Maciej 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.

Enhancement of critical temperature of superconductors implied by the local fluctuation of EDOS

Abstract Applying the formalism developed for the generalized gap equation of BCS type, the critical temperature and heat capacity leap at the phase transition point between the superconducting and normal state are derived. The calculations are carried out when the fluctuation of the electronic density of states in the Fermi surface vicinity is included. Assuming that the EDOS has the Lorentzian form superimposed symmetrically near the Fermi surface on the constant background, it is shown that the critical temperature enhances significantly even for negligible modification of quasiparticle concentration. The disclosed effects can be interpreted in terms of the Van Hove scenario.

Valuation of characteristic ratios for high-Tc superconductors with anisotropic gap in the conformal transformation method

The application of a method of conformal-like transformation of the two-dimensional momentum space is presented and discussed with regard to two-dimensional anisotropic superconductors. A new function, the kernel of the density of states, and some integral formulae are derived, interpreted and studied for a few specific cases. Evidences for the incompatibility between the Van Hove scenario and the conformal transformation method with respect to a couple of characteristic ratios, i.e. the transition temperature and the specific heat leap, are displayed in detail for d- and p-wave pairing. The established method allows us to obtain the gap equation in a standardized form common to the models of superconductivity with an arbitrary dispersion relation. Classification of superconductors with respect to the valuation of characteristic ratios is proposed and commented on.

Capability of the Free-Ion Eigenstates for Crystal-Field Splitting

Any electronic eigenstate of the paramagnetic ion open-shell is characterized by the three independent multipole asphericities for and 6 related to the second moments of the relevant crystal-field splittings by , where . The Ak as the reduced matrix elements can serve as a reliable measure of the state capability for the splitting produced by the k-rank component of the crystal-field Hamiltonian. These multipolar characteristics allow one to verify any fitted crystal-field parameter set by comparing the calculated second moments and the experimental ones of the relevant crystal-field splittings. We present the multipole characteristics Ak for the extensive set of eigenstates from the lower parts of energy spectra of the tripositive 4 f N ions applying in the calculations the improved eigenfunctions of the free lanthanide ions obtained based on the M. Reid f-shell programs. Such amended asphericities are compared with those achieved for the simplified Russell-Saunders states. Next, they are classified with respect to the absolute or relative weight of Ak in the multipole structure of the considered states. For the majority of the analyzed states (about 80%) the Ak variation is of order of only a few percent. Some essential changes are found primarily for several states of Tm3+, Er3+, Nd3+, and Pr3+ ions. The detailed mechanisms of such Ak changes are unveiled. Particularly, certain noteworthy cancelations as well as enhancements of their magnitudes are explained.

Equilibrium States and Thermodynamical Properties of D-Wave Paired HTSC in the Tight-Binding Model

A quasi two-dimensional d-wave paired superconductor in the BCS-type approach is considered. The tight-binding model corrections are taken into account by means of the conformal transformation defined in the framework of the formalism postulated recently by the authors. This method constitutes an extension of the van Hove scenario and exhibits a coincidence of the BCS-type and the Hubbard-type models, as well as makes possibility to determine equilibrium states of the system according to the symmetry rules. The results show essential changes in the critical temperature and specific heat jump for various forms of the dispersion relation. Additionally, a few examples of the energy gap contours obtained by the reverse conformal transformation are presented.

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