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Gas-surface scattering distributions according to the hard-spheroid model
Surface Science I 15(1982) 247-258 North-Holland Publishing Company
247
GAS-SURFACE SCATTERING DISTRIBUTIONS THE HARD-SPHEROID MODEL
ACCORDING
TO
Christoph STEINBRDCHEL Laboratories Received
RCA, Ltd., Badenerstrasse
28 September
198 1; accepted
569, CH-8048 Ziirich, Switzerland for publication
27 November
198 1
Gas-surface scattering distributions are calculated on the basis of the hard-spheroid model. This new model is related to the hard-cube model but incorporates surface structure by letting the surface atoms have hard spherical caps. Surface structure is found to reduce the maximum scattered intensity and to broaden the scattering distribution. The decrease in the maximum scattered intensity is most pronounced for a light gas. Broadening is asymmetric relative to the direction of maximum scattered intensity in that backscattering towards the surface is favored. Adding surface structure to the hard-cube model improves significantly the agreement between model predictions and experimental results.
1. Introduction
The scattering of an atom beam from a surface is becoming an ever more versatile technique for probing the structure of surfaces and their interaction with gases [ 1,2]. Experimental developments, such as the use of mono-energetic beams [1,3-61, are paralleled by theoretical efforts aimed at describing efficiently the many-body nature of the process [7]. To the experimenter it is of interest to have at hand a computationally simple model which enables one, at least, to correlate qualitatively experimental results. The simplest model which still gives a reasonable picture of gas-surface dynamics is the hard-cube model [8-lo]. In it the gas-surface collision is represented as an impulsive collision of a gas atom with a vibrating cube. This model has been quite successful in describing the gross features of gas-surface scattering in the absence of diffraction [2]. In addition, it has been able to account very well for gas-surface energy transfer [ 1l- 131. A major assumption of the hard-cube model, and one of its main deficiencies, is that the gas-surface interaction potential has no structure parallel to the surface. Thus the tangential momentum of a gas atom is assumed conserved during a collision. In a recent paper I proposed an extension of the hard-cube model, termed “hard-spheroid model”, which incorporates some surface structure in a simple way [ 141. That work indicated that in most cases 0039-6028/82/0000-0000/$02.75
0 1982 North-Holland
Ch. Steinbriichel
24R
/ Gas-surface
scattering distributions
surface structure would be rather unimportant for gas-surface energy transfer, but would be reflected directly in transfer of momentum parallel to the surface. Here I use the hard-spheroid model to calculate gas-surface scattering distributions for mono-energetic beams. (A similar model was used some time ago for trajectory calculations, but only with zero surface temperature [29,30].) The main conclusion will be that adding surface structure to the hard-cube model brings about a marked improvement between model predictions and experimental results.
2. Description of model In the hard-spheroid model [14] a gas atom undergoes an impulsive and purely repulsive collision with a surface atom. But in contrast to the hard-cube model, the surface atom is now assumed to have a spherical cap (fig. 1). The center-of-mass of the surface atom is taken to be in the center of curvature of the atom’s spherical cap, and all collisions are assumed to occur in the plane containing the center-of-mass. We will only consider sufficiently small surface structure, such that multiple collisions with neighboring surface atoms and shadowing can be neglected [14]. Let u and u be the gas and surface atom velocities before a collision, and let primes denote the same quantities after a collision. Also, let 8 be the macroscopic incident angle, 8’ the macroscopic reflected angle (taken negative), and + the local incident angle (fig. 1). We are interested in the probability P(P) that an atom impinging on the surface with velocity u = (a, 8) be reflected at an angle 8’:
where o = V(U, 8, 8’, <) and a* = M/Zk,T,. In (1) multiple collisions with a single spheroid are neglected [ 11,141. From fig. 1, relative to the coordinate system at the point of impact, one has u,, =usm+,
o,, = -u sint,
UI =ucos+,
ol=ucos&
(2)
with + = 8 - E. Furthermore, UX=u,,cos~+u,sin~,
u,=-u,,sin~+u,cosE,
(3)
where also u, = u sin 8 and uY = u cos 8, According to classical collision kinematics UT,=ull’
Y---l ’ --u1+ %-1”+1
2 -2) CL+1 I’
(4)
Ch. Steinbriichel / Gas-surface
249
scattering distributions
l V Fig. I. Geometry of the hard-spheroid
model.
with p= m/M the ratio of gas atom to surface atom mass. On combining eqs. (3) and (4) one obtains z@,[)
= u,
(
l-
2 -sin26 cc+1
r$(@, S) = -+j&
1
-(uY
sin6cost+u,
-o)--&sinEcosE,
(
l-
-cos2~ 2 P+l
1
+fJ-
2 cos2<. 1.1+1
(5)
One now uses (5) in the form tan 8’ = Q,
O/$(@,
g),
(6)
and solves for u as a function of U, 8, 8’, and 6: U
-=
UY
tan2t [
1+l.r (--pane’+-
‘Tme 2
+tant(l-ttanet=e’)--
)
1+cL -ttane
1-p 2 t-e’-
2
X+pt~ey’. Differentiation
I
(7) of (7) with respect to 8’ yields
-tm2~-tm~tme-x [c0s2epa E- tan et)] -‘.
1-p 2
0 +u
Y (8)
Thus, for any (u, 8,/Y) and any surface temperature T,, eqs. (7) and (8) may be used to evaluate the scattering probabilitv P(P), eq. (1). If one sets Es = imu
Ch. Steinbriichel
250
and defines
Es by Es = 2k,T,,
/ &.-surface
scattering distributions
then one can write (1) in the form
Eqs. (7) to (9) imply that P(P) depends on p, Es/E, and 8, as well as on the angle (Ycharacterizing surface structure. For (Y= 0 eqs. (7) to (9) reduce to the hard-cube result [9].
9
a)
b)
7
6
5
0
IS
30
45
60
0
15
30
45
60
75
0
15
30
45
60
0
15
30
45
60
75
SCATTERING
ANGLE
10’1 (dsg)
Fig. 2. Scattering distributions for p=O.O2: (V) designates angle of incidence; (p (flat surface); (- - -) a=7.5O; (- - - - - -) a= 12.5”. In (a) and (b) ES/Es =0.4, Es/E, =2.5.
) a=0 in (c) and (d)
Ch. Steinbriichel
/ Gas-surface
stuttering
251
distributions
3. Numerical results Results for scattering distributions obtained from numerically integrating (9) with the help of (7) and (8) are shown in figs. 2 to 4. These cover a range of values for the parameter ~1, two angles of incidence, and the incoming beam either considerably hotter or colder than the surface. Three different values for the angle (Y characterizing surface structure, or surface roughness, are compared: (Y= 0 (i.e., the hard-cube model), (Y= 7.5”, and (Y= 12.5’. Several features of the curves in figs. 2 to 4 are noteworthy with respect to the importance of surface structure. First, surface structure will broaden the scattering distribu&ns P(P) while reducing their maximum intensity Pm,. The reduction of P,, is the more pronounced the lighter the gas, the colder the surface, and the closer 8 to normal incidence. The decrease of P,,,= towards grazing incidence is made less significant by surface structure. For low Es and a fairly rough surface PmBx may even increase slightly towards grazing incidence (see, for example, figs. 2a and 2b for (Y= 7.5” or 12.5’, or figs. 3a and 3b for (Y= 12.5’). Thermal attenuation, i.e., a decrease of P,,,, with increasing Es, also becomes less significant because of surface structure. With regard to the position 18; 1 of I’,, we note that for a light gas, small surface structure has practically no effect on 18; (, but large surface structure
0
15
30
45
60
0
15
30
45
60
75
0
15
30
45
60
0
15
30
45
60
75
SCATTERING
ANGLE
18’1 (deg)
Fig. 3. Scattering distributions for p=O.l: (V) designates angle of incidence; ( (flat surface); (- - -) a=7.5’; (- - - - - -) a= 12.5’. In (a) and (b) Es/E, ~0.4, Es/E, =2.5.
) n=O in (c) and (d)
252
Ch. Sieinbriichel / Gas-&ace
0
15
30
45
SCATTERING
60
0
ANGLE
15
scattering distributions
30
45
60
75
1.9’1 (deg)
Fig. 4. Scattering distributions for p=O.2: (V) designates angle of incidence; (--) (Y=o (flat surface); (- - -) 01=7.5”; (- - - - - -) a= 12.5”. In (a) and (b) ES/Es =0.4. in (c) and (d) Es/E, =2.5.
will move I&, 1 in the backward direction (fig. 2). For a heavy gas, surface structure generally will move ItI; 1in the forward direction, particularly if Es is large (fig. 4). The shape of P(P) is affected in an interesting way by surface structure. A rough surface will broaden P( 8’) asymmetrically, enhancing backward scattering towards the surface normal, even if 18; I is moved in the forward direction (see e.g. figs. 3c, 3d, 4c and 4d). In fig. 5 results are displayed in the form of the deviation of the position
8 : . Fig. 5. Deviation of direction of scattered intensity maximum from specular direction, AB, versus pE,/E,, with 8=45”. Solid lines are for a=0 (flat surface) and thermal beam [eq. (IO)], other symbols [(0), (O), etc.] for mono-energetic beam. (0), (0) cr=O (flat surface); (a), (A) a=7.5O; (V), (V) (I= 12.5”. Open symbols: p=O.l; full symbols: ).~=0.2.
Ch. Sreinbriichel / Gas-surface
scattering distributions
253
(8; 1 of the scattering maximum from the specular direction, A0 = 0 - 18; (, versus pE,/E, and incident angle 8. (For the curves corresponding to a Maxwellian beam Eg was identified with 2k,T,.) Here one notes that surface structure makes A9 a less rapidly varying function of pEs/Eg, and this the more so the lighter the gas.
4.Discussion We wish first to relate the predictions of the hard-spheroid model to experimental results as well as to other simple model calculations, in particular to the hard-cube and the soft-cube models. Then we examine the assets and limitations of the present model, and finally we point to some unresolved issues arising from this work and to some suggestions for future work. For a comparison to experiments we rely on three papers in which monoenergetic beams were used as well as on some work employing thermal beams. Yamamoto and Stickney [3] studied scattering of rare gases with Eg = 0.065eV from W( 110) at T, = 2100 K. Subbarao and Miller [4] investigated scattering of rare gases with 0.06 G Eg G 2.5eV from epitaxially grown Ag( 111) at T, = 560K. (Here only the lower part of the range of Eg is relevant for our purpose.) In both cases the hard-cube model is able to describe qualitatively experimental trends, while quantitatively certain discrepancies are noted. The hard-cube model overestimates the magnitude of A@, the deviation of the scattering maximum from the specular direction, whereas it underestimates t9,,*, the full-width-at-half-maximum of the scattering distribution [3,15]. For both these quantities, our work shows that including surface structure leads to improved agreement with experiments (table 1). Note also that the soft-cube model [ 161, in which the gas-surface potential is not impulsive but has non-zero range, provides an improvement over the hard-cube model as regards A8 but not 13,,~ [ 151. On the other hand, a finite-lattice model improves upon the hard-cube model with respect to both AB and 8,,* [15]. Lapujoulade and Lejay [S], in a detailed study, determined scattering distributions and a few moments of the scattered-velocity distributions for He, Ne, and Ar on Cu( 100). In that work Eg = 0.065(or0.063)eV, 300 G T, 4 800 K, and 30” d 8 G 60”. The scattering distribution for He is very narrow and specular, and for Ne and Ar becomes wider and slightly backscattered. Also, for Ne and Ar, A@and 8,,, are both smaller than expected form the hard-cube or the hard-spheroid model, but one should keep in mind that certainly for Ar (CL= 0.63), and probably even for Ne (~1= 0.32), the gas-surface collision will not really be impulsive. On the other hand, the dependence of A8 and e1,* on surface roughness (comparing scattering with [lOO] and [ 1lo] azimuths of the incoming beam) is as expected on the basis of the present work. Furthermore, scattered gas atoms have, on average, lost momentum
Experimental a=7.Y.
2 0.8 2 0.8
30
I
31 37 53 29 28 35 28
Expt.
Bt/r
I I I). Figures for hard-spheroid
-I
10
5 12 16 5 -2
0 3 12 0 -4 2 0
Hard-spher. model
results
2 8 13 2 -4 6
Hard-cube model
scattering
Expt.
A@
gas-surface
data from ref. [3] for W(110) and from ref. [4] for Ag(
50
5.6
40
He/W( 1 f0) Ne/W( 110) Ar/W(I IO) Ne/Ag( 111)
G/Es
and experimental
e
of theoretical
System
Table 1 Comparison
26 37 39 23 22 31 28 model calculated
Hard-cube model 30 39 44 34 33 37 36
with surface
Hard-spher. model
roughness
parameter
B
5 ii: 8,
B F: 3. 2
4 $
f
:
i
2““.
9
parallel to the surface, and the more so the higher T,, in agreement with the hard-spheroid model [ 141. In order to compare the present results to experiments involving a thermal beam we note the expected similarity between scattering of a thermal beam at Tp and a mono-energetic beam with Eg = 2k,Ts [IO]. (Of course it would be str~~tfo~~d to integrate, say, eq. (9) over a distribution of incident velocities u.) Moreover, it should be pointed out that if in the hard-cube model one integrates over the incoming thermal velocity distribution [9,10], instead of using a mean-speed analysis [S], one obtains
Thus A8 is not a universal function of pq/T, [ 173,but rather depends on p and T,/T, separately (cf. fig. 5). It then becomes clear from fig. 5 that surface structure improves the hard-cube model in the right direction, making A&a less rapidly varying function of pT,/T,. For quasi-elastic scattering [2], typically He on Ag [ 171or W [3,18,19], fig. 2 suggests that P,,,= is a sensitive function of surface roughness and that thermal attenuation is the less pronounced the rougher the surface. Both these trends are observed experimentally [17]. For inelastic scattering 121, e.g. Ne on Ag [4,17] or W [3,19,20], attenuation and broadening do not simply correlate with surface roughness, but involve dynamical interactions of the gas with the solid lattice [17]. The hard-spheroid model combines both these features and indicates, in agreement with experimental trends [ 171, that the effect of increased surface roughness may be offset, in part, by decreased inelasticity (compare e.g. figs. 3a and 4a). The limitations of the hard-cube model are due essentially to two assumptions: an impulsive gas-surface collision, and negligible surface structure. The hard-spheroid model retains the former but overcomes the latter assumption. The ass~ption of an impulsive collision h&s been discussed extensively [2,10,21]. It should be valid for a light gas colliding with a surface of heavy atoms, as is apparent on comparing relevant surface atom dimensions and gas atom velocities. These arguments are confirmed by detailed molecular dynamics simulations [22]. One might try to make the model more realistic by including an attractive well of depth D in the gas-surface potential and/or by allowing for a gas atom to collide with a larger, effective number of surface atoms (i.e., using a reduced effective p) [13]. But clearly, these two effects tend to compensate for each other. Indeed, hard-cube calculations of energy transfer indicate that a best fit to experiments, varying both D and ,u, is rather ill-determined [ 131. Although the hard-spheroid model incorporates surface structure, it does so in a somewhat crude fashion, since a realistic hard-wall envelope of the gas-surface potential would certainly not contain any cusps as shown in fig. 1
256
Ch. Steinbriichel / Gas-surface
scattering distributions
[1,23]. Thus the hard-spheroid model cannot reproduce accurately such an effect as rainbow scattering [20,24]. But, perhaps surprisingly, the model does display the beginnings of rainbow scattering in that for small p and large surface structure the scattering distribution is very broad and flat, with the maximum displaced strongly towards the surface normal (cf. figs. 2a and 2b). On the other hand, semiclassical calculations using a more realistically shaped but static hard-wall potential also fail to account quantitatively for rainbow scattering by predicting too narrow a scattering distribution [24]. This suggests that structure and inelastic effects must be treated on an equal footing. The same conclusion follows from a comparison of the hard-spheroid, the cube-type [8-10,161, and the finite-lattice models [ 151 (see arguments above). The surface structure in the hard-spheroid model is somewhat unrealistic in another way because the two-dimensional nature of the surface topography is neglected. Yet it is clear that one needs to be careful in comparing scattered intensities, taking proper account of detector geometry and out-of-plane scattering [5,28]. Logan et al. [9] discussed briefly applying the hard-cube model to the situation where there is a distribution of incident angles, probably with a view on dealing with a macroscopically rough surface, but they did not pursue the subject any further. The results from the hard-spheroid model imply that surface structure, and also macroscopic surface roughness, will manifest itself most clearly in scattering distributions and momentum transfer parallel to the surface, less so in energy transfer. As to unresolved issues, the hard-spheroid model makes a definite, somewhat unrealistic assumption about the shape of surface structure (cf. information on surface structure derived from diffraction experiments [1,23]). But the degree of surface roughness for a particular gas-surface system, given in the hard-spheroid model by the parameter (Y(fig. l), is also not clear a priori. (If c is the relative amplitude of modulation of the hard-wall surface envelope, then a! * 4c [ 141.) For example, from measurements of momentum accommodation parallel to the surface [5,25] one would infer that aNe < oAr [14], whereas from atomic size and probably also from scattering distributions one would conclude that aNe > ok. Yet even arguments based on atomic size are somewhat contradictory (cf. refs. [ 17,23,26]). Moreover, for He scattering, experiments [l] as well as calculations of the gas-surface potential [27] imply a turning point of the gas atom rather far away from the geometrical surface. Thus surface roughness may well be a function of how deeply gas atoms penetrate the surface potential, or of how inelastic the gas-surface collision turns out to be. Neither the hard-spheroid model nor any of the other models mentioned is able to explain that for He the maximum scattered intensity increases towards grazing incidence [3,5,17,18] (cf. fig. 2). This effect can be rationalized quantum -mechanically in terms of attenuation of elastic scattering via the Debye-Waller factor [31].
Ch. Steinbrikhel
/ Gas-surface
scattering distributions
251
It will also be quite straightforward to calculate velocity distributions of scattered atoms as a function of scattering angle with the present model. Indeed, it is clear that in that respect as well the hard-spheroid model will improve upon the hard-cube model. One need only recall that because of conservation of momentum parallel to the surface, the energy of a scattered gas atom is fixed by the scattering angle in the hard-cube model. Experimentally, the mean speed of a scattered atom is found to be lower at small scattering angles (small )@‘I), and higher at large scattering angles (large l&l), than predicted by the hard-cube model [3,10]. But, for example, a small macroscopic scattering angle implies, on average, more nearly normal incidence locally on the spheroid (cf. fig. l), i.e., a more inelastic collision and, therefore, a smaller speed for the scattered gas atom. By the same token, atoms scattered backwardly will tend to have lost momentum parallel to the surface [5,14].
5. summaly
We have presented theoretical results on gas-surface scattering distributions using the hard-spheroid model. This new model derives from, and is conceptually as simple as, the hard-cube model but it also incorporates some surface structure in a simple way. Surface structure turns out to reduce the maximum scattered intensity and to broaden the scattering distribution, with backscattering towards the surface normal being favored. Thus the hard-spheroid model provides a physically transparent, qualitative picture of gas-surface scattering as a function of gas and surface atom mass, surface temperature, gas atom energy and angle of incidence, and surface structure. Since the model allows for gas-surface momentum transfer parallel to the surface, it should yield reasonable scattered-velocity distributions as well.
References [l] (21 [3] [4] [S] [6] [7] [8] [9] [lo]
K.H. Rieder and T. Bngel, Nucl. Instr. Methods 170 (1980) 483. W.H. Weinberg, Advan. Colloid Interface Sci. 4 (1975) 301. S. Yamamoto and R.E. Stickney. J. Chem. Phys. 53 (1970) 1594. D.R. Miller and R.B. Subbarao, J. Chem. Phys. 52 (1970) 425. J. Lapujoulade and Y. Lejay, J. Chem. Phys. 63 (1975) 1389. K.C. Janda, J.E. Hurst, CA. Becker, J.P. Cowin, D.J. Auerbach and L. Wharton, J. Chem. Phys. 72 (1980) 2403. J.C. Tully, Ann. Rev. Phys. Chem. 91 (1980) 319. R.M. Logan and R.E. Stickney, J. Chem. Phys. 44 (1966) 195. R.M. Logan, J.C. Keck and R.E. Stickney, in: Proc. 5th Intern. Symp. on Rarefied Gas Dynamics, Vol. 1, Ed. C.L. Brundin (Academic Press, New York, 1967) p. 49. R.E. Stickney, in: The Structure and Chemistry of Solid Surfaces, Ed. G.A. Somorjai (Wiley, New York, 1969) p. 41-1.
258
[ 1 I] [12] [13] [14] [15] [ 161 [17] [I81 [ 191 [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
Ch. Steinbriichel
/ Gas-surface
scattering distributions
Ch. Steinbrtichel, Surface Sci. 66 (1979) 13 1. J.A. Barker and D.J. Auerbach, Chem. Phys. Letters 67 (1979) 393. E.K. Grimmelmann, J.C. Tully and M.J. Cardillo, J. Chem. Phys. 72 (1980) 1039. Ch. Steinbtiichel, Chem. Phys. Letters 76 (1980) 58. J. Lorenzen and L.M. Raff, J. Chem. Phys. 54 (1971) 674. R.M. Logan and J.C. Keck, J. Chem. Phys. 49 (1968) 860. R. Sau and R.P. Merrill, Surface. Sci. 34 (1973) 268. W.H. Weinberg and R.P. Merrill, J. Chem. Phys. 56 (1972) 2881. A.G. Stoll, Jr., J.J. Ehrhardt and R.P. Merrill, J. Chem. Phys. 64 (1976) 34. R.E. White, J.J. Ehrhardt and R.P. Merrill, J. Chem. Phys. 64 (1976) 41. Ch. Steinbrtichel and L.D. Schmidt, J. Phys. Chem. Solids 34 (1973) 1379. J.A. Barker, D.R. Dion and R.P. Merrill, Surface Sci. 95 (1980) 15. N. Garcia, J. Chem. Phys. 67 (1977) 897. RI. Masel, R.P. Merrill and W.H. Miller, J. Chem. Phys. 64 (1976) 45. R.B. Subbarao and D.R. Miller, J. Chem. Phys. 58 (1973) 5247. G.G. Kleiman and U. Landman, Solid State Commun. 18 (1976) 819. N. Esbjerg and J.K. Neirskov, Phys. Rev. Letters 45 (1980) 807. H. Asada, Japan. J. Appl. Phys. 19 (1980) 2055. F.O. Goodman, Surface Sci. 7 (1967) 391. D.P. Jackson and J.B. French, in: Proc. 6th Intern. Symp. On Rarefied Gas Dynamics, Eds. L. Trilling and H.Y. Wachman (Academic Press, New York, 1969) p. 1119. [31] J. Lapujoulade, Y. Lejay and G. Armand, Surface Sci. 95 (1980) 107.
Vol. 2,
247
GAS-SURFACE SCATTERING DISTRIBUTIONS THE HARD-SPHEROID MODEL
ACCORDING
TO
Christoph STEINBRDCHEL Laboratories Received
RCA, Ltd., Badenerstrasse
28 September
198 1; accepted
569, CH-8048 Ziirich, Switzerland for publication
27 November
198 1
Gas-surface scattering distributions are calculated on the basis of the hard-spheroid model. This new model is related to the hard-cube model but incorporates surface structure by letting the surface atoms have hard spherical caps. Surface structure is found to reduce the maximum scattered intensity and to broaden the scattering distribution. The decrease in the maximum scattered intensity is most pronounced for a light gas. Broadening is asymmetric relative to the direction of maximum scattered intensity in that backscattering towards the surface is favored. Adding surface structure to the hard-cube model improves significantly the agreement between model predictions and experimental results.
1. Introduction
The scattering of an atom beam from a surface is becoming an ever more versatile technique for probing the structure of surfaces and their interaction with gases [ 1,2]. Experimental developments, such as the use of mono-energetic beams [1,3-61, are paralleled by theoretical efforts aimed at describing efficiently the many-body nature of the process [7]. To the experimenter it is of interest to have at hand a computationally simple model which enables one, at least, to correlate qualitatively experimental results. The simplest model which still gives a reasonable picture of gas-surface dynamics is the hard-cube model [8-lo]. In it the gas-surface collision is represented as an impulsive collision of a gas atom with a vibrating cube. This model has been quite successful in describing the gross features of gas-surface scattering in the absence of diffraction [2]. In addition, it has been able to account very well for gas-surface energy transfer [ 1l- 131. A major assumption of the hard-cube model, and one of its main deficiencies, is that the gas-surface interaction potential has no structure parallel to the surface. Thus the tangential momentum of a gas atom is assumed conserved during a collision. In a recent paper I proposed an extension of the hard-cube model, termed “hard-spheroid model”, which incorporates some surface structure in a simple way [ 141. That work indicated that in most cases 0039-6028/82/0000-0000/$02.75
0 1982 North-Holland
Ch. Steinbriichel
24R
/ Gas-surface
scattering distributions
surface structure would be rather unimportant for gas-surface energy transfer, but would be reflected directly in transfer of momentum parallel to the surface. Here I use the hard-spheroid model to calculate gas-surface scattering distributions for mono-energetic beams. (A similar model was used some time ago for trajectory calculations, but only with zero surface temperature [29,30].) The main conclusion will be that adding surface structure to the hard-cube model brings about a marked improvement between model predictions and experimental results.
2. Description of model In the hard-spheroid model [14] a gas atom undergoes an impulsive and purely repulsive collision with a surface atom. But in contrast to the hard-cube model, the surface atom is now assumed to have a spherical cap (fig. 1). The center-of-mass of the surface atom is taken to be in the center of curvature of the atom’s spherical cap, and all collisions are assumed to occur in the plane containing the center-of-mass. We will only consider sufficiently small surface structure, such that multiple collisions with neighboring surface atoms and shadowing can be neglected [14]. Let u and u be the gas and surface atom velocities before a collision, and let primes denote the same quantities after a collision. Also, let 8 be the macroscopic incident angle, 8’ the macroscopic reflected angle (taken negative), and + the local incident angle (fig. 1). We are interested in the probability P(P) that an atom impinging on the surface with velocity u = (a, 8) be reflected at an angle 8’:
where o = V(U, 8, 8’, <) and a* = M/Zk,T,. In (1) multiple collisions with a single spheroid are neglected [ 11,141. From fig. 1, relative to the coordinate system at the point of impact, one has u,, =usm+,
o,, = -u sint,
UI =ucos+,
ol=ucos&
(2)
with + = 8 - E. Furthermore, UX=u,,cos~+u,sin~,
u,=-u,,sin~+u,cosE,
(3)
where also u, = u sin 8 and uY = u cos 8, According to classical collision kinematics UT,=ull’
Y---l ’ --u1+ %-1”+1
2 -2) CL+1 I’
(4)
Ch. Steinbriichel / Gas-surface
249
scattering distributions
l V Fig. I. Geometry of the hard-spheroid
model.
with p= m/M the ratio of gas atom to surface atom mass. On combining eqs. (3) and (4) one obtains z@,[)
= u,
(
l-
2 -sin26 cc+1
r$(@, S) = -+j&
1
-(uY
sin6cost+u,
-o)--&sinEcosE,
(
l-
-cos2~ 2 P+l
1
+fJ-
2 cos2<. 1.1+1
(5)
One now uses (5) in the form tan 8’ = Q,
O/$(@,
g),
(6)
and solves for u as a function of U, 8, 8’, and 6: U
-=
UY
tan2t [
1+l.r (--pane’+-
‘Tme 2
+tant(l-ttanet=e’)--
)
1+cL -ttane
1-p 2 t-e’-
2
X+pt~ey’. Differentiation
I
(7) of (7) with respect to 8’ yields
-tm2~-tm~tme-x [c0s2epa E- tan et)] -‘.
1-p 2
0 +u
Y (8)
Thus, for any (u, 8,/Y) and any surface temperature T,, eqs. (7) and (8) may be used to evaluate the scattering probabilitv P(P), eq. (1). If one sets Es = imu
Ch. Steinbriichel
250
and defines
Es by Es = 2k,T,,
/ &.-surface
scattering distributions
then one can write (1) in the form
Eqs. (7) to (9) imply that P(P) depends on p, Es/E, and 8, as well as on the angle (Ycharacterizing surface structure. For (Y= 0 eqs. (7) to (9) reduce to the hard-cube result [9].
9
a)
b)
7
6
5
0
IS
30
45
60
0
15
30
45
60
75
0
15
30
45
60
0
15
30
45
60
75
SCATTERING
ANGLE
10’1 (dsg)
Fig. 2. Scattering distributions for p=O.O2: (V) designates angle of incidence; (p (flat surface); (- - -) a=7.5O; (- - - - - -) a= 12.5”. In (a) and (b) ES/Es =0.4, Es/E, =2.5.
) a=0 in (c) and (d)
Ch. Steinbriichel
/ Gas-surface
stuttering
251
distributions
3. Numerical results Results for scattering distributions obtained from numerically integrating (9) with the help of (7) and (8) are shown in figs. 2 to 4. These cover a range of values for the parameter ~1, two angles of incidence, and the incoming beam either considerably hotter or colder than the surface. Three different values for the angle (Y characterizing surface structure, or surface roughness, are compared: (Y= 0 (i.e., the hard-cube model), (Y= 7.5”, and (Y= 12.5’. Several features of the curves in figs. 2 to 4 are noteworthy with respect to the importance of surface structure. First, surface structure will broaden the scattering distribu&ns P(P) while reducing their maximum intensity Pm,. The reduction of P,, is the more pronounced the lighter the gas, the colder the surface, and the closer 8 to normal incidence. The decrease of P,,,= towards grazing incidence is made less significant by surface structure. For low Es and a fairly rough surface PmBx may even increase slightly towards grazing incidence (see, for example, figs. 2a and 2b for (Y= 7.5” or 12.5’, or figs. 3a and 3b for (Y= 12.5’). Thermal attenuation, i.e., a decrease of P,,,, with increasing Es, also becomes less significant because of surface structure. With regard to the position 18; 1 of I’,, we note that for a light gas, small surface structure has practically no effect on 18; (, but large surface structure
0
15
30
45
60
0
15
30
45
60
75
0
15
30
45
60
0
15
30
45
60
75
SCATTERING
ANGLE
18’1 (deg)
Fig. 3. Scattering distributions for p=O.l: (V) designates angle of incidence; ( (flat surface); (- - -) a=7.5’; (- - - - - -) a= 12.5’. In (a) and (b) Es/E, ~0.4, Es/E, =2.5.
) n=O in (c) and (d)
252
Ch. Sieinbriichel / Gas-&ace
0
15
30
45
SCATTERING
60
0
ANGLE
15
scattering distributions
30
45
60
75
1.9’1 (deg)
Fig. 4. Scattering distributions for p=O.2: (V) designates angle of incidence; (--) (Y=o (flat surface); (- - -) 01=7.5”; (- - - - - -) a= 12.5”. In (a) and (b) ES/Es =0.4. in (c) and (d) Es/E, =2.5.
will move I&, 1 in the backward direction (fig. 2). For a heavy gas, surface structure generally will move ItI; 1in the forward direction, particularly if Es is large (fig. 4). The shape of P(P) is affected in an interesting way by surface structure. A rough surface will broaden P( 8’) asymmetrically, enhancing backward scattering towards the surface normal, even if 18; I is moved in the forward direction (see e.g. figs. 3c, 3d, 4c and 4d). In fig. 5 results are displayed in the form of the deviation of the position
8 : . Fig. 5. Deviation of direction of scattered intensity maximum from specular direction, AB, versus pE,/E,, with 8=45”. Solid lines are for a=0 (flat surface) and thermal beam [eq. (IO)], other symbols [(0), (O), etc.] for mono-energetic beam. (0), (0) cr=O (flat surface); (a), (A) a=7.5O; (V), (V) (I= 12.5”. Open symbols: p=O.l; full symbols: ).~=0.2.
Ch. Sreinbriichel / Gas-surface
scattering distributions
253
(8; 1 of the scattering maximum from the specular direction, A0 = 0 - 18; (, versus pE,/E, and incident angle 8. (For the curves corresponding to a Maxwellian beam Eg was identified with 2k,T,.) Here one notes that surface structure makes A9 a less rapidly varying function of pEs/Eg, and this the more so the lighter the gas.
4.Discussion We wish first to relate the predictions of the hard-spheroid model to experimental results as well as to other simple model calculations, in particular to the hard-cube and the soft-cube models. Then we examine the assets and limitations of the present model, and finally we point to some unresolved issues arising from this work and to some suggestions for future work. For a comparison to experiments we rely on three papers in which monoenergetic beams were used as well as on some work employing thermal beams. Yamamoto and Stickney [3] studied scattering of rare gases with Eg = 0.065eV from W( 110) at T, = 2100 K. Subbarao and Miller [4] investigated scattering of rare gases with 0.06 G Eg G 2.5eV from epitaxially grown Ag( 111) at T, = 560K. (Here only the lower part of the range of Eg is relevant for our purpose.) In both cases the hard-cube model is able to describe qualitatively experimental trends, while quantitatively certain discrepancies are noted. The hard-cube model overestimates the magnitude of A@, the deviation of the scattering maximum from the specular direction, whereas it underestimates t9,,*, the full-width-at-half-maximum of the scattering distribution [3,15]. For both these quantities, our work shows that including surface structure leads to improved agreement with experiments (table 1). Note also that the soft-cube model [ 161, in which the gas-surface potential is not impulsive but has non-zero range, provides an improvement over the hard-cube model as regards A8 but not 13,,~ [ 151. On the other hand, a finite-lattice model improves upon the hard-cube model with respect to both AB and 8,,* [15]. Lapujoulade and Lejay [S], in a detailed study, determined scattering distributions and a few moments of the scattered-velocity distributions for He, Ne, and Ar on Cu( 100). In that work Eg = 0.065(or0.063)eV, 300 G T, 4 800 K, and 30” d 8 G 60”. The scattering distribution for He is very narrow and specular, and for Ne and Ar becomes wider and slightly backscattered. Also, for Ne and Ar, A@and 8,,, are both smaller than expected form the hard-cube or the hard-spheroid model, but one should keep in mind that certainly for Ar (CL= 0.63), and probably even for Ne (~1= 0.32), the gas-surface collision will not really be impulsive. On the other hand, the dependence of A8 and e1,* on surface roughness (comparing scattering with [lOO] and [ 1lo] azimuths of the incoming beam) is as expected on the basis of the present work. Furthermore, scattered gas atoms have, on average, lost momentum
Experimental a=7.Y.
2 0.8 2 0.8
30
I
31 37 53 29 28 35 28
Expt.
Bt/r
I I I). Figures for hard-spheroid
-I
10
5 12 16 5 -2
0 3 12 0 -4 2 0
Hard-spher. model
results
2 8 13 2 -4 6
Hard-cube model
scattering
Expt.
A@
gas-surface
data from ref. [3] for W(110) and from ref. [4] for Ag(
50
5.6
40
He/W( 1 f0) Ne/W( 110) Ar/W(I IO) Ne/Ag( 111)
G/Es
and experimental
e
of theoretical
System
Table 1 Comparison
26 37 39 23 22 31 28 model calculated
Hard-cube model 30 39 44 34 33 37 36
with surface
Hard-spher. model
roughness
parameter
B
5 ii: 8,
B F: 3. 2
4 $
f
:
i
2““.
9
parallel to the surface, and the more so the higher T,, in agreement with the hard-spheroid model [ 141. In order to compare the present results to experiments involving a thermal beam we note the expected similarity between scattering of a thermal beam at Tp and a mono-energetic beam with Eg = 2k,Ts [IO]. (Of course it would be str~~tfo~~d to integrate, say, eq. (9) over a distribution of incident velocities u.) Moreover, it should be pointed out that if in the hard-cube model one integrates over the incoming thermal velocity distribution [9,10], instead of using a mean-speed analysis [S], one obtains
Thus A8 is not a universal function of pq/T, [ 173,but rather depends on p and T,/T, separately (cf. fig. 5). It then becomes clear from fig. 5 that surface structure improves the hard-cube model in the right direction, making A&a less rapidly varying function of pT,/T,. For quasi-elastic scattering [2], typically He on Ag [ 171or W [3,18,19], fig. 2 suggests that P,,,= is a sensitive function of surface roughness and that thermal attenuation is the less pronounced the rougher the surface. Both these trends are observed experimentally [17]. For inelastic scattering 121, e.g. Ne on Ag [4,17] or W [3,19,20], attenuation and broadening do not simply correlate with surface roughness, but involve dynamical interactions of the gas with the solid lattice [17]. The hard-spheroid model combines both these features and indicates, in agreement with experimental trends [ 171, that the effect of increased surface roughness may be offset, in part, by decreased inelasticity (compare e.g. figs. 3a and 4a). The limitations of the hard-cube model are due essentially to two assumptions: an impulsive gas-surface collision, and negligible surface structure. The hard-spheroid model retains the former but overcomes the latter assumption. The ass~ption of an impulsive collision h&s been discussed extensively [2,10,21]. It should be valid for a light gas colliding with a surface of heavy atoms, as is apparent on comparing relevant surface atom dimensions and gas atom velocities. These arguments are confirmed by detailed molecular dynamics simulations [22]. One might try to make the model more realistic by including an attractive well of depth D in the gas-surface potential and/or by allowing for a gas atom to collide with a larger, effective number of surface atoms (i.e., using a reduced effective p) [13]. But clearly, these two effects tend to compensate for each other. Indeed, hard-cube calculations of energy transfer indicate that a best fit to experiments, varying both D and ,u, is rather ill-determined [ 131. Although the hard-spheroid model incorporates surface structure, it does so in a somewhat crude fashion, since a realistic hard-wall envelope of the gas-surface potential would certainly not contain any cusps as shown in fig. 1
256
Ch. Steinbriichel / Gas-surface
scattering distributions
[1,23]. Thus the hard-spheroid model cannot reproduce accurately such an effect as rainbow scattering [20,24]. But, perhaps surprisingly, the model does display the beginnings of rainbow scattering in that for small p and large surface structure the scattering distribution is very broad and flat, with the maximum displaced strongly towards the surface normal (cf. figs. 2a and 2b). On the other hand, semiclassical calculations using a more realistically shaped but static hard-wall potential also fail to account quantitatively for rainbow scattering by predicting too narrow a scattering distribution [24]. This suggests that structure and inelastic effects must be treated on an equal footing. The same conclusion follows from a comparison of the hard-spheroid, the cube-type [8-10,161, and the finite-lattice models [ 151 (see arguments above). The surface structure in the hard-spheroid model is somewhat unrealistic in another way because the two-dimensional nature of the surface topography is neglected. Yet it is clear that one needs to be careful in comparing scattered intensities, taking proper account of detector geometry and out-of-plane scattering [5,28]. Logan et al. [9] discussed briefly applying the hard-cube model to the situation where there is a distribution of incident angles, probably with a view on dealing with a macroscopically rough surface, but they did not pursue the subject any further. The results from the hard-spheroid model imply that surface structure, and also macroscopic surface roughness, will manifest itself most clearly in scattering distributions and momentum transfer parallel to the surface, less so in energy transfer. As to unresolved issues, the hard-spheroid model makes a definite, somewhat unrealistic assumption about the shape of surface structure (cf. information on surface structure derived from diffraction experiments [1,23]). But the degree of surface roughness for a particular gas-surface system, given in the hard-spheroid model by the parameter (Y(fig. l), is also not clear a priori. (If c is the relative amplitude of modulation of the hard-wall surface envelope, then a! * 4c [ 141.) For example, from measurements of momentum accommodation parallel to the surface [5,25] one would infer that aNe < oAr [14], whereas from atomic size and probably also from scattering distributions one would conclude that aNe > ok. Yet even arguments based on atomic size are somewhat contradictory (cf. refs. [ 17,23,26]). Moreover, for He scattering, experiments [l] as well as calculations of the gas-surface potential [27] imply a turning point of the gas atom rather far away from the geometrical surface. Thus surface roughness may well be a function of how deeply gas atoms penetrate the surface potential, or of how inelastic the gas-surface collision turns out to be. Neither the hard-spheroid model nor any of the other models mentioned is able to explain that for He the maximum scattered intensity increases towards grazing incidence [3,5,17,18] (cf. fig. 2). This effect can be rationalized quantum -mechanically in terms of attenuation of elastic scattering via the Debye-Waller factor [31].
Ch. Steinbrikhel
/ Gas-surface
scattering distributions
251
It will also be quite straightforward to calculate velocity distributions of scattered atoms as a function of scattering angle with the present model. Indeed, it is clear that in that respect as well the hard-spheroid model will improve upon the hard-cube model. One need only recall that because of conservation of momentum parallel to the surface, the energy of a scattered gas atom is fixed by the scattering angle in the hard-cube model. Experimentally, the mean speed of a scattered atom is found to be lower at small scattering angles (small )@‘I), and higher at large scattering angles (large l&l), than predicted by the hard-cube model [3,10]. But, for example, a small macroscopic scattering angle implies, on average, more nearly normal incidence locally on the spheroid (cf. fig. l), i.e., a more inelastic collision and, therefore, a smaller speed for the scattered gas atom. By the same token, atoms scattered backwardly will tend to have lost momentum parallel to the surface [5,14].
5. summaly
We have presented theoretical results on gas-surface scattering distributions using the hard-spheroid model. This new model derives from, and is conceptually as simple as, the hard-cube model but it also incorporates some surface structure in a simple way. Surface structure turns out to reduce the maximum scattered intensity and to broaden the scattering distribution, with backscattering towards the surface normal being favored. Thus the hard-spheroid model provides a physically transparent, qualitative picture of gas-surface scattering as a function of gas and surface atom mass, surface temperature, gas atom energy and angle of incidence, and surface structure. Since the model allows for gas-surface momentum transfer parallel to the surface, it should yield reasonable scattered-velocity distributions as well.
References [l] (21 [3] [4] [S] [6] [7] [8] [9] [lo]
K.H. Rieder and T. Bngel, Nucl. Instr. Methods 170 (1980) 483. W.H. Weinberg, Advan. Colloid Interface Sci. 4 (1975) 301. S. Yamamoto and R.E. Stickney. J. Chem. Phys. 53 (1970) 1594. D.R. Miller and R.B. Subbarao, J. Chem. Phys. 52 (1970) 425. J. Lapujoulade and Y. Lejay, J. Chem. Phys. 63 (1975) 1389. K.C. Janda, J.E. Hurst, CA. Becker, J.P. Cowin, D.J. Auerbach and L. Wharton, J. Chem. Phys. 72 (1980) 2403. J.C. Tully, Ann. Rev. Phys. Chem. 91 (1980) 319. R.M. Logan and R.E. Stickney, J. Chem. Phys. 44 (1966) 195. R.M. Logan, J.C. Keck and R.E. Stickney, in: Proc. 5th Intern. Symp. on Rarefied Gas Dynamics, Vol. 1, Ed. C.L. Brundin (Academic Press, New York, 1967) p. 49. R.E. Stickney, in: The Structure and Chemistry of Solid Surfaces, Ed. G.A. Somorjai (Wiley, New York, 1969) p. 41-1.
258
[ 1 I] [12] [13] [14] [15] [ 161 [17] [I81 [ 191 [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
Ch. Steinbriichel
/ Gas-surface
scattering distributions
Ch. Steinbrtichel, Surface Sci. 66 (1979) 13 1. J.A. Barker and D.J. Auerbach, Chem. Phys. Letters 67 (1979) 393. E.K. Grimmelmann, J.C. Tully and M.J. Cardillo, J. Chem. Phys. 72 (1980) 1039. Ch. Steinbtiichel, Chem. Phys. Letters 76 (1980) 58. J. Lorenzen and L.M. Raff, J. Chem. Phys. 54 (1971) 674. R.M. Logan and J.C. Keck, J. Chem. Phys. 49 (1968) 860. R. Sau and R.P. Merrill, Surface. Sci. 34 (1973) 268. W.H. Weinberg and R.P. Merrill, J. Chem. Phys. 56 (1972) 2881. A.G. Stoll, Jr., J.J. Ehrhardt and R.P. Merrill, J. Chem. Phys. 64 (1976) 34. R.E. White, J.J. Ehrhardt and R.P. Merrill, J. Chem. Phys. 64 (1976) 41. Ch. Steinbrtichel and L.D. Schmidt, J. Phys. Chem. Solids 34 (1973) 1379. J.A. Barker, D.R. Dion and R.P. Merrill, Surface Sci. 95 (1980) 15. N. Garcia, J. Chem. Phys. 67 (1977) 897. RI. Masel, R.P. Merrill and W.H. Miller, J. Chem. Phys. 64 (1976) 45. R.B. Subbarao and D.R. Miller, J. Chem. Phys. 58 (1973) 5247. G.G. Kleiman and U. Landman, Solid State Commun. 18 (1976) 819. N. Esbjerg and J.K. Neirskov, Phys. Rev. Letters 45 (1980) 807. H. Asada, Japan. J. Appl. Phys. 19 (1980) 2055. F.O. Goodman, Surface Sci. 7 (1967) 391. D.P. Jackson and J.B. French, in: Proc. 6th Intern. Symp. On Rarefied Gas Dynamics, Eds. L. Trilling and H.Y. Wachman (Academic Press, New York, 1969) p. 1119. [31] J. Lapujoulade, Y. Lejay and G. Armand, Surface Sci. 95 (1980) 107.
Vol. 2,