Abstract. It is shown that coordination numbers (CN) of alkali atoms can be determined in crystal structures by means of Voronoi-Dirichlet polyhedra without using crystal-chemical radii. It is found that a domain with constant volume corresponds to an alkali atom at CN 6. The conclusion about correctness of application of the model of deformable spheres for the description of alkali atoms in a crystal field is drawn.

- the structural determination was executed with R_f10%;

1. According to [4], the strength of interatomic bond is proportional to the value of solid angle (W ) of the face which is joint for VDPs of considered atoms. Obviously, when W values do not exceed the standard deviation caused by errors of structural experiment (s(W ) 1.5% of the total solid angle of 4p steradian), there are no reasons to speak about presence of strong interatomic interactions, and corresponding faces of VDP should not be taken into account.

3. It is necessary to clarify whether a set of interatomic interactions, to which the faces of VDPs with W >1.5% correspond, is identical and whether it is required to make their additional classification before the final CN calculation. In this case, the total atomic CN can be written as Sn_i, where n_i shows the number of interatomic contacts of identical (or almost identical) nature, and index i=1 to m enumerates m different types of interatomic interactions in order of decreasing their strength. Note that, usually, CN=n_l is assumed independently of the range of i. Further we shall call so-defined CN as classical (CN_cl), and use the abbreviation ‘CN’ for designation of total CN.

From the remarks indicated the first two are formalised and can be easily taken into account during the determination of CNs, whereas the classification of interatomic interactions seems to be a rather complex methodological problem. For the illustration of our proposed method of decision of this problem, we shall consider distributions of R(M-O) and corresponding W (M-O) in VDPs of M atoms.

Fig.1a,b show that in the first approximation the distributions of R(M-O) at M=Li or Na are bimodal, with maxima at R₁ 2.0 and R₂ 3.5µ or R₁ 2.4 and R₂ 3.8µ, respectively, and clearly expressed minima at R_min 2.7 or R_min 3.2µ. The values of R_min divide each distribution into two parts, first of which (R(M-O)<R_min) characterises strong M-O bonds of ionic-covalent type, and second (R(M-O)>R_min) – additional weaker M-O contacts with mainly ionic nature. Note that R_min values are in good accordance with the upper limits of R(M-O), which are frequently used in practice for determination of CNs of lithium and sodium ions without any reason (see, for example, [10]). Accepting W (M-O) value as a more accurate characteristic of strength of M-O bond, we can get on the basis of W (R) dependence that W_min 8 or 5% correspond to R(M-O) R_min. Thus, the W_min indicated were accepted as lower limits of W (M-O) values, at which the corresponding M-O contact should be taken into consideration during the calculation of CN_cl of M atoms (i.e. of n_l values). In our opinion, the total description of coordination sphere of lithium and sodium atoms also requires the indication of the second component of total CN (n₂). This component corresponds to the number of M-O contacts of mainly ionic type and is characterised by 1.5%<W (Li-O)<8% or 1.5%<W (Na-O)<5%.

The results of calculations of VDPs of potassium, rubidium and caesium atoms for 559, 142 and 166 oxygen-containing compounds correspondingly, show that R(M-O) distributions (M=K, Rb or Cs) (Fig.1c-e) are characterised only by one maximum at R₁ 2.8, 3.1 or 3.2µ, respectively. Therefore, in contrast to lithium and sodium compounds, the division of M-O contacts at M=K, Rb or Cs into several groups is not reasonable. In this connection, during the determination of CNs of potassium, rubidium and caesium ions the choice of W_min=1.5% seems to be logical (i.e. in this case total CN is equal to CN_cl). We found that R(M-O)=3.8 to 4.1µ corresponds to the W_min indicated. Thus, R(M-O) 4.1µ characterises the maximum distances, at which appreciable interatomic interaction can correspond to K-O, Rb-O or Cs-O contacts. Also, it should be noted that monomodal character of R(M-O) distributions at M=K, Rb or Cs may be interpreted as an effect of the mainly ionic nature of all M-O contacts of the same type with W>1.5%, independently of R(M-O) values.

In structure of a -Na₂Cr₂O₇ [10] according to the authors, Na1 and Na2 atoms are characterised by CN=7 with regard to Na-O contacts with R(Na-O)3.3µ. The authors do not explain the reasons of the choice of upper limit of lengths of Na-O bonds, however, calculations of VDPs (Tables 1,2) show, that at the consideration of all Na-O contacts with R(Na-O)<R_min classical coordination numbers of sodium atoms are in agreement with CN, specified in the primary source. For the calculation of total CN of Na1 and Na2 atoms, also it is necessary to consider weaker Na-O contacts with W >1.5%, in view of which total CN should be written as 7+1 and 7+2, respectively. Fig.2a,b show that O3, O6 and O7 atoms, Na-O contacts with participation of which determine n₂ values and were not considered in [10], give the appreciable contribution to the formation of atomic domains of sodium (on Fig.2 corresponding faces are shaded).

According to [11], in structure of Rb_1.67[Pt(C₂O₄)₂]Ç1.5H₂O crystallographically independent Rb1, Rb2, Rb3, Rb4 and Rb5 atoms have CN=7, 8, 8, 7 and 7, respectively, at that the Rb-O contacts with R(Rb-O)3.3µ were only in consideration. The established characteristics of VDPs (tables 3-7) show, that for the given atoms also it is necessary to take into account Rb-O contacts with R(Rb-O)=3.3 to 4.1µ and W>1.5%. Thus, on our data CNs of all Rb atoms differ from those specified in the primary source and equal 9, 10, 10, 9 and 11, respectively. Note that contacts, which are not only characterised by W <1.5%, but also are ‘indirect’, correspond to almost all Rb and Cs atoms, which take part in formation of domains of Rb2-5 atoms (tables 4-7), excepting Rb2 in the VDP of Rb2 atom. The data of tables 1-7 confirm conclusions of [9], indicating that all ‘indirect’ contacts (including Rb-O contacts) should not be considered as chemical bonds in view of the condition W<1.5%.

The results of calculation of the characteristics of VDPs for 2749 crystallographic sorts of M atoms in structure of compounds, data on which are contained in the above-mentioned database, and CNs of M atoms defined as CN_cl using the method described are given in tables 8-12. As can be seen from tables 8-12, at CN_cl 6 the hypothesis of constancy of volume of atomic domains (estimated by the volume of their VDPs (V_VDP)) for M atoms seems to be true overall. Some deviations of V_VDP from mean values (in particular, for Na atoms at CN_cl=9 and for K and Rb atoms at CN_cl=18) are not numerous. They occur because corresponding CN_cl are typical for the atoms in structure of thermodynamically metastable (under standard conditions) compounds (table 13). Features of a structure of the corresponding compounds also explain non-monotonic change of V_VDP for lithium atoms. In particular, among 14 compounds, in which lithium atoms are characterised by CN_cl=8, in 13 structures they are coordinated by the molecules of crown-ether and thus, the dimension of their atomic domains is determined by a diameter of ligand cavity. Data on unique compound SrLi₂Al₄(OH)₄(PO₄)₄ [20], in structure of which lithium atoms have CN_cl=7, cannot be sufficient for statistically significant estimation of the dimensions of atomic domains for Li at that CN_cl. The unusually low CN_cl=3, found in structures of LiBa₃A₃Ti₅O₂₁ (A=Nb, Ta or Sb) [21], is result of that the indicated compounds are intercalation phases.

,                                                  (1)

where V[P(p_i)] is the volume of the VDP (designated as P(p_i)) of a point p_i, r_i is the distance from a point of P(p_i) to the central point (p_i), n is space dimensionality (n=3) and Z is the number of basis points in unit cell of multiregular system. In particular, when Z=1 and a point p coincides with the center of gravity of VDP, G₃ is equal to the dimensionless second moment of inertia of the VDP. As was shown in [4], G₃ of a single VDP is an integral parameter of a degree of VDP distortion. Sphere is characterized by the smallest G₃ value (0.07697) therefore, the more G₃ value of a VDP the stronger its distortion (the smaller the degree of its sphericity).

The model of deformable spheres [23] may be used for alkali atoms at CN_cl 6. That is proved by indicated constancy of V_VDP and by isotropic character of distortion of their atomic domains, since the value of shift of M atom from a centre of gravity of VDP (D_A) is practically equal to zero for all CN_cl (tables 8-12). The degree of deformation of an atomic domain decreases with increase of the size of atom M, that is confirmed not only by the analysis of G₃ values, but also by the character of change of radii (R_sd) of spherical domains, volume of which is equal to V_VDP, in comparison with the values of radii [7] of corresponding ions (Fig.3). Note that, within the framework of the model of deformable spheres, R_sd can be considered as a value being proportional to the radius of an atom or an ion in the given electron state before placing it in a crystal field. For lithium and sodium atoms the significant deformation of their domains caused by formation of ionic-covalent bonds leads to appreciable increase of R_sd in comparison with ionic radii given in [7]. For K, Rb and Cs atoms the differences among R_sd and corresponding ionic radii do not exceed 0.05µ (Fig.3), that, in our opinion, also confirms nondirectivity of M-O interactions (M=K, Rb or Cs).

[10] Panagiotopoulos, N.Ch., Brown, I.D.: The crystal structure of a -Na₂Cr₂O₇ and the a –b phase transition. Acta Cryst. B29 (1973) 890-894.

[11] Kobajashi, A., Sasaki, Y., Kobajashi, H.: Structural studies of commensurate peierls state of Rb_1.67[Pt(C₂O₄)₂]Ç 1.5H₂O. Bull. Chem. Soc. Jpn. 52 (1979) 3682-3691.

Table 1. Results of calculation of VDP of Na(1) atom in structure of a -Na₂Cr₂O₇*

*The radius of sphere, volume of which is equal to volume of VDP (R_sd), the characteristic of sphericity (G₃) and the shift of sodium atom from a centre of gravity of VDP (D_A) are specified as the general parameters of VDP, besides its volume and surface area. In each line of the table the name of L atom, its coordinates (x, y, z), interatomic distance (R(Na-L)) and solid angle (W ), at which corresponding face of VDP is ‘visible’ from a point, coinciding with sodium atom (or L) are consequently given. The symbol ‘#’ marks atoms, which are ‘indirect’ neighbours according to criterion [12]. W values corresponding to atoms, which were not taken into account by the authors [13] during determination of CN of sodium atoms, are marked by '##'.

Table 2. Results of calculation of VDP of Na(2) atom in structure of a -Na₂Cr₂O₇*

Fig.1. R(M-O) distributions in oxygen-containing compounds of

a) lithium (3093 Li-O contacts); b) sodium (10081 Na-O contacts); c) potassium (11323 K-O contacts);

d) rubidium (2793 Rb-O contacts); e) caesium (3775 Cs-O contacts).

The values on the axis of ordinates are expressed in percentage of the total number of corresponding M-O contacts.

Fig.2. VDPs and environment of Na(1) (a) and Na(2) (b) atoms in the structure of a -Na₂Cr₂O₇. The faces of VDPs, corresponding to Na-O contacts with 1.5%<W <5% are shaded.

Fig.3. Dependence of R_sd (— —) or R_ion (—n —) on chemical sort of an alkali atom. The values calculated by averaging ionic radius for all CNs given in [7] are accepted as R_ion (µ), namely: 0.90 (Li⁺); 1.27 (Na⁺); 1.64 (K⁺); 1.79 (Rb⁺); 1.93 (Cs⁺).