The two-neutron transfer reactions (O, O) with O, C and Mg targets

Excitation functions, angular distributions and differential ranges were measured for the ²⁶Mg(¹⁸O, ¹⁶O)²⁸Mg reaction at ¹⁸O beam energies of 20–45 MeV. Excitation functions only were measured for the reactions ¹⁴C(¹⁸O, ¹⁹O)¹³C, ¹⁴C(¹⁸O, ¹⁶O)¹⁶C, ¹⁴C(¹⁸O, ²⁰O)¹²C, ¹⁴C(¹⁸O, ¹⁵N)¹⁷N and ¹⁸O(¹⁸O, ¹⁹O)¹⁷O, ¹⁸O(¹⁸O, ¹⁶O)²⁰O, ¹⁸O(¹⁸O, ¹⁵N)²¹F at ¹⁸O beam energies of 13–41 MeV. We have identified these as direct reactions in which a single neutron, a two-neutron cluster, a deuteron and a triton are transferred between projectile and target.

The cross sections for two-neutron transfer reactions were found to be relatively high and those for the ¹⁸O+¹⁸O and the ¹⁴C+¹⁸O reactions were higher than the ones of single-neutron transfers over most of the energy range.

Attempts were made to apply the theory of Buttle and Goldfarb for single-neutron transfer to the case of two-neutron transfer in the ²⁶Mg(¹⁸O, ¹⁶O)²⁸Mg reaction below the Coulomb barrier. It is shown that for those reactions for which the assumptions, implicit in the model, are valid, good agreement is obtained with experiment. We also tried to apply the diffraction model of Dar and Kozlovsky to the calculation of the angular distribution of these reactions. A good fit to the experimental results could be obtained if quite different sets of parameters were used in the calculations for the two bombarding energies.     

A function with prescribed initial derivatives in different Banach spaces

Let X₀ ? X₁ ? ··· ? X_p be Banach spaces with continuous injection of X_k into X_{k + 1} for 0 ? k ? p ? 1, and with X₀ dense in X_p. We seek a function u: [0, 1] → X₀ such that its kth derivative u^(k), k = 0, 1,…, p, is continuous from [0, 1] into x_k, and satisfies the initial condition u^(k)(0) = a_k?X_k. It is shown that such a function exists if and only if the initial values a₀, a₁, …, a_p satisfy a certain condition reminiscent of interpolation theory. This condition always holds when p = 1; when p ? 2, the spaces X_k (k = 0, 1, …, p) may or may not be such that the desired function exists for any given initial values a_k?X_k.     

Hyperfine structure measurements of103Rh

Hyperfine structure has been measured in 28 states of¹⁰³Rh below 34,000 cm^?1. The magnetic dipole interactionA constants in the metastable states 4d ⁸ 5s ⁴ P _5/2,3/2,² P _3/2,1/2,² D _5/2,3/2,² G _9/2,7/2 and 4d ⁷ 5s ^{2 4} F _{9/2,7/2,5/2,3/2} have been determined using the laser radio-frequency double-resonance method and in the states 4d ⁸ 5p ⁴ D _{7/2,5/2,3/2,1/2},⁴ G _7/2,5/2,⁴ F _{9/2,7/2,5/2,3/2},² G _9/2,7/2,² F _7/2,5/2 and² D _5/2,3/2 by high-resolution laser spectroscopy.     

Further characterizations and properties of exactly covering congruences

If every nonnegative integer occurs in exactly one of the m integer sequences $\text{a}{}_{\mathrm{i}}\text{n\#\&62; + b}{}_{\mathrm{i}}\text{(n ? 0.1 ? i ? m)}$, then the system a_in + b_i is called an exactly covering congruence (ECC). Three characterizations of ECC's in terms of exponential functions, Bernoulli polynomials and Evler polynomials are given, from which several properties of ECC's are deduced, including a method of obtaining from an ECC with m moduli a_i several ECC's with ? m moduli.     

Ion strength limit of computed excess functions based on the linearized Poisson–Boltzmann equation

The linearized Poisson–Boltzmann (L‐PB) equation is examined for its κ‐range of validity (κ, Debye reciprocal length). This is done for the Debye–Hückel (DH) theory, i.e., using a single ion size, and for the SiS treatment (D. Fraenkel, Mol. Phys. 2010 , 108, 1435), which extends the DH theory to the case of ion‐size dissimilarity (therefore dubbed DH–SiS). The linearization of the PB equation has been claimed responsible for the DH theory's failure to fit with experiment at > 0.1 m; but DH–SiS fits with data of the mean ionic activity coefficient, γ_± (molal), against m, even at m > 1 (κ > 0.33 Å^?1). The SiS expressions combine the overall extra‐electrostatic potential energy of the smaller ion, as central ion—Ψ_a>b(κ), with that of the larger ion, as central ion—Ψ_b>a(κ); a and b are, respectively, the counterion and co‐ion distances of closest approach. Ψ_a>b and Ψ_b>a are derived from the L‐PB equation, which appears to conflict with their being effective up to moderate electrolyte concentrations (≈1 m). However, the L‐PB equation can be valid up to κ ≥ 1.3 Å^?1 if one abandons the 1/κ criterion for its effectiveness and, instead, use, as criterion, the mean‐field electrostatic interaction potential of the central ion with its ion cloud, at a radial distance dividing the cloud charge into two equal parts. The DH theory's failure is, thus, not because of using the L‐PB equation; the lethal approximation is assigning a single size to the positive and negative ions. © 2015 Wiley Periodicals, Inc.     

A deletion game on hypergraphs

Two players alternately select either a vertex or an edge of a hypergraph H, deleting it together with all the edges containing the selected vertex or edge. The player first unable to move loses and the other player wins. We analyze the game for simple classes of hypergraphs, and discuss various related questions.     

The equilibrium geometry,electronic structure and heat of formation of ortho-benzyne

Ab initio calculations at the SCF and Cl level have been carried out for the singlet ground state of ortho-benzyne (1,2,-dehydrobenzene) at a variety of C_2v molecular geometries. The principal features of the equilibrium geometry are: (1) an “acetylenic” C₁C₂ bond (1.22 Å): (2) a C₄C₅ bond slightly elongated (1.42 Å) with respect to benzene; (3) no elongation of the C₂C₃ and C₁C₆ bonds, due to the high s-character and angular deviation of the hybrid orbitals. Extended basis SCF calculations lead to an estimate of ≈ 120 kcal/mole for the ΔH^298°_f of o-benzyne.