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Sensitivity of differential cross sections and angular correlation parameters to scattering

parameters

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1979 J. Phys. B: At. Mol. Phys. 12 3399

(http://iopscience.iop.org/0022-3700/12/20/018)

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J. Phys. B: Atom. Molec. Phys., Vol. 12, No. 20, 1979. Printed in Great Britain

Sensitivity of differential cross sections and angular correlation parameters to scattering parameters

D H Madison Drake University, Des Moines, Iowa 50311, USA

Received 11 December 1978, in final form 20 April 1979

Abstract. Over the last few years there have been several methods proposed by different investigators for calculating distorted-wave amplitudes. It is difficult to compare the relative merits of these calculations since widely different atomic potentials and wavefunc- tions have been used in the various calculations. We have examined these different approximations by calculating different distorted-wave amplitudes using a common Hartree-Fock basis for atomic potentials and wavefunctions. This procedure allows us to examine the relative effects of different types of approximations. For purposes of comparison, we have calculated differential cross sections and angular correlation parameters for electron impact excitation of the 2'P and 3lP allowed transitions of helium for incident electron energies in the range 40-200 eV. All of the calculations include both the direct and exchange amplitudes, and calculations were performed using both a static atomic distorting potential and a complex optical potential which included polarisation, exchange and absorption distortion in addition to the distortion of the static potential.

1. Introduction

Recent experimental advances in the area of electron-photon coincidence measure- ments have stimulated much interest in differential cross sections for electron impact excitation of magnetic sublevels of atoms. Many elaborate theoretical calculations for these quantities have been proposed using widely differing approaches-distorted- wave, many-body, close-coupling, second-order potential, pseudo-state expansions, multichannel eikonal, etc. Bransden and McDowell(l977,1978) have recently given a review of this work. These calculations have given differing types of results and it is difficult to judge the merits of all these works since not only are different approxima- tions used but also different atomic potentials and wavefunctions. One of the more successful approximations for agreement with experimental data is the distorted-wave (DW) approximation. Several different types of DW calculations have been proposed (see Madison and Shelton 1973, Thomas et a1 1974, Scott and McDowell 1976, Bransden and Winters 1975, 1976, Winters 1978, Madison 1978, Vanderpoorten and Winters 1979). Again it is difficult to effectively evaluate these different DW cal- culations since different techniques, atomic potentials and wavefunctions have been used. The purpose of this paper is to provide a systematic comparison of different types of DW calculations.

In the DW approximation, the transition probability for direct scattering is related to the overlap integral (&&I( v - ~ ) 1 4 ~ + b ~ ) where (La@) is the initial (final) atomic wave- function, 4a(b) is the initial (final) eigenfunction of the distorting potential U, and V is

0022-3700/79/203399 + 16$01.00 @ 1979 The Institute of Physics 3399

3400 D H Madison

the total interaction potential. This amplitude can be obtained from a number of different approaches-two-potential formulation, close-coupling expansion, Greens’ function expansion, many-body theory, etc. Each of these approaches provides differing bases for arguing about the best way to evaluate the amplitude. Some of these arguments are summarised below.

1.1. First-order distorted waves

The procedure to be used in calculating initial- and final-state distorted waves is not well defined. Thomas et a1 (1974) have argued that both 4a and 4 b should be obtained using the ground state of the atom; Madison and Shelton (1973) have argued that 4 a ( b )

should both be eigenfunctions of the same atomic potential whether it be the ground, excited or some intermediate atomic state; while Winters (1978) has argued that q5a should be calculated as an eigenfunction of an initial-channel potential and C$b be calculated as an eigenfunction of a final-channel potential consistent with the historical approach of Mott and Massey (1965).

1.2. First-order atomic wavefunctions

A wide variety of atomic wavefunctions have been used in various first-order cal- culations ranging from simple hydrogenic wavefunctions to highly correlated configuration-interaction wavefunctions. The assumption that is generally made is that improving the bound-state wavefunction improves the scattering calculation within the particular approximation that is being used. Improving the bound-state wavefunction is generally defined to mean obtaining a better wavefunction for an isolated atom.

1.3. Higher order corrections for the distorted waves

First-order distorted waves are calculated as eigenfunctions of a static atomic potential. Better distorted waves can be obtained by including effects of dynamical modifications of the atom in the calculation of the projectile wavefunction. In general, this calculation involves coupled equations and non-local potentials, and is equivalent to an elastic scattering problem since a distorted wave in a particular channel is an elastic scattering wavefunction in that channel. There has recently been considerable attention (Furness and McCarthy 1973, Byron and Joachain 1974,1977, Vanderpoorten 1975, Riley and Truhlar 1975, Bransden eta1 1976, Joachain et a1 1977, McCarthy et a1 1977, Rudge 1977) in elastic scattering research directed at obtaining a complex local central potential (optical potential) that can be used to calculate elastic scattering wavefunc- tions easily. In those works, optical potentials have been obtained for the elastic scattering problem using both phenomenological and ab initio approaches. These calculations have demonstrated that it is possible to obtain single-channel elastic scattering wavefunctions to a reasonable degree of accuracy using a complex local central potential.

The primary effects influencing the projectile in these calculations are distortion by electron exchange, atomic polarisation and absorption into other open channels. The first two effects can be represented by real potentials and the last by an imaginary potential. Baluja et a1 (1978) have recently reported a DW calculation for electron excitation of H and He+ which included a real polarisation and exchange potential in the calculation of the distorted waves. Bransden and Winters (1975, 1976) have reported

Comparison of different types of Dw calculations 3401

calculations in which only the incident-channel distorted wave was obtained as a solution of a one-channel second-order potential equation including all optical poten- tial effects. Madison (1978) recently reported a DW calculation in which both the incident- and exit-channel distorted waves were obtained using complex optical potentials.

1.4. Higher order corrections for the atomic wavefunctions

The dynamical changes of the atomic wavefunction induced by the incident electron can be reflected in $ a ( b ) . There have been several papers examining this possibility by McDowell and his colleagues (see Scott and McDowell 1976, and references therein) utilising the approximation they have called the DW polarised-orbital (DWPO) model. In this work, a correction has been added to the atomic wavefunction simulating the polarisation of the atomic wavefunction by the incident electron. Distorted waves were calculated in only one channel, however. It is now known that distorted waves must be calculated for both the incident and exit channels if reasonable agreement with large-angle experimental data is desired.

1.5. Higher order terms in the interaction operator

The previous DW 2'-matrix represents the leading term of an expansion in orders of the interaction potential. Second-order potentials and amplitudes have been calculated for s -$ s transitions in hydrogen by Winters (1978) and Vanderpoorten and Winters (1979). It was concluded that these amplitudes can contribute significantly to small-angle scattering. It is to be noted that the effects of 8 1.4 and 0 1.5 are not mutually exclusive, however. It has been pointed out by Vanderpoorten and Winters (1979) and Winters (1979, and private communication) that one of the effects included in the second-order DW amplitude is target distortion. In fact, the DWPO model of McDowell can be obtained from the second-order amplitude (Bransden and McDowell 1977).

The present study was initiated to examine the relative importance of the various effects. We have attempted to duplicate the various types of DW approximations using a common basis set for atomic wavefunctions and static atomic potentials. Differences between the various calculations then reflect differences between the approaches instead of differences between approaches plus considerable differences between atomic wavefunctions. For our calculations, we have generated atomic wavefunctions and potentials using the Hartree-Fock program of Froese-Fischer (1972). These wavefunctions were chosen since they are easily accessible and give a reasonably accurate description of the bound-state problem. For the purpose of comparison with experimental data, we have calculated differential cross sections, ratios of the m = 0 cross section to the cross section summed over magnetic sublevels ( A ) and differences between the complex phases for the m = 0 and m = 1 magnetic sublevel amplitudes (x) for excitation of the 2'P and 3'P state of helium. We have not included the effects of 0 1.4 and Q 1.5 in our calculation.

2. Calculational procedure

The DW theory for complex optical potentials has recently been given by Madison (1978) and further discussed by Vanderpoorten and Winters (1979). The basic

3402 D H Madison

formulae for the cross section, ratio of m = 0 cross section to the cross section summed over magnetic sublevels ( A ) and difference between the complex phases for the m = 0 and m = 1 magnetic sublevel amplitude (x) are contained in Calhoun et a1 (1977) and Madison (1978) and will not be repeated here. The distorted waves are solutions of the equation

( V 2 + k 2 - U ) r $ = o (1) where k 2 is the projectile energy and U is the optical potential. We use atomic units with energy expressed in rydbergs in all equations. The optical potential was taken to be

(2) where Vs is the spherical average of the static atomic potential for an isolated atom, Vpol is the polarisation potential, Vex is the exchange distortion potential and Vabs is the imaginary absorption potential. For this calculation we have used the adiabatic polarisation potential of Temkin and Lamkin (1961).

(3) where y = 1 . 3 4 1 4 ~ consistent with a polarisability of 1.39 a:. This polarisation poten- tial which was originally obtained for hydrogenic systems has also been used by Bransden et a1 (1976) and McCarthy et a1 (1977) for scattering from helium and was chosen due to ease of calculation. For the exchange potential, we have used the form first suggested by Furness and McCarthy (1973) and further discussed by Riley and Truhlar (1975):

U = vs + vp0l-t v e x + Vabs

4 3 2 4 Vpol(r) = -911 -(I +2y +2y2+3y +?;Y + & y s ) exp(-2y)I/(2y4)

2Vex= ( k 2 - U ) - [ ( k 2 - U)2+8(pl,/r)2]1’2 (4) where Pls is r times the atomic radial wavefunction for the ground state. This local potential has been shown (Bransden and McDowell 1977) to be a reasonably good approximation for the exact non-local exchange potential in the calculation of elastic scattering wavefunctions (the distorted waves r$ are elastic scattering wavefunctions). The absorption potential is pure imaginary of the form suggested by McCarthy et a1 (1977).

( 5 )

where W is an energy-dependent parameter which we have approximated from figure 1 of McCarthy et al (1977) for the energy range 40-200 eV as

Vabs= 15.01579x lo4 Wp:,/(k’- U)’

lo4 W = 0-068k2- 0.099. (6) For all the results reported here, we have calculated both the direct and electron

exchange scattering amplitudes as described by Madison and Shelton (1973). In the partial-wave expansions for the distorted waves, thirty partial waves were used in the exchange amplitude. In the direct amplitude, the DW overlap integrals had converged to the Born results by thirty partial waves. Analytic Born results were used for higher partial waves up to a total of 175. Numerical integrations were carried to 144 Bohr radii.

Atomic potentials and wavefunctions used in the calculation were obtained from the Martree-Fock code of Froese-Fischer (1972). The following three calculations were performed: (a ) a helium atom in the 1s’ configuration yielding the ground-state potential V, and ground-state 1s wavefunction Jl(1sg); ( b ) a helium atom in the

Comparison of different types of DW calculations 3403

ls2p 2lP configuration with the 1s wavefunction frozen in the ground state yielding the frozen-core potential Vf, ground-state 1s wavefunction +(lsg) and frozen-core 2p wavefunction +(2pf); and (c) a helium atom in the ls2p 2lP configuration yielding the excited-state potential V,, excited-state 1s wavefunction +(lse) and excited-state 2p wavefunction +(2pe). The following DW calculations were performed using these wavefunctions and potentials.

2.1. MB

The static atomic potential for both channels was chosen to be the ground-state potential V,. The bound-state wavefunctions used in this calculation were bound-state eigenfunctions of V, (i.e. bound-state solutions of equation (1) with U = V,). Both initial- and final-state distorted waves were calculated using (1) and (2) with V, = V,. This procedure simulates the many-body theory calculation of Thomas et al (1974). It should be noted that these wavefunctions would not be a good representation of the RPA wavefunctions of Thomas et al, but that they are consistent with the wavefunctions used in the other DW calculations and were chosen since they maintain the orthogonality requirement for the case in which U = V,. Different from the RPA wavefunctions, the 1s wavefunction used here does not give a good representation of the ground state of the helium atom +(lsg). H S Taylor, TScott and P Driessen (1978, private communication) point out that, in general, the 2p wavefunction from a frozen-core approximation would be more appropriate for a MB calculation. Interestingly, the present 2p wavefunction is similar to the frozen-core 2p wavefunction +(2pf) described above. The results of this calculation for the ,y and A parameters are in qualitative agreement with the recent many-body theory calculation of Meneses et a1 (1978).

2.2. MM

The initial-channel distorted waves are calculated using the ground-state atomic potential (i.e. in equation (2) V,= V,) and the final-channel distorted waves are calculated using the excited-state atomic potential (V, = V,). The corresponding ground +(lsg) and excited +(2pe) atomic wavefunctions were used in the evaluation of the matrix elements. This corresponds to the historical approach of Mott and Massey recently discussed by Winters (1978). In a similar fashion, Csanak et a1 (1973) formulated a higher order many-body approach for which computation of the final- channel distorted wave involved the final state of the target.

2.3, ES

This is similar to MB except that in the calculation of the distorted waves the excited- state potential V, is used for both channels instead of the ground-state potential V,. For this calculation, the bound-state wavefunctions +( lse) and +(2pe) were used in the evaluation of the matrix elements.

2.4. FC

This calculation corresponds to using an intermediate atomic potential as was proposed by Madison and Shelton (1973). It is argued that the 1s atomic wavefunction for the inactive electron does not have time to adjust to the excited state during the passage of

3404 D H Madison

the projectile electron. As a result $1, is ‘frozen’ in its ground-state configuration $(lsg) and $2p is calculated in the frozen-core approximation $(2pf). Both the initial- and final-channel distorted waves are calculated as eigenfunctions of this intermediate atomic potential (i.e. in equation (2) V, = V, for both channels).

These four approaches are fairly representative of the different first-order DW

calculations that have been reported in the literature. We have chosen not to examine cases where distortion is not treated equivalently in both channels. When comparing this calculation with the recent calculation by Winters (1978) for exciting the 2s state of hydrogen, the following correspondences can be identified; MB would correspond to his

As was mentioned earlier, one of the objectives of this study was to perform the different types of DW calculations using the same kind of bound-state wavefunctions. All of the bound-state wavefunctions which we have used are of the Hartree-Fock type. However, there are differences in the bound-state wavefunctions as dictated by the method of calculation. If one is inclined to rate bound-state wavefunctions by their ability to predict bound-state atomic parameters for the incident and exit channel, then for this calculation the wavefunctions used in MB would be the least successful and the wavefunctions used in MM would be the most successful with FC and ES holding intermediate positions.

For each of the four cases which we have studied, we have performed calculations using only a static distorting potential (U = V,) and the full optical potential which includes polarisation, exchange and absorption distortion effects. We have included the optical potential in both channels and again have not considered cases in which the optical potential is included in only one channel such as the Bransden and Winters (1975, 1976) calculations. Including the optical potential does not greatly change the results of the calculation as was recently discussed by Madison (1978). In that work, the effects of polarisation, exchange and absorption were examined individually and collectively. Of the three terms added to the static potential, it was found that overall the exchange term generally produced the largest change with absorption being relatively unimportant. In a recent calculation Winters (1978) included polarisation and absorption in the initial channel only for excitation of the 2s state of hydrogen and found that it had a relatively small effect in that case also. In that work, Winters noted that higher order corrections to the distorted waves can be expected to be small since they alter the DW amplitudes in third and higher orders but not in second order. Vanderpoorten and Winters (1979) have pointed out that use of second-order optical potentials in a first-order DW calculation such as this will produce multiple countings to third order in the interaction potential. While this effect can be important at lower incident electron energies (Winters 1979) for some transitions, it is expected that it probably plays a negligible role in most practical applications (Vanderpoorten and Winters 1979).

DW-11; MM would correspond to DWB1, and ES would correspond to DW-FF.

3. 2 l ~ results

3.1. Differential cross sections

Differential cross sections for excitation of the 2’P state obtained from the various DW calculations are compared with experimental data in figures 1-2. We have used the experimental data as presented by the original authors. The differential cross section

Comparison of different types of DW calculations 3405

0 90 180 Angle (deg l

Figure 1. Differential cross sections for electron impact excitation of the 2lP state of helium in units of ai sr-'. The theoretical curves are DW calculations described in the text. The experimental data are: U, Hall et al (1973); V, Truhlar et al (1973); 0, Chutjian and Srivastava (1975); t, Opal and Beaty (1972); and 0, Truhlar et a1 (1970).

0 -*0 Angle idegl

Figure 2. Same as figure 1. Here the experimental data are: 0, Vriens et a1 (1968); V, Suzuki andTakayanagi (1973); 0, Dillon and Lassettre (1975); and t, Opal and Beaty (1972).

data have been placed on an absolute scale using various different normalisation schemes ranging from normalising to differential elastic scattering cross sections, normalising to integrated total cross sections, or normalising to optical oscillator strengths extrapolated from experimental generalised oscillator strengths. The various normalisation procedures, discussed by Dillon and Lassettre (1975) and Truhlar et a1 (1973), have given experimental cross sections in reasonable accord with each other as can be seen from figures 1 and 2. For these two figures, static atomic potentials were used in the calculation of the continuum distorted waves and the different types of calculations and abbreviations are described in the previous section. From the figures, it is readily seen that the FC calculation is in good agreement with the experimental data except at the lowest energy.

A couple of comparisons can be made between the various DW calculations which are particularly interesting. One such comparison lies in the FC and ES calculations. When the atomic potentials and bound-state wavefunctions used in these two cal- culations are examined, it is found that they are all very similar except for the initial-channel Is wavefunction. The FC calculation uses the Hartree-Fock 1s wave- function of a neutral helium atom while the ES calculation uses a Hartree-Fock Is wavefunction for a helium atom in the 2'P state which is concentrated closer to the nucleus. The differences between the two calculations reflect the sensitivity of the DW

calculation to the initial bound-state wavefunction. As may be seen from the figures, drawing the 1s wavefunction closer to the nucleus has the effect of lowering the magnitude of the differential cross section without significantly affecting its shape.

Another interesting comparison can be made between the FC and MM calculations. When the atomic potentials and wavefunctions for these two calculations are compared, it is seen that the bound-state wavefunctions and final-channel atomic potentials are essentially identical. (The 2p wavefunction in the frozen-core approximation is similar

3406 D H Madison

to the 2p for the normal excited state.) The differences between these two calculations is then the scattering potential in the initial channel (MM uses an initial-channel potential for the ground state of the atom while FC uses an intermediate potential which includes the effects of a 2p wavefunction). The initial channel MM atomic potential is of a shorter range than the FC potential. (The MM potential drops to lo-' atomic units by 2.5 a. while the corresponding radius for the FC potential is 10 ao.) Comparison of the various differential cross sections presented in the figures reveals that MM and FC are essentially identical for small scattering angles reflecting an insensitivity of small-angle differential cross sections on initial-channel atomic potentials. At larger angles, there is a significant difference between the two calculations. Using the shorter range atomic potential of MM causes the large-angle cross section to decrease more rapidly around 40-50" and then level off and increase above the FC calculation at the back angles.

Of the four different DW calculations, only FC and MM agree with the absolute magnitude of the small-angle experimental data. It is interesting to note that these are the only two calculations which use initial- and final-channel wavefunctions appropriate for isolated atoms in the respective channels. (The MB calculation would agree with the small-angle differential cross sections if better bound-state wavefunctions had been used such as the RPA wavefunctions of Thomas et a1 (1974).) This implies that it is necessary to use 'good' initial- and final-channel bound-state wavefunctions if one wishes to get the proper magnitude for the small-angle differential cross section and that small-angle cross sections have a weak dependence on the distorting potential. This would explain why the Born approximation has historically been relatively successful for small-angle differential cross sections.

The effect of including the full optical potential with exchange, polarisation and absorption distortion is shown in figure 3 for the FC and MM calculations. The effect of the optical potential decreases with increasing energy as was discussed by Madison (1978). In general, the optical potential has a minimal effect on the small-angle differential cross section while increasing the large-angle cross section. An exception to this rule is seen in figure 3 for the MM calculation at 40 eV where the cross section has a minimum near 90" that is shifted to larger angles by the optical potential. For these calculations, the optical potential did not significantly improve agreement with experimental data. The effect of the optical potential for higher energies is not shown since it is relatively small.

I , , , * *+ 1 0 90 180

Angle ldegl

Figure 3. Same as figure 1.

Comparison of different types of DW cakulations 3407

3.2. A parameter

The various DW calculations for the A parameter are compared with experimental data in figures 4-5. Each of the DW calculations in these figures was obtained using static atomic potentials. It is seen that the FC calculation provides very good overall agreement with the small-angle data and small-angle minimum with the poorest agreement occurring for the lowest energy considered. Large-angle measurements have been made by Sutcliffe et a1 (1978) at 80 eV and Hollywood et a1 (1979) at 81.2 eV. Our calculations are in better agreement with the large-angle measurements of Sutcliffe et al. The MM calculation which was the second best for the differential cross section gives the worse overall agreement for the A parameter. The differences between the various calculations decrease with increasing energy although there are still significant differences at 200 eV particularly for larger angles. All the calculations are essentially identical at small angles reflecting the fact that the m = i 1 cross section is becoming small.

OL 60 120 d o Angle ldegl

./ 0 60 120

051 0'

Angle ldegl 0

Figure 4. Angular correlation parameter A for electron impact excitation of the 2'P state of helium. The theoretical curves are DW calculations described in the text. The experimental data are: 0, Tan et al (1977); V, Eminyan et al (1974); 0, Sutcliffe et al (1978); *, Ugbabe et al (1977); 0, Hollywood et al (1979).

Figure 5. Same as figure 4 except for higher incident electron energies.

When FC and ES are compared, it may be seen that drawing the 1s wavefunction closer to the nucleus has the net effect of broadening and lowering the small-angle minimum while increasing A at the larger angles. Comparing FC and MM reveals that using a shorter range initial-channel potential has a similar but more pronounced effect to using a shorter ranged 1s wavefunction. In this case, the small-angle minimum is much lower and the large-angle values are also lowered except at the lowest energies.

In terms of agreement with experimental data, it is interesting to note that the two calculations which contain no final-state effects in the incident channel (MB and MM) are qualitatively similar and in worse overall agreement with the data while the two calculations which include final-state effects in the incident channel are in better agreement with the data. The best agreement is obtained using appropriate initial- and

3408 D H Madison

final-channel bound-state wavefunctions and an atomic potential for both channels which includes effects of both channels.

The effect of the optical potential on the A parameter is shown in figure 6. We have again chosen the FC and MM calculations as being representative. For each energy, results obtained using both the static atomic potentials and optical potentials are presented. It may be seen from the figure that similar to the differential cross section results, the optical potential does not have a large effect on the A parameter. In general, the optical potential makes the small-angle minimum more narrow and increases the absolute value of the minimum. For the larger angles, the optical potential tended to increase the value of the A parameter particularly for the higher energies. In terms of agreement with experimental data, the optical potential made the FC calculation slightly better at the lower incident electron energies. While the change in the MM calculation is toward the experimental data, the change is much too small to bring the calculation into agreement with the data. Again, the effect of the optical potential on the A parameter at higher incident electron energies is not shown since the effect becomes small.

I 0 60 120 Id0

Angle (degl

Figure 6 . Same as figure 4.

3.3. x parameter

The present DW calculations using static atomic potentials are compared with small- angle experimental x parameters in figures 7-8(a). An examination of these figures indicates that the ES calculation gives the best overall agreement with this data. The FC

calculation which gave good agreement for the differential cross section and the A parameter is unfortunately not as good for the small-angle x parameter. This interes- ting phenomenon has been observed in other calculations of angular correlation parameters (Bransden and McDowell 1978).

Figure 8 ( b ) compares the present calculations with the very recent large-angle measurements of Hollywood et al (1979) where x is restricted to lie in the interval 0 c x G T. As may be seen from this figure, the FC calculation gives the best overall agreement with the data and is generally within experimental error for angles greater than 90". It is interesting to note that the FC calculation gives the general shape of the small-angle data fairly well but a magnitude that is too high.

Figure 8 ( b ) reveals interesting differences between the various calculations for x. The FC x goes through 7~ near 70" while MM goes through 0 at about 66" scattering. An examination of the polar nature of the complex amplitudes which give x reveals even

Comparison of different types of DW calculations 3409

Angle ldeg)

Figure 7. Angular correlation parameter ,y for electron impact excitation of the 2'P state of helium. The theoretical curves are DW calculations described in the text. The experimental values are: 0, Tan er al (1977); V, Eminyan et al (1974); and t, Ugbabe et al (1977).

0 20 LO

M M I ,-- I , --- - - - _._ -

0 60 120 180 Angle ldeg)

Figure 8. ( a ) Same as figure 7 except for higher incident electron energies. ( b ) Same as figure 7 for 80 e V incident electrons. The 0 are the experimental data of Hollywood et al (1979).

more interesting differences. For example, in the MM calculation at 80 eV both the m = 0 and m = 1 complex amplitudes are initially in the first quadrant in the complex plane with m = 0 leading m = 1 by about 6". As the scattering angle increases, both amplitudes move anticlockwise. Initially the m = 0 amplitude moves faster until a maximum separation occurs at a scattering angle of 49". At this point, the m = 1 amplitude starts moving faster and catches the m = 0 amplitude at about 66" scattering. The m = 1 amplitude continues to move faster until a scattering angle of 80" is reached. At that point, the m = 0 amplitude starts to overtake the m = 1 amplitude. The situation is very different for the FC calculation, where both amplitudes again start anticlockwise in the first quadrant with m = 0 leading by 11" and moving faster. The m = 0 amplitude continues anticlockwise for all increasing scattering angles. However, for a scattering angle of 58", the m = 1 amplitude turns around and moves clockwise causing the steep increase in x which goes through 7~ at a scattering angle of about 70".

3410 D H Madison

For a scattering angle of 85', the m = 1 amplitude again turns around and goes anticlockwise.

The effect of the optical potential on the DW calculations for the x parameter is shown in figure 9 for small-angle scattering for the FC and MM calculations. For each energy, the results using both the static and optical potential including polarisation, exchange, and absorption distortion are presented. From the figure, it is seen that the optical potential increases and also the x parameter in this angular range. For the FC and ES calculations, this behaviour makes agreement with experimental data worse. For the MM and MB calculations, the agreement is perhaps better at the low energies. This may be fortuitous, however, since these calculations rise too rapidly at the higher energies. At the large scattering angles the FC results using the full optical potential are also larger than the static potential results making agreement with experiment slightly worse.

0 20 L O Angle ldeg)

Figure 9. Same as figure 7 .

4. 3 l ~ results

The present DW calculations are compared with experimental data for excitation of the 3lP state of helium in figures 10-12 for the differential cross section, A parameter and x parameter respectively. The theoretical calculations were obtained using static atomic potentials. The behaviour of the data and the various calculations are very similar to the 2lP results and most of the previous remarks apply equally well here. The FC calculation agrees best with the experimental differential cross sections and A parameters, although the agreement is not as good as for the 2lP case. As may be seen from the work of Chutjian (1976), the present calculations are in better agreement with the differential cross section data than the multichannel eikonal calculation of Flannery and McCann (1975) or the DWPO calculation of Scott and McDowell (1976) however. For the A parameter there is a smaller difference between the FC and ES calculations than was seen for the 2lP. Probably the biggest difference between the calculations for excitation of the two different states is seen in the x parameter where the ES calculation does not agree with the data as well due to the fact that there is a smaller lowering in the value of the x parameter.

We have also performed the same calculations using the full optical potential instead of the static potential and the effects on the results were similar to those seen for the 2lP

Comparison of different types of DW calculations 3411

I , . . , . I t 0 90 180

Angle ldeg)

Figure 10. Differential cross sections for electron impact excitation of the 3'P state of helium in units of a: sr-l. The theoretical curves are DW cal- culations described in the text. The experimental data are those of Chutjian (1976).

0 60 120 180 Angle ldeg)

I I

Angle ldeg) 0 20 LO

Figure 11. Angular correlation Figure 12. Same as figure 11 parameter A for electron except for the ,y parameter. impact excitation of the 3lP state of helium. The theoretical curves are DW calculations described in the text. The experimental data are: ., Eminyan er al (1975) and V, Standage and Kleinpoppen (1976).

excitation. It is interesting that the DW calculations give results as good as they do for excitation of the 3lP state since the states of the N = 3 manifold are closer together in energy and one might expect that couplings to other channels would be important. This point has previously been discussed by Chutjian and Thomas (1975). The good agreement seen here indicates that the two channels contain most of the information about the scattering process.

5. Conclusion

We have examined different DW approximations for excitation of the 2lP and 3lP states of helium. To determine the relative importance of the different elements of a DW

calculation, we have calculated ail the approximations using the same kind of bound- state wavefunctions and atomic potentials-namely numerical Hartree-Fock. We have included distortion equivalently in the incident and exit channels and have not examined the large number of possible variations in which one channel is a plane wave, Coulomb wave, or any other possible continuum wave with lesser distortion since these variations are generally not reliable for predicting large-angle results (Bransden and McDowell1977). It is expected that the general effects observed in this work should be valid for other calculations using wavefunctions of lesser or greater accuracy.

The following observations can be made as a result of this work. To achieve agreement with the experimental absolute magnitude of the small-angle differential cross section, it was necessary to use bound-state wavefunctions in each channel that are fairly good representations of an isolated atom in that channel. The small-angle differential cross section was fairly insensitive to the distorting potential while the

3412 D H Madison

large-angle cross section is much more sensitive to the potential. The A parameter was more sensitive to the distorting potentials than the bound-state wavefunctions and only the calculations which included final-state distortion effects in the incident channel gave results which behaved like the experimental data at small angles. For the ,y parameter, the situation was not as clearly defined. However, it could be argued that similar to the A parameter, final-state distortion effects must be included in the incident channel to get agreement with the experimental data.

The different elements of a DW calculation have different relative importance on the results of the calculation. It is clear that the bound-state wavefunctions and distorting potentials influence the results more strongly than the optical potential effects of polarisation, exchange and absorption distortion. It is difficult to compare the relative effects of potentials and bound-state wavefunctions since they are different entities. For the present calculation, the changes which were made in the static distorting potentials generally produced larger changes in the results than the changes which were made in the bound-state wavefunctions. Obviously, large changes could be produced by using arbitrary bound-state wavefunctions. However, to achieve agreement with experi- mental data one must first use the proper channel potentials and wavefunctions and then apply corrections such as the optical potential effects.

The present FC calculation using a static distorting potential gives a fairly good agreement with the experimental differential cross section, good agreement with the small-angle A parameter and large-angle data of Sutcliffe et a1 (1978), good agreement with the very recent large-angle ,y measurements of Hollywood et a1 (1979), but only a fair representation of the small-angle ,y parameter. Including the full optical potential instead of just the static distorting potential made the FC value for the ,y parameter worse. The ES calculation, on the other hand, gave reasonable values for the small- angle ,y parameter, values for the A parameter that were similar to the FC calculation, particularly for the higher incident electron energies, and differential cross sections which have about the right shape but whose absolute magnitudes were too low. Including the full optical potential in the ES calculation improves the agreement between the A parameter and experimental data but makes the ,y parameter in worse agreement with the experimental data.

It is disappointing that no single calculation gives good agreement with all three physical quantities considered over the entire angular range. The fact that no single calculation gives good agreement with the data indicates that the second-order DW amplitude might have an important effect. As was noted in the introduction, the second-order amplitude contains the effects of target distortion. The primary difference between the two best calculations FC and ES is the initial-channel target wavefunction. Calculations including either target distortion or a second-order ampli- tude would be very interesting.

Acknowledgments

The author would like to thank F Lang and M Stewart for assistance with the computer work and B Lutz and his staff at the Drake Computing Center for their assistance and cooperation. He would also like to thank K B MacAdam, H S Taylor, D Truhlar and K Winters for comments on the manuscript, and J F Williams for sending the large-angle data prior to publication. Work was supported by the Research Corpora- tion.

Comparison of different types of DW calculations 3413

Note added in proof. Very recently, Baluja and McDowell (1979) published a DW calculation in which both the initial- and final-channel distorted waves were calculated using the initial-channel potential. Their calculation is very similar to the present MB calculation except that they used analytic bound-state wavefunctions, solved the adiabatic exchange equation for the continuum waves and they did not include an absorption term. In that work, a comparison was made to an earlier calculation which we performed reported by Sutcliffe er a1 (1978). In that comparison, it was assumed that the results reported by Sutcliffe et a / were the present static MB results when in fact they were the present static FC results. Consequently,the differences noted at the small-angle minimum result primarily from the different distorting potentials and not to the exchange potential or use of numerical wavefunctions. While the present MB results for ,y are in qualitative agreement with the results of Baluja and McDowell (1979) the present MB A values are in qualitative agreement with Meneses et a1 but not Baluja and McDowell.

References

Baluja K L and McDowell M R C 1979 J. Phys. B: Atom. Molec. Phys. 12 835-45 Baluja K L, McDowell M R C, Morgan L A and Myerscough V R 1978 J. Phys. B: Atom. Molec. Phys. 11

Bransden B H and McDowell M R C 1977 Phys. Rep. 30 207-303 - 1978 Phys. Rep. 46 250-294 Bransden B H, McDowell M R C, Noble C J and Scott T 1976 J. Phys. B: Atom. Molec. Phys. 9 1301-17 Bransden B H and Winters K H 1975 J. Phys. B: Atom. Molec. Phys. 8 1236-44 - 1976 J. Phys. B: Atom. Molec. Phys. 9 1115-20 Byron F W and Joachain C J 1974 Phys. Lett. 49A 306-8 - 1977 Phys. Rev. A 15 128-46 Chutjian A 1976 J. Phys. B: Atom. Molec. Phys. 9 1749-56 Chutjian A and Srivastava S K 1975 J. Phys. B: Atom. Molec. Phys. 8 2360-8 Chutjian A and Thomas L D 1975 Phys. Rev. A 11 1583-95 Csanak G, Taylor H S and Tripathy D N 1973 J. Phys. B: Atom. Molec. Phys. 6 2040-54 Dillon M A and Lassettre E N 1975 J. Chem. Phys. 62 2373-90 Eminyan M, MacAdam K B, Slevin J and Kleinpoppen H 1974 J. Phys. B: Atom. Molec. Phys. 7 1519-42 Eminyan M, MacAdam K B, Slevin J, Standage M C and Kleinpoppen H 1975 J. Phys. B: Atom. Molec. Phys.

Flannery M R and McCann K R 1975 J. Phys. B: Atom. Molec. Phys. 8 1716-33 Froese-Fischer C 1972 Comput. Phys. Commun. 4 107-16 Furness J B and McCarthy I E 1973 J. Phys. B: Atom. Molec. Phys. 6 2280-91 Hall R I, Joyez G, Mazeau J, Reinhardt J and Schermann C 1973 J. Physique 34 827-43 Hollywood M T, Crowe A and Williams J F 1979 J. Phys. B: Atom. Molec. Phys. 12 819-34 Joachain C J, Vanderpoorten R, Winters K H and Byron F W 1977 J. Phys. B: Atom. Molec. Phys. 10 227-38 Madison D W 1978 Proc. Int. Workshop on Coherence and Correlation in Atomic Collisions, London 18-20

Madison D H and Shelton W N 1973 Phys. Rev. A 7 499-513 McCarthy I E, Noble C J, Phillips B A and Turnbull A D 1977 Phys. Rev. A 15 2173-85 Meneses G D, Padial N T and Csanak G 1978 J. Phys. B: Atom. Molec. Phys. 11 L237-42 Mott N F and Massey H S W 1965 The Theory of Atomic Collisions (Oxford: Clarendon) Opal C B and Beaty E C 1972 J. Phys. B: Atom. Molec. Phys. 5 627-35 Riley M E and Truhlar D G 1975 J. Chem. Phys. 63 2182-91 Rudge M R H 1977 J. Phys. B: Atom. Molec. Phys. 10 2451-7 Scott T and McDowell M R C 1976 J. Phys. B: Atom. Molec. Phys. 9 2235-54 Standage M C and Kleinpoppen H 1976 Phys. Rev. Lett. 36 577-80 Sutcliffe V C, Haddad G N, Steph N C and Golden D E 1978 Phys. Rev. A 10 100-7 Suzuki H and Takayanagi T 1973 Proc. 8th Int. Conf. on PhysicsofElectronic andAtomic Collisions (Beograd:

Tan K H, Fryar J, Farago P S and McConkey J W 1977 J. Phys. B: Atom. Molec. Phys. 10 1073-82 Temkin A and Lamkin J C 1961 Phys. Rev. 121 788-94 Thomas L D, Csanak G, Taylor H S and Yarlagada B S 1974 J. Phys. B: Atom. Molec. Phys. 7 1719-33 Truhlar D G, Rice J K, Kupperman A, Trajmar S and Cartwright D C 1970 Phys. Rev. A 1 778-802 Truhlar D G, Trajmar S, Williams W, Ormonde S and Torres B 1973 Phys. Rev. A 8 2475-82

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Ugbabe A, Teubner P J 0, Weigold E and Arriola H 1977 J. Phys. B: Atom. Molec. Phys. 10 71-9 Vanderpoorten R 1975 J. Phys. B: Atom. Molec. Phys. 8 926-39 Vanderpoorten R and Winters K H 1979 J. Phys. B: Atom. Molec. Phys. 11 473-88 Vriens L, Simpson J A and Mielsczarek 1968 Phys. Rev. 165 7-15 Winters K H 1978 J. Phys. B: Atom. Molec. Phys. 11 149-65 - 1979 Comni. Atom. Molec. Phys. to be published