Differential Scattering in He–He and He+ -He Collisions at KeV Energies

11

ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. 25

DIFFERENTIAL SCATTERING IN He-He A N D He+-He COLLISIONS A T KeV ENERGIES R. F. STEBBINGS Department of Spare Physics and Astronomy und Thr Rice Quuntum Institute Rice Univrrsity Houston, Texus

I. Introduction . . . . . . . . .

11. He-He Collisions at Small Angles. .

A. Measured Quantities . . . . . B. Data Analysis . . . . . . . C. Results and Discussion . . . . 111. He' + He Collisions at Small Angles A. Results and Discussion . . . . IV. He-He Scattering at Large Angles . V. Conclusion. . . . . . . . . . Acknowledgments . . . . . . . References . . . . . . . . . .

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83 84 86 87 88 91 92 95 98 98 98

I. Introduction Measurements of the scattering of beams of ions and atoms by gases have provided a valuable source of information on the forces between the colliding particles at close distances of approach. Early studies as exemplified by the work of Amdur and Jordan (1 966) measured partial total cross sections for scattering outside some minimum angle. Such measurements, however, do not lead unequivocally to the intermolecular potential (Mason and Vanderslice, 1962); instead they provide the parameters for assumed forms of the potential. Consequently, such measurements have been in recent years, largely superceded by studies of the angular distribution of the scattered collision products, since these provide a much more stringent test of the theory. Numerous studies of differential scattering in ion-atom and atomatom collisions have been reported. Notable among them is the work of Fedorenko et al. (1960), Lockwood et al. (1963), Lorents and Aberth (19659, Baudon et al. ( 1968) and Wijnaendts and Los (1 979). 83 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 1-12-003825-0

84

R . F . Stebbings

Derivation of the interaction energy from measurements of differential scattering is most readily accomplished when the scattering is free from quanta1 interference patterns. In general, these are a consequence of effects that can be related semiclassically to the presence of two or more trajectories leading to scattering at the same angle and at the same final velocity. Such patterns are observed in both elastic and inelastic scattering and commonly arise from the crossing of two molecular electronic energy curves. The abundance of curve crossings in most atomic and molecular systems ensures that such effects will be observed in most differential cross section measurements. Structure also arises due to rainbow, glory, and diffraction effects, while in the case of symmetric systems, oscillations also appear due to both electronic and nuclear symmetry. Where oscillations of more than one origin are superimposed, the interpretation becomes quite complex but it is nonetheless generally possible to distinguish between the various effects and ascertain their separate contributions to the scattering. Despite the substantial body of data on differential scattering, there are only very few absolute measurements. Lorents and Aberth (1965) reported absolute cross sections for elastic scattering in He+-He collisions while Smith et al. (1970) have published data in which relative differential charge transfer cross sections were placed on an absolute footing by integrating them and normalizing the resulting cross sections to previously measured partial total cross sections. A program of measurements of absolute differential cross sections for scattering of neutral atoms and positive ions was begun at Rice University in 1983. This work was motivated both by the need for improved interaction potentials and also for cross section data for the interpretation of the atmospheric effects (Dalgarno, 1979) due, for example, to precipitation of ring current ions during times of geomagnetic disturbance. This paper presents some findings of this group.

11. He-He Collisions at Small Angles Fig. 1 shows a schematic of the apparatus used for small angle scattering. He' ions generated by electron bombardment in a low voltage, medium pressure, magnetically confined arc plasma source are extracted and focused by a three element einzel lens before passing through a pair of 60" sector confocal bending magnets. The beam is then partially neutralized by helium in the charge transfer cell (CTC) before entering the target cell (TC) where a small percentage of the beam is scattered by helium target gas admitted through a variable leak valve. Typical pressures in the target cell of a few

85

DIFFERENTIAL SCATTERING S C A T T ERE D PARTICLES

DP1

DP2 COMPUTER

FIG. 1. Schematic of the apparatus used for small angle scattering.

millitorr are measured with an MKS Baratron Model 390 capacitance manometer. Background pressure in the main vacuum chamber is typically 2x torr under operating conditions. The laser drilled apertures at the exit of the charge-transfer cell and at the entrance of the target cell at 20 pm and 30 pm in diameter, respectively, and are separated by 49 cm so that the neutral beam is collimated to less than 0.003" divergence. The TC is 0.35 cm in length and has an exit aperture 300 pm in diameter. Electrostatic deflection plates DP1 remove the residual ion beam, while plates DP2 remove charged collision products due to stripping, for example. Both the unscattered primary beam particles and the scattered collision products strike the front face of a position sensitive detector (PSD) whose properties have been discussed by Gao et al. (1984) and by Newman et al. (1985). The PSD is located 109 cm beyond the TC on the beam axis. This detector has a 2.5cm diameter active area, thereby limiting the maximum observable scattering angle to about 0.7". The PSD operates as a single-particle detector, and consists of two microchannel electron multiplier plates (MCPs) and a specially shaped resistive anode. When a particle strikes a channel wall in the first microchannel plate, secondary electrons are produced, and are accelerated down the channel by an applied voltage. When these electrons strike the walls of the channel, more secondary electrons are produced, leading to a small cascade of electrons. The small ( - lo3) cloud of electrons leaving the first MCP enters several channels on the second MCP, increasing the number of electrons to about lo6. These electrons are accelerated toward and stike the resistive anode. The charges leave the anode through four wires located at the corners of the anode, whose resistive characteristics are such that the measured amounts of charge leaving the anode on each of the wires may be utilized to determine the location of the electron cloud impact on the anode, thus determining the location of the impact of the incident particle. An LSI 11/2 microcomputer

86

R . F . Stebbings

latches and stores the output of the PSD position-encoding electronics. As each event occurs, the computer sorts the x and y impact coordinates into a 90 x 90 array and increments the appropriate array element. A. MEASUREDQUANTITIES

For thin target conditions, the differential cross section, do(& q)/dR, is related to the measured quantities by the expression

where S is the primary beam flux, AS(& q ) is the flux scattered into the solid angle AR, and 7 is the “target thickness.” In these experiments, it is not necessary that the absolute efficiency of the detector be known, because the primary and scattered particles are identical and are thus detected with equal efficiency. There is a small effect which decreases the detection efficiency in regions on the detector where the count rate is above a few hundred Hz, but it is possible to operate the detector so that detection efficiencies for regions of high count rate and low count rate are equal to within a few percent. This operating point is determined before each data run, since the condition of the microchannel plates changes as a consequence of the plate’s history of particle impacts. Newman et al. (1985) and Schafer (1987) have shown that for the cell used in this experiment, 7 is accurately given by the product nL where L is the geometric cell length and n is the number density obtained from a measurement of the gas pressure is the TC at a location far from the exit aperture. The accuracy with which At2 = sin BABAq is measured is strictly limited by the accuracy of distance measurements, the size of the primary beam and the detector’s position-finding uncertainty. The PSD spatial resolution is related to the size of the electron pulse impinging on the resistive anode. The positions of large pulses are determined with more precision than are those of small pulses, due to noise in the amplification and summing circuits. This phenomenon was studied with the use of a single-channel analyzer (SCA) to record the contributions to the electronic image of the primary beam from different portions of the pulse height spectrum. In general, it is found that the largest pulses provide good signal-to-noise ratio for the position encoding electronics and result in accurate position data (within about 60 pm). On the other hand, the smallest pulses (amounting to a few percent of the total counts) may be registered as much as 1OOOpm outside the geometricallylimited impact region. The details of the distribution depend on operating conditions: the problem is accentuated by high local count rates and by low

DIFFERENTIAL SCATTERING

87

PSD operating voltage, both of which increase the relative number of small output pulses. Under conditions appropriate to the collection of data for 0.5 keV collisions, it is found that 95 % of the counts are recorded with an error less than 0.016", 3 % with an error between 0.016" and 0.032", and the remaining 2 are distributed out to 0.05". The relatively inaccurate position assignment for these particles results in a loss of angular resolution, particularly where the measured cross section shows sharp features. The inaccuracy in position assignment also interferes with measurement of the scattered signal at the smallest angles (0 < O.OSO), where inaccurate position indexing of some primary beam counts increases the apparent diameter of the primary beam. These counts can be eliminated (and the angular resolution enhanced) by using the SCA to reject the small pulses; but in this case, absolute cross sections are not determinable since the pulse height distributions for primary and scattered particles are no longer the same. As a consequence, different detection efficiencies for primary and scattered particles result when a limited range of the pulse height spectrum is sampled. Therefore, the procedure for measuring cross sections below 0.05" is to obtain relative data using the SCA, and then to normalize these to the absolute data at angles greater than 0.05", where the effects of the spurious primary beam counts are negligible. B. DATAANALYSIS

Advantage is taken of cylindrical symmetry when analyzing the data, and the array elements are summed into annular rings co-axial with the primary beam. Data are accumulated until the statistical uncertainty in the signal at a given angle is small, typically less than 10 %. Two data sets, one with gas in the target cell and one without, are taken to permit discrimination between counts due to scattering from the target gas and counts arising from other sources, such as scattering from the background gas, scattering from edges of apertures, and dark counts. The "gas out" data provide a measure of the primary flux S, since the dark current of the detector is negligible compared to the primary beam flux. The 90 x 90 data arrays are organized into concentric rings whose widths are chosen subject to the competing demands of good angular resolution and an acceptable rate of data accumulation. Typically, the ring widths are somewhat larger than the detector's uncertainty. Although the summation into rings does not take full advantage of the information available from the detector, the technique is necessary to obtain adequate counting statistics. The ring center is determined by fitting a smoothly peaked function to the data; the peak of this function is taken to be the center of the rings. In many cases, where the cross section does not show

R . F . Stebbings

88

sharp features, the results of the analysis are not particularly sensitive to the choice of beam center, Care is taken, however, to ensure that the center is chosen optimally and both the “gas-in” and “gas-out” peaks are fitted to determine a composite “best” center for analysis. The angle B is determined to within f (0.03 8 + 0.002) degrees. This error reflects the uncertainties in PSD calibration, distance from target cell to detector, and location of beam center. One is not only interested in the value of 6, however, but also in the range 66 of physical scattering angles contributing to the signal at 8. The angular resolution 68 arises from the finite width of the primary beam, the discrete nature of the analysis rings, and electronic noise in the detector’s position encoding circuits. Counts registered in the iZhring are assigned to the angle Oi which is the average of the angles corresponding to the inner and outer radii of the ring. The scattered flux, AS(O), is obtained by subtracting the gas-out data from the gas-in data and the angular range A6 is taken to be the ring width.

c. RESULTS AND

DISCUSSION

Differential cross sections have been determined at laboratory collision energies of 0.25, 0.5, 1.5, and 5.0 keV (Nitz et al. 1987). Previous results for small angle He-He scattering were reported by Leonas and Sermyagin (1977) who measured relative cross sections at approximately an order of magnitude less resolution. He-He scattering at reduced scattering angles below 1 keV-degree is not expected to involve excited states of the collision complex. The scattering, therefore, occurs from a single potential and the differential cross section is expressed quantum mechanically as

This expression (Massey and Smith, 1930) takes account of the fact that because the projectile and the target are identical, the signal of atoms at angle 9 comprises: (1) Projectiles scattering at 9 with amplitude f(9) and (2) Target atoms recoiling at angle 9. This occurs when the projectiles are scattered at ( n - 9) with amplitude f ( n - 9). At these small values of 9, however, f ( n - 9) is negligible compared to f(9)

and the cross section is accordingly well-represented by

DIFFERENTIAL SCATTERING

89

where the scattering amplitude j ( 9 ) is given by the partial wave summation formula

where 9 is the scattering angle in the center of mass frame, k is the wave number, 6, is the phase shift of the lth partial wave, and P,(cos 9) is the lth Legendre polynomial. Conversion into laboratory coordinates is required for comparison with experimental results. Cross sections have been calculated with this equation, using phase shifts derived from various proposed forms of the interaction potential. The phase shifts are obtained using the semiclassical JWKB approximation, except in the limit of large I, when the phase shifts become small and they are then determined using the eikonal, or JB approximation. Information about the He-He interaction potential has been derived from a combination of scattering experiments, dilute gas transport experiments, and theory. Recent attention in the literature has focused on the lower repulsive wall at internuclear separations less than 1.8 A, where the potential rises above the 0.1 eV level. In this region, the results of high temperature transport experiments indicate the validity of a more steeply rising potential than do scattering measurements of integral cross sections. Of the many potentials that could be investigated, two analytic forms which provide a convenient characterization of the situation are the potentials proposed by Aziz ef al. (1 979) and by Ceperley and Partridge (1 986). The potential of Aziz et a / . has an attractive well consistent with a large body of thermal-energy data and a steep repulsive wall consistent with the high temperature (2500 K) measurements of thermal conductivity by Jody et al. (1977). Ceperley and Partridge have proposed a composite potential based on ab initio quantum Monte Carlo calculations and an extrapolation to larger r. This potential follows Aziz et al. for I > 1.828 A, but at smaller r it agrees more closely with results of Feltgen et al. (1982) and Foreman et al. (1974), who obtained potentials by inverting integral cross section data. Theoretical differential cross sections derived from the Aziz et al. and Ceperley and Partridge potentials are plotted in Fig. 2 along with the 0.5 keV experimental results. The 0.5 keV results were selected for this comparison, since cross sections at low projectile energies are particularly sensitive to the choice of potential. In general, the agreement to both shape and magnitude of the cross sections is excellent, and i t is worthwhile to mention that the calculations include no adjustable parameters. At angles less than 0. lo, the predictions are almost identical and lie within the experimental uncertainty. At angles greater than O.V, the steeper Aziz et al. potential yields undulations larger than are observed experimentally and also gives slightly lower values of the cross

R . F . Stebbings

90

Io6

lo5

v

lo4

lo3 0,OI

0 ,I L A B ANGLE ( d e g )

FIG.2. Differential cross sections for He-He scattering at 0.5 keV. The measured values are shown together with the calculations based on the potentials of Aziz e t a / .(shown dotted) and of Ceperley and Partridge (shown as the full line).

section, while the prediction based on the Ceperley and Partridge potential lies within the uncertainty of the data throughout almost the entire angular range of the experiment. Calculations have also been carried out using the exponential potential of Foreman et al. (1974) extrapolated to larger r. The resulting cross sections exhibit a slightly weaker undulation than do the data, but are otherwise in excellent agreement with the experimental results. The experimental data are thus most consistent with the less steeply rising of the He, potentials. The range of the potential probed by the 0.5 keV data can be estimated in several ways. Calculations of a classical deflection function from the 0.5 keV phase shifts indicates that the experimental scattering angles correspond to impact parameters in the range 1.2-2.0 A. In addition, the 0.5 keV partialwave series essentially converges at 1 = 1000, which translates into an impact parameter of 2.04 A. Finally, empirical tests show that the cross section predictions are insensitive to the behavior of the potential for r > 2 A. Partial-wave calculations based on the Ceperley and Partridge potential have also been performed for collision energies of 1.5 keV and 5.0 keV. The agreement between experiment and theory is generally very good except at

I

DIFFERENTIAL SCATTERING

91

the largest angles in the 5.0 keV data where the observed cross section deviates from the single-channel elastic scattering calculation. The onset of this behavior at an energy-angle product Ed of 2 keV-deg is consistent with previous observations at lower energies and corresponds to the opening of inelastic channels at internuclear separations of approximately 0.5 8, (Morgenstern et al., 1973, Brenot et al., 1975, Guayacq, 1976). The behavior of the cross section below 0.2" is of particular interest since, whereas the classical differential cross section rises monotonically and diverges as 0 --* 0, the observed behavior exhibits an undulating structure superimposed on the classical cross section and a leveling-off which varies as exp[ - cd2] at small angles. This behavior has been predicted theoretically (Mason et al. 1964) and was observed in thermal energy alkali-mercury and alkali-rate gas collisions (Berry, 1969). The undulation is referred to as the forward diffraction peak and is understood as arising from interference over a broad range of impact parameters associated with weak deflections from the tail of the potential. This phenomenon is markedly distinct from rainbow and glory scattering, which are associated with a few particular impact parameters. When observed as a function of energy, a given undulation feature (the first minimum, for example) resembles optical diffraction from a disk, moving to smaller angles as the de Broglie wavelength decreases. Beier (1973) has utilized this analogy to relate the undulation characteristics to potential parameters in the case of a screened Coulomb interaction. Depending on the collision energy and the potentials involved, the diffraction peak can be characteristic of either the attractive or the repulsive part of the potential. Partial wave calculations indicate that the influence of the weak van der Waals attraction is negligible in the present experiment.

111. He'

+ He Collisions at Small Angles

Ion-neutral collisions have also been investigated using the apparatus depicted in Fig. 1. For these measurements, the CTC is evacuated and the primary ion beam passes directly to the target cell. Both charged and neutral collision products have been investigated. The neutral collision products are measured by deflecting away the ions emerging from the target cell while the ionic collision products are determined as follows. Two files are taken with gas in the target cell; one where all energetic particles emerging from the target cell are collected (ASToT),and one where only neutral charge transfer collision products are collected (AS,,). Two files are also taken without gas in the target cell; one of the primary ion beam (AS,*)and one of the background noise (AS,,). The AS,,, file includes four contributions: 1) attenuated

92

R . F . Stebbings

primary ion beam, 2) elastically scattered ions, 3) charge transfer neutrals, and 4)brackground noise. The ion scattering signal is therefore obtained as follows:

AsIs = (ASTOT

+ AS,,)

-(

Ah

+ ASCT).

(5)

A. RESULTS AND DISCUSSION

Raw data for both charge transfer and elastic scattering at 1.5 keV are shown in Fig. 3 which depicts the contents of the 90 x 90 array. The vertical displacement indicates the number of counts at a given location on the detector. I . Elastic scattering

Elastic scattering cross sections have been obtained for 1.5 keV He' projectiles elastically scattered from neutral He, over the laboratory angular range 0.04" - 1.0" and are shown in Fig. 4 together with theoretical cross sections obtained using the potentials found in Marchi and Smith (1965). The calculations are now complicated by the fact that for He+-He collisions the Hamiltonian is symmetric and two electronic states result when a ground state helium atom and ion are brought together adiabatically, one of which is symmetric (9) and the other antisymmetric (u) in the nuclei. Scattering occurs from each of these potentials and the differential cross section for elastic scattering is given by % )

dR

1 4

(elastic) = -)$(,fI

Charge Transfer

-

f,(n - 9)

+ f&S) + f,
Elastic Scattering

FIG.3. Raw data for He+-He scattering at 1.5 keV.

(6)

DIFFERENTIAL SCATTERING

93

h

c

v)

\

cu

E

0

U

8 (degrees) FIG.4. Elastic scattering in He+-He colisions at 1.5 keV. The experimental data are shown together with the results ofa partial wave calculation using the potentials of Marchi and Smith (1 965).

Once again, because the scattering is confined to such small angles, the cross section is well-represented by the expression

which accounts for the superposition of the scattering from the gerade and the ungerade potentials. The partial wave method therefore leads to two scattering amplitudes

94

R. F . Stebbings

where Sf."is the phase shift of the lfh partial wave. The agreement between experiment and theory is quite satisfactory with the calculations showing rather more pronounced oscillations than do the experiments. 2. Charge Transfer

Measurements of charge transfer have been obtained at several energies. A difficulty arises when placing these data on an absolute footing because the primary and product particles are no longer identical and attention must be given to the possibility that they may be detected with different efficiencies. Nagy et al. (197 1) have previously recognized this problem and offered an

0101

8

0 ,I (degrees)

1'0

FIG.5. Charge transfer in Het-He collisions. The solid lines are obtained using Eq. (9) together with the potentials of Marchi and Smith (1965).

DIFFERENTIAL SCATTERING

95

ingenious solution. They noted that whereas both the elastic-scattering cross section and charge transfer cross section, which is given in its approximate form by

individually exhibit oscillatory behavior because of the interference between the two scattering amplitudes, their sum is nevertheless free from such oscillations (because the interference terms cancel) and decreases monotonically with increasing angle at angles larger than the rainbow angle. It follows that, if the detection efficiencies of the ions and the neutral atoms are different, an oscillatory structure will be observed in the combined signal whose phase will be determined by whichever of the efficiencies is the greater. No such oscillatory behavior is discernible in the data, however; the detection efficiencies of helium atoms and ions are thus identical to within experimental uncertainty. Experimental cross sections derived on this basis are plotted in Fig. 5 and are seen to be in good agreement with the theoretical cross sections obtained from Eq. (9) using the Marchi and Smith (1965) He: potentials. Furthermore, integration of these differential cross sections yields partial total cross sections that are in excellent agreement with direct measurements of total cross sections. Such comparisons are valid, in that the differential cross sections are so strongly peaked in the forward direction that only a small contribution to the total cross section results from scattering outside the range of the measurements.

IV. He-He Scattering at Large Angles The experimental techniques described above may be used, with minor modification, to investigate scattering to larger angles. Indeed, measurements have been carried out in this way to scattering angles up to approximately 10" in the laboratory system. The differential cross sections typically fall off so abruptly with increasing angle, however, that at even larger angles alternative techniques must be adopted to achieve satisfactory signal to noise. Accordingly, the apparatus shown in Fig. 6 was developed for the study of scattering in the angular range 26" < 6 c 45". This apparatus differs from that used for the study of small angle scattering by the use of a target cell of length 450 pm and exit diameter 250pm and the employment of two PSDs which permit detection of both the scattered and the recoil particles. For a scattering event to be recorded, the projectile and the recoil particle must be detected in coincidence. The sequence of events is as follows: detection of a particle on

96

R . F . Stebbings TO COMPUTER

MAGNET I

PSDl

T I T O TIMING CIRCUIT

.. r 5

R 'I

ION SOURCE

5'' INCIDENT BEAM ?

AXIS

CIRCUIT

FIG.6. Schematic of the apparatus used for the study of large angle He-He scattering.

PSDl opens a gate enabling recognition of counts from PSD2 for 0.5 psec. If PSD2 detects a particle within this interval, both the time difference between the arrival of the two particles and the position coordinates of the particle impacts are recorded. After rejection of spurious events that fail to meet certain momentum conservation requirements, the remainder are stored in an array analogous to that described earlier. Corrections to the contents of the array elements are then made to account for the fact that 1) only a fraction of atoms scattered at a given angle strike the detector, 2) particles scattered at different angles have different energies and are therefore detected with different efficiencies, and 3) the effective collision volume varies slightly with the scattering angle. The measured relative differential cross sections for 4He-4He at 3 keV are plotted in Fig. 7. Pronounced oscillations are observed which are also present in the case of 3He-3He collisions. Such oscillations were previously observed by Dhuicq et al. (1973) and by Lorents and Aberth (1965) and are explained as follows. For 4He-4He collisions, the projectiles are indistinguishable bosons and the cross section is accordingly given by

At these large scattering anglesf(n - 9) is not negligible in comparison with f ( s )and the observed oscillations are thus a consequence of the interference

97

DIFFERENTIAL SCATTERING

125

L

CJ

v

cn 5 0 0

n

0

30

33

36 39 42 LAB ANGLE ( d e g )

45

FIG.7. Relative differential scattering cross sections for 4He-4He collisions at 3 keV. Experimental data are shown together with calculations based on Eq. (10).

between the two scattering amplitudes at supplementary CM scattering angles. For the case of 3He-3He collisions involving indistinguishable fermions, the situation is analogous but even further complicated by the fact that the particles now have nuclear spin and the cross section is given by

a(9) = (lf(@l2+ If(. 2

-

S)I2> lf(W - f ( . - $)I2 2 +

(1 1)

From Eq. (10) and Eq. (ll), the amplitudes clearly interfere in different ways for bosons and fermions. In the asymmetric case 4He-3He, there is no nuclear symmetry and the differential cross section is given by Eq. (3). The scattering amplitudes, and thus the cross sections, can be calculated if the interaction potential between the colliding atoms is known. For a 3 keV collision between 4He and 4He resulting in scattering at 45", the impact parameter is approximately 1.92 x lo-'' cm and the classical distance of closest approach is approximately 4.64 x 10- l o cm. Since the Bohr radius of the ground state He atom is approximately 2.65 x lo-' cm, it is reasonable to assume that the scattering is largely determined by the Coulomb interaction between the two He nuclei:

98

R . F. Stebbings

Cross sections calculated in this manner agree very well with the experimental data. For 3 keV collisions, the best agreement between experiment and theory is achieved when the effective nuclear charge is taken to be 2 and the results are shown as the solid line in Fig. 7. For 1.5 keV collisions, Zeff= 1.8 gives the best fit, suggesting that, at this lower energy, partial shielding of the nuclei occurs.

V. Conclusion Position sensitive detectors are seen to be an excellent tool for the investigation of absolute differential scattering cross sections for heavy particles at keV energies. Theoretical cross sections for He+-He and for He-He obtained using the method of partial waves are in very good agreement with the measurements when the Marchi and Smith (1965) potentials are used for He: and the Ceperley and Partridge (1986) potentials for He,. ACKNOWLEDGMENTS The work at Rice University described in this chapter was carried out with support from the National Science Foundation, the National Aeronautics and Space Administration, and the Robert A. Welch Foundation.

REFERENCES Amdur, I., and Jordan, J. E. (1966). In Advances in Chemical Physics Vol. X (John Ross, ed.) Interscience publishers, New York. Aziz, R. A., Nain, V. P. S., Carley, J. S., Taylor, W. L., and McConville, G. T.(1979). J. Chem. Phys. 70, 4330. Baudon, J., Barat, M., and Abignoli, M. (1968). J. Phys. B. 1, 1083. Beier, H. J. (1973). J. Phys. 8. 4 683. Berry, M. V. (1969). J. Phys. B. 2, 381. Brenot, J. C., Dhuicq, D., Gauyacq, J. P., Pommier, J., Sidis, V., Barat, M., and Pollack, E. (1975). Phys. Rev. A . 11, 1254. Ceperley, D. M., and Partridge, H. (1986). J. Chem. Phys. 84, 820. Dalgarno, A. (1979). In Advances in Afomic and Molecular Physics Volume 15 (D. Bates and B. Bederson, eds.), page 37. Dhuicq, D., Baudin, J., and Barat M. (1973). J. Phys. 8.6, L1. Fedorenko, N. V., Filippenko, L. G., and Flaks, N. P. (1960). Soviet Phys.-Techn. Phys. 5 4 5 . Feltgen, R., Kirst, H., Kohler, K. A., Pauly, H., and Torello, F. (1982). J. Chem. Phys. 76, 2360. Foreman, P. B., Rol, P. K., and Coffin, K. P. (1974). J. Chem. Phys. 61, 1658.

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Gao, R. S., Gibner. P. S., Newman, J. H., Smith, K. A,, and Stebbings, R. F. (1984). Rev. Sci. Insr. 55, 1756. Guayacq, J. P. (1976). J . Phys. B. 9, 2289. Jody, B. J., Saxena, S. C., Nain, V. P. S., and Aziz, R. A. (1977). Chem. Phys. 22, 53. Leonas, V. B., and Sermyagin, A. V. (1977). Khim. Vys. Energ. 11, 296. Lockwood, G. J., Helbig, H. F., and Everhart, E. (1963). Phys. Rev. A. 132, 2078. Lorents, D. C., and Aberth, W. (1965). Phys. Rev. A. 139, 1017. Marchi, R. P., and Smith, F. T. (1965). Phys. Rev. A . 139, 1025. Mason, E. A., and Vanderslice, J. T. (1962). In Atomic and Moiecutar Processes (D. R. Bates, ed.) Academic Press, New York, New York. Mason, E. A,, Vdnderslice, J. T., and Raw, C. J. G . (1964). J. Chem. Phys. 40, 2153. Massey, H. S. W., and Smith, R. A. (1930). Proc. Roy. Soc. (London) A . 126, 259. Morgenstern, R., Barat, M., and Lorents, D. C. (1973). J. Phys. B. 6, L330. Nagy. S. W., Fernandez, S. M., and Pollack, E. (1971). Phys. Rev. A . 3,280. Newman, J. H.. Smith, K. A., Stebbings, R. F., and Chen, Y. S. (1985). J . Geophys. Res. 90, 11045. Nitz, D. E., Gao, R . S., Johnson, L. K., Smith, K. A,, and Stebbings, R. F. (1987). Phys. Reo. A . 35, 4541.

Schafer, D. A. (1987). Private communication. Smith, F. T., Fleischmann, H. H., and Young, R. A. (1970). Phys. Rev. A . 2, 379. Wijnaendts van Resandt, R. W., and Los, J. (1979). Proc XI 1.C.P.E.A.C.