###### Abstract

The supersymmetric analysis of spinning cosmic string spacetime, involving an
electron in magnetic fields, has been conducted. We examined the Dirac system
within extended special functions known as exceptional orthogonal polynomials.
Corresponding Dirac system is transformed to a relativistic system with a
nonlinear isotonic oscillator. Furthermore, new potential models that extend
the radial oscillator by adding rational terms are expressed in terms of the
exceptional orthogonal Laguerre $X_{m}$ polynomial. The necessary analyses of
the potential, energy levels, and probability density graphs are introduced
for various cosmic string topological defects and Aharonov-Bohm interaction
parameters.

, $a$ is the rotation parameter given in terms of angular momentum $J$,
$a=4J$. The interval for the parameters can be written as $-\infty<z<\infty$,
$r\geq 0$, $0\leq\varphi\leq 2\pi$. The metric is proposed as [24]

$ds^{2}=dt^{2}+2adtd\varphi-(\alpha^{2}r^{2}-a^{2})d\varphi^{2}-dr^{2}-dz^{2},$
(2)

where the $\alpha^{2}r^{2}-a^{2}<0$ a condition related to closed timelike
curves [27], [28], the spacetime admits closed timelike curves if
$r<\frac{|a|}{\alpha}$. The dynamics equation for a spin-$1/2$ particle,
considering its interaction with rotation and external magnetic fields, is
given by

$i\gamma^{\mu}(x)(\nabla_{\mu}+ieA_{\mu}(x))\Psi(x)=M\Psi(x),$ (3)

where covariant derivaive is $\nabla_{\mu}=\partial_{\mu}+\Gamma_{\mu}$. Dirac
matrices $\gamma^{\mu}(x)$ satisfy the expression

$\gamma^{\mu}(x)=e_{a}^{\mu}(x)\gamma^{a},$ (4)

where $e^{\mu}_{a}$ are the tetrad fields, $\gamma^{a}$ are the flat matrices
in Minkowski spacetime where

$\gamma^{0}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\
\end{array}\right),~{}~{}\gamma^{i}=\left(\begin{array}[]{cc}0&\sigma^{i}\\\
\sigma^{i}&0\\\ \end{array}\right).$ (5)

The anti-commutation of the Dirac matrices give metric tensor, i.e.

$\\{\gamma^{\mu}(x),\gamma^{\nu}(x)\\}=2g^{\mu\nu}(x).$ (6)

In the covariant derivative definition, the spin affine connection reads as

$\Gamma_{\mu}(x)=\frac{1}{4}\gamma^{a}\gamma^{b}e^{\mu}_{a}(x)(\partial_{\mu}e_{b\nu}(x)-\Gamma^{\sigma}_{\mu\nu}e_{b\sigma}(x)),$
(7)

where $\Gamma^{\sigma}_{\mu\nu}$ are the Christoffel symbols. It is understood
that tetrads are coefficients of components of an orthogonal basis, and the
metric tensor can be transformed from a coordinate basis to the tetrad basis.
Here, the tetrad basis is used as follows,

$e^{a}_{\mu}(x)=\left(\begin{array}[]{cccc}1&0&a&0\\\
0&\cos\varphi&-r\alpha\sin\varphi&0\\\ 0&\sin\varphi&r\alpha\cos\varphi&0\\\
0&0&0&1\\\
\end{array}\right),~{}~{}e^{\mu}_{a}(x)=\left(\begin{array}[]{cccc}1&\frac{a\sin\varphi}{r\alpha}&-\frac{a\cos\alpha}{r\alpha}&0\\\
0&\cos\varphi&\sin\varphi&0\\\
0&-\frac{\sin\varphi}{r\alpha}&\frac{\cos\varphi}{r\alpha}&0\\\ 0&0&0&1\\\
.\end{array}\right)$ (8)

Then, spinor affine connection reads as

$\Gamma_{\mu}=(0,0,\frac{i}{2}(1-\alpha)\Sigma,0)$ (9)

where

$\Sigma=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&1&0\\\
0&0&0&-1\\\ \end{array}\right).$ (10)

Because (3) includes magnetic field interactions, vector potential can be
expressed as

A $\displaystyle=$ $\displaystyle[0,A_{\varphi},0]$ (11) $\displaystyle
A_{\varphi}(r)$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\alpha
Br^{2}-\frac{\Phi}{e}$ (12)

where $\Phi$ is the magnetic flux through the solenoid. In the presence of A,
as given in (11), the wavefunction of an electron is influenced by this vector
potential, even the magnetic field
$\textbf{B}=\nabla\times\textbf{A}=B_{z}\hat{k}$ is zero. $A_{\varphi}(r)$,
exhibiting cylindrical symmetry, would affect the particle’s wavefunction in a
manner similar to that in an electron-nucleus system. Therefore, the Aharonov-
Bohm phenomenon is expected to occur around the cosmic string considered in
this study. Next, the spinor of the fermion field is written as below

$\Psi(t,r,\varphi)=\exp(-iEt)\left(\begin{array}[]{c}\psi_{1}(r,\varphi)\\\
\psi_{2}(r,\varphi)\\\ \end{array}\right)$ (13)

where

$\displaystyle\psi_{1}(r,\varphi)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}e^{im\varphi}\psi_{11}(r)\\\
ie^{i(m+1)\varphi}\psi_{12}(r)\\\
\end{array}\right),~{}~{}~{}~{}\psi_{2}(r,\varphi)=\left(\begin{array}[]{c}e^{im\varphi}\psi_{21}(r)\\\
ie^{i(m+1)\varphi}\psi_{22}(r)\\\ \end{array}\right)$ (18)

Then, the Dirac system is separated as follows,

$\displaystyle-\frac{1}{2M}u^{\prime\prime}_{11}+\left(\frac{\ell_{1}(\ell_{1}+1)}{2Mr^{2}}+V_{1}(r)\right)u_{11}(r)$
$\displaystyle=$ $\displaystyle\varepsilon^{2}u_{11}(r)$ (19)
$\displaystyle-\frac{1}{2M}u^{\prime\prime}_{12}+\left(\frac{\ell_{2}(\ell_{2}+1)}{2Mr^{2}}+V_{2}(r)\right)u_{12}(r)$
$\displaystyle=$ $\displaystyle\varepsilon^{2}u_{12}(r)$ (20)

where $\varepsilon^{2}=E^{2}-M^{2}$ and,

$\displaystyle V_{1}(r)$ $\displaystyle=$
$\displaystyle\frac{B^{2}e^{2}}{8M}r^{2}-\frac{Be}{2M}\left(\ell_{1}+\frac{3}{2}\right)$
(21) $\displaystyle V_{2}(r)$ $\displaystyle=$
$\displaystyle\frac{B^{2}e^{2}}{8M}r^{2}-\frac{Be}{2M}\left(\ell_{2}+\frac{1}{2}\right).$
(22)

Here, wavefunction transformations and angular momentum numbers are defined as
follows,

$\displaystyle u_{1i}(r)$ $\displaystyle=$
$\displaystyle\sqrt{r}\psi_{1i}(r),~{}~{}i=1,2,$ (23) $\displaystyle\ell_{1}$
$\displaystyle=$ $\displaystyle\frac{1+2m-2\alpha+2aE-2\Phi}{2\alpha},$ (24)
$\displaystyle\ell_{2}$ $\displaystyle=$
$\displaystyle\frac{-1-2m-2\alpha-2aE+2\Phi}{2\alpha}.$ (25)

The system defined by equations (19) and (20), with the elements given in (21)
and (22), is known as a three-dimensional oscillator. The solutions to this
system are already known. So, the system, as presented in [29], reads as
follows

$-y^{\prime\prime}(x)+V_{\ell}(x)y(x)=\epsilon y(x)$ (26)
$V_{\ell}(x)=\frac{1}{4}\omega^{2}x^{2}+\frac{\ell(\ell+1)}{x^{2}},$ (27)

and the solutions are given by [29]

$\epsilon^{\ell}_{n}=\omega\left(2n+\ell+\frac{3}{2}\right),~{}~{}~{}~{}n=0,1,2,...$
(28)
$y_{n}(x)=\left(\frac{\omega}{2}\right)^{\frac{1}{2}(\ell+3/2)}\sqrt{\frac{2n!}{\Gamma(n+\ell+3/2)}}~{}x^{\ell+1}e^{-\frac{1}{4}\omega
x^{2}}L^{\ell+1/2}_{n}\left(\frac{1}{2}\omega x^{2}\right),$ (29)

where $L^{\ell+1/2}_{n}(\frac{1}{2}\omega x^{2})$ are the Laguerre
polynomials. Now we can define (19) and (20) in terms of the new variable of
$x$ which is $x=\sqrt{2M}r$ and obtain ,

$\displaystyle-u^{\prime\prime}_{11}(x)+V_{eff}(x)u_{11}(x)$ $\displaystyle=$
$\displaystyle\epsilon^{2}u_{11}(x)$ (30)
$\displaystyle-u^{\prime\prime}_{12}(x)+U_{eff}(x)u_{12}(x)$ $\displaystyle=$
$\displaystyle\epsilon^{2}u_{12}(x),$ (31)

and

$\displaystyle V_{eff}(x)$ $\displaystyle=$
$\displaystyle\frac{\ell_{1}(\ell_{1}+1)}{x^{2}}+\frac{B^{2}e^{2}}{16M^{2}}x^{2}-\frac{Be}{2M}\left(\ell_{1}+\frac{3}{2}\right),$
(32) $\displaystyle U_{eff}(x)$ $\displaystyle=$
$\displaystyle\frac{\ell_{2}(\ell_{2}+1)}{x^{2}}+\frac{B^{2}e^{2}}{16M^{2}}x^{2}-\frac{Be}{2M}\left(\ell_{2}+\frac{1}{2}\right)$
(33)

If we match (26) and (19), we obtain the parameters $\\{\omega,\ell\\}$ in
terms of our parameters as

$\displaystyle\omega$ $\displaystyle=$ $\displaystyle\frac{Be}{2M}$ (34)
$\displaystyle\ell$ $\displaystyle=$
$\displaystyle\ell_{1}~{}~{}~{}~{}or~{}~{}~{}~{}\ell=-1-\ell_{1}.$ (35)

By the way, $\ell_{2}$ can be obtained from $\ell_{1}$ using the mirror
symmetry of the parameters $E,m,\Phi$, i.e,

$E\rightarrow-E,~{}~{}m\rightarrow-m,~{}~{}\Phi\rightarrow-\Phi.$ (36)

Then, we can express $E_{n}$ and $u^{\ell_{1}}_{11,n}(r)$ given as below

$\displaystyle E^{\ell_{1}}_{n}$ $\displaystyle=$
$\displaystyle\frac{-aBe}{2M\alpha}\pm\frac{\sqrt{4a^{2}B^{2}e^{2}-8M\alpha\gamma}}{4M\alpha},~{}~{}\ell=-1-\ell_{1}$
(37)

where $\gamma=Be(1+2m-\alpha-2n\alpha-2\Phi)-2M^{3}\alpha$,

$u^{\ell}_{11,n}(x)=\left(\frac{Be}{2M}\right)^{\frac{1}{2}(\ell_{1}+3/2)}\sqrt{\frac{2n!}{\Gamma(n+\ell_{1}+3/2)}}~{}x^{\ell_{1}+1}e^{-\frac{1}{4}\frac{Be}{2M}x^{2}}L^{\ell_{1}+1/2}_{n}\left(\frac{Be}{4M}x^{2}\right).$
(38)

Here, (19) and (20) are known as shape invariant potentials in the literature
[30]. In the standard SUSY QM, the superpotential is given in terms of the
derivative of the groundstate solutions
$W(r)=-\frac{u^{\prime}_{11,0}(x)}{u_{11,0}(x)}$ and the partner potentials
are expressed using the function $W(x)$,

$\displaystyle V_{eff}(x)=V_{1}(x)$ $\displaystyle=$ $\displaystyle
W^{2}(x)-\frac{dW}{dx}+E^{2}$ (39) $\displaystyle V_{2}(x)$ $\displaystyle=$
$\displaystyle W^{2}(x)+\frac{dW}{dx}+E^{2}.$ (40)

If we call the Hamiltonian operator for each system in (19) and (20) as

$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\mathcal{A}^{{\dagger}}\mathcal{A}=-\frac{d^{2}}{dx^{2}}+V_{1}(x)-E^{2}$
(41) $\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle\mathcal{A}\mathcal{A}^{{\dagger}}=-\frac{d^{2}}{dx^{2}}+V_{2}(x)-E^{2}$
(42)

and

$\mathcal{A}=\frac{d}{dx}+W(x),~{}~{}\mathcal{A}^{{\dagger}}=-\frac{d}{dx}+W(x)$
(43)

where $E$ is the factorization energy, $\mathcal{A}u_{11,0}(x)=0$,
$u_{11,0}(x)$ is the groundstate wavefunction of the corresponding $V_{1}(x)$
potential. Here, $V_{2}(x)$ shares the same energy spectrum with $V_{1}(x)$
except groundstate [30]. We can define now the superpotential as

$W(x)=\frac{Be}{4M}x+\frac{\ell_{1}}{x},$ (44)

where we use the second solution to the angular momentum $\ell=-\ell_{1}-1$.
Hence, we can continue with the extension of the radial oscillator classes of
our system. Before, let’s check the graph of potential functions and energy
levels.

Figure 1: Energy levels given in 37 with respect to parameter $\Phi$.
$M=2,m=1,e=1,n=1$ for each curve while
$(B,a,\alpha)=(100,0.1,0.1),(1,0.5,0.2),(8,0.1,0.45)$ for red/black,
green/brown and for pink and yellow curves.

In Figure $1$, it is observed that as the $\alpha$ topological deflection
parameter with the angular momentum parameter $a$ increase, the energy levels
become less negative.

Figure 2: Effective potential graph of (32)$M=2,m=1,e=1,B=10,\alpha=0.5$ for
each curve while $(\Phi,a)=(0.4,1),(5.0,2),(10,5),(20,2)$ for red, black,
green and orange correspondingly.

Figure 2 shows that for smaller values of the magnetic flux parameter $\Phi$,
the equilibrium point of the potential appears deeper. When $\Phi$ assumes
dramatically greater values, the potential curve approaches zero more rapidly.

Figure 3: Effective potential graph of
(33)$M=2,m=1,e=1,n=1,\Phi=0.10,\alpha=0.1$ for each curve while
${(B,a)}={(0.1,1.2),(0.5,10),(140,50)}$ for black, green and blue curves
correspondingly.  Figure 4: Probability density
$\rho(r)=|u_{11}|^{2}+|u_{12}|^{2}$ graphs where $u_{12}$ are the solutions of
(31). $B=100,\alpha=0.1,\Phi=10,a=0.1$ and n=0,1,2 for green, orange and black
curves correspondingly.  Figure 5: Energy levels given in 37 with respect to
parameter $\alpha$. $M=2,m=1,e=1,a=0.1$ for each curve while
$(\Phi,B)=(0.5,5),(1.3,10),(5.0,50)$ for orange/black, blue/green and for
yellow/magenta curves respectively.

The topological deflection parameter is increased to $\alpha=0.9$ in Figure 3,
which depicts the partner potential graph of (33). As observed in the graph,
as $B$ and $\alpha$ increase, the potential well becomes deeper. Figure $4$
shows the probability density graph is drawn for the solutions given in (38).

In figure $5$, as the value of $\alpha$ increases, the energy becomes more
stable.

###  2.1 Rationally extended radial oscillator model

_ansatz for the superpotential_
Let us give an ansatz for the superpotential which may both give $V_{1}(x)$ in
(39) under some parameter restrictions and also produce the partner potential
$V_{2,p}(x)$ which is different from the one given in (40). Then, we have

$W(x)=\frac{a_{1}}{x}+a_{2}x-\frac{f^{\prime}(x)}{f(x)}.$ (45)

This ansatz in (45) is given in [29], then, we can follow the straightforward
algebra in [29]. Thus, one obtains $V_{1}(x)$ and $V_{2,p}(x)$ as follows

$\displaystyle V_{1}(x)$ $\displaystyle=$ $\displaystyle
a_{2}(2a_{1}-1)+\frac{a_{1}(a_{1}+1)}{x^{2}}+a^{2}_{2}x^{2}+\frac{2a_{1}f^{\prime}(x)}{f(x)}+\frac{2a_{2}xf^{\prime}(x)}{f(x)}+\frac{2f^{\prime
2}}{f^{2}}-\frac{f^{\prime\prime}}{f}$ (46) $\displaystyle V_{2,p}(x)$
$\displaystyle=$ $\displaystyle
a_{2}(2a_{1}+1)+\frac{a_{1}(a_{1}-1)}{x^{2}}+a^{2}_{2}x^{2}+\frac{2a_{1}f^{\prime}(x)}{f(x)}+\frac{2a_{2}xf^{\prime}(x)}{f(x)}+\frac{2f^{\prime
2}}{f^{2}}+\frac{f^{\prime\prime}}{f}.$ (47)

Using (45) in (39) and (40) and $f(x)=x^{2}+c$, one gets

$V_{1}(x)=E^{2}+2a_{1}a_{2}-a_{2}+\frac{a_{1}(a_{1}+1)}{x^{2}}+a^{2}_{2}x^{2}-\frac{4a_{2}x^{2}+4a_{1}-2}{c+x^{2}},$
(48)
$V_{2,p}(x)=E^{2}+2a_{1}a_{2}+a_{2}+\frac{a_{1}(a_{1}-1)}{x^{2}}+a^{2}_{2}x^{2}-\frac{4a_{2}x^{2}+4a_{1}+2}{c+x^{2}}+\frac{8x^{2}}{(c+x^{2})^{2}}.$
(49)

We can eliminate the rational and constant terms in (48) using the parameters

$c=\frac{2a_{1}-1}{2a_{2}},~{}~{}E^{2}=-a_{2}(2a_{1}-5).$ (50)

Then, $V_{2,p}(x)$ is obtained as

$V_{2,p}(x)=V_{1}(x)+2W^{\prime}(x)=a^{2}_{2}x^{2}+\frac{a_{1}(a_{1}-1)}{x^{2}}+\frac{4}{x^{2}+c}-\frac{8c}{(x^{2}+c)^{2}}+2a_{2},$
(51)

where

$a_{1}(a_{1}-1)=\ell_{1}(\ell_{1}+1),~{}~{}~{}~{}a^{2}_{2}=\frac{B^{2}e^{2}}{16M^{2}}.$
(52)

The solutions are given by
$\\{(a_{1},a_{2})=(-\ell_{1},\frac{Be}{4M}),(-\ell_{1},-\frac{Be}{4M}),(1+\ell_{1},\frac{Be}{4M}),(1+\ell_{1},-\frac{Be}{4M})\\}$.
Next, we can focus on the solutions of (51). In the literature, (51) is known
as quantum isotonic nonlinear-oscillator potentials [31], [32]. It is noted
that (51) can be compared to Eq.$(28)$ in [31]. If we apply the supersymmetric
relationship of the wavefunctions

$u_{12}(x)=C_{2}\mathcal{A}^{{\dagger}}u_{11}=C_{2}\left(-\frac{d}{dx}+\frac{\ell_{1}+1}{x}+\frac{Be}{4M}x-\frac{2x}{x^{2}+c}\right)u_{11}(x)$
(53)

where $u_{11}(x)$ is given by (38). Using the differential identity of
Laguerre polynomials given below

$\frac{d}{dz}L^{\alpha}_{n}(z)=-L^{\alpha+1}_{n-1}(z),$ (54)

and

$L^{\alpha}_{n}(z)=L^{\alpha}_{n-1}+L^{\alpha-1}_{n}(z),$ (55)

we obtain

$u_{12}(x)=C_{2}\frac{x^{\ell_{1}+2}e^{-\frac{Bex^{2}}{8M}}}{x^{2}+c^{2}}\left((2\ell_{1}+2n+5)L^{\ell_{1}+1/2}_{n}(x^{\prime})-2(n+1)L^{\ell_{1}+1/2}_{n+1}(x^{\prime})+\frac{Bec^{2}}{4M}L^{\ell_{1}+3/2}_{n}(x^{\prime})\right)$
(56)

where $x^{\prime}=\frac{Be}{4M}x^{2}$.
Figure $6$ is drawn for (51) which also shows that a potential well can be
obtained for the smaller values of $\Phi$. Upon examining Figure $7$ it is
seen that the rationally extended potential in equation (74) differs from
those shown previously. For smaller $\Phi$ values, such as $\Phi=0.1$ there is
singularity at the points $|x|=7.797$. Figure $8$ presents the probability
densities for the solutions given in equations (33) and (56) while Figure $8$
shows the probability densities for the solutions given in equations
$V_{{}_{\textbf{m}}}(x)$ and (56).

Figure 6: Effective partner potential $V_{2,p}(x)$ graphs.
$\Phi=0.5,1.5,0.1,50$ corresponding to black, yellow,red, green curves.

##  3 Exceptional Orthogonal Polynomials

###  3.1 $X_{\textbf{m}}$ exceptional Laguerre polynomials

$X_{1}$-Jacobi and $X_{1}$\- Laguerre are known as exceptional orthogonal
polynomials which exist as two infinite sequences of polynomial eigenfunctions
of a Sturm–Liouville problem. Unlike the classical orthogonal polynomial
systems, their sequences start with a polynomial of degree one. These
polynomials $R(g(x))$ satisfy a second order differential equation which is
given by

$R^{\prime\prime}(g(x))+F(g)R^{\prime}(g(x))+G(g)R(g(x))=0.$ (57)

Here, $F(g)$ and $G(g)$ functions are given as

$\displaystyle F(g)$ $\displaystyle=$
$\displaystyle\frac{1}{g}\left((\alpha^{\prime}+1-g)-2g\frac{L^{\alpha^{\prime}}_{\textbf{m}-1}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}}(-g)\right)$
(58) $\displaystyle G(g)$ $\displaystyle=$
$\displaystyle\frac{1}{g}\left(n-2\alpha^{\prime}\frac{L^{\alpha^{\prime}}_{\textbf{m}-1}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}(-g)}\right),~{}~{}\textbf{m}\geq
0,~{}~{}n\geq\textbf{m},~{}~{}\alpha^{\prime}>0.$ (59)

The norm integral is given by

$\int^{\infty}_{0}(L^{\alpha^{\prime}}_{n,\textbf{m}}(g))^{2}W^{\alpha^{\prime}}_{\textbf{m}}(g)dg=\frac{(\alpha^{\prime}+n)\Gamma(\alpha^{\prime}+n+\textbf{m})}{(n-\textbf{m})!},$
(60)

where

$W^{\alpha^{\prime}}_{n}(g)=\frac{g^{\alpha^{\prime}}e^{-g}}{(L^{\alpha^{\prime}-1}_{\textbf{m}})^{2}}$
(61)

and $X_{m}$ Laguerre polynomials can be written as

$L^{\alpha^{\prime}}_{n,\textbf{m}}=L^{\alpha^{\prime}}_{\textbf{m}}(-g)L^{\alpha^{\prime}-1}_{n-\textbf{m}}+L^{\alpha^{\prime}-1}_{\textbf{m}}(-g)L^{\alpha^{\prime}}_{n-\textbf{m}-1},~{}~{}n\geq\textbf{m}.$
(62)

$X_{m}$-exceptional orthogonal Laguerre polynomials satisfies the differential
equation below

$\frac{d^{2}L^{\alpha^{\prime}}_{n,\textbf{m}}}{dx^{2}}+\frac{1}{g}\left((\alpha^{\prime}+1-g)-2g\frac{L^{\alpha^{\prime}}_{\textbf{m}-1}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}(-g)}\right)\frac{dL^{\alpha^{\prime}}_{n,\textbf{m}}(g)}{dx}+\frac{1}{g}\left(n-2\alpha^{\prime}\frac{L^{\alpha^{\prime}}_{\textbf{m}-1}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}}\right)L^{\alpha^{\prime}}_{n,\textbf{m}}(g)=0.$
(63)

###  3.2 initial equation

Let us mention (57) again. We need to find an equation of our system which
fits (57). When we consider the system (30), there should be a transformation
in order to make (30)-(31) comparable with (57). So we apply
$u_{12}(x)=x\chi(x)$, (30) turns into

$\chi^{{}^{\prime\prime}}(x)+\frac{2}{x}\chi^{{}^{\prime}}(x)+(\varepsilon^{2}-V_{eff}(x))\chi(x)=0.$
(64)

Then, we proceed with the point canonical transformation $\chi(x)=f(x)R(g(x))$
which is substituted into equation (64) [36], yielding

$f(x)=x^{-1}(g^{{}^{\prime}}(x))^{-1/2}\exp\left(\frac{1}{2}\int
Q(x)dx\right),$ (65)
$\varepsilon^{2}-V_{eff}(x)=\frac{1}{2}\frac{g^{\prime\prime\prime}}{g^{\prime}}-\frac{3}{4}\frac{g^{\prime\prime
2}}{g^{\prime 2}}+g^{\prime
2}\left(R(g)-Q^{\prime}(g)-\frac{Q^{2}}{4}\right).$ (66)

Now $Q(g)$ and $R(g)$ correspond to

$\displaystyle Q(g)$ $\displaystyle=$
$\displaystyle\frac{1}{g}\left((\alpha^{\prime}+1-g)-2g\frac{L^{\alpha^{\prime}}_{\textbf{m}-1}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}(-g)}\right),$
(67) $\displaystyle R(g)$ $\displaystyle=$
$\displaystyle\frac{1}{g}\left(n-2\alpha^{\prime}\frac{L^{\alpha^{\prime}}_{\textbf{m}-1}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}}\right).$
(68)

Hence, one can use $Q(g)$ and $R(g)$ in (66), and get

$\begin{split}\varepsilon^{2}-V_{eff,\textbf{m}}(x)&=\frac{1}{2}\frac{g^{\prime\prime\prime}}{g^{\prime}}-\frac{3}{4}\frac{g^{\prime\prime
2}}{g^{\prime 2}}+\\\ &g^{\prime
2}\left(-\frac{1}{4}+\frac{2n+\alpha^{\prime}+1}{2g}-\frac{\alpha^{\prime
2}-1}{4g^{2}}+\frac{L^{\alpha^{\prime}+1}_{\textbf{m}-2}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}}-\frac{\alpha^{\prime}+g-1}{g}\frac{L^{\alpha^{\prime}}_{\textbf{m}-1}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}}-2\left(\frac{L^{\alpha^{\prime}}_{\textbf{m}-1}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}(-g)}\right)^{2}\right).\end{split}$
(69)

On the left handside of (69), the constant $\varepsilon^{2}$ may be obtained
using

$\frac{g^{\prime 2}}{g}=C,$ (70)

where $C$ is a constant. Since a term that is constant and aligns with
$\varepsilon^{2}$, is required, the proposed form for $g(x)$ is presented in
equation (70). Then, $g(x)=\frac{1}{4}Cx^{2}$ can be obtained and
$\varepsilon^{2}$ may be taken as

$\varepsilon^{2}=nC,~{}~{}n=0,1,2,...$ (71)

using $n\rightarrow n+\textbf{m}$. Consequently, $V_{\textbf{m}}(x)$ is
expressed as follows

$V_{\textbf{m}}(x)=\frac{C^{2}}{16}x^{2}+\frac{\alpha^{\prime
2}-1/4}{x^{2}}-\frac{C^{2}x^{2}}{4}\frac{L^{\alpha^{\prime}+1}_{\textbf{m}-2}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}(-g)}+C\left(\alpha^{\prime}+\frac{Cx^{2}}{4}-1\right)\frac{L^{\alpha^{\prime}}_{\textbf{m}-1}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}(-g)}+\frac{C^{2}x^{2}}{2}\left(\frac{L^{\alpha^{\prime}}_{\textbf{m-1}}(-g)}{L^{\alpha^{\prime}-1}_{\textbf{m}}(-g)}\right)^{2}-\frac{C}{2}(2\textbf{m}+\alpha^{\prime}+1).$
(72)

And, in our system, the constants $C$ and $\alpha^{\prime}$ stand for

$C=\frac{Be}{M},~{}~{}~{}~{}\alpha^{\prime}=\frac{1}{2}(1+2\ell_{2}).$ (73)

Rewriting $V_{\textbf{m}}(x)$ in terms of the parameters of our system yields,

$\begin{split}V_{\textbf{m}}(x)&=\frac{B^{2}e^{2}}{16}x^{2}+\frac{\ell_{2}(\ell_{2}+1)}{x^{2}}-\frac{B^{2}e^{2}}{4M^{2}}x^{2}\frac{L^{\ell_{2}+3/2}_{\textbf{m}-2}(-g)}{L^{\ell_{2}-1/2}_{\textbf{m}}(-g)}+\frac{Be}{M}\left(\ell_{2}-\frac{1}{2}+\frac{Be}{4M}x^{2}\right)\frac{L^{\ell_{2}+1/2}_{\textbf{m}-1}}{L^{\ell_{2}-1/2}_{\textbf{m}}}+\\\
&\frac{B^{2}e^{2}}{2M^{2}}\left(\frac{L^{\ell_{2}+1/2}_{\textbf{m}-1}}{L^{\ell_{2}-1/2}_{\textbf{m}}}\right)-\frac{Be}{2M}\left(2\textbf{m}+\ell_{2}+\frac{3}{2}\right).\end{split}$
(74)

By substituting $Q(g)$ and $g(x)$ into (65) and employing this in the
expression $u_{12}(x)=x\chi(x)$, one obtains:

$u_{12}(x)=N_{n,\textbf{m}}\frac{x^{\ell_{2}+1}e^{-\frac{Bex^{2}}{16M}}}{L^{\ell_{2}-1/2}_{\textbf{m}}(-\frac{Bex^{2}}{8M})}L^{\ell_{2}+1/2}_{n+\textbf{m},\textbf{m}}\left(\frac{Bex^{2}}{8M}\right).$
(75)

Figure 7: Effective partner potential $V_{\textbf{m}}(x)$ in (74) graphs.
$\Phi=0.1,0.5,0.95$ corresponding to black, green, blue curves. Figure 8:
Probability density $\rho(r)=|u_{11}|^{2}+|u_{12}|^{2}$ graphs where $u_{12}$
is given by (56). $a=0.1,B=10,M=2,m=1,\alpha=0.2,\Phi=0.9,e=1$ and $n=0,1,3$
corresponding to black, blue, orange curves. Figure 9: Probability density
$\rho(r)=|u_{11}|^{2}+|u_{12}|^{2}$ graphs where $u_{12}$ is given by (75).
$a=0.1,B=10,M=2,m=1,\textbf{m}=1,\alpha=0.2,\Phi=0.5,e=1$ and $n=0,1,2$
corresponding to green, black and blue curves correspondingly. Figure 10:
Energy levels given in (37) as a function of $B$ and $\Phi$.
$a=0.2,B=10,M=2,m=1,\alpha=0.6,n=2,e=1$.

Finally, the probability density for the solutions in (75) and (38), and
energy as a function of magnetic field strength and flux of the solenoid, are
given in Figures $9$ and $10$, respectively.

##  4 Conclusions

This study demonstrates how the system of a spinning cosmic string spacetime,
interacting with external fields via Aharonov-Bohm interaction, is adapted to
rationally extended potential systems. Within these systems, the solutions of
the partner Dirac Hamiltonian are found using SUSY QM and are expressed in
terms of $X_{m}$ exceptional orthogonal Laguerre polynomials. The newly
generated family of rationally expanded potentials has been identified as
belonging to the class of nonlinear isotonic potentials. According to the
analytical solutions, a sudden drop in magnetic field strength can lead to
shifts and transitions in energy levels in Figure $1$ , which also agree with
the results in [14]. It is also observed that when $\alpha<0.45$ and at
smaller values of the magnetic field strength and flux, there are fewer
positive energy levels compared to the energy of the antiparticles in Figure
$5$. Moreover, the probability density graphs, which are obtained only for the
smaller magnetic field strength and magnetic flux, are presented in Figures 8
and 9, in contrast to the probability density graph depicting greater magnetic
field strength and magnetic flux in Figure 4. When we examine the change in
energy levels with respect to $B$ and $\Phi$ in the Figure $10$, we find that
the antiparticle’s eigenvalues are more numerous than those of the particle.
By expressing wavefunction solutions in terms of exceptional $X_{m}$ Laguerre
polynomials, we have both the advantage of deriving new exactly solvable
potentials in (74) in terms of the string and magnetic field parameters, and
it may also allow us to study systems with spectra that have ”gaps” or missing
levels, which can be related to the study of supersymmetric quantum mechanics.

##  5  Data Availability Statement

Data sets generated during and/or analyzed during the current study are
available from the corresponding author upon reasonable request. The data is
housed within a secure repository to ensure compliance with ethical guidelines
and privacy laws applicable to the study. Interested researchers may contact
the corresponding author, providing a detailed explanation of their request
and intended use of the data. Approval for data access will be subject to an
assessment of the request’s alignment with the conditions under which the data
was collected and any ethical or legal restrictions.

## References

  * [1] T. W. B. Kibble, J. Phys. A: Math. Gen., Topology of cosmic domains and strings 9 1387–1398 1976.
  * [2] A. Vilenkin, Cosmic strings and domain walls, Phys. Rep. 121(5) 263-315 1985.
  * [3] I. Yu. Rybaka and L. Sousa, Emission of gravitational waves by superconducting cosmic strings, J. Cos. Astr. Phys. 11 024 2022.
  * [4] K. Kuijken, X. Siemens and T. Vachaspati, Monthly Notices of the Royal Astronomical Society, 384(1) 11 2008.
  * [5] K. Dimopoulos, Primordial magnetic fields from superconducting cosmic strings, Phys. Rev. D 57 4629-4641 1998.
  * [6] S. Nayak, S. Sau and S. Sanyal, Evolution of magnetic fields in cosmic string wakes, Astr. Phys. 146 102805 2023.
  * [7] T. Vachaspati and A. Vilenkin, Large-scale structure from wiggly cosmic strings, Phys. Rev. Lett. 67 1057 1991.
  * [8] Y. A. Sitenko, V. M. Gorkavenko and M. S. Tsarenkova, Magnetic flux in the vacuum of quantum bosonic matter in the cosmic string background, Phys. Rev. D 106 105010 2022.
  * [9] B. Chakraborty, K. S. Gupta and S. Sen, Topology, cosmic strings and quantum dynamics - a case study with graphene, Journal of Physics: Conference Series 442 012017 2013.
  * [10] M. M. Cunha and E. O. Silva, Self-Adjoint Extension Approach to Motion of Spin-1/2 Particle in the Presence of External Magnetic Fields in the Spinning Cosmic String Spacetime, Universe, 6 203 2020.
  * [11] E. R. Bezerra de Mello, V. B. Bezerra, A. A. Saharian, and H. H. Harutyunyan, Vacuum currents induced by a magnetic flux around a cosmic string with finite core, Phys. Rev. D 91 064034 2015.
  * [12] E. R. Figueiredo Medeiros and E. R. Bezerra de Mello, Relativistic quantum dynamics of a charged particle in cosmic string spacetime in the presence of magnetic field and scalar potential ,The Eur. Phys. J. C, 72 2051 2012.
  * [13] M. Hosseinpour, H. Hassanabadi and M. de Montigny, The Dirac oscillator in a spinning cosmic string spacetime, Eur. Phys. J. C 79 311 2019\.
  * [14] M. M. Cunha and E. O. Silva, Relativistic Quantum Motion of an Electron in Spinning Cosmic String Spacetime in the Presence of Unifor Magnetic Field and Aharonov-Bohm Potential, Adv. High En. Phys. 2021 6709140 2021.
  * [15] D S Citrin, The Aharonov–Bohm effect in quadratic Gauss quantum rings, Phys. Scr., 99 025914 2024.
  * [16] K. Bhattacharya, Demystifying the nonlocality problem in Aharonov–Bohm effect, Phys. Scr. 96 084011 2021.
  * [17] P. Auclair, K. Leyde and D. A. Steer, A window for cosmic strings, J. of Cosm. and Astr. Phys., 04 005 2023.
  * [18] W. Buchmüller, V. Domcke and K. Schmitz, Metastable cosmic strings, 11 020 2023.
  * [19] J. D. Garcia-Munoz and A. Raya, Supersymmetric quantum potentials analogs of classical electrostatic fields, Int. J of Geom. Meth. in Mod. Phys. https://doi.org/10.1142/S021988782450052.
  * [20] J. D. García-Munoz, D. J. Fernández and F. Vergara-Méndez, Supersymmetric quantum mechanics, multiphoton algebras and coherent states, Phys. Scr. 98 105243 2023.
  * [21] D. Gómez-Ullate, N. Kamran, R. Milson, An extension of Bochner’s problem: exceptional invariant subspaces. J. Approx. Theory 162(5), 987–1006 2010.
  * [22] M. A. Garcia-Ferrero, D. Gomez-Ullate and R. Milson, Exceptional Legendre Polynomials and Confluent Darboux Transformations, Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 016 2017.
  * [23] S. Yadav, A. Khare, B. P. Mandal, Supersymmetry and shape invariance of exceptional orthogonal polynomials, 444 169064 2022.
  * [24] K. Bakke, Rotating effects on the Dirac oscillator in the cosmic string spacetime, Gen. Relativ. Gravit. 45 1847 2013.
  * [25] M. G. Alford and F. Wilczek, Aharonov-Bohm Interaction of Cosmic Strings with Matter, Phys. Rev. Lett. 62(10) 1989.
  * [26] M. M. Cunha, E. O. Silva, Relativistic quantum motion of an electron in spinning cosmic string spacetime in the presence of uniform magnetic field and Aharonov-Bohm potential, 2021 6709140 2021.
  * [27] K. S. Thorne, Closed timelike curves, 13th Int. Conf. on General Relativity and Gravitation 295 Cordoba, Argentina, 1992.
  * [28] M. Ringbauer, M. A. Broome, C. R. Myers, A. G. White and T. C. Ralph , Experimental simulation of closed timelike curves, Nature Communications 5 4145 2014.
  * [29] C. Quesne, Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics, Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 084 2009.
  * [30] F. M. Cooper, A. Khare and U. P. Sukhatme, Supersymmetry in Quantum Mechanics, World Scientific Publishing Company (July 3, 2001.
  * [31] R. L. Hall, N. Saad and Ö. Yeşiltaş, Generalized quantum isotonic nonlinear oscillator in d dimensions , J. Phys. A: Math. Theor. 43 465304 2010.
  * [32] J. Sesma, The generalized quantum isotonic oscillator , J. Phys. A: Math. Theor. 43 185303 2010.
  * [33] ] D. Gomez-Ullate, N. Kamran, R. Milson, An extended class of orthogonal polynomials defined by a Sturm–Liouville problem, J. Math. Anal. Appl. 359 352 2009.
  * [34] D. Gomez-Ullate, N. Kamran and R. Milson, Exceptional orthogonal polynomials and the Darboux transformation, J. Phys. A: Math. Theor. 43 434016 2010.
  * [35] R. K. Yadav, N. Kumar, A. Khare and B. P. Mandal, Rationally extended shape invariant potentials in arbitrary D-dimensions associated with exceptional $X_{m}$ polynomials, Acta Polytechnica 57(6) 477 2017.
  * [36] A. Bhattacharjie and E. C. G. Sudarshan, A Class of Solvable Potentials, Nuovo Cimento 25 864 1962.


* Nussbaumer and Storey (1979) Nussbaumer, H.; Storey, P.J.  N III lines for solar diagnostics.  Astron. Astrophys. 1979, 71, L5–L8.
  * Stafford et al. (1993) Stafford, R.P.; Hibbert, A.; Bell, K.L.  Accurate transition probabilities for spectral lines of N III.  Mon. Not. R. Astron. Soc. 1993, 260, L11–L14.  doi:black10.1093/mnras/260.1.L11.
  * Stafford et al. (1994a) Stafford, R.P.; Bell, K.L.; Hibbert, A.  Electron impact excitation of N III - Fine-structure collision strengths and Maxwellian-averaged rate coefficients.  Mon. Not. R. Astron. Soc. 1994, 266, 715.  doi:black10.1093/mnras/266.3.715.
  * Stafford et al. (1994b) Stafford, R.P.; Bell, K.L.; Hibbert, A.; Wijesundera, W.P.  Electron Impact Excitation of NII - Fine Structure Collision Strengths and Maxwellian-Averaged Rate Coefficients.  Mon. Not. R. Astron. Soc. 1994, 268, 816.  doi:black10.1093/mnras/268.3.816.
  * Bhatia et al. (1986) Bhatia, A.K.; Feldman, U.; Seely, J.F.  Atomic Data and Spectral Line Intensities for the Boron Isoelectronic Sequence (Ar XIV through Kr XXXII).  At. Data Nucl. Data Tables 1986, 35, 319.  doi:black10.1016/0092-640X(86)90025-2.
  * Zhang et al. (1994) Zhang, H.L.; Graziani, M.; Pradhan, A.K.  Atomic Data from the Iron Project: III. Rate Coefficients for Electron Impact Excitation of Boron-like Ions: NE VI, MG VIII, AL IX, SI X, S XII, AR XIV, CA XVI and Fe XXII.  Astron. Astrophys. 1994, 283, 319.
  * Sampson et al. (1986) Sampson, D.H.; Weaver, G.M.; Goett, S.J.; Zhang, H.; Clark, R.E.H.  Collision Strengths and Line Strengths for All Transitions among the Levels of the 1s22s22p, 1s22s2p2, and 1s22p3, Configurations of Boron-like Ions.  At. Data Nucl. Data Tables 1986, 35, 223.  doi:black10.1016/0092-640X(86)90023-9.
  * Sampson et al. (1990) Sampson, D.H.; Zhang, H.L.; Fontes, C.J.  Relativistic Distorted-Wave Collision Strengths and Oscillator Strengths for the 71 Na-like Ions with 22 <= Z <= 92.  At. Data Nucl. Data Tables 1990, 44, 209.  doi:black10.1016/0092-640X(90)90014-B.
  * Bell et al. (1995) Bell, K.L.; Hibbert, A.; Stafford, R.P.  Transition probabilities for some spectral lines of singly ionised nitrogen.  Phys. Scripta 1995, 52, 240–251.  doi:black10.1088/0031-8949/52/3/004.
  * Aggarwal (1983) Aggarwal, K.M.  Electron excitation rates among fine structure levels in O III.  Astrophys. J. Suppl. Ser. 1983, 52, 387–398.  doi:black10.1086/190873.
  * Aggarwal (1985) Aggarwal, K.M.  Line intensity ratios for transitions in O III.  Astron. Astrophys. 1985, 146, 149–158.
  * Bhatia and Kastner (1993) Bhatia, A.K.; Kastner, S.O.  Collision Strengths and Transition Rates for O III.  At. Data Nucl. Data Tables 1993, 54, 133–164.  doi:black10.1006/adnd.1993.1011.
  * Lennon and Burke (1994) Lennon, D.J.; Burke, V.M.  Atomic data from the IRON project. II. Effective collision strength S for infrared transitions in carbon-like ions.  Astron. Astrophys. Suppl. Ser. 1994, 103, 273–277.
  * Aggarwal (1984) Aggarwal, K.M.  Electron impact excitation rates for fine-structure transitions in NE V and SI IX - an R-matrix approach.  Astrophys. J. Suppl. Ser. 1984, 54, 1–15.  doi:black10.1086/190916.
  * Bhatia and Doschek (1993a) Bhatia, A.K.; Doschek, G.A.  Atomic Data and Spectral Line Intensities for C-Like Si IX.  At. Data Nucl. Data Tables 1993, 55, 281–314.  doi:black10.1006/adnd.1993.1023.
  * Bhatia and Doschek (1993b) Bhatia, A.K.; Doschek, G.A.  Atomic Data and Spectral Line Intensities for C-like Ne V.  At. Data Nucl. Data Tables 1993, 55, 315–348.  doi:black10.1006/adnd.1993.1024.
  * Bhatia and Doschek (1995) Bhatia, A.K.; Doschek, G.A.  Atomic Data and Spectral Line Intensities for C-like Mg VII.  At. Data Nucl. Data Tables 1995, 60, 145.  doi:black10.1006/adnd.1995.1006.
  * Bhatia and Kastner (1987) Bhatia, A.K.; Kastner, S.O.  Atomic parameters for carbon-like S XI.  J. Quant. Spectros. Radiat. Transf. 1987, 37, 527–529.  doi:black10.1016/0022-4073(87)90055-0.
  * Dere et al. (1979) Dere, K.P.; Mason, H.E.; Widing, K.G.; Bhatia, A.K.  XUV electron density diagnostics for solar flares.  Astrophys. J. Suppl. Ser. 1979, 40, 341–364.  doi:black10.1086/190590.
  * Mason and Bhatia (1978) Mason, H.E.; Bhatia, A.K.  Theoretical intensity ratios for the UV lines of Mg VII, Si IX and S XI.  Mon. Not. R. Astron. Soc. 1978, 184, 423–437.  doi:black10.1093/mnras/184.3.423.
  * Froese Fischer and Saha (1985) Froese Fischer, C.; Saha, H.P.  Multiconfiguration Hartree-Fock Results with Breit-Pauli Corrections for Transitions in the Carbon Sequence.  Phys. Scripta 1985, 32, 181–194.  doi:black10.1088/0031-8949/32/3/004.
  * Aggarwal et al. (1991a) Aggarwal, K.M.; Berrington, K.A.; Burke, P.G.; Kingston, A.E.; Pathak, A.  Electron collision cross sections at low energies for all transitions between the n=1, 2, 3, 4 and 5 levels of atomic hydrogen.  J. Phys. B At. Mol. Phys. 1991, 24, 1385–1410.  doi:black10.1088/0953-4075/24/6/024.
  * Aggarwal et al. (1991b) Aggarwal, K.M.; Berrington, K.A.; Keenan, F.P.  Effective collision strengths among fine-structure levels of CA XV: erratum.  Astrophys.J. Suppl. Ser. 1991, 77, 441–449.  doi:black10.1086/191612.
  * Bhatia and Mason (1980) Bhatia, A.K.; Mason, H.E.  Theoretical atomic structure and electron scattering data for ions in the nitrogen isoelectronic sequence - MG VI, SI VIII, S X, A XII and CA XIV.  Mon. Not. R. Astron. Soc. 1980, 190, 925–929.  doi:black10.1093/mnras/190.4.925.
  * Bhatia et al. (1989) Bhatia, A.K.; Seely, J.F.; Feldman, U.  Atomic Data and Spectral Line Intensities for the Nitrogen Isoelectronic Sequence (Ar XII through Kr XXX).  At. Data Nucl. Data Tables 1989, 43, 99.  doi:black10.1016/0092-640X(89)90016-8.
  * Mason et al. (1979) Mason, H.E.; Bhatia, A.K.; Doschek, G.A.; Feldman, U.  Fe XXI as an electron density diagnostic in solar flares.  Astron. Astrophys. 1979, 73, 74–81.
  * Aggarwal (1991) Aggarwal, K.M.  Collision Strengths and Excitation Rate Coefficients for Transitions in Fe XXI.  Astrophys. J. Suppl. Ser. 1991, 77, 677.  doi:black10.1086/191619.
  * Bhatia et al. (1979) Bhatia, A.K.; Feldman, U.; Doschek, G.A.  New atomic data for Si/6+/, S/8+/ and Ar/10+/.  Astron. Astrophys. 1979, 80, 22–26.
  * McLaughlin and Bell (1994) McLaughlin, B.M.; Bell, K.L.  Analytical FITS for Electron Collisionally Excited Effective Collision Strengths of O II.  Astrophys. J. Suppl. Ser. 1994, 94, 825.  doi:black10.1086/192091.
  * Loulergue et al. (1985) Loulergue, M.; Mason, H.E.; Nussbaumer, H.; Storey, P.J.  Fe XIX transitions within the N = 2 complex.  Astron. Astrophys. 1985, 150, 246–250.
  * Blackford and Hibbert (1994) Blackford, H.M.S.; Hibbert, A.  Transitions in Fluorine-Like Ions.  At. Data Nucl. Data Tables 1994, 58, 101–164.  doi:black10.1006/adnd.1994.1025.
  * Butler and Zeippen (1994) Butler, K.; Zeippen, C.J.  Atomic data from the IRON Project. V. Effective collision strengths for transitions in the ground configuration of oxygen-like ions.  Astron. Astrophys. Suppl. Ser. 1994, 108, 1–8.
  * Sampson et al. (1991) Sampson, D.H.; Zhang, H.L.; Fontes, C.J.  Relativistic Distorted-Wave Collision Strengths and Oscillator Strengths for F-like Ions with 22 <= Z <= 92.  At. Data Nucl. Data Tables 1991, 48, 25.  doi:black10.1016/0092-640X(91)90023-W.
  * Cornille et al. (1992) Cornille, M.; Dubau, J.; Loulergue, M.; Bely-Dubau, F.; Faucher, P.  Theoretical analysis of the Fe XVIII X-ray spectrum and application to solar coronal observations.  Astron. Astrophys. 1992, 259, 669–681.
  * Bhatia et al. (1985) Bhatia, A.K.; Feldman, U.; Seely, J.F.  Atomic Data and Spectral Line Intensities for the Neon Isoelectronic Sequence (Si V through Kr XXVII).  At. Data Nucl. Data Tables 1985, 32, 435.  doi:black10.1016/0092-640X(85)90020-8.
  * Hibbert et al. (1993) Hibbert, A.; Ledourneuf, M.; Mohan, M.  Energies, Oscillator Strengths, and Lifetimes for Neon-like Ions Up to Kr XXVII.  At. Data Nucl. Data Tables 1993, 53, 23–112.  doi:black10.1006/adnd.1993.1002.
  * Saraph and Tully (1994) Saraph, H.E.; Tully, J.A.  Atomic data from the IRON project. IV. Electron excitation of the 2̂P̂0̂_̂3/2_-2̂P̂0̂_̂1/2_ fine structure transition in fluorine-like ions.  Astron. Astrophys. Suppl. Ser. 1994, 107, 29–38.
  * Mohan et al. (1988) Mohan, M.; Baluja, K.L.; Hibbert, A.  Electron-impact excitation of MG IV using the R-matrix method.  Phys. Scripta 1988, 38, 699–701.  doi:black10.1088/0031-8949/38/5/008.
  * Zhang et al. (1987) Zhang, H.; Sampson, D.H.; Clark, R.E.H.; Mann, J.B.  Collision Strengths and Oscillator Strengths for Excitation to the n=3 and 4 Levels of Neon-like Ions.  At. Data Nucl. Data Tables 1987, 37, 17.  doi:black10.1016/0092-640X(87)90003-9.
  * Bhatia and Doschek (1992) Bhatia, A.K.; Doschek, G.A.  Atomic Data and Spectral Line Intensities for Ne-like Fe XVII.  At. Data Nucl. Data Tables 1992, 52, 1.  doi:black10.1016/0092-640X(92)90007-5.
  * Mohan and Le Dourneuf (1990) Mohan, M.; Le Dourneuf, M.  Electron-impact excitation from the Si(5+) ground state, using the R-matrix method.  Astron. Astrophys. 1990, 227, 285–288.
  * Mohan et al. (1987) Mohan, M.; Hibbert, A.; Baluja, K.L.  Electron impact excitation of S VIII from its ground state.  J. Phys. B At. Mol. Phys. 1987, 20, 2565–2570.  doi:black10.1088/0022-3700/20/11/023.
  * Dufton et al. (1983) Dufton, P.L.; Hibbert, A.; Kingston, A.E.; Doschek, G.A.  SI III line ratios in the Sun.  Astrophys. J. 1983, 274, 420–428.  doi:black10.1086/161458.
  * Mason (1975) Mason, H.E.  The excitation of several iron and calcium lines in the visible spectrum of the solar corona.  Mon. Not. R. Astron. Soc. 1975, 170, 651–689.  doi:black10.1093/mnras/170.3.651.
  * Christensen et al. (1986) Christensen, R.B.; Norcross, D.W.; Pradhan, A.K.  Electron-impact excitation of ions in the magnesium sequence. II. S v, Ar vii, Ca ix, Cr xiii, and Ni xvii.  Phys. Rev. A 1986, 34, 4704–4715.  doi:black10.1103/PhysRevA.34.4704.
  * Lanzafame (1994) Lanzafame, A.C.  SiII resonance multiplets in the Sun.  Astron. Astrophys. 1994, 287, 972–981.
  * Mohan et al. (1990) Mohan, M.; Hibbert, A.; Berrington, K.A.; Burke, P.G.  Electron impact excitation of neon-like S VII from its ground state.  J. Phys. B At. Mol. Phys. 1990, 23, 989–996.  doi:black10.1088/0953-4075/23/6/007.
  * Bhatia et al. (1980) Bhatia, A.K.; Doschek, G.A.; Feldman, U.  Atomic data for S IV and solar observations of the 3x/2/3p /2/P-3s3p/2/ /4/P multiplet.  Astron. Astrophys. 1980, 86, 32–35.
  * Dufton et al. (1982) Dufton, P.L.; Hibbert, A.; Kingston, A.E.; Doschek, G.A.  S IV emission-line ratios in the sun.  Astrophys. J. 1982, 257, 338–344.  doi:black10.1086/159992.
  * Fischer and Liu (1986) Fischer, C.F.; Liu, B.  Energy Levels, Lifetimes, Wavelengths, and Transition Probabilities of Fe XIV.  At. Data Nucl. Data Tables 1986, 34, 261.  doi:black10.1016/0092-640X(86)90004-5.
  * Czyzak and Krueger (1966) Czyzak, S.J.; Krueger, T.K.  On the Excited Levels of Fe VIII.  Astrophys. J. 1966, 144, 381.  doi:black10.1086/148613.
  * Fawcett (1989) Fawcett, B.C.  Computed Oscillator Strengths and Energy Levels for Potassium-like Ions V V, Cr VI, Mn VII, Fe VIII, Co IX, Ni X, and Cu XI with Calculated Wavelengths and Wavelengths Derived from Established Data.  At. Data Nucl. Data Tables 1989, 43, 71.  doi:black10.1016/0092-640X(89)90015-6.
  * Dufton and Kingston (1989) Dufton, P.L.; Kingston, A.E.  Effective collision strengths for Si III.  Mon. Not. R. Astron. Soc. 1989, 241, 209–214.  doi:black10.1093/mnras/241.2.209.
  * Nussbaumer et al. (1982) Nussbaumer, H.; Storey, P.J.; Storey, P.J.  Forbidden emission lines of Fe VII.  Astron. Astrophys. 1982, 113, 21–26.
  * Dufton and Kingston (1991) Dufton, P.L.; Kingston, A.E.  Effective collision strengths for Si II.  Mon. Not. R. Astron. Soc. 1991, 248, 827–828.  doi:black10.1093/mnras/248.4.827.
  * Bhadra and Henry (1980) Bhadra, K.; Henry, R.J.W.  Oscillator strengths and collision strengths for S IV.  Astrophys. J. 1980, 240, 368–373.  doi:black10.1086/158241.
  * Fawcett and Mason (1991) Fawcett, B.C.; Mason, H.E.  Collision Strengths and Oscillator Strengths for Fe IX.  At. Data Nucl. Data Tables 1991, 47, 17.  doi:black10.1016/0092-640X(91)90017-X.
  * Mason (1994) Mason, H.E.  An Assessment of Theoretical Electron Excitation Data for Fe IX-Fe XIV.  At. Data Nucl. Data Tables 1994, 57, 305–328.  doi:black10.1006/adnd.1994.1020.
  * Pindzola et al. (1989) Pindzola, M.S.; Griffin, D.C.; Bottcher, C.  Coupling effects for electron-impact excitation in the potassium isoelectronic sequence.  Phys. Rev. A 1989, 39, 2385–2391.  doi:black10.1103/PhysRevA.39.2385.
  * Fawcett and Mason (1989) Fawcett, B.C.; Mason, H.E.  A Slater Parameter Optimization Method Applied to the Computation of Collision Strengths for Fe XIII.  At. Data Nucl. Data Tables 1989, 43, 245.  doi:black10.1016/0092-640X(89)90002-8.
  * Chen and Pradhan (1999) Chen, G.X.; Pradhan, A.K.  Atomic data from the Iron Project. XXXVII. Electron impact excitation collision strengths and rate coefficients for Fe VI.  Astron. Astrophys. Suppl. Ser. 1999, 136, 395–403.  doi:black10.1051/aas:1999464.
  * Eissner et al. (1999a) Eissner, W.; Galavís, M.E.; Mendoza, C.; Zeippen, C.J.  Atomic data from the IRON Project. XXXIV. Electron impact excitation of Fe XVI.  Astron. Astrophys. Suppl. Ser. 1999, 136, 385–394.  doi:black10.1051/aas:1999466.
  * Eissner et al. (1999b) Eissner, W.; Galavís, M.E.; Mendoza, C.; Zeippen, C.J.  Atomic data from the IRON Project. XXXVIII. Electron impact excitation of the fine-structure transitions in the n=3 complex of Fe XV.  Astron. Astrophys. Suppl. Ser. 1999, 137, 165–173.  doi:black10.1051/aas:1999243.
  * Garstang (1962) Garstang, R.H.  Transition probabilities for forbidden lines of Fe II.  Mon. Not. R. Astron. Soc. 1962, 124, 321.  doi:black10.1093/mnras/124.4.321.
  * Nussbaumer and Storey (1988) Nussbaumer, H.; Storey, P.J.  Transition probabilities for Fe II infrared lines.  Astron. Astrophys. 1988, 193, 327–333.
  * Giridhar and Arellano Ferro (1995) Giridhar, S.; Arellano Ferro, A.  A Critical Compilation of Oscillator Strengths for Fe II Lines.  Rev. Mex. Astron. Astrofis. 1995, 31, 23–37.
  * Pelan and Berrington (1995) Pelan, J.; Berrington, K.A.  Atomic data from the IRON Project. IX. Electron excitation of the 2̂P̂0̂_̂3/2_-2̂P̂0̂_̂1/2_ fine-structure transition in chlorine-like ions, from AR II to NI XII.  Astron. Astrophys. Suppl. Ser. 1995, 110, 209.
  * Berrington and Tully (1997) Berrington, K.A.; Tully, J.A.  Atomic data from the IRON Project. XXIV. Electron excitation of Li-like Fe XXIV between the N = 2 and n’ = 2, 3, 4 fine-structure levels.  Astron. Astrophys. Suppl. Ser. 1997, 126, 105–111.  doi:black10.1051/aas:1997384.
  * Garstang (1958) Garstang, R.H.  Energy levels and transition probabilities of Fe IV.  Mon. Not. R. Astron. Soc. 1958, 118, 572.  doi:black10.1093/mnras/118.6.572.
  * Kono and Hattori (1984) Kono, A.; Hattori, S.  Accurate oscillator strengths for neutral helium.  Phys. Rev. A 1984, 29, 2981–2988.  doi:black10.1103/PhysRevA.29.2981.
  * Goett et al. (1983) Goett, S.J.; Sampson, D.H.; Clark, R.E.H.  Intermediate-Coupling Collision Strengths for Transitions between Doubly Excited Levels Produced by Electron Impact on Highly Charged He-like Ions.  At. Data Nucl. Data Tables 1983, 28, 279.  doi:black10.1016/0092-640X(83)90018-9.
  * Kaufman and Sugar (1986) Kaufman, V.; Sugar, J.  Forbidden Lines in ns2npk Ground Configurations and nsnp Excited Configurations of Beryllium through Molybdenum Atoms and Ions.  J. Phys. Chem. Ref. Data 1986, 15, 321–426.  doi:black10.1063/1.555775.
  * Nussbaumer and Rusca (1979) Nussbaumer, H.; Rusca, C.  Forbidden transitions in the C I sequence.  Astron. Astrophys 1979, 72, 129–133.
  * Mendoza (1983) Mendoza, C.  Recent advances in atomic calculations and experiments of interest in the study of planetary nebulae.  Planetary Nebulae; Aller, L.H., Ed., 1983, Vol. 103, pp. 143–172.
  * Zeippen (1982) Zeippen, C.J.  Transition probabilities for forbidden lines in the 2p3 configuration.  Mon. Not. R. Astron. Soc. 1982, 198, 111–125.  doi:black10.1093/mnras/198.1.111.
  * Nahar (1996) Nahar, S.N.  Photoionization cross sections and oscillator strengths for Fe III.  Phys. Rev. A 1996, 53, 1545–1552.  doi:black10.1103/PhysRevA.53.1545.
  * Butler and Zeippen (1984) Butler, K.; Zeippen, C.J.  N I forbidden lines revisited.  Astron. Astrophys. 1984, 141, 274–278.
  * Bautista (1999a) Bautista, M.A.  Atomic data from the Iron Project. XXXIX. Photoionization cross sections and oscillator strengths for NI II.  Astron. Astrophys. Suppl. Ser. 1999, 137, 529–535.  doi:black10.1051/aas:1999491.
  * Bautista (1999b) Bautista, M.A.  Continuum Fluorescence Excitation of [N I] and [O I] Lines in Gaseous Nebulae.  Astrophys. J. 1999, 527, 474–478.  doi:black10.1086/308086.
  * Moore (1985) Moore, C.E.  Selected Tables of Atomic Spectra. Atomic Energy Levels and Multiplet Table O III; US Government Printing Office: Washington, DC, USA, 1985.
  * Pequignot and Aldrovandi (1976) Pequignot, D.; Aldrovandi, S.M.V.  Effective collision strengths for the forbidden emission lines of neutral carbon, nitrogen and oxygen.  Astron. Astrophys. 1976, 50, 141–144.
  * Johnson et al. (1987) Johnson, C.T.; Burke, P.G.; Kingston, A.E.  Electron scattering from the fine structure levels within the 1s22s22p2 3Pe ground state of C I.  J. Phys. B At. Mol. Phys. 1987, 20, 2553–2563.  doi:black10.1088/0022-3700/20/11/022.
  * Thomas and Nesbet (1975) Thomas, L.D.; Nesbet, R.K.  Low-energy electron scattering by atomic nitrogen.  Phys. Rev. A 1975, 12, 2369–2377.  doi:black10.1103/PhysRevA.12.2369.
  * Dopita et al. (1976) Dopita, M.A.; Mason, D.J.; Robb, W.D.  Atomic nitrogen as a probe of physical conditions in the interstellar medium.  Astrophys. J. 1976, 207, 102–109.  doi:black10.1086/154472.
  * Berrington and Burke (1981) Berrington, K.A.; Burke, P.G.  Effective collision strengths for forbidden transitions in e-N and e-o scattering.  Planet. Space Sci. 1981, 29, 377–381.  doi:black10.1016/0032-0633(81)90026-X.
  * LeDourneuf and Nesbet (1976) LeDourneuf, M.; Nesbet, R.K.  Electron impact excitation of fine-structure levels of atomic oxygen.  J. Phys. B At. Mol. Phys. 1976, 9, L241–L243.  doi:black10.1088/0022-3700/9/9/007.
  * Johnson and Kingston (1987) Johnson, C.T.; Kingston, A.E.  Electron excitation of the 1s(2)2s(2)2p(5) (2)P(0)3/2-(2)P(0)1/2 transition and analysis of the 1s(2)2s2p(6) nl resonances for Ne II and Mg IV.  J. Phys. B At. Mol. Phys. 1987, 20, 5757–5769.  doi:black10.1088/0022-3700/20/21/023.
  * Zeippen et al. (1987) Zeippen, C.J.; Butler, K.; Le Bourlot, J.  Effective collision strengths for fine-structure forbidden transitions in the 3p3 configuration of AR IV.  Astron. Astrophys. 1987, 188, 251–257.
  * Krueger and Czyzak (1970) Krueger, T.K.; Czyzak, S.J.  Excitation of Forbidden Lines in Gaseous Nebulae. II. Calculations for 3pq Ions.  Proc. R. Soc. Lon. Ser-A 1970, 318, 531–539.  doi:black10.1098/rspa.1970.0159.
  * Mendoza and Zeippen (1982) Mendoza, C.; Zeippen, C.J.  Transition probabilities for forbidden lines in the 3p3 configuration.  Mon. Not. R. Astron. Soc. 1982, 198, 127–139.  doi:black10.1093/mnras/198.1.127.
  * Hayes (1986) Hayes, M.A.  The S III intercombination multiplet - Radiative and electron excitation data.  J. Phys. B At. Mol. Phys. 1986, 19, 1853–1862.  doi:black10.1088/0022-3700/19/12/023.
  * Mendoza and Zeippen (1982) Mendoza, C.; Zeippen, C.J.  Transition probabilities for forbidden lines in the 3p2 configuration -II.  Mon. Not. R. Astron. Soc. 1982, 199, 1025–1032.  doi:black10.1093/mnras/199.4.1025.
  * Fuhr and Wiese (1990) Fuhr, J.R.; Wiese, W.L.  RCR Handbook of Chemistry and Physics (71st Edition); CRC Press: Boca Raton, FL, USA, 1990.
  * Ramsbottom et al. (1996) Ramsbottom, C.A.; Bell, K.L.; Stafford, R.P.  Effective Collision Strengths for Electron Impact Excitation of Singly Ionized Sulfur.  At. Data Nucl. Data Tables 1996, 63, 57.  doi:black10.1006/adnd.1996.0009.
  * Cai and Pradhan (1993) Cai, W.; Pradhan, A.K.  Excitation Rate Coefficients and Line Ratios for the Optical and Ultraviolet Transitions in S II.  Astrophys. J. Suppl. Ser. 1993, 88, 329.  doi:black10.1086/191824.
  * Storey et al. (2000) Storey, P.J.; Mason, H.E.; Young, P.R.  Atomic data from the IRON Project. XL. Electron impact excitation of the Fe XIV EUV transitions.  Astron. Astrophys. Suppl. Ser. 2000, 141, 285–296.  doi:black10.1051/aas:2000319.
  * Saraph and Storey (1996) Saraph, H.E.; Storey, P.J.  Atomic data from the IRON Project. XI. The 2̂P̂ô_̂1/ 2_-2̂P̂ô_̂3/2_ fine-structure lines of AR VI, K VII and CA VIII.  Astron. Astrophys. Suppl. Ser. 1996, 115, 151.
  * Zhang (1996) Zhang, H.  Atomic data from the Iron Project. XVIII. Electron impact excitation collision strengths and rate coefficients for Fe III.  Astron. Astrophys 1996, 119, 523–528.
  * Zhang and Pradhan (1997) Zhang, H.L.; Pradhan, A.K.  Atomic data from the Iron Project. XXVII. Electron impact excitation collision strengths and rate coefficients for Fe IV.  Astron. Astrophys. Suppl. Ser. 1997, 126, 373–378.
  * García et al. (2009) García, J.; Kallman, T.R.; Witthoeft, M.; Behar, E.; Mendoza, C.; Palmeri, P.; Quinet, P.; Bautista, M.A.; Klapisch, M.  Nitrogen K-Shell Photoabsorption.  Astrophys. J. Suppl. Ser. 2009, 185, 477–485.  doi:black10.1088/0067-0049/185/2/477.
  * Palmeri et al. (2006) Palmeri, P.; Mendoza, C.; Bautista, M.A.; García, J.; Kallman, T.R.  K lines in the sulphur isonuclear sequence.  Rad. Phys. Chem. 2006, 75, 1465–1470.  doi:black10.1016/j.radphyschem.2005.07.009.
  * Fernley et al. (1987) Fernley, J.A.; Seaton, M.J.; Taylor, K.T.  Atomic data for opacity calculations. VII. Energy levels, f values and photoionisation cross sections for He-like ions.  J. Phys. B At. Mol. Phys. 1987, 20, 6457–6476.  doi:black10.1088/0022-3700/20/23/032.
  * Peach et al. (1988) Peach, G.; Saraph, H.E.; Seaton, M.J.  Atomic data for opacity calculations. IX. The lithium isoelectronic sequence.  J. Phys. B At. Mol. Phys. 1988, 21, 3669–3683.  doi:black10.1088/0953-4075/21/22/006.
  * Tully et al. (1990) Tully, J.A.; Seaton, M.J.; Berrington, K.A.  Atomic data for opacity calculations. XIV. The beryllium sequence.  J. Phys. B At. Mol. Phys. 1990, 23, 3811–3837.  doi:black10.1088/0953-4075/23/21/016.
  * Fernley et al. (1999) Fernley, J.A.; Hibbert, A.; Kingston, A.E.; Seaton, M.J.  Atomic data for opacity calculations: XXIV. The boron-like sequence.  J. Phys. B At. Mol. Phys. 1999, 32, 5507–5522.  doi:black10.1088/0953-4075/32/23/307.
  * Luo and Pradhan (1989) Luo, D.; Pradhan, A.K.  Atomic data for opacity calculations. XI. The carbon isoelectronic sequence.  J. Phys. B At. Mol. Phys. 1989, 22, 3377–3395.  doi:black10.1088/0953-4075/22/21/005.
  * Hibbert and Scott (1994) Hibbert, A.; Scott, M.P.  Atomic data for opacity calculations. XXI. The neon sequence.  J. Phys. B At. Mol. Phys. 1994, 27, 1315–1323.  doi:black10.1088/0953-4075/27/7/007.
  * Butler et al. (1993) Butler, K.; Mendoza, C.; Zeippen, C.J.  Atomic data for opacity calculations. XIX. The magnesium isoelectronic sequence.  J. Phys. B At. Mol. Phys. 1993, 26, 4409–4423.  doi:black10.1088/0953-4075/26/23/013.
  * Mendoza et al. (1995) Mendoza, C.; Eissner, W.; LeDourneuf, M.; Zeippen, C.J.  Atomic data for opacity calculations. XXIII. The aluminium isoelectronic sequence.  J. Phys. B At. Mol. Phys. 1995, 28, 3485–3504.  doi:black10.1088/0953-4075/28/16/006.
  * Nahar and Pradhan (1993) Nahar, S.N.; Pradhan, A.K.  Atomic data for opacity calculations. XVIII. Photoionization and oscillator strengths of Si-like ions Si0, S2+, Ar4+, Ca6+.  J. Phys. B At. Mol. Phys. 1993, 26, 1109–1127.  doi:black10.1088/0953-4075/26/6/012.
  * Bautista (1996) Bautista, M.A.  Atomic data from the Iron Project. XVI. Photoionization cross sections and oscillator strengths for Fe V.  Astron. Astrophys. Suppl. Ser 1996, 119, 105–110.
  * Bautista and Pradhan (1997) Bautista, M.A.; Pradhan, A.K.  Atomic data from the Iron Project. XXVI. Photoionization cross sections and oscillator strengths for Fe IV.  Astron. Astrophys. Suppl. Ser 1997, 126, 365–371.  doi:black10.1051/aas:1997386.
  * Nahar and Pradhan (1996) Nahar, S.N.; Pradhan, A.K.  Atomic data from the Iron Project. XVII. Radiative transition probabilities for dipole allowed and forbidden transitions in Fe III.  Astron. Astrophys. Suppl. Ser. 1996, 119, 509–522.
  * Nahar and Pradhan (1994) Nahar, S.N.; Pradhan, A.K.  Atomic data for opacity calculations: XX. Photoionization cross sections and oscillator strengths for Fe II.  J. Phys. B At. Mol. Phys. 1994, 27, 429–446.  doi:black10.1088/0953-4075/27/3/010.
  * Bautista (1997) Bautista, M.A.  Atomic data from the IRON Project. XX. Photoionization cross sections and oscillator strengths for Fe I.  Astron. Astrophys. Suppl. Ser 1997, 122, 167–176.  doi:black10.1051/aas:1997327.
  * Gorczyca (2000) Gorczyca, T.W.  Auger decay of the photoexcited 1s-1np Rydberg series in neon.  Phys. Rev. A 2000, 61, 024702.  doi:black10.1103/PhysRevA.61.024702.
  * Witthoeft et al. (2013) Witthoeft, M.C.; Bautista, M.A.; García, J.; Kallman, T.R.; Mendoza, C.; Palmeri, P.; Quinet, P.  Photoionization and photoabsorption cross sections for the aluminum iso-nuclear sequence.  At. Data Nucl. Data Tables 2013, 99, 53–61.  doi:black10.1016/j.adt.2012.04.003.