Spectral Line Shapes in Plasmas

This document is intended to define the particulars of the workshop submissions. In the sections below we define the case problems, the comparison quantities which we require and the detailed format of the data files that we will be expecting. The webpage of the meeting is at http://plasma-gate.weizmann.ac.il/slsp/. The submission files are to be uploaded to the same server using a web interface with userid and password. Details will be announced separately.


Introduction
This document is intended to define the particulars of the workshop submissions. In the sections below we define the case problems, the comparison quantities which we require and the detailed format of the data files that we will be expecting.
The webpage of the meeting is at http://plasma-gate.weizmann.ac.il/slsp/. The submission files are to be uploaded to the same server using a web interface with userid and password. Details will be announced separately. Timeline:

Statement of cases
We have selected a number of transitions to consider, given in Table 1. For each transition we are requesting results on a grid of electron densities (n e ) and temperatures (T = T e = T i ). For each case, the atomic and plasma models are specified, and for some cases, there are more than one atomic or plasma model suggested.
(Unless specified otherwise, plasma is assumed quasi-neutral, consisting of electrons and a single type of ions. For example, the plasma model in Case 3 is described as "Plasma ions: protons", meaning the plasma consists of electrons and protons of an equal density.) In addition, some cases are further detailed by specifying extra parameters, such as the magnetic field. Therefore, subcases for each case are defined on a four-dimensional grid. Each calculation will be referenced by its subcase name. The subcase name is of the form Case ID.N.T.M.F, where Case ID is from the first column of Table 1, and the N, T, M, and F correspond, respectively, to the n e , T , model, and external-field indices, each counting from 1. For example, 1.2.3.1.1 identifies H Lyman-α, calculated for n e = 10 18 cm −3 and T = 100 eV, assuming only electrons as perturbers. Similarly, 11.2.1.1.3 stands for D Balmer-β in a deuterium plasma with n e = 10 15 cm −3 and T = 1 eV, and magnetic field 10 T.
The models suggested are limited -some by design, others by necessity, to make them manageable without too much computational resources and human time spent. If you feel that the best suggested model for a particular case is still too far from reality, you are encouraged to submit a separate result using an alternative model you see fit best, using "0" as the model index. E.g., 3.2.1.0.1 would designate the H n = 6 → 5 transition calculated for n e = 2 × 10 16 cm −3 and T = 1 eV including, say, all discrete levels up to n = ∞. Submissions of all such results should include an adequate description of the model used in the <comments> field of the file (see Sec. 4).

Justification of cases and details 2.1 Reference cases
These cases are not necessarily realistic, but good for basic comparison and understanding what is wrong/different if there is a significant scatter in the results of more advanced cases below. There are quite a few sub-cases, however the models are simplest: ideal plasma (straight path trajectories and infinite Debye length for MD, or Holtsmark distribution for analytical approaches) and pure linear Stark effect (interactions between states with ∆n = 0 ignored and no fine structure). In order to assess influence of electrons and ions (protons), we ask to calculate the broadening assuming e and p acting separately and together, i.e., three variants in total for each pair of n e and T .
1. Hydrogen Lyman-α in an ideal plasma is a classical ion-dynamics test.

2.
A relatively high-n line. For the plasma parameters selected, this is a test of the transition for electrons from dynamic to almost static regime.
Please note that all cases below do NOT assume an ideal plasma, unless explicitly said so.

High-n ∆n = 1 transitions
A representative of n, n 1, ∆n n class of transitions. Radio-frequency lines which are of great interest for astrophysics belong to it. We do not want to go to really high n, however, due to the computational costs. Nevertheless, we shall deal with the coupling between states with ∆n = 0.
3. Hydrogen n = 6 → n = 5 transition. We ask to calculate this case using three atomic models: (i) no ∆n = 0 coupling accounted for, (ii) n = 5 and n = 6 states couple, and (iii) n = 5, 6, and 7 states included in the Hamiltonian and allowed to mix.
and three values of the temperature. Only widths (FWHM) are requested. The plasma model for these cases consists only of electrons, and we ask to assume either straight path trajectories or more realistic quasi-classical hyperbolic trajectories (due to the Coulomb interaction with the radiator). 4. Be II is the first non-neutral species of the sequence. 5. N V -something in between. 6. Ne VIII is about the highest Z for which the 3s-3p broadening can be reliably measured.
In connection with the three previous cases, we consider another isolated line for which quantum effects are not expected to be so significant (larger matrix elements and cross-sections), i.e., a higher-n line.
7. Al III 4s-4p. In addition to the width, we also want to compare shifts, therefore, calculations with only electrons and e+ions are requested.

Intermediate case between isolated and degenerate regimes
8. He-like Si XIII 3 → 1 transitions. No inter-combination lines, only singlet levels. At the low density, only 1s-3p (He-β proper) is seen, then 1s-3d and 1s-3s appear as well, approaching Lyman-β-like shape at the highest density. Plasma ions are protons.

External fields
The external macro fields (both electric and magnetic) are always assumed to be parallel to the z axis.
9. Al XIII Lyman-α under external harmonic perturbation, e.g., a laser. The functional dependence of the electric field is F cos (ωt), with ω and F given in Table 1. The two plasma densities correspond to laserdominated and plasma-dominated line shapes. Two variants of the atomic model: with and without fine structure taken into account.
10. D Balmer-α in the presence of magnetic field, parameters typical for tokamaks. We want to investigate the difference between dynamic and (quasi)static ions (for MD codes, set the ion and radiator masses as large as possible).

High-n merging with continuum
These cases are rather advanced. It is interesting to see what different approaches use when several discrete levels start to overlap between themselves and continuum states. A broad spectral region will be analyzed covering both discrete and continuum spectrum and a transition region in between.
12. D Balmer series at T = 1 eV and three densities, the lowest one about corresponding to tokamak conditions [2] (but no magnetic field in this case). The higher densities are typical for white dwarf photospheres [3].

Influence of particle correlations on electric micro-fields
In this set of cases the focus will be on analyzing the correlation properties of plasma fields on line shapes [4,5].
To this end, a set of statistical properties of the micro-fields will be compared in each calculation case. These are (i) distributions of magnitudes of the "slow" and "fast" components of the total micro-field F (t) = F e (t)+ F i (t), defined as and respectively, and (ii) correlations between directionalities of micro-field components where and the indices a and b represent either electrons (e) or ions (i).
13. H Balmer-α in two plasma model variants: without interactions between the plasma particles (ideal plasma) and with such interactions.

Satellite broadening
Another advanced case, satellites from the previous charge state with a spectator electron.
15. Ar XVII He-β and its Li-like satellites. The argon He-β composite spectral feature is observed in inertial confinement fusion implosion core plasmas when a tracer amount of argon is added to the deuterium gas fill to diagnose the plasma conditions. This spectral feature is comprised of the n = 1 to n = 3 line transition in He-like Ar and satellite line transitions in Li-like Ar. It is temperature and density sensitive through the density dependence of the Stark-broadened line shapes and the temperature and density dependence of the atomic level populations. In implosion core dense plasmas the Stark broadening effect dominates the line shapes. The details of these line shapes and their overlapping, impact the photon-energy dependent emissivity and opacity that, in turn, determine the emergent intensity distribution of the spectral feature and its diagnostic properties [6].
Assume an equilibrium (i.e., LTE) distribution of population within the initial (upper) energy levels of the transitions within a given line shape.

Atomic data
In all cases, we assume the dipole approximation both for the radiation (E1) and the perturbation due to the plasma micro-fields. The relevant matrix elements are The reduced radius-vector matrix elements (αj|r|α j ), relevant for the cases considered, are given below.

Hydrogen-like
For hydrogen (Z = 1) and hydrogen-like cases, the data are to be calculated analytically. For cases where the fine structure is neglected, the binding energies to be assumed are (in atomic units E H ≈ 27.211 eV, corresponding to ≈ 2.1947 × 10 5 cm −1 ) When the fine structure is asked to be accounted for, the energies are where α ≈ 7.2974 × 10 −3 is the fine-structure constant. Reduced matrix elements of radius-vector are where > = max ( , ) and for diagonal terms (e.g., Eq. (63.5) in [7]) and for off-diagonal ones (Eq. (63.2) in [7]). Here, F 21 is the Gauss hypergeometric function and n r = n − − 1, n r = n − are the radial quantum numbers of the two states. For convenience, the reduced matrix elements up to n = 5 are given in Table 2.

Non-hydrogen
The data are taken from the NIST on-line compilation [8] 1 . The level energies, averaged over the fine-structure components for > 0, are given in Table 3. The absolute values of the matrix elements are obtained from the respective multiplet-averaged absorption oscillator strengths f according to  and sign as in respective H-like from Eqs. (8 -10). The data are summarized in Table 4. For Si XIII, hydrogenlike matrix elements (Z = 13) between the n = 3 states should be used, which are accurate to within ∼ 1%.

Data for cases 15*
Atomic physics files computed with R. Cowan's atomic structure code [10] are provided for the four line shape calculations, namely the argon He-β line transition, and the Li-like Ar transitions with spectator electron in n = 2, n = 3 and n = 4. Each atomic physics data file contains the list of fine structure energy levels for the set of upper and lower configurations including energy value, parity and total angular momentum J, and reduced electric-dipole matrix elements between the levels. Due to the vast amount of data for these cases, they are provided as attachments, formatted as follows: Ar XVII: Ar XVI n = 2 satellites: Ar XVI n = 3 satellites: Ar XVI n = 4 satellites: (Click on the *.txt links above to download the data attached. If your PDF viewer is unable to display the attachments, you can download all data files in a single archive, http://plasma-gate.weizmann.ac. il/uploads/slsp/he_beta_data.zip.)

Submission format
We use an XML-based format for submissions, with an example shown schematically in Listing 1.
Everything is included between the <slsp> and </slsp> tags. The meaning of other tags is described below: <case> The subcase identification in the Case ID.N.T.M.F format, see Sec. 1.
<contributor> The person who submits these results.
<code> Name of the code/approach. <spectrum> For all cases except those concerned with isolated lines (4 -7), we ask to provide entire line shapes on a reasonably dense grid, typically ∼ 1000 points (see Table 5). When the spectral range is symmetric (± something), it means relative to the unperturbed position ω 0 , calculated as a difference between the weighted-average energies of the initial and final levels: The spectral windows and distances between the consecutive abscissas defined are recommended values. The relatively wide spectral windows are defined on purpose, to investigate far wings of the spectral lines. You can use denser and/or wider grids as you see fit. It is suggested to use equidistant grids. The units are cm −1 . The optional unit attribute allows for scaling the abscissas, e.g., by using unit="8065.5" one can output spectra in eV's. Where the spectra are requested and external fields specified (cases 9, 10, and 11), the π (∆M = 0) and σ (∆M = ±1) polarizations will be needed separately (to be provided as the second and third columns, respectively):

It is assumed that
No normalization condition is imposed (you can use peak-or area-normalized profiles).
<field distribution> Quasi-static field distribution (normalized) used for the calculation (due to all plasma particles, but excluding external fields, if any). The fields are in V/cm. The optional unit attribute allows for scaling the field strength values conveniently, e.g., by setting it to the Holtsmark normal field strength F 0 one obtains the distribution of the reduced field strengths. The distributions should be calculated on an equidistant grid covering at least 0 − 10 with a step not exceeding 0.1 (in units of F 0 ). In addition, for cases 13 and 14, distributions of magnitudes of the "slow" and "fast" microfield components, defined by Eqs. (1) and (2)   <field correlation> For cases 13 and 14 only. Correlations between directionalities of micro-field components, see Eq. (3). We ask for all three C ee , C ii , and C ei over τ from 10 −16 s to 10 −11 s on a logarithmic grid of 100 points at least: . . . . . . < f i e l d c o r r e l a t i o n > t a u 1 C ee ( t a u 1 ) C i i ( t a u 1 ) C e i ( t a u 1 ) t a u 2 C ee ( t a u 2 ) C i i ( t a u 2 ) C e i ( t a u 2 ) . . . . . . t a u N C ee ( t a u N ) C i i ( t a u N ) C e i ( t a u N ) </ f i e l d c o r r e l a t i o n > . . . . . .