Higher-order terms in the dielectric constant of ionic crystals

1959 ◽  
Vol 252 (1269) ◽  
pp. 217-235 ◽  
Keyword(s):  
Dielectric Constant ◽  
Dipole Moment ◽  
Higher Order ◽  
Second Order ◽  
Static Dielectric Constant ◽  
Order Moment ◽  
Ionic Crystals ◽  
Third Order ◽  
The Third ◽  
Infra Red

The infra-red absorption of ionic crystals differs in important details from the predictions of the theory based on first approximations. It is known that this discrepancy may be due to two effects which are neglected in such a theory, namely, to the anharmonic terms in the potential energy and to those terms in the dipole moment which are of higher order than the first in the displacement co-ordinates. These higher-order terms in the dipole moment arise from the deformation of the electron shells. The present paper develops in a systematic way the influence of these higher-order effects on the static dielectric constant. Because of the dispersion relations, the terms occurring in the static dielectric constant must also appear in the infra-red absorption spectrum . It is found that the third- and the fourth-order potential, the second- and the third-order dipole moment, and cross-terms between the second-order moment and the third-order potential, all con­tribute terms in the same order to the static dielectric constant. It is also found that the third-order potential contains important contributions from the long-range dipolar inter­action. These dipolar contributions are proportional to the product of the first- and second-order dipole moments, and it follows that in ionic crystals a large second-order moment automatically results in a large third-order potential. It is suggested that these dipolar contributions to the third-order potential may be responsible for the fact that in the infra-red spectra of different ionic crystals not only the intensity of the side band but also the width of the main band varies in the same way as the deformability of the electron shells.

The infra-red spectra of crystals

1960 ◽  
Vol 258 (1294) ◽  
pp. 377-401 ◽  
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Order Moment ◽  
Order Effects ◽  
Ionic Crystals ◽  
Third Order ◽  
Main Band ◽  
The Third ◽  
Cross Terms ◽  
Infra Red

The higher-order effects in the intrinsic infra-red absorption of crystals are investigated in a systematic way. In agreement with a previous paper which dealt with the static dielectric constant, it is found that in the case of ionic crystals the third- and fourth-order potential, the second- and the third-order dipole moment, and the cross-terms between the second-order moment and the third-order potential, all contribute terms of the same order to the infra-red spectrum. In the lowest approximation, the third-order moment and the fourth-order potential only affect the absorption in the immediate neighbourhood of the maximum and hence have little effect on the shape of the spectrum. The broadening of the main band is due mainly to the third-order potential, while the side bands may be caused by the second-order moment as well as by the third-order potential and by cross-terms between the two. But due to an internal field effect, in strongly ionic crystals a large second-order moment automatically leads to a large third-order potential; thus a large second-order moment may increase the width of the main band as well as the intensity of the side bands. Although the intrinsic infra-red absorption of valency crystals, such as diamond or germaniam, is due to the second-order moment only, nevertheless, there is a strong similarity between the expressions for the infra-red absorption of valency crystals and for the side-band absorption of ionic crystals. This similarity suggests that the spectra of all ionic crystals should exhibit a number of secondary maxima. The available experimental evidence does not seem sufficient to decide whether this suggestion is correct.

A perturbation calculation of properties of the 2 pπ state of HeH 2+

1957 ◽  
Vol 240 (1221) ◽  
pp. 274-283 ◽  
Keyword(s):  
Dipole Moment ◽  
Total Energy ◽  
Second Order ◽  
Wave Functions ◽  
Quadrupole Moments ◽  
Perturbation Calculation ◽  
Third Order ◽  
The Third ◽  
Fifth Order ◽  
Kinetic And Potential Energies

A perturbation calculation, valid in the limit of large separations, of various properties of the 2 pπ state of HeH 2+ is carried out. The total energy and the kinetic and potential energies are calculated to the fifth order, the dipole moment to the third order and the quadrupole moments to the second order and the results compared with those obtained using exact and variationally determined two-centre wave functions. Some results are also given for the 2 pπ u and 3 dπ g states of H + 2 and the influence of nuclear symmetry at large separations is briefly discussed.

scholarly journals 02. Social Pathologies, Reflexive Pathologies, and the Idea of Higher-Order Disorders

2015 ◽  
Author(s):  
Arto Laitinen
Keyword(s):  
Social Reality ◽  
Social Criticism ◽  
Higher Order ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
The Social ◽  
Order Reflection

This paper critically examines Christopher Zurn’s suggestion mentioned above that various social pathologies (pathologies of ideological recognition, maldistribution, invisibilization, rationality distortions, reification and institutionally forced self-realization) share the structure of being ‘second-order disorders’: that is, that they each entail ‘constitutive disconnects between first-order contents and secondorder reflexive comprehension of those contents, where those disconnects are pervasive and socially caused’ (Zurn, 2011, 345-346). The paper argues that the cases even as discussed by Zurn do not actually match that characterization, but that it would be premature to conclude that they are not thereby social pathologies, or that they do not have a structure in common. It is just that the structure is more complex than originally described, covering pervasive socially caused evils (i) in the social reality, (ii) in the first order experiences and understandings, (iii) in the second order reflection as discussed by Zurn, and also (iv) in the ‘third order’ phenomenon concerning the pre-emptive silencing or nullification of social criticism even before it takes place 

Solvatochromism and Electroabsorption Studies of Drug Carriers

2014 ◽  
Vol 807 ◽  
pp. 169-176
Author(s):  
R. Jothilakshmi ◽  
R. Rajeswari ◽  
E. Thanikaivelan
Keyword(s):  
Drug Delivery ◽  
Dipole Moment ◽  
Electric Field ◽  
Applied Electric Field ◽  
Spectral Properties ◽  
Dipole Moments ◽  
Drug Carriers ◽  
Higher Order ◽  
Third Order ◽  
The Third

2, 6-Diaminoanthraquinone is used in drug delivery. The absorption and fluorescence spectral properties of 2,6-Diaminoanthraquinone (DAAQ) have been investigated in a series of organic solvents with different polarity functions ranging from 0.207 to 0.762.The different solvents used are Tetrahydrofuran (THF) (0.207), Dimethylformamide (DMF) (0.404), ethanol (0.654) and Methonal (0.762).The fluorescence shifts are more pronounced than the absorption shifts indicating that the change in dipole moment is positive. The change in dipole moment and the ratio of the dipole moments are determined and discussed by Solvatochromism method. The third order nonlinearity of 2,6-DAAQ in PVA dissolved in DMF solution is determined using Electroabsorption. Higher order nonlinearities can be deduced on successful recording of the stark spectrum at the higher order of the applied electric field.

Magnetic circular dichroism and moments for A2 → E transition in C3ν defect centers

1970 ◽  
Vol 48 (22) ◽  
pp. 2769-2779 ◽  
Author(s):  
R. Sati ◽  
M. Inoue ◽  
S. Wang
Keyword(s):  
Circular Dichroism ◽  
Line Shape ◽  
Magnetic Circular Dichroism ◽  
Second Order ◽  
Order Moment ◽  
Third Order ◽  
Closed Expression ◽  
The Third ◽  
Jahn Teller ◽  
Circular Dichroïsm

The spectral line shape of the magnetic circular dichroism (MCD) for the A2 → E transition is investigated. The dynamic Jahn–Teller effect is taken into account by two different methods, i.e. (i) in Franck–Condon and adiabatic approximation and (ii) method of moments. The line-shape function for the magnetic circular dichroism is calculated without using the rigid shift approximation. A comparison of this line shape with the one obtained in the rigid shift shows the two line shapes to be different. General closed expression for the nth order moment is obtained, in the absence and the presence of an externally applied magnetic field, by making use of (i). Zeroth-, first-, and second-order and the change in the third-order moments have been obtained using (ii). The moments up to the second order and the change in the third-order moment have the same expressions as obtained by (i) and (ii). The MCD line shape and the different order moments have also been obtained for the 3A2 → 3E electronic transition in the R′ center.

scholarly journals Dielectric Saturation in Ionic Crystals. II. Deformable Ions

1968 ◽  
Vol 21 (4) ◽  
pp. 439
Author(s):  
J Oitmaa
Keyword(s):  
Dielectric Constant ◽  
Shell Model ◽  
Ionic Crystal ◽  
Higher Order ◽  
Static Dielectric Constant ◽  
Ionic Crystals ◽  
Dependent Term ◽  
Field Dependent ◽  
Dielectric Saturation

The lowest order field-dependent term in ~he static dielectric constant is calculated for an ionic crystal with deformable ions, and is evaluated numerically for NaI using a simple shell model. Some terms of higher order are also calculated.

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.

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.

A SPECTRAL/hp NONLINEAR FINITE ELEMENT ANALYSIS OF HIGHER-ORDER BEAM THEORY WITH VISCOELASTICITY

2012 ◽  
Vol 04 (01) ◽  
pp. 1250010 ◽  
Author(s):  
V. P. VALLALA ◽  
G. S. PAYETTE ◽  
J. N. REDDY
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Virtual Work ◽  
Constitutive Relations ◽  
Beam Theory ◽  
Element Model ◽  
Higher Order ◽  
Time Step ◽  
Third Order ◽  
The Third

In this paper, a finite element model for efficient nonlinear analysis of the mechanical response of viscoelastic beams is presented. The principle of virtual work is utilized in conjunction with the third-order beam theory to develop displacement-based, weak-form Galerkin finite element model for both quasi-static and fully-transient analysis. The displacement field is assumed such that the third-order beam theory admits C0 Lagrange interpolation of all dependent variables and the constitutive equation can be that of an isotropic material. Also, higher-order interpolation functions of spectral/hp type are employed to efficiently eliminate numerical locking. The mechanical properties are considered to be linear viscoelastic while the beam may undergo von Kármán nonlinear geometric deformations. The constitutive equations are modeled using Prony exponential series with general n-parameter Kelvin chain as its mechanical analogy for quasi-static cases and a simple two-element Maxwell model for dynamic cases. The fully discretized finite element equations are obtained by approximating the convolution integrals from the viscous part of the constitutive relations using a trapezoidal rule. A two-point recurrence scheme is developed that uses the approximation of relaxation moduli with Prony series. This necessitates the data storage for only the last time step and not for the entire deformation history.

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