# [Transitional strength under plasma] Precise estimations of astrophysically
relevant electromagnetic transitions of Ar7+, Kr7+, Xe7+, and Rn7+ under
plasma atmosphere

Swapan Biswas1, Anal Bhowmik2,3, Arghya Das4, Radha Raman Pal1, Sonjoy
Majumder4
1Department of Physics, Vidyasagar University, Midnapore-721102, West Bengal,
India.
2 Department of Physics, University of Haifa, Haifa 3498838, Israel.
3 Haifa Research Center for Theoretical Physics and Astrophysics, University
of Haifa, Haifa 3498838, Israel.
4Department of Physics, Indian Institute of Technology Kharagpur,
Kharagpur-721302, India

###### Abstract

The growing interest in atomic structures of moderately-stripped alkali-like
ions in diagnostic study and modeling of astrophysical and laboratory plasma
makes an accurate many-body study of atomic properties inevitable. This work
presents transition line parameters in the absence or presence of plasma
atmosphere for astrophysically important candidates, Ar7+, Kr7+, Xe7+, and
Rn7+. We employ relativistic coupled-cluster (RCC) theory, a well-known
correlation exhaustive method. In the case of a plasma environment, we use
Debye Model. Our calculations agree with experiments available in the
literature for ionization potentials, transition strengths of allowed and
forbidden selections, and lifetimes of several low-lying states. The unit
ratios of length and velocity forms of transition matrix elements are the
critical estimation of the accuracy of the transition data presented here,
especially for a few presented first time in the literature. We do compare our
findings with the available recent theoretical results. Our reported data can
be helpful to the astronomer in estimating the density of the plasma
environment around the astronomical objects or in the discovery of
observational spectra corrected by that environment. The present results
should be advantageous in the modeling and diagnostics laboratory plasma,
whereas the calculated ionisation potential depression parameters reveal
important characteristics of atomic structure.

###### keywords:

Atomic data, transition probability, oscillator strength, lifetime, and plasma
density

††journal: Atoms

Atomic spectroscopy for Ar7+, Kr7+, Xe7+ and Rn7+ ions with high accuracy

Plasma screened ionization potential, atomic transition amplitudes and rates

ionisation potential depression parameters

##  1 Introduction

Barlow et al. [1] first observed noble gas molecules in the interstellar
medium. The other detections of noble gas elements, either in diatomic [2, 3,
4, 5] or ionic forms [6] in space at UV and IR spectrum, motivate further
observations of these species in the universe. It is well known that the
atomic and spectroscopic processes are valuable diagnostics for plasma
atmosphere in the laboratory or Astronomy. Noble gas atoms are known to be
chemically inactive and require high energy to ionize. But, once ionized,
their reaction rates are rather fast. Over the years, spectroscopic properties
of ionized noble gas atoms have become popular, and observers have started to
detect them in space [7]. On the other hand, alkali-like ions have emerged as
the standard test beds for detailed investigation of current relativistic
atomic calculations due to their adequately simple but highly correlated
electronic structures [8, 9, 10, 11, 12]. Accurate theoretical and
experimental determinations of the transition line parameters and excited-
state lifetimes of highly stripped ions are collaborative with the astronomer
to investigate dynamics, chemical compositions, opacity, density, and
temperature distributions of the distant galaxy [13], planetary nebulae, and
even entire the interstellar medium [14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24]. Furthermore, one requires the accurate atomic data of different isotopes
of noble gas elements to understand the production of heavy elements in the
stellar medium by radiative r- and s-processes [25, 26]. The data of energy
spectra of moderate to high-stripped ions are required for precise
astrophysical and laboratory plasma modelling. All these physical facts and
figures motivate us to investigate the transition line parameters and
lifetimes of septuple ionized astrophysically pertinent inert gases, such as
Ar7+, Kr7+, Xe7+, and Rn7+.

In the series, Ar7+ is well-studied in literature. Berry et al. [27] observed
74 lines of Ar7+ using the beam-foil technique. In 1982, Striganov and
Odintsova [28] published the observed lines of Ar+ through Ar8+. [29, 30, 31]
applied the multi-configuration Dirac-Fock (MCDF) method to calculate the
autoionization spectrum, energy levels, transition rates, oscillator
strengths, and lifetimes of Ar7+. Saloman [32] identified the energy spectra
of Ar+ to Ar17+, which he studied from the year 2006 to 2009 employing beam
foil Spectroscopy (BFS), electron beam ion trap (EBIT), laser-excited plasmas,
fusion devices, astronomical observations, and ab initio calculations with
quantum electrodynamic corrections.

Similarly, krypton ion spectra were detected in the interstellar medium [33,
34], the galactic disc [35], and the planetary nebulae [36]. Fine structure
intervals, fine structure inversions, and core-polarization study of the Kr7+
ion were performed by different groups [37, 38, 39] including third-order
many-body perturbation theory and Møller-Pleset perturbation theory [40, 41]
for the energy levels.

It is found that Cu I isoelectronic sequence ions are prominent impurities at
high-temperature magnetically confined plasmas [42], and their emission
spectra are observed under the spark sources [43, 44, 45, 46] of the laser-
produced plasma [47, 48] and in the beam-foil excitations [49, 50, 51]. The
abundance of photospheric lines of trans-iron group elements in the emission
spectra of the white dwarfs opens a new way of studying their radiative
transfer mechanism [52]. The presence of the spectral lines of Cu-like ions
motivates more accurate determination of atomic data of the radiative
properties of these ions for modelling the chemical abundances. These studies
are essential for deducing the stellar parameters necessary to investigate the
environmental condition of the white dwarfs. There are studies of electronic
properties for Xe7+ using various many-Body methods [53, 54, 55, 56, 57].
Dimitrijevc et al. [24] identified the importance of Stark broadening at the
spectral lines observed in extremely metal-poor halo PNH4-1 in primordial
supernova [58]. However, we study Ar+7, Kr7+, and Xe7+ here again to mitigate
the lack of all-order many-body calculations or precise experiments and to
estimate their spectroscopic properties under a plasma environment. Recent
past, one of the present authors [59] studied Xe7+ exclusively as a single
valance system without the plasma screening effect.

Unlike other noble gas ions, studies of radon ions are rare. However, there
are a few many-body calculations on Rn+ [60], and Rn2+ [61]. The observation
of several forbidden lines of Kr and Xe ions in the planetary nebula NGC 7027
was reported recent past [62]. For Rn7+, only Migdalek [63] computed a few
energy levels and oscillator strengths of allowed transitions using the Dirac-
Fock method corrected by the core-polarisation effect.

The aim of the paper is to estimate (a) energies of the ground and low-lying
excited states, (b) the oscillator strengths of electromagnetically allowed
transitions, (c) transition probabilities of the forbidden transitions, and
(d) lifetimes for a few excited states of Ar7+, Kr7+, Xe7+, and Rn7+ using the
relativistic coupled-cluster (RCC) method [64, 65, 66]. The accuracies of the
RCC calculations are well-established by our group for different applications
[67, 68, 69, 70, 71, 72, 73, 74, 75, 76]. Our special effort here is to study
the plasma screening effect on the radiative transition parameters. It is
obvious that the nuclear attraction to the bound electrons of atoms or ions
immersed in plasma is screened by the neighbouring ions and the free
electrons. The essential feature to note is that the electron correlation of
atomic systems in this environment is remarkably different from their
corresponding isolated candidate. Therefore, the screening estimations on the
transition parameters play a crucial role in the precise diagnostics of plasma
temperature and density in the emitting region. In the plasma environment, the
ionization potentials decrease gradually with the increasing strength of
plasma screening [77] till they become zero at some critical parameter of
plasma. Beyond these critical values of plasma, the states become a continuum
state. The corresponding ionization potential beyond which instability occurs
is known as ionisation potential depression (IPD) according to the
Stewart–Pyatt (SP) model [78]. Accurate determination of the IPD can infer
much useful information about the plasma atmosphere, such as providing the
proper equation of the state, estimating radiate opacity of stellar plasma,
and internal confinement fusion plasma etc. We have investigated the change in
spectroscopic properties of Ar7+, Kr7+, Xe7+, and Rn7+ in the plasma
environment.

##  2 Theory

Precise generation of wave functions is important for accurately estimating
the atomic properties of few-electron monovalent ions presented in this paper.
Here we employ a non-linear version of the well-known RCC theory, a many-body
approach which exhaustively pools together correlations. Initially, we solve
the Dirac-Coulomb Hamiltonian $H$ satisfying eigenvalue equation
$H|\Phi\rangle=E_{0}|\Phi\rangle$ to generate closed-shell atomic wave
function under the potential of $(N-1)$ electrons where

$H=\sum_{i}\left(c{\bf\alpha}_{i}\cdot{\bf
p}_{i}+(\beta_{i}-1)c^{2}+V_{\mathrm{nuc}}({\bf
r}_{i})+\sum_{j<i}\left(\frac{1}{{\bf r}_{ij}}\right)\right).$

Here standard notations are used for all the variables. A single valence
reference state for the RCC calculation is generated by adding a single
electron in the v-th orbital following Koopman’s theorem [79]. In RCC
formalism, the single valence correlated state $|\Psi_{\text{v}}\rangle$ is
connected with single valence reference state $|\Phi_{\text{v}}\rangle$ as

$|\Psi_{\text{v}}\rangle=e^{T}\\{1+S_{\text{v}}\\}|\Phi_{\text{v}}\rangle,\hskip
5.69046pt\text{where}\
|\Phi_{\text{v}}\rangle=a^{\dagger}_{\text{v}}|\Phi\rangle.$ (1)

The operator $T$ deals with the excitations from core orbitals and can
generate core-excited configurations from closed-shell Dirac-Fock state
$|\Phi\rangle$. Whereas $S_{\text{v}}$ excites at least one electron from the
valence orbital, giving rise to valence and core-valence excited
configurations [64]. The operator $S_{\text{v}}$ can yield the valence and
core-valence excited configurations with respect to the open-shell Dirac–Fock
state $|\Phi_{\text{v}}\rangle$ [69]. Here we generate single- and double-
excited correlated configurations from Eq. (1). The amplitudes of these
excitations are solved from the energy eigenvalue equations of the closed-
shell and open-shell systems, which are
$He^{T}|\Phi\rangle=Ee^{T}|\Phi\rangle$ and
$H_{\text{v}}e^{T}|\Phi_{\text{v}}\rangle=E_{\text{v}}e^{T}|\Phi_{\text{v}}\rangle$,
respectively [80]. In the present method, these amplitudes are solved
following the Jacobi iteration scheme, which is considered all-ordered. The
initial guesses of the single- and double-excitation amplitudes are made
consistent with the first order of the perturbation theory [81]. In the
present version of RCC theory, we also consider some important triple
excitations and hence the abbreviation is used RCCSD(T).

The matrix elements of an arbitrary operator can be written as

$\displaystyle O_{ki}$ $\displaystyle=$
$\displaystyle\frac{\langle{\Psi_{k}|\hat{O}|\Psi_{i}}\rangle}{\sqrt{\langle{\Psi_{k}|\Psi_{k}}\rangle{\langle{\Psi_{i}|\Psi_{i}}\rangle}}}$
(2) $\displaystyle=$
$\displaystyle\frac{\langle{\Phi_{k}|\\{1+S^{\dagger}_{k}\\}e^{T^{\dagger}}{\hat{O}}e^{T}\\{1+S_{i}\\}|\Phi_{i}}\rangle}{\sqrt{{\langle{\Phi_{k}|\\{1+S^{\dagger}_{k}\\}e^{T^{\dagger}}e^{T}\\{1+S_{k}\\}|\Phi_{k}}\rangle\langle\Phi_{i}|\\{1+S^{\dagger}_{i}\\}e^{T^{\dagger}}e^{T}\\{1+S_{i}\\}|\Phi_{i}}\rangle}}.$

The detailed derivations and explanations of the matrix elements associated
with electric dipole ($E_{1}$), electric quadrupole ($E_{2}$), and magnetic
dipole ($M_{1}$) transitions can be found in the literature[82]. Emission
transition probabilities (s-1) for the $E_{1}$, $E_{2}$, and $M_{1}$ from
$|\Psi_{k}\rangle$ to $|\Psi_{i}\rangle$ state are [83]

$\hskip 28.45274ptA^{E_{1}}_{k\rightarrow i}=\frac{2.0261\times
10^{-6}}{\lambda^{3}(2J_{k}+1)}S^{E_{1}},$ (3) $\hskip
28.45274ptA^{E_{2}}_{k\rightarrow i}=\frac{1.12\times
10^{-22}}{\lambda^{5}(2J_{k}+1)}S^{E_{2}},$ (4) $and\hskip
28.45274ptA^{M_{1}}_{k\rightarrow i}=\frac{2.6971\times
10^{-11}}{\lambda^{3}(2J_{k}+1)}S^{M_{1}}.$ (5)

Where, $\lambda$ is in cm and $S$ is the square of the transition matrix
elements of $O$ (corresponding transition operator) in atomic unit of
$e^{2}a_{0}^{2}$($e$ is the charge of an electron and $a_{0}$ is the Bohr
radius). The oscillator strength for the $E1$ transition is related to the
corresponding transition probability (s-1) with following equation [84]

$f^{osci}_{k\rightarrow i}=1.4992\times 10^{-16}A_{k\rightarrow
i}\frac{g_{k}}{g_{i}}\lambda^{2},$ (6)

where $g_{k}$ and $g_{i}$ are the degeneracies of the final and initial
states, respectively. The lifetime of the $k$-th state is calculated by
considering all transition probabilities to the lower energy states ($i$-th)
and is given by

$\tau_{k}=\frac{1}{\sum_{i}{A_{k\rightarrow i}}}.$ (7)

In order to incorporate the plasma screening effect on the atomic
spectroscopic properties, the Dirac-Coulomb potential takes the form as

$H^{D}_{\mathrm{eff}}=H+V^{D}_{\mathrm{eff}}({\bf r}_{i})$ (8)

Here $V^{D}_{\mathrm{eff}}({\bf r}_{i})$ is the effective potential of the
nucleus on the $i$-th electron due to the presence of the plasma environment.
The Debye-Hückle potential is considered to examine the effect of screening of
nuclear coulomb potential due to the presence of ions and free electrons in
plasma [85, 86]. In the case of a weekly interacting plasma medium, the
effective potential experienced by the $i$-th electron is given as

$V^{D}_{\mathrm{eff}}({r}_{i})=\frac{Ze^{-\mu{r}_{i}}}{{r_{i}}}.$ (9)

Where Z is the nuclear charge and $\mu$ is the Debye screening parameter,
which is related to the ion density $n_{ion}$ and plasma temperature T through
the following relation

$\mu=\left[{\frac{4\pi(1+Z)n_{ion}}{K_{B}T}}\right]^{2},$ (10)

where $k_{B}$ is the Boltzmann constant. Therefore, a given value of $\mu$
represents a range of plasma conditions with different ion densities and
temperatures.

##  3 Results and discussions

The single-particle Dirac-Fock (DF) wavefunctions are the building blocks of
the RCC calculations yielding the many-electron correlation energies and
correlated wavefunctions. We calculate the bound Dirac-Hartree Fock orbitals
as accurately as possible using a sophisticated numerical approach, GRASP92
[87]. Further, we apply the basis-set expansion technique [88] in the self-
consistent field approach to obtain Gaussian-type DF orbital (GTO) used in the
RCC calculations. The radial part of each basis function has two parameters,
$\alpha_{0}$, and $\beta$, as exponents [89] to be optimized. The parameters
are required to optimize due to the finite size of the basis set. The exponent
parameters are optimized compared to the DF bound orbitals obtained from
GRASP92, discussed in detail in our old papers [12, 66]. In the basis
optimization method, we consider 33, 30, 28, 25, 21, and 20 basis functions
for $s$, $p$, $d$, $f$, $g$, and $h$ symmetries, respectively. This basis set
is considered for all the ions. But the choice of the active orbitals in the
RCC calculation relies on the convergence of the correlation contribution to
the closed-shell energy with the increasing number of the orbitals [90, 66].
Therefore, the active orbitals for the converged correlation contribution to
the closed-shell energy are found to be distinct for different ions
investigated in this work.

In this article, we calculate the ionization potential of Ar7+, Kr7+, Xe7+,
and Rn7+ using the RCC method and compare them in Table 1 with results
published in National Institute of Standards and Technology (NIST) [91]
wherever available. NIST estimations are considered to have the best accuracy.
We find that our calculated ground state energies of Ar7+, Kr7+, and Xe7+ are
in excellent agreement with NIST results, and deviations are estimated to be
$-0.01\%$, $0.45\%$, and $-0.03\%$, respectively. Table I presents the
ionisation potential of the low-lying excited states of these ions with
average deviations around $-0.08\%$, $0.42\%$, and $0.30\%$, respectively. In
these cases, the maximum difference is $-0.23\%$ occurred for $5p_{3/2}$ state
of Ar7+, $0.60\%$ for $5g_{7/2,9/2}$ of Kr7+, and $1.2\%$ for $6d_{3/2,5/2}$
of Xe7+.

Our calculated energies agree well with estimations by Fischer et al. [31],
who computed energy levels of Ar7+ using the core polarization effect on the
Dirac Hartree-Fock (CP-DHF) theory. Cheng and Kim [37] tabulated the energy
levels of Kr7+ from the relativistic Hartree-Fock (RHF) calculations. As
expected, our RCC calculated results are found to be better in agreement with
the NIST values. For Rn7+, we have not found any experimental measurement in
the literature nor NIST compiled values. Only one theoretical calculation
based on the CP-DHF method by [63] is available with an average deviation of
IP is $0.66\%$ from our calculations.

The percentage of electron correlation correction, i.e., $\frac{(\text{RCC-
DF})\times 100\%}{\text{DF}}$ in IP of the ground state is monotonically
increases from Ar7+ to Rn7+ with the values $0.39\%$, $0.52\%$, $1.81\%$, and
$1.87\%$, respectively.

Figure 1: The plasma contribution in IP of low-lying septuple ionized atoms: %
variation of IP with plasma screening strength. Energy levels are indicated as
$nL(2J+1)$. Results are calculated from
$\frac{\text{IP}_{\mu=0}-\text{IP}_{\mu>0}}{\text{IP}_{\mu=0}}\times 100$.

Now we investigate the impact of the plasma screening potential on the energy
levels of the considered ions. Table I shows that IP monotonically decreases
with the increase of the $\mu$ value. The bold values for each ion in the
table represent the limiting case beyond which the system becomes unbound.
Figure 1 presents the plasma screening contribution in IP for a few low-lying
states, such as ground state $S_{1/2}$, excited $P_{1/2,3/2}$, and
$D_{3/2,5/2}$ states of Ar7+, Kr7+, Xe7+, and Ra7+ ions. The panels of the
figure show the plasma screening contribution increases from the ground to
higher excited states, as the latter states are less bound by the Coulomb
attraction. For Xe7+ and Rn7+, we could plot the effect up to a certain value
of $\mu$ as most of the states become continuum states beyond that. We observe
that the plasma screening effect is practically strong for fine structure
levels for Ar7+ and Kr7+ ions and weak for Xe7+, and Ra7+ ions.

Figure 2: The contribution of plasma screening in E1 matrix elements of Ar7+
of transitions ${}^{2}S_{1/2}\rightarrow$ ${}^{3}P_{1/2,3/2}$. The figures
display % variation of the E1 matrix elements vs. plasma screening strength.

We present the electric dipole matrix elements for the ions in the plasma
medium in Table 2. The table also displays our computed DF values of the
matrix elements to reveal the correlation contributions. The separate
presentation of the DF and RCC values over the span of the plasma screening
parameter, $\mu$, in the table is an intentional move. Here we want to
highlight that the plasma screening impacts the DF and the RCC correlation
parts differently. In the case of Ar7+, the average changes in matrix element
due to the increasing values of plasma screening parameter is less than 1$\%$.
However, Figure. 2 shows significant changes for $3s_{1/2}\rightarrow
4p_{1/2,3/2}$ and $3s_{1/2}\rightarrow 5p_{1/2,3/2}$ for finite values of
$\mu$, especially for $\mu=0.05$. This is true for any $n\ ^{2}S_{1/2}$
$\rightarrow$ $n^{\prime}\ {}^{2}P_{1/2,3/2}$ of this ionic series. However,
apart from such a few transitions, the plasma effects lie between 1 $\%$ to 2
$\%$ for most of the other transitions in the series. Table 2 shows that the
average correlations (for the non-plasma environment) in the transition
amplitudes for Ar7+, Kr7+, Xe7+, and Rn7+ are $0.3858\%$, $4.0831\%$,
$7.6074\%$, $11.6379\%$ apart from $4d_{3/2}$$\rightarrow$ $5f_{5/2}$
transition where the correlation is $67.11\%$.

For observational astronomy and laboratory spectroscopy, we present a
tabulation of a list of our computed oscillator strengths ($f_{\text{RCC}}$)
of $E1$ transitions along with their previously reported theoretical and
experimental values in Table 3. Most of the transitions fall in the far and
mid-UV regions of the electromagnetic spectrum. $f_{\text{RCC}}$ is calculated
using the RCC transition amplitudes in length gauge [92] form presented in
Table 2 and the NIST [91] wavelengths, wherever available (in the case of
Rn7+, our computed RCC wavelengths are used). The ratios between the length
and velocity gauge amplitudes of our calculated $E1$-transitions are also
displayed in table to show the accuracy of our RCC wavefunctions, which is
close to unity for all the cases confirming the accuracy of our correlated
atomic wavefunctions. However, we find that the ratio is almost two for
${}^{2}F\rightarrow^{2}D$ transitions of Xe7+ and Rn7+. Point to note that
this disagreement is also available in the ratio at the DF level, where we
also employed the numerically accurate GRASP92 Code [87]. One of the reasons
for this outcome is due to the strong correlation effect from d- and f-states,
so as in similar alkali systems [93]. In addition, the consistency of accuracy
of our calculations can be drawn from the approximate consistency of the
ratios, 3:2:1 among the transition matrix estimations of
${}^{2}P_{3/2}\rightarrow$ ${}^{2}D_{5/2}$: ${}^{2}P_{1/2}\rightarrow$
${}^{2}D_{3/2}$: ${}^{2}P_{3/2}\rightarrow$ ${}^{2}D_{3/2}$ [94].

The $E1$ Oscillator strengths for Ar7+ are well studied in literature [95, 96,
97, 98, 99, 100, 101, 102, 31, 103], and they are in good agreement with our
estimations based on correlation exhaustive RCC method. Table 3 shows that the
same is true for Kr7+. For Ar7+, our calculations for $f_{\text{RCC}}$ are
almost as accurate as those found from other sophisticated theoretical
approaches, such as the relativistic many-body perturbation theory [100], and
for the most latest theoretical results employing the multiconfigurational
Dirac-Hartree-Fock approximation [31]. To the best of our knowledge, in the
case of Kr7+, we could not find any correlation-exhaustive many-body result of
$E1$ transition. There have been experiments, mostly using beam-foil
experiments, on the $E1$ transition from the ground state to the first excited
states of Ar7+ [95], and Kr7+ [50, 105, 51, 104, 49]. Our estimations are well
within the uncertainty limit of the latest experiments. We also see that some
of the old calculations either underestimate or overestimate the oscillator
strength values due to non-appropriate considerations of correlations and
relativistic effects.

Over the last two decades, a few of the low-lying $E1$ transitions of Xe7+ are
estimated using core-polarization or model potential as an effective means of
correlation calculations, apart from third order perturbation calculations
[55]. It is known that our RCC method is an all-order extension of many-body
perturbation theory [81]. Further, it includes most of the correlation
features, including core correlation, pair correlation, and higher order
correlation effects [69] for a given level of excitation. For Rn7+, we find
only one theoretical endeavour [63] using model potential. The presence of
$d$\- and $f$-orbitals for Xe7+ and Rn7+ ions in the core makes these two ions
highly correlated. Because of the large atomic number and highly stripped
configurations, we expect a strong relativistic effect in their spectroscopy.
Therefore, it is necessary to do relativistic ab initio correlation exhaustive
calculations for them and our computations exactly mitigate that requirement.
In Table 3, we also present the effect of the plasma atmosphere on the
oscillator strengths for the observational and laboratory spectroscopy. The
oscillator strengths for $\mu>0$ are calculated using the $E1$ matrix elements
presented in Table 2 and the corresponding transition wavelengths computed
from RCC theory. The table exhibits the significant effects of plasma
screening parameters on the oscillator strengths.

Table 4 and Table 5 present transition probabilities for the relatively strong
forbidden transitions governed by the electric quadruple ($E2$) and the
magnetic dipole ($M1$) moments. Similar to oscillator strength in Table 3,
here we use NIST wavelengths for the transition probability wherever
available. For Rn7+, we use the RCC calculated transition wavelengths. We do
not find any estimation of the forbidden transitions in the literature of this
ionic series which fall either in the ultraviolet or in the near infra-red
regions of the electromagnetic spectrum. Transitions falling in the
ultraviolet region are significant in astronomical observation and plasma
research [106, 107, 108, 109]. While the infra-red transitions have
applications in astronomy using space-based telescopes ([110]). Moreover,
infrared spectroscopy provides major information about cool astronomical
regions in space, like, interstellar medium [111] and planetary nebulae [112].
It is found that $5p_{1/2}$ $\rightarrow$ $4f_{5/2}$ of Kr7+ and $5p_{1/2}$
$\rightarrow$ $5p_{3/2}$ of Xe7+ emit orange and green lights, respectively,
which can be used in laser spectroscopy.

It is found from Table 5 that the $M1$ transition probability is stronger
among fine-structure levels than the $E2$ transition. Table 4 reveals that the
maximum $M1$ transition probability, $A_{RCC}^{M1}$, occurs for the transition
$3p_{1/2}$ $\rightarrow$ $3p_{3/2}$ of Ar7+, $4p_{1/2}$ $\rightarrow$
$4p_{3/2}$ of Kr7+, $4f_{7/2}$ $\rightarrow$ $5f_{7/2}$ of Xe7+ and $6p_{1/2}$
$\rightarrow$ $6p_{3/2}$ of Rn7+, and they have values 0.17949, 8.4000, 158.33
and 1137.7, respectively. Moreover, our estimations of $M1$ transition
probability for the $4f_{5/2}$ $\rightarrow$ $4f_{7/2}$ transition of Xe7+ has
excellent agreement with the calculations using the multi-configuration Dirac-
Hartree-Fock method [113, 114]. Table 6 presents lifetime of the low-lying
states of this series. We compare our results with other experimental and
theoretical estimations wherever available and find good agreement with the
recent endeavors. We provide lifetimes of many excited states calculated the
first time in literature to our knowledge.

The comparisons of our computed results with the other estimations obtained
from correlation exhaustive ab initio theoretical computations or precise
experiments are the one of the measures of accuracy of our calculations.
Further, the differences between the calculated matrix elements in the length
and velocity gauge forms are the characteristic of the preciseness of our
calculations. A recent literature [93] also claims that the difference in
length gauge and velocity gauge is a measure of accuracy. Another factor of
accuracy in ab initio calculations arises from the DF wavefunctions used for
correlation calculations. Also, we should consider the uncertainty that arises
from the other correlation terms (which we did not consider in this article)
and the quantum electrodynamics effect, which is at most $2\%$ in total.
Taking all these into account, the maximum calculated uncertainties for Ar7+,
Kr7+, Xe7+, and Rn7+ are about $5.6\%$, $5.37\%$, $5\%$ and $5.01\%$,
respectively.

To understand the critical effect of the plasma atmosphere on the ionization
potential of the ions, we highlight the IPD values in bold fonts in Table 1
for different values of screening length, $\mu$. These IPD values reflect
critical electron or plasma density at a particular temperature for the ionic
system when a few of the bound ionic states are elevated to continuum states.

Table 1 also reveals that the fine structure splittings (FSS) are suppressed
as the screening strength increases from $\mu$=0 to 1.0. For example, the
energy differences between $4p_{3/2}$ and $4p_{1/2}$ of Kr7+ are evaluated as
9774 a.u, 9732 a.u, 9608 a.u., and 9407 a.u. for $\mu$ = 0, 0.025, 0.05, 0.075
a.u., respectively. This phenomenon is consistent with earlier calculations
for sodium D line [115], and hydrogen-like atoms [116]. The suppression of the
transition rate among the fine-structure levels are mainly arising from the
energy quench.

From Figure 3, we pictorially estimate the critical values of plasma screening
strength ($\mu_{c}$) where the ionisation potential becomes zero for
particular atomic state. We also tabulate these value in Table 7. The critical
screening strength is essential in photo-ionisation cross section, which
increases with increasing $\mu$ till $\mu=\mu_{c}$. This increment is obvious
due to the decrease of bound state energy leading to the increase of radial
expansion of bound state wavefunction [116]. This phase shift of bound state
to continuum state is induced by the plasma atmosphere, and ionization
threshold decreases with the Debye screening length ($\mu^{-1}$). In terms of
photo-ionization cross-section [117], plasma decreases the threshold cross-
section, and the discrete bound wavefunctions become diffused. Therefore,
critical screening strength plays a important role in atomic structure.
However, we have not found any spectroscopic data in literature for these ions
in plasma medium to compare with our results.

##  4 conclusion

The continuous progress in astrophysical and astronomical observations demands
accurate theoretical transition data in a realistic environment. In many
cases, the experiment is difficult to extract the data used to estimate the
abundance of the ions in the stellar chemical composition. Here, the highly
correlated relativistic coupled-cluster theory is applied to precisely
determine the excitation energies of a few low-lying states of astrophysically
relevant such as Ar7+, Kr7+ and Xe7+ and Rn7+. Furthermore, we calculate
various properties of allowed and forbidden transitions, such as transition
probabilities, oscillator strengths and lifetimes, and compare them with
previously reported data in the literature. We found an overall good agreement
between our results with the other theoretical and experimental results.
Moreover, the concurrence between the length and velocity gauge allowed
transition amplitudes signifies the exact calculations of our correlated
wavefunctions. We found that most of the transitions shown here fall in the
ultraviolet region of the electromagnetic spectrum, useful for astrophysical
plasma research and telescope-based astronomy. A few transitions, such as
$4d_{3/2}-4f_{5/2}$, $4d_{5/2}-4f_{5/2}$ and $4d_{5/2}-4f_{7/2}$ of Ar7+,
$5p_{1/2}-4f_{5/2}$ of Kr7+ and $5p_{1/2}-5p_{3/2}$ of Xe7+ emit the visible
light which can have application in laser spectroscopy. Our presented
transition line parameters of Rn7+ may help the astronomer identify the ion’s
unknown lines. To the best of our knowledge, some of the oscillator strengths
of allowed transitions and most of the transition rates of the forbidden
transitions are reported here for the first time in the literature.

The main focus of this paper is to evaluate the above spectroscopic properties
under a realistic astronomical atmosphere. We showed the variation of our
results for different values of Debye screening lengths and ionisation
potential depression values for each atomic state useful for atomic structure
characterization.

Table 1: Comparison of our RCC ionization potential (in $cm^{-1}$) with NIST data and our estimations of plasma screening effect on them. Estimations for $5g$ states of Rn7+ were not available in the literature (a) [63]. Plasma screening strength ($\mu$) is in a.u. unit. Energy levels are indicated as $nL(2J+1)$. The bold values indicate that beyond which the system becomes unbound.  state | NIST |  $\mu$=0 |  $\mu$=0.025 |  $\mu$=0.05 |  $\mu$=0.075 |  $\mu$=0.1 |
---|---|---|---|---|---|---|---
Ar7+ |  |  |  |  |  |  |
3s2 | 1157056 | 1157201 | 1059866 | 965330 | 873513 | 784345 |
3p2 | 1016961 | 1016995 | 919704 | 825299 | 733697 | 644828 |
3p4 | 1014248 | 1014184 | 916898 | 822505 | 730924 | 642084 |
3d4 | 824447 | 824210 | 726923 | 632529 | 540956 | 452148 |
3d6 | 824302 | 824027 | 726741 | 632349 | 540779 | 451974 |
4s2 | 581098 | 581069 | 485070 | 394397 | 308793 | 228054 |
4p2 | 528815 | 528528 | 432658 | 342356 | 257356 | 177460 |
4p4 | 527813 | 527428 | 431565 | 341287 | 256323 | 176473 |
4d4 | 459524 | 459475 | 363709 | 273714 | 189240 | 110134 |
4d6 | 459435 | 459386 | 363620 | 273627 | 189156 | 110055 |
4f6 | 440204 | 440190 | 344137 | 253339 | 167618 | 86891 |
4f8 | 440181 | 440159 | 344107 | 253309 | 167590 | 86865 |
5s2 | 349750 | 349752 | 255504 | 169770 | 91961 | 21742 |
5p2 | 324795 | 323193 | 229236 | 144332 | 67862 | 557 |
5p4 | 324307 | 322543 | 228611 | 143769 | 67376 |  |
5d4 | 291782 | 291699 | 197867 | 113352 | 37637 |  |
5d6 | 291778 | 291652 | 197818 | 113302 | 37591 |  |
5f6 | 281727 | 281736 | 187657 | 102487 | 25846 |  |
5f8 | 281707 | 281720 | 187642 | 102473 | 25833 |  |
5g8 | 281051 | 281015 | 186389 | 99684 | 20685 |  |
5g10 | 281037 | 281015 | 186380 | 99676 | 20678 |  |
Kr7+ |  |  |  |  |  |  |
4s2 | 1014665 | 1010099 | 815902 | 628237 | 446944 | 271892 |
4p2 | 870969 | 867027 | 673080 | 486149 | 306059 | 132689 |
4p4 | 861189 | 857253 | 663348 | 476541 | 296652 | 123557 |
4d4 | 640619 | 636965 | 443698 | 258762 | 81891 | 86678 |
4d6 | 639288 | 635514 | 442257 | 257352 | 80623 |  |
5s2 | 524578 | 523198 | 331842 | 152413 | 15482 |  |
5p2 | 467984 | 465448 | 274592 | 96629 |  |  |
5p4 | 464221 | 461406 | 270630 | 92891 |  |  |
4f6 | 451900 | 450180 | 257831 | 75655 |  |  |
4f8 | 451934 | 450186 | 257837 | 75660 |  |  |
5d4 | 373589 | 371922 | 182111 | 7230 |  |  |
5d6 | 373048 | 371273 | 181476 | 6635 |  |  |
6s2 | 322147 | 321635 | 133957 | 34847 |  |  |
5f6 | 289666 | 288637 | 100173 |  |  |  |
5f8 | 289661 | 288634 | 100170 |  |  |  |
5g8 | 281574 | 281387 | 92121 |  |  |  |
5g10 | 281572 | 281379 | 92114 |  |  |  |
Xe7+ |  |  |  |  |  |  |
5s2 | 854769 | 854995 | 564858 | 286731 | 20272 |  |
5p2 | 738302 | 737059 | 447391 | 170643 | 93515 |  |
5p4 | 719717 | 718263 | 428759 | 152487 |  |  |
4f6 | 589608 | 588730 | 297599 | 16588 |  |  |
4f8 | 589058 | 588088 | 296970 | 16085 |  |  |
5d4 | 544881 | 543506 | 255203 | 17518 |  |  |
5d6 | 541953 | 540549 | 252295 |  |  |  |
6s2 | 459272 | 455364 | 170155 |  |  |  |
6p2 | 411391 | 406318 | 121973 |  |  |  |
6p4 | 403996 | 398801 | 114717 |  |  |  |
5f6 | 357190 | 356376 | 70683 |  |  |  |
5f8 | 356751 | 355922 | 70245 |  |  |  |
6d4 | 327344 | 323450 | 41320 |  |  |  |
6d6 | 325975 | 322176 | 40102 |  |  |  |
7s2 | 289473 | 276902 | 451 |  |  |  |
5g8 | 284501 | 283609 |  |  |  |  |
5g10 | 284501 | 283617 |  |  |  |  |
Rn7+ | (a) |  |  |  |  |  |
6s2 | 834624 | 839362 | 377923 | 63552 |  |  |
6p2 | 712821 | 718015 | 257667 |  |  |  |
6p4 | 661071 | 665627 | 205616 |  |  |  |
5f6 | 536092 | 531079 | 69545 |  |  |  |
5f8 | 534720 | 529291 | 68106 |  |  |  |
6d4 | 498636 | 501235 | 43424 |  |  |  |
6d6 | 491240 | 493699 | 36116 |  |  |  |
7s2 | 446847 | 445577 | 8041 |  |  |  |
7p2 | 397883 | 398535 |  |  |  |  |
7p4 | 377435 | 377332 |  |  |  |  |
5g8 |  | 287997 |  |  |  |  |
5g10 |  | 288252 |  |  |  |  |

Table 2:  Our DF and RCC matrix element (a.u.), in length gauge, of electric dipole ($E1$) transitions in plasma medium. Energy levels are indicated as $nL(2J+1)$.  | $\mu=0$ | $\mu=0.025$ | $\mu=0.05$ | $\mu=0.075$ | $\mu=0.1$
---|---|---|---|---|---
Ar7+ |  |  |  |  |  |  |  |  |  |
transitions | DF | RCC | DF | RCC | DF | RCC | DF | RCC | DF | RCC
3s2$\rightarrow$3p2 | 0.9617 | 0.9341 | 0.9623 | 0.9346 | 0.9638 | 0.9390 | 0.9663 | 0.9405 | 0.9697 | 0.9436
3s2$\rightarrow$3p4 | 1.3619 | 1.3228 | 1.3626 | 1.3236 | 1.3648 | 1.3299 | 1.3683 | 1.3317 | 1.3732 | 1.3355
3s2$\rightarrow$4p2 | 0.2002 | 0.2093 | 0.1999 | 0.2089 | 0.1989 | 0.2210 | 0.1973 | 0.2181 | 0.1950 | 0.2164
3s2$\rightarrow$4p4 | 0.2770 | 0.2898 | 0.2765 | 0.2893 | 0.2751 | 0.3059 | 0.2728 | 0.3028 | 0.2698 | 0.3012
3s2$\rightarrow$5p2 | 0.1011 | 0.1062 | 0.1006 | 0.1057 | 0.0992 | 0.0995 | 0.0972 | 0.1105 | 0.0944 | 0.1007
3s2$\rightarrow$5p4 | 0.1397 | 0.1470 | 0.1390 | 0.1463 | 0.1372 | 0.1215 | 0.1345 | 0.1541 | 0.1307 | 0.1528
4s2$\rightarrow$4p2 | 1.8958 | 1.8813 | 1.8996 | 1.8851 | 1.9105 | 1.9059 | 1.9284 | 1.8987 | 1.9537 | 1.9012
4s2$\rightarrow$4p4 | 2.6827 | 2.6622 | 2.6879 | 2.6675 | 2.7033 | 2.7006 | 2.7286 | 2.6857 | 2.7644 | 2.6844
3p2$\rightarrow$4s2 | 0.3604 | 0.3646 | 0.3608 | 0.3649 | 0.3619 | 0.3419 | 0.3637 | 0.3437 | 0.3661 | 0.3432
3p4$\rightarrow$4s2 | 0.5172 | 0.5231 | 0.5178 | 0.5236 | 0.5193 | 0.4898 | 0.5219 | 0.4943 | 0.5253 | 0.4952
3p2$\rightarrow$5s2 | 0.1237 | 0.1269 | 0.1240 | 0.1271 | 0.1246 | 0.1275 | 0.1252 | 0.1144 | 0.1253 | 0.1274
3p4$\rightarrow$5s2 | 0.1771 | 0.1816 | 0.1775 | 0.1819 | 0.1783 | 0.2021 | 0.1792 | 0.1653 | 0.1793 | 0.1659
3p2$\rightarrow$3d4 | 1.3534 | 1.3174 | 1.3547 | 1.3186 | 1.3583 | 1.3207 | 1.3642 | 1.3218 | 1.3725 | 1.3305
3p4$\rightarrow$3d4 | 0.6060 | 0.5899 | 0.6066 | 0.5905 | 0.6082 | 0.5916 | 0.6109 | 0.5918 | 0.6146 | 0.5956
3p4$\rightarrow$3d6 | 1.8184 | 1.7700 | 1.8201 | 1.7720 | 1.8250 | 1.7736 | 1.8330 | 1.7750 | 1.8442 | 1.7795
3p2$\rightarrow$4d4 | 0.3857 | 0.3957 | 0.3842 | 0.3942 | 0.3800 | 0.3925 | 0.3730 | 0.3926 | 0.3629 | 0.3551
3p4$\rightarrow$4d4 | 0.1756 | 0.1801 | 0.1750 | 0.1794 | 0.1731 | 0.1770 | 0.1699 | 0.1791 | 0.1654 | 0.1645
3p4$\rightarrow$4d6 | 0.5262 | 0.5394 | 0.5242 | 0.5375 | 0.5185 | 0.5321 | 0.5091 | 0.5372 | 0.4956 | 0.5357
4p2$\rightarrow$4d4 | 2.7891 | 2.7687 | 2.7966 | 2.7761 | 2.8181 | 2.7737 | 2.8536 | 2.7737 | 2.9038 | 2.8032
4p4$\rightarrow$4d4 | 1.2496 | 1.2405 | 1.2529 | 1.2438 | 1.2626 | 1.2470 | 1.2785 | 1.2420 | 1.3010 | 1.2498
4p4$\rightarrow$4d6 | 3.7490 | 3.7217 | 3.7590 | 3.7316 | 3.7880 | 3.7880 | 3.8357 | 3.7255 | 3.9033 | 3.7279
3d4$\rightarrow$4p2 | 0.5411 | 0.5434 | 0.5426 | 0.5449 | 0.5470 | 0.5616 | 0.5542 | 0.5407 | 0.5644 | 0.5701
3d4$\rightarrow$4p4 | 0.2395 | 0.2406 | 0.2402 | 0.2412 | 0.2487 | 0.2450 | 0.2454 | 0.2395 | 0.2499 | 0.2530
3d6$\rightarrow$4p4 | 0.7193 | 0.7225 | 0.7213 | 0.7244 | 0.7271 | 0.7193 | 0.7368 | 0.7201 | 0.7504 | 0.7184
3d4$\rightarrow$4f6 | 1.7707 | 1.7378 | 1.7705 | 1.7377 | 1.7702 | 1.7384 | 1.7691 | 1.7401 | 1.7663 | 1.7778
3d6$\rightarrow$4f6 | 0.4734 | 0.4671 | 0.4734 | 0.4646 | 0.4733 | 0.4649 | 0.4730 | 0.4653 | 0.4722 | 0.4658
3d6$\rightarrow$4f8 | 2.1172 | 2.078 | 2.1171 | 2.0779 | 2.1166 | 2.0793 | 2.1154 | 2.0768 | 2.1121 | 2.0799
4d4$\rightarrow$4f6 | 3.1546 | 3.1495 | 3.1686 | 3.1634 | 3.2100 | 3.1508 | 3.2796 | 3.1504 | 3.3822 | 3.1414
4d6$\rightarrow$4f6 | 0.8430 | 0.8416 | 0.8468 | 0.8454 | 0.8578 | 0.8420 | 0.8764 | 0.8419 | 0.9087 | 0.8423
4d6$\rightarrow$4f8 | 3.7706 | 3.7644 | 3.7874 | 3.7812 | 3.7652 | 3.7644 | 3.9201 | 3.7778 | 4.0427 | 3.7790
Kr7+ |  |  |  |  |  |  |  |  |  |
| DF | RCC | DF | RCC | DF | RCC | DF | RCC | DF | RCC
4s2$\rightarrow$4p2 | 1.1348 | 1.0794 | 1.1363 | 1.1017 | 1.1405 | 1.1223 | 1.1475 | 1.1088 | 1.1575 | 1.1245
4s2$\rightarrow$4p4 | 1.6095 | 1.5314 | 1.6115 | 1.5632 | 1.6175 | 1.5900 | 1.6275 | 1.5724 | 1.6416 | 1.6085
4s2$\rightarrow$5p2 | 0.1488 | 0.1655 | 0.1479 | 0.1584 | 0.1451 | 0.1537 |  |  |  |
4s2$\rightarrow$5p4 | 0.1812 | 0.2057 | 0.1799 | 0.1944 | 0.1761 | 0.1839 |  |  |  |
4p2$\rightarrow$4d4 | 1.7539 | 1.6812 | 1.7569 | 1.7412 | 1.7657 | 1.7506 | 1.7804 | 1.7422 |  |
4p4$\rightarrow$4d4 | 0.7940 | 0.7616 | 0.7954 | 0.7734 | 0.7995 | 0.7836 | 0.8064 | 0.7771 |  |
4p4$\rightarrow$4d6 | 2.3818 | 2.2847 | 2.3860 | 2.3218 | 2.3984 | 2.3511 | 2.4190 | 2.3301 |  |
4p2$\rightarrow$5s2 | 0.4884 | 0.4934 | 0.4894 | 0.4892 | 0.4925 | 0.4838 |  |  |  |
4p4$\rightarrow$5s2 | 0.7316 | 0.7379 | 0.7331 | 0.7325 | 0.7377 | 0.7272 |  |  |  |
4p2$\rightarrow$6s2 | 0.1655 | 0.1686 | 0.1654 | 0.1664 | 0.1646 | 0.1624 |  |  |  |
4p4$\rightarrow$6s2 | 0.2447 | 0.2486 | 0.2445 | 0.2460 | 0.2427 | 0.2424 |  |  |  |
4d4$\rightarrow$4f6 | 2.8243 | 2.7466 | 2.8340 | 2.7759 | 2.8623 | 2.7985 |  |  |  |
4d6$\rightarrow$4f6 | 0.7565 | 0.7358 | 0.7591 | 0.7436 | 0.7668 | 0.7499 |  |  |  |
4d4$\rightarrow$5p2 | 1.1740 | 1.1630 | 1.1809 | 1.1776 | 1.2015 | 1.1742 |  |  |  |
4d4$\rightarrow$5p4 | 0.5090 | 0.5124 | 0.5120 | 0.5107 | 0.5212 | 0.5168 |  |  |  |
4d6$\rightarrow$5p4 | 1.5415 | 1.5522 | 1.5507 | 1.5464 | 1.5786 | 1.5435 |  |  |  |
4d6$\rightarrow$4f8 | 3.3836 | 3.2916 | 3.3953 | 3.3257 | 3.4297 | 3.3529 |  |  |  |
4d4$\rightarrow$5f6 | 0.0596 | 0.0427 | 0.0582 | 0.0445 |  |  |  |  |  |
4d6$\rightarrow$5f6 | 0.0133 | 0.0091 | 0.0175 | 0.0089 |  |  |  |  |  |
4d6$\rightarrow$5f8 | 0.0592 | 0.0418 | 0.0779 | 0.0394 |  |  |  |  |  |
4f6$\rightarrow$5g8 | 3.8576 | 3.7993 | 3.8688 | 3.8112 |  |  |  |  |  |
4f8$\rightarrow$5g8 | 0.7422 | 0.7311 | 0.7444 | 0.7340 |  |  |  |  |  |
4f8$\rightarrow$5g10 | 4.3914 | 4.3256 | 4.4042 | 4.3380 |  |  |  |  |  |
4f6$\rightarrow$5d4 | 2.0421 | 2.0387 | 2.0728 | 2.0318 | 2.1683 | 2.0704 |  |  |  |
4f6$\rightarrow$5d6 | 0.5430 | 0.5420 | 0.5512 | 0.5401 | 0.5766 | 0.5498 |  |  |  |
4f8$\rightarrow$5d6 | 2.4274 | 2.4232 | 2.4641 | 2.4137 | 2.5780 | 2.4567 |  |  |  |
5p2$\rightarrow$5d4 | 3.1683 | 3.1275 | 3.1867 | 3.1373 | 3.2401 | 3.1494 |  |  |  |
5p4$\rightarrow$5d4 | 1.4407 | 1.4226 | 1.4489 | 1.4271 | 1.4728 | 1.4325 |  |  |  |
5p4$\rightarrow$5d6 | 4.3172 | 4.2629 | 4.3417 | 4.2781 | 4.4131 | 4.2912 |  |  |  |
5p2$\rightarrow$6s2 | 1.0220 | 1.0239 | 1.0277 | 1.0215 |  |  |  |  |  |
5p4$\rightarrow$6s2 | 1.5278 | 1.5310 | 1.5347 | 1.5276 |  |  |  |  |  |
|  |  |  |  |  |  |  |  |  |
Xe7+ | DF | RCC | DF | RCC | DF | RCC | DF | RCC | DF | RCC
5s2$\rightarrow$5p2 | 1.3758 | 1.1736 | 1.3791 | 1.1769 | 1.3889 | 1.1866 |  |  |  |
5s2$\rightarrow$5p4 | 1.9516 | 1.6705 | 1.9562 | 1.6752 | 1.9700 | 1.6890 |  |  |  |
5p2$\rightarrow$5d4 | 2.1135 | 1.8629 | 2.1197 | 1.8697 |  |  |  |  |  |
5p4$\rightarrow$5d4 | 0.9768 | 0.8646 | 0.9799 | 0.8679 |  |  |  |  |  |
5p4$\rightarrow$5d6 | 2.9252 | 2.5911 | 2.9344 | 2.6011 |  |  |  |  |  |
5p2$\rightarrow$6s2 | 0.5866 | 0.6026 | 0.5893 | 0.6046 |  |  |  |  |  |
5p4$\rightarrow$6s2 | 0.9449 | 0.9582 | 0.9494 | 0.9619 |  |  |  |  |  |
4f6$\rightarrow$5d4 | 1.7952 | 1.5958 | 1.8177 | 1.6155 |  |  |  |  |  |
4f6$\rightarrow$5d6 | 0.4761 | 0.4237 | 0.4822 | 0.4291 |  |  |  |  |  |
4f8$\rightarrow$5d6 | 2.1392 | 1.9075 | 2.1663 | 1.9313 |  |  |  |  |  |
4f6$\rightarrow$5g8 | 1.8079 | 1.5869 | 1.7885 | 1.5731 |  |  |  |  |  |
4f8$\rightarrow$5g8 | 0.3495 | 0.3073 | 0.3457 | 0.3046 |  |  |  |  |  |
4f8$\rightarrow$5g10 | 2.0684 | 1.819 | 2.0462 | 1.8032 |  |  |  |  |  |
5d4$\rightarrow$5f6 | 2.9732 | 2.8328 | 2.9687 | 2.8347 |  |  |  |  |  |
5d6$\rightarrow$5f6 | 0.8050 | 0.7670 | 0.8040 | 0.7676 |  |  |  |  |  |
5d6$\rightarrow$5f8 | 3.5922 | 3.4233 | 3.5876 | 3.4252 |  |  |  |  |  |
6s2$\rightarrow$6p2 | 2.7436 | 2.5326 | 2.5640 | 2.5931 |  |  |  |  |  |
6s2$\rightarrow$6p4 | 3.3346 | 3.5052 | 3.6133 | 3.6569 |  |  |  |  |  |
6p2$\rightarrow$6d4 | 3.7253 | 3.5930 | 3.7520 | 3.6089 |  |  |  |  |  |
6p4$\rightarrow$6d4 | 1.7277 | 1.6725 | 1.7399 | 1.6762 |  |  |  |  |  |
6p4$\rightarrow$6d6 | 5.1635 | 4.9966 | 5.1994 | 5.0147 |  |  |  |  |  |
5f6$\rightarrow$5g8 | 5.5926 | 5.3626 |  |  |  |  |  |  |  |
5f6$\rightarrow$5g8 | 1.0772 | 1.0330 |  |  |  |  |  |  |  |
5f8$\rightarrow$5g10 | 6.3725 | 6.1114 |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |  |
Rn7+ | DF | RCC | DF | RCC | DF | RCC | DF | RCC | DF | RCC
6s2$\rightarrow$6p2 | 1.4159 | 1.1344 | 1.4210 | 1.1414 |  |  |  |  |  |
6s2$\rightarrow$6p4 | 1.9902 | 1.6147 | 1.9969 | 1.6264 |  |  |  |  |  |
6p2$\rightarrow$7s2 | 0.6184 | 0.6284 | 0.6189 | 0.6678 |  |  |  |  |  |
6p4$\rightarrow$7s2 | 1.2512 | 1.2162 | 1.2657 | 1.2024 |  |  |  |  |  |
6p2$\rightarrow$6d4 | 2.0887 | 1.7562 | 2.0978 | 1.7629 |  |  |  |  |  |
6p4$\rightarrow$6d4 | 1.0493 | 0.8953 | 1.0556 | 0.9008 |  |  |  |  |  |
6p4$\rightarrow$6d6 | 3.1200 | 2.6596 | 3.1380 | 2.6691 |  |  |  |  |  |
5f6$\rightarrow$6d4 | 2.3362 | 2.0900 | 2.3794 | 2.0911 |  |  |  |  |  |
5f6$\rightarrow$6d6 | 0.6118 | 0.5500 | 0.6236 | 0.5478 |  |  |  |  |  |
5f8$\rightarrow$6d6 | 2.7709 | 2.4962 | 2.8234 | 2.4931 |  |  |  |  |  |
5f6$\rightarrow$5g8 | 2.7042 | 2.3829 |  |  |  |  |  |  |  |
5f8$\rightarrow$5g8 | 0.5263 | 0.4650 |  |  |  |  |  |  |  |
5f8$\rightarrow$5g10 | 3.1174 | 2.7455 |  |  |  |  |  |  |  |
7s2$\rightarrow$7p2 | 2.5750 | 2.4195 |  |  |  |  |  |  |  |
7s2$\rightarrow$7p4 | 3.5648 | 3.3636 |  |  |  |  |  |  |  |
6d4$\rightarrow$7p2 | 2.0098 | 1.9415 |  |  |  |  |  |  |  |
6d4$\rightarrow$7p4 | 0.7525 | 0.7381 |  |  |  |  |  |  |  |
6d6$\rightarrow$7p4 | 2.4112 | 2.3331 |  |  |  |  |  |  |  |

Table 3:  Our RCC oscillator strengths of electric dipole transitions. We compare our results with other estimations available in recent literature (experimental endeavours are highlighted with ”exp” subscript). Our results (”RCC”) are obtained using the RCC calculations, except NIST wavelengths are used for $\mu=0$ wherever available. Transition states are designated with the outermost orbital followed by $(2J+1)$ of the state. Values at the parenthesis in the second column are ratios between length- and velocity-gauged dipole matrix elements.  |  $\mu$=0 |  $\mu$=0.025 |  $\mu$=0.050 |  $\mu$=0.075 |  $\mu$=0.1 |
---|---|---|---|---|---|---
Transition | $RCC$ | Other |  |  |  |  |
Ar7+ |  |  |  |  |  |  |
3s2 $\rightarrow$ 3p2 | 0.1857(0.99) | 0.183(4)${}^{a_{1}}_{exp}$, $0.188^{b_{1}}$, | 0.1859 | 0.1875 | 0.1878 | 0.1887 |
|  |  $0.193^{c_{1}}$, $0.186^{d_{1}}$, $0.1864^{e_{1}}$, |  |  |  |  |
|  |  $0.185^{f_{1},i_{1}}$, $0.187^{g_{1}}$, $0.196^{h_{1}}$ |  |  |  |  |
3s2 $\rightarrow$ 3p4 | 0.3795(0.99) | 0.398(10)${}^{a_{1}}_{exp}$, 0.385${}^{b_{1}}$, 0.394${}^{c_{1}}$, | 0.3804 | 0.3837 | 0.3840 | 0.3854 |
|  | 0.381${}^{d_{1},g_{1}}$, $0.3811^{e_{1}}$, $0.379^{f_{1}}$, |  |  |  |  |
|  |  $0.401^{h_{1}}$, $0.378^{i_{1}}$ |  |  |  |  |
3s2 $\rightarrow$ 4p2 | 0.0418(1.00) |  $0.0414^{b_{1}}$, $0.0401^{c_{1}}$, $0.0376^{d_{1}}$ | 0.0416 | 0.0462 | 0.0445 | 0.0432 |
|  |  $0.0415^{e_{1}}$, $0.0385^{h_{1}}$, $0.0432^{i_{1}}$ |  |  |  |  |
3s2 $\rightarrow$ 4p4 | 0.0803(1.00) |  $0.0829^{b_{1}}$, $0.0766^{c_{1}}$, $0.0751^{d_{1}}$, | 0.0793 | 0.0877 | 0.0859 | 0.0838 |
|  |  $0.0798^{e_{1}}$, $0.0739^{h_{1}}$, $0.0836^{i_{1}}$ |  |  |  |  |
3s2 $\rightarrow$ 5p2 | 0.0143(0.98) | $0.0146^{b_{1},e_{1}}$ | 0.0141 | 0.0123 | 0.0149 | 0.0121 |
3s2 $\rightarrow$ 5p4 | 0.0271(1.00) |  $0.0292^{b_{1}}$, $0.0284^{e_{1}}$ | 0.0270 | 0.0184 | 0.0291 | 0.0278 |
4s2 $\rightarrow$ 4p2 | 0.2810(1.01) |  $0.2821^{e_{1}}$, $0.2824^{i_{1}}$ | 0.2829 | 0.2871 | 0.2816 | 0.2777 |
4s2 $\rightarrow$ 4p4 | 0.5736(1.01) |  $0.5755^{e_{1}}$, $0.5759^{i_{1}}$ | 0.5782 | 0.5883 | 0.5748 | 0.5645 |
3p2 $\rightarrow$ 4s2 | 0.0880 (1.00) |  $0.0876^{e_{1}}$, $0.0884^{i_{1}}$ | 0.0879 | 0.0765 | 0.0763 | 0.0760 |
3p4 $\rightarrow$ 4s2 | 0.0900(1.01) |  $0.0896^{e_{1}}$, $0.08947^{i_{1}}$ | 0.0899 | 0.0780 | 0.0783 | 0.0783 |
3p2 $\rightarrow$ 5s2 | 0.0163(1.01) | $0.0161^{e_{1}}$ | 0.0163 | 0.0162 | 0.0128 | 0.0154 |
3p4 $\rightarrow$ 5s2 | 0.0166(1.01) | $0.0164^{e_{1}}$ | 0.0166 | 0.0202 | 0.0133 | 0.0130 |
3p2 $\rightarrow$ 3d4 | 0.5074(0.96) |  $0.532^{c_{1}}$, $0.5097^{e_{1}}$, $0.5074^{g_{1}}$ | 0.5091 | 0.5107 | 0.5115 | 0.5180 |
|  |  $0.508^{i_{1}}$, $0.47^{j_{1}}$ |  |  |  |  |
3p4 $\rightarrow$ 3d4 | 0.0502(0.96) |  $0.0527^{c_{1}}$, $0.0504^{e_{1}}$, $0.0501^{g_{1}}$ | 0.0503 | 0.0505 | 0.0505 | 0.0512 |
|  |  $0.0502^{i_{1}}$, $0.046^{j_{1}}$ |  |  |  |  |
3p4 $\rightarrow$ 3d6 | 0.4556(0.96) |  $0.475^{c_{1}}$, $0.4539^{e_{1}}$, $0.4517^{g_{1}}$ | 0.4534 | 0.4542 | 0.4549 | 0.4571 |
|  |  $0.452^{i_{1}}$, $0.42^{j_{1}}$ |  |  |  |  |
3p2 $\rightarrow$ 4d4 | 0.1326(1.05) |  $0.1310^{e_{1}}$, $0.1344^{i_{1}}$ | 0.1312 | 0.1291 | 0.1275 | 0.1024 |
3p4 $\rightarrow$ 4d4 | 0.0137(1.03) |  $0.0135^{e_{1}}$, $0.0136^{i_{1}}$ | 0.0135 | 0.0131 | 0.0132 | 0.0109 |
3p4 $\rightarrow$ 4d6 | 0.1226(1.03) |  $0.1212^{e_{1}}$, $0.1228^{i_{1}}$ | 0.1214 | 0.1180 | 0.1187 | 0.1159 |
4p2 $\rightarrow$ 4d4 | 0.8067(0.95) | $0.8085^{i_{1}}$ | 0.8070 | 0.8020 | 0.7930 | 0.8035 |
4p4 $\rightarrow$ 4d4 | 0.0798(0.98) | $0.0796^{i_{1}}$ | 0.0797 | 0.0798 | 0.0786 | 0.0787 |
4p4 $\rightarrow$ 4d6 | 0.7192(0.98) | $0.7177^{i_{1}}$ | 0.7185 | 0.7215 | 0.7079 | 0.7009 |
3d4 $\rightarrow$ 4p2 | 0.0663(0.98) |  $0.0657^{e_{1}}$, $0.0663^{i_{1}}$ | 0.0663 | 0.0695 | 0.0630 | 0.0678 |
3d4 $\rightarrow$ 4p4 | 0.0130(1.00) |  $0.0129^{e_{1}}$, $0.0131^{i_{1}}$ | 0.0130 | 0.0133 | 0.0124 | 0.0134 |
3d6 $\rightarrow$ 4p4 | 0.0784(0.97) |  $0.0778^{e_{1}}$, $0.0786^{i_{1}}$ | 0.0784 | 0.0762 | 0.0747 | 0.0720 |
3d4 $\rightarrow$ 4f6 | 0.8812(1.00) | $0.8776^{i_{1}}$ | 0.8777 | 0.8702 | 0.8584 | 0.8762 |
3d6 $\rightarrow$ 4f6 | 0.0424(1.00) | $0.0418^{i_{1}}$ | 0.0418 | 0.0415 | 0.0409 | 0.0401 |
3d6 $\rightarrow$ 4f8 | 0.8397(1.00) | $0.8360^{i_{1}}$ | 0.8364 | 0.8297 | 0.8149 | 0.7796 |
Kr7+ |  |  |  |  |  |  |
4s2 $\longrightarrow$ 4p2 | 0.2543(1.00) | 0.25(1)${}^{a_{2},f_{2}}_{exp}$, 0.24(2)${}^{b_{2}}_{exp}$, 0.246${}^{c_{2}}$, | 0.2633 | 0.2718 | 0.2705 | 0.2673 |
|  | 0.2781${}^{d_{2}}$, 0.278${}^{e_{2}}$, $0.28^{g_{2}}$, |  |  |  |  |
|  | 0.2578${}^{h_{2}}$, 0.220${}^{i_{2}}$, 0.2448${}^{j_{2}}$ |  |  |  |  |
4s2 $\longrightarrow$ 4p4 | 0.5466(0.97) | 0.53$(2)^{a_{2}}_{exp}$, 0.47$(4)^{b_{2}}_{exp}$, 0.526${}^{c_{2}}$, | 0.5661 | 0.5825 | 0.5833 | 0.5829 |
|  | 0.5965${}^{d_{2}}$, 0.60${}^{e_{2}}$, 0.59$(9)^{f_{2}}_{exp}$, |  |  |  |  |
|  |  $0.59^{g_{2}}$, $0.554^{h_{2}}$, $0.473^{i_{2}}$ |  |  |  |  |
|  | 0.5265${}^{j_{2}}$ |  |  |  |  |
4s2 $\longrightarrow$ 5p2 | 0.0227(1.01) | $0.0176^{d_{2}}$ | 0.0206 | 0.0191 |  |  |
4s2 $\longrightarrow$ 5p4 | 0.0354(1.00) | $0.0265^{d_{2}}$ | 0.0313 | 0.0275 |  |  |
4p2 $\rightarrow$ 5s2 | 0.1281(1.02) | $0.1212^{d_{2}}$ | 0.1261 | 0.1187 |  |  |
4p4 $\rightarrow$ 5s2 | 0.1392(1.02) | $0.1321^{d_{2}}$ | 0.1351 | 0.1302 |  |  |
4p2 $\rightarrow$ 4d4 | 0.9888(1.00) | $1.057^{d_{2}}$ | 1.0562 | 1.0583 | 1.1068 |  |
4p4 $\rightarrow$ 4d4 | 0.0972(1.00) | $0.1038^{d_{2}}$ | 0.0998 | 0.1015 | 0.1045 |  |
4p4 $\rightarrow$ 4d6 | 0.8796(1.00) | $0.9395^{d_{2}}$ | 0.9051 | 0.9200 | 0.9480 |  |
4p2 $\rightarrow$ 6s2 | 0.0237(1.06) | $0.0208^{d_{2}}$ | 0.0227 | 0.0181 |  |  |
4p4 $\rightarrow$ 6s2 | 0.0253(1.07) | $0.0220^{d_{2}}$ | 0.0243 | 0.0197 |  |  |
4d4 $\rightarrow$ 4f6 | 1.0811(0.98) | $1.126^{d_{2}}$ | 1.0876 | 1.0890 |  |  |
4d6 $\rightarrow$ 4f6 | 0.0514(0.98) | $0.0535^{d_{2}}$ | 0.0508 | 0.0509 |  |  |
4d6 $\rightarrow$ 4f8 | 1.0277(0.98) | $1.0700^{d_{2}}$ | 1.0326 | 1.0340 |  |  |
4d4 $\rightarrow$ 5p2 | 0.1828(1.00) | $0.1784^{d_{2}}$ | 0.1781 | 0.1697 |  |  |
4d4 $\rightarrow$ 5p4 | 0.0352(1.03) | $0.0344^{d_{2}}$ | 0.0345 | 0.0329 |  |  |
4d6 $\rightarrow$ 5p4 | 0.2135(1.00) | $0.2088^{d_{2}}$ | 0.2078 | 0.1984 |  |  |
4d4 $\rightarrow$ 5f6 | 0.0001(1.09) | $0.0001^{d_{2}}$ | 0.0005 |  |  |  |
4f6 $\rightarrow$ 5d4 | 0.1647(0.98) | $0.1638^{d_{2}}$ | 0.1582 | 0.1485 |  |  |
4f6 $\rightarrow$ 5d6 | 0.0117(0.98) | $0.0117^{d_{2}}$ | 0.0113 | 0.0106 |  |  |
4f8 $\rightarrow$ 5d6 | 0.1759(0.98) | $0.1750^{d_{2}}$ | 0.1689 | 0.1582 |  |  |
4f6 $\rightarrow$ 5g8 | 1.2447(1.00) | $1.261^{d_{2}}$ | 1.2185 |  |  |  |
4f8 $\rightarrow$ 5g8 | 0.0346(1.00) | $0.0350^{d_{2}}$ | 0.0339 |  |  |  |
4f8 $\rightarrow$ 5g10 | 1.2103(1.00) | $1.226^{d_{2}}$ | 1.1841 |  |  |  |
5p2 $\rightarrow$ 5d4 | 1.4023(1.00) | $1.401^{d_{2}}$ | 1.3825 | 1.3467 |  |  |
5p4 $\rightarrow$ 5d4 | 0.1393(1.01) | $0.1383^{d_{2}}$ | 0.1369 | 0.1335 |  |  |
5p4 $\rightarrow$ 5d6 | 1.2582(1.01) | $1.249^{d_{2}}$ | 1.2391 | 1.2062 |  |  |
4d6 $\rightarrow$ 5f6 | 0.00002(0.28) | $0.00003^{d_{2}}$ | 0.00001 |  |  |  |
4d6 $\rightarrow$ 5f8 | 0.0003(0.29) | $0.0007^{d_{2}}$ | 0.0003 |  |  |  |
5p2 $\rightarrow$ 6s2 | 0.2322(1.07) | $0.2023^{d_{2}}$ | 0.2229 |  |  |  |
5p4 $\rightarrow$ 6s2 | 0.2529(1.08) | $0.2179^{d_{2}}$ | 0.2422 |  |  |  |
Xe7+ |  |  |  |  |  |  |
5s2 $\rightarrow$ 5p2 | 0.2436(1.01) |  $0.294^{a_{3}}$, $0.234^{b_{3}}$, $0.242^{c_{3}}$ | 0.2471 | 0.2482 |  |  |
|  |  $0.253^{d_{3}}$, $0.237^{e_{3}}$, $0.237^{f_{3}}$ |  |  |  |  |
|  |  $0.232^{g_{3}}$, $0.223^{h_{3}}$, $0.232^{i_{3}}$ |  |  |  |  |
5s2 $\rightarrow$ 5p4 | 0.5724(1.01) |  $0.697^{a_{3}}$, $0.550^{b_{3}}$, $0.569^{c_{3}}$ | 0.5801 | 0.5816 |  |  |
|  |  $0.596^{d_{3}}$, $0.560^{e_{3}}$, $0.563^{f_{3}}$, |  |  |  |  |
|  |  $0.543^{g_{3}}$, $0.522^{h_{3}}$, $0.537^{i_{3}}$ |  |  |  |  |
5p2 $\rightarrow$5d4 | 1.0195(1.02) |  $1.189^{a_{3}}$, $0.977^{b_{3}}$, $1.020^{c_{3}}$ | 1.0204 |  |  |  |
|  |  $1.025^{d_{3}}$, $1.003^{e_{3}}$, $1.000^{f_{3}}$ |  |  |  |  |
|  | $1.057^{i}$ |  |  |  |  |
5p4 $\rightarrow$5d4 | 0.0992(1.02) |  $0.095^{b_{3}}$, $0.089^{c_{3}}$, $0.099^{d_{3}}$ | 0.0993 |  |  |  |
|  |  $0.097^{e_{3}}$, $0.097^{f_{3}}$, $0.095^{i_{3}}$ |  |  |  |  |
5p4 $\rightarrow$5d6 | 0.9064(1.02) |  $0.523^{a_{3}}$, $0.868^{b_{3}}$, $0.904^{c_{3}}$ | 0.9066 |  |  |  |
|  |  $0.907^{d_{3}}$, $0.889^{e_{3}}$, $0.886^{f_{3}}$ |  |  |  |  |
|  | $0.875^{i_{3}}$ |  |  |  |  |
5p2 $\rightarrow$ 6s2 | 0.1539(1.05) |  $0.160^{c_{3}}$, $0.156^{d_{3}}$, $0.155^{e_{3}}$ | 0.1539 |  |  |  |
|  |  $0.153^{f_{3}}$, $0.199^{i_{3}}$ |  |  |  |  |
5p4 $\rightarrow$ 6s2 | 0.1816(1.05) |  $0.188^{c_{3}}$, $0.186^{d_{3}}$, $0.184^{e_{3}}$ | 0.1817 |  |  |  |
|  |  $0.182^{f_{3}}$, $0.186^{i_{3}}$ |  |  |  |  |
4f6 $\rightarrow$5d4 | 0.0577(2.35) |  $0.130^{a_{3}}$, $0.058^{b_{3}}$, $0.060^{i_{3}}$ | 0.0560 |  |  |  |
4f6 $\rightarrow$5d6 | 0.0043(2.12) | $0.0044^{b_{3}}$ | 0.0042 |  |  |  |
4f8 $\rightarrow$5d6 | 0.0651(2.19) |  $0.075^{a_{3}}$, $0.065^{b_{3}}$, $0.068^{i_{3}}$ | 0.0633 |  |  |  |
4f6 $\rightarrow$ 5g8 | 0.3890(1.02) |  $0.3646^{b_{3}}$, $0.354^{i_{3}}$ |  |  |  |  |
4f8 $\rightarrow$ 5g8 | 0.0109(1.02) | $0.0102^{b_{3}}$ |  |  |  |  |
4f8 $\rightarrow$ 5g10 | 0.3826(1.02) |  $0.3595^{b_{3}}$, $0.343^{i_{3}}$ |  |  |  |  |
5d4 $\rightarrow$5f6 | 1.1437(1.04) | $1.099^{i_{3}}$ | 1.1259 |  |  |  |
5d6 $\rightarrow$5f6 | 0.0550(1.04) | $0.052^{i_{3}}$ | 0.0813 |  |  |  |
5d6 $\rightarrow$5f8 | 1.0981(1.04) | $1.032^{i_{3}}$ | 1.0812 |  |  |  |
5f6 $\rightarrow$ 5g8 | 1.0583(0.99) | $1.071^{i_{3}}$ |  |  |  |  |
5f8 $\rightarrow$ 5g8 | 0.0293(0.99) | $0.030^{i_{3}}$ |  |  |  |  |
5f8 $\rightarrow$ 5g10 | 1.0246(0.99) | $1.035^{i_{3}}$ |  |  |  |  |
6s2 $\rightarrow$ 6p2 | 0.4778(1.13) |  | 0.4921 |  |  |  |
6s2 $\rightarrow$ 6p4 | 1.0555(1.13) |  | 1.1260 |  |  |  |
6p2 $\rightarrow$ 6d4 | 1.6248(1.11) |  | 1.5953 |  |  |  |
6p4 $\rightarrow$ 6d6 | 1.4527(1.11) |  | 1.4249 |  |  |  |
6p4 $\rightarrow$ 6d4 | 0.1601(1.11) |  | 0.1566 |  |  |  |
Rn7+ |  |  |  |  |  |  |
6s2 $\rightarrow$ 6p2 | 0.2372(1.00) | $0.234^{a_{4}}$ | 0.2379 |  |  |  |
6s2 $\rightarrow$ 6p4 | 0.6880(1.02) | $0.689^{a_{4}}$ | 0.6922 |  |  |  |
6p2 $\rightarrow$ 7s2 | 0.1634(1.06) | $0.173^{a_{4}}$ |  |  |  |  |
6p4 $\rightarrow$ 7s2 | 0.2472(1.07) | $0.259^{a_{4}}$ |  |  |  |  |
6p2 $\rightarrow$6d4 | 1.0154(1.01) | $1.059^{a_{4}}$ | 1.0113 |  |  |  |
6p4 $\rightarrow$6d4 | 0.1001(1.01) | $0.103^{a_{4}}$ | 0.0999 |  |  |  |
6p4 $\rightarrow$6d6 | 0.9234(1.01) | $0.956^{a_{4}}$ | 0.9170 |  |  |  |
5f6 $\rightarrow$ 6d4 | 0.0660(2.13) | $0.0753^{a_{4}}$ | 0.0578 |  |  |  |
5f8 $\rightarrow$ 6d6 | 0.0842(1.83) | $0.0981^{a_{4}}$ | 0.0755 |  |  |  |
5f6 $\rightarrow$ 6d6 | 0.0057(1.73) | $0.0069^{a_{4}}$ | 0.0051 |  |  |  |
5f6 $\rightarrow$ 5g8 | 0.6982(1.00) |  |  |  |  |  |
7s2 $\rightarrow$ 7p2 | 0.4182(1.11) | $0.414^{a_{4}}$ |  |  |  |  |
7s2 $\rightarrow$ 7p4 | 1.1727(1.12) | $1.126^{a_{4}}$ |  |  |  |  |
6d4 $\rightarrow$ 7p2 | 0.2940(1.05) | $0.292^{a_{4}}$ |  |  |  |  |
6d4 $\rightarrow$ 7p4 | 0.0513(1.04) | $0.052^{a_{4}}$ |  |  |  |  |
6d6 $\rightarrow$ 7p4 | 0.3207(1.05) | $0.33^{a_{4}}$ |  |  |  |  |

${a_{1}}$$\implies$Beam-foil technique [95]

$b_{1}$ $\implies$Calculations are based on high level methods such as the
R-matrix method and asymptotic techniques developed by seaton [96]

$c_{1}$$\implies$ Single Configuration Interaction Hartree-Fock method using a
pseudopotential [97]

$d_{1}$ $\implies$Nonrelativistic WKB approaches(Klein-Gordon dipole matrix)
[98]

$e_{1}$ $\implies$Single configuration Dirac-Fock method [99]

$f_{1}$ $\implies$Relativistic many-body perturbation theory [100]

${g_{1}}$ $\implies$Realistic model potential [101]

$h_{1}$ $\implies$Relativistic Hartree Fock method [102]

$i_{1}$ $\implies$multiconfiguration Dirac Hartee-Fock approximation. [31]

$j_{1}$ $\implies$Relativistic effective orbital quantum number [103]

$a_{2}$ $\implies$Jointly analyzed decay curves: Beam-foil [104]

$b_{2}$ $\implies$Multiexponential fits: Beam-foil [104]

$c_{2}$ $\implies$Non-Relativistic Multi Configuration Hartree-Fock
approximation [118]

$d_{2}$ $\implies$ Relativistic Hartree-Fock [37]

$e_{2}$ $\implies$Hartree-Fock oscillator strength using the Dirac correction
factor [119]

$f_{2}$ $\implies$Arbitrarily Normalized Decay curve method for cascade-
correction in beam-foil [105]

$g_{2}$ $\implies$Hartree-Fock with relativistic correction [120]

$h_{2}$ $\implies$Semi-empirical Coulomb approximation [121]

$i_{2}$ $\implies$model potential [122]

$j_{2}$ $\implies$Hartree-slater method [23]

$a_{3}$ $\implies$RPTMP [57]

$b_{3}$ $\implies$RMBPT(3) [55]

$c_{3}$ $\implies$DF+CP [54]

$d_{3}$ $\implies$DX+CP method with SCE model potential [54]

$e_{3}$ $\implies$DX+CP method with CAFEGE model potential [54]

$f_{3}$ $\implies$DX+CP method with HFEGE model potential [54]

$g_{3}$ $\implies$CIDF method with integer occupation number [56]

$h_{3}$ $\implies$CIDF(q) method with non-occupation number [56]

$i_{3}$ $\implies$HFR+CP method [53]

$a_{4}$$\implies$Relativistic core-polarization corrected Dirac-Fock
method(DF+CP) [63]

Table 4: Transition rate (in $s^{-1}$) of E2 ($A_{RCC}^{E2}$) in plasma screened and unscreened medium. Here we have used our RCC matrix element (in a.u) and RCC wavelength ( in Å). Note that the notation $P(Q)$ in the case of transition rates means $P\times 10^{Q}$. Transitions |  $\mu$=0 |  $\mu$=0.025 |  $\mu$=0.050 |  $\mu$=0.075 |  $\mu$=0.1 |
---|---|---|---|---|---|---
Ar7+ |  |  |  |  |  |
3p4 $\rightarrow$ 4f6 | 1.4501(+06) | 1.4349(+06) | 1.3927(+06) | 1.3250(+06) | 1.2333(+06) |
3p4 $\rightarrow$ 4f8 | 6.5288(+06) | 6.4605(+06) | 6.2705(+06) | 5.9658(+06) | 5.5526(+06) |
3s2$\rightarrow$ 3d4 | 2.1840(+05) | 2.2032(+05) | 2.2179(+05) | 2.2417(+05) | 2.2742(+05) |
3s2$\longrightarrow$3d6 | 2.1919(+05) | 2.2126(+05) | 2.2256(+05) | 2.2494(+05) | 2.2819(+05) |
3d4 $\rightarrow$ 5g8 | 2.7225(+06) | 2.6258(+06) | 2.3685(+06) | 1.9874(+06) |  |
3d6 $\rightarrow$ 5g8 | 3.0252(+05) | 2.9179(+05) | 2.6352(+05) | 2.2077(+05) |  |
3d6 $\rightarrow$ 5g10 | 3.0267(+06) | 2.9190(+06) | 2.6362(+06) | 2.2085(+06) |  |
Kr7+ |  |  |  |  |  |
4s2$\longrightarrow$ 4d4 | 8.9514(+05) | 8.7868(+05) | 8.6208(+05) | 8.3473(+05) |  |
4s2$\longrightarrow$ 4d6 | 9.1180(+05) | 8.9640(+05) | 8.7912(+05) | 8.5066(+05) |  |
5p2 $\rightarrow$ 4f6 | 7.0537(-01) | 8.9779(-01) | 3.0592(00) |  |  |
5p4 $\rightarrow$ 4f8 | 2.3536(-01) | 2.9840(-01) | 1.4702(00) |  |  |
5d6 $\rightarrow$ 5g8 | 2.7840(+03) | 2.6100(+03) |  |  |  |
5d6 $\rightarrow$ 5g10 | 2.9352(+04) | 2.6114(+04) |  |  |  |
Xe7+ |  |  |  |  |  |
5s2$\rightarrow$ 5d4 | 6.6714(+05) | 6.7116(+05) |  |  |  |
5s2$\rightarrow$ 5d6 | 6.9674(+05) | 7.0069(+05) |  |  |  |
5p2 $\rightarrow$ 5p4 | 6.0045(-01) | 6.1555(-01) | 5.6058(-01) |  |  |
5p2 $\rightarrow$ 4f6 | 8.5467 (+03) | 2.7288(+04) | 3.3098(+04) |  |  |
5p4 $\rightarrow$ 4f6 | 1.2882 (+03) | 2.0657(+03) | 2.6049(+03) |  |  |
5p4 $\rightarrow$ 4f8 | 5.9866 (+03) | 1.2836(+04) | 1.6523(+04) |  |  |
5p2$\rightarrow$ 5f6 | 1.6471(+06) | 1.5379(+06) |  |  |  |
5p4$\rightarrow$ 5f6 | 4.3300(+05) | 4.0395(+05) |  |  |  |
5p4$\rightarrow$ 5f8 | 1.9479(+06) | 1.8169(+06) |  |  |  |
4f6 $\rightarrow$ 5f6 | 5.9296 (+04) | 5.4103(+04) |  |  |  |
4f6 $\rightarrow$ 5f8 | 7.4168 (+03) | 6.7680(+03) |  |  |  |
4f8 $\rightarrow$ 5f6 | 9.9224 (+03) | 9.0350(+03) |  |  |  |
4f8 $\rightarrow$ 5f8 | 6.2129(+04) | 5.6576(+04) |  |  |  |
5d4$\rightarrow$ 6s2 | 5.3813(+03) |  |  |  |  |
5d4 $\rightarrow$ 5g8 | 1.1670 (+06) |  |  |  |  |
5d6 $\rightarrow$ 5g8 | 1.2597 (+05) |  |  |  |  |
5d6$\rightarrow$ 5g10 | 1.2601(+06) |  |  |  |  |
5d6$\rightarrow$ 6s2 | 7.0259 (+03) |  |  |  |  |
Rn7+ |  |  |  |  |  |
6s2 $\rightarrow$ 6d4 | 1.0243(+06) | 9.8377(+05) |  |  |  |
6s2 $\rightarrow$ 6d6 | 1.0371(+06) | 9.9749(+05) |  |  |  |
6p2 $\rightarrow$5f6 | 5.0885(+04) | 5.4520(+04) |  |  |  |
6p4 $\rightarrow$5f6 | 3.2019(+03) | 3.5290(+03) |  |  |  |
6p4 $\rightarrow$5f8 | 1.5773(+04) | 1.7127(+04) |  |  |  |
6d4 $\rightarrow$ 5g8 | 6.3782(+05) |  |  |  |  |
6d6 $\rightarrow$ 5g8 | 6.2707(+04) |  |  |  |  |
6d6 $\rightarrow$ 5g10 | 6.2238 (+05) |  |  |  |  |
6p2 $\rightarrow$ 6p4 | 1.2855 (+02) | 1.2763(+02) |  |  |  |
7p2 $\rightarrow$ 7p4 | 1.5299 (+01) |  |  |  |  |
6p4 $\rightarrow$ 7p4 | 2.3245 (+05) |  |  |  |  |
5f6 $\rightarrow$ 7p2 | 1.4611 (+04) |  |  |  |  |
5f6 $\rightarrow$ 7p4 | 3.0317 (+03) |  |  |  |  |
5f8 $\rightarrow$ 7p4 | 1.7756 (+04) |  |  |  |  |
Table 5: Magmnetic dipole transition rate (in $sec^{-1}$) in plasma screened and unscreened medium. Notes: The notation $P(Q)$ in the case of transition rates means $P\times 10^{Q}$. For $4f_{5/2}\ \rightarrow\ 4f_{7/2}$ (Xe7+), transition rates 1.9227 (-03) and 1.9277 (-03) are available in the literature (a) using multiconfiguration Dirac-Fock method without and with Breit interaction plus quantum electrodynamics effect, respectively . Transition |  $\mu$=0 |  $\mu$=0.025 |  $\mu$=0.050 |  $\mu$=0.075 |  $\mu$=0.1
---|---|---|---|---|---
Ar7+ |  |  |  |  |
3p2 $\rightarrow$ 3p4 | 1.7951(-01) | 1.9857(-01) | 1.9603(-01) | 1.9165(-01) | 1.8570(-01)
3p4 $\rightarrow$ 4f6 | 3.7845(-02) | 3.7248(-02) | 3.5575(-02) | 3.2958(-02) | 2.9561(-02)
3d6 $\rightarrow$ 5g8 | 1.0634 (-02) | 1.0124(-02) | 8.8719(-03) | 7.0637(-03) |
Kr7+ |  |  |  |  |
4s2$\rightarrow$ 4d4 | 9.0675(-03) | 7.2173(-03) | 8.2040(-03) | 6.7390(-03) |
4p2 $\rightarrow$ 4p4 | 8.4015(00) | 8.2789(00) | 7.9664(00) | 7.4776(00) | 6.8397(00)
4d4 $\rightarrow$ 4d6 | 2.5436(-02) | 3.2234(-02) | 3.0177(-02) | 2.7000(-02) |
5p2 $\rightarrow$ 5p4 | 4.7841(-01) | 5.3517(-01) | 4.6911(-01) |  |
5d4 $\rightarrow$ 5d6 | 1.7080(-03) | 1.0904(-03) | 2.2682(-03) |  |
Xe7+ |  |  |  |  |
5p2 $\rightarrow$ 5p4 | 5.7437(+01) | 5.7875(+01) | 5.3500(+01) |  |
5p4 $\rightarrow$ 5f6 | 6.3588(-03) | 5.7557(-03) |  |  |
4f6 $\rightarrow$ 5f6 | 5.0459(+01) | 4.9605(+01) |  |  |
4f6 $\rightarrow$ 5f8 | 4.9212(-01) | 5.3389(-01) |  |  |
4f8 $\rightarrow$ 5f6 | 1.1142(+01) | 1.0620(+01) |  |  |
4f8 $\rightarrow$ 5f8 | 1.5833 (+02) | 1.5517(+02) |  |  |
5d4 $\rightarrow$ 5d6 | 2.7070(-01) | 2.6538(-01) | 2.2900(-01) |  |
Rn7+ |  |  |  |  |
6p2 $\rightarrow$ 6p4 | 1.2442 (+03) | 1.2212(+03) |  |  |
6d4 $\rightarrow$ 6d6 | 4.5873(00) | 4.1847(00) |  |  |
5f6 $\rightarrow$ 5f8 | 6.6129 (-02) | 3.4422(-02) |  |  |
6s2 $\rightarrow$ 7s2 | 1.0113 (00) |  |  |  |
7p2 $\rightarrow$ 7p4 | 8.1938 (+01) |  |  |  |
6p2 $\rightarrow$ 7p2 | 7.2065 (-02) |  |  |  |
6p4 $\rightarrow$ 7p4 | 2.4557 (00) |  |  |  |
5f6 $\rightarrow$ 7p4 | 2.4281 (-03) |  |  |  |
a[113, 114]
Table 6: Lifetimes in ns of few low-lying states.  Level | present work | other work(experiment) | other work(theory)
---|---|---|---
Ar7+ |  |  |
3p2 | 0.411 | 0.417 $\pm 0.010^{a}$, 0.423 $\pm 0.040^{b}$, |  $0.413^{c}$, $0.407^{d}$, $0.397^{e}$, $0.409^{f}$
|  | 0.48$\pm 0.05^{g}$, 0.49$\pm 0.05^{h}$ |  $0.389^{i}$, $0.408^{j}$, $0.4121^{k}$
|  | 0.55$\pm 0.03^{l}$, 0.53$\pm 0.11^{m}$ |
3p4 | 0.387 | 0.389$\pm 0.010^{a}$, 0.421$\pm 0.030^{b}$, |  $0.389^{c}$, $0.382^{d}$, $0.373^{e}$, $0.386^{f}$
|  | 0.428$\pm 0.027^{g}$, 0.48$\pm 0.06^{h}$ |  $0.366^{i}$, $0.388^{j}$, $0.3872^{k}$
|  | 0.54$\pm 0.0.02^{l}$, 0.527$\pm 0.018^{m}$ |
3d4 | 0.132 | 0.170 $\pm 0.0.010^{a}$, 0.130 $\pm 0.005^{b}$, |  $0.127^{e}$, $0.134^{f}$, $0.133^{j}$, $0.1318^{k}$
|  | 0.158$\pm 0.008^{g}$, |
3d6 | 0.137 | 0.166 $\pm 0.008^{a}$, 0.131 $\pm 0.005^{b}$, |  $0.131^{e}$, $0.138^{f,j}$, $0.1361^{k}$
|  | 0.160$\pm 0.008^{g}$ |
4f6 | 0.003 |  |
4f8 | 0.002 |  |
Kr7+ |  |  |
4p2 | 0.293 | 0.41 $\pm 0.04^{n}$, $0.291\pm 0.012^{o}$ |  $0.282^{p}$, $0.29653^{q}$
|  |  $0.290\pm 0.015^{g}$, $0.401\pm 0.018^{l}$ |
4p4 | 0.235 | 0.33$\pm 0.03^{n}$, $0.243\pm 0.01^{o}$ |  $0.230^{p}$, $0.24176^{q}$
|  |  $0.218\pm 0.033^{g}$, $0.331\pm 0.011^{l}$ |
4d4 | 0.048 |  | $0.05019^{q}$
4d6 | 0.052 | $0.048\pm 0.004^{o}$ | $0.05388^{q}$
4f6 | 0.055 |  |
4f8 | 0.055 |  |
Xe7+ |  |  |
5p2 | 0.45 |  $0.52(3)^{r}$, 0.50 $\pm 0.05^{s}$, |  $0.37^{t}$, $0.47^{u}$, $0.48^{v}$, $0.53^{w}$
|  | $0.380\pm 0.040^{g}$ |
5p4 | 0.29 |  $0.35(2)^{r}$, 0.33$\pm 0.03^{s}$, |  $0.23^{t}$, $0.30^{u}$, $0.31^{v}$, $0.33^{w}$
|  | $0.272\pm 0.037^{g}$ |
5d4 | 0.08 | $0.10(2)^{r}$ |  $0.07^{t}$, $0.07^{v}$, $0.06^{w}$
5d6 | 0.08 | $0.14(2)^{r}$ |  $0.14^{t}$, $0.08^{v}$, $0.07^{w}$
Rn7+ |  |  |
6p2 | 0.429 |  |
6p4 | 0.144 |  |
6d4 | 0.056 |  |
6d6 | 0.082 |  |

$a$$\implies$Beam-Foil technique [95]

$b$$\implies$[123]

$c$$\implies$Third order many-Body pertrubation theory [124]

$d$$\implies$ R-matrix theory [96]

$e$$\implies$Single Configuration interaction Hartree-Fock method using a
pseudo potential [97]

$f$$\implies$ Relastic model potential [101]

$g$$\implies$Arbitrarily Normalized Decay curve method for cascade-correction
in beam-foil [105]

$h$$\implies$Beam-Foil technique in the vacuum u.v [125]

$i$$\implies$Multiconfiguration Dirac-Fock method [102]

$j$$\implies$Charge expansion technique [126]

$k$$\implies$Multiconfiguration Dirac Hartree-Fock theory including core
polarisation [31]

$l$$\implies$Beam-Foil [50]

$m$$\implies$Beam-Foil [127]

$n$$\implies$Beam-Foil [49]

$o$$\implies$Foil excitation [104]

$p$$\implies$Coulomb approximation [121]

$q$$\implies$Hartree-Slater method [23]

$r$$\implies$Beam-foil spectroscopy [53]

$s$$\implies$Relativistic Hartree-Fock method [128]

$t$$\implies$Relativistic perturbation theory with a zero approximation model
potential [57]

$u$$\implies$Relativistic Many-Body perturbation theory(RMBPT(3)) [55]

$v$$\implies$Relativistic HFR+CP [53]

$w$$\implies$Relativistic MCDF [53]

Table 7: Critical values of plasma screening strength ($\mu_{c}$) in a.u for the following ions.  Kr7+ | Xe7+ | Rn7+
---|---|---
state | $\mu_{c}$ | state | $\mu_{c}$ | state | $\mu_{c}$
4d4 | 0.087152 | 5s2 | 0.076987 | 5s2 | 0.046401
4d6 | 0.086956 | 5p2 | 0.066149 | 6p2 | 0.039652
|  | 5p4 | 0.064474 | 6p4 | 0.036763
|  | 4f6 | 0.051526 | 6d4 | 0.027521
|  | 4f8 | 0.051486 | 6d6 | 0.027100
|  | 5d4 | 0.048395 |  |
|  | 5d6 | 0.048139 |  |
Figure 3: Determination of the value of critical plasma screening strength

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