Hydrogen Lyman β, γ, and Balmer β Spectral Lines in Strong Uniform Magnetic and Electric Fields

Understanding the behavior of atoms in external magnetic and electric fields is of fundamental physics interest. Such an atomic system has attracted considerable attention since the early 1970s. Demkov et al. (1970) first formulated the energy level splitting of hydrogen in magnetic and electric fields of arbitrary mutual orientation within the framework of perturbation theory, and the symmetry of the four-dimensional Fock rotation group was adopted in their formulation. Soon afterward, a quasiclassical calculation was also reported for atomic states of H-like systems in crossed electric and magnetic fields (Burkova et al. 1976). Relevant experimental progress lagged behind theoretical work due to the difficulty of experimental techniques. Up until the 1980s, high-resolution laser spectroscopy setups were being constructed to study the problem of atoms in the presence of magnetic and electric fields (Cacciani et al. 1986). Photoexcitation spectra of lithium in parallel external fields below the field-free ionization threshold were recorded, and the structures of energy spectra belonging to high-n manifolds of lithium were systematically investigated as a function of the intensity of one of the two fields, with the other one being kept constant in Cacciani et al. (1986, 1988).

The development of the experimental technique mentioned above further stimulated interest in atoms in external fields. This atomic system may serve as a potential tool to investigate chaotic phenomena because of its unique characteristics. Main & Wunner (1992) was dedicated to this type of study by selecting photoionization processes of hydrogen in crossed magnetic and electric fields. A quantum calculation of photoionization cross sections was performed using the complex-coordinate-rotation method, and strong Ericson fluctuations, a characteristic feature of chaotic scattering, were found in the calculated photoionization spectra. This theoretical prediction was first observed by Stania & Walther (2005) in a photoionization experiment of rubidium atoms in strong crossed fields. Hydrogen in two kinds of external fields with an arbitrary mutual orientation is a more general system than that in crossed magnetic and electric fields, and hence photoionization for such a system was selected by Main et al. (1998) to further deeply understand chaotic phenomena. The dependence of photoionization spectra on the mutual orientation of two kinds of external fields was studied. Their calculations of eigenenergies show strong level repulsions around the Stark saddle point, namely the so-called avoided crossings. The large avoided crossings are believed to be a quantum manifestation of the classical intramanifold chaos.

To explain the experimental photoabsorption spectra of multielectronic atoms, as reported in Cacciani et al. (1986, 1988), the R-matrix method combined with the quantum-defect theory was formulated by Santos et al. (1999) to describe the behavior of multielectronic atoms in parallel electric and magnetic fields. This theoretical approach can be regarded as an extension of the previous adiabatic-basis-expansion method, which was originally developed by Mota-Furtado & O'Mahony (1991) to calculate the photoionization of atoms and molecules in a pure magnetic field with terrestrial laboratory strengths, and later applied to calculations of continuum spectra of atoms in white dwarf magnetic field strengths (Mota-Furtado & O'Mahony 2007; Zhao et al. 2021; Zhao 2021a). Using this approach, Santos et al. (1999) presented photoionization spectra for Li and Rb in parallel external fields and gave a detailed analysis of nonhydrogenic characters of the spectra for Li and Rb. Their computational spectra were compared to high-precision experimental ones, and good agreement was found.

The discovery of strong magnetic fields in extremely dense celestial objects, the so-called magnetic white-dwarf and neutron stars (Garstang 1977), further strengthened interest in the problem of atoms in magnetic and electric fields. The magnetic fields of these celestial objects can be determined from Zeeman spectral lines calculated and astronomically observed in their atmospheres, while the magnetic fields are crucial to understanding their evolution from normal stars to white-dwarf and neutron stars (Garstang 1977). A variety of theoretical studies have been hitherto performed to calculate the atomic structures and spectra in a wide range of magnetic fields (see Ruder et al. 1994 and references therein). Substantial headway was made on the simulation of spectra for hydrogen in an arbitrary magnetic field (see, e.g., Forster et al. 1984; Rösner et al. 1984; Baye et al. 2008a, 2008b; Schimeczek & Wunner 2014a, 2014b; Zhao & Liu 2019; Liu & Zhao 2020), and thus the identification of discrete spectral lines for any transition of magnetized hydrogen has become practicable in white-dwarf and neutron stars with hydrogen-dominated atmospheres.

As more and more white dwarfs were classified as magnetic (Ferrario et al. 2015, 2020), some magnetized multielectronic atoms were identified in the atmospheres of white dwarfs. For example, discrete spectral lines of Na i, Mg i, Al i, Ca ii, and Fe i in the cool magnetic white dwarf NLTT 7547 (Kawka et al. 2019) and lines of Na i, Mg i, Ca i, and Ca ii were observed in the magnetic DZ white dwarf LHS 2534 (Reid et al. 2000). In the current stage, however, electron correlation is effectively treated only in several few-electron atoms in magnetic fields of white-dwarf and/or neutron stars, such as He and Li (see, e.g., Becken et al. 1999; Zhao 2018, and references therein), and calculations of magnetized multielectronic atoms depend mainly on the Hartree–Fock theory, which is not able to produce spectral data with the sufficient accuracy required in astronomy and astrophysics. The progress on the observation of spectral lines in magnetic white dwarfs poses a big challenge to theorists, who are inevitably confronted with the difficult problem of electron correlation in a strong magnetic field.

Notwithstanding significant success in the simulation of atomic spectral lines in a pure magnetic field with arbitrary strength, distinct discrepancies were noticed between the computed and observed spectra in some magnetic white dwarfs. For example, absorption spectra for two stationary transitions, 2s₀ → 3p₀ and 2s₀ → 4f₀, observed in the hydrogen-dominated atmosphere of Grw+70.8247 were found to shift about 5–6 Å with respect to the computed spectra (Friedrich et al. 1994). The reasons that give rise to the distinct discrepancies have been analyzed by Friedrich et al. (1994) and Fassbinder & Schweizer (1996a). Their analyses show that an additional electric field, which may be caused by free electrons and ions in the stellar atmosphere or motional Stark effects, should be responsible for the discrepancies. The electric field strength was believed to be of the order of 10⁶–10⁹ V m⁻¹.

Fassbinder & Schweizer (1996a) theoretically studied the problem of hydrogen in strong magnetic and electric fields with arbitrary mutual orientation using a discrete variable representation (DVR) method. The DVR method was first developed by Melezhik (1993) to calculate atomic structures of hydrogen in external magnetic and electric fields, and then soon extended by Fassbinder & Schweizer (1996a) to simulate discrete spectral lines of this system observed in white-dwarf stars. It was found that in the range of white-dwarf field strengths, the influence of the perpendicular component of an arbitrarily oriented electric field on wavelengths and oscillator strengths is negligible. This important conclusion makes it much easier to describe hydrogen in magnetic and electric fields because the problem of hydrogen in two kinds of fields with arbitrary mutual orientation is able to be simplified to that in parallel external fields. Soon afterward, Fassbinder & Schweizer (1996b) reduced their three-dimensional discretization method to a two-dimensional one and calculated stationary spectral lines in parallel magnetic and electric fields. Wavelengths and oscillator strengths of Lyman α, and Balmer α and Balmer β lines were presented as a function of white-dwarf-strength magnetic and electric fields.

Guan & Zhang (2004) formulated a two-dimensional discretization pseudospectral method to understand the influence of an additional electric field on Zeeman spectral lines of hydrogen atoms in a magnetic field. These two fields are parallel with each other, and their field strengths are taken in the regions with magnetic fields B ≤ 470 MG and electric fields F ≤ 10⁸ V m⁻¹. Oscillator strength spectra were reported for Balmer α transitions of hydrogen atoms in parallel fields as a function of magnetic and electric field strengths. Based on the pseudospectral discretization technique, Guan (2006) developed two independent methods to study the more general problem of hydrogen in crossed magnetic and electric fields with arbitrary mutual orientation, and spectra of the Balmer α series in crossed fields were computed using the two methods. Special attention was paid to the effect of electric fields with arbitrary orientation on hydrogen Zeeman spectral lines in Guan (2006), and the perpendicular component of an electric field was found to give rise to a weaker coupling between atomic states belonging to different subspaces of magnetic quantum numbers. This conclusion is similar to that obtained by Fassbinder & Schweizer (1996a).

To the best of our knowledge, theoretical results available in the literature are far from satisfying the requirements for astronomical application. A comprehensive compilation of atomic spectral data containing all transitions in magnetic and electric fields is obviously essential. Recently, the two-dimensional B-spline approach, which was previously developed to describe lithium atoms in a pure magnetic field (Zhao 2020, 2021b), was extended to calculate hydrogen in parallel magnetic and electric fields (Zhao & Liu 2021). Wavelengths and oscillator strengths were presented for spectral lines of 14 Balmer α transitions as a function of white-dwarf-strength magnetic and electric fields. In this paper, we will apply the extended approach to calculations of spectral lines of Lyman α, β, γ, and Balmer β transitions of hydrogen atoms in parallel magnetic and electric fields. However, we will not list here the atomic data of Lyman α lines because the influence of an electric field with strength F ≤ 10⁸ eV on their wavelengths and oscillator strengths is trivial.

The remainder of this paper is organized as follows. Section 2 outlines the two-dimensional B-spline approach utilized in this work, including the theoretical description of hydrogen atoms in parallel magnetic and electric fields, the numerical solution of the two-dimensional Schrödinger equation, and the calculations of the spectral lines. Section 3 is devoted to a detailed presentation of the calculated spectral lines for a total of 31 transitions of hydrogen atoms in parallel magnetic and electric fields. Our spectral data are also compared with available results in the literature in this section. The main results of this work are summarized, and concluding remarks concerning the influence of an additional electric field on spectral lines of hydrogen atoms in pure magnetic fields are given in Section 4. Atomic units are used throughout unless otherwise noted.

We recapitulate in this section the two-dimensional B-spline approach utilized in the current work. This approach was previously developed to calculate the Zeeman spectra of lithium atoms in a pure white-dwarf-strength magnetic field (Zhao 2020, 2021b), following the formulation of Schimeczek & Wunner (2014a). Very recently, it has been extended to describe hydrogen atoms in parallel magnetic and electric fields (Zhao & Liu 2021).

2.1. Hamiltonian and Symmetries of Hydrogen in Parallel Fields

Let us take into account an atomic system of hydrogen in the presence of external magnetic B and electric fields F, with the directions of both fields selected to point toward the direction of the z-axis. Our theoretical description of this system begins by assuming that the nuclear mass is infinite and its relativistic effects, such as spin–orbit coupling, are negligible. In this assumption, the Hamiltonian of this system can be written in cylindrical coordinates as

$$\begin{eqnarray}\begin{array}{rcl}\widehat{H} & = & -\displaystyle \frac{1}{2}\left(\displaystyle \frac{{\partial }^{2}}{\partial {\rho }^{2}}+\displaystyle \frac{1}{\rho }\displaystyle \frac{\partial }{\partial \rho }+\displaystyle \frac{1}{{\rho }^{2}}\displaystyle \frac{{\partial }^{2}}{\partial {\phi }^{2}}+\displaystyle \frac{{\partial }^{2}}{\partial {z}^{2}}\right)-\displaystyle \frac{1}{\sqrt{{\rho }^{2}+{z}^{2}}}\\ & & +\,\displaystyle \frac{\gamma }{2}\left({\hat{{\ell }}}_{z}+2{\hat{s}}_{z}\right)+\displaystyle \frac{1}{8}{\gamma }^{2}{\rho }^{2}+{\mathscr{F}}z,\end{array}\end{eqnarray} \tag{ 1 }$$

where γ = B/B₀ and ${\mathscr{F}}=F/{F}_{0}$ denote the magnetic and electric field strengths with ${B}_{0}={m}_{e}^{2}{e}^{3}/{{\hslash }}^{3}\approx 2.35\times {10}^{5}$ T and ${F}_{0}={m}_{e}^{2}{e}^{5}/{{\hslash }}^{4}\approx 5.14\times {10}^{11}$ V m⁻¹, and ${\hat{{\ell }}}_{z}$ and ${\hat{s}}_{z}$ are the z components of the orbital and spin angular momenta, respectively. In Equation (1), the third term linear in γ is the paramagnetic potential, the fourth term quadratic in γ is the diamagnetic potential, and the term ${\mathscr{F}}z$ represents the interaction of the hydrogen atom with the electric field. Note that energies in Equation (1) are measured in units of Hartrees. For the hydrogen atom in a pure magnetic field, i.e., ${\mathscr{F}}z$ in Equation (1) is removed, ${\hat{{\ell }}}_{z}$, ${\hat{s}}_{z}$, and the parity in the z direction π_z are constants of motion for the Hamiltonian.

It is easily seen that if an additional parallel electric field is added to a hydrogen atom exposed to a magnetic field, the parity symmetry of this atomic system is no longer maintained, but ${\hat{{\ell }}}_{z}$ and ${\hat{s}}_{z}$ are still conserved quantities. As a consequence of the conservation of the z component of the orbital angular momentum, the three-dimensional Schrödinger equation for the Hamiltonian (1) can be reduced to a two-dimensional problem, while the two-dimensional Schrödinger equation is no longer separable due to the existence of magnetic fields, and thus it becomes inevitable that two-dimensional differential equations will need to be solved.

2.2. Solution of the Two-dimensional Schrödinger Equation

We select a finite basis set from B-spline functions to solve the two-dimensional Schrödinger equations. Concerning the B-spline functions, many types of computer programs have been written for the purpose of calculations. This kind of function was initially introduced in the 1950s, and a surge of applications of B-splines in many fields appeared in the 1990s with the advent of powerful computers (Bachau et al. 2001). Their properties have been detailed in de Boor (2001), and a FORTRAN code to produce B-spline functions with any order is given in that book. In particular, applications of B-splines in atomic and molecular physics have been reviewed by Bachau et al. (2001). Details on computing magnetized atoms using B-splines are also available in Zhao & Stancil (2007).

Our procedure for determining the eigenvalues and eigenfunctions of the Hamiltonian (1) starts from expanding the wave function Ψ(ρ, z, ϕ) in the ρ − z plane of the cylindrical coordinate system in terms of B-spline functions B_i,k of order k,

$$\begin{eqnarray}&&{\rm{\Psi }}(\rho ,z,\phi )=\displaystyle \frac{{e}^{{im}\phi }}{\sqrt{2\pi }}\sum _{i,j}{C}_{{ij}}{B}_{i,k}(\rho ){B}_{j,k}(z),\end{eqnarray} \tag{ 2 }$$

where m represents the magnetic quantum number corresponding to ${\hat{{\ell }}}_{z}$, and B_i,k (z) and B_i,k (ρ) are located in limited regions with $-{z}_{\max }\leqslant z\leqslant {z}_{\max }$ and $0\leqslant \rho \leqslant {\rho }_{\max }$, respectively. Substituting Equation (2) into the Schrödinger equation for Hamiltonian (1), multiplying the resulting equation by ${B}_{i^{\prime} ,k}(\rho ){B}_{j^{\prime} ,k}(z)$ for each $i^{\prime}$ and each $j^{\prime}$, and then integrating each of the obtained equations over ρ and z, we obtain a generalized eigenvalue equation set. Using a compact matrix notation, the generalized eigenvalue equation set is written as

$$\begin{eqnarray}&&{HC}=E{ \mathcal N }C,\end{eqnarray} \tag{ 3 }$$

where H and ${ \mathcal N }$ are the Hamiltonian and overlap matrices. The expressions of their matrix elements have been given in Zhao (2020) and hence are not listed here again. We would like to emphasize that the overlap matrix ${ \mathcal N }$ is reduced to an identity matrix if the nonorthogonal B-spline basis is replaced by an orthogonal basis set.

The solution of any eigenvalue or generalized eigenvalue problem inevitably involves applying suitable boundary conditions. The details to enforce the physical boundary conditions of the problem given in Equation (3) have been illustrated in Zhao & Liu (2021) and hence are not repeated here. We adopt the Gaussian quadratures (Press et al. 1992) to compute each matrix element of H and ${ \mathcal N }$. Once numerical calculations of all matrix elements are finished, we utilize the Cholesky matrix factorization technique to transfer the generalized eigenvalue problem into an eigenvalue problem and then solve the resulting eigenvalue equations. Note that we will not directly factorize ${ \mathcal N }$ and instead factorize two small-size overlap submatrices belonging to the ρ and z coordinates, ${{ \mathcal N }}_{\rho }$ and ${{ \mathcal N }}_{z}$, in order to save computing time. It can be demonstrated that ${ \mathcal N }$ is the Kronecker product of ${{ \mathcal N }}_{\rho }$ and ${{ \mathcal N }}_{z}$, i.e., ${ \mathcal N }={{ \mathcal N }}_{\rho }\otimes {{ \mathcal N }}_{z}$, while it is much faster to compute the Cholesky matrix factorization of ${ \mathcal N }$ from the two submatrices ${{ \mathcal N }}_{\rho }$ and ${{ \mathcal N }}_{z}$ than directly factorize ${ \mathcal N }$.

2.3. Oscillator Strengths for Dipole Transitions in Parallel Fields

Using energy eigenvalues and eigenvectors obtained by solving the generalized eigenvalue problem, Equation (3), wavelengths and oscillator strengths of spectral lines for any transition of hydrogen in parallel magnetic and electric fields can be computed. Considering the fact that the m degeneracy in the field-free case is entirely removed when an atom is placed in an external magnetic field, an averaging procedure over m in the initial and final atomic states is no longer required, and therefore the oscillator strength for a dipole transition from an initial state Ψ_i to a final state Ψ_f is written in the form

$$\begin{eqnarray}&&{f}_{{if}}=2({E}_{f}-{E}_{i})| \langle {{\rm{\Psi }}}_{f}| D| {{\rm{\Psi }}}_{i}\rangle {| }^{2},\end{eqnarray} \tag{ 4 }$$

where E_i and E_f denote the energy levels for the initial and final states, respectively, and D is the dipole operator, defined in the cylindrical coordinate system as

$$\begin{eqnarray}D=\left\{\begin{array}{cl}z & \mathrm{for}\ \mathrm{linearly}\ \mathrm{polarized}\ \mathrm{light}\\ \mp \displaystyle \frac{1}{\sqrt{2}}\rho {e}^{\pm i\phi } & \mathrm{for}\ \mathrm{circularly}\ \mathrm{polarized}\ \mathrm{light},\end{array}\right.\end{eqnarray} \tag{ 5 }$$

specifying the interaction of atoms with the radiation field.

A crucial step of atomic structure calculations with a finite basis set from B-spline functions is to choose a suitable knot sequence {t_i } in a limited region, according to the distribution of wave functions. For the system of hydrogen atoms in parallel magnetic and electric fields, a scheme to distribute the knot sequences on both the ρ- and z-axes with a linearly increasing spacing was found to be an efficient scheme (Zhao & Liu 2021), which was first introduced by Schimeczek & Wunner (2014a) to describe hydrogen in a pure magnetic field. This distribution scheme of the knot sequences on both the ρ- and z-axes is adopted in the current work, and ρ and z are supposed to be located in a limited range, [0, ${\rho }_{\max }]$ and $[-{z}_{\max }$, z_max], respectively.

For the system of atoms in a pure magnetic field, the integration in the z coordinate involved in calculations of all matrix elements of H and ${ \mathcal N }$ can be restricted to z ≥ 0, as done in Schimeczek & Wunner (2014a, 2014b) and Zhao (2021b). It is reasonable to impose this restriction because the parity under the reflection of the wave function about the z = 0 plane is a conserved quantity in this case. However, if an additional electric field is added, the parity symmetry in the z direction is broken, and as a consequence, such a restriction is no longer valid. In such a case, one has to perform integration from $z=-{z}_{\max }$ to z_max. The feature that the parity symmetry no longer remains for the system of atoms in external magnetic and electric fields leads to greatly increasing the number of the B-spline basis and the size of the Hamiltonian matrix, and thus computing time is remarkably increased compared to the case of atoms in a pure magnetic field.

Our aim in the current work lies in the Lyman α, β, γ, and Balmer β spectral lines of hydrogen in parallel magnetic and electric fields, as these lines are relevant to the atomic states in the n = 1–4 manifolds. Note that the hydrogen Balmer α spectral lines in parallel fields have already been reported (Zhao & Liu 2021), and thus it is not required to recalculate their atomic states in the n = 2 and 3 manifolds related to these Balmer α lines. Only atomic states in the n = 1 and 4 manifolds are to be calculated. These states will be optimized based on the following steps. For each atomic state of a given magnetic field and a given electric field, we vary ${\rho }_{\max }$, z_max, and the knot sequences on both the ρ- and z-axes to seek the optimized eigenvalue of the atomic state. Furthermore, we also check the convergence of the current problem by means of increasing the number N_ρ and order k_ρ of B-spline functions on the ρ-axis, and the number N_z and order k_z of B-spline functions on the z-axis.

The present optimal calculations of atomic structures begin from selecting reasonable initial values of the following parameters, ${\rho }_{\max }$, z_max, N_ρ , and N_z . Doing so can save a lot of computing time. Small and large initial values of these parameters should be taken, respectively, for low- and high-lying atomic states. For example, for the high-lying $4{f}_{0}^{{\prime} }$ state at γ = 0.01 au and F = 10⁸ V m⁻¹, we took these initial values be ${\rho }_{\max }=100$, z_max = 100, N_ρ = 32, and N_z = 62. By increasing the four parameters up to 120, 150, 72, 152, we found that a convergent energy eigenvalue of eight significant digits, −0.038259232 au, is obtained at ${\rho }_{\max }=100$, z_max = 100, N_ρ = 52, and N_z = 102. It should be mentioned that convergence with only five significant digits is reached using the initial values of these parameters.

We make use of the notation of Fassbinder & Schweizer (1996b), $n{{\ell }}_{m}^{{\prime} }$, to specify atomic states in parallel magnetic and electric fields throughout this paper. This notation with the prime distinguishes that in a pure magnetic field, n ℓ_m (see, e.g., Rösner et al. 1984). We optimized atomic states in the n = 1 and 4 manifolds, $1{s}_{0}^{{\prime} }$, $4{p}_{0,-1}^{{\prime} }$, $4{d}_{0,-1,-2}^{{\prime} }$, and $4{f}_{0,-1,-2}^{{\prime} }$, for given magnetic and electric field strengths. The field strengths range, respectively, from 23.5 to 2350 MG, and from 0 to 10⁸ V m⁻¹. Notice that we did not include $4{s}_{0}^{{\prime} }$ because ionization of this state begins to be significant for not very low electric field strengths. It is well known that an atomic system placed in an electric field may ionize by tunneling through the potential barrier. If the energy level of this system approaches, and in particular is above, the classical saddle point $-2\sqrt{{\mathscr{F}}}$, its ionization becomes so fast that it is imperative to include continuum channels in the wave function. In such a case, it is inappropriate to treat the atomic state as a bound state anymore. However, it is not the aim of this work to treat tunneling through the potential barrier, so our current calculations drop those spectral lines related to $4{s}_{0}^{{\prime} }$.

To understand the spectral line shifts caused by electric fields, it is obviously helpful to explore the influence of electric fields on the atomic states of hydrogen in a magnetic field. We illustrate the variation of atomic states of magnetized hydrogen in n = 4 manifolds with electric fields. Contour plots of probability density distributions for two states $4{f}_{0}^{{\prime} }$ and $4{d}_{-1}^{{\prime} }$ are shown in Figures 1 and 2. We selected three magnetic field strength values of γ = 0.01, 0.1, and 1.0 au and drew plots of F = 0, 10⁶, 10⁷, and 10⁸ V m⁻¹ for each γ value. From these two figures, one can see a definite symmetry of atomic states with respect to the inversion of z for each γ value of these two states at F = 0 V m⁻¹, but such a definite symmetry is clearly broken by a strong electric field. For a given γ, the asymmetry of atomic states becomes more and more remarkable as electric fields increase.

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The probability density distributions for the two n = 4 manifold states at γ = 1.0 au and F = 10⁸ V/cm are discovered to obviously deviate from symmetry with respect to the inversion of z. This case is different from that of the n = 3 manifold states. The influence of electric fields on the probability density distributions for $3{d}_{0}^{{\prime} }$ was noticed to be imperceptible at the same field strengths (Zhao & Liu 2021). This is attributed to their different energy levels. The atomic states $4{f}_{0}^{{\prime} }$ and $4{d}_{-1}^{{\prime} }$ of higher energy levels are more easily affected than the $3{d}_{0}^{{\prime} }$ of lower energy levels. For a given strong electric field, the variation of electron cloud distributions with γ is also visible. One can see from the last column of Figure 1 that the electron clouds are distributed almost under the z = 0 plane at γ = 0.01 au but are primarily above the z plane at γ = 0.1 au, and again primarily under the z = 0 plane at γ = 1.0 au. Such a variation is the result of competition among the Coulombic force, the Lorentz force, and the electrostatic field force. A similar phenomenon was also found by Paradis et al. (2013). They show that the calculated normalized electron wave function probabilities of Rb atoms in parallel magnetic and electric fields are distributed not only under but also above the z = 0 plane under different conditions.

Using atomic structure data in the n = 1–4 manifolds obtained, spectral lines of two transitions, $1{s}_{0}^{{\prime} }\to 2{p}_{0}^{{\prime} }$ and $1{s}_{0}^{{\prime} }\to 2{p}_{-1}^{{\prime} }$, are first calculated. Our current calculations show that the influence of electric fields with strengths F ≤ 10⁸ V m⁻¹ on Zeeman spectral lines of Lyman α transitions is trivial. The very same conclusion was also drawn by Fassbinder & Schweizer (1996b). They discovered that only for very strong electric fields do the wavelengths of spectral lines of Lyman α transitions show an observable variation. For example, an electric field of strength F = 10⁹ V m⁻¹ was found to give rise to a redshift of about 0.4 Å for the transition $1{s}_{0}^{{\prime} }\to 2{p}_{-1}^{{\prime} }$. In view of the rather small influence of electric fields on Lyman α lines, it is superfluous to tabulate Lyman α lines in electric fields of strengths F ≤ 10⁸ V m⁻¹.

Let us then move on to Lyman β, γ, and Balmer β spectral lines in parallel fields. Wavelengths and oscillator strengths for a total of 31 transitions of these three spectral series are calculated and presented in Tables 1–31 as a function of magnetic and electric field strengths. The 31 transitions include 16 special transitions, which are dipole-forbidden in a pure magnetic field, such as $1{s}_{0}^{{\prime} }\to 3{s}_{0}^{{\prime} }$ and $2{p}_{-1}^{{\prime} }\to 4{f}_{0}^{{\prime} }$. Only when an electric field is added do these dipole-forbidden transitions in a pure magnetic field become allowed. Our oscillator strengths calculated for 16 special transitions are found to increase by several orders of magnitude when increasing electric fields from F = 10⁶ to 10⁸ V m⁻¹. From the current results and those reported in Zhao & Liu (2021), it can be concluded that only in the case of relatively high electric fields are spectral lines for dipole-forbidden transitions in a pure magnetic field strong enough to be visible in celestial observations. It is worth mentioning that in the scope of field strengths illustrated in these tables, the Lyman β and γ spectral lines lie in the ultraviolet region, while the Balmer β lines in the ultraviolet and visible-light regions.