Allowance for more realistic trajectories of plasma electrons in the Stark broadening of hydrogenlike spectral lines

The theory of the Stark broadening of hydrogenlike spectral lines by plasma electrons, developed by Griem and Shen [1] and later presented also in books [2, 3], is usually referred to as the Conventional Theory, hereafter CT, also known as the standard theory. (Further advances in the theory of the Stark broadening of hydrogenlike spectral lines by plasma electrons can be found, e.g., in books [4, 5] and references therein.) In the CT, the perturbing electrons are considered moving along hyperbolic trajectories in the Coulomb field of the effective charge Z − 1 (in atomic units), where Z is the nuclear charge of the radiating ion. In other words, in the CT there was made a simplifying assumption that the motion of the perturbing electron can be described in frames of a two-body problem, one particle being the perturbing electron and the other 'particle' being the charge Z − 1.

However, in reality one have to deal with a three-body problem: the perturbing electron, the nucleus, and the bound electron. Therefore, trajectories of the perturbing electrons should be more complicated.

In the present paper we take this into account by using the standard analytical method of separating rapid and slow subsystems—see, e.g., book [6]. The characteristic frequency of the motion of the bound electron around the nucleus is much higher than the characteristic frequency of the motion of the perturbing electron around the radiating ion. Therefore the former represents the rapid subsystem and the latter represents the slow subsystem. This approximate analytical method allows a sufficiently accurate treatment in situations where the perturbation theory fails—see, e.g., book [6].

By applying this method we obtain more accurate analytical results for the electron broadening operator than in the CT. We show by examples of the electron broadening of the Lyman lines of He II that the allowance for this effect increases with the electron density N_e, becomes significant already at N_e ∼ 10¹⁷ cm⁻³ and very significant at higher densities.

In the CT the electron broadening operator is expressed in the form (see, e.g., paper [1])

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{ab}\equiv 2\pi \,v\,{N}_{e}{\int }^{}d\rho \,\rho \{{S}_{a}{{S}_{b}}^{* }-1\},\end{eqnarray} \tag{ 1 }$$

where N_e, $v,$ and $\rho$ are the electron density, velocity, and impact parameter, respectively; ${S}_{a}(0)$ and ${S}_{b}(0)$ are the S matrices for the upper (a) and lower (b) states involved in the radiative transition, respectively; {...} stands for the averaging over angular variables of vectors v and ρ. Further in the CT, the collisions are subdivided into weak and strong. The weak collisions are treated by the time-dependent perturbation theory. The impact parameter, at which the formally calculated expression {S_aS_b* − 1} for a weak collision starts violating the unitarity of the S-matrices, serves as the boundary between the weak and strong collisions and is called Weisskopf radius ρ_We.

So, in the CT the integral over the impact parameter diverges at small ρ. Therefore in the CT this integral is broken down in two parts: from 0 to ρ_We (strong collisions) and from ρ_We to ρ_max for weak collisions. The upper cutoff ρ_max (typically chosen to be the Debye radius ρ_D = [T/(4πe²N_e)]^1/2, where T is the electron temperature) is necessary because this integral diverges also at large ρ.

In the CT, after calculating the S matrices for weak collisions, the electron broadening operator becomes (in atomic units)

$$\begin{eqnarray}&&{{{\rm{\Phi }}}_{ab}}^{weak}\equiv C\displaystyle {\int }_{{\rho }_{we}}^{{\rho }_{\max }}d\rho \,\rho {\sin }^{2}\displaystyle \frac{{\rm{\Theta }}(\rho )}{2}=\displaystyle \frac{C}{2}\displaystyle {\int }_{{{\rm{\Theta }}}_{\min }}^{{{\rm{\Theta }}}_{\max }}d{\rm{\Theta }}\displaystyle \frac{d{\rho }^{2}}{d{\rm{\Theta }}}{\sin }^{2}\displaystyle \frac{{\rm{\Theta }}}{2},\end{eqnarray} \tag{ 2 }$$

where ${\rm{\Theta }}$ is the scattering angle for the collision between the perturbing electron and the radiating ion (the dependence between ${\rm{\Theta }}$ and ρ being discussed below) and the plasma electron and the operator $C$ is

$$\begin{eqnarray}&&C=-\displaystyle \frac{4\,\pi }{3}{N}_{e}\left[\displaystyle {\int }_{0}^{\infty }dv\,{v}^{3}f(v)\right]\displaystyle \frac{{m}^{2}}{{(Z-1)}^{2}}{({{\boldsymbol{r}}}_{a}-{{{\boldsymbol{r}}}_{b}}^{* })}^{2}.\end{eqnarray} \tag{ 3 }$$

Here f(v) is the velocity distribution of the perturbing electrons, $m$ is the reduced mass of the system 'perturbing electron—radiating ion'; r is the radius-vector operator of the bound electron (which scales with Z as 1/Z), its subscripts a or b correspond to this operator acting on the upper or lower states, respectively, involved in the radiative transition. The expression (r_a−r_b*)² stands for the scalar product (also known as the dot-product) of the operator (r_a−r_b*) with itself. In the theory of the dynamical Stark broadening of spectral lines in plasmas by electrons, the corresponding matrix elements are calculated with respect to the unperturbed wave functions.

In the CT the scattering occurs in the effective Coulomb potential, so that the trajectory of the perturbing electron is hyperbolic and the relation between the impact parameter and the scattering angle is given by

$$\begin{eqnarray}&&{\rho }^{(0)}=\displaystyle \frac{Z-1}{m\,{v}^{2}}\,\cot \,\displaystyle \frac{{\rm{\Theta }}}{2}.\end{eqnarray} \tag{ 4 }$$

Here the superscript (0) in ${\rho }^{(0)}$ refers to hyperbolic trajectories.

In the present paper we consider the realistic situation where trajectories of the perturbing electrons are more complicated because the perturbing electron, the nucleus, and the bound electron should be more accurately treated as the three-body problem. We use the standard analytical method of separating rapid and slow subsystems—see, e.g., book [6]. It is applicable here because the characteristic frequency v_Te/ρ_We of the variation the electric field of the perturbing electrons at the location of the radiating ion is much smaller than the frequency Ω_ab of the spectral line (the latter, e.g., in case of the radiative transition between the Rydberg states would be the Keppler frequency or its harmonics)—more details are presented in appendix.

The first step in this method is to 'freeze' the slow subsystem (perturbing electron) and to find the analytical solution for the energy of the rapid subsystem (the radiating ion) that would depend on the frozen coordinates of the slow subsystem (in our case it will be the dependence on the distance R of the perturbing electron from the radiating ion). To the first non-vanishing order of the R-dependence, the corresponding energy in the parabolic quantization is given by

$$\begin{eqnarray}&&{E}_{nq}(R)=-\displaystyle \frac{{Z}^{2}}{{n}^{2}}+\displaystyle \frac{3\,n\,q}{2\,Z\,{R}^{2}},\end{eqnarray} \tag{ 5 }$$

where $n$ and $q$ = n₁−n₂ are the principal and electric quantum numbers, respectively, of the Stark state of the radiating ion; n₁ and n₂ are the parabolic quantum numbers of that state.

The next step in this method is to consider the motion of the slow subsystem (perturbing electron) in the 'effective potential' V_eff(R) consisting of the actual potential plus E_nq(R). Since the constant term in equation (5) does not affect the motion, the effective potential for the motion of the perturbing electron can be represented in the form

$$\begin{eqnarray}&&{V}_{eff}(R)=-\displaystyle \frac{\alpha }{R}+\displaystyle \frac{\beta }{{R}^{2}},\,\alpha =Z-1.\end{eqnarray} \tag{ 6 }$$

For the spectral lines of the Lyman series, since the lower (ground) state b of the radiating ion remains unperturbed (up to/including the order ∼1/R²), the coefficient β is

$$\begin{eqnarray}&&\beta =\displaystyle \frac{3\,{n}_{a}{q}_{a}}{2\,Z}.\end{eqnarray} \tag{ 7 }$$

For other hydrogenic spectral lines, for taking into account both the upper and lower states of the radiating ion, the coefficient β can be expressed as

$$\begin{eqnarray}&&\beta =\displaystyle \frac{3\,({n}_{a}{q}_{a}-{n}_{b}{q}_{b})}{2\,Z}.\end{eqnarray} \tag{ 8 }$$

The motion in the potential from equation (6) allows an exact analytical solution. In particular, the relation between the scattering angle and the impact parameter is no longer given by equation (4), but rather becomes (see, e.g., book [7])

$$\begin{eqnarray}&&{\rm{\Theta }}=\pi -\displaystyle \frac{2}{\sqrt{1+\displaystyle \frac{2\,m\,\beta }{{M}^{2}}}}\arctan \sqrt{\displaystyle \frac{4\,E}{{\alpha }^{2}}\left(\beta +\displaystyle \frac{{M}^{2}}{2m}\right)}.\end{eqnarray} \tag{ 9 }$$

Here $E$ and $M$ are the energy and the angular momentum of the perturbing electron, respectively. We can rewrite the angular momentum in terms of the impact parameter $\rho$ as

$$\begin{eqnarray}&&M=m\,v\,\rho \end{eqnarray} \tag{ 10 }$$

Then a slight rearrangement of equation (9) yields

$$\begin{eqnarray}&&\tan \left(\displaystyle \frac{\pi -{\rm{\Theta }}}{2}\sqrt{1+\displaystyle \frac{2\beta }{m\,{v}^{2}{\rho }^{2}}}\right)=\displaystyle \frac{v}{\alpha }\sqrt{{m}^{2}{v}^{2}{\rho }^{2}+2m\,\beta }.\end{eqnarray} \tag{ 11 }$$

After solving equation (11) for $\rho$ and substituting the outcome in equation (2), a more accurate expression for the electron broadening operator can be obtained. However, equation (11) does not have an exact analytic solution for $\rho$ so that this could be done only numerically.

In the present paper, to get the message across in the simplest form, we will provide an approximate analytical solution of equation (11) by expanding it in powers of β. This yields (keeping up to the first power of β)

$$\begin{eqnarray}&&\tan \left(\displaystyle \frac{\pi -{\rm{\Theta }}}{2}\right)+\left(\displaystyle \frac{\pi -{\rm{\Theta }}}{2}\right)\left[1+{\tan }^{2}\left(\displaystyle \frac{\pi -{\rm{\Theta }}}{2}\right)\right]\displaystyle \frac{\,\beta }{m\,{v}^{2}{\rho }^{2}}\approx \displaystyle \frac{m\,{v}^{2}\rho }{\alpha }+\displaystyle \frac{\beta }{\alpha \,\rho }.\end{eqnarray} \tag{ 12 }$$

We seek the analytical solution for $\rho$ in the form $\rho \approx {\rho }^{(0)}+{\rho }^{(1)},$ where ${\rho }^{(0)}$ corresponds to $\beta =0$ (and was given by equation (4)) and ${\rho }^{(1)}\ll {\rho }^{(0)}.$ Substitution of $\rho \approx {\rho }^{(0)}+{\rho }^{(1)}$ into equation (12) yields the expression

$$\begin{eqnarray}&&\displaystyle \frac{(\pi -{\rm{\Theta }})\,\beta }{2m\,{v}^{2}{\rho }^{(0)2}{\sin }^{2}\displaystyle \frac{{\rm{\Theta }}}{2}}-\displaystyle \frac{\beta }{\alpha \,{\rho }^{(0)}}\approx \displaystyle \frac{m\,{v}^{2}{\rho }^{(1)}}{\alpha }.\end{eqnarray} \tag{ 13 }$$

After solving equation (13) for ${\rho }^{(1)},$ we get the expression for $\rho :$

$$\begin{eqnarray}&&\rho \approx \displaystyle \frac{\alpha }{m\,{v}^{2}}\,\cot \,\displaystyle \frac{{\rm{\Theta }}}{2}+\displaystyle \frac{\beta }{\alpha }\left(\displaystyle \frac{\pi -{\rm{\Theta }}}{2{\cos }^{2}\displaystyle \frac{{\rm{\Theta }}}{2}}-\,\tan \,\displaystyle \frac{{\rm{\Theta }}}{2}\right).\end{eqnarray} \tag{ 14 }$$

As a reminder, our goal is to perform the integration in equation (1) for obtaining a more accurate analytical result for the electron broadening operator. This can be more easily accomplished by performing the integration over ${\rm{\Theta }}$ instead of $\rho .$ For this purpose, first we square equation (14)

$$\begin{eqnarray}&&{\rho }^{2}\approx \displaystyle \frac{{\alpha }^{2}}{{m}^{2}\,{v}^{4}}{\cot }^{2}\displaystyle \frac{{\rm{\Theta }}}{2}+\displaystyle \frac{\beta }{m\,{v}^{2}}\left(\displaystyle \frac{\pi -{\rm{\Theta }}}{\sin \,\displaystyle \frac{{\rm{\Theta }}}{2}\,\cos \,\displaystyle \frac{{\rm{\Theta }}}{2}}-1\right)\,,\end{eqnarray} \tag{ 15 }$$

where only the first order terms in $\beta$ have been kept for consistency. To make formulas simpler, we denote $\phi ={\rm{\Theta }}/2.$ After differentiating equation (15) with respect to $\phi ,$ we obtain

$$\begin{eqnarray}&&\displaystyle \frac{d{\rho }^{2}}{d\phi }\approx -\displaystyle \frac{{\alpha }^{2}}{{m}^{2}\,{v}^{4}}\displaystyle \frac{2\,\cot \,\phi }{{\sin }^{2}\phi }-\displaystyle \frac{2\beta }{m{v}^{2}}\left[\left(\displaystyle \frac{1}{\sin \,\phi \,\cos \,\phi \,}\right)+\left(\displaystyle \frac{\pi }{2}-\phi \right)\left(\displaystyle \frac{1}{{\sin }^{2}\phi \,}-\displaystyle \frac{1}{{\cos }^{2}\phi \,}\right)\right]\end{eqnarray} \tag{ 16 }$$

After substituting in the utmost right side of equation (2) first ${\rm{\Theta }}=2\phi$ and then $\displaystyle \frac{d{\rho }^{2}}{d\phi }$ from equation (16), the contribution of the weak collisions to the electron broadening operator becomes

$$\begin{eqnarray}&&{{{\rm{\Phi }}}_{ab}}^{weak}=-C\left[\displaystyle \frac{{\alpha }^{2}}{{m}^{2}{v}^{4}}\displaystyle {\int }_{{\phi }_{\min }}^{{\phi }_{\max }}\cot \phi d\phi +\displaystyle \frac{\beta }{m\,{v}^{2}}\displaystyle {\int }_{0}^{\displaystyle \frac{\pi }{2}}\tan \phi d\phi +\displaystyle \frac{\beta }{m\,{v}^{2}}\displaystyle {\int }_{0}^{\displaystyle \frac{\pi }{2}}\left(\displaystyle \frac{\pi }{2}-\phi \right)(1-{\tan }^{2}\phi )d\phi \right].\end{eqnarray} \tag{ 17 }$$

In equation (17), in the two correction terms proportional to $\beta ,$ we extended the integration over the full range of the variation of the angle $\phi .$ The corresponding minor inaccuracy would not contribute significantly to the electron broadening operator, since the terms involving $\beta$ are considered to be a relatively small correction to the first term in equation (17).

Performing the integrations in equation (17) we obtain:

$$\begin{eqnarray}&&{{{\rm{\Phi }}}_{ab}}^{weak}=-\displaystyle \frac{4\,\pi }{3}{N}_{e}{({{\boldsymbol{r}}}_{a}-{{{\boldsymbol{r}}}_{b}}^{* })}^{2}\left[\displaystyle {\int }_{0}^{\infty }dv\,\displaystyle \frac{f(v)}{v}\right]\left[\mathrm{log}\,\displaystyle \frac{\sin \,{\phi }_{\max }}{\sin \,{\phi }_{\min }}+\displaystyle \frac{m{v}^{2}\beta }{{(Z-1)}^{2}}\left(\displaystyle \frac{{\pi }^{2}}{4}-1\right)\right].\end{eqnarray} \tag{ 18 }$$

Now we add the CT estimate for the contribution of strong collisions

$$\begin{eqnarray}&&{{{\rm{\Phi }}}_{ab}}^{strong}\approx \pi \,v\,{N}_{e}\,{{\rho }_{We}}^{2},\end{eqnarray} \tag{ 19 }$$

where ${\rho }_{We}$ corresponds to ${\phi }_{\max }.$ Expressions for ${\phi }_{\max }$ and ${\phi }_{\min }$ are given in paper [1] (in their equations (9) and (10a)) as follows

$$\begin{eqnarray}&&\sin \,{\phi }_{\max }=\sqrt{\displaystyle \frac{3}{2}}\displaystyle \frac{Z(Z-1)}{({n}_{a}^{2}-{n}_{b}^{2})m\,v},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\sin \,{\phi }_{\min }=\displaystyle \frac{\displaystyle \frac{Z-1}{m\,{v}^{2}{\rho }_{D}}}{\sqrt{1+\displaystyle \frac{{(Z-1)}^{2}}{{m}^{2}{v}^{4}{\rho }_{D}^{2}}}}\,\end{eqnarray} \tag{ 21 }$$

It should be emphasized that the factor $({n}_{a}^{2}-{n}_{b}^{2})$ in the denominator of the right side of equation (20) was an approximate allowance by the authors of paper [1] for the contribution of the lower level b while estimating the operator (${{\boldsymbol{r}}}_{a}-{{{\boldsymbol{r}}}_{b}}^{* }$) for hydrogenic lines of spectral series other than the Lyman lines. However, for the Lyman lines the lower (ground) level does not contribute to electron broadening operator, so that for the Lyman lines equation (20) should be simplified as follows:

$$\begin{eqnarray}&&\sin \,{\phi }_{\max }=\sqrt{\displaystyle \frac{3}{2}}\displaystyle \frac{Z(Z-1)}{{n}_{a}^{2}m\,v}.\end{eqnarray} \tag{ 22 }$$

We also note that at relatively small velocities of perturbing electrons, the right side of equations (20) or (22) could exceed unity. In this case one should set $\sin \,{\phi }_{\max }=1,$ what corresponds to ρ_min = 0, so that there would be no contribution from strong collisions. Typically, the range of such small velocities has a very low statistical weight in the electron velocity distribution.

After substituting the above formulas for $\sin \,{\phi }_{\max }$ and $\sin \,{\phi }_{\min }$ into equation (17), and combining the contributions from weak and strong collisions, we obtain the final results for the electron broadening operator:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{ab}(\beta )=-\displaystyle \frac{4\,\pi }{3}{N}_{e}{({{\boldsymbol{r}}}_{a}-{{{\boldsymbol{r}}}_{b}}^{* })}^{2}\left[\displaystyle {\int }_{0}^{\infty }dv\,\displaystyle \frac{f(v)}{v}\right]\left\{\displaystyle \frac{1}{2}\left[1-\displaystyle \frac{3}{2}\displaystyle \frac{{Z}^{2}{(Z-1)}^{2}}{{({n}_{a}^{2}-{n}_{b}^{2})}^{2}{m}^{2}\,{v}^{2}}\right]\right.\\ &&\left.\,\,\,\,+\,\mathrm{log}\left[\sqrt{\displaystyle \frac{3}{2}}\displaystyle \frac{Z\,v\,{\rho }_{D}}{({n}_{a}^{2}-{n}_{b}^{2})}\sqrt{1+{\left(\displaystyle \frac{Z-1}{m{v}^{2}{\rho }_{D}}\right)}^{2}}\right]+\displaystyle \frac{m{v}^{2}\beta }{{(Z-1)}^{2}}\left(\displaystyle \frac{{\pi }^{2}}{4}-1\right)\right\}\,\end{eqnarray} \tag{ 23 }$$

for the non-Lyman lines and

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{ab}(\beta )=-\displaystyle \frac{4\,\pi }{3}{N}_{e}{({{\boldsymbol{r}}}_{a}-{{{\boldsymbol{r}}}_{b}}^{* })}^{2}\left[\displaystyle {\int }_{0}^{\infty }dv\displaystyle \frac{f(v)}{v}\right]\left\{\displaystyle \frac{1}{2}\left[1-\displaystyle \frac{3}{2}\displaystyle \frac{{Z}^{2}{(Z-1)}^{2}}{{n}_{a}^{4}{m}^{2}\,{v}^{2}}\right]\right.\\ &&\,\,\,\,+\,\left.\mathrm{log}\left[\sqrt{\displaystyle \frac{3}{2}}\displaystyle \frac{Z\,v\,{\rho }_{D}}{{n}_{a}^{2}}\sqrt{1+{\left(\displaystyle \frac{Z-1}{m{v}^{2}{\rho }_{D}}\right)}^{2}}\right]+\displaystyle \frac{m{v}^{2}\beta }{{(Z-1)}^{2}}\left(\displaystyle \frac{{\pi }^{2}}{4}-1\right)\right\}\end{eqnarray} \tag{ 24 }$$

for the Lyman lines. Here and below log[...] stands for the natural logarithm.

In order to determine the significance of the effect of non-hyperbolic trajectories, it is necessary to evaluate the ratio of the third term in braces in equation (24) to the first two terms in the same braces

$$\begin{eqnarray}&&ratio=\displaystyle \frac{\displaystyle \frac{3}{2}\displaystyle \frac{m{v}^{2}({n}_{a}{q}_{a}-{n}_{b}{q}_{b})}{{(Z-1)}^{2}}\left(\displaystyle \frac{{\pi }^{2}}{4}-1\right)}{\displaystyle \frac{1}{2}\left[1-\displaystyle \frac{3}{2}\displaystyle \frac{{Z}^{2}{(Z-1)}^{2}}{{({n}_{a}^{2}-{n}_{b}^{2})}^{2}{m}^{2}\,{v}^{2}}\right]+\,\mathrm{log}\left[\sqrt{\displaystyle \frac{3}{2}}\displaystyle \frac{Z\,v\,{\rho }_{D}}{({n}_{a}^{2}-{n}_{b}^{2})}\sqrt{1+{\left(\displaystyle \frac{Z-1}{m{v}^{2}{\rho }_{D}}\right)}^{2}}\right]}\end{eqnarray} \tag{ 25 }$$

for the non-Lyman lines or the ratio

$$\begin{eqnarray}&&ratio=\displaystyle \frac{\displaystyle \frac{3}{2}\displaystyle \frac{m{v}^{2}{n}_{a}{q}_{a}}{{(Z-1)}^{2}}\left(\displaystyle \frac{{\pi }^{2}}{4}-1\right)}{\displaystyle \frac{1}{2}\left[1-\displaystyle \frac{3}{2}\displaystyle \frac{{Z}^{2}{(Z-1)}^{2}}{{n}_{a}^{4}{m}^{2}\,{v}^{2}}\right]+\,\mathrm{log}\left[\sqrt{\displaystyle \frac{3}{2}}\displaystyle \frac{Z\,v\,{\rho }_{D}}{{n}_{a}^{2}}\sqrt{1+{\left(\displaystyle \frac{Z-1}{m{v}^{2}{\rho }_{D}}\right)}^{2}}\right]}\end{eqnarray} \tag{ 26 }$$

for the Lyman lines.

Below we present numerical examples for several Lyman lines. As it is customary in the Stark broadening theory, instead of the integration over velocities, for the numerical examples we use the mean thermal velocity v_T of the perturbing electrons. In atomic units, the mean thermal velocity v_T, the Debye radius ρ_D, and the reduced mass can be expressed as follows

$$\begin{eqnarray}&&{v}_{T}=0.1917\sqrt{\displaystyle \frac{T({\rm{eV}})}{m}}\,{\rho }_{D}=1.404\times {10}^{11}\sqrt{\displaystyle \frac{T({\rm{eV}})}{{N}_{e}({{\rm{cm}}}^{-3})}}\,\,\,\,m=\displaystyle \frac{1+\displaystyle \frac{{m}_{e}}{A\,{m}_{p}}}{1+\displaystyle \frac{2\,{m}_{e}}{A\,{m}_{p}}},\end{eqnarray} \tag{ 27 }$$

where ${m}_{e}$ is the electron mass, ${m}_{p}$ is the proton mass, and $A$ is the atomic number of the radiating ion $(A\approx 2Z).$

Table 1 presents the values of the ratio from equation (26) for several Lyman lines of He II at the temperature T = 8 eV and the electron density ${N}_{e}=2\times {10}^{17}$ cm⁻³.

Table 1.  Ratio from equation (26) for the Stark components of several Lyman lines of He II at the temperature T = 8 eV and the electron density ${N}_{e}=2\,\times \,{10}^{17}$ cm⁻³.

n	∣q∣	$ratio$
2	1	0.3261
3	1	0.3748
3	2	0.7496
4	1	0.5156
4	2	1.0311
4	3	1.5467

Figure 1 shows the ratio from equation (26) versus the electron density N_e for the Stark components of the electric quantum number ∣q∣ = 1 of Lyman-alpha (n = 2), Lyman-beta (n = 3), and Lyman-gamma (n = 4) lines of He II at the temperature T = 8 eV.

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It is seen that for the electron broadening of the Lyman lines of He II, the allowance for the effect under consideration indeed becomes significant already at electron densities N_e ∼ 10¹⁷ cm⁻³ and increases with the growth of the electron density. It should be noted that when the ratio, formally calculated by equation (26), becomes comparable to unity, this is the indication that the approximate analytical treatment based on expanding equation (11) up to the first order of parameter β, is no longer valid. In this case the calculations should be based on solving equation (11) with respect to ρ without such approximation.

In this paper we considered the electron broadening of hydrogenlike spectral lines in plasmas more accurately than in the CT. In distinction to the CT, we treated it as a three-body problem involving the perturbing electron, the nucleus, and the bound electron. We employed the standard analytical method of separating rapid and slow subsystems by using the fact that the characteristic frequency of the motion of the bound electron around the nucleus is much higher than the characteristic frequency of the motion of the perturbing electron around the radiating ion.

With the help of this method we obtained more accurate analytical results for the electron broadening operator compared to the CT. By examples of the electron broadening of the Lyman lines of He II, we demonstrated that the allowance for this effect becomes significant at electron densities N_e ∼ 10¹⁷ cm⁻³ and very significant at higher densities.

It is important to emphasize that we were able to obtain the above analytical results primarily due to the underlying fundamental symmetry of the class of potentials V(R) = –A/R + B/R², where A and B are constants. Namely, this class of potentials possesses an additional conserved quantity M_eff² = M² + 2 mB, where M is the angular momentum and m is the mass of a particle, so that M_eff is the effective angular momentum. As for the impact approximation, it was not crucial to our work—we used it only for the following two purposes: first, to get the message across in a simple form, and second, for the comparison with the CT (in which the impact approximation was crucial), so that we would compare 'apples to apples' rather than 'apples to oranges'. A brief outline of the impact approximation is presented in appendix B.

We also mention that in 1981, Baryshnikov and Lisitsa [8] published very interesting results for the electron broadening of hydrogenlike spectral lines in plasmas (also presented later in book [9]) in frames of the quantum theory of the dynamical Stark broadening, while we obtained our results in frames of the semiclassical theory of the dynamical Stark broadening, just as in the CT. (For clarity: in the semiclassical theory, the radiating atom/ion is treated quantally, while perturbing electrons classically; in the quantum theory both the radiating atom/ion and perturbing electrons are treated quantally.) Both in paper [8] and in our paper, there was used the underlying symmetry of the class of potentials V(R) = −A/R + B/R² for obtaining analytical solutions.

A specific result for the line width Baryshnikov and Lisitsa [8] obtained for Lyman lines in the classical limit using the impact approximation, as presented in their equations (4.5) and (4.6). We compared their results from equations (4.5) and (4.6) with the CT [1] for He II Lyman lines. It turned out that for N_e ∼ (10¹⁷–10¹⁸) cm⁻³, i.e. for the range of electron densities, in which the overwhelming majority of measurements of the width of He II lines were performed, Baryshnikov-Lisitsa's line width exceeds the CT line width by two orders of magnitude or more. In view of the fact that the width of He II lines, measured by various authors in benchmark experiments (i.e., experiments where plasma parameters were measured independently of the line widths), never exceeded the CT width by more than a factor of two (see, e.g., benchmark experiments [10–12]), this seems to indicate that equations (4.5) and (4.6) from paper [8] are inapplicable for He II lines (indeed, the validity condition for equations (4.5) and (4.6) from paper [8], which is Z ≫ n², is not met). In distinction, the corrections to the CT that we introduced in the present paper, do not exceed the factor of two for He II lines in the range of N_e ∼ (10¹⁷–10¹⁸) cm⁻³.

The characteristic frequency of the motion of the perturbing electron around the radiating ion in the process of the Stark broadening of spectral lines is the so-called Weisskopf frequency

$$\begin{eqnarray}&&{\omega }_{We}=\displaystyle \frac{{v}_{T}}{{\rho }_{We}}\sim \displaystyle \frac{Z\,m\,{v}_{T}^{2}}{({n}_{a}^{2}-{n}_{b}^{2})\hslash }\sim \displaystyle \frac{Z\,T}{({n}_{a}^{2}-{n}_{b}^{2})\hslash }.\end{eqnarray} \tag{ A.1 }$$

The characteristic frequency of the motion of the bound electron around the nucleus is the frequency of the spectral line

$$\begin{eqnarray}&&{\rm{\Omega }}=\displaystyle \frac{{Z}^{2}{U}_{H}}{\hslash }\left(\displaystyle \frac{1}{{n}_{b}^{2}}-\displaystyle \frac{1}{{n}_{a}^{2}}\right),\end{eqnarray} \tag{ A.2 }$$

where U_H is the ionization potential of hydrogen. The ratio of these two frequencies is

$$\begin{eqnarray}&&\displaystyle \frac{{\omega }_{{\rm{We}}}}{{\rm{\Omega }}}\sim \left(\displaystyle \frac{{\rm{T}}}{{\rm{Z}}\,{U}_{H}}\right)\left[\displaystyle \frac{{{\rm{n}}}_{{\rm{a}}}^{2}{n}_{b}^{2}}{{({n}_{a}^{2}-{n}_{b}^{2})}^{2}}\right].\end{eqnarray} \tag{ A.3 }$$

For the simplicity of estimating this ratio, let us consider n_a ≫ n_b, so that

$$\begin{eqnarray}&&\displaystyle \frac{{\omega }_{{\rm{We}}}}{{\rm{\Omega }}}\sim \left(\displaystyle \frac{{\rm{T}}}{{\rm{Z}}\,{{\rm{n}}}_{{\rm{a}}}^{2}{U}_{H}}\right)\ll 1\end{eqnarray} \tag{ A.4 }$$

as long as

$$\begin{eqnarray}&&{\rm{T}}({\rm{eV}})\ll (13.6\,{\rm{eV}})\,{\rm{Z}}\,{{\rm{n}}}_{{\rm{a}}}^{2}.\end{eqnarray} \tag{ A.5 }$$

For example, for Z = 2 the above validity condition becomes

$$\begin{eqnarray}&&{\rm{T}}({\rm{eV}})\ll (27.2\,{\rm{eV}}){{\rm{n}}}_{{\rm{a}}}^{2}\end{eqnarray} \tag{ A.6 }$$

and is satisfied for a broad range of temperatures, at which He II spectral lines are observed in plasmas.

The dynamical broadening of spectral lines in plasmas by electrons is effective if the number ν_We of perturbing electrons in the sphere of the electron Weisskopf radius ρ_We is much smaller than unity (see, e.g., review by Lisitsa [13]): ν_We = 4πN_eρ_We³/3 ≪ 1, where ρ_We ∼ n² ħ/(m_ev_Te), n is the principal quantum number of the radiator energy level involved in the radiative transition, and v_Te is the mean thermal velocity of plasma electrons. Under this condition, for the overwhelming majority of perturbing electrons, the characteristic frequency of the variation of the electron microfield Ω_e ∼ v_Te/ρ_We is much greater than the instantaneous Stark splitting in the electron microfield. Physically the electron Weisskopf radius is related to the impact parameters ρ ∼ ρ_We that contribute most effectively to the dynamical Stark broadening of spectral lines by electrons in plasmas [13].

The gist of dynamical effects in the Stark broadening of spectral lines in plasmas by electrons is the following. Collisions with plasma electrons cause virtual transitions mostly within the upper (n) and lower (n') multiplets during the radiative transition

n ↔ n'. The primary outcome is a decrease of the lifetime of the states n' and/or n of the radiator, thus leading to the broadening of the corresponding spectral line¹ .

The fact that virtual transitions occur mostly within the upper and lower multiplets conventionally leads to so-called no-quenching approximation, in which virtual transitions between states of different principal quantum numbers are totally disregarded. This approximation allows to introduce a line space: a direct product of the Hilbert space, spanned on the basis vectors of the n-shell, with the Hilbert space, spanned on the (complex-conjugated) basis vectors of the n'-shell.

Both the impact formalism (developed by Baranger [14] and then by Kolb and Griem [15]) and the key features of the unified formalism (developed by Vidal, Cooper, and Smith [16]) can originate from the same sequence of mathematical operations—see, e.g., review by Sahal-Brechot [17]. The primary difference between them is the following. The impact formalism considers only completed collisions, while the unified formalism allows also for incomplete collisions. (Another distinction is that the unified formalism allows in principle a transition to the nearest-neighbor quasistatic result in the wings of the spectral line—this is a less important distinction because numerically the unified theory does not always yield such transition correctly.)

The most important step toward the impact formalism is the introduction of a coarse-grained time scale Δt, chosen such that

$$\begin{eqnarray}&&\rho /{{\rm{v}}}_{{\rm{e}}}\ll {\rm{\Delta }}{\rm{t}}\ll {[\max (\gamma ,{\rm{\Delta }}\omega ,{\omega }_{{\rm{pe}}})]}^{-1}.\end{eqnarray} \tag{ B.1 }$$

Here γ is the inversed lifetime of the radiator (the impact width of the spectral line is of the order of γ), Δω is the detuning from the unperturbed frequency ω₀ of the spectral line, ω_pe = (4πN_ee²/m_e)^1/2 is the plasma electron frequency. Physically, the coarse-grained time scale means that one gives up details of the evolution of the radiator during the time ∼ρ/v_e (which is the characteristic time of the individual collision). Instead, the focus is at the evolution of the radiator during the time intervals ∼Δt defined by equation (B.1). The possibility of introducing the coarse-grained time scale controls the limits of validity of the impact formalism. We note that in the unified formalism, the left strong inequality of equation (B.1) is relaxed to: ρ/v_e ↕ Δt.

The electron broadening operator Φ_ab is defined via the time evolution operators U_a and U_b for the upper (a) and lower (b) multiplets in the line space, respectively, as follows (see, e.g., review [13]):

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{ab}}}=[{{\rm{U}}}_{{\rm{a}}}({\rm{t}},\,{\rm{t}}+{\rm{\Delta }}{\rm{t}}){{\rm{U}}}_{{\rm{b}}}* ({\rm{t}},\,{\rm{t}}+{\rm{\Delta }}{\rm{t}})-1]/{\rm{\Delta }}{\rm{t}}.\end{eqnarray} \tag{ B.2 }$$

Under the impact approximation Δt ≫ ρ/v_e, within the interval (t, t + Δt) the time evolution operators in equation (B.2) are replaced by the corresponding scattering matrices S_a and S_b, leading to equation (1) of the main text. We note that the terminology of having not one, but two scattering matrices S_a and S_b is related to the concept of the line space and is used in the entire extensive scope of literature on the impact broadening of spectral lines in plasmas.

Finally we note that the dynamical Stark broadening of any hydrogenic spectral lines (whether Lyman or non-Lyman) in plasmas by electrons occurs in the overlapping regime: the Stark components of any hydrogenic spectral line become overlapping. This is true even for the simplest hydrogenic line Lyman-alpha, as shown in detail, e.g., by Strekalov and Burshtein [18].