scholarly journals The accuracy of atomic co-ordinates derived from Fourier series in X-ray structure analysis IV. The two-dimensional projection of oxalic acid

1947 ◽  
Vol 190 (1023) ◽  
pp. 490-496 ◽  
Keyword(s):  
Experimental Data ◽  
Fourier Series ◽  
Oxalic Acid ◽  
Recent Work ◽  
Two Dimensional ◽  
X Ray ◽  
Oxalic Acid Dihydrate ◽  
Theoretical Predictions ◽  
Experimental Errors ◽  
Extensive Experimental Data

The methods, developed in previous papers of this series, have been applied to an examina­tion of the errors in the atomic co-ordinates derived from the ( h , 0, l ) projection of oxalic acid dihydrate. It is shown that the experimental errors are of the order ±0.01 A, and that the finite sum­mation errors are slightly larger—in agreement with the theoretical predictions of the former papers. Recent work, using more extensive experimental data, is discussed, and it is concluded that,owing to the introduction of an artificial temperature factor, the results are unlikely to be of greater accuracy than the originals.

Experimental estimation of uncertainties in powder diffraction intensities with a two-dimensional X-ray detector

2016 ◽  
Vol 31 (3) ◽  
pp. 216-222 ◽  
Author(s):  
Takashi Ida
Keyword(s):  
Powder Diffraction ◽  
Rietveld Method ◽  
Two Dimensional ◽  
X Ray ◽  
Lattice Constants ◽  
Cell Dimensions ◽  
Fine Quartz ◽  
Experimental Errors ◽  
Estimation Of Uncertainties ◽  
Unit Cell Dimensions

A method to obtain both one-dimensional powder diffraction intensities I(2θ) and statistical uncertainties σ(2θ) from the data collected with a flat two-dimensional X-ray detector is proposed. The method has been applied to analysis of the diffraction data of fine quartz powder recorded with synchrotron X-ray. The profile and magnitude of the estimated uncertainties σ(2θ) have shown that the effects of propagation of the errors in 2θ are dominant as the uncertainties about the observed intensity values I(2θ). The powder diffraction intensity data I(2θ), including nine reflection peaks have been analyzed by the Rietveld method incorporating the experimentally estimated uncertainties σ(2θ). The observed I(2θ) data have been reproduced with a symmetric peak profile function (Rwp = 0.84 %), and no significant peak shifts from calculated locations have been detected as compared with the experimental errors. The optimized values of the lattice constants of the quartz sample have nominally been estimated at a = 4.9131(4) Å and c = 5.4043(2) Å, where the uncertainties in parentheses are evaluated by the Rietveld optimization based on the estimated uncertainties σ(2θ) for intensities I(2θ). It is likely that reliability of error estimation about unit-cell dimensions has been improved by this analytical method.

scholarly journals The accuracy of atomic co-ordinates derived from Fourier series in X-ray crystallography. V

1948 ◽  
Vol 193 (1034) ◽  
pp. 305-310 ◽  
Keyword(s):  
Fourier Series ◽  
Closed Form ◽  
Experimental Error ◽  
Fourier Coefficients ◽  
Probable Error ◽  
X Ray ◽  
X Ray Crystallography ◽  
Experimental Errors

In the first paper of this series, the effect of experimental errors in the Fourier coefficients, upon the derived atomic co-ordinates was investigated. The assumption was made that the probable errors were independent of the magnitudes of their parent Bragg reflexions. It has been suggested that a more accurate assumption would be to take the probable error of any coefficient as being proportional to the magnitude of that coefficient. The present paper develops the theory on this basis, and a solution in closed form is obtained. The question of how the experimental error in the co-ordinates varies with the position of the atom in relation to its neighbours is also investigated, and it is shown that the variation is much smaller than that previously derived for finite summation errors.

scholarly journals Determinations of the signs of the fourier terms in complete crystal structure analysis

1933 ◽  
Vol 141 (843) ◽  
pp. 188-193 ◽  
Keyword(s):  
Crystal Structure ◽  
Fourier Series ◽  
Structure Analysis ◽  
Convenient Method ◽  
Unit Area ◽  
Crystal Structure Analysis ◽  
Two Dimensional ◽  
X Ray Diffraction ◽  
Dimensional Representation ◽  
X Ray

The Fourier representation of the results of X-ray diffraction by crystals was first suggested by W. H. Bragg,** and later it was used by Havighurst and Compton† to obtain the distribution of scattering matter in crystals. W. L. Bragg‡ developed the two-dimensional representation of the Fourier series into a very convenient method of crystal analysis. He considered crystals having a centre of symmetry where it can be shown that if a projection of the atoms on the yz plane be considered the density of scattering matter per unit area of projection at the point y , z is given by

scholarly journals OBSERVATION OF THE THERMAL CASIMIR FORCE IS OPEN TO QUESTION

2011 ◽  
Vol 03 ◽  
pp. 515-526 ◽  
Author(s):  
G. L. KLIMCHITSKAYA ◽  
M. BORDAG ◽  
E. FISCHBACH ◽  
D. E. KRAUSE ◽  
V. M. MOSTEPANENKO
Keyword(s):  
Experimental Data ◽  
Casimir Force ◽  
Drude Model ◽  
Plasma Model ◽  
Theoretical Predictions ◽  
Proximity Force Approximation ◽  
Experimental Errors ◽  
Recent Claim ◽  
Model Approach

We discuss theoretical predictions for the thermal Casimir force and compare them with available experimental data. Special attention is paid to the recent claim of the observation of that effect, as predicted by the Drude model approach. We show that this claim is in contradiction with a number of experiments reported so far. We suggest that the experimental errors, as reported in support of the observation of the thermal Casimir force, are significantly underestimated. Furthermore, the experimental data at separations above 3 μm are shown to be in agreement not with the Drude model approach, as is claimed, but with the plasma model. The seeming agreement of the data with the Drude model at separations below 3 μm is explained by the use of an inadequate formulation of the proximity force approximation.

scholarly journals The accuracy of atomic co-ordinates derived from Fourier series in X-ray structure analysis. III

1947 ◽  
Vol 190 (1023) ◽  
pp. 482-489 ◽  
Keyword(s):  
Fourier Series ◽  
Power Law ◽  
Three Dimensional ◽  
Thermal Motion ◽  
One Dimensional ◽  
X Ray ◽  
Order Of Magnitude ◽  
Quantitative Examination ◽  
Experimental Errors ◽  
Dimensional Series

Further work on the problems considered in the previous papers of this series has resulted in a more satisfactory treatment of finite summation errors in the three-dimensional diatomic case. The results are extended to the two- and one-dimensional series, and the interesting result emerges that finite summation errors are of the same order of magnitude whatever the dimensions of summation. Using the new results a more quantitative examination of the effects of real thermal motion becomes possible. It is shown that the relative accuracies of parameters in structures, the higher order reflexions from which are suppressed by thermal motion, follows a simple power law in the corresponding reciprocal spacings. These considerations lead to an examination of the artificial temperature factor method of securing convergence, and it is shown that this produces greater errors due to overlapping than those it is designed to eliminate. A method of correcting these distortions is suggested. Finally, the treatment of the effect of experimental errors is extended to two and one dimensions, and it is shown that the three-dimensional summation is least affected by experimental inaccuracy. The errors for three-, two- and one-dimensional summation, in a particular case, are calculated to be in the ratio 1: 3: 10.

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