A description of fine-structure patterns in nonpenetrating high-L Rydberg atoms and ions is derived in a perturbative model in which the energy denominators occurring in the second-order perturbation theory are evaluated using the adiabatic expansion. The patterns of Rydberg energies that result are dominated by the expectation value of an effective potential containing a range of tensor orders and increasing negative powers of the Rydberg electron's radial coordinate. The coefficients of each term in the effective potential are expressed in terms of matrix elements and energies of the free ion at the core of the Rydberg system. Smaller corrections to these patterns due to application of the effective potential in second order and due to relativistic and spin contributions are also described. The effective potential provides a framework for extracting ion properties from measurements of high-L fine-structure patterns. )] 2 .The two terms proportional to r −6 agree with the results of Clark, Greene, and Miecznik [12]. The other terms are new. This completes the list of terms which contribute to E [2] proportional to r −s with s 8, as long as J c < 3. This is sufficient to account for all cases studied experimentally to date. The full effective potential to this point consists of the sum of all the second-order terms listed above plus the two first-order terms from Sec. II A. C. Rydberg intermediate statesThe expression for the second-order perturbation energy derived in Sec. II B excluded the contributions to E [2] from intermediate states where the core was in its ground electronic state. The number of such states depends on the ion in question. For an ion with 042505-8 1 2in which α D,2 (J c ) is given by Eq. (77) and

Binding energies of 23 Rydberg fine-structure levels of Ni with n = 9 and L 5 were measured using resonant excitation Stark ionization spectroscopy. From this spectrum, the quadrupole moment and the scalar and tensor dipole polarizability of the 2 D 5/2 ground state of Ni + were determined to be Q = −0.474(2), α d,0 = 7.92(6), and α d,2 = 1.15(14) a.u. The electric hexadecapole moment was determined to be −0.33(21) a.u.

Binding energies of high-L Rydberg states (L 7) of Th 2+ with n = 27-29 were studied using the resonant excitation Stark ionization spectroscopy (RESIS) method. The core of the Th 2+ Rydberg ion is the Fr-like ion Th 3+ whose ground state is a 5 2 F 5/2 level. The large-core angular momentum results in a complex Rydberg fine-structure pattern consisting of six levels for each value of L that is only partially resolved in the RESIS excitation spectrum. The pattern is further complicated, especially for the relatively-low-L levels, by strong nonadiabatic effects due to the low-lying 6d levels. Analysis of the observed RESIS spectra leads to determination of five properties of the Th 3+ ion: its electric quadrupole moment Q = 0.54(4); its adiabatic scalar and tensor dipole polarizabilities α d ,0 = 15.42(17) and α d ,2 = -3.6(1.3); and the dipole matrix elements connecting the ground 5 2 F 5/2 level to the low-lying 6 2 D 3/2 and 6 2 D 5/2 levels, | 5 2 F 5/2 ||D||6 2 D 3/2 | = 1.435(10) and | 5 2 F 5/2 ||D||6 2 D 5/2 | = 0.414(24). All are in atomic units. These are compared with theoretical calculations.

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The complete pattern of Rydberg binding energies of the 18 n = 9 levels of nickel with L = 6, 7, and 8 was measured using microwave plus resonant-excitation Stark-ionization spectroscopy. The measured pattern is consistent with the form predicted with the effective potential model, showing significant structure proportional to scalar products of tensor operators of order 0-4. The variation of the structure with L separates the various contributing terms and provides determinations of several properties of the Ni + core ion. These include the quadrupole moment, Q = −0.469 78(9) a.u., the hexadecapole moment, = 0.36(5) a.u., the scalar and tensor dipole polarizabilities, α D,0 = 7.949(2) a.u., α D,2 = 0.905(12) a.u., the scalar quadrupole polarizability, α Q,0 = 55(8) a.u., the g value, g J = 1.257 (14), and the vector hyperpolarizability, β D,1 = 0.454 (24) a.u. + A 3 ( X [3] (J c ) · T [3] (r) ) + A 4 X [4] (J c ) · C [4] (r)