On the role of unitarity in statistical theories of nuclear reactions

It is argued that the approximate statistical S-matrix with energy independent background S-matrix, residues, poles, and with fixed dimensions need not satisfy analytic unitarity but should in general obey the average unitarity condition. The freedom obtained by relaxing analytic unitarity allows a representation where level-level correlations are not present. Different approaches to statistical theories of nuclear reactions employing the pole decomposition of the S-matrix are compared. It is seen that any such approach is characterized by the assumed form of two (matrix) parameters. A model is developed which gives the expected results for the compound cross sections in the limits of strong absorption and weak absorption with statistically equivalent channels, and interpolates between the two extremes. The model depends, however, on the parameter $\text{πΓ}\text{D}$. The possibility of extracting the value of this parameter from experimental data for the variance of cross sections is also investigated.     

Theory of the Trojan-Horse method

The Trojan-Horse method is an indirect approach to determine the energy dependence of S factors of astrophysically relevant two-body reactions. This is accomplished by studying closely related three-body reactions under quasi-free scattering conditions. The basic theory of the Trojan-Horse method is developed starting from a post-form distorted wave Born approximation of the T-matrix element. In the surface approximation the cross-section of the three-body reaction can be related to the S-matrix elements of the two-body reaction. The essential feature of the Trojan-Horse method is the effective suppression of the Coulomb barrier at low energies for the astrophysical reaction leading to finite cross-sections at the threshold of the two-body reaction. In a modified plane wave approximation the relation between the two- and three-body cross-sections becomes very transparent. The appearing Trojan-Horse integrals are studied in detail.     

Extraction of Astrophysical Cross Sectionsin the Trojan-Horse Method

 The Trojan-horse method has been proposed to extract S-matrix elements of a two-body reaction at astrophysical energies from a related reaction with three particles in the final state. This should be useful in cases where the direct measurement of the two-body reaction at the necessary low energies is experimentally difficult. The formalism of the Trojan-horse method for nuclear reactions is developed in detail from basic scattering theory including spin degrees of freedom of the nuclei and we specify the necessary approximations. The energy dependence of the three-body reaction is determined by characteristic functions that represent the theoretical ingredients for the method. In a plane-wave Born approximation of the T-matrix the differential cross section assumes a simple structure. Received August 31, 1999; revised June 14, 2000; accepted for publication June 30, 2000     

Analyticity and ergodicity in the maximum-entropy approach to statistical nuclear reactions

In the maximum-entropy approach to statistical nuclear reactions one imposes naturally the constraints of unitarity and symmetry of theS-matrix, and of a fixed expectation value ofS. We show that the analytical structure of theS-matrix and the requirement that the problem be ergodic (so that energy averages can be replaced by ensemble averages) impose certain restrictions on the distribution of statisticalS-matrices. Some of these additional constraints are then imposed numerically in a two-channel calculation, and are shown to improve the results for the fluctuation cross sections, the elastic enhancement factor, etc.     

Three-body scattering in Poincaré-invariant quantum mechanics

The relativistic three-nucleon problem is formulated by constructing a dynamical unitary representation of the Poincaré group on the three-nucleon Hilbert space. Two-body interactions are included that preserve the Poincaré symmetry, lead to the same invariant two-body S-matrix as the corresponding non-relativistic problem, and result in a three-body S-matrix satisfying cluster properties. The resulting Faddeev equations are solved by direct integration, without partial waves for both elastic and breakup reactions at laboratory energies up to 2?GeV.     

$\mathcal{PT}$-Symmetric Quantum Electrodynamics—$\mathcal{PT}$QED

The construction of PT\mathcal{PT}-symmetric quantum electrodynamics is reviewed. In particular, the massless version of the theory in 1+1 dimensions (the Schwinger model) is solved. Difficulties with unitarity of the S-matrix are discussed.     

Modification of Hauser-Feshbach calculations by direct-reaction channel coupling

If open channels are strongly coupled by direct reactions, the traditional Hauser-Feshbach method of calculating fluctuation cross sections is invalid, because of non-statistical correlations which the direct channel-coupling induces between resonance partial widths in different channels. The fluctuation cross sections can still be computed from the optical S-matrix elements, however, and the formulas necessary for doing so are obtained here with the aid of an “optical background” representation of the full S-matrix. The resulting compound-elastic cross section is increased over the Hauser-Feshbach expression by a factor of 2(Γ ? D) or 3(Γ ? D) in the large-N limit, and compound-reaction cross sections are increased by roughly a factor of $\text{(N + 1)}\text{N}$, where N is the number of directly-coupled open channels.     

Comments on massive and massless Yang-Mills lagrangians with a quartic coupling of Faddeev-Popov ghosts

The complicated Lagrangians for massless as well as massive Yang-Mills fields and their BRS and anti-BRS transformations proposed by Curci and Ferrari are simplified by the help of an auxiliary field. In the massive non-Abelian case, the mechanism of the breakdown of physicalS-matrix unitarity is clarified and contrasted to the situation in the Abelian case.     

Reactions leading to the 7Li and 8Li systems: Shell model and R-matrix calculations

Level energies and reduced width amplitudes obtained from shell model calculations for ⁷Li and ⁸Li are incorporated into multilevel, multichannel R-matrix calculations of cross sections for reactions leading to these compound systems. With changes to a very few of the large number of parameters obtained from the model calculations, cross sections for eight reactions leading to the ⁷Li system and for five reactions leading to the ⁸Li system are calculated. For reactions where data are available, the calculated cross sections are in good agreement with experimental measurements. Comparisons of the present work to other structure studies are made and suggestions as to possible improvements in the model calculations are given. The applicability of the technique used here to other nuclear structure studies is discussed.     

Interaction potentials and energy transfer cross sections for collisions of metastable helium and neon I: He(23 S)+Ne

Differential cross sections have been measured for He(2³ S)+Ne at kinetic energies between 28 and 370 meV. For energies above 90meV the elastic cross sections show Stückelberg oscillations from curve crossings, which lead to the energy exchange process: He(2³ S)+Ne→He(1¹ S)+Ne(2p ⁵ 4s,3d,4p). Differential cross sections for this inelastic process could be measured above 200 meV. A fit to the data gives the potentials for He(2³ S)+Ne and, less accurately, for He+Ne^*. These results offer a simple explanation, why the exothermic pumping process of the infrared lines of the HeNe laser has a threshold of about 80 meV and a small cross section.     

Discording power of Hamiltonian interactions

In this paper, we report the coherent scattering cross sections of some lanthanides at low momentum transfer in four angular ranges of (0°?4°), (0°?6°), (0°?8°) and for ²⁴¹Am (59.54 keV) and ¹³⁷Cs (661.6 keV) gamma rays. The coherent scattering cross sections were derived by subtracting the small contribution of the corresponding angle integrated incoherent scattering cross sections from the experimentally measured total (coherent + incoherent) scattering cross sections for the elements and energies of interest. The coherent scattering cross sections were found to agree with the corresponding theoretical cross sections within the range of experimental errors. The theoretical coherent scattering cross sections were computed by numerically integrating the S-matrix data of the elements in the angular ranges of interest. The incoherent scattering cross sections were based on the compilations which make use of the non-relativistic Hartree-Fock (NRHF) model for the atomic charge distribution.     

Imprints of compound-nucleus and direct-interaction processes in energy-averaged cross sections and durations of nuclear reactions

New analytical expressions for the energy-averaged S-matrix and nuclear-reaction cross sections are obtained on the basis of the unitary S-matrix parametrization. Some approximations are analyzed when compound-nucleus cross sections are described with and without the Hauser-Feshbach formula with the partial-width-fluctuation correction. The connection of the S-matrix parameters in the Simonius representation with the dynamics of compound-nucleus processes is clarified. An explicit expression for the averaged duration of the nuclear reaction is obtained for the range of closely situated resonances. Possible methods of obtaining information on some compound-nucleus and direct-interaction characteristics from the averaged cross sections and durations are examined.     

The energy dependence of Delbrück scattering investigated at Z = 73, 82 and 92

Using neutron capture γ-rays from a ¹⁴⁰CeO₂ source installed in the Grenoble high-flux reactor, differential cross sections for the elastic scattering of photons by Ta, Pb and U through θ = 120° have been measured for E = 4.291 and 4.767 MeV. These data have been supplemented by measuring elastic differential cross sections for U, θ = 120° and energies ranging from 0.279 to 1.332 MeV, using radioactive sources. The experimental differential cross sections below 1 MeV confirm the predicted Rayleigh amplitudes based on the second-order S-matrix within 3%. An excellent agreement between experiment and lowest-order Delbrück theory is observed between 1.0 and 1.4 MeV, showing that Coulomb corrections are small close to the threshold for pair production. At 4.291 and 4.767 MeV experiment and lowest-order Delbrück theory agree within ~12%     

Separable energy-dependent approximation to local potentials

An extension of the single-pole separable approximation of a two-body t-matrix in which the effects of several poles are included is made. The simple form for the t-matrix derived from a single separable potential is retained. However, the separable potential is constructed using an energy-dependent superposition of the states corresponding to the various poles. The energy dependence is chosen so as to obtain the correct residue of both the on-shell and off-shell t-matrices at each of these poles, while preserving unitarity. The formalism is specialized to the case of s-wave scattering from an attractive square well. Comparison to the exact s-wave cross section gives good results.     

Excitation of Jπ=2+ resonances in 24Mg by the 23Na(p, 12C)12C reaction

Angular distributions of the ²³Na(p, ¹²C_g.s.)¹²C_g.s. reaction were measured at energies on top of resonance-like structures or close to them in the range E_p=8−18 MeV and were analysed using parametrized S-matrix elements. Partial cross sections proceeding through compound nuclear states of ²⁴Mg with various spin J were derived. In was found that the resulting cross sections for J=2 exhibit correlated structures with the “fissioning resonances” recently observed by the ¹²C(¹²C, λ₀) and ²⁴Mg(α, α') ¹²C_g.s + ¹²C_g.s. reactions.     

The 18O(p, α)15N cross section at low energies

The ¹⁸O(p, α)¹⁵N reaction cross section has been measured over the energy range 661 keV > E_c.m > 223 keV. The S-function was extrapolated to energies of astrophysical interest using the R-matrix theory. The S-factor, S₀, is estimated to be 46 MeV · b which is a factor of 3 larger than the value used in a recent tabulation of nuclear reaction rates. The effects of broad levels near the proton threshold are discussed.     

Analysis of elastic and inelastic α + <Superscript>24</Superscript>Mg scattering within the <Emphasis Type="Italic">S</Emphasis> matrix model involving regge poles

A generalization of the phenomenological S-matrix model taking into account isolated Regge poles is proposed for simultaneously describing refractive effects in the cross sections for the elastic and inelastic scattering of light nuclei. The cross sections for elastic α + ²⁴Mg scattering are analyzed at energies of 50, 54, 65.7, 81, and 119 MeV. The analysis of the cross sections for elastic scattering is supplemented with an analysis of the inelastic scattering of alpha particles that is accompanied by the excitation of the first excited state (2⁺) of ²⁴Mg nuclei. It is shown that the proposed model makes it possible to describe satisfactorily all of the aforementioned cases of elastic and inelastic scattering, correctly reproducing the refraction Airy structures and anomalous large-angle scattering that are observed at large scattering angles.     

Yang-Mills theories in space-like axial and planar gauges

The quantization of Yang-Mills theories according to a canonical procedure is studied first in the axial gauge A₃^a ≡ 0. We show that the perturbative S-matrix cannot be expressed in terms of well-defined distributions in a Hilbert space involving only physical states without conflicting with unitarity. We then resort to the space-like planar gauge and show that it is possible to define a perturbative S-matrix at the price of introducing a set of free ghost fields. The S-matrix is unitary in the subspace of the physically acceptable states on which A₃^a vanish.     

Physical unitarity in the Lagrangian Sp(2)-symmetric formalism

The structure of state vector space for a general (non-anomalous) gauge theory is studied within the Lagrangian version of the Sp(2)-symmetric quantization method. The physical S-matrix unitarity conditions are formulated. The general results are illustrated on the basis of simple gauge theory models.