The vertex constant for the virtual decay of a nucleus into two charged particles within the effective range theory

Explicit expressions for the virtual a → b + c decay vertex constant (or the asymptotic normalization factor) for charged particles b and c are obtained for the first time using the effective range function K(k ²) for the standard effective range expansion and for the case where K(k ²) has a pole. The bound and resonant states of the ²He nucleus and the resonant states of the ²He and ⁸Be nuclei are considered as physical examples. A trajectory in the complex momentum plane is constructed for the transition from the resonant state to the virtual state while the Coulomb interaction gradually decreases.     

Doublet coulomb-nuclear scattering length and other parameters of the effective-range function for proton-deuteron scattering from an analysis of present-day data

Parameters of the effective-range function K(k ²) having a pole are found from the results that were obtained by calculating the S-wave phase shifts for doublet proton-deuteron scattering and the binding energies of three-nucleon nuclei on the basis of Faddeev equations and within the N/D method and which are known from the literature. The convergence of the expansion of K(k ²) in powers of the momentum is studied. The energy of the proton-deuteron resonance corresponding to the virtual triton state is calculated.     

Effective-range expansion for two coupled channels and properties of bound states

The S matrix and the scattering-amplitude matrix (F matrix) are considered for the case of two coupled elastic-scattering channels differing by the values of the orbital angular momentum (l ₁ and l ₂ = l ₁ + 2). The matrix elements of the S and F matrices in the absence of Coulomb interaction are expressed in terms of the matrix elements of the matrix K ⁻¹ inverse to the reaction K matrix. The elements of the K ⁻¹ matrix are written in the form of expansions that are generalizations of the single-channel effective-range expansion. If there is a bound state in the system of colliding particles, then an analytic continuation of these expansions to the region of negative energies makes it possible to obtain both the position of the pole corresponding to this bound state and the residues of scattering amplitudes at this pole, the respective vertex constants and asymptotic normalization coefficients being expressed in terms of these residues. By way of example, the developed formalism is applied to describing triplet neutron-proton scattering.     

The characteristics of the <Superscript>8</Superscript>Be ground state in the effective-range approximation

We find the vertex constant for the synthesis α + α → ⁸Be and the corresponding asymptotic normalization coefficient of the Gamov wave function for ⁸Be in the ground state. We use modern data on the position and width of the narrow resonance of this state as well as the energy dependence of the αα scattering phase shift in the s-wave known from the literature. The effective-range theory was applied with the Coulomb interaction taken into account. The parameters of the standard effective-range function expansion up to the member with k ⁴ (k being the relative momentum) were found.     

Effective-range function for doublet nd scattering from an analysis of modern data

The parameters of the generalized effective-range function K(k ²) having a pole are found by using the results that were obtained by calculating the S-wave phase shift δ(E) for doublet nd scattering and the triton binding energy on the basis of Faddeev equations and within the N/D method and which were presented in the literature. The convergence of the expansion of K(k ²) in powers of momentum is studied. The binding energy of the virtual triton and the residues of the partial-wave scattering amplitudes at the poles corresponding to the bound and virtual states are calculated. Correlations between the binding energies of the bound and virtual states of the triton, on one hand, and the doublet scattering length for nd interaction, on the other hand, are considered. The function K(k ²) is also calculated within a two-body model featuring various potentials.     

Description of the low-energy doublet neutron-deuteron scattering in terms of parameters characterizing bound and virtual triton states

Low-energy doublet neutron-deuteron scattering is described in terms of the energies of the bound and virtual triton states and the nuclear vertex constants for these states. For the first time, the van Oers-Seagrave formula is derived from the Bargmann representation of the S matrix for a system having two states. The presence of a pole in this formula is shown to be a direct corollary of the existence of a low-energy virtual triton state. Simple explicit expressions for the nd scattering length and for the pole of the function kcotδ are obtained in terms of the parameters of the bound and virtual triton states. The low-energy parameters of nd scattering are calculated numerically, whereby it is shown that these parameters are highly sensitive to variations in the asymptotic normalization constant C _v ² for the virtual state. The C _v ² value fitted in our model to the experimental result for the nd scattering length is 0.0592.     

Zerfallende Zustände als physikalisch nichtisolierbare Teilsysteme

Presently the investigations of decaying quantum mechanical systems lack a well-founded concept, which is reflected by several formal difficulties of the corresponding mathematical treatment. In order to clarify in some respect the situation, we investigate, within the framework of nonrelativistic quantum mechanics, the resonant scattering of an initially well localized partial wave packet ϕ_l(r, t). If the potential decreases sufficiently fast for r → ∞, ϕ_l(r, t) can be expressed at sufficiently long time after the scattering has taken place, as ϕ_l(r, t) = I(r, t) + ∑ N_iϕl(K_i, r) exp {–iK_i² t/2M} × Θ(k_i – γ_i – Mr/t), ϕ_l(K_i, r) being the resonant solution with complex “momentum” K_i = k_i – iγ_i. From this heuristic relation one can deduce not only the probability for the creation of unstable particles but also obtain some hints to a connection between decaying states and physically nonisolable partial systems. On the other hand, this connection can perhaps display the inadequacy of attempts which suggest to solve the problem of decaying states within the usual Hilbert space methods.     

Allowing for Coulomb effects in the effective range expansion for two coupled channels

A Coulomb-modified matrix of scattering amplitudes (an [(F)\tilde]\tilde F matrix) is considered for the case of two coupled channels of elastic scattering of charged particles with different orbital angular momenta (l ₁ and l ₂ = l ₁ + 2). Matrix elements of the [(F)\tilde]\tilde F matrix are expressed in terms of the matrix elements of a [(K)\tilde] ^{- 1}\tilde K^{ - 1} matrix inverse to a modified reaction K matrix. The elements of the [(K)\tilde] ^{- 1}\tilde K^{ - 1} matrix are written as expansions that are generalizations of single-channel effective range expansion with allowance for the Coulomb interaction. If a system of colliding particles involves a bound state, the analytic continuation of these expansions into the region of negative energies makes it possible to obtain both the position of the pole corresponding to the bound state and the scattering amplitude residues in this pole, in terms of which the corresponding vertex constants and asymptotic normalization coefficients are expressed.     

On the normalization of the Gamov resonant wave function in the configuration space

The regularization of the normalization integral for the resonant wave function, proposed by Zeldovich, is valid only when |Req _res| > |Imq _res|. A new normalization procedure is proposed and implemented, which is valid when this condition fails. First, an arbitrarily normalized vertex function g(k) is calculated using the formula with the potential V(r) in the integrand. This Fourier integral converges for a potential with the asymptotics V(r) → constr ^?n exp(?μr) if |Imq _res| < μ/2. Then the function g(k) is normalized using the generalized normalization rule, which is independent of the resonance pole position. The proposed method is approved by the example of calculation for a virtual triton.     

An investigation of the influence of the pairing correlations on the properties of the isobar analog resonances inA = 208 nuclei

Within the quasi-particle random phase approximation (QRPA), the method of the self-consistent determination of the isovector effective interaction which restores a broken isotopic symmetry for the nuclear part of the Hamiltonian is given. The effect of the pairing correlations between nucleons on the following quantities were investigated for theA = 208 nuclei: energies of the isobar analog 0⁺ states, the isospin admixtures in the ground state of the even-even nuclei, and the differential cross-section for the²⁰⁸Pb(³He,t)²⁰⁸Bi reaction atE(³He)=450 MeV. Both couplings of the excitation branches withT ^z = T⁰ ± 1, and the analog state with isovector monopole resonance (IVMR) in the quasi-particle representation were taken into account in our calculations. As a result of these calculations, it was seen that the pairing correlations between nucleons have no considerable effect on theT = 23 isospin admixture in the ground state of the²⁰⁸Pb nucleus, and they cause partially an increase in the isospin impurity of the isobar analog resonance (IAR). It was also established that these correlations have changed the isospin structure of the IAR states, and shifted the energies of the IVMR states to the higher values.     

High-Resolution Infrared Spectra of the ν2, ν3, ν4, and 2ν3 Bands of SO3

New measurements are reported for the infrared spectrum of sulfur trioxide, ³²S¹⁶O₃, with resolutions ranging from 0.0015 cm⁻¹ to 0.0025 cm⁻¹. Rovibrational constants have been measured for the fundamentals ν₂, ν₃, and ν₄ and the overtone band 2ν₃. Comparisons are made with the earlier high-resolution measurements on SO₃, and the high correlation among some of the constants related to the Coriolis coupling of the ν₂ and ν₄ levels is discussed in order to understand the areas of disagreement with the earlier work. Splittings of some of the levels are observed and the splitting constant for K=3 of the ground state is determined for the first time. Other observed splittings include the K=1 levels of 2ν₃ (l=2), the K=2 levels of ν₃ and ν₄, and the K=3 levels of ν₂. The analysis shows that there are level crossings between the l=0 and l=2 states of 2ν₃ that allow one to determine the separation of the subband centers for these two states even though access to the l=0 state from the ground state is electric-dipole forbidden. This is a generalized phenomenon that should be found for many other molecules with the same symmetry. The l-type resonance constant, q₃, that causes the splitting of the l₃=±1, k=±1 levels of ν₃ also couples the l₃=0 and 2 states of 2ν₃.     

The contribution of the (t(τ)+N) component to the α-particle wave function from data on the reactions p(α,pp)t and p(α,pn)τ at incident α-particle momentum 5 GeV/c

The 2-meter liquid-hydrogen bubble chamber is exposed to a separated beam of α particles from the ITEP synchrotron. The momentum of the incident ⁴He nuclei, averaged over the fiducial volume of the chamber, is 5 GeV/c (the kinetic energy of initial protons in the nucleus rest frame is T _p =620 MeV). The spectral functions of decays α→tp and α→τn) are extracted in the 4π geometry (the latter is extracted for the first time) from the exclusive reactions αp→tpp and αp→τpn in the spectator momentum region 0<q<0.3 GeV/c. The pole dominance criteria are carefully checked. Extrapolation of the nuclear vertex function to the pole gives a lower bound on the nuclear vertex constant. Our experimental data are compared with the results of other studies and with theoretical calculations. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 2, 69–74 (25 July 1997) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Vertex function and coupling constant for the virtual decay of7Li

The alpha-triton relative wave function with various nucleon exchange contributions and their asymptotic normalization are considered in the framework of the generator coordinate method (GCM). The asymptotic normalization of relative wave function provide the estimate of the coupling constant. The relative wave function is also used to obtain⁷Li-α-t vertex function in the virtual decay of⁷Li. The extrapolation of vertex function for negative values ofq ² up to the alpha-triton pole provide the vertex constant, which is compared with the experimentally determined estimates 0.67 FM and 0.72 FM. Our calculated value is 0.656 FM which is in close agreement with the above estimates.     

Coherence transfer between nuclear spins in paramagnetic systems: effects of nucleus—electron dipole—dipole cross-correlation

In nuclear magnetic resonance of paramagnetic systems, cross-correlations between the fluctuations of a nucleus—nucleus dipole—dipole coupling I^k I^l and a nucleus—electron dipole coupling I^kS induces cross-relaxation and makes it possible to generate bilinear terms in the density matrix of the type 2I^k _xI^l _z from coherence I^k _x that can lead to ‘relaxation-allowed’ coherence transfer between two nuclei I^k and I^l . In this paper these effects are demonstrated in a complex involving a fragment of double-stranded DNA and two chromomycin molecules complexing a paramagnetic cobalt ion. Analytical expressions are given for the cross-correlation rates in particular conditions, while the extension to anisotropic g tensors or zero field splittings are addressed. It is shown that relaxation-allowed coherence transfer leads to characteristic signals in double-quantum filtered correlation spectroscopy (DQF—COSY), but not in total correlation spectroscopy (TOCSY). Analytical expressions are unable to reproduce the observed cross-peak patterns. A careful numerical study reveals that in the high spin Co(II) complex studied here, the cross-correlation dynamic shift contribution is of the same order of magnitude as the cross-correlation rate, a value much larger than what can be computed assuming isotropic Brownian motion and complete separation between the electron spin and the lattice.     

Two-Channel Effective Range Expansion and Bound-State Properties

The S matrix and the f matrix amplitudes are considered for the case of two coupled elastic scattering channels, which differ in values of orbital angular momenta. Matrix elements of S and f matrices are parametrized in terms of scattering phases ?? _i (i?=?1, 2) and a mixing parameter ${\epsilon}$ and are expressed in terms of matrix elements c _ij = (K ^?1) _ij where K is the reaction K matrix. Quantities ${g_{ij}(k)=k^{l_i+l_j+1}c_{ij}(k)}$ are expanded in powers of k ², k being the relative momentum of colliding particles B and C. Then functions g _ij (k) and c _ij (k) are continued analytically to the pole of amplitudes f _ij corresponding to the bound state A of colliding particles. This procedure allows to get the position of the pole as well as the residues of amplitudes f _ij at that pole which are related directly to vertex constants and asymptotic normalization coefficients corresponding to the vertex A ?? B?+?C.     

Intercepts of the momentum correlation functions in the $ \mu$ -Bose gas model and their asymptotics

The so-called μ-deformed oscillator (or μ-oscillator) introduced by A. Jannussis, though it possesses rather exotic properties with respect to other better known deformed oscillator models, also has a good potential for diverse physical applications. In this paper, the corresponding μ-Bose gas model based on μ-oscillators is developed. Within this model, the intercepts l⁽²⁾ \lambda^{{(2)}}_{}(K) and l⁽³⁾ \lambda^{{(3)}}_{}(K) of two- and three-particle momentum correlation functions are calculated with the goal of possible application for modeling the non-Bose-type behavior of the intercepts of two- and three-pion correlations, observed in the experiments on relativistic heavy-ion collisions. In the derivation of intercepts, a fixed order of approximation in the deformation parameter μ is assumed. For the asymptotics of the intercepts l⁽²⁾ \lambda^{{(2)}}_{}(K) and l⁽³⁾ \lambda^{{(3)}}_{}(K) , we derive exact analytical formulas. The results for l⁽²⁾ \lambda^{{(2)}}_{}(K) are compared with experimental data, and with earlier known results drawn using other deformed Bose gas models.     

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.     

Evaluating emission widths of giant dipole resonances in a Fermi liquid model

It is shown in the theory of finite Fermi systems that giant resonances can be considered as zero-sound excitations in concrete nuclei. Solutions to the zero-sound dispersion equation in symmetric nuclear matter, ωs(k), are obtained. Hydrodynamic models determine the wave vector k _A corresponding to giant resonance in a nucleus with atomic number A. The real and imaginary parts of ωs(k _A ) compare to the energy and escape width of giant resonance. Calculations are performed for giant dipole resonances.     

The resonance pole and the residue of the scattering amplitude while the Coulomb forces gradually decrease

Trajectories for the S-wave pole of the scattering amplitude are constructed in the complex planes of the resonance momentum and of the renormalized vertex constant squared. They correspond to transitions from the resonance states to the virtual states of the singlet deuteron and the doublet triton while the Coulomb interaction gradually decreases to zero. The effective-range theory and a Yukawa-potential model are used. Physical reasons for the differences between the corresponding trajectories for Np and Nd systems are found. The results obtained explain why the resonances in the doublet Nd scattering are more difficult to detect in experiments in comparison with the well-studied singlet NN resonances.