O₄ -Symmetry-Based Non-Perturbative Analytical Calculations of the Effect of the Helical Trajectories of Electrons in Strongly Magnetized Plasmas on the Width of Hydrogen/Deuterium Spectral Lines

Abstract

The effects of the helical trajectories of the perturbing electrons in magnetized plasmas on the dynamical Stark width of hydrogen or deuterium spectral lines have been studied analytically in our previous two papers—specifically in the situation where the magnetic field B is so strong that the dynamical Stark width of these lines reduces to the so-called adiabatic Stark width because the so-called nonadiabatic Stark width is completely suppressed. This situation corresponds, for example, to DA and DBA white dwarfs. We obtained those analytical results by using the formalism of the so-called conventional (or standard) theory of the impact Stark broadening: namely, by performing calculations in the second order of the Dyson perturbation expansion. The primary outcome was that the dynamical Stark broadening was found to not depend on the magnetic field B (for sufficiently strong B). In the present paper, we use the O₄ symmetry of hydrogen atoms for performing the corresponding non-perturbative analytical calculations equivalent to accounting for all orders of the Dyson perturbation expansion. The results, obtained by using the O₄ symmetry of hydrogen atoms, differ from our previous ones not only quantitatively, but—most importantly—qualitatively. Namely, the dynamical Stark broadening does depend on the magnetic field B, even for strong B. These results should be important for revising the interpretation of the hydrogen Balmer lines observed in DA and DBA white dwarfs. We also address confusion in the literature on this subject.

1. Introduction

Among the analytical theories of the dynamical Stark broadening of hydrogen (or deuterium) spectral lines in plasmas, the most practically useful are the semiclassical theories that started from the pioneering works by Baranger [1,2]. These were then further developed in paper [3] into what was later called the conventional or standard theory, where it was assumed that the ion microfield was quasistatic (from the point of view of radiating atoms), while the electron microfield was dynamical and treated in the impact approximation. The primary feature of the impact approximation is treating the electron microfield as a sequence of completed binary collisions of the electrons with the radiating atom. The allowance for incomplete collisions was later made in papers [4,5,6], calling this version the unified theory.

In the intervening dozens of years, many further advances have been made, especially in treating the dynamical part of the ion microfield. We refer the readers to books [7,8,9,10,11,12,13,14,15,16], listed in chronological order, and the references therein.

The effects of the helical trajectories of the perturbing electrons in magnetized plasmas on the dynamical Stark width of hydrogen or deuterium spectral lines has been studied analytically for the first time in papers [17,18]—specifically in the situation where the magnetic field B is so strong that the dynamical Stark width of these lines reduces to the so-called adiabatic Stark width because the so-called nonadiabatic Stark width is completely suppressed. This situation corresponds, for example, to DA and DBA white dwarfs. The analytical results were obtained in papers [17,18] by using the formalism of the so-called conventional or standard theory of the impact Stark broadening [3,19]: namely, by performing calculations in the second order of the Dyson perturbation expansion. The primary outcome was that the dynamical Stark broadening was found not to depend on the magnetic field B (for sufficiently strong B).

In the present paper, we use the O₄ symmetry of hydrogen atoms for performing the corresponding non-perturbative analytical calculations equivalent to accounting for all orders of the Dyson perturbation expansion. The results, obtained by using the O₄ symmetry of hydrogen atoms, differ from those obtained in papers [17,18] not only quantitatively, but—most importantly—qualitatively: namely, the dynamical Stark broadening does depend on the magnetic field B, even for strong B.

2. Summary of Starting Formulas from Papers [17,18]

1. The radius vector R(t) of a perturbing electron traveling on a helical trajectory in a magnetized plasma is

R(t) = v_ztB/B + ρ[1 + (r_Bp/ρ) cos(ω_Bt + φ)] + ρ × B [r_Bp/(ρB)] sin(ω_Bt + φ).

(1)

In Equation (1), ρ is the impact parameter vector and ρ × B is its cross-product (vector product) with the magnetic field B; v_z is the component of the electron velocity parallel to B (the z-axis is chosen parallel to B). Other notations are presented as follows:

r_Bp = v_p/ω_B, ω_B = eB/(m_ec).

(2)

In Equation (2), ω_B is the Larmor frequency and v_p is the component of the electron velocity perpendicular to B.

2. The electric field created by the perturbing electron at the location of the radiating hydrogen or deuterium atom:

E(t) = eR(t)/[R(t)]³.

(3)

The z-projection of this electric field is

E_z(t) = e(v_zt)/[ρ² + v_z²t² + v_p²/ω_B² + 2(ρv_p/ω_B) cos(ω_Bt + φ)]^3/2,

(4)

3. The nonadiabatic contribution to the dynamical Stark width, i.e., the contribution caused by the component of the field E(t) perpendicular to B, is completely suppressed if

B > B_cr = 2T_e/(3|X_αβ|eλ_c) = 9.2 × 10²T_e(eV)/|X_αβ| Tesla,

(5)

where λ_c = ħ/(m_ec) = 2.426 × 10⁻¹⁰ cm is the Compton wavelength of electrons. In Equation (5), T_e is the electron temperature, and

X_αβ = |n_a(n₁ − n₂)_α − n_b(n₁ − n₂)_β|.

(6)

In Equation (6), n is the principal quantum number, and n₁ and n₂ are the parabolic quantum numbers of the upper (aα) and lower (bβ) Stark states involved in the radiative transition.

4. The adiabatic part Φ_ad of the electron broadening operator Φ_ab (and the corresponding Stark width) is controlled by E_z(t), given by Equation (4).

5. For the helical trajectories of perturbing electrons, the starting formula for the adiabatic Stark width Γ_αβ = − Re[_αβ(Φ_ad)_βα] for the line component, corresponding to the radiative transition between the upper Stark sublevel α and the lower Stark sublevel β, is

$${\Gamma }_{\mathsf{\alpha }\mathsf{\beta },\mathrm{h}\mathrm{e}\mathrm{l}}={\mathrm{N}}_{\mathrm{e}}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\mathrm{d}{\mathrm{v}}_{\mathrm{z}}{\mathrm{f}}_1\left({\mathrm{v}}_{\mathrm{z}}\right)}{\displaystyle \underset{0}{\overset{\infty }{\int }}\mathrm{d}{\mathrm{v}}_{\mathrm{p}}{\mathrm{f}}_2\left({\mathrm{v}}_{\mathrm{p}}\right)}{\left({{\mathrm{v}}_{\mathrm{z}}}^2+{{\mathrm{v}}_{\mathrm{p}}}^2\right)}^{1/2}\mathsf{\sigma }\left({\mathrm{v}}_{\mathrm{z}},{\mathrm{v}}_{\mathrm{p}}\right).$$

(7)

In Equation (7), f₂(v_p) is the two-dimensional Maxwell distribution and f₁(v_z) is its one-dimensional counterpart. The operator σ(v_z, v_p) discussed below has the physical meaning of the cross-section of the “optical” collisions causing the dynamical Stark broadening of the spectral line.

3. Non-Perturbative Analytical Calculations

Let us start in a general way. We consider an atomic system described by the following Hamiltonian:

H(t) = H₀ + V(t), V(t) = −dE(t).

(8)

In Equation (8), H₀ is the unperturbed Hamiltonian, d is the electric dipole moment operator, and dE(t) is its scalar product (also known as the dot-product) with a time-dependent electric field E(t).

The central idea is to split the interaction V(t) into two parts

V(t) = V₁(t) + V₂(t)

(9)

and to rewrite the Hamiltonian from Equation (8) as follows:

H(t) = H₁(t) + V₁(t), H₁(t) = H₀ + V₁(t),

(10)

where the extended “unperturbed” Hamiltonian H₁(t) can be diagonalized analytically at any instant of time (at least, in a subspace of a fixed principal quantum number n). In particular, for hydrogenic atoms or ions, the analytical diagonalization of the extended “unperturbed” Hamiltonian H₁(t) is possible due to the higher than geometrical symmetry (O₄ symmetry) of the system described by H₁(t).

We remind that one of the consequences of the O₄ symmetry of hydrogenic atoms or ions is that the separation of variables is possible in several different types of coordinates: in the spherical coordinates, in the parabolic coordinates, and in the elliptical coordinates. For the present paper, the most convenient is the utilization of the parabolic coordinates, in which the projection d_z of the dipole moment operator on the magnetic field can be diagonalized within any subspace of the fixed principal quantum number n.

We choose the z-axis along the magnetic field B and use the following specific breakdown of the interaction V(t) into two parts: the adiabatic part

V₁(t) = −d_zE_z(t),

(11)

and the nonadiabatic part

V₂(t) = −d_perpE_perp(t),

(12)

where d_perp and E_perp are the components of the electric dipole moment and of the electric field, respectively, perpendicular to the magnetic field B. Under the condition (5), the nonadiabatic part of the interaction is completely suppressed, so that only the adiabatic part V₁(t) remains.

We calculate the adiabatic contribution to the dynamical Stark broadening of hydrogenic spectral lines exactly, nonperturbatively (rather than in the second order of the Dyson perturbation expansion) [20,21,22]. This is achieved along the lines of the so-called old adiabatic theory—see, e.g., papers [23,24] and Appendix A of the present paper. In the formalism of the old adiabatic theory, the cross-section σ of the optical collisions has the following form:

$$\mathsf{\sigma }=2\mathsf{\pi }{\displaystyle \underset{0}{\overset{\infty }{\int }}\mathrm{d}\varrho \varrho }〈1-\cos \left[{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\mathrm{dtd}}_{\mathrm{z}}{\mathrm{E}}_{\mathrm{z}}(\mathrm{t})/\mathrm{ħ}}\right]〉.$$

(13)

In Equation (13), the symbol <…> stands for the average over angular or phase variables; d_z is the matrix element of the z-projection of the electric dipole moment:

d_z/ħ = 3X_αβħ/(2m_ee).

(14)

The integral over time in Equation (13) vanishes for the odd part of E_z(t). Thus, it suffices to perform the integration only for the even part E_z,even(t) of E_z(t):

E_z(t)_even = [E_z(t) + E_z(−t)]/2.

(15)

The latter was calculated in papers [17,18] as follows.

The integral over the impact parameter ρ was broken into two parts

σ = σ₁ + σ₂,

(16)

σ₁ corresponding to the integral from ρ₀ to infinity and σ₂ corresponding to the integral from zero to ρ₀, where

ρ₀ = v_p/ω_B.

(17)

For calculating σ₁, E_z,even(t) was expanded in terms of the small parameter ρ₀/ρ = v_p/(ω_Bρ). After keeping the first non-vanishing term of the expansion, the following was obtained:

E_z(t)_even = (sin φ) (3eρv_pv_z/ω_B) t[sin(ω_Bt)]/(ρ² + v_z²t²)^5/2.

(18)

For calculating σ₂, E_z,even(t) was expanded in terms of the small parameter ρ/ρ₀ = ω_Bρ/v_p. After keeping the first non-vanishing term of the expansion, the following was obtained:

E_z(t)_even = (sin φ) (3eρv_pv_z/ω_B) t[sin(ω_Bt)]/(v_p²/ω_B² + v_z²t²)^5/2.

(19)

Using expressions (18) and (19), obtained in papers [17,18], we proceed now to calculating

$${\mathsf{\sigma }}_1=2\mathsf{\pi }{\displaystyle \underset{{\varrho }_0}{\overset{\infty }{\int }}\mathrm{d}\varrho \varrho }〈1-\cos \left[{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\mathrm{dtd}}_{\mathrm{z}}{\mathrm{E}}_{\mathrm{z}}{(\mathrm{t})}_{\mathrm{even}}/\mathrm{ħ}}\right]〉$$

(20)

and

$${\mathsf{\sigma }}_2=2\mathsf{\pi }{\displaystyle \underset{0}{\overset{{\varrho }_0}{\int }}\mathrm{d}\varrho \varrho }〈1-\cos \left[{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\mathrm{dtd}}_{\mathrm{z}}{\mathrm{E}}_{\mathrm{z}}{(\mathrm{t})}_{\mathrm{even}}/\mathrm{ħ}}\right]〉.$$

(21)

For σ₁, after calculating the integral over time, the expression within the symbol <…> in Equation (20) becomes

1 − cos[(sinφ)3X_αβħv_pω_BK₁(ω_Bρ/|v_z|)/(m_e|v_z|³)],

(22)

where K₁(s) is the modified Bessel function of the 2nd kind. Then, the averaging of the expression (22) over φ yields

<…> = 1 − J₀[3X_αβħv_pω_BK₁(ω_Bρ/|v_z|)/(m_e|v_z|³)],

(23)

where J₀(u) is the Bessel function. For the range of ρ under consideration, the Bessel function K₁(ω_Bρ/|v_z|) is exponentially small, so that the argument of the Bessel function J₀[…] is much smaller than unity. Therefore, Equation (23) can be simplified to the following:

<…> ≈ [3X_αβħv_pω_BK₁(ω_Bρ/|v_z|)/(2m_e|v_z|³)]².

(24)

Then, the integration over impact parameters in Equation (20) yields the following:

σ₁ ≈ (π/4)^3/2[3X_αβħv_p/(m_ev_z²)]² MeijerG[{{},{3/2}}, {{0, 0, 2},{}}, v_p²/v_z²],

(25)

where MeijerG[…] is the Meijer G-function.

Equation (24) for σ₁ coincides with σ₁ from papers [17,18]. It does not depend on the magnetic field B (provided that B satisfies the condition (5)). However, below it is shown that our nonperturbative result for σ₂ significantly differs—both quantitatively and qualitatively—from the perturbatively obtained result for σ₂ from papers [17,18]. In particular, our nonperturbative result depends on the magnetic field B, while the perturbative result from papers [17,18] did not depend on B.

For σ₂, after calculating the integral over time, the expression within the symbol <…> in Equation (21) becomes

1 − cos[(sinφ)3X_αβħv_pω_B²K₁(v_p/|v_z|) ρ/(m_e|v_z|³)].

(26)

Then, the averaging of the expression (26) over φ yields

<…> = 1 − J₀[3X_αβħω_B²K₁(v_p/|v_z|) ρ/(m_e|v_z|³)].

(27)

The subsequent integration over impact parameters in Equation (27) leads to the following result for σ₂:

σ₂ = (πv_p²/ω_B²)[1 − J₁(w)/w], w = 3X_αβħv_pω_BK₁(v_p/|v_z|)/(m_e|v_z|³).

(28)

Obviously, σ₂ depends on ω_B and thus on the magnetic field B.

For presenting the results in plots, we calculate the effective average values <v_p> and <|v_z|>, as well as their ratio, as follows (compare to Equation (7)):

$$<{\mathrm{v}}_{\mathrm{p}}>={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\mathrm{d}{\mathrm{v}}_{\mathrm{z}}{\mathrm{f}}_1\left({\mathrm{v}}_{\mathrm{z}}\right)}{\displaystyle \underset{0}{\overset{\infty }{\int }}\mathrm{d}{\mathrm{v}}_{\mathrm{p}}{\mathrm{f}}_2\left({\mathrm{v}}_{\mathrm{p}}\right)}{\left({{\mathrm{v}}_{\mathrm{z}}}^2+{{\mathrm{v}}_{\mathrm{p}}}^2\right)}^{1/2}{\mathrm{v}}_{\mathrm{p}},$$

(29)

$$<{\mathrm{v}}_{\mathrm{p}}>={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\mathrm{d}{\mathrm{v}}_{\mathrm{z}}{\mathrm{f}}_1\left({\mathrm{v}}_{\mathrm{z}}\right)}{\displaystyle \underset{0}{\overset{\infty }{\int }}\mathrm{d}{\mathrm{v}}_{\mathrm{p}}{\mathrm{f}}_2\left({\mathrm{v}}_{\mathrm{p}}\right)}{\left({{\mathrm{v}}_{\mathrm{z}}}^2+{{\mathrm{v}}_{\mathrm{p}}}^2\right)}^{1/2}|{\mathrm{v}}_{\mathrm{z}}|.$$

(30)

As a result, we find

<v_p> = 1.60 <v_T>, <|v_z|> = 0.665 <v_T>, <v_p>/<|v_z|> = 2.41,

(31)

where v_T = (2T_e/m_e)^1/2 is the mean thermal velocity of plasma electrons. Then, we use these effective average values in Equation (25) for σ₁ and in Equation (28) for σ₂ in the subsequent plots where all quantities will be in atomic units.

Figure 1 shows the dependence of the ratio Γ/(N_ev_T), where Γ is the adiabatic Stark width calculated by combining Equations (7), (25), and (28), on the Larmor frequency ω_B and on the mean thermal velocity of electrons v_T for X_αβ = 6, X_αβ being defined in Equation (9). We note that X_αβ = 6 corresponds to either n_a = 3, (n₁ − n₂)_α = 2, n_b = 1 (which is a component of the Ly-beta line), or to n_a = 4, (n₁ − n₂)_α = 2, n_b = 2, (n₁ − n₂)_β = 1, (which is a component of the Balmer-beta line), or to n_a = 4, (n₁ − n₂)_α = 3, n_b = 3, (n₁ − n₂)_β = 2, (which is a component of the Paschen-alpha line), or to n_a = 6, (n₁ − n₂)_α = 1, n_b = 1, (which is a component of Lyman-epsilon line), or to n_a = 6, (n₁ − n₂)_α = 2, n_b = 3, (n₁ − n₂)_β = 2, (which is a component of the Paschen-gamma line), or to n_a = 4, (n₁ − n₂)_α = 2, n_b = 3, (n₁ − n₂)_β = 2, (which is a component of the Paschen-alpha line), or to n_a = 6, (n₁ − n₂)_α = 3, n_b = 4, (n₁ − n₂)_β = 3, (which is a component of the Brackett-beta line). Thus, the value of X_αβ = 6 represents Stark components of seven spectral lines, which is why we chose this value for Figure 1.

Figure 2 displays the dependence of the ratio Γ/(N_ev_T) on the Larmor frequency ω_B for the mean thermal velocity of electrons v_T = 0.2 a.u., corresponding to 2.7 eV, for X_αβ = 6 (solid line). The corresponding perturbative result from papers [17,18] is shown by the dashed line.

From Figure 2, it is seen that the perturbative calculation from papers [17,18] overestimated the Stark width. It also illustrates that the perturbative calculation from papers [17,18] did not produce any dependence of the Stark width on the magnetic field, while the nonperturbative (more accurate calculation) demonstrates a significant dependence of the Stark width on the magnetic field.

The physical reason for the decrease in the dynamical Stark width as the magnetic field becomes stronger is the following. The dynamical Stark width is Γ~N_ev_Tσ, where σ is the cross-section of the optical collisions responsible for Stark broadening of hydrogen or deuterium lines [9]. In its turn, σ~(v_T/Ω)², where Ω is the characteristic frequency of the variation of the electric field of the perturbing ions at the location of the radiating atom. Thus, the dynamical Stark width Γ is inversely proportional to Ω². In non or weakly magnetized plasmas, Ω~ω_W~T_e/(n²ħ), where ω_W is the Weisskopf frequency [24]. In strongly magnetized plasmas, as the Larmor frequency ω_B starts exceeding ω_W, then the characteristic frequency of the variation of the electric field of the perturbing ions at the location of the radiating atom becomes Ω~ω_B and the dynamical Stark width decreases with the growth of the magnetic field.

As for the dependence of the dynamical Stark width Γ on the mean thermal velocity of plasma electrons v_T, physically it is the competition of the following two factors. On the one hand, at v_T = 0, the dynamical Stark width vanishes because it is dynamical, while there no dynamics at v_T = 0. On the other hand, at relatively large v_T, the dynamical Stark width Γ decreases as v_T increases. This is because at relatively large v_T, so that the Weisskopf frequency exceeds the Larmor frequency, the dynamical Stark width Γ, being proportional to v_Tσ(v_T), becomes inversely proportional to v_T. Consequently, somewhere between v_T = 0 and relatively large v_T, Γ reaches a maximum, as it is seen in Figure 1 at a fixed ω_B.

4. Discussion of the Results

We showed how far the analytical semiclassical theory of the dynamical Stark broadening of hydrogen or deuterium lines in plasmas has progressed. While the pioneering works [1,2,3] presented the most simple theory, lots of subsequent improvements were made in the intervening dozens of years, as reflected, e.g., in works [4,5,6,7,8,9,10,11,12,13,14,15,16]. However, it was not until the year 2017 that the effect of helical rather than rectilinear trajectories of perturbing electrons in strongly magnetized plasmas on the dynamical Stark width of hydrogen or deuterium spectral lines was studied analytically for the first time, in paper [17].

Specifically, in papers [17,18], the situation was studied where the magnetic field B is so strong that the dynamical Stark width of these lines reduces to the so-called adiabatic Stark width because the so-called nonadiabatic Stark width is completely suppressed. The analytical calculations in papers [17,18] were perturbative: they were performed in the second order of the Dyson perturbation expansion.

In distinction, in the present paper we used the O₄ symmetry of hydrogen or deuterium atoms for performing non-perturbative analytical calculations equivalent to accounting for all orders of the Dyson perturbation expansion in the spirit of the generalized theory of the dynamical Stark broadening [20,21,22].

The results of the present paper differ from those obtained in papers [17,18], not only quantitatively, but—most importantly—qualitatively. Namely, the dynamical Stark broadening does depend on the magnetic field B even for strong B, while in papers [17,18] the dynamical Stark broadening did not depend on B. In addition to this, the non-perturbative calculations yield the dynamical Stark width by about an order of magnitude smaller than the perturbative calculations from papers [17,18], as seen, e.g., in Figure 2.

The results of the present paper should be important for revising the interpretation of the hydrogen Balmer lines observed from DA and DBA white dwarfs. According to observations, the magnetic field in plasmas of white dwarfs can range from 10³ Tesla to 10⁵ Tesla (see, e.g., papers [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42], listed in chronological order and the references therein), thus easily exceeding the critical value from Equation (5). Indeed, for the typical electron temperature T_e~1 eV in the white dwarfs plasma emitting hydrogen lines, Equation (5) yields B_cr~10³/|X_αβ| Tesla < 10³ Tesla.

Finally, we note that in the literature there is a confusion concerning the effect that the helical trajectories of the perturbing electrons have on the dynamical Stark width of hydrogen spectral lines. We clarify this confusion in Appendix B of the present paper.