1064 L E T T E R S T O T H E E D I T O R Vol. 54

Penetration Effects in the 2F-Series of CsI† ANDERS FRÖMAN,* JAN LINDERBERG, AND YNGVE ÖHRN

Quantum Chemistry Group, Uppsala University, Uppsala, Sweden (Received 6 May 1964)

AN unusual effect in the 2F terms of Cs I1 has been observed by Bockasten.2 A theoretical description is given in this

letter. Hydrogen-like terms in the level scheme of the alkali atoms occur when the orbit of the series electron does not penetrate the ion core, and a simple polarization formula is applicable to de­scribe term energies.3 Penetration of the series electron into the core generally causes all term levels to deviate considerably from the hydrogenic values. We will show here that two different pene­tration effects can be distinguished, and that it is reasonable to expect that the 4 ƒ-level of Cs I exhibits such an effect while the other n ƒ-levels are practically hydrogenic. We are using the phase integral or WKB method4 and assume that the potential experi­enced by the outer electron can be represented by means of an effective nuclear charge Z(r) which depends on the distance r from the nucleus. The quantum condition for the binding energy E (in atomic units) of an electron with the orbital angular mo­mentum quantum number l is4

where

and where a and b are the minimum and maximum values of r for which p(r) is real, viz., the turning points of the classical motion. Equation (1) leads to the ordinary Rydberg formula

when Z(r) = 1 in the interval (a,b). Penetration effects in all levels are found when Z(r) differs significantly from unity in the range of integration.

The situation is quite different for the ƒ levels in heavier atoms, because the effective radial potential has then two negative or allowed regions, which are accessible for a bound electron, and which are separated by a rather high and wide potential barrier as shown in Fig. 1. A new quantization rule is required and has

FIG. 1. Effective radial potential for ƒ electrons in Cs from Ref. 8.

been derived by Wu.5 Three phase integrals enter his expression, one referring to the inner, allowed interval (a,b), one taken over the barrier (b,c), and one being connected with the outer permis­sible region (c,d)· Wu's formulas are

The three phase integrals all contain E as a parameter. The eigen­values cannot readily be expressed in terms of the principal quantum number n, but if we assume that Z(r) = 1 in the interval (c,d), we obtain the simple quantum defect formula5

A small quantum defect Δ can be accommodated within the polari­zation formula provided it is of the form6

and we can compare this with Eq. (5) which gives Δ as a more complicated function of E. We can use a Taylor series for the right-hand side of Eq. (5) and find then that the validity of the simple formula is dependent on the magnitude of the second derivative of Δ with respect to the parameter E. The convergence of the Taylor series is poor when A is such that cosA~0 and this will give rise to noticeable deviations in the term values from those predicted with the formula. When A =π/2, the quantum condition for an inner 4ƒ level like those in the rare earths is fulfilled.7 We expect that for an atom like cesium, which is close to the lantha-nides in the periodic system, A would be somewhat smaller but still close to the critical value π/2. A calculation confirms this expectation, and with the use of an effective potential for neutral Cs computed by Herman and Skillman8 we have obtained values ranging from 0.433 π for n = 4 and 0.441 π for n=10, where E= — yn -2 . This variation and the associated changes in B give a deviation from the polarization formula value for the 4 ƒ-level of — 1.4 cm - 1 which compares well with Bockasten's semi-empirical value —0.2643 cm-1.

Exchange effects have been exaggerated in the calculations of Herman and Skillman. Hartree9 has given an effective Z(r) for Cs+ which neglects exchange completely, and with his potential we have computed the value —0.014 cm - 1 for the deviation, and a more accurate description of exchange and short range correla­tions should lead to an improvement of the theoretical calculations relative to the experimental observations. The calculated devia­tions for other ƒ levels are relatively large when the potential of

August 1964 L E T T E R S T O T H E E D I T O R 1065

Herman and Skillman is used (<0.09 cm-1) but are smaller than the experimental uncertainties when no exchange is allowed for (<0.0006 cm -1).

The approximations in the calculations are such that the argu­ment must be considered merely to indicate a plausible explanation as to why the 4ƒ level does not fall within the applicability of the polarization formula, while at the same time penetration effects are not evident in the rest of the 2F terms in Cs I. Similar observa­tions might be done in the Ba I I spectrum but there is a possibility that more than one level is affected there.

We are indebted to Dr. K. Bockasten for drawing our attention to this problem and to Mr. Bertil Jansson who helped us with the numerical calculations.

† The research in this paper was sponsored in part by the King Gustaf VI Adolf's 70-Years Fund for Swedish Culture, the Knut and Alice Wallen­berg's Foundation, and in part by the Aeronautical Research Laboratory, OAR, through the European Office, Aerospace Research, United States Air Force.

* Present address: Research Institute of National Defence, Stockholm 80, Sweden. 1 H. Kleiman, J. Opt. Soc. Am. 52, 441 (1962). 2 K. Bockasten, J. Opt. Soc. Am. 54, 1065 (1964). 3 B. Edlén, Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1964), Vol. 27, p. 125. 4 B. S. Jeffreys, Quantum Theory, edited by D. R. Bates (Academic Press Inc., New York, 1961), Vol. 1, p. 245. 5 T. Y. Wu, Phys. Rev. 44, 727 (1933). 6 See Ref. 3, p. 127. 7 M. G. Mayer, Phys. Rev. 60, 184 (1941). 8 F. Herman and S. Skillman, Atomic Structure Calculations (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1963), pp. 6-101. 9 D. R. Hartree, Proc. Roy. Soc. (London) A143, 506 (1934).