A systematic QED treatment of electron correlation is presented for ions along the lithium isoelectronic sequence. We start with the zeroth-order approximation that accounts for a part of the electron-electron interaction by a local model potential introduced into the Dirac equation. The residual electron correlation is treated by perturbation theory. Rigorous QED evaluation is presented for the first two terms of the perturbative expansion; the third-order contribution is calculated within the many-body perturbation theory. We report accurate numerical results for the electronic-structure part of the ionization potential for the n=2 states of all Li-like ions up to uranium.
A rigorous QED theory of the spectral line profiles is applied to transition probabilities in few-electron highly charged ions. Interelectron interaction corrections are included as well as radiative corrections. Parity nonconserving (PNC) amplitudes with effective weak interactions between the electrons and nucleus are also considered. QED and interelectron interaction corrections to the PNC amplitudes are derived.
Accurate QED calculation of transition probabilities for the low-lying two-electron configurations of multicharged ions is presented. The calculation is performed for the nondegenerate states $(1s2s) 3S1$, $(1s2p_{3/2}) 3P2$ ($M 1$ and $M 2$ transitions, respectively) and for the quasidegenerate states $(1s2p) 1P1$, $(1s2p) 3P1$ ($E 1$ transitions) decaying to the ground state $(1s1s) 1S0$. Two-electron ions with nuclear charge numbers $Z=10-92$ are considered. The line profile approach is employed for the description of the process in multicharged ions within the framework of QED.
The screened QED corrections to the g factor of Li-like ions are evaluated utilizing an effective potential approach. The evaluation is performed within the range of nuclear charge numbers Z=32-92. The results obtained serve for improving the theoretical predictions for the g factor of heavy Li-like ions.
The quantum electrodynamics (QED) corrections are directly incorporated into the most accurate treatment of the correlation corrections for ions with complex electronic structure of interest to metrology and tests of fundamental physics. We compared the performance of four different QED potentials for various systems to access the accuracy of QED calculations and to make prediction of highly charged ion properties urgently needed for planning future experiments. We find that all four potentials give consistent and reliable results for ions of interest. For the strongly bound electrons the nonl…
The current status of bound state quantum electrodynamics calculations of transition energies for few-electron ions is reviewed. Evaluation of one and two body QED correction is presented, as well as methods to evaluate many-body effects that cannot beevaluated with present-day QED calculations. Experimental methods, their evolution over time, as well as progress in accuracy are presented. A detailed, quantitative, comparison between theory and experiment is presented for transition energies in few-electron ions. In particular the impact of the nuclear size correction on the quality of QED tes…
The present status of tests of QED with highly charged ions is reviewed. The theoretical predictions for the Lamb shift and the transition energies are compared with available experimental data. Recent achievements in studies of the hyperfine splitting and the $g$-factor isotope shift with highly charged ions are reported. Special attention is paid to tests of QED within and beyond the Furry picture at strong-coupling regime. Prospects for tests of QED at supercritical fields that can be created in low-energy heavy-ion collisions are discussed as well.
Mass-dependent and field shift components of the isotopic shift are determined to high accuracy for the ground $1^1S-$states of some light two-electron Li$^{+}$, Be$^{2+}$, B$^{3+}$ and C$^{4+}$ ions. To determine the field components of these isotopic shifts we apply the Racah-Rosental-Breit formula. We also determine the lowest order QED corrections to the isotopic shifts for each of these two-electron ions.
The FAC, GRASP2K, and MCDFGME codes are compared in three case study of radiative transitions occurring in tungsten ions: (i) Ni1 and Ni2 lines in Ni-like tungsten, (ii) $3p_{3/2}{-}3p_{1/2}$ fine splitting in Cl-like tungsten, and (iii) $Kα_{1}$ and $Kα_{2}$ lines in W VIII. Various approaches to including the Breit interaction term and QED corrections in atomic calculations are examined. Electron correlation effects are also investigated and compared to the Breit and QED contributions. The data presented here may be used to estimate theoretical uncertainties relevant to interpretation of hig…
Accurate calculations of the binding energies, the hyperfine splitting, the bound-electron g-factor, and the parity nonconservation effects in heavy few-electron ions are considered. The calculations include the relativistic, quantum electrodynamic (QED), electron-correlation, and nuclear effects. The theoretical results are compared with available experimental data. A special attention is focused on tests of QED in a strong Coulomb field.
Large-scale {\em ab initio} QED calculations are performed for the $2p_{3/2}$--$2p_{1/2}$ fine-structure interval of Li-like ions with nuclear charges $Z = 5\,$--$\,92$. Improved theoretical predictions are obtained by combining together two complementary theoretical methods, namely, the approach that accounts for all orders in the binding nuclear strength and the nonrelativistic QED approach that accounts for all orders in the nonrelativistic electron-electron interaction. The resulting unified approach provides theoretical predictions which are more accurate than the available experimental r…
Ab initio calculations of QED radiative corrections to the $^2P_{1/2}$ - $^2P_{3/2}$ fine-structure transition energy are performed for selected F-like ions. These calculations are nonperturbative in $αZ$ and include all first-order and many-electron second-order effects in $α$. When compared to approximate QED computations, a notable discrepancy is found especially for F-like uranium for which the predicted self-energy contributions even differ in sign. Moreover, all deviations between theory and experiment for the $^2P_{1/2}$ - $^2P_{3/2}$ fine-structure energies of F-like ions, reported rec…
We perform ab initio QED calculations of energy levels for the $n=1$ and $n=2$ states of He-like ions with the nuclear charge in the range $Z = 12$-100. The complete set of two-electron QED corrections is evaluated to all orders in the parameter $\aZ$. Uncalculated contributions to energy levels come through orders $α^3 (\aZ)^2$, $α^2 (\aZ)^7$, and higher. The calculation presented is the first treatment for excited states of He-like ions complete through order $α^2 (\aZ)^4$. A significant improvement in accuracy of theoretical predictions is achieved, especially in the high-$Z$ region.
A rigorous evaluation of the complete gauge-invariant set of the screened one-loop QED corrections to the hyperfine structure and g factor in lithiumlike heavy ions is presented. The calculations are performed in both Feynman and Coulomb gauges for the virtual photon mediating the interelectronic interaction. As a result, the most accurate theoretical predictions for the specific difference between the hyperfine splitting values of H- and Li-like Bi ions as well as for the g factor of Li-like Pb ion are obtained.
Calculations of various corrections to the g factor of Li-like ions are presented, which result in a significant improvement of the theoretical accuracy in the region Z = 6-92. The configuration-interaction Dirac-Fock method is employed for the evaluation of the interelectronic-interaction correction of order 1/Z^2 and higher. This correction is combined with the 1/Z interelectronic-interaction term derived within a rigorous QED approach. The one-electron QED corrections of first in αare calculated to all orders in the parameter αZ. The screening of QED corrections is taken into account to the…
QED corrections to the $g$ factor of Li-like and B-like ions in a wide range of nuclear charges are presented. Many-electron contributions as well as radiative effects on the one-loop level are calculated. Contributions resulting from the interelectronic interaction, the self-energy effect, and most of the terms of the vacuum-polarization effect are evaluated to all orders in the nuclear coupling strength $Zα$. Uncertainties resulting from nuclear size effects, numerical computations, and uncalculated effects are discussed.
A systematic investigation of the nuclear-polarization effects in one- and few-electron heavy ions is presented. The nuclear-polarization corrections in the zeroth and first orders in $1/Z$ are evaluated to the binding energies, the hyperfine splitting, and the bound-electron g factor. It is shown, that the nuclear-polarization contributions can be substantially canceled simultaneously with the rigid nuclear corrections. This allows for new prospects for probing the QED effects in strong electromagnetic field and the determination of fundamental constants.
We develop {\it ab initio} relativistic QED theory for elastic electron scattering on hydrogen-like highly charged ions for impact energies where, in addition to direct (Coulomb) scattering, the process can also proceed via formation and consequent Auger decay of autoionizing states of the corresponding helium-like ions. Even so the primary goal of the theory is to treat electron scattering on highly charged ions, a comparison with experiment shows that it can also be applied for relatively light ions covering thus a very broad range of the scattering systems. Using the theory we performed cal…
We calculated spectra of the first six members of the Mg-like isoelectronic sequence using different approximations. The most accurate results were obtained with the configuration interaction + all order method (CI+AO), which provided relative accuracy for transition energies on the level of 0.1\%, or better. On this level of accuracy the Breit and QED corrections become important for the systems with nuclear charge $Z\gtrsim 20$. The retardation part of the Breit interaction and QED corrections for the partial waves with $l\ne 0$ are still too small to be important.
Results of a calculation valid to all orders in the nuclear-strength parameter Zαare presented for the two-loop Lamb shift, notably for the two-loop self-energy correction, for the ground and first excited states of ions with the nuclear charge numbers Z=60-100. A detailed comparison of the all-order calculation with earlier investigations based on an expansion in the parameter Zαis given. The calculation removes the largest theoretical uncertainty for the 2p_j-2s transition energies in heavy Li-like ions and is important for interpretation of experimental data.
Self-energy corrections to the ionization energies in sodium-like ions: comparison of the \textit{ab initio} QED and model-QED-operator approaches
Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences（中国科学院精密测量科学与技术创新学院）；University of Chinese Academy of Sciences（中国科学院大学）；Department of Physics, St. Petersburg State University（圣彼得堡国立大学物理系）；Petersburg Nuclear Physics Institute named by B.P. Konstantinov of National Research Center “Kurchatov Institute”（库尔恰托夫国家研究中心B.P.康斯坦丁诺夫彼得堡核物理研究所）；School of Physics and Engineering, ITMO University（ITMO大学物理与工程学院）
Ab initio QED calculations of the ground-state binding energies of berylliumlike ions are performed for the wide range of the nuclear charge number: Z=18-96. The calculations are carried out in the framework of the extended Furry picture starting with three different types of the screening potential. The rigorous QED calculations up to the second order of the perturbation theory are combined with the third- and higher-order electron-correlation contributions obtained within the Breit approximation by the use of the large-scale configuration-interaction Dirac-Fock-Sturm method. The effects of n…
A rigorous QED evaluation of the two-photon exchange corrections to the g factor of lithiumlike ions is presented. The screened self-energy corrections are calculated for the intermediate-$Z$ region and its accuracy for the high-$Z$ region is essentially improved in comparison with that of previous calculations. As a result, the theoretical accuracy of the g factor of lithiumlike ions is significantly increased. The theoretical prediction obtained for the g factor of $^{28}$Si$^{11+}$ ${\rm g}_{\rm th} = 2.000\,889\,892(8)$ is in an excellent agreement with the corresponding experimental value…
Fully relativistic approach to evaluate the correlation effects in highly charged ions is presented. The interelectronic-interaction contributions of first and second orders in $1/Z$ are treated rigorously within the framework of bound-state quantum electrodynamics, whereas the calculations of the third- and higher-order contributions are based on the Dirac-Coulomb-Breit Hamiltonian. The developed approach allows one to deal with single as well as degenerate or quasi-degenerate states. We apply this approach to the calculations of the correlation contributions to the $n=1$ and $n=2$ energy lev…
A calculation of two-electron QED effects to all orders in the nuclear binding strength parameter $Zα$ is presented for the ground and $n = 2$ excited states of helium-like ions. After subtracting the first terms of the $Zα$ expansion from the all-order results, we identify the higher-order QED effects of order $mα^7$ and higher. Combining the higher-order remainder with the results complete through order $mα^6$ from [V. A. Yerokhin and K. Pachucki, Phys. Rev. A 81, 022507 (2010)], we obtain the most accurate theoretical predictions for the ground and non-mixing $n=2$ states of helium-like ion…
The screened QED corrections of the first orders in αand 1/Z to the g factor and the hyperfine splitting of lithiumlike ions are evaluated within ab initio quantum electrodynamical approach. The complete gauge-invariant set of the two-electron self-energy diagrams in the presence of the magnetic field and a dominant part of the two-electron vacuum-polarization diagrams are calculated. The most accurate values of the g factor of Li-like lead and uranium are presented. The theoretical prediction for the specific difference of the hyperfine splittings of H- and Li-like bismuth is improved.
Ab initio QED calculations of the four x-ray transitions from the $L$ to $K$ shell in heliumlike argon, titanium, iron, copper, and krypton are performed. The binding energies for all the $n=1$ and $n=2$ states are evaluated as well. The calculation approach combines the rigorous QED treatment in the first two orders of the perturbation theory constructed within the extended Furry picture with the third- and higher-order correlation effects evaluated in the Breit approximation. The obtained results are compared with the previous evaluations and available experimental data.
The multiconfiguration Dirac-Hartree-Fock theory is used to calculate the 3d9 2D3/2 - 2D5/2 transition energy for Co-like ions with Z = 28-100. We investigate how electron correlation, frequency-independent and -dependent Breit interactions, as well as QED corrections, vary along the sequence. The well-understood frequency-independent Breit contribution has the largest contribution for all ions. Among the corrections to this, correlation decreases rapidly with Z, the frequency-dependent Breit contribution is important especially for high-Z ions, and the self-energy contribution to the QED beco…
The quantum electrodynamical theory of the three-photon transitions in hydrogen-like ions is presented. Emission probabilities of various three-photon decay channels for $ 2p_{3/2} $, $ 2p_{1/2} $ and $ 2s_{1/2} $ states are calculated for the nuclear charge $Z$ values $1\leqslant Z \leqslant 95$. The results are given in two different gauges. The fully relativistic three-photon decay rates of hydrogen-like ions with half-integer nuclear spin are given for transitions between fine structure components. The results can be applied to the tests of the Bose-Einstein statistics for the multiphoton …
We perform ab initio calculations of the QED effects of order $mα^7$ for the $2^3S$ and $2^3P$ states of He-like ions. The computed effects are combined with previously calculated energies from [V. A. Yerokhin and K. Pachucki, Phys. Rev. A 81, 022507 (2010)], thus improving the theoretical accuracy by an order of magnitude. The obtained theoretical values for the $2^3S$-$2^3P_{0,2}$ transition energies are in good agreement with available experimental results and with previous calculations performed to all orders in the nuclear binding strength parameter $Zα$. For the ionization energies, howe…
The current status of tests of quantum electrodynamics with heavy ions is reviewed. The theoretical predictions for the Lamb shift and the hyperfine splitting in heavy ions are compared with available experimental data. Recent achievements and future prospects in studies of the $g$ factor with highly charged ions are also reported. These studies can provide precise determination of the fundamental constants and tests of QED within and beyond the Furry picture at the strong-coupling regime. Theoretical calculations of the electron-positron pair creation probabilities in low-energy heavy-ion col…
The shielding of the nuclear magnetic moment by the bound electron in hydrogen-like ions is calculated ab initio with inclusion of relativistic, nuclear, and quantum electrodynamics (QED) effects. The QED correction is evaluated to all orders in the nuclear binding strength parameter and, independently, to the first order in the expansion in this parameter. The results obtained lay the basis for the high-precision determination of nuclear magnetic dipole moments from measurements of the g-factor of hydrogen-like ions.
The magnetic dipole (M1) line strength between the fine-structure levels of the ground configurations in B-, F-, Al- and Cl-like ions are calculated for the four elements argon, iron, molybdenum and tungsten. Systematically enlarged multiconfiguration Dirac-Hartree-Fock~(MCDHF) wave functions are employed to account for the interelectronic interaction with the Breit interaction included in first-order perturbation theory. The QED corrections are evaluated to all orders in $αZ$ utilizing an effective potential approach. The calculated line strengths are compared with the results of other theori…
A model-operator approach to fully relativistic calculations of the nuclear recoil effect on energy levels in many-electron atomic systems is worked out. The one-electron part of the model operator for treating the normal mass shift beyond the Breit approximation is represented by a sum of semilocal and nonlocal potentials. The latter ones are constructed by employing the diagonal and off-diagonal matrix elements rigorously evaluated for hydrogenlike ions to first order in the electron-to-nucleus mass ratio. The specific mass shift beyond the lowest-order relativistic approximation has a form …
The relativistic nuclear recoil, higher-order interelectronic-interaction, and screened QED corrections to the transition energies in Li-like ions are evaluated. The calculation of the relativistic recoil effect is performed to all orders in 1/Z. The interelectronic-interaction correction to the transition energies beyond the two-photon exchange level is evaluated to all orders in 1/Z within the Breit approximation. The evaluation is carried out employing the large-scale configuration-interaction Dirac-Fock-Sturm method. The rigorous calculation of the complete gauge invariant sets of the scre…
In [Li ${\it et \, al.}$, Phys. Rev. A ${\bf 98}$, 020502(R) (2018)] it was claimed that the model-potential computations of the Lamb shift on the $^2P_{1/2}-{}^2P_{3/2}$ fine structure in fluorinelike uranium lead to a discrepancy between theory and experiment. Later, it was reported by [Volotka ${\it et \, al.}$, Phys. Rev. A ${\bf 100}$, 010502(R) (2019)] that {\it ab initio} QED calculation, including the first-order one-electron QED contributions and the related effects of two-electron screening, yields the result which restores the agreement between theory and experiment and strongly dis…
A rigorous evaluation of the two-photon exchange corrections to the hyperfine structure in lithiumlike heavy ions is presented. As a result, the theoretical accuracy of the specific difference between the hyperfine splitting values of H- and Li-like Bi ions is significantly improved. This opens a possibility for the stringent test of the many-electron QED effects on a few percent level in the strongest electromagnetic field presently available in experiments.
High-precision QED calculations of the ground-state ionization energies are performed for all boronlike ions with the nuclear charge numbers in the range $16 \leqslant Z\leqslant 96$. The rigorous QED calculations are performed within the extended Furry picture and include all many-electron QED effects up to the second order of the perturbation theory. The contributions of the third- and higher-order electron-correlation effects are accounted for within the Breit approximation. The nuclear recoil and nuclear polarization effects are taken into account as well. In comparison with the previous e…
Formulas and expectation values which are need to determine the lowest-order QED corrections ($\sim α^3$) and corresponding recoil (or finite mass) corrections in the two-electron helium-like ions are presented. Other important properties of the two-electron ions are also determined to high accuracy, including the expectation values of the qausi-singular Vinti operator and $\langle r^{-2}_{eN} \rangle$ and $\langle r^{-2}_{ee} \rangle$ expectation values. Elastic scattering of fast electrons by the two-electron ions in the Born approximation is considered. Interpolation formulas are derived fo…
An overview is presented of laser spectroscopy experiments with cold, trapped, highly-charged ions, which will be performed at the HITRAP facility at GSI in Darmstadt (Germany). These high-resolution measurements of ground state hyperfine splittings will be three orders of magnitude more precise than previous measurements. Moreover, from a comparison of measurements of the hyperfine splittings in hydrogen- and lithium-like ions of the same isotope, QED effects at high electromagnetic fields can be determined within a few percent. Several candidate ions suited for these laser spectroscopy studi…
The paper presents the current status of the theory of bound-electron g factor in highly charged ions. The calculations of the relativistic, QED, nuclear recoil, nuclear structure, and interelectronic-interaction corrections to the g factor are reviewed. Special attention is paid to tests of QED effects at strong coupling regime and determinations of the fundamental constants.
Dielectronic recombination (DR) of few-electron ions has evolved into a sensitive spectroscopic tool for highly charged ions. This is due to technological advances in electron-beam preparation and ion-beam cooling techniques at heavy-ion storage rings. Recent experiments prove unambiguously that DR collision spectroscopy has become sensitive to 2nd order QED and to nuclear effects. This review discusses the most recent developments in high-resolution spectroscopy of low-energy DR resonances, experimental studies of KLL DR of very heavy hydrogenlike ions, isotope shift measurements of DR resona…
Comparison of precision frequency measurements to quantum electrodynamics (QED) predictions for Rydberg states of hydrogen-like ions can yield information on values of fundamental constants and test theory. With the results of a calculation of a key QED contribution reported here, the uncertainty in the theory of the energy levels is reduced to a level where such a comparison can yield an improved value of the Rydberg constant.
We report an ab initio calculation of the shielding of the nuclear magnetic moment by the bound electron in hydrogen-like ions. This investigation takes into account several effects that have not been calculated before (electron self-energy, vacuum polarization, nuclear magnetization distribution), thus bringing the theory to the point where further progress is impeded by the uncertainty due to nuclear-structure effects. The QED corrections are calculated to all orders in the nuclear binding strength parameter and, independently, to the leading order in the expansion in this parameter. The res…
The quantum electrodynamical theory of the two-photon transitions in hydrogenlike ions is presented. The emission probability for 2s1/2 -> 2E1+1s1/2 transitions is calculated and compared to the results of the previous calculations. The emission probabilities 2p12 -> E1E2+1s1/2 and 2p1/2 -> E1M1+1s1/2 are also calculated for the nuclear charge Z values 1-100. This is the first calculation of the two latter probabilities. The results are given in two different gauges.
The electron-electron interaction correction of first order in $1/Z$ to the one-electron part of the nuclear recoil effect on binding energies in atoms and ions is considered within the framework of the rigorous QED approach. The calculations to all orders in $αZ$ are performed for the $1s^2$ state in heliumlike ions and the $1s^2 2s$ and $1s^2 2p_{1/2}$ states in lithiumlike ions in the range $Z=5$--$100$. The results obtained are compared with the Breit-approximation values. The performed calculations complete a systematic treatment of the QED nuclear recoil effect up to the first order in $…
Theoretical calculations of the interelectronic-interaction and QED corrections to the $g$ factor of the ground state of boronlike ions are presented. The first-order interelectronic-interaction and the self-energy corrections are evaluated within the rigorous QED approach in the effective screening potential. The second-order interelectronic interaction is considered within the Breit approximation. The nuclear recoil effect is also taken into account. The results for the ground-state $g$ factor of boronlike ions in the range $Z$=10--20 are presented and compared to the previous calculations.
Theoretical \emph{g}-factor calculations for the first excited \exst state of boronlike ions in the range $Z$=10--20 are presented and compared to the previously published values. The first-order interelectronic-interaction contribution is evaluated within the rigorous QED approach in the effective screening potential. The second-order contribution is considered within the Breit approximation. The QED and nuclear recoil corrections are also taken into account.
Accurate QED calculations of the interelectron interaction corrections for the $(1s2p)2 {}^1 P_1$, $(1s2p)2 {}^3 P_1$ two-electron configurations for ions with nuclear charge numbers $10\le Z \le 92$ are performed within the line profile approach. Total energies of these configurations are evaluated. Employing the fully relativistic treatment based on the {$j$--$j$} coupling scheme these energy levels become quasi-degenerate in the region $Z\le 40$. To treat such states within the framework of QED we utilize the line profile approach. The calculations are performed within the Coulomb gauge.
The present status of quantum electrodynamics (QED) theory of heavy few-electron ions is reviewed. The theoretical results are compared with available experimental data. A special attention is focused on tests of QED at strong fields and on determination of the fundamental constants. A recent progress on calculations of the QED corrections to the parity nonconserving 6s-7s transition amplitude in neutral Cs is also discussed.
Recent progress in quantum electrodynamics (QED) calculations of highly charged ions is reviewed. The theoretical predictions for the binding energies, the hyperfine splittings, and the g factors are presented and compared with available experimental data. Special attention is paid to tests of bound-state QED at strong field regime. Future prospects for tests of QED at the strongest electric and magnetic fields as well as for determination of the fine structure constant and the nuclear magnetic moments with heavy ions are discussed.
The hyperfine structure (HFS) of a bound electron is modified by the self-interaction of the electron with its own radiation field. This effect is known as the self-energy correction. In this work, we discuss the evaluation of higher-order self-energy corrections to the HFS of bound P states. These are expressed in a semi-analytic expansion involving powers of Zalpha and ln(Zalpha), where Z is the nuclear charge number and alpha is the fine-structure constant. We find that the correction of relative order alpha (Zalpha)^2 involves only a single logarithm ln(Zalpha) for P_1/2 states [but no ter…
The hyperfine splitting of the ground state of selected B-like ions within the range of nuclear charge numbers $Z=49-83$ is investigated in detail. The rigorous QED approach together with the large-scale configuration-interaction Dirac-Fock-Sturm method are employed for the evaluation of the interelectronic-interaction contributions of first and higher orders in $1/Z$. The screened QED corrections are evaluated to all orders in $αZ$ by using an effective potential. The influence of nuclear magnetization distribution is taken into account within the single-particle nuclear model.
Hyperfine induced $1s2s ^1S_0 \to 1s^2 ^1S_0$ M1 transition probabilities of He-like ions have been calculated from relativistic configuration interaction wavefunctions including the frequency independent Breit interaction and QED effects. Present results for {$^{151}$}Eu and {$^{155}$}Gd are in good agreement with previous calculations [Phys. Rev. A {\bf 63}, 054105 (2001)]. Electronic data are given in terms of a general scaling law in $Z$ that, given isotopic nuclear spin and magnetic moment, allows hyperfine induced decay rates to be estimated for any isotope. The results should be helpful…
