PHYSICAL REVIEW A VOLUME 28, NUMBER 3 SEPTEMBER 1983 Double electron excitation in heliumlike ions B. Padhy, R. Srivastava, and D. K. Rai Department of Physics, Banaras Hindu University, Varanasi 221005, India (Received 16 August 1982) A Coulomb-Born-Oppenheimer approach to the calculation of cross sections for electron- impact excitation of the doubly excited 2p I' state in heliumlike ions is presented. It is shown that closed-form expressions can be obtained for both differential and total cross sections. The analysis is applied to calculate the total cross section for the process e + Li+(1s~ tS) ~ e +Li+(2p 3P). Excited atomic and molecular states in which two electrons have been promoted from the orbitals occu- pied by them in the ground state of the system have been known spectroscopically for quite a long time. ' Their fundamental role in an understanding of atomic and molecular structure has, however, been realized more recently as finer experimental and theoretical techniques have been developed. In fact, such states not only in neutral species but also of ions have been identified as of great importance in astrophysical plas- ma investigations. This recent upsurge of interest in doubly excited states has resulted in a few studies of the electron-impact excitation of such states in neu- tral atoms. However, as compared with the vast amount of literature existing on theoretical and ex- perimental studies of an electron-impact excitation of singly excited states for atoms and ions, such inves- tigations are only in the nascent state. In the present paper, we have presented for the first time a theoreti- cal computation for electron-impact two-electron ex- citation in an ion (He-like iona). The theoretical treatment is carried out in the Coulomb-Born-Oppenheimer (CBO) approximation and, for the present, we consider only the 1s' 'S 2p P excitation which is a parity unfavored transi- tion. We do not use the prevalent partial-wave analysis (whose limitations are well known'o) of in- cident and (or) scattered electrons and we obtain closed-form expressions for differential and total cross sections. As a test case, we have applied the analysis to predict the cross section for the following transition: e + Li+(1s 'S) e + Li+(2p~ P ) d /d cT0 = (3k'/k;) g)g (0, Q) ) (2) The g (0, @) is the exchange amplitude and the sub- script m refers to various components of a degenerate ionic target state. In the CBO approximation, the ex- change amplitude g can be written as The choice of this specific ion, Li+, is based on the fact that this is the only heliumlike ion for which ac- curate experimental double excitation energies are known. " For the above process [Eq. (1)] with only Coulomb interactions, the direct amplitude is zero so that only the exchange amplitude is to be computed. Let us denote the incident electron by 1 with a momentum vector k; and the two target (ion) elec- trons in the initial state @;( r q, r 3) by 2 and 3. The outgoing electron is numbered as 3 having a final momentum vector kf and the final target state is $f( r q, r t). (Atomic units will be used throughout and the direction of k; is chosen as the axis for space quantization and is along the z axis. ) For an unpolar- ized electron beam the differential cross section for the process (1), summed over all final spin states, is g = (2sr) ' d rtd rqd r3+t, ( r 3)@I'( r q, r t) V( r t, r q, r 3)@;( r q, r 3)+t, + ( r t) (3) where Wqt+)( r ) and Wqt )( r ) are the full Coulomb wave functions corresponding to the residual charge P = Z — 2 (Z being the nuclear charge) of the target ion; we write i k. ~ r Ot, + ( r ) =NtI+ e ' tFt(ia;1;i (kr — k; r )) (4) and ikf r ( r ) =Nt, e tFt( — ib;1; — i (kIf + kf' r )), where a =P/k;, b =P/kI and Nt, = e r (1+ib) f The interaction potential V in Eq. (3) is given, in its 28 1825 1983 The American Physical Society