3 Computations

The R-matrix calculations begin with the target wavefunction obtained through configuration interaction atomic structure calculation. Present target wavefunctions are obtained from the atomic structure calculations using SUPERSTRUCTURE (Eissner et al. 1974). The wavefunctions of Ar XIII and Fe XXI are represented by expansions of 15 fine structure levels of the target or the core ion, Ar XIV and Fe XXII, belonging to the 8 lowest LS terms: $2{\rm s}^22{\rm p}(^2{\rm P^o})$, $2{\rm s}2{\rm p}^2(^4{\rm P},^2{\rm P},^2{\rm D},^2{\rm S})$, and $2{\rm p}^3(^4{\rm S^o},^2{\rm D^o},^2{\rm P^o})$. The 15 fine structure levels alongwith their energies are listed in Table 1.

 

Table 1: The 15 fine structure levels and their relative energies of the target (core) ions Ar XIV and Fe XXII in the close coupling eigenfunction expansions of Ar XIII and Fe XXI. The list of spectroscopic and correlation configurations, and the scaling parameter ($\lambda$) for each orbital are given below the table

	Term	$J_{\rm t}$	$E_{\rm t}$(Ry)		Term	$J_{\rm t}$	$E_{\rm t}$(Ry)		Term	$J_{\rm t}$	$E_{\rm t}$(Ry)		Term	$J_{\rm t}$	$E_{\rm t}$(Ry)
Ar XIV								Fe XXI
1	${\rm 2s^22p(^2P^o)}$	1/2	0.0	9	${\rm 2s2p^2(^2P^e)}$	1/2	4.9685	1	${\rm 2s^22p(^2P^o)}$	1/2	0.0	9	${\rm 2s2p^2(^2S^e)}$	1/2	8.91420
2	${\rm 2s^22p(^2P^o)}$	3/2	0.20647	10	${\rm 2s2p^2(^2P^e)}$	3/2	5.0544	2	${\rm 2s^22p(^2P^o)}$	3/2	1.07776	10	${\rm 2s2p^2(^2P^e)}$	3/2	9.04241
3	${\rm 2s2p^2(^4P^e)}$	1/2	1.7925	11	${\rm 2p^3(^4S^o)}$	3/2	6.2455	3	${\rm 2s2p^2(^4P^e)}$	1/2	3.68653	11	${\rm 2p^3(^4S^o)}$	3/2	11.44278
4	${\rm 2s2p^2(^4P^e)}$	3/2	1.8709	12	${\rm 2p^3(^2D^o)}$	3/2	7.1115	4	${\rm 2s2p^2(^4P^e)}$	3/2	4.19365	12	${\rm 2p^3(^2D^o)}$	3/2	12.72512
5	${\rm 2s2p^2(^4P^e)}$	5/2	1.9756	13	${\rm 2p^3(^2D^o)}$	5/2	7.1361	5	${\rm 2s2p^2(^4P^e)}$	5/2	4.67717	13	${\rm 2p^3(^2D^o)}$	5/2	13.00269
6	${\rm 2s2p^2(^2D^e)}$	3/2	3.7387	14	${\rm 2p^3(^2P^o)}$	1/2	8.0301	6	${\rm 2s2p^2(^2D^e)}$	3/2	6.71166	14	${\rm 2p^3(^2P^o)}$	1/2	14.30352
7	${\rm 2s2p^2(^2D^e)}$	5/2	3.7471	15	${\rm 2p^3(^2P^o)}$	3/2	8.0711	7	${\rm 2s2p^2(^2D^e)}$	5/2	6.92217	15	${\rm 2p^3(^2P^o)}$	3/2	14.83288
8	${\rm 2s2p^2(^2S^e)}$	1/2	4.6878					8	${\rm 2s2p^2(^2P^e)}$	1/2	7.77748
Ar XIV:
Spectroscopic: $\rm 2s^22p,\ 2s2p^2,\ 2p^3$
Correlation: $\rm 2s^23s,\ 2s^23p,\ 2s^23d$, $\rm 2s3p^2,\ 2s2p3s,\ 2s2p3p,\ 2s2p3d,\ 2p^23p,\ 2p^23d,\ 2p3p3d$
$\lambda$: 2.45(1s), 1.34(2s), 1.44(2p), -0.76(3s), -0.83(3p), -1.5(3d)
Fe XXI:
Spectroscopic: $\rm 2s^22p,\ 2s2p^2,\ 2p^3$
Correlation: $\rm 2s^23s,\ 2s^23p,\ 2s^23d$, $\rm 2s3d^2,\ 2s2p3s,\ 2s2p3p,\ 2s2p3d,\ 2p^23s,\ 2p^23p,\ 2p^23d$
$\lambda$: 1.38259(1s), 1.16738(2s), 1.09097(2p), -1.18466(3s), -1.18082(3p), -1.50447(3d)

The energies are the observed ones (Kelly, NIST, for Ar XIV, and Sugar & Corliss 1985 for Fe XXII) for improved accuray. The correlation and spectroscopic configurations and the values of the scaling parameter ($\lambda$) in the Thomas-Fermi potential for each orbital of Ar XIV and Fe XXII in the atomic structure calculations are also listed in Table 1. Each target ion was optimized separately with configurations contributing significantly and, hence, does not correspond to the same set. For example, $2{\rm p}^23{\rm s}$ does not effect the optimization of Ar XIV, and is not included. The optimization was carried out to achieve close agreement with the observed energies and between the length and velocity f-values. The bound-channel term, the second expansion in the wavefunction, Eq. (1), includes all possible configurations upto $2{\rm p}^4$, $3{\rm s}^2$, $3{\rm p}^2$, and $3{\rm d}^2$ for both Ar XIII and Fe XXI.

The one- and two-electron radial integrals are computed by STG1 of the BPRM codes using the one-electron target orbitals generated by SUPERSTRUCTURE. The number of continuum R-matrix basis functions is 12 for each ion. The calculations consider all possible bound levels for $0 \leq J\leq 7$ with $n < 10, \ \ell \leq n-1$, $0\leq L \leq$ 9, and (2S+1)=1, 3, 5, even and odd parities. The intermediate coupling calculations are carried out on recoupling the LS symmetries in a pair-coupling representation in stage RECUPD. The (e + core) Hamiltonian matrix is diagonalized for each resulting $J\pi$ in STGH.

STGB of the BPRM codes calculates the fine structure energy levels and their wavefunctions. As fine structure causes large number of closely spaced energy levels, STGB requires to use an energy mesh of effective quantum number, $\Delta \nu =0.001$, an order of magnitude finer than needed in the LS coulping case to avoid levels missing. This increases the computation time considerably.

The oscillator strengths (f-values) are computed using STGBB of the BPRM codes. STGBB computes the transition matrix elements using the bound wavefuncitons created by STGB and angular algebra for the dipole moment calculated by STGH. The STGBB computations are also considerably CPU time extensive due to large number of dipole allowed and intercombination transitions among the fine structure levels.

The BPRM method, which uses the collision theory, describes the energy levels with channel identification for a given total $J\pi$. The information is not adequate for spectroscopic identification of the levels. A level identification procedure as described in Nahar & Pradhan (2000) is implemented to identify the levels of Ar XIII and Fe XXI. The identification scheme is based on the analysis of quantum defects and percentage weights of the channel wavefunctions similar to that under the Opacity Project (Seaton 1987). Each level is associated with a numer of collision channels. The analysis is carried out over the contributing channels with maximum channel percentage weight, i.e., the dominant channels that determine the proper configurations and terms of the core and the outer electrons. The levels are finally designated with possible identification of $C_{\rm t}(S_{\rm t}L_{\rm t}\pi_{\rm t})J_{\rm t} nlJ(SL)\pi$ where $C_{\rm t}$, $S_{\rm t} L_{\rm t}\pi_{\rm t}$, $J_{\rm t}$ are the configuration, LS term and parity, and total angular momentum of the core or target, nl are the principle and orbital quantum numbers of the outer or the valence electron, J and $SL\pi$ are the total angular momentum, possible LS term and parity of the (N+1)-electron system.

The principle quantum number, n, of the outer electron of a level is determined from its effective quantum number, $\nu$. For each partial wave (l), $\nu$ of the lowest member (level with the lowest principle quantum number of the valence electron) is determined from quantum defect analysis of all the computed levels with same l. The lowest partial wave (e.g. s) has the highest quantum defect ($\mu$). A check is maintained to differentiate the quantum defect of a "s'' electron from that of an equivalent electron state which has typically a large $\mu$ in a close coupling calculation. Once the lowest $\nu$ is determined, the levels of the corresponding Rydberg series is identified through $\Delta \nu$ which is approximately equal to 1 for the consecutive members. Such pattern for a series can be seen in the complete table for level energies (available electronically). For example, $\nu$ for a 3s electron is about 2.77 and for a 4s electron is about 3.66 for Ar XIII (Table 2a, $J\pi$ = 0$^{\rm o}$), and it is 2.85 for a 3s electron and 3.85 for a 4s electron for Fe XXI (Table 3a, $J\pi$ = 0$^{\rm o}$).

 

Table 2: a. Calculated BPRM fine structure energy levels of Ar XIII with spectrocopic identification. N_J= total number of levels of symmetry $J\pi$. $SL\pi$ lists possible set of LS terms

i	$C_{\rm t}$	$S_{\rm t} L_{\rm t}\pi_{\rm t}$	$J_{\rm t}$	nl	J	$E({\rm Ry})$	$\nu$	$SL\pi$
NJ = 56,     $J\pi$ = $0^{\rm e}$
1	${\rm 2s22p2}$				0	-5.04159E+01		${\rm ^3P e}$
2	${\rm 2s22p2}$				0	-4.89054E+01		${\rm ^1S e }$
3	${\rm 2p4}$				0	-4.14331E+01		${\rm ^3P e}$
4	${\rm 2p4}$				0	-3.96913E+01		${\rm ^1S e }$
5	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3p}$	0	-2.05345E+01	2.87	${\rm ^3P e}$
6	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3p}$	0	-1.97019E+01	2.91	${\rm ^1S e }$
7	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3s}$	0	-1.93963E+01	2.82	${\rm ^3P e}$
8	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3d}$	0	-1.74680E+01	2.96	${\rm ^5D e}$
9	${\rm 2s2p2}$	${\rm ^4P^e}$	5/2	${\rm 3d}$	0	-1.71405E+01	2.97	${\rm ^3P e}$
10	${\rm 2s2p2}$	${\rm ^2S^e}$	1/2	${\rm 3s}$	0	-1.70355E+01	2.79	${\rm ^1S e }$
11	${\rm 2s2p2}$	${\rm ^2P^e}$	1/2	${\rm 3s}$	0	-1.67478E+01	2.79	${\rm ^3P e}$
12	${\rm 2s2p2}$	${\rm ^2D^e}$	3/2	${\rm 3d}$	0	-1.53551E+01	2.97	${\rm ^3P e}$
13	${\rm 2s2p2}$	${\rm ^2D^e}$	5/2	${\rm 3d}$	0	-1.49520E+01	3.01	${\rm ^1S e }$
14	${\rm 2s2p2}$	${\rm ^2P^e}$	3/2	${\rm 3d}$	0	-1.41433E+01	2.97	${\rm ^3P e}$
15	${\rm 2p3}$	${\rm ^4S^o}$	3/2	${\rm 3p}$	0	-1.36805E+01	2.91	${\rm ^3P e}$
16	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 4p}$	0	-1.31449E+01	3.59	${\rm ^3P e}$
NJ = 53,     $J\pi$ = $0^{\rm o}$
1	${\rm 2s2p3}$				0	-4.58522E+01		${\rm ^3P o}$
2	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3s}$	0	-2.20575E+01	2.77	${\rm ^3P o}$
3	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3d}$	0	-1.90794E+01	2.96	${\rm ^3P o}$
4	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3p}$	0	-1.86737E+01	2.87	${\rm ^5D o}$
5	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3p}$	0	-1.80112E+01	2.91	${\rm ^3P o}$
6	${\rm 2s2p2}$	${\rm ^2D^e}$	3/2	${\rm 3p}$	0	-1.64710E+01	2.89	${\rm ^3P o}$
7	${\rm 2s2p2}$	${\rm ^2S^e}$	1/2	${\rm 3p}$	0	-1.59211E+01	2.86	${\rm ^3P o}$
8	${\rm 2s2p2}$	${\rm ^2P^e}$	3/2	${\rm 3p}$	0	-1.57212E+01	2.85	${\rm ^3P o}$
9	${\rm 2s2p2}$	${\rm ^2P^e}$	1/2	${\rm 3p}$	0	-1.54559E+01	2.88	${\rm ^1S o}$
10	${\rm 2p3}$	${\rm ^2P^o}$	1/2	${\rm 3s}$	0	-1.33819E+01	2.81	${\rm ^3P o}$
11	${\rm 2p3}$	${\rm ^4S^o}$	3/2	${\rm 3d}$	0	-1.30580E+01	2.96	${\rm ^5D o}$
12	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 4s}$	0	-1.26019E+01	3.66	${\rm ^3P o}$
13	${\rm 2p3}$	${\rm ^2D^o}$	5/2	${\rm 3d}$	0	-1.21319E+01	2.96	${\rm ^3P o}$
14	${\rm 2p3}$	${\rm ^2D^o}$	3/2	${\rm 3d}$	0	-1.17041E+01	3.00	${\rm ^1S o}$
15	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 4p}$	0	-1.14540E+01	3.57	${\rm ^5D o}$
16	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 4p}$	0	-1.11632E+01	3.60	${\rm ^3P o}$
NJ = 131,     $J\pi$ = $1^{\rm e}$
1	${\rm 2s22p2}$				1	-5.03260E+01		${\rm ^3P e}$
2	${\rm 2p4}$				1	-4.14770E+01		${\rm ^3P e}$
3	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3p}$	1	-2.08448E+01	2.85	${\rm ^1P e}$
4	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3p}$	1	-2.07028E+01	2.84	${\rm ^3SPD e}$
5	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3p}$	1	-2.05068E+01	2.86	${\rm ^3SPD e}$
6	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3p}$	1	-2.04340E+01	2.88	${\rm ^3SPD e}$
7	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3s}$	1	-1.99821E+01	2.79	${\rm ^5P e}$
8	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3s}$	1	-1.93348E+01	2.82	${\rm ^3P e}$
9	${\rm 2s2p2}$	${\rm ^2D^e}$	3/2	${\rm 3s}$	1	-1.81570E+01	2.78	${\rm ^3D e}$
10	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3d}$	1	-1.76801E+01	2.95	${\rm ^5PDF e}$
11	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3d}$	1	-1.74609E+01	2.96	${\rm ^5PDF e}$
12	${\rm 2s2p2}$	${\rm ^4P^e}$	5/2	${\rm 3d}$	1	-1.72186E+01	2.97	${\rm ^5PDF e}$
13	${\rm 2s2p2}$	${\rm ^4P^e}$	5/2	${\rm 3d}$	1	-1.71594E+01	2.97	${\rm ^3PD e}$
14	${\rm 2s2p2}$	${\rm ^4P^e}$	5/2	${\rm 3d}$	1	-1.71434E+01	2.97	${\rm ^3PD e}$
15	${\rm 2s2p2}$	${\rm ^2S^e}$	1/2	${\rm 3s}$	1	-1.69176E+01	2.80	${\rm ^3S e}$
16	${\rm 2s2p2}$	${\rm ^2P^e}$	3/2	${\rm 3s}$	1	-1.67240E+01	2.79	${\rm ^3P e}$
NJ = 136,     $J\pi$ = $1^{\rm o}$
1	${\rm 2s2p3}$				1	-4.65272E+01		${\rm ^3D o}$
2	${\rm 2s2p3}$				1	-4.58436E+01		${\rm ^3P o}$
3	${\rm 2s2p3}$				1	-4.45972E+01		${\rm ^3S o}$
4	${\rm 2s2p3}$				1	-4.39490E+01		${\rm ^1P o}$
5	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3s}$	1	-2.20006E+01	2.76	${\rm ^3P o}$
6	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3s}$	1	-2.17248E+01	2.79	${\rm ^1P o}$
7	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3d}$	1	-1.92584E+01	2.96	${\rm ^3PD o}$
8	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3d}$	1	-1.91117E+01	2.96	${\rm ^3PD o}$
9	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3d}$	1	-1.87724E+01	2.98	${\rm ^1P o}$
10	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3p}$	1	-1.86813E+01	2.87	${\rm ^3SPD o}$
11	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3p}$	1	-1.86458E+01	2.88	${\rm ^5PD o}$
12	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3p}$	1	-1.84751E+01	2.88	${\rm ^5PD o}$
13	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3p}$	1	-1.82444E+01	2.90	${\rm ^3SPD o}$
14	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3p}$	1	-1.79813E+01	2.92	${\rm ^3SPD o}$
15	${\rm 2s2p2}$	${\rm ^2D^e}$	3/2	${\rm 3p}$	1	-1.66700E+01	2.88	${\rm ^1P o}$
16	${\rm 2s2p2}$	${\rm ^2D^e}$	5/2	${\rm 3p}$	1	-1.66046E+01	2.88	${\rm ^3PD o}$

 

Table 2: b. Calculated BPRM fine strucuture energy levels of Ar XIII, identified and ordered. Nlv= total number of levels expected for the possible set of LS terms listed and Nlv(c) = number of calculated levels. $SL\pi$ lists the possible LS terms for each level. (See texts for details)

$C_{\rm t}$	$S_{\rm t} L_{\rm t}\pi_{\rm t}$	$J_{\rm t}$	nl	J	E(cal)	$\nu$	$SL\pi$
Eqv electron/unidentified levels, parity: e
2p4				2	-4.16174E+01		3  P e
2p4				1	-4.14770E+01		3  P e
2p4				0	-4.14331E+01		3  P e
$Nlv{\rm (c)}= 3$: set complete
Nlv=3,  3,o:	P ( 2 1 0 )
2s22p	(2Po)	1/2	3s	0	-2.20575E+01	2.77	3  P o
2s22p	(2Po)	3/2	3s	1	-2.20006E+01	2.76	3  P o
2s22p	(2Po)	3/2	3s	2	-2.18454E+01	2.77	3  P o
Nlv(c)=  3  : set complete
Nlv= 1, 1, o:	P ( 1 )
2s22p	(2Po)	1/2	3s	1	-2.17248E+01	2.79	1  P o
Nlv(c)=1: set complete
Nlv=3,  1,e:	S ( 0 ) P ( 1 ) D ( 2 )
2s22p	(2Po)	1/2	3p	1	-2.08448E+01	2.85	1  P e
2s22p	(2Po)	3/2	3p	2	-2.06875E+01	2.84	1  D e
2s22p	(2Po)	3/2	3p	0	-1.97019E+01	2.91	1  S e
Nlv(c)=  3  : set complete
Nlv=7,  3,e:	S ( 1 ) P ( 2 1 0 ) D ( 3 2 1 )
2s22p	(2Po)	3/2	3p	1	-2.07028E+01	2.84	3  SPD e
2s22p	(2Po)	3/2	3p	3	-2.05487E+01	2.85	3  D e
2s22p	(2Po)	1/2	3p	0	-2.05345E+01	2.87	3  P e
2s22p	(2Po)	3/2	3p	1	-2.05068E+01	2.86	3  SPD e
2s22p	(2Po)	1/2	3p	1	-2.04340E+01	2.88	3  SPD e
2s22p	(2Po)	3/2	3p	2	-2.03885E+01	2.86	3  PD e
2s22p	(2Po)	3/2	3p	2	-2.00647E+01	2.89	3  PD e
Nlv(c)=7: set complete
Nlv= 3,  5,e:	P ( 3 2 1 )
2s2p2	(4Pe)	1/2	3s	1	-1.99821E+01	2.79	5  P e
2s2p2	(4Pe)	3/2	3s	2	-1.99038E+01	2.79	5  P e
2s2p2	(4Pe)	5/2	3s	3	-1.98012E+01	2.79	5  P e
Nlv(c)= 3: set complete
Nlv=3,  1,o:	P ( 1 ) D ( 2 ) F ( 3 )
2s22p	(2Po)	1/2	3d	2	-1.95754E+01	2.94	1  D o
2s22p	(2Po)	3/2	3d	3	-1.87869E+01	2.98	1  F o
2s22p	(2Po)	3/2	3d	1	-1.87724E+01	2.98	1  P o
$Nlv{\rm (c)}= 3$: set complete
Nlv= 9,  3,o:	P ( 2 1 0 ) D ( 3 2 1 ) F ( 4 3 2 )
2s22p	(2Po)	1/2	3d	3	-1.94634E+01	2.95	3  DF o
2s22p	(2Po)	3/2	3d	2	-1.94284E+01	2.93	3  PDF o
2s22p	(2Po)	3/2	3d	4	-1.93579E+01	2.94	3  F o
2s22p	(2Po)	1/2	3d	1	-1.92584E+01	2.96	3  PD o
2s22p	(2Po)	3/2	3d	2	-1.92077E+01	2.95	3  PDF o
2s22p	(2Po)	3/2	3d	3	-1.91768E+01	2.95	3  DF o
2s22p	(2Po)	3/2	3d	2	-1.91492E+01	2.95	3  PDF o
2s22p	(2Po)	3/2	3d	1	-1.91117E+01	2.96	3  PD o
2s22p	(2Po)	3/2	3d	0	-1.90794E+01	2.96	3  P o
$Nlv{\rm (c)}=~9$: set complete
Nlv= 9,  3,e:	G ( 5 4 3 ) H ( 6 5 4 ) I ( 7 6 5 )
2s22p	(2Po)	1/2	6h	4	-4.69457E+00	6.00	3  GH e
2s22p	(2Po)	1/2	6h	5	-4.69457E+00	6.00	3  GHI e
2s22p	(2Po)	1/2	6h	6	-4.69455E+00	6.00	3  HI e
2s22p	(2Po)	3/2	6h	7	-4.48800E+00	6.00	3  I e
2s22p	(2Po)	3/2	6h	3	-4.48795E+00	6.00	3  G e
$Nlv{\rm (c)}=5$, Nlv= 9: set incomplete, level missing: 5 4 6 5
Nlv=3,  1,o:	P ( 1 ) D ( 2 ) F ( 3 )
2s22p	(2Po)	3/2	6d	2	-4.58862E+00	5.94	1  D o
2s22p	(2Po)	3/2	6d	3	-4.54791E+00	5.96	1  F o
2s22p	(2Po)	3/2	6d	1	-4.54451E+00	5.96	1  P o
$Nlv{\rm (c)}= 3$: set complete

 

Table 3: a. Calculated BPRM fine structure energy levels of Fe XXI with spectrocopic identification. N_J= total number of levels of symmetry $J\pi$. $SL\pi$ lists possible set of LS terms

i	$C_{\rm t}$	$S_{\rm t} L_{\rm t}\pi_{\rm t}$	$J_{\rm t}$	nl	J	$E{\rm (Ry)}$	$\nu$	$SL\pi$
NJ= 64,     $J\pi$ = $0^{\rm e}$
1	${\rm 2s22p2}$				0	-1.24292E+02		${\rm ^3P e}$
2	${\rm 2s22p2}$				0	-1.20849E+02		${\rm ^1S e }$
3	${\rm 2p4}$				0	-1.08406E+02		${\rm ^3P e}$
4	${\rm 2p4}$				0	-1.05511E+02		${\rm ^1S e }$
5	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3p}$	0	-5.21862E+01	2.91	${\rm ^3P e}$
6	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3s}$	0	-5.03119E+01	2.86	${\rm ^3P e}$
7	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3p}$	0	-4.99106E+01	2.94	${\rm ^1S e }$
8	${\rm 2s2p2}$	${\rm ^2P^e}$	1/2	${\rm 3s}$	0	-4.66313E+01	2.85	${\rm ^3P e}$
9	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3d}$	0	-4.62816E+01	2.96	${\rm ^5D e}$
10	${\rm 2s2p2}$	${\rm ^2S^e}$	1/2	${\rm 3s}$	0	-4.52695E+01	2.85	${\rm ^1S e }$
11	${\rm 2s2p2}$	${\rm ^4P^e}$	5/2	${\rm 3d}$	0	-4.52395E+01	2.97	${\rm ^3P e}$
12	${\rm 2s2p2}$	${\rm ^2D^e}$	3/2	${\rm 3d}$	0	-4.30183E+01	2.98	${\rm ^3P e}$
13	${\rm 2s2p2}$	${\rm ^2D^e}$	5/2	${\rm 3d}$	0	-4.22934E+01	2.99	${\rm ^1S e }$
14	${\rm 2s2p2}$	${\rm ^2P^e}$	3/2	${\rm 3d}$	0	-4.08816E+01	2.97	${\rm ^3P e}$
15	${\rm 2p3}$	${\rm ^4S^o}$	3/2	${\rm 3p}$	0	-4.01067E+01	2.92	${\rm ^3P e}$
16	${\rm 2p3}$	${\rm ^2D^o}$	3/2	${\rm 3p}$	0	-3.85891E+01	2.93	${\rm ^3P e}$
NJ= 61,     $J\pi$ = $0^{\circ}$
1	${\rm 2s2p3}$				0	-1.15895E+02		${\rm ^3P o}$
2	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3s}$	0	-5.48009E+01	2.84	${\rm ^3P o}$
3	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3d}$	0	-4.90612E+01	2.97	${\rm ^3P o}$
4	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3p}$	0	-4.89298E+01	2.89	${\rm ^5D o}$
5	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3p}$	0	-4.73308E+01	2.93	${\rm ^3P o}$
6	${\rm 2s2p2}$	${\rm ^2D^e}$	3/2	${\rm 3p}$	0	-4.50107E+01	2.92	${\rm ^3P o}$
7	${\rm 2s2p2}$	${\rm ^2P^e}$	1/2	${\rm 3p}$	0	-4.46865E+01	2.90	${\rm ^3P o}$
8	${\rm 2s2p2}$	${\rm ^2P^e}$	3/2	${\rm 3p}$	0	-4.35887E+01	2.89	${\rm ^1S o}$
9	${\rm 2s2p2}$	${\rm ^2S^e}$	1/2	${\rm 3p}$	0	-4.31686E+01	2.91	${\rm ^3P o}$
10	${\rm 2p3}$	${\rm ^2P^o}$	1/2	${\rm 3s}$	0	-3.98818E+01	2.85	${\rm ^3P o}$
11	${\rm 2p3}$	${\rm ^4S^o}$	3/2	${\rm 3d}$	0	-3.89417E+01	2.96	${\rm ^5D o}$
12	${\rm 2p3}$	${\rm ^2D^o}$	3/2	${\rm 3d}$	0	-3.74513E+01	2.96	${\rm ^3P o}$
13	${\rm 2p3}$	${\rm ^2D^o}$	5/2	${\rm 3d}$	0	-3.66926E+01	2.98	${\rm ^1S o}$
14	${\rm 2p3}$	${\rm ^2P^o}$	3/2	${\rm 3d}$	0	-3.53176E+01	2.96	${\rm ^3P o}$
15	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 4s}$	0	-2.97182E+01	3.85	${\rm ^3P o}$
16	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 4d}$	0	-2.69365E+01	3.97	${\rm ^3P o}$
NJ= 153,      $J\pi = 1^{\rm e}$
1	${\rm 2s22p2}$				1	-1.23615E+02		${\rm ^3P e}$
2	${\rm 2p4}$				1	-1.08363E+02		${\rm ^3P e}$
3	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3p}$	1	-5.28040E+01	2.89	${\rm ^3SPD e}$
4	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3p}$	1	-5.23926E+01	2.90	${\rm ^3SPD e}$
5	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3p}$	1	-5.15048E+01	2.90	${\rm ^1P e}$
6	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3p}$	1	-5.13231E+01	2.90	${\rm ^3SPD e}$
7	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3s}$	1	-5.09584E+01	2.84	${\rm ^5P e}$
8	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3s}$	1	-4.96425E+01	2.86	${\rm ^3P e}$
9	${\rm 2s2p2}$	${\rm ^2D^e}$	3/2	${\rm 3s}$	1	-4.78489E+01	2.84	${\rm ^3D e}$
10	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3d}$	1	-4.68999E+01	2.95	${\rm ^5PDF e}$
11	${\rm 2s2p2}$	${\rm ^2S^e}$	1/2	${\rm 3s}$	1	-4.65180E+01	2.82	${\rm ^3S e}$
12	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3d}$	1	-4.62583E+01	2.96	${\rm ^5PDF e}$
13	${\rm 2s2p2}$	${\rm ^2P^e}$	1/2	${\rm 3s}$	1	-4.58680E+01	2.87	${\rm ^3P e}$
14	${\rm 2s2p2}$	${\rm ^4P^e}$	5/2	${\rm 3d}$	1	-4.55835E+01	2.96	${\rm ^5PDF e}$
15	${\rm 2s2p2}$	${\rm ^2P^e}$	3/2	${\rm 3s}$	1	-4.53921E+01	2.85	${\rm ^1P e}$
16	${\rm 2s2p2}$	${\rm ^4P^e}$	5/2	${\rm 3d}$	1	-4.52726E+01	2.97	${\rm ^3PD e}$
NJ= 157,      $J\pi = 1^{\circ}$
1	${\rm 2s2p3}$				1	-1.17168E+02		${\rm ^3D o}$
2	${\rm 2s2p3}$				1	-1.15815E+02		${\rm ^3P o}$
3	${\rm 2s2p3}$				1	-1.14241E+02		${\rm ^3S o}$
4	${\rm 2s2p3}$				1	-1.12695E+02		${\rm ^1P o}$
5	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3s}$	1	-5.45579E+01	2.81	${\rm ^3P o}$
6	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3s}$	1	-5.36838E+01	2.87	${\rm ^1P o}$
7	${\rm 2s22p}$	${\rm ^2P^o}$	1/2	${\rm 3d}$	1	-4.98528E+01	2.97	${\rm ^3PD o}$
8	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3d}$	1	-4.92676E+01	2.96	${\rm ^3PD o}$
9	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3p}$	1	-4.89841E+01	2.89	${\rm ^5PD o}$
10	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3p}$	1	-4.87720E+01	2.90	${\rm ^5PD o}$
11	${\rm 2s22p}$	${\rm ^2P^o}$	3/2	${\rm 3d}$	1	-4.84796E+01	2.98	${\rm ^1P o}$
12	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3p}$	1	-4.79949E+01	2.91	${\rm ^3SPD o}$
13	${\rm 2s2p2}$	${\rm ^4P^e}$	3/2	${\rm 3p}$	1	-4.78167E+01	2.91	${\rm ^3SPD o}$
14	${\rm 2s2p2}$	${\rm ^4P^e}$	1/2	${\rm 3p}$	1	-4.69872E+01	2.95	${\rm ^3SPD o}$
15	${\rm 2s2p2}$	${\rm ^2D^e}$	3/2	${\rm 3p}$	1	-4.57908E+01	2.90	${\rm ^1P o}$
16	${\rm 2s2p2}$	${\rm ^2D^e}$	5/2	${\rm 3p}$	1	-4.51108E+01	2.91	${\rm ^3PD o}$

 

Table 3: b. Calculated BPRM fine structure energy levels of Fe XXI, identified and ordered. Nlv= total number of levels expected for the possible set of LS terms listed and $Nlv{\rm (c)} =$ number of calculated levels. $SL\pi$ lists the possible LS terms for each level. (See texts for details)

$C_{\rm t}$	$S_{\rm t} L_{\rm t}\pi_{\rm t}$	$J_{\rm t}$	nl	J	E(cal)	$\nu$	$SL\pi$
Nlv= 3,  3,o:	P ( 2 1 0 )
2s22p	(2Po)	1/2	3s	0	-5.48009E+01	2.84	3  P o
2s22p	(2Po)	3/2	3s	1	-5.45579E+01	2.81	3  P o
2s22p	(2Po)	3/2	3s	2	-5.37314E+01	2.84	3  P o
$Nlv{\rm (c)}= 3$: set complete
Nlv=1,  1,o:	P ( 1 )
2s22p	(2Po)	1/2	3s	1	-5.36838E+01	2.87	1  P o
$Nlv{\rm (c)}= 1$: set complete
Nlv= 7,  3,e:	S ( 1 ) P ( 2 1 0 ) D ( 3 2 1 )
2s22p	(2Po)	1/2	3p	1	-5.28040E+01	2.89	3  SPD e
2s22p	(2Po)	1/2	3p	1	-5.23926E+01	2.90	3  SPD e
2s22p	(2Po)	1/2	3p	2	-5.23595E+01	2.90	3  PD e
2s22p	(2Po)	1/2	3p	0	-5.21862E+01	2.91	3  P e
2s22p	(2Po)	3/2	3p	3	-5.14268E+01	2.90	3  D e
2s22p	(2Po)	3/2	3p	1	-5.13231E+01	2.90	3  SPD e
2s22p	(2Po)	3/2	3p	2	-5.12455E+01	2.90	3  PD e
$Nlv{\rm (c)}= 7$: set complete
Nlv= 3,  1,e:	S ( 0 ) P ( 1 ) D ( 2 )
2s22p	(2Po)	3/2	3p	1	-5.15048E+01	2.90	1  P e
2s22p	(2Po)	3/2	3p	2	-5.07154E+01	2.92	1  D e
2s22p	(2Po)	3/2	3p	0	-4.99106E+01	2.94	1  S e
$Nlv{\rm (c)}= 3$: set complete
Nlv= 3,  5,e:	P ( 3 2 1 )
2s2p2	(4Pe)	1/2	3s	1	-5.09584E+01	2.84	5  P e
2s2p2	(4Pe)	3/2	3s	2	-5.05218E+01	2.84	5  P e
2s2p2	(4Pe)	5/2	3s	3	-5.00672E+01	2.84	5  P e
Nlv(c)= 3: set complete
Nlv= 3,  1,o:	P ( 1 ) D ( 2 ) F ( 3 )
2s22p	(2Po)	1/2	3d	2	-5.05889E+01	2.95	1  D o
2s22p	(2Po)	3/2	3d	3	-4.85779E+01	2.98	1  F o
2s22p	(2Po)	3/2	3d	1	-4.84796E+01	2.98	1  P o
$Nlv{\rm (c)}= 3$: set complete
Nlv=3,  3,e:	P ( 2 1 0 )
2s2p2	(4Pe)	1/2	3s	0	-5.03119E+01	2.86	3  P e
2s2p2	(4Pe)	3/2	3s	1	-4.96425E+01	2.86	3  P e
2s2p2	(4Pe)	5/2	3s	2	-4.91481E+01	2.86	3  P e
$Nlv{\rm (c)}= 3$: set complete
Nlv= 9,  3,o:	P ( 2 1 0 ) D ( 3 2 1 ) F ( 4 3 2 )
2s22p	(2Po)	1/2	3d	3	-5.00116E+01	2.97	3  DF o
2s22p	(2Po)	1/2	3d	2	-4.99604E+01	2.97	3  PDF o
2s22p	(2Po)	1/2	3d	1	-4.98528E+01	2.97	3  PD o
2s22p	(2Po)	3/2	3d	2	-4.94634E+01	2.95	3  PDF o
2s22p	(2Po)	3/2	3d	4	-4.94577E+01	2.95	3  F o
2s22p	(2Po)	3/2	3d	3	-4.94017E+01	2.96	3  DF o
2s22p	(2Po)	3/2	3d	1	-4.92676E+01	2.96	3  PD o
2s22p	(2Po)	3/2	3d	2	-4.92368E+01	2.96	3  PDF o
2s22p	(2Po)	3/2	3d	0	-4.90612E+01	2.97	3  P o
$Nlv{\rm (c)}= 9$: set complete
Nlv= 9,  5,o:	S ( 2 ) P ( 3 2 1 ) D ( 4 3 2 1 0 )
2s2p2	(4Pe)	1/2	3p	1	-4.89841E+01	2.89	5  PD o
2s2p2	(4Pe)	1/2	3p	0	-4.89298E+01	2.89	5  D o
2s2p2	(4Pe)	1/2	3p	2	-4.88377E+01	2.90	5  SPD o
2s2p2	(4Pe)	1/2	3p	1	-4.87720E+01	2.90	5  PD o
2s2p2	(4Pe)	3/2	3p	3	-4.85537E+01	2.89	5  PD o
2s2p2	(4Pe)	3/2	3p	2	-4.82586E+01	2.90	5  SPD o
2s2p2	(4Pe)	5/2	3p	4	-4.77166E+01	2.90	5  D o
2s2p2	(4Pe)	5/2	3p	3	-4.76095E+01	2.90	5  PD o
2s2p2	(4Pe)	5/2	3p	2	-4.72496E+01	2.91	5  SPD o
$Nlv{\rm (c)}= 9$: set complete
Nlv=3,  3,e:	D ( 3 2 1 )
2s2p2	(2De)	3/2	3s	1	-4.78489E+01	2.84	3  D e
2s2p2	(2De)	5/2	3s	2	-4.77709E+01	2.84	3  D e
2s2p2	(2De)	5/2	3s	3	-4.76659E+01	2.84	3  D e
$Nlv{\rm (c)}= 3$: set complete

It may be noted that due to weaker correlation effects quantum defects are more well defined for higher levels, than the lower ones. where the difference, $\Delta \nu$, approaches unity.

Two levels with the same configuration and set of quantum numbers can actually be two independent levels due to outer electron spin addition/subtraction to/from the parent spin angular momentum, i.e. $S_{\rm t}\pm s = S$. The lower energies are normally assigned with the higher spin multiplicity while the higher levels are with the lower spin multiplicity. However, the energies and effective quantum numbers ($\nu$) of levels of higher and lower spin multiplicity can be very close to each other, in which case the observation may show them otherwise.

Following level identification, a direct correspondence is made with standard spectroscopic designations that follow different coupling schemes, such as between LS and JJ. The correspondence provides the check for completeness of calculated set of levels or the levels missing. A computer program, PRCBPID, is developed to carry out the identification scheme (Nahar & Pradhan 2000).

The level identification procedure involves considerable manipulation of the bound level data and, although it has been encoded for general applications, still requires analysis and interpretation of problem cases of highly mixed levels that are difficult to identify. The mixed states are dominated by a number of channels. The uncertainty in their identification can be large.