A pion-nucleus optical potential containing effects of genuine pion absorption and emission

We derive in two ways a lowest-order optical potential $\text{V}$_πA using the projectile and a pair of target nucleons as the basic interacting unit. The first derivation—a heuristic one—uses a multiple-scattering theory, while the second employs a field theory where pions can be absorbed and emitted as well as scattered. The lowest-order terms of $\text{V}$_πA contain the two-particle ground state density and the πNN → πNN scattering matrix. In the same field-theoretical model one analyzes the latter in standard multiple-scattering contributions and genuine absorption corrections. The form of these corrections and their relation to often ad hoc assumed forms is discussed.     

Toroidal hydromagnetics

The complete set of hydromagnetic equations is transformed into Poisson equations and equations of motion for flux densities and their associated variables. The toroidal components of the vector potential A and of the momentum density $\text{a}\text{≠}\text{πv}$ are represented by the po loidal flux densities Ψ and Ψ, respectively, for which the equations of motion are derived. The poloidal components A_⊥ and a_⊥ are represen ed by the potentials atΦ, U and φ, u, for which we obtain Poisson equations in the poloidal plane. Thus one has to solve two Dirichlet and two von Neumann problems at every time step. The source terms of the four Poisson equations define the remaining four variables, namely, $\text{Λ = ▽ ·}\text{A}$,$\text{Ω=(▽×}\text{A}\text{)ζ/R}$, $\text{λ=?·}\text{a}$, and $\text{ω=(?×}\text{a}\text{)ζ/R}$, for which equations of motion are also derived. In the limit of small toroidicity ? we look fo r a selfconsistent scaling of the equations with v_⊥～ε. But the curl of v_⊥×B in Faraday's law creates a toroidal plasma component of B which is one order of magnitude larger than in the case of a low β equilibrium; therefore, the motion becomes fully three-dimensional. Finally, an artificial pressure law is needed to balance the lowest order of the Lorentz force. The conclusion is then that the scaling laws previously used are not applicable for toroidal geometry, and that the effort to obtain numerical solutions is not dramatically higher than without using any scaling law.     

Quantizing a classically ergodic system: Sinai's billiard and the KKR method

Sinai's “billiards on a torus,” i.e., free motion of a particle in a plane amongst reflecting discs of radius R centred on points of the unit square lattice, is a classically ergodic system with two freedoms, parametrized by R. Quantal energy levels E_n are given by the vanishing of the Korringa-Kohn-Rostoker (KKR) determinant of solid state theory. This gives a rapid computational scheme for computing E_n as functions of R. Except for the integrable case R = 0, no degeneracies are found, illustrating the theorem that two parameters, not one, are required to make levels cross in a generic system. The same theorem leads to the prediction that the probability distribution of the spacings S of neighbouring levels is $\text{O}$(S) as S → 0, in good agreement with computation. The KKR determinant is transformed analytically to give the level density d(E) semiclassically (i.e., as ? → 0) as the sum of a steady contribution $\text{d}\text{?}\text{(E)}$ and an oscillatory contribution d_osc(E). $\text{d}\text{?}$ is $\text{O}$(?^?2) and is given by the Weyl “area” formula plus “edge,” “corner” and “curvature” corrections, in excellent agreement with computation. d_osc is given by a sum over classical closed orbits (all unstable). Nonisolated closed orbits (not hitting discs) contribute terms with $\text{O}$$\text{(?}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext><mtext>)</mtext>}}$ to d_osc, while isolated closed orbits (bouncing between discs) contribute terms with $\text{O}$(?^?1) to d_osc. The isolated orbits are vastly more numerous than the nonisolated orbits and their contributions cannot be neglected. As a means of calculating the individual E_n (rather than the smoothed spectrum), the KKR method is much more efficient than the classical path sum.     

Partial-wave analysis of the low-mass (π+π−p) system produced in K−p interactions at 4.2 GeV/c

A partial-wave analysis of the low-mass (π⁺π^?p) system produced in the reaction K^?p → K^?(π⁺π^?p) at 4.2 GeV/c incident momentum is performed in order to study the two (π⁺π^?p) enhancements around 1500 and 1700 MeV. It is found that the low-mass (π⁺π^?p) system can be described using the spin-parity states $\text{J}{}^{\mathrm{P}}\text{=}\text{1}\text{2}{}^{\mathrm{+}}\text{,}\text{3}\text{2}{}^{\mathrm{?}}$ and $\text{5}\text{2}{}^{\mathrm{+}}$ only. In the 1500 MeV region contributions are observed from the $\text{1}\text{2}{}^{\mathrm{+}}$ wave decaying into p? and the $\text{3}\text{2}{}^{\mathrm{?}}$ wave decaying into Δ⁺⁺π^?; in the 1700 MeV region contributions are found from the $\text{1}\text{2}{}^{\mathrm{+}}$ wave decaying into Δ⁺⁺π^?, the $\text{3}\text{2}{}^{\mathrm{?}}$ wave decaying into p?, and the $\text{5}\text{2}{}^{\mathrm{+}}$ wave decaying into p?.     

The electronic spectrum of Dewar benzene

The electronic spectrum of Dewar benzene [bicyclo (2.2.0) hexa-2,5-diene] consists of two transitions above 185 nm. The lower energy transition is structureless and broad with a maximum at 205 nm. The other system begins around 200 nm and shows initially well-defined vibronic structure on a background of increasing continuous absorption, but becomes structureless by 185 nm. Arguments are presented that the two excited states involved in the spectrum are ¹A₂ arising from a $\text{π-π}{}^1$ excitation, and ¹B₁ from a $\text{σ-π}{}^1$ excitation involving the strained σ-bridge bond in Dewar benzene. The interrelationships both photochemical and photophysical of the (CH)₆ potential surface are discussed.     

Nonexponential decay law

The decay of a nonstationary state usually starts as a quadratic function of time and ends as an inverse power law (possibly with oscillations). Between these two extremes, the familiar exponential decay law may be approximately valid. The main purpose of this paper is to find the conditions which must be satisfied by the Hamiltonian and by the initial state, for the exponential law to have a significant domain of validity. It is shown that the evolution of a nonstationary state is governed by a nonnegative function W(E), having the dimensions of an energy. Among its properties are: the energy uncertainty is given by $\text{(ΔH)}{}^2\text{= ?W(E)dE}$, and the inverse lifetime by Γ = 2πW(E₀), where E₀ is the expectation value of H. The detailed shape of W(E) defines two characteristic times between which the exponential decay law is a good approximation: roughly speaking, the smoother W(E), the larger the domain of validity of the exponential law. For instance, if W(E) is very smooth ($\text{|}\text{dW}\text{dE}\text{| ? 1}$) except for a sharp threshold at E = E_thr, the transition from quadratic to exponential decay occurs for $\text{t ?}\text{1}\text{(E}{}_0\text{? E}{}_{\mathrm{<mtext>thr</mtext>}}\text{)}$, and the transition from exponential to inverse power law when $\text{Γt ?}\text{log}\text{[}\text{(E}{}_0\text{? E}{}_{\mathrm{<mtext>thr</mtext>}}\text{)}\text{Γ}\text{]}$.     

Production of A20and ω1 (1675) in the reaction π−p → π+π−π0n at 12 and 15 GeV/c

Data on the reaction π^?p → π⁺π^?π⁰ have been taken at 12 and 15 GeV/c with the CERN Omega multiparticle spectrometer. In a 3-pion partial-wave analysis strong production of $\text{A}{}_2{}^0\text{(1310)}\text{and}\text{ω}{}^1\text{(1675)}$ is observed. Total and differential cross sections are determined and density matrix elements presented as a function of t in the t- and s-channel frames. The energy dependence of A₂⁰ production is studied, and a comparison of $\text{ω(780),}\text{A}{}_2{}^0\text{(1310)}\text{and}\text{ω}{}^1\text{(1675)}$ production is made.     

Positive cones in enveloping algebras

The concept of strong ordering on enveloping algebras of finite-dimensional Lie algebras is introduced and studied as a generalization of the corresponding notion for the commutative polynomial algebra. A linear functional f on an enveloping algebra $\text{E}$ (G) is called strongly positive if f(x) ? 0 for all x ? $\text{E}$(G) which are mapped on positive operators for all G-integrable irreducible representations of $\text{E}$(G). We prove that for each real connected Lie group G ≠ R₁ there are positive, not strongly positive, linear functionals on $\text{E}$(G). A non-commutative problem of moments is defined. It has a solution iff the corresponding linear functional is strongly positive.     

Multiparameter study of 1H, 3H and 4He in fast neutron induced fission of 235U

The emission probabilities per fission of α-particles, tritons and protons have been measured in fast neutron induced fission of ²³⁵U. The measurements were carried out at neutron energies of 120, 180, 230 and 550 keV. AΔE-E semiconductor detector telescope was used to identify different light charged particles and the fission fragments were detected with an ionization chamber. The three-parameter data corresponding to the pulse heights from the ΔE-E detectors and the ion-chamber were recorded event by event on magnetic tape and were analyzed off-line by computer. No significant variation in the most probable energy ($\text{E}$) and the standard deviation (σ_E) of the energy spectra of different light charged particles with incident neutron energy was observed, although $\text{E}{}_{\mathrm{\alpha }}$ was seen to have a slightly higher value beyond E_n = 230 keV. The yield of α-particles in fission induced by neutrons of E_n ～ 200 keV was found to be higher by about 20 % than that in thermal neutron induced fission. The yields of tritons and protons were found to increase significantly with neutron energy.     

Structure factor for a Bose fluid

The structure factor is calculated for a Bose fluid using an expression for S(k), which is approximately valid for all values of k. Marked fluctuations appear for $\text{k = 2.15}\text{A}\text{?}{}^{-1}$, near the roton dip in E_k, and at $\text{k = 3.65}\text{A}\text{?}{}^{-1}$ near the dip in the second branch of E_k. Our calculations are indirectly a theoretical justification for the existence of a second branch in E_k. The results presented here agree fairly well with the experimental values for $\text{k >; 3.0}\text{A}\text{?}{}^{-1}$ along with a dip at $\text{k = 3.67}\text{A}\text{?}{}^{-1}$ which has not been reported earlier. It also suggests the existence of two waves, phonons and rotons, for $\text{k >; 3.0}\text{A}\text{?}{}^{-1}$.     

Pion production by 600 MeV protons on light nuclei

The forward cross section at E_p=600 MeV is calculated for the reaction $\text{p}\text{+(N, Z)→(N+1, Z)}{}^1\text{+π}{}^{\mathrm{+}}$ using a single-nucleon mechanism, distorted pion and proton wave functions and bound-state nucleon wave functions with realistic asymptotic behavior. Good agreement with experiment is found for ⁴He, ¹²C, and ¹⁴N.     

Isospin analysis of the reactions pp→N(Nπ) at 12 and 24 GeV/c

The reaction pp → nucleon + nucleon + pion at 12 and 24 GeV/c is analyzed in terms of the isospin amplitudes for the production of the (Nπ) system. The energy dependence of the $\text{I(}\text{N}\text{π) =}\text{1}\text{2}$ amplitude is weak, while the $\text{I(}\text{N}\text{π) =}\text{3}\text{2}$ contribution shows the strong energy dependence known from meson exchange reactions. The slope parameter B of the dσ/dt′ distributions of the $\text{I(}\text{N}\text{π)=}\text{1}\text{2}$ contribution is a strong function of the (Nπ) mass, decreasing sharply from about 12 GeV^?2 at threshold to about 4 GeV^?2 above 1700 MeV. Comparing our results for the $\text{I(}\text{N}\text{π) =}\text{1}\text{2}$ cross section with those of similar investigations in πp and Kp reactions, we find that factorisation is valid within experimental errors. The results support the conclusion that the $\text{I(}\text{N}\text{π) =}\text{1}\text{2}$ contribution is dominated by diffraction dissociation of the proton.     

Elastic scattering of positive pions from 12C at 30, 40 and 50 MeV

We have measured the angular distribution of π⁺ elastically scattered from ${}^{12}\text{C at}\text{E}\text{?}{}_{\mathrm{\pi }}\text{= 28.4, 38.6}\text{and}\text{48.9}\text{MeV}$. The angular range was between 13° and 165° lab. Scattered pions were detected with two counter telescopes which required pion identification through the observation of the decay muon. Data agree well with calculations derived from the free πn scattering amplitudes when absorption effects have been included.     

The Schwinger model: A view from the temporal gauge

We present a new operator solution of the Schwinger model, i.e., of massless quantum electrodynamics in 1 + 1 dimensions in the temporal gauge A⁰ = 0. This gauge is well-suited for the treatment of static external charges. The energy functional reflects the immediate onset of pair creation of massless fermions. We show that every point charge is screened completely by a Yukawa-like polarization charge cloud of radius $\text{π}\text{e}$, e the coupling constant.     

Evidence for assignment of 14.0 MeV state in 11B from 10B(n,n)10B

The differential cross section and polarization for neutrons scattered from ¹⁰B have been measured at E_n = 2.63 MeV (E_x = 13.85 MeV). The results of this experiment and other available neutron scattering data in the range 1 < E_n < 4 MeV are interpreted through a single-level R-matrix calculation over the region 12 < E_x < 15 MeV. Based on this analysis the most probable J^π assignment for the 14.0 MeV level in ¹¹B is $\text{11}\text{2}{}^{\mathrm{+}}$. The anomaly near E_x = 13.1 MeV can only be explained in terms of two overlapping levels having assignments of ($\text{5}\text{2}$, $\text{7}\text{2}$)^? and ($\text{3}\text{2}$, $\text{5}\text{2}$, $\text{7}\text{2}$)⁺.     

Ericson fluctiations in the π±p systems in the energy range 2 to 3.5 GeV

Existing data on π^±p total cross-sections and 180° elastic differential cross-sections in the energy range 2<E_cm<3.5 GeV are used to estimate the fluctuating differential cross-section within the statistical models of Ericson and Frautschi. A mass spectrum of the form $\text{?(m)=a(}\text{m}\text{m}{}_0\text{)}{}^{\mathrm{b}}$ exp (cm) describes correctly the fluctuations observed in total cross-sections and in 180° elastic differential cross-sections. The mass spectrum is found to be denser for π^?p than for π⁺p.     

Das Niveauschema von 47V aus dem γ-zerfall von Analogresonanzen

A high-accuracy investigation of the level scheme of ⁴⁷V has been performed using the ⁴⁶Ti(p, γ)⁴⁷V reaction. The γ-decay schemes of the strong (p, γ) resonances at E_p = 1546, 1549, 1565 and 1572 keV lead to 17 new energy levels in ⁴⁷V with excitation energies between 2.7 and 5.1 MeV. From the (p,γ) angular distributions mixing ratios of the primary γ-transitions and J^π values of the resonances and of many states populated in the γ-decay have been determined. The total width of the E_p = 1549, 1565 and 1572 keV resonances for γ-decay are found to be Γ_γ = 0.12, 0.15 and 0.03 eV, respectively. The Q-value of the ⁴⁶Ti(p,γ)⁴⁷V reaction is found to be 5168.6 keV. The two resonances at E_p = 1549 and 1565 keV, which have $\text{J}{}^{\mathrm{\pi }}\text{=}\text{3}\text{2}{}^{\mathrm{?}}$, are interpreted as fine structure components of the analogue state of the $\text{E}{}^1\text{= 2.545}\text{MeV}\text{J}{}^{\mathrm{\pi }}\text{=}\text{3}\text{2}{}^{\mathrm{?}}$ level in ⁴⁷Ti while the ($\text{7}\text{2}$) resonance at E_p = 1546 keV might correspond to the $\text{E}{}^1\text{= 2.615}\text{MeV}\text{7}\text{2}{}^{\mathrm{?}}$ parent state in ⁴⁷Ti. The analogue-antianalogue M1 transition strength of the split $\text{3}\text{2}{}^{\mathrm{?}}$ analogue state is 0.01 single-particle units and fits well into our systematics of IAS → AIAS transitions in fp-shell nuclei.