Noninteger Numbers Research Articles

DIMENSIONAL CONTINUATION WITHOUT PERTURBATION THEORY

A formula is proposed for continuing physical correlation functions to noninteger numbers of dimensions, expressing them as infinite weighted sums over the same correlation functions in arbitrary integer dimensions. The formula is motivated by studying the strong coupling expansion, but the end result makes no reference to any perturbation theory. It is shown that the formula leads to the correct dimension dependence in weak coupling perturbation theory at one-loop.

Coherent Interface Structures Observed in Co/Au(001) Epitaxial Superlattices

A series of Co/Au(001) superlattices were prepared on GaAs(001) single-crystal substrates using an electron beam deposition method in an ultrahigh vacuum. X-ray diffraction spectra were measured with the scattering vector normal to the film surface, and diffraction peaks from the Co/Au(001) superlattices were observed throughout the entire measured range from low-angle to high-angle. For samples with noninteger numbers of atomic planes in one Co/Au bilayer unit, in the intermediate range (2θ: 20°∼31°) X-ray diffraction peaks indexed from the low angle side did not merge with those satellite peaks indexed from the Co/Au(002) fundamental peak in the high-angle range. This observation indicates a high crystalline quality for Co/Au(001) superlattices with a negligible fluctuation in layer thicknesses and a coherent interface structure.

Eigenmoments for multifragmentation

Linear rate equations are used to describe the cascading decay of an initial heavy cluster into fragments. Using a procedure inspired by the similar, but continuous case of jet fragmentation in QCD, this discretized process may be analyzed into eigenmodes, corresponding to moments of the distribution of multiplicities. The orders of these moments are usually noninteger numbers. The resulting analysis can be made time independent and is applicable to various phenomenological multifragmentation processes, in which case it leads to new approximate finite-size scaling relations for the spectrum of fragments.

Fluorine fluorosulfate revisited

Fluorine fluorosulfate, FOSO 2F, appears to be ideally suited to introduce both fluorine and fluorosulfate into a molecule and a number of such oxidative addition reactions have been reported early on 1). Once the highly explosive nature of FOSO 2F was recognized 2), almost all synthetic work stopped. A number of these addition reactions with the fluorides S 2F 2, SF 4, SeF 4, AsF 3, SbF 3 and MoF 5 and the halogen Br 2 and I 2 have been re-investigated or studied for the first time. The following conclusions are reached: (i) All reactions of FOSO 2F are vigorous and often explosive with FOSO 2F acting primarily as fluorinating agent. (ii) Byproducts observed include always bis(fluorosulfuryl) peroxide, S 2O 6F 2, and sometimes SO 2F 2 and O 2. (iii) Except for additions to SF 4 and SeF 4 where SF 5SO 3F and SeF 5SO 3F form in low yield, and are studied by heteronuclear NMR and vibrational spectroscopy, the reaction products are frequently non-stoichiomenric eg. IF n(SO 3F) 5−n, or BrF(SO 3F) 3−nwith n noninteger numbers. (iv) In most instances safe alternative routes are explored to avoid the use of FOSO 2F.

Thin-target cross sections of proton-induced reactions on barium and solar cosmic ray production rates of xenon-isotopes in lunar surface materials

Thin-target excitation functions for the proton-induced production of radionuclides and stable Xe-isotopes from barium were measured at the JULIC isochronous cyclotron of the IKP/KFA Jülich up to E p=45MeV using the stacked-foil technique. Additionally, thin-target cross sections for the production of radionuclides from Ba were determined at 147.5 MeV using the synchrocyclotron of the IPN Orsay. For stable Xe-isotopes the thin-target data are in severe disagreement with earlier measurements. The reasons for these discrepancies are discussed. For the experimental excitation functions a detailed hybrid model analysis was performed using the code ALICE LIVERMORE 87, which in contrast to earlier formulations of the model takes into account experimental mass data, multiple preequilibrium nucleon emission and non-integer initial exciton numbers. In general, the a priori calculations satisfactorily reproduce the experimental excitation functions. Some discrepancies have to be attributed to the neglect of preequilibrium emission of alpha-particles and to the use of average nuclear parameters in the calculations for 138Ba, which has a magic neutron configuration. Using the new experimental excitation functions up to 45 MeV and calculated ones between 45 to 200 MeV, depth-dependent production rates for the production of stable Xe isotopes from Ba by solar cosmic ray protons in the lunar surface are calculated and compared with the results of earlier investigations. The improvements still necessary to come to a complete and adequate description of cosmogenic Xe isotopes in extraterrestrial matter are outlined.

The phase formalism for the one-dimensional eigenvalue problem and its relation with the quantum Bohr-Sommerfeld rule

A quantum phase formalism is offered for searching for eigenvalues of the one-dimensional Schrodinger equation. Its application in the shooting method appreciably sped up convergence and made it absolute. The introduced quantum phase functions are close to their semiclassical analogues, unlike those used before, and transform into them in the semiclassical limit. A form of quantum analogue of the Bohr-Sommerfeld quantisation rule, allowing it to operate with non-integer quantum numbers within the framework of precise quantum consideration is obtained.

Ionization potentials of cobalt and nickel ions in the local-spin-density approximation.

In this article we report on the ionization potentials of all configurations of the ${\mathrm{Co}}^{\mathit{n}+}$ and ${\mathrm{Ni}}^{\mathit{n}+}$ ions obtained via transition-state calculations using local-spin-density (LSD) potentials. The calculations were performed numerically with and without modifications of the local exchange potential for fractional occupation numbers. When the exchange potential is corrected for noninteger occupation numbers, a more consistent picture of the ionization process is obtained than that given by the LSD Kohn-Sham exchange. The agreement with experimental results is also significantly improved.

Valence-band levels and optical transitions in quantum wells in a parallel magnetic field

The magnetic hole levels of GaAs/${\mathrm{Al}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Ga}}_{\mathrm{x}}$As quantum wells in a parallel field are computed in the effective-mass approximation, by means of an extension of the Luttinger oscillator formalism to noninteger quantum numbers. A detailed analysis of the valence subbands and their dependence on the well width, magnetic field, and aluminum composition is performed. We also compute the transition energy for different conduction valence subbands in GaAs/${\mathrm{Al}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Ga}}_{\mathrm{x}}$As quantum wells in parallel fields, and compare with recent photoconductivity experiments.

Electronic properties of certain chromium spinels

Some electronic properties of the following spinel series:Zn 1-xCu xCr 2Se 4,Zn 1-xGa 2x 3 Cr 2Se 4 and Co 0.55Cu 0. 45Cr 2S 4-ySe y are presented. Using Slater's ideas on non-integer occupation numbers, the numbers of 3d holes (as teh carriers of magnetism in these spinels) per Cr atom with spins up and down have been obtained. From the condition of self-consistency for the magnetic moment the number of occupied states at the Fermi level with given spin projection and the Pauli spin paramagnetic susceptibility have been calculated and compared with other terms of the total high field susceptibility. According to the Stoner-Edwards-Wohlfarth theory a set of important exchange and band parameters of these compounds was calculated for the first time.

The semiclassical surface-of-section method in three dimensions

The semiclassical surface-of-section method is extended to treat a three-dimensional model problem. Integration of classical trajectories generates four-dimensional phase-space surfaces; two-dimensional interpolation on each hypersurface provides an exact Poincaré surface of section. Numerical quadrature provides classical actions and semiclassical quantum numbers. The energy, a function of three non-integer quantum numbers, is interpolated to integer values of the quantum numbers to obtain the semiclassical eigenvalue spectrum. Results for a model problem are compared with quantum variational calculations. Twenty-two eigenvalues were obtained with 17 trajectories.

Magnetic hole levels in quantum wells in a parallel field

The energy levels of holes in GaAs quantum wells in the presence of a parallel magnetic field are computed, based on a generalization to non-integer oscillator quantum numbers of the Luttinger Hamiltonian.

Semiclassical determination of eigenvalues in nonbending ozone

The authors have determined all the vibrational eigenvalues below 16,000 cm/sup -1/ for a two-mode ozone potential, using the surface of section semiclassical method combined with interpolation to integer quantum numbers; with this approach more than 80 eigenvalues were determined from only 21 arbitrary classical trajectories. The ozone potential surface is represented by a 20 x 20 grid of points obtained at the 6-31G* level with electron correlation included via a fourth-order Moeller-Plesset perturbation calculation. The forces needed to integrate the classical equations of motion are found by using a two-dimensional spline fit to the potential points. Classical trajectories are then generated; each gives tow surfaces of section from which classical actions and noninteger quantum numbers are extracted. A two-dimensional interpolation of the energy surface expressed as a function of these quantum numbers then rapidly locates the eigenvalues. These semiclassical eigenvalues generally agree with variationally determined quantum eigenvalues to within a few wavenumbers, for the first five states of each parity; this agreement holds even for states with eigenvalues corresponding to trajectories located in a classical resonance region.

Measurement and hybrid model analysis of integral excitation functions for proton-induced reactions on vanadium, manganese and cobalt up to 200 MeV

Integral excitation functions for p-induced reactions on V, Mn and Co were measured for p-energies between 45 and 200 MeV. 30 excitation functions for the production of radionuclides with 43 ⩽ A ⩽ 60 were determined. For most of the reactions, ranging from (p, n) to (p, 7p9n), scarcely no experimental data existed before now in this energy region. Together with earlier measurements the new data provide a consistent set of excitation functions of p-induced reactions from 0 to 200 MeV for these elements. A detailed hybrid model analysis was performed using the recent code ALICE LIVERMORE 82, which in contrast to earlier formulations of the model takes into account experimental mass data, multiple preequilibrium nucleon emission and non-integer initial excitation numbers. In general, the a priori calculations satisfactorily reproduce the experimental excitation functions. Some discrepancies have to be attributed to the neglect of direct processes and of preequilibrium emission of α-particles. The reasons for a complete failure of the calculations in some exceptional cases are still not clear.

Blocking Probability of Noninteger Groups with Trunk Reservation

A model for computing the blocking probability in circuitswitched networks operating with trunk reservation is extended to the case of noninteger trunk numbers. This generalization is required if the model is to be used in network optimization algorithms, which are generally evaluated over the reals. A simple extension is first presented, and shown to be numerically unstable over some values of the parameters. A recursive approach is then explained that is both robust and efficient for an integer number of reserved trunks and arbitrary parameters. The model is finally extended to arbitrary trunk numbers, and an efficient and robust algorithm is proposed.

Electron densities in search of Hamiltonians

By utilizing the knowledge that a Hamiltonian is a unique functional of its ground-state density, the following fundamental connections between densities and Hamiltonians are revealed: Given that ${\ensuremath{\rho}}_{\ensuremath{\alpha}}, {\ensuremath{\rho}}_{\ensuremath{\beta}},\dots{},{\ensuremath{\rho}}_{\ensuremath{\omega}}$ are ground-level densities for interacting or noninteracting Hamiltonians ${H}_{1}, {H}_{2},\dots{},{H}_{M}$ ($M$ arbitrarily large) with local potentials ${v}_{1}$,${v}_{2}$,$\dots{}$,${v}_{M}$, but given that we do not know which $\ensuremath{\rho}$ belongs with which $H$, the correct mapping is possible and is obtained by minimizing $\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}} [{v}_{1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\alpha}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+{v}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+\ensuremath{\cdots}{v}_{M}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\omega}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})]$ with respect to optimum permutations of the $\ensuremath{\rho}$'s among the $v$'s. A tight rigorous bound connects a density to its interacting ground-state energy via the one-body potential of the interacting system and the Kohn-Sham effective one-body potential of the auxiliary noninteracting system. A modified Kohn-Sham effective potential is defined such that its sum of lowest orbital energies equals the true interacting ground-state energy. Moreover, of all those effective potentials which differ by additive constants and which yield the true interacting ground-state density, this modified effective potential is the most invariant with respect to changes in the one-body potential of the true Hamiltonian. With the exception of the occurrence of certain linear dependencies, $a$ density will not generally be associated with any ground-state wave function (is not wave function $v$ representable) if that density can be generated by a special linear combination of three or more densities that arise from a common set of degenerate ground-state wave functions. Applicability of the approach to density-functional theory is emphasized for non-$v$-representable as well as for $v$-representable densities. In fact, a particular constrained ensemble search is revealed which provides a general sufficient condition for non-$v$ representability by a wave function. The possible appearance of noninteger occupation numbers is discussed in connection with the existence of non-$v$ representability for some Kohn-Sham noninteracting systems.

An investigation of the1 B 1 excited states of water

The first seven1 B 1 excited states of water are investigated within the framework of the single centre method. Energies, wave-functions and electronic densities are found using one basis set of 25 Slater type orbitals and one of 40 orbitals (both with non-integer principal quantum numbers). All singly excited configurations of1 B 1 symmetry are taken into account. For each state the orbital parameters (n's andζ's) are carefully optimized. The energy differences between neighboring excited states are, when the comparison can be made, in good agreement with experiment as is also the major contributing configuration of each state. Particular emphasis is placed on the electronic rearrangement that occurs during excitation and this is explicitly brought out by the contour maps of the electronic density. These maps show “satellite-clouds” in the excited states.

Investigation of the A11 Excited States of Water

The first six A11 excited states of water are investigated within the framework of the single-center method. Energies, wavefunctions, and electronic densities are found using a basis set which consists of 29 Slater-type orbits (with noninteger principal quantum numbers). For each state the orbital parameters (n's and ζ's) are carefully optimized. The ordering of the states and the major contributing configuration of each state is in agreement with experiment. Particular emphasis is placed on the electronic rearrangement that occurs during excitation, and this is explicitly brought out by the contour maps of the electronic density.

Computations for Two-center Transition Moments Concerning Slater Orbitals with Non-integer Principal Quantum Numbers

The precision of the parabolic interpolation technique which have been usually applied for the computations bf the two-center transition moment integrals with Slater Orbitals of non-integer effective principal quantum numbers is ascertained tolerable by comparing with the results of the McLaurin series expansion technique.

A theoretical investigation of the water molecule

A single centre wave function for H2O consisting of 19 determinants is obtained. Each determinant is composed of Slater type orbitals having non-integer principal quantum numbers. Up to f orbitals are considered and in the final wave function there are 19 linear and 36 non-linear parameters; these are determined variationally. The total molecular energy is found to be -75·8611 a.u. (-76·4606), the bond distance to be 1·714 a.u. (1·81) and the breathing force constant to be 1·13 a.u. (1·07), where the values in parentheses are those determined experimentally.

Theoretical Investigation of H3O+

A single-center molecular wavefunction for H3O+ consisting of nine terms is obtained. Each term is a symmetry-adapted combination of Slater determinants and each determinant is composed of Slater-type orbitals having noninteger principal quantum numbers. Up to and including f orbitals are considered and in the final wavefunction there are nine linear and 24 nonlinear parameters in all; these are determined variationally. The molecule is predicted to be a rather flat pyramid (H–O–H angle: 114°26′) having a bond distance of 0.95 Å; this is in accord with experimental observation.