Positron—Gas Scattering Experiments

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS,VOL. 18

POSITRON-GAS SCATTERING EXPERIMENTS TALBERT S . STEIN and WALTER E. KAUPPILA Department of Physics and Astronomy Wuyrie State University Detroit, Michigan

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Positron-Beam Production . . . . . . . . . . . . . . . . . . . B. Total Cross-Section Experiments . . . . . . . . . . . . . . . . 111. Total Cross-Section Results . . . . . . . . . . . . . . . . . . . . A. Inert Gases at Low Energies . . . . . . . . . . . . . . . . . . B. Inert Gases at Intermediate Energies . . . . . . . . . . . . . . I Introduction

11 Experimental Techniques for Total Cross-Section Measurements

C. Positron and Electron Comparisons for the Inert Gases . . . . . . D. Tests of the Sum Rule . . . . . . . . . . . . . . . . . . . . . E. Molecular Gases . . . . . . . . . . . . . . . . . . . . . . . IV. Differential Scattering Cross Sections . . . . . . . . . . . . . . . . V. Inelastic Scattering Investigations . . . . . . . . . . . . . . . . . A. Positronium Formation Cross Sections . . . . . . . . . . . . . B. Excitation and Ionization Cross Sections . . . . . . . . . . . . VI . Resonance Searches . . . . . . . . . . . . . . . . . . . . . . . VII. Possible Future Directions for Positron Scattering Experiments . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 55 55

60 64 64 76 79 80 82 84 86 86 89 91 92 93

I. Introduction Although the first electron-atom total cross section (QT) measurements were reported in the early 1920s (Ramsauer, 1921a), and the positron has been known to exist since the early 1930s (Anderson, 1933),there were no direct measurements of positron-atom total cross sections until the early 1970s (Costello et a f . , 1972a). One of the main obstacles to performing such measurements earlier was the difficulty of producing a sufficiently intense e+ beam of well-defined low energy. Positron-atom scattering experiments are of interest because they involve interactions of antimatter with matter, and also because they can help provide a better understanding of the scattering of electrons by atoms and molecules, a subject of great importance to many different fields of 53 Copyright @ 1982 by Academic Press, Lnc. All rights of reproduction in any form reserved. ISBN 0-12-003818-8

54

Talberr S. Stein and Walter E. Kauppila

science and technology such as plasma physics, laser development, gaseous electronics, astrophysics, and studies of the earth’s upper atmosphere. During the past decade, additional interest in positron-atom (molecule) collisions has been stimulated by the discoveries of 0.51 MeV e + annihilation gamma rays coming from solar flares (Chupper d., 1973) and from the direction of the center of our galaxy (Leventhal et nl., 1978). Such annihilation gamma rays can provide considerable information on the type of environment which exists at the site of their origin if sufficient information can be obtained on the ways in which positrons interact with H, H2, and other atoms and molecules of astrophysical interest (Crannell et d., 1976; Bussard et d.. 1979). Comparisons between e+-atom (molecule) and e--atom (molecule) scattering reveal some interesting differences and similarities. The static interaction (associated with the Coulomb field of the undistorted atom) is attractive for electrons and repulsive for positrons, while the polarization interaction (resulting from the distortion of the atom by the passing charged projectile) is attractive for both projectiles. The exchange interaction contributes to e- scattering (due to the indistinguishability of the projectile and electrons in the target atoms) but does not play a role in e+ scattering. The combined effect of the static and polarization interactions is that they add to each other in e- scattering, whereas there is a tendency toward cancellation in e + scattering. This results in smaller total scattering cross sections, in general, for positrons than for electrons at low energies. As the projectile energy is increased, the polarization and exchange interactions eventually become negligible compared with the static interaction (which has the same magnitude for positrons and electrons). This results in a merging of the corresponding e+ and e- scattering cross sections at sufficiently high projectile energies. Two scattering processes which occur for positrons (but not for electrons) are annihilation and positronium (Ps) formation (real and virtual). Annihilation is not expected (Massey, 1976) to be a significant effect at the energies which have been used in e + scattering experiments (>0.2 eV). On the other hand, Ps formation has been found to be an important factor in e+-gas collision studies. A compact survey of the e+-gas scattering measurements reported during the first decade of activity in this area is provided in Tables IA and IB. Total scattering cross sections have been measured for the inert gases and a variety of molecules by several different groups. However, Table IA indicates that experimental areas beyond QT measurements, such as differential cross sections, elastic, excitation, ionization, and Ps-formation cross sections, and searches for resonances (temporary bound states) are in very early stages of exploration. Positron scattering experiments have been discussed in two recent review articles (Griffith and Heyland, 1978;

POSITRON-GAS SCATTERING EXPERIMENTS

55

Griffith, 1979) and in recent progress reports (Kauppila and Stein, 1982; Stein and Kauppila, 1982),but the field is evolving sufficiently rapidly that there are several different experiments which have been reported since the review articles were written. Our main goals in writing this article are to ( I ) point out what we feel are the most significant developments in the first decade of e+-gas scattering experiments, (2) search for some consistent patterns in the experimental results in cases where several different groups have investigated the same collision processes, (3) present some puzzling questions raised by the new generation of experiments which go beyond QT measurements, and (4) indicate some experimental areas of e+-gas scattering which we feel would be interesting and feasible to investigate in the near future.

11. Experimental Techniques for Total Cross-Section Measurements A. POSITRON-BEAM PRODUCTION

The difficulty and expense of producing intense, monochromatic e+ beams stands in sharp contrast to the relative ease with which one can produce such beams of electrons. Table I1 summarizes the characteristics of the e+ beams used by various groups that have reported QT measurements. The experiment of Costello et d.(1972a) utilized a %-MeV electron linear accelerator to produce positrons by pair production (PP). All of the other laboratories listed in Table I1 use commercially available 22Naas e+ sources, except for the Detroit group, which uses the proton beam of a 4.75-MeV Van de G r a d accelerator to produce an llC e+ source by the reaction “B(p, n)”C (Stein et d.,1974). A variety of moderators used in either a backscattering ( B ) or transmission ( T ) mode, have been found to yield low-energy positrons with relatively narrow energy distributions when exposed to the high-energy, broad energy-width fluxes resulting from p+ decay. Two properties of moderators which are of particular interest in e+ scattering experiments are ( I ) the energy width of the emitted slow e+ energy distribution, AEFWH”(“full width at half-maximum”), and (2) the “conversion efficiency” ( 6 ) defined as the ratio of the slow e+ emission rate of the moderator to the total rate of positron production by the radioactive source. For the moderators which have been used in e+-gas scattering experiments, E ranges from less than lo-’, in the case of the Au-plated mica transmission moderator used by Jaduszliwer et al. (1972) to lop5 in the

TABLE IA A

Gas

SURVEY OF THE

Energy range (eV) ~

He

0.3-1000

Ne

0.25-1000

Ar

0.4-1000

Kr

REPORTED POSITRON-GAS SCATTERING MEASUREMENTS

Gas

Groupsa

~

~

Groups"

~

G I , T I , LI, L2, T2, T4, L3, T5,SI, L7, S2, Al, D3, BI, A3, L9, B2, D5 L2, L3, T3, T6, L7, D3, S3, A3, L9, B2, D5 L1, L2, L3, T3, T6, DI, L7, S3, L9, A4, B2, D5 L2, L3, L7, D4, B2, D6

0.35-960

Energy range (eV)

Xe H2 D2

Nz 0,

co

COP CHI

0.35-800 1-600 4-400 0.5-1000 2-600 2-400 0.5-600 19-600

L3, D4, A4, 8 2 , D6 L4, D2, L8, LIO, D8 L4 L4, D2, LIO, S4, D8 L6, L10 L4 L6, D2, L10, D8 L10

I

Differentid cross sections

Gas

Energy range (eV)

Range of angles

Group

Ar

2.2-8.7

20-60"

A2

Gas

Energy range (eV)

He Ar

14-34 7- I8

Energy range (eV)

Gas

4.5-20 4-18

Gas

Energy range (eV)

Group

Total inelastic cross sections

50-170

He

L5

Partictl excitation plus ionization cross sections

23-50 20-40 15-40

He Ne Ar

A5, A6 A6 A6

Resonctnce searches ~~

~~

Gas

Energy ranges (eV)

Ar

8.5-9.5, 11.0-12.0 ~~~

~~

Energy increments (eV)

Beam-energy width (eV)

Group

0.025


D7

~~

Groups are listed in chronological order. See Table IB for an explanation of researchgroup codes and reference codes.

POSITRON-GAS SCATTERING EXPERIMENTS

57

TABLE IB KEYT O RESEARCH GROUPS A N D REFERENCES LISTED I N T A B L E1A Code

Research group

A

University of Texas at Arlington

B

University of Bielefeld Wayne State University (Detroit)

D

G

L

Gulf Energy and Environment, Inc. University College London

Reference code A1 A2 A3 A4 A5 A6

B1

B2 DI D2 D3 D4 D5 D6 D7 D8 GI

LI L2 L3 L4

L5

S

T

University College of Swansea University of Toronto

L6 L7 L8 L9 L10 LI 1 S1

s2

s3 s4 TI T2 T3 T4 T5 T6

Reference Burciaga et [ I / . (1977) Coleman and McNutt (1979) Coleman et cil. (1979) Coleman et n / . (1980a) Coleman and Hutton (1980) Coleman er ril. (1981) Wilson (1978) Sinapius el id. (1980) Kauppila et a / . (1976a) Kauppila et cil. ( I977a) Stein et a / . (1978) Dababneh el c d . (1980) Kauppila et ril. (1981a) Kauppila et ril. ( I98 1b) Stein ef [ I / . (1981) Hoffman er ril. (1982) CosteUo et cil. (1972a)

Canter et etl. (1972) Canter et ril. (1973) Canter et N I . (1974a) Coleman er rr/. (1974) Coleman et a / . (l975b) Coleman et a/. ( 1 9 7 5 ~ ) Coleman er e l / . (1976a) Griffith and Heyland (1978) Griffith er ( I / . (1979a) Charlton et ril. (1980b) Charlton er ril. (1980~) Dutton et cil. (1975) Brenton et ril. (1977) Brenton er 01. (1978) Dutton et c r / . (1981) Jaduszliwer et a / . (1972) Jaduszliwer and Paul (1973) Jaduszliwer and Paul (1974a) Jaduszliwer and Paul (1974b) Jaduszliwer et a / . (1975) Tsai et a/. (1976)

TABLE I1 POSITRON-BEAM CHARACTERISTICS FOR TOTALCROSS-SECTION EXPERIMENTS Laboratory"

Source

Moderator*

Gulf London Toronto Swansea Detroit Texas Bielefeld

PP 22Na '%a 22Na "C 22Na 22Na

T: AulAI B: MgO/Au T Admica' B: MgOIAu T Boron B: MgOIss B: Cu(0FHC)

(I

Energy analysis TOF TOF !WE

4

ER

TOF TOF

AEFWHM

(eV) 1-2 -1

-1

0.14E co.1

1.5 0.4 at 6 eV

References can be found in Table IB. T refers t o transmission, B t o backscattering, Au/Al to gold over aluminum, etc. Other moderators used include T: Ni, B: MgO/brass, B: MgO/Au.

Energy range (eV) 1-26 2-1000

4-300

13-1000 0.3-800 2-50 1-6

Scattering region

I0

Detector

(no./ sec)

B ,I

2Y CEM 2Y 2Y CEM

-FF

CEM

Few 100 Few
Ell

4

II

4

4,

y.

Y

POSITRON-GAS SCATTERING EXPERIMENTS

59

case of the MgO-coated Au “venetian blind” backscattering moderator was used by Canter et al. (1972). A conversion efficiency of 3 x reported for a MgO-coated Au backscattering moderator by Canter et al. (1974b) in an experiment which detected efficient Ps formation on various solid surfaces. Although the MgO-coated Au moderator has a relatively high conversion efficiency, it has the disadvantage of emitting a rather broad e+ energy distribution (AEFWHM 2= 1-2 eV) possibly due to the charging of the insulating powder grains. The existence of a narrow, low-energy peak in the e+ energy spectrum emitted by a successful moderator was attributed in the early stages of moderator development (Costello et al., 1972b; Tong, 1972) to the thermalization of high-energy positrons in the moderator, and their subsequent ejection from the surface due to a “negative” work function for positrons in certain metals. However, typical energy widths of the lowenergy e+ peaks observed for the first few successful moderators were relatively broad (-1 eV), until Stein et al. (1975) observed a very narrow energy width (AEFWHM < 0.1 eV), low-energy peak in the energy spectrum of positrons emitted from their boron moderator which did, in fact, appear to be consistent with a thermal energy distribution at room temperature. Throughout the development of the moderators listed in Table 11, investigators have recognized that the conversion efficiencies of various moderators were sensitive to their surface conditions, but none of the experiments referred to in Table I1 was done with well-characterized surfaces. In the first experiment using atomically clean (submonolayer contaminated surfaces) in an ultrahigh vacuum (-1O-lO Torr), Mills et al. (1978) demonstrated that positrons implanted in a clean single-crystal target can diffuse back to the surface and be emitted as Ps or as free positrons. The positrons are emitted predominantly in a forward lobe (Murray and Mills, 1980), with a maximum energy interpreted as the positron negative work function of the surface (Murray et al., 1980). These observations by Mills and co-workers have led to the fabrication of new e+ moderators with very high conversion efficiencies and narrow energy widths. Mills (1980) has found that a clean single-crystal Cu( 11 1) moderator which has been exposed to H2S in situ, and cooled to 100 K, combined with a low self~ source has an efficiency of 1.5 x the highest absorption 5 s Ce+ reported efficiency for an e+ moderator. The narrowest energy width ( < O . 1 eV) achieved by Mills et al. (1978) with their single-crystal Al( 100) moderator is about the same as that achieved by Stein et al. (1975) with their boron-moderated “C source. The single-crystal moderators referred to above may be somewhat difficult to adapt to e+-gas scattering experiments, since the conversion efficiencies of such moderators may be affected by their exposure to thc

60

Talbert S . Stein and Walter E. Kauppila

various target gases admitted to the scattering region. There have been some recent advances in moderator technology in moderate vacuums (-lo-’ Torr) which could also be useful in future e+-gas scattering experiments. As an example, Dale et a / . (1980) find that a tungsten vane moderator (prepared by heating to 2200°C in a vacuum) is stable in air, and if used with a better geometry (a larger solid angle for stopping fast positrons), the estimated conversion efficiency would be 0.7 x However, its energy width is relatively broad (1-2 eV). As a result of the relatively low intensities of the e + beams used in scattering experiments, most groups use a rather long scattering region and an axial magnetic field (sometimes curved) to transport the e+ beam through that region. The laboratories (listed in Table 11) which do not use an axial magnetic field in the scattering region are Swansea, where the positrons move along a circular path in a transverse magnetic field (a Ramsauer type of system), and Bielefeld, where the positrons move in a field-free (FF) scattering region (except for a single axial magnetic focusing lens). For energy-analysis of the e+ beams, the Gulf, London, Texas, and Bielefeld groups use time of flight (TOF), Toronto uses a 90” electrostatic analyzer ( W E ) , Swansea uses a transverse magnetic field with beamdefining apertures (BJ, and Detroit uses a retarding electrostatic field (I&). In all cases except for Toronto, the method of energy analysis also provides some discrimination against the detection of scattered positrons. The e+ energies used in Q T measurements range from 0.3 eV at Detroit to of the slow 1000 eV at Swansea and London. The energy widths, AEFWHM, e + beams used in scattering experiments are typically about 1 eV or more, except for the Detroit energy width of less than 0.1 eV. Three methods used for detecting positrons are to ( I ) observe the two coincident annihilation gamma rays (2y) with two NaI scintillation counters, (2) observe one or both annihilation gamma rays (y) with a single NaI well counter, and (3) use a Channeltron electron multiplier (CEM). The detected primary beam currents (Z,) range from less than l/sec to more than 1OO/sec.

B. TOTALCROSS-SECTION EXPERIMENTS The basic experimental method used by all of the groups which have measured QT is to study the attenuation of the e+ beam as it passes through a gas scattering region. Under “ideal” experimental conditions, QT can be obtained from the expression

I =

~ o ~ ~ - ’ ~ L Q ~

(1)

61

POSITRON-GAS SCATTERING EXPERIMENTS

where lo is the detected beam intensity with no gas in the scattering region,l is the detected beam intensity with gas of number densityn in the scattering region, and L is the path length of the e+ beam through the scattering region. The Bielefeld and Detroit systems (shown in Figs. 1 and 2, respectively) represent two significantly different experimental approaches to measuring QT. However, they share a feature that distinguishes them from the other experiments listed in Table 11; namely, in each case, absolute total cross sections for positrons and electrons have been measured in the same apparatus, using the same technique. In the Bielefeld experiment (Sinapius et a / . , 1980) the fast positrons from a 22Na source strike an OFHC Cu tube moderator used in a backscattering mode, yielding slow positrons and secondary electrons, which are used for the respective QT measurements. When a fast positron passes through a thin scintillator foil on its way to the Cu moderator, light is emitted which, when detected by a photomultiplier tube (PMT), provides one of the timing marks needed for the TOF measurement. A second timing mark is provided by the detection of the projectiles (with a CEM) which have traveled through the straight, approximately field-free scattering region. The use of a weak magnetic lens helps to focus divergent projectiles through the exit aperture without increasing their path length in the scattering region by more than 3%. The pressure in the scattering region is measured by an ionization gauge. Sinapius et 01. (1980) have recognized the need to apply an average 18% downward correction to the e+-He measurements of Wilson (1978) due to scattering in the vicinities of the moderator and the accelerating exit lens. In the Detroit experiments (Kauppila et al., 1977b), the electrons for the e- QT measurements are produced by a type-B Philips cathode, which

,Scattering reglon

Extraction grldr 1 IDeflectlon Dhtes

-

I

\ Moderator

Source

/

/

Magnetic rhleld

Exlt

Ion gauge -c

I

lenr

FIG. I . Bielefeld experimental setup for measuring total scattering cross sections. (From Wilson, 1978.)

62

Talbert S . Stein and Walter E. Kauppila

Magnetic Shielding

limating Apertures

Retarding Element Channeltron

Detector

FIG. 2. Detroit experimental setup for measuring total scattenng cross sections. (From Kauppila el crl , 1976a.)

replaces the boron target used to generate the positrons for the e+ QT measurements. The gas number density in the scattering region is determined from pressure and temperature measurements using a capacitance manometer (MKS Baratron) and thermocouples, respectively. The increase in the effective path length of projectiles due to spiraling in the axial magnetic field of the scattering region has been estimated to be a maximum of 1% (Kauppila et nl., 1977b). The Detroit group has also checked for the possibility that the target gases being studied could directly affect e+ or e- emission from the respective sources and have found no measurable direct effects of this type with the gases for which Detroit has reported QT values.

POSITRON-GAS SCATTERING EXPERIMENTS

63

In total cross-section experiments, care must be taken to minimize the number of scattered positrons which reach the detector, since inadequate discrimination against scattered positrons during the measurement of I [in Eq. ( l ) ] will result in measured cross sections being too low. In TOF experiments, some discrimination against scattered positrons is provided by their longer flight times compared with unscattered positrons. The angular discrimination achieved by the TOF method depends on the geometry of the experiment (e.g., the length of the scattering region and the distance between the scattering region and the detector), the beam energy and energy width, and the timing resolution of the electronics. Griffith et ul. (1978) in a reassessment of the angular discrimination capabilities of the original London total scattering system, have determined that an additional source of angular discrimination arises from the fact that appreciably stronger axial magnetic fields exist in their positron-detection region than in the major portion of their scattering region. This arrangement of magnetic fields results in a “magnetic reflection” of positrons scattered at sufficiently.large angles in the forward direction. Griffith et al. (1978) estimate that at 2 eV, the experiment of Canter et al. (1972) was able to discriminate against all positrons elastically scattered at angles greater than lo”, rather than the estimate by Canter et al. (1973) of 40-55”. We suspect that the recent reassessment by Griffith et al. (1978) is incomplete due to their neglect of the effect of positrons which scatter in their 150mm-long detector magnetic field region (refer to Fig. 1 of Coleman et al., 1973), which accounts for more than 15% of the total length of their scattering region. Since positrons scattered in the detector region are very close to the detector, the ability of their TOF approach to discriminate against such scattered positrons would be significantly worse, and could appreciably degrade their overall angular discrimination. The Texas group (Coleman et d.,1980a) has estimated an upper limit of the angular discrimination of their TOF system to be about 20”. In non-TOF experiments, there are several different techniques which have been used to discriminate against scattered positrons. One method is to use a retarding electrostatic field between the scattering region and the detector which serves as a potential “hill” for the axial energy of the positrons. If the retarding potential is set sufficiently close to the actual positron beam energy, it is possible to have 100% discrimination against inelastic scattering for many target atoms (e.g., the inert gases) and partial discrimination against small-angle elastic scattering, depending on the retarding field and the e+ beam energy distribution. Another method of discriminating against scattered positrons is to use a well-collimated e+ beam that must pass through a small aperture between the scattering region and the detector. The geometry of the beam and exit aperture will

64

Talbert S . Stein and Walter E . Kauppila

then provide discrimination against small-angle scattering. The Toronto group (Jaduszliwer and Paul, 1973) relied on the use of collimators in their beam scattering region, variations in their axial magnetic field strength, and Monte Carlo calculations to compensate for small-angle elastic scattering, and the use of a retarding field to discriminate against inelastically scattered positrons. The group at Swansea (Brenton et NI., 1977) used a retarding field and beam-collimating apertures for discrimination in their Ramsauer-type experiment. Although the Bielefeld group (Sinapius ef d., 1980) used TOF for analyzing their beam energy, they employed the geometrical approach (i.e., a collimated beam with a small exit aperture) to obtain an angular discrimination estimated to be 7". The ability of the Detroit group to discriminate against scattered positrons by using a retarding field and well-defined e+ beam with a small exit aperture from the scattering region has recently been analyzed by Kauppila et ( I / . (1981a). Since their retarding potentials are generally within a few tenths of a volt of their e + beam energy for energies less than 100 eV and within a few volts for energies up to 800 eV, the Detroit group should have 100% discrimination against inelastic scattering for the inert gases where the minimum energy lost by inelastically scattered positrons is more than 5 eV. For e+energies from 1 to 20 eV, Dababneher 01. (1980) have estimated angular discriminations of 15-20" for their Kr and Xe measurements, while for energies above 100 eV, Kauppilaer al. (1981a) estimate angular discriminations of less than lo". The Detroit group (Kauppilaet a/., 1981a) obtains better angular discriminations (typically 5" for all energies) for their recent e- measurements than for their e+ measurements at the same energy in the same apparatus because the more intense (by a factor of 10 or more), and nondecaying e- beams permit better overall tuning conditions to be achieved. However, the earlier Detroit low-energy e- measurements (for He and Ar by Kauppila et d . , 1976a, 1977b) were made with tuning conditions similar to those used for the low-energy positron work of Dababneh et al. (1980) and should have similar angular discriminations.

111. Total Cross-Section Results A. INERT GASESAT Low ENERGIES

The QT measurements for e+-He collisions are shown in Fig. 3, along with the results of several theoretical calculations. All of these measurements are absolute except the normalized ( n ) values of Sinapius el d.

65

POSITRON-GAS SCATTERING EXPERIMENTS

o ~ ~ ' " 5 ' ' " ' ' 10" ' " ' ' 15" "

20

Positron Energy (eV)

F I G .3. Low-energy e+-He total cross-section measurements (symbols) and theoretical calculations (curves). The thresholds for inelastic processes are indicated by the labeled arrows: "n" refers to normalized results; only the name of the lead author for each set of reported results is shown in the figure [e.g., Coleman (1979) refers to Coleman t'i NI. (1979).1 (Unless otherwise specified, the same conventions will be used in all of the following figures.) (From Stein and Kauppila, 1982.)

(1980). The values reported by Sinapius et al., are the results of Wilson (1978) corrected for scattering in the moderator and accelerator regions of their apparatus, which lowered Wilson's results by an average of 18%. Most of the measurements lie in a fairly narrow band in Fig. 3. One of the interesting qualitative features is the observation of a RamsauerTownsend effect (a minimum in QT) by Stein et al. (1978) and Sinapius et d.(1980) near 2 eV, with a steeply rising cross section at the lowest energies. All of the theoretical results in Fig. 3 indicate the existence of a cross-section minimum. Cross-section minima such as this were first observed by Ramsauer (1921b, 1923), Townsend and Bailey (1922), and Ramsauer and Kollath (1929) for low-energy e--Ar, Kr, and Xe collisions. These minima were so deep that it appeared that the target gases were nearly transparent to the projectile electrons. The RamsauerTownsend minima arise from quantum mechanical effects associated with a net attractive interaction between the projectile and the target atoms. Another interesting feature of most of the experimental data in Fig. 3 is the noticeable increase in Q T as the energy is increased above the Ps formation threshold, which shows up most clearly in the narrow energywidth measurements of Stein ef a/. (1978). The e+ experimental QTresults for Ne, Ar, Kr, and Xe are displayed in

66

Talbert S . Stein and Walter E. Kauppila

Positron Energy (eV) F I G . 4. Low-energy e+-Ne total cross section results. UN refers to the (2p-d) unnormalized results and N refers to the (2s-p) + (2p-d) normalized results of Montgomery and LaBahn (1970). (From Stein and Kauppila, 1982.)

Figs. 4-7, along with the results of several theories. In a qualitative sense, the shapes of the theoretical curves at low energies are quite similar to some of the experimental results. Stein et al. (1978) observe a rather deep Ramsauer-Townsend effect in Ne near 0.6 eV. Kauppila et al. (1976a) observe a shallow cross-section minimum in Ar near 2 eV. There are

McEachran (1979)

_--- Schrader (1979) _- Montgomery (1970) -__Massey (1966) Positron Energy (eV)

FIG.5 . Low-energy e+-Ar total cross-section results. (From Stein and Kauppila, 1982.)

POSITRON-GAS SCA'ITERING EXPERIMENTS

67

. . V

0

Dobobnrh (1980) Sinapiu, (1980)n Canter (1973) Mc Eochron (1980) Schrader (1979) Mmey (1966)

-_0

"

.

"

*

"

"

.

.

"

t

10 20 Positron Energy (eV)

30

FIG.6. Low-energy e+-Kr total cross-section results. (FromStein and Kauppila, 1982.)

dramatic increases in the measured QT values above the Ps formation thresholds for all the inert gases, illustrated most clearly in the narrow energy-width measurements made in Detroit (Kauppilaer ul., 1976a; Stein et d.,1978; Dababneh et ul., 1980). The theoretical cross section curves which extend above the Ps formation thresholds in Figs. 3-7 do not account for inelastic scattering, and thus are not expected to show the

Dobabneh (1980) (1980)n o Coleman (198Oa) v Canter (19740) McEochran (1980) ---- Schrodrr (1979)

+ Sinopiur

-

0

0 0

"

.

.

t

"

.

10

.

"

'

*

"

20

Positron Energy (eV)

30

FIG.7. Low-energy e+-Xe total cross-section results. (From Stein and Kauppila, 1982.)

68

Talbert S . Stein and Walter E. Kauppila

abrupt increases near the Ps formation thresholds which show up in the experimental results. The qualitative features of the low-energy e+ QT curves for the inert gases are summarized in Fig. 8 with the measurements of Stein rt (11. (1978) for He and Ne, Kauppilaet al. (1976a) for Ar, and Dababneh rt d. (1980) for Kr and Xe. For comparison, the corresponding e- Q.,.curves are also provided in Fig. 8, where the results of Ramsauer (1921b, 1923) and Ramsauer and Kollath (1929) for Ne, Ar, Kr, and Xe, and Milloy and Crompton (1977) for He, are used for energies below a few electron volts, and the results of Kauppilart crl. (1976a, 1977b) for Ar and He, Stein et a/. (1978) for Ne, and Dababneh et al. (1980) for Kr and Xe are displayed at the higher energies. It is interesting to note that the situation regarding the existence of Ramsauer-Townsend effects in the inert gases is nearly reversed for positrons compared with electrons in the sense that positrons exhibit Ramsauer-Townsend minima only for the lighter inert gases (He, Ne, and possibly a shallow minimum for Ar), whereas electrons exhibit Ramsauer-Townsend minima only for the heavier inert gases (Ar, Kr, and Xe). Another interesting observation that is apparent when comparing the general nature of the e + and e- curves shown in Fig. 8 is the dramatic

OO

5

I0

15

20

Porllron Energy (eV1

25

10

20

3

Electron Energy (eV1

FIG.8. Total cross-section curves for low-energy e+-inert gas and e--inert gas scattering. The arrows (in the order of increasing energy) refer to the thresholds for positronium formation, atomic excitation and ionization for e + scattering, and atomic excitation and ionization f o r e - scattering. (From Stein and Kauppila, 1982.)

POSITRON-GAS SCATTERING EXPERIMENTS

69

increase in Q T at each of the lowest energy inelastic thresholds (due to Ps formation) for e+ scattering and the lack of any noticeable changes in QTat the lowest inelastic thresholds (due to atomic excitation) for electrons. The e+-He curve also indicates a noticeable increase in the slope at the threshold energy for atomic excitation by e+ impact which was first pointed out by Coleman et al. (1975a) in an analysis of the data of Canter et al. (1973). When making these comparisons, it is important to realize that narrow resonances (not shown in Fig. 8) exist near several of the inelastic thresholds for e- scattering, while none have yet been observed for e+ scattering (see Section VI). The e + curves in Fig. 8 can be used to obtain estimates of the Ps formation cross sections (Qps) (crosshatched regions) for e+ energies between the thresholds for Ps formation and atomic excitation, assuming that the elastic scattering cross sections are smoothly varying as the e+ energy increases through the Ps formation thresholds. (For a further discussion of QB refer to Section V,A.) It is not a simple matter to ascertain which e+ QT measurements are the most reliable. Two recent review articles (Griffith and Heyland, 1978; Griffith, 1979) discuss errors in several of the experiments represented in Figs. 3-7. Although experimental groups will often make estimates of potential systematic errors in their experiments, the estimated magnitudes of such errors may be incorrect, and there may be systematic errors which are overlooked that are as large or larger than those which are painstakingly considered. An approach initiated by the Detroit group (Kauppila et ul., 1976a) is to make the corresponding measurements for each gas with electrons and positrons using the same experimental approach and system. An advantage of this approach is that most of the potential systematic errors should equally affect the e- and e + measurements. The Detroit e--He results (Kauppila et al., 1977b) are within a few percentage points of several other sets of absolute measurements (Milloy and Crompton, 1977; Kennerly and Bonham, 1978; Blaauw et al., 1980) and theoretical calculations (Callaway et al., 1968; Yau et al., 1978; Nesbet, 1979; Fon et al., 1981). The Detroit e- measurements for the other inert gases (Kauppila et d.,1976a; Stein et al., 1978; Dababneh et al., 1980) have been compared with other e- measurements and appear to be reliable. The absolute e--inert gas QT measurements of Sinapius et al. (1980) and some normalized e--He and e--Ar Q T measurements of Charlton et al. (1980a) are also in quite good agreement with the results discussed above. It is informative to examine the Q T results for low-energy e+-inert gas collisions, keeping in mind the e--inert gas results referred to above. For the e+-He case, Fig. 3 indicates that there are several calculations that agree quite well with the experiments in the vicinity of the RamsauerTownsend minimum. The variational calculation of Campeanu and Hum-

70

Talbert S.Stein and Walter E . Kauppila

berston ( 1977) and an exchange adiabatic approximation calculation of Massey et a/. (1966) remain in quite good agreement with several of the experiments u p to the highest energies of overlap. Since the calculation of Campeanu and Humberston is likely to be the most elaborate of all the above calculations, it is of interest to consider comparisons between this theory and the various experiments. Considerable attention (Campeanu and Humberston, 1977; Humberston, 1978; Humberston and Campeanu, 1980) has been devoted to trying to explain the discrepancy between their calculations and the measurements made in Detroit (Stein et a/., 1978; Kauppila et d.,1976b) and Texas (Burciaga et a/., 1977; Coleman et ul., 1979) for energies below 6 eV, where the experiments are lower. Humberston (1978) contends that the Detroit and Texas results are both too low below 6 eV due to the neglect of positrons scattered through small angles and that the measurements of Canter et d.(1973), which are claimed to agree with Campeanu and Humberston (1977) below 6 eV, are more accurate in the vicinity of 2 eV. Humberston (1978) deduces that at 2 eV, the discrepancy between the results of Campeanu and Humberston (1977) and those of Stein et a / . (1978) could be explained by an angular discrimination of 12", while an angular discrimination of 20" could explain the discrepancy with the results of Burciaga et a / . (1977). In order to determine if differing angular discriminations could provide a consistent explanation for the discrepancies between the results of Campeanu and Humberston (1977) and the various experimental results, we have calculated the percent error introduced into the total elastic for e+-He and e--He collisions due to variscattering cross section ous values of angular discrimination at various energies. The results for positrons, summarized in Fig. 9, were obtained using the s-wave phase shifts of Campeanu and Humberston (1977), the p-wave phase shifts of Humberston and Campeanu (1980), the lowest set of d-wave phase shifts of Drachman (1966), and the higher phase shifts (up to 1 = 20) from the formula of O'Malley e f a/. (1961):

(eel)

61

=

~k'P/((21+ 3)(21 + 1)(21 - 1))ao

(2)

whereP is the static dipole polarizability (1.3830:) of the target atom (He) and k is the wavenumber of the projectile. The corresponding percent errors for e--He collisions, also shown in Fig. 9, were obtained using the s-, p-, and d-wave phase shifts of Callaway et a / . (1968) (from their "EP" calculation) and the higher phase shifts (up to 1 = 20) from the formula of O'Malley et d. (1961) given above. Figure 9 clearly illustrates that the percent error in Qe,(and therefore in QT)for a particular angular discrimination reaches a maximum for e+-He collisions in the region of the Ramsauer-Townsend minimum (about 2 eV). An experiment with a large

POSITRON-GAS SCATTERING EXPERIMENTS

71

40 s wove IHIB-Cornpconu (1977)

Q .-

g 30

c

-

p wove lH5)-Hurnberston (1980) d wow-Drachman 11966) (3-201- OMolley (1961)

-

).

t,p,d wove Colloway (1968) I:(3--2O)-O'Malley (1961)

5

0 c .L

2

20

-

fi

8

10

value for the angular discrimination could thus be expected to show a deeper Ramsauer-Townsend effect than actually exists. The angular discrimination in the experiment of Stein et al. (1978) was initially estimated to be about 13" at 2 eV. Figure 9 indicates that an angular discriminatioii of about 17"could explain the difference between the e+-He results of Stein et NI. (1978) and the theoretical results of Campeanu and Humberston (1977) near 2 eV shown in Fig. 3. The more recent and more complete analysis of the angular discrimination for low-energy positrons by Dababneh ef a / . (1980) (using the same apparatus as that used by Stein et a/., 1978), which gave an angular discrimination of 15-20' for e+ energies between 1 and 20 eV, is consistent with the 17" value required to explain the difference between the measurements of Stein et al. (1978) and the values of Campeanu and Humberston (1977) in the vicinity of 2 eV. The results of Canter et al. (1973) agree with those of Campeanu and Humberston (1977) below 3.5 eV, but are measurably higher than the curve of Campeanu and Humberston (1977) between 3.5 and 6 eV, and this is inconsistent with any explanation based on angular discrimination alone. Coleman et NI. (1980a) estimate that the angular discrimination of their apparatus is less than 20", which, taken with the percent error results shown in Fig. 9, would suggest a consistency of the e+-He results of Coleman et a/. (1979) with those of Campeanu and Humberston (1977),

72

Talbert S . Stein and Walter E . Kauppila

except for the lowest energies (<3 eV) where Coleman et d.41979) would still be too low. Of all the measurements, the corrected, normalized values reported by Sinapius et nl. (1980), who estimate an angular discrimination of 7, are in the best agreement with Campeanu and Humberston between 1 and 6 eV. A major part of the discrepancy between the e+-He measurements of Sinapius et 01. (1980) and Stein et d . (1978) could be due to the differing angular discriminations of the respective experiments. An observation which at first glance seems inconsistent with an angular discrimination argument is that the measurements of Stein et d.(1978) for e+-He above 7 eV are measurably higher than the curve of Campeanu and Humberston (1977), shown in Fig. 3. However, we found that the curve in Fig. 3 (obtained from the H14 curve of Fig. 1 of Campeanu and Humberston, 1977) was noticeably lower at energies above the RamsauerTownsend minimum than one which we obtained using the s-wave phase shifts of Campeanu and Humberston (1977), the p-wave phase shifts of Humberston and Campeanu (1980), the lowest set of d-wave phase shifts of Drachman (1966), and higher phase shifts (up to 1 = 20) from the formula of O’Malley et (11. (1961) referred to above, with only part of the discrepancy between the two theoretical curves being accounted for by the inclusion of the higher partial waves. The resulting theoretical curve is shown in Fig. 10, along with the measurements of Canter et (11. (1973) and Stein et a / . (1978). The percent error estimates shown in Fig. 9 indicate

FIG. 10. Low-energy e+-He total cross-section measurements of Canter i’f ctl. (1973) and Stein rt ( I / . (1978) compared with a theoretical curve obtained by using the phase shifts referred to in Fig. 9. (From Stein and Kauppila, 1982.)

POSITRON-GAS SCAmERING EXPERIMENTS

73

that a 15-20" angular discrimination (as estimated by Dababneh et al., 1980) would result in a remarkable consistency between the theoretical results in Fig. 10 and the measurements of Stein et al. (1978) not only at 2 eV, but also for all energies above 2 eV. The measurements of Canter et al. (1973) agree with the theoretical curve below 3.5 eV, are higher than the curve between 3.5 and 6 eV and lower than the curve between 7 and 13.6 eV. It is interesting to note from the percent error curves in Fig. 9 that a 15-20' angular discrimination for the low-energy e--He QT measurements would introduce less than a 5% error in those measurements from 0 to 13.6 eV in contrast to the e+-He case, which suggests that the excellent agreement of the Detroit e--He results with several other experiments and theories may be an indication that systematic errors in the Detroit measurements, other than that due to angular discrimination, are relatively small. (For some additional considerations of the e+-He situation, the reader is referred to Wadehra et al., 1981.) We have used calculations of phase shifts available in the literature to make percent error (in Qel) estimates (Table 111) for various angular discriminations and projectile energies for positrons colliding with the heavier inert gases (Ne, Ar, Kr, and Xe) in order to search for some consistent patterns in the measurements of QTfor these gases. Although we have only displayed the results obtained from one particular set of theoretical phase shifts for each gas in Fig. 9 and in Table 111, we computed percent errors using different sets of theoretical phase shifts and found that the general trends and magnitudes of the percent errors were surprisingly insensitive to the particular set of theoretical phase shifts which we used, provided that we used ten or more phase shifts, supplementing the phase shifts obtained from a particular theoretical calculation with the higher phase shifts up to I = 10 or more obtained from the formula of O'Malleyer al. (1961) referred to above. Thus, the conclusions that we draw in the following discussion using the percent error results in Fig. 9 and in Table 111 would not be significantly affected by our particular choice of the sets of available theoretical phase shifts. There is a very intriguing consistency between the Bielefeld and Detroit experimental results when the percent error data in Fig. 9 and Table I11 are taken into account. With the exception of Ne, the measurements of Sinapius et al. (1980) are consistently higher than those of Detroit (Kauppila et al., 1976a; Stein et al., 1978; Dababneh et al., 1980) for each of the heavier inert gases, as was the case for He. Our percent error estimates suggest that a significant part of the discrepancies between the measurements of Sinapius et d.(1980) and Detroit for Ar (Kauppila et d.,1976a), Kr, and Xe (Dababneh et al., 1980) could be related to the differing angular discriminations of the respective experiments (-7" for Bielefeld and

Talbert S . Stein and Walter E . Kauppila

74

15-20" for Detroit) as was the case for He. Furthermore, our percent error estimates for the e+-Ne case indicate that the differing angular discriminations of the Bielefeld and Detroit experiments would not result in a noticeable discrepancy between their measurements above 2 eV. Figure 4 indicates that the Bielefeld e+-Ne results are in reasonably good agreement with the Detroit results (Stein rt nl., 1978)above 2 eV but are lower at the lowest energies. If the Detroit e+-Ne QTresults should be shifted upward TABLE 111 PERCENTAGE ERRORS I N Q P , FOR QT MEASUREMENTS D U ETO F I N I T E A N G U L AD RI S C R I M I N A T I O N " Error in System et-Ne

e+-Ar

e+-Kr

QPI

(5%) with angular discrimination (degrees)

Energy (eV)

5"

10

15

20

25

30

45

60

90

0.14 0.54 1.22 2.18 3.40 4.90 6.66 8.70 11.02 13.60

0.6% 1.8 1.1 0.6 0.4 0.3 0.3 0.2 0.2 0.2

2.4 6.8 3.8 2.0 1.3 I .o 0.9 0.8 0.7 0.7

5.2 13.9 7.5 3.8 2.4 1.8

8.7 22.2 11.4 5.5 3.3 2.4 1.9 1.6 1.5 1.4

13.0 31.2 15.3 7.0 4.1 2.8 2.2 1.9 1.8 1.8

17.6 40.4 18.7 8.1 4.6 3.1 2.4 2.1 2.2 2.5

33.3 64.4 25.2 9.5 5.0 3.7 3.7 4.5 5.8 7.5

49.1 79.7 26.7 9.8 6.4 6.8 8.6 11.4 14.7 18.4

75.1 88.3 30.3 20.6 22.7 27.5 33.3 39.4 45.4 51.0

0.14 0.54 1.22 2.18 3.40 4.90 6.66 8.70 11.02 13.60

0.3 0.8 1.7 2.3 2.5 2.5 2.6 2.6 2.5 2.5

1.3 3.2 6.5 8.2 8.7 8.8 8.7

2.9 6.8 13.3 16.4 16.8 16.5

8.5

8. I 7.8

14.8 13.8 12.8

5.0 11.4 21.5 25.6 25.4 24.1 22.3 20.3 18.3 16.4

7.6 16.7 30.4 35.2 33.9 31.0 27.8 24.5 21.4 18.7

10.5 22.6 39.7 44.4 41.3 36.5 31.6 27.0 23.1 20.0

21.5 41.2 65.1 65.9 55.3 44.6 35.9 29.7 26.1 24.5

34.2 58.8 83.2 76.4 59.2 45.8 37.9 34.6 34.8 37.2

60.3 83.7 %.8 79.3 62.0 55.7 57.0 61.3 66.1 70.1

0.14 0.54 1.22 2.18 3.40 4.90 6.66 8.70 11.02 13.60

0.3 0.9 1.8 2.4 2.8 3.1 3.3 3.3 3.4 3.3

1.3 3.5 6.8 8.6 9.8 10.6 10.9 10.9 10.7 10.3

2.8 7.4 13.8 17.2 18.9 19.7 19.6 18.9 17.9 16.7

4.9 12.3 22.3 26.8 28.4 28.5 27.4 25.5 23.4 21.2

7.4 18.0 31.6 36.6 37.5 36.4 33.7 30.3 26.9 23.9

10.3 24.2 41.0 45.9 45.4 42.4 38.0 33.2 28.9 25.3

21.1 43.8 66.5 67.0 59.5

33.7 61.8 83.8 76.5 62.8 51.8 45.2 42.9 43.8 46.6

59.8 86.1 95.6 78.9 66.6 64.0 66.7 70.6 74.0 76.3

1.5

1.2 1.1 1.1

15.8

50.5

42.3 36.3 32.7 31.5

75

POSITRON-GAS SCATTERING EXPERIMENTS TABLE 111 (continued)

System et-Xe

"

Error in Qel (%) with angular discrimination (degrees)

Energy (eV)

5"

10

15

20

25

30

45

60

90

0.14 0.54 1.22 2. I8 3.40 4.90 6.66 8.70 11.02 13.60

0.3%) 1.0 1.9 2.6 3.3 3.8 4.2 4.5 4.6 4.7

1.2 3.9 7.0 9.4 11.6 13.2 14.1 14.5 14.6 14.4

2.6 8.2 14.3 18.5 22.0 24.2 25.0 24.8 24.1 22.9

4.6 13.6 22.9 28.6 32.9 34.7 34.5 33.1 31.0 28.5

7.0 19.9 32.4 38.9 43.0 43.7 41.9 38.8 35.2 31.6

9.8 26.6 41.9 48.5 51.5 50.4 46.6 41.9 37.3 33.3

20.1 47.7 67.2 69.3 65.7 58.5

32.3 66.4 83.5 77.8 68.4 60.0 55.2 54.3 55.9 58.6

58.2 89.4 93.1 79.7 73.8 74.6 77.5 79.9 81.0 81.1

51.0

45.5 42.4 41.6

Sources of phase shifts used in these calculations:

e+-Ne: ef-Ar: e+-Kr: e+-Xe:

/ I

= =

/ = I =

0 t o / = 3, Schrader (1979); 0 to I = 6, McEachran ct d.(1979); 0 t o / = 6, McEachran ef c d . (1980): 0 to / = 6, McEachran et e l / . (1980);

/ = 4 to / = I = 7 to / = I = 7 to/ = I = 7 to / =

20, O'Malley et crl. ( l % l ) . 20, O'Malley et c d . ( l % l ) . 20, O'Malley ct id. ( l % l ) . 20, O'Malley et e l / . (1961).

in energy by a few tenths of an electron volt (as discussed in Section V,A), this would make the agreement between Bielefeld and Detroit even better. The measurements of London (Canter et al., 1973) for the heavier inert gases do not appear to fit any consistent pattern in relation to the other experiments. The results of the Texas group (Colemanet a/., 1979, 1980a) are in rather good agreement with Detroit for He, Ne (Stein et d . , 1978), and Ar (Kauppilaet d . , 1976a), and with London (Canter et al., 1974a)for Xe . On the basis of the consistency of the low-energy Detroit and Bielefeld e+ QTresults for the inert gases when the angular discriminations of those experiments are taken into account, we feel that there is a reasonable chance that the Bielefeld results (having an estimated angular discrimination of about 7") may be the closest of all the measurements to being correct within their restricted energy range (1-6 eV). However, it should be recalled that the Bielefeld measurements can be subject to an appreciable correction (Sinapius et a/., 1980) when QT depends strongly on the positron energy, and this could affect their reliability. The Detroit experiments should yield reliable QT values if a correction is made for their angular discrimination (estimated to be 15-20" for the low-energy e+ experiments). Using Fig. 9 and Table I11 to make such corrections, one finds, for instance, that at 2.2 and 13.6 eV in He, the results of Stein ef d.(1978) could be about 19 and 5% low, respectively; at 2.2 and 13.6 eV in Ne, the

76

Talbert S . Stein and Walter E. Kauppila

results of Stein et d.(1978) could be about 5 and 1% low, respectively; at 2.2 and 8.7 eV in Ar, the results of Kauppila et a / . (1976a) could be about 21 and 18% low, respectively; at 1.2 and 6.7 eV in Kr, the results of Dababneh et d.( I 980) could be about 18and 23% low, respectively; and at 1.2 and 4.9 eV in Xe, the results of Dababneh et ti/. (1980) could be about 19 and 29% low, respectively. Similar corrections for the Bielefeld work would range from less than 1% for Ne to about 8% for Xe. B. INERT GASESAT INTERMEDIATEENERGIES Experimental results for He, Ne, and Ar at intermediate energies are shown in Figs. 11-13 along with several theoretical calculations. The measurements reported by Griffith c't d.(1979a) were obtained by normalization of their results to the measurements of Coleman et d . (1976a) between 30 and 100 eV. The other measurements in Figs. 11-13 are absolute. Above 50 eV, the measurements of Toronto (Jaduszliwer et d., 1975; Tsai et d . , 1976), Swansea (Brenton ct d.,1977, 1978), and Detroit (Kauppila rt d . . 1981a) are in good agreement (generally within 10% of each other) for these gases. The normalized measurements of Griffith ct d.(1979a) are also in good agreement with the other results for He and Ne, except from 1

2

3

4

I

I

I

k (iia,,)

5

6

7

8

I

I

I

I

9

Koupp~Ia(198101

\

\\

o Brenlon 11977) A Coleman (19760) + Jodurzlwer (19751 v Conler (1973) -.-Byron (197B)EBS Dewongen (1977)DW Byron (1977MM lnahull (19751B*G,mel. Dcrongen 11977)DW,e lnohuli (1974)EB

-

I &-A 0'

I " ' "

20 50

' " ' I

100

'

' '

I

'

'

I

'

'

I

400 700 1000 Positron Energy (eV) Fib. I I . Intermediate-energy e+-He total cross-section results. The results displayed for Jaduszliwer C I trl. (1975) are interpolated values. The codes for the theoretical results are identified in the text. except for the Bethe-Born (BB) theory of Inokuti and McDowell (1974). (From Stein and Kauppila, 1982.) 200

77

POSITRON-GAS SCATTERING EXPERIMENTS

-

1

2

k l

.

E

I

2-

* g o .

0

E

2

c

2

"

? I

2

:o

I-

4

I

1

k (i/ao)

5

6

7

8

I

1

I

I

.

0 A

. O

+

f

0

..

- ------ Dewangen (1977)DW Byron (197730M 0

-.-.-

c 0

g

lnokuti (1975)B.G,lnel; Dewangen (1977)DW,el

.

0

A

.

'

1

k I

2

3

4

I

I

I

'#

01

9

Kauppila (1981a) Coleman (1979) Griffith (1979a)n 0 Brenton (1978) 4 Coleman (1976a) + Tsai (1976) v Canter (1973)

4

N -

y

3

100

5

6

7

I

1

I

8 1

Kauppila (1981a) Coleman (1980a) A Griffith (1979a)n 0 Brenton (1978) 4 Coleman (1976a) + Tsai (19761 v Canter (1973) Joachain ( 1 9 7 n O M I

.

' *.*'

' ' * I * n

20 50

k(t/a,)

'

'

___-

'

'

"

200 400 Positron Energy (eV)

'

*

I

700

.

'

IOOC

F I G . 13. Intermediate-energy e+-Ar total cross-section results. OM1 refers to the method I "optical model'' calculation of Joachain ct ol. (1977). (From Stein and Kauppila, 1982.)

78

Talbert S . Stein and Walter E. Kauppila

200-400 eV for He, where Griffith et a / . are 10-15% lower. For Ar, the measurements of Griffith et d . (1979a) are about 10% higher than Tsai vt ( I / . (1976) and Kauppilaet cil. (1981a) below 100 eV, and more than 10% lower than Tsai et nl. (1976), Kauppila et t i / . (198la), and Brenton et r i l . (1978) above 200 eV. The higher energy results of Canter et r i / . (1973, 1974a) and Coleman et cil. (1976a) are lower than the other measurements, and this has been attributed to their inadequate discrimination against small-angle scattering. On the basis of these comparisons, it seems that the most reliable measurements for energies above 50 e V are those of Toronto (Jaduszliwer ef d . , 1975; Tsai et d.,1976), Swansea (Brenton ef N/., 1977, 1978), and Detroit (Kauppila et N I . . 1981a). The Detroit group has also made the corresponding e- measurements on the same gases (using the same apparatus and technique), which are in remarkable agreement with the recent measurements of Blaauw et r i l . (1980) and Wagenaar and de Heer (1980). The only significant differences between the e- measurements of these two laboratories could be explained by their differing angular discriminations for elastic scattering estimated to be less than I" for the Amsterdam group and typically 5-6" for Detroit. For positrons at energies above 100 eV, the estimated angular discriminations for the Detroit work are generally 6-8" for He, Ne, and Ar (Kauppila et d . , 1981a). Hence, the measurements made in Detroit are subject to corrections, which would increase their values by an amount depending on the nature of the differential elastic scattering of positrons, and the contribution of elastic scattering to QT. Using the e+-He differential elastic cross sections calculated by Byron and Joachain (1977) [using an optical model (OM) formalism], Kauppilaer t i / . (1981a) estimate that their e+-He measurements may be an average of 1% too low due to the neglect of small-angle elastic scattering from 100500 eV. In surveying the various e+-He theoretical results, Kauppila et t i / . (l981a) found that by adding the inelastic cross sections that can be calculated from the Bethe theory with an additional "gamma" term (related to the number of electrons in the target atom) of Inokuti et a / . (1975) and Kim and Inokuti (1971), and the elastic scattering cross sections calculated by Dewangen and Walters (1977) using a distorted wave second Born approximation (DW), QTvalues (designated B + G, DW in Fig. 1 I ) are obtained which agree to within 2% of their measurements above 200 eV. Brenton et t i / . (1977) are in good agreement with this composite theory up to their highest energy, 1000 eV. The theoretical calculations for Ne and Ar, shown in Figs. 12 and 13 are reasonably close to the measurements, but do not merge with them at the highest energies of overlap. For Kr and Xe, the measurements of Canter et nl. (1973, 1974a) and Coleman et a/. (1976a) are

POSITRON-GAS SCATTERING EXPERIMENTS

79

significantly lower than the recent results reported by Kauppila et NI. (1981b). C. POSITRON A N D ELECTRON COMPARISONS FOR THE INERT GASES Comparisons have been made by Kauppila et d.(1981a,b) of the e+inert gas and e--inert gas total cross sections up to 800 eV. For the He comparisons (up to 650 eV) shown in Fig. 14, the Detroit measurements 1981a) have been used for all e+ ener(Stein et a / . , 1978; Kauppila er d., gies and for e- energies above 2 eV, while the measurements of Milloy and Crompton (1977) were used for e- energies below 2 eV. Since the e+-He and e--He measurements (from 2 to 600 eV) shown in Fig. 14 have been made with the same experimental apparatus and techniques, most of the potential systematic errors should equally affect the e+ and e- measurements. The partial neglect of small-angle elastic scattering, on the other 0 2 10

50

Energy (eV)

100

200

400

60C

F I G . 14. Comparison of measured e+-He and e--He total cross sections. The lowest inelastic thresholds for each projectile are indicated by arrows. (From Kauppila cf c d . , 1981a.)

80

Tulbeit S . Stein and Walter E . Kuuppila

hand, is a source of error which, in general, does not affect the e + and emeasurements equally, since it depends on the angular discrimination (estimated to be roughly 5" for all e- energies and ranging from 15-20" at low energies to 6- 10" at higher energies for positrons) and on the differential elastic cross sections. The e'-He comparison in Fig. 14 provides a striking illustration of some of the differences and similarities in e+ and escattering. At low energies the e + cross section is about two orders of magnitude smaller than the e- cross section. This is consistent with the fact that the static and polarization interactions are both attractive in the e- case, whereas there is a tendency toward cancellation of these interactions in the e+ case. In sharp contrast to the vastly different cross sections at low energies, there is an observed merging (to within 2%) of the e- and e + results above 200 eV. Kauppila e f d.(1981a) estimate that the maximum amounts by which their cross-section measurements could be too low due to the neglect of small-angle elastic scattering are 2% for electrons and 1% for positrons. The merging of the QT curves was not expected to occur at such low energies. The e +and e- distorted wave second Born approximation (DW) calculations of Dewangen and Walters ( 1977) do not merge (to within 2%) until 2000 eV, while the composite (B + G, DW) calculations of Inokuti r t t i / . (1979, Kim and Inokuti (l971), and Dewangen and Walters referred to in Section II1,B merge (to within 2%) at 1000 eV. At 200 eV, the DW calculations for electrons are 21% higher than the corresponding e+ calculations, while there is a 14% difference for the B + G , DW composite theory and a 21% difference for the eikonal Born series (EBS) calculations of Byron (1978). The comparison e' measurements of the Detroit group for Ne, Ar, Kr, and Xe (Kauppila et d . , 1981a,b) do not indicate any merging of the cross sections at the highest energies studied. D. TESTSOF

THE

S U MR U L E

An aspect of total scattering of positrons and electrons by atoms that has received considerable attention in recent years is the question concerning the validity of the sum rule, which is based on the forward dispersion relations of Gerjuoy and Krall (1960). For scattering by the inert gases, in which the projectile and the target atoms do not form bound states, the sum rule has the form (Bransden and McDowell, 1969)

POSITRON-GAS SCATTERING EXPERIMENTS

81

whereA is the scattering length,fi andfEBare the first Born elastic scattering amplitudes in the forward direction for direct and exchange scattering, I, is the projectile wave number, all in atomic units, and Q T is the total scattering cross section in units of m i ; . By using available e+-He and e+-Ne QT measurements (which at the highest energies are extrapolated to the Born approximation) to evaluate the integral in Eq. (3) and theoretical results for the other terms in Eq. (3), it has been determined by Bransden and Hutt (1979, Byron rf (11. (1979, de Heer ef al. (1976), Hutt rt d.(1976), Tsai et (11. (1976), Brenton et (11. (1977), Griffith et al. (1979a), and Kauppila et d.(1981a) that the sum rule is valid when applied to e+-He, Ne scattering and not valid for e--He, Ne scattering. The sum rule tests by Kauppila et (11. (1981a) displayed in Table IV provide a summary of the current situation for e*-He, Ne, Ar scattering. In the latter work we feel there are possible uncertainties on the order of a few percentage points in the evaluation of the integral term for each projectile-target combination, which is based on a fit to the QT measurements of Kauppila et d.(1976a, 1977b, 1981a) and Stein ef d . (1978). Similarly there are small uncertainties associated with some of the other terms in Eq. (3) which get progressively larger as one goes from He to Ar (where there is no known calculation of J'E). From Table IV it is seen that the sum rule also appears to be valid for e+-Ar scattering as was first suggested by Tsai et (11. (1976). The apparent validity of the sum rule for positron-atom scattering and its invalidity for electron-atom scattering has been studied by Byron et al. (1975), de Heer ef (11. (1976), Hutt et NI. (1976), and Tip (1977), and is understood to arise from the nature of singularities in the exchange ampliTABLE IV SUM-RULE TESTSOR e'-He, Ne,

System e--He e+-He e--Ne e+-Ne e--Ar e+-Ar

.t'E

A

+ 1.178b - 0.48*

+O.P -0.614" 1.7h -3.0 to -4.W ~

AND

+0.796c -0.796' +3.21' -3.21' +9.7' -9.7'

Ar SCATTERING"

-A - .f!

+ 3.943'

+.tE

I .969 1.276 I .891 3.824

0 +5.32If 0 Unknown 0

-

12.7-13.7 ~

2.56 I .25 5.10 3.88 14.1 13.4

~~

From Kauppila ei id. (1981a). O'Malley (1977). Ho (1977). Campeanu and Humberston (1977). Naccache and McDowell(1974). Hutt C I d.(1976). McEachraner crl. (1978). O'Malley (1963). Tsai et ( I / . (1976). Hara and Fraser (1975). a

J

82

Talbert S . Stein and Walter E. Kauppila

tude which causes the sum rule to fail for electron scattering. No such difficulties arise for positron scattering since there are no exchange effects between the incident positrons and target electrons. E. MOLECULAR GASES The low-energy (up to 40 eV) QTmeasurements for e+-Hz, N2, and CO, scattering are shown in Figs. 15-17, along with a few available theoretical results. Earlier measurements by Coleman et al. (1974) for Hzare not included because they have been remeasured and reported by Griffith and Heyland (1978). The measurements of Charlton er al. (1980b) are normalized. The measurements for H2 shown in Fig. 15 indicate the existence of a broad minimum below 9 eV and a dramatic increase near the Ps formation threshold. Another process (besides Ps formation) which could contribute to the large increase is dissociation of Hz, which becomes energetically favorable near 8.8 eV (the H2dissociation energy is 4.48 eV). A fixed-nuclei approximation calculation of the elastic scattering cross section by Hara (1974) is in quite good agreement with the measurements. An adiabatic nuclei approximation calculation by Baille et d.(1974) is somewhat lower than the measurements but similar in shape. The QT measurements of Hoffman er d.(1982) for N2 (Fig. 16) also show a broad minimum with increasing cross sections at the lowest energies and near the Ps formation threshold. The measurements of Coleman et cil. (1975~) for N, are generally lower than Hoffman et a/. and do not indicate an I

I

I

I

I

1

I

I

I

I

A.

i

I

0

i 01 0

cn

.V

2 2-

*.

2

0

FIG.15.

0

..v

.c 31

0

0

"

"t

Hoffman (1982) Charlton (1980b)n v Coleman 0976b) -Hara (19741 --- Baille 0974)

IHt

A

I

4

8

12

I

1

I

20 24 28 32 Positron Energy (eV) 16

I

I

36

40

Low-energy e+-HZtotal cross-section results. (From Hoffman et

(I/.,

1982.)

83

POSITRON-GAS SCATTERING EXPERIMENTS

-6 8 l

N

.II

s

7

1

I

I

1

1

/ N E * A A

4

“0

8

12

1

.

1

16 20 24 28 Positron ‘Energy (eV)

I

1

32 36 40

FIG.16. Low-energy e+-N, total cross-section results. (From Hoffman et a / . , 1982.)

increase at the lowest energies. For e+-CO, scattering (Fig. 17) the meaare much lower (20-40%) than the surements of Coleman et a/. (1975~) results of Hoffman et a/. (1982),with the latter measurements revealing an interesting bump just above the Ps formation threshold. Comparisons of e*-H, total cross sections by Hoffman et al. (1982) are shown in Fig. 18. The e*-H, results merge for energies above 200 eV, but the merging could be fortuitous if there is appreciable small angle elastic scattering for Hz. Similar comparisons for e2-N, by Hoffman et al.’(1982)

-

15

N

:n I

O

5 s

.*

010 c

.,”

-

- ..

2 -In

roz* IPS :. I

I

.%...

.

..p

A

1

0

*

A

I

’

.

vvvv

V

V

Hoffman 0 9 8 2 ) Charltan (1980b)n v Coleman ( 1 9 7 5 ~ ) A

-

0

I - -

0

I

1

A

vv vvvvv~v~vvvv

c

1

*.*.

a”.*”

8- :

1

co,

e*

-0

Y

1

1

1

1

I

1

1

1

1

1

I

84

Talbert S . Stein and Walter E . Kuuppilu

-0

1

2

3

4

5

6

7

k(l/ao)

FIG.18. Comparison of measured e+-Hzand e--Hz total cross sections. (From Hoffman et 01.. 1982.)

indicate that the e--N, results remain above the e+-N2 results up to the highest energies studied (700 eV).

IV. Differential Scattering Cross Sections Coleman and McNutt (1979) have recently reported the first measurements of differential cross sections for the elastic scattering of 2-9 eV positrons by Ar for angles from 20-60”. A schematic diagram of their 1980b) is shown in Fig. 19. In their experiment, apparatus (Coleman et d., slow positrons pass through a I-cm-long gas cell and then travel approximately 25 cm through an evacuated straight flight tube in a strong axial magnetic field to a detector (Channeltron electron multiplier). The larger the angle through which a positron is scattered in the gas cell, the longer its TOF will be in the axial magnetic field. By alternately admitting gas into and then evacuating the gas cell, TOF histograms such as those shown in Fig. 20 are obtained. A “tail” can be observed on the long-time side of the “gas” TOF spectrum, which is associated with detected posi-

POSITRON-GAS SCATTERING EXPERIMENTS

c

85

Solenoid

FIG. 19. Texas time-of-flight spectrometer for measuring differential cross sections for elastic scattering. (From Coleman rr u l . , 1980b.)

trons which have undergone forward elastic scattering. The “difference” spectrum is obtained by subtracting an appropriately adjusted “vacuum” TOF spectrum from the “gas” spectrum. Differential cross sections can be obtained by correlating the “difference” TOF spectrum with various angles of forward elastic scattering. The differential cross sections measured by Coleman and McNutt (1979) for e+-Ar collisions are compared

GAS 0

VACUUM

’DIFFERENCE

0- 200

i e

40“

* \

TCF (nsec)

FIG.20. TOF histogram with and without argon in the gas cell to illustrate the increase in flight time corresponding to elastic scattering at various angles. (From Coleman and McNutt, 1979.)

86

Talbert S . Stein and Walter E. Kauppila

10

SCATTERING ANGLE

FIG.21. Differential e+-Ar scattering results for different positron mean energies. The error bars represent statistical standard deviations. (From Coleman and McNutt, 1979.)

with the calculations of Schrader (1979) (solid lines) and “scaled-down’’ calculations of McEachran et 01. (1979) (broken lines) in Fig. 21. The agreement between experiment and theory is reasonable. Coleman et crl. (1980b) have also measured differential scattering cross sections for e--Ar collisions in the same apparatus and using the same technique as used for their positron studies and obtain reasonable agreement with other recent e--Ar results.

V. Inelastic Scattering Investigations A. POSITRONIUM FORMATION CROSS SECTIONS The crosshatched regions of the e+ QT curves illustrated in Fig. 8 can be used to provide estimates of the Ps formation cross sections (Qps)in the

POSITRON-GAS SCATTERING EXPERIMENTS

87

inert gases. Such estimates assume a smooth extrapolation of the elastic scattering cross sections from below the Ps formation thresholds and also depend on the e+ energy calibration. Recent resonance searches by the Detroit group in Ar (Stein et al., 1981) described in Section VI indicate that the energy calibration used by the Detroit group could be low by a few tenths of an electron volt. This appears to be supported by the locations of the sharp onsets of Ps formation in the inert gases shown in Fig. 8, since the abrupt increases in QT occur at assigned energies which are a few tenths of an electron volt below the predicted Ps formation thresholds (indicated by arrows in Fig. 8). In order to compare the Detroit estimates of Qpswith other experimental and theoretical results, we have extracted QB curves for He, Ne, and Ar from the crosshatched regions in Fig. 8, but we have shifted the energy calibrations used by the Detroit group upward (0.2-0.4 eV) in order to match the observed and predicted thresholds for Ps formation. The Detroit QB results for He, Ne, and Ar, based upon the QT measurements of Stein et al. (1978) and Kauppila et al. (1976a), are shown in Fig. 22, where they are compared with several other sets of experimental and theoretical results. Charlton et al. (1980~)have recently made the first direct measurements of the energy dependence of the orrho-positronium formation cross section (Q,,-ps)in He, Ar, Hz, and CH, by passing a slow e+ beam through a scattering chamber and counting triple coincidences from the 3y decay of 0-Ps. In order to compare their Qo-Psenergy dependence with the other results shown in Fig. 22, we have normalized the relative He and Ar results of Charlton et al. (1980~)by scaling their values to match the Detroit curves at the respective excitation thresholds of He and Ar. We have also set the positron energy by adding 1.8 and 2.0 V to the “applied voltage” quoted by Charlton et al. for He and Ar, respectively, in order to match the predicted Ps formation thresholds with the observed onsets. Implicit in this normalization scheme is the assumption that the Qo-ps energy dependence would be the same as the Q, energy dependence. It is interesting that the Qo-psvalues indicated by Charlton ef al. (1980~)increase only up to roughly 4 eV above the respective Ps formation thresholds for He and Ar and then begin to diminish. Based on their direct measurements, Charlton et al. (1980~)have estimated the ratio of Q,,.ps(Ar)/Q,,-p,(He) for the peak values to be 6.5. The normalized Q,,-ps results of Charlton et al. (1980~)shown in Fig. 22 indicate a ratio of QB(Ar)/Qps(He)for the peak values of about 25, which is much larger than the ratio based on the direct measurements of the London group. The Qpsvalues of Coleman et al. (1975a) for He (based upon e+ lifetime and QT measurements) are in good agreement with the QPsvalues based upon the QT measurements of Stein et al. (1978) (and with the normalized values of

88

Talbert S . Stein and Walter E. Kauppila

FIG.22. Positronium formation cross sections for e+-He, e+-Ne, and e+-Ar scattering. The S and K curves are Q,., values based on the Qr measurements of Stein rt a / . (1978) and Kauppila el ( I / . (1976a), respectively: the closed circles are the QePj results of Charltonrt t i / . (1980~)normalized as described in the text; the open squares are estimates from Coleman t t id. (1975c), based on the QTmeasurements of Canter er a ( . (1974a); C refers to estimates by Coleman E I ( I / . (1975a), based on e + lifetime and QT measurements; the FBA (first Born approximation) and FOEA (first-order exchange approximation) calculations (for Ps formed in the IS state) are from Mandal er a/. (1980), where the suffixes “a” and “b” refer to “postinteraction” and “prior interaction,” respectively: the DWA (distorted-wave approximation) calculation is from Mandal et a / . (1979); F refers to a calculation by Fels and Mittleman (1969) using a projection operator technique and the Temkin form of the polarization potential; B, P, and NP refer to the Born approximation, distorted-wave approximation (with polarization), and distorted-wave approximation (without polarization) calculations, respectively, by Gillespie and Thompson (1977).

Charltonrr al. (1980~)up to about 22 eV), but end up being about twice as large as the normalized values of Charlton et al. (19804 at 24.5 eV, while their results (Coleman et id., 1975a) for Ne and Ar are much lower than the QPs values based upon the QTmeasurements of Stein et al. (1978) and Kauppila et id. (1976a), respectively. In a recent TOF investigation (not represented in Fig. 22) of inelastic e+-He scattering, Griffith et (11. (1979b) made estimates of QB (based on numerous assumptions) which indicate that Qpsreaches a peak value of about 0.48 x 10-l“ cm2,roughly five times the peak value indicated by the normalized results of Charlton et a/. (1980~).The position of the peak indicated by Griffith et a/. (1979b) is 10-15 eV above the Ps formation threshold as compared with 4 e V above that threshold for the results of Charlton er al. (1980~). None of the theories shown in Fig. 22 has a shape which matches the results of Charlton et al. (1980~)for He. It is also interesting to note the

POSITRON-GAS SCATTERING EXPERIMENTS

89

poor agreement of the different theories with each other and with the experiments in the case of He. One indication of the extent of the disagreements between theories and experiments in the case of He is that none of the available theories could be placed in reasonable locations on the graph of the experimental results without the use of the indicated scaling factors ranging from A5 to 10. Results of a first Born approximation calculation of QPs for He by Massey and Moussa (1961) have not been included in Fig. 22 because Mandal et 611. (1975) have indicated that numerical errors were made in that calculation. The Born approximation results of Gillespie and Thompson (1977) for Ne and Ar remain above all of the experimental values. The Born values of Qps for Ar are still rising at the highest energies studied by Gillespie and Thompson (8-9 eV above the Ps formation threshold) in contrast to the Q,,-ps observations of Charlton et d.(1980~).The distorted wave approximation (P, NP) calculations of Gillespie and Thompson (1977) for Ne indicate a threshold behavior similar to the QPsvalues based on the QTmeasurements of Stein et a/. (1978) but begin to level off just above 15 eV whereas the values of Stein et al. continue a rapid increase up to the excitation threshold of Ne (16.6 eV). For Ar the distorted wave approximation (P)calculation of Gillespie and Thompson (1977) indicates an energy dependence roughly similar to the Q,,+? results obtained by Charlton ef a/. (1980~)but the theoretical QPs values are much lower than the normalized values of Charlton et a/. ( 1980~). B.

EXCITATION A N D IONIZATION

CROSS SECTIONS

Using the TOF apparatus shown in Fig. 19 (the same apparatus used for measurements of differential cross sections), Coleman and Hutton (1980) have obtained lower bounds on total excitation cross sections for 23-31 eV e+-He collisions. In this energy range, well-defined secondary peaks were observed in the TOF spectra corresponding to positrons which have lost 20.6 eV of energy and have been scattered in the forward direction at angles less than 70". A TOF spectrum for an incident e+ energy of 25.8 e V is shown in Fig. 23. At incident e+ energies above 30 eV, a secondary peak associated with ionization overlaps the excitation peak, making it difficult to assign excitation cross sections. The secondary peak corresponding to a 20.6 eV energy loss indicates that in the projectile energy range from 23 to 31 e V the total excitation cross section is dominated by excitation of the 2lS state, and that there is appreciable small angle scattering associated with this excitation process. This work has recently been extended to Ne and Ar (Coleman et al., 1981),

90

Talbert S . Stein and Walter E. Kauppila

t

25.8 eV

1

* - I

TOF (nrrc) FIG.23. TOF spectrum showing primary (unscattered) and secondary (excitation) peaks for positrons with a mean incident energy of 25.8 eV colliding with He. (From Coleman and Hutton, 1980.)

and lower bounds on “excitation plus ionization” cross sections have also been measured for He, Ne, and Ar. In Fig. 24 we summarize the current state of affairs regarding the partitioning of e+ total scattering cross sections for He and Ar between the elastic and inelastic scattering channels. The curves shown in Fig. 24 are portions of the QT and the extrapolated elastic cross section curves from Fig. 8 [based on the results of Stein et a / . (1978) and Kauppilaet a / . (1976a)l. The increase in the Detroit results due to inelastic scattering should be reliable as a consequence of the ability of the Detroit experiments to discriminate 100% against inelastically scattered positrons. The Qo.psresults of Charlton et a / . (19804 (normalized as discussed in Section V,A), and the partial excitation and partial “excitation plus ionization” cross sections of Coleman et a / . (1981) have been added to the respective extrapolated elastic cross section curves. It is clear from Fig. 24 that if the extrapolations of the elastic scattering cross sections above the Ps formation thresholds are valid, then a large part of the inelastic scattering cross section for He and Ar is unaccounted for by the normalized 0-Psformation cross section measurements of Charlton et al. (1980~)and the partial excitation and partial “excitation plus ionization” cross-section measurements of Coleman at a / . (1981). Griffith et a / . (1979b) have also used their TOF system with a localized scattering region and a long flight tube to investigate intermediate energy e+-He inelastic scattering. From studies of the shapes and positions of secondary peaks observed in their “gas-in’’ TOF spectra, Grfith et a / .

POSITRON-GAS SCATTERING EXPERIMENTS

91

-’.O Positron Energy (eV) FIG.24. Inelastic cross sections for e+-He and e+-Ar scattering. The solid and broken curves are the total cross section and extrapolated elastic cross section curves from Fig. 8, respectively. The open circles represent the relative ortho-positronium formation measurements of Charlton er d . (198Oc) normalized in the same manner as described for Fig. 22 and added to the broken curves. The asterisks and plus signs represent the partial “excitation” and ‘‘excitation-plus-ionization” cross-section measurements, respectively, by Coleman and Hutton (1980) and Colemaner al. (1981), added to the broken curves.

have deduced that ionization is the dominant inelastic process in e+-He scattering between 100 and 500 eV.

VI. Resonance Searches Resonances have been predicted to exist for e+-H scattering just below then = 2 atomic excitation threshold (Doolen et al., 19781, and associated with the first excited state of Ps in the e+-H system (Doolen, 1978), and some possible theoretical evidence of a resonance in the e+-He system near 20.375 eV (above the Ps formation threshold and just below the first singlet excitation of He) has been provided by Ho and Fraser (1976). Up to the present time, there have not been any experimental observations of e+ scattering resonances. Stein et a/.(1981) are currently using their narrow energy width (<0.1 eV) e+ beam in a transmission experiment to search for e+ scattering resonances. Figure 25 shows measurements of the transmitted beam current versus the voltage applied to the e+ source over 1.0-V ranges centered near the Ps formation threshold (9.0 eV) and the

92

Talbert S . Stein and Walter E. Kauppila

50K

90K Lo

c

c

0

u)

V

4-

e

0

V

(b)

(a)

801

.5

9.0 Applied Volts

9.

40t

I

.o

11.5 Applied Volts

L

12.

FIG.25. Transmitted positron beam currents in Ar versus voltage applied to the positron source. The error bars represent statistical uncertainties. (From Stein et ( I / . , 1981.)

lowest atomic excitation threshold (1 1.5 eV) for Ar. These data were taken at 25-mV intervals, with the primary beam attenuated by about 50%. In the data for Ar shown in Fig. 25 there is no convincing evidence of a resonance (which would be expected to manifest itself as a relatively narrow structure in the transmitted beam current). There is also no evidence of resonances in preliminary searches with He and H2. However, there is still useful information that is obtained from these studies. An abrupt change in the slope of the transmitted current curve appears in Fig. 25a which should correspond to the abrupt increase in the total cross section at the Ps formation threshold (see Fig. 8). The slope change in Fig. 25a occurs at 8.8 V, whereas the Ps formation threshold is known to be 9.0 eV. This implies that an applied voltage of 8.8 V corresponds to an e+ energy of 9.0 eV in the scattering region, which is consistent with the observation that the Detroit e+ QTcurves in Fig. 8 appear to be starting their abrupt increases a few tenths of an electron volt below the predicted Ps formation thresholds. The curve in Fig. 25b does not show an appreciable change in its slope as the energy is increased through the atomic excitation threshold (1 1.5 eV) and this is supported by the corresponding QT curve for Ar in Fig. 8, which is quite smooth between I 1 and 12 eV.

VII. Possible Future Directions for Positron Scattering Experiments A glance at Table I conveys the feeling that there are many feasible, interesting e+ scattering experiments still waiting to be done for the first

POSITRON-GAS SCATTERING EXPERIMENTS

93

time. Even in the area of Q T measurements, there are a large number of obvious candidates for target atoms and molecules which have not been studied at all. Atomic hydrogen, and the alkali atoms would be of particular theoretical interest as target atoms due to their relatively simple structure. We have discussed several “second generation” experiments which have gone beyond Q T measurements. Although such efforts demonstrate the feasibility of investigating e+differential cross sections, inelastic scattering cross sections, and resonances, it seems that these areas can still be regarded as essentially “wide open” for improved experimental techniques and for the first measurements on many different collision systems. As an example, in the area of differential cross-section measurements, only one gas (Ar) has been studied, and it has only been studied in a very limited energy range (2.2-8.7 eV) and over a very restricted range of angles (20-60”). The studies of inelastic scattering up to the present time may have raised more questions than they have answered, and indicate a need for additional direct measurements of Ps formation, excitation and ionization cross sections.

ACKNOWLEDGMENTS We would like to thank Mr. Kevin Hoffman, Mr. Paul Felcyn, and Mr. Diab Jerius for their helpful assistance, and Ms. Evelyn Williams for typing the manuscript. We acknowledge, with gratitude, the support of the National Science Foundation for our Research program.

REFERENCES Anderson, C. D. (1933). Phys. Reit. 43, 491. Aulenkamp, H., Heiss, P., and Wichmann, E. (1974). Z. Phys. 268, 213. Baille, P., Darewych, J. W., and Lodge, J. G. (1974). Cun. J . P h y s . 52, 667. Blaauw, H. J., Wagenaar, R. W., Barends, D. H., and de Heer, F. J. (1980).J. Phys. B 13, 359. Bransden, B. H., and Hutt, P. K. (1975). J . Phys. E 8, 603. Bransden, B. H., and McDowell, M. R. C. (1%9). J . Phys. E 2, 1187. Brenton, A. G., Dutton, J., Harris, F. M., Jones, R. A., and Lewis, D. M. (1977).J. Phys. B. 10, 2699. Brenton, A. G., Dutton, J., and Hams, F. M. (1978). J . P h y s . E 11, L15. Burciaga, J. R., Coleman, P. G., Diana, L. M., and McNutt, J. D. (1977). J. Phys. B 10, L569. Bussard, R. W., Ramaty, R., and Drachman, R. J. (1979). Astrophys. J . 228, 928. Byron, F. W., Jr. (1978).Phys. Rev. A 17, 170.

Talbert S . Stein and Walter E. Kauppila

94

Byron, F. W., Jr., and Joachain, C. J. (1977). Phys. Rev. A 15, 128. Byron, F. W., Jr., de Heer, F. J., and Joachain, C. J. (1975). Phys. Re\*. Lerr. 35, 1147. Callaway, J., LaBahn, R. W., Pu,R. T., and Duxler, W. M. (1968). Phys. Re\,. 168, 12. Campeanu, R. I., and Dubau, J. (1978). J. Phgs. B 11, L567. Campeanu, R. I., and Humberston, J. W. (1977). J . Phys. B 10, L153. Canter, K. F., Coleman, P. G., Griffith, T. C., and Heyland, G. R.(1972).J. Phps. B 5, L167. Canter, K. F., Coleman, P. G., Griflith, T. C., and Heyland, G. R.(1973).J. Phps. B 6, L201. Canter, K. F., Coleman, P. G., Griffith, T. C., and Heyland, G. R. (1974a). Appl. Phys. 3, 249. Canter, K. F.,Mills, A. P., Jr., and Berko, S. (1974b). Phys. Re),. Lett. 33, 7. Charlton, M., Griffith, T. C., Heyland, G. R., and Twomey, T. R. (1980a). J. Phys. B 13, L239. Charlton, M., Griffith,T. C., Heyland, G. R., and Wright, G. L. (1980b). J . Phys. B 13, L353. Charlton, M., Griffith,T. C., Heyland, G. R., Lines, K. S . , and Wright, G. L. (1980~).J. f'hys. B 13, L757. Chupp, E. L., Forrest, D. J., Higbie, P. R.,Suri, A. N., Tsai, C., and Dunphy, P. P. (1973). Nfitiire (Loridori) 241, 333. Coleman, P. G., and Hutton, J. T. (1980). Phys. Re\-. Lett. 45, 2017. Coleman, P. G., and McNutt, J. D. (1979). Phys. ReLv. Lett. 42, 1130. Coleman, P. G., Griffith, T. C., and Heyland, G. R. (1973).Proc. R . Soc. L O ~ I ~Ser. O N A 331, 561. Coleman, P. G., Griffth, T. C., and Heyland, G. R. (1974). Appl. Phys. 4, 89. Coleman, P. G., Griffith, T. C., Heyland, G. R.,and Killeen, T. L. (1975a). J. Phys. B 8, L185. Coleman, P. G., Griffith, T. C., Heyland, G. R.,and Killeen, T. L. (1975b). J. Phys. B 8, L454. Coleman, P. G., Griffith, T. C., Heyland, G. R.,and Killeen, T. L. (1975~). Atom. Phps. 4, 355.

Coleman, P. G., Griffith, T. C., Heyland, G. R., and Twomey, T. R. (1976a). Appl. Phys. 11, 321. Coleman, P. G., Griffith,T. C., Heyland, G. R.,and Twomey, T. R. (1976b). Private communication as reported by Griffith and Heyland (1978). Coleman, P. G., McNutt, J. D., Diana, L. M., and Burciaga, J. R. (1979). Phys. Re\,. A 20, 145. Coleman, P. G., McNutt, J. D., Diana, L. M., and Hutton, J. T. (1980a). Phys. Re\,. A 22, 2290. Coleman, P. G., McNutt, J. D., Hutton, J. T., Diana, L. M., and Fry, J. L. (1980b).Reis. Sci. 1ri.striirn. 51, 935. Coleman, P. G., Hutton, J. T., Cook, D. R., Diana, L. M., and Sharma, S. C. (1981). Proc. l i l t . C q f . Phys. Electron. Atom. Collisions, 12th Abstr., p. 426. Costello, D. G., Groce, D. E., Herring, D. F., and McGowan. J. W. (1972a). Ctrri. J. Phys. 50, 23. Costello, D. G., Groce, D. E., Herring, D. F., and McGowan, J. W. (1972b). Phys. Rev. B 5, 1433. Crannell, C. J., Joyce, G., Ramaty, R.,and Werntz, C. (1976). Astrophys. J. 210, 582. Dababneh, M. S., Kauppila, W. E., Downing, J. P., Laperriere, F., Pol, V., Smart, J. H., and Stein, T. S . (1980). Phys. Rev. A 22, 1872. Dale, J. M., Hulett, L. D., and Pendyala, S . (1980). Surf. Interface Anal. 2, 199. Darewych, J. W., and Baille, P. (1974).J . f h y s . B 7, L1.

POSITRON-GAS SCATTERING EXPERIMENTS

95

de Heer, F. J., Wagenaar, R. W., Blaauw, H. J., and Tip, A. (1976). J . Phys. B 9, L269. Dewangen, D. P., and Walters, H. R. J. (1977).J. Phys. B 10, 637. Doolen, G. D. (1978). I n t . J. Quotit. Chem. 14, 523. Doolen, G. D., Nuttall, J., and Wherry, C. J. (1978). Phys. Rev. Lett. 40, 313. Drachman, R. J. (1966). f h y s . Rev. 144, 25. Dutton, J., Harris, F. M., and Jones, R. A. (1975). J . Phys. B 8, L65. Dutton, J., Evans, C. J., and Mansour, H. L. (1981). Proc. I n t . ConJ Phys. Electron. Atom. Collisions. 12th Abstr., p. 428. Fels, M. F., and Mittleman, M. H. (1969). PhyA. Rev. 182, 77. Fon, W. C., Berrington, K. A., and Hibbert, A. (1981). J. Phys. B 14, 307. Gerjuoy, E., and Krall, N. A. (1960). Phys. Rev. 119, 705. Gillespie, E. S., and Thompson, D. G. (1975). J . f h y s . B 8, 2858. Gillespie, E. S., and Thompson, D. G. (1977). J. Phys. B 10, 3543. Griffith, T. C. (1979). Adv. Atom. M o l . Phys. 15, 135. Griffith, T. C., and Heyland, G. R. (1978). Phys. Rep. 39, 169. Griffith, T. C., Heyland, G. R., Lines, K. S., and Twomey, T. R. (1978).J. Phys. B 11, L635. Griffith, T. C., Heyland, G. R., Lines, K. S., and Twomey, T. R. (1979a). Appl. Phys. 19, 431.

Griffith, T. C., Heyland, G. R., Lines, K. S., and Twomey, T. R. (1979b). J . f h y s . B 12, L747. Hara, S. (1974). J . Phys. B 7 , 1748. Hara, S., and Fraser, P. A. (1975). J . Pliys. B 8, 219. Ho, Y. K. (1977). J . Phys. B 10, L149. Ho, Y. K., and Fraser, P. A. (1976). J . Phys. B 9, 3213. Hoffman, K. R., Dababneh, M. S., Hsieh, Y.-F., Kauppila, W. E.,Pol, V., Smart, J. H., and Stein, T. S. (1982). Phys. Rev. A 25, 1393. Humberston, J. W. (1978). J. Pliys. B 1 1 , L343. Humberston, J. W., and Campeanu, R. I. (1980). J . Phys. B 13, 4907. Hutt, P. K., Islam, M. M., Rabheru, A., and McDoweU, M. R. C. (l976).J. Phys. B 9,2447. Inokuti, M., and McDowell, M. R. C. (1974). J. Phys. B 7 , 2382. Inokuti, M., Saxon, R. P.,and Dehmer, J. L. (1975). I n t . J. Rodiat. P h p . Chem. 7 , 109. Jaduszliwer, B., and Paul, D. A. L. (1973). Can. J. Phys. 51, 1565. Jaduszliwer, B., and Paul, D. A. L. (1974a). Can. J . Phys. 52, 272. Jaduszliwer, B., and Paul, D. A. L. (1974b). Can. J. Phys. 52, 1047. Jaduszliwer, B., Keever, W. C., and Paul, D. A. L. (1972). Can. J. Phys. 50, 1414. Jaduszliwer, B., Nakashima, A., and Paul, D. A. L. (1975). Can. J . f h y s . 53, %2. Joachain, C. J., Vanderpoorten, R., Winters, K. H., and Byron, F. W., Jr. (1977). J. Phys. B 10, 227. Kauppila, W. E., and Stein, T. S. (1982). Con. J. Phys. 60,471. Kauppila, W. E., Stein, T. S., and Jesion, G. (1976a). Phys. Rev. Lett. 36, 580. Kauppila, W. E., Stein, T. S . , Pol, V., and Jesion, G. (1976b). Proc. I n t . Conf. Positron Annihilation, 4th (Helsingor A b s t r . ) p. 25. Kauppila, W. E., Stein, T. S., Smart, J. H., and Pol, V. (1977a). Proc. I n t . Cotif. Phys. Electron. Atom. Collisions, 10th Abstr., p. 826. Kauppila, W. E., Stein, T. S., Jesion, G., Dababneh, M. S.,and Pol, V. (1977b). Rev. Sci. lnstrrim. 48, 822. Kauppila, W. E . , Stein, T. S. , Smart, J. H., Dababneh, M. S.,Ho, Y. K.,Downing, J. P., and Pol, V. (1981a).fhys. Rev. A 24, 725. Kauppila, W. E., Dababneh, M. S., Hoffman, K. R., Hsieh, Y.-F.,Pol, V., and Stein, T. S. (1981b). Proc. I n t . Conf. Phys. Electron. Atom. Collisions, 12th Abstr., p. 422.

96

Talbert S . Stein and Walter E . Kauppila

Kennedy, R. E., and Bonham, R. A. (1978). Phys. Rev. A 17, 1844. Kim, Y.-K., and Inokuti, M. (1971). Phys. Re\,. A 3, 665. Leventhal, M . , MacCallum, C. J., and Stang, P. D. (1978). Astrophys. J . 225, L1 I . McEachran, R. P., Morgan, D. L., Ryman, A. G., and Stauffer, A. D. (1977).J. Pliys. B 10, 663. McEachran, R. P.,Ryman, A. G., and Stauffer, A. D. (1978). J . Phys. B 11, 551. McEachran, R. P., Ryman, A. G., and Stauffer, A. D. (1979). J. Pliys. B 12, 1031. McEachran, R. P., Stauffer, A. D., and Campbell, L. E. M. (1980). J. Phys. B 13, 1281. Mandal, P., Ghosh, A. S., and Sil, N. C. (1975). J. Phys. B 8, 2377. Mandal, P., Guha, S., and Sil, N. C. (1979). J. P/iys. B 12, 2913. Mandal, P., Guha, S., and Sil, N. C. (1980). Phys. Rev. A 22, 2623. Massey, H. S. W. (1976). Phys. Todcry 29 (3), 42. Massey, H. S. W., and Moussa, A. H. (1961). Proc. Pliys. Soc. 77, 811. Massey, H. S. W., Lawson, J., and Thompson, D. G. (1966).In “Quantum Theory of Atoms, Molecules, Solid State” (P.-0. Lowdin, ed.), p. 203. Academic Press, New York. Milloy, H. B., and Crompton, R. W. (1977). Phys. Rev. A 15, 1847. Mills, A. P., Jr. (1980). Appl. Phys. L e t t . 37, 667. Mills, A. P., Jr., Platzman, P. M., and Brown, B. L. (1978). Phyh. Rev. Lett. 41, 1076. Montgomery, R. E., and LaBahn, R. W. (1970). C m . J . f l i p s . 48, 1288. Total cross section values were provided by R. E. Montgomery and R. W. LaBahn, (private communication). Murray, C. A., and Mills, A. P., Jr. (1980). Solid Stccte Cornmrrn. 34, 789. Murray, C. A., Mills, A. P., Jr., and Rowe, J. E. (1980). S i y f . Sci. 100, 647. Naccache, P. F., and McDowell, M. R. C . (1974). J . Phys. B 7, 2203. Nesbet, R. K . (1979). J. Phvs. B 12, L243. O’Malley, T. F. (1963). Phys. Re\,. 130, 1020. O’Malley, T. F. (1977). Phys. Lett. 64A, 1%. O’Malley, T. F., Spruch, L., and Rosenberg, L . (1961). J. M d i . Pliys. 2, 491. Ramsauer, C. (1921a). A n n . Phys. ( L e i p z i ~ 64, ) 513. Ramsauer, C. (1921b). Ann. Phys. ( L c i p z i g ) 66, 546. Ramsauer, C. (1923). Ann. Phys. (Leipzig) 72, 345. Ramsauer, C., and Kollath, R. (1929). Ann. Phys. ( L e i p z i g ) 3, 536. Schrader, D. M. (1979). Phys. Re\*. A 20, 918. Sinapius, G., Raith, W., and Wilson, W. G. (1980). J . Phys. B 13, 4079. Stein, T. S., and Kauppila, W. E. (1982). I n “Physics of Electronic and Atomic Collisions” (S. Datz, ed.), p. 311. North-Holland Publ., Amsterdam. Stein, T. S., Kauppila, W. E., and Roellig, L. 0. (1974). Reiq. Sci. Instrrrm. 45, 951. Stein, T. S . , Kauppila, W. E., and Roellig, L. 0. (1975). PIiys. Lett. SlA, 327. Stein, T. S . , Kauppila, W. E., Pol, V., Smart, J. H., and Jesion, G. (1978). Phys. Re\’. A 17, 1600. Stein, T. S., Laperriere, F., Dababneh, M. S. , Hsieh, Y-F., Pol, V., and Kauppila, W. E . (1981). Proc. 1nt. Coi!f: Pkys. Electron. Atom. Collisions, l2th Abstr., p. 424. Tip, A. (1977).J. Phys. B 10, LII. Tong, B. Y. (1972). Phys. Rein. B 5, 1436. Townsend, J. S., and Bailey, V. A. (1922). Philos. M n g . 43, 593. Tsai, J-S., Lebow, L., and Paul, D. A. L. (1976). Ctrn. J . PAys. 54, 1741. Wadehra, J. M.. Stein, T. S., and Kauppila, W. E. (1981). J. Pltys. B 14, L783. Wagenaar, R. W., and de Heer, F. J. (1980). J . Phys. B 13, 3855. Wilson, W. G. (1978). J. Pliys. B 11, L629. Yau, A . W., McEachran, R. P., and Stauffer, A. D. (1978). J . Phys. B 11, 2907.