Jacek Mulak

On a complementary scale of crystal-field strength

A new measure of the crystal-field strength, complementary to the conventional one, is defined. It is based on spherical averages |Bk0|av or |∑kBk0|av, k = 2, 4, 6, of the crystal-(ligand)-field Hamiltonian parametrizations, i.e. on the axial crystal-field (CF) parameters modules averaged over all the reference frame orientations. It is proved that they are equal to and , respectively. While the traditional CF strength measure has been established on the parametrization modules, it means on the second moment of the CF energy levels, the new introduced scale employs rather the first moment of the energy modules and reveals a better resolving power. This complementary measure enables us to differentiate the strength of various iso-modular parametrizations according to the classes of rotationally equivalent ones. Using both the compatible CF strength scales one may draw more accurate conclusions about the Stark levels arrays and, in particular, their total splitting.

Symmetries of the oxygen ligands of Ni(II) and Co(II) ions in synthetic A, X, and Y zeolites

Abstract The magnetic susceptibility of Ni 2+ and Co 2+ ions in fully and partly dehydrated X, Y, and A zeolites has been measured over the temperature range from 4.2 to 300 K. Zeolites of various exchange levels (11–69%) have been investigated. A comparison of ionic radii of Ni 2+ , Co 2+ , and of O 2− shows that Ni 2+ and Co 2+ ions in dehydrated zeolites can assume C 3 v and C 4 v symmetries as well as the previously proposed (11, 12) O h and T d coordinations. The temperature dependences of the magnetic susceptibility of Ni 2+ and Co 2+ in C 3 v and C 4 v O 2− ligand symmetries were calculated from crystal field theory, using Van Vlecks formula. By comparison of the experimentally found χ M ( T ) dependences with those theoretically calculated for C 3 v and C 4 v as well as with those already known for O h and T d symmetries, we have tried to determine a sequence of occupation and distribution of Ni 2+ , Co 2+ ions in zeolites as depending on temperature, dehydration time and dehydration level. It was established that in fully dehydrated Y zeolite there is a preference for Ni 2+ ions to occupy C 4 v sites and for Co 2+ to occupy C 3 v sites, whereas in X and A zeolites no such preference to occupy specific sites is found. No relation between the quantity of exchanged ions and distribution or sequence of occupation of exchange positions seems to exist for A and X zeolites. In partly dehydrated zeolites the change of the χ M −1 ( T ) slope with the degree of dehydration is similar to that described in other papers. It seems that the main factors controlling the process of occupation and distribution of the ion exchange sites are the parameters of the dehydration process and ion migration in the zeolites. In fully dehydrated zeolites the exchange sites of low coordination number and C 3 v or C 4 v symmetries are most probable on geometrical, electrostatic and chemical grounds.

On standardization of crystal-field Hamiltonians parametrization: triclinic symmetry case

The maps of magnitudes of the axial crystal-field parameters, Bk0, for k = 2, 4, 6, which enter the parametrizations as functions of the z-axis spherical coordinates of the relevant reference frames are identical for all equivalent parametrizations with accuracy to the definite rotations of these frames. Therefore, it is possible to reduce all tested parametrizations to the one common reference frame providing that one can find for all these parametrizations the same distinguished space direction. This condition is fulfilled by the three z-axes of the frames for which the axial parameter, Bk0 (where k = 2, 4, 6 corresponds to the component multipoles), reaches its maximal value max Bk0 ≤ [∑m|Bkm|2]1/2. These maxima can serve as convenient discriminants of the entire classes of equivalent parametrizations. Based on the distinguished directions and transformational properties of parametrizations with respect to the reference frame rotations, the paper presents the method how to effectively verify the equivalence of these parametrizations and postulates the way of their standardization. This method can be applied to all point symmetries of the central ion, although it seems to be particularly useful and recommended for triclinic symmetry (C1, Ci).

Multipole characteristics of the open‐shell electron eigenstates

The second moment of the sublevels within the initial state |αSLJ) which constitues a natural and adequate measure of the crystal-field (CF) effect can be redefined as σ 2 = 1/(2J+1)Σ k S 2 k A 2 k , where S k = (1/(2j+1)Σ q |B kq | 2 ) is the so-called 2 k -pole CF strength, whereas A k = the reduced matrix element of the k-rank spherical tensor operator. Therefore, the CF effect depends on the sum of products of the two factors representing the identical multipole components of two different charge distributions. The term A k expresses the asphericity of the central ion open-shell, whereas the term S k the asphericity of its surroundings. When these two distributions do not fit each other the observed CF splitting can be unexpectedly weak even for considerable values of the total S = (Σ k S 2 k ) and A = (Σ k A 2 k ) ½ . The tabulated quantities of the Ak (|αSLJ>) as the 2 k -pole type asphericities, are the intrinsic characteristics of the electron states revealing their multipolar structure and hence their potential susceptibility to CF splitting separately for each effective multipole. For any chosen pair of a central-ion and CF potential the relevant A k and S k magnitudes, respectively, allow us to predict the scale of the related splitting. We can also compare the CF splitting of various states in the same CF potentials or the splitting of the same state in various CF potentials. Having the model σ 2 and their experimentally available counterparts we can evaluate the degree of admixing of the free-ion states. Since the independent quantities S k and A k occur as the scalar product in the formula for σ 2 the use of the total S and A notions should be critically considered.

The limits of the total crystal-field splittings

The crystal-fields causing

The quadrupolar term in the equivalent crystal-field Hamiltonians for various central-ion point symmetries

|J>

The total energy splitting of ionic eigenstates in the axial crystal fields

electron states splittings of the same second moment

Chapter 17 – Analysis of the experimental data. Interpretation of crystal field parameters with additive models

\sigma^{2}

Chapter 19 – Extension of the crystal field potential beyond the one-electron model

can produce different total splittings

Virtual bound state contribution to the crystal field potential

\Delta E