2022

[1]Sultana Nahar These authors contributed equally to this work.

[1]Astronomy Department, The Ohio State University, 140 W. 18th Ave, Columbus,
43210 , OH , USA

2]Physics Department, University of Azad Jammu and Kashmir, , Muzzafarabad,
13100, Kashmir , Pakistan

# Collisional- and photo-excitations of Ca IV including strong 3.2 $\mu$m
emission line

<EMAIL_ADDRESS>Bilal Shafique<EMAIL_ADDRESS>*  [

###### Abstract

We report a detailed study of features of electron-impact excitation (EIE) of
Ca IV (Ca IV + e $\rightarrow$ Ca IV* + $e^{\prime}\rightarrow$ Ca IV +
h$\nu+e^{\prime}$), for the first time using the relativistic Breit-Pauli
R-Matrix method with a large close coupling wavefunction expansion of 54 fine
structure levels belonging to n=2,3,4 complexes. Calcium lines in the infrared
(IR) are expected to be observed by the high resolution James Webb Space
Telescope. Our study predicts presence of a strong 3.2 $\mu$m emission line in
IR formed due to EIE of $3p^{5~{}2}P^{o}_{3/2}-3p^{5~{}2}P^{o}_{1/2}$ in Ca
IV. The EIE collision strength ($\Omega$) for the transition shows extensive
resonances with enhanced background resulting in an effective collision
strength ($\Upsilon$) of 2.2 at about 104 K that increases to 9.66 around
3$\times 10^{5}$ K. The present results include $\Omega$ of all 1431
excitation among the 54 levels and $\Upsilon$ for a limited number of
transitions of possible interest. We have found extensive resonances in the
low energy region of $\Omega$, convergence of the resonances and of the
partial waves with the 54 levels wavefunction. At high energy $\Omega$
decreases beyond the resonance region for forbidden transitions, is almost
constant or decreases slowly for dipole allowed transitions with low
oscillator strengths (f-values), and rises with Bethe-Coulomb behavior of
ln(E) to almost a plateau for transitions with high f-values. The wavefunction
of Ca IV was obtained from optimization of 13 conﬁgurations $3s^{2}3p^{5}$,
$3s3p^{6}$, $3s^{2}3p^{4}3d$, $3s^{2}3p^{4}4s$, $3s^{2}3p^{4}4p$,
$3s^{2}3p^{4}4d$, $3s^{2}3p^{4}4f$, $3s^{2}3p^{4}5s$, $3s3p^{5}3d$,
$3s3p^{5}4s$, $3s3p^{5}4p$, $3p^{6}3d$, $3s3p^{4}3d^{2}$, each with the core
configuration of $1s^{2}2s^{2}2p^{6}$, using the atomic structure program
SUPERSTRUCTURE. They produce 387 fine structure levels. We report transition
parameters - oscillator strengths, line strength ($S$) and $A$-values for a
total of 93296 electric dipole (E1), quadrupole (E2), octupole (E3), magnetic
dipole (M1) and quadrupole (M2) transitions among these levels. Lifetimes of
these levels are also presented.

###### keywords:

Collisional excitation, Photoexcitation, 3.2 micron line of Ca IV

##  1 Introduction

Calcium is one of most abundant biogenic elements which is created during
supernova (SN) explosions. Lines of Ca I and II are seen in cool stars. The H
and K lines and forbidden lines of Ca II are commonly detected in SNe (.e.g.
setal11 ). The lines can determine the abundances of the element in the
object. Asplund et al aetal09  obtained Ca abundance in the Sun with respect
to hydrogen, in logarithmic abundances, to be log[e(Ca)] = 6.34$\pm$0.04 where
log[e(Ca])=12 + log[N(Ca)/N(H)], H is defined as log H = 12.00, N(Ca) and N(H)
are the number densities of elements Ca and H, respectively. Ca-rich
supernova, such as, SN 2019ehk in the star forming spiral galaxy Messier 100,
about 55 million light years away from the earth, first by Shepherd js19  and
analysis by Jacobson-Galan et al galenetal20 , brought much attention
recently. Jacobson-Galan galenetal20  found that the calcium-rich SN belongs
to CaST class.Ca-IV lines in infrared (IR) was observed in NGC 7027 nebula by
Feuchtgruber et al. fetal97 . 3.2 $\mu$m line was observed in three Wolf-Rayet
stars and was used to determine abundance and wind speed by Ignance et al
ietal01 . Furthermore, emission lines of two calcium ions, Ca-V and Ca-VII,
were observed in the NGC 2707 and NGC 6302 nebulae fetal01 , and Ca II lines
in the high-temperature environment of white dwarfs zetal03 .

The present work reports study of two atomic processes of Ca IV that produce
spectral lines, electron-impact excitation (EIE) and photo-excitation. EIE is
one of the most common atomic process in near empty space or in astrophysical
plasmas with or without a radiative source. A traveling electron interacts
with an ion and transfers part of its energy and promotes the ion to an
excited state. It is followed by the target ion de-exciting by emission of a
photon, (e.g. aas11 )

${X^{+z}+e\rightarrow X^{+z*}+e^{\prime}\rightarrow X^{+z}+h\nu+e^{\prime}}$
(1)

where $X^{+z}$ is the target ion of charge $z$. The emitted photon can form an
observable line depending on the radiative transfer and density of the plasma.
The projectile electron can also form a quasi-bound state by exciting the
target ion to a higher energy state, $E^{**}$, while attaching itself to an
orbit $\nu l$ which happens very commonly in nature, matching the energy of a
doubly excited autoionizing Rydberg state

$E_{x}\nu l=E^{**}(X^{+z}\nu l)=E_{x}-z^{2}/\nu^{2}$ (2)

before going out free. This intermediate state introduces a resonance in the
collisional scattering. $E_{x}$ is the excited energy of the ion, $\nu l$ are
the effective quantum number and angular momentum of the scattered electron.
The present study implements close coupling (CC) wavefunction expansion that
produces the autoionizing resonances automatically.

The other process, photo-excitation/ de-excitation, forms a line as the ion
absorbs or emits a photon

${X^{+z}+h\nu\leftrightarrow X^{+z*}}$ (3)

This process occurs most commonly when there is a radiative source, such as, a
star shining the plasma.

Among the low ionization stages of Ca, Ca IV has been the least studied ion.
There is no study on electron impact excitation of it found in the literature.
EIE of Ca IV is the main focus of the present study.

Among the past studies on energies and transitions, Sugar and Corliss sc85
compiled the experimentally measured energy levels of Ca IV which are
available at NIST website nist . The radiative transition rates of Ca IV were
reported by Naqvi n51 , Varsavsky v61 , Fawcett and Gabriel fg66 , Huang et al
h83 , Wilson et al. w00 , Gabriel et. al getal66  who identified several lines
generated from observed UV transitions. The probability of detection of Ca IV
lines has increased considerably with high resolution observation of James
Webb Space Telescope (JWST) in the infra-red region. The present work reports
collisional excitation and photo-excitations for many levels of Ca IV which
include collision strengths ($\Omega$) and the Maxwellian averaged collision
strengths or effective collision strengths ($\Upsilon$), and parameters $f$-,
$S-$ and $A-$ values for radiative transitions.

##  2 Theoretical Approximation

We give a brief outline of the theoretical background for electron impact
excitation and radiative photo-excitations below as guidance for the readers.
More details can be found, e.g. in Pradhan and Nahar aas11 . We have treated
EIE of Ca IV for collision strengths with relativistic Breit-Pauli R-matrix
(BPRM) method, as developed under the Iron Project (IP, ip ; betal95 ). We
used a wavefunction expansion in close-coupling (CC) approximation that
includes excitation to n=2,3,4 levels in the target and obtained collision
strengths. We obtained radiative transition parameters for an extensive set of
transition using relativistic Breit-Pauli approximation implemented in atomic
structure program SUPERSTRUCTURE (SS, ejn74 ; necp03 ).

Although two approaches are used for collision and photo-excitation, the
computations are related. BPRM calculations are initiated with the
wavefunction expansion of the target ion, e.g. Ca IV, generated by program
SUPERSTRUCTURE (SS). We discuss the outlines of collisional excitation first
and then photo-excitations or radiative transitions.

###  2.1 Breit-Pauli R-matrix (BPRM) calculations for EIE

BPRM Hamiltonian as adopted under the Iron Project ip ; betal95  in atomic
Rydberg unit is given by

$H_{N+1}^{\rm
BP}=\sum_{i=1}^{N+1}\left\\{-\nabla_{i}^{2}-\frac{2Z}{r_{i}}\right\\}+\sum_{j>i}^{N+1}\frac{2}{r_{ij}}+H_{N+1}^{\rm
mass}+H_{N+1}^{\rm Dar}+H_{N+1}^{\rm so}.$ (4)

where the first three terms belong to the non-relativistic Hamiltonian and
last three terms are the 1-body relativistic corrections which are mass,
Darwin, and spin-orbit interaction terms respectively. BPRM codes include all
of them and part of the two-body correction terms of the Breit-interaction
(e.g. aas11 ). One Rydberg (Ry) is half of a Hartree giving the factor 2 in
the terms.

BPRM calculations start with the target ion wavefunction generated by SS and
calculates the wavefunction of the total atomic system of the target ion and
the interacting electron in the close coupling (CC) approximation. In CC
approximation, the wavefunction of (e+ion) in a state $SL\pi J$, where $S$ is
total spin, $L$ is the orbital, and $J$ is the total angular momenta, is
expressed as

$\Psi_{E}(e+ion)=A\sum_{i}^{n}\chi_{i}(ion)\theta_{i}+\sum_{j}c_{j}\Phi_{j}(e+ion)$
(5)

In the first term $\chi_{i}(ion)$ is the wavefunction expansion of the target
ion, $\theta_{i}$ is that of the interacting electron, in channel
$S_{t}L_{t}\pi_{t}J_{t}k_{i}^{2}l(SL\pi J)$ where $S_{t}L_{t}\pi_{t}J_{t}$ is
the target ion state interacting with the projectile electron of energy
$k_{i}^{2}$ and orbital angular momentum $l$. The sum represents ground and
various excited states of the target ion. $A$ is the anti-symmetrization
operator. In the second term, $\Phi_{j}(e+ion)$ represents the
(target+electron) wavefunction, basically part of the first term separated out
to show the orthogonality condition of the interacting electron and short
range interaction. Close-coupling wave function expansion which includes
target ion excitations enables producing the resonances inherently. The
interference of the bound states of the target ion and the projectile electron
continuum wavefunction in the transition matrix introduces the resonances.
Substitution of the CC expansion in the Schrodinger equation with the Breit-
Pauli Hamiltonian results in a set of coupled equations. The R-matrix method
is used to solve this set of equations for the energy and wavefunctions of the
(e+ion) system.

The scattering matrix for transition of the target ion from state $i$ to state
$k$ by collision, ${\bf S}_{SL\pi
J}(S_{i}L_{i}J_{i}l-S_{k}L_{k}J_{k}l^{\prime})$ where $SL\pi J$ is the (e+ion)
state, $l$ and $l^{\prime}$ are the incident and scattered partial waves of
the free electron, is derived from the reactant matrix of the incident wave
(e.g. br75 ; sb80 ; st82 ; aas11 ). The collision strength $\Omega$ for
electron impact excitation (EIE) is given by,

$\Omega(S_{i}L_{i}J_{i}-S_{k}L_{k}J_{k})={1\over 2}\sum_{SL\pi
J}\sum_{l,l^{\prime}}(2J+1)\lvert r{\bf S}_{SL\pi
J}(S_{i}L_{i}J_{i}l-S_{k}L_{k}J_{k}l^{\prime})\rvert^{2}$ (6)

$\Omega$ reveals the detailed features with resonances of the collision. The
plasma models use the temperature dependent quantity, the effective collision
strengths $\Upsilon(T)$ which is obtained by averaging $\Omega$ over
Maxwellian distribution function of the electrons at temperature $T_{e}$ as

$\Upsilon_{ij}(T_{e})=\int_{0}^{\infty}\Omega_{ij}(E)\exp(-E/kT_{e})d(E/kT_{e}),$
(7)

where $k$ is the Boltzmann constant and $E$ is the energy of the projectile
electron after the excitation, that is, the energy of the scattered electron.
The excitation rate coefficient ($q_{ij}(T_{e})$) is related to the effective
collision strength $\Upsilon_{ij}$ as

$q_{ij}(T_{e})=\frac{8.63\times
10^{-6}}{g_{i}T_{e}^{1/2}}e^{-E_{ij}/kT_{e}}\Upsilon_{ij}(T_{e})cm^{3}/s,$ (8)

where $g_{i}$ is the statistical weight of the initial level, T is in K,
$E_{ij}$ is the transition energy in Rydberg, and (1/kT = 157885/T).

The high energy background of collision strength shows certain general
behaviors depending on the type of excitation. The background of $\Omega$ is
the smooth curve at the base of resonant features. For forbidden transitions,
$\Omega$ decreases to almost zero with higher energy. For dipole allowed
transitions, using Born approximation with Coulombic wavefunction, $\Omega$
shows a high energy limiting behavior and is given by Coulomb-Bethe
approximation

$\Omega_{ij}(E)={4g_{i}f_{ij}\over E_{ij}}ln{E\over E_{ij}},$ (9)

where $f_{ij}$ is the oscillator strength for a dipole allowed transition, and
E is the incident electron energy. In the high energy limit,

$\Omega_{ij}(E)\sim_{E\rightarrow\infty}d_{ij}ln(E)$ (10)

where $d_{ij}$ is proportional to the oscillator strength. The logarithmic
function will increase with increase in electron energy, but the the rising
trend of the function slows down with very high values of the argument. Hence,
for a low value of $d$, $\Omega$ is may not change as it is multiplied by a
small number. For high value of $d$, $\Omega$ will increase but will lead
toward a plateau.

###  2.2 Atomic structure calculations for radiative transitions

Theoretical details for obtaining radiative transition parameters through
atomic structure calculations using program SUPERSTRUCTURE can be found, for
example, in aas11 ; necp03 . The Hamiltonian includes relativistic mass,
Darwin, spin-orbit interaction correction terms, full 2-body Breit interaction
and some additional two body terms. The interacting electron and core ion
potential, implemented in program SS, is represented by Thomas-Fermi-Amaldi-
Dirac potential. The program uses configuration interaction wavefunction
expansion, which for a symmetry $J\pi$ can be expressed as

$\Psi(\mbox{$J\pi$})=\sum_{i=1}^{N}a_{i}\psi[C_{i}(\mbox{$J\pi$})]$ (11)

where $a_{i}$s is the amplitude or the mixing coefficient of wavefunction of
configuration $C_{i}$, $\psi[C_{i}(\mbox{$J\pi$})]$ with symmetry $J\pi$, and
the sum is over all $N$ configurations that can produce a level of symmetry
$J\pi$. The wavefunction will result in having $N$ number of eigenvalues from
the Hamiltonian matrix and each eigenvalue will correspond to energy of one
level of the symmetry. Accuracy of the energy of a level may depend on the
size of the expansion and identification on the value of the mixing
coefficient (e.g. aas11 .)

The transition matrix element, for example, for electric dipole allowed
transition (E1), is given by $<\Psi_{B}\lvert\lvert{\bf
D}\rvert\rvert\Psi_{B^{\prime}}>$ where $\Psi_{B}$ and $\Psi_{B^{\prime}}$ are
the initial and final state bound wavefunctions, ${\bf D}=\sum_{i}{\bf r}_{i}$
is the dipole operator where the sum is over the number of electrons. The line
strength $\bf S$ is obtained from the mod squared of the transition matrix,

${\bf
S=\lvert\left\langle{\mit\Psi}_{f}|\sum_{j=1}^{N+1}r_{j}|{\mit\Psi}_{i}\right\rangle\rvert^{2}}$
(12)

where $\Psi_{i}$ and $\Psi_{f}$ are the initial and final wavefunctions. The
transition parameters, oscillator strength ($f_{ij}$) and radiative decay rate
($A$) can be obtained from line strength as

$f_{ij}={E_{ji}\over{3g_{i}}}{\bf S},~{}~{}A_{ji}(sec^{-1})=\left[0.8032\times
10^{10}{E_{ji}^{3}\over{3g_{j}}}\right]{\bf S}$ (13)

Transition probabilities for electric quadrupole (E2), magnetic dipole (M1),
electric octupole (E3), magnetic quadrupole (M2) transition parameters can be
obtained from their respective line strengths,(e.g. aas11 ; necp03 ).

Lifetime of an excited level can be obtained from the inverse of the sum of
all transition probabilities to lower levels,

$\tau_{i}(s)=1/[\sum_{j}A_{ji}(s^{-1})]$ (14)

In atomic unit of time $\tau_{0}=2.4191\times 10^{-17}$s , the transition
probabilities or the radiative decay rate can be expressed as
$A_{ji}(s^{-1})={A_{ji}(a.u.)/\tau_{0}}$.

##  3 Computation

The R-matrix calculations start with the target wavefunctions as an input.
These wavefunctions are obtained from atomic structure calculations, mainly
using program SUPERSTRUCTURE (SS) ejn74 ; necp03 . Ca IV wavefunction,
energies and the relevant radiative transition parameters were obtained from
an optimized a set of 13 configurations of the ion, $3s^{2}3p^{5}$,
$3s3p^{6}$, $3s^{2}3p^{4}3d$, $3s^{2}3p^{4}4s$, $3s^{2}3p^{4}4p$,
$3s^{2}3p^{4}4d$, $3s^{2}3p^{4}4f$, $3s^{2}3p^{4}5s$, $3s3p^{5}3d$,
$3s3p^{5}4s$, $3s3p^{5}4p$, $3p^{6}3d$, $3s3p^{4}3d^{2}$, with the same core
configuration $1s^{2}2s^{2}2p^{6}$ for each, using SS. The set of optimized
Thomas-Fermi orbital scaling parameters are 1.26865(1s), 1.0395(2s),
1.04288(2p), 1.1(3s), 1.1(3p), 1.1(3d), 1.083(4s), 1.079(4p), 1.031(4d), 1.1
(4f), 1.1 (5s) respectively. The configurations set provided 387 fine
structure levels, 54 of which were considered for the collisional excitation
in the present study and hence used in the wavefunction expansion of Ca IV.
Selecting a set of levels of the target or core ion from a large set is the
standard for a R-matrix calculation. Computations of collision strengths,
which depend on the size of the wavefunction expansion, needed several
hundreds of CPU hours on the high performance computers at the Ohio
Supercomputer Center.

The purpose of having an optimized set of configurations, which produced a
large set of levels, is to ensure contributions from configuration
interactions are included in the wavefunctions of levels, and thus can provide
a set of levels, starting from the ground level, of higher accuracy for the CC
wavefunction expansion. Inclusion of all levels, 387 in the present case, in
the calculations will require considerably large computational time, and
expected to be computationally prohibitive which is very common, and hence
impractical to consider. The set of excited levels considered for a R-matrix
calculation is typically based on the expected physics to be revealed, mainly
the resonant features. No new physics was expected from levels higher than 54
levels since the resonances would have converged and hence those belonging to
higher ones would no longer be strong but have weakened to converge to the
background. This was also our finding as demonstrated in the Results section.

Table 1 presents the 54 fine structure levels of Ca IV included in the
wavefunction expansion and compares the calculated energies from SS with
experimental values tabulated by the National Institute of Standards and
Technology (NIST) (www.nist.gov). The comparison shows that the present
computed energies are generally within a few percent of the observed values.
The accuracy of a level energy depends on the how well the wavefunction
expansion is representing the level through interaction of the given set of
configurations. For more precise energy positions of the resonances in EIE
collision strength, we have replaced the calculated energies with the
available observed energies in the BPRM calculations.

Table 1: Comparison of the present calculated energies for the 54 fine structure levels of Ca IV with those (Sugar and Corliss sc85 ) available in the compilation table of the NIST nist  K | Configuration | Term |  Epresent |  ENIST | % diff
---|---|---|---|---|---
|  |  | (Ry) | (Ry) |
1 | $3s^{2}3p^{5}$ | ${}^{2}P^{o}_{3/2}$ | 0 | 0 | 0.0
2 | $3s^{2}3p^{5}$ | ${}^{2}P^{o}_{1/2}$ | 0.0302 | 0.0284 | 5.9
3 | $3s3p^{6}$ | ${}^{2}S_{1/2}$ | 1.3031 | 1.3891 | 6.2
4 | $3s^{2}3p^{4}3d$ | ${}^{4}D_{7/2}$ | 1.8197 | 1.8362 | 0.9
5 | $3s^{2}3p^{4}3d$ | ${}^{4}D_{5/2}$ | 1.8218 | 1.8385 | 0.9
6 | $3s^{2}3p^{4}3d$ | ${}^{4}D_{3/2}$ | 1.8243 | 1.8410 | 0.9
7 | $3s^{2}3p^{4}3d$ | ${}^{4}D_{1/2}$ | 1.8263 | 1.8431 | 0.9
8 | $3s^{2}3p^{4}3d$ | ${}^{4}F_{9/2}$ | 1.9901 | 1.9865 | 0.2
9 | $3s^{2}3p^{4}3d$ | ${}^{4}F_{7/2}$ | 1.9999 | 1.9964 | 0.2
10 | $3s^{2}3p^{4}3d$ | ${}^{2}P_{3/2}$ | 2.0047. | 1.9965 | 0.4
11 | $3s^{2}3p^{4}3d$ | ${}^{2}P_{1/2}$ | 2.0070 | 2.0036 | 0.2
12 | $3s^{2}3p^{4}3d$ | ${}^{4}F_{5/2}$ | 2.0115 | 2.0083 | 0.2
13 | $3s^{2}3p^{4}3d$ | ${}^{4}F_{3/2}$ | 2.0225. | 2.0149 | 0.4
14 | $3s^{2}3p^{4}3d$ | ${}^{4}P_{5/2}$ | 2.0705. | 2.0473 | 0.2
15 | $3s^{2}3p^{4}3d$ | ${}^{4}P_{3/2}$ | 2.0761 | 2.0536 | 1.1
16 | $3s^{2}3p^{4}3d$ | ${}^{2}D_{3/2}$ | 2.0817 | 2.0732 | 0.4
17 | $3s^{2}3p^{4}3d$ | ${}^{4}P_{1/2}$ | 2.0840 | 2.0616 | 1.0
18 | $3s^{2}3p^{4}3d$ | ${}^{2}D_{5/2}$ | 2.0970 | 2.0890 | 0.4
19 | $3s^{2}3p^{4}3d$ | ${}^{2}F_{7/2}$ | 2.1076 | 2.1074 | 0.02
20 | $3s^{2}3p^{4}3d$ | ${}^{2}F_{5/2}$ | 2.1310 | 2.1297 | 0.06
21 | $3s^{2}3p^{4}3d$ | ${}^{2}G_{7/2}$ | 2.1369 | 2.1573 | 0.95
22 | $3s^{2}3p^{4}3d$ | ${}^{2}G_{9/2}$ | 2.1382 | 2.1571 | 0.88
23 | $3s^{2}3p^{4}3d$ | ${}^{2}F_{7/2}$ | 2.2990 | 2.3051 | 0.26
24 | $3s^{2}3p^{4}3d$ | ${}^{2}F_{5/2}$ | 2.3046 | 2.3109 | 0.27
25 | $3s^{2}3p^{4}3d$ | ${}^{2}D_{5/2}$ | 2.4602 | 2.4937 | 1.36
26 | $3s^{2}3p^{4}3d$ | ${}^{2}D_{3/2}$ | 2.4653 | 2.4993 | 1.34
27 | $3s^{2}3p^{4}3d$ | ${}^{2}S_{1/2}$ | 2.6559 | 2.6979 | 1.8
28 | $3s^{2}3p^{4}4s$ | ${}^{4}P_{5/2}$ | 2.6688 | 2.7714 | 3.7
29 | $3s^{2}3p^{4}4s$ | ${}^{4}P_{1/2}$ | 2.6701 | 2.7972 | 4.5
30 | $3s^{2}3p^{4}4s$ | ${}^{2}P_{3/2}$ | 2.6780 | 2.7974 | 4.24
31 | $3s^{2}3p^{4}4s$ | ${}^{4}P_{3/2}$ | 2.6818 | 2.7857 | 3.6
32 | $3s^{2}3p^{4}4s$ | ${}^{2}P_{1/2}$ | 2.6895 | 2.8098 | 4.3
33 | $3s^{2}3p^{4}3d$ | ${}^{2}D_{5/2}$ | 2.7449 | 2.8566 | 4.2
34 | $3s^{2}3p^{4}3d$ | ${}^{2}P_{3/2}$ | 2.7495 | 2.8976 | 5.2
35 | $3s^{2}3p^{4}3d$ | ${}^{2}P_{1/2}$ | 2.7666 | 2.9135 | 4.9
36 | $3s^{2}3p^{4}4s$ | ${}^{2}D_{3/2}$ | 2.7689 | 2.8794 | 3.8
37 | $3s^{2}3p^{4}4s$ | ${}^{2}D_{5/2}$ | 2.8479 | 2.9941 | 4.7
38 | $3s^{2}3p^{4}3d$ | ${}^{2}D_{3/2}$ | 2.8491 | 2.9961 | 4.7
39 | $3s^{2}3p^{4}4p$ | ${}^{4}P^{o}_{5/2}$ | 3.0145 | 3.1268 | 3.8
40 | $3s^{2}3p^{4}4p$ | ${}^{4}P^{o}_{3/2}$ | 3.0179 | 3.1318 | 3.5
41 | $3s^{2}3p^{4}4p$ | ${}^{4}P^{o}_{1/2}$ | 3.0251 | 3.1386 | 3.5
42 | $3s^{2}3p^{4}4p$ | ${}^{4}D^{o}_{7/2}$ | 3.0539 | 3.1829 | 4.1
43 | $3s^{2}3p^{4}4p$ | ${}^{4}D^{o}_{5/2}$ | 3.0610 | 3.1913 | 4.0
44 | $3s^{2}3p^{4}4p$ | ${}^{4}D^{o}_{3/2}$ | 3.0706 | 3.1996 | 4.0
45 | $3s^{2}3p^{4}4p$ | ${}^{2}P^{o}_{1/2}$ | 3.0729 | 3.2128 | 4.3
46 | $3s^{2}3p^{4}4p$ | ${}^{4}D^{o}_{1/2}$ | 3.0751 | 3.2039 | 4.0
47 | $3s^{2}3p^{4}4p$ | ${}^{2}D^{o}_{5/2}$ | 3.0824 | 3.2185 | 4.0
48 | $3s^{2}3p^{4}4s$ | ${}^{2}S_{1/2}$ | 3.0828 | 3.2573 | 5.4
49 | $3s^{2}3p^{4}4p$ | ${}^{2}D^{o}_{3/2}$ | 3.0888 | 3.2376 | 4.0
50 | $3s^{2}3p^{4}4p$ | ${}^{2}P^{o}_{3/2}$ | 3.1017 | 3.2209 | 3.7
51 | $3s^{2}3p^{4}4p$ | ${}^{4}S^{o}_{3/2}$ | 3.1217 | 3.2577 | 4.0
52 | $3s^{2}3p^{4}4p$ | ${}^{2}S^{o}_{1/2}$ | 3.1249 | 3.2598 | 4.0
53 | $3s^{2}3p^{4}4p$ | ${}^{2}F^{o}_{7/2}$ | 3.2091 | 3.3629 | 4.5
54 | $3s^{2}3p^{4}4p$ | ${}^{2}F^{o}_{5/2}$ | 3.2133 | 3.3591 | 4.5
|  |  |  |  |

It is important to ensure convergence in contributions by the number of
partial waves and (e+ion) symmetries to the collision strengths $\Omega$. We
computed $\Omega$ several times by varying different sets of partial waves $l$
going up to 22 and (e-ion) symmetries J going up to $\leq 22$ of even and odd
parities. We found that i) $\Omega$ background is converged with the highest
values of $l$ = 20 and $J\pi$ = 11, ii) larger number of $l$ and $J\pi$ than
these introduced computational instability which gives NaN (Not a Number) for
the collision strengths at various electron energies.

Getting ”NaN” for $\Omega$ values is a known problem for a R-matrix
calculation. They are introduced when the computation goes through very small
numbers. Usually the NaN points are deleted from the sets of $\Omega$ values.
However, we made attempts to get approximate values $\Omega$ at energies of
NaN values obtained with large $l$ and $J$ values by extrapolating or
interpolating the neighboring $\Omega$ points. Smooth background can be
extrapolated but resonances are not reproduced in $\Omega$ as can be seen in
the blue (with extrapolation) and red curves (no extrapolation) in Figure 1
representing the same excitation of the ion. The resonant peaks of the blue
curve, which has many NaN points and interpolation was carried out to replace
the NaNs with real numbers, are lower than the red curves.

Figure 1 also demonstrates the test of convergence of $\Omega$ for a few sets
of highest values of $l$ and $J$ for the excitations of a)
${}^{2}P^{o}_{3/2}-^{2}P^{o}_{1/2}$ and b) ${}^{2}P^{o}_{1/2}-^{2}S_{1/2}$
computed at a coarse energy mesh. The red curves correspond to use of $l$ = 0
- 20 and $J\pi$ = 0- 11, the blue ones to $l$ = 0 - 22 and $J\pi$ = 0 - 21 and
the magenta ones to $l$ = 0 - 11 and $J\pi$ = 0 - 10. $\Omega$ in magenta ($l$
= 0- 11 and $J\pi$= 0- 10) are considerably lower than the other curves
indicating that convergence of contributions from 12 partial waves has not
been reached. The blue and the red curves have about the same background
indicating convergence in contributions of partial waves has reached. Hence
specifications for the red curve, $l$ = 0 - 20 and $J\pi$ = 0- 11, have been
used for the computation of $\Omega$. Final $\Omega$ values were obtained at
fine energy meshes. We used a very fine energy mesh of $\Delta E<10^{-6}$
Rydberg to resolve the near-threshold resonances.

Figure 1: Demonstration of convergence of contributions of partial waves with
various $l$ and $J$ values to $\Omega$. The x-axis corresponds to the energy
of the scattered electron after the excitation which starts with zero energy.
Red curves indicate the best convergent condition.

The above discussion concerns impact of partial waves that are noticeable.
Beyond it, top-up which gives some additional contributions is added. As top-
up contributions, we included contributions of higher multipole potentials to
$\Omega$ using the option ipert=1 in STGF of the R-matrix codes. The other
top-up contribution can come from higher partial waves, beyond the specified
ones, using the option chosen as ipert=2, in STGF. The approximation
incorporated in the R-matrix code STGF is most probably based on the treatment
of Burgess and Tully bt78  of higher partial waves. However, computation of
contributions of the higher partial waves, not only takes much longer time, it
is not done for most cases as it stops due to numerical issues except for a
few excitations and at some energies. We computed contributions of higher
partial waves when it is possible to test them and found negligible
contributions in increasing the $\Omega$ values to affect the average
collision strengths $\Upsilon$. The problem is compensated, as done in the
present case, by including larger number of partial waves without the
approximation.

Program ”ecs-omg.f” snn13  was used to calculate the effective collision
strengths $\Upsilon$, Eq.(7), at various temperatures where $\Omega$ is
integrated over the energy of the scattered electron from zero to a high
value. The high energy limit is chosen to a value at which $\Omega$ has
diminished to a near zero value or has reached a plateau and the exponential
factor of $\Upsilon$ has approached a near zero value. $\Omega$ points between
the highest electron energy computed by the BPRM codes to the highest energy
limit of the $\Upsilon$ integral are obtained using the logarithmic behavior
of Coulomb-Bethe approximation of Eq.(10).

The radiative data of $f-$ and $A-$values for dipole allowed photo-excitations
(E1) have been reprocessed with experimental energies using code PRCSS (e.g.
snn14). This allows to obtain the transition parameters at observed
wavelengths. For the reprocessing, the transition energies were obtained from
the experimental level energies and then multiplied, following Eq. 13. to the
calculated line strengths from code SUPERSTRUCTURE. For the levels for which
no observed or measured values are available, calculated energies were used.

##  4 Results and Discussions

We present atomic parameters for electron impact excitation of (e + Ca IV
$\rightarrow$ Ca IV∗ \+ e’ $\rightarrow$ Ca IV + h$\nu$ \+ e’) and photo-
excitation of (Ca IV + h$\nu~{}\leftrightarrow$ Ca IV∗). The results for the
collisional excitation are reported for the first time, as indicated by
literature search. They are described below first followed by those for
photoexcitation.

###  4.1 Collisional excitation of Ca IV

We discuss the characteristic features of collisional excitation with
illustrative examples. The first excitation of the target is usually of
particular interest because of its high probability through EIE and the
emitted photon, typically of low energy, can travel for a long time without
being absorbed. If the corresponding emission line is strong, it can be
detected easily in low density plasmas and be used for identification of the
ion and environmental diagnostics. The first excitation in Ca IV,
$3p^{5}~{}^{2}P^{o}_{3/2}-3p^{5}~{}^{2}P^{o}_{1/2}$, within the ground
configuration is of particular importance since the wavelength of the emission
line, 3.207 $\mu$m, is well within the high resolution IR wavelength detection
range, 0.6 - 28.3 $\mu$m, of JWST, and could be used for diagnostics,
abundances (e.g. ietal01 ).

Figure 2 upper panel presents collision strength for the first excitation
$\Omega(^{2}P^{o}_{3/2}-^{2}P^{o}_{1/2})$ of the target ion Ca IV with respect
to scattered electron energy after the excitation. The electron energy is
relative to the excitation threshold and hence starts at zero. We note that
the $\Omega$ for the collisional excitation is quite strong as it shows
extensive resonances with enhanced background in the energy region between the
first excited level ${}^{2}P^{o}_{1/2}$ and the next one ${}^{2}S_{1/2}$
(pointed by arrows), and continues to be strong beyond it up to
${}^{2}D_{3/2}$ level, the one before the last dipole allowed transition in
the 54 level wavefunction expansion. Beyond, ${}^{2}D_{3/2}$ the transitions
are forbidden except one and the resonances become weaker. There are 29 dipole
allowed levels in total that exist in this energy range. Each excitation of
the target ion corresponds to a Rydberg series of resonances. Typically, the
resonances corresponding to a dipole allowed excitation are visible while
others are suppressed. The resonances belonging to very high energy levels are
found to become weaker. Such weakening trend indicates convergence of resonant
contributions to the collisional parameters.

Figure 2 lower panel presents effective collision strength ($\Upsilon$) for
$\Omega[(^{2}P^{o}_{3/2}-^{2}P^{o}_{1/2})$. Starting low at lower energy,
$\Upsilon$ forms a shoulder bump at 104 K. Then it rises relatively quickly
reaching to the high peak value 9.66 at about 3$\times 10^{5}$ K. The peak
indicates existence of a strong line of the transition in Ca IV which can be
detected. The intensity of the line will depend on the plasma environment.

Figure 2: Upper panel: EIE collision strength ($\Omega$) of Ca IV for the
first excitation
$3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-3s^{2}3p^{5}~{}^{2}P^{o}_{1/2}$ of the ground
level with respect to scattered electron energy in Ry unit. Extensive
resonances with enhanced background can be noted within the energy region of
dipole allowed transitions from the ground level
$3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}$ up to ${}^{2}D^{o}_{1/2}$. Lower panel:
Effective collision strength for
$3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-3s^{2}3p^{5}~{}^{2}P^{o}_{1/2}$ excitation in
Ca IV showing a high peak value at about 3$\times 10^{5}$ K indicating high
probability of detection of the line by JWST.

We present illustrative examples of forbidden transitions in Ca IV. Figure 3
presents collision strengths for two forbidden excitations in Ca IV, a)
$\Omega(3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-3s^{2}3p^{4}4s^{4}D_{7/2})$ and b)
$\Omega(3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-3s^{2}3p^{4}4s^{4}F_{9/2})$. Both
transitions show presence of strong resonances in the lower energy region. The
resonances become weaker in higher energy region indicating contribution of
resonances is converging. This is the typical trend of $\Omega$ for forbidden
transitions. Both of these transitions lie in the extreme ultraviolet region
with wavelengths of 496 and 458 $\AA$ respectively.

Figure 3: Collision strength for forbidden excitation a)
$\Omega(3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-3s^{2}3p^{4}4s~{}^{4}D_{7/2})$ and b)
$\Omega(3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-3s^{2}3p^{4}4s~{}^{4}F_{9/2})$ in
extreme ultraviolet, 496 $\AA$ and 458 $\AA$ respectively, illustrating
resonant features in the low energy region and decreasing background in the
high energy region.

Collision strengths for dipole allowed transitions may show a different trend
in the high energy background from those of forbidden transitions. The dipole
in the target can affect the partial waves of the incident electron and
contribute to the collision strength. The contribution depends on the
oscillator strength for the dipole transition. For stronger transitions,
inclusion of larger number of partial waves is important for converged
contributions.

Figure 4: Features of $\Omega(EIE)$ for two weaker dipole allowed transitions
(oscillator strengths are given in Table 2): a)
$3s^{2}3p^{5}~{}^{2}P^{o}_{1/2}-3s3p^{6}~{}^{2}S_{1/2}$ (levels 2-3) and b)
relatively higher transition
$3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-3s^{2}3p^{4}3d^{2}D_{5/2}$ (levels 1 -18).
Resonances have converged to a smooth background at high energy where the
background remains almost constant or decreases slowly with increase of
energy, typical for weak transitions.

Figure 4 presents features of $\Omega$ (EIE) for two dipole allowed
transitions to low lying excited levels, a)
$3s^{2}3p^{5}~{}^{2}P^{o}_{1/2}-3s3p^{6}~{}^{2}S_{1/2}$ (levels 2-3) at 670
$\AA$ and b) $3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-3p^{4}3d~{}^{2}D_{5/2}$ (levels
1-18) at 435 $\AA$. $\Omega$ for these transitions show presence of prominent
resonances in the lower energy regions which become weaker converging to
smooth background at higher energy. It can be noted that the high energy
background is decreasing very slowly or remaining almost constant. These
transitions are much weaker compared to others as indicated by their smaller
values for the oscillator strengths and $A$-values presented in Table 2.

Figure 5: Features of $\Omega$ for excitation to high lying dipole allowed levels, a) $3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-(3s^{2}3p^{4}3d)^{2}D_{5/2}$ and b) $3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-(3s^{2}3p^{4}4s)~{}^{2}P_{3/2}$ with stronger or larger values for oscillator strengths illustrating (a) convergence of resonances and (b) rising Coulomb-Bethe $ln(E)$ behavior of $\Omega$ at higher energy. The background of $\Omega$ rises toward forming a plateau.  Table 2: Dipole allowed transitions between levels $i$ and $j$, their $f$ and A-values, and transition wavelength $\lambda_{ij}$ in Angstrom unit for the illustrated examples of $\Omega$ of Ca IV in Figures 4 and 5. The level indices $i$ and $j$ correspond to those in Table 1. Figure | i - j | $SL\pi J$ | $\lambda_{ij}$ | $f_{ij}$ |  $A_{ji}$(s${}^{-1)})$
---|---|---|---|---|---
|  | $i$ | $j$ | ($\AA$) |  |
4a | 2 - 3 | ${}^{2}P^{o}_{1/2}$ | ${}^{2}S_{1/2}$ | 670 | 1.55E-02 | 2.01E+08
4b | 1 -18 | ${}^{2}P^{o}_{3/2}$ | ${}^{2}D_{5/2}$ | 435 | 8.14E-03 | 1.98E+08
5a | 1 -33 | ${}^{2}P^{o}_{3/2}$ | ${}^{2}D_{5/2}$ | 319 | 2.60E+00 | 1.14E+11
5b | 1 -30 | ${}^{2}P^{o}_{3/2}$ | ${}^{2}P_{3/2}$ | 332 | 1.09E+00 | 7.10E+10
|  |  |  |  |  |

Figure 5 presents $\Omega$ for dipole allowed excitation of a)
$3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-(3s^{2}3p^{4}3d)^{2}D_{5/2}$ at 340 $\AA$ and
b) $3s^{2}3p^{5}~{}^{2}P^{o}_{3/2}-(3s^{2}3p^{4}4s)~{}^{2}P_{3/2}$ at 332
$\AA$ in Ca IV. The high peak resonances are converging near 1 Ry. Although
there are much more core ion excitations included in the wavefunction
expansion, the Rydberg series of resonances belonging to them have become weak
with almost no contributions to $\Omega$. This indicates convergence in the
wavefunction expansion for generating resonant features. However, the
background of $\Omega$ is rising with energy toward forming a plateau. This
trend is in agreement with the expected Coulomb-Bethe behavior of
$\Omega_{if}(E)\sim_{E\rightarrow\infty}dln(E)$ (Eq.10) at high energy for
transitions with stronger or larger values for oscillator strength or
$A$-values. The $f$\- and $A$-values for for the two transitions are given in
Table 2. As discussed above, for these transitions, larger number of partial
waves needs to be considered in calculating $\Omega$. The rising trend toward
a plateau may not result in with inadequate number of partial waves. In such
case, $\Omega$ is often extrapolated with Coulomb-Bethe form.

Astrophysical models require temperature dependent effective collision
strength ($\Upsilon$). Table 3 presents $\Upsilon$ for some excitations to
levels that could be of importance for astrophysical applications. $\Upsilon$
for any other excitation of the 1431 transitions can be computed by averaging
$\Omega$ over Maxwellian distribution function at any temperature. The
$\Omega$ values, and $\Upsilon$ for a number of excitations, in addition to
the ones presented here, are available electronically at NORAD-Atomic-Data
norad .

Table 3:  $\Upsilon$ at various temperatures for different excitations of levels 1-2 to 1-18 in Ca IV.  logT | T (K) |  $\Upsilon$: 1- 2 | 1- 3 | 2- 3 | 1- 4 | 1- 5
---|---|---|---|---|---|---
2.00 | 1.000E+02 | 9.312E-01 | 1.038E+00 | 5.709E-01 | 3.274E+01 | 2.158E-01
2.10 | 1.265E+02 | 9.339E-01 | 1.061E+00 | 5.674E-01 | 3.199E+01 | 2.395E-01
2.20 | 1.600E+02 | 9.382E-01 | 1.075E+00 | 5.587E-01 | 3.104E+01 | 2.827E-01
2.31 | 2.024E+02 | 9.439E-01 | 1.077E+00 | 5.447E-01 | 2.986E+01 | 3.545E-01
2.41 | 2.560E+02 | 9.507E-01 | 1.065E+00 | 5.261E-01 | 2.841E+01 | 4.608E-01
2.51 | 3.237E+02 | 9.581E-01 | 1.040E+00 | 5.037E-01 | 2.669E+01 | 6.014E-01
2.61 | 4.095E+02 | 9.654E-01 | 1.002E+00 | 4.791E-01 | 2.474E+01 | 7.688E-01
2.71 | 5.179E+02 | 9.722E-01 | 9.568E-01 | 4.537E-01 | 2.261E+01 | 9.507E-01
2.82 | 6.551E+02 | 9.785E-01 | 9.081E-01 | 4.294E-01 | 2.042E+01 | 1.134E+00
2.92 | 8.286E+02 | 9.851E-01 | 8.608E-01 | 4.081E-01 | 1.826E+01 | 1.315E+00
3.02 | 1.048E+03 | 9.948E-01 | 8.192E-01 | 3.909E-01 | 1.623E+01 | 1.503E+00
3.12 | 1.326E+03 | 1.014E+00 | 7.854E-01 | 3.784E-01 | 1.440E+01 | 1.724E+00
3.22 | 1.677E+03 | 1.054E+00 | 7.604E-01 | 3.702E-01 | 1.284E+01 | 2.002E+00
3.33 | 2.121E+03 | 1.127E+00 | 7.436E-01 | 3.654E-01 | 1.156E+01 | 2.335E+00
3.43 | 2.683E+03 | 1.242E+00 | 7.345E-01 | 3.634E-01 | 1.055E+01 | 2.690E+00
3.53 | 3.393E+03 | 1.395E+00 | 7.321E-01 | 3.635E-01 | 9.865E+00 | 3.018E+00
3.63 | 4.292E+03 | 1.572E+00 | 7.354E-01 | 3.655E-01 | 9.610E+00 | 3.292E+00
3.73 | 5.429E+03 | 1.752E+00 | 7.433E-01 | 3.694E-01 | 9.976E+00 | 3.553E+00
3.84 | 6.866E+03 | 1.920E+00 | 7.551E-01 | 3.750E-01 | 1.113E+01 | 3.947E+00
3.94 | 8.685E+03 | 2.064E+00 | 7.705E-01 | 3.844E-01 | 1.312E+01 | 4.689E+00
4.04 | 1.099E+04 | 2.185E+00 | 7.909E-01 | 4.056E-01 | 1.577E+01 | 5.927E+00
4.14 | 1.389E+04 | 2.294E+00 | 8.220E-01 | 4.587E-01 | 1.869E+01 | 7.609E+00
4.24 | 1.758E+04 | 2.411E+00 | 8.747E-01 | 5.748E-01 | 2.139E+01 | 9.464E+00
4.35 | 2.223E+04 | 2.560E+00 | 9.640E-01 | 7.800E-01 | 2.341E+01 | 1.113E+01
4.45 | 2.812E+04 | 2.759E+00 | 1.101E+00 | 1.073E+00 | 2.448E+01 | 1.232E+01
4.55 | 3.556E+04 | 3.016E+00 | 1.285E+00 | 1.419E+00 | 2.455E+01 | 1.288E+01
4.65 | 4.498E+04 | 3.353E+00 | 1.497E+00 | 1.762E+00 | 2.373E+01 | 1.282E+01
4.76 | 5.690E+04 | 3.841E+00 | 1.708E+00 | 2.047E+00 | 2.222E+01 | 1.224E+01
4.86 | 7.197E+04 | 4.576E+00 | 1.886E+00 | 2.241E+00 | 2.024E+01 | 1.130E+01
4.96 | 9.103E+04 | 5.599E+00 | 2.007E+00 | 2.331E+00 | 1.803E+01 | 1.015E+01
5.06 | 1.151E+05 | 6.818E+00 | 2.060E+00 | 2.323E+00 | 1.574E+01 | 8.907E+00
5.16 | 1.456E+05 | 8.028E+00 | 2.045E+00 | 2.234E+00 | 1.353E+01 | 7.674E+00
5.27 | 1.842E+05 | 8.996E+00 | 1.975E+00 | 2.088E+00 | 1.147E+01 | 6.514E+00
5.37 | 2.330E+05 | 9.559E+00 | 1.865E+00 | 1.907E+00 | 9.610E+00 | 5.464E+00
5.47 | 2.947E+05 | 9.661E+00 | 1.733E+00 | 1.711E+00 | 7.982E+00 | 4.541E+00
5.57 | 3.728E+05 | 9.347E+00 | 1.592E+00 | 1.516E+00 | 6.583E+00 | 3.745E+00
5.67 | 4.715E+05 | 8.718E+00 | 1.453E+00 | 1.332E+00 | 5.398E+00 | 3.072E+00
5.78 | 5.964E+05 | 7.893E+00 | 1.324E+00 | 1.165E+00 | 4.408E+00 | 2.509E+00
5.88 | 7.543E+05 | 6.978E+00 | 1.207E+00 | 1.018E+00 | 3.589E+00 | 2.044E+00
5.98 | 9.541E+05 | 6.058E+00 | 1.105E+00 | 8.908E-01 | 2.917E+00 | 1.663E+00
6.08 | 1.207E+06 | 5.187E+00 | 1.017E+00 | 7.837E-01 | 2.370E+00 | 1.352E+00
6.18 | 1.526E+06 | 4.398E+00 | 9.435E-01 | 6.943E-01 | 1.927E+00 | 1.101E+00
6.29 | 1.931E+06 | 3.705E+00 | 8.822E-01 | 6.205E-01 | 1.570E+00 | 8.981E-01
6.39 | 2.442E+06 | 3.112E+00 | 8.317E-01 | 5.602E-01 | 1.283E+00 | 7.353E-01
6.49 | 3.089E+06 | 2.612E+00 | 7.906E-01 | 5.113E-01 | 1.053E+00 | 6.050E-01
6.59 | 3.907E+06 | 2.198E+00 | 7.572E-01 | 4.718E-01 | 8.700E-01 | 5.010E-01
6.69 | 4.942E+06 | 1.857E+00 | 7.303E-01 | 4.400E-01 | 7.238E-01 | 4.181E-01
6.80 | 6.251E+06 | 1.579E+00 | 7.087E-01 | 4.146E-01 | 6.075E-01 | 3.521E-01
6.90 | 7.906E+06 | 1.354E+00 | 6.914E-01 | 3.942E-01 | 5.151E-01 | 2.997E-01
7.00 | 1.000E+07 | 1.173E+00 | 6.776E-01 | 3.780E-01 | 4.417E-01 | 2.580E-01
logT | T (K) |  $\Upsilon$: 1- 6 | 1- 7 | 1-12 | 1-13 | 1-18
2.00 | 1.000E+02 | 1.307E+00 | 4.810E-01 | 4.444E-01 | 8.154E-01 | 9.113E-01
2.10 | 1.265E+02 | 1.221E+00 | 4.936E-01 | 4.540E-01 | 8.560E-01 | 9.043E-01
2.20 | 1.600E+02 | 1.164E+00 | 5.075E-01 | 4.647E-01 | 9.006E-01 | 8.956E-01
2.31 | 2.024E+02 | 1.136E+00 | 5.222E-01 | 4.758E-01 | 9.425E-01 | 8.845E-01
2.41 | 2.560E+02 | 1.131E+00 | 5.374E-01 | 4.866E-01 | 9.756E-01 | 8.705E-01
2.51 | 3.237E+02 | 1.147E+00 | 5.526E-01 | 4.963E-01 | 9.969E-01 | 8.528E-01
2.61 | 4.095E+02 | 1.175E+00 | 5.670E-01 | 5.042E-01 | 1.008E+00 | 8.304E-01
2.71 | 5.179E+02 | 1.210E+00 | 5.797E-01 | 5.102E-01 | 1.014E+00 | 8.023E-01
2.82 | 6.551E+02 | 1.246E+00 | 5.902E-01 | 5.195E-01 | 1.028E+00 | 7.675E-01
2.92 | 8.286E+02 | 1.285E+00 | 5.983E-01 | 5.546E-01 | 1.077E+00 | 7.260E-01
|  |  |  |  |  |

Table 3 continues
logT T (K) $\Upsilon$: 1- 6 1- 7 1-12 1-13 1-18 3.02 1.048E+03 1.335E+00
6.038E-01 6.781E-01 1.206E+00 6.795E-01 3.12 1.326E+03 1.414E+00 6.065E-01
1.021E+00 1.485E+00 6.321E-01 3.22 1.677E+03 1.533E+00 6.080E-01 1.804E+00
2.005E+00 5.891E-01 3.33 2.121E+03 1.690E+00 6.163E-01 3.326E+00 2.872E+00
5.558E-01 3.43 2.683E+03 1.873E+00 6.615E-01 5.914E+00 4.173E+00 5.359E-01
3.53 3.393E+03 2.079E+00 8.171E-01 9.770E+00 5.920E+00 5.317E-01 3.63
4.292E+03 2.336E+00 1.199E+00 1.478E+01 7.988E+00 5.465E-01 3.73 5.429E+03
2.688E+00 1.911E+00 2.039E+01 1.011E+01 5.866E-01 3.84 6.866E+03 3.169E+00
2.955E+00 2.576E+01 1.198E+01 6.590E-01 3.94 8.685E+03 3.769E+00 4.202E+00
3.003E+01 1.332E+01 7.674E-01 4.04 1.099E+04 4.427E+00 5.432E+00 3.264E+01
1.402E+01 9.097E-01 4.14 1.389E+04 5.047E+00 6.429E+00 3.343E+01 1.406E+01
1.080E+00 4.24 1.758E+04 5.531E+00 7.061E+00 3.259E+01 1.357E+01 1.273E+00
4.35 2.223E+04 5.813E+00 7.294E+00 3.053E+01 1.268E+01 1.480E+00 4.45
2.812E+04 5.870E+00 7.175E+00 2.768E+01 1.154E+01 1.685E+00 4.55 3.556E+04
5.718E+00 6.789E+00 2.446E+01 1.029E+01 1.865E+00 4.65 4.498E+04 5.401E+00
6.230E+00 2.117E+01 9.026E+00 1.993E+00 4.76 5.690E+04 4.969E+00 5.576E+00
1.802E+01 7.809E+00 2.053E+00 4.86 7.197E+04 4.469E+00 4.890E+00 1.514E+01
6.679E+00 2.038E+00 4.96 9.103E+04 3.942E+00 4.216E+00 1.258E+01 5.656E+00
1.956E+00 5.06 1.151E+05 3.420E+00 3.583E+00 1.036E+01 4.749E+00 1.825E+00
5.16 1.456E+05 2.926E+00 3.009E+00 8.470E+00 3.957E+00 1.663E+00 5.27
1.842E+05 2.474E+00 2.502E+00 6.885E+00 3.277E+00 1.487E+00 5.37 2.330E+05
2.072E+00 2.064E+00 5.571E+00 2.700E+00 1.312E+00 5.47 2.947E+05 1.722E+00
1.691E+00 4.490E+00 2.215E+00 1.145E+00 5.57 3.728E+05 1.423E+00 1.378E+00
3.608E+00 1.812E+00 9.941E-01 5.67 4.715E+05 1.170E+00 1.118E+00 2.892E+00
1.479E+00 8.604E-01 5.78 5.964E+05 9.593E-01 9.047E-01 2.314E+00 1.207E+00
7.448E-01 5.88 7.543E+05 7.849E-01 7.303E-01 1.849E+00 9.842E-01 6.467E-01
5.98 9.541E+05 6.418E-01 5.888E-01 1.477E+00 8.038E-01 5.646E-01 6.08
1.207E+06 5.252E-01 4.744E-01 1.179E+00 6.582E-01 4.965E-01 6.18 1.526E+06
4.307E-01 3.824E-01 9.408E-01 5.410E-01 4.407E-01 6.29 1.931E+06 3.544E-01
3.087E-01 7.513E-01 4.471E-01 3.953E-01 6.39 2.442E+06 2.931E-01 2.497E-01
6.007E-01 3.719E-01 3.585E-01 6.49 3.089E+06 2.440E-01 2.027E-01 4.809E-01
3.119E-01 3.288E-01 6.59 3.907E+06 2.047E-01 1.652E-01 3.859E-01 2.641E-01
3.050E-01 6.69 4.942E+06 1.734E-01 1.354E-01 3.106E-01 2.261E-01 2.859E-01
6.80 6.251E+06 1.485E-01 1.117E-01 2.508E-01 1.959E-01 2.707E-01 6.90
7.906E+06 1.287E-01 9.292E-02 2.035E-01 1.719E-01 2.586E-01 7.00 1.000E+07
1.129E-01 7.802E-02 1.661E-01 1.529E-01 2.490E-01

###  4.2 Energy levels and radiative transition parameters of Ca IV

We present about 93000 radiative transition rates for Ca IV obtained from
atomic structure calculations in Breit-Pauli approximation implemented in code
SUPERSTRUCTURE. The 13 configurations of Ca IV, listed in Computation section,
resulted in 387 fine structure energy levels, and 93,296 radiative transitions
of types allowed (E1) and forbidden (E2, E3, M1, M2).

Our calculated energies for Ca IV have been compared with the measured values
in Table 1 in the Computation section. They are in good agreement with the
observed energies available at NIST table. As seen in Table 1, the differences
between calculated and measured energies are within 5% for most of the levels
except for the highest lying ${}^{4}D^{o}$ and ${}^{4}F^{o}$ states where the
differences are about 8%.

The A-values have been benchmark with limited number of available data. The
present transition probabilities are compared, in Table 4, with the only 3
available A-values in the NIST compilation, and with those available from
other sources. Transition $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow
3s^{2}3p^{5}(^{2}P^{o}_{1/2})$ can be both M1 and E2 types. The present
A-value for the M1 transition, 0.545, is in excellent agreement with 0.543 of
Naqvi n51 . The present value is also in good agreement with, 0.379 by Huang
et al h83  given the typical magnitude of A-values of the order of 108 s-1 or
higher for E1 transitions. Similarly for the very weak E2 transition, the
present $A$-value, 3.82e-05 sec-1 is also in very good agreement with,
1.906E-05 sec-1, of Huang et al h83 . For the first dipole allowed transitions
$3s^{2}3p^{5}(^{2}P^{o}_{3/2,1/2})\rightarrow 3s3p^{6}(^{2}S_{1/2})$ present
A-values agree very well with those of Wilson et al w00 , but differ from
Huang et al and Varsavsky v61  who also differ from each other and from Wilson
et al. As Table 4 shows, the present results are in agreement with those of
Wilson et al for other transitions.

We are providing two calculated A-values for the dipole allowed transitions in
order to compare the accuracy, the first A-value has been obtained using the
experimental transition energy and the second one, below it, using the
calculated energy. We can see that they themselves do not differ significantly
from each other. Some differences among the results are expected due to use of
different optimization of the configurations included in each calculations and
the accuracy in the methods considered for calculations of the transition
parameters.

Table 4: Comparison of present A-values for Ca-IV with those available in literature. For the E1 transitions, the first A-value from the present work represents use of experimental transition energy while the second one (below it) of calculated energy. K, KP are the initial and final transitional energy level indices (as given in Table 1), $\lambda$ is the transition wavelength in $\AA$ unit. The references are given in superscripts.  K | KP | $\lambda$ | Transition | A(Present) | A(Others)
---|---|---|---|---|---
|  |  ($\AA$) |  |  (sec-1) |  (sec-1)
1 | 2 |  | $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow 3s^{2}3p^{5}(^{2}P^{o}_{1/2})$ | M1:0.545 |  M1:0.541${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{n51}{\@@citephrase{(}}{\@@citephrase{)}}}}}$,0.3786${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{h83}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
1 | 2 |  | $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow 3s^{2}3p^{5}(^{2}P^{o}_{1/2})$ | E2:3.82e-5 |  E2:1.906E-5${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{h83}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
1 | 3 | 656 | $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow 3s3p^{6}(^{2}S_{1/2})$ | 5.11E+8 |  7.425E+8${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$,1.20E+10${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{v61}{\@@citephrase{(}}{\@@citephrase{)}}}}}$,
|  |  |  | 4.22E+8 |  1.09e+10${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{h83}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
2 | 3 | 669.7 | $3s^{2}3p^{5}(^{2}P^{o}_{1/2})\rightarrow 3s3p^{6}(^{2}S_{1/2})$ | 2.466E+8 |  3.529E+8${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$, 5.4E+9${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{v61}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
|  |  |  | 2.01E+8 |
1 | 10 | 454.6 | $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow 3s^{2}3p^{4}3d(^{4}F_{5/2})$ | 1.84E+6 |  1.692E+6${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
1 | 12 | 543 | $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow 3s^{2}3p^{4}3d(^{2}P_{1/2})$ | 9.696E+6 |  7.889E+6${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
|  |  |  | 9.57E+6 |
2 | 12 | 459.5 | $3s^{2}3p^{5}(^{2}P^{o}_{1/2})\rightarrow 3s^{2}3p^{4}3d(^{2}P_{1/2})$ | 3.930E+7 |  3.503E+7${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
|  |  |  | 3.87E+7 |
1 | 14 | 440 | $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow 3s^{2}3p^{4}3d(^{4}P_{1/2})$ | 1.350E+7 |  1.243E+7${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
|  |  |  | 1.31E+7 |
1 | 15 | 439 | $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow 3s^{2}3p^{4}3d(^{4}P_{3/2})$ | 3.90E+6 |  5.367E+6${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
|  |  |  | 3.77E+6 |
1 | 16 | 437.3 | $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow 3s^{2}3p^{4}3d(^{4}P_{5/2})$ | 2.740E+6 |  2.754E+6${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
|  |  |  | 2.66E+6 |
1 | 17 | 437.8 | $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow 3s^{2}3p^{4}3d(^{2}D_{3/2})$ | 6.530E+7 |  5.201E+7${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
|  |  |  | 6.45E+7 |
1 | 18 | 434.6 | $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow 3s^{2}3p^{4}3d(^{2}D_{5/2})$ | 1.920E+8 |  1.576E+8${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
|  |  |  | 1.90E+8 |
1 | 27 | 341.3 | $3s^{2}3p^{5}(^{2}P^{o}_{3/2})\rightarrow 3s^{2}3p^{4}3d(^{2}S_{1/2})$ | 6.23E+10 |  4.264E+10${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$,6.543E+10${}^{\mbox{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{w00}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
|  |  |  | 6.43E+10 |
|  |  |  |  |

The present work provides lifetimes of all 386 excited levels in a file that
is available electronically at NORAD-Atomic-Data norad . Lifetime of an
excited level can be calculated if the A-values or the radiative decay rates
of the level to the lower levels are known. Lifetimes are measurable at
experimental set-ups. Table 5 presents lifetimes of a few levels to illustrate
their values. For each excited level in the table, the level number,
configuration number, spectroscopic notation and energy are given. This line
is followed by the A-values of the level decaying to lower levels. The
A-values are added and the sum is inverted to obtain the lifetime. Levels
decaying through the forbidden transitions (E2,E3,M1,M2) have longer lifetimes
than those through dipole allowed transitions (same spin E1d and
intercombination E1i). No lifetimes for Ca IV levels were found in literature
for comparison. However, their accuracies are related to those of the A-values
which have been discussed for Table 4.

Table 5: Lifetimes of a few levels illustrating the complete table of lifetimes of all 386 levels. Radiative decay rates of level j to various lower levels $j\rightarrow i$ are given. Notation C is for configuration, g for statistical weight factor, lv for level, f for f-value for an E1 transition or S-values for E2, E3, M1, M2 transitions, A for A-value and E for transition energy.  Type | LSi | Ci | gi | lvi | LSj | Ci | gj | lvj | f(E1)/S(E2, | Aji | Eij
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  | E3,M1,M2) |  (s-1) |  ($\AA$)
lifetime: sslevel j= 2, Cf= 1, 2P${}^{o}_{1/2}$ [E= 3.019E-02 Ry=
3.3134852E+03 /cm]
E2 | ${}^{2}P^{o}$ | 1 | 4 | 1 | ${}^{2}P^{o}$ | 1 | 2 | 2 | 1.54E+00 | 5.180E-05 | 3.0180E+04
M1 | ${}^{2}P^{o}$ | 1 | 4 | 1 | ${}^{2}P^{o}$ | 1 | 2 | 2 | 1.33E+00 | 6.540E-01 | 3.0180E+04
Summed A-values: Af (forbidden)= 6.541E-01, Aa (allowed)= 0.000E+00 s-1
Total Sum(Af+Aa)= Aji ( 2 transitions) to the level= 6.541E-01 s-1
Lifetime (=1/Aji)= 1.529E+00 s
lifetime: sslevel j= 7, cf= 3, 44D1/2 [E= 1.826E+00 Ry= 2.0041544E+05 /cm]
E1i | ${}^{2}P^{o}$ | 1 | 4 | 1 | ${}^{4}D$ | 3 | 2 | 7 | 1.01E-06 | 5.420E+04 | 4.9896E+02
E1i | ${}^{2}P^{o}$ | 1 | 2 | 2 | ${}^{4}D$ | 3 | 2 | 7 | 3.96E-06 | 1.030E+05 | 5.0735E+02
E2 | ${}^{2}S$ | 2 | 2 | 3 | ${}^{4}D$ | 3 | 2 | 7 | 0.00E+00 | 0.000E+00 | 1.7417E+03
M1 | ${}^{2}S$ | 2 | 2 | 3 | ${}^{4}D$ | 3 | 2 | 7 | 3.18E-07 | 8.120E-04 | 1.7417E+03
E2 | ${}^{4}D$ | 3 | 6 | 5 | ${}^{4}D$ | 3 | 2 | 7 | 1.15E-03 | 2.890E-12 | 2.0196E+05
E2 | ${}^{4}D$ | 3 | 6 | 5 | ${}^{4}D$ | 3 | 2 | 7 | 1.15E-03 | 2.890E-12 | 2.0196E+05
E2 | ${}^{4}D$ | 3 | 4 | 6 | ${}^{4}D$ | 3 | 2 | 7 | 2.69E-03 | 1.120E-13 | 4.5789E+05
M1 | ${}^{4}D$ | 3 | 4 | 6 | ${}^{4}D$ | 3 | 2 | 7 | 5.97E+00 | 8.380E-04 | 4.5789E+05
Summed A-values: Af (forbidden)= 1.650E-03, Aa (allowed)= 1.572E+05 s-1
Total Sum(Af+Aa)= Aji( 8 transitions) to the level= 1.572E+05 s-1
Lifetime (=1/Aji)= 6.361E-06 s
|  |  |  |  |  |  |  |  |  |  |

##  5 Conclusion

We have studied collision strength of Ca IV using a 54-levels close coupling
wave function expansion that corresponds to target ion excitations to high
lying levels. This ensures inclusion of converged contributions of resonances
generated by all the levels. We have demonstrated the effect of number of
partial waves in the collision strength $\Omega$ and showed convergence of
partial waves contributing to collision strengths.

Features of $\Omega$ show resonances in the low energy region but they
converge to the background much before reaching the highest 54th excitation of
Ca IV. We find that $\Omega$ of the emission line of 3.2 $\mu$m due to
collisional excitation of ${}^{2}P^{o}_{3/2}-^{2}P^{o}_{1/2}$ of ground
configuration $3s^{2}3p^{5}$ has extensive resonances with enhanced background
in the low energy region. This has resulted in a strong effective collision
strength $\Upsilon$ with a peak around 3 $\times 10^{5}$ K indicating a
distinct presence of an emission line when the environmental plasma effects
are low. The 3.2 $\mu$m line is within the wavelength range of JWST.

The present $\Omega$ has shown expected features at high electron energy, such
as, decaying background for the forbidden transitions, slow decay or almost
constant value for weak dipole transitions and rising trend of Coulomb Bethe
ln(E) behavior toward a plateau for strong dipole allowed transitions.

We present a set of over 93,000 radiative transitions among 387 energy levels
with orbitals going up to 5s in Ca IV. Results include lifetimes of all 386
excited levels.

The present results are expected to be accurate and large enough for the two
processes to provide a complete astrophysical modeling for all practical
purposes.

Data availability

All atomic data for energies, radiative transitions, collisional excitations,
and effective collision strengths of a set of transition are available online
at the NORAD-Atomic-Data database at the Ohio State University at:
https://norad.astronomy.osu.edu/

Acknowledgments

All computations were carried on the high performance computers of the Ohio
Supercomputer Center. BS acknowledges of IRSIP fellowship from the Government
of Pakistan to carry out the research at the Ohio State University.

## Declarations

Both authors, S.N. Nahar and B. Shafique, contributed equally on the contents
of the paper. While SNN trained BS, set up the project, wrote necessary
program, and remained engaged in studying the project, BS picked up all
aspects of computations, carried out computations, and was engaged in the
analysis.

## References

  * (1) Shore SN, Augusteijn T, Ederoclite A, and Uthas H 2011 A&A 533 L8
  * (2) Asplund M, Grevesse N, Sauval AJ, and Scott P, 2009 Annu. Rev. Astron. Astrophys. 47, 481–52
  * (3) Shepherd J 2019, https://www.astrobin.com/full/406485/0/
  * (4) Jacobson-Galan W V et al, 2020, Astrophys. J. 898 166
  * (5) Feuchtgruber H et al 1997 ApJ 487 962
  * (6) Ignanc R, Cassinelli JP, Quigley M, Babler B 2001 ApJ 558 771-779
  * (7) Feuchtgruber H, Lutz D, Beintema D A 2001 ApJS 136 221
  * (8) Zuckerman B, Koester D, Reid I N, and Hunsch M 2003 ApJ 596 477
  * (9) Pradhan A K and Nahar S N 2011 Atomic Astrophysics and Spectroscopy (Cambridge University press, New York)
  * (10) Sugar J and Corliss C 1985 J. Phys. Chem. Ref. Data 14, Suppl. 2 1-664
  * (11) https://physics.nist.gov/PhysRefData/ASD/levels_form.html
  * (12) Naqvi A M 1951 PhD Thesis Harvard University 192pp (data available at NIST)
  * (13) Varsavsky C M 1961 Astrophys. J., Suppl. Ser. 6 75-107
  * (14) Fawcett B C and Gabriel A H 1966 Proc. Phys. Soc. 88 262
  * (15) Gabriel A H, Fawcett B C, Jordan C 1966 Proc. Phys. Soc. 87 825
  * (16) Huang K -N, Kim Y -K, Cheng K T and Desclaux J P 1983 ADNDT 28 355-377
  * (17) Wilson N J, Hibbert A and Bell K L, 2000 Phys.Scr. 61 603-610
  * (18) Hummer D G, Berrington K A, Eissner W, Pradhan A K, Saraph H E, and Tully J A 1993 Astron Astrophys 279 298-309
  * (19) Berrington K A, Eissner W, and Norrington P H 1995 Comput Phys Commun 92 290-420
  * (20) Eissner W, Jones M and Nussbaumer H 1974 Comput. Phys. Commun. 8, 270-306
  * (21) Nahar S N, Eissner W, Chen G X and Pradhan A K 2003 A&A 408 789
  * (22) Burgess A and Tully JA 1978 J. Phys. B 11 4271
  * (23) Burke P G and Robb W D 1975 Adv. At. Mol. Phys. 11 143-214
  * (24) Scott N S and Burke P G 1980 J. Phys. B 12 4299
  * (25) Scott N S and Taylor K T 1982 Comput. Phys. Commun. 25 347
  * (26) Nahar S N, written in 2013, unpublished
  * (27) NORAD-Atomic-Data: https://norad.astronomy.osu.edu/