Renormalization and effective lagrangians

This article is meant as a summary and introduction to the ideas of effective field theory as applied to gravitational systems, ideas which provide the theoretical foundations for the modern use of general relativity as a theory from which precise predictions are possible.

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Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory

A Calculation of Atomic Structures. By D. R. Hartree Pp. xiii, 181. 40s. 1957. (Wiley, New York)

Bogoliubov’s method of renormalization is formulated in momentum space. The convergence of the rcnonnalizcd Fcynman integrand is proved by an application of the power counting theorem.

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Convergence of Bogoliubov's method of renormalization in momentum space

Aims. The goal of the CHIANTI atomic database is to provide a set of atomic data for the interpretation of astrophysical spectra emitted by collisionally dominated, high temperature, optically thin sources. Methods. A complete set of ground level ionization and recombination rate coefficients has been assembled for all atoms and ions of the elements of H through Zn and inserted into the latest version of the CHIANTI database, CHIANTI 6. Ionization rate coefficients are taken from the recent work of Dere (2007, A&A, 466, 771) and recombination rates from a variety of sources in the literature. These new rate coefficients have allowed the calculation of a new set of ionization equilibria and radiative loss rate coefficients. For some ions, such as Fe viii and Fe ix, there are significant differences from previous calculations. In addition, existing atomic parameters have been revised and new atomic parameters inserted into the database. Results. For each ion in the CHIANTI database, elemental abundances, ionization potentials, atomic energy levels, radiative rates, electron and proton collisional rate coefficients, ionization and recombination rate coefficients, and collisional ionization equilibrium populations are provided. In addition, parameters for the calculation of the continuum due to bremsstrahlung, radiative recombination and two-photon decay are provided. A suite of programs written in the Interactive Data Language (IDL) are available to calculate line and continuum emissivities and other properties. All data and programs are freely available at http://wwwsolar.nrl.navy.mil/ chianti

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CHIANTI - an atomic database for emission lines. IX. Ionization rates, recombination rates, ionization equilibria for the elements hydrogen through zinc and updated atomic data

Extensive theoretical and experimental studies have been carried out for the past 20 years on electron - ion recombination processes, as they are applied to the analysis of astrophysical and laboratory plasmas. We review the basic understanding gained through these efforts, with emphasis on some of the more recent progress made in recombination theory as the recombining system is affected by time-dependent electric fields and plasma particles at low temperature. Together with collisional ionization and excitation processes, recombination is important in determining ionization balance and excited-state population in non-equilibrium plasmas. The radiation emitted by plasmas is usually the principal medium with which to study the plasma condition, as it is produced mainly during the recombination and decay of excited states of ions inside the plasma. This is especially true when the plasma under study is not readily accessible by direct probes, as in astrophysical plasmas. Moreover, external probes may sometimes cause undesirable disturbances of the plasma. Electron-ion recombination proceeds in several different modes. The direct modes include three-body recombination (TBR) and one-step radiative recombination (RR), all to the ground- and singly-excited states of the target ions. By contrast, the indirect resonant mode is a two-step dielectronic recombination (DR), which proceeds first with the formation of doubly-excited states by radiationless excitation/capture. The resonant states thus formed may relax by autoionization and/or radiative cascades. For more exotic modes of recombination, we consider off-shell dielectronic recombination (radiative DR = RDR), in which an electron capture is accompanied by simultaneous radiative emission and excitation of the target ion. Some discussion on attachment of electrons to neutral atoms, resulting in the formation of negative ions, is also given. When resonance states involve one or more electrons in high Rydberg states, presence of an external or intrinsic electric field in the vicinity of the target ions can seriously affect the ionic states involved and the resulting reaction rates. Such perturbative fields can be intrinsic, as in the case of the plasma ion field, or externally imposed. A proper theoretical treatment of this difficult problem is crucial in understanding the recombination process which takes place in a field contaminated environment. The simple off-shell dressing procedure of high Rydberg states by a time-dependent field is reviewed, and the possibility of an anomalously large enhancement in the rates, due to the momentum coherence effect (MCE), is discussed. The presently available data on recombination rates are summarized, and several important deficiencies and future directions for further research are pointed out. Based on the detailed calculations for a number of cases, several empirical rate formulae for RR and DR processes have been generated to summarize the data for ready applications. As the collection of atoms is cooled to very low temperatures, , and the bound electrons are ionized by laser irradiation to states of very precisely controlled energies, the prospect for accurate experimental measurements of very-low-energy recombination rates is considered, where the electron temperature can be very low. Therefore, it is of interest to reconsider theoretically some new phenomena which may occur at such cold environments, in which the electron de Broglie wavelength can be very large, and both the density and coherent effects, as well as possible field effects, must be properly taken into account. Finally, a broader understanding of the various recombination processes may be achieved by studying their relationships to other reactions initiated by electron, ion and photon impact.

https://iopscience.iop.org/article/10.1088/0034-4885/60/7/001/pdf

Electron - ion recombination processes - an overview

An electron-impact-ionization mechanism for ions is considered in which the projectile is first captured by the target to form a doubly excited resonant intermediate state. Subsequently, this state decays in two steps by double Auger emission. The effect upon the ionization cross section is large near the excitation threshold and effectively extends the apparent threshold for scattering excitation single Auger ionization to lower energies. A detailed calculation of the ionization cross section and rate coefficient is presented for ${\mathrm{Fe}}^{15+}$ targets.

Electron impact ionization of Fe 1 5 + by resonant excitation double Auger ionization

Publisher Summary Dielectronic recombination (DR) is by now a well-known process in astrophysical and laboratory fusion plasmas. Together with the radiative decay of collisionally excited states, DR and direct radiative recombination (RR) are important modes by which high-temperature plasmas cool radiatively. Precise determination of the behavior of high-temperature, low-density plasmas require an accurate estimate of rate coefficients for the atomic processes for any different ions. Much progress has been made both experimentally and theoretically in understanding DR and its implications in various physical phenomena. In particular, availability of high-temperature tokamak plasmas and highly stripped ion beams for collision experiments have allowed systematic studies of several ionization reactions. The theory of DR also includes a variety of higher-order processes and also enables identifying various simplifying approximations employed in actual calculations, so that systematic improvements can be made as situations dictate.

Theory of Dielectronic Recombination

Low energy electron /or positron/ scattering by spherically symmetric atoms, discussing long range interactions

Dominant nonadiabatic contribution to the long-range electron-atom interaction.

A single-channel scattering process with the set of quantum numbers $C$ is completely characterized by a phase shift, ${\ensuremath{\eta}}_{C}$. A common approximation in the determination of ${\ensuremath{\eta}}_{C}$ for the scattering of a particle by a compound system is to assume that, apart from the possibility of an exchange of the incident particle with an identical target particle, the target is unaffected by the incident particle. The incident particle is then scattered by the static potential generated by the target in its ground state. The phase shift determined in this approximation, to be called ${{\ensuremath{\eta}}_{C}}^{P}$, can be calculated for a number of scattering processes. Let $H$ represent the Hamiltonian of the entire system, incident particle plus target, let ${E}_{T0}$ be the ground-state energy of the target, and let ${{{E}^{\ensuremath{'}}}_{C}}^{Q}$ be the smallest energy for which ${{{E}^{\ensuremath{'}}}_{C}}^{Q}+{E}_{T0}\ensuremath{-}H$ is a negative definite operator in the space in which the given quantum numbers are $C$ and in which the ground state of the target is projected out. Utilizing the generalized optical potential formalism due to Feshbach and others, it can then be shown that ${\ensuremath{\eta}}_{C}g{{\ensuremath{\eta}}_{C}}^{P}$ if the incident energy is less than ${{{E}^{\ensuremath{'}}}_{C}}^{Q}$. (The bound is probably valid for higher energies, perhaps for all energies for which the process remains a single-channel process. If so, however, the difference between ${\ensuremath{\eta}}_{C}$ and ${{\ensuremath{\eta}}_{C}}^{P}$ for these higher energies will generally be large, of the order of a multiple of $\ensuremath{\pi}$, and the bound will not be immediately useful. We will, therefore, be concerned primarily with incident energies ${E}^{\ensuremath{'}}$ less than ${{{E}^{\ensuremath{'}}}_{C}}^{Q}$.) Furthermore, as one allows for more and more virtual excitation of the target system, the approximate phase shift is guaranteed to improve if ${E}^{\ensuremath{'}}$ is less than ${{{E}^{\ensuremath{'}}}_{C}}^{Q}$, and ${{{E}^{\ensuremath{'}}}_{C}}^{Q}$ will itself increase. Applications are given to the scattering of electrons and of positrons by hydrogen atoms.

Yukap Hahn

Papers

Electron - ion recombination processes - an overview

Electron impact ionization of Fe 1 5 + by resonant excitation double Auger ionization

Theory of Dielectronic Recombination

Dominant nonadiabatic contribution to the long-range electron-atom interaction.

Static Approximation and Bounds on Single-Channel Phase Shifts