Theoretical study on behaviors of 1s22s-1s2np transition for Co24+ ion in the whole energy region

Non-relativistic energies of 1s²2s and 1s² np (n⩽9) states for Co²⁴⁺ ion are calculated by using the full-core plus correlation method. Our results of 1s²2s and 1s²2p states agree well with the high-precision results of Yan et al. Based on calculating the first-order corrections to the energy from relativistic and mass-polarization effects, we estimate the higher-order relativistic contribution and QED correction to the energy under a hydrogenic approximation. The transition energies, wavelengths and oscillator strengths for the 1s²2s−1s² np (n⩽9) transitions of this ion are calculated. The results agree with the experimental data available in literature satisfactorily. By combining with quantum defect theory, our theoretical predictions on the energy and oscillator strength of this ion are extrapolated to the whole energy region including continuous states.     

The generalized dynamical equation and the vacuum polarization effect in hydrogenlike atoms

Based on the generalized dynamical equation, vacuum polarization effects are studied within the scope of the bound state theory in quantum electrodynamics. We find a vacuum-polarization correction to the Lamb shift for the 1S state of the hydrogen atom on the order of (α/π)²(Zα)⁴ that is not considered in the standard theory of bound states in quantum electrodynamics.     

X ray spectroscopic intercomparison between hydrogenic ions of different Z as a method for determining Lamb shifts

Novel methods for determining the 1s Lamb shift of hydrogenic ions are outlined. In one method the energy of a 2–1 transition in the hydrogenic ion of nuclear charge Z is compared to a 4–2 transition in the hydrogenic ion of charge 2Z. A second method exploits close coincidences between Lyman series transitions in ions of similar Z. This revised version was published online in August 2006 with corrections to the Cover Date.     

Systematic trends in the line strengths of E1 transitions in the oxygen isoelectronic sequence

Line strengthsS for the dipole allowed transitions within then=2 complex of the oxygen isoelectronic sequence have been fitted in the formZ ² S=A+B/(Z −C), whereZ is the nuclear charge of a particular ion. The constantsA, B andC are determined by using a non-linear least square method. The data forS are taken from the configuration interaction calculations which included internal, semi-internal and all external type correlations for ions in the rangeZ=8 − 25. It is shown that the values ofA obtained from the fit for all the transitions are in excellent accord with the ab-initio values obtained in the hydrogenic limitZ → ∞ provided near degeneracy effects are included in the ground state multiplet 1s²2s²2p^{4 1}S.     

Hybrid model analysis of the excitation function for alpha induced reaction on121Sb and123Sb

Excitation functions for the reactions¹²¹Sb(α, xn)^125−x I,¹²³Sb(α, xn)^127−x I and¹²¹Sb(α, p3n)¹²¹Te were obtained from the measurements of the residual activity of stacked foils of antimony trioxide evaporated on Al backings from threshold to 60 MeV. The excitation functions for the production of¹²¹I,¹²³I,¹²⁴I,¹²⁶I and¹²¹Te are presented. The experimental data are compared with calculations considering equilibrium as well as pre-equilibrium reaction mechanism according to the hybrid model of Blann. The high energy part of the excitation functions are dominated by the pre-equilibrium reaction mechanism. Calculations were done using a priori calculational method of Overlaid Alice Code of Blann. Most of the excitation functions in the energy range mentioned above could very well be fitted with the hybrid model calculation for exciton numbern=4 withn _n=2 andn _p=2. The overall agreement with the theory is good. Certain discrepancies for example¹²¹Sb(α, p3n)¹²¹Te excitation function, indicate that the production mechanism is different from the one presumed for the calculation.     

Fine and hyperfine structure of the muonic <Superscript>3</Superscript>He ion

On the basis of quasipotential approach to the bound state problem in QED we calculate the vacuum polarization, relativistic, recoil, structure corrections of orders α ⁵ and α ⁶ to the fine structure interval ΔE ^fs = E(2P _3/2) − E(2P _1/2) and to the hyperfine structure of the energy levels 2P _1/2 and 2P _3/2 in muonic ₂³He ion. The resulting values ΔE ^fs = 144 803.15 μeV, Δ$ \tilde E $ \tilde E ^hfs(2P _1/2) = −58 712.90 μeV, Δ$ \tilde E $ \tilde E ^hfs(2P _3/2) = −24 290.69 μeV provide reliable guidelines in performing a comparison with the relevant experimental data.     

Semi-analytic approach to higher-order corrections in simple muonic bound systems: Vacuum polarization,self-energy and radiative-recoil

The current discrepancy of theory and experiment observed recently in muonic hydrogen necessitates a reinvestigation of all corrections to contribute to the Lamb shift in muonic hydrogen (μH), muonic deuterium (μD), the muonic \hbox{³He{}^3{\rm He}} 3 He ion (denoted here as μ ³He⁺), as well as in the muonic \hbox{⁴He{}^4{\rm He}} 4 He ion (μ ⁴He⁺). Here, we choose a semi-analytic approach and evaluate a number of higher-order corrections to vacuum polarization (VP) semi-analytically, while remaining integrals over the spectral density of VP are performed numerically. We obtain semi-analytic results for the second-order correction, and for the relativistic correction to VP. The self-energy correction to VP is calculated, including the perturbations of the Bethe logarithms by vacuum polarization. Subleading logarithmic terms in the radiative-recoil correction to the 2S–2P Lamb shift of order α(Zα)⁵ μ ³ln(Zα) / (m _μ m _N ) are also obtained. All calculations are nonperturbative in the mass ratio of orbiting particle and nucleus.     

Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms

We study the reduced Bogoliubov-Dirac-Fock (BDF) energy which allows to describe relativistic electrons interacting with the Dirac sea, in an external electrostatic potential. The model can be seen as a mean-field approximation of Quantum Electrodynamics (QED) where photons and the so-called exchange term are neglected. A state of the system is described by its one-body density matrix, an infinite rank self-adjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator (the Dirac sea). We study the minimization of the reduced BDF energy under a charge constraint. We prove the existence of minimizers for a large range of values of the charge, and any positive value of the coupling constant α. Our result covers neutral and positively charged molecules, provided that the positive charge is not large enough to create electron-positron pairs. We also prove that the density of any minimizer is an L ¹ function and compute the effective charge of the system, recovering the usual renormalization of charge: the physical coupling constant is related to α by the formula α_phys ≃ α(1 + 2α/(3π) log Λ)⁻¹, where Λ is the ultraviolet cut-off. We eventually prove an estimate on the highest number of electrons which can be bound by a nucleus of charge Z. In the nonrelativistic limit, we obtain that this number is  ≤  2Z, recovering a result of Lieb. This work is based on a series of papers by Hainzl, Lewin, Séré and Solovej on the mean-field approximation of no-photon QED.     

A re-examination of fermi's hyperfine contact interaction

Summary The hyperfine coupling between spin of electrons ins states and nuclear spin is generally represented by a contact Hamiltonian in which a δ(r) factor appears. Utilizing relativistic equations and considering pointlike nuclei, we show that the δ(r) factor must be replaced by a steeply decreasing radial function of half-maximum width δr=5.8·10⁻¹⁴Z cm. For hydrogen, the correction with respect to the contact Hamiltonian turns out to be small, but for high-Z nuclei this correction acquires substantial importance. For iron 1s states, it rises up to 9.6%.     

Double atomic photoeffect in the relativistic domain: Angular and energy distributions of photoelectrons

We study the double ionization of the atomic K-shell by a single photon in the relativistic energy domain. The differential and total cross sections of the process are calculated. It is shown that the ratio of the cross sections of double and single ionization increases with the photon energy, tending to the limit 0.34/Z ², where Z is the atomic number or the nuclear charge. The formulas are found to be valid for Z≫1 and αZ≪1, where α=1/137 is the fine-structure constant. Zh. éksp. Teor. Fiz. 114, 1537–1554 (November 1998)     

Bound state solutions of the two-electron Dirac-Coulomb equation

We present a variational method for solving the two-electron Dirac-Coulomb equation. When the expectation value of the Dirac-Coulomb Hamiltonian is made stationary for all possible variations of the different components of a well-behaved trial function one obtains solutions representative of the physical bound state wave functions. The ground state wave function is derived from the application of a minimax principle. Since the trial function remains well-behaved, the method remains safe from the twin demons of variational collapse and continuum dissolution. The ground state wave function thus derived can be interpreted as a linear combination of different configurations. In particular, the admixing of intermediate states having one (two) electron(s) deexcited to a negative-energy orbital (orbitals) contributes a second-order level shiftE ₀₋ ⁽²⁾ which can be identified with the second-order shift due to the Pauli blocking of the production of one (or two) virtual electron-positron pair(s). Thus the minimax solution corresponds to the renormalized ground state in quantum electrodynamics, with deexcitations to negative-energy orbitals taking the place of the avoidance of virtual pairs. If one extends the relativistic configuration interaction (RCI) treatment by additionally including negative-energy and mixed-energyeigenvectors of the Dirac-Hartree-Fock hamiltonian matrix in the two-electron basis, the calculated energy will be shifted from the conventional RCI value by an amount that is much smaller thanE ₀₋ ⁽²⁾ . For two-electron atoms, we have derived expressions for the all-spinor limit (δE) and thes-spinor limit (δE _s) of this shift in leading orders. The all-spinor limit (δE) is of orderα ⁴ Z ^{4 1/3} whereas thes-spinor limit (δE _s) is of orderα ⁴ Z ^{3 2/3}. leading components are related to the 1-pair component ofE ₀₋ ⁽²⁾ in a simple way, and the relationships offer the possibility of computing energy due to virtual pairs. Numerical results are discussed.     

Wavelength and oscillator strength of dipole transition 1s<Superscript>2</Superscript>2p—1s<Superscript>2</Superscript><Emphasis Type="Italic">n</Emphasis>d for Mn<Superscript>22+</Superscript> ion

The transition energies, wavelengths and dipole oscillator strengths of 1s²2p—1s² nd (3⩽n⩽9) for Mn²²⁺ ion are calculated. The fine structure splittings of 1s² nd (n</9) states for this ion are also evaluated. In calculating energy, the higher-order relativistic contribution is estimated under a hydrogenic approximation. The quantum defect of Rydberg series 1s² nd is determined according to the quantum defect theory. The results obtained in this paper excellently agree with the experimental data available in literatures. Supported by the National Natural Science Foundation of China (Grant No. 10774063)     

The Primakoff reaction study for pion polarizability measurement at COMPASS

The electromagnetic structure of charged pions can be described by the electric (α_π) and magnetic (β_π) polarizabilities that depend on the rigidity of pion’s internal structure as a composite particle. It is shown that the values of α_π and β_π can be precisely measured via the Primakoff reaction π⁻ + (A, Z) → π⁻ + (A, Z) + γ in the COMPASS experiment at CERN.     

Measurement of the gj factor of hydrogenic ions: a sensitive test of bound state QED

Theg _j factor measurement of hydrogenic ions in the 1s ground state is with an expected accuracy of 10^–7 a sensitive test of bound state QED. We expect to determine the deviations from the free electron value, caused by relativistic and radiative corrections, up to the order/4(Z) ² with an accuracy of 1%. As a first step, light ions like C⁵⁺ will be investigated. Later on, heavier hydrogenic ions up to U⁹¹⁺ will be examined using the accelerator facilities at GSI in Darmstadt.     

Systematic trends in the line strengths and radial matrix elements of E1 transitions in the Be isoelectronic sequence

Line strengthS and radial matrix elementσ for the dipole allowed transitions withinn=2 complex of ions in the Be isoelectronic sequence have been fitted in the formsZ ² S=A+B/(Z − C) andZσ=A′ + B′/(Z − C′). The constantsA, B, C andA′, B′, C′ have been calculated by employing a non-linear least square method. The relevant data forS andσ have been taken from calculations which includes correlation effects. It is shown that the fitted yalues ofA andA′ are in excellent accord with their hydrogenic values (Z →α) provided that we express the zeroth-order wavefunction of the ground state 1s²2s^{2 1} S as a quantum-mechanical admixture of the Hartree-Fock (HF) state 1s²2s^{2 1} S and the near-degenerate state 1s²2p^{2 1} S.     

Scott Correction for Large Atoms and Molecules in a Self-Generated Magnetic Field

We consider a large neutral molecule with total nuclear charge Z in non-relativistic quantum mechanics with a self-generated classical electromagnetic field. To ensure stability, we assume that Z α ² ≤ κ ₀ for a sufficiently small κ ₀, where α denotes the fine structure constant. We show that, in the simultaneous limit Z → ∞, α → 0 such that κ = Z α ² is fixed, the ground state energy of the system is given by a two term expansion c ₁ Z ^7/3 + c ₂(κ) Z ² + o(Z ²). The leading term is given by the non-magnetic Thomas-Fermi theory. Our result shows that the magnetic field affects only the second (so-called Scott) term in the expansion.     

Dynamics of α-clusters in N = Z nuclei

The systematics for binding energies per α-particle in N = Z nuclei, E _Bα/N _α, are studied up to ¹⁶⁴Pb. It is shown that, although a geometrical model can be used to explain the systematics for light nuclei, the binding energy per α-particle exhibits structures which are due to the well-known shells of the mean field of nucleons in nuclei. The overall dependence of E _Bα/N _α on N _α in N = Z nuclei (for the ground-state masses) can be described in a liquid-drop model of α-particles. Conditions for a phase change with the formation of an α-particle condensate, a dilute Bose gas in excited compound nuclei are discussed for E _Bα/N _α = 0, at the thresholds. This is achieved when the binding energy per nucleon in nuclei is equal to or smaller than in the α-cluster. At somewhat smaller excitation energies the appearance of a Bose gas with a closed-shell core (N = Z, e.g. of ⁴⁰Ca) is proposed within the same concept. The experimental observation of the decay of such condensed α-particle states is proposed with the coherent emission of several correlated α-particles not described by the Hauser-Feshbach approach for compound-nucleus decay. This decay will be observed by the emission of unbound resonances in the form of ⁸Be and ¹²C ^* (0⁺ ₂) clusters.     

Measurement and analysis of alpha-induced reactions on Ta,Ag and Co

Excitation functions for the reaction¹⁸¹Ta (α,xn)^185−x Re,^107,109Ag (α, ypxn) and⁵⁹Co (α, ypxn) were obtained from measurements of residual activity of stacked foils from threshold to 60 MeV. The excitation functions for the production of¹⁸¹Re,¹⁸²Re,¹⁸³Re,¹⁸⁴Re,¹⁰⁵Ag,¹¹¹In,⁵⁴Mn,⁵⁶Co,⁵⁸Co, and⁶⁰Co, are being presented. The experimental data are compared with calculations considering equilibrium as well as pre-equilibrium reactions according to the hybrid model of Blann. High energy part of the excitation functions is dominated by the pre-equilibrium reaction mechanism. Calculations were done using a priori calculational method of Overlaid Alice Code of Blann. Most of the excitation functions in the energy range mentioned above could be very well fitted with the hybrid model calculation for exciton numbern=4 withn _p=2 andn _n=2. The overall agreement with theory is good. Certain discrepancies, however, indicate the necessity to revise the hybrid model with respect to emission of complex particles.     

Uncertainties on <Emphasis Type="Italic">α</Emphasis><Subscript><Emphasis Type="Italic">S</Emphasis></Subscript> in global PDF analyses and implications for predicted hadronic cross sections

We determine the uncertainty on the strong coupling α _S due to the experimental errors on the data fitted in global analysis of hard-scattering data, within the standard framework of leading-twist fixed-order collinear factorisation in the [`(MS)]\overline{\mathrm{MS}} scheme, finding that α _S (M _Z ²)=0.1202_−0.0015^+0.0012 at next-to-leading order (NLO) and α _S (M _Z ²)=0.1171_−0.0014^+0.0014 at next-to-next-to-leading order (NNLO). We do not address in detail the issue of the additional theory uncertainty on α _S (M _Z ²), but an estimate is ±0.003 at NLO and at most ±0.002 at NNLO. We investigate the interplay between uncertainties on α _S and uncertainties on parton distribution functions (PDFs). We show, for the first time, how both these sources of uncertainty can be accounted for simultaneously in calculations of cross sections, and we provide eigenvector PDF sets with different fixed α _S values to allow further studies by the general user. We illustrate the application of these PDF sets by calculating cross sections for W, Z, Higgs boson and inclusive jet production at the Tevatron and LHC.     

Measurement and analysis of excitation functions for alpha induced reactions on iodine and cesium

The excitation functions for the reactions¹²⁷I(α, 2n)¹²⁹Cs,¹²⁷I(α, 4n)¹²⁷Cs,¹³³Cs(α, 2n)¹³⁵La and¹³³Cs(α, 4n)¹³³La have been measured up to ≈50 MeVα-particle energy using the stacked foil activation technique. Measured excitation functions are compared with pre-equilibrium geometry dependent hybrid model calculations. It has been found that theoretical calculations using an initial exciton numbern ₀=4 (2p+2n+0h) give good agreement with experimental excitation functions.