Crystallographic Statistics in Chemical Physics

An ideal crystal is usually understood to be a regular array of ions or atoms. However, if we speak of positions of these particles with respect to their crystallographic origin, physical reality forces us to make some approximations. An experiment based on the diffraction of X-ray or neutron radiation by crystal makes it possible to obtain information on the positions of atoms or ions in the form of coordinates of local maxima of electron densities.

Table 2.1 presents empirical estimates of the first two central moments of interatomic vector lengths with the described properties. In this case terminal monatomic entities belong to a certain type. The dispersion of lengths of these vectors expressed by the second central moment m ₂ shows the value of 7.383 Ų for the sample of vector lengths of Zn²⁺ → 0, this value being much higher than 5.192 Ų for the sample of vector lengths for Zn²⁺ → S. In the case of such great sample sizes the variance equality of these data sets can be tested using sample characteristics of \(F = S_1^2/S_2^2:\) We will use the estimate of the above characteristics F ₁ = m₂(Zn^{2 +} → 0)/m ₂(Zn^{2 +} → S) = 1.422. The critical Fvalue on the significance level of α = 0.05 for v₁ = n(Zn^{2 +} → 0) — 1 and v₂ = n(Zn^{2 +} →S) — 1 is 1.011 [1]. Since F > F_crit the hypothesis H ₀ of the equality of dispersions can be rejected. The vector lengths of Co³⁺ → Cl and Co³⁺ → N also show a significant difference between their dispersions. In both cases of central monatomic entities one could explain the differences between the dispersions of interatomic vector lengths with different qualities of their terminal entities. The values of m ₂ of the samples of vectors of Zn²⁺ → S and Zn²⁺ → N exhibit, however, only a small difference (0.139 Ų). Similarly, the estimates of the second central moments of samples Cu²⁺ →Cl and Cu²⁺ → N are comparatively near.

Nonrigid properties of monatomic entities manifest themselves in the whole geometry of their environs. Until now we studied the statistics of these geometries using radial distributions of interatomic vectors. The classification of surroundings of a certain central monatomic entity M^Z+ can be based also on their symmetry. It has group — theoretical properties and it can be described by means of non — crystallographic symmetry point groups, of which there are infinitely many. There is, however, a finite amount of crystallographic point groups, viz. 2 one-dimensional, 10 two-dimensional and 32 three-dimensional point groups. The latest of them also describe the site symmetry of monatomic entity in the crystal structure and thus also the symmetry of its surroundings. Surroundings of monatomic entity means here, however, not only the near surroundings usually consisting of coordinating ligands, but the complete surroundings containing all other non coordinating entities of the structure. The symmetry of an ideal crystal structure — of discontinuum, is described by one of 230 space groups. The site symmetry of each monatomic entity of the structure thus has a certain connection with the relative positions of entities which may be classified based on the conception of lattice complexes.

The search for the factors causing distortions of surroundings of central entities is the subject of interest especially of chemists. It is the region of interatomic vector lengths, which are less than the most probable minimum value a ₀′ = 4.25 Å, where it is already difficult to suppose a fitting of some known statistical model. The non-rigid properties of central entities in this region, let us call it the coordination sphere of entity M^z+, apparently manifest themselves also in distortions of the geometry of arrangement of monatomic ligand entities. In agreement with the commonly used terms for describing crystal structures of compounds we will call monatomic entities of ligands “ligand atoms” resp. atoms. The number of bonds formed by the central entity is the coordination number. The bond lengths between the central entity and ligand atoms will be signed M-L.

Analysis of empirical distribution of interatomic vector lengths with central monatomic entities (M^Z+) of transition elements showed their different non-rigid behavior in condensed matter. This intrinsic property of central entities can be expressed by the variance of interatomic vector lengths with common origin in the site of central entity of a certain type (M^Z+), when the extrinsic variance (σ² _ε) has no significant contribution. It is apparent that this quantitative measure does not express completely the non-rigid behavior of central monatomic entities. In statistical approach, however, the variance of lengths of the described vectors with significantly zero extrinsic component sufficiently expresses the ability of entity M^Z+ in condensed matter to form environments with the minimum number of equidistant interatomic vectors in the sense of VEP. Empirical estimates of the first moments (m ₁) (Table 1.1.3), of interatomic vector length distributions with significant averaging of extrinsic factors (Table 2.1.1), are in the range of 3.168 (Mn²⁺) to 7.396 (Mn⁴⁺) Å which may be considered to be the region of lengths, where the principle of vector equilibrium applies in crystal structures of coordination compounds. From the point of view of this principle (Chap. 1.2) the variance of interatomic vector lengths with insignificant extrinsic component expresses the ability of a certain central monatomic entity to deviate from the vector equilibrium.