Probability Distribution Of X-ray Intensities Research Articles

Intensity statistics. I

Two methods for evaluating the probability distribution of X-ray intensities are presented. The first is an exact series representation in powers of the atomic form factors. Overall thermal-displacement and scale parameters are shown to be dependent only on the ratio of intensity to σ2. The second method is a maximum-entropy distribution based on the first two moments of the intensity distribution. It is shown to be a robust and accurate method of incorporating heavy-atom effects and results in a simple generalization of the Wilson distribution.

Probability distribution of X-ray intensities in a noncentrosymmetric crystal with a degree of centrosymmetry

The expression for the probability distribution of X-ray intensities is obtained for a non-centrosymmetric crystal possessing an approximate centre of symmetry and from this the acentric and centric distributions of Wilson are deduced as limiting cases. Theoretical expressions for the variance, test ratio q and moments of the normalized intensity z are also obtained in terms of the parameter 〈|Δr|〉 which is a measure of the degree of centrosymmetry. From the theoretical results it appears that the non-centrosymmetric nature of such a crystal could be detected by statistical tests, provided accurate z values are available and reflexions with high (sin θ/λ) values are used for the test.

The probability distribution of X-ray intensities. VIII. A note on the compensation for excess average intensity

The probability distribution of X-ray intensities. VIII. A note on the compensation for excess average intensity

The probability distribution of X-ray intensities: the effect of one heavy atom in a triclinic cell containing a number of light atoms

The probability distribution of X-ray intensities: the effect of one heavy atom in a triclinic cell containing a number of light atoms

The probability distribution of X-ray intensities: atoms in special positions

The probability distribution of X-ray intensities: atoms in special positions

The probability distribution of X-ray intensities. VII. Some sesquisymmetric distributions

The probability distribution of X-ray intensities. VII. Some sesquisymmetric distributions

Some observations on the probability distribution of X-ray intensities

Some observations on the probability distribution of X-ray intensities

The probability distribution of X-ray intensities. V. A note on some hypersymmetric distributions

The probability distribution of X-ray intensities. V. A note on some hypersymmetric distributions

X-ray determination of centrosymmetry in three felspars

Three statistical tests, depending upon the probability distribution of X-ray intensities, have been applied to orthoclase, sanidine, and albite. It is concluded that these three felspars posses centres of symmetry, justifying the assumption of holohedral symmetry made in earlier structure analyses.

The probability distribution of X-ray intensities. IV. New methods of determining crystal classes and space groups

The probability distribution of X-ray intensities. IV. New methods of determining crystal classes and space groups

The probability distribution of X-ray intensities. III. Effects of symmetry elements on zones and rows

The probability distribution of X-ray intensities. III. Effects of symmetry elements on zones and rows

The probability distribution of X-ray intensities. II. Experimental investigation and the X-ray detection of centres of symmetry

The probability distribution of X-ray intensities. II. Experimental investigation and the X-ray detection of centres of symmetry

The probability distribution of X-ray intensities

In a crystal without symmetry elements and containing a sufficiently large number of atoms the probability of the hkl reflexion having an intensity between I and I + dI is P(I) dI, where P(I) = Σ−1 exp{−I/Σ}, and Σ is the sum of the squares of the scattering factors of the atoms. In a centrosymmetric crystal the probability of the structure amplitude of the hkl reflexion lying between F and F + dF is P(F) dF, where P(F) = (2πΣ)½ exp{−F2/2Σ}, a result noticed empirically. In a centred crystal (k−1)/k of the reflexions are zero, and the remaining 1/k of them are distributed like those of an uncentred crystal with parameterkΣ, where k is 4 for face-centring and 2 for end- or body-centring. Other symmetry elements do not produce important effects on the general reflexions, but may make a zone or line of intensities behave as if centred or centrosymmetric. The mean value of I is Σ, a fact that can be used to put relative intensities on an absolute basis. The mean values of |F| or I2 can also be used, but the mean value of I is the only one independent of the symmetry. The difference between the ratios of 〈|F|〉2 or 〈I〉 for centrosymmetric and noncentrosymmetric crystals may serve for the purely X-ray determination of a centre of symmetry.