Calculate Transition Matrix Elements Research Articles

On the adiabatic approximation for the laser-dressed atom

The adiabatic approximation is analyzed for non-perturbative interaction of an atomic system with an intense low-frequency laser field. We find that the well-known adiabatic ansatz, which consists in using the DC-based Green’s and wave functions for an atom in the instantaneous AC field to calculate transition matrix elements, exceeds the level of accuracy of the adiabatic approximation and requires further theoretical revision. It is shown that the non-analyticity in the field strength of the DC-based Green’s and wave functions leads to violation of the adiabaticity constrains and the appearance of rapidly oscillating components in the transition matrix elements. We suggest an improved adiabatic approach utilizing the analytical parts in the DC-based Green’s and wave functions. We illustrate our revised adiabatic approach for analysis of atomic polarizability in two-component laser field with an intense infrared and perturbative extreme ultraviolet components within the zero-range potential model.

Multi-photon above threshold ionization of multi-electron atoms and molecules using the R-matrix approach

We formulate a computationally efficient time-independent method based on the multi-electron molecular R-matrix formalism. This method is used to calculate transition matrix elements for the multi-photon ionization of atoms and molecules under the influence of a perturbative field. The method relies on the partitioning of space which allows us to calculate the infinite-range free-free dipole integrals analytically in the outer region, beyond the range of the initial bound wave function. This approach is valid for an arbitrary order, that is, any number of photons absorbed both in the bound and the continuum part of the spectrum (below- and above-threshold ionization). We calculate generalized multi-photon cross sections and angular distributions of different systems (H, He, hbox {H}_{{2}}, hbox {CO}_{{2}}) and validate our approach by comparison with data from the literature.

Multipole transitions to determine lifetimes and polarizabilities in Mg-like ions from Si2+ to Fm88+

The relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate the multipole ($E1,M1,E2,M2$, and $E3$) matrix elements to determine the $3s3p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{2}$ lifetime and multipole polarizabilities in Mg-like ions. The electric multipole matrix elements are determined in length and velocity forms. The calculations start from a $1{s}^{2}2{s}^{2}2{p}^{6}$ Dirac-Fock potential. First-order RMBPT is used to obtain intermediate coupling coefficients, and second-order RMBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order multipole matrix elements to ensure gauge independence of transition amplitudes. The details of our calculations of the multipole polarizabilities are illustrated for Mg-like $\mathrm{Si}{}^{2+}$, $\mathrm{Fe}{}^{14+}$, $\mathrm{Kr}{}^{24+}$, $\mathrm{Mo}{}^{30+}$, $\mathrm{Xe}{}^{42+}$, and $\mathrm{W}{}^{62+}$ ions. Our RMBPT results are compared with available theoretical results and experimental measurements. Trends of the line strengths, transition rates, and contributions in the ground-state multipole polarizabilities as functions of $Z$ are illustrated graphically in Mg-like ions with $Z=14--100$.

SymbMat: Symbolic computation of quantum transition matrix elements

SymbMat: Symbolic computation of quantum transition matrix elements

Relativistic many-body calculations of the oscillator strengths, transition rates and polarizabilities in Zn-like ions

Transition rates, oscillator strengths and line strengths are calculated for electric-dipole (E1) transitions between even-parity 4s2, 4p2, 4s4d, 4d2, 4p4f and 4f2 states and odd-parity 4s4p, 4s4f, 4p4d and 4d4f states in Zn-like ions with the nuclear charges ranging from Z = 32 to 100. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a 1s22s22p63s23p63d10 Dirac–Fock potential. First-order RMBPT is used to obtain intermediate coupling coefficients and second-order RMBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order E1 matrix elements to ensure the gauge independence of transition amplitudes. Transition energies used in the calculation of oscillator strengths and transition rates are from second-order RMBPT. Ground state scalar α0(4s2 1S0) polarizabilities are calculated for Zn-like ions from Z = 33 to 100. To evaluate the α0(4s2 1S0) polarizabilities, we calculate RMBPT energies for the odd-parity 4l5l′ complex with J = 1 and line strengths between the even-parity 4l4l′ complex with J = 0 and the odd-parity 4l5l′, 4l6l′ complexes with J = 1.

Relativistic many-body calculations of excitation energies and transition rates from core-excited states in silverlike ions

Energies of [Kr]4d94f2, [Kr]4d94f5l, and [Kr]4d95l5l′ states (with l = s, p, d, f) for Ag-like ions with Z = 50–100 are evaluated to second order in relativistic many-body perturbation theory (RMBPT) starting from a Pd-like Dirac–Fock potential ([Kr]4d10). Second-order Coulomb and Breit–Coulomb interactions are included. Correction for the frequency dependence of the Breit interaction is taken into account in lowest order. The Lamb-shift correction to energies is also included in lowest order. Intrinsic particle–particle–hole contributions to energies are found to be 20–30% of the sum of the one- and two-body contributions. Transition rates and line strengths are calculated for the 4d–4f and 4d–5l electric-dipole (E1) transitions in Ag-like ions with nuclear charge Z = 50–100. RMBPT including the Breit interaction is used to evaluate retarded E1 matrix elements in length and velocity forms. First-order RMBPT is used to obtain intermediate coupling coefficients and second-order RMBPT is used to calculate transition matrix elements. A detailed discussion of the various contributions to the dipole matrix elements and energy levels is given for silverlike tungsten (Z = 74). The transition energies included in the calculation of oscillator strengths and transition rates are from second-order RMBPT. Trends of the transition rates as functions of Z are illustrated graphically for selected transitions. Additionally, we perform calculations of energies and transition rates for Ag-like W by the Hartree–Fock relativistic method (Cowan code) and the Multiconfiguration Relativistic Hebrew University Lawrence Atomic Code (HULLAC code) to compare with results from the RMBPT code. These atomic data are important in modeling of N-shell radiation spectra of heavy ions generated in various collision as well as plasma experiments. The tungsten data are particularly important for fusion application.PACS Nos.: 31.15.A–, 31.15.ag, 31.15.am, 31.15.aj

Relativistic many-body calculations of electric-dipole lifetimes, rates and oscillator strengths of Δn = 0 transitions between 3l−14l′ states in Ni-like ions

Transition rates, oscillator strengths and line strengths are calculated for electric-dipole (E1) transitions between odd-parity 3s23p63d94l1, 3s23p53d104l2 and 3s3p63d104l1 states and even-parity 3s23p63d94l2, 3s23p53d104l1 and 3s3p63d104l2 (with 4l1 = 4p, 4f and 4l2 = 4s, 4d) in Ni-like ions with the nuclear charges ranging from Z = 34 to 100. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a 1s22s22p63s23p63d10 Dirac–Fock potential. First-order RMBPT is used to obtain intermediate coupling coefficients and second-order RMBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order E1 matrix elements to ensure the gauge independence of transition amplitudes. Transition energies used in the calculation of oscillator strengths and transition rates are from second-order RMBPT. Lifetimes of the 3s23p63d94d levels are given for Z = 34–100. These atomic data are important in modelling of M-shell radiation spectra of heavy ions generated in electron beam ion trap experiments and in M-shell diagnostics of plasmas.

Relativistic many-body calculations of electric-dipole lifetimes, transition rates and oscillator strengths for 2l−13l′ states in Ne-like ions

Transition rates, oscillator strengths and line strengths are calculated for electric-dipole (E1) transitions between odd-parity 2s22p53s, 2s22p53d and 2s2p63p states and even-parity 2s22p53p, 2s2p63s and 2s2p63d states in Ne-like ions with the nuclear charges ranging from Z = 14 to 100. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a 1s22s22p6 Dirac–Fock potential. First-order RMBPT is used to obtain intermediate coupling coefficients and second-order RMBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order E1 matrix elements to ensure the gauge independence of transition amplitudes. Transition energies used in the calculation of oscillator strengths and transition rates are from second-order RMBPT. Lifetimes of the 16 even-parity and 18 odd-parity levels are given for Z = 14–100. Transition rates, line strengths and oscillator strengths are compared with critically evaluated experimental values and with results from other recent calculations. These atomic data are important in the modelling of L-shell radiation spectra of heavy ions generated in electron beam ion trap experiments and in L-shell diagnostics of plasmas.

Antisymmetrized Molecular Dynamics with Coherent State Pions and Its Application to Excited Spectrum of 12C

We introduce coherent state neutral pions into Antisymmetrized Molecular Dynamics. With the aid of coherent state technique, it becomes possible to calculate transition matrix elements of the pion field operator and to study excited states containing pions. For a large pion-nucleon coupling, ऒπN ≳ 1.6, pions have a finite expectation value and cause a large energy gain in 12C. We discuss two aspects of pionic effects in spectroscopy, an LS-like interaction effect and the mixing of different nucleon parity states, which would modify low energy nuclear levels.

Relativistic many-body calculations of excitation energies and transition rates from core-excited states in copperlike ions

Energies of (3s2 3p6 3d9 4l4l'), (3s2 3p5 3d10 4l4l'), and (3s 3p6 3d10 4l4l') states for Cu-like ions with Z = 30 -100 are evaluated to second order in relativistic many-body perturbation theory (RMBPT) starting from a Ni-like Dirac-Fock potential. Second-order Coulomb and Breit-Coulomb interactions are included. Correction for the frequency-dependence of the Breit interaction is taken into account in lowest order. The Lamb shift correction to energies is also included in lowest order. Intrinsic particle-particle-hole contributions to energies are found to be 20-30% of the sum of one- and two-body contributions. Transition rates and line strengths are calculated for the 3l-4l' electric-dipole (E1) transitions in Cu-like ions with nuclear charge Z = 30 - 100. RMBPT including the Breit interaction is used to evaluate retarded E1 matrix elements in length and velocity forms. First-order RMBPT is used to obtain intermediate coupling coefficients and second-order RMBPT is used to calculate transition matrix elements. A detailed discussion of the various contributions to the dipole matrix elements and energy levels is given for copperlike tungsten (Z = 74). The transition energies used in the calculation of oscillator strengths and transition rates are from second-order RMBPT. Trends of the transition rates as functions of $Z$ are illustrated graphically for selected transitions. Comparisons are made with available experimental data. These atomic data are important in modeling of M-shell radiation spectra of heavy ions generated in electron beam ion trap experiments and in M-shell diagnostics of plasmas.

Relativistic many-body calculations of electric-dipole lifetimes, transition rates, and oscillator strengths for n=3 states in Al-like ions

Transition rates, oscillator strengths, and line strengths are calculated for electric-dipole (E1) transitions between even-parity 3 s3 p 2, 3 s 23 d, 3 p 23 d, 3 d 23 s, and 3 d 3 states and odd-parity 3 s 23 p, 3 p 3, 3 s3 p3 d, and 3 d 23 p states in Al-like ions with the nuclear charges ranging from Z=15 to 100. Relativistic many-body perturbation theory (MBPT), including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a 1 s 22 s 22 p 6 Dirac–Fock potential. First-order MBPT is used to obtain intermediate coupling coefficients and second-order MBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order E1 matrix elements to ensure gauge-independence of transition amplitudes. Transition energies used in the calculation of oscillator strengths and transition rates are from second-order MBPT. Transition rates, line strengths, and oscillator strengths are compared with critically evaluated experimental values and with results from other recent calculations. We present data for selected transitions between 10 of the possible 73 even-parity 3 s3 p 2, 3 s 23 d states and 29 of the possible 75 odd-parity 3 s 23 p, 3 p 3, and 3 s3 p3 d states in Al-like ions. Trends of the transition rates as functions of Z are illustrated graphically for 220 of the 3220 possible transitions. Lifetimes of the 10 even-parity lower levels and the 27 odd-parity upper levels are given for Z=15–100.

Relativistic many-body calculations of transition rates from core-excited states in sodiumlike ions

Rates and line strengths are calculated for the 2s22p53l3l' - 2s22p63l'' and 2s2p63l3l' - 2s22p63l'' electric-dipole (E1) transitions in Na-like ions with nuclear charges ranging from Z = 14 to 100. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a 1s22s22p6 Dirac-Fock potential. First-order RMBPT is used to obtain intermediate coupling coefficients and second-order RMBPT is used to calculate transition matrix elements. A detailed discussion of the various contributions to dipole matrix elements is given for sodiumlike copper (Z = 29). Transition energies used in the calculation of transition rates are from second-order RMBPT. Trends of transition rates as functions of Z are shown graphically for selected transitions.

Lepton flavor violation: Constraints from the exotic μ−-eμ− conversion

The exotic neutrinoless $\mu^- - e^-$ nuclear conversion is studied within the conventional extensions of the standard model as well as in the minimal supersymmetric (SUSY) models with and without R-parity conservation. The dependence of the $\mu^- - e^-$ conversion rates on the nucleon and nuclear structure is consistently taken into account. Using our calculated transition matrix elements and the available experimental data on the branching ratio $R_{\mu e^-}$ for $^{48}$Ti and $^{208}$Pb as well as the expected experimental sensitivity for $^{27}$Al employed as a target in the planned at Brookhaven $\mu^--e^-$ conversion (MECO) experiment, we extract very severe constraints for the flavor violation parameters. We especially emphasize on the constraints resulting for SUSY R-parity violating parameters.

Relativistic many-body calculations of electric-dipole transitions betweenn=2states in B-like ions

Transition rates, oscillator strengths, and line strengths are calculated for the 49 possible electric-dipole transitions between the eight even-parity 2s2p{sup 2} states and the seven odd-parity 2s{sup 2}2p and 2p{sup 3} states in boronlike ions with nuclear charges ranging from Z=6 to 100. Relativistic many-body perturbation theory (MBPT), including the Breit interaction, is used to evaluate retarded E{sub 1} matrix elements in length and velocity forms. The calculations start from a 1s{sup 2} Dirac-Fock potential. First-order MBPT is used to obtain intermediate coupling coefficients and second-order MBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order E{sub 1} matrix elements to ensure gauge independence of transition amplitudes. The transition energies used in the calculation of oscillator strengths and transition rates are from second-order MBPT. Transition rates, line strengths, and oscillator strengths are compared with critically evaluated experimental values and with results from other recent calculations. Lifetimes of the five possible odd-parity upper levels and the five possible even-parity upper levels are given for Z=6{endash}100. Trends of the transition rates as functions of {ital Z} are illustrated graphically for selected transitions. thinsp {copyright} {ital 1999} {ital The American Physical Society}

Identification of room temperature photoluminescence in pseudomorphic modulation-doped AlGaAs/InGaAs/GaAs quantum wells

Room temperature photoluminescence in δ-doped pseudomorphic AlGaAs/InGaAs/GaAs quantum wells is investigated. The electron and hole subband structure and optical transition matrix elements are calculated by a self-consistent theoretical method. Separations of the calculated conduction subband levels and valence subband levels reveal that the dominant emissions are due to the transitions from the second electron subband to the first heavy-hole subband, the first electron subband to the first heavy-hole subband, and the first electron subband to the second heavy-hole subband. The calculation also predicts that the transition energies shift to lower energies with the increase of the δ-doping level. This is accounted for by the lowering of the confined levels originating from electrostatically induced band bending. The calculated transition matrix elements demonstrate that transitions with different electron and hole subband indices, i.e., forbidden pairs, are preferred. The relative magnitudes of the squared optical matrix elements for the transitions from the first and second electron subbands to the first heavy-hole subband are analyzed as functions of separations of the first two electron subbands and the first two heavy-hole subbands, respectively. The results show that the 2-1 pair has a larger oscillator strength than the 1-1 and 1-2 pairs in agreement with observations.

Scaling properties of resonant electron transfer in ion—metal-surface interactions

Scaling properties of resonant electron transfer in the interaction of atoms and positive ions with metal surfaces are revealed by examining the dependence of numerically calculated transition matrix elements and (first-order) transition rates upon the scaled ion-surface distance ∼ D = D/ D n , where D n is the classical threshold distance for electron transfer involving ionic stat principal quantum number n. For zero orbital angular momentum and fixed energy of the ionic states, the n-dependence of the rates at ∼ D = 1 is found to approach, in the large- n limit, a simple power law. A scaling law is established that connects, in the range ∼D ⩾ 1, transition rates for arbitrary (large) principal quantum numbers.

Scaling properties of resonant electron transfer in ion-metal-surface interactions

Scaling properties of resonant electron transfer in the interaction of atoms and positive ions with metal surfaces are revealed by examining the dependence of numerically calculated transition matrix elements and (first-order) transition rates upon the scaled ion-surface distance D ̃ = D D n , where D n is the classical threshold distance for electron transfer involving ionic states with principal quantum number n. For zero orbital angular momentum and fixed energy of the ionic states, the n-dependence of the rates at D ̃ = 1 is found to approach, in the large- n limit, a simple power law. A scaling law is established that connects, in the range D ̃ ≥ 1 , transition rates for arbitrary (large) principal quantum numbers.

Quadrupole collective correlations and the depopulation of the superdeformed bands in mercury

Fully self-consistent generator coordinate method calculations have been performed on a basis of quadrupole constrained Hartree-Fock plus BCS wave functions for the five even mercury isotopes 190–198Hg. The gcm results support conclusions drawn from previous HF + BCS calculations. The predicted evolution of superdeformed band head (shape isomer) properties as a function of the neutron number is consistent with the data. Using calculated transition matrix elements, we evaluate in-band versus out-of-band quadrupole decay and explain the sudden termination of the super-deformed band.

Spectroscopy and nonradiative phenomena of rare earths in glasses: Future applications

The radiative transition probabilities of rare earth ions in glasses can be calculated from the measured intensity parameters and calculated transition matrix elements using the Judd-Ofelt theory. The Judd-Ofelt parameters in oxide, flouride and chalcogenide glasses were obtained for a variety of rare earth ions. These intensity parameters depend on the chemical bonding between the rare earths and the glass matrix and the viscosity of the medium. The non-radiative transition probabilities are calculated using a phenomenological theory, experimentally measured fluorescence lifetimes and quantum efficiencies. The nonradiative transition probabilities show an exponential dependence on the phonon energy of the network-forming material. Experimental results according to this theory show that Er(III) has the highest quantum efficiency in fluoride glasses. Technological implementation of the above theory will be discussed for the lasers, sources for fiber waveguides and luminescent solar concentrators.

Spectroscopy and nonradiative phenomena of rare earths in glasses: Future applications

The radiative transition probabilities of rare earth ions in glasses can be calculated from the measured intensity parameters and calculated transition matrix elements using the Judd-Ofelt theory. The Judd-Ofelt parameters in oxide, fluoride and chalcogenide glasses were obtained for a variety of rate earth ions. These intensity parameters depend on the chemical bonding between the rare earths and the glass matrix and the viscosity of the medium. The nonradiative transition probabilities are calculated using a phenomenological theory, experimentally measured fluorescence lifetimes and quantum efficiencies. The nonradiative transition probabilities show an exponential dependence of the phonon energy of the network forming material. Experimental results according to this theory show that Er 3+ has the highest quantum efficiency in fluoride glasses. Technological implementation of the above theory will be discussed for glass lasers, sources for fiber waveguides and luminescent solar concentrators.