The Third Non-LTE (Local Thermodynamic Equilibrium) Kinetics Workshop (NLTE-3)

 

December 1-5, 2003

NIST, Gaithersburg MD

 

 

Submission of Calculations

 

 

This document is intended to define the particulars of the workshop submissions. In the sections below, we will define the case problems, the comparison quantities which we require, and the detailed format of the data files which we will be expecting. The case problems are of two types:  steady-state and time-dependent. These problems are completely defined by a specification of the electron temperature and electron or ion density. For the time-dependent cases, these quantities are provided as a time history. With the large number of case studies included in these two categories, we have avoided additional variations on the general NLTE problem, i.e. we specifically request that there be no consideration of optical depth, radiative transfer, or heavy-particle interactions. (The non-Maxwellian electron effects, however, are included in one of the cases.)  These interesting topics are best deferred to later meetings.

 

An ftp/http server will be set up early summer 2003 to serve our needs for this workshop, and an email containing the relevant details will be distributed among the potential participants. Although the specific data transfer mode is still under consideration (ftp uploading being the primary candidate), it would be most convenient if the contributor created an archive file containing all the individual result files. Submissions employing such data compression techniques as zip/gzip/bzip2/arj/lha/rar along with the Unix tar archiving utility will be accepted. Please contact Yuri Ralchenko (email: yuri.ralchenko@nist.gov) if other compression/archiving options are necessary. An example submission file will be provided on the ftp server for comparisons.

 

Timeline:

 

1. June  2003 – web server is set up
2. October 1 2002 – submission deadline
3. December 1 2003 – workshop opens
4. December 5 2003 – workshop closes

 

---

I.       STATEMENT OF STEADY-STATE CASES

 

Notations: electron temperature T_e, electron density N_e, ion density N_i, hot electron temperature T_hot, radiation field temperature T_rad 

We have selected a number of atoms to consider, and for each atom are requesting results on a grid of electron temperatures and electron or ion (for the Xe case only) densities.  In the following, temperatures are given in eV (1 eV = 1.602 176 462 · 10^-19 J) and particle densities in cm^-3. 

 

The following problems have been established for the steady-state cases:

 

 

 

 

                        Table I.  Steady-state case definitions.

 

The grid of plasma temperatures and densities is given in Table I. If your calculation requires an ion temperature, then you should assume it is identical to the electron temperature.

 

In case of argon calculations, 10% of the hot electrons means 10% of the total electron density given by N_e. Thus, for instance, for N_e = 10¹⁸ cm^-3 the density of hot electrons would be 10¹⁷ cm^-3 while the density of the bulk thermal electrons would be 9×10¹⁷ cm^-3. The hot electrons are to be presented as a second Maxwellian with the temperature of 10 000 eV.

 

Each calculation will be referenced by a case name, which is to be given in the submissions data file (as described further below). The case name is constructed by appending a suffix to the Case_ID shown in the preceding table. The suffix consists of a digit, a letter and, for Ar and GeTr cases only, a digit. The first digit identifies the problem temperature, while the letter indicates the electron density. For the Ar case, the final digit is 1 for 0% of hot electrons and 2 for 10% of hot electrons. For the GeTr case, the final digit is 1, 2 or 3 for T_rad = T_e/2, T_e or 300, respectively. Thus, the case name Al3b refers to the third temperature and second density of the aluminum steady-state grid, namely 100 eV and 10²² cm^-3. The name GeTr4a2 refers to germanium at T_e = 600 eV, N_e = 3×10²² cm³, and T_rad = T_e.

 

The quantities to be computed for each case are described below. The Xe case additionally requires calculation of emission spectra. The emissivity of the xenon plasma is to be calculated in the wavelength range of 9-120 Å[1].

 

In all the above cases, the calculations are to be solved in steady-state, at the specified electron densities and temperatures. In all but the GeTr cases the radiation field should be zero − only spontaneous radiative decays and radiative recombination shall be included –  while for the GeTr case we request the photoinduced processes (photoionization, photoexcitation, stimulated emission, etc.) to be added as well.

 

For both steady-state and time-dependent cases the submissions file should be named as <case>.<contributor_name>.<code_name>, so that Dr. A. Einstein’s calculations for one of the hot-electron cases in Argon with his code GTOE would be in the file ar2c2.einstein.gtoe (case insensitive).

---

II.                 JUSTIFICATIONS OF THE STEADY-STATE CASES

 

1. C

 

Since the NLTE-1 comparisons for carbon cases have shown quite a good agreement for lower densities of 10¹⁶ and 10²⁰ cm^-3, here we would like to study the highest-density case for which the codes produced the mostly divergent results.

 

2. Al

 

Aluminum cases for NLTE-1 have showed a wide spread for temperatures below 200 eV and all densities. Correspondingly, the number of low-T_e points is increased here.

 

3. Ar

 

For argon, a rather good agreement was found in NLTE-1, mostly since the high temperatures shift the ion distribution towards He- and H-like ions. Here we, first, add lower temperatures and second, ask for calculations with hot electrons.

 

4. Ge

 

To move the comparisons toward higher atomic number we have attempted to take cases motivated by experiment. Thus we choose this Ge (Z=32) steady-state case to provide further information on the same T_e and N_e range that is used in the TD-Ge case which was motivated schematically by the transient collisional laboratory X-ray laser scheme. We have done this in the hopes of a) reducing the number of kinetics models that need to be constructed and b) providing further insight in the TD-Ge case.

 

5. GeTr

 

To test the various aspects of the code with respect to radiation fields we have taken the simplest form of the problem.  That is, we impose a Planckian radiation field at the specified radiation temperature T_rad.

 

6.      Xe

 

This is the most “experiment-related” case inspired by the recent both published (Phys. Rev. E 65, 046418, 2002) and yet unpublished measurements by C. Chenais-Popovics et al. The measurements were aimed “…at validating recent calculation techniques for the emission properties of medium- and high-Z multicharged ions in hot plasmas." The fundamental plasma parameters, such as electron temperature, electron density and average charge state, were deduced from electronic and ionic Thomson scattering spectra, so that the calculated spectra could be directly compared to experimental data.

 

7. Au

 

Here we would like to explore the rapid changes in ion distribution which seems to strongly depend not only on T_e but on N_e as well.

---

III.       STATEMENT OF TIME-DEPENDENT CASES

 

Three simple cases have been defined by temperature and density histories:

 

TD-C            Impurity transport in a reversed-field pinch.

TD-Ar           Implosion of argon plasma in a capillary discharge.

TD-Ge         Germanium X-ray laser plasma.

 

For all time-dependent cases the time grid is linear starting from t = 0. The exact time grids as well as the electron temperature and density histories for each case are provided in the Appendix. The calculated data are to be provided at each time step.

 

We would like to emphasize that although the proposed cases are motivated by some particular experiments, the presented time histories are not necessarily identical to those determined in various measurements. This was done intentionally in order to enhance some or to diminish other physical processes taking place in the plasmas.

 

The TD-C case of the NLTE-1 was devoted to recombination of high-density plasma of carbon. Here we propose a low-density case which is related to the reversed-field pinch experiment. The impurity emission during its transport across the experimental device is widely used for diagnostics of this type of plasma. The major process here is also recombination although at a much lower density. The electron temperature and density are presented in Fig. 1. This calculation should be carried out to t = 200 ms. The initial condition is that all the population is in the bare ion. Output should be provided at t = 0 and thereafter at 40 uniformly spaced times spanning the range 5 to 200 ms.  The intensities of the Ly_α and He_α (W) and intercombination (Y) lines in C V and C VI are requested as well.

 

The TD-Ar case is inspired by the implosion of argon inside capillary discharge used in the studies of the collisional X-ray laser. However, the N_e and T_e are, correspondingly, lower and higher than those in the X-ray laser capillaries. For this case all population at time t=0 is in the ground state of Ar VI or, only if Ar VI is not included in the model, in the ground state of the lowest-charge ion. The history of the electron temperature and density is presented in Fig. 2; the calculation should be carried out to t = 50 ns.

 

In Figure 3 we show the time history of T_e (left hand axis) and N_e (right hand axis) for the TD-Ge case, which is motivated by the germanium X-ray laser experiments. This calculation should be carried out to t = 1.975 ns, and the non-uniform time grid along with the corresponding values of T_e and N_e are presented in the Appendix.  The initial condition is LTE. 

 

Case	Initial Condition	# Output Times	Output Δ	Stop Time
TD-C	bare ion	41	5 ms	200 ms
TD-Ar	Ar VI g.s.	51	1 ns	50 ns
TD-Ge	LTE	83	variable	1.975 ns

Table III.  Summary of parameters for time-dependent calculations.

 

Fig 1. Time history of plasma parameters for TD-C.

---

Fig. 2. Time history of plasma parameters for TD-Ar.

---

Fig. 3. Time history of plasma parameters for TD-Ge.

---

IV.    SUBMISSION FILE DESCRIPTION

 

We are asking for a fairly large amount of information by comparison with the opacity workshops. To simplify the specification for the contributors we have adopted a keyword approach within the submissions file. In this approach, all quantities are space delimited. In Section V, we give a schematic of the file format.  For clarity we will use the courier font to indicate the keywords (the actual submissions should be unformatted text). The user-supplied data which are problem-dependent are indicated by a bold-face parameter name in brackets (e.g., <pop_frac>). We anticipate that not all information will be provided by every user.  However, since the information is space delimited, and not fixed in a particular column, then some value must be given for each field. The best default value is to put a zero. The longer records, such as the ion and elev lines, may continue over several lines at the contributor’s discretion.  Do not break a line in the middle of a keyword or a number. Blank lines may be used anywhere within the file to make the text more readable. While some of the names we use suggest integer quantities, please use decimal values if appropriate to your calculation. For floating point numbers an e11.4 format is generally adequate. The exact definitions of the quantities requested, including units, are given in Section V.

 

The submissions file is structured in 5 sections. These sections are identified by keywords. In some cases, an integer follows the section keyword to indicate the number of records which follow for that section. Some codes will not be able to provide information for every section. Thus, an entire section may be omitted. If all information is provided, then there will be a certain amount of redundancy. This redundancy is intentional and has at least two uses. First, it can be used to detect errors in the file formatting. Second, it is often possible to compute overall quantities more accurately internal to the kinetics code than by post-processing the results.

 

The initial section provides general problem identification information. This section begins with the keyword data.

 

The sections 2-4 constitute a set of output data which is to be generated at each time step for the time-dependent problems. Naturally, only one such set should be produced for the steady-state cases.

 

The second section gives overall quantities describing the plasma population and energy distribution. This section is signaled by the keyword summary_quantities. Note there are no spaces in the keywords.

 

The third section gives information by ionization stage. This section is signaled by the keyword ion_stages. Within this section, information for each ionization stage begins with the keyword ion. As mentioned above, multiple lines may be used if desired (we intentionally used multiple lines in the schematic file listing below to improve its readability). Important note: we use <Nbound>, the number of bound electrons, not the ion stage charge, to label the ion stages.

 

The fourth section gives information by energy level (keyword energy_levels). Since many codes employ some form of continuum lowering and/or moving calculational windows, we require that energy level definitions be provided for every case. The shell occupation numbers (<occK>, <occL> etc.) as defined for each elev record will be used to compare codes for the cross-over from a ladder-like de-excitation regime to one which is in Saha-Boltzmann equilibrium with the continuum.

 

Finally, the fifth section contains spectra intensities, either for a prescribed spectral range (steady-state) or for some specific lines (time-dependent).

 

A relational database tool will be used to manage the data during the course of the workshop. To simplify the data access, the database layout will parallel the file formats specified here.

 

If necessary, extra clarifications regarding the submission format will be provided at the Workshop’s web site.

---

V.     SUBMISSION FILE FORMAT

 

The text that follows is a schematic of a submissions file:

 

 

 

For the cases where we request spectra, the output format is different for Xe and TD-C cases. The spectral information will be given in this same text file, following the information above. The Xe spectrum will be in the format:

 

 

 

The TD-C spectrum will be in the format:

 

 

 

 

---

VI.    DEFINITIONS OF REQUESTED QUANTITIES

 

Before proceeding to a detailed description of the requested quantities, we would like to comment on the ion density. For all but the TD-C case the electron and ion densities are directly related via the plasma neutrality condition, i.e., . Thus, for the Xe case the electron density is to be determined from the plasma average charge and the known ion density. For the TD-C case, where carbon is an impurity in the reversed-field pinch plasma, we accept that the carbon ion density is .

 

 

In section 1, the identification section, the following quantities are requested:  

data           Calculation identifier and user comment line. Comment should be limited to this one line only and should include the contributor’s name, institution, the version of the code, and the date at which calculation was run. This can be invaluable in maintaining order in large number of submissions.

case           All steady-state and time-dependent calculations will have a case identification of the form Au1b, C6a or the like (see Section I). These identifiers are assigned in the section below where the specific calculations are called out.

code           An identifier for each contributor's code which may be chosen by the contributors. For convenience in post-processing and tabulation the names should not be excessively long. The names will be used in all tables and graphs of comparisons, and must be the same from case to case.

atom           Identifies the atom under study. The field <name> is a convenience for the contributor. In many cases, calculations are driven by atomic data found in a file. The file <name> may be used to specify that name. The field <Znuc> is the nuclear charge of the atom.

 

 

As has already been explained above, the group of sections summary_quantities, ion_stages and energy_levels repeats for each time step for the time-dependent cases.

 

 

In section 2, the summary_quantities section, the following items are requested:

plasma      This record specifies the plasma conditions used in this calculation. The electron temperature is in units of eV. The electron density is in units of cm^-3. For time-dependent cases both T_e and N_e may be arbitrary.

time           The output time for time-dependent cases or zero for steady-state cases.

zbar           The average charge of the plasma.

m2                The second central moment of the charge state distribution.

m3                The third central moment of the charge state distribution.

m4                The fourth central moment of the charge state distribution.

eint           The specific internal energy of the atom.

deintdt    The "specific heat" of the atom.

pfn              The "partition function" of the atom.

nmax_eff For this calculation, the principal quantum number of the outermost electron in any bound state. We will be interested in sensitivity of comparison quantities to the highest bound states accounted for by the model. This quantity will also be used as a measure of continuum lowering.

 

The central moments are defined as: 

 

 

where y_j is the fractional population of ion stage j,  q_j is the ion charge, and is the average charge.

 

The specific internal energy is the sum of level populations, n_j, multiplied by their energy value, E_j, divided by the total ion density N_i:

 

.

 

The energy reference is the ground state of the neutral atom. We recognize that a kinetics model may not include all ionization stages of the atom – the ground state of the most neutral ion is the most reasonable substitute. For intercomparisons, this quantity will likely need zero point shifts. Units are eV/atom.

 

The specific heat is the derivative with respect to electron temperature of the specific internal energy of the atom. Units are eV/atom/eV. If computed by finite difference, the step size is to be chosen by the contributor.

 

The partition function is defined as the classical partition function: 

 

,

 

where g_j is the statistical weight of level j and E_j is the energy of the level, with respect to the ground state of the most neutral ion.

 

Note that many of the "thermodynamic" quantities are intentionally sensitive to continuum lowering models. Possibly affected quantities are <eint>, <deintdt>, and <pfn>. If your continuum lowering model alters the energy levels or statistical weights, please include these effects in the appropriate "thermodynamic" quantities.

 

High-lying bound states can be included in the population kinetics in a variety of ways. The field nmax_eff is intended to give information on the highest-lying bound state, which is affecting the calculation of the populations.  It is thus an "effective" principal quantum number.  If a code includes a level, which accounts for more than one n value, then for this field we recommend giving the largest value that is being modeled.

 

 

In section 3, the ion_stages section, the following quantities are requested:

 

<Nbound>       The number of bound electrons in this ionization stage.

<pop_frac>     The fraction of atoms in this ionization stage. Sum over all ions should be normalized to unity.

<nouter>         The principal quantum number of the outermost electron for any state in this ion stage.

<S_tot>               The total ionization rate out of this ion stage, averaged over all initial states in this ion stage (weighted by the fractional populations of the initial states), and summed over all final states.  This quantity is further summed over all ionization processes.

<f_S_coll>          The fractional contribution of electron collisional ionization processes to <S_tot>.

<f_S_photo>         The fractional contribution of photo-ionization processes to S_tot>.

<f_S_auto>          The fractional contribution of auto-ionization processes to <S_tot>.

<α_tot>             The total recombination rate out of this ion stage, averaged over all initial states in this ion stage (weighted by the fractional populations of the initial states), and summed over all final states.  This quantity is further summed over all recombination processes.

<f_α_coll>           The fractional contribution of three-body recombination to the total <α_tot>.

<f_α_photo>      The fractional contribution of radiative-recombination to the total <α_tot>.

<f_α_auto>       The fractional contribution of dielectronic capture processes to the total <α_tot>.

 

We note that the total ionization and recombination rates are rates, and not rate coefficients.  It is also important to be precise about the direction of these total rates.  <S_tot> is the total rate out of the indicated ion.  Similarly, <α_tot> is the total rate out of the ion.

 

The definitions of S_tot and α_tot are best clarified through an example.  Consider a three-ion stage problem consisting of levels in Li-like, He-like, and H-like ions.  For the He-like ion, S_tot is the sum of all ionization rates out of He-like, weighted by the appropriate He-like initial state populations, and summed over all final states in the H-like ion. The averaging over initial states is completed by dividing the above sum by the total population of the He-like ion. α_tot for the He-like ion is the sum of all recombination rates out of He-like, weighted by the appropriate He-like initial state populations, and summed over all final states in the Li-like ion.  The averaging over initial states is completed by dividing the above sum by the total population of the He-like ion.  With these definitions, we can define a set of ionization rate equations.  In the case of the He-like ion, we write:

 

 

.

 

Our primary interest is the collisional radiative modeling of highly-charged ions and so we have not requested detailed information for contributions due to other processes, such as charge-exchange and heavy ion collisions.  Units of <S_tot> and <α_tot> are s^-1.

 

In section 4, the energy_levels section, the following quantities are requested:

<Nbound>           Identifies the ionization stage to which this energy level belongs.  As always, this quantity is the number of bound electrons in the level.

<level>                A sequential level number within this ionization stage.  This index begins at 1 within each ionization stage for use as a label in model comparisons.  The ground state of each ion stage will be identified by locating the state of lowest energy within the ion stage.

<stwt>                 The statistical weight of this energy level.

<energy>             The energy of the level relative to the overall model. Ionization potentials will be obtained by subtraction of successive ground state energies.  Units are in eV. The overall energy reference is the ground state of the most neutral ion in the problem.

<population>       The ion density of this level.  Units are cm^-3.

<Γ_tot>                  The total destruction rate of this level.  This is the absolute value of the corresponding term in the rate matrix diagonal.  Units are s^-1.

<f_Γ_collBB>           The fractional contribution of electron collision excitation/de-excitation processes to <Γ_tot>.

<f_Γ_photoBB>         The fractional contribution of bound-bound radiation processes to <Γ_tot>.

<f_Γ_collBF>           The fractional contribution of electron collision ionization-recombination processes to <Γ_tot>.

<f_Γ_photoBF>         The fractional contribution of photo-ionization-recombination to <Γ_tot>.

<f_Γ_auto>             The fractional contribution of auto-ionization/dielectronic recombination processes to <Γ_tot>.

<occK>                Occupation number:  for this energy level, the number of electrons in the K shell.  Users of configuration interaction codes might wish to use the dominant configuration to assign this value.

<occL>                The number of electrons in the L shell.

...

<nouter>             The principal quantum number of the outermost electron in that energy level.

 

 

The shell occupation numbers (<occK>, <occL>, etc.) could be variable in number for each code, plasma condition, and energy level.  Contributors are not constrained on this point:  they may specify as many shells as necessary, and as relevant to their calculational approach.  The final entry for this energy level record should be the principal quantum number of the outermost electron in that level.  In the case of highly-excited levels, the shell occupation numbers may be simplified by only specifying the core, <Nbound>-1, electrons.  In this case the field <nouter> will be used to set the location of the remaining electron.  We will be using the values given in this section to compute some of the quantities given in section 2 for consistency checks.

 

In section 5, the spectrum session, the data format is different for the Xe and TD-C cases. For the Xe case, the required data are bound-bound <ε_bb>, free-bound <ε_fb>, free-free <ε_ff> and total <ε_tot> emissivities  in units of erg/atom/s/Å (= 10^-7 J/atom/s/Å). The field <count> specifies the number of (wavelengths, emissivities) rows which follow. The minimum and maximum wavelengths to be covered are 9 and 120 Å, respectively, and the wavelength mesh should consist of 2221 equally-spaced points (Δλ = 0.05 Å, λ₁ = 9 Å, λ₂ = 9.05 Å … λ₂₂₂₀ = 119.95 Å, λ₂₂₂₁ = 120 Å). The broadening should be natural+Doppler only. The uniform gridding will greatly simplify the spectrum comparison and post-processing operations (e.g., convolution with experimental broadening function).

 

For the TD-C case, the data required are the time-dependent intensities emitted per unit volume of the Ly_α, W and Y lines in H- and He-like C. The intensity emitted per unit volume is defined as:

 

 

in units of erg/cm³/s (= 10^-7 J/cm³/s), where A_ij is the radiative transition probability (emission) and ΔE is the transition energy. No opacity effects are to be taken into account when calculating the intensities. The field <count> in this case would be 41 (number of time steps for TD-C).

---

APPENDIX.   TIME HISTORIES FOR TIME-DEPENDENT CASES

 

 

 

1. TD-C

 

Step #	Time (s)	Te	Ne
0	0.000E+00	3.000E+02	1.000E+13
1	5.000E-03	2.990E+02	1.790E+13
2	1.000E-02	2.970E+02	2.700E+13
3	1.500E-02	2.950E+02	3.730E+13
4	2.000E-02	2.920E+02	4.840E+13
5	2.500E-02	2.900E+02	6.070E+13
6	3.000E-02	2.870E+02	7.310E+13
7	3.500E-02	2.840E+02	8.050E+13
8	4.000E-02	2.810E+02	8.360E+13
9	4.500E-02	2.770E+02	8.460E+13
10	5.000E-02	2.720E+02	8.440E+13
11	5.500E-02	2.660E+02	8.380E+13
12	6.000E-02	2.590E+02	8.280E+13
13	6.500E-02	2.500E+02	8.130E+13
14	7.000E-02	2.380E+02	7.950E+13
15	7.500E-02	2.250E+02	7.790E+13
16	8.000E-02	2.090E+02	7.620E+13
17	8.500E-02	1.920E+02	7.430E+13
18	9.000E-02	1.740E+02	7.220E+13
19	9.500E-02	1.570E+02	7.030E+13
20	1.000E-01	1.420E+02	6.810E+13
21	1.050E-01	1.280E+02	6.580E+13
22	1.100E-01	1.140E+02	6.330E+13
23	1.150E-01	1.010E+02	6.080E+13
24	1.200E-01	8.950E+01	5.820E+13
25	1.250E-01	7.930E+01	5.530E+13
26	1.300E-01	6.990E+01	5.250E+13
27	1.350E-01	6.110E+01	4.980E+13
28	1.400E-01	5.330E+01	4.730E+13
29	1.450E-01	4.680E+01	4.480E+13
30	1.500E-01	4.140E+01	4.230E+13
31	1.550E-01	3.640E+01	3.980E+13
32	1.600E-01	3.190E+01	3.730E+13
33	1.650E-01	2.820E+01	3.510E+13
34	1.700E-01	2.520E+01	3.310E+13
35	1.750E-01	2.310E+01	3.120E+13
36	1.800E-01	2.150E+01	2.950E+13
37	1.850E-01	2.030E+01	2.800E+13
38	1.900E-01	1.940E+01	2.670E+13
39	1.950E-01	1.890E+01	2.560E+13
40	2.000E-01	1.880E+01	2.490E+13

 

2.TD-Ar

 

Step #	Time (s)	Te	Ne
0	0.00E+00	2.000E+01	1.000E+17
1	1.00E-09	2.463E+01	2.027E+17
2	2.00E-09	3.210E+01	3.083E+17
3	3.00E-09	4.431E+01	4.157E+17
4	4.00E-09	6.309E+01	5.239E+17
5	5.00E-09	9.319E+01	6.372E+17
6	6.00E-09	1.386E+02	7.566E+17
7	7.00E-09	1.666E+02	8.759E+17
8	8.00E-09	1.856E+02	9.887E+17
9	9.00E-09	1.971E+02	1.085E+18
10	1.00E-08	2.000E+02	1.164E+18
11	1.10E-08	1.991E+02	1.231E+18
12	1.20E-08	1.958E+02	1.290E+18
13	1.30E-08	1.897E+02	1.346E+18
14	1.40E-08	1.811E+02	1.399E+18
15	1.50E-08	1.695E+02	1.447E+18
16	1.60E-08	1.560E+02	1.489E+18
17	1.70E-08	1.433E+02	1.522E+18
18	1.80E-08	1.315E+02	1.546E+18
19	1.90E-08	1.212E+02	1.559E+18
20	2.00E-08	1.119E+02	1.562E+18
21	2.10E-08	1.031E+02	1.552E+18
22	2.20E-08	9.478E+01	1.528E+18
23	2.30E-08	8.716E+01	1.488E+18
24	2.40E-08	8.036E+01	1.432E+18
25	2.50E-08	7.436E+01	1.365E+18
26	2.60E-08	6.896E+01	1.289E+18
27	2.70E-08	6.424E+01	1.210E+18
28	2.80E-08	6.029E+01	1.132E+18
29	2.90E-08	5.719E+01	1.056E+18
30	3.00E-08	5.500E+01	9.846E+17
31	3.10E-08	5.355E+01	9.195E+17
32	3.20E-08	5.220E+01	8.601E+17
33	3.30E-08	5.093E+01	8.059E+17
34	3.40E-08	4.974E+01	7.564E+17
35	3.50E-08	4.863E+01	7.111E+17
36	3.60E-08	4.761E+01	6.698E+17
37	3.70E-08	4.668E+01	6.321E+17
38	3.80E-08	4.584E+01	5.979E+17
39	3.90E-08	4.510E+01	5.667E+17
40	4.00E-08	4.445E+01	5.384E+17
41	4.10E-08	4.390E+01	5.127E+17
42	4.20E-08	4.336E+01	4.894E+17
43	4.30E-08	4.281E+01	4.681E+17
44	4.40E-08	4.227E+01	4.487E+17
45	4.50E-08	4.176E+01	4.309E+17
46	4.60E-08	4.128E+01	4.146E+17
47	4.70E-08	4.085E+01	3.994E+17
48	4.80E-08	4.048E+01	3.852E+17
49	4.90E-08	4.020E+01	3.717E+17
50	5.00E-08	4.000E+01	3.587E+17

 

 

3. TD-Ge

 

 

Step #	Time (s)	Te	Ne
0	0.0000E+00	6.5000E+00	6.3500E+19
1	5.2150E-12	6.5008E+00	6.3374E+19
2	1.0594E-10	6.4371E+00	6.1460E+19
3	2.1674E-10	6.3692E+00	5.9467E+19
4	3.3861E-10	2.4395E+01	5.7396E+19
5	3.5993E-10	2.7260E+01	1.2616E+20
6	3.7093E-10	2.8521E+01	1.4297E+20
7	3.9237E-10	3.1415E+01	1.5421E+20
8	4.1281E-10	3.3754E+01	1.5344E+20
9	4.3415E-10	3.5964E+01	1.6586E+20
10	4.5211E-10	3.8761E+01	1.8080E+20
11	4.7441E-10	8.2758E+01	2.2823E+20
12	4.9214E-10	1.3384E+02	2.3100E+20
13	5.1014E-10	1.7085E+02	2.3522E+20
14	5.3137E-10	2.0025E+02	2.4230E+20
15	5.5393E-10	2.4130E+02	2.4981E+20
16	5.7102E-10	2.7437E+02	2.5455E+20
17	5.9100E-10	3.1680E+02	2.5793E+20
18	6.1008E-10	3.5932E+02	2.5855E+20
19	6.3010E-10	4.0613E+02	2.5655E+20
20	6.5534E-10	4.6734E+02	2.5237E+20
21	6.7186E-10	5.0507E+02	2.5003E+20
22	6.9363E-10	5.6072E+02	2.4833E+20
23	7.1017E-10	6.0676E+02	2.4784E+20
24	7.2306E-10	6.4286E+02	2.4730E+20
25	7.3048E-10	6.6373E+02	2.4664E+20
26	7.3865E-10	6.8685E+02	2.4546E+20
27	7.5738E-10	7.4003E+02	2.4056E+20
28	7.6811E-10	7.7007E+02	2.3649E+20
29	7.7991E-10	8.0246E+02	2.3137E+20
30	7.9116E-10	8.3152E+02	2.2640E+20
31	8.0316E-10	8.6198E+02	2.2157E+20
32	8.1635E-10	8.9456E+02	2.1741E+20
33	8.2791E-10	9.1915E+02	2.1501E+20
34	8.4773E-10	9.5533E+02	2.1355E+20
35	8.5754E-10	9.7175E+02	2.1373E+20
36	8.6577E-10	9.8360E+02	2.1408E+20
37	8.7344E-10	9.9369E+02	2.1440E+20
38	8.8016E-10	1.0017E+03	2.1454E+20
39	8.9118E-10	1.0132E+03	2.1430E+20
40	8.9932E-10	1.0206E+03	2.1363E+20
41	9.0820E-10	1.0274E+03	2.1236E+20
42	9.1836E-10	1.0339E+03	2.1028E+20
43	9.2794E-10	1.0387E+03	2.0786E+20
44	9.3900E-10	1.0431E+03	2.0480E+20
45	9.4798E-10	1.0455E+03	2.0235E+20
46	9.5969E-10	1.0471E+03	1.9948E+20
47	9.7018E-10	1.0472E+03	1.9739E+20
48	9.7876E-10	1.0465E+03	1.9610E+20
49	9.8920E-10	1.0453E+03	1.9500E+20
50	9.9795E-10	1.0433E+03	1.9442E+20
51	1.0088E-09	1.0399E+03	1.9396E+20
52	1.0186E-09	1.0357E+03	1.9363E+20
53	1.0297E-09	1.0302E+03	1.9313E+20
54	1.0382E-09	1.0258E+03	1.9253E+20
55	1.0494E-09	1.0197E+03	1.9132E+20
56	1.0589E-09	1.0140E+03	1.8990E+20
57	1.0705E-09	1.0063E+03	1.8774E+20
58	1.0844E-09	9.9604E+02	1.8467E+20
59	1.0920E-09	9.8994E+02	1.8291E+20
60	1.1174E-09	9.6775E+02	1.7727E+20
61	1.1363E-09	9.4980E+02	1.7386E+20
62	1.1564E-09	9.2887E+02	1.7098E+20
63	1.1768E-09	9.0607E+02	1.6847E+20
64	1.1971E-09	8.8197E+02	1.6591E+20
65	1.2207E-09	8.5382E+02	1.6242E+20
66	1.2343E-09	8.3720E+02	1.6013E+20
67	1.2657E-09	7.9913E+02	1.5443E+20
68	1.2837E-09	7.7694E+02	1.5123E+20
69	1.3036E-09	7.5275E+02	1.4803E+20
70	1.3254E-09	7.2667E+02	1.4496E+20
71	1.3493E-09	6.9878E+02	1.4203E+20
72	1.3757E-09	6.6969E+02	1.3906E+20
73	1.4044E-09	6.3949E+02	1.3578E+20
74	1.4356E-09	6.0889E+02	1.3194E+20
75	1.4696E-09	5.7821E+02	1.2745E+20
76	1.5065E-09	5.4807E+02	1.2250E+20
77	1.5465E-09	5.1851E+02	1.1754E+20
78	1.5887E-09	4.9210E+02	1.1314E+20
79	1.6781E-09	4.4295E+02	1.0952E+20
80	1.7710E-09	4.0369E+02	1.1632E+20
81	1.8742E-09	3.6957E+02	1.2569E+20
82	1.9749E-09	3.4240E+02	1.2526E+20