Lifshits Tails Caused by Anisotropic Decay: The Emergence of a Quantum-Classical Regime

We investigate Lifshits-tail behaviour of the integrated density of states for a wide class of Schrödinger operators with positive random potentials. The setting includes alloy-type and Poissonian random potentials. The considered (single-site) impurity potentials f: ?^d→[0,∞[ decay at infinity in an anisotropic way, for example, \(f(x_{1},x_{2})\sim (|x_{1}|^{\alpha_{1}}+|x_{2}|^{\alpha_{2}})^{-1}\) as |(x₁,x₂)|→∞. As is expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponent d/2 if both α₁ and α₂ are big enough, and there is a so-called classical regime with Lifshits exponent depending on α₁ and α₂ if both are small. In addition to this we find two new regimes where the Lifshits exponent exhibits a mixture of quantum and classical behaviour. Moreover, the transition lines between these regimes depend in a nontrivial way on α₁ and α₂ simultaneously.     

Energy levels of ions of silver and rhodium isoelectronic sequences with <Emphasis Type="Italic">Z</Emphasis> < 86

The method of extrapolation of the parameter of a model potential for states of one electron (4f, 5s, 5p, 5d, 5f) above the core 1s ²2s ²2p ⁶3s ²3p ⁶3d ¹⁰4s ²4p ⁶4d ¹⁰ and one vacancy (4d ⁹) in the same core is applied to calculate the energy levels in the silver and rhodium isoelectronic sequences with the maximum nuclear charge Z = 86. The energy levels of Ag-and Rh-like ions were used for the calculation of the energies of resonance transitions to the ground state ¹ S ₀ in Pd-like ions. Good agreement between the theoretical and calculated energies of the resonance transitions in Pd-like ions indicates the reliability of the results obtained.     

The Weak Localization for the Alloy-Type Anderson Model on a Cubic Lattice

We consider alloy type random Schrödinger operators on a cubic lattice whose randomness is generated by the sign-indefinite single-site potential. We derive Anderson localization for this class of models in the Lifshitz tails regime, i.e. when the coupling parameter λ is small, for the energies E≤?Cλ ².     

Untersuchungen von (d,p)- und (d, α)-Reaktionen an B10 und B11

Investigating reaction mechanisms, angular distributions and cross sections of the reaction B¹⁰(d, p) B¹¹ have been measured in the energy interval from 1,4 to 3,3 MeV of deuteron energy. More detailed measurements than until known have shown, that besides the well known stripping mechanism withl _n =1 contributions of compound nucleus formation are not neglectable. Especially atE _d =2,3 MeV,E _X (C¹²)=27,1 MeV, the effect of a single resonance contributes a great deal to the cross section of the groupsp ₁ andp ₃ . Further angular distributions and yield curves between 1,4 and 3,3 MeV have been measured in the (d, α)-reactions on B¹⁰ and B¹¹, showing quite different behaviour for both target nuclei.     

Efficient Quantum Algorithm for the Parity Problem of a Certain Function

Based on a particular mathematical structure of a certain function f(x) under our attention, we present a novel quantum algorithm. The algorithm allows one to determine the property of a certain function. In our study, it is f(x) = f(?x). Therefore, there would be a question here, “How fast can we succeed in this?” All we need to do is only the evaluation of a single quantum state \(|\overbrace {0,0,\ldots ,0,1}^{N}\rangle \) (N ≥?2). Only using that with a little amount of information, we can derive the global property f(x) = f(?x). Our quantum algorithm overcomes a classical counterpart by a factor of the order of 2^N.     

Observation of coherent <Emphasis Type="Italic">π</Emphasis><Superscript>0</Superscript> electroproduction on deuterons at large momentum transfer

The first experimental results for the coherent π⁰ electroproduction on a deuteron, e+d → e+d+π⁰, at large momentum transfer, are reported. The experiment was performed at Jefferson Laboratory at an incident electron energy of 4.05 GeV. A large pion production yield has been observed in the kinematical region 1.1<Q²<1.8 GeV², from the threshold to 200-MeV excitation energy in the dπ⁰ system. The Q² dependence is compared with theoretical predictions.     

Casimir Energy in Contact-Interaction Models

The problem of Casimir interaction between two δ ^d -like (d = 1, 2, and 3) sources in Minkowski space is examined on the basis of the ln det formalism. The result obtained for the case of two semitransparent plates (d = 1) coincides with the earlier result based on an alternative approach. The earlier assertion that there is no vacuum interaction between linear (d = 2) sources is disproved. An expression for the Casimir energy for two pointlike (d = 3) sources is obtained.     

Radiative Lifetimes of the Energy Levels of Palladium-like Ions Pr XIV and Nd XV

Radiative lifetimes for the 4d⁹4f, 4d⁹5p, 4d⁹5d, and 4d⁹6s levels in the spectra of palladium-like ions Pr XIV and Nd XV are calculated in the electric-dipole approximation by the intermediate coupling semiempirical method with the use of experimental energy levels available in the literature. The radial integrals of transitions necessary for calculating absolute values of transition probabilities are obtained in the length form with Hartree–Fock functions.     

Excitation of even levels of erbium atoms without a change in the number of 4<Emphasis Type="Italic">f</Emphasis> electrons

The excitation of even levels of erbium atoms by slow electrons that occurs without a change in the number of electrons in the 4f shell is experimentally studied. The levels investigated belong to the 4f ¹²6s7s, 4f ¹²5d6s, 4f ¹²6s6d configurations. The cross sections measured at an electron energy of 30 eV lie within the range (0.2–18) × 10^?18 cm².     

Hyperfeinstruktur-Untersuchungen im Rhenium-I-Spektrum

By optical means (i.e. hollow cathode, Fabry-Perot-interferometer and spectrograph) the hyperfine structure of 15 transitions in the Re-I spectrum was investigated. This permits to state the hfs coupling constantsA, B for 15 levels belonging to the configurations (5d ⁵) 6s ², (5d ⁵) 6s 6p, (5d ⁵) 6s 7s, (5d ⁶) 6s, (5d ⁶) 6p and (5d ⁴) 6s ² 6p. It was found that the (5d ⁵) 6s ^{2 6} S _5/2 ground state, too, shows a small hfs. Of its magnetic splitting both size and sign could be determined (A _0,185,187=?2,2±0,4 mK); for a possible quadrupole splitting upper limits were obtained. Estimates of the quadrupole moments resulted inQ _185,187?2,9b. — In the case of two levels (x ⁶ P _7/2 andy ⁶ F _11/2) the hfs analysis raises doubts as to their classification.     

The probabilities of radiative transitions and lifetimes of levels in the spectra of Cu XX,La IV

The probabilities of radiative electric dipole transitions 2p ⁵3p, 2p ⁶–2p ⁵3s, and 2p ⁵3d in the spectrum of neon-like copper Cu XX and 5p ⁵6p, 5p ⁶–5p ⁵6s, 5p ⁵7s, and 5p ⁵5d in the spectrum of xenon-like lanthanum La IV are calculated. The wave functions of the intermediate coupling were found from the wellknown experimental energy levels by the least-squares method (LSM). To transfer to an absolute scale the radial integrals for the transitions calculated in the form of a length on the Hartree–Fock functions have been used. By summing the calculated probabilities of the transitions, the lifetimes of the levels of configurations 2p ⁵3s, 2p ⁵3p, and 2p ⁵3d in the spectrum of Cu XX and levels of configurations 5p ⁵6s, 5p ⁵6p, and 5p ⁵7s in the spectrum of La IV have been obtained.     

Calculation of Spectra of Neutrons and Charged Particles Produced in a Target of a Neutron Generator

An algorithm for calculating the spectra of neutrons and associated charged particles produced in the target of a neutron generator is detailed. The products of four nuclear reactions ³H(d,n)⁴He, ²H(d,n)³He, ²H(d,p)³H, and ³He(d,p)⁴He are analyzed. The results of calculations are presented in the form of neutron spectra for several emission angles and spectra of associated charged particles emitted at an angle of 180° for a deuteron initial energy of 0.13 MeV.     

Spectra of rubidium-like Pd X-Sn XIV ions

Palladium, silver, and cadmium spectra excited in a low-inductive vacuum spark have been recorded on a vacuum grazing-incidence spectrograph with a grating of 3600 lines/mm and 3 m in radius, in the range 120–350 Å. The 4p ⁶4d-(4p ⁶5p + 4p ⁶4f + 4p ⁵4d ²) transitions in rubidium-like Pd X-Cd XII spectra have been investigated. Twenty two new spectral lines have been identified from the Dirac-Fock and Hartree-Fock calculations and the known lines were measured with high accuracy. The fine-structure intervals for the 4p ⁶4d ² D ground state and the energies of six, seven, and ten energy levels of excited configurations in Pd X, Ag XI, and Cd XII, respectively, were determined. On the basis of the analysis of isoelectronic regularities in the position of the energy levels, the previous identifications for the In XIII and Sn XIV spectra were refined.     

Theory of the muonic hydrogen-muonic deuterium isotope shift

Corrections of the α³, α⁴, and α⁵ orders are calculated for the Lamb shift of the 1S and 2S energy levels of muonic hydrogen μp and muonic deuterium μd. The nuclear structure effects are taken into account in terms of the charge radii of the proton r _p and deuteron r _d for one-photon interaction, as well as in terms of the electromagnetic form factors of the proton and deuteron for the case of one-loop amplitudes. The μd-μp isotope shift for the 1S-2S splitting is found to be equal to 101003.3495 meV, which can be treated as a reliable estimate when conducting the corresponding experiment with an accuracy of 10^?6. The fine-structure intervals E(1S)-8E(2S) in muonic hydrogen and muonic deuteron are calculated.     

4<Emphasis Type="Italic">p</Emphasis><Superscript>5</Superscript>4<Emphasis Type="Italic">d</Emphasis><Superscript>2</Superscript> configuration in the spectra of rubidium-like Ru VIII and Rh IX ions

The spectra of ruthenium and rhodium excited in a low-inductive vacuum spark have been obtained using a vacuum grazing-incidence spectrograph with a grating of 3600 lines/mm 3 m in radius in the range of 180–350 Å. The bidium-like spectra of Ru VIII and Rh IX were analyzed. Ten spectral lines of the 4p ⁶4d-4p ⁵4d ² transitions are identified and six energy levels in the 4p ⁵4d ² configuration with an excited internal electron were found for each ion for the first time.     

Unique Continuation Principle for Spectral Projections of Schrödinger Operators and Optimal Wegner Estimates for Non-ergodic Random Schrödinger Operators

We prove a unique continuation principle for spectral projections of Schrödinger operators. We consider a Schrödinger operator H = ?Δ + V on ${{\rm L}^2(\mathbb{R}^d)}$ L 2 ( R d ) , and let H _Λ denote its restriction to a finite box Λ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the type χ _I (H _Λ ) W χ _I (H _Λ ) ≥ κ χ _I (H _Λ ) with κ > 0 for appropriate potentials W ≥ 0 and intervals I. As an application, we obtain optimal Wegner estimates at all energies for a class of non-ergodic random Schrödinger operators with alloy-type random potentials (‘crooked’ Anderson Hamiltonians). We also prove optimal Wegner estimates at the bottom of the spectrum with the expected dependence on the disorder (the Wegner estimate improves as the disorder increases), a new result even for the usual (ergodic) Anderson Hamiltonian. These estimates are applied to prove localization at high disorder for Anderson Hamiltonians in a fixed interval at the bottom of the spectrum.     

Effect of spin crossovers on the Mott-Hubbard transition at high pressures

The effect of spin crossovers in d _n terms on the effective Hubbard parameter (U _eff) determining the gap between the lower and upper Hubbard bands is analyzed using a many-electron approach to describe the electron structure of Mott insulators. A new mechanism of the insulator-metal transition is established for d ⁵ ions, which is related to a decrease in U _eff caused by spin crossover. For other ions, the U _eff value is either independent of pressure (d ², d ⁴, d ⁷) or it exhibits nonmonotonic growth (d ³, d ⁶, d ⁸).     

Manifestation of the Coster-Kronig effect in the near-threshold electron-impact excitation of the Cd<Superscript>+</Superscript> ion

The near-threshold portions of the energy dependences of the effective excitation cross sections of the resonance transition 4d¹⁰5p²P_1/2° → 4d¹⁰5s²S_1/2 and the two-electron forbidden transition 4d⁹5s²²D_5/2 → 4d¹⁰5p²P_3/2° in the spectrum of the Cd⁺ ion were investigated by the spectroscopic method in crossed electron and ion beams. In the region of energy splitting of the ²P° and ²D levels, a significant resonance contribution of the autoionizing states of cadmium (decaying during the Coster-Kronig process) to the effective excitation cross sections of the noted transitions was revealed for the first time. It is found that the resonance contribution manifests itself much more strongly for the forbidden transition in comparison with the more intense resonance transition; i.e., the manifestation of the Coster-Kronig effect in the electron excitation of ions depends strongly on the cross section of the direct process. It is ascertained that, during the Coster-Kronig process, the main contribution to the resonance excitation of both the resonance and the two-electron forbidden spectral transitions is from the low-lying terms of the series of autoionizing states 4d¹⁰5p(²P_3/2°)ns, md and 4d⁹(²D_3/2)5s²ns, md, which are in the splitting region of the ²P_{1/2, 3/2}° and ²D_{5/2, 3/2} levels, rather than from the high-lying atomic autoionizing states of cadmium, which are located near the ionization limits (corresponding to the and ²P_3/2° and ²D_3/2 levels).     

Polarisationseffekte in der Hyperfeinstruktur von65Cu I, 3d 94s 2 m 2 D

The hyperfinestructure of the transition⁶⁵CuI, 3d ¹⁰4p ² P _3/2—3d ⁹4s ² m ² D _5/2 was investigated by optical interferometric methods. For the hfs-factors the following values were found:A(⁶⁵Cu,m ² D _5/2)=26,79(3) mK andB(⁶⁵Cu,m ² D _5/2)=5,81(10) mK. The core polarization of them ² D-terms and the Sternheimer corrections in the calculation of the quadrupolmoment of copper 65 from theB-factors of the terms 3d ⁹ 4s² m ² D and 3d ¹⁰ 4p ² P _3/2. were discussed.     

<Emphasis Type="Italic">nd</Emphasis> scattering at low energies in the two-body potential model

The S-wave phase shift δ(E) for the spin-doublet nd scattering at low energy E is calculated in the framework of the two-body approach. The effective-range-theory formula k cot δ = (1+k²/k ₀ ² )^?1(?1/α+C₂k²+C₄k⁴) is used to obtain approximate analytical results with different potentials. The corresponding coefficients C₂ and C₄ are obtained from our previous calculations of the asymptotic normalization parameter function C _t ² (aκ), where κ is the triton wave number and a is the doublet nd scattering length. The model reasonably describes δ(E), the results being quite sensitive to the choice of the effective nd potential.