Improving the Method of Measuring the Electron Density via the Asymmetry of Hydrogenic Spectral Lines in Plasmas by Allowing for Penetrating Ions

Abstract

There was previously proposed and experimentally implemented a new diagnostic method for measuring the electron density N_e using the asymmetry of hydrogenic spectral lines in dense plasmas. Compared to the traditional method of deducing N_e from the experimental widths of spectral lines, the new method has the following advantages. First, the traditional method requires measuring widths of at least two spectral lines (to isolate the Stark broadening from competing broadening mechanisms), while for the new diagnostic method it is sufficient to obtain the experimental profile of just one spectral line. Second, the traditional method would be difficult to implement if the center of the spectral lines was optically thick, while the new diagnostic method could still be used even in this case. In the theory underlying this new diagnostic method, the contribution of plasma ions to the spectral line asymmetry was calculated only for configurations where the perturbing ions were outside the bound electron cloud of the radiating atom/ion (non-penetrating configurations). In the present paper, we take into account the contribution to the spectral line asymmetry from penetrating configurations, where the perturbing ion is inside the bound electron cloud of the radiating atom/ion. We show that in high-density plasmas, the allowance for penetrating ions can result in significant corrections to the electron density deduced from the spectral line asymmetry.

1. Introduction

In medium-density plasmas, profiles of hydrogenic spectral lines look symmetric, but in high-density plasmas, they become asymmetric. This asymmetry is caused primarily by the nonuniformity of the ion microfield, as noted by Sholin and his co-workers in papers [1,2,3]—for the latest advances in the theory of the asymmetry we refer to papers [4,5] and the references therein, of which we especially note papers [6,7]. (There are also secondary sources of the asymmetry, as discussed in more detail below in the first paragraph of Section 2). Often, the blue maximum of the spectral line is higher than the red maximum, and the positions of the intensity maxima are asymmetrical with respect to the unperturbed line center.

A new diagnostic method for measuring the electron density using the asymmetry of hydrogenic spectral lines in dense plasmas was proposed and implemented in paper [8]. In that paper, in particular, from the experimental asymmetry of the C VI Lyman-delta line emitted by a vacuum spark discharge, the electron density was deduced to be N_e = 3 × 10²⁰ cm⁻³. This value of N_e was in good agreement with the electron density determined from the experimental widths of C VI Lyman-beta and Lyman-delta lines.

Later, this diagnostic method was also employed in the experiment presented in paper [9]. In that laser-induced breakdown spectroscopy experiment, the electron density N_e ~ 3 × 10¹⁷ cm⁻³ was determined from the experimental asymmetry of the H I Balmer-beta (H-beta) line.

This new diagnostic method has the following advantages compared to the method of deducing N_e from the experimental widths of spectral lines. First, the latter, traditional, method requires measuring widths of at least two spectral lines—because the widths are affected not only by the Stark broadening, but also by competing broadening mechanisms, such as, e.g., Doppler broadening. In distinction, to use the new diagnostic method, it is sufficient to obtain the experimental profile of just one spectral line—because the Doppler broadening does not cause the asymmetry.

Second, the traditional method based on experimental widths would be difficult to implement if the center of the spectral lines were optically thick. In distinction, the new diagnostic method can still be used even if the spectral line is optically thick in its central part. This is because the overwhelming contribution to the asymmetry originates from the wings of the spectral line, the wings usually being optically thin. More details can be found in Section 1.6 of [10]1.

In the theory underlying this new diagnostic method, the contribution of plasma ions to the spectral line asymmetry was calculated only for configurations where the perturbing ions were outside the “atomic sphere”, i.e., outside the bound electron cloud of the radiating atom/ion (non-penetrating configurations). In the present paper, we take into the contribution to the spectral line asymmetry from penetrating configurations, i.e., from configurations where the perturbing ion is inside the bound electron cloud of the radiating atom/ion (hereafter, radiator). We show that, in high-density plasmas, the allowance for penetrating ions can result in significant corrections to the electron density deduced from the spectral line asymmetry.

2. Allowance for Penetrating Ions

Let us first present a brief overview of the underlying theory for non-penetrating configurations. The dipole interaction of the radiator with perturbing ions outside the bound electron cloud, being calculated in the first order of the perturbation theory, splits the spectral line into Stark components symmetrically with respect to the unperturbed frequency or wavelength—in terms of both positions and intensities of the Stark components. The quadrupole interactions of the radiator with perturbing ions outside the bound electron cloud, being calculated in its first nonvanishing order, causes the asymmetry of the Stark splitting—in terms of both positions and intensities of the Stark components. The latter is the primary source of the asymmetry: other sources of the asymmetry—such as, but not limited to, e.g., the dipole interaction in the second order (known as the quadratic Stark effect), the quadrupole interaction in the second order, the octupole interaction in the first order—add to the asymmetry only higher-order corrections in terms of the corresponding small parameter n²a₀N_e^1/3/Z^4/3, where n is the principal quantum number of the upper level involved in the radiative transition, a₀ is the Bohr radius, Z is the charge of plasma ions and the nuclear charge of the radiating ion.2

However, in paper [11], it was shown that the quadrupole interaction, despite causing the asymmetric splitting of the spectral line into Stark components, does not shift the center of gravity of the line profile. Therefore, in the new diagnostic method presented in paper [8], first the center of gravity of the experimental profile was determined, and then it was taken as the reference point. Then, with respect to this point, the integrated intensities of the blue (I_B) and red (I_R) wings of the experimental profile were found. After that, the experimental degree of asymmetry, defined as

$${\rho }_{quad}=\frac{I_B-I_R}{0.5\left[I_B+I_R\right]},$$

(1)

was determined and then compared with the corresponding theoretical value given below.

The theoretical intensities of the blue and red wings, resulting from dipole and quadrupole interactions of the radiator with perturbing ions outside the bound electron cloud, can be expressed as follows (see paper [8]):

$$I_B={\displaystyle \sum }_{k>0}{I}_k^{\left(0\right)}\left(1+\frac{a_o}{Z_r{R}_o}{ϵ}_k^{\left(1\right)}\langle \frac{R_0}{\mathrm{R}}\rangle \right),$$

(2)

and

$$I_R={\displaystyle \sum }_{k<0}{I}_k^{\left(0\right)}\left(1+\frac{a_o}{Z_r{R}_o}{ϵ}_k^{\left(1\right)}\langle \frac{R_0}{\mathrm{R}}\rangle \right),$$

(3)

where Z_p is the charge of perturbing ions, Z_r is the nuclear charge of the radiator, $a_o$ is the Bohr radius, and $R_o=$ [(4$\pi$/3)N_p]^−1/3 is the mean interionic distance, N_p = N_e/Z_p being the perturbing ion density. Here, $I_k^{\left(0\right)}$ and ${ϵ}_k^{\left(1\right)}$ are the unperturbed intensity and the quadrupole correction to the intensity, respectively, the subscript k being the label of Stark components of the spectral line; k > 0 and k < 0 correspond to the blue-shifted and red-shifted components, respectively (the values of $I_k^{\left(0\right)}$ and ${ϵ}_k^{\left(1\right)}$ for several Lyman and Balmer lines were tabulated in paper [2]). The quantity <R₀/R> is the scaled inverse distance between the perturbing ion and the radiator averaged over the distribution of such distances.

Finally, the theoretical degree of asymmetry was presented in paper [8] in the form:

$${\rho }_{quad}=0.46204{\left(\frac{N_e \left[c{m}^{-3}\right]}{{10}^{21}}\right)}^{\frac{1}{3}}\frac{1}{Z_p^{\frac{1}{3}}{Z}_r} {\displaystyle \sum }_{k>0}{I}_k^{\left(0\right)}{ϵ}_k^{\left(1\right)},$$

(4)

Then the electron density N_e was determined in paper [8] by substituting the experimental degree of asymmetry into the left side of Equation (4).

In the present paper, we consider the contribution of penetrating ions to the spectral line asymmetry in order to refine this diagnostic method. For simplicity, we limit ourselves below to the practically important case $Z_p=Z_r=Z$. The energy shifts due to penetrating ions can be calculated by the perturbation theory on the basis of the spherical wave functions of the so-called “united atom” of the nuclear charge 2Z.

The perturbed energy shifts (counted from the unperturbed energies) for the orbital quantum number $l>0$ are given by (see, e.g., Equations (6) and (7) from paper [12] or Equations (5.11) and (5.12) from book [13]):

$$E_{nlm}=-\frac{8\left[\mathrm{l}\left(\mathrm{l}+1\right)-3{\mathrm{m}}^2\right] Z^4{R}^2 e^2}{{{\mathrm{a}}_{\mathrm{o}}}^3{\mathrm{n}}^3\mathrm{l}\left(\mathrm{l}+1\right)\left(2\mathrm{l}-1\right)\left(2\mathrm{l}+1\right)\left(2\mathrm{l}+3\right)}.$$

(5)

For the case of $l=0$, the calculated energy shift is:

$$E_{n00}=\frac{8 Z^4{R}^2{e}^2}{3 {{\mathrm{a}}_{\mathrm{o}}}^3{n}^3}.$$

(6)

We note that Equation (6) can also be obtained from Equation (5), first by setting m = 0, and then by canceling out l(l + 1) in the numerator and denominator, and by setting l = 0. (This was mentioned in book [12], but in Equation (5.11) from [12] corresponding to our Equation (6), there was a typographic error in the sign.)

The frequency change of an individual Stark component is thus given by

$$\Delta {\omega }_k=-\frac{Z^2{e}^2{\Delta }_k^1}{2 \hslash {\mathrm{a}}_{\mathrm{o}}{}^3}{R}^2,$$

(7)

where

$${\Delta }_k^1=16 Z^2\left[\frac{l\left(l+1\right)-3 m^2}{n^{3}l\left(l+1\right)\left(2l-1\right)\left(2l+1\right)\left(2l+3\right)}-\frac{l^{\prime \left(l^{\prime }+1\right)}- 3 {m^{\prime }}^2}{{n^{\prime }}^3{l}^{\prime }\left(l^{\prime }+1\right)\left(2l^{\prime }-1\right)\left(2l^{\prime }+1\right)\left(2l^{\prime }+3\right)}\right].$$

(8)

For the specific case where either l = 0 or $l^{\prime }=0$, Equation (8) reduces to

$${\Delta }_k^1=\{\begin{array}{c}16 Z^2\left[\frac{1}{3 n^3}-\frac{l^{\prime }\left(l^{\prime }+1\right)- 3 {m^{\prime }}^2}{{n^{\prime }}^3{l}^{\prime }\left(l^{\prime }+1\right)\left(2l^{\prime }-1\right)\left(2l^{\prime }+1\right)\left(2l^{\prime }+3\right)}\right],\hspace{1em}l=0;l^{\prime }\ne 0\\ 16 Z^2\left[\frac{l\left(l+1\right)- 3 m^2}{n^{3}l\left(l+1\right)\left(2l-1\right)\left(2l+1\right)\left(2l+3\right)}-\frac{1}{3 {n^{\prime }}^3}\right],\hspace{1em}{l}^{\prime }=0;l\ne 0\end{array}.$$

(9)

Then, the quasi-static profile of each Stark component can be represented in the form:

$$S_k\left(\Delta \lambda \right)={{\displaystyle \int }}_0^{u_{max}}W\left(u\right)\left[I_k^{\left(0\right)}+I_k^{\left(1\right)}\right]\delta \left(\Delta \lambda -\frac{Z^2{e}^2{\Delta }_k^1 {{\lambda }_0}^2}{4 \pi c \hslash a_o{}^3}u\right)du.$$

(10)

Here, $u$ $\equiv$ $R^2$, and the probability of finding the perturbing ion a distance $u$ away from the radiating atom is taken to be the binary distribution. For simplifying the integration, we use the expansion of the distribution in powers $u$/R₀² and keep the terms up to ~$u^2$:

$$W\left(u\right)du=\frac{3 \sqrt{u}}{2 R_o^3}\exp \left(-\frac{{\sqrt{u}}^3}{R_o^3}\right)du \approx \frac{3 \sqrt{u}}{2 R_o^3}-\frac{3 u^2}{2 R_o^6}.$$

(11)

For the case of a hydrogenic radiator under the presence of a penetrating ion, the relative intensities of each line component can be best calculated analytically using the robust perturbation theory developed by Oks and Uzer [14]. A more detailed explanation of this procedure is outlined in Appendix A. The relative intensities of each component can be written as

$$I_k={\Delta }_{Ik}^0+Z^2{\Delta }_{Ik}^1{u}^2,$$

(12)

where ${\Delta }_{Ik}^0$ and ${\Delta }_{Ik}^1$ are tabulated in Appendix B for each component of the spectral line Balmer-alpha, considered here as an example. These coefficients represent corrections to the intensity of the line, calculated from the perturbation theory briefly mentioned above.

The upper limit $u_{max}$ of the integration in Equation (10) should be the smallest of the following two “candidates”. One candidate for $u_{max}$ is the root mean square size of the bound electron cloud, which depends on the sublevel in consideration:

$${\mathrm{r}}_{\mathrm{rms}}=\sqrt{\frac{n^2}{2 Z^2}\left[5 n^2+1-3 l \left(l+1\right)\right]} .$$

(13)

The other candidate for $u_{max}$ is defined by the limit of the applicability of the perturbation theory. Of course, this would ensure that formally calculated corrections to the energy and intensities of the spectral line would remain relatively small.

The allowance for penetrating ions shifts the center of gravity of the spectral line, as shown in paper [15]. (This is the only contribution to the shift of the center of gravity, since the dipole and quadrupole interactions of the radiator with perturbing ions outside the bound electron cloud do not shift the center of gravity, as shown in paper [11] and mentioned above). For the He II Balmer-alpha line, which we use as an example, the center of gravity shift due to penetrating ions was calculated analytically in paper [15] to be

$$\Delta {\lambda }_{PI}\left(mÅ\right)=17\frac{N_e\left(c{m}^{-3}\right)}{{10}^{17}}.$$

(14)

So, with the allowance for penetrating ions, the reference point for calculating the integrated intensities of the blue and red wings must be shifted by the amount given by Equation (14).

After carrying out the integration in Equation (10), the profile reduces to

$$S_k\left(\Delta \lambda \right)={\left(\frac{Z^2{e}^2{\Delta }_k^1 {{\lambda }_0}^2}{4 \pi c \hslash a_o{}^3}\right)}^{-1}{I}_k\left(u_0\right)\left(\frac{3u_0{}^{\frac{1}{2}}}{2 R_o^3}-\frac{3u_0{}^2}{2 R_o^6}\right)\Theta \left[u_{max}\frac{Z^2{e}^2{\Delta }_k^1 {\lambda }_0{}^2}{4 \pi c \hslash a_o{}^3}-\left|\Delta \lambda \right|\right],$$

(15)

where $\Theta [\dots ]$ is the Heaviside step function and $u_0$ is the root of the delta function, given by

$$u_0=\frac{4 \pi c \hslash a_o{}^3}{Z^2{e}^2{\Delta }_k^1 {\lambda }_0{}^2}\Delta \lambda .$$

(16)

Thus, for the contributions of the penetrating ions to the integrated intensities of the blue and read parts of the line profile, we get

$${\mathrm{I}}_{\mathrm{PI},\mathrm{B}}={\displaystyle \sum }_{k<0}{{\displaystyle \int }}_{-\Delta {\lambda }_{max}}^{\Delta {\lambda }_{PI} }{S}_k\left(\Delta \lambda \right)d\Delta \lambda$$

(17)

and

$${\mathrm{I}}_{\mathrm{PI},\mathrm{R}}={\displaystyle \sum }_{k>0}{{\displaystyle \int }}_{\Delta {\lambda }_{PI}}^{\Delta {\lambda }_{max}}{S}_k\left(\Delta \lambda \right)d\Delta \lambda ,$$

(18)

respectively. Here

$$\Delta {\lambda }_{max}=u_{max}\frac{Z^2{e}^2{\Delta }_k^1 {\lambda }_0{}^2}{4 \pi c \hslash a_o{}^3},$$

(19)

which is obtained by equating to zero the argument of the Heaviside step function. Additionally, what is meant in Equations (17) and (18) by k < 0 (or k > 0) is the inclusion of only those components which involve corrections to the energy that are positive (or negative), implying blue-shifted (or red-shifted) components of the spectral line.

By combining the above result with the contribution of the quadrupole interaction (the interaction of the radiator with perturbing ions outside the bound electron cloud) to the integrated intensities of the blue and read parts of the profile, we obtain our final result for the degree of asymmetry

$${\rho }_{act}=\frac{I_B+{\mathrm{I}}_{\mathrm{PI},\mathrm{B}}-I_R-{\mathrm{I}}_{\mathrm{PI},\mathrm{R}}}{0.5\left[I_B+{\mathrm{I}}_{\mathrm{PI},\mathrm{B}}+I_R+{\mathrm{I}}_{\mathrm{PI},\mathrm{R}}\right]},$$

(20)

where subscript act stands for actual—in distinction to ρ_quad.

The combination of Equations (4) and (20) connects the degree of asymmetry with the electron density N_e and thus allows a more accurate determination of the electron density from the experimental asymmetry. We illustrate this below by the example of the He II Balmer-alpha line.

Table 1. The relative error in determining the electron density N_e from the experimental asymmetry degree while disregarding the contribution of the penetrating ions for the He II Balmer-alpha line. The physical quantities in Table 1 are explained in the text directly above Table 1.

Ne,act/(1018 cm−3)	ρact	ρquad	Ne,quad/(1018 cm−3)	|Ne,quad − Ne,act|/Ne,act
2	0.0925	0.0955	1.82	9.03%
4	0.114	0.120	3.42	14.5%
6	0.128	0.138	4.86	19.1%
8	0.139	0.152	6.16	23.1%
10	0.147	0.163	7.33	26.7%

presents the following quantities for the He II Balmer-alpha line at five different values of the actual electron density:

-

the theoretical degree of asymmetry ρ_act calculated with the allowance for penetrating ions,

-

the theoretical degree of asymmetry ρ_quad calculated without the allowance for penetrating ions,

-

the electron density N_e,quad that would be deduced from the experimental asymmetry degree while disregarding the contribution of the penetrating ions,

-

the relative error |N_e,quad – N_e,act|/N_e,act in determining the electron density from the experimental asymmetry degree while disregarding the contribution of the penetrating ions.

It is seen that in high-density plasmas, the allowance for penetrating ions can indeed result in significant corrections to the electron density deduced from the spectral line asymmetry.

3. Conclusions

To improve the diagnostic method for measuring the electron density using the asymmetry of spectral lines in dense plasmas, we took into account the contribution to the spectral line asymmetry from penetrating configurations, i.e., from the configurations where the perturbing ion is inside the bound electron cloud of the radiating atom/ion. After performing the corresponding analytical calculations, we demonstrated that in high-density plasmas, the allowance for penetrating ions can result in significant corrections to the electron density deduced from the spectral line asymmetry.

It is worth clarifying why we took into account the shift of the line as a whole due to penetrating ions, but did not take into account other mechanisms shifting the line as a whole, such as, e.g., plasma polarization shift and the shift by plasma electrons. The experimental integrated intensities of the blue (I_B) and red (I_R) parts of the profile are calculated with respect to the experimental center of gravity of the profile. The latter shifts of the line as a whole do not contribute to the asymmetry, and thus should not affect the experimental values of I_B and I_R. The reason why we took into account the shift of the line as a whole by penetrating ions is that penetrating ions contribute simultaneously to both the asymmetry and the shift of the line as the whole. Since these two effects of penetrating ions are two sides of the same coin, both of them should be taken into account.

We mention in passing that the potential transition of electrons into the quasistatic regime is practically irrelevant to the asymmetry. Indeed, as Demura and Sholin wrote in paper [3], for the quadrupole interaction U ~ Q/R³, which controls the asymmetry of hydrogenic spectral lines, electrons can become quasistatic at the frequency detuning from the line center Δω ~ v_Te^3/2/Q^1/2, where v_Te is the mean thermal velocity of plasma electrons—according to Holstein [16] (see also Sobelman book [17]). However, such detuning significantly exceeds the mean separation between spectral lines—both for the Lyman and Balmer series, as noted by Demura and Sholin [3], thus making the potential transition of electron into the quasistatic regime irrelevant to the problem of asymmetry of hydrogenic spectral lines. A similar conclusion was drawn also in paper [18].

Finally, we note that the electron densities N_e ~ (10¹⁸—10¹⁹) cm⁻³, which we used in the illustrative example of the He II Balmer-alpha line, are achievable in plasma spectroscopy. Examples include experiment [19], with a hydrogen plasma, and experiment [20], with a helium plasma.