- Email: [email protected]
Two-target-atom model for calculating cross sections of direct recoils
127
Surface Science 255 (1991) 127-134 North-Holland
Two-target-atom Shiladitya Department
model for calculating cross sections of direct recoils
Chaudhury of Physics and Solid State Science Center, UCLA, Los Angeles, CA 90024-1547,
USA
and R. Stanley Department Received
Williams
of Chemistry and Biochemistry
22 January
1991; accepted
and Solid State Science Cenfer, UCLA, Los Angeles,
for publication
21 March
CA 90024-1569,
USA
1991
A new two-target-atom model for calculating the intensity of particles that have been directly recoiled from a surface has been developed to examine the effects of shadowing and blocking on the recoils. This model is similar to one that was recently introduced for calculating the intensity of the primary particles scattered back from a surface. Important surface structure information can be obtained by comparing calculated recoil intensities as a function of incident ion identity and kinetic energy, target orientation, and recoil angle and energy to experimentally measured quantities.
1. Introduction Direct recoil spectroscopy (DRS) is an emerging technique for determining structural features, such as bond angles and interatomic distances, at crystalline surfaces. The technique is an extension of impact-collision ion scattering spectroscopy (ICISS) [l] inasmuch as the intensity of the particles detected at a particular energy is measured as a function of the angle between the incident ion beam and the surface being investigated. In DRS the measured particles are surface atoms that have recoiled directly after collision with an incident ion. This ion scattering technique is especially useful for examining surfaces on which a lower atomic mass species is adsorbed on a higher mass substrate. The yield of recoiled particles increases or decreases as near-surface atoms are exposed to variations in the ion flux caused by the shadowing and blocking effects of other surface atoms. Rabalais and coworkers [2,3] have already shown that interatomic distances can be determined from DRS experiments by established analysis of the 0039-6028/91/$03.50
0 1991 - Elsevier
Science Publishers
shape of shadow and blocking cones [4]. The two-atom model presented here was developed to make such determinations more quantitative. Recently, Williams et al. [5] introduced a new formalism to calculate three-dimensional crosssections for an ion to scatter sequentially and classically from two atoms. This method has been applied to the interpretation of ICISS experimental data. In this paper, a similar formalism has been developed to calculate the three-dimensional cross section for an ion to recoil one of two target atoms from a surface. The collisions are classical and involve sequential forward scattering and recoil events that are confined to a plane into a fixed total scattering angle. Classical trajectory simulations that have traditionally been employed to assist in recoil spectroscopy data analysis can be very CPU intensive [2,6,7]. The model described here is computationally very fast and includes two cases for a given interatomic spacing of the target atoms: the case where the first event is a recoil, followed by the scattering of the recoiled particle from a second
B.V. (North-Holland)
S. Chaudhuty, R.S. Williams / Two-target-atom
128
atom; and where the first event is a scattering event followed by a recoil. The vibrational motions of the target atoms and the energy resolution of the detector are taken into consideration to calculate the intensity of the recoiled ions as a function of the angle between the incoming ion beam and the line joining the two atoms. In the next section, the mathematical details of the two-atom model are presented. The computer implementation is discussed in section 3. Results from the simulation programs are discussed in section 4. Section 5 contains the conclusions of the paper.
model for calculating cross sections of direct recoils
2.2. Cross sections for scattering/recoiling
For scattering from some arbitrary three-dimensional target into an element of solid angle dS2, for a particular incident flux I, and a particular orientation of the scatterer, one can write the generalized cross-section formula (Williams et al. [J]):
(4) where D is the Jacobian
2. Mathematical
details
D=
For a particle of energy E, and mass MP colliding elastically with a particle of mass M, at rest, the principles of conservation of energy and momentum allow us to describe the final energies of the particles and their directions relative to the original angle of incidence (Marion [8]). If the energy after collision of the incident particle is E,, and that of the recoil particle is E,, then the following relations for the energy ratios I&,/E, and E,/E, hold:
X [cOS 0,+
J( M,/M,)2
ace,, kd ah, +d
2.1. Classical binary collisions
- sin28,]’
(1)
in three
dimensions
determinant
_ ab ---ah,,,
as, ah
ah. ah
a+, as,.
(5)
All angles in this treatment are measured in the laboratory coordinate system. For an isolated atom we need to consider only a single binary collision that is characterized by axial symmetry. In this case &9,/a+, = 0 and a&,,,,/a+, = 0 SO that the expressions for the cross-sections may now be written as:
(6) for the scattered particle, and
for the recoiled particle.
and E 1=
4$3f,
Eo
(Mp+MJ2
cos”e,,
where eP and 6, are the laboratory scattering and recoil angles respectively (see fig. 1). In eq. (1) the + sign should be used if MP > M,, otherwise the + sign should be used. This work uses only the + sign, since we intentionally chose the mass of the scattering center to be larger than the projectile. A further relation between the angles 0, and 0, can be written as (Marion [S]): sin
8, =
MPEPsin e,. ME
J
r r
x EP
”
eP
__-_---
MP
“&__>___.
‘0,
Mr \t
Er
Binary Elastic Collision Fig. 1. Schematic drawing showing relevant parameters of a binary elastic collision. A particle with mass Mp and energy E,, scatters at an impact parameter of s from a particle of mass M, initially at rest. The scattering and recoil angles are O,, and O,, respectively.
129
S. Chaudhury, R.S. Williams / Two-target-atom model for calculating cross sections of direct recoils
Scattered Projectile Recoiled Target
_
4$ 0 L 0
200.0
0.2 .
Fig. 2,. Dependence on impact parameter of (a) the scattering angle and scattering cross section for a lighter projectile from a heavier target and (b) the recoil angle and recoil cross section for a heavier projectile from a lighter target in binary collisions. In (a) Arf is incident upon W and in (b) Ar+ recoils 0.
The cross sections for each of these cases is evaluated by first determining the deflection function @,(s,) or el(s,) from the scattering potential and then calculating the derivative with respect to the impact parameter. Calculating the cross-section for the scattering case allows us to easily obtain the cross-section for the recoil case as is derived in standard texts [8,9],
(8) For a strictly repulsive potential and A4r < M,, the deflection functions are monotonic and the derivatives are well behaved. The conditions for the collision for which we present the cross section in fig. 2a were as follows: MP = 39.9 amu (Ar), M, = 183.85 amu (W) and the incident particle energy was 5 keV. At zero impact parameter, a lighter projectile scatters back at 180” from a heavier target with a small cross section, and the cross section increases as the scattering angle decreases to zero. For fig. 2b the projectile was also 5 keV Ar, but this time M, = 16 amu (0). Fig. 2b shows the dependence of the recoil scattering angle and cross section on the impact parameter: the recoil angle increases from zero to a limiting value of 90 o as the cross section increases [lO,ll]. These are the types of interactions that will be examined for the case of a projectile interacting with two target atoms.
Figs. 3a and 3b illustrate the cases where the final detected particle is the result of two sequential events - one scattering and one recoil. In fig. 3a, the first event is a forward scattering event followed by a recoil, which we call a shadowing trajectory. The total recoil angle obeys the condition 4 = @&l) + @,(%L
(9)
where 8, is the angle of the recoil detector with respect to the incident ion beam. The angle through which the incident particle is scattered, In,, is a function of the impact parameter at the first target atom, and @,(s,) is the scattering angle of the recoiled particle, as a function of the impact parameter at the second target atom. In fig. 3b, the first event is a recoil event, followed by the scattering of the recoiled particle from a second atom, which we call a blocking trajectory. This event satisfies the condition 4_ = 4M
+ e,,(+).
00)
In this instance, the dependence of the total recoil angle B, is on the recoil angle of the first target atom &(s,), and the angle through which it is scattered at the second target atom t9,,(+). In both these cases axial symmetry is broken, so &#+,,,,/a+, # 1, but for an experimental setup
130
S. Chaudhury, R.S. Williams / Two-target-atom
Shadowing
Trajectory
model for calculating cross sections of direct recoils
sional cross-section
for planar scattering,
In the following discussion, we shall develop the cross section for the shadowing trajectory; the development for the blocking trajectory is similar. In DRS, as with ICISS, the total scattering angle 8, is fixed ad the cross section is desired as a function of (Y, the angle between the incident ion trajectory and the line connecting the two target atoms. The factor &$,,,,/&#B, in the Jacobian is determined from geometrical considerations and can be shown [5] to be: X %U, a+, =G
Isin4 I I sin 8, cos f3, sm8,
sin 8,
’
(12)
where X is the line segment in fig. 3a, X=R
Blocking
Trajectory
M Awn1
cMjon
sin~+s2cos$,.
(13)
This planar configuration of the atoms ignores both vibrational motion of the atoms perpendicular to the scattering plane and any interplanar (zig-zag) scattering. Following the methods set forth previously [5], the expression (Y= eP + sin-’
s2
[
+
s1
cos
R
ep I
is evaluated as a function of sr. The derivative in the Jacobian determinant, eq. (ll), can be evaluated numerically for a fixed value of (Y, and a(B,, a) can then be computed.
Fig. 3. Illustration of (a) shadowing and (b) blocking trajectories. In both cases the incoming ion (Ar+ ) mass is between the masses of the target atoms. In (a) atom 1 is the heavier atom (W) while in (b) the heavier atom is atom 2. The relevant parameters necessary to calculate (Yas a function of s, are O,, R, s2 and O,, as indicated.
where the ion beam, detector and the two atoms are coplanar, then it is still true that %3,/a+, = 0 [5]. Thus, we can still write down a three-dimen-
3. Computer implementation The calculations are performed using a suite of programs adapted from the routines of Williams et al. [5]. The first program calculates single-atom deflection functions, i.e., laboratory scattering angle e as a function of impact parameter for a particular projectile energy, and writes a table to a disk file. A recoil deflection table is then calculated from eq. (3) using the first table as input. The scattering potential used most often is the Moliere [12] with the Firsov [13] screening con-
S. Chaudhury, R.S. Williams / Two-target-atom
stant modified through experience gained from simulations of ICISS data. The only input parameters needed for these deflection tables are the atomic number and mass of the projectile and target, the laboratory energy of the projectile, and a constant to scale the Firsov screening length. Tables for each of the different binary interactions are required to fully simulate both the shadowing and blocking trajectories. A program that calculates (Y versus s1 and the total u versus a! distribution for each trajectory is then called, with the appropriate scattering tables as input. The other inputs to these programs are the total scattering angle (the detector angle) and R, the interatomic distance. The divergence in the calculated cross section at the critical angle for recoiling is avoided by utilizing the same type of semiclassical approximation discussed by Daley et al. [14]. The u versus (Yand the E,/E, versus (Ydistributions (figs. 5 and 6) then become the inputs to a program that includes thermal vibrations and the energy resolution of the detector [l]. In order to correctly account for the dispersion in the energy of the detected particle, a two-dimensional thermal broadening approach was developed. First, the values of the cross sections for each trajectory at specific values of the polar angle 1y (up to the laboratory scattering angle) were tabulated against the specific energies at which they would be detected. These energy spectra were then broadened with a Gaussian function that accounts for the energy resolution of the detector. The effects of thermal vibrations were incorporated by choosing a value of the energy and broadening the corresponding 1y distribution of the intensity with a Gaussian that represents the range of relative thermal displacements between the atoms [5]. Thus one can simulate a DRS angular distribution for a given energy or obtain an energy spectrum of the recoiled particle at a given value of the polar angle.
4. Results and discussion Calculations are presented for what promises to be the most useful of the various mass configura-
131
model for calculating cross sections of direct recoils 60
^
40
E s
20
0.0
0.1
0.2
0.3
0.4
0.5
S,(A) Fig. 4. Plot of a versus s, for 5 keV Ar+ scattering from a W and an 0 atom 3 A apart shown for (a) shadowing and (b) blocking trajectories. The laboratory scattering angle was 60 O. The minimum allowed value of a in (a) is the shadowing critical angle while the maximum allowed value in (b) is the blocking critical angle.
tions involving the two target atoms and the incident ion beam: MP > M, > M,; where M, and M, are the masses of the two target atoms and MP is the mass of the incident ion. In particular, we have calculated the cross sections for both shadowing and blocking trajectories for 5 keV Ar+ ions incident on a W-O atom pair separated by 3 A. The 0 atom is recoiled into a detector at a fixed angle of 60 o with respect to the incident ion direction. A schematic figure showing the various parameters for each trajectory is presented in figs. 3a and 3b. Since the trajectories investigated here are a combination of binary scattering and recoil events, calculations for the cross sections of the single binary events were first performed to ensure the validity of the computer programs. Figs. 2a and 2b represent the results of these calculations for the case of Ar+ scattering from W and 0 recoiling from Ar. The first part of the two atom calculations are presented in fig. 4. Here, (Y,the angle between the incident particle and the axis of the two target atoms, is plotted against si, the initial impact
S. Chaudhury, R.S. Williams / Two-target-atom
132
parameter (see figs. 3a and 3b), for both trajectories. For the shadowing trajectory, there is a ~nimum value of a at s, = 0.9 A, whereas for the blocking brajectory there is a maximum in QI at s, = 0.21 A. The value of CYat these critical values of the impact parameter give a measure of the size of the shadowing and blocking cones, respectively. Figs. 5 and 6, which address the shadowing and blocking cases, respectively, show the angular dependence of a number of interesting relevant quantities that the calculation yields. The topmost plot tracks the variation of the final energy of the detected particle as a fraction of E,, with the polar angle (Y. For both trajectories there is a clearly defined critical value of a, which is identical to that obtained from fig. 4, and two different possible energies for each allowed value of CY(except for the critical value). These two energies correspond to the two branches of the cross section (middle plot}. The lower branch of the cross
1.0 0.8
10
20
30
40
50
10 20 30 40 50 60 o(polar angle In degrees) Fig. 6. Blocking trajectory kinematic factor EC/E0 (top), calculated cross sections (middle) and DRS distribution (bottom) at 1064 eV as a function of a for the system in fig. 3b.
Shadowing Trajectory
0
model for calculating cross sections of direct recoils
60
Cx (polar angte in degrees)
Fig. 5. Shadowing trajectory kinematic factor E,/Ec (top), calculated cross sections (middle) and two-atom DRS distribution (bottom) at 1064 eV as a function of a for the system described in fig. 3a. The upper and lower branches of the cross section correspond to the lower and higher branches of the kinematic factor respectively.
section is for a value of the impact parameter increasing from the minimum to the critical value, and the higher branch for the region from the critical value outward. At the extremum value of CX,the derivative dat/ds, = 0, which gives rise to a classical rainbow in the scattering. Since the energies of the two branches are very different away from the critical angle, the two branches have not been added together as is usually done for the backscattered ion case [S]. The lower energy branch (almost constant with a) corresponds to the larger values of the impact parameter. The major contribution to the cross section also comes from this regime of the impact parameter. An additional validation of the calculation is provided by the fact that in the limit of large st, i.e., when the two-collision process is essentially reduced to a single recoil event (and the trajectories become identical), both the energy values and the cross sections calculated here do in fact reduce to the binary collision values. Thus the kinematic factor approaches 0.2 and the cross section approaches
S. Chauakuy,
R.S. Williams / Two-target-atom
0.12 A2 for the Ar+-W-O system presented in fig. 3. The existence of a rainbow in the scattering for both trajectories leads to a divergence in the classical cross sections at the respective extrema in (Y(figs. 5 and 6, middle plot). In order to eliminate this unphysical divergence, a quantum correction has been made to locate a cut-off value in LYand correspondingly si, beyond which the classical cross section is not computed. For example, the blocking trajectory in fig. 4b has a critical value of CYof 38.02 ‘. Following the methods of Daley et al. [14], there is a cut-off imposed on the cross section at a ma~mum value of a = 37.4”. A similar cut-off is imposed on the shadowing trajectory cross section, though here there is a minimum value of OL where the cutoff is to be calculated. &,= 60
133
model for calculating cross sections of direct recoils
800
1200
1600
2000
Energy (eV) Fig. 8. DRS simulated energy spectra for a = 16 o and 36 ’ in the shadowing case and for the blocking case. The solid lines are energy spectra for LYvalues close to the critical angles for each case, and show a distinctive two-peak structure corresponding to the two branches of the cross sections shown in figs. 5 and 6. All parameters are the same as in fig. 7.
0
Fig. 7. Two-atom DRS energy distributions for recoiled 0 as a function of a. The top figure is for the shadowing trajectory, and the lower is for the blocking trajectory. The intensity of the detected flux of recoiled particles for a total scattering angle of 60° is shown for 5 keV ArC incident on the W-O atom pair described in fig. 3. These distributions include an energy resolution broadening of 80 eV (FWHM Gaussian distribution) and an angular broadening corresponding to a thermal vibrational amplitude of 0.15 A for the 0 atom.
134
S. &hau&uty, R.S. WiIiiams / Two-target-atom
The lowermost plots in figs. 5 and 6 represent DRS angular distributions for a given energy. In this case, an energy of 1064 eV was chosen to provide a contribution to the scattering over the entire range of CYoutside the critical angle. The distributions were obtained by taking a cross-sectional slice of the three-dimensional distribution presented in fig. 7 at the selected energy. Both the energy resolution of the detector and thermal vibrations of the target atoms have been accounted for by methods described in section 3 of this paper. Fig. 8 shows sample energy spectra for each trajectory at fixed values of the polar angles close to the critical angle in each case. These spectra are obtained from the distributions of fig. 7 for the specific values of a chosen. Thus, once a double differential cross section of the type of fig. 7 is generated, one can easily obtain both energy spectra and angular intensity distributions. To illustrate the applicability of the methods described in this paper to ascertain interatomic distances and orientations, in fig. 9 we show combined DRS shadowing and blocking distributions for a fixed energy of 1064 eV. Fig. 9b illustrates the case of the recoil 0 atom initially sitting at the midpoint of the interatomic axis of two W atoms separated by 6 A. This ~st~bution is obtained by simply multiplying the broadened shadowing and blocking distributions from figs. 5 and 6 [15]. Figs. 9a and 9c shows the effect on the DRS distribution if the recoil atom is displaced from the interatomic axis below or above by 0.2 A, respectively. This results in a co~esponding shift in the respective critical angles by 7.6*. It becomes clear from this figure that the DRS distributions are very sensitive to relative positions of atoms on a surface.
5. Conclusions We have developed a new two-atom method for calculating energy- and angle-dependent recoil cross sections to assist in the interpretation of
mode/for
caiculoting cross sections of direct recoils
DRS angular distributions. A group of computer programs has been written and tested for this purpose. The method is computationally very fast, thus opening up the possibility for essentially real time analysis of experimental data. On a SUN 4fllO workstation the CPU time required to generate a full angle- and energy-dependent distribution similar to that in fig. 7 is 30 s.
This work was supported in part by the National Science Foundation. We also wish to gratefully acknowledge the extensive help in programming by R.S. DaIey.
References HI M. Aono, C. Oshima, S. Zaima, S. Otani and Y. Ishizawa, Jpn. J. Appl. Phys. 20 (1981) L829. I21 J.W. Rabaiais, 0. Grizzi, M. Shi and H. Bu, Phys. Rev. Lett. 63 (1989) 51. [31 0. Grizzi, M. Shi, H. Bu, J.W. Rabalais and P. Hochmann, Phys. Rev. B 40 (1989) 127. [41 O.S. Oen, Surf. Sci. 131 (1983) L407. 151 R.S. Williams, M. Kato, R.S. Daley and M. Aono, Surf. Sci. 225 (1990) 355. [61 S.R. Kasi, M.A. Kilbum, H. Kang, J.W. Rabalais, L. Tavernini and P. Hochmann, J. Chem. Phys. 88 (1988) 5902. [71 S.R. Kasi, H. Kang, C.S. Sass and J.W. Rabalais, Surf. Sci. Rep. 10 (1989). [81 J.B. Marion, Classical Dynamics (1970) pp. 291-310. 191 H. Goldstein, Classical Mechanics (Addison-Wesley, Palo Alto, 1950) pp. 58-89. HOI J.W. Rabalais, CRC Critical Rev. Solid State Mater. Sci. 14 (1988) 330. 1111 E.S. Mashkova and V.A. Molchanov, Medium Energy Ion Reflection from Solids (North-Holland, Amsterdam, 1985) pp. 125-152. 1121 G. Moliere, Z. Naturforsch. A 2 (1947) 133. 1131 0. Firsov, Zh. Eksp. Teor. Fiz. 33 (1958) 696; Sov. Phys.JETP 6 (1958) 534. [141 R.S. Daley, D. Farrelly and RX Williams, Surf. Sci. 234 (1990) 355. [151 R.S. Daley, J.H. Huang and R.S. Williams, Surf. Sci. 215 (1989) 281.
Surface Science 255 (1991) 127-134 North-Holland
Two-target-atom Shiladitya Department
model for calculating cross sections of direct recoils
Chaudhury of Physics and Solid State Science Center, UCLA, Los Angeles, CA 90024-1547,
USA
and R. Stanley Department Received
Williams
of Chemistry and Biochemistry
22 January
1991; accepted
and Solid State Science Cenfer, UCLA, Los Angeles,
for publication
21 March
CA 90024-1569,
USA
1991
A new two-target-atom model for calculating the intensity of particles that have been directly recoiled from a surface has been developed to examine the effects of shadowing and blocking on the recoils. This model is similar to one that was recently introduced for calculating the intensity of the primary particles scattered back from a surface. Important surface structure information can be obtained by comparing calculated recoil intensities as a function of incident ion identity and kinetic energy, target orientation, and recoil angle and energy to experimentally measured quantities.
1. Introduction Direct recoil spectroscopy (DRS) is an emerging technique for determining structural features, such as bond angles and interatomic distances, at crystalline surfaces. The technique is an extension of impact-collision ion scattering spectroscopy (ICISS) [l] inasmuch as the intensity of the particles detected at a particular energy is measured as a function of the angle between the incident ion beam and the surface being investigated. In DRS the measured particles are surface atoms that have recoiled directly after collision with an incident ion. This ion scattering technique is especially useful for examining surfaces on which a lower atomic mass species is adsorbed on a higher mass substrate. The yield of recoiled particles increases or decreases as near-surface atoms are exposed to variations in the ion flux caused by the shadowing and blocking effects of other surface atoms. Rabalais and coworkers [2,3] have already shown that interatomic distances can be determined from DRS experiments by established analysis of the 0039-6028/91/$03.50
0 1991 - Elsevier
Science Publishers
shape of shadow and blocking cones [4]. The two-atom model presented here was developed to make such determinations more quantitative. Recently, Williams et al. [5] introduced a new formalism to calculate three-dimensional crosssections for an ion to scatter sequentially and classically from two atoms. This method has been applied to the interpretation of ICISS experimental data. In this paper, a similar formalism has been developed to calculate the three-dimensional cross section for an ion to recoil one of two target atoms from a surface. The collisions are classical and involve sequential forward scattering and recoil events that are confined to a plane into a fixed total scattering angle. Classical trajectory simulations that have traditionally been employed to assist in recoil spectroscopy data analysis can be very CPU intensive [2,6,7]. The model described here is computationally very fast and includes two cases for a given interatomic spacing of the target atoms: the case where the first event is a recoil, followed by the scattering of the recoiled particle from a second
B.V. (North-Holland)
S. Chaudhuty, R.S. Williams / Two-target-atom
128
atom; and where the first event is a scattering event followed by a recoil. The vibrational motions of the target atoms and the energy resolution of the detector are taken into consideration to calculate the intensity of the recoiled ions as a function of the angle between the incoming ion beam and the line joining the two atoms. In the next section, the mathematical details of the two-atom model are presented. The computer implementation is discussed in section 3. Results from the simulation programs are discussed in section 4. Section 5 contains the conclusions of the paper.
model for calculating cross sections of direct recoils
2.2. Cross sections for scattering/recoiling
For scattering from some arbitrary three-dimensional target into an element of solid angle dS2, for a particular incident flux I, and a particular orientation of the scatterer, one can write the generalized cross-section formula (Williams et al. [J]):
(4) where D is the Jacobian
2. Mathematical
details
D=
For a particle of energy E, and mass MP colliding elastically with a particle of mass M, at rest, the principles of conservation of energy and momentum allow us to describe the final energies of the particles and their directions relative to the original angle of incidence (Marion [8]). If the energy after collision of the incident particle is E,, and that of the recoil particle is E,, then the following relations for the energy ratios I&,/E, and E,/E, hold:
X [cOS 0,+
J( M,/M,)2
ace,, kd ah, +d
2.1. Classical binary collisions
- sin28,]’
(1)
in three
dimensions
determinant
_ ab ---ah,,,
as, ah
ah. ah
a+, as,.
(5)
All angles in this treatment are measured in the laboratory coordinate system. For an isolated atom we need to consider only a single binary collision that is characterized by axial symmetry. In this case &9,/a+, = 0 and a&,,,,/a+, = 0 SO that the expressions for the cross-sections may now be written as:
(6) for the scattered particle, and
for the recoiled particle.
and E 1=
4$3f,
Eo
(Mp+MJ2
cos”e,,
where eP and 6, are the laboratory scattering and recoil angles respectively (see fig. 1). In eq. (1) the + sign should be used if MP > M,, otherwise the + sign should be used. This work uses only the + sign, since we intentionally chose the mass of the scattering center to be larger than the projectile. A further relation between the angles 0, and 0, can be written as (Marion [S]): sin
8, =
MPEPsin e,. ME
J
r r
x EP
”
eP
__-_---
MP
“&__>___.
‘0,
Mr \t
Er
Binary Elastic Collision Fig. 1. Schematic drawing showing relevant parameters of a binary elastic collision. A particle with mass Mp and energy E,, scatters at an impact parameter of s from a particle of mass M, initially at rest. The scattering and recoil angles are O,, and O,, respectively.
129
S. Chaudhury, R.S. Williams / Two-target-atom model for calculating cross sections of direct recoils
Scattered Projectile Recoiled Target
_
4$ 0 L 0
200.0
0.2 .
Fig. 2,. Dependence on impact parameter of (a) the scattering angle and scattering cross section for a lighter projectile from a heavier target and (b) the recoil angle and recoil cross section for a heavier projectile from a lighter target in binary collisions. In (a) Arf is incident upon W and in (b) Ar+ recoils 0.
The cross sections for each of these cases is evaluated by first determining the deflection function @,(s,) or el(s,) from the scattering potential and then calculating the derivative with respect to the impact parameter. Calculating the cross-section for the scattering case allows us to easily obtain the cross-section for the recoil case as is derived in standard texts [8,9],
(8) For a strictly repulsive potential and A4r < M,, the deflection functions are monotonic and the derivatives are well behaved. The conditions for the collision for which we present the cross section in fig. 2a were as follows: MP = 39.9 amu (Ar), M, = 183.85 amu (W) and the incident particle energy was 5 keV. At zero impact parameter, a lighter projectile scatters back at 180” from a heavier target with a small cross section, and the cross section increases as the scattering angle decreases to zero. For fig. 2b the projectile was also 5 keV Ar, but this time M, = 16 amu (0). Fig. 2b shows the dependence of the recoil scattering angle and cross section on the impact parameter: the recoil angle increases from zero to a limiting value of 90 o as the cross section increases [lO,ll]. These are the types of interactions that will be examined for the case of a projectile interacting with two target atoms.
Figs. 3a and 3b illustrate the cases where the final detected particle is the result of two sequential events - one scattering and one recoil. In fig. 3a, the first event is a forward scattering event followed by a recoil, which we call a shadowing trajectory. The total recoil angle obeys the condition 4 = @&l) + @,(%L
(9)
where 8, is the angle of the recoil detector with respect to the incident ion beam. The angle through which the incident particle is scattered, In,, is a function of the impact parameter at the first target atom, and @,(s,) is the scattering angle of the recoiled particle, as a function of the impact parameter at the second target atom. In fig. 3b, the first event is a recoil event, followed by the scattering of the recoiled particle from a second atom, which we call a blocking trajectory. This event satisfies the condition 4_ = 4M
+ e,,(+).
00)
In this instance, the dependence of the total recoil angle B, is on the recoil angle of the first target atom &(s,), and the angle through which it is scattered at the second target atom t9,,(+). In both these cases axial symmetry is broken, so &#+,,,,/a+, # 1, but for an experimental setup
130
S. Chaudhury, R.S. Williams / Two-target-atom
Shadowing
Trajectory
model for calculating cross sections of direct recoils
sional cross-section
for planar scattering,
In the following discussion, we shall develop the cross section for the shadowing trajectory; the development for the blocking trajectory is similar. In DRS, as with ICISS, the total scattering angle 8, is fixed ad the cross section is desired as a function of (Y, the angle between the incident ion trajectory and the line connecting the two target atoms. The factor &$,,,,/&#B, in the Jacobian is determined from geometrical considerations and can be shown [5] to be: X %U, a+, =G
Isin4 I I sin 8, cos f3, sm8,
sin 8,
’
(12)
where X is the line segment in fig. 3a, X=R
Blocking
Trajectory
M Awn1
cMjon
sin~+s2cos$,.
(13)
This planar configuration of the atoms ignores both vibrational motion of the atoms perpendicular to the scattering plane and any interplanar (zig-zag) scattering. Following the methods set forth previously [5], the expression (Y= eP + sin-’
s2
[
+
s1
cos
R
ep I
is evaluated as a function of sr. The derivative in the Jacobian determinant, eq. (ll), can be evaluated numerically for a fixed value of (Y, and a(B,, a) can then be computed.
Fig. 3. Illustration of (a) shadowing and (b) blocking trajectories. In both cases the incoming ion (Ar+ ) mass is between the masses of the target atoms. In (a) atom 1 is the heavier atom (W) while in (b) the heavier atom is atom 2. The relevant parameters necessary to calculate (Yas a function of s, are O,, R, s2 and O,, as indicated.
where the ion beam, detector and the two atoms are coplanar, then it is still true that %3,/a+, = 0 [5]. Thus, we can still write down a three-dimen-
3. Computer implementation The calculations are performed using a suite of programs adapted from the routines of Williams et al. [5]. The first program calculates single-atom deflection functions, i.e., laboratory scattering angle e as a function of impact parameter for a particular projectile energy, and writes a table to a disk file. A recoil deflection table is then calculated from eq. (3) using the first table as input. The scattering potential used most often is the Moliere [12] with the Firsov [13] screening con-
S. Chaudhury, R.S. Williams / Two-target-atom
stant modified through experience gained from simulations of ICISS data. The only input parameters needed for these deflection tables are the atomic number and mass of the projectile and target, the laboratory energy of the projectile, and a constant to scale the Firsov screening length. Tables for each of the different binary interactions are required to fully simulate both the shadowing and blocking trajectories. A program that calculates (Y versus s1 and the total u versus a! distribution for each trajectory is then called, with the appropriate scattering tables as input. The other inputs to these programs are the total scattering angle (the detector angle) and R, the interatomic distance. The divergence in the calculated cross section at the critical angle for recoiling is avoided by utilizing the same type of semiclassical approximation discussed by Daley et al. [14]. The u versus (Yand the E,/E, versus (Ydistributions (figs. 5 and 6) then become the inputs to a program that includes thermal vibrations and the energy resolution of the detector [l]. In order to correctly account for the dispersion in the energy of the detected particle, a two-dimensional thermal broadening approach was developed. First, the values of the cross sections for each trajectory at specific values of the polar angle 1y (up to the laboratory scattering angle) were tabulated against the specific energies at which they would be detected. These energy spectra were then broadened with a Gaussian function that accounts for the energy resolution of the detector. The effects of thermal vibrations were incorporated by choosing a value of the energy and broadening the corresponding 1y distribution of the intensity with a Gaussian that represents the range of relative thermal displacements between the atoms [5]. Thus one can simulate a DRS angular distribution for a given energy or obtain an energy spectrum of the recoiled particle at a given value of the polar angle.
4. Results and discussion Calculations are presented for what promises to be the most useful of the various mass configura-
131
model for calculating cross sections of direct recoils 60
^
40
E s
20
0.0
0.1
0.2
0.3
0.4
0.5
S,(A) Fig. 4. Plot of a versus s, for 5 keV Ar+ scattering from a W and an 0 atom 3 A apart shown for (a) shadowing and (b) blocking trajectories. The laboratory scattering angle was 60 O. The minimum allowed value of a in (a) is the shadowing critical angle while the maximum allowed value in (b) is the blocking critical angle.
tions involving the two target atoms and the incident ion beam: MP > M, > M,; where M, and M, are the masses of the two target atoms and MP is the mass of the incident ion. In particular, we have calculated the cross sections for both shadowing and blocking trajectories for 5 keV Ar+ ions incident on a W-O atom pair separated by 3 A. The 0 atom is recoiled into a detector at a fixed angle of 60 o with respect to the incident ion direction. A schematic figure showing the various parameters for each trajectory is presented in figs. 3a and 3b. Since the trajectories investigated here are a combination of binary scattering and recoil events, calculations for the cross sections of the single binary events were first performed to ensure the validity of the computer programs. Figs. 2a and 2b represent the results of these calculations for the case of Ar+ scattering from W and 0 recoiling from Ar. The first part of the two atom calculations are presented in fig. 4. Here, (Y,the angle between the incident particle and the axis of the two target atoms, is plotted against si, the initial impact
S. Chaudhury, R.S. Williams / Two-target-atom
132
parameter (see figs. 3a and 3b), for both trajectories. For the shadowing trajectory, there is a ~nimum value of a at s, = 0.9 A, whereas for the blocking brajectory there is a maximum in QI at s, = 0.21 A. The value of CYat these critical values of the impact parameter give a measure of the size of the shadowing and blocking cones, respectively. Figs. 5 and 6, which address the shadowing and blocking cases, respectively, show the angular dependence of a number of interesting relevant quantities that the calculation yields. The topmost plot tracks the variation of the final energy of the detected particle as a fraction of E,, with the polar angle (Y. For both trajectories there is a clearly defined critical value of a, which is identical to that obtained from fig. 4, and two different possible energies for each allowed value of CY(except for the critical value). These two energies correspond to the two branches of the cross section (middle plot}. The lower branch of the cross
1.0 0.8
10
20
30
40
50
10 20 30 40 50 60 o(polar angle In degrees) Fig. 6. Blocking trajectory kinematic factor EC/E0 (top), calculated cross sections (middle) and DRS distribution (bottom) at 1064 eV as a function of a for the system in fig. 3b.
Shadowing Trajectory
0
model for calculating cross sections of direct recoils
60
Cx (polar angte in degrees)
Fig. 5. Shadowing trajectory kinematic factor E,/Ec (top), calculated cross sections (middle) and two-atom DRS distribution (bottom) at 1064 eV as a function of a for the system described in fig. 3a. The upper and lower branches of the cross section correspond to the lower and higher branches of the kinematic factor respectively.
section is for a value of the impact parameter increasing from the minimum to the critical value, and the higher branch for the region from the critical value outward. At the extremum value of CX,the derivative dat/ds, = 0, which gives rise to a classical rainbow in the scattering. Since the energies of the two branches are very different away from the critical angle, the two branches have not been added together as is usually done for the backscattered ion case [S]. The lower energy branch (almost constant with a) corresponds to the larger values of the impact parameter. The major contribution to the cross section also comes from this regime of the impact parameter. An additional validation of the calculation is provided by the fact that in the limit of large st, i.e., when the two-collision process is essentially reduced to a single recoil event (and the trajectories become identical), both the energy values and the cross sections calculated here do in fact reduce to the binary collision values. Thus the kinematic factor approaches 0.2 and the cross section approaches
S. Chauakuy,
R.S. Williams / Two-target-atom
0.12 A2 for the Ar+-W-O system presented in fig. 3. The existence of a rainbow in the scattering for both trajectories leads to a divergence in the classical cross sections at the respective extrema in (Y(figs. 5 and 6, middle plot). In order to eliminate this unphysical divergence, a quantum correction has been made to locate a cut-off value in LYand correspondingly si, beyond which the classical cross section is not computed. For example, the blocking trajectory in fig. 4b has a critical value of CYof 38.02 ‘. Following the methods of Daley et al. [14], there is a cut-off imposed on the cross section at a ma~mum value of a = 37.4”. A similar cut-off is imposed on the shadowing trajectory cross section, though here there is a minimum value of OL where the cutoff is to be calculated. &,= 60
133
model for calculating cross sections of direct recoils
800
1200
1600
2000
Energy (eV) Fig. 8. DRS simulated energy spectra for a = 16 o and 36 ’ in the shadowing case and for the blocking case. The solid lines are energy spectra for LYvalues close to the critical angles for each case, and show a distinctive two-peak structure corresponding to the two branches of the cross sections shown in figs. 5 and 6. All parameters are the same as in fig. 7.
0
Fig. 7. Two-atom DRS energy distributions for recoiled 0 as a function of a. The top figure is for the shadowing trajectory, and the lower is for the blocking trajectory. The intensity of the detected flux of recoiled particles for a total scattering angle of 60° is shown for 5 keV ArC incident on the W-O atom pair described in fig. 3. These distributions include an energy resolution broadening of 80 eV (FWHM Gaussian distribution) and an angular broadening corresponding to a thermal vibrational amplitude of 0.15 A for the 0 atom.
134
S. &hau&uty, R.S. WiIiiams / Two-target-atom
The lowermost plots in figs. 5 and 6 represent DRS angular distributions for a given energy. In this case, an energy of 1064 eV was chosen to provide a contribution to the scattering over the entire range of CYoutside the critical angle. The distributions were obtained by taking a cross-sectional slice of the three-dimensional distribution presented in fig. 7 at the selected energy. Both the energy resolution of the detector and thermal vibrations of the target atoms have been accounted for by methods described in section 3 of this paper. Fig. 8 shows sample energy spectra for each trajectory at fixed values of the polar angles close to the critical angle in each case. These spectra are obtained from the distributions of fig. 7 for the specific values of a chosen. Thus, once a double differential cross section of the type of fig. 7 is generated, one can easily obtain both energy spectra and angular intensity distributions. To illustrate the applicability of the methods described in this paper to ascertain interatomic distances and orientations, in fig. 9 we show combined DRS shadowing and blocking distributions for a fixed energy of 1064 eV. Fig. 9b illustrates the case of the recoil 0 atom initially sitting at the midpoint of the interatomic axis of two W atoms separated by 6 A. This ~st~bution is obtained by simply multiplying the broadened shadowing and blocking distributions from figs. 5 and 6 [15]. Figs. 9a and 9c shows the effect on the DRS distribution if the recoil atom is displaced from the interatomic axis below or above by 0.2 A, respectively. This results in a co~esponding shift in the respective critical angles by 7.6*. It becomes clear from this figure that the DRS distributions are very sensitive to relative positions of atoms on a surface.
5. Conclusions We have developed a new two-atom method for calculating energy- and angle-dependent recoil cross sections to assist in the interpretation of
mode/for
caiculoting cross sections of direct recoils
DRS angular distributions. A group of computer programs has been written and tested for this purpose. The method is computationally very fast, thus opening up the possibility for essentially real time analysis of experimental data. On a SUN 4fllO workstation the CPU time required to generate a full angle- and energy-dependent distribution similar to that in fig. 7 is 30 s.
This work was supported in part by the National Science Foundation. We also wish to gratefully acknowledge the extensive help in programming by R.S. DaIey.
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