Error analysis of 2π powder data for cubic or uniaxial phases

1978 ◽  
Vol 11 (3) ◽  
pp. 184-189 ◽  
Author(s):  
L. K. Frevel
Keyword(s):  
Error Analysis ◽  
Correction Factor ◽  
Square Root ◽  
Lattice Constants ◽  
Self Consistent ◽  
Consistent Error

The likely error in the 2θ measurement of a particular powder reflection can be bracketed by a self-consistent error analysis. To a specific observed θm value one assigns arbitrary increments {Δθm } and calculates iteratively the corresponding increments of any other θn according to the expression: \Delta \theta_{n} = \sin^{-1} {{{q \sin (\theta_{m} + \Delta \theta_{m})-\sin \theta_{n} \cos \Delta \theta_{n}}\over{\cos \theta_{n}}}}where q is the product of the square root of the ratio of the quadratic factors for (hmkmlm ) and (hnknln ) and a correction factor for refraction. By considering special coincidences for which the various Δθn 's are 0° or nil (0.000X°), one arrives at a likely bracketing interval for Δθm . Continuing this process one computes the indicated errors for the remaining reflections. The intent of the error analysis is to induce the experimenter to seek the cause(s) of the indicated errors in the 2θ determinations, and to delimit more precisely the accuracy of lattice constants of cubic or uniaxial phases.

Electronic properties of austenite and martensite Fe-9%Mn alloys

Open Physics ◽  
2008 ◽  
Vol 6 (4) ◽  
Author(s):  
Ercan Uçgun ◽  
Hamza Ocak
Keyword(s):  
Total Energy ◽  
Electronic Properties ◽  
Lattice Relaxation ◽  
The Self ◽  
Full Potential ◽  
Generalized Gradient ◽  
Plane Wave Method ◽  
Lattice Constants ◽  
Generalized Gradient Approximation ◽  
Self Consistent

AbstractWe calculate the electronic properties of austenite and martensite Fe-9%Mn alloys using the self consistent full-potential linearized-plane-wave method under the generalized gradient approximation full lattice relaxation. By minimizing total-energy, the lattice constants in their ground states were determined. We discuss the total energy dependence of the volume, and density of states (DOS).

scholarly journals Self-consistent statistical error analysis ofππscattering

2015 ◽  
Vol 91 (7) ◽  
Author(s):  
R. Navarro Pérez ◽  
E. Ruiz Arriola ◽  
J. Ruiz de Elvira
Keyword(s):  
Error Analysis ◽  
Statistical Error ◽  
Self Consistent

Modeling of Growth and Dissolution of Grain Boundary Cementite in Vacuum Carburizing Process

2011 ◽  
Vol 172-174 ◽  
pp. 1177-1182
Author(s):  
Hideaki Ikehata ◽  
Kouji Tanaka ◽  
Hiroyuki Takamiya ◽  
Hiroyuki Mizuno
Keyword(s):  
Retained Austenite ◽  
Calphad Method ◽  
Metastable Equilibrium ◽  
Low Alloy Steels ◽  
Square Root ◽  
Vacuum Carburizing ◽  
Lattice Constants ◽  
Moving Interface ◽  
Alloy Steels ◽  
Carburizing Process

In order to predict microstructures during vacuum carburizing, the model which simulates not only the carbon(C) diffusion but also growth/dissolution of cementite(θ) is required. For development of a new model we applied vacuum carburizing to low alloy steels and analyzed the distribution of C and θ by GD-OES and image analysis of microstructures. The C in retained austenite(γ) phase after carburizing was also measured by lattice constants obtained from XRD. We also simulated multi-component diffusion with γ matrix and θ layer to analyze a velocity of the moving interface. The new carburizing model was proposed based on the findings, which suggest that C in γ phase at the carburizing surface is supersaturated and corresponds to C concentration for metastable equilibrium condition to graphite. The growth and dissolution of the θ follow a square root of time with the coefficients controlled by diffusion of Si in γ and Cr in θ respectively. The required parameters such as diffusivity coefficients are obtained by the CALPHAD method. The calculated C distributions and volume fractions of θ represent the experimental results.

Microfields Induced by Random Compensated Charge Pairs in Ferroelectric Materials

2002 ◽  
Vol 720 ◽  
Author(s):  
Frank J. Crowne ◽  
Steven C. Tidrow ◽  
Daniel M. Potrepka ◽  
Arthur Tauber
Keyword(s):  
Dielectric Constant ◽  
Mean Field ◽  
Negative Ions ◽  
Charge Compensation ◽  
Ferroelectric Materials ◽  
Second Moment ◽  
Lattice Constants ◽  
Large Fluctuations ◽  
Self Consistent ◽  
Random Placement

AbstractThe dc and microwave responses of the BaxSr1-x (X,Y)yTi1-yO3 family of ferroelectric compounds with various substitutional additives X3+, Y5+ are analyzed by combining the random-field technique with the mean-field (Landau-Devonshire) theory of ferroelectricity, along with a self-consistent computation of the dielectric constant of the host material in the presence of the impurity fields. The fields in the material are assumed to arise from charge compensation at the Ti4+ sites, leading to permanent dipoles made up of the resulting positive and negative ions separated by a few lattice constants. It is shown that whereas completely random placement of positive and negative ions generates a Holtsmark distribution of electric field, with infinite second moment and hence extremely large fluctuations in field strength, the association of ionized impurities into permanent dipoles leads to much lower fluctuations in field and a distribution with finite second moment, which makes a self-consistent dielectric constant meaningful.

CONSTRAINED HAMILTONIANS AND LOCAL-SQUARE-ROOT ACTIONS

2002 ◽  
Vol 17 (20) ◽  
pp. 2717-2720 ◽  
Author(s):  
NIALL Ó. MURCHADHA
Keyword(s):  
General Relativity ◽  
Phase Space ◽  
Lagrange Multiplier ◽  
Configuration Space ◽  
Lagrange Multipliers ◽  
Square Root ◽  
Square Roots ◽  
Self Consistent ◽  
Lapse Function

The configuration space of general relativity is superspace, the space of Riemannian three-geometries and the Hamiltonian is just a sum of constraints, with Lagrange multipliers. One can go from this Hamiltonian, via a Legandre transformation, back and forth to the Lagrangian. The Lagrange multiplier (the lapse function) can be eliminated from the Lagrangian and one is left with an action which is a product of square roots. This is the Baierlein-Sharp-Wheeler action for general relativity. This action is unique in that all other square root actions are not self-consistent. This paper shows how to express this result in phase space. The only selfconsistent constrained Hamiltonian on superspace which is ultralocal and quadratic in the momenta in the ADM Hamiltonian for general relativity.

Total Energy Calculation in α-SiO2 and β-Si3N4

1987 ◽  
Vol 105 ◽  
Author(s):  
Yongnian Xu ◽  
W. Y. Ching
Keyword(s):  
First Principles ◽  
Electronic Structures ◽  
Local Density ◽  
Energy Calculation ◽  
Atomic Orbitals ◽  
Lattice Constants ◽  
Total Energy Calculation ◽  
Self Consistent ◽  
Total Energies ◽  
Cohesive Energies

AbstractThe self-consistent electronic structures and total energies for the crystalline α-SiO2 and β-Si3N4 are studied by means of the first-principles orthogonalized linear combination of atomic orbitals method. The calculated band structures are compared with the earlier non-self consistent results. The total energies in the local density approximation are evaluated as a function of lattice parameters. Reasonable values of equilibrium lattice constants and cohesive energies are obtained but the bulk modulus for α-SiO2 is overestimated.

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