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PROBLEMS WITH IPA LATTER TAIL POTENTIALS IN NEUTRAL AND IONIC PHOTOEFFECT
CALCULATIONS
Y. Kuang, R. Pratt, Y.-J. Wu, J. Stein, I. Goldberg, A. Ron
To cite this version:
Y. Kuang, R. Pratt, Y.-J. Wu, J. Stein, I. Goldberg, et al.. PROBLEMS WITH IPA LATTER
TAIL POTENTIALS IN NEUTRAL AND IONIC PHOTOEFFECT CALCULATIONS. Journal de
Physique Colloques, 1987, 48 (C9), pp.C9-527-C9-530. �10.1051/jphyscol:1987986�. �jpa-00227407�
JOURNAL DE PHYSIQUE
Colloque C9, supplement a u n012, Tome 4 8 , dgcembre 1 9 8 7
PROBLEMS WITH IPA LATTER TAIL POTENTIALS IN NEUTRAL AND IONIC PHOTOEFFECT CALCULATIONS
Y. KUANG, R.H. PRATT, Y.-J. WU, J. STEIN*, I.B. GOLDBERG' and A. RON*
Department of Physics and Astronomy, University
ofPittsburgh, Pittsburgh.
PA 15260. U . S . A .' ~ a c a h Institute of Physics, Hebrew University of Jerusalem,
IL-91000Jerusalem, Israel
Abstract. - It has been asserted that oscillations and other structures observed in photoionization cross sections are due to the use of Coulombic Latter tails and to independent-particle-approximation (IPA) calculations. We show that suitable IPA calculations do not have these features, as for example if a vacancy potential is used.
1. Introduction
In a previous study [ l ] of photoionization of ions as a function of their degree of ionization, for which we used Dirac-Fock-Slater Latter tail potentials, we found that the cross section ratio, of ion to neutral atom cross sections of various subshells in neon and carbon, showed oscillation effects as a function of photon energy. These oscillations in some cases were even larger than the differences between the various ion and neutral cross sections. Here we first discuss the origins of this structure, which appears to be unphysical, and then present neutral atom and ion results using a potential which does not suffer from these particular defects: a vacancy potential. In the IPA (independent-particle-approximation) model for photoeffect, we usually calculate all the bound states and continuum states in one potential, so that these states are orthogonal to each other. We usually obtain that potential self-consistently, as due to the charge density of all the electrons in the initial state. Since the potential seen by any one of these electrons will be Coulombic far away from the atom or ion, a (Zi+l)/r Latter tail is imposed as the large distance behavior, where Zi is the ionic charge. This approach suffers from two defects: (1) discontinuities in derivatives of the poten- tial where the Latter tail is joined, and (2) self screening in the potential seen by an electron due to its own charge distribution. Both of these defects are removed in a vacancy potential, which has no Latter tail and removes the contribu- tion of the charge distribution of the active electron. Of course, since vacancy, potentials for different active electrons are different, the states of these elec- trons will not be orthogonal. This can be better tolerated in IPA calculations than when higher corrections are being included.
2. Unphysical Structures and the Latter Tail
Calculations of inner shell photoeffect cross sections (e.g. K-shell) have exhibited some structure or oscillations when plotted vs. photon energy. The oscillation effect is more obvious when for example taking the ion-neutral cross
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987986
C9-528 JOURNAL
DE
PHYSIQUEsection ratios, as noted above. However threshold structures have also been seen in the neutral atom cross sections themselves, varying in detail with occupation numbers of various excited states. Such features have also been reported by Manson and Inokuti [ 2 ] and by Holland et al. 1 3 1 . In their study of the Cu K shell differential optical oscillator strength df/dE for ionization, proportional to the photoioniza- tion cross section, Manson and Inokuti stated "it is seen that dfld~ is clearly non- monotonic in E, unlike the screened hydrogenic result
. . . the curve for df/dE has
two maxima,
... ."
Holland explained some near threshold structure (near the K edge) in EXAFS experiment in terms of this atomic origin [ 3 ] . In our calculation of neon subshell cross section ratios of ionic to neutralapecies, we observed oscillation effects both for the K-shell and for L-shells.Recently it has been reported that these near threshold structure effects are not present in Hartree-Fock calculations, and it was suggested that they are an artifact of the independent particle approach. Tulkki and Aberg [ 4 ] discussed the Ar K-shell case, where they found that using the Dirac-Fock method, and including relaxation, the near threshold structure disappeared. They said " these and similar results for other noble gases indicate that the deviations from the monotonically decreasing behavior reported by Manson and Inokuti (1980) are due to their calcula- tional procedure for obtaining the continuum wavefunction." Recently Amus'ya et al.
[5] also argued that the Latter tail is responsible for these effects, but rather through its influence on the bound state wave function.
3. Further Studies of these Structures
We have studied these photoionization structures further and in some cases find a somewhat different interpretation of their origin. We agree that the switch to a Latter tail can lead to oscillations in cross sections. We have verified that such effects of an IPA calculation can be removed if one goes beyond IPA and considers additional effects of the residual interaction. However we also find that such effects are not present in IPA calculations based on a more appropriate potential choice. Such choices include (1) smoothing the joining of the Latter tail or (2) utilizing a vacancy potential from which the contribution of the active electron has been removed. We illustrate two such cases in Fig. 1 and Fig. 2. Evidently smoothing does tend to reduce oscillations in cross sections, but use of a vacancy potential is more effective.
We have also studied how these Latter tail oscillations are produced in a matrix element. One main mechanism appears to be through the discontinuity in derivatives of wave functions which results from the discontinuity in derivatives of the potential. (Of course a rapid variation with large derivatives would have similar consequences as an actual mathematical discontinuity.) A matrix element has somewhat the character of a Fourier transform, and the transform of a function with such a discontinuity exhibits oscillations. A second mechanism involves oscillations in the continuum normalization which result from these discontinuities in the continuum wave function. Figure 1 illustrates a situation where the oscil- lations are in the continuum normalization. Figure 2 illustrates a situation in which bound state discontinuities are dominant. The relative amplitude of these sources of oscillation depends on the amplitude of the discontinuity in the state in comparison to the magnitude of the matrix element arising from its dominant region. Larger oscillations occur for valence electrons than for the inner shell electrons where the phenomena was first noticed, since the discontinuity in poten- tial occurs where the wave function is much larger. In carbon L-shell oscillation ratios were several percent, K-shell oscillation ratios only some tenths of a per- cent.
We also are not prepared to agree that all structure effects result from the Latter tail problem. We refer to our earlier studies of high Z inner shells, which showed structures influenced by choice of outer electron occupation numbers. These structures could be identified with the continuum normalization, and the correspond- ing reduced matrix elements exhibitedino structure.
Neon K-sheLL Cross Section R a t i o
0 : r a t L o using smoothed L a t t e r t a i L potentiaL : r a t i o using L a t t e r t a i l potentiaL
0 : r a t i o using vacancy potentiaL : r o t i o using mixed caLcuLotion
Figure 1. Ion to neutral atom photoionization cross section ratios of Ne K-shell vs. photoelectron energy for different potentials: 1) smoothed Latter tail potential, 2) Latter tail potential, 3) vacancy potential and
4 ) mixed calculations using a Latter tail potential for the bound state
and a.vacancy potential for the continuum state.
4 . Ion Results
The vacancy potential may be a better choice of potential than the Latter tail potential. In addition to giving a smoother energy dependence for cross section ratios, it a1.so gives a reasonable ordering of the magnitude of K-shell photoioniza- tion cross sections as a function of degree of ionization. Using the Latter tail potential overestimates screening effects, especially when only K-shell electrons are left in an ion. In the neon case, with a Latter tail potential the K-shell cross section increases from neutral to seven-times-ionized neon, while for eight- times-ionized neon the cross section is less than neutral. Using a vacancy poten:
tial the eight-times-ionized neon K-shell cross section indeed lies above the other cases.
J O U R N A L D E PHYSIQUE
Neon L2-sheLL Cross Section R a t i o
Figure 2. Same as above but for Ne L2 shell. Here the mixed calculations use a vacancy potential for the bound state and a Latter tail potential for the continuum state.
References
[ I ] L. Wang and R. H. Pratt, Bull. Am. Phys. Soc. 31, (1986), 936.
[ 2 ] S. T. Manson and M. Inokuti, J. Phys. B
13,
( ~ Z O ) , L323.[3] B. W. Holland, J. B. Pendry, R. F. Pettifer and J. Bordas, J. Phys. C
11,
(1978), 633.
[ 4 ] J. Tulkki and T. Aberg, J. Phys. B
18,
(1985), L489.[ 5 J M. Y a . Amus'ya, I. M. Band, V. K. Ivanov, V. A. Kupchenko and M. B.
Trzhaskovskaya, Bull. Acad. Sci. USSR
