The temperature factor of an atom in a rigid vibrating molecule. II. Anisotropic thermal motion

1970 ◽  
Vol 26 (2) ◽  
pp. 260-262 ◽  
Author(s):  
G. S. Pawley ◽  
B. T. M. Willis
Keyword(s):  
Rigid Body ◽  
Temperature Factor ◽  
Thermal Motion ◽  
Mean Square ◽  
First Order ◽  
Translational Displacement ◽  
Librational Motion ◽  
The Mean ◽  
A Site ◽  
Body Theory

The analysis in part I is generalized to any crystal containing rigid molecules which undergo anisotropic translational and librational motion about a site fixed by symmetry. The treatment is correct to terms in (ui 2)2 and (ωi 2)2, where (ui 2) is the mean-square translational displacement of the molecule along the ith axis and (ωi 2) is the mean-square angular libration about the same axis. The first-order treatment to terms in (ui 2) and (ωi 2) is shown to be equivalent to the rigid-body theory in current use.

The temperature factor of an atom in a rigid vibrating molecule. I. Isotropic thermal motion

1970 ◽  
Vol 26 (2) ◽  
pp. 254-259 ◽  
Author(s):  
B. T. M. Willis ◽  
G. S. Pawley
Keyword(s):  
Temperature Factor ◽  
Thermal Motion ◽  
Subsequent Paper ◽  
Mean Square ◽  
Cumulant Expansion ◽  
Cubic Crystal ◽  
Translational Displacement ◽  
The Mean ◽  
Expansion Model ◽  
Body Parameters

The temperature factor of an atom in a molecular crystal has been derived in terms of the rigid-body parameters (u 2) and (ω2), where (u 2) is the mean-square amplitude of translational displacement of the molecule in any direction and (ω 2) is the mean-square amplitude of angular (librational) displacement about any axis through the centre of inertia of the molecule. The analysis, which is appropriate to a cubic crystal containing rigid molecules undergoing isotropic thermal motion, is correct to the second power of (ω 2); the second-order treatment is necessary for interpreting accurate Bragg intensity data such as those discussed in a subsequent paper. Substitution of the temperature factor into the structure-factor equation yields an expression containing terms which can be identified with the first, second and third cumulants of the `cumulant expansion' model of thermal motion.

scholarly journals Diffraction Measurements and Equilibrium Parameters

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Victor A. Sipachev
Keyword(s):  
Low Frequency ◽  
Mean Square ◽  
Structural Studies ◽  
Free Rotation ◽  
Theory Analysis ◽  
First Order ◽  
Internuclear Distances ◽  
The Mean ◽  
Asymmetry Parameters ◽  
First Order Perturbation

Structural studies are largely performed without taking into account vibrational effects or with incorrectly taking them into account. The paper presents a first-order perturbation theory analysis of the problem. It is shown that vibrational effects introduce errors on the order of 0.02 Å or larger (sometimes, up to 0.1-0.2 Å) into the results of diffraction measurements. Methods for calculating the mean rotational constants, mean-square vibrational amplitudes, vibrational corrections to internuclear distances, and asymmetry parameters are described. Problems related to low-frequency motions, including torsional motions that transform into free rotation at low excitation levels, are discussed. The algorithms described are implemented in the program available from the author (free).

scholarly journals A Study on the Chain Ratio-Type Estimator of Finite Population Variance

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Yunusa Olufadi ◽  
Cem Kadilar
Keyword(s):  
Finite Population ◽  
Numerical Comparison ◽  
Mean Square ◽  
Auxiliary Variables ◽  
Population Variance ◽  
Two Phase ◽  
First Order ◽  
Phase Sampling ◽  
The Mean ◽  
Two Phase Sampling

We suggest an estimator using two auxiliary variables for the estimation of the unknown population variance. The bias and the mean square error of the proposed estimator are obtained to the first order of approximations. In addition, the problem is extended to two-phase sampling scheme. After theoretical comparisons, as an illustration, a numerical comparison is carried out to examine the performance of the suggested estimator with several estimators.

Distortion of sonic bangs by atmospheric turbulence

1969 ◽  
Vol 37 (3) ◽  
pp. 529-563 ◽  
Author(s):  
S. C. Crow
Keyword(s):  
Scattering Theory ◽  
Critical Time ◽  
Observation Point ◽  
Inertial Subrange ◽  
Pressure Jump ◽  
Mean Square ◽  
Pressure Perturbation ◽  
Surface Integral ◽  
First Order ◽  
The Mean

Recorded pressure signatures of supersonic aircraft often show intense, spiky perturbations superimposed on a basicN-shaped pattern. A first-order scattering theory, incorporating both inertial and thermal interactions, is developed to explain the spikes. Scattering from a weak shock is studied first. The solution of the scattering equation is derived as a sum of three terms: a phase shift corresponding to the singularity found by Lighthill; a small local compression or rarefaction; a surface integral over a paraboloid of dependence, whose focus is the observation point and whose directrix is the shock. The solution is found to degenerate at the shock into the result given by ray acoustics, and the surface integral is identified with the scattered waves that make up the spikes. The solution is generalized for arbitrary wave-forms by means of a superposition integral. Eddies in the Kolmogorov inertial subrange are found to be the main source of spikes, and Kolmogorov's similarity theory is used to show that, for almost all timestafter a sonic-bang shock passes an observation point, the mean-square pressure perturbation equals$(\Delta p)^2 (t_c/t)^{\frac{7}{6}}$, where Δpis the pressure jump across the shock andtcis a critical time predicted in terms of meteorological conditions. For an idealized model of the atmospheric boundary layer,tcis calculated to be about 1 ms, a figure consistent with the qualitative data currently available. The mean-square pressure perturbation just behind the shock itself is found to be finite but enormous, according to first-order scattering theory. It is conjectured that a second-order theory might explain the shock thickening that actually occurs.

Pseudo Two-Hop Distributed Consensus with Adaptive Quantization

2014 ◽  
Vol 889-890 ◽  
pp. 662-665
Author(s):  
Huan Xin Peng ◽  
Wen Kai Wang ◽  
Bin Liu
Keyword(s):  
Convergence Rate ◽  
Consensus Algorithm ◽  
Quantization Scheme ◽  
Mean Square ◽  
Distributed Consensus ◽  
First Order ◽  
Average Consensus ◽  
Adaptive Quantization ◽  
The Mean ◽  
Mean Square Errors

In order to improve the accuracy and the convergence rate of distributed consensus under quantized communication, in the paper, based on adaptive quantization scheme, we propose the pseudo two-hop distributed consensus algorithm. By analyses and simulations, Results show that the pseudo two-hop distributed consensus algorithm based on adaptive quantization can reach an average consensus, and its convergence rate is higher than that of the first-order adaptive quantized distributed consensus algorithm, moreover, the mean square errors are smaller within the finite steps.

Predicting some Effects of Lattice Vibrations on Proton Scattering Patterns

1970 ◽  
Vol 14 ◽  
pp. 1-10
Author(s):  
C. S. Barrett
Keyword(s):  
Temperature Factor ◽  
Lattice Vibrations ◽  
Proton Scattering ◽  
Mean Square ◽  
X Ray Diffraction ◽  
X Ray ◽  
Structure Factors ◽  
Integrated Intensity ◽  
The Mean ◽  
Thermal Vibrations

AbstractA method of predicting the approximate relative intensities of lines in proton blocking patterns recently proposed, which is based on summing the squares of structure factors for the various orders of reflection of a plane, is found to predict certain effects of lattice vibrations on the lines in some recently reported patterns. The mean square amplitude of vibration enters the calculations through a Debye-Waller temperature factor like that used in X-ray diffraction. When patterns are compared for groups of crystals that are nearly identical except for this temperature factor, the qualitative predictions by this method agree with the observations. If it is also arbitrarily assumed that the integrated intensity dip at a spot where lines intersect is approximated by summing the calculated Integrated intensity dips for all of the lines crossing at the spot, one has a simple and convenient method of predicting relative spot intensities. Such calculations have been successful in establishing the order of decreasing intensity for most of the spots along a given line, with several different kinds of crystals. This method also serves to predict qualitatively how prominent the spots appear relative to the lines, in general, in patterns of crystals that differ appreciably only in the amplitude of the thermal vibrations.

On the Molecular Motion in the Orthorhombic Solid Acetylene-d2

1975 ◽  
Vol 30 (7-8) ◽  
pp. 554-560
Author(s):  
Harri K. Koski
Keyword(s):  
Rigid Body ◽  
Thermal Neutron ◽  
Powder Diffraction ◽  
Molecular Motion ◽  
Neutron Powder Diffraction ◽  
Mean Square ◽  
Molecular Dimension ◽  
Thermal Displacements ◽  
The Mean ◽  
Lattice Modes

The mean square thermal displacements of the carbon and deuterium atoms in crystalline orthorhombic acetylene-d2, C2D2, derived from earlier thermal neutron powder diffraction studies were interpreted as rigid-body translational and librational motions of the C2D2 molecule. An attempt to assign the derived amplitudes of the angular vibrations to the Raman-active rotational lattice modes was not unambiguous. Finally, a calculation was made to correct the apparent shortening in the bond lengths and the molecular dimension.

First-Passage Time in a Random Vibrational System

1966 ◽  
Vol 33 (1) ◽  
pp. 187-191 ◽  
Author(s):  
A. H. Gray
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Time To Failure ◽  
Mean Square ◽  
First Passage ◽  
One Dimensional ◽  
First Order ◽  
Vibrational System ◽  
The Mean ◽  
First Passage Problem

The first-passage problem in a random vibrational system is solved by means of approximations which convert the problem to a first-order one-dimensional Markov process. Laplace transforms are used to evaluate the mean square time to failure.

scholarly journals An Improved Exponential Type Estimator Of Population Mean of Sensitive Variable Using Optional Randomized Response Technique

2019 ◽  
pp. 49-59
Author(s):  
Lovleen Kumar Grover ◽  
Amanpreet Kaur
Keyword(s):  
Mean Square Error ◽  
Exponential Type ◽  
Mean Square ◽  
Order Of Approximation ◽  
Randomized Response Technique ◽  
Randomized Response ◽  
Population Mean ◽  
First Order ◽  
The Mean ◽  
Sensitive Variable

In this paper, we improve the efficiency of  Koyuncu et al (2014)’s estimator of population mean of sensitive variable by replacing Traditional Randomized response technique with Optional Randomized response technique as suggested by Gupta et al (2014). The mean square error of proposed estimator is obtained, up to first order of approximation, and is compared with mean square error of various existing estimators theoretically as well as numerically.

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