2. High-nl term energies

2.1. The Ritz formula

In 1s²2s²nl (C II), 1s²2snl (C III) and 1s²nl (C IV) configurations, the outer nl electron moves in the resultant field of the nucleus and the inner electrons. In such cases, the deviations between the observed term energies and the exact hydrogen-like values, caused by small core-penetration and polarization effects, are accurately described by the Ritz formalism. In this semi-empirical technique, discussed in detail by Edlén (1964) and more recently by Drake (1994), the term energies are written as

where is the ionization energy, Ry is the reduced mass Rydberg constant (Ry = 109732.3 cm^-1 for carbon), is the net charge of the core ( = 2, 3, 4 for C II, C III and C IV respectively) and n^* is the effective principal quantum number given by

(n,l) is the so-called quantum defect which is strongly dependent on the l-quantum number but only weakly on n and which can be expressed, in the Ritz formalism, by

2.2. The polarization formula

The energies of high l (non-penetrating) states can also be given to a good approximation by the polarization formula (Edlén 1964):

The hydrogenic, , and relativistic, , terms are given by

where is the fine structure constant. The polarization term, , can be written as

with

and

In Eq. (7), the factors A and k can be expressed respectively by

and

where is the dipole polarizability of the atomic core in units of a₀³ and represents the quadrupole polarizability of the core in units of a₀⁵.

2.3. Predicted term energies

Using the values 196664.7 and 386241.0 cm^-1 (Moore 1970) for the ionization energies of C II and C III respectively and the observed term energies deduced from the compilation of Bashkin & Stoner (1975), new Ritz formulae were derived for the 1s²2s²nf, 1s²2s²ng (C II) and 1s²2sng (C III) series. The Ritz parameters (a, b, c) obtained in the present work as well as the standard deviations between the experimental term energies and those calculated from the Ritz formulae are given in Table 1 (click here). For the 1s²2s²nh-ni (C II), 1s²2snh-nk (C III) and 1s²nf-nk (C IV) series, the polarization Eq. (4) was used. In the cases of C II and C III, only the dipole contribution was retained (k was set equal to zero) with the dipole polarizabilities = 3.77 and 3.46 a₀³ respectively. These values were derived from the oscillator strengths published for C III and C IV using the following formula:

where (E_i-E₀/Ry) are the transition energies (in Rydbergs) for lines connecting the ground level in C III and C IV and f_0i the corresponding oscillator strengths taken from the Opacity Project (the Opacity Project Team 1995) and corrected with experimental wavelengths by Verner et al. (1994). In the case of C IV, we used the polarization Eq. (4) including both dipole and quadrupole polarizabilities published recently by Tunklev et al. (1997), i.e. a₀³ and = 0.00111 a₀⁵, and the ionization energy = 520175.9 cm^-1 obtained by the same authors.

 

Ion Series a b c standard deviation

C II 0.028958 -0.221105  0.828474 0.50 cm^-1

C II ng ²G 0.006837 -0.040060 -0.242426 0.32 cm^-1

C III ng ^1,3G 0.026138 -0.672472  6.720485 0.41 cm^-1

 

The term energies calculated in our work are compared with available experimental values in Table 2 (click here). Examining this table, it is seen that the discrepancies between both sets of values are very small (the mean difference being 0.8 cm^-1) if we except the 6h ²H term belonging to C II. For this term, the difference between our value, calculated using the polarization formula, and the measured energy (Bashkin & Stoner 1975) reaches 64.6 cm^-1 which probably means that this particular term energy should be redetermined experimentally.

Tables 3 (click here), 4 (click here) and 5 (click here) contain the term energies adopted in the present study for the nf, ng, nh, ni series in C II, for the ng, nh, ni, nk series in C III and for the nf, ng, nh, ni, nk series in C IV respectively for n up to 30. Following the hereabove discussion, the mean uncertainty affecting our predicted values in these series should not exceed 1 cm^-1.

 

Ion Term Experiment This work

C II 168978.8^a 168979.3^c -0.5

C II 178956.4^a 178956.5^c -0.1

C II 184376.5^a 184376.5^c 0.0

C II 187642.0^a 187643.3^c -1.3

C II 5g ²G 179073.5^a 179073.5^c 0.0

C II 6g ²G 184449.7^a 184449.7^c 0.0

C II 7g ²G 187691.8^a 187691.8^c 0.0

C II 8g ²G 189794.6^a 189795.9^c -1.3

C II 184529.9^a 184465.3^d 64.6

C II 187701.0^a 187702.2^d -1.2

C III 5g ¹G 346579.3^a 346579.9^c -0.6

C III 5g ³G 346579.3^a 346579.9^c -0.6

C III 6g ¹G 358692.2^a 358692.6^c -0.4

C III 6g ³G 358692.3^a 358692.6^c -0.3

C III 7g ³G 365998.4^a 365998.6^c -0.2

C III 358776.3^a 358775.8^d 0.5

C IV 410431.5^b 410431.3^d 0.2

C IV 449941.3^b 449939.8^d 1.5

C IV 471402.4^b 471401.3^d 1.1

C IV 484341.9^b 484341.8^d 0.1

C IV 492740.8^b 492740.7^d 0.1

C IV 5g ²G 449945.2^b 449945.0^d 0.2

C IV 6g ²G 471405.8^b 471404.5^d 1.3

C IV 7g ²G 484345.6^b 484343.9^d 1.7

C IV 8g ²G 492743.8^b 492742.1^d 1.7

C IV 471406.8^a 471405.3^d 1.5

C IV 484345.8^a 484344.4^d 1.4

C IV 492743.8^a 492742.4^d 1.4

C IV 498501.4^a 498500.1^d 1.3

C IV 7i ²I 484346.0^a 484344.7^d 1.3

C IV 8i ²I 492743.9^a 492742.6^d 1.3

C IV 9i ²I 498501.4^a 498500.2^d 1.2

 

 

n ng ²G ni ²I

4 168978.8^a

5 178956.4^a 179073.5^a

6 184376.5^a 184449.7^a 184529.9^a

7 187642.0^a 187691.8^a 187701.0^a 187705.1^c

8 189762.2^b 189794.6^a 189803.0^c 189805.1^c

9 191214.0^b 191238.2^b 191243.3^c 191244.8^c

10 192251.8^b 192269.8^b 192273.6^c 192274.6^c

11 193019.2^b 193032.9^b 193035.8^c 193036.6^c

12 193602.6^b 193613.3^b 193615.5^c 193616.1^c

13 194056.4^b 194064.9^b 194066.6^c 194067.1^c

14 194416.3^b 194423.1^b 194424.6^c 194425.0^c

15 194706.6^b 194712.2^b 194713.3^c 194713.7^c

16 194944.1^b 194948.7^b 194949.7^c 194949.9^c

17 195140.9^b 195144.7^b 195145.5^c 195145.7^c

18 195305.7^b 195309.0^b 195309.6^c 195309.8^c

19 195445.2^b 195448.0^b 195448.5^c 195448.7^c

20 195564.3^b 195566.6^b 195567.1^c 195567.3^c

21 195666.7^b 195668.8^b 195669.2^c 195669.3^c

22 195755.5^b 195757.3^b 195757.6^c 195757.7^c

23 195832.9^b 195834.5^b 195834.8^c 195834.9^c

24 195900.9^b 195902.2^b 195902.5^c 195902.6^c

25 195960.8^b 195962.0^b 195962.3^c 195962.4^c

26 196014.0^b 196015.1^b 196015.3^c 196015.4^c

27 196061.3^b 196062.3^b 196062.5^c 196062.6^c

28 196103.7^b 196104.6^b 196104.8^c 196104.8^c

29 196141.8^b 196142.5^b 196142.7^c 196142.8^c

30 196176.1^b 196176.8^b 196176.9^c 196177.0^c

 

 

n ng ^1,3G ni ^1,3I

5 346579.3^a

6 358692.2^a 358776.3^a

7 365998.4^a 366063.8^c 366077.5^c

8 370743.2^b 370794.1^c 370803.6^c 370807.1^c

9 373997.4^b 374037.0^c 374043.8^c 374046.4^c

10 376325.4^b 376356.4^c 376361.5^c 376363.5^c

11 378047.8^b 378072.5^c 378076.3^c 378077.8^c

12 379357.8^b 379377.6^c 379380.6^c 379381.7^c

13 380377.1^b 380393.2^c 380395.5^c 380396.5^c

14 381185.8^b 381198.9^c 381200.9^c 381201.6^c

15 381838.1^b 381849.0^c 381850.6^c 381851.2^c

16 382371.8^b 382381.0^c 382382.3^c 382382.8^c

17 382814.1^b 382821.8^c 382822.9^c 382823.3^c

18 383184.7^b 383191.3^c 383192.2^c 383192.6^c

19 383498.3^b 383503.9^c 383504.7^c 383505.0^c

20 383766.0^b 383770.9^c 383771.5^c 383771.8^c

21 383996.3^b 384000.6^c 384001.1^c 384001.4^c

22 384195.9^b 384199.6^c 384200.1^c 384200.3^c

23 384370.1^b 384373.3^c 384373.8^c 384373.9^c

24 384522.9^b 384525.8^c 384526.1^c 384526.3^c

25 384657.7^b 384660.3^c 384660.6^c 384660.7^c

26 384777.2^b 384779.5^c 384779.8^c 384779.9^c

27 384883.8^b 384885.8^c 384886.1^c 384866.2^c

28 384979.0^b 384980.9^c 384981.1^c 384981.2^c

29 385064.6^b 385066.3^c 385066.5^c 385066.6^c

30 385141.8^b 385143.3^c 385143.5^c 385143.6^c

 

 

n ng ²G ni ²I

4 410431.5^a

5 449941.3^a 449945.2^a

6 471402.4^a 471405.8^a 471406.8^b

7 484341.9^a 484345.6^a 484345.8^b 484346.0^b

8 492740.8^a 492743.8^a 492743.8^b 492743.9^b 492742.7^c

9 498498.8^c 498499.8^c 498501.4^b 498501.4^b 498500.3^c

10 502617.6^c 502618.3^c 502618.5^c 502618.6^c 502618.6^c

11 505665.0^c 505665.5^c 505665.7^c 505665.7^c 505665.8^c

12 507982.7^c 507983.2^c 507983.3^c 507983.3^c 507983.4^c

13 509786.5^c 509786.8^c 509786.9^c 509787.0^c 509787.0^c

14 511217.7^c 511218.0^c 511218.1^c 511218.1^c 511218.1^c

15 512372.4^c 512372.6^c 512372.6^c 512372.7^c 512372.7^c

16 513317.3^c 513317.5^c 513317.6^c 513317.6^c 513317.6^c

17 514100.5^c 514100.7^c 514100.7^c 514100.7^c 514100.7^c

18 514756.8^c 514756.9^c 514757.0^c 514757.0^c 514757.0^c

19 515312.2^c 515312.4^c 515312.4^c 515312.4^c 515312.4^c

20 515786.5^c 515786.6^c 515786.6^c 515786.6^c 515786.6^c

21 516194.6^c 516194.6^c 516194.7^c 516194.7^c 516194.7^c

22 516548.3^c 516548.3^c 516548.4^c 516548.4^c 516548.4^c

23 516856.9^c 516856.9^c 516856.9^c 516857.0^c 516857.0^c

24 517127.7^c 517127.7^c 517127.8^c 517127.8^c 517127.8^c

25 517366.7^c 517366.7^c 517366.7^c 517366.7^c 517366.7^c

26 517578.6^c 517578.7^c 517578.7^c 517578.7^c 517578.7^c

27 517767.5^c 517767.5^c 517767.5^c 517767.5^c 517767.5^c

28 517936.4^c 517936.4^c 517936.5^c 517936.5^c 517936.5^c

29 518088.2^c 518088.2^c 518088.2^c 518088.2^c 518088.2^c

30 518225.1^c 518225.1^c 518225.1^c 518225.1^c 518225.1^c