Physica Scripta. Vol. 45, 450-453, 1992. Lifetimes of the Singly Excited Be I1 and Be 111 States with n<ll Constantine E. Theodosiou Department of Physics and Astronomy, The University of Toledo, Toledo, Ohio 43606 Received September 20,1991: accepted in revised form December 2,1991 Abstract The oscillator strengths of all allowed transitions between the singly excited states of Be II and Be 111 with principal quantum number up to n = 11 were calculated within the Coulomb approximation with a Hartree-Slater core (CAHS) approach. The results were subsequently combined to obtain the radiative lifetimes of all excited states and are presented here. 1. Introduction and review of the method A compilation is presented of calculated lifetimes for the Be II ls2nI2L and Be 111 lsnl 's3L states using a numerical Coulomb approximation with a Hartree-Slater potential (CAHS) to represent the ionic core. The approach has been extensively and successfully employed [ 1-81 in alkali-like systems, e.g. Li, Na, K, Rb, Cs, Ca', Li-, Na-, and Cu- isoelectronic sequences, and for He I and Li 11. The method has been sufficiently described elsewhere, so that only points will be covered here of relevance to the specific cases treated. The philosophy of the Coulomb approximation (CA) [9] is the fact that the Rydberg state wavefunctions, at least for medium to high principal quantum number n, are deter- mined by their energies, or equivalently, their effective quantum numbers n*. Apart from its effect on the n*, i.e. the introduction of a "quantum defect", the ion core affects the wavefunctions only at small distances and for a dipole tran- sition this is usually a small effect. Therefore the wavefunc- tions can be approximated either by analytic Whittaker wavefunctions with the appropriate n*, or by inward numerical integration and a judicial cutoff of the integration at small distances. The CA approach has been augmented by this author [l] to obtain the radial wavefunction by inward integration of the Schrodinger equation where the potential V(r) is obtained from a Hartree-Slater self-consistent field calculation using standard programs [lo]. This approach was developed to treat alkali-like and helium-like systems where the Rydberg series are not strongly perturbed [l-81, but was also extended with success to atoms with strong perturbations like A1 [ll] and Sr [123. The transition probability and absorption oscillator strength between two states 1 yJ) amd I yJ') are given by a3 AE S(yJ, y'J') 6 0 R 2J+1 A(yJ + y'J') = - - and 1 AE S(yJ, y'J') 3R w'+1 f(y'J' -+ yJ) = - - (3) where SbJ, Y'J') = I (YJ II D II r'J> I* (4) is the line strength, D is the dipole operator, a is the fine structure constant, R is the Rydberg energy constant, and AE = E(yJ) - E(y'J') is the transition energy. The line strength is calculated within the LS-coupling scheme and it is given by S(ySLJ, y'S'LJ') = ds,(23 + 1)(2J' + 1X2L + 1) x (2L: + 1) max (1, 1') {1 L L}2{ 1 1 l'}2 S J ' J L,LL 2 x ([drP"dr)rP".,.(r)) * (5) L, is the total angular momentum of the core after removal of the active electron with initial and final orbital angular momentum quantum numbers 1 and 1', respectively. The energy levels and ionization potentials, used as input in the calculation, were taken from Be 11 for the measure- ments and analysis of Johansson [13] and Holmstrom and Johansson [14] whereas for Be 111 we used the measure- ments of Lofstrand [ls]. In both cases, and where neces- sary, levels not observed experimentally were obtained from extrapolation and fitting lower levels. The exclusion of Be I from this calculation is due to the existence of doubly excited states quite low in energy within the Rydberg series of singly excited states. The mixing between series and states becomes in this case an important component that has to be explicitly treated. Work is in progress in that direction but it is too early to report results. 2. Results and discussion 2.1 Be 11 Is% states Several calculations have been made for the oscillator strengths of this system but, other than the work of Lindgird and Nielsen [16] no oscillator strengths or life- times have been reported for an extensive set of states. The present paper provides such information. The accuracy of the calculations is expected to be comparable to the ones for Li for which our predicted [l, 81 oscillator strengths and lifetimes are of high accuracy and, if not better than, com- Physica Scripta 45