Suric, T

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Indian J. P h ys . 71B (3), 415-420 (1997)I J P  B— an imemotional journalA simple procedure for the calculation of the shake-off ratio of double- to-single ionization of a helium-like ion by photoabsorption of a high-en­ergy photonT. SuricRugjer Boskovic InstituteP. O. Box 1016, 10001 Zagreb, CROATIAAbstract: A simplified, but accurate procedure is proposed for the cal­culation of the shake-off ratio of double-to-single ionization of a helium-like ion by photoabsorption of a high-energy photon. The pro­cedure is based on the observations that: i) the high-energy shake-off ratio of double-to-single ionization for photoeffect is determined by the behavior of a two-electron wave function in the region where one elec­tron is near the nucleus, and ii) the two-electron wave function, in this region can be determined employing only the monopole term of the electron-electron interaction (i.e replacing l/lrr r 2l of the exact Hamil­tonian of a He-like system by 1 /^  where r> is a larger of r| and r2). The proposed procedure might be helpful in evaluating double-to-single photoabsorption ratios for more complex systems.Keywords: double photonization, photoeffect, HeliumPACS numbers: 32.80.Fb, 32.30.RjIn recent years the double ionization of two-electron atoms by one pho­ton has attracted significant attention both experimentally and theoretically I !]■ The study has particularly focused on the ratio R = a +V a + of double- to-single ionization. Measurement and prediction of this observable for the© 1997 IACS416 TSuricsimplest many-body system is considered to provide a good test of our under­standing of electron-electron correlation in complex systems. Generally, both Compton scattering and photoeffect contribute to ionization at high photon energies [2,3], with Compton scattering becoming dominant over the photo­effect for energies above 6 keV for Helium. It is now becoming possible to measure these contributions separately [4,5]. Here we are concerned with the photoeffect contribution.The first experimental studies of R using He atoms as a target [6] were performed with relatively low energy photons (©). At such energies, for which the incoming photon momentum (k = co/c )  is several times smaller than the average momentum of the bound electrons (pav), the dipole approx­imation for photoeffect was assumed to be sufficient for the theoretical calcu­lations. With the advent of modem synchrotron light sources, experiments measuring the ratio R have become available [7] which extend into a higher energy region.The calculation of the asymptotic high-energy limit of the ratio R stems from works beginning in the 1960’s [8-10], using what is generally called the sudden approximation [11], leading to the shake-off mechanism. In these pio­neering works it was shown that, in the dipole approximation using velocity or acceleration gauge, the limit as co isX I J V f r i > ' |, i<r i . 0> * i |21 1 ^ 1 , 0 ) 1 ^ ,Here ^ ( r  (, r 2) represents the initial state wave function, Og(r]) is a bound state hydrogen-like electron wave function (in the potential of charge Z), and the summation is over all bound states. Using the best available correlated two electron wave functions this formula gives for He R = 0.0166 (1.66%) [10]. Experimental observations in accord with this prediction were reported for photon energies above 2.5 keV [4,5]. Note that the evaluation of the shake- off ratio from the Eq. (1) is separate from the issue of calculation of the pho­toionization cross sections themselves.The shake-off approach assumes that in most cases one electron takes almost all the photon energy with the other electron of low energy. In such a situation, the recoiled nucleus takes the rest of the momentum, which is approximately equal and opposite to the momentum of the fast electron. Recently, Drukarev [12] has called attention to an additional mechanism of double photoeffect which would first manifest itself as a linear rise of the ratioA simple procedure for the calculation o f the shake-off ratio etc 417with energy, causing roughly a 10% correction to the shake-off limit at about 12 keV. However, even though the shake-off ratio may not be the high-energy limit of the photoeffect ratio, it is a physically well defined quantity which we may measure [4].The usual procedure in evaluating Eq. (1) is straightforward, although somewhat cumbersome. It requires evaluation, usually using variational method, of the ground-state He-like wave function in, generally, six dimen­sional configuration space. After obtaining this function we put r2 = 0 and substitute it in the Eq. (1). Here we propose a different approach based on the results of Altick [13] and recent results of Suric et al. [14], and demonstrate that Altick’s one parameter ground-state wave function can be used for accu­rate evaluation of the photoionization shake-off ratio.In his work Altick simplified the exact Hamiltonian for two electron atom by just keeping the first term (monopole) in the multipole expansion of the electron-electron interaction, that is1h - r 2i (2)where r> is a larger of r , and . Altick has chosen, as an approximation to the ground state of the Hamiltonian with monopole electron-electron interac­tion, the wave function of the form(3)where the variables r ,  and r> are the lesser and greater of r , and r 2; Z is the atomic number, p is an unspecified parameter and a  is fixed by the requirement that 4* have continuous first derivatives at rl = r2. Using vari­ational method, treating (3 as variational parameter and fixing its value by minimizing the energy Altick has obtained the following values for (3: (3 = 0.42 for Z = 1, |3 = 1.47 for Z = 2, and |3 = 2.48 for Z = 3. For our discussion here the parameter [3 is of primary interest and we shall discuss this parameter from the point of view of the recent results of Suric, Pisk and Pratt [14],In their recent paper Suric, Pisk and Pratt [14] consider charge depen­dence of the ratio of double-to-single ionization of a helium-like ion by Compton scattering of a high-energy photon. They introduce the concept of the effective shake-off charges which, according to their discussion, deter­418 T Suricmine ratios of double-to-single photoionization (photoabsorption or Compton scattering). Their discussion on effective shake-off charge for photoabsorp­tion, made in Coulomb gauge and in velocity form, is relevant for our discus­sion here.The shake-off mechanism for ionization of two-electron system assumes that one of the two electrons is suddenly removed. The other electron feels a sudden change of potential, from the potential of two electrons in a field of charge Z to the potential of one electron in a field of charge Z. According to Suric, Pisk and Pratt, the state in which this remaining electron is found can roughly be described by some effective charge which they call the effective shake-off charge Z2ePeff for photoeffect. They evaluate Z2cPeff through the procedure beginning with highly correlated He-like wave func­tions KP((r1, r2) . In the high-energy limit the fast electron is removed from the origin and the other electron is left in the state described by the function 4/i(r], r2) . They normalize this function and obtain <t>(r,), the wave function of the remaining electron. They try, as a simplest approach, to find the Cou- lombic ground-state wave function in a field of an effective charge which has the largest overlap with <J)(r,). The effective charge found in such way is Z2ePefr • They find that for all Z the double photoeffect effective shake-off charge can, within few percent accuracy, be written asZPPC ff = Z -0 .5 3 . *(4)Suric, Pisk and Pratt interpret the number 0.53 as the screening which the slow electron secs when the fast electron has left the system from the origin. They have used Z2ePeff to predict the double ionization ratio [14],Repeating the procedure of Suric, Pisk and Pratt we obtain z2ePeff = 043 for Z = 1, Z2ePeff = 1.47 for Z = 2 , and Z2ePeff = 2.48 for Z = 3.Comparing these values for Z2ePeff with the Altick parameter P we may conclude that they are the same (within the numerical accuracy).Following the procedure of Ref. [14], we project hydrogenic ground- state wave functions of the charges given above (0.43,1.47 and"2.48), on the hydrogenic Z = 1, Z = 2 , and Z = 3 scattering states, and obtain results for R which agree exactly (within the numerical accuracy) with those obtained with Hylleraas wave functions and Eq. (1) [14].Therefore, we may conclude that: i) the concept of the effective shake- off charge allows us to calculate the high-energy shake-off photoabsorption ratio exactly without need to know any other information about the ground state of a helium-like ion, and ii) the effective shake-off charge can beA simple procedure for the calculation o f the shake-off ratio etc 419obtained through the variational method using only the monopole term of the electron-electron interaction.In this paper we presented a simplified procedure for the calculation of the high-energy shake-off photoabsorption ratio for helium-like ions. It is demonstrated that this ratio is completely determined by the effective photo­effect shake-off charge, as defined by Suric, Pisk and Pratt [14]. To obtain this effective charge a simple variational procedure of Altick [13], using a one parameter wave function, is sufficient and, therefore it is not necessary to employ highly-correlated, many-parameter, ground-state wave functions. High-energy shake-off photoabsorption ratio (for helium-like systems), there­fore, contains the information on monopole, ground-state electron-electron interaction only.References[1] See the recent extensive review: J. H. McGuire, N. Berrah, R. J. Bar­tlett, J. A. R. Samson, J. A. Tanis, C. L. Cocke and A. S. Schlachter, J. Phys. B 28, 913 (1995).|2] J. A. R. Samson, C. H. Greene and R. J. Bartlett, Phys. Rev. Lett. 71, 201 (1993).[3] K. Hino, P. M. Bergstrom and J. H. Macek, Phys. Rev. Lett. 72, 1620(1994) .[4] L. Spielberger, 0 . Jagutzki, R. Domer, J. Ullrich, U. Meyer, V. Mergel, M. Unvcrzagt, M. Damrau, T. Vogt, 1. Ali, Kh. Khayyat, D. Bahr, H. G. Schmidt, R. Frahm and H. Schmidt-Bocking, Phys. Rev. Lett. 74,4615(1995) .15] M. Sagurton, R. J. Bartlett, J. A. R. Samson, Z. H. He and D. Morgan (submitted to Phys. Rev. A).[61 T. A. Carlson, Phys. Rev. 156, 142 (1967); V. Schmidt, N. Sander, H. Kuntzemuller, P. Dhez, F. Wuilleumier and E. Kallne, Phys. Rev. A 13, 1748 (1976); D. M. P. Holland, K. Codling, J. B. West and G. V. Marr, J. Phys. B 18,2465 (1979); H. Kossmann, V. Schmidt and T. Andersen, Phys. Rev. Lett. 60,1266 (1988); R. Wehlitz, F. Heiser, O. Hemmers, B. Langer, A. Menzel and U. Becker, Phys. Rev. Lett. 27,3764 (1991);[7] J. C. Levin, D. W. Lindle, N. Keller, R. D. Miller, Y. Azuma, N. Berrah Mansour, H. G. Berry and I. A. Sellin, Phys. Rev. Lett. 67,968 (1991); J. C. Levin, I. A. Sellin, B. M. Johnson, D. W. Lindle, R. D. Miller, N. Berrah, Y. Azuma, H. G. Berry and D. H. Lee, Phys. Rev. A 47, R16 (1993); R. J. Bartlett, P. J. Walsh, Z. X. He, Y. Chung, E-M Lee and J. A. R. Samson, Phys. Rev. A 46, 5574 (1992); N. Berrah, F. Heiser, R. Wehlitz, J. Levin, S. B. Whitfield, J. Viefhaus, I. A. Sellin and U.420 T SuricBecker, Phys. Rev. A 48, R1733 (1993).[8] A. Dalgamo and A. L. Stewart, Proc. Phys. Soc. London 76 ,49  (1960).[9] F. W. Byron, Jr., and C. J. Joachain, Phys. Rev. 164,1 (1967).[10] T. Aberg, Phys. Rev. A 2 ,1726  (1970).[11] T. Aberg, Ann. Acad. Sci. Fenn. AVI 308,1 (1969). T. Aberg, in Pho­toionization and Other Probes of Many Electron Interactions, edited byF. J. W uilleumier (Plenum, New York, 1976).[12] E. G. Drukarev, Phys. Rev. A 51, R2684 (1995).[13] P. L. Altick, J. Phys. B 5 ,1095 (1972).[14] T. Suric, K. Pisk and R. H. Pratt, Phys. Lett. A 211,289 (1996)

A Simple Procedure for the Calculation of the Shake-Off Ratio of Double-To- Single Ionization of a Helium-Like Ion by Photoabsorption of a High- Energy Photon

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A Simple Procedure for the Calculation of the Shake-Off Ratio of Double-To- Single Ionization of a Helium-Like Ion by Photoabsorption of a High- Energy Photon

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