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Specular reflection of helium and hydrogen molecular beams from the (111) plane of silver
Surface Science 81 (1979) 386-408 0 North-Holland Publishing Company
SPECULAR REFLECTION OF HELIUM AND HYDROGEN MOLECULAR BEAMS FROM THE (111) PLANE OF SILVER * Hiromu ASADA
Received 7 August 1978; manuscript received in final form 11 October 1978
The Debye-Wailer (DW) factor in the specular reffection intensity of He and Us molecular beams from the Ag (111) plane has been studied experiment~y and theoretically. A new expression for the DW factor corrected for a stationary part of the gas-surface interaction potential is derived kinematically and semi-classically by the use of a Morse potential. An analysis of the experimental data through the above DW factor yields a surface Debye temperature of 251 _+20 K, which is unusually high, and potential depths of 1.5 + 1.0 meV for He and 6.4 + 2.9 meV for Hz, which seem slightly too small. These results are discussed on the basis of the nature of gas-surface interactions and in comparison with the results deduced from the conventional DW factor corrected for a constant attractive potential depth.
1. Introduction Many discussions [l-17] have been made concerning the Debye-Waher (DW) factor in the specular reflection intensity of molecular beams from solid surfaces. One of the important conclusions is that the kinetic energy of incident molecules during collision and reflection on the surface should be corrected for a stationary attractive potential in front of the surface. The corrected DW factor, e-2w, has been proposed by Hoinkes et al. [9] in the following form e-2w = exp[-{(Ask)’
t S~nD/fi~‘)<3;~>],
(1-l)
where Ak is the change in the normal component of the wave vector of the molecular beam, evaluated at the infinite distance from the surface, m the mass of the molecule, D the attractive potential depth, A the Planck constant devided by 27r, and (t21 the mean square displacement, perpendicular to the surface, of the surface atom due to thermal vibration. The (t2) is conveniently expressed in terms of the Debye temperature Or, by means of the isotropic Debye model for crystals. When the temperature is rather higher than OD, the expression is
* Taken from the Ph.D. thesis submitted to Hokkaido University, 1977. 386
H. Asada /Specular Cc*) = 3t2*Ts/MkBO~
reflection of He and Hz from Ag(ll1)
,
381
(l-2)
where M is the mass of the solid atom, kB the Boltzmann constant, and T, the surface temperature. The corrected DW factor has succeeded in describing very well the observed dependence of the specular reflection intensity on the surface temperature and the incident angle of the beam. But on all the surfaces investigated [9-161, eq. (1 .l) together with eq. (1.2) has been found to yield higher Debye temperatures than those observed in LEED [18] or those expected theoretically [ 19-211; the surface Debye temperature is roughly estimated to be the bulk one divided by 42. The result has been ascribed to a collision of a gas molecule simultaneously with more than one surface atom due to their finite sizes [9,13]. In the present paper the DW factor in the specular reflection of thermal molecular beams of He and Hz from the (111) plane of silver is investigated. It will be shown that He and H2 beams give different Debye temperatures if the specular reflection intensity is analysed by eq. (1.1) together with eq. (1.2). Instead of eq. (l.l), a new expression for the correction on the DW factor will be proposed from the kinematical theory of diffraction based on semi-classical mechanics. The surface Debye temperature and the depths of the gas-surface interaction potentials will be deduced through the new DW factor and discussed on the basis of a nature of gas-surface interactions and in comparison with the results through eqs. (1.1) and (1.2).
2. Experimental The experimental apparatus, shown schematically in fig. 1, has been previously described elsewhere [22]. The beam is generated by a Pyrex glass nozzle with an aperture of 40-50 pm diameter, which can be cooled down to ca. 150 K. The gas beam passes through a skimmer (0.62 mm diam.) at a distance of 6-8 mm from the nozzle, and it is collimated by two circular orifices. The direct beam, which is ob-
TO PUMP
Fig. 1. Schematic drawing of the experimental apparatus. N = nozzle, S = skimmer, 01 = 1st collimating orifice, 02 = 2nd collimating orifice, C = chopper wheel rotated by a synchronous motor, E = evaporator, T = test surface, and D = detector.
388
H. Asada / Specular reflection of He and H, from Ag(ll1)
served downstream along the incident beam with the test surface removed, has a full width 0.9 + 0.1’ at half maximum in terms of the rotation angle of the detector around the test surface. The - - wave number k of the incident beam has been determined by observing the (1, 1) diffraction of the beam from a cleaved (001) surface of LiF: The wave number distributionj(k) in the incident beam, deduced from the broadening of the (i, i) peak, has been found to be well described by - /~,,)~/a’] dk .
f(k) dk = const. k3 exp[-(k
(2.1)
The constants, kO and a, are listed in table 1 with other related quantities . The spatial distributions of He and H2 scattered by a test surface are observed with an electron-impact ionization detector, whose output is measured by means of phasesensitive detection with the incident beam modulated at about 140 Hz. The output signal is proportional to the beam flux flowing into the detector. The silver surfaces have been prepared in the apparatus by evaporating 99.99% Ag onto cleaved mica surfaces at temperatures 500-550 K. Transmission electron diffraction revealed that the deposited films are oriented with the (111) axis perpendicular to the surface due to epitaxial growth. After the deposition of Ag was completed, the film was annealed at 550-600 K for more than 10 min before scattering experiments. The experiments have been performed on four films at surface temperatures of 450-600 K and at background pressures of 4-10 X 10e6 Pa. The specular reflection intensity of the molecular beams, which has been proved to be a sensitive monitor of surface cleanliness [23], has been found not to be changed significantly for several hours after deposition but to be reduced to half or less if the
transverse 4 plane
Fig. 2. Definitions
of the coordinate
system
and various
angles.
H. Asada / Specular reflection of He and H, from Ag(l I I)
389
H. Asada / Specular rejlection of He and H2 from Ag(il I)
390
test surface has been left in the apparatus without pumping for a night. Additional deposition of Ag has restored a great part of the specular reflection intensity and enabled one to continue the experiments. In fig. 2 the coordinate system and the definitions of angles are illustrated. The z-axis is a surface normal. The incident angle Bi is measured from the z-axis. The plane which contains the incident beam and the z-axis is called the incident plane (I-plane}. A directional vector of scattering makes an angle ~9~with the I-plane, and its projection on the I-plane makes an angle 8, with the z-axis. A plane perpendicular to the l-plane is called a transverse plane {T-plane), which is specified by the angle 0 r
.
3. Experimental results A typical trace of the spatial distribution of He scattered by a Ag surface is shown in fig. 3. A similar distribution is observed also for Hz. The distribution consits of a sharp specular reflection peak and a broad skirt around it. This broad skirt,
(al
I
-
l
..*** . .
10
l
..*
. . ..l* . l
.
x5 ‘_ x5 . -. *... b
‘.
.
l. 80
f
90
-5
l** 20
0 h
cdk”e )
Fig. 3. Typical trace of a spatial distribution of He scattered from a Ag(lll) surface; E = 31 meV, Bi = 70”, T, = 500 K: (a) in the I-plane; (b) in the T-plane with 0r = 70”. The inelastic scattering part is enlarged by a factor of 5. The inelastic background level which should be subtracted in the evaluation off,, is indicated by “b.g.“.
H. Asada / Specular rejlection of He and Ii, from Ag(iI I)
391
which is regarded as a result of inelastic scattering by phonons of Ag, has been discussed in a previous paper [22]. The specular reflection peak is found to have a greater width than that of the incident beam by l-3’ in the I-plane, but to be scarcely broadened in the T-plane. The broadening of the peak may however be attributed to bending and/or large scale roughness of the test surface [ 171. The intensity of specular reflection, IsP, is defined as the peak area in the I-plane after subtracting the inelastic scattering background whose level is indicated in fig. 3 as an example. The ISp is normalized with respect to the direct beam intensity which is also defined as an area of the profile in the I-plane. In fig. 4, the dependence of ISr, on (Ak)2 at constant surface temperature is exemplified for He and H2 beams. The Ak is given by Ak = 2ki
COS 8i
(3.1)
,
where ki, the wave number of the incident beam, is equated with k. in eq. (2.1). The reason for choosing kO rather than the mean value k for ki is that molecules with k < k contribute to the specular reflection intensity more than those with k > % due to the effect of the DW factor. The value of k. is smaller than k by several percent (see table 1). The plot indicates a good linear relationship between In Is,., and (Ak)2. In fig. 5, the dependence of ZSp on the surface temperature I;; ranging from 450 to 600 K at fixed incident angles is exemplified. The plot also follows a straight line. At temperatures lower than 450 K a decrease in f,, is observed, which is probably attributed to adsorption of some of the background gases. The good linear dependence of In IS,, and (A/C)’ and T, allows one to represent the specular reflection intensity by the conventional DW factor, Is,, = const. exp[-3(tzAk)2Ts/&ikB02]
.
(3.2)
It should be kept in mind that the Debye temperature 0 in eq. (3.2) does not necessarily represent a constant characteristic of the solid surface, as shown below. For convenience, two apparent Debye temperatures, OT and 00, are defined; the former is calculated from the observed linear relationship between In ISp and (Ak)2 at constant T,, and the latter calculated from that between In ISp and T, at fixed Bi. The suffixes T and B signify measurements at constant temperature and at fixed incident angle, respectively. In fig. 6, the values of OT obtained are plotted against T,. Though OT has not been necessarily well reproduced for four test surfaces, the following results may be seen: Firstly, the beam of the same kind of gas yields the same value of OT almost ~dependen~y of the incident energy. This result supports the expectation that ISp is well described only by the DW factor. Secondly, the values of OT observed with He are in the range of 190-220 K,
H. Asada / Specular reflection of He and H, from Ag(lI1)
392
z”r
20
70 II, 0.1’
0
I
65
60 a 100
55 I I
h
50 I ’
I
r
’ 200
(ak)' (A-')
surface Fig. 4. Semi-log plots of the specular reflection intensity Is* versus (Ak)’ at constant temperatures written in the figure. The incident angles Bi are also shown on the abscissa. The straight lines are fitted by the least-squares method. (a) He beam with /CCJ= 9.9 A-‘. (b) He beam with k,~ = 7.5 A-‘. (c) Hz beam with ka = 7.3 a-‘. (d) Hz beam with ko = 5.5 A-‘.
393
(a) He, ko=9.9h-’
(b) He, ko=7.5?Y’
'OF
o1* 450
500
550 Ts (K)
600
olI * 450
500
550 Ts (K)
600
Cc) HP. ko-7.3&-l
,.
E
T
70l
60Q
-.-‘::: .
0
l
.
5o”
0
0
.
4oa .
Ode
30°
600
Ts (K)
\ 0
t
ool. .
450
500
Ts (K)
550
600
Fig. 5. Semi-log plots of Isp versus T, at fixed incident angles written in the figure. The straight lines are fitted by the least-sqaures method. (a) He beam with ko = 9.9 A-‘. (b) He beam with ko = 7.5 A-‘. (c) Hz beam with k. = 7.3 A-‘. (d) H2 beam with k. = 5.5 A-‘.
H. Asadu f Specular rejlection of Ne and H, from AgJll I)
394
250-
(a)
He
250-
Y
-
;;
0'
_
IO.
(b)
H2
-
200-
200"
I,ot-----450
500
550
600
Ts (K)
,,oL 450
500
550
600
T, (Kl
Fig. 6. Dependence of the apparent Debye temperature @T on Ts; the symbols 0, n,*, and v, whether open or closed, refer to the test surfaces of No. 1, 2, 3, and 4, respectively: (a) by use of He beam with k. = 9.9 A-’ (open symbols) or 7.5 A-I (closed ones); (b) by use of Hz beam with ko = 7.3 A-” (open symbols) or 5.5 A-’ (closed ones).
while those with Hz are in the range of 180-200 K, obviously lower than the former by lo-20 K. In other words, the apparent vibrat~onai ~plitude of the surface probed by He beams is smaller than that by H2 beams. The result cannot be explained by eq. (1.1) together with eq. (1.2), which always yields &-equal to the true Debye temperature (see eq. (5.6)). Finally, 0~ decreases with increasing Ts. The decrement of OT is rather large, lo-20 K per an increment of T, of 100 K. If it is wholly attributed to an anharmonicity of surface lattice vibration perpendicular to the surface, the anharmonicity should be anomalously large. A similar result has been found by Armand et al. [ 131 on the (001) plane of Cu. In fig. 7, the values of Oe are plotted against Ak. Although 00 is not necessarily well reproduced for four test surfaces, the following results are seen: Firstly, the beam of each gas seems to yield nearly the same value of Oe on each test surface when Ak is identical for different incident energies. This result also gives support to eq. (3.2). Secondly, Oe is found to decrease with increasing Ak (i.e., with decreasing ei). Finally, the He beam gives a greater value of 00 than Hz does by 20-40 K at the same value of Ak.
4. Kinematical theory of the DW factor Since the corrected DW factor of eq. (1 .I) has been four& to fail in interpreting the discrepancy in @T observed with He and Hz molecular beams, one should con-
H. Asada / Specular reflection of He and Hz from Ag(l I I)
(a)
200-
He
5
Ak
(b)
200
395
(A-‘,
IO
15
Hz
r
I
I
5
IO
15
Ak ( 8’)
Fig. 7. The apparent Debye temperature 00 plotted against Ak; the symbols are as in fig. 6: (a)
by use of He beam; (b) by use of Hz beam.
sider the DW factor in the molecular beam scattering intensity from a basic viewpoint. At first, it should be noted that atomic and molecular beams have unique features, compared with X-rays, electrons and neutrons: The beam particle itself has a size of the order of an atom or a molecule, and it interacts with solid atoms through not only a short-range but also a long-range force. Owing to the first feature, the incident particle even has a chance of colliding simultaneously with more than one surface atom. The second feature may also allow the incident particle to interact with more than one solid atom in the vicinity of the surface. The simultaneous
396
H. Asada /Specular
reflection of He and H, from Ag(l1 I)
interaction brought about by these two features makes it difficult to treat the scattering by the kinematical theory which in general assumes a single collision between an incident particle and solid atoms. But this dif~culty is overcome by regard~g the scattering events as caused by a potential fiefd at the surface rather than by an assembly of scattering centers. According to this idea, consider an interference of the waves which are accompanied by the molecule and scattered by the potential field affected by lattice vibration. The assumed potential k’is V(x, y, 2, t) = L)(e-2”(z-r)
- 2 e-“‘)
,
(4.1)
where { = {(x, y, r) is the displacement of the “surface” due to lattice vibration perpendicular to the surface at the position (x, y) and at the time instant t. The D and K represent the depth and the reciprocal range of the potential, respectively. The attractive part of the potential given by the second term is assumed to include no effect of thermal vibration, since many atoms in the vicinity of the surface contribute to the attractive potential through the Van der Waals long-range force to make it rather stationary on the average. The displacement is assumed so that (f2) is represented by eq. (1.2) at any point in a surface unit cell, which is equivalent to regarding the surface as a continuum. The bracket ( > refers to a statistical ensemble average. The scattering ~plitude F of the wave accompanied by the molecular beam is given by the following integral over the surface with an area S, F = const. jjexp s
[i@(x, y. t)] d.x d.y ,
(4.2)
where #(x, y, t) is the phase of the wave scattered at the position (x, y) and the time c. Other factors in F except the phase factor are insigni~c~t in the calculation of the DW factor and assumed to be constant. The intensity I of the scattered wave is given by I = (IFi’) = const. (
exp [i#(x, y, t) - @(xi, y’, t’)f d.x dy dx’ dy’> ~~ s S
=const.exp[-&jjfi(@-$1)2)dxdyti’dyj s s (4.3) where 4(x, y, t) and @(xl, y’, t’) are abbreviated to @and $‘, respectively. The phase @ is calculated by the following phase integral along the classica trajectory of the molecule in the semi-classical approximation,
391
H. Asada / Specular reflection of He and H, from Ag(l I I)
4(x, Y, 0
=$j- (2m [E -
I+, y, z, t)]}“’
ds ,
(4.4)
c.t. where ds is an element of the distance along the trajectory. When { is small enough, one may expand the integrand in a power series of { and neglect the higher order terms, i.e., @(x, y, t) =$ s
[2m(E - V,)] 1’2 ds -;
c.t.
s
r(aV’a’)‘=or,2
c.t . k@
- Vo>l
ds
,
(4.5)
where V&Z) = D(e-2Kz - 2 eeKZ) .
(4.6)
The first constant term in eq. (4.5) may however be omitted, since it has no significance in the calculation of I. It is further assumed that the vibration of the surface is so slow compared with the speed of the incident molecule that one may regard the displacement 5 as independent of time t during the gas molecule-surface collision. More precisely, this assumption is valid provided that the period of the surface atom vibration is sufficiently longer than the time necessary for the molecule to pass through the region with a significant effect of thermal vibration. This condition is formulated as 2/KVg<
27&J,
(4.7)
where us is the speed of the molecule and w is the angular frequency of the surface atom vibration. When relevant values, o = 2 X 10’ 3 s-l, vg = 1 X 1O’cm/s, and K = 5 X IO8 cm-’ are substituted, the left and right hand sides have values of 4 X IO-l4 s and 3 X lo-l3 s, respectively, i.e., the relation (4.7) holds rather well. Then, the integral of the second term of eq. (4.5) can be calculated exactly for specular reflection, after { is put out of the integral. The resultant equation (see Appendix I) is @(x>Y, t) = -Ak&)
Rx, Y, 0 ,
(4.8)
where g(t) = 1 + (rr - tan-‘E)/$
,
g = (:Ez/D)1'2 = tzAk/(8mD)1’2
(4.9) ,
(4.10)
and E, is the incident energy of the molecule associated with the z-directed motion. The intensity I is now calculated by substituting eq. (4.8) in eq. (4.3). The first term in the exponent of eq. (4.3) is $ ljQ2) 5
dx dy = (Ak)’ (g(.$))‘Q2,
.
(4.11)
398
H. Asadu / Specular reflection
of He and H, from Ag(lI 1)
The second term is
(4.12)
where T is a time interval (x sin ei)[us during which a wave front of the molecular beam proceeds from the position (0,O) to (x, y) on the surface. Eq. (4.12) contains the correlation between { at different positions, averaged over the whole surface. This quantity is ho&ever much smaller than (c2), so that the second term in the exponent of eq. (4.3) can be neglected in comparison with the first term. Consequently, the specular reflection intensity Is, is given by f,, = const. exp[-(Ak)‘({‘) =
const.
exp _ [
{g(#“]
?$fS?J.{g(g))*
.
1
(4.13)
The difference between the present DW factor and the conventional one is a factor k(t)}” in the exponent, arising from the stationary attractive part of the Morse potential. For convenience, the model used in the derivation of eq. (4.13) is called a Morse potential model or an MP model. The corrected DW factor of eq. (1.1) can be derived in a similar way, assuming that a stationary attractive potential in front of the vibrating surface has a constant depth D near the surface. This model is referred to as a constant potential model or a CP model (see Appendix 2).
5. Potential
depth and true Debye temperature
According to eq. (4.13) a plot of In I,, versus T, gives a straight line, while that of In ISp versus (Ak)’ does not. The deviation from the straight line is however very small, as illustrated in fig. 8, so that the apparent Debye temperature 0~ can be determined experimentally without difficulty and ambiguity. The apparent Debye temperatures, OT and 0 e , are calculated from the following equations,
and
H. Asada / Specular reflection of He and Hz from Ag(I I I)
399
Therefore, %-I%
= [(I + ?)/~*g(U
@&D
= l/g(g) .
l’* ,
64)
The ratio of the two apparent
Incidentally,
(5.3)
Debye temperatures
is
in the CP model, the apparent Debye temperatures
are given by
(5.6)
o*=oD, 2
O&D = 0&3T =
l/2
)
(5.7)
where Sj is given by eq. (4.10) so far as D is the potential
0.011 0
’
’
3
’
’
100
’
’
( ak)' (
3
A-‘)
’
depth. In fig. 9,@,
Or,
1
2oo
of Ak, calculated from eq. (4.13). The Fig. 8. Specular reflection intensity Isp as a function mean sqaure displacement (<* ) of the surface and the depth of the Morse potential for He are assumed as written in the figure. For Hz, the potential depth should be only doubled.
400
H. Asada / Specular reflection of He and Hz from Ag(ll
I)
Fig. 9. The apparent Debye temperatures, @T and 00, and the ratio @e/@T as functions of [, calculated by use of the MP model. The apparent Debye temperatures calculated by use of the CP model are also shown by dotted curves. Typical ranges of c at the actual measurements are indicated for the beams of (I) He with /co = 9.9 A-‘, (II) He with ko = 7.5 A-‘, (III) Ha with ke = 7.3 A-‘, and (IV) Ha with ko = 5.5 A-‘.
and @e/@T in the two models are illustrated as functions of $Y.Both models give the dependence of OB on ,$ (i.e., on Ak), which agrees qualitatively with the experimental results (fig. 7). Also in fig. 10, the calculated ratios OO/OT are compared between the models as functions of Ak for various values of D. The two models exhibit simiD CmeV)
I Or
O0
c.
5
IO
, 15
Ak (ii-', Fig. 10. The ratios @e/@T as functions of Ak, calculated by use of the MP model (solid curves) and the CP model (dashed curves). Assumed potential depths are written in the figure.
H. Asada / Specular reflection of He and Hz from Agfll
(a)
l.O-
401
I)
He
0’ \ 0” 0.5
0
-
I 5
0
I IO
I 15
Ak(Ii-‘1
(b)
HP
l.O-
*
0’
P
1
0
5
IO
15
Ak (A-‘, Fig. 11. Experimentally (a) by use of He beam;
determined ratios @e/@T plotted (b) by use of H2 beam.
against
Ak; symbols
are as in fig. 6;
lar curves, but the D in the CP model is several times as large as that in the MP model. The experimentally determined ratio, Oe/O~, is shown in fig. 11. The mean value of OT over all the surface temperatures is taken as the value of OT to be used in the evaluation of the ratio, and listed in the sixth column of table 2. The dependence of the ratio on Ak can be relatively well interpreted by each of the models. The value of D, which reproduces these data best, is calculated by the help of the least-squares method, and summarized in table 2. It is reasonable that the obtained values of D are nearly independent of the incident energy of the beam within
402
H. Asada /Specular
reflection of He and Hz from Ag(l I I)
Table 2 Results on four test surface Beam GoI
He (9.9 A-‘>
He (7.5 A-‘)
(H7f3A-‘)
H2
(5.5 A-‘)
Test surface
No. 1 No. 2 No. 3 No. 4 Mean No. 1 No. 2 No. 3 No. 4 Mean
Potential depth D (meV) CP model
MP model
lo*
3
0.9 + 0.4
8+ 9i: 12+ 222
1 3 58
1.0 * 0.2 1.1 k 0.4 1.6 ?: 1.2 i 3.2 * 0.8
46-162
13* 1.52
2
;I;
26-
8+ ilk 12i
2 3 4
;I;:
: i.5” 1 8.5 }
26-169
1.5 r 0.7
No. 3 No. 4 Mean
41+ 4 54r 4 22+ 7 43* 5 42 f 1.5
8.2 9.5 3.7 6.6 7.0
+. 0.7 f 0.8 f 0.3 * 1.0 i 2.6
No. 1 No. 2 No. 3 No. 4 Mean
31i 4 58 + 4 15t 2 29? 2 33r 18
5,3 10.2 2.4 4.9 5.7
t 1.0 F 1.0 i 0.4 f 0.4 f 3.4
No. 21
93
25-
88
25-160 14-
50
14-
91
From @T
From @e
10 13 7 12 8
242 f 15 252 ?: 18 260212 247 f 17 250+ 20
238 253 260 246 249
* * 2 c f
19 19 14 15 21
203 ? 8 211 f 12 217k 8 202~ 7 208 + 13
260 f. 10 267 fr 17 261*16 253+13 260 +_17
261 272 254 245 258
+ + + ? *
13 26 26 15 27
187+ 183 + 202+ 188 + 190*
12 10 11 11 16
243i: 238 ?: 256 k 241 + 245 f
16 13 15 1.5 19
246 241 257 242 247
i f5 + +
17 13 13 14 18
199 f 195+ 193t 1902 194 2
18 8 4 6 13
255 r 244t 241+ 244i: 246 +
24 14 10 8 19
268 253 243 244 252
+ 40 + 8 k 16 r 9 t 28
203 f 199 i 218k 205 + 206 f
the experimental error. The D values averaged over the test surfaces and over the incident energies are, in the MP model, D(He) = 1.5 + I .O meV ,
D(Hz) = 6.4 t 2.9 meV ,
while in the CP model , D(He)=12+6meV,
D(Hz)=37+16meV.
The latter values agree rather well with those deduced by Weinberg [3] through the DW factor given by eq. (1 .I) with neglection of (Ak)‘. The D values should be compared with physisorption energies, but unfortunately they have not yet been reported. Theoretical calculation of the physisorption energy of He on silver has been tried by a few authors: Kleiman et al. [24] have reported a potential depth of 4.9 meV, and Zaremba et al. [25], one of 5.0 meV. Van Himbergen et al. [26] also have obtained a value of the same order. Compared with their results, the MP model gives slightly too small values, while the CP model slightly too large ones. Similar relation between a theoretical value and an experimental one obtained from the CP
H. Asada j Specular reflection of He and Hz from Ag(ll I)
403
model has been found by Armand et al. [13,14] for He on various principal planes of Cu. They have reported the values of D ranging from 9.4-12.5 meV, which is greater than the theoretical values of 3.8 meV by Kleiman et al. and 5.1 meV by Zaremba et al. On the other hand, Hoinkes et al. [9] have found a potential depth of 17.8 meV of the CP model for H atom on a LiF (001) surface, which is consistent with that determined from an observation of selective adsorption [27]. It is difficult to see whether the MP or CP model gives a more reasonable potential depth, since there are few experimental and theoretical data on gas-surface interaction potentials. Once the potential depth is known, the true Debye temperature 0, can now be calculated from the observed apparent Debye temperatures by the help of eqs. (5.3) and (5.4). The term true means only that the Debye temperature is corrected for the effect due to the stationary part of the gas-surface interaction potential. When 0, is calculated from OT it should be taken into account that Or is not constant in the range of 5 where OT is observed, as shown in fig. 9. Therefore, 0~ is calculated by multiplying Or, which is averaged over all the temperatures, by a factor [E’g(t)/ (1 +t ’)I l/2 calculated at the mean value of g at the measurement, The deviation of the factor in the range of [ is allowed to contribute to the error in 0,. From the observed value of Oe, 0, is calculated by use of eq. (5.4) without any modification, and the obtained 0, is averaged over all the incident angles. The values of 0~ evaluated through such procedures are listed in the last two columns of table 2. At first a close agreement between 0, evaluated from OT and that from 00 should be noted. This means that the analysis starting from eq. (S.5) has been consistent. Secondly, He beams and H, beams are found to give 0~ with smaller discrepancy than that in O,, which should be equal to the true Debye temperature in the CP model. The values of 0, observed with He and H2 beams are 254 + 20 K and 247 f 19 K, respectively, after being averaged over all of the obtained values from OT and 00. They agree relatively well with each other. This result is regarded as a success of the MP model, which interpretes the discrepancy in Or between He and Hz in terms of the range of $ at the measurement of OT, as illustrated in fig. 9. After being further averaged over He and Hz, the true Debye temperature On is 0,=251
+20K.
Thirdly, 0, is found to be reproduced fairly well for four test surfaces, in contrast to the poor reproducibility of D. The D is smallest on the test surface of No. 2 and largest on No. 3. This result corresponds to the fact that the apparent Debye temperatures, OT and 00, are highest on the test surface of No. 2 and lowest on No. 3 in most cases. It follows that the difference in the apparent Debye temperatures between the surfaces results mainly from a difference in the gas-surface interaction potential rather than that in the true Debye temperature. The potential is possibly affected by the topology of the surface and/or adsorbed impurities. They should also have influences on the surface lattice vibrations, but the effect on the
404
H. Asada / Specular rejlection of He and H, from Ag(l1 I)
true Debye temperature might not be so significant as to be recognized. Finally, the obtained true Debye temperature of the surface is found to be unusually high: Jones et al. [28] have observed the surface Debye temperature of 155 K on the Ag(l11) surface in LEED, and Goodman [29] has also obtained the same value in a theoretical analysis of inelastic scattering distributions of He beams. The present value is even higher than the bulk one, 225 K [30]. Unusually high surface Debye temperatures observed with molecular beams have been reported for many systems [9-161, and they have been interpreted from a viewpoint of simultaneous collision of the incident molecule with more than one surface atom. Hoinkes et al. [9] have estimated this effect by the help of plane geometry based on the sizes of the incident particle and the surface atom. They have introduced a correction factor in the exponent of the DW factor. But their correction factor brings a conflict with the experimental results, as discussed below: From their point of view, a large molecule is expected to probe a larger surface area and therefore to observe a smaller vibrational amplitude, because the amplitude felt by the molecule is that averaged over the probed area. The size of the molecule can be estimated from the second virial coefficient (in the power series of pressure) of the gas at high temperature, which is proportional to the volume of the molecule in the first approximation. The coefficient at 2OO’C [31] is 0.65 X low3 atm-’ for He and 0.92 X 10e3 atm-’ for Hz. Then, the correction leads to lower @-for He than H,, contrary to the observed results. On the other hand, according to the theory of Armand et al. [13] on the effect of the simultaneous collision, the displacement c, probed by molecular beams, of the surface vibration perpendicular to the surface is given by ~=$~“i~
(5.8)
independently of the kind of the gas molecule, where Ui is the displacement of the i-th atom perpendicular to the surface, and i runs over N atoms constituting a surface unit cell (see fig. 12). Assuming an isotropic Debye solid with the Debye tem-
Fig. 12. Schematic
drawing
of a surface
unit cell of the (111) plane of Ag.
H. Asada / Specular reflection of He and H, from Ag(l1 I)
405
perature O,, <{‘) is calculated (see Appendix 3) as (t2) = 0.516 X 3A2Ts/Mk&, which yields the true Debye temperature
(5.9) Oo,
OD= 180+ 14K. Although this value agreees moderately well with that observed in LEED, one should not place too much emphasis on this result, since eq. (5.8) is too crude to evaluate quantitatively the effect of simultaneous collisions. One should deal more precisely with the simultaneous interaction of the molecule with more than one solid atom through not only a short-range but also a long-range force. The latter force (i.e., the Van der Waals force) is assumed also in the present study to constitute a completely stationary potential of attraction, which might not be a good approximation. Moreover, in the case of metal surface, Smoluchowski’s smoothing effect [32] should be taken into account [33] : Owing to this effect, the electronic surface of the metal, roughened by thermal lattice vibration, is smoothed to some extent. The effect should share the responsibility for the high surface Debye temperature observed with molecular beams.
6. Conclusions In the present paper the specular reflection intensity of He and H, molecular beams from the Ag(ll1) plane has been observed as a function of the surface temperature and incident angle. The Debye temperature of the Ag(ll1) surface observed with He beams and that with H2 beams have been found to be different from each other, if the DW factor in the CP model is applied to the specular reflection intensity. On the other hand, the DW factor in the MP model has been successful in yielding nearly the same Debye temperature whether He of H, beams are used. The obtained Debye temperature is however too high, which is presumably attributed to the effect of the simultaneous interaction of the gas molecule with more than one solid atom in the vicinity of the surface and to Smoluchowski’s smoothing effect on the electronic surface. The potential depth obtained in the CP model is several times as large as that in the MP model. This result implies that the depth is deduced with strong dependence on the potential assumed in the analysis, and requires more precise theoretical investigations on the DW factor.
Acknowledgements The author expresses his sincere thanks to Professor T. Toya for valuable discussions and encouragement in the course of this work. He also thanks very much Professors T. Matsui and T. Nakamura, for helpful discussions, Mr. T. Yamazaki for
406
H. Asada / Specular reflection of He and H, from Agfl I I)
examining the test surfaces A. Hiratsuka for preparing by a grant from the Japan Aid for Scientific Research
Appendix
by means of transmission electron diffraction, and Miss. all of the figures. He acknowledges support for this work Society of the Promotion of Science and by a Grant-infrom the Ministry of Education of Japan.
1
The integral of eq. (4.5) is rewritten
as that along the z-axis, i.e.,
(A.11 where z. is the turning point determined by V,(z) = E,. A constant term -n/2 which arises from reflection of the wave at the turning point [34] has also been omitted in the above equation. The variable z is transformed into w defined as w = [(a - eeKZ)/@ t eKKZ)]“* ,
(A-2)
where p = -1 t (1 t g*)r’* .
(Y= 1 + (1 t [*)I’* )
(A.3)
Then, the phase $ is calculated as ~=
_4(2tTZ)"*KD{ A
Appendix
(a'p)1'2 @ _ 0~2 -~ s 0
dw=_Ak{
(1 + w*)*
.
(A.4)
2
In the CP model, the interaction
potential
V is assumed to be
V= Vr(z - {) f V,(z).
(A-5)
The first repulsive term oscillates together with the lattice. The second attractive term is stationary and has a constant value -D at z z,, V, is assumed to have a negligible contribution to V. Then, the phase 4(x. y, t) is calculated by use of eq. (A.l) as
@(x,Y,
(2m)“*< t> =
A
s
h,
(A@
=0
where
vo(z>= V,(z) + V,(z).
(A.7)
401
H. Asada / Specular reflection of He and Hz from Ag(l I I)
Since the integrand
Substituting
is nearly equal to zero when z > z,, eq. (A.6) is rewritten
as
eq. (A.8) in eq. (4.3) yields the DW factor given by eq. (1 .l).
Appendix 3 According to eq. (5.8), the mean square displacement
(c2) is given by
Each term is calculated as below. The displacement Ui of the i-th atom whose equilibrium resented in terms of the normal modes of the Debye solid as Ui(t) = C (2knTJNJ4CLl’)“’ Q
COS(Ut
- 4 'li t 84) ,
position
is ri is rep-
(A.lO)
where o, q and 6, are a circular frequency, wave vector and phase of one of the normal modes, respectively, and NC is the number of atoms in the crystal. The summation is carried over NC normal modes with a polarization vector perpendicular to the surface. The correlation between the displacements of the i-th andj-th atoms is then given by (A.1 1) The summation the q-space :
over q is replaced by an integration
‘Uiui’;jma’i~~~2cos~q.(ri-?)]RldPsinBded~.
over q with the state density in
(A.12)
0
By use of the assumed relations, o = u,q and 0, = Qqmax/kn, where u, is the constant group velocity of the lattice waves, the integration is carried out as
WmaxIri - rjl)
(Milli) =
4max
Iri - ql
,
(A.13)
where Si is a sine integral defined by
Si(x)= _/Fdy. 0
(A.14)
H. Asada / Specular reflection
408
of He and Hz from A&l I I)
Whenqmax is assumed to be 2n(3/4~~,.)‘/~ , i.e., 1.51 A-‘, where u, is the atomic volume of the solid, eq. (A.9) is 1
5 Si(4.36)
1 Si(7.57)
4
8
8
_+__++_
4.36
7.57
1
3A2T, _.
-_ Mk&
.
516 3A2T, ____. Mk&
(A.15)
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [ 121 [ 131 [14] [15] [ 161 [17] [ 181 [19] [ 201 [21] (221 [23] [24] [25] [26] [ 271 [28] [29] [ 301 [ 311 [32] [33] [34]
J.L. Beeby, J. Phys. C4 (1971) L359. W.H. Weinberg, J. Phys. CS (1972) 2098. W.H. Weinberg, J. Chem. Phys. 57 (1972) 5463. G. Cornsa, J. Phys. C6 (1973) 2648. J.L. Beeby, Proc. 2nd Intern. Conf. on Solid Surface, 1974, Japan; Japan. J. Appl. Phys., Suppl. 2, Pt. 2 (1974) 537. F.O. Goodman, Surface Sci. 46 (1974) 118. J.D. Doll, Chem. Phys. 3 (1974) 257. F.O. Goodman, Surface Sci. 65 (1977) 37. H. Hoinkes, H. Nahr and H. Wilsch, Surface Sci. 33 (1972) 516;40 (1973) 457. J.R. Bledsoe and S.S. Fisher, Surface Sci. 46 (1974) 129. B. Wood, B.F. Mason and B.R. Williams, J. Chem. Phys. 61 (1974) 1435. A.G. Stall, Jr., J.J. Ehrhardt and R.P. Merrill, J. Chem. Phys. 64 (1976) 34. G. Armand, J. Lapujoulade and Y. Lejai, Surface Sci. 63 (1977) 143. G. Armand, J. Lapujoulade and Y. Lejai, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, Vol. 2 (1977) p. 1361. G. Boato, P. Cantini and R. Tatarek, ibid, p. 1377. Y. Hamauzu and H. Asada, J. Cryst. Sot. Japan 20 (1978) 102 (in Japanese). R. Sau and R.P. Merrill, Surface Sci. 34 (1973) 268. G.A. Somorjai and H.H. Farrell, Advan. Chem. Phys. 20 (1972) 215. R.F. Wallis, B.C. Clark and R. Herman, Phys. Rev. 167 (1968) 652. R.E. Allen and F.W. de Wette, Phys. Rev. 188 (1969) 1320. D.P. Jackson, Surface Sci. 43 (1974) 431. H. Asada, J. Res. Inst. Catalysis, Hokkaido Univ. 25 (1977) 175. D.L. Smith and R.P. Merrill, J. Chem. Phys. 52 (1970) 5861. G.G. Kleiman and U. Landman, Phys. Rev. B8 (1973) 5484. E. Zaremba and W. Kohn, Phys. Rev. B15 (1977) 1769. J.E. Van Himbergen and R. Silbey, Solid State Commun. 23 (1977) 623. H. Hoinkes, H. Nahr and H. Wilsch, Surface Sci. 30 (1972) 363. E.R. Jones, J.T. Mckinney and M.B. Webb, Phys. Rev. 151 (1977) 476. F.O. Goodman, Surface Sci. 30 (1972) 1. American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, 1972). II.Band, l.teil, Mechanisch-Thermisch Landolt-Bornstein Zahlenwerte und Funktionen, Zustandsgrossen (Springer, 1971). R. Smoluchowski, Phys. Rev. 60 (1941) 661. T. Toya, private communication. L.D. Landau and E.M. Lifschitz, Quantum Mechanics - Non-Relativistic Theory, 2nd ed. (State Publisher of Physico-Mathematical Literatures, Moscow, 1963).
SPECULAR REFLECTION OF HELIUM AND HYDROGEN MOLECULAR BEAMS FROM THE (111) PLANE OF SILVER * Hiromu ASADA
Received 7 August 1978; manuscript received in final form 11 October 1978
The Debye-Wailer (DW) factor in the specular reffection intensity of He and Us molecular beams from the Ag (111) plane has been studied experiment~y and theoretically. A new expression for the DW factor corrected for a stationary part of the gas-surface interaction potential is derived kinematically and semi-classically by the use of a Morse potential. An analysis of the experimental data through the above DW factor yields a surface Debye temperature of 251 _+20 K, which is unusually high, and potential depths of 1.5 + 1.0 meV for He and 6.4 + 2.9 meV for Hz, which seem slightly too small. These results are discussed on the basis of the nature of gas-surface interactions and in comparison with the results deduced from the conventional DW factor corrected for a constant attractive potential depth.
1. Introduction Many discussions [l-17] have been made concerning the Debye-Waher (DW) factor in the specular reflection intensity of molecular beams from solid surfaces. One of the important conclusions is that the kinetic energy of incident molecules during collision and reflection on the surface should be corrected for a stationary attractive potential in front of the surface. The corrected DW factor, e-2w, has been proposed by Hoinkes et al. [9] in the following form e-2w = exp[-{(Ask)’
t S~nD/fi~‘)<3;~>],
(1-l)
where Ak is the change in the normal component of the wave vector of the molecular beam, evaluated at the infinite distance from the surface, m the mass of the molecule, D the attractive potential depth, A the Planck constant devided by 27r, and (t21 the mean square displacement, perpendicular to the surface, of the surface atom due to thermal vibration. The (t2) is conveniently expressed in terms of the Debye temperature Or, by means of the isotropic Debye model for crystals. When the temperature is rather higher than OD, the expression is
* Taken from the Ph.D. thesis submitted to Hokkaido University, 1977. 386
H. Asada /Specular Cc*) = 3t2*Ts/MkBO~
reflection of He and Hz from Ag(ll1)
,
381
(l-2)
where M is the mass of the solid atom, kB the Boltzmann constant, and T, the surface temperature. The corrected DW factor has succeeded in describing very well the observed dependence of the specular reflection intensity on the surface temperature and the incident angle of the beam. But on all the surfaces investigated [9-161, eq. (1 .l) together with eq. (1.2) has been found to yield higher Debye temperatures than those observed in LEED [18] or those expected theoretically [ 19-211; the surface Debye temperature is roughly estimated to be the bulk one divided by 42. The result has been ascribed to a collision of a gas molecule simultaneously with more than one surface atom due to their finite sizes [9,13]. In the present paper the DW factor in the specular reflection of thermal molecular beams of He and Hz from the (111) plane of silver is investigated. It will be shown that He and H2 beams give different Debye temperatures if the specular reflection intensity is analysed by eq. (1.1) together with eq. (1.2). Instead of eq. (l.l), a new expression for the correction on the DW factor will be proposed from the kinematical theory of diffraction based on semi-classical mechanics. The surface Debye temperature and the depths of the gas-surface interaction potentials will be deduced through the new DW factor and discussed on the basis of a nature of gas-surface interactions and in comparison with the results through eqs. (1.1) and (1.2).
2. Experimental The experimental apparatus, shown schematically in fig. 1, has been previously described elsewhere [22]. The beam is generated by a Pyrex glass nozzle with an aperture of 40-50 pm diameter, which can be cooled down to ca. 150 K. The gas beam passes through a skimmer (0.62 mm diam.) at a distance of 6-8 mm from the nozzle, and it is collimated by two circular orifices. The direct beam, which is ob-
TO PUMP
Fig. 1. Schematic drawing of the experimental apparatus. N = nozzle, S = skimmer, 01 = 1st collimating orifice, 02 = 2nd collimating orifice, C = chopper wheel rotated by a synchronous motor, E = evaporator, T = test surface, and D = detector.
388
H. Asada / Specular reflection of He and H, from Ag(ll1)
served downstream along the incident beam with the test surface removed, has a full width 0.9 + 0.1’ at half maximum in terms of the rotation angle of the detector around the test surface. The - - wave number k of the incident beam has been determined by observing the (1, 1) diffraction of the beam from a cleaved (001) surface of LiF: The wave number distributionj(k) in the incident beam, deduced from the broadening of the (i, i) peak, has been found to be well described by - /~,,)~/a’] dk .
f(k) dk = const. k3 exp[-(k
(2.1)
The constants, kO and a, are listed in table 1 with other related quantities . The spatial distributions of He and H2 scattered by a test surface are observed with an electron-impact ionization detector, whose output is measured by means of phasesensitive detection with the incident beam modulated at about 140 Hz. The output signal is proportional to the beam flux flowing into the detector. The silver surfaces have been prepared in the apparatus by evaporating 99.99% Ag onto cleaved mica surfaces at temperatures 500-550 K. Transmission electron diffraction revealed that the deposited films are oriented with the (111) axis perpendicular to the surface due to epitaxial growth. After the deposition of Ag was completed, the film was annealed at 550-600 K for more than 10 min before scattering experiments. The experiments have been performed on four films at surface temperatures of 450-600 K and at background pressures of 4-10 X 10e6 Pa. The specular reflection intensity of the molecular beams, which has been proved to be a sensitive monitor of surface cleanliness [23], has been found not to be changed significantly for several hours after deposition but to be reduced to half or less if the
transverse 4 plane
Fig. 2. Definitions
of the coordinate
system
and various
angles.
H. Asada / Specular reflection of He and H, from Ag(l I I)
389
H. Asada / Specular rejlection of He and H2 from Ag(il I)
390
test surface has been left in the apparatus without pumping for a night. Additional deposition of Ag has restored a great part of the specular reflection intensity and enabled one to continue the experiments. In fig. 2 the coordinate system and the definitions of angles are illustrated. The z-axis is a surface normal. The incident angle Bi is measured from the z-axis. The plane which contains the incident beam and the z-axis is called the incident plane (I-plane}. A directional vector of scattering makes an angle ~9~with the I-plane, and its projection on the I-plane makes an angle 8, with the z-axis. A plane perpendicular to the l-plane is called a transverse plane {T-plane), which is specified by the angle 0 r
.
3. Experimental results A typical trace of the spatial distribution of He scattered by a Ag surface is shown in fig. 3. A similar distribution is observed also for Hz. The distribution consits of a sharp specular reflection peak and a broad skirt around it. This broad skirt,
(al
I
-
l
..*** . .
10
l
..*
. . ..l* . l
.
x5 ‘_ x5 . -. *... b
‘.
.
l. 80
f
90
-5
l** 20
0 h
cdk”e )
Fig. 3. Typical trace of a spatial distribution of He scattered from a Ag(lll) surface; E = 31 meV, Bi = 70”, T, = 500 K: (a) in the I-plane; (b) in the T-plane with 0r = 70”. The inelastic scattering part is enlarged by a factor of 5. The inelastic background level which should be subtracted in the evaluation off,, is indicated by “b.g.“.
H. Asada / Specular rejlection of He and Ii, from Ag(iI I)
391
which is regarded as a result of inelastic scattering by phonons of Ag, has been discussed in a previous paper [22]. The specular reflection peak is found to have a greater width than that of the incident beam by l-3’ in the I-plane, but to be scarcely broadened in the T-plane. The broadening of the peak may however be attributed to bending and/or large scale roughness of the test surface [ 171. The intensity of specular reflection, IsP, is defined as the peak area in the I-plane after subtracting the inelastic scattering background whose level is indicated in fig. 3 as an example. The ISp is normalized with respect to the direct beam intensity which is also defined as an area of the profile in the I-plane. In fig. 4, the dependence of ISr, on (Ak)2 at constant surface temperature is exemplified for He and H2 beams. The Ak is given by Ak = 2ki
COS 8i
(3.1)
,
where ki, the wave number of the incident beam, is equated with k. in eq. (2.1). The reason for choosing kO rather than the mean value k for ki is that molecules with k < k contribute to the specular reflection intensity more than those with k > % due to the effect of the DW factor. The value of k. is smaller than k by several percent (see table 1). The plot indicates a good linear relationship between In Is,., and (Ak)2. In fig. 5, the dependence of ZSp on the surface temperature I;; ranging from 450 to 600 K at fixed incident angles is exemplified. The plot also follows a straight line. At temperatures lower than 450 K a decrease in f,, is observed, which is probably attributed to adsorption of some of the background gases. The good linear dependence of In IS,, and (A/C)’ and T, allows one to represent the specular reflection intensity by the conventional DW factor, Is,, = const. exp[-3(tzAk)2Ts/&ikB02]
.
(3.2)
It should be kept in mind that the Debye temperature 0 in eq. (3.2) does not necessarily represent a constant characteristic of the solid surface, as shown below. For convenience, two apparent Debye temperatures, OT and 00, are defined; the former is calculated from the observed linear relationship between In ISp and (Ak)2 at constant T,, and the latter calculated from that between In ISp and T, at fixed Bi. The suffixes T and B signify measurements at constant temperature and at fixed incident angle, respectively. In fig. 6, the values of OT obtained are plotted against T,. Though OT has not been necessarily well reproduced for four test surfaces, the following results may be seen: Firstly, the beam of the same kind of gas yields the same value of OT almost ~dependen~y of the incident energy. This result supports the expectation that ISp is well described only by the DW factor. Secondly, the values of OT observed with He are in the range of 190-220 K,
H. Asada / Specular reflection of He and H, from Ag(lI1)
392
z”r
20
70 II, 0.1’
0
I
65
60 a 100
55 I I
h
50 I ’
I
r
’ 200
(ak)' (A-')
surface Fig. 4. Semi-log plots of the specular reflection intensity Is* versus (Ak)’ at constant temperatures written in the figure. The incident angles Bi are also shown on the abscissa. The straight lines are fitted by the least-squares method. (a) He beam with /CCJ= 9.9 A-‘. (b) He beam with k,~ = 7.5 A-‘. (c) Hz beam with ka = 7.3 a-‘. (d) Hz beam with ko = 5.5 A-‘.
393
(a) He, ko=9.9h-’
(b) He, ko=7.5?Y’
'OF
o1* 450
500
550 Ts (K)
600
olI * 450
500
550 Ts (K)
600
Cc) HP. ko-7.3&-l
,.
E
T
70l
60Q
-.-‘::: .
0
l
.
5o”
0
0
.
4oa .
Ode
30°
600
Ts (K)
\ 0
t
ool. .
450
500
Ts (K)
550
600
Fig. 5. Semi-log plots of Isp versus T, at fixed incident angles written in the figure. The straight lines are fitted by the least-sqaures method. (a) He beam with ko = 9.9 A-‘. (b) He beam with ko = 7.5 A-‘. (c) Hz beam with k. = 7.3 A-‘. (d) H2 beam with k. = 5.5 A-‘.
H. Asadu f Specular rejlection of Ne and H, from AgJll I)
394
250-
(a)
He
250-
Y
-
;;
0'
_
IO.
(b)
H2
-
200-
200"
I,ot-----450
500
550
600
Ts (K)
,,oL 450
500
550
600
T, (Kl
Fig. 6. Dependence of the apparent Debye temperature @T on Ts; the symbols 0, n,*, and v, whether open or closed, refer to the test surfaces of No. 1, 2, 3, and 4, respectively: (a) by use of He beam with k. = 9.9 A-’ (open symbols) or 7.5 A-I (closed ones); (b) by use of Hz beam with ko = 7.3 A-” (open symbols) or 5.5 A-’ (closed ones).
while those with Hz are in the range of 180-200 K, obviously lower than the former by lo-20 K. In other words, the apparent vibrat~onai ~plitude of the surface probed by He beams is smaller than that by H2 beams. The result cannot be explained by eq. (1.1) together with eq. (1.2), which always yields &-equal to the true Debye temperature (see eq. (5.6)). Finally, 0~ decreases with increasing Ts. The decrement of OT is rather large, lo-20 K per an increment of T, of 100 K. If it is wholly attributed to an anharmonicity of surface lattice vibration perpendicular to the surface, the anharmonicity should be anomalously large. A similar result has been found by Armand et al. [ 131 on the (001) plane of Cu. In fig. 7, the values of Oe are plotted against Ak. Although 00 is not necessarily well reproduced for four test surfaces, the following results are seen: Firstly, the beam of each gas seems to yield nearly the same value of Oe on each test surface when Ak is identical for different incident energies. This result also gives support to eq. (3.2). Secondly, Oe is found to decrease with increasing Ak (i.e., with decreasing ei). Finally, the He beam gives a greater value of 00 than Hz does by 20-40 K at the same value of Ak.
4. Kinematical theory of the DW factor Since the corrected DW factor of eq. (1 .I) has been four& to fail in interpreting the discrepancy in @T observed with He and Hz molecular beams, one should con-
H. Asada / Specular reflection of He and Hz from Ag(l I I)
(a)
200-
He
5
Ak
(b)
200
395
(A-‘,
IO
15
Hz
r
I
I
5
IO
15
Ak ( 8’)
Fig. 7. The apparent Debye temperature 00 plotted against Ak; the symbols are as in fig. 6: (a)
by use of He beam; (b) by use of Hz beam.
sider the DW factor in the molecular beam scattering intensity from a basic viewpoint. At first, it should be noted that atomic and molecular beams have unique features, compared with X-rays, electrons and neutrons: The beam particle itself has a size of the order of an atom or a molecule, and it interacts with solid atoms through not only a short-range but also a long-range force. Owing to the first feature, the incident particle even has a chance of colliding simultaneously with more than one surface atom. The second feature may also allow the incident particle to interact with more than one solid atom in the vicinity of the surface. The simultaneous
396
H. Asada /Specular
reflection of He and H, from Ag(l1 I)
interaction brought about by these two features makes it difficult to treat the scattering by the kinematical theory which in general assumes a single collision between an incident particle and solid atoms. But this dif~culty is overcome by regard~g the scattering events as caused by a potential fiefd at the surface rather than by an assembly of scattering centers. According to this idea, consider an interference of the waves which are accompanied by the molecule and scattered by the potential field affected by lattice vibration. The assumed potential k’is V(x, y, 2, t) = L)(e-2”(z-r)
- 2 e-“‘)
,
(4.1)
where { = {(x, y, r) is the displacement of the “surface” due to lattice vibration perpendicular to the surface at the position (x, y) and at the time instant t. The D and K represent the depth and the reciprocal range of the potential, respectively. The attractive part of the potential given by the second term is assumed to include no effect of thermal vibration, since many atoms in the vicinity of the surface contribute to the attractive potential through the Van der Waals long-range force to make it rather stationary on the average. The displacement is assumed so that (f2) is represented by eq. (1.2) at any point in a surface unit cell, which is equivalent to regarding the surface as a continuum. The bracket ( > refers to a statistical ensemble average. The scattering ~plitude F of the wave accompanied by the molecular beam is given by the following integral over the surface with an area S, F = const. jjexp s
[i@(x, y. t)] d.x d.y ,
(4.2)
where #(x, y, t) is the phase of the wave scattered at the position (x, y) and the time c. Other factors in F except the phase factor are insigni~c~t in the calculation of the DW factor and assumed to be constant. The intensity I of the scattered wave is given by I = (IFi’) = const. (
exp [i#(x, y, t) - @(xi, y’, t’)f d.x dy dx’ dy’> ~~ s S
=const.exp[-&jjfi(@-$1)2)dxdyti’dyj s s (4.3) where 4(x, y, t) and @(xl, y’, t’) are abbreviated to @and $‘, respectively. The phase @ is calculated by the following phase integral along the classica trajectory of the molecule in the semi-classical approximation,
391
H. Asada / Specular reflection of He and H, from Ag(l I I)
4(x, Y, 0
=$j- (2m [E -
I+, y, z, t)]}“’
ds ,
(4.4)
c.t. where ds is an element of the distance along the trajectory. When { is small enough, one may expand the integrand in a power series of { and neglect the higher order terms, i.e., @(x, y, t) =$ s
[2m(E - V,)] 1’2 ds -;
c.t.
s
r(aV’a’)‘=or,2
c.t . k@
- Vo>l
ds
,
(4.5)
where V&Z) = D(e-2Kz - 2 eeKZ) .
(4.6)
The first constant term in eq. (4.5) may however be omitted, since it has no significance in the calculation of I. It is further assumed that the vibration of the surface is so slow compared with the speed of the incident molecule that one may regard the displacement 5 as independent of time t during the gas molecule-surface collision. More precisely, this assumption is valid provided that the period of the surface atom vibration is sufficiently longer than the time necessary for the molecule to pass through the region with a significant effect of thermal vibration. This condition is formulated as 2/KVg<
27&J,
(4.7)
where us is the speed of the molecule and w is the angular frequency of the surface atom vibration. When relevant values, o = 2 X 10’ 3 s-l, vg = 1 X 1O’cm/s, and K = 5 X IO8 cm-’ are substituted, the left and right hand sides have values of 4 X IO-l4 s and 3 X lo-l3 s, respectively, i.e., the relation (4.7) holds rather well. Then, the integral of the second term of eq. (4.5) can be calculated exactly for specular reflection, after { is put out of the integral. The resultant equation (see Appendix I) is @(x>Y, t) = -Ak&)
Rx, Y, 0 ,
(4.8)
where g(t) = 1 + (rr - tan-‘E)/$
,
g = (:Ez/D)1'2 = tzAk/(8mD)1’2
(4.9) ,
(4.10)
and E, is the incident energy of the molecule associated with the z-directed motion. The intensity I is now calculated by substituting eq. (4.8) in eq. (4.3). The first term in the exponent of eq. (4.3) is $ ljQ2) 5
dx dy = (Ak)’ (g(.$))‘Q2,
.
(4.11)
398
H. Asadu / Specular reflection
of He and H, from Ag(lI 1)
The second term is
(4.12)
where T is a time interval (x sin ei)[us during which a wave front of the molecular beam proceeds from the position (0,O) to (x, y) on the surface. Eq. (4.12) contains the correlation between { at different positions, averaged over the whole surface. This quantity is ho&ever much smaller than (c2), so that the second term in the exponent of eq. (4.3) can be neglected in comparison with the first term. Consequently, the specular reflection intensity Is, is given by f,, = const. exp[-(Ak)‘({‘) =
const.
exp _ [
{g(#“]
?$fS?J.{g(g))*
.
1
(4.13)
The difference between the present DW factor and the conventional one is a factor k(t)}” in the exponent, arising from the stationary attractive part of the Morse potential. For convenience, the model used in the derivation of eq. (4.13) is called a Morse potential model or an MP model. The corrected DW factor of eq. (1.1) can be derived in a similar way, assuming that a stationary attractive potential in front of the vibrating surface has a constant depth D near the surface. This model is referred to as a constant potential model or a CP model (see Appendix 2).
5. Potential
depth and true Debye temperature
According to eq. (4.13) a plot of In I,, versus T, gives a straight line, while that of In ISp versus (Ak)’ does not. The deviation from the straight line is however very small, as illustrated in fig. 8, so that the apparent Debye temperature 0~ can be determined experimentally without difficulty and ambiguity. The apparent Debye temperatures, OT and 0 e , are calculated from the following equations,
and
H. Asada / Specular reflection of He and Hz from Ag(I I I)
399
Therefore, %-I%
= [(I + ?)/~*g(U
@&D
= l/g(g) .
l’* ,
64)
The ratio of the two apparent
Incidentally,
(5.3)
Debye temperatures
is
in the CP model, the apparent Debye temperatures
are given by
(5.6)
o*=oD, 2
O&D = 0&3T =
l/2
)
(5.7)
where Sj is given by eq. (4.10) so far as D is the potential
0.011 0
’
’
3
’
’
100
’
’
( ak)' (
3
A-‘)
’
depth. In fig. 9,@,
Or,
1
2oo
of Ak, calculated from eq. (4.13). The Fig. 8. Specular reflection intensity Isp as a function mean sqaure displacement (<* ) of the surface and the depth of the Morse potential for He are assumed as written in the figure. For Hz, the potential depth should be only doubled.
400
H. Asada / Specular reflection of He and Hz from Ag(ll
I)
Fig. 9. The apparent Debye temperatures, @T and 00, and the ratio @e/@T as functions of [, calculated by use of the MP model. The apparent Debye temperatures calculated by use of the CP model are also shown by dotted curves. Typical ranges of c at the actual measurements are indicated for the beams of (I) He with /co = 9.9 A-‘, (II) He with ko = 7.5 A-‘, (III) Ha with ke = 7.3 A-‘, and (IV) Ha with ko = 5.5 A-‘.
and @e/@T in the two models are illustrated as functions of $Y.Both models give the dependence of OB on ,$ (i.e., on Ak), which agrees qualitatively with the experimental results (fig. 7). Also in fig. 10, the calculated ratios OO/OT are compared between the models as functions of Ak for various values of D. The two models exhibit simiD CmeV)
I Or
O0
c.
5
IO
, 15
Ak (ii-', Fig. 10. The ratios @e/@T as functions of Ak, calculated by use of the MP model (solid curves) and the CP model (dashed curves). Assumed potential depths are written in the figure.
H. Asada / Specular reflection of He and Hz from Agfll
(a)
l.O-
401
I)
He
0’ \ 0” 0.5
0
-
I 5
0
I IO
I 15
Ak(Ii-‘1
(b)
HP
l.O-
*
0’
P
1
0
5
IO
15
Ak (A-‘, Fig. 11. Experimentally (a) by use of He beam;
determined ratios @e/@T plotted (b) by use of H2 beam.
against
Ak; symbols
are as in fig. 6;
lar curves, but the D in the CP model is several times as large as that in the MP model. The experimentally determined ratio, Oe/O~, is shown in fig. 11. The mean value of OT over all the surface temperatures is taken as the value of OT to be used in the evaluation of the ratio, and listed in the sixth column of table 2. The dependence of the ratio on Ak can be relatively well interpreted by each of the models. The value of D, which reproduces these data best, is calculated by the help of the least-squares method, and summarized in table 2. It is reasonable that the obtained values of D are nearly independent of the incident energy of the beam within
402
H. Asada /Specular
reflection of He and Hz from Ag(l I I)
Table 2 Results on four test surface Beam GoI
He (9.9 A-‘>
He (7.5 A-‘)
(H7f3A-‘)
H2
(5.5 A-‘)
Test surface
No. 1 No. 2 No. 3 No. 4 Mean No. 1 No. 2 No. 3 No. 4 Mean
Potential depth D (meV) CP model
MP model
lo*
3
0.9 + 0.4
8+ 9i: 12+ 222
1 3 58
1.0 * 0.2 1.1 k 0.4 1.6 ?: 1.2 i 3.2 * 0.8
46-162
13* 1.52
2
;I;
26-
8+ ilk 12i
2 3 4
;I;:
: i.5” 1 8.5 }
26-169
1.5 r 0.7
No. 3 No. 4 Mean
41+ 4 54r 4 22+ 7 43* 5 42 f 1.5
8.2 9.5 3.7 6.6 7.0
+. 0.7 f 0.8 f 0.3 * 1.0 i 2.6
No. 1 No. 2 No. 3 No. 4 Mean
31i 4 58 + 4 15t 2 29? 2 33r 18
5,3 10.2 2.4 4.9 5.7
t 1.0 F 1.0 i 0.4 f 0.4 f 3.4
No. 21
93
25-
88
25-160 14-
50
14-
91
From @T
From @e
10 13 7 12 8
242 f 15 252 ?: 18 260212 247 f 17 250+ 20
238 253 260 246 249
* * 2 c f
19 19 14 15 21
203 ? 8 211 f 12 217k 8 202~ 7 208 + 13
260 f. 10 267 fr 17 261*16 253+13 260 +_17
261 272 254 245 258
+ + + ? *
13 26 26 15 27
187+ 183 + 202+ 188 + 190*
12 10 11 11 16
243i: 238 ?: 256 k 241 + 245 f
16 13 15 1.5 19
246 241 257 242 247
i f5 + +
17 13 13 14 18
199 f 195+ 193t 1902 194 2
18 8 4 6 13
255 r 244t 241+ 244i: 246 +
24 14 10 8 19
268 253 243 244 252
+ 40 + 8 k 16 r 9 t 28
203 f 199 i 218k 205 + 206 f
the experimental error. The D values averaged over the test surfaces and over the incident energies are, in the MP model, D(He) = 1.5 + I .O meV ,
D(Hz) = 6.4 t 2.9 meV ,
while in the CP model , D(He)=12+6meV,
D(Hz)=37+16meV.
The latter values agree rather well with those deduced by Weinberg [3] through the DW factor given by eq. (1 .I) with neglection of (Ak)‘. The D values should be compared with physisorption energies, but unfortunately they have not yet been reported. Theoretical calculation of the physisorption energy of He on silver has been tried by a few authors: Kleiman et al. [24] have reported a potential depth of 4.9 meV, and Zaremba et al. [25], one of 5.0 meV. Van Himbergen et al. [26] also have obtained a value of the same order. Compared with their results, the MP model gives slightly too small values, while the CP model slightly too large ones. Similar relation between a theoretical value and an experimental one obtained from the CP
H. Asada j Specular reflection of He and Hz from Ag(ll I)
403
model has been found by Armand et al. [13,14] for He on various principal planes of Cu. They have reported the values of D ranging from 9.4-12.5 meV, which is greater than the theoretical values of 3.8 meV by Kleiman et al. and 5.1 meV by Zaremba et al. On the other hand, Hoinkes et al. [9] have found a potential depth of 17.8 meV of the CP model for H atom on a LiF (001) surface, which is consistent with that determined from an observation of selective adsorption [27]. It is difficult to see whether the MP or CP model gives a more reasonable potential depth, since there are few experimental and theoretical data on gas-surface interaction potentials. Once the potential depth is known, the true Debye temperature 0, can now be calculated from the observed apparent Debye temperatures by the help of eqs. (5.3) and (5.4). The term true means only that the Debye temperature is corrected for the effect due to the stationary part of the gas-surface interaction potential. When 0, is calculated from OT it should be taken into account that Or is not constant in the range of 5 where OT is observed, as shown in fig. 9. Therefore, 0~ is calculated by multiplying Or, which is averaged over all the temperatures, by a factor [E’g(t)/ (1 +t ’)I l/2 calculated at the mean value of g at the measurement, The deviation of the factor in the range of [ is allowed to contribute to the error in 0,. From the observed value of Oe, 0, is calculated by use of eq. (5.4) without any modification, and the obtained 0, is averaged over all the incident angles. The values of 0~ evaluated through such procedures are listed in the last two columns of table 2. At first a close agreement between 0, evaluated from OT and that from 00 should be noted. This means that the analysis starting from eq. (S.5) has been consistent. Secondly, He beams and H, beams are found to give 0~ with smaller discrepancy than that in O,, which should be equal to the true Debye temperature in the CP model. The values of 0, observed with He and H2 beams are 254 + 20 K and 247 f 19 K, respectively, after being averaged over all of the obtained values from OT and 00. They agree relatively well with each other. This result is regarded as a success of the MP model, which interpretes the discrepancy in Or between He and Hz in terms of the range of $ at the measurement of OT, as illustrated in fig. 9. After being further averaged over He and Hz, the true Debye temperature On is 0,=251
+20K.
Thirdly, 0, is found to be reproduced fairly well for four test surfaces, in contrast to the poor reproducibility of D. The D is smallest on the test surface of No. 2 and largest on No. 3. This result corresponds to the fact that the apparent Debye temperatures, OT and 00, are highest on the test surface of No. 2 and lowest on No. 3 in most cases. It follows that the difference in the apparent Debye temperatures between the surfaces results mainly from a difference in the gas-surface interaction potential rather than that in the true Debye temperature. The potential is possibly affected by the topology of the surface and/or adsorbed impurities. They should also have influences on the surface lattice vibrations, but the effect on the
404
H. Asada / Specular rejlection of He and H, from Ag(l1 I)
true Debye temperature might not be so significant as to be recognized. Finally, the obtained true Debye temperature of the surface is found to be unusually high: Jones et al. [28] have observed the surface Debye temperature of 155 K on the Ag(l11) surface in LEED, and Goodman [29] has also obtained the same value in a theoretical analysis of inelastic scattering distributions of He beams. The present value is even higher than the bulk one, 225 K [30]. Unusually high surface Debye temperatures observed with molecular beams have been reported for many systems [9-161, and they have been interpreted from a viewpoint of simultaneous collision of the incident molecule with more than one surface atom. Hoinkes et al. [9] have estimated this effect by the help of plane geometry based on the sizes of the incident particle and the surface atom. They have introduced a correction factor in the exponent of the DW factor. But their correction factor brings a conflict with the experimental results, as discussed below: From their point of view, a large molecule is expected to probe a larger surface area and therefore to observe a smaller vibrational amplitude, because the amplitude felt by the molecule is that averaged over the probed area. The size of the molecule can be estimated from the second virial coefficient (in the power series of pressure) of the gas at high temperature, which is proportional to the volume of the molecule in the first approximation. The coefficient at 2OO’C [31] is 0.65 X low3 atm-’ for He and 0.92 X 10e3 atm-’ for Hz. Then, the correction leads to lower @-for He than H,, contrary to the observed results. On the other hand, according to the theory of Armand et al. [13] on the effect of the simultaneous collision, the displacement c, probed by molecular beams, of the surface vibration perpendicular to the surface is given by ~=$~“i~
(5.8)
independently of the kind of the gas molecule, where Ui is the displacement of the i-th atom perpendicular to the surface, and i runs over N atoms constituting a surface unit cell (see fig. 12). Assuming an isotropic Debye solid with the Debye tem-
Fig. 12. Schematic
drawing
of a surface
unit cell of the (111) plane of Ag.
H. Asada / Specular reflection of He and H, from Ag(l1 I)
405
perature O,, <{‘) is calculated (see Appendix 3) as (t2) = 0.516 X 3A2Ts/Mk&, which yields the true Debye temperature
(5.9) Oo,
OD= 180+ 14K. Although this value agreees moderately well with that observed in LEED, one should not place too much emphasis on this result, since eq. (5.8) is too crude to evaluate quantitatively the effect of simultaneous collisions. One should deal more precisely with the simultaneous interaction of the molecule with more than one solid atom through not only a short-range but also a long-range force. The latter force (i.e., the Van der Waals force) is assumed also in the present study to constitute a completely stationary potential of attraction, which might not be a good approximation. Moreover, in the case of metal surface, Smoluchowski’s smoothing effect [32] should be taken into account [33] : Owing to this effect, the electronic surface of the metal, roughened by thermal lattice vibration, is smoothed to some extent. The effect should share the responsibility for the high surface Debye temperature observed with molecular beams.
6. Conclusions In the present paper the specular reflection intensity of He and H, molecular beams from the Ag(ll1) plane has been observed as a function of the surface temperature and incident angle. The Debye temperature of the Ag(ll1) surface observed with He beams and that with H2 beams have been found to be different from each other, if the DW factor in the CP model is applied to the specular reflection intensity. On the other hand, the DW factor in the MP model has been successful in yielding nearly the same Debye temperature whether He of H, beams are used. The obtained Debye temperature is however too high, which is presumably attributed to the effect of the simultaneous interaction of the gas molecule with more than one solid atom in the vicinity of the surface and to Smoluchowski’s smoothing effect on the electronic surface. The potential depth obtained in the CP model is several times as large as that in the MP model. This result implies that the depth is deduced with strong dependence on the potential assumed in the analysis, and requires more precise theoretical investigations on the DW factor.
Acknowledgements The author expresses his sincere thanks to Professor T. Toya for valuable discussions and encouragement in the course of this work. He also thanks very much Professors T. Matsui and T. Nakamura, for helpful discussions, Mr. T. Yamazaki for
406
H. Asada / Specular reflection of He and H, from Agfl I I)
examining the test surfaces A. Hiratsuka for preparing by a grant from the Japan Aid for Scientific Research
Appendix
by means of transmission electron diffraction, and Miss. all of the figures. He acknowledges support for this work Society of the Promotion of Science and by a Grant-infrom the Ministry of Education of Japan.
1
The integral of eq. (4.5) is rewritten
as that along the z-axis, i.e.,
(A.11 where z. is the turning point determined by V,(z) = E,. A constant term -n/2 which arises from reflection of the wave at the turning point [34] has also been omitted in the above equation. The variable z is transformed into w defined as w = [(a - eeKZ)/@ t eKKZ)]“* ,
(A-2)
where p = -1 t (1 t g*)r’* .
(Y= 1 + (1 t [*)I’* )
(A.3)
Then, the phase $ is calculated as ~=
_4(2tTZ)"*KD{ A
Appendix
(a'p)1'2 @ _ 0~2 -~ s 0
dw=_Ak{
(1 + w*)*
.
(A.4)
2
In the CP model, the interaction
potential
V is assumed to be
V= Vr(z - {) f V,(z).
(A-5)
The first repulsive term oscillates together with the lattice. The second attractive term is stationary and has a constant value -D at z
@(x,Y,
(2m)“*< t> =
A
s
h,
(A@
=0
where
vo(z>= V,(z) + V,(z).
(A.7)
401
H. Asada / Specular reflection of He and Hz from Ag(l I I)
Since the integrand
Substituting
is nearly equal to zero when z > z,, eq. (A.6) is rewritten
as
eq. (A.8) in eq. (4.3) yields the DW factor given by eq. (1 .l).
Appendix 3 According to eq. (5.8), the mean square displacement
(c2) is given by
Each term is calculated as below. The displacement Ui of the i-th atom whose equilibrium resented in terms of the normal modes of the Debye solid as Ui(t) = C (2knTJNJ4CLl’)“’ Q
COS(Ut
- 4 'li t 84) ,
position
is ri is rep-
(A.lO)
where o, q and 6, are a circular frequency, wave vector and phase of one of the normal modes, respectively, and NC is the number of atoms in the crystal. The summation is carried over NC normal modes with a polarization vector perpendicular to the surface. The correlation between the displacements of the i-th andj-th atoms is then given by (A.1 1) The summation the q-space :
over q is replaced by an integration
‘Uiui’;jma’i~~~2cos~q.(ri-?)]RldPsinBded~.
over q with the state density in
(A.12)
0
By use of the assumed relations, o = u,q and 0, = Qqmax/kn, where u, is the constant group velocity of the lattice waves, the integration is carried out as
WmaxIri - rjl)
(Milli) =
4max
Iri - ql
,
(A.13)
where Si is a sine integral defined by
Si(x)= _/Fdy. 0
(A.14)
H. Asada / Specular reflection
408
of He and Hz from A&l I I)
Whenqmax is assumed to be 2n(3/4~~,.)‘/~ , i.e., 1.51 A-‘, where u, is the atomic volume of the solid, eq. (A.9) is 1
5 Si(4.36)
1 Si(7.57)
4
8
8
_+__++_
4.36
7.57
1
3A2T, _.
-_ Mk&
.
516 3A2T, ____. Mk&
(A.15)
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [ 121 [ 131 [14] [15] [ 161 [17] [ 181 [19] [ 201 [21] (221 [23] [24] [25] [26] [ 271 [28] [29] [ 301 [ 311 [32] [33] [34]
J.L. Beeby, J. Phys. C4 (1971) L359. W.H. Weinberg, J. Phys. CS (1972) 2098. W.H. Weinberg, J. Chem. Phys. 57 (1972) 5463. G. Cornsa, J. Phys. C6 (1973) 2648. J.L. Beeby, Proc. 2nd Intern. Conf. on Solid Surface, 1974, Japan; Japan. J. Appl. Phys., Suppl. 2, Pt. 2 (1974) 537. F.O. Goodman, Surface Sci. 46 (1974) 118. J.D. Doll, Chem. Phys. 3 (1974) 257. F.O. Goodman, Surface Sci. 65 (1977) 37. H. Hoinkes, H. Nahr and H. Wilsch, Surface Sci. 33 (1972) 516;40 (1973) 457. J.R. Bledsoe and S.S. Fisher, Surface Sci. 46 (1974) 129. B. Wood, B.F. Mason and B.R. Williams, J. Chem. Phys. 61 (1974) 1435. A.G. Stall, Jr., J.J. Ehrhardt and R.P. Merrill, J. Chem. Phys. 64 (1976) 34. G. Armand, J. Lapujoulade and Y. Lejai, Surface Sci. 63 (1977) 143. G. Armand, J. Lapujoulade and Y. Lejai, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, Vol. 2 (1977) p. 1361. G. Boato, P. Cantini and R. Tatarek, ibid, p. 1377. Y. Hamauzu and H. Asada, J. Cryst. Sot. Japan 20 (1978) 102 (in Japanese). R. Sau and R.P. Merrill, Surface Sci. 34 (1973) 268. G.A. Somorjai and H.H. Farrell, Advan. Chem. Phys. 20 (1972) 215. R.F. Wallis, B.C. Clark and R. Herman, Phys. Rev. 167 (1968) 652. R.E. Allen and F.W. de Wette, Phys. Rev. 188 (1969) 1320. D.P. Jackson, Surface Sci. 43 (1974) 431. H. Asada, J. Res. Inst. Catalysis, Hokkaido Univ. 25 (1977) 175. D.L. Smith and R.P. Merrill, J. Chem. Phys. 52 (1970) 5861. G.G. Kleiman and U. Landman, Phys. Rev. B8 (1973) 5484. E. Zaremba and W. Kohn, Phys. Rev. B15 (1977) 1769. J.E. Van Himbergen and R. Silbey, Solid State Commun. 23 (1977) 623. H. Hoinkes, H. Nahr and H. Wilsch, Surface Sci. 30 (1972) 363. E.R. Jones, J.T. Mckinney and M.B. Webb, Phys. Rev. 151 (1977) 476. F.O. Goodman, Surface Sci. 30 (1972) 1. American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, 1972). II.Band, l.teil, Mechanisch-Thermisch Landolt-Bornstein Zahlenwerte und Funktionen, Zustandsgrossen (Springer, 1971). R. Smoluchowski, Phys. Rev. 60 (1941) 661. T. Toya, private communication. L.D. Landau and E.M. Lifschitz, Quantum Mechanics - Non-Relativistic Theory, 2nd ed. (State Publisher of Physico-Mathematical Literatures, Moscow, 1963).