Auger width of metastable states in antiprotonic helium II

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1 «Избранные вопросы теоретической физики и астрофизики». Дубна: ОИЯИ, С Auger width of metastable states in antiprotonic helium II J. Révai a and A. T. Kruppa b a Research Institute for Particle and Nuclear Physics H-1525 Budapest, P.O.B. 49, Hungary b Institute of Nuclear Research of the Hungarian Academy of Sciences H-4001 Debrecen, P.O.B. 51, Hungary Our previously proposed method for calculating the Auger-rates of metastable states in antiprotonic helium was generalized by the inclusion of closed electron channels. Adding the extended channel part to the simple Born-Oppenheimer wave function reasonable Auger-rates can be derived. I cordially dedicate this paper to my old friend Vladimir B. Belyaev on occasion of his 70th birthday. J.Révai I. INTRODUCTION In our first paper [1] devoted to the Auger-decay probabilities of long-lived states in antiprotonic helium we shall refer to it as I) we proposed the so called "minimal extension"of a square integrable metastable state wave function to account for possible electron emission from it. This goal was achieved by adding to the given bound state wave function a single term channel part consisting of a product of an unknown electron wave function and a 4 He p bound state function. The channel part quantum numbers correspond to the lowest possible orbital momentum of the Auger-electron. As for the bound state part we used the simplest Born-Oppenheimer BO) wave function [2] and some not too sophisticated variational functions. The results can be summarized as follows: The proposed method can be applied for calculation of decay width of a given bound state configuration from which energy conditions allow particle emission. As for the Auger-widths of the considered bound state functions, both give unsatisfactory results. The BO wave functions, which give correct energies up to digits yield Auger-rates which are 1-2 orders of magnitude smaller, than the expected values.

2 154 Révai, Kruppa The simple Korobov-type functions, used in I give correct order of magnitude, but their numerical values are highly unstable. The main conclusion was, that the Auger-probabilities are governed by very small components of the bound state part of the wave function, the accurate determination of which requires a highly sophisticated and complete variational calculation, with energyconvergence up to 8-9 digits Indeed, later on, such variational calculations were performed [3, 4, 5]. Unfortunately, Augerrates were calculated only in [3], the high precision wave functions of the other calculations were not used for determination of Auger-probabilities. The aim of the present work is to extend the method proposed in I in such a way, that, on one hand, preserve the simplicity of the wave function with the possibility of transparent physical interpretation of its components which is not the case for large basis variational calculations), and, on the other hand, obtain reasonable, % accurate Auger-decay probabilities. II. CALCULATION METHOD Similarly to the approach followed in I, the solution of the Schrödinger equation 1 is sought in the form E H)Ψ E r,r) = 0 1) Ψ E r,r) = Ψ 1E r,r) + Ψ 2E r,r), 2) however, the two components Ψ 1E and Ψ 2E have now different meaning. For Ψ 1E r,r) we use the BO type wave function Ψ 1E r,r) = λ E Φ JMmi BO r,r) = λ E χ J mr) R DJP MmΦ,Θ,0)ϕ im r;r) 3) The BO wave functions are built upon the two-center functions ϕ im r;r) corresponding to electron motion in the field of two Coulomb potentials separated by a fixed vector R: 1 2µ r ) ϕ im r;r) = ε im R)ϕ im r;r), 4) r 2 r 1 where m is the projection of electron orbital momentum in the R direction and ε im R) is the energy eigenvalue. The heavy particle relative motion wave function χ J mr) and the BO energy EBO Jmi are obtained from the equation 1 [ d 2 ] JJ + 1) 2m2 2M dr2 R 2 2 ) R + ε imr) EBO Jmi χ J mr) = 0 5) 1 For the details of notation see I

3 Auger width of metastable states in antiprotonic helium II 155 The properly normalized and symmetrized Wigner D-function [ D JP Mm = 2J π δ m0 ) ] 1/2 D J Mm + P 1) J m D J ) M, m 6) ensures correct total angular momentum J and parity P of the wave function 3). In most cases the BO approximation is built upon the lowest two-center state with m = 0, the so called 1sσ term. The present approach differs from the one used in I basically in the definition of the second term Ψ 2E r,r) in eq.2): Ψ 2E r,r) = ψ r) Φ NL R) [ Y l ˆr)Y L ˆR) ] JM, 7) r where, as before, Φ NL R) denotes a bound state wave function of the 4 He p system and [..] JM stands for vector coupling. Considering a sum over channel indices = lnl) allows to extend our previous approach in two respects: a) more open electron channels can be included in addition to the lowest possible electron orbital momentum l 0 allowed for a given Jv) state by energy and momentum conservation; b) by adding closed channels with l < l 0 we allow for the modification of the square integrable part of the total wave function. Both extensions a) and b) imply a modification of the basic attitude of I, namely the "minimal extension"of a given bound state calculation to account for its weak coupling to a given particle emission channel. While the inclusion of open channels with l > l 0 turned out to have negligible effect on Auger-decay probabilities, the inclusion of the closed electron channels proved to be able to fill the 1-2 order-of-magnitude gap between the Auger-rates given by the BO wave function and more sophisticated variational wave functions. It has to be noted, that a method, somewhat similar to the one outlined above, has been used for the calculation of Auger-rates in [6]. The unknown quantities in the wave function 2) are the c-number λ E and the set of functions ψ r) and the equations defining them can be derived from the conditions 2 : Φ BO r,r)e H)Ψ Er,R)drdR = 0 8) Φ NL R) [ Y l ˆr)Y L ˆR) ] JM E H)ΨE r,r)d ˆrdR = 0 9) With the notations Φ NL R) [ Y l ˆr)Y L ˆR) ] JM H 1 ) ) β r)/r Φ BO r,r)d ˆrdR = ρ r)/r 10) 2 The conserved quantum numbers JMmi will be omitted from the equations where no confusion can occur.

4 156 Révai, Kruppa Φ NL R) [ Y l ˆr)Y L ˆR) ] JM 1 r 1 2 r 2 ) Φ N L R)[ Y l ˆr)Y L ˆR) ] JM d ˆrdR = v r) 11) Φ BO r,r)hφ BOr,R)drdR = E BO + E BO = E BO 12) β r) E ρ r) = γ r); E E NL = ε 13) we get the following set of equations [ 1 d 2 2µ dr 2 l ) ] l + 1) r 2 ε ψ r) + v r)ψ r) + λ E γ r) = 0 14) λ E E BO E) + γ r)ψ r)dr = 0 15) for the quantities λ E and ψ r). The eq.12) among the notations expresses the fact, that the BO energy E BO as defined in eq.5) is not equal to Φ BO H Φ BO. When the BO approximation is used "alone"the correction E BO is usually neglected, since, on one hand, it is small, and on the other, it acts in the wrong direction. The eqs. 14),15), which can be conveniently rewritten as h 0 ε ) ψ + v ψ + λ E γ = 0 16) λ E E BO E) + γ ψ = 0, 17) are usual coupled-channel scattering equations, which via their inhomogeneity are also coupled to a specific bound state. As we mentioned before, we consider only one open channel ε 0 > 0), all the other channels are closed ε < 0, 0 ) The boundary conditions are ψ 0) = 0; k r) 1 ψ r) r j l k r)δ 0 +t h 1) l k r) 18) with k = 2µε. The solution ψ can be expressed in terms of the solutions of the homogeneous and inhomogeneous versions of eq.16): h 0 ε ) u f satisfying appropriate boundary conditions as ) ) ) u 0 + v = f γ 19) ψ = u + f γ u E E BO γ f 20)

5 Auger width of metastable states in antiprotonic helium II 157 The first term of this expression corresponds to "background"elastic scattering of an electron with energy ε 0 on a bound state Φ N0,L 0 of the 4 He p system, while the second one describes resonance coupling to the bound state Φ BO. The resonance energy E r = E 0 i Γ 2 21) is given by the zero of the denominator of the second term: E r = E BO + γ f 22) and Γ is the Auger-width of the state Ψ JM E. Since both γ and f depend on energy, eq.22) is a transcendental equation for E r. It is also complex due to the outgoing boundary conditions. We solved it by iteration and, due to the smallness of Γ, we neglected the imaginary part of the energy when solving the inhomogeneous equation 19) for f. It has to be noted, that for the convergence of the iteration the inclusion of E BO in E BO was essential. III. RESULTS AND DISCUSSION We have calculated the Auger transition rates as a function of included closed channels. Only one open channel was considered for a given state J,v), the one with the lowest possible orbital momentum l 0 of the electron. Since we are considering a Coulomb system, the number of possible closed channels is infinite. Different strategies can be considered for systematic increasing of their number. We have chosen the following procedure. There are l 0 possible values of the electron orbital momentum l with l < l 0, l = 0,1,...,l 0 1. Assuming "maximally stretched"angular coupling J = l + L each l has its own L value: L = J l, and for these l 0 pairs l,l) we systematically increased their possible N values: N = L + 1,L + 2,... Thus the total number N of included closed channels was varying in steps of l 0 : N = l 0,2l 0,3l 0,... Our results are summarized in Table I, where also a comparison is made with the two existing Auger-rate calculations. The numbers in Table I do not allow to make unique and overall conclusions about the Auger-rates of the considered states. In some cases all three calculations give very similar results very probably the correct ones. When the numbers differ by % it is hard to say, which of them is the "best". If the necessity of decision occurred, further calculations with high quality wave functions should be done. Finally, there are one or two cases, where the results are very different, furthermore, for these states the individual calculations show very bad or no convergence at all. This could be the result of some peculiarity in the structure of these states missing from the conventional calculations. In ref.[6] a mechanism was proposed for explanation of this situation, unfortunately no detailed proof of the idea was performed.

6 158 Révai, Kruppa TABLE I: Auger-rates of selected antiprotonic helium states J = 32 J = 33 J = 34 J = 35 N v,l 0 ) v,l 0 ) v,l 0 ) v,l 0 ) 1, 3 2, 3 3, 2 1, 4 2, 3 3, 3 4, 2 1, 4 2, 4 3, 3 4, 3 2, 4 3, 4 4, 4 l 0 1.6[8] 4.9[9] 4.5[11] 1.4[5] 1.8[8] 6.0[10] 2.4[11] 8.7[3] 1.6[5] 9.4[7] 5.9[8] 9.0[3] 5.4[4] 1.5[5] 2l 0 2.3[8] 3.5[9] 4.5[11] 2.2[5] 2.4[7] 5.6[10] 2.5[11] 2.0[4] 3.4[5] 9.2[7] 6.6[8] 1.5[4] 3.4[4] 1.6[5] 3l 0 2.3[8] 5.2[9] 1.1[12] 2.2[5] 1.8[8] 4.9[10] 1.9[11] 2.0[4] 1.5[4] 1.4[8] 3.9[6] 1.8[4] 7.3[4] 1.5[5] 4l 0 2.3[8] 5.3[9] 3.4[11] 2.2[5] 1.9[8] 1.4[11] 2.8[11] 2.0[4] 8.0[4] 1.6[8] 6.9[8] 1.8[4] 8.1[4] 1.3[5] 5l 0 2.3[8] 5.3[9] 4.0[11] 2.2[5] 1.9[8] 2.5[10] 3.2[11] 2.0[4] 8.6[4] 1.7[8] 7.8[8] 1.8[4] 8.1[4] 1.3[5] KS 2.2[8] 3.7[9] 5.8[11] 2.1[5] 2.4[8] 5.7[9] 3.1[11] 2.0[4] 1.9[5] 1.3[8] 7.7[8] 1.1[4] 7.0[4] 4.7[5] KFM 2.3[8] 7.5[9] 6.1[11] 2.4[5] 2.4[8] 2.0[10] 3.0[11] 2.4[4] 2.3[5] 1.4[8] 1.0[9] 1.4[4] 8.0[4] Auger-rates are given in sec 1, the numbers in [ ] are powers of 10. N is the total number of included closed channels, l 0 is the orbital momentum of the Auger-electron. The rows labelled by KS and KFM are the results of the two existing Auger-rate calculations [3] and [6] Acknowledgments The authors gratefully acknowledge the financial support from the OTKA grants T and T [1] J. Révai and A.T. Kruppa, Phys. Rev. A 57, ) [2] I. Shimamura, Phys.Rev. A 46, ) [3] V. I. Korobov and I. Shimamura, Phys. Rev. A 56, ) [4] N. Elander, and E. Yarevsky, Phys. Rev. A 56, ); ibid 57, ) [5] N. Yamanaka, Y. Kino, H. Kudo, and M. Kamimura, Phys. Rev. A 63,012518,2001) [6] O. I. Kartavtsev, D. E. Monakhov and S. I. Fedotov, Phys. Rev. A 61, )