Second-order Relativistic Many-body Perturbation Theory Research Articles

Theoretical determination of energies, wavelengths, and transition probabilities for EUV and SXR spectral lines in Rb XXXIV, Sr XXXV, Zr XXXVII, and Nb XXXVIII

Energy levels, radiative processes and lifetimes are studied among the 98 lowest levels of the 1s22snl and 1s22pnl (n = 2–4, l = 0–3) configurations of Rb XXXIV, Sr XXXV, Zr XXXVII, and Nb XXXVIII. The GRASP2K (General-purpose Relativistic Atomic Structure Package) is adopted, in which the orbital set is systematically expanded within the orbital space up to n = 9, and the Breit interaction and leading quantum electrodynamical corrections are considered. Results are provided for four types of transitions:electric dipole, electric quadrupole, magnetic dipole, and magnetic quadrupole. Similar data for the 166 lowest states belonging to the n ≤ 5 configurations are also determined using the second-order relativistic many-body perturbation theory to assess the accuracy of the calculations, taking Sr XXXV as an example. Good agreement is found between results of this work and available values from the Atomic Spectra Database of the National Institute of Standards and Technology and other theoretical reports. The accuracy for energies and radiative rates is estimated to be about 0.02% and 5%, respectively. These accurate theoretical data can aid line identifications in extreme ultraviolet (EUV) and soft X-ray (SXR) spectra. The present work should be useful for diagnostics of hot plasmas in fusion applications.

Calculations with spectroscopic accuracy: Energies and transition parameters for lines of fusion interest in Mo XXXIX

We have performed high-accuracy calculations of energy levels, wavelengths, lifetimes, line strengths and radiative rates among the 250 lowest levels of the (1s2)2s2, (1s2)2s2p, (1s2)2p2, (1s2)2snl (n=3,4,5,6; l=0,1,2,3,4,5), and (1s2)2pnl (n=3,4,5,6; l=0,1,2,3,4,5) configurations of Mo XXXIX. The second-order relativistic many-body perturbation theory (MBPT) is adopted and data are provided for all electric/magnetic-dipole (E1/M1) and electric/magnetic-quadrupole (E2/M2) transitions. Similar data for the 98 lowest levels of the n ≤ 4 configurations are also determined using fully relativistic multi-configuration Dirac-Fock (MCDF) wave functions in the active space approximation with the inclusion of a finite nuclear size, the Breit interaction, self-energy, and vacuum polarization to assess the accuracy of the calculations. Based on direct comparisons between the results obtained from these two independent methods, MBPT and MCDF, as well as with the other available theoretical values and measurements, our energies are assessed to be accurate to better than 0.02%. The accuracy for strong radiative rates is estimated to be about 5%. For Mo XXXIX, only a few transition lines have been established experimentally. The present new results considerably expand the existing data sets for this ion, and should be beneficial in analysis of experimental spectra and useful for spectroscopic diagnostics and modeling of hot plasmas, in particular in fusion devices.

Theoretical determination of energies, wavelengths, and transition rates for the Y[formula omitted]–Y[formula omitted] spectra of fusion interest

We report a set of theoretical data for the energy levels, wavelengths, transition rates of electric/magnetic-dipole (E1/M1) and electric/magnetic-quadrupole (E2/M2) transitions for the L-shell charge states of Y30+, Y31+, Y32+, Y33+, Y34+, and Y36+ and the similar spectroscopic data of the 166 fine-structure levels arising from the 2s2, 2s2p, 2p2, 2snl (n=3,4,5; l=0,1,2,3,4), and 2pnl (n=3,4,5; l=0,1,2,3,4) configurations of Y35+, obtained using the second-order relativistic many-body perturbation theory (MBPT). The overall quality of the calculations is assessed by comparisons with the other available theoretical values and the measured spectra in the Electron Beam Ion Trap (EBIT) of the National Institute of Standards and Technology (NIST). For these yttrium ions, only a few levels have been experimentally established. The accuracy of our calculations is however high enough to facilitate identifications of observed lines. The present new data, fill in a gap in the available data of the extreme ultraviolet (EUV) transitions for these ions particularly important for fusion research where their spectra can provide diagnostic information on hot plasmas, in particular in fusion devices.

High accuracy theoretical calculation of wavelengths and transition probabilities in Se- through Ga-like ions of tungsten

The Ab initio multiconfiguration Dirac-Hartree-Fock (MCDHF) calculations of atomic parameters have been carried out with high accuracy for four highly charged tungsten ions from Se-like W40+ to Ga-like W43+. The second-order relativistic many-body perturbation theory (MBPT) was used to confirm the accuracy of the MCDHF calculations. Excitation energies, wavelengths and transition rates of E1, M1, E2, M2, E3 and lifetimes are presented. The core-valence electron correlation effects arising from deep subshells 3d and 3p, have been taken into account along with relativistic effects. The calculated wavelengths are in good agreement with available experimental results.

Responsivity calibration of the extreme ultraviolet spectrometer in the range of 175-435 Å

We reported the relative responsivity calibration of the grazing-incidence flat-field EUV spectrometer between 175 and 435 Å by means of two methods. The first method is implemented by measuring the diffraction efficiency of the grating with synchrotron radiation light source. Considering the transmission efficiency and quantum efficiency of the other optical components in the spectrometer, the total responsivity was then obtained. The second one was carried out by measuring line emissions from C3+, N4+ and O3+ ions at Shanghai high temperature super conductor electron beam ion trap (SH-HtscEBIT). The EUV spectra were also simulated theoretically via a collisional radiative model. In the calculation, the second-order relativistic many-body perturbation theory approach based on the flexible atomic code was used to calculate the energy levels and transition rates; the close-coupling R-matrix approach and relativistic distorted wave method were utilized to calculate the collision strength of electron impact excitation. In comparison with the spectroscopic measurements at EBIT device, the differences between the measured and simulated relative line intensities were obtained. The responsivity calibration for the spectrometer was then achieved by a 3rd degree polynomial function fitting. Our measurement shows that the responsivity between 175 and 435 Å varies by factor of ∼ 46. The two results of calibration demonstrated a consistency within an average deviation of 24%. In addition, an evaluation of our calculations on C iv, N v and O iv line emissions in this wavelength region was given.

Relativistic many-body calculations on wavelengths and transition probabilities for forbidden transitions within the ground configurations in Co- through K-like ions of hafnium, tantalum, tungsten and gold

Wavelengths and transition probabilities have been computed for forbidden lines within the 3dk(k = 1-9) ground configurations in ions of hafnium, tantalum, tungsten and gold, employing the second-order relativistic many-body perturbation theory (RMBPT). For comparison, the relativistic configuration-interaction (RCI) calculations have also been performed. Comparing to the previously published theoretical wavelengths and the present RCI ones, the present RMBPT results are in much better agreement with the experimental values measured recently by using the electron beam ion trap facility, reproducing these observed wavelengths to within 0.2% for all transitions only with one exception. In addition, it removes dramatically the systematic underestimation/overestimation for shorter/longer wavelengths existing in the present RCI and other calculations.

Investigation of transitions between metastable levels of the first excited configuration of palladium-like tungsten

Spectroscopy in the wavelength region of 330 nm to 400 nm for highly charged tungsten was performed using the High Temperature Super Conducting Electron Beam Ion Trap (SH-HtscEBIT) at Fudan University. Three lines from palladium-like tungsten (W28+) were identified as transitions between metastable levels in the first excited configuration ([Kr]4d94f). A second-order relativistic many-body perturbation theory approach and a simple collisional radiative model were used to theoretically study the fine structure levels of the 4d94f configuration. The calculated results show qualitative agreement with experiment. We conclude that some levels in the 4d94f excited configuration have extremely long lifetimes and may exhibit extraordinarily high populations, possibly leading to indirect ionization in, for example, fusion plasmas.

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 $Z=14--100$.

Oscillator strengths and lifetimes of Zn I and Ga II

We have applied the second-order relativistic many-body perturbation theory to calculate the oscillator strengths for the spin-allowed electric-dipole (E1) transitions in Zn I and Ga II. We have studied 11 transitions among the first 12 levels of Zn I and Ga II. Radiative lifetimes of some levels in Zn I and Ga II are also evaluated. In particular, the present study provides the first relativistic calculation for the transitions in Zn I whose oscillator strengths are sensitive to the interplay between the relativistic effects and the bound–continuum correlations. The transition amplitudes obtained in different gauges agree within 2%. The present results agree well with experiment.

Relativistic many-body calculations of excitation energies, oscillator strengths, transition rates, and lifetimes in samariumlike ions

The unique atomic properties of samariumlike ions, not yet measured experimentally, are theoretically predicted and studied in this paper. Excitation energies, oscillator strengths, transition probabilities, and lifetimes are calculated for $(5{s}^{2}+5{p}^{2}+5{d}^{2}+5s5d+5s5g+5p5f)$--$(5s5p+5s5f+5p5d+5p5g)$ electric dipole transitions in Sm-like ions with nuclear charge $Z$ ranging from 74 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 $1{s}^{2}2{s}^{2}2{p}^{6}3{s}^{2}3{p}^{6}3{d}^{10}4{s}^{2}4{p}^{6}4{d}^{10}4{f}^{14}$ Dirac-Fock potential. First-order perturbation theory is used to obtain intermediate coupling coefficients, and the second-order RMBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order $E1$ matrix elements to achieve agreement between length-form and velocity-form amplitudes. The resulting transition energies and transition probabilities, and lifetimes for Sm-like W${}^{12+}$ are compared with results obtained by the relativistic Hartree-Fock approximation (cowan code) to estimate contributions of the $4f$-core-excited states. Trends of excitation energies and oscillator strengths as the function of nuclear charge $Z$ are shown graphically for selected states and transitions. This work provides a number of yet unmeasured atomic properties of these samariumlike ions for various applications and as a benchmark for testing theory.

Correlation and relativistic effects within the trivalent Lu-like sequence

Energies of the three-electron-above the closed-shell lutetium-like ions ([Xe]4f145d3, [Xe]4f145d26s, [Xe]4f145d26p and [Xe]4f145d6s6p states) with Z = 74–100 are determined using second-order relativistic many-body perturbation theory. Our calculations start from a Er-like V(N − 3) Dirac–Fock potential ([Xe]4f14, where [Xe] = 1s22s22p63s23p63d104s24p64d105s25p6). Second-order Coulomb and Breit–Coulomb interactions are included. Correction for the frequency dependence of the Breit interaction is taken into account in the lowest order. The Lamb shift correction to energies is also included in the lowest order. The three-electron contributions to the energy are compared with the one- and two-electron contributions. They are found to contribute about 10–20% of the total second-order energy. The ratios of the third- and second-order corrections to the one-electron contributions are found to be about 5–10%. A detailed discussion of the various contributions to the energy levels is given for Lu-like tungsten (Z = 74). Comparisons are made with available experimental data. Trends of excitation energies including splitting of the doublet and quartet terms as functions of nuclear charge Z = 73–100 are illustrated graphically for some states. These are the first ab initio calculations of excitation energies in Lu-like ions. The results for the W3 + ion are particularly important for fusion application.

Correlation and relativistic effects in actinide ions

Wavelengths, line strengths, and transition rates are calculated for the multipole ($E1$, $M1$, $E2$, $M2$, $E3$, and $M3$) transitions between the excited $6{s}^{2}6{p}^{5}nl$ and $6s6{p}^{6}nl$ states and the ground $6{s}^{2}6{p}^{6}$ state in Ac${}^{3+}$, Th${}^{4+}$, and U${}^{6+}$ Rn-like ions. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate energies and transition rates for multipole transitions in these hole-particle systems. The RMBPT method agrees with multiconfigurational Dirac-Fock (MCDF) calculations in lowest order, includes all second-order correlation corrections, and includes corrections from negative-energy states. The calculations start from a [Xe]$4{f}^{14}5{d}^{10}6{s}^{2}6{p}^{6}$ Dirac-Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the matrix elements. Evaluated multipole matrix elements for transitions from excited states to the ground states are used to determine the line strengths, transition rates, and multipole polarizabilities. This work provides a number of yet unmeasured properties of these actinide ions for various applications and for benchmark tests of theory and experiment.

Correlation and relativistic effects for the4f−nland5p−nlmultipole transitions in Er-like tungsten

Wavelengths, transition rates, and line strengths are calculated for the multipole ($E1$, $M1$, $E2$, $M2$, $E3$, and $M3$) transitions between the excited [Cd]$4{f}^{13}5{p}^{6}\mathit{nl}$, [Cd]$4{f}^{14}5{p}^{5}\mathit{nl}$ configurations and the ground [Cd]$4{f}^{14}5{p}^{6}$ state in Er-like ${\mathrm{W}}^{6+}$ ion ([Cd]=[Kr]$4{d}^{10}5{s}^{2}$). In particular, the relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate energies and transition rates for multipole transitions in this hole-particle system. This method is based on the relativistic many-body perturbation theory that agrees with multiconfiguration Dirac-Fock (MCDF) calculations in lowest order, and includes all second-order correlation corrections and corrections from negative-energy states. The calculations start from a [Cd]$4{f}^{14}5{p}^{6}$ Dirac-Fock (DF) potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the multipole matrix elements needed for calculations of other atomic properties such as line strengths and transition rates. In addition, core multipole polarizability is evaluated in random-phase and DF approximations. The comparison with available data is demonstrated.

Determination of molecular hyperfine-structure constant using the second-order relativistic many-body perturbation theory

The spin-rotational Hamiltonian parameters ${A}_{\ensuremath{\parallel}}$ and ${A}_{\ensuremath{\perp}}$ for the BaF molecule are calculated using four-component relativistic spinors at the second-order many-body perturbation theory (MBPT) level via the Z-vector technique. The second-order MBPT is applied to assess the accuracy of the computed hyperfine-structure constants before studying the problem with the state-of-the-artcoupled cluster with single and double excitations (CCSD) method which is highly accurate but computationally more expensive than MBPT. The hyperfine-structure constants $A$ and ${A}_{d}$ resulted from these calculations agree favorably well with experimental findings and with other correlated calculations. The convergence behavior of $A$ and ${A}_{d}$ with respect to the number of active orbitals used in the perturbative calculations suggests that our estimated $A$ and ${A}_{d}$ values should be accurate.

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.

Wavelengths and transition rates for nl–n′l′ transitions in Be-, B-, Mg-, Al-, Ca-, Zn-, Ag- and Yb-like tungsten ions

We present a comprehensive theoretical study of atomic characteristics of eight isoelectronic sequences of tungsten ions in a broad range of wavelengths and transitions. In particular, excitation energies, oscillator strengths and transition probabilities are calculated for nl–n′l′ transitions in W70 +, W69 +, W62 +, W61 +, W54 +, W44 +, W27 + and W4 + ions. Atomic structure and radiative characteristics of Be-like ([He]2lnl′, n = 2, 3), B-like ([He]2l2l′2l″), Mg-like ([Ne]3l3l′), Al-like ([Ne]3l3l′3l″), Ca-like ([Ar]3d4l), Zn-like ([Ni]4l4l′), Ag-like [Kr]4d10nl) and Yb-like ([Xe]4f145l6l′) tungsten ions are computed by the relativistic many-body perturbation theory (RMBPT) method. The calculations start from a 1s2 Dirac–Fock potential for Be- and B-like W, from the 1s22s22p6 Dirac–Fock potential for Mg- and Al-like W; from the 1s22s22p63s23p6 and 1s22s22p63s23p63d10 Dirac–Fock potentials for Ca- and Zn-like W, respectively. Evaluation of properties of Ag-like and Yb-like ions starts from a Dirac–Fock potential with only partially filled n = 4 shell (1s22s22p63s23p63d104s24p64d10) and n = 5 shell (1s22s22p63s23p63d104s24p64d104f145s25p6), respectively. First-order perturbation theory is used to obtain intermediate coupling coefficients, and second-order RMBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order E1 matrix elements to achieve agreement between length-form and velocity-form amplitudes. Our calculations present benchmark data for many yet unmeasured properties of tungsten ions and are in particular important in diagnostics of tungsten plasma of a broad range of temperatures as well as for future ITER plasmas.

Correlation and relativistic effects for the4f−nlmultipole transitions in Yb III ions

Wavelengths, transition rates, and line strengths are calculated for the multipole (E1, M1, E2, M2, and E3) transitions between the excited $[\mathrm{Xe}]4{f}^{13}ns$, $[\mathrm{Xe}]4{f}^{13}nd$, and $[\mathrm{Xe}]4{f}^{13}np$ and the ground $[\mathrm{Xe}]4{f}^{14}$ state in Yb III ion with the nuclear charge $Z=70$ $([\mathrm{Xe}]=[\mathrm{Ni}]4{s}^{2}4{p}^{6}4{d}^{10}5{s}^{2}5{p}^{6})$. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate energies and transition rates for multipole transitions in this hole-particle system. This method is based on the relativistic many-body perturbation theory that agrees with multiconfiguration Dirac-Fock calculations in lowest-order, includes all second-order correlation corrections, and includes corrections from negative energy states. The calculations start from a $[\mathrm{Xe}]4{d}^{14}$ Dirac-Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the matrix elements. Evaluated multipole matrix elements for transitions from excited states into the ground states and transitions between excited states are used to determine the lifetime of the 20 $[\mathrm{Xe}]4{f}^{13}5d(J)$ levels, four $[\mathrm{Xe}]4{f}^{13}6s(J)$ levels, 12 $[\mathrm{Xe}]4{f}^{13}6p(J)$ levels, and four $[\mathrm{Xe}]4{f}^{13}7s(J)$ levels.

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 atomic properties in Pd-like ions

Wavelengths, transition rates, and line strengths are calculated for the 85 possible multipole transitions between the excited 4p6 4d9 4f, 4p6 4d9 5l, 4p5 4d10 4f, and 4p5 4d10 5l states and the ground 4p6 4d10 state in Pd-like ions with the nuclear charges ranging from Z = 47 to 100. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate energies and transition rates for multipole transitions in hole–particle systems. This method is based on the relativistic many-body perturbation theory, agrees with MCDF calculations in lowest order, includes all second-order correlation corrections, and includes corrections from negative energy states. The calculations start from a [Zn]4p64d10 Dirac–Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order multipole matrix elements. The resulting transition energies and transition rates are compared with experimental values and with results from other recent calculations. Trends of the transitions rates for the selected multipole transitions as function of Z are illustrated graphically. The Z dependence of the energy splitting for all triplet terms of the 4p64d9 4f and 4p64d9 5l configurations are shown for Z = 47–100. PACS Nos.: 31.15.Ar, 31.15.Md, 32.70.Cs, 32.30.Rj, 31.25.Jf

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.