Volume 39, Issue 8

We explicitly construct normalized solutions of the Klein-Gordon equation for various types of attractive potentials. Our computed binding energies of several short- and long-range potentials are compared with results published by different authors. Special emphasis is laid on the determination of critical potential strengths, i.e. the onset of spontaneous pair creation of spin-0 particles. In the vicinity of these critical values the single-particle interpretation breaks down. The phenomenon of simultaneous particle and antiparticle binding in a short-range potential is investigated.

The non-relativistic motion of a particle in a central field with 1/r potential, e.g. the motion of an electron in the Coulomb field of a charged nucleus at rest, is described by the equation of motion (non-relativistic Kepler problem) m x″ = α · x /r 3 with α = e z e (product of the charges of the central body e z and the electron e). From this equation of motion, three statements of conservation can be derived: in respect of the energy E, of the angular momentum L and of the Lenz vector Λ = m {x′× L + α ·x/r}. The geometric meaning of Λ is that of a vector pointing in the direction of the perihelion of the particle orbits (conic sections). It will be demonstrated that also at the relativistic Kepler problem, which is based on the equation of motion an analogous Lenz vector exists. It represents a quantity of conservation - in the same way as the relativistic energy and the relativistic angular momentum. For the transitional case → ∞, where the relativistic problem turns into the non-relativistic problem, the relativistic Lenz vector also turns into the non-relativistic Lenz vector. The generalised (relativistic) Lenz vector has also a geometric meaning. Its direction coincides with the oriented axis of symmetry of the orbits (rosettes, spirals, hyperbola-type curves etc.). The quantity of conservation Λ occupies a special position in respect of the quantities of conservation energy and angular momentum. Whereas the energy and the angular momentum correspond with a symmetry of time and space, the Lenz quantity of conservation corresponds with a symmetry of the orbits. The fact that the Lenz vector can relativistically be generalised touches thereby on principal aspects.

The Lenz vector Λ= m {x′ × L + α · x/r} is a special integral of motion of the non-relativistic Kepler problem. Accordingly, there also exists a special integral of motion Λ (x, x') of the relativistic Kepler problem as pointed out in Part I of this paper. In the following Part II, the system of equations is compiled to the extent required for the relativistic generalisation of the Lenz vector and its interpretation. Starting point are the statements of conservation for the energy and the angular momentum. From these two quantities, it is possible to derive the complete system of solutions x(t) in accordance with standardised methods. Some details and some modifications of the approach are of special importance for the purposes of this paper. In the centre of the consideration are the construction and interpretation of the standardised polar angle θ as a function of r, r'. The explicit time dependance of the functions r(t), θ(t) is not required for constructing the relativistic Lenz vector in the form Λ (x, x'). In the literature, one does Find only few, not sufficient for our purposes, data on the relativistic orbits (rosettes, spirals, hyperbola-type curves etc.). It is, therefore, necessary to discuss these orbits in detail. For this purpose, a series of relationships are derived which apparantly have not been published before.

Lattice parameters and thermal expansion of the three crystallographic modifications (α, β and β') of gadolinium molybdate have been determined. The α-β and β'-β transitions have also been investigated by means of DSC and DTA measurements. For the first one a transition enthalpy of 7.5 ± 0.5 kcal/mol was evaluated, while it was observed that factors such as the thermal history, the particle size and the heating/cooling rate affect the peak areas of the second one. The results can be explained on the basis of the model proposed by Jeitschko for the β'-β transition provided that the necessary kinetic aspects are taken into account.

The refractive indexes of molten LaCl 3 -KCl, LaCl 3 -NaCl, and LaCl 3 -CaCl 2 mixtures were measured goniometrically for nine wavelengths and represented by empirical formulas as functions of both temperature and wavelength. According to the obtained formulas and the molar volume data, the molar refractivities and the electronic polarizabilities were estimated.

Internal mobility ratios in the molten binary system (Rb-Cs)NO 3 have been measured with the Klemm method. From these and the available data on the densities and conductivities, the internal mobilities of Rb + and Cs + have been calculated. Above (below) 420 °C, Cs + is found to be more (less) mobile than Rb + . The internal mobilities of these ions in various binary alkali nitrates are compared.

The electrolytic conductivity of pure molten silver chloride and of molten silver chloride doped with up to 2mole% of CaCl 2 , CdCl 2 , CsCl, LiCl has been measured in the temperature range 460°C to 620°C. The changes in the conductivity of silver chloride when doped with ionic salts are found to be analogous to the conductivity changes found in binary alkali halide mixtures.

Graphite has been oxidized to graphite hydrogensulfate with (NH 4 ) 2 S 2 Og in concentrated H 2 SO 4 . The stage 2 compound formed very rapidly, but further oxidation to stage 1 could be followed conveniently. X-ray diffraction of the intermediate phases showed nonintegral (00l)-reflections indicative of partial interstratification disorder. The interlayer spacing of the stage 1 compound increased with continued oxidation. The degree of oxidation as function of oxidation time has been determined from the position of the minima in the reflectance spectra. The density of the compounds and the ratio of acid molecules to anions in the intercalated layers has been estimated using these data and those of buoyancy measurements. There is a minimum in packing density early in the formation of the first stage. The kinetics of intercalation seems to be first order with respect to the free interlayer spaces.

The crystal structure of ThS 2 has been refined starting from single crystal diffractometry data. The accurate determination of the positional parameters allows us to reexamine the SPM analysis of the second degree crystal field parameters for Gd 3+ in ThS 2 matrix. The fitting of the EPR data is still possible by means of a suitable two exponents power law for the intrinsic parameter only if the substitutional Gd 3+ ion displaces in the mirror (bc) plane with respect to the undistorted Th 4+ site. The resulting increase of the mean metal-ligand distances agrees with the expected expansion of the ligand cage around the trivalent Gd ion. The applicability of the proposed method to examine the impurity induced distortions in an host crystal is discussed too.

On the basis of the rigid ion model and polarizable ion model, lattice dynamical calculations have been carried out for the fluorohalides BaFCl and BaFBr and for the oxyhalides LaOCl, LaOBr, YOCl, SmOCl and GdOCl which crystallize with the tetragonal Matlockite-type structure (PbFCl-type). The results of these calculations: short range force constants, effective dynamical charges and polarizabilities are communicated and discussed. Bonding in the fluorohalides is much more ionic than in the oxyhalides.

In a series of ortho- and para-substituted phenoxyls the coupling of the β-nuclei (H and P) is caused by hyperconjugation and π-σ-mechanisms, leaving open the signs of the combined HFS constants. By examining the ENDOR and TRIPLE spectra these couplings turned out to be negative. Thus the sign of the well known temperature gradient of these parameters, due to hindered rotation of the substituted group, is positive, leading to a planar conformation of the radicals at low temperature.

Three recursion relations for the algebraic structure count are deduced, which are closely analogous to the well-known recursion relation for the number of Kekulé structures. An efficient graphical method for the calculation of algebraic structure count is proposed.

Some properties of Hosoya's topological index Z and the modified topological index Z͂ of benzenoid hydrocarbons are examined. A simple relation between Z͂ and Z has been established, namely Z͂= Z 1+a , where a = 0.02.

The melting pressure curves of 2,2-dimethylpropane, 2,2-dimethylpropionitrile and 2-methylpropanol- 2 and the pressure dependence of the hex ⇔ fee transition of 2,2-dimethylpropane have been determined in the pressure range between 0.1 and 200 MPa. The self diffusion coefficients D in the liquid state of the three compounds have been determined as function of pressure (p max ≧ 120 MPa) from the melting pressure curve up to ca. 450 K. At high temperatures and high pressures the self diffusion coefficients of the three compounds become rather similar. The influence of the hydroxylgroup upon the diffusivity of (CH 3 ) 3 COD is most clearly seen at the lowest temperatures. The isobars of C(CH 3 ) 4 and (CH 3 ) 3 CCN are straight lines with identical slopes. For (CH 3 ) 3 COD all isobars show a curvature. From the data in the Arrhenius representation activation volumes and activation energies are derived. For C (CH 3 ) 4 the density dependence of D can be described with the rough hard sphere model.