Experimentelle Untersuchungen am Elektroneneinfang des85Strontium und88Yttrium

The electron capture decay of⁸⁵Sr and⁵⁵Y has been investigated using NaI(Tl)-detectors. Spectroscopic-coincidence techniques have been employed to examine theK-electron capture rates. The experimental results areP _K ω _K (Sr)=.5959 ± 0.0035 andP _K ω _K (Y)=0.6340± 0.0032. Using theK-capture probabilities, derived from the capture ratios ofBrysk andRose with the exchange correction ofBahcall:P _K (Sr)=0.880± 0.008 andP _K (Y)=0.883 ± 0.008, one can derive from these results the fluorescence yieldsω _K (Rb)=0.677± 0.009 andω _K (Sr)=0.712± 0.010. Functions ofω _K (Z) were fitted to recent experimental data and tabulated forZ=5–90. In addition to these results the half life of the 514 keV state of⁸⁵Rb and the branching ratios and the positron rate of the⁸⁸Y-decay have been measured with the same techniques. The data obtained partly disagree from measurements of other authors.     

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.     

Nichtsenkrechte Interbandübergänge in Graphit durch unelastische Elektronenstreuung

Characteristic energy loss measurements are carried out on natural graphite single crystals. The frequency and wave vector dependent dielectric constant ?(ω,q) forq⊥c-axis is determined by Kramers Kronig Analysis. The results are interpreted in terms of non vertical interband transitionsE(k)→E(k+q) between the π-bands within the two dimensional approximation.     

The magnetic moments and halflives of the 5/2− states in199Hg and197Hg and the discrepancies in the experimental hyperfine fields at Hg in Fe

The Larmor-precession frequencies for¹⁹⁷Hg in Fe have been determined to beω _L = 1291 ±25 MHz at 293 K andω _L = 1334±25 MHz at 105 K. For¹⁹⁹Hg in Feω _L =1372±50 MHz has been measured at 293 K. The half-lives of the 5/2^? states in¹⁹⁷Hg and¹⁹⁹Hg have been remeasured asT _1/2(¹⁹⁷Hg)=8.1±0.16 nsec andT _1/2(¹⁹⁹Hg)=2.45±0.05 ns, respectively. The magnetic moment of the 5/2^? 158keV state in¹⁹⁹Hg was redetermined by the integral perturbed angular correlation method in an external magnetic field of 47kG asμ(5/2 ^? ) = 0.905±0.091 nm. With this new value consistency for the magnetic hyperfine fields at Hg in Fe measured with the TDPAC-method and with the NMR/ON-method is obtained. This fact is used to determine theg-factors of the 5/2^? states in¹⁹⁷Hg and¹⁹⁹Hg more precisely fromω _L -values given above:g(¹⁹⁷Hg)=0.342(6);g(¹⁹⁹Hg)=0.352(13). The magnetic moments of the first excited 2⁺ states in^198–204Hg isotopes which rely on calibrations with the¹⁹⁹Hg-g-factor, are revised.     

The gamma decay of high spin states of39K

First will be exhibited, for the stationary case, a connection between the probability, to find a particle in the region of interaction, and the derivative of the scattering phase shift for the momentum. From the idea, that in stationary scattering a resonance is linked with an appreciable increase of this probability, one obtains new and quantitative criteria for the behavior ofδ _l(k). For instance, the nonresonant behavior can be characterised by the condition 2k dδ _l(k)/dk<1. The maximum of probability for the particle to be in the region of interaction, is considered in accordance with the criterium of maximal change of the phase shift, as a function ofk. This characterises the location of the resonance. Wigner's lower limit fordδ _l (k)/dk allows to formulate a resonance condition directly for the cross section. Furthermore will be discussed some connections with the pole approximations of theS-matrix and the “collision lifetime” of Smith.     

Extension to velocity dependent gauge transformations

In previous papers it was shown that in phase space a generator of gauge transformations for a singular Lagrangian \(L(q,\dot q)\) is given byG[ε^α]=ε _k ^α φ _α ^k (q,p,t) where φ _α ^k are first class constraints andG is subjected to a stationarity condition. A non-trivial extension from velocity (or momentum) independent gauge transformations to velocity dependent ones is realized, by replacing gauge functions ε^α(t) with momentum dependent functions ε^α(q,p,t), as long asG satisfies the stationarity condition. Inversely, it is proved in a classical framework that, within velocity dependent gauge transformations, all generators of gauge transformations can be expressed in terms of the linear combinations of φ _α ^k .     

On the temperature behavior of second sound and Poiseuille flow

The linearized Peierls equation for the phonon densityN (k λ,r t) is solved by replacing the collision operator in the subspace orthogonal to the collision invariants byk-dependent relaxation rates. For the normal process relaxation time the behaviorτ _N (k λ)∝|k|^?p for smallk is assumed. Taking into account thisk-dependence ofτ _N explicitly and avoiding an expansion with respect toΩτ _N (kλ) before performing the necessary integration overk yields new, non-analytic, terms in the hydrodynamic equations describing second sound and Poiseuille flow. It is shown that this may lead to a temperature dependence of second sound damping and thermal conductivity in the Poiseuille flow region differing from the usual theoretical predictions and in better agreement with experiments.     

Self-consistent calculation of the optical potential and ground-state properties of nuclear matter

On the basis of the Green-function formalism, we performed a self-consistent calculation of the self-energy ∑(k, ω) of a particle interacting with the infinite nuclear medium. The function ∑(k, ω) was mapped out in the energy-momentum plane, and the single-particle energy ω(k), momentum distribution ?(k) and the “on-shell” part of the self-energy, ∑(k, ω(k)), were defined, from which all physical properties followed. In particular we investigated the ground-state properties of nuclear matter in two Λ-approximations of the T-matrix. In one, the intermediate two-particle propagator, Λ₀₀, represented free-particle propagation; in the other, called Λ₁₁, intermediate states included both interacting particles and holes. Pauli principle effects were included in both approximations. The second approximation was expected to be conserving because it included a large part of the rearrangement effects which, we found, contributed ～6 MeV per particle to the average energy and ～28 MeV to the singleparticle energy at zero momentum. The Hugenholtz-van Hove theorem was nearly satisfied, with only 1 MeV separating the chemical potential from the average energy. We also studied, in the Λ₀₀-approximation, the optical potential for the scattering of a particle by a large nucleus; it was directly related to the “on-shell” part of the self-energy. It was found that, below 100 MeV, the real part varied as (?90 + 0.584E) [MeV], and the imaginary part as (2.4 + 0.009 E) [MeV].     

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 ^∞      

Experimentelle Untersuchungen am Elektroneneinfang des Mangan 54 und Zink 65

The electron capture decay of Mn 54 and Zn 65 was investigated using Na I(Tl)-detectors. Spectroscopic-coincidence techniques were employed to examine theK-electron capture rates of Mn 54 and Zn 65, the branching ratio and positron rate of Zn 65. The experimental results are:P _k ω _k (Mn)=0.2492±0.0017,P _k ω _k (Zn)=0.3927±0.0026,EC ^* /EC ⁰(Zn)=53.2±1.0/46.8±1.0,γ/β ⁺=35.1±1.7,K ⁰/β ⁺=27.7±1.5. Thus electron capture to the excited state of Cu 65 occurs to the extent of 52.4±1.0% while electron capture and positron decay to the ground state were found to be associated with 46.1±1.0% and 1.49±0.05% of the disintegrations. These data partly disagree significantly from measurements of other authors. Using the fluorescence yieldsω _k (Cr)=0.279 andω _k (Cu)=0.4425, deduced from other measurements, values ofL+M/K of electron capture to the excited state could be derived:L+M/K(Mn 54)=0.120±0.015 andL+M/K(Zn 65)=0.127±0.015. These ratios agree with the calculations ofBahcall. The half-lives of Mn 54 and Zn 65 have been determined to beT _1/2(Mn 54)=312±5d,T _1/2(Zn 65)=243±4d The energy of the Zn 65-γ-rays was measured with a Ge(Li)-counter:E ^γ=1115.5±0.5 keV.     

On the nilpotent * - Fourier transform

LetG be a nilpotent Lie group. The adapted nilpotent Fourier transform was introduced by D. Arnal and J. C. Cortet,:L(G) C (V,L(^2d )), whereL(G) is the Schwartz space ofG andV × ^2k is aG-invariant Zariski open set ing ^* the dual of the Lie algebra ofG. We prove the surjectivity of this transformation, which allows us to extend it to distribution spaces.     

Moments of the spectra for rotational transitions induced by collisions or by external perturbations

Considered first is the cross sectiond σ(J′←J)/dω for the rotational transitionJ′←J of a linear rotator in collisions. The set of cross sections for a givenJ and for allJ′ defines a discrete spectral distribution as a function of the transition energy ΔE=E(J′)?E(J). In both the classical and the quantum mechanical sudden approximations the moments \(S(\mu ;J) = \sum\limits_{J'} {(\Delta {\rm E}} )^\mu d\sigma (J' \leftarrow J)/d\omega \) of ordersμ=2n andμ=2n+1 (n=0,1,...) are polynomials of degreen in the rotational energyE(J). The special cases forn=0 indicate thatS(0;J) andS(1;J) are independent ofJ. This is a theorem proved elsewhere. Explicit expressions forS(μ; J) of low orders are presented. They are derived on the assumption of equal relative velocitiesv andv′ before and after the collision. Correction onS(μ;J) for the difference betweenv andv′ is made to the lowest order in terms ofS(μ+1;J) forv=v′. The quantum mechanical results for linear rotators are extended to spherical-top and symmetric-top rotators. These results apply not only to the collision cross section but also to a physical quantity expressible in the form ¦〈Γ′¦F¦Γ〉¦² of a transition probability withany F that is independent of the initial and final rotational states ¦Γ〉 and ¦Γ′〉.     

Dependency of control parameters on a rate coefficient giving the potential curvature in a reaction cascade model

Many chemical reactions in vivo are self-controlled by fluxes of chemical energy and matter through biological systems, so the induction of such reactions can be governed by changes in the control parameters of the rate equation. A potential of a system is assumed to be given by Gibbs' functionG(T, P, x), which is continuously differentiable, and the rate equation can be derived from the differential (–G/x) of Taylor's expansion ofG _(T,P)(x) for the order parameterx, which corresponds to the product number, at around the critical pointC(T _C, P_C). The equation is described bydx/dt=(x)–k₁x–k₂x³, andk ₂>0. In this equation,k ₁ andk ₂ are functions of the control parameters, temperatureT and pressureP, andk ₁ is allowed to have a positive or negative values as (T, P). Thenk ₁ is an important factor that decides the induction conditions of the reactions with a phase transition in the steady statex=0. Because bothk ₁ (the transition parameter) andG are the quantity of state, they are given by the total differential, and functions that decideG andk ₁ are related to a mutual inverse function. From the above relation, the rate of change ink ₁ by G, which corresponds to the reaction energy of the system, is uniquely determined by a function ofk ₁, [f(k ₁ ^± )] andf(k ₁ ^± ) is described approximately by ±₁ k ₁ ^± in the transient process thatk ₁ approaches zero, where ₁ implies 1/RT. These results indicate that internal driving forces caused by a stimulus in a system are proportional tok ₁ ^± and that the system is regulated by competition of the forces. an approximate function fork ₁ in the transient process is described by tanh (G/RT) and Arrhenius' law is elucidated from this theory.Decreased January 19, 1992     

Analytic structure and explicit solution of an important implicit equation

The equationz=2G(z)?expG(z)+1 (and similar ones obtained from it by substitutions) appears in connection with a variety of problems ranging from pure mathematics (combinatorics; some first order, nonlinear differential equations) over statistical thermodynamics to renormalization theory. It is therefore of interest to solve this equation forG(z) explicitly. It turns out, after study of the complex structure of thez andG planes, that an explicit integral representation ofG(z) can be given, which may be directly used for numerical calculations of high precision.     

Perturbed angular distributions in the presence of isotropic paramagnetic relaxation

The effect of paramagnetic relaxation on perturbed angular distributions is treated for nuclei interacting with their electronic shells via isotropic hyperfine interaction. The conditions are given under which Blume's analytical stochastic-model result for the nuclear perturbation factorsG _k (t) can be derived quantum mechanically. Systems with arbitrary nuclear spin, but electronic spinS=1/2 may be calculated without resorting to the assumption necessary forS>1/2. Explicit closed expressions forG _k (t) can be found for this particular case.     

Influence of gravity on surface melting

A chemical network system for use in the study of reaction cascades is described by the nonlinear rate equation , in which (x) is derived from the –G/x of Taylor's expansion of the order parameterx of the thermodynamic potential, Gibbs' functionG(T,P,x), at about the critical pointC(T _C ,P _C ) of the control variables (parameters)T andP. The behavior of the system around the stable pointx=0 is analyzed only with the sign ofk ₁(T, P), becausek ₂(T, P) is always positive. The system is not closed and is affected by physical and chemical changes in the neighbor systems. WhenT andP fluctuate ( ) through the changes andk ₁ passes zero, the system in the steady state becomes instable andk ₁ jumps at the reaction threshold. Then, the products are formed at the number of moleculesx=(|k ₁|/k ₂)^1/2. To describe such a transition,k ₁ giving theG curvature atx=0 is represented phenomenologically by an approximate function, tanh(G/RT), for a metastable state with a relatively smallk ₁(>0). The reaction takes place in a cascade in accordance with a cubic state equation obtained from tanh(G/RT), which describes a jump of the reaction energy G at the reaction threshold.     

空间羣算符的矩阵元

本文首先通过对波失星直接乘积的分析指出,如果 ^*K″∈^*K?^*K′,则波矢峯G_k″可以有下面四种情况:A)G_k同G_k′,是G_k″的子峯,B)G_k=G_k′=G_k″,C)G_k″=G_s, D)G_s是G_k″的子峯。根据上述四种情况,分别考虑积分A=μ″^i″(k″)|f_μⁱ(k)|f_μ′^i′(k′)>的简约。并且证明,对所有四种情况,积分A的值均可由短阵U的矩阵元统一地表达出来。矩阵U是使可约表示Г′简约的转换矩阵,对四种情况,可约表示Г′分别是:A)(Г_kⁱ?Г_k′^i′)^(?),B)Гⁱ?Г_k′^i′,C)Г_k^i(s)?Г_k′^i′(s),D)(Г_k^i(s)?Г_k′^i′(s))^(?)。最后,利用准投影算符的方法,对矩阵U进行了计算,导出空间峯Wigner-Eckart定理的表式。     

Free-field realization of theWKP(N) algebra

TheW _KP ^(N) algebra has been identified with the second Hamiltonian structure in theNth Hamiltonian pair of the KP hierarchy. In this Letter, by constructing the Miura map that decomposes the second Hamiltonian structure in theNth pair of the KP hierarchy, we show thatW _KP ^(N) can also be decomposed toN independent copies ofW _KP ⁽¹⁾ algebras, therefore its free-field realization can be worked out by constructing free fields for each copy ofW _KP ⁽¹⁾ . In this way, the free fields may consist ofN + 2n number of bosons, among them, 2n are in pairs, wheren is an arbitrary integer between 1 andN. We also express the currents ofW _KP ^(N) in terms of the currents ofN —n copies of U(1) andn copies of SL(2,R) _k algebras with levelk = 1. By reductions, we give similar results forW ^(N) andW ₃ ⁽²⁾ algebra.