Roughly, the conditions are that , the variance of the sum, should tend to infinity and , where is the variance of the j th random variable, should tend to zero for all j as n tends to infinity. The central-limit theorem, under certain conditions, remains valid even when the variables summed in equation 2. The conditions have been investigated by Bernstein , ; roughly they amount to requiring that the variables should not be too closely correlated. The theorem applies, in particular, when each is related to a finite number, , of its neighbours, when the x 's are said to be dependent.
Indexing grazing-incidence X-ray diffraction patterns of thin films: lattices of higher symmetry
The dependence seems plausible for crystallographic applications, since the positions of atoms close together in a structure are closely correlated by interatomic forces, whereas those far apart will show little correlation if there is any flexibility in the asymmetric unit when unconstrained. Long-range stereochemical effects, as in pseudo-graphitic aromatic hydrocarbons, would presumably produce long-range correlations and make dependence less plausible. If Bernstein's conditions are satisfied, the central-limit theorem would apply, but the actual value of would have to be used for the variance, instead of the sum of the variances of the random variables in 2.
Because of the correlations the two values are no longer equal. In applications of the central-limit theorem, and its extensions, to intensity statistics the 's of equation 2. These trigonometric expressions for all the space groups, and general Wyckoff positions, are given in Tables A1. One cannot, of course, conclude that the magnitudes of the structure factor always have a normal distribution — even if the structure is homoatomic; one must look at each problem and see what components of the structure factor can be put in the form 2.
Ordinarily the components are the real and the imaginary parts of the structure factor; the structure factor is purely real only if the structure is centrosymmetric, the space-group origin is chosen at a crystallographic centre and the atoms are non-dispersive. The ideal acentric distributions are obtained by applying the central-limit theorem to the real and the imaginary parts of the structure factor, as given by equation 2.
Consider first a crystal with no rotational symmetry space group P 1. The real part, A , of the structure factor is then given by where N is the number of atoms in the unit cell and is the phase angle of the j th atom. The central-limit theorem then states that A tends to be normally distributed about its mean value with variance equal to its mean-square deviation from its mean.
The asymptotic form of the distribution of A is therefore given by A similar calculation, with sines instead of cosines, gives an analogous distribution for the imaginary part B , so that the joint probability of the real and imaginary parts of F is Ordinarily, however, we are more interested in the distribution of the magnitude, , of the structure factor than in the distribution of A and B.
Using polar coordinates in equation 2. If has been put on an absolute scale see Section 2. Distributions resulting from noncentrosymmetric crystals are known as acentric distributions; those arising from centrosymmetric crystals are known as centric. These adjectives are used to describe distributions , not crystal symmetry.
When a non-dispersive crystal is centrosymmetric, and the space-group origin is chosen at a crystallographic centre of symmetry, the imaginary part B of its structure amplitude is zero. In the simplest case, space group , the contribution of the j th atom plus its centrosymmetric counterpart is. The calculation of goes through as before, with allowance for the fact that there are pairs instead of N independent atoms, giving or equivalently or.
In addition, in noncentrosymmetric space groups, the distribution of certain groups of reflections becomes centric, though the general reflections remain acentric. The changes are summarized in Tables 2.
The values of S are integers for lattice centring, glide planes and those screw axes that produce absences, and approximate integers for rotation axes and mirror planes; the modulations of the average intensity in reciprocal space outlined in Section 2. It should be noted that if intensities are normalized to the average of the group to which they belong, rather than to the general average, the distributions given in equations 2. The distributions just derived are asymptotic, as they are limiting values for large N. They are the only ideal distributions, in this sense, when there is only strict crystallographic symmetry and no dispersion.
However, other ideal asymptotic distributions arise when there is noncrystallographic symmetry, or if there is dispersion. The ideal subcentric and bicentric distributions are expressed in terms of known functions, but the higher hypercentric and the sesquicentric distributions have so far been studied only through their moments and integral representations.
Certain hypersymmetric distributions can be expressed in terms of Meijer's G functions Wilson, b. When only the intrinsic probability distributions are being considered, it does not greatly matter whether the variable chosen is the intensity of reflection I , or its positive square root, the modulus of the structure factor , since both are necessarily real and non-negative. In an obvious notation, the relation between the intensity distribution and the structure-factor distribution is or Statistical fluctuations in counting rates, however, introduce a small but finite probability of negative observed intensities Wilson, a , a and thus of imaginary structure factors.
This practical complication is treated in IT C , Parts 7 and 8. Both the ideal centric and acentric distributions are simple members of the family of gamma distributions, defined by where n is a parameter, not necessarily integral, and is the gamma function. Thus the ideal acentric intensity distribution is and the ideal centric intensity distribution is The properties of gamma distributions and of the related beta distributions, summarized in Table 2. If are independent gamma-distributed variables with parameters , their sum is a gamma-distributed variable with.
Shmueli | International Tables for Crystallography, Volume B | | Reciprocal Space | B
If x and y are independent gamma-distributed variables with parameters p and q , then the ratio has the distribution. With the same notation, the ratio has the distribution. Differences and products of gamma-distributed variables do not lead to simple results. The integral of the probability density function from the lower end of its range up to an arbitrary value x is called the cumulative probability distribution, or simply the distribution function, , of x.
It can always be written if the lower end of its range is not actually one takes as identically zero between and the lower end of its range. For the distribution of A [equation 2. In such cases, equation 2. The cumulative forms of the ideal acentric and centric distributions Howells et al. For the acentric distribution of [equation 2.
International Tables for Crystallography, Vol.B: Reciprocal Space
We thus need to consider the probability density functions of sums like and averages like where is the intensity of the i th reflection. The probability density distributions are easily obtained from a property of gamma distributions: If are independent gamma-distributed variables with parameters , their sum is a gamma-distributed variable with parameter p equal to the sum of the parameters. The sum of n intensities drawn from an acentric distribution thus has the distribution the parameters of the variables added are all equal to unity, so that their sum is p.
Similarly, the sum of n intensities drawn from a centric distribution has the distribution each parameter has the value of one-half.
The corresponding distributions of the averages of n intensities are then for the acentric case, and for the centric. In both cases the expected value of Y is and the variances are and , respectively, just as would be expected. Ratios like where is given by equation 2.
There are three main cases:.
see Aside from the scale factor, in case i and will differ chiefly through relatively small statistical fluctuations and uncorrected systematic errors, whereas in case ii the differences will be relatively large because of the inherent differences in the intensities. Here we are concerned only with cases ii and iii ; the practical problems of case i are postponed to IT C There is little in the crystallographic literature concerning the probability distribution of sums like 2. In case ii and independent , acentric distribution, Table 2. The expected value of is then with variance One sees that is a biased estimate of the scaling factor between two sets of intensities and the bias, of the order of , depends only on the number of intensities averaged in the denominator.
This may seem odd at first sight, but it becomes plausible when one remembers that the mean of a quantity is an unbiased estimator of itself, but the reciprocal of a mean is not an unbiased estimator of the mean of a reciprocal. The mean exists only if and the variance only for. In the centric case, the expression for the distribution of the ratio of the two means Y and Z becomes with the expected value of equal to and with its variance equal to For the same number of reflections, the bias in and the variance for the centric distribution are considerably larger than for the acentric. For both distributions the variance of the scaling factor approaches zero when n and m become large.
These biases are readily removed by multiplying by or. Wilson a has given reasons for supposing that the bias of the ratio 2. Equations 2. When the 's are a subset of the 's, the beta distributions of the second kind are replaced by beta distributions of the first kind, with means and variances readily found from Table 2. The distribution of such a ratio is chiefly of interest when Y relates to a single reflection and Z relates to a group of m intensities including Y.
This corresponds to normalizing intensities to the local average. Its distribution is in the acentric case, with an expected value of of unity; there is no bias, as is obvious a priori.
Unlike the variance of the scaling factor, the variance of the normalized intensity approaches unity as n becomes large. Similar considerations apply to intensities normalized to in the usual way, since they are equal to those normalized to multiplied by.