Study of the relationship of structural and symmetric features of crystals with their physical properties is an urgent problem of condensed-matter physics. Point symmetry determines the set of possible physical properties of crystals, primarily, in correspondence with the Neumann principle. The symmetric features of atomic structures of crystals can be characterized more completely taking into account the phenomenon of pseudosymmetry, which makes it possible to establish finer relationships of the structure-property type. Fedorov pseudosymmetry of crystals [1] is the phenomenon of invariance of a considerable part of the crystal atomic structure (part of electron density and (or) subsystem of atomic nuclei) with respect to some group of symmetry operations compatible with the lattice (with respect to some supergroup of the symmetry space group of crystal).

The pseudosymmetry of a specific structure can quantitatively be characterized by the degree of invariance (degree of pseudosymmetry) of its total electron density r(r) with respect to some isometric operation [1,2]:

Integration in Equation (1) is performed over the volume V of crystal unit cell. If is not a symmetry operation for the function r(r), the degree of pseudosymmetry < 1; however, if r(r) is symmetric with respect to the operation, reaches a maximally possible value: unity.

The second-order susceptibility of crystal determines the intensity of generation of the second optical harmonic and is a structureand symmetry-sensitive property of crystal. For centrosymmetric crystals, the second-order susceptibility should be zero. One might suggest that reduction of symmetry will lead to some dependence of the second-order susceptibility of crystal on the degree of invariance of crystal structure with respect to inversion.

The family of crystals with potassium titanyl phosphate (KTP) structure includes more than 100 compounds [3,4]. Their general formula can be written as MM’OXO₄, where M = K, Rb, Na, Cs, Tl, NH₄; M’ = Ti, Sn, Sb, Zr, Ge, Al, Cr, Fe, V, Nb, Ta, Ga; X = P, As, Si, Ge. Interoctahedral (M’-O-M’) oxygen atoms can be replaced with OHand F-; the resulting compounds with the general formula MM’(F,OH)XO4 also belong to the KTP family. The structure of KTP crystals is described by the space group Pna2₁.

We considered 118 crystals belonging to the KTP family, including 29 pure and 89 doped ones. Information about the nonlinear optical characteristics of 108 crystals was found in the corresponding publications. All crystals under consideration were separated into three groups with respect to the available data on their structure and nonlinear optical properties; the relationship between these groups is clearly shown in Figure 1.

The characteristics of nonlinear optical properties of KTP crystals are listed in Table 1, where the parameter I/I_reference is the ratio of second harmonic intensities from a sample under study studied and a powder sample of reference crystal.

The characteristics of nonlinear optical properties were determined in [8,10,11,24] by the Kurtz-Perry method [26] and in [3,5-7,12,14,19,22] by the method described in [27]. The components of the second-order susceptibility tensor were found in [16,17,20] using the Maker fringe technique [28,29].

With allowance for the results of our analysis of the corresponding publications, we can select crystals whose characteristics of nonlinear optical properties are comparable with those for KTiOPO₄ crystal (K_0.5Rb_0.5TiOPO₄, RTA, K_0.966Ti_0.966Nb_0.034OPO₄, K_0.921Ti_0.921Nb_0.079OPO₄, RTP, K_0.99Ti_0.99Sb_0.01OPO₄, KTi_0.96Zr_0.04OPO₄, TTP, KTi_0.9975V_0.0025OPO₄, K_0.5Ti_0.5Nb_0.5OPO₄, CTA, K_0.5Ta_0.5Ti_0.5OPO₄, KTiO(PO₄)_0.5(AsO₄)_0.5, TTA, KTi_0.7Nb_0.3OP_0.7Si_0.3O₄, KTi_0.65Nb_0.35OP_0.65Si_0.35O₄, KTi_0.6Nb_0.4OP_0.6Si_0.4O₄, RbTi_0.98Nb_0.02OPO₄, Na_0.87K_0.13TiOAsO₄, KTi_0.7Nb_0.3OAs_0.7Si_0.3O₄, KTi_0.6Nb_0.4OAs_0.6Si_0.4O₄ RbTi_0.927Nb_0.056Er_0.017OPO₄) and crystals with characteristics of nonlinear optical properties exceeding those of KTiOPO₄ crystal (KTA, K_0.98Ti_0.98Nb_0.02OPO₄, K_0.96Ti_0.96Nb_0.04OPO₄, K_0.97Ti_0.97Sb_0.07OPO₄, K_0.88Ti_0.98Zr_0.06OP_0.99O₄, K_0.88Ti_0.93Zr_0.11OP_0.99O₄, K_0.97Ti_0.99OAs_0.53P_0.49O₄, KTi_0.9Nb_0.1OP_0.9Si_0.1O₄, K_0.80Ti_0.26Zr_0.78OAs_1.01O₄, KTi_0.9Nb_0.1OAs_0.9Si_0.1O₄, K_0.94Nb_0.12Ti_0.91OAs_0.89Ge_0.09O₄, KTi_0.8Nb_0.2OAs_0.8Si_0.2O₄, K_1.02Nb_0.25Ti_0.76OAs_0.75Ge_0.23O₄, K_0.68Rb_0.32TiOPO₄, Cs_0.5K_0.5TiOAsO₄, KTi_0.97Zr_0.03OPO₄, K_0.54Li_0.46TiOAsO₄, K_1.03Nb_0.52Ti_0.48OAs_0.48Ge_0.51O₄, Rb_0.855Ti_0.955Nb_0.045OPO₄, KNb_0.52Ti_0.48OAs_0.48Ge_0.51O₄, RbTi_0.96Nb_0.04OPO₄, K_0.98Nb_0.46Ti_0.56OAs_0.58Ge_0.39O₄).

There are data in the literature on KTiOPO₄ crystals doped with Nb [30-36], Ge [37], Sn [33,38-40], Zr [41- 43], Sb [35,44,45], Ta [35], Fe [46], Hf [47], and Zn [48] and RTP crystals doped with Cs [49] and Zr [50,51]. It was indicated in [39] that an increase in the Sn content leads to a dramatic decrease in the output second-harmonic intensity to zero. The intensity of second-harmonic generation (SHG) for KTi_1-xZr_xOPO₄ crystals reaches a maximum at x = 0.28, where it is more than doubled in comparison with the KTiOPO₄ sample [41]. The SHG intensity increases with an increase in the zirconium content in RbTi_1-xZr_xOPO₄ crystals; at x = 0.034, it rises by 40% [50]. The SHG intensity increases by approximately 35% in comparison with pure KTiOPO₄ samples after replacement of 6% titanium atoms with hafnium [47]. RbTi_1-xTa_xOPO₄ and RbTi_1-xNb_xOPO₄ crystals were investigated in [52], as well as Yb-doped RbTi_1-xTa_xOPO₄ crystals and RbTi_1-xNb_xOPO₄ crystals doped with Yb, Ln and Er. KTiOPO₄ crystals doped with transition metals and RTA crystals doped with lanthanides were studied in [53]. A number of compounds (RbTi_0.98Er_0.01Nb_0.01OAsO₄, RbTi_0.96Er_0.02Nb_0.02OAsO₄, and KTi_0.98Cr_0.02O_0.98F_0.02PO₄, KTi_0.99Fe_0.01O_0.99F_0.01PO₄ RbTi_0.98Er_0.02O(AsO₄)_0.98(SO₄)_0.02) exhibited an increase in the SHG intensity in comparison with RTA and KTP crystals, respectively.

The complete characteristic of pseudosymmetry of any crystal under study with respect to inversion is a threedimensional distribution map of the degree of structural invariance (electron density) with respect to this operation (hereinafter, pseudoinversion), calculated for different positions of inversion points within their unit cell. These maps were obtained with scanning steps over the unit-cell axes a, b, and c chosen to be 0.05 of the cor responding unit-cell parameters. For 118 crystals (Figure 1) with known structure, we calculated three-dimensional pseudoinversion maps using equation (1). The calculations were performed using the computer program and technique described in [54]. Within this approach the electron density function is expanded in a Fourier series in structural amplitudes ([1], see Formulas (5) and (6)).

Figure 2 presents typical examples of cross sections of three-dimensional distribution maps of the degree of pseudoinversion for the structures of KTiOPO₄, KSnOPO₄, KTiOAsO₄, Cs_0.625K_0.375TiOAsO₄ crystals (cuts by the plane z = 0.25). For the structures presented in Figure 2, the origin of coordinates is chosen on the two fold screw axis, and the coordinates of pseudoinversion peaks on the x and y axes are 0.25. We chose cuts by the plane z = 0.25 in Figure 2 because the z coordinate of the pseudoinversion peaks for the structures of the aforementioned crystals is also 0.25.

This situation is typical of most structures under study; in KTP crystals is accompanied by a phase transition to the centrosymmetric space group Pnan. Indeed, having

added inversion to the set of symmetry operations of the space group Pna2₁, which describes the structure of KTP crystals at room temperature, we obtain the group Pnan, where the inversion centre with respect to the twofold screw axis has coordinates (0.25, 0.25).

Thus, in the polar phase of KTP structures, the pseudoinversion peaks are located specifically at inversion centres of these crystals in their high-symmetry nonpolar phase. With allowance for this circumstance, we will characterize the pseudosymmetry of the electron density in each crystal with a known structure by the maximum value of pseudoinversion h_max in the three-dimensional map, and the point with the coordinates corresponding to the found h_max values will be referred to as pseudoinversion centres.

Since the origin of coordinates is arbitrarily chosen in X-ray diffraction analysis, the coordinates of the pseudoinversion centres may differ from 0.25. In the Pna2₁ group [55], the origin of coordinates on the z axis can be chosen at any point, while in the directions of the x and y axes it may lie either on the twofold axis or at the intersection of mirror planes; therefore, the x and y coordinates of pseudoinversion centres can be either (0.25, 0.25) or (0, 0).

To refine the coordinates of pseudoinversion centres and h_max values, we additionally calculated the distribution of the degree of pseudoinversion with a relative scanning step of 0.025 over the unit cell axes. Fixed refined x and y coordinates of pseudoinversion centres were used for repeated calculation of pseudoinversion distribution along the z axis with a relative scanning step of 0.001.

Table 2 contains the maximum pseudoinversion values h_max and coordinates of pseudoinversion centres z(h_max) for a number of KTP structures.

Figures 3(a) and 3(b) show the distribution histograms for the degree of pseudoinversion h_max for pure and doped KTP crystals.

The distribution of pure KTP crystals over pseudoinversion is fairly uniform. As is indicated in Table 2, the mean value < h_max > for them is 0.606. The situation for doped crystals is different: the pronounced maximum in the histogram in Figure 3(b), which amounts to 31%, lies in the range of pseudoinversion values of 0.4 - 0.5, which is followed by a sharp falloff. Therefore, the fraction of pseudo-centrosymmetric structures among doped KTP crystals is very small. The mean value < h_max > for doped crystals is 0.490. Thus, doped KTP crystals are “less symmetric” with respect to inversion than pure compositions.

For the crystals listed in Table 2, along with the calculations of the pseudoinversion of their structures as a whole, pseudoinversion extrema for sublattices of individual types of atoms (h_max(sublattices)) were also calculated. Pseudoinversion was calculated for the pure sublattices of all 118 crystals in Table 2; the distribution histogram of the corresponding extrema is shown in Figure 3(c). For 89 doped crystals in Table 2, the results of similar calculations for Mand M’-type sublattices containing doped atoms are presented as histograms in Figure 3(d). The histograms in Figure 3(c) indicate that the sublattices of Х, О, and F atoms are most pseudocentrosymmetric, sublattices of M-type atoms are least pseudo-centrosymmetric, and the pseudoinversion of the M’ sublattice is intermediate (< h_max(X) ≥ 0.857, < h_max(O) ≥ 0.720, < h_max(F) ≥ 0.870, < h_max(M ) ≥ 0.395, < h_max(M’) ≥ 0.661).

In the presence of impurities, the general view of the histogram for the M’ sublattice (Figure 3(d)) barely differs from that in Figure 3(c); its characteristic maximum shifts to higher pseudoinversion values and the mean value < h_max(M’) > becomes 0.700.

The pseudoinversion histogram for the M-type sublettice changes more radically: the pronounced peak in the range of 0.3 - 0.4 in Figure 3(c) disappears in Figure 3(d), and the distribution becomes more uniform in a wider pseudoinversion range; the fraction of crystals with ultimately acentric M sublattices increases. The mean pseudo inversion < h_max(M) > becomes 0.384; i.e., it barely changes in comparison with < h_max(M) > for M sublattices without impurities. Thus, the analysis of the pseudo inversion of individual sublattices suggests that the reductions of pseudoinversion of structures as a whole at a transition to doped KTP compositions, which is noted in Table 2 and Figure 3(b), is related to a great extent to the higher sensitivity of the pseudoinversion of M-type sublattice to the presence of doped atoms. Note that pseudosymmetry was previously studied [58] by the atomic displacement method [1] for 11 KTP-type structures. In particular, it was established that the potassium sublattice is less centrosymmetric in comparison with the TiO₆-PO₄ subsystem, and its pseudosymmetry relative to inversion is more sensitive to introduction of impurities.

A model was proposed in [8], according to which the second-order susceptibility of crystals is related to the symmetry of KTP-type structures and their pseudoinversion as follows:

As can be seen in Table 1, the SHG data with respect to the reference sample (powder of pure SiO₂ crystal) differ by an order of magnitude in different studies for KTP [3,5-7], and KTA [3,6,7] crystals. Based on this fact, we illustrated equation (2) by selecting a group of com-

positions (from the aforementioned set of crystals) for which experimental SHG data were obtained either by the powder method [26], or directly with respect to a powder of pure KTiOPO₄ crystal, or the data can be recalculated with respect to it based on a specific publication. In addition, since most sources yield data on the ratio of second-harmonic intensities for the studied and reference samples (I/I_KTP = I_2w/I_2w(KTP)), they were additionally recalculated into estimated values of the relative effective second-order susceptibility (it will be denoted as c_2w/c_2w(KTP)). In the first approximation, one can assume that where I_2w is the second harmonic intensity and I_w is the primary radiation intensity. Therefore, the desired ratio c_2w/c_2w(KTP) was estimated to be

.

Figure 4 shows the dependence of the set of c_2w/c_2w (KTP) values for KTiOPO₄ (CSD-№ 20970), RbTiOPO₄ (CSD-№ 281379, [10, 11]), KTiOAsO₄ (CSD-№ 202158, [14, 15]), K_0.565Li_0.34TiOPO₄ (CSD-№ 83482, [14]), RbTi_0.927Nb_0.056Er_0.017OPO₄ (CSD-№ 96408, [11]), Rb_0.855Ti_0.955Nb_0.045OPO₄ ([11]), K_0.534Li_0.34TiOAsO₄ (CSD-№ 83483, [14]), Na_0.87K_0.13TiOAsO₄ (CSD-№ 67541, [14]), K_0.89Nb_0.11Ti_0.89OPO₄ (CSD-№ 250046, [8]), K_0.96Nb_0.04Ti_0.96OPO₄ (CSD-№ 91556, [8]), K_0.99Ti_0.988Sb_0.0125OPO₄ (CSD-№ 250298, [8]), K_0.874Ti_0.927Sb_0.074OPO₄ (CSD-№ 250299, [8]), K_0.893Ti_0.833Sb_0.166OPO₄ (CSD-№ 250300, [8]), KTi_0.97Zr_0.03OPO₄ (CSD-№ 173235, [8]), and KTi_0.96Zr_0.04OPO₄ (CSD-№ 173233, [8]) crystals on the pseudoinversion h = h_max of their atomic structures in the coordinates. The linear approximation of the dependence presented in Figure 4 within the model described in [8], is characterized by a correlation coefficient of 0.91, and the confidence interval boundaries are (0.76, 0.97) at a confidence probability of 0.95.

Equation (2) can be more pronounced within the concentration series of samples of the same qualitative composition. For example, the SHG intensity decreases with an increase in the tin fraction in the KTi_1-xSn_xOPO₄ series, and the calculation of pseudoinversion for a series of known structures of this composition indicates a monotonic increase in the latter (Figure 5).

The boundary-composition crystal KSnOPO₄ has almost zero SHG intensity and the largest (in the KTi_1-xSn_xOPO₄ series) pseudoinversion: 0.886 (Table 1, no. 7; Table 2, no. 4). This fact is in agreement with the

data of Godfrey et al. [58], who established that the KSnOPO₄ structure can be partially described (in good approximation) by the Pnan group; exact description is obtained within the Pna2₁ group.

To date, despite the numerous publications on the structure and properties of KTP crystals, the question of the structural conditionality of the behavior of their nonlinear optical properties has not been completely clarified.

In this paper, we reported the results of studying the pseudosymmetric features of known structures of KTP crystals with respect to inversion and tried to analyze the entire set of known nonlinear optical parameters of these crystals in view of the obtained pseudosymmetric characteristics. In particular, it was shown that doped structures of KTP crystals have on average a lower degree of pseudoinversion than “pure” compositions; in some cases this feature adequately correlates with the increase in the relative intensity of the second optical harmonic. This correlation may manifest itself within the concentration series samples of the same qualitative composition.

We believe that, in order to establish the fundamental correlations between the structural and symmetric features of crystals (in particular, those belonging to the KTP family) and their nonlinear optical properties, for example, having the degree of pseudoinversion as a symmetric characteristic, it is necessary to primarily calculate this characteristic for the entire structure. This thesis is justified by the fundamental principles of symmetry in physical crystallography. The Neumann princeple, which sets a relationship between the symmetry of a medium (crystal) and the set of physical properties that are forbidden or allowed in this medium, deals with specifically the symmetry of the medium as a whole rather than the symmetry of its individual structural fragments within the unit cell. This approach was applied both in [8] and in this study. However, this does not depreciate the validity of the analysis of the characteristics of sublattices of individual types of atoms. Due to this analysis one can find sublattices with pseudosymmetric characteristics exhibiting a more significant sensitivity, for example, at a transition to doped compositions, and therefore, can determine to greater extent the behavior of the pseudosymmetric characteristics of crystal structures, as whole and physical properties of crystals.

This work was supported financially by the Ministry of Education and Science of the Russian Federation, project 14.B37.21.1158.