Effect of higher-order multipole moments on the Stark line shape

Spectral line shapes are sensitive to plasma conditions and are often used to diagnose electron density of laboratory plasmas as well as astrophysical plasmas. Stark line-shape models take into account the perturbation of the radiator's energy structure due to the Coulomb interaction with the surrounding charged particles. Solving this Coulomb interaction is challenging and is commonly approximated via a multipole expansion. However, most models include only up to the second term of the expansion (the dipole term). While there have been studies on the higher-order terms due to one of the species (i.e., either ions or electrons), there is no model that includes the terms beyond dipole from both species. Here, we investigate the importance of the higher-order multipole terms from both species on the $\mathrm{H}\beta$ line shape. First, we find that it is important to include higher-order terms consistently from both ions and electrons to reproduce measured line-shape asymmetry. Next, we find that the line shape calculated with the dipole-only approximation becomes inaccurate as density increases. It is necessary to include up to the third (quadrupole) term to compute the line shape accurately within 2%. Since most existing models include only up to the dipole terms, the densities inferred with such models are in question. We find that the model without the quadrupole term slightly underestimates the density, and the discrepancy becomes as large as 12% at high densities. While the case of study is limited to $\mathrm{H}\beta$, we expect similar impact on other lines.

Figure 1

The double-peaked $\mathrm{H}\beta$ line shape. The asymmetry of the core is defined as the fractional difference of the intensity between the blue peak and red peak, $(I_b-I_r)/I_b$, where the subscripts $b$ and $r$ denote the blue ($\mathrm{\Delta }E>0$) and the red peaks ($\mathrm{\Delta }E<0$), respectively.

Figure 2

Asymmetry calculations (lines) with different approximations: dipole only interaction (black); dipole + ion-quadrupole (blue); dipole + electron- and ion-quadrupole moments (red). Dipole only calculation becomes too asymmetric above $3\times {10}^{17}e/{\mathrm{cm}}^3$; including $Q_i$ further increases the asymmetry and does not agree well with data; including both $Q_i$ and $Q_e$ reduces the asymmetry above $3\times {10}^{17}e/{\mathrm{cm}}^3$ and agrees well with data.

Figure 3

Area-normalized $\mathrm{H}\beta$ profiles at various levels of approximation over several different conditions. Profiles calculated at electron densities of ${10}^{17}$, ${10}^{18}$, and ${10}^{19}e/{\text{cm}}^3$ are shown (all at 1-eV temperature). All moments include contributions from both ions and electrons. The green line shows the dipole approximation of Eq. (4); the blue line shows up to the quadrupole term; the orange line shows up to the octupole term; and lastly, the red line shows up to the sedecapole $(k\le 4)$ term, indistinguishable from the octupole profile. The profiles that include up to the quadrupole term match reasonably well to the sedecapole profile.

Figure 4

(a) The highest-density measured spectrum from the Wiese et al. [27] experiment, focused on $\mathrm{H}\beta$ (dots). The fits for the calculated dipole, quadrupole, and sedecapole profiles are identical, yielding fit differences less than the fit errors. (b) The measured $\mathrm{H}\beta$ spectrum by Carlhoff et al. [23] (dots). The fits for the calculated dipole, quadrupole, and sedecapole profiles yield differences in density of 12%, which is greater than the fit errors.