3d Ions in Solids and Microscopic Crystal Field Effects: Theoretical Analysis and Relations with Experimental Spectroscopic Data

The transition metal ions with unfilled 3d electron shell are important activator ions for solid-state lasing and solid-state lighting. Since their spectroscopic properties depend significantly on the nearest environment, studies of the microscopic crystal field effects (influence of variation of the impurity center geometry) acquire additional importance. In the present paper, methods of calculating the crystal field strength 10Dq and its dependence on the interionic distances R are described. Relation between the 10Dq(R) functions and experimentally observed quantities such the Stokes shift is highlighted. The results of performed 10Dq(R) calculations for several crystals doped with various 3d ions are summarized and discussed.


Introduction
It is a well-known fact that the highly degenerated energy levels of free ions with unfilled d-and felectron shells are split into a number of sublevels, if such ions are placed into crystalline solids. The number of the formed energy levels and their properties are determined by the crystal field created by the crystal lattice ions, whose geometrical arrangement and electrical charges are the most important factors affecting the optical properties of impurity ions [1].
The ions with the unfilled 3d electron shell are excellent probes for the crystal field effects, since their open external 3d shell makes them very sensitive even to small variations of the interionic distances and/or angles between the chemical bonds. In an ideal octahedral crystal field the five-fold degenerated 3d orbitals are split into two sets: three orbitals with the t2g symmetry and two orbitals with the eg symmetry. The energy interval between them is called the crystal field strength and is denoted by 10Dq. The value of 10Dq increases with shortening interionic separations R and decreases otherwise. The distance dependence of 10Dq on R can be written in general as where A and n are some constants. The value of n if determined from the point charge model (PCM) of crystal field should be 5 [2]. However, numerical estimations of 10Dq for real systems with such n value do not yield good agreement with the experimental data because of oversimplified PCM assumptions, which completely neglects the quantum nature of the impurity ions and nearest neighbors, overlap of their wave functions etc. Various quantum chemical calculations allow for theoretical estimations of the n value; it appears to vary in a wide range from 3.5 to 7.3 for various systems [3][4][5][6][7]. It is also possible to estimate the n value from the experimental measurements of the absorption peaks shifts with pressure [8,9].
Knowledge of n gives a deeper insight into the microscopic structure of the impurity ions and relations between the geometry of the impurity center and its energy level scheme. It also allows to estimate the experimentally observed quantities, such as Stokes shift (difference between the maxima of the absorption and emission bands related to the same electronic transition), thus linking together the microscopic crystal field effects with macroscopic characteristics of impurity centers.
In Section 2 the basic theoretical foundations needed for the microscopic crystal field effects studies are described briefly, and a short summary of the calculated results for a number of crystals containing the ions with unfilled 3d shell is given in Section 3. The paper is concluded with a short summary.

Calculations of 10Dq values: theoretical background
If the values of A and n in Eq. (1) are known, then it is possible to evaluate the constants of the electron-vibrational interaction with the a1g and eg normal modes of the octahedral complex as follows: These estimated constants can be used for calculations of the energetical Stokes' shift ( ), where i denoted the a1g and eg normal modes: with M standing for the mass of a single ligand and Si being the non-dimensional Huang-Rhys factor for ith normal mode with the frequency . The total Stokes shift, which is a result of interaction with both a1g and eg normal modes can be taken as a simple sum of the Stokes shifts coming from each mode separately [10,11]. Moreover, if both pressure and distance dependences of 10Dq are known, it is possible to evaluate the bulk modulus B of a considered crystal from the following equation: It is possible to use the crystal field theory or various ab initio methods for calculations of the 10Dq values. In the case of the ions with the d 1 (d 9 ) electronic configurations the 10Dq value is simply the separation between the t2g and eg states. If the ions with the d 2 (d 8 ) electronic configurations are considered, 10Dq equals to the energy interval between the 3 A2g and 3 T2g states originating from the ground 3 F term. If the ions with the d 3 (d 7 ) electronic configurations are studied, then 10Dq is the separation between the 4 A2g and 4 T2g states coming from the ground 4 F term [2].
The method of finding the distance dependence of 10Dq in Eq. (1) can be formulated as follows: i) The energy levels of an impurity ion in a given crystal are calculated for different interionic distance with a small step (it is sufficient to take an equilibrium "impurity ion -ligand distance" R0 and consider the interval from 0.9R0 to 1.1R0 with a step of 0.01R0). ii) The extracted 10Dq values are plotted as a function of distance and then are fitted to the power law given by Eq. (1). iii) Eqs. (2)-(4) can be used to estimate the Stokes shift and the bulk modulus, which then can be compared with the experimental data (if available) to check the validity of the performed calculations.

Results and discussion
The results of calculations of the 10Dq dependence on distance for a number of solids doped with the 3d transition metal ions are presented in this section. Different calculating techniques have been employed, e.g. discrete variational multielectron method (DVME) [12], exchange charge model (ECM) of crystal field [13], and plane-wave based CASTEP software [14]. The calculations were performed for varying interionic distances and the obtained 10Dq numerical values were approximated to the power laws, like in Eq. (1). The calculated 10Dq(R) distances are summarized below in Table 1. In all equations the distance R should be taken in Å, then the 10Dq value will be in cm -1 .   Table 1. Figure 1 illustrates the calculated 10Dq(R) dependences for some of systems from Table 1. The decreasing trend of the crystal field strength with increasing interionic separation is clearly seen. The calculated 10Dq values are shown by symbols, they were approximated by the power law functions, whose equations can be found in Table 1 as well. The vast majority of the considered systems are the Cr 3+ -bearing materials, which is explained by their importance for many applications (such as solidstate lasers, phosphors for the solid-state lighting etc) and availability of the experimental data on the absorption and emission spectra of these crystals. The data in Table 1 show considerably wide range of the obtained n values, from about 4.3 to about 6.55. The deviation of n from the point charge value of 5 is explained by the influence of covalent effects, overlap of the wave functions of the impurity ions and ligands, formation of the molecular orbitals (rather than pure atomic states) etc. The distance dependence of 10Dq(R) appears to be essentially host-and impurity ion-dependent. It can be also seen from Table 1, that the value of n increases with increased charge of an impurity ion. This fact can be attributed to more pronounced variations of the electron density around highly charged ions, since they stronger attract the electron density of the s-and p-states of ligands.
The calculated Stokes shifts in Table 1 agree well with the experimental data. Additionally, using the 10Dq(R) dependence for Cs2NaYCl6:Cr 3+ , its compressibility was estimated to be 9.88×10 −4 kbar -1 , that is very close to the experimental value of 9.70×10 −4 kbar -1 [19], which serves as an additional argument confirming validity of the performed analysis of the microscopic field effects.

Conclusions
Several examples of calculations of the crystal field strength 10Dq for the transition metal ions with the unfilled 3d electron shell are given in the present paper. Importance of knowledge of the 10Dq(R) dependence is emphasized by the possibility of extracting the experimentally observed value of the Stokes shift and host's crystal compressibility. If the experimental and theoretical Stokes shifts are in good agreement, this circumstance can give an opportunity of predicting the Stokes shift for other systems, which may be of high importance for assessing the application perspectives of optical materials.