Abstract: A modified atomic calculations program enhanced with a Hydrogenic model is introduced and tested based on the recently published Hamasha-Tahat Atomic Code (HTAC). Modified HTAC program generates atomic data for plasma and astrophysics applications without going through the complications of atomic theories or the programming languages. It includes extra features that enable producing accurate atomic data for radiative transition rates, radiative recombination rates, photoionization cross sections, radiative recombination cross sections and the two photon decay rates. The program enables access to many parameters that determine the calculation accuracy and allows inputting the parameters through a user friendly interface.
INTRODUCTION
Theoretical atomic data is important for plasma modeling under variety of physical conditions as well as many other atomic physics studies and applications. Some of the widely used atomic physics computer programs are based on non-relativistic approximations and some are fully relativistic codes while other codes consider the first and the second orders of relativistic corrections. CIV3 code (Gupta and Msezane, 2009), SUPER STRUCTURE code (Zanna et al., 2005) and COWAN code (Cowan, 1981) are based on non-relativistic approximations. The Relativistic Many Body Perturbation Theory code (RMBPT) accounts for the first-and the second-order relativistic energy corrections (Safronova et al., 2006). Other programs such as HULLAC atomic code (Busquet et al., 2006). SZ (Zhang et al., 1989), ATOM package (Amusia and Chernysheva, 1997) and the Flexible Atomic Code (FAC) (Gu, 2008) are fully relativistic codes based on solving Dirac equations. FAC is a public-freely powerful atomic code that could be downloaded from the web: (http://sprg.ssl.berkeley.edu/ ~mfgu/fac/fac.tar.gz). It has modified the numerical calculation methods of several fully relativistic codes in order to encompass their strengths. However, using FAC has been limited to those who are familiar with Python programming language which is not commonly used by most of atomic physics researchers.
The available atomic codes are often complicated, difficult to use and most of them works under UNIX or other operation systems that require commands typing. There is a real need for a user-friendly atomic code that could be used through Windows-based operating systems (Hamasha et al., 2011; Hamasha and Tahat, 2010). From this point Hamasha-Tahat Atomic Code (HTAC) is developed as a utility program to be used in various atomic calculations without going through complications of the atomic theories or the programming languages. It will be available for users in public domain at no cost upon their requests.
HTAC performs calculations for atomic structure and transition rates for allowed and forbidden transitions. It is based on three independent calculation techniques; the fully relativistic configuration interaction method (CI) that follows FAC methodology (Gu, 2008), the multi-reference many-body perturbation theory (MR-MBPT) that follows the method outlined by Vilkas et al. (2007) and the advanced R-matrix calculation method. The current modified HTAC edition includes a hydrogenic ions model that contains improved features for calculating atomic data for astrophysics and plasma physics applications. In addition to its advanced atomic calculation methods, HTAC has advanced plotting capabilities. (Descouvemont and Baye, 2010).
HTAC produces theoretical data for energy levels, radiative transition rates, wave functions and basis and mixing coefficients. Produced atomic data is stored in output files that have the same format regardless of calculation method which could be readily recognized from the extensions of the files (Hamasha et al., 2011).
The hydrogenic-ions model assumes a one electron in the system in n1 complex with an effective charge state. It contains features that are particularly important in astrophysics and plasma research. It performs calculations for radiative transition rates (TRRrate), radiative recombination rates (RRRate), photoionization cross sections (PICross), radiative recombination cross sections (RRCross) of H-like ions, in addition to the two-photon decay rate of H-like and He-like transitions (2s1/2-1s1/2 and 1s2s S0-1s2 S0).
This study described how HTAC is used in order to perform atomic structure calculations and the hydrogenic-ions model. The findings of some case studies are also presented and validated against published experimental and theoretical data when it is available.
PERFORMING ATOMIC STRUCTURE CALCULATIONS
The structure of HTAC_atomic structure program with its three calculation methods along with the extensions of their output files is shown in Fig. 1. The default method is fully relativistic Configuration Interaction (CI) method, The MBPT and R-matrix check boxes are embedded in HTAC main tab. By pointing the chosen method checkbox, a new main atomic structure panel pops up. The MBPT method has the options to generate output files for single excitation (S) and double excitation (D) or both (SD-excitation). The new panel enables accessing several parameters that determine calculation accuracy. The user may specify an option from the print table menu and press the calculation button (Fig. 2). Then the atomic structure program generates required output files and save them under user-specified names with the method of calculation extension (e.g., *.RMATRIX for R-matrix calculations).
Output files are four files: energy levels table, transition rates table, basis with mixing coefficients table and wave functions details table.
PLOTTING CAPABILITIES ARE ADDED TO HTAC
After performing atomic calculations, HTAC enables the user to produce colored plots for calculated data. Several produced data files could be added to the same plot. Plots are very important because they lend atomic molder an easy and straight forward tool to readily check out and assess the accuracy of produced atomic data. HTAC used, Biggles, to create the graphs, where Biggles is high-level scientific plotting module for Python for creating publication-quality 2D scientific plots. It supports multiple output formats (postscript, x11, png, svg, gif), understands simple TeX and sports a high-level, elegant interface. It is intended for technical users with sophisticated plotting needs.
| Fig. 1: | The structure of HTAC atomic structure program with the three calculation methods and the extensions of their output files |
| Fig. 2: | The atomic structure calculations methods main panel |
| Fig. 3: | Plot tab menu and two plots for the Delta E versus the weighted oscillator strength calculated using the fully relativistic CI (line a) and MBPT (line b), are depicted on a one chart |
HTAC provides a series of graphs to illustrate the atomic data that has been tabulated in the output files. User must run the program atomic structure to produce the transition rate table for a specific atomic number that stored in a specific file. HTAC extracts the required input data automatically to be plotted by clicking the item plot from HTAC standard bar (Fig. 3). Plot menu provides several options, clicking any of them means specifying the input data to be extracted from the output files. The graphs are stored in a permanent catalog by default (files folder). Three options are available for the plot type; line, doted and dashed.
Figure 3 shows plot tab window and a plot for the energy difference of transitions (delta E) versus the weighted oscillator strength of the transitions of Fe19+ ion. In this plot two output data files are used at the same time, line a for the full relativistic configuration interaction CI output data and line b for the multi-reference many body perturbation MBPT output data. The shown plot represents the spectrum of the selected ion.
THE HYDROGENIC MODEL
In order to run the Hydrogenic model, the user clicks on “Atomic calculations” button from HTAC standard tools bar, then a window pops up showing “Atomic structure” bar and an atomic structure main panel appears (Fig. 4). Only fields that appear in the Hydrogenic atoms tab must be filled; all other tabs and parameters have to be inactive. Ion charge has to be specified in its designated field. The output menu provides five options: TRRate, RRRate, PICross, RRCross and Two photon (Fig. 5).
| Fig. 4: | The hydrogenic model as appear in the Main panel of HTAC_atomic structure program |
| Fig. 5: | The Hydrogenic-atoms tab in HTAC_atomic structure program |
| Fig. 6: | Radiative transition rate (TRRate) output file for H-like carbon ion |
Each option has to be filled with different parameters in order to generate output data.
The calculated atomic data contains the following:
Radiative Transition rates (TRRate): The output file has the extension (htr)., the charge (Z), initial state and final state principle and angular quantum numbers should be specified in their fields before pressing the button (calculate). Figure 6 shows a sample output file for H-like carbon ion. The active output window provides three services: save, save as and print.
Table 1 presents atomic data generated through the option TRRate (Radiative transition rates) for ions with charge Z-1 in the range 1≤Z-1≤100, for transitions from state (n0 = 3, l0 =2) to state (n1=2, l1 =1).
| Table 1: | Output data for H-like ions with charge Z-1 in the range 1≤Z-1≤100 produced using the TRRate option in HTAC |
Output data includes the electric dipole (E1) radiative transition rates, weighted oscillator strength (gf) and the radial integral value (R-integral).
It could be readily learned from Table 1 that as ion charge (Z-1) increases, the radiative transition rate increases but the electric dipole radial integral decreases.
Table 2 shows calculated atomic data for H-like Promethium (Pm, Z = 61) at different transitions levels. It is clear that when the state quantum numbers (n, l) of initial and final states increases the weighted oscillator strength increases but the radiative transition rate decreases.
Radiative Recombination Rates (RRRate): This option allows calculating the radiative recombination rates for H-like ions with ion charge (Z-1) at any temperature T. HTAC calculates rate coefficients in several ways based on the fitting formulas used. The ion charge, the temperature (°K) and the principle quantum number n (QNumber runs from 0 to 999) of the recombined electron should be specified first. An output file window that is called (Hydrogenic) with the extension (hrrtr) appears after clicking the calculate button. Two rates are calculated: the radiative recombination rate into all states with principle quantum number equals to inserted n and the radiative recombination rate into all states with principle quantum numbers greater than n. Table 3 presents produced data for a hypothetical ion with a charge equals to 107 at different temperatures and for different states. It is a demo example of output collection of radiative recombination rates for quantum numbers n = 1 and 30. For same quantum number, both rates are decreasing with temperature increase. For same temperature the radiative recombination rates decreasing with quantum number increase. But in general the radiative recombination rates into all states with principle quantum number equals to inserted n is less than the radiative recombination rate into all states with principle quantum numbers greater than n.
| Table 2: | Radiative transition rates for H-like Pm at different transition levels calculated through the hydrogenic approximation for E1 multipole integrals in HTAC |
| Table 3: | Calculated data for a hypothetical ion having a charge equals to 107 at different temperatures for different quantum numbers. aRRRrate is the rate at n=specified Qnumber, bRRRrate is the rate at n> specified Qnumber |
| a: Radiative Recombination Rrate at n = specified Qnumber, b: Radiative Recombination Rrate at n> specified Qnumber | |
THE FITTING OF THE RADIATIVE RECOMBINATION COEFFICIENTS
Several subroutines were written in the c++ language based on fitting formulas written by Verner et al. (1996) to fit the recombination rate coefficients of H-like, He-like, Li-like and Na-like ions, for all elements from H (Z = 1) to Zn (Z = 30). The fits are valid over a wide temperature range. There are several fittings:
| • | Radiative recombination rates for H-like, He-like, Li-like and Na-like ions over a wide range of temperature (3°K to 109°K) (http://www.pa.uky.edu/~verner/rec.html) |
The fitting uses the following formula which ensures correct asymptotic behavior for the rate coefficients at low and at high temperatures:
where, a, b, T0, T1 are the fitting parameters:
| • | Power-law fits (Aldrovandi and Pequignot, 1973), fttp://gradj.pa.uky.edu//dima//rec//pl.txt): |
We wrote a C++ subroutine to calculate the radiative recombination rates through this fitting formula
| • | Seaton asymptotic expansion method which is recommended by Arnaud and Raymond (1992) for the hydrogenic species. The formula is: |
where, λ = 157890Z2/T (K), however, this formula is not valid for high temperature (T>106Z2).
Table 4 presents a comparison between calculated data for radiative recombination rate coefficient by the Hydrogenic model of HTAC with a previous work by Gould (1978) for the selected ions in the table. The new data by HTAC are less than that of Gould’s but both are close.
| • | Partial Radiative recombination rates for H-like, He-like, Li-like and Na-like ions over a broad range of temperature |
Separating the partial recombination rates of the ground state from the recombination rates of excited states may be required for astrophysical application. In order to use all above fitting formula in the calculations, the proper subroutine is called easily by clicking the button “data” in HTAC main panel. A window as the one shown in Fig. 7 appears.
User only needs to select the ion from the drop down menu (selected ion), then insert the temperature in Kelvin, HTAC will produce the radiative rate coefficient by using several formulas at the same time. The user may copy desired results from their box and paste them into a table similar to Table 5.
PHOTOIONIZATION CROSS SECTION FUNCTION (PICROSS)
This function is accessed by selecting the option “PICross” from the output menu, It allows calculating of the photoionization cross section for H-like ion with nuclear charge Z at photon energy E for the non-relativistic subshell nl. The result is in unit of 10-20 cm2. The state n0 and l0 counters have to be specified before pressing the “calculate” button. If the user forgets to specify the photon energy, the program will generate a warning message asking for it.
| Table 4: | Radiative recombination coefficients at T = 104 °K (in cm3 s-1) |
| Fig. 7: | HTAC main panel for the radiative recombination rates subroutines |
| Table 5: | H-like Cl (Z = 17) radiative recombination rates using the fitting formulas from parts a, b, c and d (the partial RRRate) |
| aRadiative recombination rates in part a with correct asymptotic behavior. bRadiative recombination rates by power-law fits. cRadiative recombination rates by Seaton asymptotic expansion. dPartial Radiative recombination rates | |
| Table 6: | Photoionization cross section for Pt (Nuclear charge = 78) for different quantum numbers at different photon energies |
Produced data file appears in a separate window and is saved under the name: Hydrogenic.hpics. As an example, the photoionization cross section atomic data calculated for the H-like Platinum (Z = 78) at different photon energies are presented in Table 6.
THE RADIATIVE RECOMBINATION CROSS SECTIONS (RRCROSS)
This option allows calculating the radiative recombination cross section for bare ion of a nuclear charge Z at an electron of energy E, into the non-relativistic sub shell nl in 10-20 cm2. The output file is saved under the extension hrrcs. Table 7 presents a set of output data obtained through the RRCross option for several H-like ions (1≤Z≤100), into the non-relativistic sub shell (n2, l1) at electron energies 1, 2 and 3 keV, respectively. It is evident that the radiative recombination cross section decreases with the electron energy increase for the same ion. For example the radiative recombination cross section *10-20 cm2 for Z = 1 at different energies E = 1000, 2000, 4000 eV, respectively are: 2.81 *10-9, 2.28*10-10 and 2.63*10-11. With same quantum numbers (n, l) and same electron energy the radiative recombination cross section increases rapidly with the charge of the ions nuclei increase. For example: when the electron energy equal to 1000 eV, the radiative recombination cross section in terms of 10-20 cm2 changes from 2.8*10-9 to 0.532, as Z varies from 1 to 78.
| Table 7: | Radiative recombination cross section atomic data for some H-like ions at different electron energies and different nuclei charges |
| Table 8: | Output data for two-photon decay rate of H-like and He-like for different ions |
TWO PHOTON DECAY
This option allows calculating the rate of the two-photon decay for transitions 2s1/2-1s1/2 and 1s2s S0-1s2 S0 in a H-like or a He-like ion with a nucleus charge Z. Output file is saved under a name followed by the extinction “2photon”. Examples of produced two-photon decay rate of H-like and He-like data for different ions are presented in Table 8. It can be seen that the two photon decay rate for both H-like ions and He like ions increases with the charge of the ions nuclei increase. The two photon decay rate of He-like ions is smaller than that of H-like ions within Z<10 but larger than that of H-like ions within Z =10.
CONCLUSION
Recently a complete program (HTAC) for calculating atomic structure and spectra data have been developed and tested. HTAC performs calculations based on three advanced methods; fully relativistic configuration interaction method, multi-reference many body perturbation theory and R-Matrix method. Produced data include energy levels, radiative transition rates, details of wave functions and basis with mixing coefficients for all allowed and forbidden transitions. HTAC also is improved to include new features like plotting capabilities and a Hydrogenic model that contains improved program for calculating atomic data for radiative transition rates, radiative recombination rates, photoionization cross sections, radiative recombination cross sections and the two photon decay rates which are very important for plasma and astrophysics applications.
HTAC provides a user friendly interface works under windows in the dialog mode. It provides a Graphical User Interface (GUI) that clarifies the analysis steps and handles the file manipulations required by the used model. Furthermore it provides viewing and plotting capabilities of the produced atomic data. It has been designed to be easy to run and to produce tabulated atomic data files that can be interfaced with other programs easily. It works for all elements in the periodic table and their ions.