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Neutralization of Ne+ at GaAs(110)
444
Nuclear
NEUTRALIZATION A. RICHARD Physikalisches
Instruments
and Methods
in Physics Research B2 (1984) 444-447 North-Holland, Amsterdam
OF Ne + AT GaAs(ll0)
and H. ESCHENBACHER
Institut der Universitiit Wiirzburg Am Hubland, D - 8700 Wiirzburg, Germany
We measured the ion yield of Ne+ scattered in a single collision from GaAs(ll0) in the energy range between 300 and 1200 eV. The energy dependence of the ion yield divided by the differential cross section for elastic scattering could not be explained by a simple Hagstrum model for neutralization. Exact integration of a local transition rate over the whole trajectory gives excellent agreement with the measurement.
1. Introduction
perpendicular
Low energy ion scattering (LEIS) for the study of the composition and structural arrangement of solid surfaces has developed as a powerful tool during the past few years [l]. The collision can be well described by a binary collision mode1 [2]. The positions of the peaks in LEISspectra are determined by the mass of the target atoms while the intensities of the peaks are determined by the cross section for elastic scattering and the widely unknown ion survival probability P,. The neutralization of the impinging noble gas ions has been mainly explained by a simple Hagstrum mode1 [3]. The physics of the neutralization is described by the transition rate R,(s) which can be expressed in a first order approximation by Fermi’s golden rule [4]
R,(s)
=~l(4W)12~r~
where the perturbation Hamiltonian V describes the physical process of the neutralization. The ion survival probability P, is finally obtained by [3] P, = exp( - jtmR,(s)dt), --m
where the time integral is taken along the trajectory of the incoming particle. One usually makes the approximations dt = ds/v,
, vI = constant.
r, = 0, of the projectile where vI is the velocity component perpendicular to the surface and rc the classical turning point. Together with an exponential dependence of the transition rate
R,(s)=A exp(-ys) the well known dependence of the ion survival probability on the incoming and outgoing velocity components 0168-583X/84/$03.00 Q Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
to the surface is obtained
[S]
This description seems to be valid for a whole class of ion-target combinations [6]. In fig. 3 the dependency of P, as described with this model is shown as a dashed curve with A/y = 2 X 10'cm/s. For this calculation a scattering of Ne at As was assumed with a scattering angle 9 = 90’ and a polar incidence angle of Z/J= 45’ referred to the surface. If there is a quasiresonant level a strong local interaction was assumed [7]. In certain cases a neutralization behaviour was observed which could not be explained by one of the mentioned processes [6]. Our measurements and model calculations in the following sections could give some answers to those problems.
2. Experimental approach The measurements were carried out under ultra high vacuum conditions with a base pressure of about 2 X lo-’ Pa. The UHV-chamber is equipped with a sputter gun and a flash filament for preparing the target, a quadrupole mass filter for monitoring the vacuum conditions and with an electrostatic 127”-analyser which was used to prove the cleanliness of the target with AES and LEIS. Neon ions were produced in a Bayard-Alpert-like electron impact source. The isotopes were not mass separated. The scattering angle was fixed at 90’. The polar incidence angle IJ was varied. The azimuthal angle Q, was set to 15’ relative to the [OOl]-direction. Scattered ions were analysed with an electrostatic 127’analyser with a channeltron at the exit. The analyser was operated with constant energy resolution and the transmitted ions were postaccelerated to a constant energy such that no correction was necessary for any
A. Richard, H. Eschenbacher
of Ne + at GaAs(Il0)
/ Neutralization
variation of the detection efficiency. At each analyser energy the measured current was integrated up to 20 x 10e9 C. The spectra were finally corrected due to the secondary electron emission. Prior to each measurement the GaAs crystal was sputtered with 1 keV, 10m3 C Ne+ at nearly glancing incidence and then annealed to 550°C for 10 min. This procedure gives a well ordered surface [8]. With AES neither oxygen nor carbon contaminations could be detected.
445
Ne -GaAs(llO) -
0.5 -
E. = 1200eV 9 = 900 $ = 800
3. Experimental results 0.50 Fig. 1 shows a typical LEIS-spectrum of Ne+ scattered from GaAs(ll0). The experimental condition of a nearly glancing exit angle allowed a distinction between Ne scattered at As (atomic mass 75) and Ne scattered at Ga (atomic mass about 70). The spectra were analysed with a least squares fit which takes into account the masses of all isotopes involved in the scattering. The functional form of the single scattering peak of each isotope pair was assumed to be Gaussian. The measured ion yield divided by the differential cross section for elastic scattering for which we used a TFM potential with a screening length according to Firsov [9] is shown in fig. 2 for four different projectile energies. The energy dependence cannot be explained with the above mentioned Hagstrum model. The unexpected high yield at low energy lets us assume that the neutralization was more effective if the ion approached closer to the surface.
0.54
0.56
0.62
E/E.
Fig. 1. A LEIS-spectrum of Ne scattered from GaAs(ll0) at a nearly glancing exit angle can be well resolved into single scattering peaks of all isotope projectile target pairs. The sum ofallNe-rAs(.-.-.)andNe-,Ga(---)singlescattering peaks is shown as a solid curve and agrees well with the measurements. The azimuth angle was 15’ relative to the [OOl]-direction to minimize blocking effects.
The parameters (Y and fi were adjusted to a TFM potential with a screening length according to Firsov [9]. Calculations have shown that the matched potential agrees excellently with the TFM potential, at least for large angle scattering. The advantage of this potential is that all relevant quantities of the scattering can be calculated analytically. With this potential the ion survival probability turns out, to a very good approximation, to be
4. Model of local neutralization K, the modified Bessel function of the first kind and rE the classical turning point. A fit of this function with
In this model
the time integral
of a local transition
rate R,(r)=Aexp(-yr) is taken along the trajectory. Here the distance r between projectile and target atom is used rather than the distance s between projectile and the surface. In the following the resulting integral is called the local path integral. We integrated exactly with the substitution dr = dr/u. The velocity was calculated classically
where rrt is the reduced mass of projectile and target, E the kinetic energy in the center of mass system, p the impact parameter and V the interaction potential between projectile and target atom for which we used a matched potential of the form [lo]
Y(r)=;[(f)‘-11.
1.01
_I
Ne+-
‘h
GaAs
(110)
I
300
600 projectile
energy
900 teV)
1200
Fig. 2. The ion yield divided by the cross section for elastic scattering (0) was fitted with the local ion survival probability function (full curve). The error bars are due to the counting statistics. The agreement between experiment and fit is encouraging. VI. LOW ENERGY
SURFACE
INTERACTIONS
446
A. Richard H. Eschenbacher / Neutralization of Ne + at GaAs(ll0)
Table 1 # fro 16s-1 1 Ne-As
80” 60” 80’ 60’
Ne-Ga
2.1 3.0 2.4 2.9
2.8 2.9 2.5 2.8
to our experimental data gives excellent agreement (see fig. 2). The fit parameters A and y are summarized in table 1. l/y = 0.36 A as obtained from the fits indicates a very local neutralization behaviour. Woodruff [ll] and Preuss [12] suggested, that a local neutralization model could be responsible for some of their experimental findings. Therefore we made some model calculations to investigate the influence of y and A to the ion survival probability. Fig. 3 shows the result of these calculations. A/y was kept constant because in
y = 3.10s cm-l A/y
= 5.107 cm/s
= const. 2 lOa cm-l
locals,
/ / ,
\ “7; /
global path integral A= 1.5 .10’6s-’
.<3 --
.lOs cm-’ --_
/ / /
Aly = 2.107 cm/s “Hagstrummodel”
I
300
600 900 projectile energy (eV)
1200
Fig. 3. A model calculation with the local path integral shows a drastic dependence of P, on y. The trajectory was arbitrarily chosen to describe a Ne + As scattering with a scattering angle 9 of 90’. For comparison a plot of the global path integral (dashed-dotted line), calculated with I/J= 70” and a plot of the commonly used “Hagstrum model” (dashed line), calculated with $ = 45O are included.
the simple Hagstrum model the ion survival probability should be independent of A/y. The energy dependence of P, changes drastically with increasing l/y and finally reaches a form which can explain some of the earlier experiments [13]. Included in fig. 3 are the results of two other model calculations. The first is a plot of formula (2) denoted as the “Hagstrum model” in fig. 3. A/y was reduced to 2 X 10’ cm/s due to the different boundary conditions in the integrations to give a “reasonable” result. The calculation was performed for Ne As scattering with 6 = 90”, # = 45”. In the second calculation we integrated the global transition rate R,(s)=A
exp(-ys)
along the trajectory. This integral is called global path integral in the following. As the scattering parameters we chose 9 = 90’ and I$ = 70”. The transition rate parameters A and y were chosen such that the survival probability obtained with the global path integral should be compared with the topmost curve of fig. 3. The curvatures of both curves are nearly identical indicating that the curvature is determined by the slowing down of the projectile and the neutralization in the vicinity of the target atom. However, the survival probabilities calculated with the global path integral for I,!J= 80’ are a factor lo5 smaller than for 4 = 60’ whereas the measured ion yield was almost the same for both angles of incidence. Some comments should be made concerning the validity of the path integrals used in the calculations. A commonly accepted assumption was that for projectile energies between some hundred eV and some keV for large scattering angles the electron clouds of projectile and target atoms interpenetrate deeply before the actual scattering process takes place and the impinging projectile loses its memory of the initial charge state [14]. Contrary to this assumption, Buck et al. concluded from charge state measurements of neon scattered from nickel that the projectile after neutralization on the incoming path is re-ionized only if the distance of closest approach is smaller than the Ni M-shell radius [15]. This assumption can be supported if one uses neuttals as impinging particles. Verhey et al. determined an onset of the ionization probability of neutral He scattered at copper between 2 and 3 keV [16]. The distances of closest approach for 30’ scattering in this energy region are 0.24-0.3 A. This is just inside the Ni M-shell which has a radius of 0.34 A [17]. In our experiments the classical turning point was always greater than 0.52 A which is larger than the radii of the Ga or As M-shells of about 0.29 A [17]. So re-ionization in the close collision should be negligible. A further question is the spatial distribution of the electrons involved in the neutralization process. The local density of states calculations of Chadi [18] show that the valence 4s and 4p electrons of Ga and As are
A. Richard, H. Eschenbacher strongly ute
localized;
to
the
arguments zation
but
only
neutralization together
support
those of
the
electrons
can
incoming
the outlined
local
/ Neutralization
contrib-
ion.
Both
neutrali-
model.
5. Conclusion
We measured the ion yield of Ne+ scattered from GaAs(ll0) in the energy range between 300 and 1200 eV. The ion yield divided by the differential cross section for elastic scattering decreased with increasing energy. This anomalous behaviour could be explained with a local neutralization model in which a local transition rate was integrated exactly along the trajectory. Comparison of the classical turning points of the trajectories with shell radii of the target atoms and different model calculations support the suggestion, that the local neutralization model gives a reasonable description of the experimental findings. The published experimental results should be re-examined to give a better understanding of the neutralization of noble gas ions at solid surfaces.
References [l] E. Taglauer and W. Heiland, Appl. Phys. 9 (1976) 261. [2] E. Hulpke and U. Gerlach-Meyer, Vakuum-Technik 25 (1977) 233.
441
of Ne + at GaAs(Il0)
in: Electron and ion spectroscopy of [31 H.D. Hagstrum, solids, eds., L. Fiermans, J. Vennik and W. Dekeyser (Plenum Press, New York, 1978) p. 273. [41 C.A. Moyer and K. Orvek, Surf. Sci. 114 (1982) 295. 151 D.P. Woodruff, Nucl. Instr. and Meth. 194 (1982) 639. 161 T.W. Rusch and R.L. Erickson, in: Proc. 2nd Intern. Conf. on Inelastic ion surface collisions, eds., N.H. ToIk, J.C. Tully, W. Heiland and C.W. White (Academic Press, New York, San Francisco, London, 1977) p. 73. [71 N.H. Tolk and J. Kraus, in: Coherence and correlation in atomic collisions, eds., H. Kleinpoppen and J.F. Williams (Plenum, New York, 1980) p. 533. PI A.U. MacRae and G.W. Gobeli, J. Appl. Phys. 35 (1964) 1629. [91 M.T. Robinson and I.M. Torrens, Phys. Rev. B9 (1974) 5008. Phys. Rev. 134 (1964) WI C. Lehmann and M.T. Robinson, A37. Surf. Sci. 105 (1981) 1111 D.J. Godfrey and D.P. Woodruff, 438. WI E. Preuss, Surf. Sci. 110 (1981) 287. and A. Richard, these Proceedings (131 H. Eschenbacher (ICACS-10) p. 411. [I41 H.H. Brongersma and T.M. Buck, Nucl. Instr. and Meth. 132 (1976) 559. 1151 T.M. Buck, G.H. Wheatley and L.K. Verhey, Surf. Sci. 90 (1979) 635. 1161 L.K. Verhey, B. Poelsema and A.L. Boers, Nucl. Instr. and Meth. 132 (1976) 565. theory of matter, (MacGraw-Hill, v71 J.C. Slater, Quantum New York, 1968) p. 150. WI D.J. Chadi, Phys. Rev. B18 (1978) 1800.
VI. LOW ENERGY
SURFACE
INTERACTIONS
Nuclear
NEUTRALIZATION A. RICHARD Physikalisches
Instruments
and Methods
in Physics Research B2 (1984) 444-447 North-Holland, Amsterdam
OF Ne + AT GaAs(ll0)
and H. ESCHENBACHER
Institut der Universitiit Wiirzburg Am Hubland, D - 8700 Wiirzburg, Germany
We measured the ion yield of Ne+ scattered in a single collision from GaAs(ll0) in the energy range between 300 and 1200 eV. The energy dependence of the ion yield divided by the differential cross section for elastic scattering could not be explained by a simple Hagstrum model for neutralization. Exact integration of a local transition rate over the whole trajectory gives excellent agreement with the measurement.
1. Introduction
perpendicular
Low energy ion scattering (LEIS) for the study of the composition and structural arrangement of solid surfaces has developed as a powerful tool during the past few years [l]. The collision can be well described by a binary collision mode1 [2]. The positions of the peaks in LEISspectra are determined by the mass of the target atoms while the intensities of the peaks are determined by the cross section for elastic scattering and the widely unknown ion survival probability P,. The neutralization of the impinging noble gas ions has been mainly explained by a simple Hagstrum mode1 [3]. The physics of the neutralization is described by the transition rate R,(s) which can be expressed in a first order approximation by Fermi’s golden rule [4]
R,(s)
=~l(4W)12~r~
where the perturbation Hamiltonian V describes the physical process of the neutralization. The ion survival probability P, is finally obtained by [3] P, = exp( - jtmR,(s)dt), --m
where the time integral is taken along the trajectory of the incoming particle. One usually makes the approximations dt = ds/v,
, vI = constant.
r, = 0, of the projectile where vI is the velocity component perpendicular to the surface and rc the classical turning point. Together with an exponential dependence of the transition rate
R,(s)=A exp(-ys) the well known dependence of the ion survival probability on the incoming and outgoing velocity components 0168-583X/84/$03.00 Q Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
to the surface is obtained
[S]
This description seems to be valid for a whole class of ion-target combinations [6]. In fig. 3 the dependency of P, as described with this model is shown as a dashed curve with A/y = 2 X 10'cm/s. For this calculation a scattering of Ne at As was assumed with a scattering angle 9 = 90’ and a polar incidence angle of Z/J= 45’ referred to the surface. If there is a quasiresonant level a strong local interaction was assumed [7]. In certain cases a neutralization behaviour was observed which could not be explained by one of the mentioned processes [6]. Our measurements and model calculations in the following sections could give some answers to those problems.
2. Experimental approach The measurements were carried out under ultra high vacuum conditions with a base pressure of about 2 X lo-’ Pa. The UHV-chamber is equipped with a sputter gun and a flash filament for preparing the target, a quadrupole mass filter for monitoring the vacuum conditions and with an electrostatic 127”-analyser which was used to prove the cleanliness of the target with AES and LEIS. Neon ions were produced in a Bayard-Alpert-like electron impact source. The isotopes were not mass separated. The scattering angle was fixed at 90’. The polar incidence angle IJ was varied. The azimuthal angle Q, was set to 15’ relative to the [OOl]-direction. Scattered ions were analysed with an electrostatic 127’analyser with a channeltron at the exit. The analyser was operated with constant energy resolution and the transmitted ions were postaccelerated to a constant energy such that no correction was necessary for any
A. Richard, H. Eschenbacher
of Ne + at GaAs(Il0)
/ Neutralization
variation of the detection efficiency. At each analyser energy the measured current was integrated up to 20 x 10e9 C. The spectra were finally corrected due to the secondary electron emission. Prior to each measurement the GaAs crystal was sputtered with 1 keV, 10m3 C Ne+ at nearly glancing incidence and then annealed to 550°C for 10 min. This procedure gives a well ordered surface [8]. With AES neither oxygen nor carbon contaminations could be detected.
445
Ne -GaAs(llO) -
0.5 -
E. = 1200eV 9 = 900 $ = 800
3. Experimental results 0.50 Fig. 1 shows a typical LEIS-spectrum of Ne+ scattered from GaAs(ll0). The experimental condition of a nearly glancing exit angle allowed a distinction between Ne scattered at As (atomic mass 75) and Ne scattered at Ga (atomic mass about 70). The spectra were analysed with a least squares fit which takes into account the masses of all isotopes involved in the scattering. The functional form of the single scattering peak of each isotope pair was assumed to be Gaussian. The measured ion yield divided by the differential cross section for elastic scattering for which we used a TFM potential with a screening length according to Firsov [9] is shown in fig. 2 for four different projectile energies. The energy dependence cannot be explained with the above mentioned Hagstrum model. The unexpected high yield at low energy lets us assume that the neutralization was more effective if the ion approached closer to the surface.
0.54
0.56
0.62
E/E.
Fig. 1. A LEIS-spectrum of Ne scattered from GaAs(ll0) at a nearly glancing exit angle can be well resolved into single scattering peaks of all isotope projectile target pairs. The sum ofallNe-rAs(.-.-.)andNe-,Ga(---)singlescattering peaks is shown as a solid curve and agrees well with the measurements. The azimuth angle was 15’ relative to the [OOl]-direction to minimize blocking effects.
The parameters (Y and fi were adjusted to a TFM potential with a screening length according to Firsov [9]. Calculations have shown that the matched potential agrees excellently with the TFM potential, at least for large angle scattering. The advantage of this potential is that all relevant quantities of the scattering can be calculated analytically. With this potential the ion survival probability turns out, to a very good approximation, to be
4. Model of local neutralization K, the modified Bessel function of the first kind and rE the classical turning point. A fit of this function with
In this model
the time integral
of a local transition
rate R,(r)=Aexp(-yr) is taken along the trajectory. Here the distance r between projectile and target atom is used rather than the distance s between projectile and the surface. In the following the resulting integral is called the local path integral. We integrated exactly with the substitution dr = dr/u. The velocity was calculated classically
where rrt is the reduced mass of projectile and target, E the kinetic energy in the center of mass system, p the impact parameter and V the interaction potential between projectile and target atom for which we used a matched potential of the form [lo]
Y(r)=;[(f)‘-11.
1.01
_I
Ne+-
‘h
GaAs
(110)
I
300
600 projectile
energy
900 teV)
1200
Fig. 2. The ion yield divided by the cross section for elastic scattering (0) was fitted with the local ion survival probability function (full curve). The error bars are due to the counting statistics. The agreement between experiment and fit is encouraging. VI. LOW ENERGY
SURFACE
INTERACTIONS
446
A. Richard H. Eschenbacher / Neutralization of Ne + at GaAs(ll0)
Table 1 # fro 16s-1 1 Ne-As
80” 60” 80’ 60’
Ne-Ga
2.1 3.0 2.4 2.9
2.8 2.9 2.5 2.8
to our experimental data gives excellent agreement (see fig. 2). The fit parameters A and y are summarized in table 1. l/y = 0.36 A as obtained from the fits indicates a very local neutralization behaviour. Woodruff [ll] and Preuss [12] suggested, that a local neutralization model could be responsible for some of their experimental findings. Therefore we made some model calculations to investigate the influence of y and A to the ion survival probability. Fig. 3 shows the result of these calculations. A/y was kept constant because in
y = 3.10s cm-l A/y
= 5.107 cm/s
= const. 2 lOa cm-l
locals,
/ / ,
\ “7; /
global path integral A= 1.5 .10’6s-’
.<3 --
.lOs cm-’ --_
/ / /
Aly = 2.107 cm/s “Hagstrummodel”
I
300
600 900 projectile energy (eV)
1200
Fig. 3. A model calculation with the local path integral shows a drastic dependence of P, on y. The trajectory was arbitrarily chosen to describe a Ne + As scattering with a scattering angle 9 of 90’. For comparison a plot of the global path integral (dashed-dotted line), calculated with I/J= 70” and a plot of the commonly used “Hagstrum model” (dashed line), calculated with $ = 45O are included.
the simple Hagstrum model the ion survival probability should be independent of A/y. The energy dependence of P, changes drastically with increasing l/y and finally reaches a form which can explain some of the earlier experiments [13]. Included in fig. 3 are the results of two other model calculations. The first is a plot of formula (2) denoted as the “Hagstrum model” in fig. 3. A/y was reduced to 2 X 10’ cm/s due to the different boundary conditions in the integrations to give a “reasonable” result. The calculation was performed for Ne As scattering with 6 = 90”, # = 45”. In the second calculation we integrated the global transition rate R,(s)=A
exp(-ys)
along the trajectory. This integral is called global path integral in the following. As the scattering parameters we chose 9 = 90’ and I$ = 70”. The transition rate parameters A and y were chosen such that the survival probability obtained with the global path integral should be compared with the topmost curve of fig. 3. The curvatures of both curves are nearly identical indicating that the curvature is determined by the slowing down of the projectile and the neutralization in the vicinity of the target atom. However, the survival probabilities calculated with the global path integral for I,!J= 80’ are a factor lo5 smaller than for 4 = 60’ whereas the measured ion yield was almost the same for both angles of incidence. Some comments should be made concerning the validity of the path integrals used in the calculations. A commonly accepted assumption was that for projectile energies between some hundred eV and some keV for large scattering angles the electron clouds of projectile and target atoms interpenetrate deeply before the actual scattering process takes place and the impinging projectile loses its memory of the initial charge state [14]. Contrary to this assumption, Buck et al. concluded from charge state measurements of neon scattered from nickel that the projectile after neutralization on the incoming path is re-ionized only if the distance of closest approach is smaller than the Ni M-shell radius [15]. This assumption can be supported if one uses neuttals as impinging particles. Verhey et al. determined an onset of the ionization probability of neutral He scattered at copper between 2 and 3 keV [16]. The distances of closest approach for 30’ scattering in this energy region are 0.24-0.3 A. This is just inside the Ni M-shell which has a radius of 0.34 A [17]. In our experiments the classical turning point was always greater than 0.52 A which is larger than the radii of the Ga or As M-shells of about 0.29 A [17]. So re-ionization in the close collision should be negligible. A further question is the spatial distribution of the electrons involved in the neutralization process. The local density of states calculations of Chadi [18] show that the valence 4s and 4p electrons of Ga and As are
A. Richard, H. Eschenbacher strongly ute
localized;
to
the
arguments zation
but
only
neutralization together
support
those of
the
electrons
can
incoming
the outlined
local
/ Neutralization
contrib-
ion.
Both
neutrali-
model.
5. Conclusion
We measured the ion yield of Ne+ scattered from GaAs(ll0) in the energy range between 300 and 1200 eV. The ion yield divided by the differential cross section for elastic scattering decreased with increasing energy. This anomalous behaviour could be explained with a local neutralization model in which a local transition rate was integrated exactly along the trajectory. Comparison of the classical turning points of the trajectories with shell radii of the target atoms and different model calculations support the suggestion, that the local neutralization model gives a reasonable description of the experimental findings. The published experimental results should be re-examined to give a better understanding of the neutralization of noble gas ions at solid surfaces.
References [l] E. Taglauer and W. Heiland, Appl. Phys. 9 (1976) 261. [2] E. Hulpke and U. Gerlach-Meyer, Vakuum-Technik 25 (1977) 233.
441
of Ne + at GaAs(Il0)
in: Electron and ion spectroscopy of [31 H.D. Hagstrum, solids, eds., L. Fiermans, J. Vennik and W. Dekeyser (Plenum Press, New York, 1978) p. 273. [41 C.A. Moyer and K. Orvek, Surf. Sci. 114 (1982) 295. 151 D.P. Woodruff, Nucl. Instr. and Meth. 194 (1982) 639. 161 T.W. Rusch and R.L. Erickson, in: Proc. 2nd Intern. Conf. on Inelastic ion surface collisions, eds., N.H. ToIk, J.C. Tully, W. Heiland and C.W. White (Academic Press, New York, San Francisco, London, 1977) p. 73. [71 N.H. Tolk and J. Kraus, in: Coherence and correlation in atomic collisions, eds., H. Kleinpoppen and J.F. Williams (Plenum, New York, 1980) p. 533. PI A.U. MacRae and G.W. Gobeli, J. Appl. Phys. 35 (1964) 1629. [91 M.T. Robinson and I.M. Torrens, Phys. Rev. B9 (1974) 5008. Phys. Rev. 134 (1964) WI C. Lehmann and M.T. Robinson, A37. Surf. Sci. 105 (1981) 1111 D.J. Godfrey and D.P. Woodruff, 438. WI E. Preuss, Surf. Sci. 110 (1981) 287. and A. Richard, these Proceedings (131 H. Eschenbacher (ICACS-10) p. 411. [I41 H.H. Brongersma and T.M. Buck, Nucl. Instr. and Meth. 132 (1976) 559. 1151 T.M. Buck, G.H. Wheatley and L.K. Verhey, Surf. Sci. 90 (1979) 635. 1161 L.K. Verhey, B. Poelsema and A.L. Boers, Nucl. Instr. and Meth. 132 (1976) 565. theory of matter, (MacGraw-Hill, v71 J.C. Slater, Quantum New York, 1968) p. 150. WI D.J. Chadi, Phys. Rev. B18 (1978) 1800.
VI. LOW ENERGY
SURFACE
INTERACTIONS