Energies of 1snl (l≥3) Configurations in He⁴ i as Calculated from Polarization Theory

Abstract

The calculated singlet-triplet mean energies are given for all He⁴ i configurations having 3 ≤ l ≤ n — 1 from 4f to 8k. Since only Stark-shifted transitions have been observed for such configurations with l ≥ 4, the calculated positions are the most accurate available. The polarization energy (to the quadrupole approximation) for two-electron ions may be evaluated for higher nl or Z from the simple formulas. The calculated and observed positions of the Isnf configurations in He⁴ I and Li⁷ II agree within the experimental uncertainties.

1. Introduction

In a list of He⁴ I energy levels published in 1960 [1],¹ positions were given for several configurations 1s5l, 6l, 7l (l ≥ 4) that had been observed only in Stark-effect investigations [2]. An attempt was made to derive the approximate undisplaced positions for these configurations from Foster’s measurements [2]. However, Foster [3] had already given a more accurate position for the 5g configuration relative to the other levels for n = 5. This position was not derived from his observations but from the polarization theory of term defects due to Born and Heisenberg [4] and Waller [5]. (The “term defect” was the difference between the actual term value and the Bohr energy R/n².) A simple expression for the polarization defect for Isnl configurations for l ≥ 3 is applied here to obtain positions for several such configurations in He⁴ I.

2. Polarization Formulas

Polarization theory [4–9] is now frequently used to describe the deviations of non-penetrating terms T in alkali-like atoms from corresponding hydrogenic term values T₀. For a non-penetrating electron outside a closed-shell core [9]:

$$T-T_0={\Delta }_p=R{\alpha }_d\langle r^{-4}\rangle +R{\alpha }_q\langle r^{-6}\rangle$$ (1)

where R is the Rydberg constant and α_d and α_q are the electric dipole and quadrupole polarizabilities of the core in units of $a_0^3$ and $a_0^5$, respectively (α₀ is the Bohr radius). The expectation values 〈r⁻⁴〉 and 〈r⁻⁶〉 have been tabulated [8] and Edlén [9] has given a convenient table for use with the expression ²

$${\Delta }_p=A(Z)P(n,l)[1+k(Z)q(n,l)].$$ (2)

Here

$$A(Z)=Z_c^4{\alpha }_{d}andk(Z)=Z_c^2{\alpha }_q/{\alpha }_d$$ (3)

where Z_c is the net charge of the core; P(n, l) and q(n, l) are the quantities tabulated by Edlén.

The polarization energy for an atom with a core having a partly filled shell will in general contain terms not appearing for the closed-shell core. Bethe [10] gives an expression for the polarization quantum-defect for a two-electron atom (Isnl) that reduces to the dipole value for large Z·r.

It is useful to note that Waller’s dipole calculation agrees with the current best observations for Isnf in both He⁴ I and Li⁷ II to within the experimental accuracies. Weiss [11] has now shown that the simple expression (1) is valid for Isnl configurations of two-electron ions if l ≥ 3. In this case T is the arithmetic mean of the singlet and triplet terms for Isnl. This result allows an accurate calculation of Δ_p for such configurations since the Is core polarizabilities are just the hydrogenic values [5, 7]: α_d=9/(2Z⁴) and α_q = 15/Z⁶. For helium-like ions Z_c =Z − 1, and we have from (3):

$$A(Z)=9{(Z-1)}^4/\left(2Z^4\right)$$ (4)

$$k(Z)=10{(Z-1)}^2/\left(3Z^2\right).$$ (5)

With (4), (5), and (2) the polarization energies for Isnl (l ≥ 3) configurations may be evaluated to the quad-rupole approximation for any two-electron ion.

In the case of hydrogenic spectra, T₀ in eq (1) includes the appropriate relativistic correction to the Bohr term [9]. Relativistic corrections for excited states in two-electron ions are not generally available, but calculations for Isnp terms (n = 2, 3, 4) in He I have been reported [12]. The singlet-triplet mean corrections for these terms are of the same order as the relativistic corrections for the corresponding np terms in hydrogen. Since the hydrogenic corrections for l ≥ 3 are less than 0.01 cm⁻¹, the relativistic corrections for He are probably negligible to the accuracies of the calculations given here. Bethe and Salpeter [13] indicate that the mass-polarization correction is negligible for terms having l ≥ 2 in two-electron ions. Calculations by Mayer and Mayer [6] show that the effect due to penetration of the Is core by the excited wave function would also be negligible to the accuracy needed here. We thus adopt the uncorrected Bohr energy R/n² for T₀, the mean of the singlet and triplet terms without the Δ_p correction.

3. Term Defects

An accurate value for the He II (²S_1/2) series limit is needed in order to obtain (from calculated terms) the positions of the l ≥ 3 configurations relative to the S, P, and D levels. Seaton [14] applied a least-squares fitting procedure to several observed series and obtained a weighted mean of 198310.76 ±0.01 cm⁻¹ for the limit (relative to the He⁴I ¹S₀ ground level at 0.00±0.15 cm⁻¹) [15].³ His method is a refinement of the usual procedure in that it takes the relation between the quantum defect and the term value explicitly into account in the weighting. He did not, however, give a specific weighting according to the different experimental accuracies. The 3d, 4d, 5d ¹D and 3d-6d ³D levels [1, 16] are known to better than 0.01 cm⁻¹, and it should be possible to describe the D series with a simple two-parameter formula. If the 3d, 4d, and 5d baricenters are fitted to a Ritz formula n — n* = a + βT, a limit of 198310.741 cm⁻¹ is obtained. (Here T=R/n*² is the absolute term value, and R for He⁴ is taken as 109722.267 cm⁻¹.) Table 1 gives the results of a Ritz-formula fit⁴ based on Seaton’s value for the limit. Comparison of the observed and calculated baricenters for the accurately known lowest three terms shows them to be consistent with this limit. Seaton’s method showed convincingly that the earlier values given for the limit [1, 15] were too high, and his result is adopted here.

Table 1. Term values for the Isnd (n = 3 to 8) baricenters in He⁴ I.

Calculated values were determined from a Ritz formula with the constants (see text): R = 109722.267 cm⁻¹; α = 2.6865 × 10⁻³; β = −5.00 × 10⁻⁸. Observed values are based on levels from Ref. [1] referred to a limit of 198310.76 cm⁻¹.

Isnd	n* (calc)	Term (calc)	Term (obs)	O–C
		cm–1	cm–1	cm–1
3d	2.9979239	12208.254	12208.254	0.000
4d	3.9976568	6865.683	6865.679	‒.004
5d	4.997533	4393.225	4393.228	.003
6d	5.997466	3050.417	3050.42	.00
7d	6.997426	2240.878	2240.92	.04
8d	7.997399	1715.526	1715.49	‒.04

Table 2 gives the results of an evaluation of (2) for the He⁴ I configurations Isnl (l ≥ 3) to n = 8. From (4) and (5) the constants are: A = 9/32; k = 5/6. The limitation on the principal quantum number is arbitrary and the results may be extended to any Isnl configuration with l ≥ 3. It seems unlikely that the approximations in the method give an error as large as 0.1 cm⁻¹ even for 4f(the largest value of T). The deviations from the observed positions for 5f and 6f are within the experimental uncertainties. Any reasonable fitting of the nf series by one of the usual series formulas shows the observations for the lowest three members to be inconsistent on the scale of ~ 0.1 cm⁻¹. It could well be that the 5f and 6f positions are mainly at fault, as indicated by the deviations in table 2.

Table 2. Calculated polarization defects and term values for some configurations in He⁴ I.

The term values and positions are singlet-triplet means. T₀= (109722.267/n²) cm⁻¹, and T=T₀ + Δ_p.

1snl	T0	Δp	T	Positiona	O‒Cb
	cm−1	cm−1	cm−1	cm−1	cm−1
4f	6857.642	1.168	6858.810	191451.950	−0.01
5f	4388.891	0.673	4389.564	193921.196	.14
6f	3047.8408	.413	3048.254	195262.506	.08
7f	2239.2299	.269	2239.499	196071.261	.00
8f	1714.4104	.184	1714.594	196596.166	.00
5g	(see above)	.157	4389.048	193921.712
6g		.101	3047.942	195262.818
7g		.068	2239.298	196071.462
8g		.047	1714.457	196596.303
6h		.032	3047.873	195262.887
7h		.022	2239.252	196071.508
8h		.016	1714.426	196596.334
7i		.009	2239.239	196071.521
8i		.006	1714.416	196596.344
8k		.003	1714.413	196596.347

^aRelative to the Is^{2 1}S₀ ground level at 0.00 ±0.15 cm⁻¹. (Position) = 198310.760 cm⁻¹ — T; this limit is thought to be accurate to± 0.01 cm⁻¹.

^bDifference between the observed [Ref. 1] and calculated positions.

The nf ¹F° — ³F° separations obtained from the experimental positions for 4f and 5f (0.28 and 0.06 cm⁻¹, respectively) [1] are almost surely due to observational error (perhaps including source effects). The calculated separations [17] of only 0.011 and 0.009 cm⁻¹, respectively, should be accurate to much better than an order of magnitude. Thus at least one of the 4f terms is in error by more than 0.1 cm⁻¹, and an error of 0.2 cm⁻¹ in the mean seems reasonable.

If the various terms are known experimentally to about the same accuracy (in cm⁻¹), the most sensitive comparisons with polarization calculations for l ≥ 3 are obtained with the lowest nf configurations. The “O—C” results for these configurations in table 2 thus confirm the dipole contribution to the term defect to about 20 percent. They are not accurate enough to check the quadrupole contribution, however, which amounts to only about 2 percent of the dipole correction.

Any errors in T due to the approximations become smaller (in cm⁻¹) for higher values of n. Term values with more than three decimal places would be required in table 2 to give some separations of the higher configurations to the probable accuracy of the method.

The positions (relative to the ground level) previously given by the author [1] for the g, h, and i configurations in He are too high by up to 0.8 cm⁻¹ (for 5g). They are, in fact, unphysical in that they are above the corresponding Bohr positions. The best values now available for these configurations, and those of still higher l, are obtained from the polarization theory.

The calculated polarization energy for l = 3 is confirmed by the low F terms [18] in Li⁷ II to a somewhat better percentage accuracy than the available measurements allow for He⁴ I. In table 3 we see that the term defects for Li⁷ II nf configurations are about three times the corresponding values for He⁴ I. The experimental uncertainty⁵ for the lower F terms in Li II is probably about 0.2 cm⁻¹. The calculated and experimental term defects thus agree to within the experimental accuracy, and calculated values different by more than ~ 10 percent would not agree with the observations. The values given for T₀ in table 3 are themselves uncertain by at least 5 percent, because relativistic effects are not included.⁶ The total uncertainty in T₀ — T_obs for these terms is thus about three times the ~ 4 percent quadrupole contribution to Δ_p.

Table 3. Calculated and observed term defects for 1 snf configurations in Li⁷ II.

The polarization constants for Δ_p are A = 8/9 and k = 40/27. Term values are singlet-triplet means (see footnote 5). T₀ = 4R/n² = (438914.91/n²) cm⁻¹.

1snf	T0	Tobs	T0−Tobs	Δp
	cm‒1	cm‒1	cm‒1	cm‒1
4f	27432.18	27435.78	3.60	3.74
5f	17556.60	17558.69	2.09	2.16
6f	12192.08	12193.35	1.27	1.33
7f	8957.45	8958.32	0.8	0.87
8f	6858.05	6858.6g	.63	.59
9f	5418.70	5419.18	.48	.42
10f	4389.15	4389.33	.18	.31

Note added in proof: Another effect neglected here is also probably comparable in magnitude to the quadrupole-polarization energy, even in He I. A very recent paper giving polarization energies in He I [C. Deutsch, Phys. Rev. A 2, 43 (1970)] includes a nonadiabatic correction calculated to be slightly larger than, and opposite in sign to, the quadrupole-polarization contribution.

We see from eq (5) for k(Z) that the quadrupole contribution rises with Z; for high Z, k(Z) → 10/3 and A(Z) → 9/2. The polarization energies for a particular term thus approach asymptotic values with increasing Z. The quadrupole contribution for 4f approaches 6.9 percent, and the total polarization defect approaches (18.37 + 1.28) cm⁻¹ = 19.65 cm⁻¹. However, the realivistic corrections to T₀ are expected to increase as (Z — l)⁴. These corrections (see footnote 6) are probably already of the same order as the quadrupole energy for Isnf in Li II. It appears that verification of the quadrupole term for Isnf in two-electron ions would require a calculation of the relativistic effects.