Recombination Lines of C II in the Spectra of

arXiv:1011.4670v1 [physics.atom-ph] 21 Nov 2010

Recombination Lines of C II in the Spectra of Planetary Nebulae

Taha Sochi1

Technical Report

2008

1 University

College London, Department of Physics & Astronomy, Gower Street, London, WC1E 6BT. Email: [email protected].

Abstract The current report presents the work carried out by the author to investigate the recombination lines of C II in the spectra of planetary nebulae. Two C III targets were prepared and used to generate theoretical data required in the investigation of recombination lines that arise from collisions between electrons and ions in thin plasma found in planetary nebulae and other astrophysical objects. One of these targets contains 9 atomic terms while the other contains 26 terms. For each one of these targets, theoretical data concerning bound and autoionizing states were generated in the intermediate coupling approximation by R-matrix and Autostructure codes and compared to experimental data. The comparison revealed very good agreement. These theoretical data were then used to generate emissivity data and compare it to the carbon recombination lines found in the observational line list of Zhang et al [2005] on the planetary nebula NGC 7027. The main tool used in this analysis is the ‘Emissivity’ code which is a program developed by the author [2010] in the course of this investigation to calculate the emissivity of transition lines from resonances and subsequent cascade decay. The results of the analysis from both targets using a least-squares optimization technique indicate a temperature of 10200-10500 K of the line-emitting region.

1

Acknowledgement I would like to acknowledge the substantial contributions of Professor Peter Storey to the work presented in this report, especially on the theoretical side.

2

Contents Abstract . . . . . . . . Acknowledgements and Contents . . . . . . . . List of Figures . . . . . List of Tables . . . . . Nomenclature . . . . .

. . . . . . . Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

1 Introduction 1.1 Recombination Processes . . . . . . . . . . . 1.1.1 Radiative Recombination . . . . . . . 1.1.2 Dielectronic Recombination . . . . . 1.2 Literature Review . . . . . . . . . . . . . . . 1.2.1 Theory of Recombination Lines . . . 1.2.2 Recombination Lines of Carbon . . . 1.3 Recombination vs. Forbidden Lines Analysis

. . . . . .

. . . . . . .

. . . . . .

. . . . . . .

2 Physics of Scattering and Recombination 2.0.1 Close Coupling Approximation . . . . . 2.0.2 R-matrix Method . . . . . . . . . . . . . 2.0.2.1 Inner and Outer Regions . . . . 2.0.2.2 Matching of Solutions . . . . . 2.0.2.3 Bound States . . . . . . . . . . 2.0.2.4 Resonances . . . . . . . . . . . 2.0.2.5 Oscillator Strengths . . . . . . 2.0.2.6 Photoionization Cross Sections 2.0.2.7 Relativistic Effects . . . . . . . 2.0.3 RMATRX-I Code . . . . . . . . . . . . .

3

. . . . . .

. . . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . .

. . . . . . . . . .

. . . . . .

1 2 3 6 7 9

. . . . . . .

15 17 18 19 21 21 24 28

. . . . . . . . . .

32 33 36 37 38 38 39 39 40 40 41

3 Investigating Recombination Lines of C II 3.1 Theoretical Background . . . . . . . . . . . . . . . . . 3.2 Autoionizing and Bound States Calculations . . . . . . 3.2.1 Methods for Investigating Resonances . . . . . . 3.2.1.1 QB Method . . . . . . . . . . . . . . . 3.2.1.2 Time-Delay Method . . . . . . . . . . 3.2.1.3 K-Matrix Method . . . . . . . . . . . 3.2.2 Preparing Target . . . . . . . . . . . . . . . . . 3.2.3 Developing and Implementing K-matrix Method 3.2.4 Calculations . . . . . . . . . . . . . . . . . . . . 3.3 f -Values Calculations . . . . . . . . . . . . . . . . . . . 3.3.1 f -Values for Free-Free Transitions . . . . . . . . 3.3.2 f -Values for Free-Bound Transitions . . . . . . 3.3.3 f -Values for Bound-Bound Transitions . . . . . 3.4 Emissivity Calculations . . . . . . . . . . . . . . . . . . 3.4.1 Practical Aspects . . . . . . . . . . . . . . . . . 3.4.2 Emissivity Analysis . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

45 46 52 52 53 54 56 58 58 62 66 66 67 72 73 73 76

4 Conclusions and Future Work 4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . .

83 83 84

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

Bibliography

85

A Using Lifetime Matrix to Investigate Single-Channel Resonances 92 B Emissivity Program Documentation B.1 ‘Input’ Data File . . . . . . . . . . B.2 ‘ResEmis’ Input Data File . . . . . B.3 ‘ELEVEmis’ Input Data File . . . . B.4 ‘Transitions’ Output Data File . . . B.5 TESTS . . . . . . . . . . . . . . . . C Input Data for R-matrix C.1 R-matrix Input Data . C.1.1 STG1 . . . . . C.1.2 STG2 . . . . . C.1.3 STGJK . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

and Autostructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

97 98 105 106 107 108

. . . .

109 109 109 110 112

C.1.4 STG3 . . . . . . . . . . C.1.5 STGF . . . . . . . . . . C.1.6 STGB . . . . . . . . . . C.1.7 STGBF . . . . . . . . . C.1.8 STGBB . . . . . . . . . C.1.9 STGQB . . . . . . . . . C.2 Autostructure Input Data . . . C.2.1 Polarizability . . . . . . C.2.2 Transition Probabilities . C.2.3 f -values . . . . . . . . . D Tables Bound States . . . . . . . . Resonances from K-matrix Resonances from QB . . . 9-term FB f -values . . . . 26-term FB f -values . . . .

. . . . .

. . . . .

5

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

114 114 114 114 114 115 115 115 115 116

. . . . .

117 119 124 126 128 136

List of Figures 1.1

Two prominent planetary nebulae, NGC 7027 and Cat’s eye . . . .

16

2.1

Flow chart of the RMATRX-I suite . . . . . . . . . . . . . . . . . .

43

3.1 3.2

Examples of single-channel K-matrix poles in the background . . . Schematic diagram of the low-lying autoionizing quartet and doublet states and their interaction in IC scheme . . . . . . . . . . . . . . . Examples of photoionization cross sections in Mb (y-axis) of the indicated bound states over the displayed range of photon energy in Ryd (x-axis) on log-linear graphs for the 9-term target. . . . . . . Examples of photoionization cross sections in Mb (y-axis) of the indicated bound states with respect to the designated autoionizing states versus photon energy in Ryd (x-axis) on linear-linear graphs for the 26-term target. . . . . . . . . . . . . . . . . . . . . . . . . . Graph of χ2 of normalized emissivity (relative to CII 426.7 nm line) as a function of temperature for the emissivity model compared to the observational data of Zhang et al [2005] for the 9-term target. The confidence interval limits for 68% confidence level and the temperature at minimum χ2 are also displayed. . . . . . . . . . . . . . Graph of χ2 of normalized emissivity (relative to CII 426.7 nm line) as a function of temperature for the emissivity model compared to the observational data of Zhang et al [2005] for the 26-term target. The confidence interval upper limit for 68% confidence level and the temperature at minimum χ2 are also displayed. . . . . . . . . . . .

57

3.3

3.4

3.5

3.6

6

63

68

69

81

81

List of Tables 3.1 3.2

The seven orbitals used to construct the C III targets and their STO parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The configurations, terms and levels used to build the C III targets

D.1 The available experimental data from NIST for the bound states of C II below the C III 1 Se0 threshold alongside the theoretical results from R-matrix for the 9-term and the 26-term targets with 0.1 ≤ nf ≤ 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 The available experimental data from NIST for the resonance states of C II above the C III 1 Se0 threshold alongside the theoretical results as obtained by K-matrix method for the 9-term and the 26-term targets with n < 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3 The available experimental data from NIST for the resonance states of C II above the C III 1 Se0 threshold alongside the theoretical results as obtained by QB method for the 9-term and the 26-term targets with n < 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4 FB f -values for bound symmetry J π = 1/2e using 9-term target. . D.5 FB f -values for bound symmetry J π = 3/2e using 9-term target. . D.6 FB f -values for bound symmetry J π = 5/2e using 9-term target. . D.7 FB f -values for bound symmetry J π = 7/2e using 9-term target. . D.8 FB f -values for bound symmetry J π = 9/2e using 9-term target. . D.9 FB f -values for bound symmetry J π = 1/2o using 9-term target. . D.10 FB f -values for bound symmetry J π = 3/2o using 9-term target. . D.11 FB f -values for bound symmetry J π = 5/2o using 9-term target. . D.12 FB f -values for bound symmetry J π = 7/2o using 9-term target. . D.13 FB f -values for bound symmetry J π = 9/2o using 9-term target. . D.14 FB f -values for bound symmetry J π = 11/2o using 9-term target. . D.15 FB f -values for bound symmetry J π = 1/2e using 26-term target. . D.16 FB f -values for bound symmetry J π = 3/2e using 26-term target. . 7

59 60

119

124

126 128 129 130 131 131 132 133 134 135 135 135 136 137

D.17 FB f -values for bound symmetry J π = 5/2e using 26-term target. . D.18 FB f -values for bound symmetry J π = 7/2e using 26-term target. . D.19 FB f -values for bound symmetry J π = 9/2e using 26-term target. . D.20 FB f -values for bound symmetry J π = 1/2o using 26-term target. . D.21 FB f -values for bound symmetry J π = 3/2o using 26-term target. . D.22 FB f -values for bound symmetry J π = 5/2o using 26-term target. . D.23 FB f -values for bound symmetry J π = 7/2o using 26-term target. . D.24 FB f -values for bound symmetry J π = 9/2o using 26-term target. . D.25 FB f -values for bound symmetry J π = 11/2o using 26-term target. D.26 Zhang et al [2005] observational data for the C II recombination lines that we used in this investigation. . . . . . . . . . . . . . . . . . . .

8

138 139 139 140 141 142 143 143 143 144

Nomenclature and Notation α Γa Γr Γrul δ δ 0 δ ∆χ2 ∆E ∆r ∆E i o ε εul εN o N εt ζ η θ ϑ λ Λ µ ν ξi π %c %f

fine structure constant (= e 2 /(~c 4πo ) ' 7.2973525376 × 10−3 ) autoionization transition probability (s−1 ) radiative transition probability (s−1 ) radiative transition probability from upper state u to lower state l (s−1 ) eigenphase sum eigenphase background eigenphase gradient of K-matrix with respect to energy (J−1 ) change in goodness-of-fit index full width at half maximum for energy distribution (J) width of resonance (J) energy difference (J) energy of colliding electron (J) permittivity of vacuum (' 8.854187817 × 10−12 F.m−1 ) emissivity of transition line (J.s−1 .m−3 ) emissivity of transition line from state u to state l (J.s−1 .m−3 ) normalized observational emissivity normalized theoretical emissivity effective charge (C) number of degrees of freedom function of colliding electron Coulomb function wavelength (m) eigenvalue reduced mass (kg) frequency (s−1 ) exponent in Slater Type Orbital formulation parity recombination coefficient for capture process (m3 .s−1 ) effective recombination coefficient (m3 .s−1 )

9

σ τ τo φ Φ χ2 χ ψ Ψ

photoionization cross section (m2 ) lifetime of state (s) atomic time unit (= ~/Eh ' 2.418884326505 × 10−17 s) vector coupled function function of bound-state type in CC expansion goodness-of-fit index spin and angle functions of colliding electron function of N -electron target wavefunction of Schrödinger equation

a a0 A A ˚ A b bu B c c Ci Cp D e e− E Eh Ei E0 Ep Er f ful

scaling parameter in χ2 confidence interval procedure Bohr radius (' 5.2917720859 × 10−11 m) anti-symmetrization operator inner region radius in R-matrix formulation (m) angstrom departure coefficient departure coefficient of upper state matrix in QB method speed of light in vacuum (299792458 m.s−1 ) coefficient in CC expansion coefficient in Slater Type Orbital formulation contribution to polarizability dipole operator elementary charge (' 1.602176487 × 10−19 C) electron energy (J) Hartree energy (' 4.35974394 × 10−18 J) energy point in K-matrix method (J) position of K-matrix pole on energy mesh (J) photon energy (J) energy position of resonance (J) oscillator strength oscillator strength of transition between upper state u and lower state l

10

F F g g h ~ H Hbp Hnr Hrc i I J k K K Ki Ko L m me M M Mmax MJ n nf N Ne Ni NS O P

radial function vector of radial functions parameter in K-matrix method (J) statistical weight in coupling schemes (= 2J + 1 for IC) Planck’s constant (' 6.6260693 × 10−34 J.s) reduced Planck’s constant (= h/2π ' 1.0545717 × 10−34 J.s) Hamiltonian of the whole system Breit-Pauli Hamiltonian non-relativistic Hamiltonian relativistic correction Hamiltonian √ imaginary unit ( −1) identity matrix total angular momentum quantum number Boltzmann’s constant (' 1.3806505 × 10−23 J.K−1 ) reactance matrix in R-matrix theory single-element K-matrix scalar value of single-element K-matrix at energy Ei background contribution to K-matrix total orbital angular momentum quantum number mass (kg) mass of electron (' 9.10938215 × 10−31 kg) lifetime matrix single-element M-matrix maximum value of single-element M-matrix solar mass (' 1.99 × 1030 kg) principal quantum number effective quantum number number density (m−3 ) number density of electrons (m−3 ) number density of ions (m−3 ) Saha population (m−3 ) number of observations number of fitting parameters

11

Pnl (r) Pi Q R r r S S S S T Te Vii0 w Xn+ X(n−1)+ ∗ X(n−1)+

radial function in Slater Type Orbital formulation power of r in Slater Type Orbital formulation matrix in QB method resonance matrix in R-matrix theory radius (m) radius vector scattering matrix in R-matrix theory single-element S-matrix total spin angular momentum quantum number line strength (J−1 ) temperature (K) electron temperature (K) multipole potentials (J.C−1 ) statistical weight in least squares procedure ion of effective positive charge n bound state of recombined ion with effective positive charge n − 1 doubly-excited autoionizing state with effective positive charge n − 1

yc yo z Z ∇ ∇2

calculated value observed value residual charge atomic number del operator Laplacian operator

Abbreviations [.]∗ au BB C CC CEL CI DR E1

complex conjugate of matrix atomic unit Bound-Bound Coulomb Close Coupling Collisionally Excited Lines Configuration Interaction Dielectronic Recombination Electric dipole

12

F FB FF FWHM IC In. ID IUE J K kg kpc LS LTE (.)l (.)L m Mb (.)max (.)min NASA NIST ORL PN RR Ryd s STO TE UV (.)u (.)ul vs.

Faraday Free-Bound Free-Free Full Width at Half Maximum Intermediate Coupling Index Integro-Differential International Ultraviolet Explorer Joule Kelvin kilogram kiloparsec (' 3.1 × 1019 meters) Least Squares Local Thermodynamic Equilibrium lower length form meter Mega barn (= 10−22 m2 ) maximum minimum National Aeronautics and Space Administration National Institute of Standards and Technology Optical Recombination Lines Planetary Nebula Radiative Recombination Rydberg second Slater Type Orbital Thermodynamic Equilibrium Ultra Violet upper upper to lower versus

13

(.)V ∀

velocity form for all

Note: Units, when relevant, are given in the SI system. The physical constants are obtained from the National Institute of Standards and Technology (NIST) website [NIST]. Vectors are marked with boldface. Some symbols may rely on the context for unambiguous identification. Most subscripts (e.g. l and u) are dummy variables, and hence they are subject in their interpretation to the context.

14

Chapter 1 Introduction A Planetary Nebula (PN) is an expanding and usually symmetrical cloud of gas that has been ejected from a dying star, most are believed to come from red giant stars, shed as a result of instabilities late in their evolution. The term ‘planetary nebula’ refers to the resemblance of theses objects to planetary disks, rather than point-like stars, under low magnification. At first, astronomers thought they were planets like Uranus. A planetary nebula is believed to form as part of the normal evolution of single stars with masses between 0.8-8 solar mass (MJ ). A PN is formed when the star can no longer support itself by fusion reactions in its core. Consequently the gravity forces the inner parts to condense and heat up while the high temperature central region drives the outer parts away in a stellar wind. The PN eventually fades and disperses into the interstellar medium while the degenerate core becomes a white dwarf. The envelope is ejected over a period of about 104 years, while the star is near the tip of the asymptotic giant branch. Ejection of a succession of planetary nebula shells during the lifetime of the star reduces the mass of the star until the core is only about 0.6 MJ . The PNe expand at a speed of a few tens of km.s−1 . The gas cloud is ionized by the highly energetic radiation from the compact hot stellar core (progenitor) that remains in the center of the 15

CHAPTER 1. INTRODUCTION

16

cloud. The cloud is detected by virtue of its emitted radiation. PNe often show spectroscopic evidence of helium, carbon, nitrogen and other elements produced by nuclear reactions in the parent stars. The range of temperature in PNe is 103 2 × 104 K with a typical value of 104 K, while the range of electron density is 109 -1011 m−3 with a typical value of 1010 m−3 . Figure (1.1) shows two prominent examples of planetary nebulae.

Figure 1.1: Two prominent planetary nebulae, NGC 7027 (left) and Cat’s eye (right) [from NASA]. It is estimated that about one planetary nebula is born in the Milky Way per year, leading to an inventory of about 104 nebulae, of which about 1000-1500 have been cataloged. The complex shapes of PNe are due to a combination of effects from companions, rotation, and collisions of the primary wind with material ejected earlier and later. The PNe are essential for astronomical and astrophysical studies because they are relatively abundant in our galaxy and hence provide imperative case studies. In other galaxies, they may be used to acquire useful information about the chemical composition and the thermodynamic conditions. Unlike H II regions, which cannot be used to probe elliptical galaxies, planetary nebula can be used to probe the dynamics and metallicity of any type of galaxy. Via accurate radial velocity measurements, extragalactic PNe are being used as dynamical

1.1 Recombination Processes

17

mass probes of galaxies. Nowadays, PNe are routinely used to study the internal dynamics and mass distributions of both early and late type galaxies. For cosmic processes, the PNe are very important objects because they play a crucial role in the chemical evolution of the galaxies, returning materials enriched in heavy elements to the interstellar medium (Illingworth and Clark [2000], Matzner [2001], Barlow [2006]). In this report we investigate the recombination lines of C II in the spectra of the celebrated nebula NGC 7027. NGC 7027 is a compact, bright, young, high excitation planetary nebula with one of the hottest central stars known for a PN. Among the observed planetary nebulae, it has one of the richest lines emission which spans most parts of the electromagnetic spectrum and has been a challenge for observers as well as theoreticians. NGC 7027 is located about 0.9 kpc from the sun in the direction of the constellation Cygnus in the Milky Way. It is generally believed that a significant amount of dust coexists with the ionized and neutral gas in NGC 7027. In spite of obscuration by interstellar and local dust, the nebula is sufficiently bright and compact to be easily observed. It has been a reference object and the subject of several spectral surveys, and probably it is the best studied PN. There are several indications that the progenitor of NGC 7027 is a carbon-rich star (Péquignot and Baluteau [1988], Zhang et al [2005]).

1.1

Recombination Processes

The low density plasma of PNe heated by energetic photons from their progenitors provides an ideal environment for many physical processes. These include radiative recombination, dielectronic recombination, three body recombination and charge transfer. Recombination is believed to play a vital role in the physical processes that occur in PNe, and hence a deep understanding of its role is required for mod-

1.1.1 Radiative Recombination

18

eling nebular spectra. The principal electron-ion recombination processes which are responsible for the emission of recombination lines are Radiative Recombination (RR) and Dielectronic Recombination (DR). Recombination of an incoming electron to a target ion may occur through a non-resonant background continuum known as radiative recombination, or through a recombination process involving doubly-excited states (resonances) known as dielectronic recombination. The latter can lead either to autoionization, a radiationless transition to a lower state of the ion with the ejection of a free electron, or to stabilization by radiative decay of the core to a lower bound state, possibly the ground state, with a bound electron. The RR and DR processes are closely linked and the difference between them may therefore be described as artificial. Quantum mechanically, dielectronic and radiative recombinations are indistinguishable processes. Recombination is mainly due to dielectronic processes which are especially important at high temperatures when the autoionizing levels are energetically accessible to the free electrons (Péquignot et al [1991], Nahar et al [2000], Badnell [2006]). In the following sections we give a brief account of these two important recombination processes.

1.1.1

Radiative Recombination

Radiative recombination (RR) is the time inverse of direct photoionization where a single electron is captured by an ion to a true bound state with the subsequent emission of photon(s). Symbolically, the RR process is given by

Xn+ + e−

→

X(n−1)+ + hν

(1.1)

where Xn+ is an ion of effective positive charge n, e− is an electron, X(n−1)+ is a bound state of the recombined ion with effective positive charge (n−1), h is the Planck’s constant and ν is the frequency of radiation (Aldrovandi and Péquignot

1.1.2 Dielectronic Recombination

19

[1973] and [1973b]). RR is the dominant recombination process at low temperatures, and hence is especially important in the very cold ionized plasmas found in old nova envelopes and some supernova remnants. RR also works mainly on single electron systems such as H and He+ .

1.1.2

Dielectronic Recombination

Dielectronic recombination (DR) is a two step process that involves the capture of a colliding continuum electron by an ion to form a doubly excited state. This can be followed either by autoionization to the continuum with probability Γa , or by stabilization through a radiative decay of the captured electron or an electron in the parent to a true bound state of the excited core ion with probability Γr . The latter, which is the dielectronic recombination process, is responsible for the production of many recombination lines. Depending on the resonance state and the ambient conditions, the probabilities of these competing processes can go in either direction, that is Γa > Γr or Γa < Γr . Symbolically, DR can be described by the relation

Xn+ + e−

↔

∗ X(n−1)+

→

X(n−1)+ + hν

(1.2)

∗ is a doubly-excited autoionizing state with effective positive charge where X(n−1)+

(n−1). Such an excited autoionizing system gives rise to resonances in the photoionization cross sections. The photoionization cross sections typically exhibit resonances due to photoexcitation of the core electrons followed by autoionization. These resonance states are generally very fragile and decay predominantly by autoionization rather than radiative decay (Aldrovandi and Péquignot [1973] and [1973b]). Dielectronic recombination dominates at high temperatures as it requires sufficiently energetic electrons to excite the target, though it is still highly signifi-

1.1.2 Dielectronic Recombination

20

cant at low temperatures by recombination through low-lying resonances near the ionization threshold. As the energy of the resulting system in dielectronic recombination is greater than the binding energy of the entire system, it is usually described as an autoionizing state as the state spontaneously autoionizes in most cases. In fact, the DR autoionization process is so fast that these two processes can hold the population of the autoionizing level at its local thermodynamic equilibrium (LTE) value relative to the ion-free electron system. At high temperatures, appropriate for a collisionally ionized plasma, free electrons can reach many autoionizing levels and no one state dominates the process. However, at low temperatures, typical of a photoionized gas, the electrons can enter only a small number of autoionizing levels and hence only a few autoionizing states contribute significantly (Ferland [2003]). DR is a fundamental atomic process that has many applications in astrophysics and fusion plasmas. These applications include planetary nebulae and laboratory laser-induced plasmas. More specifically, the recombination lines produced by such processes can be used to determine element abundances and electron temperatures in planetary nebulae and to diagnose plasma conditions in laboratory. These processes are also important for evaluating the degree of ionization as well as the energy balance in plasmas. They play an important role in determining both the level populations and the ionization balance of both high and low temperature plasmas over a wide range of electron densities. Dielectronic recombination is the dominant electron-ion recombination process in many astronomical and laboratory systems. This includes both photoionized and electron-collisional plasmas. DR can take place via many intermediate autoionizing states; possibly the entire Rydberg series (Badnell et al [2006]).

1.2 Literature Review

1.2

21

Literature Review

In this section we present a short literature review to some important developments in the theory of recombination lines, followed by a brief account on the previous work on the carbon lines in the spectra of planetary nebulae.

1.2.1

Theory of Recombination Lines

Recombination is a fundamental atomic process that has a strong impact on many physical systems. Recombination lines produced by such a process can be used as a diagnostic tool to determine element abundances and electron temperatures in planetary nebulae. In the laboratory, these lines can be used to probe the conditions of laser induced plasmas and provide information that is essential for plasma modeling and fusion conditions. Therefore, there is a huge amount of literature on all aspects of this important phenomenon. In this section we present a non-comprehensive overview on some developments in the theory of recombination lines. As dielectronic recombination is the principal recombination process in most systems of interest to us and is the only one that we considered in our emissivity model, this overview will be dedicated to this process. Dielectronic recombination was first discussed by Massey and Bates [1942] and Bates and Massey [1943] in connection with the Earth’s atmosphere using intuitive non-rigorous arguments. Their concern was with recombination in plasmas at fairly low electron temperatures. They therefore considered low-lying doubly-excited states with energies just above the energy of the ion ground state. Burgess [1964] investigated dielectronic recombination in the solar corona and demonstrated that this process in low density hot plasma is possibly far more important at high temperatures than it was thought previously and could be the dominant electron-ion recombination process at coronal temperature. He showed

1.2.1 Theory of Recombination Lines

22

that DR can be particularly important because it can take place via complete Rydberg series of resonances converging to the energy of an excited state of the recombining ion. Moreover, the rate of dielectronic recombination can greatly exceed that of direct radiative recombination. Goldberg (1968) emphasized the importance of dielectronic recombination at low temperatures to explain some line intensities in planetary nebulae. His results, however, were limited to two temperatures with no possibility of extrapolation to other temperatures (Aldrovandi and Péquignot [1973]). The general theory of electron-ion collisions were developed by Seaton [1969] in a study to analyze resonance structures on the basis of quantum defect theory using the R-matrix formulation. In another study, Davies and Seaton [1969] investigated the process of emission of radiation in the optical continuum due to radiative capture of an electron by a positive ion. They used a theory similar to the radiation damping for transitions between bound states, and hence generalized this theory to the case of an optical continuum. Their techniques are capable of solving the problem of overlapping resonances found in DR. The results obtained from this investigation contributed to the subsequent development of a rigorous quantum mechanical theory of dielectronic recombination. Presnyakov and Urnov [1975] examined the problem of excitation of positive ions by electron impact using the analytical properties of Green functions for the Coulomb field. They investigated the scattering of an electron in the field of a positive ion which leads to resonance structures in the cross sections when some channels are closed. Their investigation revealed that the DR processes can lead to significant reductions in rate coefficients for collisional excitation of ions. Beigman and Chichkov [1980] examined dielectronic recombination through the forbidden levels and found that this process essentially increases the rate of total re-

1.2.1 Theory of Recombination Lines

23

combination for low temperatures. They noticed that this process is of importance for a low density plasma with a rather low electron temperature and high degree of ionization. The importance of dielectronic recombination in general at low temperatures, typical of PNe, for the low-lying autoionizing states was confirmed by Storey [1981] and Nussbaumer and Storey (e.g. [1983]). Bell and Seaton [1985] developed an ab initio theory for dielectronic recombination making extensive use of quantum defect theory of Seaton and the generalized radiation damping theory of Davies and Seaton [1969] to allow for overlapping resonances and their interaction with the radiation field. Harmin [1986] extended Bell and Seaton’s ab initio theory of dielectronic recombination to include the influence of arbitrary electric fields in the S-matrix. Rydberg series of resonances were treated in quantum-defect-Stark theory, which led to the investigation of field enhancement of DR rate over the whole Rydberg spectrum. In a series of papers, mainly on the second-row elements of the periodic table, Nussbaumer and Storey (e.g. [1987]) developed the theory of DR rate coefficients calculations which is widely used nowadays in the computational work and simulation of recombination processes. Their approach is based on the use of experimental data for energy levels for autoionizing states, and hence is limited to the species with such data. The use of theoretical data obtained from atomic structure codes such as Autostructure is a possibility though the accuracy of results may be limited (Ferland [2003]). LaGattuta et al [1987] presented a theoretical method for determining the effects of static applied electric fields on the dielectronic recombination cross section in which the distorted wave theory and the isolated resonance approximation are invoked.

1.2.2 Recombination Lines of Carbon

24

Dielectronic recombination processes under the influence of an electric field were theoretically investigated by Sakimoto [1987] using a multichannel quantum defect theory of the Stark effect. He derived simple formulae for calculating the DR cross sections so that the effect of an applied electric field is taken into account. Nahar and Pradhan [1994, 1995] presented a unified ab initio treatment of electron-ion recombination that incorporates both the radiative recombination and the dielectronic recombination processes within the close coupling approximation of atomic collision theory. Their method enables the calculation of a single, total recombination rate coefficient that may be employed for a variety of astrophysical applications.

1.2.2

Recombination Lines of Carbon

There is an overwhelming literature on the applications of recombination lines theory. However, in this section we only report some work on the recombination lines of carbon relevant to the PNe applications. We would like to emphasize that this is a short non-comprehensive overview intended to highlight some contributions related to our research. Leibowitz [1972] developed a theory for the calculation of the energy level population of an ion in planetary nebulae in which radiative excitation is taken specifically into account. The theory is applied in the computation of the spectrum of the ion C IV under different excitation conditions and yields measurable differences between a pure recombination spectrum of this ion and a spectrum of radiatively excited C IV. The theory was applied to the intensity ratios of C IV lines in several PNe including NGC 7027. In another study [1972b], he investigated the polarization of C IV emission lines in planetary nebulae. Investigating the structure of the Orion Nebula and the neutral medium around


25

it, Balick et al [1974] mapped the C 85α carbon recombination line with high spatial resolution throughout the nebula to determine the distribution of the ionized carbon and the relationship to the H II region. Nussbaumer and Storey [1975] examined a number of recombination lines of carbon in the context of investigating the influence of the observed solar radiation field on the ionization balance with a view to application in the solar chromospherecorona transition region. In a subsequent study [1984] they calculated effective dielectronic recombination coefficients for selected lines and ground and metastable terms of ions of carbon and other elements. The effective recombination coefficients were fitted to a convenient function of temperature in a range found in PNe and similar astrophysical objects. Boughton [1978] investigated several carbon recombination lines observed in the direction of the Orion Nebula to construct two-cloud (foreground and background) models which can reproduce the observed intensities of the carbon lines and the velocities of the emitting regions. In a series of papers, Harrington et al [1980] discussed the recombination processes in the context of investigating the carbon abundance in the low excitation planetary nebula IC 418 using ultraviolet spectra observations. In another study, Harrington et al [1981] investigated some recombination lines of carbon using in their analysis several results obtained earlier by Storey. In another paper of the same series, Clavel et al [1981] discussed the interpretation of observations of the C II λ1335˚ A dielectronic recombination multiplet in the planetary nebula IC 418. Clegg et al [1983] investigated the optical and IUE spectra of a bright region of nebulosity in the low-excitation planetary nebula NGC 40 for an analysis of the physical conditions and abundances in the nebula. They identified new recombination lines of C II in the IUE spectra and analyzed their intensities in NGC 40 and


26

IC 418. Hayes and Nussbaumer [1984] examined the effect of dielectronic recombination process on the emissivity of the C II λ1335˚ A multiplet observed in various astronomical objects including PNe, and compared this to the effect of collisional excitation. Investigating the carbon abundance in planetary nebulae, Kholtygin [1984] calculated the intensities of the recombination lines of C III ion and used the intensities of the observed lines of several carbon ions to determine their abundances and the total carbon abundance in 46 planetary nebulae. In a later study [1998] he used the calculated and observed intensities of carbon recombination and intercombination lines to find the effect of small temperature and density fluctuations on the line intensities in the spectra of PNe. Bogdanovich et al [1985] calculated the relative intensities of the recombination lines of the C II and N III ions, and used the observed intensities of these lines in the spectra of planetary nebulae to determine the abundances of the C III and N IV ions in 12 nebulae. Badnell [1988] used the Autostructure program to calculate configuration mixing intermediate coupling effective dielectronic recombination rate coefficients for the recombined C+ ion at temperatures T = 103 − 6.3 × 104 K applicable to planetary nebulae. He tabulated the results for the quartet-quartet transition lines and some selected doublet-doublet lines. He also compared the total LS-coupling and intermediate coupling DR rate coefficients and direct RR rate coefficients. Péquignot et al [1991] computed effective and total radiative recombination coefficients for important optical and UV transitions of ions of carbon and other elements that produce most recombination lines observed in nebulae. They presented their results in the form of four parameter fits valid at low electron density


27

and over a wide range of electron temperature. In their investigation to the carbon abundance in planetary nebulae and the long standing problem of discrepancy between recombination and forbidden lines results, Rola and Stasi´ nska [1994] examined the C II λ4267˚ A recombination lines and analyzed the observational data obtained from various sources for many planetary nebulae. Baluteau et al [1995] examined several recombination lines of carbon in the course of studying the far-red spectrum of the bright and spectrum-rich NGC 7027 planetary nebula. Davey et al [2000] presented effective recombination coefficients for C II transitions between doublet states in LS-coupling using R-matrix approximation. The calculations were carried out in the temperature range 500-20000 K and for an electron density of 104 cm−3 relevant to planetary nebulae. Investigating the chemical abundances of planetary nebulae from optical recombination lines, Liu et al [2004] examined several recombination lines of carbon observed in a number of planetary nebulae. Peng et al [2004] used the R-matrix method to carry out theoretical calculations for the recombination line λ8794˚ A of C II, which is important for estimating the abundance of carbon in planetary nebulae. In another study [2005] they carried out a theoretical investigation using R-matrix method on dielectronic recombination processes of C2+ ions in planetary nebulae. Several such recombination lines were examined in the course of this investigation. Zhang et al [2005] thoroughly examined the rich spectra of the planetary nebula NGC 7027. The extensive line list presented in this study comprises over 1100 transition lines from many atoms and ions of various elements. This includes many recombination lines of carbon and its ions. Their data is used as the main source

1.3 Recombination vs. Forbidden Lines Analysis

28

of observational data in our investigation, as will be examined subsequently in this report.

This overview highlights the fact that relatively little work has been done on the C II recombination lines. From the perspective of atomic physics, the most comprehensive of these studies and the most relevant to our investigation are those of Badnell [1988] and Davey et al [2000]. Despite the fact that Badnell has performed his calculations in both LS and intermediate coupling schemes and over a wide temperature range, he used the Autostructure code which may be inferior to R-matrix code in this regard. On the other hand, although Davey et al have performed their calculations using the R-matrix code utilizing a comprehensive recombination theory, they followed an LS-coupling scheme and hence their results were limited to the doublet states. The aim of the current study is to build on these investigations and elaborate on them by using the R-matrix code with an intermediate coupling scheme. However, our investigation will be limited to the dielectronic recombination concentrating our attention on the low-lying autoionizing states. On the observational perspective, most of the previous studies have focused on a few recombination lines. Moreover, the quality of some of those observations may be questionable. The advantage that we have in the current study is the availability of the extensive and recently obtained NGC 7027 line list of Zhang et al [2005] which is the most thorough list of its kind in existence. Moreover, it is potentially of high-quality from observational side.

1.3

Recombination vs. Forbidden Lines Analysis

In this section we briefly discuss the long standing problem in nebular physics of discrepancy between recombination and forbidden lines results and the implication


29

of this on our investigation. Resolving this problem is extremely important not only for understanding the nebular conditions but also for identifying possible flaws in the underlying physical models. The collisionally excited lines (CEL), also collectively known in this context as forbidden lines, are produced by collision between energetic electrons and atoms and ions with a subsequent excitation and decay, while the optical recombination lines (ORL) are produced when electrons combine with ions by one of the aforementioned recombination mechanisms. These lines are detected either in emission or absorption modes. Both CELs and ORLs can in principle provide information about the physical and chemical conditions of the emitting and absorbing regions such as temperature, number density, pressure, elemental abundance, chemical composition and so on. Until recently, the main means used in the planetary nebula investigation is the CELs. One reason is the relative ease of observing and analyzing these lines as they are more abundant and intense than the ORLs. However, the ORLs are also used in the elemental abundance determination studies by using their relative intensities to the hydrogen lines (Kisielius et al [1998], Ferland [2003], Tsamis et al [2003b], Tsamis et al [2004]). Although the CELs are much stronger than the ORLs, they are highly dependent on temperature and density structure and hence can lead to large systematic errors when used for instance in ionic abundance estimation. On the other hand, the faint recombination lines can be easily contaminated by radiation from other excitation processes such as fluorescence. Despite all these differences, there is a common feature between the results obtained from these lines; that is for all the ionic species investigated so far (mainly from the second row of the periodic table such as C, N, O and Ne) the recombination lines in planetary nebulae produce higher ionic abundances than the corresponding values obtained from the forbidden lines. The ratio of the ORL to the CEL abundances is case dependent and can vary by a factor ranging between 2-30. This has called into question the validity


30

of the previously accepted CELs analysis results. The systematic nature of this problem can rule out the contamination of ORLs by resonance fluorescence or the inaccuracies in recombination coefficients as possible reasons, as may have been suggested. This problem may be correlated to the dichotomy between the temperature obtained from the Balmer jump of H I recombination lines and that from the collisionally excited forbidden lines where the latter is systematically higher than the former (Kholtygin [1998], Liu [2002], Tsamis et al [2007]). Several explanations have been proposed to justify these discrepancies individually or collectively, though no one seems to be satisfactory or universally accepted. One explanation is the sensitivity of the CELs to temperature and structure fluctuations (e.g. exponential temperature dependence of emissivities of forbidden lines) which amplifies the errors that are intrinsically associated with these parameters. However, this reason alone cannot explain why the CELs always produce higher values. Another explanation is that the large temperature and density fluctuations within the nebular structure result in systematic underestimation of the heavy element abundances from the forbidden lines. The existence of knots depleted of hydrogen with high heavy element contents within the nebular gas has been proposed as a possible reason for these fluctuations and subsequent implications. The temperature inside these knots of high metallicity, and hence high opacity to stellar UV emissions and large cooling rates, is expected to be too low for efficient production of CELs though it is still sufficiently high for the emission of ORLs. Consequently, the CELs and ORLs originate in different regions of the nebular gas with different elemental abundances. However, the existence and composition of these knots and their effect on the selectivity of emission lines is unsettled issue and a matter of debate. Moreover, in some cases the discrepancy between the CELs and the ORLs abundances is too large to be explained by temperature fluctuation alone although it may be partly responsible (Liu et al [1995], Garnett and


31

Dinerstein [2001], Tsamis et al [2003], Liu et al [2004]). Amid the uncertainties highlighted by the CELs vs. ORLs problem, it is extremely important to do more research especially on the recombination lines theory to find a breakthrough in this long-standing problem. We hope that our research on carbon recombination lines will be a valuable contribution in this direction.

Chapter 2 Physics of Scattering and Recombination In this chapter, and for the purpose of completeness, we present a concise account of the theoretical background that our work is based upon. The main body of this account is no more than a summary of extracts obtained from the cited references, presented in what we believe as a logical order. The interested reader should consult these references for more thorough and precise theory. In the recombination processes, the scattering system consists of an atomic target with N electrons and an additional colliding electron. It is assumed that the behavior of this system is determined solely by the electromagnetic interaction between the charged particles. It is therefore required to find solutions Ψ, with appropriate boundary conditions, for the time independent Schrödinger equation

HΨ = EΨ

(2.1)

where H is the Hamiltonian of (N + 1)-electron system and E is the total energy of the system. For light atoms and ions, where relativistic effects can be neglected, 32

2.0.1 Close Coupling Approximation

33

the (N + 1)-electron Hamiltonian in atomic units is given by

H=

N +1 X n=1

N +1 X Z 1 1 2 + − ∇n − 2 rn m>n rnm

! (2.2)

where rn is an electronic radius vector drawn from the atomic nucleus with atomic number Z and rnm = |rn − rm | an inter-electronic distance. It is assumed that the nucleus is infinitely heavy and is a point charge. The first two terms on the right-hand side of Equation (2.2) are a sum over the electron kinetic energy and electron-nucleus Coulomb attraction: the so-called one-electron terms. The final term on the right is a summation over the Coulomb repulsion between pairs of electrons: the so-called two-electron term (Berrington et al [1995]). One way for solving this system is to use the close coupling (CC) method which provides a powerful technique for calculations involving collisions between an electron and an N -electron target. The method can also be used to obtain wavefunctions for bound states of the entire (N + 1)-electron system and hence to calculate quantities such as oscillator strengths and photoionization cross sections.

2.0.1

Close Coupling Approximation

The basic expansion of the close coupling method is

Ψ=A

X

ψi θi

(2.3)

i

where A is an anti-symmetrization operator, ψi are functions of the N -electron target, and θi are functions of the colliding electron given by 1 θi = χi Fi r

(2.4)


34

where χi are the spin and angle functions and Fi are the radial functions of the colliding electron. The target functions ψi are calculated using standard methods of configuration interaction (CI) theory with a common set of radial functions. These radial functions are of two types: spectroscopic orbitals of the type which would occur in a central-field model, and correlation orbitals included to improve the accuracy. On introducing functions φi which are vector-coupled products of ψi and θi and imposing an orthogonality condition, the CC expansion becomes

Ψ=A

I X i=1

J

X 1 cj Φj φi Fi + r j=1

(2.5)

where cj are coefficients in this expansion and Φj are functions of bound-state type for the whole system. Since (2.5) is a truncated expansion, Ψ is not a solution of the Schrödinger equation given in (2.1). The equations used in CC theory can be derived from the variational condition (δΨ | H − E | Ψ) = 0

(2.6)

where δΨ is any variation in Ψ due to variations of the functions Fi and coefficients cj . This gives a system of integro-differential (ID) equations to be satisfied by Fi and cj (Seaton [1985], Berrington et al [1987]). If E is the total energy, Ei is the energy for target state i and Ee− is the energy of the added electron, then E = Ei + Ee−

(2.7)

At energies giving bound states for the whole system the condition Ee− < 0 is satisfied for all i and the radial functions Fi go to zero exponentially in the limit of r → ∞. The collision states are such that Ee− > 0 for some values of i. Channel i is said to be open if Ee− > 0 and closed if Ee− < 0. Let the Ei be ordered such


35

that E1 < E2 < ... < EI . For a given E there are Io open channels and

Ee− > 0

for

i = 1

to Io

Ee− < 0

for

i = (Io + 1) to I

(2.8)

When some channels are open, use is made of a second subscript on the radial functions to specify a boundary condition. For numerical work it is first required to calculate real functions Fii0 satisfying the reactance matrix boundary conditions Fii0 (r) ∼ si (r)δ(i, i0 ) + ci (r)K(i, i0 )

for

i=1

Fii0 (r) ∼

for

i = (Io + 1) to I

r→∞

r→0

0

to Io (2.9)

with i0 = 1 to Io . The si (r) and ci (r) are the components of the Coulomb functions with a sinusoidal asymptotic form. The Coulomb functions ϑ± are defined by

ϑ± (r) = c ± is

(2.10)

The S-matrix functions F− (r) have asymptotic forms 1 0 Fii−0 (r) ∼ {ϑ− (r)δ(i, i0 ) − ϑ+ i (r)S(i, i )} 2 i

for

i=1

Fii−0 (r) ∼ 0

for

i = (Io + 1) to I

to Io (2.11)

It can be shown that S=

I + iK I − iK

(2.12)

and that −iF− F = I − iK −

(2.13)

The functions F+ used for final states in photoionization calculations are the com-

2.0.2 R-matrix Method

36

plex conjugates of the functions F− (Berrington et al [1987]).

2.0.2

R-matrix Method

One way of solving the Schrödinger equation of electron-atom collision in the CC approximation is the R-matrix method which is a computational method based on the theory developed by Burke and Seaton and provides an excellent basis for describing many types of atomic processes. The R-matrix method can be considered to be a computationally efficient way of solving the resulting close coupling equations. Its main feature is the partitioning of the configuration space into two physically-distinctive regions: inner and outer. The two regions are linked by the R-matrix on the boundary. The main computational effort is in obtaining solutions for the inner region. The R-matrix starts by choosing a value A for the radial variable r such that the functions φi and Φj are small for r ≥ A. The part of space with r ≤ A is called the inner region while the rest of space is called the outer region. The R-matrix method provides a technique for obtaining solutions in the inner region. When the incident electron is within the inner region a many-body Schrödinger equation is solved, and hence the most mathematically complex part of the interaction is confined to a finite region. The collision wavefunction is expressed in terms of a discrete complete set of basis functions. A first step is to obtain solutions which satisfy a fixed condition on the boundary at r = A. These solutions Ψ = ψn exist for a discrete set of energies E = en and contain radial functions Fi (r) = fin (r). 0 Different choices of boundary condition can be made. One choice is that fin (A) = 0

where the prime indicates a derivative with respect to r. The functions ψn are normalized to (ψn |ψn0 )I = δ(n, n0 )

(2.14)

2.0.2.1 Inner and Outer Regions

37

where matrix elements (...)I are evaluated in the inner region. For any value of E the complete wavefunction ΨE with radial functions FiE (r) can be expanded as ΨE =

X

ψn AnE

(2.15)

n

where AnE is given by AnE = (en − E)−1

X

0 fin (A)FiE (A)

(2.16)

i 0 where FiE (A) is taken to be such that ΨE satisfies required normalization and

boundary conditions. On substituting (2.16) in (2.15) with some manipulation the following relation can be obtained X

Rii0 (E)Fi00 E (A)

(2.17)

fin (A)(en − E)−1 fi0 n (A)

(2.18)

FiE (A) =

i0

where Rii0 (E) =

X n

is the R-matrix. The functions ψn are calculated using expansions of the radial functions fin (r) with the inclusion of orthogonality conditions to obtain good convergence. The expansion coefficients are obtained on diagonalizing the matrix of the Hamiltonian with a truncation process. The Buttle method is then used to correct for the truncation (Berrington et al [1987]).

2.0.2.1

Inner and Outer Regions

The inner region, known as the R-matrix box, is identified as a sphere of radius A centered on the target nucleus and encloses all target electrons, while the outer

2.0.2.2 Matching of Solutions

38

region is the rest of the configuration space. The R-matrix method provides a technique for obtaining solutions in the inner region where the system is considered a many-body problem, as outlined in the previous section. In the outer region, where exchange and correlation effects can be neglected, the system reduces to a two-body problem and the solution can be obtained using an asymptotic expansion or by using perturbation theory. In this region, the ID equations reduce to ordinary differential equations of the form

X li (li + 1) 2z d2 + F + Vii0 Fi0 = 0 − + i i dr2 r2 r 0 i

(2.19)

where z is the charge on the ion, i is the energy of the added electron and the Vii0 are multipole potentials (Berrington et al [1987, 1995]).

2.0.2.2

Matching of Solutions

All equations in the R-matrix method should be matched at the boundary r = A. Depending on the nature of the available channels (some open or all closed), several matrices are defined and used to obtain a solution. In this stage of R-matrix calculations the reactance matrix K is evaluated. This matrix is real and symmetric, and represents the asymptotic form of the entire wavefunction, containing information from both internal and external regions (Berrington et al [1995]).

2.0.2.3

Bound States

Once the computations of the inner-region R-matrix solutions are carried out and the basic R-matrix data fin (A) and en are obtained, as outlined already, fast calculations can be made with minimal additional cost to obtain the energies and wavefunctions for bound states of atoms and positive ions. The technical details are given by Seaton [1985].

2.0.2.4 Resonances 2.0.2.4

39

Resonances

On obtaining the reactance matrix K and the scattering matrix S which is given by Equation (2.12), resonances can be obtained following one of several approaches. Some of these are outlined in § 3.2.1. An advantage of using the close coupling approximation is that resonance effects are automatically included, as the interaction between open and closed channels is already considered.

2.0.2.5

Oscillator Strengths

The dipole operators in length and velocity form are given by

DL =

X

rn

DV = −

n

X

∇n

(2.20)

n

where the summation is over all atomic electrons. Once these operators are obtained, the line strength for a dipole transition between two states a and b of energy Ea and Eb can be defined in the length and velocity formulations: L Sba = |(b k DL k a)|2 V Sba = 4(Eb − Ea )−2 |(b k DV k a)|2

(2.21)

where the energies are in Ryd. The oscillator strength is the dimensionless quantity fba defined by fba =

(Eb − Ea )Sba 3ga

(2.22)

where ga is the statistical weight or degeneracy of the initial state given by (2Sa + 1)(2La + 1) for LS-coupling and (2Ja + 1) for intermediate coupling. fba can be in length or velocity form depending on the line strength Sba . Use of exact functions would give S L = S V and f L = f V . With approximate functions the differences

2.0.2.6 Photoionization Cross Sections

40

between the length and velocity forms give an indication of the accuracy achieved although agreement between f L and f V does not guarantee accuracy (Berrington et al [1987]).

2.0.2.6

Photoionization Cross Sections

Using the dipole operators, a generalized line strength similar to Equations (2.21) for a transition from an initial bound state to a final free state can be defined. The photoionization cross section is then given by

σ=

4π 2 a20 αEp S 3g

(2.23)

where the photon energy Ep is in Ryd and g is the statistical weight of the initial bound state. Again, in this formulation either the length or the velocity operator can be used (Berrington et al [1995]).

2.0.2.7

Relativistic Effects

To account for relativistic effects, the non-relativistic R-matrix method can be extended to include relativistic terms from the Breit-Pauli Hamiltonian defined by

Hbp = Hnr + Hrc

(2.24)

where Hnr is the non-relativistic Hamiltonian and Hrc is the relativistic correction terms which include the one body mass, Darwin and spin-orbit terms, the two body fine structure terms and the two body non-fine structure terms. In this approach, the Hamiltonian matrices are first calculated in LS-coupling and then transformed to Intermediate Coupling (IC) through a unitary transformation. In the RMATRXI code only some of these non-relativistic terms (i.e. the mass-correction term, the

2.0.3 RMATRX-I Code

41

one-electron Darwin term and the spin-orbit term) are explicitly retained (Berrington et al [1995]).

2.0.3

RMATRX-I Code

The R-matrix theory for atomic scattering is implemented in a computer code called RMATRX-I, which is a general program to calculate atomic continuum processes including electron-atom and electron-ion scattering, and radiative processes such as bound-bound transitions, photoionization and polarizabilities. Calculations can be made either in LS-coupling, or in the intermediate coupling scheme by including the one electron relativistic terms from the Breit-Pauli Hamiltonian, i.e. the mass-correction term, the one-electron Darwin term and the spin-orbit term. The program consists of sequentially-coupled stages and is used in this study as the main tool for generating theoretical data for scattering and subsequent atomic processes. These stages fall into two major categories: inner region and outer region. The inner region code is composed of the following stages (Berrington et al [1987, 1995], Badnell [2002])

• STG1 to calculate orbital basis and radial integrals. It reads radial functions for the target and evaluates all required radial integrals. The target functions can be obtained either from Autostructure or from CIV3 codes, as will be discussed later. • STG2 to calculate angular algebra. This stage reads radial integrals from STG1 and evaluates matrices of the Hamiltonian and of the dipole operators. It has an option to diagonalize the matrix of the Hamiltonian for the target. • STGJK which is an optional stage to recouple the Hamiltonian in the case of intermediate coupling scheme by transforming the Hamiltonian matrices

2.0.3 RMATRX-I Code

42

from LS-coupling to pair-coupling by means of a unitary transformation. It includes relativistic effects in the Breit-Pauli approximation, and is run after module STG2 and before STG3. • STG3 to diagonalize the Hamiltonian matrix in the continuum basis and produce the essential R-matrix data sets which are needed for running the outer region stages. The outer region code consists of the following stages • STGB to calculate wavefunctions and energy levels for bound states and produce a data set required for bound-bound and bound-free radiative calculations. • STGF to calculate free states wavefunctions and collision strengths for inelastic collisions and produce a data set required for bound-free and free-free radiative calculations. • STGBB to calculate oscillator strengths for bound-bound transitions. • STGBF to calculate photoionization cross sections for bound-free transitions. An extra stage called STGQB for finding and analyzing resonances was added to the R-matrix code by Quigley and co-workers [1998]. In Figure (2.1) a flow chart of the RMATRX-I suite and its companions is presented. RMATRX-I needs an input data to define the atomic target. This can be provided either by CIV3 of Hibbert [1975] or by Autostructure which incorporates the Superstructure code of Eissner et al [1974] with the modifications of Nussbaumer and Storey [1978]. CIV3 is a general multi-configuration atomic structure code that calculates configuration interaction (CI) wavefunctions in LS and intermediate coupling schemes.

2.0.3 RMATRX-I Code

43

Autostructure OR CIV3: Prepare target stg1: Evaluate radial integrals stg2: Evaluate angular algebra IC?

No

Yes

stgjk: Recouple Hamiltonians

Stg3: Diagonalise Hamiltonian

Stgb: Energy levels

Stgqb: Resonances

Stgbb: Oscillator strengths

Stgf: Collision strengths

Stgbf: Photoionisation cross-sections

Figure 2.1: Flow chart of the RMATRX-I suite. The wavefunctions may then be used to calculate parameters for atomic systems such as oscillator strengths. CIV3 uses Slater Type Orbital (STO) specification to represent the CI functions. The Slater Type Orbitals are defined by

Pnl (r) =

X

Ci rPi e−ξi r

(2.25)

i

where Pnl (r) is a radial function for the nl orbital, Ci is a coefficient and Pi and ξi are indicial parameters in this specification, r is the radius, and i is a counting index that runs over the orbitals of interest. Autostructure (Eissner et al [1974], Nussbaumer and Storey [1978], Badnell [2008]) is a general purpose automatic atomic structure code based on the Thomas-

2.0.3 RMATRX-I Code

44

Fermi-Dirac formulation. It calculates radial functions using central potentials

V (r) = −

ζ(r) r

(2.26)

where ζ(r) is a radius-dependent effective charge. The program produces a wide range of atomic data particularly suitable for astrophysical applications. The input required by Autostructure is the minimum needed to specify the atomic system and the level of approximation. Among its many applications, it can be used to calculate the target functions for the R-matrix code and configuration mixing autoionization rates in LS or intermediate couplings, where the relativistic corrections are made by means of Breit-Pauli approximation. Autostructure can also be used to calculate term energies, atomic and ionic energy levels, term-coupling coefficients, radiative data which includes permitted and forbidden transition probabilities, and cascade coefficients. The program uses multi-configuration type expansions. In our investigation of the C II recombination lines, we used target data generated by CIV3 for the scattering calculations as an input to STG1 of the R-matrix code, as presented in Table (3.1) and § C.1.1, while we used Autostructure to elaborate and investigate related issues. Autostructure was also used to find the oscillator strengths for the free-free transitions between resonance states since the available RMATRX-I code does not contain a free-free stage. In this approach the free-free transitions were treated as bound-bound transitions. Autostructure was also used for generating f -values for free-bound and bound-bound transitions involving the 8 topmost bound states, namely the 4 Fo and 4 Do levels, as these quartets with their large effective quantum number for the outer electron are out of range of the R-matrix code validity. The Autostructure input data files for these calculations are given in § C.2 in Appendix C.

Chapter 3 Investigating Recombination Lines of C II Carbon recombination lines are an important diagnostic tool that can provide valuable information on the physical conditions in various astrophysical environments such as planetary nebulae and the interstellar medium. Because the recombination rate is a function of electron temperature Te and number density Ne and the effective quantum number nf , the observed variation of the carbon line intensity with nf can be used in principle to determine the temperature and density in the line-emitting region. In this study we investigate the recombination lines of C II, that is the emission lines produced by the following recombination process

C2+ + e−

→

C+

(3.1)

and the subsequent radiative cascade decay which continues until the atom reaches the ground or a metastable state. Our investigation includes all the autoionizing resonance states above the threshold of C III 1 Se with a principal quantum number 45

3.1 Theoretical Background

46

n < 5 for the captured electron as an upper limit. This condition was adopted mainly because of computational limitations. Moreover, the contribution to the total emissivity of the levels with higher n is minor at typical nebular temperature. As we are only concerned with states of 2s2p(3 Po ) parentage which have no significant radiative recombination contribution at nebular temperatures, in our theoretical model we considered only the contribution to emissivity due to dielectronic recombination. In total, 64 autoionizing states (27 doublets and 37 quartets) with this condition have been theoretically found by the R-matrix method. Of these 64 resonances, 58 are experimentally observed according to the NIST database [NIST]. The major work in our investigation of the C II recombination lines can be split into three main parts

1. Autoionizing and bound states calculations to generate the required theoretical data for the resonances and bound states. 2. Calculation of f -values for various transitions. 3. Emissivity calculations and analysis to find the emissivity of various transition lines and compare the theoretical emissivity model to the observational data.

In the next section we provide a general theoretical background. This will be followed by three sections in which we outline the theory and procedures that we followed in this investigation.

3.1

Theoretical Background

In a thermodynamic equilibrium situation an excited atomic state is populated by recombination and radiative cascade from higher states, and depopulated by autoionization and radiative decay to lower states. Many recombination lines arise


47

from radiative decay and subsequent cascade of strongly autoionizing resonance states near the ionization limit. These lines are dominated by low temperature dielectronic recombination. The populations of such resonance states are determined by the balance between autoionization and radiative decay. When autoionization dominates, the populations are then given by the Saha equation for thermodynamic equilibrium

NX(n−1)+ = Ne NXn+

gX(n−1)+ 2gXn+

h2 2πme kTe

3/2

−∆Et /kTe

e

(3.2)

where ∆Et is the energy of the recombined electron in the X(n−1)+ state relative to the ionization threshold, and the other symbols have their usual meaning as given in Nomenclature. The Saha equation, which describes the ratio of different stages of ionization, is based on the assumption of Local Thermodynamic Equilibrium (LTE) in a gas where collision dominates other physical processes. Consequently, the local velocity and energy distributions of particles are given by the Maxwell and Boltzmann distributions respectively and a temperature can be defined locally. The Saha equation is therefore strictly applicable only if elastic collisions are responsible for establishing the energetic distribution of particles. In many practical cases, however, atomic processes such as radiative transitions or dynamic effects are more important than elastic collisions and the assumption of LTE is not justified within the whole energy range. In these cases explicit detailed equilibrium calculations are required to determine the velocity and energy distributions of particles over the various energy levels (Benoy et al [1993], Davey [1995]). To measure the departure of the state from thermodynamic equilibrium a departure coefficient, bu , is defined as the ratio of autoionization probability to the sum of radiative and autoionization probabilities, as given by Equation (3.8). The


48

value of this coefficient is between 0 for radiative domination (Γa Γr ) and 1 for thermodynamic equilibrium (Γa Γr ). A large value of bu (' 1) is then required to justify the assumption of a thermodynamic equilibrium and apply the relevant physics. In thermodynamic equilibrium (TE) the rate of radiationless capture equals the rate of autoionization, giving (Seaton and Storey [1976])

Ne Ni %c = NuS Γau

(3.3)

where Ne and Ni are the number density of electrons and ions respectively, %c is the recombination coefficient for the capture process, NuS is the Saha population of the doubly-excited state and Γau is the autoionization transition probability of that state. In non-TE situation, the balance is given by

Ne Ni %c = Nu (Γau + Γru )

(3.4)

where Nu is the non-TE population of the doubly-excited state and Γru is the radiative transition probability of that state. The departure coefficient bu is a measure of the departure from TE, and hence is the ratio of the non-TE population to the Saha population. Comparing Equations (3.3) and (3.4) gives

bu =

Nu Γau = NuS Γau + Γru

(3.5)

In this section we summarize the theoretical background for our emissivity calculations as implemented in the ‘Emissivity’ code [2010], where we assumed that recombination is largely dominated by dielectronic recombination with no significant contribution from other recombination processes. Though this is a restricted condition, it is applicable in many practical situations. In particular it is a reason-


49

able assumption for the cases of primary interest to us. In the following points we outline how our ‘Emissivity’ code works and the general theoretical background on which it is based

• The program starts by obtaining the radiative transition probability Γrul for all free-free transitions as given by

Γrul =

α3 Ep2 gl ful 2gu τo

(3.6)

where the photon energy Ep is in Ryd, and the other symbols are defined in Nomenclature. This is followed by obtaining the radiative transition probability Γrul for all free-bound transitions, as in the case of free-free transitions. • The total radiative transition probability Γru for all resonances is then found. This is the probability of radiative decay from an upper resonance state to all accessible lower resonances and bound states. This probability is found by summing up the individual probabilities Γrul over all lower free and bound states l for which a transition is possible according to the electric dipole rules, that is Γru =

X

Γrul

(3.7)

l

• The departure coefficient, bu , for all resonances is then obtained

bu =

Γau Γau + Γru

(3.8)

where Γau and Γru are the autoionization and radiative transition probabilities of state u, and Γa is given by

Γa =

∆r ~

(3.9)


50

where ∆r is the resonance width and ~ is the reduced Planck’s constant. • The next step is to calculate the population of resonances by summing up two components: the Saha capture, and the radiative decay from all upper levels. In thermodynamic equilibrium (TE) the rate of radiationless capture equals the rate of autoionization, giving

Ne Ni %c = NlS Γal

(3.10)

where Ne and Ni are the number density of electrons and ions respectively, %c is the recombination coefficient for the capture process, NlS is the Saha population of the doubly-excited state and Γal is the autoionization transition probability of that state. In non-TE situation, the population and depopulation of the autoionizing state due to radiative decay from upper states and to lower states respectively should be included, and hence the balance is given by Ne Ni %c +

X

Nu Γrul = Nl (Γal + Γrl )

(3.11)

u

where Nl is the non-TE population of the doubly-excited state, Γrl is the radiative transition probability of that state, Γrul is the radiative transition probability from an upper state u to the autoionzing state l, and the sum is over all upper states that can decay to the autoionzing state. Combining (3.10) and (3.11) yields

NlS Γal +

X u

Nu Γrul = Nl (Γal + Γrl )

(3.12)


51

On manipulating (3.12) the following relation can be obtained

Nl

X Nu Γrul Γal + = Γal + Γrl Γrl + Γal u X Nu Γr ul = NlS bl + a r + Γ Γ l l u NlS

(3.13)

where bl is the departure coefficient of the autoionizing state. • The next step is to calculate Γrul for the bound-bound transitions. In these calculations the f -values can be in length form or velocity form, though the length values usually converge fairly quickly and hence are likely to be more reliable. The ‘Emissivity’ therefore reads these values from the f -values file produced by the R-matrix code. This is followed by finding Γru for the bound states by summing up Γrul over all lower bound states l, as given earlier by (3.7) for the case of resonances. • The population of the bound states is then obtained

Nl =

X Nu Γr

ul

u

Γrl

(3.14)

where u includes all upper free and bound states. • Finally, all possible free-free, free-bound and bound-bound transitions are found. The emissivity of all recombination lines that arise from a transition from an upper state u to a lower state l is then computed using the relation

εul = Nu Γrul hν

where ν is the frequency of the transition line.

(3.15)

3.2 Autoionizing and Bound States Calculations

3.2

52

Autoionizing and Bound States Calculations

The main computing tools that we used in this part to generate theoretical data is RMATRX-I code and its companions (i.e. STGQB code and K-matrix implementation in stage STGF) though we occasionally used Autostructure to accomplish complementary tasks. In the first part of this section we present three methods for finding and analyzing resonances. We will focus on the two methods that we used in our investigation of the recombination lines of C II. The remaining parts will be dedicated to the practical aspects.

3.2.1

Methods for Investigating Resonances

The resonance phenomenon is a highly important feature in atomic collisions as it plays a vital role in electron scattering with atoms, ions and molecules. The Auger effect is one of the early examples of the resonance effects that have been discovered and extensively investigated. There are several methods for finding and analyzing resonances that arise during the recombination processes. In the following sections we outline three of these methods; two of which are used in the current study to find and analyze resonances of C II. These two are the QB method of Quigley and Berrington [1996], as implemented in the STGQB code of Quigley and coworkers [1998] as an extension to the R-matrix code, and the K-matrix method of P.J. Storey [private communication] which we implemented by modifying stage STGF of the R-matrix code as part of this study. The third method is the Time-Delay of Stibbe and Tennyson [1998] which is implemented in a computer code called TIMEDEL. We tried to implement and use this method in an early stage of our investigation before developing the K-matrix method, but this attempt was abandoned in favor of K-matrix.

3.2.1.1 QB Method 3.2.1.1

53

QB Method

A major approach for finding and analyzing resonances is to use a fitting procedure based on the energy variation of the reactance K-matrix or its eigenphase near an isolated resonance. For an isolated resonance which interacts with several continua (a number of open channels), fitting the energy-dependent K-matrix directly is difficult because K-matrix has a pole in the resonance region. Therefore it is easier to fit the arctan of K-matrix as used by Bartschat and Burke [1986]. The eigenphase sum is defined by

δ=

N X

arctan Λi

(3.16)

i=1

where Λi is an eigenvalue of K-matrix and the sum runs over all open channels. The eigenphase is normally fitted to a Breit-Wigner form as used by Tennyson and Noble [1984] δ = δ + arctan

∆r /2 Er − E

(3.17)

where Er is the resonance energy, ∆r is the resonance width and δ is the eigenphase background. In theoretical calculations, a resonance manifests itself by a rapid increase by π radians in the total eigenphase sum δ, superimposed on a background. Because a resonance has a finite width, the increase in the eigenphase sum may not be exactly π due to the background variation. Therefore, a more useful definition of a resonance position is the energy at which the eigenphase sum increases most rapidly by having a maximum gradient with respect to energy, i.e. dδ/dE (Quigley and Berrington [1996]). The QB method of Quigley and Berrington [1996] is a procedure for analyzing resonances in atomic and molecular collision theory using eigenphase fitting. The merit of the QB method over other eigenphase fitting procedures is that it exploits

3.2.1.2 Time-Delay Method

54

the analytic properties of the R-matrix theory to determine the energy variation of the eigenphase analytically rather than numerically. This overcomes a possible drawback in the numerical fitting procedures when locating the K-matrix poles and carrying out the arctan operation. The energy derivative of the K-matrix near an isolated resonance is then used in a fitting procedure to find the resonance position and width. According to QB, two matrices, Q and B, are defined in terms of the asymptotic solutions in the outer region of the R-matrix such that dK = B−1 Q dE

(3.18)

From this, the eigenphase gradients of the K-matrix with respect to energy can be obtained. Resonance positions, Er , are defined at the points of maximum gradient, while resonance widths, ∆r , are related to the eigenphase gradient at resonance, 0

δ (Er ), by the relation ∆r =

2 δ (Er ) 0

(3.19)

The widths of a series of resonances are calculated using (3.19) as a first approximation. The width of each resonance is then corrected by evaluating the background due to overlapping resonances to obtain a more accurate width. The QB method was implemented in the STGQB code of Quigley and coworkers [1998] as an extension to the R-matrix code. In this study we used STGQB as a supplementary tool for finding and investigating resonances.

3.2.1.2

Time-Delay Method

The Time-delay method of Stibbe and Tennyson [1998] is based on the time-delay theory of Smith [1960]. According to this theory, the time-delay matrix M is

3.2.1.2 Time-Delay Method

55

defined in terms of the scattering matrix S by

M = −i ~ S∗

dS dE

(3.20)

where i is the imaginary unit, ~ is the reduced Planck’s constant, S∗ is the complex conjugate of S, and E is the energy. Smith has demonstrated that the largest eigenvalue of the M-matrix represents the longest time-delay of the incident particle. The time-delay has a Lorentzian form with a maximum exactly at the resonance position. By calculating the time-delay as a function of energy and fitting the resultant time-delay to Lorentzians, the resonances can be located and their widths are identified. This method, as implemented in the TIMEDEL program of Stibbe and Tennyson [1998], takes the K-matrix from a scattering calculation, either read from a file or calculated on a dynamically adjusted grid. The S-matrix is then formed using the relation S=

I + iK I − iK

(3.21)

where I is the identity matrix, and the derivative of the S-matrix is computed numerically. The time-delay M-matrix is then calculated and diagonalized to find the longest time-delay experienced by the scattering particle. Resonances are located approximately by finding maxima in the time-delay, and the widths are estimated from the Lorentzian fit. The TIMEDEL program determines, from the overlap of adjacent resonances, whether each resonance should be fitted on its own or jointly with one or more other resonances. It finally outputs the positions and widths of the resonances to data files.

3.2.1.3 K-Matrix Method 3.2.1.3

56

K-Matrix Method

Resonances arise from poles in the scattering matrix S which varies slowly with energy. The K-matrix method used here is based on the fact that for the low-lying resonances just above the 1 Se atom ionization threshold, the scattering matrix S has only one channel, and hence the reactance matrix, K, is a real scalar with a pole near the resonance position at the energy mesh. According to the collision theory of Smith [1960], the lifetime matrix M is related to the S-matrix by Equation (3.20). Now, a K-matrix with a pole at energy E0 superimposed on a background Ko satisfies the relation Ki = Ko +

g Ei − E0

(3.22)

where Ki is the value of K-matrix at energy Ei and g is a physical parameter with dimension of energy. In Appendix A we revealed that in the case of single-channel scattering the M-matrix is real with a value given by

M=

(1 +

K 2 )(E o

−2g − E0 + 2Ko g(E − E0 ) + g2 )2

(3.23)

In line with the fact demonstrated by Smith [1960] that the lifetime of the state is the expectation value of M , it can be shown from Equation (3.23) that the position of the resonance Er is given by

Er = E0 −

Ko g 1 + Ko2

(3.24)

while the full width at half maximum ∆E is given by

∆E =

|2g| 1 + Ko2

(3.25)

The two parameters of primary interest to our investigation are the resonance

3.2.1.3 K-Matrix Method

57

position Er , and the resonance width ∆r which equals the full width at half maximum ∆E . However, for an energy point Ei with a K-matrix value Ki , Equation (3.22) has three unknowns, Ko , g and E0 , which are needed to find Er and ∆r . Hence, three energy points at the immediate neighborhood of E0 are required to identify these unknowns. As the K-matrix changes sign at the pole, the neighborhood of E0 is located by testing the K-matrix value at each point of the energy mesh for sign change. Consequently, the three points are obtained and used to find Er and ∆r . Figure (3.1) presents an authentic example of the K-matrix poles which are observed during resonance search by K-matrix method. Complete derivation of the K-matrix method can be found in Appendix A.

K-matrix

1.5E+15

Energy

-1.5E+15 Figure 3.1: Examples of single-channel K-matrix poles in the background. The two discontinuities in the curve are due to narrow resonances which do not appear on this relatively coarse mesh.

3.2.2 Preparing Target

3.2.2

58

Preparing Target

The first step in the R-matrix is to have a target for the scattering calculations. In our case, the C2+ target was obtained from Berrington et al [1977]. This target includes the 2s2 1 Se , 2s2p 3 Po , 1 Po and 2p2 3 Pe , 1 De , 1 Se eigenstates represented in terms of seven orthogonal orbitals; three physical and four pseudo orbitals. These orbitals are: 1s, 2s, 2p, 3s, 3p, 3d and 4f, where the bar marks the pseudo orbitals. The purpose of including pseudo orbitals is to represent electron correlation effects and to improve the target wavefunctions. The radial parts of these orbitals were obtained from the literature where they have been generated by the CIV3 program of Hibbert [1975] in Slater Type Orbital specification, as given in Table (3.1). Using Autostructure, the target was elaborated and extended by adding more terms from other configurations, and hence attaining several other targets with varying complexity. Of these, the main two which we used most in our calculations are the 9-term and the 26-term targets. These two, alongside the original 6-term target, are presented in Table (3.2). As seen, the 26-term target is obtained by including all the terms of the nine most important configurations of the C III. To obtain the three extra terms for the 9-term target, Autostructure was used to decide which terms are the most important by having the largest polarizability. This is based on the fact that the contribution to polarizability is proportional to the ratio of the oscillator strength of transition between levels u and l to the square of the 2 corresponding energy difference (i.e. Cp ∝ ful /∆Eul ). The Autostructure input

data file for this calculation is given in § C.2.1 in Appendix C.

3.2.3

Developing and Implementing K-matrix Method

In the early stages of our theoretical investigation to C II resonances, we experienced numerical instabilities, convergence difficulties and failures from the QB method

3.2.3 Developing and Implementing K-matrix Method

59

as implemented in the STGQB code. It was necessary, therefore, to verify the results that we obtained from QB by an independent method and fill the gaps left by the failure of STGQB to converge in some cases due to limitation on the width of resonances as it was established afterwards. This motivated the development of K-matrix approach which we implemented in the STGF stage by searching for sign-change in the K-matrix at the poles. This came after an attempt to implement the time-delay method of Stibbe and Tennyson [1998] once within STGF stage and another time as a stand-alone program that processes the data produced by STGF. However, this attempt was abandoned in favor of the K-matrix approach which

Table 3.1: The seven orbitals that we used to construct the C III STO parameters. The bar marks the pseudo orbitals. Orbital Ci Pi 1s 21.28251 1 6.37632 1 0.08158 2 -2.61339 2 -0.00733 2

targets and their ξi 5.13180 8.51900 2.01880 4.73790 1.57130

2s

-5.39193 -1.49036 5.57151 -5.25090 0.94247

1 1 2 2 2

5.13180 8.51900 2.01880 4.73790 1.57130

3s

5.69321 -19.54864 10.39428

1 2 3

1.75917 1.75917 1.75917

2p

1.01509 3.80119 2.75006 0.89571

2 2 2 2

1.47510 3.19410 1.83070 9.48450

3p

14.41203 -10.88586

2 3

1.98138 1.96954

3d

5.84915

3

2.11997

4f

9.69136

4

2.69086


60

Table 3.2: The configurations, terms and levels that we used to build the C III targets. A tick indicates the inclusion of the related states. 26-Term target Configuration Term J 1 Se 1s2 2s2 0 3 o 2 P 1s 2s 2p 0, 1, 1 Po 1 3 e 2 2 P 1s 2p 0, 1, 1 De 2 1 Se 0 2 3 e 1s 2s 3s S 1 1 Se 0 3 Po 1s2 2s 3p 0, 1, 1 Po 1 2 3 1s 2s 3d 1, 2, De 1 De 2 2 1 Po 1s 2p 3s 1 3 Po 0, 1, 1 Pe 1s2 2p 3p 1 3 De 1, 2, 3 Se 1 3 Pe 0, 1, 1 De 2 1 Se 0 3 Fo 1s2 2p 3d 2, 3, 1 Do 2 3 Po 0, 1, 3 Do 1, 2, 1 Fo 3 1 Po 1

2 2

9-Term target

Original target

X X X X X X X

X X X X X X

2 3

X

2 3

X

2

4 2 3

relies on the same physical principle as the time-delay method. To improve the performance of the K-matrix method, which proved to be very successful and computationally efficient, an interactive graphical technique was also developed to read the K-matrix data file directly and plot it against energy or arbitrary ordinal number while searching for poles. With a reasonably smooth mesh, a positive or negative pulse in the graph appears even when the background is large enough to prevent sign-change. As soon as this is detected, the search is stopped and resumed at higher resolution by focusing on a very narrow energy band using a fine mesh, and hence very small number of energy points are needed to


61

find the resonance. The strategy is to start the search with a coarse mesh over the suspected energy range. If the glitch failed to appear, the search is repeated with smoother mesh until the glitch is observed. This graphical technique was essential for finding resonances in reasonably short time compared to the time required by STGQB. In a later stage, non-graphical tools for poles search were developed and used. The purpose of these tools is to search for any sudden increase or decrease in the background of the K-matrix. As soon as this is detected, a search for poles with a finer mesh is resumed. These non-graphic tools proved to be more efficient and time-saving compared to the graphic tools, and hence helped substantially in finding most of the resonances in very short time. It is noteworthy that the Kmatrix method as implemented in STGF stage is completely automatic and can find and identify resonances without user interaction. The purpose of these graphic and non-graphic tools is to speed up the search and save the computing resources by reducing the required CPU time. In this paragraph we present a brief comparison between these two methods (i.e. K-matrix and QB) and assess their roles in this study and any possible future investigation. The main results of these methods are presented in Tables (D.2) and (D.3). As seen, K-matrix and QB produce identical results in most cases. However, K-matrix is computationally superior in terms of the required computational resources, mainly CPU time, especially when coupled with the above-mentioned interactive graphic and non-graphic techniques by first applying a coarse energy mesh and then zooming with a fine mesh over a very limited energy band as soon as a sign of resonance appears, i.e. a glitch in the smoothly-varying background is detected. In fact we obtained most of the QB results guided by the K-matrix results which were obtained earlier. Without K-matrix it would be extremely hard and time consuming to obtain results for some resonances, especially the very narrow ones, by QB directly. In general, the purpose of obtaining QB results is to

3.2.4 Calculations

62

check the K-matrix results and compare the two methods. Another advantage of the K-matrix method is that it has a wider range of validity with regard to the resonance width, that is in principle it can be used to find resonances with any width. The QB method fails to converge when the resonance width falls below a certain limit, whereas the only observable constraint on the K-matrix method is numerical precision. Nevertheless, the QB method is more general as it deals with multi-channel resonances, as well as single-channel resonances, while the K-matrix method in its current formulation is restricted to single open-channel resonances.

3.2.4

Calculations

Using the 9-term and the 26-term targets, extensive calculations were carried out in intermediate coupling scheme by including the spin-orbit interaction terms of the Breit-Pauli Hamiltonian. The IC was achieved by utilizing stage STGJK of the Rmatrix code with the levels and Jπ-symmetries identification. Sample calculations in LS-coupling were also performed in an early stage for completeness and check. The IC is required for a comprehensive C II recombination lines investigation as many states and transitions do not exist within LS-coupling. An important case is the quartets and some doublet states which have particular significance to our investigation. The requirement of IC scheme arises from the fact that in LScoupling the conserved quantities are LSπ and hence only the doublet states that conserve these quantities, such as 2 Se and 2 Po , can autoionize. Therefore, in LScoupling no autoionization is allowed for the quartets or other doublet states, such as 2 So and 2 Pe . However, in intermediate coupling what is conserved is Jπ, therefore the LS-allowed and LS-forbidden states with the same Jπ do mix giving access to channels that were formerly inaccessible, and hence these states can exist and autoionize. This is schematically depicted in Figure (3.2).

3.2.4 Calculations

63

Figure 3.2: Schematic diagram of the low-lying autoionizing quartet and doublet states and their interaction in IC scheme. In total, 150 bound states belonging to 11 symmetries (1,3,5,7,9 half even and 1,3,5,7,9,11 half odd) and 64 resonances belonging to 11 symmetries (1,3,5,7,9,11 half even and 1,3,5,7,9 half odd) were investigated. The input data files for the R-matrix calculations for the 9-term and the 26-term targets are given in § C.1 in Appendix C. To investigate the convergence and numerical stability, the number of continuum basis orbitals used to express the wavefunction in the inner region (MAXC in STG1) was varied between 6-41 and the results were analyzed. It was noticed that increasing the number of basis functions, with all ensuing computational costs, does not necessarily improve the results and the convergence instability may persist. It was decided therefore to use MAXC = 16 in all calculations as a compromise between the computational resources requirement and accuracy. The effect of varying the size of the inner region radius A was also investigated and a value of 10 au was chosen on the basis of numerical stability and convergence, that is the resulting values do not change on increasing the radius.

3.2.4 Calculations

64

Concerning the investigation of resonances, initially all computational work was carried out using the STGQB code since this was the only available tool before developing and implementing the K-matrix approach. Several parameters of the QB method were systematically inspected. The effect of varying the accuracy parameter, AC, in STGQB was investigated and a value of 10−6 was chosen for all calculations as a deal between accuracy and speed. The effect of modifying the step size for the effective quantum number mesh was studied and beside missing some resonances when the step is not appropriately small as expected, it is observed that the resonance position and width may depend on the step size. Including the first order perturbation in the long-range coupling potentials when calculating the reactance matrix was examined by turning ‘IPERT’ on and off (i.e. IPERT = 1 and 0) but no tangible change was observed. We therefore concluded that the perturbative treatment is not implemented in the STGQB code. The development of the K-matrix method provided a way for checking and filling the gaps of the QB results. In the initial formulation of the K-matrix the background was ignored, and although there was good agreement between QB and K-matrix in some cases where the background is small, there was significant differences in the other cases. Consequently, the current formulation which takes into account the effect of background was developed after some checking using Autostructure. The results produced by the K-matrix method, which are presented in Table (D.2), are perturbation-free, i.e. with ‘IPERT = NO’ in the stage STGF. Theoretical data with perturbative treatment was also produced for some resonances with both 9-term and 26-term targets at an early stage, but this work was abandoned. The main reason is the highly time-demanding nature of the perturbative calculations. It is estimated that the perturbative calculations are slower than the non-perturbative by a factor of 5-10 (Badnell [2002]). Our personal experience confirms this estimation.

3.2.4 Calculations

65

With regard to sampling the three points for the K-matrix calculations, it was noticed that sampling the points very close to the pole makes the energy and width of resonances susceptible to fluctuations and instabilities. We therefore adopted a sampling scheme where the points are selected from a broad range not too close to the pole. This approach was implemented by generating two meshes, coarse and smooth, around the pole when it is found. To check the results, several dozens of different three-point combinations for each resonance were used to find the position and width of the resonance. In each case, the results from these different combinations were compared. In all cases, the stability of the results were astonishing as they were almost identical within acceptable numerical errors. We therefore concluded that the sampling scheme is reliable. The results of QB, presented in Table (D.3), confirm this conclusion as they agree with the K-matrix results as presented in Table (D.2). As seen in Tables (D.1-D.3), the theoretical data for both bound and resonance states agree well with the available experimental data both in energy levels and in fine structure splitting. Experimental energies are not available for the very broad resonances as they are difficult to find experimentally. The maximum discrepancy between experiment and theory in the worst cases do not exceed a few percents and hence the results are very satisfactory. Furthermore, the ordering of the energy levels is also the same between the theory and experiment in most cases. Order reversal in some cases is indicated by a minus sign in the fine structure splitting. An interesting fact is that while the theoretical energy for all 9-term states are higher than the experimental, for the 26-term states the energy goes both ways. This indicates that the 26-term results are generally closer to experiment and hence better. However, this is not the case always particularly for the fine structure splitting where the 9-term results are better in some cases.

3.3 f -Values Calculations

3.3

66

f -Values Calculations

In the following sections we outline the procedures that we followed to produce the f -values for FF, FB and BB transitions.

3.3.1

f -Values for Free-Free Transitions

As we are not aware of the existence of a free-free stage for the R-matrix code, we could not produce the f -values for the FF transitions by RMATRX-I. We therefore produced these values by Autostructure in intermediate coupling scheme. Consequently, this data is the same for both the 9-term and the 26-term targets. Moreover, the quality of Autostructure data are not as good as the other data generated by RMATRX-I. Because Autostructure produces data only to a certain level of approximation (i.e. no data is output when the value falls below a certain limit) some of these data were missing from the ‘olg’ file of the Autostructure output and hence we substituted zero for these missing entries. In the Autostructure input we included 60 configurations, which are: 2s2 nl (2p≤ nl ≤7s), 2s2p nl (2p≤ nl ≤7s), 2p3 , and 2p2 nl (3s≤ nl ≤7s). An iterative procedure was followed to find the scaling parameters (λ’s) for Autostructure before producing the required data. The Autostructure input data file for generating the f -values for the FF transitions is given in § C.2.3 in Appendix C. It should be remarked that this input data file was also used with Autostructure to generate f -values for the FB transitions and the BB transitions for the 8 topmost bound states, namely the 4 Fo and 4 Do levels, as these states have large effective quantum number and hence are not accessible to RMATRX-I code.

3.3.2 f -Values for Free-Bound Transitions

3.3.2

67

f -Values for Free-Bound Transitions

The f -values for more than 2500 FB transitions for each target (9-term and 26term) were computed by integrating the peaks of the photoionization cross sections (in Mb) over the photon energy (in Ryd). This is based on the fact demonstrated by Storey [1994] that the contribution of a resonance to the recombination rate depends on the area under it, which is also a measure of the oscillator strength between the resonance and the interacting state, while the width of a resonance depends on the strength of the interaction with the accessible continuum states. The first step therefore is to obtain the photoionization cross section of a particular transition from a lower bound-state l to the continuum, σ(l→u) . Since the resonance contribution to a particular radiative process is proportional to the area beneath σ for that process, by integrating σ over energy the f -value for a transition from an upper resonance state u to a lower bound state l is found. The photoionization cross sections as a function of photon energy are generated by stage STGBF of the R-matrix code where they are stored in ‘XSECTN’ file. Several graphs of these peaks are displayed in Figures (3.3) and (3.4). Figure (3.3) displays a number of randomly selected examples of photoionization cross sections of the indicated bound states over the displayed range of photon energy on loglinear graphs, while Figure (3.4) displays a number of randomly selected examples of photoionization cross sections of the indicated bound states with respect to the designated autoionizing states versus photon energy on linear-linear graphs. In the following points we outline the general procedure that we followed to compute the oscillator strengths for the FB transitions:

• RESMESH program of P.J. Storey [private communication] was used to generate an energy mesh that maps the resonances in the most accurate way.


1E+12 1E+8

68

1E+4

1E+12

1E+2

1E+8

1E+0

1E+4

1E-2

1E+0

1E+4 1E+0 1E-4 1E-8 1.40

1.45

1.50

1.55

2 4

(a) 2s2p

1.60

1.65

1E-4 0.35

e

P1/2

0.40

0.45

0.50

0.55

2

2 e

0.60

1E-4 0.75

0.85

0.90

2 2

(b) 2s 4s S1/2

(c) 2s2p

1E+10

1E+10

0.80

0.95

1.00

0.34

0.39

e

P3/2

1E+10

1E+8 1E+6

1E+6 1E+6 1E+4

1E+2

1E+2

1E+2 1E-2

1E-2 1E+0

1E-6 0.14

0.19

0.24

3

0.29

0.34

o

4

0.39

1E-2 1.10

e

1.15

1.20

1.25

2 2

(d) 2s2p( P )3p D3/2

(e) 2s2p

1E+1

1.30

1.35

1E-6 0.14

e

0.19

0.24

3

D5/2

0.29

o

4

e

(f) 2s2p( P )3p D5/2 1E+0

1E+10 1E+6

1E+0

1E-1

1E+2 1E-2

1E-1

1E-2

1E-6 1E-2 0.17

0.22

0.27

0.32

2

2

0.37

0.42

1E-10 0.14

e

0.19

0.24

3

(g) 2s 5g G7/2

0.29

o

0.34

4

0.39

1E-3 0.12

e

0.22

0.27

2

(h) 2s2p( P )3p D7/2

1E+8

0.17

2

0.32

0.37

0.80

0.85

0.47

0.52

e

(i) 2s 6g G9/2 1E+7

1E+12 1E+8

1E+5

1E+4 1E+3 1E+0

1E+2

1E-4 1E-1 1.80

1.85

1.90

1.95

2.00

2.05

1E-8 0.27

(j) 2s2 2p 2 Po1/2

0.32

0.37

0.42

0.47

0.52

1E-1 0.60

(k) 2s2p(3 Po )3s 4 Po1/2 1E+10

1E+8

1E+0

1E+6

1E+4

1E-3

1E+2

1E+0

1E-6

1E-2

1E-4

0.55

0.60

0.65

0.70

0.75

1E-6 0.41

3 4 o

(m) 2p

0.46

0.51

3 2

S3/2

(n) 2p

0.56

0.61

0.66

1E-8 0.27

o

0.32

0.75

0.37

3

D5/2

0.42

o

4

o

(o) 2s2p( P )3s P5/2

1E+3

1E+6

0.70

(l) 2s2 3p 2 Po3/2

1E+3

1E-9 0.50

0.65

1E+0

1E+4

1E-1 1E+1

1E+2

1E-2 1E-1

1E+0 1E-2 0.25

1E-3

0.30

0.35

2

0.40

2

o

(p) 2s 4f F7/2

0.45

0.50

1E-3 0.12

0.17

0.22

2

0.27

2

0.32

o

(q) 2s 6h H9/2

0.37

1E-4 0.09

0.14

0.19

2

0.24

2

0.29

0.34

o

(r) 2s 7h H11/2

Figure 3.3: Examples of photoionization cross sections in Mb (y-axis) of the indicated bound states over the displayed range of photon energy in Ryd (x-axis) on log-linear graphs for the 9-term target.


69

2.5E+8

2.0E+8

1.5E+8

1.0E+8

5.0E+7

0.0E+0 1.422259535

1.422259540

2 4

(a) 2s2p

1.422259545

e

1.422259550

3

o

4

o

2

o

1.422259555

P1/2 bound with 2s2p( P )3d P1/2 resonance.

5.0E-1

4.0E-1

3.0E-1

2.0E-1

1.0E-1

0.0E+0 1.6280

1.6281

2 4

(b) 2s2p

1.6282

1.6283

3

e

o

1.6284

1.6285

P3/2 bound with 2s2p( P )4d F5/2 resonance.

1.5E+5

1.2E+5

9.0E+4

6.0E+4

3.0E+4

0.0E+0 0.575353

0.575354

0.575355

0.575356

0.575357

0.575358

(c) 2p3 2 Do5/2 bound with 2s2p(3 Po )4p 2 Pe3/2 resonance.

Figure 3.4: Examples of photoionization cross sections in Mb (y-axis) of the indicated bound states with respect to the designated autoionizing states versus photon energy in Ryd (x-axis) on linear-linear graphs for the 26-term target.


70

The input data required by RESMESH are the position and width of resonances, the energy range of the mesh and an integral error index. In our case, the positions and widths of resonances were obtained by K-matrix method as implemented in stage STGF of the R-matrix code. • The mesh generated by RESMESH is then used as an input to STGF to create F-files which contain the required data for resonances. • Stage STGBF is then run to create ‘XSECTN’ file which contains the data for photoionization cross sections (in Mb) versus photon energies (in Ryd). • The peaks of photoionization cross sections are extracted by graphic techniques. The numerical data are then used to obtain the f -values by integrating the photoionization cross sections over photon energy using a 3-point Simpson’s rule. A piece of code from P.J. Storey [private communication] was used to perform the integration with the required scaling. Visual inspection to each peak during the extraction of data and computing the oscillator strengths was carried out to check the goodness of profile and verify that it contains sufficient number of points. In general, the data points that define the peak profile and used for the integrated cross sections were chosen to be as close as possible to the background on both sides considering some sort of symmetry and avoiding mixing with other resonances. The possibility of mixing between neighboring resonances was eliminated later on by producing photoionization cross sections data (‘XSECTN’ files) for each resonance independently with the adoption of one-resonance mesh approach, as will be discussed next. A refinement process was used to improve the positions of the resonances by inputting the results of one run to the next during mesh generation and data creation. The purpose of refinement is to optimize the resonances positions by


71

having the best possible profile with sharp and well defined peaks with sufficient data points. In the refinement process the RESMESH cycle, as outlined in the previous steps, was repeated several times (about five) where the results of one cycle was fed to the next. To find the positions in each cycle, we first find all the peaks belonging to the resonance of interest from ‘XSECTN’ file created in that cycle. We then take the energy position of the largest peak as an input to RESMESH in the next run. As the refinement progresses, the situation steadily improves in terms of peaks height and shape until it stabilizes where no improvement is observed on performing more iterations. Most of this process was carried out by automated procedures using codes and other computing tools, and the results were thoroughly inspected and checked. It should be remarked that although we select the position of the largest peak as an input to the next cycle, the situation in general improves for almost all the peaks for that resonance not only for the largest peak. Initially we used a single mesh to optimize the position of all resonances in a single run. We started with the assumption that the results will improve as refinement progresses from one cycle to the next. However, we noticed that although the results generally do improve for some resonances they deteriorate for others. Moreover, there was a considerable overlap between some resonances especially for those from multiplets belonging to different symmetries. We then started a new approach of refinement by continuing the refinement for the ones that continue to improve while holding the deteriorated ones at the position of their best photoionization cross section value. However, the results did not move in the direction that we anticipated and similar behavior to the previous (i.e. deterioration for some resonances) was observed. The apparent conclusion is that the resonances within a mesh interact, and therefore it is difficult to control the individual resonances within the mesh. Another serious problem with the single mesh is the size of the data files and the confusion created by the simultaneous

3.3.3 f -Values for Bound-Bound Transitions

72

presence of many resonances for each bound state data set. Moreover, when some resonances are too broad or too close to each other they mix up creating more sources of confusion and error. Another problem with the single-mesh approach is that the data files are too bulky and confusing and hence they are very difficult to manage and extract the data from them. These factors made processing the data without serious errors unfeasible. We therefore adopted a different approach by generating a mesh for each individual resonance. Although this seems more demanding in terms of work and computer time, it proved to be better than the single-mesh approach even in these aspects. Regarding the amount of work and quality of results, the one-resonance mesh approach requires much less work and produces very clear and reliable results. Moreover, the extraction of data can be highly automated. However, the one-resonance mesh requires more preparation work since preparing the data and running the code are repeated as many times as the number of resonances, but this is more than compensated for by the other advantages. An important advantage of the one-resonance mesh approach is that for the multiplets of the very broad resonances, such as the doublet 2s2p(3 Po )4s 2 Po1/2 and 2 Po3/2 , we obtained a separate data set for each level, whereas only one data set which combines these two levels was obtained in the previous attempts using the single-mesh approach.

3.3.3

f -Values for Bound-Bound Transitions

No special effort was required to compute the f -values for the BB transitions as these calculations were performed using stage STGBB of the R-matrix code. The ‘FVALUE’ data files which contain the BB f -values were used with no modification (apart from adding extra data as will be discussed next) as an input to ‘Emissivity’ program (documented in Appendix B) whose main function is to find the emissivity

3.4 Emissivity Calculations

73

of transitions and other relevant quantities. The input data files for all R-matrix stages, including STGBB, for both targets are given in § C.1 in Appendix C. It should be remarked that the BB f -values for the 8 uppermost bound states, namely the 4 Fo and 4 Do levels, were generated by Autostructure and then added manually to ‘FVALUE’ files that contain the data for the other bound states. The reason for this special treatment is that these states have very large effective quantum number and hence are out of range of the R-matrix code validity. The input data file for Autostructure to generate these f -values is given in § C.2.3 in Appendix C.

3.4

Emissivity Calculations

The physical parameter of primary interest to our investigation is the emissivity (power per unit volume) of nebulae to a particular recombination line by radiative decay from an upper autoionizing state u to a lower state l and subsequent cascades. Emissivity calculations are the third major part of our work on the C II recombination lines. A relatively detailed account of this part will be given in the next subsections.

3.4.1

Practical Aspects

The main tool that we used in this part is the ‘Emissivity’ code [2010] which is documented in Appendix B. Other tools were also used to carry out extensive tests. These include manual calculations and parallel implementations by code fragments and spreadsheets. Systematic checks were performed throughout all stages of writing ‘Emissivity’ code and producing the emissivity results. Some of these checks are presented below

3.4.1 Practical Aspects

74

• The emissivity calculations were implemented in spreadsheets in various stages of developing ‘Emissivity’ code. The results of these two different implementations are found to be identical. • Random checks using manual calculations was carried out throughout all stages of writing and using the code to check the results. • Two physical tests, presented in § B.5, to check the physical sensibility of the results were also implemented within the ‘Emissivity’ code and all emissivity calculations passed these tests. These checks and tests assisted in debugging the ‘Emissivity’ code and identifying sources of error and malfunctioning. Consequently, the program and its results passed all these checks and tests. Moreover, the emissivity calculations from three versions of ‘Emissivity’ program produced by different compilers were compared and found to be identical. Regarding the input data, all theoretical data for the energy of resonances and bound states were replaced with experimental data from NIST when experimental data were available. This is based on the approach of Nussbaumer and Storey which we mentioned in Section (1.2.1). The use of experimental positions removes the uncertainty and errors which are usually introduced by ab initio calculations, although the quality of the wavefunctions and the data may deteriorate. As a consequence of this replacement, the energies for the 9-term and the 26-term targets are identical in most cases. Another point is that for the f -values that were produced by Autostructure (i.e. for the FF transitions and for the transitions involving the 4 Fo and 4 Do states) the same data were used in the emissivity calculations for both targets. As part of our astronomical investigation, we surveyed a large amount of potential data sources in the literature and databases on the spectra of planetary

3.4.1 Practical Aspects

75

nebulae and recombination lines of C II. This includes

• Extensive survey to the Planetary Nebula Emission Line Catalog (ELCAT). • Inspecting a representative sample comprising several hundreds of scientific papers that were collected under suitable criteria. • Extensive investigation to the spectrum of the planetary nebula NGC 7027 where several data sets were collected.

Although we found some interesting data sets, the results in general were either limited or out of date. The main exception is the data set of Zhang et al [2005] on the spectrum of the planetary nebula NGC 7027. This data set was found to be the most thorough and up to date, and contains a large number of recombination lines. We therefore decided to take this as the sole data source for our investigation of C II recombination lines. The C II lines that we extracted from Zhang et al [2005] and used in this investigation are presented in Table (D.26) in Appendix § D. The Table contains 35 lines. These are the C II lines after removing the ones that are blended with non-C II lines. However, the lines that are blended with other C II lines are included in this list. All these 35 lines satisfy the n < 5 resonance condition and come from transitions where at least one of the interacting states have an excited core. It is noteworthy that all these 35 observational lines were identified in our theoretical line list which is generated by ‘Emissivity’ code and contains 6187 optically-allowed transitions. It should be remarked that all transitions in our calculations are subject to the electric dipole (E1) rules. Another remark is that the autoionization and radiative probabilities in the emissivity calculations as outlined in § 3.1 can also be obtained from Autostructure. In the early stages of our research we adopted such an approach by producing some data by RMATRX-I and some by Autostructure.

3.4.2 Emissivity Analysis

76

According to the old approach, the widths of resonances are first obtained by the K-matrix method, and from which the autoionization transition probabilities, Γa , are computed. The corresponding radiative transition probabilities, Γru , are then found by Autostructure. The configurations that were used in the radiative calculations are the five most important ones for C II: 1s2 2s2 3d, 1s2 2s2p2 , 1s2 2s2p 3p, 1s2 2s2p 3d and 1s2 2s2p 4s. The Autostructure input data file for these calculations is given in § C.2.2 in Appendix C. The departure coefficients, bu , are then calculated and the states with bu 1 are excluded from further investigation because of their very low likelihood of existence. However, this approach was abandoned afterwards because it is an approximation as the quality of Autostructure data is inferior to the RMATRX-I data. Moreover, it became redundant as we were able to produce all the required data by RMATRX-I. The merit of this approach is that it is less demanding in terms of time and computational resources since producing data by Autostructure is easier and faster than RMATRX-I. This approach was also used to check the R-matrix results by comparing the departure coefficients from RMATRX-I to those from Autostructure. Good agreement were found in most cases between the two methods on a sample data set.

3.4.2

Emissivity Analysis

The main method for generalizing and summarizing a set of data is by fitting it to a model that depends on adjustable parameters. A figure-of-merit function that measures the agreement between the data and the model for a particular choice of parameters is then chosen or designed so that small values represent close agreement. The model parameters are then adjusted to achieve a minimum in the merit function, yielding best fit parameters. Least squares (LS) minimization is probably the most popular method in the physical sciences. It is widely used to determine the best set of parameters in a model to fit a set of observational data.


77

The objective of the least-squares technique is to minimize the difference between the observed data and the calculated model. The goodness-of-fit index χ2 is one of the most meaningful and generally accepted figures of merit. It is widely used as an indicator of the overall fit between a theoretical model and a set of observational data. In general terms, it is given by

2

χ =

P

i

wi (yio − yic )2 η

(3.26)

where yio and yic are the observed and calculated values at step i, and wi is the corresponding weight. η is the number of degrees of freedom given by η = O − P where O and P are the numbers of observations and fitted parameters in the calculated model, respectively. The summation index i runs over all observed data points. In our investigation a least squares procedure was used to compare our theoretical model on the emissivity of recombination lines in planetary nebula to observations. In our LS calculations we have a single fitting parameter, which is the temperature of the line-emitting regions in NGC 7027 nebula, while the observations are the emissivity data of the C II recombination lines that we obtained from Zhang et al [2005]. The purpose of this procedure is to find the optimal temperature that gives the best fit of the theoretical model to the observational data, and hence identifying the temperature of the line-emitting regions in the nebula. We also used χ2 index as a statistical indicator to measure the goodness-of-fit. A procedure for finding the confidence level interval was implemented in the ‘Emissivity’ program. Here we outline how the least squares minimization and the confidence level interval procedures were implemented:

• Only ‘good’ theoretical resonances are used in the least squares procedure and χ2 analysis. The criterion for being good is that the percentage difference in


78

the departure coefficient, bu between the 9-term and the 26-term targets does not exceed a certain limit, say 25%. Out of 64 resonances found theoretically, only 45 of them were classified as good according to this criterion. • Only ‘good’ observational resonances are used in the least squares procedure and χ2 analysis. The criterion for being good is based on their profile and segregation from neighboring resonances in the observed spectra. From the 35 selected observational lines only 14 were qualified to be good according to this criterion. • The theoretical lines are mapped to the observational lines using a 1-angstrom rule, that is all theoretical lines within 1˚ A on both sides of the observational line were added up and mapped to the observational line. This mapping scheme is justified by the resolution of the observational data where all lines within a certain limit on both sides are blended and cannot be identified separately by observational means. The 1˚ A resolution of the observational data was estimated from the graphs of spectra as given in Zhang et al [2005]. • All blended C II lines in the observational list were combined by considering them as a single line with a single emissivity. • The theoretical emissivities are normalized to the theoretical emissivity of the C II 426.7 nm line, which is the most intense line of this ion in the visible part of the spectrum, while the observational emissivities are normalized to the observational emissivity of the C II 426.7 nm line. As the observational C II 426.7 nm emission consists of two blended lines, the combined emissivity was used for normalization. The observational emissivities were obtained from Zhang et al, while the theoretical emissivities were obtained from Davey et al [2000] in the form of case B effective recombination coefficients, and option ‘4’ of the ‘NormalizationChoice’ of the ‘Emissivity’ code was used, as explained


79

in Appendix B. • The normalized theoretical emissivities corresponding to a particular observational emissivity were added up when multiple mapping occurs. • The sum of squares of differences between the observational and theoretical normalized emissivities as a function of temperature is computed. This sum is given by S=

X

N εN o − εt

2

(3.27)

∀ lines N where εN o and εt are the normalized observational and theoretical emissivities

respectively. • The minimum sum of the squares of differences between the normalized theoretical emissivities and their normalized observational counterparts is found. • The minimum of χ2 is obtained from the degrees of freedom, (i.e. χ2min = η = O − P ). • A scaling parameter, a, is found by dividing the second norm at the minimum least squares residual by the minimum χ2 , that is

a=

!

1

X

χ2min

i

wi (yio

−

yic )2

(3.28) min

where the weighted square residual sum is the minimum of these sums and wi = 1/yio . • χ2 as a function of temperature is computed from the relation

2

χ =

P

i

wi (yio − yic )2 a

(3.29)

• From the tabulated data of ∆χ2 as a function of confidence level and de-


80

grees of freedom the confidence interval for temperature is found by linear interpolation. The main conclusions that can be extracted from the least squares procedure and the goodness-of-fit index analysis are • No high quality confidence interval with a smooth χ2 profile and deep valley were obtained, as this can be seen in Figures (3.5) and (3.6). Also a difference between the 9-term and the 26-term results was observed. One possible source of discrepancy between the 9-term and the 26-term results is the inclusion and exclusion of theoretical lines according to the 1˚ A rule. This is because the theoretical energy data that are used when experimental data is unavailable differ between these two targets. Another possible reason is that as we did not find three of the resonances for the 26-term target we used 10−16 as a value for their width. This could have an impact on the results and exacerbated the difference between the 9-term and 26-term. • Though the effect of bad theoretical resonances has been reduced, their effect can not be completely eliminated as long as they are present, and hence contribute indirectly to the population and decay processes. Total elimination of these resonances by removing them from the list is not an option as this will remove an important part of the states that contribute to the population and decay processes and block a large part of the decay routes. • In one experiment to investigate the bad resonances, we replaced the widths of the 9-term with the widths of the 26-term and vice versa, to identify the source of discrepancy between the results of these targets. By doing this, almost all resonances became good even after reducing the relative marginal error substantially. This reveals that the main reason for discrepancy is not the f -values data but the resonances data from the R-matrix code.


81

30

25

χ

2

20

15

10

5

0 0

5000

10000

15000

20000

Temperature (K)

Figure 3.5: Graph of χ2 of normalized emissivity (relative to CII 426.7 nm line) as a function of temperature for the emissivity model compared to the observational data of Zhang et al [2005] for the 9-term target. The confidence interval limits for 68% confidence level and the temperature at minimum χ2 are also displayed. 30

25

χ

2

20

15

10

5

0 0

5000

10000

15000

20000

Temperature (K)

Figure 3.6: Graph of χ2 of normalized emissivity (relative to CII 426.7 nm line) as a function of temperature for the emissivity model compared to the observational data of Zhang et al [2005] for the 26-term target. The confidence interval upper limit for 68% confidence level and the temperature at minimum χ2 are also displayed.


82

• However, one important conclusion is that the results of both targets agree on the temperature of the line-emitting region (about 10200 − 10500 K) within acceptable numerical errors.

Chapter 4 Conclusions and Future Work

4.1

Conclusions

Some results and conclusions that can be drawn from this work are: • Our theoretical results for energy and fine structure splitting for both targets agree very well with the available experimental data for both resonances and bound states. This is another example for the success of R-matrix approach and RMATRX-I code. • In general the results of the 26-term target are superior to the 9-term target, though this is not true in some cases especially for the fine structure splitting. • Another method for finding and analyzing resonances, namely the K-matrix method of P.J. Storey, was developed in the course of this investigation. • The QB and K-matrix methods agree very well taking into account numerical errors and approximations. The QB method as implemented in STGQB code seems to suffer from shortcomings which may deserve further investigation in the future. As far as the search for the low-lying autoionizing states 83

4.2 Recommendations for Future Work

84

is concerned, the K-matrix method is superior to the QB method in terms of numerical stability, computational viability and comprehensiveness. However, the big advantage of QB is that it is more general and can be used for multi-channel as well as single-channel resonances. • The emissivity investigation revealed some discrepancy between the 9-term and the 26-term targets. The most likely reason is the discrepancy between the two targets in the width of resonances which affects autoionization probability. Some of the conducted tests suggest that the FB f -values are not responsible for this discrepancy. • The χ2 test converges to a unique value of temperature for both targets within acceptable error tolerance (10500 K for the 9-term and 10200 K for the 26term), although the interval for the 68% confidence level is broad, as seen in Figures (3.5) and (3.6).

4.2

Recommendations for Future Work

Some prospective aspects of the future work are • Including theoretical treatment of radiative recombination process and incorporating this into ‘Emissivity’ code. • Extending the K-matrix method of P.J. Storey to multi-channel resonances. • Pushing the limit of the principal quantum number for resonances above n < 5. This extension becomes more important at higher temperatures, as the contribution of resonances arising from the n ≥ 5 levels increases. • Investigating the reason for the difference between the 9-term and 26-term results more thoroughly.

Bibliography Aldrovandi S.M.V. and Péquignot D. (1973) Radiative and dielectronic recombination coefficients for complex ions. Astronomy and Astrophysics 25: 137-140. 18, 19, 22 Aldrovandi S.M.V. and Péquignot D. (1973) Ionization and Heating of a Low Density Plasma by Energetic Particles. Astronomy and Astrophysics 26: 33-43. 19 Badnell N.R. (1988) The influence of core fine-structure interactions on dielectronic recombination at low temperatures: B-like C, N and O recombined ions. Journal of Physics B 21: 749-767. 26, 28 Badnell N.R. (2002) RMATRX-I writeup on the world wide web. URL: http://amdpp.phys.strath.ac.uk/UK RmaX/codes/serial/WRITEUP. 41, 64, 109 Badnell N.R., O’Mullane M.G., Summers H.P., Altun Z., Bautista M.A., Colgan J., Gorczyca T.W., Mitnik D.M., Pindzola M.S. and Zatsarinny O. (2006) Dielectronic recombination data for dynamic finite-density plasmas. Astronomy & Astrophysics. 18, 20 Badnell N.R. (2008) Autostructure writeup on the world wide web. URL: http://amdpp.phys.strath.ac.uk/autos/ver/WRITEUP. 43, 109 Balick B., Gammon R.H. and Doherty L.H. (1974) The Structure of the Orion Nebula. I. Observations of the C 85α Recombination Line. Astrophysical Journal 188: 45-52. 25 Baluteau J.P., Zavagno A., Morisset C. and Péquignot D. (1995) The 654-1046 nm line spectrum of the planetary nebula NGC 7027. Astronomy and Astrophysics 303: 175203. 27 Barlow M.J. (2006) Planetary Nebulae Beyond the Milky Way - Historical Overview. ESO Astrophysics Symposia, pp. 3-14. 17 Bartschat K. and Burke P.G. (1986) Resfit - A multichannel resonance fitting program. Computer Physics Communications 41(1): 75-84. 53

85

D.R. Bates and H.S.W. Massey (1943) The Negative Ions of Atomic and Molecular Oxygen. Philosophical Transactions of the Royal Society of London, Series A 239(806) 269-304. 21 Beigman I.L. and Chichkov B.N. (1980) Dielectronic recombination through the forbidden levels. Journal of Physics B 13: 565-569. 22 Bell R.H. and Seaton M.J (1985) Dielectronic recombination I. General theory. Journal of Physics B 18: 1589-1629. 23 Benoy D.A., Mullen J.A.M. and Schram D.C. (1993) Radiative energy loss in a nonequilibrium argon plasma. Journal of Physics D 26(9): 1408-1413. 47 Berrington K.A., Burke P.G., Dufton P.L. and Kingston A.E. (1977) Electron collisional excitation of C III and O V. Journal of Physics B 10(8): 1465-1475. 58 Berringtont K.A., Eissner W.B., Saraph H.E., Seaton M.J. and Storeys P.J. (1987) A comparison of close-coupling calculations using UCL and QUB codes. Computer Physics Communications 44: 105-119. 34 Berringtont K.A., Burke P.G., Butler K., Seaton M.J., Storeys P.J., Taylor K.T. and Yu Yan (1987) Atomic data for opacity calculations: II. Computational methods. Journal of Physics B 20: 6379-6397. 36, 37, 38, 40, 41 Berrington K.A., Eissner W.B. and Norrington P.H. (1995) RMATRX1: Belfast atomic R-matrix codes. Computer Physics Communications 92(2): 290-420. 33, 38, 40, 41 Bogdanovich P.O., Nikitin A.A., Rudzikas Z.B. and Kholtygin A.F. (1985) Lines of carbon, nitrogen, and oxygen ions in the spectra of planetary nebulas. II. Intensities of the C II and N III recombination lines and abundances of the C III and N IV ions. Astrofizika 23(2): 427-435. 26 Boughton W.L. (1978) Carbon Recombination Lines and the Neutral Hydrogen Clouds Near the Orion Nebula. The Astrophysical Journal 222: 517-526. 25 Burgess A. (1964) Delectronic recombination and the temperature of the solar corona. Astrophysical Journal 139: 776-780. 21 Clavel J., Flower D.R. and Seaton M.J. (1981) Ultraviolet spectra of planetary nebulae - V. The C II λ1335 dielectronic recombination lines in IC 418. Monthly Notices of Royal Astronomical Society 197: 301-311. 25 Clegg R.E.S., Seaton M.J., Peimbert M. and Torres-Peimbert S. (1983) Analysis of nebulosity in the planetary nebula NGC 40. Monthly Notices of Royal Astronomical Society 205: 417-434. 25

86

Davey A.R. (1995) The recombination spectrum of carbon II. PhD thesis, University College London. 47 Davey A.R., Storey P.J. and Kisielius R. (2000) Recombination coefficients for C II lines. Astronomy and Astrophysics Supplement Series 142: 85-94. 27, 28, 78 Davies P.C.W and Seaton M.J. (1969) Radiation damping in the optical continuum. Journal of Physics B 2(7): 757-765. 22, 23 Eissner W., Jones M. and Nussbaumer H. (1974) Techniques for the calculation of atomic structures and radiative data including relativistic corrections. Computer Physics Communications 8(4): 270-306. 42, 43 Ferland G.J. (2003) Quantitative Spectroscopy of Photoionized Clouds. Annual Review of Astronomy & Astrophysics 41: 517-54. 20, 23, 29 Garnett D.R. and Dinerstein H.L. (2001) The O II Recombination Line Abundance Problem in Planetary Nebulae. Revista Mexicana de Astronom´ıa y Astrof´ısica (Serie de Conferencias), 10: 13-18. 31 Harmin D.A. (1986) Precise Theory of Field Enhancement of Dielectronic Recombination. Physical Review Letters 57(13): 1570-1573. 23 Harrington J.P., Lutz J.H., Seaton M.J. and Stickland D.J. (1980) Ultraviolet spectra of planetary nebulae - I. The abundance of carbon in IC 418. Monthly Notices of Royal Astronomical Society 191: 13-22. 25 Harrington J.P., Lutz J.H. and Seaton M.J. (1981) Ultraviolet spectra of planetary nebulae - IV. The C III λ2297 dielectronic recombination line and dust absorption in the C IV λ1549 resonance doublet. Monthly Notices of Royal Astronomical Society 195: 21P-26P. 25 Hayes M.A. and Nussbaumer H. (1984) The C II infrared and ultraviolet lines. Astronomy and Astrophysics 134(1): 193-197. 26 Hibbert A. (1975) CIV3 - A general program to calculate configuration interaction wave functions and electric-dipole oscillator strengths. Computer Physics Communications 9(3): 141-172. 42, 58 Illingworth V. and Clark J. [Editors] (2000) Collins Dictionary of Astronomy. HarperCollinsPublishers. 17 Kholtygin A.F. (1984) Carbon Abundance in planetary nebulas. Astrofizika 20(3): 503511. 26

87

Kholtygin A.F. (1998) Inhomogeneous planetary nebulae: carbon and oxygen abundances. Astronomy and Astrophysics 329: 691-703. 26, 30 Kisielius R., Storey P.J., Davey A.R. and Neale L.T. (1998) Recombination coefficients for Ne II lines at nebular temperatures and densities. Astronomy and Astrophysics Supplement 133: 257-269. 29 LaGattuta K., Nasser I. and Hahn Y. (1987) The effect of static electric fields on dielectronic recombination: I. Basic theory. Journal of Physics B 20: 1565-1576. 23 Leibowitz E.M. (1972) The emission spectrum of the ion C IV in planetary nebulae. Monthly Notices of Royal Astronomical Society 157: 97-102. 24 Leibowitz E.M. (1972) Polarization of C IV emission lines in planetary nebulae. Monthly Notices of Royal Astronomical Society 157: 115-120. 24 Liu, X.-W., Storey P.J., Barlow M.J. and Clegg R.E.S. (1995) The rich O II recombination spectrum of the planetary nebula NGC 7009: new observations and atomic data. Monthly Notices of the Royal Astronomical Society 272(2) 369-388. 30 Liu X.-W. (2002) Optical Recombination Lines and Temperature Fluctuations. In: Proceedings of the Conference to Celebrate the 60th Birthdays of Silvia Torres-Peimbert and Manuel Peimbert, Mexico City, November 21-24, 2000. Vol. 12, pp. 70-76. 30 Liu Y., Liu X.-W., Barlow M.J. and Luo S.-G. (2004) Chemical abundances of planetary nebulae from optical recombination lines - II. Abundances derived from collisionally excited lines and optical recombination lines. Monthly Notices of the Royal Astronomical Society 353(4): 1251-1285. 27, 31 Massey H.S.W and Bates D.R. (1942) The properties of neutral and ionized atomic oxygen and their influence on the upper atmosphere. Reports on Progress in Physics 9(1): 62-74. 21 Matzner R.A. [Editor] (2001) Dictionary of Geophysics, Astrophysics and Astronomy. CRC Press. 17 Nahar S.N. and Pradhan A.K. (1994) Unified treatment of electron-ion recombination in the close-coupling approximation. Physical Review A 49(3): 1816-1835. 24 Nahar S.N. and Pradhan A.K. (1995) Unified Electron-Ion Recombination Rate Coefficients of Silicon and Sulfur Ions. Astrophysical Journal 447: 966-979. 24 Nahar S.N., Pradhan A.K. and Zhang H.L. (2000) Electron-ion recombination rate coefficients and photoionization cross sections for astrophysically abundant elements. IV. Relativistic calculations for C IV and C V for ultraviolet and X-ray modeling. The Astrophysical Journal Supplement Series 131: 375-389. 18

88

National Institute of Standards and Technology (NIST). URL: http://www.nist.gov. 14, 46 Nussbaumer H. and Storey P.J. (1975) The Ionization Balance of C0 to C+4 . Astronomy and astrophysics 44: 321-327. 25 Nussbaumer H. and Storey P.J. (1978) The C III transition probabilities. Astronomy and Astrophysics 64(1-2): 139-144. 42, 43 Nussbaumer H. and Storey P.J. (1983) Dielectronic recombination at low temperatures. Astronomy and astrophysics 126(1): 75-79. 23 Nussbaumer H. and Storey P.J. (1984) Dielectronic recombination at low temperatures. II. Recombination coefficients for lines of C, N, O. Astronomy and Astrophysics Supplement Series 56: 293-312. 25 Nussbaumer H. and Storey P.J. (1987) Dielectronic recombination at low temperatures. IV. Recombination coefficients for neon. Astronomy and Astrophysics Supplement Series 69: 123-133. 23 Peng Y.-L., Wang M.-S., Han X.-Y. and Li J.-M. (2004) Theoretic Study of C II Recombination Line. Chinese Physics Letters 21(9): 1723-1726. 27 Peng Y.-L., Han X.-Y., Wang M.-S. and Li J.-M. (2005) A theoretical study of dielectronic recombination processes of C2+ ions in planetary nebulae. Journal of Physics B 38(21): 3825-3839. 27 Péquignot D. and Baluteau J.P. (1988) The 680-1050 nm recombination spectrum of hydrogen and helium in the planetary nebula NGC 7027. Astronomy and Astrophysics 206(2): 298-315. 17 Péquignot D., Petitjean P. and Boisson C. (1991) Total and effective radiative recombination coefficients. Astronomy and Astrophysics 251(2): 680-688. 18, 26 Presnyakov L.P. and Urnov A.M. (1975) Asymptotic approach to the theory of excitation of multiply-charged ions by electron impact. Journal of Physics B 8(8): 1280-1288. 22 Press W.H., Teukolsky S.A., Vetterling W.T. and Flannery B.P. (2002) Numerical Recipes in C++ The Art of Scientific Computing. Cambridge University Press, 2nd Edition. 102 Quigley L. and Berrington K. (1996) The QB method: analysing resonances using Rmatrix theory. Applications to C+ , He and Li. Journal of Physics B 29: 4529-4542. 52, 53

89

Quigley L., Berrington K. and Pelan J. (1998) The QB program: analysing resonances using R-matrix theory. Computer Physics Communications 114(1-3): 225-235. 42, 52, 54 Rola C. and Stasi´ nska G. (1994) The carbon abundance problem in planetary nebulae. Astronomy and Astrophysics 282(1): 199-212. 27 Sakimoto K. (1987) Dielectronic recombination in electric fields: a quantum-defect theory approach. Journal of Physics B 20(4): 807-819. 24 Seaton M.J. (1969) Quantum defect theory VII. Analysis of resonance structures. Journal of Physics B 2(1): 5-11. 22 Seaton M.J. and Storey P.J. (1976) Di-electronic recombination. In: Atomic Processes and Applications. North-Holland Publishing Company. 48 Seaton M.J. (1985) Use of the R-matrix method for bound-state calculations: I. General theory. Journal of Physics B 18: 2111-2131. 34, 38 Smith F.T. (1960) Lifetime Matrix in Collision Theory. Physical Review 118(1): 349-356. 54, 56, 92, 93 Sochi T. (2010) Emissivity: A program for atomic transition calculations. Communications in Computational Physics, 7(5): 1118-1130. 1, 48, 73, 97 Storey P.J. (1981) Dielectronic recombination at nebular temperatures. Monthly Notices of Royal Astronomical Society 195: 27P-31P. 23 Storey P.J. (1994) Recombination coefficients for O II lines at nebular temperatures and densities. Astronomy and Astrophysics 282(3): 999-1013. 67 Stibbe D.T. and Tennyson J. (1998) TIMEDEL: A program for the detection and parameterization of resonances using the time-delay matrix. Computer Physics Communications 114: 236-242. 52, 54, 55, 59 Tennyson J. and Noble C.J. (1984) RESON-A program for the detection and fitting of Breit-Wigner resonances. Computer Physics Communications 33(4): 421-424. 53 Tsamis Y.G., Barlow M.J., Liu X.-W., Danziger I.J. and Storey P.J. (2003) Heavy elements in Galactic and Magellanic Cloud HII regions: recombination-line versus forbidden-line abundances. Monthly Notice of the Royal Astronomical Society 338(3): 687-710. 31 Tsamis Y.G., Barlow M.J., Liu X.-W., Danziger I.J. and Storey P.J. (2003) A deep survey of heavy element lines in planetary nebulae I. Observations and forbidden-line

90

densities, temperatures and abundances. Monthly Notices of the Royal Astronomical Society 345(1) 186-220. 29 Tsamis Y.G., Barlow M.J., Liu X.-W., Storey P.J. and Danziger I.J. (2003) A deep survey of heavy element lines in planetary nebulae - II. Recombination-line abundances and evidence for cold plasma. Monthly Notices of the Royal Astronomical Society 353(3): 953-979. 29 Tsamis Y.G., Walsh, J.R., Péquignot D., Barlow, M.J., Liu, X.-W. and Danziger I.J. (2007) Integral-field Spectroscopy of Galactic Planetary Nebulae with VLT FLAMES. The Messenger 127: 53-57. 30 Zhang Y., Liu X.-W., Luo S.-G., Péquignot D. and Barlow M.J. (2005) Integrated spectrum of the planetary nebula NGC 7027. Astronomy & Astrophysics 442: 249-262. 1, 6, 8, 17, 27, 28, 75, 77, 78, 81, 99, 118, 144

91

Appendix A

Using Lifetime Matrix to Investigate SingleChannel Resonances In this Appendix we present the method proposed and derived by P.J. Storey (with some modifications by the author) of using the lifetime matrix M expressed in terms of the reactance matrix K to investigate single-channel resonances. This method is used in this study to investigate the low-lying autoionizing states of C II. According to Smith [1960], the M-matrix is given, in terms of the scattering matrix S, by M = −i ~ S∗

dS dE

(A.1)

where S-matrix is defined, in terms of K-matrix, by

S=

I + iK I − iK

(A.2)

with I being the identity matrix. In the case of single-channel states, M, S and K are one-element matrices. To indicate this fact we annotate them with M , S and K. From (A.1) and (A.2), the following relation may be derived

M=

2 dK 1 + K 2 dE 92

(A.3)

where ~ = 1 in atomic units. It is noteworthy that since K is real, M is real as it should be. Smith [1960] has demonstrated that the expectation value of M is the lifetime of the state, τ . Now if K has a simple pole at energy E0 , given by

K=

g E − E0

(A.4)

where g is a parameter with dimensions of energy, then from (A.3) we obtain

M (E) = =

−2g (1 +

K 2 )(E

− E0 )2

−2g (E − E0 )2 + g2

(A.5)

and hence the maximum value of M (E) is M (E0 ) = −2/g. On substituting the half maximum, −1/g, into Equation (A.5) we find

E − E0 = ±g

(A.6)

and therefore, the full width at half maximum is

∆E = |2g|

(A.7)

According to Smith [1960], the lifetime of the state τ , and the FWHM ∆E are related by τ=

1 ∆E

(au)

(A.8)

Hence, for a K-matrix pole given by (A.4) the autoionization probability, Γa , which by definition equals the reciprocal of the lifetime, is given in atomic units by

Γa =

1 = ∆E = |2g| τ 93

(au)

(A.9)

Now if we consider a K-matrix with a pole superimposed on a background Ko

K = Ko +

g E − E0

(A.10)

then from (A.3) we find

M (E) =

−2g

− E0 )2 −2g = 2 2 (1 + Ko )(E − E0 ) + 2Ko g(E − E0 ) + g2 (1 +

K 2 )(E

(A.11)

The maximum value of M (E) occurs when the denominator has a minimum, that is when Ko g 1 + Ko2

(A.12)

2(1 + Ko2 ) g

(A.13)

E = E0 − and hence Mmax = −

This reveals that by including a non-vanishing background the peak of M is shifted relative to the pole position, E = E0 , and the peak value is modified. If we now calculate the FWHM, ∆E , by locating the energies where M = 21 Mmax from solving the quadratic g2 (1 − K2o ) (1 + Ko )(E − E0 ) + 2Ko g(E − E0 ) − =0 (1 + K2o ) 2

2

(A.14)

we find ∆E =

|2g| 1 + Ko2

(A.15)

As FWHM and autoionization probability are numerically equal in atomic units, we obtain Γa =

|2g| 1 + Ko2

94

(au)

(A.16)

The two parameters of primary interest are the resonance position Er and the resonance width ∆r . However, for an energy point Ei with a K-matrix value Ki , Equation (A.10) has three unknowns, Ko , g and E0 which are needed to find Er and ∆r , and hence three energy points at the immediate neighborhood of E0 are required to identify the unknowns. Since the K-matrix changes sign at the pole, the neighborhood of E0 is located by testing the K-matrix value at each point of the energy mesh for sign change and hence the three points are obtained accordingly. Now, if we take the three consecutive values of K

Ki = Ko +

g Ei − E0

(i = 1, 2, 3)

(A.17)

and define

∆Ejk = Ej − Ek

∆Kjk = Kj − Kk

&

(A.18)

then from (A.17) we obtain ∆K12 = g ∆K23 = g

1 1 − ∆E10 ∆E20

1 1 − ∆E20 ∆E30

=g =g

∆E21 ∆E10 ∆E20

∆E32 ∆E20 ∆E30

(A.19)

(A.20)

and ∆K12 ∆E30 ∆E21 = ∆K23 ∆E10 ∆E32

(A.21)

∆E30 E3 − E0 ∆K12 ∆E32 = = ∆E10 E1 − E0 ∆K23 ∆E21

(A.22)

Therefore

On algebraically manipulating (A.22) we find

E0 =

E1 ∆K12 ∆E32 − E3 ∆K23 ∆E21 ∆K12 ∆E32 − ∆K23 ∆E21 95

(A.23)

Having located the pole position, E0 , the following relation can be obtained from (A.19) g=

∆K12 ∆E10 ∆E20 ∆E21

(A.24)

Similarly, from (A.17) we obtain

Ko = K1 −

g ∆E10

(A.25)

Finally, Er and ∆r can be computed from Equation (A.12) and Equation (A.15) respectively.

96

Appendix B

Emissivity Program Documentation ‘Emissivity’ code [2010] is a command line program that was developed during this investigation to implement the emissivity model. Its main functionality is to calculate the emissivity of the transition lines from dielectronic recombination and cascade decay all the way down to the ground or a metastable state, that is all free-free, bound-free and bound-bound transitions. ‘Emissivity’, which consists of about 3000 lines of code, is written in C++ computer language and mixes procedural with object oriented programming. The program was compiled successfully with no errors or warnings using Dev-C++ compiler on Windows, and g++ compiler on Cygwin and Red Hat Linux. Sample results produced by these three versions were compared and found to be identical. Elaborate checks were carried out in all stages of writing and debugging the program and the output was verified. Thorough independent checks on sample emissivity data produced by ‘Emissivity’ were performed and found to be consistent. The program reads from plain text input files and write the results to a main plain text output file called ‘Transitions’. The structure of this output file is explained in section (B.4). Other secondary output data files are also produced for particular purposes. The required input text files are ‘Input’, ‘ResEmis’, ‘ELEVEmis’

97

and ‘FVALUE’. The structure of these files, apart from the ‘FVALUE’ file which is produced by the STGBB stage of the R-matrix code and is used by ‘Emissivity’ program with no modification, apart from possible inclusion of additional data, is outlined below. Two other input data files are also required if comparison and analysis to observational data are needed. One of these is a data file that contains astronomical observations while the other includes mapping information of the observational lines to their theoretical counterparts using their indices.

B.1

‘Input’ Data File

The program uses a keyword based input file called ‘Input’ which controls the program flow and its main parameters. All keywords are optional with no order required. If keywords are omitted, default values will be used. Comments can be inserted in any part of the input file as long as they do not intrude between a keyword and its parameter(s) or between the parameters. No commentary line should be initiated with a keyword. Each keyword must occur first on its line of text followed by its parameter(s) in the order that will be given. There is no restriction on using spaces and new lines as long as they obey the aforementioned rules.

1. ‘Temperature(K)’ The first entry after this keyword is a boolean flag to choose one of the two available schemes, followed by the number of temperature entries in K. If the flag is on (i.e. ‘1’), this must be followed by the initial temperature and the interval between the temperature points in this order; and if it is off, it should be followed by the temperature points. The first scheme is most suitable for a large and regularly spaced temperature vector, while the second is more suitable for small or irregularly spaced temperature vector. If this keyword is omitted, the default is a single 98

temperature entry of 10000.

2. ‘Ni(mˆ -3)’ This is the number density of the ions in m−3 . If this keyword is omitted, the default value is 1010 .

3. ‘Ne(mˆ -3)’ This is the number density of the electrons in m−3 . If this keyword is omitted, the default value is 1010 .

4. ‘ResidualCharge’ This is the residual charge of the ion. It is required for scaling some of the data obtained from the R-matrix code output files. If this keyword is omitted, the default value is 2.

5. ‘AstronomicalData’ This keyword requires a boolean flag to read astronomical data from a data file (‘0’ for ‘No’ and ‘1’ for ‘Yes’). If the flag is on, a second entry, which is the name of the data file to read the data from, will be required. The structure of the astronomical data file requires the file to start with a comment that can occupy any number of text lines but it should be terminated with ‘EndOfComment’ string to mark the end. This should be followed by the transition lines data organized in sets preferably each set on a single line of text. When the normalization is on, as will be explained later on, the program will normalize the astronomical data to the first data set. This facilitates the normalization as the user can simply normalize to any data set in the astronomical data by copying that data set and paste it at the front. For the astronomical data of Zhang et al [2005], each data set consists

99

of the following entries arranged in this order: the observed wavelength after correcting for Doppler shifts determined from H I Balmer lines (λobs ), the observed flux before correcting for reddening (F (λ)), the observed flux after correcting for dust extinction (I(λ)), the ionic identification given as a single string, the laboratory wavelength (λlab ), multiplet number (with a prefix ‘V’ for permitted lines, ‘F’ for forbidden lines, and ‘H’ and ‘P’ for hydrogen Balmer and Paschen lines, respectively), the lower spectral term of the transition, the upper spectral term of the transition, the statistical weight of the lower level (g1 ) and the statistical weight of the upper level (g2 ). All fluxes are normalized such that F (Hβ) = I(Hβ) = 100. If the user chooses to read astronomical data, the data lines will be inserted between the theoretical lines in the ‘Transitions’ output file according to their observed wavelength. If this keyword is omitted, the default value is 0.

6. ‘TestsFlag’ This keyword requires one entry which is a flag for carrying the first (in-out) and second (metastable) consistency tests. These tests are described in section (B.5) of this manual. The options for this flag are: ‘1’ for running the first test only, ‘2’ for running the second test only and ‘3’ for running both tests. Any other value means that no test will be carried out. If this keyword is omitted, the default value is 0.

7. ‘NormalizationChoice’ This keyword controls the generation and writing of normalization data for theoretical and observational lines to the ‘Transitions’ output file alongside the original emissivity data. The options for this keyword are • ‘0’ for no normalization. In this case no normalization data will be written to the ‘Transitions’ file. Any invalid normalization choice will also be treated 100

like zero. • ‘1’ for internal normalization with one of the transition lines produced by the program. In this case this should be followed by the index of the normalization line. • ‘2’ for normalizing with respect to an outside set of emissivity values corresponding to the temperatures included under the ‘Temperature(K)’ keyword. The emissivity values should be on the next line(s) of text and in the SI units. • ‘3’ for normalizing with respect to an outside set of emissivity data in the form of effective recombination coefficients corresponding to the specified temperatures. In this case the recombination coefficients should be followed by the wavelength of the line to be normalized to. The effective recombination coefficient %f (λ) is defined such that the emissivity ε(λ) of a transition line with wavelength λ is

ε(λ) = Ne Ni %f (λ)

hc λ

(J.m−3 .s−1 )

(B.1)

where the symbols have their usual meanings as given in Nomenclature. The recombination coefficients in the input data file should be in cm3 .s−1 while the wavelength, which follows the recombination coefficients, should be in nm. The reason behind this choice is to avoid potential error in conversion as the available data in the literature are mostly given in these units rather than SI units. These values should be entered on the next line(s) of text in the data file. • ‘4’ for normalizing with respect to an outside set of emissivity data in the form of effective recombination coefficients corresponding to a set of temperatures that may be different from those included under the ‘Temperature(K)’

101

keyword. In this case, the recombination coefficients corresponding to the temperature values of the ‘Temperature(K)’ keyword are obtained by interpolation or extrapolation using ‘polint’ algorithm from the Numerical Recipes (Press et al [2002]). This algorithm is a polynomial interpolation routine which implements Nevilles interpolating method and is regarded as an improvement to the classical Lagrange’s formula for polynomial interpolation. For this Option, the normalization choice should be followed by the number of points which identify the order of interpolation (e.g. ‘3’ for 3-point second order interpolation and so on). The recommended value for the number of points is ‘3’ or ‘4’. Going beyond this should be for a good reason, otherwise the interpolation may fail to produce reasonable results. The details should be sought in the literature of numerical methods including the Numerical Recipes of Press et al. An initial investigation suggests that the optimal number of points may depend on the proximity to the tabulated data points and the nature of the process, i.e. interpolation vs. extrapolation. It may be a good idea to have a trial run inspecting the estimated errors which can be found in the ‘Transitions’ output file before choosing the order of interpolation. The required data entries, which should follow the number of points, are the number of data pairs (i.e. temperature in K and recombination coefficient in cm3 .s−1 ) followed by the data pairs followed by the wavelength of the line of normalization in nm. These values should be included on the next line(s) of text in the data file. If this keyword is omitted, the default value is 0.

8. ‘RecCoefficient’ This keyword controls the production and writing of the effective recombination coefficients, %f , which are equivalent to the given emissivities. Vacuum wavelength 102

of the transition line will be used as the air wavelength is not available for λ < 2000˚ A. The keyword should be followed by a boolean flag, i.e. ‘1’ for ‘Yes’ and ‘0’ for ‘No’. Any other choice will be treated as zero. The written recombination coefficients to the ‘Transitions’ file will be in the SI units (m3 .s−1 ). If this keyword is omitted, the default value is 0.

9. ‘TheoAstroLS’ This keyword controls the algorithm to minimize the sum of least square deviations of emissivity between the observational lines and their theoretical counterparts over the input temperature range. This sum is given by

S=

X

N εN o − εt

2

(B.2)

∀ lines

N where εN o and εt are the normalized observational and theoretical emissivities re-

spectively. This algorithm also offers the possibility of computing the temperature confidence interval and its relevant parameters. The keyword requires a boolean flag to carry out minimization or not. If the flag is on, the name of the file that maps the indices of the theoretical and observational lines should follow. The third entry is a flag to write to ‘Transitions’ file the individual square differences for each one of the least squares and their percentage difference which is given by

N εN o − εt εN o

× 100

(B.3)

Another flag to find the confidence interval for the goodness-of-fit index χ2 is required. If this flag is on, it should be followed by the degrees of freedom η and the change in the goodness-of-fit index ∆χ2 . The structure of the mapping file requires that the index of the observational line 103

should be followed by the index of its theoretical counterpart. If an observational line corresponds to more than one theoretical line, which occurs when observational lines from the same multiplet are blended or when theoretical lines fall within 1˚ A range of an observational line, the mapping should be repeated independently for each theoretical line. The rest of the file can be used for commenting or storing other data if the index mapping is terminated with ‘-999’. The results of the minimization algorithm will be written to a file called ‘TNV’. This file contains information on η, ∆χ2 , a, temperature at minimum χ2 , and the temperature limits for the confidence interval. This is followed by the temperature array with the corresponding least squares residuals, the aχ2 and the χ2 arrays as functions of temperature.

10. ‘DecayRoutes’ This keyword is for finding the decay routes to a particular state, bound or free, from all upper states. The parameters required are a boolean flag to run this algorithm or not (‘1’ for ‘Yes’ and ‘0’ for ‘No’). In the first case, the configuration, term and 2J of the state should follow. As the number of decay routes can be very large (millions and even billions) especially for the low bound states, another parameter is used to control the maximum number of detected routes, and this parameter should follow. The results (number of decay routes and the routes themselves grouped in complete and non-complete) will be output to ‘DecayRoutes’ file. If the number of detected routes exceeds the maximum number, say n, only the first n routes will be written to the ‘DecayRoutes’ file. The scheme is that the program keeps detecting the routes as long as the total number of complete and uncomplete routes detected at any point during the program execution does not exceed the maximum number. As the algorithm follows the routes from the given state upwards, the number of detected routes increases as iteration progresses.

104

Also the number of complete routes increases progressively. Consequently, to find a substantial number of complete routes or all of them if this is feasible, the maximum number should be set to a sufficiently large value. It is recommended that for the bound states and low resonances the maximum number should be increased in steps until the output satisfies the user need. If the maximum number was set to a very large value the program may fail or take very long time to complete; moreover the data produced will become virtually useless thanks to its massive size. The states of each decay route are identified by their configuration, term and 2J. For resonances, the Saha capture term and the radiative decay term are also given when relevant. These temperature-dependent data are given for a single temperature, that is the first value in the temperature vector.

B.2

‘ResEmis’ Input Data File

This is the file containing the data for resonances. The file contains four main sections: 1. The first line in the file is a comment line. This is followed by the number of resonances followed by a number of text lines matching the number of resonances. Each one of these lines contains an index identifying the resonance, the energy position of the resonance in z-scaled Ryd, the width in z-scaled Ryd, the configuration, term, 2J, parity, a flag for marking the energy position data as experimental (1) or theoretical (0), and a flag marking the resonances as ‘good’ (1) or not (0). Each of the configuration and term should be a single string with no space in between. 2. After the resonances data, a comment is expected. This comment, which can span any number of text lines, should be terminated by the string ‘EndOfComment’ to mark the end. Next, the number of the bound-state symmetries 105

is expected. This is followed, possibly, by a comment to the end of the line. The oscillator strengths (f -values) for the free-bound transitions then follow in sections according to the symmetries of the bound states, so the number of the f -values sections is the same as the number of bound state symmetries. Each section of the f -values is headed by a line containing 2J, parity and the number of bound states in that symmetry. This is followed by a twodimensional array of f -values where the columns stand for the resonances as ordered previously, whereas the rows are for the bound states of the given symmetry. For the transitions which are forbidden by the electric dipole rules for J and parity, the f -values should be set to zero. 3. Next a comment is expected which should be terminated by the ‘EndOfComment’ string to mark the end. The data for the photon energy at resonance position which correspond to the f -values data in the previous section, then follow, sectioned and formatted as for the f -values data. The photon energy data, which are obtained from the ‘XSECTN’ file produced by stage STGBF of the R-matrix code, are in Ryd. 4. The last section starts with a comment terminated by the ‘EndOfComment’ string. The f -values data for the free-free transitions are then included in a two-dimensional array where the columns stand for the odd resonances while the rows stand for the even ones. This section is required only if the resonances are of mixed parity.

B.3

‘ELEVEmis’ Input Data File

This file contains the bound states data. The file starts with a comment terminated by the ‘EndOfComment’ marker. Next, the number of bound states symmetries are expected. This is followed by a data block for each symmetry. Each block is headed 106

by a line containing 2J, parity and the number of bound states in that symmetry. Next, a number of text lines as the number of bound states in that symmetry are expected. Each line contains (in the following order) an index identifying the state, the energy of the bound state in z-scaled Ryd, the effective quantum number, the configuration, term and a flag for marking the energy data as experimental (1) or theoretical (0). Each of the configuration and term should be a single string with no space in between. The effective quantum number, though not used in the program, is included because the original data come from the ‘ELEV’ data file generated by the STGB stage of the R-matrix code. It is kept for possible use in the future.

B.4

‘Transitions’ Output Data File

This is the main output file of ‘Emissivity’ program. The file starts with a number of text lines summarizing the input data used and the output results, followed by a few commentary lines explaining the symbols and units. This is followed by a number of data lines matching the number of transitions. The data for each transition includes (in the following order) an index identifying the transition, status (FF, FB or BB transition), two joined boolean flags describing the experimental state of the energy data of the upper and lower levels (‘0’ for theoretical data and ‘1’ for experimental), the attributes of the upper and lower levels (configuration, term, 2J and parity), wavelength in vacuum, wavelength in air (only for λ ≥ 2000˚ A), the emissivities corresponding to the given temperatures, the normalized emissivities and the effective recombination coefficients corresponding to the given emissivities. Writing the normalized emissivities and the effective recombination coefficients is optional and can be turned off, as described under ‘NormalizationChoice’ and ‘RecCoefficient’ keywords. As mentioned earlier, if the user chooses to read the astronomical data, the observational lines will be inserted in between the theoretical lines according to their lab wavelength. 107

B.5

TESTS

Apart from the normal debugging and testing of the program components to check that they do what they are supposed to do, two physical tests are incorporated within the program to validate its functionality and verify that no serious errors have occurred in processing and producing the data. These tests are the populationbalance test and the metastable test. The first test relies on the fact that the population of each state should equal the depopulation. This balance is given by the relation X

Nj Γrji = Ni

X

j>i

Γrik

(B.4)

ki

where i is an index for metastable states and j is an index for resonances.

108

(B.5)

Appendix C Input Data for R-matrix and Autostructure In this appendix we include the input data files that we used to generate our theoretical results from R-matrix and Autostructure. This is for the purpose of completion and to enable the interested researcher to generate and check our results. It should be remarked that the structure of the R-matrix and Autostructure data files is fully explained in the write-up of these codes as given by Badnell [2002, 2008].

C.1

R-matrix Input Data

In this section we present our input data files for each stage of R-matrix.

C.1.1

STG1

For both targets (i.e. 9-term and 26-term) the input data file is: STO&STG1A RAD=’YES’ RELOP=’YES’ ISMITN=1 &END &STG1B NZED=6 NELC=4 MAXORB=7 MAXLA=3 MAXLT=10 MAXC=16 ISMIT(1)=30 ISMIT(2)=31 ISMIT(3)=32 ISMIT(4)=43 MAXE=8 IBC=1 &END 1 0 2 0 2 1 3 0 3 1 3 2 4 3 5 1 1 2 2 2 5.13180 8.51900 2.01880 4.73790 1.57130 21.28251 6.37632 0.08158 -2.61339 -0.00733 5 1 1 2 2 2

109

5.13180 -5.39193 3 1 2 3 1.75917 5.69321 4 2 2 2 1.47510 1.01509 2 2 3 1.98138 14.41203 1 3 2.11997 5.84915 1 4 2.69086 9.69136 10.0000000 5 1 1 2 5.13180 21.28251 5 1 1 2 5.13180 -5.39193 3 1 2 3 1.75917 5.69321 4 2 2 2 1.47510 1.01509 2 2 3 1.98138 14.41203 1 3 2.11997 5.84915 1 4 2.69086 9.69136 10.0000000

C.1.2

8.51900 -1.49036

2.01880 5.57151

4.73790 -5.25090

1.57130 0.94247

1.75917 -19.54864

1.75917 10.39428

2 3.19410 3.80119

1.83070 2.75006

9.48450 0.89571

2 2 8.51900 6.37632

2.01880 0.08158

4.73790 -2.61339

1.57130 -0.00733

2 2 8.51900 -1.49036

2.01880 5.57151

4.73790 -5.25090

1.57130 0.94247

1.75917 -19.54864

1.75917 10.39428

2 3.19410 3.80119

1.83070 2.75006

1.96954 -10.88586

0.0

9.48450 0.89571

1.96954 -10.88586

0.0

STG2

For the 9-term target the input data file is: STO&STG2A RAD=’YES’ RELOP=’YES’ &END &STG2B MAXORB=7 NELC=4 NAST=9 INAST=0 MINST=2 MAXST=4 MINLT=0 MAXLT=8 &END 1 0 2 0 2 1 3 0 3 1 3 2 4 3 18 2 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 2 0 2 0 0 0 0 0 2 1 0 1 0 0 0 0 2 0 0 2 0 0 0 0 2 0 1 0 1 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 0 2 0 0 2 1 1 0 0 0 0 0 2 1 0 0 1 0 0 0 2 0 0 1 1 0 0 0 2 0 1 1 0 0 0 0 2 0 1 0 0 1 0 0 2 0 0 0 1 1 0 0 2 0 0 0 0 1 1 0 2 1 0 0 0 1 0 0 2 0 0 1 0 1 0 0 2 0 1 0 0 0 1 0 2 0 0 0 1 0 1 0 0 1 0 1 1 1 1 3 1 1 3 0

110

2 0 0 2 2 48 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 2 2 2 2 2 2 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0

1 1 3 3 3 0 3 1 0 0 0 0 2 3 2 2 2 2 1 0 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0

0 2 0 1 0 0 0 0 0 1 0 0 0 1 2 1 1 1 2 2 2 2 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1

0 0 0 0 0 0 3 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 1 2 1 1 2 2 3 2 2 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0

0 3 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 2 2 2 2 3 2 1 0 1 0 1 0 1 0 1 1 1 1

0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

For the 26-term target the input data file is: STO&STG2A RAD=’YES’ RELOP=’YES’ &END &STG2B MAXORB=7 NELC=4 NAST=26 INAST=0 MINST=2 MAXST=4 MINLT=0 MAXLT=8 &END 1 0 2 0 2 1 3 0 3 1 3 2 4 3 18 2 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 2 0 2 0 0 0 0 0 2 1 0 1 0 0 0 0 2 0 0 2 0 0 0 0 2 0 1 0 1 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 0 2 0 0 2 1 1 0 0 0 0 0 2 1 0 0 1 0 0 0 2 0 0 1 1 0 0 0 2 0 1 1 0 0 0 0 2 0 1 0 0 1 0 0 2 0 0 0 1 1 0 0 2 0 0 0 0 1 1 0 2 1 0 0 0 1 0 0 2 0 0 1 0 1 0 0 2 0 1 0 0 0 1 0 2 0 0 0 1 0 1 0 0 1 0 1 3 1 1 1 1 1 3 0 2 1 0

111

0 0 0 1 1 1 2 1 1 2 2 0 3 2 1 1 2 2 3 1 0 48 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 2 2 2 2 2 2 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0

1 3 1 1 3 3 3 1 1 1 3 3 3 1 3 3 3 1 1 1 1 0 3 1 0 0 0 0 2 3 2 2 2 2 1 0 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0

0 2 0 1 0 0 0 0 0 1 0 0 0 1 2 1 1 1 2 2 2 2 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1

0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 0 0 3 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 1 2 1 1 2 2 3 2 2 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0

C.1.3

0 3 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 2 2 2 2 3 2 1 0 1 0 1 0 1 0 1 1 1 1

0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

STGJK

For the 9-term target the input data file is: S.S. &STGJA RAD=’YES’ &END &STGJB JNAST=17 IJNAST=12 &END 0 0 0 1 2 1 4 1 2 1 0 0

112

2 4 4 0 2 2 4 6 2 4 6

0 0 0 0 0 0 0 0 0 0 0 1 3 5 7 9 11 1 3 5 7 9 11

0 0 0 0 0 0 1 1 1 1 1 1

For the 26-term target the input data file is: S.S. &STGJA RAD=’YES’ &END &STGJB JNAST=46 IJNAST=12 &END 0 0 0 1 2 1 4 1 2 1 0 0 2 0 4 0 4 0 0 0 2 0 0 0 2 1 0 1 2 1 4 1 2 0 4 0 6 0 0 1 2 1 4 1 4 0 2 1 2 0 4 1 6 1 8 1 2 0 4 0 6 0 4 1 2 0 0 1 2 1 4 1 2 1 4 1 6 1 0 0 2 0 4 0 4 0 2 1 6 1 0 0 1 0 3 0 5 0 7 0 9 0 11 0 1 1 3 1 5 1 7 1 9 1 11 1

113

C.1.4

STG3

For both targets the input data file is: S.S. &STG3A RAD=’YES’ &STG3B &END

C.1.5

&END

STGF

The parameters in the input data file for both targets with a sample data is: &STGF IMODE = 0 IPRKM = 1 IQDT = 0 IRD0 = 99 IMESH = 1 IEQ = -1 PERT = ’NO’ LRGLAM = -1 IBIGE = 0 IPRINT = 3 IRAD = 1 IOPT1 = 2 &END &MESH1 MXE = 1000 E0 = 0.0298042 EINCR = 5.0E-8 &END 0 3 1 -1 -1 -1

C.1.6

STGB

For both targets the input data file is: &STGB IPERT=0 IRAD=1 &END 0 1 0 0.1 13.0 0.001 0 3 0 0.1 13.0 0.001 0 5 0 0.1 13.0 0.001 0 7 0 0.1 13.0 0.001 0 9 0 0.1 13.0 0.001 0 1 1 0.1 13.0 0.001 0 3 1 0.1 13.0 0.001 0 5 1 0.1 13.0 0.001 0 7 1 0.1 13.0 0.001 0 9 1 0.1 13.0 0.001 0 11 1 0.1 13.0 0.001 -1 -1 -1

C.1.7

STGBF

For both targets default values are used, that is: &STGBF

&END

C.1.8

STGBB

For both targets the input data file is:

114

&STGBB IPRINT=0 IBUT=0 0 1 0 0 1 1 0 1 0 0 3 1 0 3 0 0 1 1 0 3 0 0 3 1 0 3 0 0 5 1 0 5 0 0 3 1 0 5 0 0 5 1 0 5 0 0 7 1 0 7 0 0 5 1 0 7 0 0 7 1 0 7 0 0 9 1 0 9 0 0 7 1 0 9 0 0 9 1 0 9 0 0 11 1 0 11 0 0 9 1 0 11 0 0 11 1 -1 -1 -1 -1 -1 -1

C.1.9

&END

STGQB

The parameters in the input data file for both targets with a sample data is: 0 0 0 1 0 3 1 2.945

4.1

C.2

1.0E-8

Autostructure Input Data

Autostructure code was used in various stages to produce required theoretical atomic data or to check the results of R-matrix. In the next sections we present some of these input data files.

C.2.1

Polarizability

The following input data file was used to decide which terms of the target are the most important ones by having the largest polarizability, as explained in § 3.2.2. S.S. 123456789 22 12513 23 12514 12515 12516 13514 13515 13516 10 20 21 &SALGEB RUN=’ ’ RAD=’YES’ CUP=’IC’ TITLE=’C III_9’ KORB1=1 KORB2=1 KUTSO=0 &END &SMINIM NZION=6 INCLUD=6 PRINT=’FORM’ NLAM=6 RADOUT=’YES’ &END 1.00000 1.00000 1.00000 -1.0000 -1.0000 -1.0000

C.2.2

Transition Probabilities

The following input data file was used to compute the radiative transition probabilities, as discussed in § 3.1. 115

S.S. 123456789 12513516 12513515 12523 22516 12513517 10 20 21 30 31 32 &SALGEB RUN=’ ’ RAD=’YES’ CUP=’IC’ TITLE=’Resonance’ KORB1=1 KORB2=1 KUTSO=0 &END &SMINIM NZION=6 INCLUD=6 PRINT=’FORM’ NLAM=6 RADOUT=’YES’ &END 1.00000 1.00000 1.00000 1.0000 1.0000 1.0000

C.2.3

f -values

The following input data file was used to generate f -values for all FF transitions for both targets. It was also used to generate f -values for the FB and BB transitions for the topmost bound states, namely the 4 Fo and 4 Do levels, as these quartets with their large effective quantum number are out of range of RMATRX-I validity. S.S. 123456789 22513 22514 22515 22516 22517 22518 22519 2251A 2251B 2251C 2251D 2251E 2251F 2251G 2251H 2251I 2251J 2251K 2251L 2251M 12523 12513514 12513515 12513516 12513517 12513518 12513519 1251351A 1251351B 1251351C 1251351D 1251351E 1251351F 1251351G 1251351H 1251351I 1251351J 1251351K 1251351L 1251351M 33 23514 23515 23516 23517 23518 23519 2351A 2351B 2351C 2351D 2351E 2351F 2351G 2351H 2351I 2351J 2351K 2351L 2351M 10 20 21 30 31 32 40 41 42 43 50 51 52 53 60 61 62 63 70 71 72 73 &SALGEB RUN=’ ’ RAD=’YES’ CUP=’IC’ TITLE=’Resonance’ KORB1=1 KORB2=1 KUTSO=0 &END &SMINIM NZION=6 INCLUD=500 PRINT=’FORM’ NLAM=22 NVAR=4 RADOUT=’YES’ &END 1.43268 1.43367 1.39649 1.24476 1.19191 1.35343 1.24238 1.18708 1.34947 1.41364 1.24459 1.19001 1.35214 1.41413 1.46134 1.25098 1.20315 1.36250 1.41483 1.41448 2.30700 1.26820 19 20 21 22

116

Appendix D

Tables In this appendix, we tabulate the main data that we produced or used during this investigation. In Table (D.1) the theoretical results for the bound states of C II below the 1 Se0 threshold are given alongside the available experimental data from the NIST database. Similarly, Table (D.2) presents the data for the resonances as obtained by the K-matrix method while Table (D.3) presents the data as obtained by the QB method. In these tables, a negative fine structure splitting indicates that the theoretical levels are in reverse order compared to their experimental counterparts. It is noteworthy that due to limited precision of figures in these tables, some data may appear inconsistent, e.g. a zero fine structure splitting from two levels with different energies. Full-precision data in electronic format are available from the author on request. Regarding the bound states, all levels with effective quantum number nf between 0.1-13 for the outer electron (142 states) are sought and found by R-matrix for both the 9-term and the 26-term targets. The 8 uppermost bound states in Table (D.1), i.e. the levels of 4 Fo and 4 Do , have quantum numbers higher than 13 and hence are out of range of the R-matrix approximation validity; therefore only experimental data are included for these states. Concerning the resonances, we

117

searched for all states with n < 5 where n is the principal quantum number of the active electron. The statistics for the resonances is as follow: 64 levels are found by K-matrix method with the 9-term target; 61 by K-matrix with the 26-term; 57 by QB method with the 9-term; and 55 by QB with the 26-term. Tables (D.4-D.14) present the f -values for the free-bound transitions as obtained by integrating photoionization cross section over photon energy where these data are obtained from ‘XSECTN’ files produced by stage STGBF of RMATRX-I code using the 9-term target. Likewise, tables (D.15-D.25) are for the 26-term target. The columns in these tables stand for the bound states identified by their indices as given in Table (D.1) while the rows stand for the resonances represented by their indices as given in Table (D.2). The superscript denotes the power of 10 by which the number is to be multiplied. An entry of ‘0’ in the f -value tables indicates that no peak was observed in the R-matrix results. In this appendix we present only the f -values for the free-bound transitions as they require considerable effort to produce. However, the f -values for the 4 Fo and 4 Do levels are not included in the following FB tables because these f -values were generated by Autostructure as these states have very high effective quantum number and hence are out of range of the R-matrix code validity. The f -values for the bound-bound transitions are not given here because they are too bulky to present. Moreover, they can be obtained directly from the ‘FVALUE’ file generated by the R-matrix code using the input data files as given in § C.1 in Appendix C. Similarly, the f -values for the free-free transitions are not given in this appendix as they can be generated by Autostructure using the input data file provided in § C.2.3 in Appendix C. Finally, Table (D.26) contains the C II recombination lines data of Zhang et al [2005] which we used in our investigation to the spectra of planetary nebula NGC 7027.

118

Table D.1: The available experimental data from NIST for the bound states of C II below the C III 1 e S0 threshold alongside the theoretical results from R-matrix for the 9-term and the 26-term targets with 0.1 ≤ nf ≤ 13. In.a

Config.b

Level

NEEWc

NEERd

1

2

2s 2p

2

o

P1/2

0

-1.792141

2

2

2s 2p

2

o

P3/2

63.42

-1.791563

3

2s2p2

4

Pe1/2

43003.3

-1.400266

-1.399207

-1.401292

4

2s2p

2

4

e

P3/2

43025.3

-1.400065

-1.399154

-1.401090

5

2s2p

2

4

e

P5/2

43053.6

-1.399807

6

2s2p2

2

De5/2

74930.1

-1.109327

7

2s2p

2

2

e

74932.62

-1.109304

8

2s2p2

2 e

S1/2

96493.74

-0.912825

-0.894918

-0.899439

9

2s2p2

2

Pe1/2

110624.17

-0.784059

-0.766714

-0.772978

2

2

e

110665.56

-0.783682

D3/2

P3/2

FSSe

TER-9f

FSS

-1.789592 63.4

50.3

-1.789078

-1.398760

41.4

-1.101157

-0.766421

56.3

49.0

-1.791994

-1.400755

8.5

32.1

-1.105266

-0.772570

2s2p

11

2s2 3s

2 e

S1/2

116537.65

-0.730171

-0.723373

-0.727698

12

2s2 3p

2

Po1/2

131724.37

-0.591780

-0.590494

-0.592081

13

2

2s 3p

2

o

131735.52

-0.591678

14

2p3

4 o

S3/2

142027.1

-0.497894

15

2s2 3d

2

De3/2

145549.27

-0.465798

16

2s2 3d

2

De5/2

145550.7

-0.465785

17

2p3

2

Do5/2

150461.58

-0.421034

18

2p

3

2

o

150466.69

-0.420987

19

2s2 4s

2 e

S1/2

157234.07

-0.359318

-0.358065

-0.358955

20

2s2 4p

2

Po1/2

162517.89

-0.311169

-0.310120

-0.310784

21

2

2s 4p

2

o

P3/2

162524.57

-0.311108

22

2s2p(3 Po )3s

4

Po1/2

166967.13

-0.270624

-0.267851

-0.268893

23

3

o

2s2p( P )3s

4

o

P3/2

166990.73

-0.270409

-0.267486

-0.268645

24

3

o

2s2p( P )3s

4

o

P5/2

167035.71

-0.269999

25

2s2 4d

2

De3/2

168123.74

-0.260084

26

2

2s 4d

2

e

D5/2

168124.45

-0.260078

27

2p3

2

Po1/2

168729.53

-0.254564

28

2p

3

2

o

P3/2

168748.3

-0.254393

29

2s2 4f

2

Fo5/2

168978.34

-0.252297

30

2

2s 4f

2

o

168978.34

-0.252297

31

2s2 5s

2 e

173347.84

-0.212479

D3/2

F7/2 S1/2

119

11.1

-0.590506

-1.3

-0.487154

-0.464383

6.7

68.6

-0.410234

-0.310087

-0.267213

-0.259476

-2.9

3.6

70.0

-0.243048

1.0

-0.252009 -0.211949

-0.466005

2.1

-0.412388

-0.310723

-0.268229

-3.1

6.6

72.9

-0.260074

1.1

-0.245329 26.5

-0.252011 0.0

11.1

-0.260084

-0.243290 18.8

44.8

-0.412359

-0.259485 0.7

-0.591980

-2.1

-0.466024 1.9

-0.410208 5.1

59.0

-0.487491

-0.464400 1.4

63.3

-1.105247

10

P3/2

FSS

-1.792571

-1.101234 2.5

TER-26g

-0.245083

27.1

-0.252162 0.2

-0.252160 -0.212278

0.2

Table D.1: continued. In.

Config.

Level

NEEW

NEER

32

2s2 5p

2

Po1/2

175287.39

-0.194804

33

2

2s 5p

2

Po3/2

175294.75

-0.194737

34

2s2p(3 Po )3s

2

Po1/2

177774.59

-0.172139

35

3

2s2p( P )3s

2

o

P3/2

177793.54

-0.171967

36

2s2 5d

2

De3/2

178495.11

-0.165573

37

2

2s 5d

2

e

D5/2

178495.71

-0.165568

38

2s2 5f

2

Fo5/2

178955.94

-0.161374

39

2

2s 5f

2

o

F7/2

178955.94

-0.161374

40

2s2 5g

2

Ge7/2

179073.05

-0.160307

41

2

2s 5g

2

e

179073.05

-0.160307

42

2s2 6s

2 e

S1/2

181264.24

-0.140339

-0.140070

-0.140231

43

2s2p(3 Po )3p

4

De1/2

181696.66

-0.136399

-0.135646

-0.136848

44

3

o

2s2p( P )3p

4

e

D3/2

181711.03

-0.136268

-0.135497

-0.136699

45

3

o

2s2p( P )3p

4

e

D5/2

181736.05

-0.136040

-0.135258

-0.136454

46

3

o

2s2p( P )3p

4

e

D7/2

181772.41

-0.135709

47

2s2p(3 Po )3p

2

Pe1/2

182023.86

-0.133417

48

3

2s2p( P )3p

2

e

P3/2

182043.41

-0.133239

49

2s2 6p

2

Po1/2

182993.23

-0.124584

50

2

2s 6p

2

o

P3/2

182993.66

-0.124580

51

2s2 6d

2

De3/2

184074.59

-0.114730

52

2s2 6d

2

De5/2

184075.28

-0.114723

53

2s2 6f

2

Fo5/2

184376.06

-0.111982

54

2

2s 6f

2

o

F7/2

184376.06

-0.111982

55

2s2 6g

2

Ge7/2

184449.27

-0.111315

56

2

2s 6g

2

e

G9/2

184449.27

-0.111315

57

2s2 6h

2

Ho9/2

184466.5

-0.111158

58

2

2s 6h

2

o

184466.5

-0.111158

59

2s2p(3 Po )3p

4 e

184690.98

-0.109113

-0.107477

-0.108410

60

2s2 7s

2 e

S1/2

185732.93

-0.099618

-0.099445

-0.099537

61

2s2p(3 Po )3p

4

Pe1/2

186427.35

-0.093290

-0.091514

-0.092097

62

2s2p(3 Po )3p

4

Pe3/2

186443.69

-0.093141

-0.091418

-0.091950

63

3

2s2p( P )3p

4

e

P5/2

186466.02

-0.092937

64

2s2 7p

2

Po1/2

186745.9

-0.090387

o

o

o

G9/2

H11/2 S3/2

120

FSS

TER-9

FSS

TER-26

-0.8

-0.190072

-0.189481 7.4

-0.189488

-0.190099

-0.163628 19.0

-0.163458

-0.165296

18.7

-0.161216

0.7

75.8

-0.160168

-0.134931

0.1

-0.131231

0.0

78.5

-0.124181

20.3

-0.114532

1.9

-0.111877

-0.111251

0.1

38.7

-0.111121

-0.091163 -0.090233

-0.136115

0.0

80.4

-0.132896

21.8

-0.124351

2.7

-0.114729

1.0

-0.111924

0.1

-0.111264 0.0

-0.111121 0.0

-0.160184

-0.111924

-0.111251 0.0

0.1

-0.114739 0.9

-0.111878 0.0

-0.161295

-0.124376

-0.114540 0.7

0.8

-0.133095

-0.124198 0.4

-0.165599

-0.160184

-0.131416 19.6

20.5

-0.161296

-0.160168 0.0

-0.165143 -0.165606

-0.161217 0.0

3.0

-0.165330

-0.165303 0.6

FSS

-0.111264

0.0

-0.111122 0.0

38.5

-0.111122

-0.091717 -0.090313

0.0

41.8


Config.

Level

NEEW

NEER

FSS

TER-9

FSS

TER-26

FSS

65

2

2s 7p

2

P3/2

186746.3

-0.090383

0.4

-0.090228

0.6

-0.090302

1.1

66

2s2 7d

2

De3/2

187353

-0.084854

67

2

2s 7d

2

e

D5/2

187353

-0.084854

68

2s2 7f

2

Fo5/2

187641.6

-0.082225

69

2

2s 7f

2

Fo7/2

187641.6

-0.082225

70

2s2 7g

2

Ge7/2

187691.4

-0.081771

71

2s2 7g

2

Ge9/2

187691.4

-0.081771

72

2s2 7h

2

Ho9/2

187701

-0.081683

73

2

2s 7h

2

o

187701

-0.081683

74

2s2 8s

2 e

S1/2

—

—

-0.074264

-0.074324

75

2s2p(3 Po )3p

2

De3/2

188581.25

-0.073662

-0.070858

-0.071901

76

3

2s2p( P )3p

2

e

D5/2

188615.07

-0.073354

77

2s2 8p

2

Po1/2

—

—

-0.068301

78

2

2s 8p

2

o

P3/2

—

—

-0.068298

79

2s2 8f

2

Fo5/2

—

—

-0.062853

80

2

2s 8f

2

o

F7/2

—

—

-0.062853

81

2s2 8g

2

Ge7/2

189794.2

-0.062609

82

2s2 8g

2

Ge9/2

189794.2

-0.062609

83

2s2 8h

2

Ho11/2

—

—

-0.062510

84

2

2s 8h

2

o

H9/2

—

—

-0.062510

85

2s2 8d

2

De3/2

—

—

-0.062442

86

2

2s 8d

2

e

—

—

-0.062393

87

2s2 9s

2 e

S1/2

—

—

-0.057594

-0.057638

88

2s2 9p

2

Po1/2

—

—

-0.053458

-0.053490

89

2

2s 9p

2

o

P3/2

—

—

-0.053457

90

2s2 9d

2

De3/2

—

—

-0.049894

91

2

2s 9d

2

e

D5/2

—

—

-0.049888

92

2s2 9f

2

Fo5/2

—

—

-0.049637

93

2

2s 9f

2

o

F7/2

—

—

-0.049636

94

2s2 9g

2

Ge7/2

—

—

-0.049437

95

2s2 9g

2

Ge9/2

—

—

-0.049437

96

2s2 9h

2

Ho9/2

—

—

-0.049391

97

2

2

o

—

—

-0.049391

2s 9h

o

o

H11/2

D5/2

H11/2

121

-0.084555 0.0

-0.084526

-0.084804 3.2

-0.082142 0.0

-0.082141

-0.081734

33.8

-0.081644

-0.070587

-0.062573

0.0

-0.081744

0.0

-0.081645 0.0

29.7

-0.081645

-0.071582

0.0

34.9

-0.068349 0.3

-0.068343

0.7

-0.062873 0.0

-0.062573 0.0

-0.082171 -0.081744

0.0

-0.081644 0.0

4.4

-0.082171 0.0

-0.081734 0.0

-0.084764

-0.062872

0.0

-0.062580 0.0

-0.062580

0.0

-0.062511 0.0

-0.062511

0.0

-0.062612 5.5

0.2

-0.062564

-0.053486

5.3

0.4

-0.049942 0.7

-0.049935

0.8

-0.049651 0.0

-0.049650

0.0

-0.049441

0.0

-0.049441 0.0

-0.049392 0.0

-0.049392

0.0


Config.

Level

NEEW

NEER

98

2s2 10s

2 e

S1/2

—

—

-0.046010

-0.046050

99

2s2 10p

2

Po1/2

—

—

-0.042967

-0.042990

o

P3/2

—

—

-0.042966

2

FSS

TER-9

100

2s 10p

2

101

2s2 10d

2

De3/2

—

—

-0.040457

e

D5/2

—

—

-0.040455

2

102

2s 10d

2

103

2s2 10f

2

Fo5/2

—

—

-0.040188

o

F7/2

—

—

-0.040188

2

104

2s 10f

2

105

2s2 10g

2

Ge7/2

—

—

-0.040041

e

G9/2

—

—

-0.040041

2

106

2s 10g

2

107

2s2 10h

2

Ho9/2

—

—

-0.040007

o

—

—

-0.040007

2

FSS

0.1

TER-26

-0.042987

0.3

-0.040484

0.0

-0.040198

0.0

-0.040044

2s2 11s

2 e

S1/2

—

—

-0.037684

-0.037746

110

2s2 11p

2

Po1/2

—

—

-0.035283

-0.035300

o

P3/2

—

—

-0.035282

2s 11p

112

2s2 11d

2

De3/2

—

—

-0.033428

e

D5/2

—

—

-0.033427

2

113

2s 11d

2

114

2s2 11f

2

Fo5/2

—

—

-0.033201

o

F7/2

—

—

-0.033201

2

115

2s 11f

2

116

2s2 11g

2

Ge7/2

—

—

-0.033089

e

G9/2

—

—

-0.033089

2

117

2s 11g

2

118

2s2 11h

2

Ho9/2

—

—

-0.033063

o

—

—

-0.033063

2

0.0

0.1

-0.040007

-0.035298

0.1

-0.033448

0.0

-0.033209

0.0

-0.033092

2s2 12s

2 e

S1/2

—

—

-0.031714

-0.031953

121

2s2 12p

2

Po1/2

—

—

-0.029488

-0.029501

o

—

—

-0.029488

0.0

-0.033064

122

2s 12p

2

123

2s2p(3 Po )3p

2 e

S1/2

—

—

-0.028295

-0.028972

124

2s2 12d

2

De3/2

—

—

-0.028074

-0.028090

e

D5/2

—

—

-0.028073

125

2s 12d

2

126

2s2 12f

2

Fo5/2

—

—

-0.027889

o

F7/2

—

—

-0.027889

127

2s 12f

2

128

2s2 12g

2

Ge7/2

—

—

-0.027802

2

e

—

—

-0.027802

129

2

2

2s 12g

G9/2

122

0.0

0.0

-0.033064

120

2

0.2

-0.033092

2s 11h

P3/2

0.2

-0.033209

119

2

0.0

-0.033449

2

H11/2

0.0

-0.040007

109

111

0.0

-0.040044

2s 10h

2

0.3

-0.040198

108

2

0.3

-0.040487

2

H11/2

FSS

0.1

0.1

-0.029499

-0.028089

0.0

0.2

0.1

-0.027895 0.0

-0.027895

0.0

-0.027804 0.0

-0.027804

0.0


Config.

Level

NEEW

NEER

130

2s2 12h

2

Ho9/2

—

—

-0.027782

Ho11/2

—

—

-0.027782

2

FSS

TER-9

FSS

TER-26

0.0

-0.027783

-0.027783

131

2s 12h

2

132

2s2 13s

2 e

S1/2

—

—

-0.025562

-0.025758

133

2s2 13p

2

Po1/2

—

—

-0.025012

-0.025021

o

P3/2

—

—

-0.025011

2

134

2s 13p

2

135

2s2 13d

2

De3/2

—

—

-0.023907

e

D5/2

—

—

-0.023907

2

136

2s 13d

2

137

2s2 13f

2

Fo5/2

—

—

-0.023757

o

F7/2

—

—

-0.023757

2

138

2s 13f

2

139

2s2 13g

2

Ge7/2

—

—

-0.023688

e

G9/2

—

—

-0.023688

2

140

2s 13g

2

141

2s2 13h

2

Ho9/2

—

—

-0.023672

o

H11/2

—

—

-0.023672

2

142

2s 13h

2

143

2s2p(3 Po )3d

4

Fo3/2

195752.58

-0.008312

144

2s2p(3 Po )3d

4

Fo5/2

195765.85

-0.008191

145

3

o

2s2p( P )3d

4

o

F7/2

195785.74

-0.008010

146

3

o

2s2p( P )3d

4

o

F9/2

195813.66

-0.007755

147

2s2p(3 Po )3d

4

Do1/2

196557.87

-0.000974

148

3

o

2s2p( P )3d

4

o

D3/2

196563.41

-0.000923

149

3

o

2s2p( P )3d

4

o

D5/2

196571.82

-0.000846

150

3

o

4

o

196581.96

-0.000754

2s2p( P )3d

D7/2

FSS

0.0

-0.025020

0.0

0.1

-0.023920 0.1

-0.023919

0.1

-0.023762 0.0

-0.023762

0.0

-0.023690 0.0

-0.023690

0.0

-0.023673 0.0

-0.023673

0.0

61.1

24.1

a

Index.

b

Configuration. The 1s2 core is suppressed from all configurations.

c

NIST Experimental Energy in Wavenumbers (cm−1 ) relative to the ground state.

d

NIST Experimental Energy in Ryd relative to the C III 1 Se0 limit.

e

Fine Structure Splitting in cm−1 . The minus sign indicates that the theoretical levels are in reverse order compared to the experimental.

f

Theoretical Energy in Ryd for the 9-term target relative to the C III 1 Se0 limit.

g

Theoretical Energy in Ryd for the 26-term target relative to the C III 1 Se0 limit.

123

Table D.2: The available experimental data from NIST for the resonance states of C II above the C III 1 Se threshold alongside the 0 theoretical results as obtained by K-matrix method for the 9-term and the 26-term targets with n < 5. In.a

Config.b

Level

NEEWc

NEERd

1

2s2p(3 Po )3d

2

Do

198425.43

0.016045

2

2s2p(3 Po )3d

2

Do

198436.31

0.016144

3

2s2p(3 Po )3d

4

Po

198844

0.019859

0.021324

4

2s2p(3 Po )3d

4

Po

198865.25

0.020053

0.021514

5

2s2p(3 Po )3d

4

Po

198879.01

0.020178

6

2s2p(3 Po )3d

2

Fo

199941.41

0.029860

7

2s2p(3 Po )3d

2

Fo

199983.24

0.030241

3/2 5/2

5/2 3/2 1/2

5/2 7/2

FSSe

TER-9f

FSS

0.018009 10.9

0.018113

35.0

0.021598

41.8

0.033024

24.7

0.054952

TER-26g

FWHM-9h

FWHM-26i

7.36E-10

6.00E-10

2.91E-09

2.61E-09

0.020655

2.10E-10

2.17E-11

0.020849

1.24E-10

1.26E-09

4.73E-09

5.10E-10

6.52E-05

5.61E-05

6.71E-05

5.79E-05

7.76E-06

1.06E-05

8.22E-06

1.12E-05

FSS

0.017012 11.4

30.1

0.032632

0.017124

0.020968

12.3

34.3

0.031693 43.0

0.032109

45.6

8

2s2p(3 Po )3d

2

Po

202179.85

0.050258

9

2s2p(3 Po )3d

2

Po

202204.52

0.050483

10

2s2p(3 Po )4s

4

Po

209552.36

0.117441

0.118995

0.118682

1.49E-07

2.55E-07

11

2s2p(3 Po )4s

4

Po

209576.46

0.117661

0.119282

0.118938

7.12E-07

6.25E-07

12

2s2p(3 Po )4s

4

Po

209622.32

0.118079

1.75E-12

Recombination Lines of C II in the Spectra of

Recombination Lines of C II in the Spectra of

Suggest Documents

physical conditions derived from o ii recombination lines in ... - SciELO

On the line shift and oscillator strength of Xe ii lines in the spectra of ...

WIDbNED LINES IN SUNSPOT SPECTRA

Reference lines in the optogalvanic spectra of uranium and thorium ...

Gravitational Redshift of Emission Lines in the AGN Spectra

Galactic nebular lines in the fiber spectra of background QSOs ...

Reference lines in the optogalvanic spectra of ... - OSA Publishing

Zero-phonon lines in the photoluminescence spectra of MgO:Mn2

Discovery of Carbon Radio Recombination Lines in absorption ...

Interface-induced suppression of the Auger recombination in type-II ...

An investigation of Fe emission lines in solar flare spectra

Exciton Lines in Luminescence Spectra of NixZn1-xO ... - ScienceDirect

Photoionized lines in X-ray Spectra of SMC X-1

Carbon radio recombination lines in the Orion Bar

Exciton Lines in Luminescence Spectra of NixZn1 ... - ScienceDirect.com

Interpretation of the Ligand Field Spectra of Nickel (II

Recombination in Hepatitis C Virus - MDPI

Radiative recombination and optical gain spectra in biaxially strained

A search for broad infrared recombination lines in NGC1068

Faint recombination lines in Galactic PNe with a [WC] nucleus

Characterisation of hepatitis C virus recombination in Cameroon using ...

Faint recombination lines in Galactic PNe with a [WC] nucleus

Evidence of structural genomic region recombination in Hepatitis C virus

C-NMR spectra of polycyclic compounds and the stereochemistry of