Table of Spin-Orbit Energies for p-Electrons in Neutral Atomic (core)np Configurations

Abstract

Data are given in support of the recently proposed formula ζ_npn*³=0.450Z^2.33 cm⁻¹ (Z ⩾ 5). The table includes evaluations of this formula, which are probably accurate to ~ 15 percent for all atoms Z = 10 to 90. For many (core)np configurations, the accuracy of ζ_np from the formula is greater than that to be expected from fitting the observed levels to intermediate coupling theory.

1. Introduction

Values of the spin-orbit coupling parameter ζ_np for configurations of the type (core)np in neutral atoms were recently shown to be fairly accurately reproduced for atomic numbers Z ⩾ 5 by the formula [1]¹

$${\zeta }_{np}n{*}^3=0.450Z^{2.33}{\text{cm}}^{-1},$$ (1)

where n* is the effective principal quantum number of the np electron. The relation of the Z-dependence of the spin-orbit interaction to its physical origin was discussed in [1]. This note gives in more detail some data supporting the formula, and evaluations of it for Z ⩽ 90.

2. Values of ζ_npn*³

The dependence ζ_np ∞ 1/n*³ is known to be a good approximation for simple spectra [2, 3, 4]. The change in n* corresponding to the fine structure splitting is thus approximately constant for a series with structure proportional to ζ_np (np ²P and snp ³P₂₋₀ intervals) [2, 4]. Observations of unperturbed series of this type usually show Δn* to be constant within ~ 2 percent, except for the lowest series member or two.

The constants in (1) were determined by graphically fitting the data from such series in ten spectra,² indicated by stars after the element symbols in table 1. The series members that determined the value of ζ_pn*³ for each of these spectra are listed under the heading “np Data.” These ten spectra range in atomic number from 11 to 81, with an average disagreement of 6 percent between the observed value of ζ_pn*³ and the value from (1). The maximum disagreement is 12 percent.

Table 1. Values of ζn*³ for np-electrons in configurations of the type (core)np.

The data are from neutral atoms with the core-configurations shown. A core with electrons of different nl indicates that configuration interaction was included in the calculation. Where more than one np electron is shown in the fourth column, the value in the fifth column is an average. References for data or calculations are indicated by letters in parentheses in the fourth column. The other experimental values of ζn*³ are based on data from Atomic Energy Levels [C. E. Moore, Nat. Bur. Stand. (U.S.), Circ. 467, Vol. I (1949); Vol. II (1952); Vol. III (1958)].

z	Atom	Core	np Data	ζn*3 Data	ζn*3 Formula
				(cm−1)	(cm−1)
1	H		np	1.94	0.45
2	He	1s	2p	5.16	2.26
3	Li	1s2	2p	1.68	5.8
4	Be	2s	2p	6.00	11.4
5	B	2s2	2, 3p	19.3	19.1
6	C	2p			29.3
7	N	2p2			41.9
8	O	2p3			57
9	F	2p4			75
10	Ne	2p5			96
11	Na*	2p6	4–8p(a)	117	120
12	Mg*	3s	4–6p(b)	149	147
13	Al*	3s2	4–7p(c)	200	177
14	Si	3p			211
15	P	3p2			247
16	s	3p3			288
17	Cl	3p4			331
18	Ar	3p5			378
				(102 cm−1)	(102 cm−1)
19	K*	3p6	5–10p(a)	4.39	4.29
20	Ca*	4s	5–8p(d)	5.1	4.84
21	Sc	(3d + 4s)2			5.4
22	Ti	(3d+4s)3			6.0
23	V	(3d + 4s)4			6.7
24	Cr	(3d + 4s)r)			7.4
25	Mn	(3d + 4s)6			8.1
26	Fe	(3d + 4s)7			8.9
27	Co	(3d + 4s)8			9.7
28	Ni	(3d + 4s)9	4p(e)	13.0	10.6
29	Cu	(3d + 4s)10	4p(f)	12.5	11.5
30	Zn*	4s	5–8p(g)	11.0	12.4
31	Ga*	4s2	5–7p(h)	14.0	13.4
32	Ge	4p	5p(i)	15.9	14.5
33	As	4p2			15.5
34	Se	4p3			16.7
35	Br	4p4			17.8
36	Kr	4p5	6p(i)	21.8	19.0
				(103 cm−1)	(103 cm−1)
37	Rb*	4p6	7–11p(j)	1.92	2.03
38	Sr	5s	5p	2.55	2.16
39	Y	(4d + 5s)2			2.29
40	Zr	(4d + 5s)3			2.43
41	Nb	(4d + 5s)4			2.58
42	Mo	(4d + 5s)5			2.73
43	Tc	(4d + 5s)6			2.88
44	Ru	(4d +5s)7			3.04
45	Rh	(4d + 5s)8			3.20
				(103 cm−1)	(103 cm−1)
46	Pd	(4d + 5s)9			3.37
47	Ag	4d10	6p	3.47	3.54
48	Cd	5s	6p	3.58	3.72
49	In	5s2	6, 7p(h)	3.99	3.90
50	Sn	5p			4.09
51	Sb	5p2			4.28
52	Te	5p3			4.48
53	I	5p4			4.69
54	Xe	5p5	6p(k)	4.79	4.89
55	Cs*	5p6	8–13p	4.74	5.1
56	Ba	6s	6p	6.1	5.3
57	La	(5d + 6s)2			5.6
58	Ce				5.8
59	Pr				6.0
60	Nd	4f46s			6.3
61	Pm	4f56s			6.5
62	Sm	4f66s			6.8
63	Eu	4f76s	6p(l)	8.3	7.0
64	Gd	4f75d6s			7.3
65	Tb				7.5
66	Dy	4f106s			7.8
67	Ho	4f116s			8.1
68	Er	4f126s			8.4
69	Tm	4f136s			8.7
70	Yb	6s	6p	10.5	9.0
71	Lu	(5d + 6s)2			9.3
72	Hf	(5d + 6s)3			9.6
73	Ta	(5d + 6s)4			9.9
74	W	(5d + 6s)5			10.2
75	Re	(5d + 6s)6			10.5
76	Os	(5d + 6s)7			10.9
77	Tr	(5d + 6s)8			11.2
78	Pt	(5d + 6s)9			11.5
79	Au	5d10	6p	14.4	11.9
80	Hg	6s	6p	17.8	12.2
81	Tl*	6s2	7–11p	13.4	12.6
82	Pb	6p			13.0
83	Bi	6p2			13.3
84	Po	6p3			13.7
85	At	6p4			14.1
86	Rn	6p5			14.5
87	Fr	6p6			14.9
88	Ra	7s	7, 8p	18.2	15.3
89	Ac	(6d + 7s)2			15.7
90	Th				16.1

^aRisberg, P., Arkiv Fysik 10, 583 (1956).

^bRisberg, G., Arkiv Fysik 28, 381 (1965).

^cEriksson, K. B. S., and Isberg, H. B. S., Arkiv Fysik 23, 527 (1963).

^dRisberg, G., Arkiv Fysik 37, 231 (1968).

^eRoth, C., J. Res. Nat. Bur. Stand. (U.S.), 74A (Phys. and Chem.) 715–722 (Sept.–Oct. 1970).

^fLandman, D. A., Levin, L. A., and Lurio, A., J. Opt. Soc. Am. 59, 962 (1969).

^gJohansson, I., and Contreras, R., Arkiv Fysik 37, 513 (1968).

^hJohansson, I., and Litzén, U., Arkiv Fysik 34, 573 (1967).

ⁱCowan, R. D., and Andrew, K. L., J. Opt. Soc. Am. 55, 502 (1965).

^jJohansson, I., Arkiv Fysik 20, 135 (1961).

^kLiberman, S., J. phys. 30, 53 (1969).

^lSmith, G., and Wybourne, B. G., J. Opt. Soc. Am. 55, 121 (1965).

The other data in the table are less reliable.³ Most configurations of the type (core)np with the core including a partly filled p or d subshell appear to be distorted by perturbations that are significant compared to ζ_np. The data were usually taken from the lowest such configuration, to minimize the relative error in ζ_p. The ζ_pn*³ values for these lowest configurations are systematically larger than the limiting constant values (from eq (1)) by ~ 15 percent. At least part of this effect is understood [3, 1]. Part of the remaining scatter in these values is probably due to configuration interactions neglected in the calculations. In some spectra there may also be uncertainty in obtaining properly corresponding values of ζ_p and n* [1].

It seems likely that the formula values of ζ_pn*³ (table 1) are correct to ~ 15 percent for (core)np configurations (n greater than any principal quantum number in the core) in all atoms Z = 10 to 90. This accuracy must of course not be expected if perturbation has affected the value of n*³ by more than a few percent. Any other large percentage deviations from the formula are suspect and probably unphysical. In many cases rather complete and detailed calculations would be necessary to obtain values of ζ_p by the level-fitting method as accurate as the formula values. It would thus be advantageous in some instances to fix this parameter.

Some examples of these various points may be inferred from table 2, where values of ζ_p obtained by different methods for several rare-gas configurations are compared. The smaller values of ζ_p, as in Ne, are relatively more sensitive to distortion by perturbations in level-fitting determinations.

Table 2.

Values of spin-orbit energy for np electrons in some rare-gas configurations. The “formula” values of ζ_np (from ζn*³ in table 1) are probably accurate to 15 percent.

Core		np	n*	ζnp, cm−1
				Formula	Level-fitting Calculations
Ne	2p5	3p	2.154	9.6	a40.0,b13 ± 60
		4p	3.163	3.0	b1 ± 12
		5p	4.165	1.33	a8.1
		6p	5.165	0.70	a 5.4
Ar	3p5	4p	2.279	32	b28 ± 29,c32.6
		5p	3.306	10.5	a44.1,b16 ± 7,c13.2
		6p	4.310	4.7	a 7.1
Kr	4p5	5p	2.322	152	a246,b159 ± 25
		6p	3.356	50	a49.0,b58 ± 9
Xe	5p5	6p	2.383	361	a374,d353.7

^aCondon, E. U., and Shortley, G. H., The Theory of Atomic Spectra, pp. 306–312 (Cambridge, 1953). Some of these values were quoted from earlier calculations, with references given.

^bCowan. R. D., and Andrew, K. L., J. Opt. Soc. Am. 55, 502 (1965).

^cGarstang, R. H., and Van Blerkom, J., J. Opt. Soc. Am. 55, 1054 (1965).

^dLiberman, S., J. Phys. 30, 53 (1969). Calculation includes interaction with 5p⁵7p.

The core configurations listed in table 1 are the lowest for each element. (A few of the cores that have low levels of both parities are not listed.) However, the value of ζ_np is only weakly affected by core excitation for most configurations observed in optical spectroscopy. Values of n* for a particular np (n greater than for any core electron) but based on different core configurations will normally differ by fairly small amounts if the configurations involved are unperturbed. The most reliable experimental values of n* are usually obtained from terms based on the core configurations in the table.

The available data would probably yield useful relations for the Z-dependence of ζ_pn*³ in ionized spectra. Both this and the probably more troublesome extension to higher l [1] would merit investigation.