The determination by an independent method ofP k electron capture probabilities in133Ba and139Ce decays gamma decay of133Ba

The electron capture decays of¹³³Ba (10.4 y) and¹³⁹Ce (140 d) were investigated with high resolution Ge(Li) detectors. By x-ray — γ-ray coincidences values of the quantityP _k ω _k ? _k for various transitions in the decay of¹³³Ba and¹³⁹Ce are obtained. The factor ω _k ? _k was measured independently by means ofK-conversion electron-K x-ray coincidences and was then used to determineP _k , theK-capture probability. The independent measurement of the product ω _k ? _k , together with recent accurate values of ω _k , provides a new method for the accurate calibration of semiconductor x-ray detectors. In the decay of¹³⁹Ce, a value ofP _k =0.78±0.03 is found (where the error represents twice the standard deviation) from whichQ _ec =326 _?30 ⁺⁷⁰ keV to the¹³⁹La ground-state is found by use of theory. In¹³³Ba decay, values are found ofP _k 1=0.72±0.04 for theEC transition to the 437 keV level in¹³³Cs,P _k 2=0.80±0.07 to the 384 keV level, and from an independent measurement, the ratioP _k 1/P _k 2=0.87±0.07. From these results the ground-state value ofQ _ec =522 _?10 ⁺²⁰ keV is derived from theory for¹³³Ba decay. The gamma spectrum of¹³³Ba also was remeasured. From the present gamma intensities and previous conversion electron intensities, new values forK-shell conversion coefficients are obtained. Previously reported γ-rays at 35 and 391 keV are not confirmed.     

Asymptotic inverse spectral problem for anharmonic oscillators

We study perturbationsL=A+B of the harmonic oscillatorA=1/2(??²+x ²?1) on ?, when potentialB(x) has a prescribed asymptotics at ∞,B(x)～|x|^?α V(x) with a trigonometric even functionV(x)=Σa _mcosω _m x. The eigenvalues ofL are shown to be λ _k =k+μ _k with small μ _k =O(k ^?γ), γ=1/2+1/4. The main result of the paper is an asymptotic formula for spectral fluctuations {μ _k }, $$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$ whose leading term \(\tilde V\) represents the so-called “Radon transform” ofV, $$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$ as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ^?α|V(x) ofB from asymptotics of {µ _k }. ₁ ^∞      

Kramers Kronig analysis of the optical properties of small silver particles

In order to complete a preceding paper the dielectric constant? of the particle material of small silver particles with diameters between 210 and 25 Å is computed in the wavelength region 365≦λ≦455 nm from the measured spectra of thesmall particle plasma reasonance absorption. For this purpose a properKramers Kronig relation is derived, and is checked by applying to particles with Drude free electron gas. The results, concerning the silver particles, are that the real part of? changes slightly, whereas the imaginary part is markedly enhanced (up to the ten-fold of the bulk values) if the particle size decreases. This size dependence of? can quantitatively be described with thefree path effect within the accuracy of the measured values. Conversely, thebulk dielectric constant of silver is obtained by applying the free path effect to the measured dielectric constant of the small particles.     

The Markov branching random walk and systems of reaction-diffusion (Kolmogorov-Petrovskii-Piskunov) equations

A general model of a branching random walk inR ¹ is considered, with several types of particles, where the branching occurs with probabilities determined by the type of a parent particle. Each new particle starts moving from the place where it was born, independently of other particles. The distribution of the displacement of a particle, before it splits, depends on its type. A necessary and sufficient condition is given for the random variable $$X^0 = \mathop {\sup max}\limits_{ n \geqq 0 1 \leqq k \leqq N_n } X_{n,k} $$ to be finite. Here,X _{n, k} is the position of thek ^th particle in then ^th generation,N _n is the number of particles in then ^th generation (regardless of their type). It turns out that the distribution ofX ⁰ gives a minimal solution to a natural system of stochastic equations which has a linearly ordered continuum of other solutions. The last fact is used for proving the existence of a monotone travelling-wave solution to systems of coupled non-linear parabolic PDE's.     

A multi-channel scattering theory for some time dependent Hamiltonians,charge transfer problem

Scattering theory for time dependent HamiltonianH(t)=?(1/2) Δ+ΣV _j (x?q _j (t)) is discussed. The existence, asymptotic orthogonality and the asymptotic completeness of the multi-channel wave operators are obtained under the conditions that the potentials are short range: |V _j (x)|≦C _j (1+|x|)^?2?ε, roughly spoken; and the trajectoriesq _j (t) are straight lines at remote past and far future, and |q _j (t)?q _k (t)| → ∞ ast → ± ∞ (j≠k).     

The decay of theT=1 isospin triplet inA=12 systems: II. The induced interactions in the anisotropy of beta-rays

Nuclear Structure: calculation of the asymmetryα _? ofβ rays emitted from polarized¹²B and¹²N nuclei. Comparison with experimental values. Formulae for the asymmetry parametersα _? ofβ rays emitted from polarized nuclei is derived using the formalism developed by Behrens-Bühring. The induced interactions are studied in detail. It is demonstrated that the expressions forα _? are independent of the induced pseudoscalar interaction. It was found that the corrections to the lowest order approximation are small. Experimental data of the Louvain and Osaka laboratories are analyzed. We should point out that the data of the asymmetry of theβ ^? decay of¹²B obtained in these two laboratories are quite different. Although the data reported by the Osaka Group require the existence of the induced pseudotensor (IPT) interaction as it has also been stated by other authors, if we combine data of¹²N from Osaka and of¹²B from Lauvain such experimental results can be understood without including the IPT interaction. Thus, we found some evidence that the IPT interaction could be not necessary to explain the asymmetries in the decay of members of theA=12 system.     

Asymptotic perturbation expansion in theP(φ)2 quantum field theory

In theP(φ)₂ model it is proved that the perturbation series for the infinite volume Schwinger functionsS(λ) are asymptotic in the limit as the coupling constant λ goes to zero. We also give conditions which imply smoothness ofS(λ) at arbitrary λ.     

Anionische Silberzentren („B-Zentren“) in Alkalihalogeniden

It is proposed that negatively charged silver ions on anion sites are responsible for theB bands found in silver doped alkalihalides. Experimental investigations confirming this model of theB center are presented. Optical absorption and emission ofB centers were measured in seven alkalihalides in the temperature range from 450 to 4 °K. Configuration coordinate diagrams were obtained for KCl, KBr, and KI. A chemical method was used to determine the charge of theB center. Thermal or optical excitation causes theB centers to dissociate into neutral silver atoms andF centers. The Ag⁰ centers are bound to interstitial positions at low temperature. At high temperature they collect together forming colloidal centers. TetragonalB _A centers were formed during optical bleaching of theB band in mixed crystals of the type KCl+ΔNaCl. Their absorption and emission spectra, optical orientation, and polarized emission were investigated. These properties are shown to be similar to those ofF _A centers.B _A fluorescence polarized almost completely in the [100] direction was observed. A Jahn-Teller splitting of theB absorption band was resolved in RbCl at low temperature. The splitting confirms the configuration5s² for the Ag^? ion. The results are compared with those for the isoelectronic centers In⁺ and Sn⁺⁺. Thin films of alkalihalides containing small amounts of silver, copper, or thallium were condensed simultaneously with alkali vapor. New bands in the UV region were found, possibly due to Cu^? and Tl^? centers.     

Degenerate asymptotic perturbation theory

Asymptotic Rayleigh-Schrödinger perturbation theory for discrete eigenvalues is developed systematically in the general degenerate case. For this purpose we study the spectral properties ofm×m—matrix functionsA(κ) of a complex variable κ which have an asymptotic expansion εA _k κ ^k as τ→0. We show that asymptotic expansions for groups of eigenvalues and for the corresponding spectral projections ofA(κ) can be obtained from the set {A _κ} by analytic perturbation theory. Special attention is given to the case whereA(κ) is Borel-summable in some sector originating from κ=0 with opening angle >π. Here we prove that the asymptotic series describe individual eigenvalues and eigenprojections ofA(κ) which are shown to be holomorphic inS near κ=0 and Borel summable ifA _k ^* =A _k for allk. We then fit these results into the scheme of Rayleigh-Schrödinger perturbation theory and we give some examples of asymptotic estimates for Schrödinger operators.     

Quantum graph vertices with permutation-symmetric scattering probabilities

Boundary conditions in quantum graph vertices are generally given in terms of a unitary matrix U. Observing that if U has at most two eigenvalues, then the scattering matrix S(k) of the vertex is a linear combination of the identity matrix and a fixed Hermitian unitary matrix, we construct vertex couplings with this property: For all momenta k, the transmission probability from the j-th edge to ?-th edge is independent of (j,?), and all the reflection probabilities are equal. We classify these couplings according to their scattering properties, which leads to the concept of generalized δ- and δ^′-couplings.     

Gentile statistics and restricted partitions

In a recent paper (Tranet al, Ann. Phys.311, 204 (2004)), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We generalise these results to obtain an asymptotic formula for therestricted or coloured partitionsp _k ^s (n), which is the number of partitions of an integern into the summand of sth powers of integers such that each power of a given integer may occur utmostk times. While the method is not rigorous, it reproduces the well-known asymptotic results fors = 1 apart from yielding more general results for arbitrary values ofs.     

Study of the40Ca(π−, 2n)38K reaction at rest

The (π ^?,2n) reaction on⁴⁰Ca has been investigated via the determination of the momenta for the two neutrons emitted simultaneously following the absorption ofπ ^?. The³⁸K excitation spectrum, opening-angle distributions, and recoil momentum distribution are determined. The direct mechanism is established for transitions to the lower part of the³⁸K excitation spectrum.     

Theory of the local field correction in an electron gas

The wave-vector- and frequency-dependent dielectric function ?(k,ω) of an electron gas can be expressed in terms of Lindhard's function and a complex local field correctionG(k,ω) which incorporates all the effects of dynamic exchange and correlation in the system. The general properties ofG(k,ω) are discussed, in particular the static and high-frequency limits. It is shown that for smallk, bothG(k, 0) andG(k, ∞) vary ask ², with different coefficients, but both determined by the average kinetic and potential energies per particle. For largek,G(k, ∞) varies again ask ² and it is argued that the same holds true forG(k, 0), with both coefficients (though different) determined by the average kinetic energy per particle. General formulas for the plasma dispersion relation and damping, involving, respectively, the real and imaginary parts ofG(k,ω), are given. The term in the plasma frequency which is proportional tok ² is given directly in terms of the average kinetic and potential energies per particle, a result true at all temperatures. A calculation of the frequency dependence ofG(k,ω), starting from the exact equation of motion for the particle-hole operator and employing a decoupling approximation introduced previously by Toigo and Woodruff, is presented. Explicit results forG(k,ω) are obtained for smallk and allω. The complete expressions forG(k, 0) andG(k, ∞) in this approximation have been obtained and are plotted.     

Nuclear orientation study of97, 103, 105Ru in Fe

Angular distributions have been measured forγ-rays emitted following the decays of^{97, 103, 105}Ru oriented in an iron matrix at temperatures down to 2.8mK. From the temperature dependence of theγ-anisotropies the magnetic hyperfine splitting frequenciesν _M =|gμ _N B _HF/h| of^{97, 103, 105}RuFe were found to be 110(7), 57(15) and 80 _?50 ⁺¹⁷ MHz, respectively. With the known hyperfine fieldB _HF=?489.6(4.0) kG the nuclearg-factors are derived as ∣g(⁹⁷Ru;j ^π=5/2⁺)∣=0.29(2), ∣g(¹⁰³Ru;j ^π=3/2⁺)∣=0.15(4) and ∣g(¹⁰⁵Ru;j ^π=3/2⁺)∣=0.21 _-0.13 ^+0.05 . The analysis for¹⁰³RuFe has been performed with the assumption ofj ^π=3/2⁺ and 5/2⁺ for the ground state of¹⁰³Ru. Taking into account experimentally knowng-factors of 3/2⁺ and 5/2⁺ states in this mass region, our data strongly favour the assignmentj ^π=3/2⁺ for the¹⁰³Ru ground state.     

Irreducible kernels and nonperturbative expansions in a theory with purem →m interaction

Recent results on the structure of theS matrix at them-particle threshold (m≧2) in a simplifiedm→m scattering theory with no subchannel interaction are extended to the Green functionF on the basis of off-shell unitarity, through an adequate mathematical extension of some results of Fredholm theory: local two-sheeted or infinite-sheeted structure ofF arounds=(mμ)² depending on the parity of (m?1)(ν?1) (where μ>0 is the mass and ν is the dimension of space-time), off-shell definition of the irreducible kernelU which is the analogue of theK matrix in the two different parity cases (m?1)(ν?1) odd or even, and related local expansion ofF, for (m?1)(ν?1) even, in powers of σ_β ln σ(σ=(mμ)²?s). It is shown that each term in this expansion is the dominant contribution to a Feynman-type integral in which each vertex is a kernelU. The links between the kernelU and Bethe-Salpeter type kernelsG of the theory are exhibited in both parity cases, as also the links between the above expansion ofF and local expansions, in the Bethe-Salpeter type framework, ofF _λ in terms of Feynman-type integrals in which each vertex is a kernelG and which include both dominant and subdominant contributions.     

Fourth forbidden beta decays of50V

The half lives ofβ ^? decay ande ^? capture in theΔJ=4 transitions of⁵⁰V were calculated in the formulation of Behrens and Schülke. The expression of formulas for these half lives are divided into two parts. The ones are common factors which depend on the lepton wave functions only. The other parts are nuclear parameters which change their values by choosing suitable nuclear models. We tabulated the numerical values of these common factors. As an example, we evaluated these parameters in a single particle shell model, and obtained the half lives 9.04×10¹⁷ years and 1.51×10¹⁶ years for theβ ^? decay ande ^? capture, respectively. The recent experimental data measured by A.J. Pape et al. aret _1/2>7.0×10¹⁷ years for theβ ^? decay andt _1/2>8.8×10¹⁷ years for thee ^? capture. In the case of thee ^? capcure, the screening effects of the other orbital electrons were properly taken into account.     

Aussagen der Theorie von Kraichnan für die isotrope hydromagnetische Turbulenz

The theory ofKraichnan is applied to quasi-stationary isotropic hydromagnetic turbulence. The average infinitesimal-impulse-response functionsg(k, τ), g _m (k, τ) and the time-correlationsr(k, τ), r _m (k, τ) are evaluated by the non-local direct-interaction approximation within the inertial range. For the range of ohmic but no viscous dissipation it is found that the magnetic energy spectrumE _m (k) obeys aE(k)k ^?2-law in accordance with results ofGolitsyn andMoffatt.     

Parity effects in radiative Ω−-decays

It is shown that, for values of theΩ ^? magnetic moment near that predicted bySU(6), parity effects in radiativeΩ ^?-decays are expected to be small and, contrary to the situation in hyperon decays, more pronounced in decays with heavy charged particles. They depend, however, sensitively on the magnetic moment ofΩ ^? and are insensitive to the hyperon moments. This is due to the dominance of a “contact” term which has to be introduced in the amplitude in order to ensure gauge invariance.     

Sum rules for Sαβ(k,ω) for two diffusing species

Starting from the Smoluchowski equation without hydrodynamic interactions for two species of spherical diffusing particles, sum rules are derived here for the first three moments of S_αβ(k,ω), i.e., for the initial value of the first, second and third time-derivatives of F_αβ(k, t) (the time-dependent correlations between the fluctuations in the local concentration of diffusing particles of species α and β). These sum rules are written in terms of the potential of interaction u_αβ(r) between the diffusing particles and the two- and three-particles distribution functions. This derivation is motivated by its potential use in the study of counterion effects on the diffusion of highly charged colloidal particles. Thus, we propose to approximate the memory function involved in the time evolution equation for F_αβ(k, t by a two-parameter model, with its (k-dependent) parameters being determined by the sum rules derived here. This procedure, along with Kirkwood's superposition approximation, reduces the dynamical problem to the knowledge of the radial distribution functions g_αβ(r).