by ES Chang · 1966 — pH-7f/5dx. (ACCESSION NUMBER). -. /. (PAGES). (C6DE). < ? (NASA OR OR TMX OR AD NUMBER). I. *Research supported in part by the National Science Foundation.
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A Modified Optical Potential Approach To Low-energy Electron-Helium * Scattering to Robert T, Pu and Edward S. Chang Department of Physics University of California, Riverside . GPO PRICE S CFSTI PRICE(S) S Hard copy (HC) 9r pr, – Microfiche (MF) I K 0 L (THRUI :: M6 37323 L pH-7f/5dx (ACCESSION NUMBER) – / (PAGES) (C6DE) < ? (NASA OR OR TMX OR AD NUMBER) I *Research supported in part by the National Science Foundation and the National Aeronautical and Space Agency.
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ABSTRACT A modified optical potential approach is introduced for electron- atom scattering at low energies whereby the formal optical potential is used directly in a variational expression for scatteriq phase shifts. This approach has the advantage that one may include the effect of second order optical potential without recourse to the usual adiabatic approximation. The diagramatic approach associated with the present method makes it possible to identify different contributing terms with different physical effects, and thus to assess the relative importance of various physical effects involved in the scattering process. To test the approach as a practical method for low energy electron-atom scattering, we applied it to the case of electron-helium scattering for energy range 1.2 ev. to 16.4 ev. Good agreement with available experimental data has been obtained. The contributions of various multipole components in the second Or- der optical potential are examined. In particular, the effect of exchange in second order optical potential, usually neglected in most calculations, was found to be very significant.
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INTRODUCTION In ee theoretical calculation of electron-atom scatterings at low energies, the difficulty is well-known to be one of complexity. That is, the problem one faces is to make suitable approximations to the solution of the complicated, but known, many-body Schrodinger equation such that good results may be obtained with reasonable effort, From a physical point of view, the approximation scheme must take into account two important physical effects, the exchange effect and the distortion effect. The exchange effect arises from the Pauli exclu- sion principle between the incident electron and the atomic electro_rsi. In general this is taken into account in theoretical calculations by explicitly antisymmetrizing the trial solution. The distorticn effect, or the polarization effect, arises from the distortion experienced by the atomic electrons in the presence of the incident electron's Cou- lomb field. The distortion or the polarization of the target atom in turn produces a potential on the scattering electron. When the scat- tering electron is stationary, or moving slowly, the atomic electrons will polarize and adjust adiabatically to the position of the scat- tering electron. At large distances the dominant polarization poten- tial is the dipole potential - a 4. where a, is the polarizability of the atom. This is the familiar adiabatic condition usually assumed r 1 for low energy scattering processes. condition for low-energy electron-atom It has been shown2 that in the case of tering the incident electron, given to The validity of the adiabatic scattering is rather dubious. electeon-hydrogen atom scat- be at rest at infinity, would be accelerated by the attractive adiabatic potential such that it will acquire speeds comparable to that of the atomic electrons while still several atomic distances away from the target atom. For atoms such as alkali atoms where the polarizability is large, the validity of
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+ -2- . adiabaticity can be expected to be even worse, The non-adiabatic effezt will be considerable. The actual potential as seen by the s-attering electron is therefore a very complicated non-local (ve- locity dependent) one. In practice the conventional theoretical met%ods are less able to cope with the above mentioned distortion effect. The familiar close-coupling method' does include some non- adiabatic effects but the complexity of the resulting close-coupled integro-differential equations severely limits the number of atomic states one is able to close-couple- This in turn will give wrong asymptotic values for the effscti.6.e potential, In addition, the close-coupling method as applkd to e-H scattering showed that the convergence is poor as the number of close-coupled states is in- ~reased.~ close-coupling approximation is the fact that it requires a knowledge of the Wave functions of excited atomic states. This makes the method much less general in practise than it appears. There are other methods such as the variational approach and Temkin's non-adiabatic approac5 which do take non-adiabatic effects into account more completelyo these methods are either developed for special cases or become diffi- cuBt for complex atom cases and are therefore restrictive in their practical applications. 4 A more serious practical difficulty associated with the 6 7 But Another general approach is the optical potential method where the effect of the target atom on the scattering particle is repre- sented by an equivalent one-body potential. The optical potential approach was first applied to atomic scattering problems by Mittlelrran and Watson8 and others. where the incident particle is an electron introduces some additional complications. This was set on a more rigorous basis by Bell and Squires' who used basis wave functions and a diagramatic approach The Pauli principle effect for the cases
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-3- similar to the Bruckner-Goldstone'B linked-cluster perturbation expansion &ichwas successfully applied by Kelly to the atomic II correlation energy calculations. Formally, this optical poten- Y) tial does contain all the non-adiabatic effects, as previously described, through the propagators which contains operators for the scattering electron. Conventionally, after obtaining the formal optical potential expression, one proceeds to calculate the scattering wave function by solving the one-body Schrodinger equa- tion with the appropriate optical potential. However, the fact that operators for the scattering electrons are contained in the propa- gator makes the optical potential extremely difficult to evaluate and one is forced to make the adiabatic approximation. Moreover, even the adiabatically approximated second-order optical potential can only be evaluated in its asymptotic region, yielding the ex- dipole polarization potential. To remedy pected dominant the divergent behavior at small r, some ad hoc cut-off parameters 4 -+ must be introduced such as the parameter d, in the Buckingham type potential - criterion for choosing the parameter d. . Unfortunately, there is no consistant 04 12 (rz+ d2.Y To avoid this difficulty, we suggest a modification of the conventional optical potential approach. Instead of trying to solve the optical potential expression and then trying to solve the sub- sequent Schrodinger equation, we use the optical potential directly in a variational expression for the scattering phase shifts. The associated diagramatic approach in enumerating different perturba- tion terms in the optical potential expression has two advantages, First, it enables one to improve the phase shift as one includes higher order optical potential in a systematic and tractable fashion,
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-4- . Secondly, it is possible to associate different physical effects with different diagrams. Thus one is able to evaluate the individual con- tributions of the direct and the exchange part of the optical poten- tial for each multipole component. Of course, our main aim is to obtain a general method that is also practical. As in common with any perturbational approach, the convergence of the optical potential expression depends on the basis wave functions one uses, which in turn depend on the "Single-particle Potential" one chooses to generate them. For a well-chosen single particle V , one hopes to obtain good results with the inclusion of only up to the second order optical potential. The second order op- S tical potential contribution to the phase shift can be then evaluated without recourse to adiabatic approximations or the introduction of any ad hoc parameters. In this paper, we have adopted the above pro- cedure in a calculation of electron-helium scattering for energies from 1.2 ev. to 16.4 ev. with gratifying result. In Section I1 we review the single particle potential and the result of the formal optical potential, first derived by Bell and Squires. The variational expression for phase shifts in terms of optical potential is given. The application to the e-He scattering is carried out with numerical procedure described in Section 111. Results and discussions are presented in Section IV. Concluding remarks are given in Section V. Section 11: REVIEW OF THE FORMAL OPTICAL POTENTIAL The formal optical potential for a system of identical fermions was first derived by Bell and Squires' in the context of nuclear scattering problems. The result is of course applicable to electron-
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. To construct a the solution 9 can -6- basis from which a perturbational expansion for be obtained we first approximate the effect of the interacting particles by a single particle potential Vs so that the total system is approximated by an unperturbed system 9 with a Hamiltonian 0 / k "he choice of the single particle potential Vs, at this point, is completely arbitrary except that it should be Hermitian so that the single-particle wave functions yh satisfying form a complete orthonormal set. wave function is a Slater determinant formed from (2 + 1) single particle states Z states in should represent the ground state of the atom. This demands that the Vs should generate a complete set of that the lowest 2 states coincide with the Hartree-Fock states of The unperturbed Z + 1 particle . Physical condition makes it desirable that _sp, % 5 such 2 the ground state atom. The complete set of 5 are used as the basis for perturbation expansion. x In treating a system of identical fermions, it is desirable to use second quantization formalism since the commutation relations of the creation and destruction operators for single-particle state automatically take care of the Pauli principle between electrons. Using the basis just defined, the Equations (2.5) become, in the second quantization formalism,
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. - 7- + and are the usual creation and destruction opera- . They obey the Fermi-Dirae anti- L tors for single particle state commutation relations. The exact expression for the matrix elements The summation of the matrix elements is over distinct elements only, e.g. < /r/nn) is not distinct from the matrix element f < f V 1% > Let us designate m,(z) as the Hartree-Fock ground state of atom. The number Z is used to remind us that the wave function is a Z electron function. Following Goldstonefl, the single particle states occupied in ${z] are called unexcited states while the rest 0 are called excited states. An unoccupied unexcited state K3 is called a hole, and an occupied excited state is called a particle, The unperturbed scattering system $e (Z+/) is +e 7; le> (2-81 Goldstone1™ showed that the true ground state of the atom $ is
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-8- . and The true ground state 6 , through Wick’s theorem, may be represented by a sum of distinct diagrams where a “particle” is represented by a line directed upwards while a “hole” is represented by a line directed downwards. The unperturbed ground-state Hartree- Fock atom is the “Vacuum” state. The matrix elements (pf/”/mfi> and rf///mc) are represented as graphs shown in Fig, 1. Carrying out the time integration, one obtains L (2.11) *ere the sum is over linked diagrams only. In general, the dia- grams representing $ but has a maximum of 2 z lines at the top, z particle lines and z hole lines. has no particle or hole lines at the bottom Similarly the true solution for the entire scattering system is – X( / – E-H,+t & The diagrams representing $ has bottom (incoming electron ko) and a the top, z 3. 1 particle lines and z %+ (2-12) H9 ‘I 4 I$) D only one particle line at the maximum of ZZ+/ lines at hole lines.
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. . -9 L To obtain the true scattering electron wave function (2-13) In terms of diagrams, is the sum of -all linked diagrams where a particle line of ko is directed upward at the bottom and only one “particle” line at the top as illustrated in Fig. 2. The optical potential for this particle, as first derived by Bell and Squires, is then (2.14) Where the symbol LP means that one sum diagrams only that are linked and proper, using the designation of Bell and Squires. By 9 ”proper” they mean those linked diagrams which are not linked by one particle line at any intermediate state. The diagram Fig. 3a is a proper diagram while the diagram Fig. 3b is not. The reason for the requirement of the “proper” diagrams in the optical potential expression can be explained as follows. If there is only one “par- ticle” line at some intermediate state of the diagram, it means that out of the (Z + 1) electron system there are Z electrons in the un- excited states, i.e. the atom is in its ground state. Thus the restriction on “pr6per” diagFms is equivalent to the restriction in conventional optical potential- formulation that the ground state of atom can not occur in the intermediate state. –< 8 Since the optical potential is defined for the basis states v , the scattering electron satisfied the one-particle -%
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