of the Lambert Problem

2014 ◽  
Author(s):  
Nitin Arora ◽  
Ryan P. Russell ◽  
Nathan J. Strange
Keyword(s):  
Partial Derivatives ◽  
Lambert Problem ◽  
Derivatives Of

Perturbation theory and finite Markov chains

1968 ◽  
Vol 5 (2) ◽  
pp. 401-413 ◽  
Author(s):  
Paul J. Schweitzer
Keyword(s):  
Perturbation Theory ◽  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Fundamental Matrix ◽  
Transition Probabilities ◽  
Semi Group ◽  
Partial Derivatives ◽  
Finite Markov Chains ◽  
Derivatives Of

A perturbation formalism is presented which shows how the stationary distribution and fundamental matrix of a Markov chain containing a single irreducible set of states change as the transition probabilities vary. Expressions are given for the partial derivatives of the stationary distribution and fundamental matrix with respect to the transition probabilities. Semi-group properties of the generators of transformations from one Markov chain to another are investigated. It is shown that a perturbation formalism exists in the multiple subchain case if and only if the change in the transition probabilities does not alter the number of, or intermix the various subchains. The formalism is presented when this condition is satisfied.

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