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The effect of the electronic surface response on sticking of He atoms on metallic substrates
231
Surface Science 139 (1984) 231-238 North-Holland, Amsterdam
THE EFFECT OF THE ELECTRONIC SURFACE RESPONSE STICKING OF He ATOMS ON METALLIC SUBSTRATES B. GUMHALTER
ON
*
Quantum Theory Group, Departments Waterloo, Ontario, Canada N2L 3GI
of Applied Mathematics and Physics, University of Waterloo,
and i. CRLJEN Rudjer BoSkoviC Institute, 41001 Zagreb, Yugoslavia Received 24 October 1983; accepted for publication 19 December 1983
Using a quantum model for the atom dissipative interactions with metallic surfaces developed earlier [Surface Sci. 117 (1982) 116; 126 (1983) 6661, we investigate the effect of the dynamic electronic response of a clean surface on sticking of He in the limit of very low incoming energies. We show within a perturbational approach that the electronic polarization and overlap induced He-surface interactions, although giving rise to a considerable increase of sticking probability in the low energy limit, cannot explain the sticking probabilities of the order of unity observed recently at very low energies [Sinvani et al., Phys. Rev. Letters 51 (1983) 1881.
1. Introduction Beam scattering experiments utilizing thermalized helium atoms [l] indicate that the sticking probability s for He impinging onto a metal surface is negligible for incident beam energies exceeding 20 meV. On the other hand, recent determination of s at beam temperatures in the range lo-20 K (- 1 mev) has shown dramatic change in the value of s [2], the latter lying in the range 3 ,< s < 1 and being unaffected by a submonolayer coverage of preadsorbed helium. These interesting findings raise many questions among which two of them are particularly controversial. First, there is a question of the energy loss mechanism involved in a highly quantum scattering regime in which the atom incident kinetic energy is of the order or lower than the well depth of the atom-surface potential. Second, due to the large increase of s as the incident atom energy is lowered, one may also ask oneself whether the inelastic scattering mechanism giving rise to sticking exhibits any peculiar behaviour in the energy interval below, say, 20 K. In this note we aim to * On leave from: Institute of Physics of the University, 41001 Zagreb, Yugoslavia
0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
232
B. Gumhalter,
2. Crljen / EJJect of electronic surJace response
contribute to the discussion of some aspects of the sticking processes caused by the atom inelastic scattering on the electronic degrees of freedom of a metal surface. The studies of sticking involve usually two distinct loss mechanisms responsible for atom inelastic scattering. These are the excitation of the polarization modes of the substrate lattice, i.e. phonons, and the polarization modes of the substrate electron gas, the electron-hole (e-h) pairs. The polarization modes are excited by the external perturbations brought about by the atomic motion in the vicinity of the surface. These perturbations are due to the potentials which the atomic structure of the impinging atom exerts on the substrate surface. In the case of a He atom we may, in general, distinguish two types of interactions between its closed shell structure and the substrate surface. The first is due to the polarization forces and its reactive, dynamic component is responsible for the long range Van der Waals interaction between the atom and the surface. The polarization induced distortion of the atom orbitals gives also rise to the formation of a static induced dipole moment on the atom [3], which in turn interacts with the electrons of the metal. This may lead to a dissipative interaction between the atom and the surface and ultimately to inelastic scattering and sticking [4]. The second type of interaction is due to the overlap induced repulsive pseudopotential fi [5] between inert He atoms and the electrons in the metal. This shows up as a short range atom-surface potential [6] which is responsible also for diffractive He scattering from metallic surfaces. The pseudopotential l? may as well act as a perturbation on the substrate electronic degrees of freedom in a way similar as does the atomic dipole moment, leading to excitation of the eeh pairs in the substrate [7,8]. Since both types of interactions may induce surface electronic charge density fluctuations, which in a real solid are coupled to the vibrational degrees of freedom of the ion cores, they may serve as mediators also for the atom-surface phonon interactions. In refs. [4] and [7] (hereafter to be referred to as I and II, respectively) we have developed a strictly quantum model for description of the inelastic scattering of low energy He atoms on the electronic charge density fluctuations characterizing the response of a metal surface. Here we shall apply this model to discuss within a perturbative approach the role of the above mentioned interactions (also to be referred to as I and II, respectively) on sticking of low energy He atoms on metallic substrates.
2. General properties of the decay rates induced by surface electronic density fluctuations Kinetics of the atoms in the vicinity of a metal surface may be characterized by the atom momentum K parallel to the surface and a quantum number n
B. Gumhalter, .?. Crljen / Effect of electronic surface response
233
which describes the perpendicular-to-the-surface motion in the atom-surface potential V(z). Here the z-coordinate denotes the direction normal to the surface. The corrugation of the surface charge density is neglected as we are dealing with low incoming atom energies for which the classical turning points are far outside the first layer of atoms. V(z) exhibits a potential well near the surface and this may accommodate a limited number of bound states whose number is characteristic of a particular substrate [6b]. Hence the spectrum of V(z) is discrete for E,, < 0, where En = 0 is the energy of a He atom at rest at infinite separation from the metal. The shape of V(z) for a metal characterized by the electronic density parameter rs = 2, the energies of the bound states and the corresponding wave functions are shown in figs. 2 and 1 of I and II, respectively. Inelastic scattering of an atom from the surface is characterized by a change of quantum numbers K and n, namely by a transition 1K, n) -+ IK - Q, n’), where Q is the wavevector of the surface mode excited in the inelastic collision. To each quantum transition corresponds the probability of the transition per unit time *
= ~r,,n-,~,,~(E)~
P ~,n-dE)
(1)
where E = fr2K2/2M + E,, and M stand for the initial energy of the atom and its mass, respectively, and rK,n_KJ,n, (E) is the decay rate per unit time characterizing this transition [9]. The total decay rate per unit time out of the state IK, n) is given by: IL(E)
=
c
G,,-,+n.,@>.
(2)
Q,n’
The summation in (2) is carried out for discrete n’ (E,,, < 0), and n’ belonging to the continuous spectrum (E,,, > 0). The former sum gives the inelastic scattering decay rate per unit time rb, due to sticking processes and the latter, I”, due to the inelastic rebounding from the surface. Clearly one has I”,,(E) = $,(E)
+ G,,(E).
In I and II we have calculated the transition rates Tb and I’” for the scattering off a metal surface within the second order distorted wave Born approximation for the two interactions considered in section 1, by assuming inelastic energy transfer to substrate e-h pairs. The results correspond to the bulk electronic density parameter rS = 2 and the latter may be considered as representative of the free electron properties of a wide range of transition metals like W, Pd, Pt, etc. * Note here that the definitions of r in I (eq. (4)) and II (eq. (5)) differ by a factor of 2. The factor l/2 multiplying r and I-, in fig. 4 of II is erroneous and should be omitted. Throughout the present paper we adopt the definition of II, viz. r = - Im 8.
234
B. Gumhalter,
.k Crljen / Effect of electronic surface response
The magnitude of r depends on the strength of the matrix elements of the interaction potential and the density of e-h excitations in the spectrum of the surface response (cf. eqs. (4) and (5) of I and II, respectively). The matrix elements themselves depend on the normalization of the atom wavefunctions. In I we used the normalization of G,(r) in which the particle wavevector k, at infinity is chosen to he appropriate quantum number, i.e., n = k,. In this case: (nln’)=t+-n’)=s(k,-k:,).
(3)
On the other hand, in II we used the normalization in which the energy of the perpendicular-to-the-surface motion is a good quantum number, namely: (filfi’) = S(E,
- E,,).
(4)
The passage from one type of normalization to the other is achieved by making use of the equality En = h2ki/2m. A typical behaviour of r and rb for normal He atom incidence within the normalization (4) is sketched in fig. 1. This r corresponds to the decay
E (meV) 10
0
20
I
I
I
/ 2.0
------____ I
0.5
I
____ I
EC 10-3Kyd J
1.0
Fig. 1. Decay rates r and Tb induced by scattering mechanism II (see section 1) for a He atom impinging normally with energy E onto a surface of a metal characterized by rS= 2. Results obtained within the normalization given by eq. (4). E, and E, denote the energies of two lowest bound states of V(z).
B. Gumhalter, .k. Crljen / Effect of electronic surface response
235
processes caused by the overlap induced pseudopotential 0 (cf. II) which induces nonadiabatic excitation of e-h pairs in an rs = 2 metal. A qualitatively similar picture has also been obtained for the polarization induced scattering potential discussed in I. Fig. 1 reveals several interesting features of r and fb at low incoming energies. Firstly, in the limit E = E,, -+ 0 we find that Tb gives a dominant contribution to r. This is in fact hardly surprising since at low enough atom incident energies the available phase space for inelastic scattering would consist mainly of bound states. Here in accord with the Heisenberg time-energy uncertainty relations, the major contribution to r comes from the two lowest bound states. This is so because the transition into them may involve the largest energy uncertainty. As the energy is increased, continuum to continuum transitions set in and Tb appears to be only a small fraction of r. At 20 meV it already becomes negligible (see also next section). This property can be understood in terms of a decreasing overlap between the itinerant and localized wavefunctions belonging to the spectrum of V{ z), and the interaction potential. Secondly, by using the numerical exact wavefunctions corresponding to V(Z) we have found that lim T,,,(E)#O,
E-+0
K=O,
(5)
even at energies several orders of magnitude lower than the potential well depth of v(z). This seems to be in accord with findings of Goodman [lo] who treated a similar problem of He sticking induced by phonons. In our work, we have not been able to reach the results of Brenig and co-workers [ll] who claim different behaviour, namely r(O) = 0 even for long range potentials. The only noticeable difference between F’s due to overlap and polarization induced dissipation within the same normalization appears to be in the overall order of magnitude of the decay rates for each particular scattering channel. In the case of polarization induced dissipative interaction, r and rb are about two orders of magnitude smaller than the corresponding overlap induced decay rates.
3. Sticking probabilities The sticking probability within the present perturbational approach may be formulated as the transition probability per unit time normalized to the unit current in the direction normal to the surface: 0)
= WME).
(6)
This definition of s rests on the assumption of an infinitely extended surface, i.e., all the particles for whichj, # 0 will eventually hit the surface with which
236
B. Gumhalter,
2. Crljen / Effect of electronic surface response
they may interact in either elastic or inelastic fashion [12]. The correctness of such a procedure was also checked by a semi-classical treatment of the particle-surface collision event and this revealed a one-to-one correspondence between the two approaches in the semiclassical limit [13]. The magnitude and the dimension of the current density depends also on the normalization used in the calculation of r and thereby P. Within the normalization utilized in I, one has dim[ P’] = dim[time]-‘. The current is then obtained from the equation:
and dim[ j,‘] = dim[ time] - ’ since dim[ $,,( z)] * = dim[length]-‘. Hence s defined by eq. (6) is a dimensionless quantity as it should be. On the other hand, within the normalization utilized in II we get dim[ P”] = dim[tz-‘1 and the current density is given by [14]: j,”
=
1/27rA,
(8)
which again yields a dimensionless quantity for s. Eqs. (6)-(g) enable us now to make use of the results of I and II to calculate the corresponding sticking probabiIities. With respect to experiments, of particular interest are the sticking probabilities for the atoms in beams of energy around 20 meV which is a typical nozzle beam energy for He [l] and at energies below 1 meV (- 10 K) utilized recently [2]. The results of calculations for s(E) induced by mechanisms I and II are summarized in table 1. They show that, as regards the nonadiabatic surface electronic response as a dissipative mechanism, the overlap induced potential gives rise to a much more efficient inelastic interaction then the one brought about by the polarization forces. (In a recent work, Sols et al. [15] have derived a similar result pertaining to mechanism I.)
Table 1 Sticking probabilities for He atoms approaching a metal surface at normal incidence for two typical experimental energies; I and II refer to the models of atom surface dissipative interactions of refs. [4] and [7], respectively; surface electronic response properties corresponding to an rs = 2 metal Dissipative
mechanism
s(E) E=l
I
II
meV
0.85~10-~ 1.9x1o-4
E=20meV 0.38x10-’ 1.2x1o-5
B. Gumhalter,
.k. Crljen / Effectof electronicsurfaceresponse
237
4. Conclusion Even though the overlap induced potential as a dominating interaction gives rise to a rapid increase of s by more than one order or magnitude as the energy of the incoming beam is lowered from 20 meV to below 1 meV, the sticking probability originating from this mechanism saturates at the value of the order 10e4. As this value was derived from a parameter free, strictly quantum model, it cannot be improved upon introducing any adjustable quantities into our calculations. Thus our conclusion is that neither polarization induced nor overlap induced dissipation of the atomic kinetic energy to the surface electronic density fluctuations is sufficiently effective to explain the sticking probability of the order of unity observed by Sinvani et al. [2]. Our results, although obtained for a perfectly flat surface, should not depend much on the corrugation of the tails of the electronic density profile extending out onto the vacuum. In any case, the corrugation of the low index metal surfaces is weak in the region of the classical turning points, for energies considered in table 1. A natural question which arises in connection with our results is whether the phonon induced dissipation could be responsible for the sticking probabilities observed. Goodman [lo] treated phonon induced dissipation for He/W system and obtained values for s which would be too low to explain recent experiments [2]. A possibility which offers itself in the direction of overcoming this controversy is to study the non-linear electron-hole or phonon excitations or to include the effects of the surface roughness and contamination into the consideration of atom scattering and sticking at metallic surfaces.
Acknowledgements One of the authors (B.G.) wishes to thank the Natural Sciences and Engineering Research Council of Canada for the award of a grant under its International Scientific Exchange Programme.
References [l] For more recent works on the scattering on solid surfaces, see e.g.: G. Boato, P. Cantini and R. Tatarek, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, Eds. R. Dobrozemsky et al. (Vienna, 1977) p. 1337; D.R. Miller and J.M. Horne, ibid., p. 1385; W. Cole and D.R. Frank], Surface Sci. 70 (1978) 585; H.U. Finzel, H. Frank, H. Hoinkes, H. Luschka, H. Nahr, H. Wilsch and U. Wonka, Surface Sci. 49 (1975) 577; G. Brusdeylins, R.B. Doak and J.P. Toennies, Phys. Rev. Letters 44 (1980) 1417; K.H. Rieder and T. Engel, Phys. Rev. Letters 47 (1980) 824;
238
[2] [3] [4] (51 [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15]
B. Gumhalter,
.k. Crljen / Effectof electronrcsurfaceresponse
B. Feuerbacher and R.F. Willis, Phys. Rev. Letters 47 (1981) 526; B.F. Mason and B.R. Williams, Phys. Rev. Letters 46 (1981) 1138; J. Lapujoulade, J. Perreau and A. Kara, Surface Sci. 129 (1983) 59; J. Lapujoulade, Y. Lejay and G. Armand, Surface Sci. 95 (1980) 107; 63 (1977) 143; B. Poelsema, R.L. Palmer, G. Mechtersheimer and G. Comsa, Surface Sci. 117 (1982) 60; 126 (1983) 641. M. Sinvani, M.W. Cole and D.L. Goodstein, Phys. Rev. Letters 51 (1983) 188. P.R. Antoniewicz, Phys. Rev. Letters 32 (1974) 1424; E. Zaremba, Phys. Letters 57A (1976) 156. Z. CrIjen and B. Gumhalter, Surface Sci. 117 (1982) 116. N.R. Kestner, J. Jortner, M.H. Cohen and S.A. Rice, Phys. Rev. 140 (1975) A56. (a) E. Zaremba and W. Kohn, Phys. Rev. B13 (1976) 2270; (b) E. Zaremba and W. Kohn, Phys. Rev. B15 (1977) 1769. B. Gumhalter and Z. Crljen, Surface Sci. 126 (1983) 666. 0. Gunnarsson and K. Schiinhammer, Phys. Rev. B25 (1982) 2514. A. Messiah, Quantum Mechanics, Vol. II (North-Holland, Amsterdam, 1965) ch. XVII, section 4. F.O. Goodman, Surface Sci. 111 (1981) 279. J. Boheim, W. Brenig and J. Stutzki, Z. Physik B48 (1982) 43. B. Gumhalter and D.M. Newns, Surface Sci. 50 (1975) 465. J.P. Muscat, Solid State Commun. 18 (1976) 1089. L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1965) section 21. F. Sols, F. Flores and N. Garcia, preprint, private communication.
Surface Science 139 (1984) 231-238 North-Holland, Amsterdam
THE EFFECT OF THE ELECTRONIC SURFACE RESPONSE STICKING OF He ATOMS ON METALLIC SUBSTRATES B. GUMHALTER
ON
*
Quantum Theory Group, Departments Waterloo, Ontario, Canada N2L 3GI
of Applied Mathematics and Physics, University of Waterloo,
and i. CRLJEN Rudjer BoSkoviC Institute, 41001 Zagreb, Yugoslavia Received 24 October 1983; accepted for publication 19 December 1983
Using a quantum model for the atom dissipative interactions with metallic surfaces developed earlier [Surface Sci. 117 (1982) 116; 126 (1983) 6661, we investigate the effect of the dynamic electronic response of a clean surface on sticking of He in the limit of very low incoming energies. We show within a perturbational approach that the electronic polarization and overlap induced He-surface interactions, although giving rise to a considerable increase of sticking probability in the low energy limit, cannot explain the sticking probabilities of the order of unity observed recently at very low energies [Sinvani et al., Phys. Rev. Letters 51 (1983) 1881.
1. Introduction Beam scattering experiments utilizing thermalized helium atoms [l] indicate that the sticking probability s for He impinging onto a metal surface is negligible for incident beam energies exceeding 20 meV. On the other hand, recent determination of s at beam temperatures in the range lo-20 K (- 1 mev) has shown dramatic change in the value of s [2], the latter lying in the range 3 ,< s < 1 and being unaffected by a submonolayer coverage of preadsorbed helium. These interesting findings raise many questions among which two of them are particularly controversial. First, there is a question of the energy loss mechanism involved in a highly quantum scattering regime in which the atom incident kinetic energy is of the order or lower than the well depth of the atom-surface potential. Second, due to the large increase of s as the incident atom energy is lowered, one may also ask oneself whether the inelastic scattering mechanism giving rise to sticking exhibits any peculiar behaviour in the energy interval below, say, 20 K. In this note we aim to * On leave from: Institute of Physics of the University, 41001 Zagreb, Yugoslavia
0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
232
B. Gumhalter,
2. Crljen / EJJect of electronic surJace response
contribute to the discussion of some aspects of the sticking processes caused by the atom inelastic scattering on the electronic degrees of freedom of a metal surface. The studies of sticking involve usually two distinct loss mechanisms responsible for atom inelastic scattering. These are the excitation of the polarization modes of the substrate lattice, i.e. phonons, and the polarization modes of the substrate electron gas, the electron-hole (e-h) pairs. The polarization modes are excited by the external perturbations brought about by the atomic motion in the vicinity of the surface. These perturbations are due to the potentials which the atomic structure of the impinging atom exerts on the substrate surface. In the case of a He atom we may, in general, distinguish two types of interactions between its closed shell structure and the substrate surface. The first is due to the polarization forces and its reactive, dynamic component is responsible for the long range Van der Waals interaction between the atom and the surface. The polarization induced distortion of the atom orbitals gives also rise to the formation of a static induced dipole moment on the atom [3], which in turn interacts with the electrons of the metal. This may lead to a dissipative interaction between the atom and the surface and ultimately to inelastic scattering and sticking [4]. The second type of interaction is due to the overlap induced repulsive pseudopotential fi [5] between inert He atoms and the electrons in the metal. This shows up as a short range atom-surface potential [6] which is responsible also for diffractive He scattering from metallic surfaces. The pseudopotential l? may as well act as a perturbation on the substrate electronic degrees of freedom in a way similar as does the atomic dipole moment, leading to excitation of the eeh pairs in the substrate [7,8]. Since both types of interactions may induce surface electronic charge density fluctuations, which in a real solid are coupled to the vibrational degrees of freedom of the ion cores, they may serve as mediators also for the atom-surface phonon interactions. In refs. [4] and [7] (hereafter to be referred to as I and II, respectively) we have developed a strictly quantum model for description of the inelastic scattering of low energy He atoms on the electronic charge density fluctuations characterizing the response of a metal surface. Here we shall apply this model to discuss within a perturbative approach the role of the above mentioned interactions (also to be referred to as I and II, respectively) on sticking of low energy He atoms on metallic substrates.
2. General properties of the decay rates induced by surface electronic density fluctuations Kinetics of the atoms in the vicinity of a metal surface may be characterized by the atom momentum K parallel to the surface and a quantum number n
B. Gumhalter, .?. Crljen / Effect of electronic surface response
233
which describes the perpendicular-to-the-surface motion in the atom-surface potential V(z). Here the z-coordinate denotes the direction normal to the surface. The corrugation of the surface charge density is neglected as we are dealing with low incoming atom energies for which the classical turning points are far outside the first layer of atoms. V(z) exhibits a potential well near the surface and this may accommodate a limited number of bound states whose number is characteristic of a particular substrate [6b]. Hence the spectrum of V(z) is discrete for E,, < 0, where En = 0 is the energy of a He atom at rest at infinite separation from the metal. The shape of V(z) for a metal characterized by the electronic density parameter rs = 2, the energies of the bound states and the corresponding wave functions are shown in figs. 2 and 1 of I and II, respectively. Inelastic scattering of an atom from the surface is characterized by a change of quantum numbers K and n, namely by a transition 1K, n) -+ IK - Q, n’), where Q is the wavevector of the surface mode excited in the inelastic collision. To each quantum transition corresponds the probability of the transition per unit time *
= ~r,,n-,~,,~(E)~
P ~,n-dE)
(1)
where E = fr2K2/2M + E,, and M stand for the initial energy of the atom and its mass, respectively, and rK,n_KJ,n, (E) is the decay rate per unit time characterizing this transition [9]. The total decay rate per unit time out of the state IK, n) is given by: IL(E)
=
c
G,,-,+n.,@>.
(2)
Q,n’
The summation in (2) is carried out for discrete n’ (E,,, < 0), and n’ belonging to the continuous spectrum (E,,, > 0). The former sum gives the inelastic scattering decay rate per unit time rb, due to sticking processes and the latter, I”, due to the inelastic rebounding from the surface. Clearly one has I”,,(E) = $,(E)
+ G,,(E).
In I and II we have calculated the transition rates Tb and I’” for the scattering off a metal surface within the second order distorted wave Born approximation for the two interactions considered in section 1, by assuming inelastic energy transfer to substrate e-h pairs. The results correspond to the bulk electronic density parameter rS = 2 and the latter may be considered as representative of the free electron properties of a wide range of transition metals like W, Pd, Pt, etc. * Note here that the definitions of r in I (eq. (4)) and II (eq. (5)) differ by a factor of 2. The factor l/2 multiplying r and I-, in fig. 4 of II is erroneous and should be omitted. Throughout the present paper we adopt the definition of II, viz. r = - Im 8.
234
B. Gumhalter,
.k Crljen / Effect of electronic surface response
The magnitude of r depends on the strength of the matrix elements of the interaction potential and the density of e-h excitations in the spectrum of the surface response (cf. eqs. (4) and (5) of I and II, respectively). The matrix elements themselves depend on the normalization of the atom wavefunctions. In I we used the normalization of G,(r) in which the particle wavevector k, at infinity is chosen to he appropriate quantum number, i.e., n = k,. In this case: (nln’)=t+-n’)=s(k,-k:,).
(3)
On the other hand, in II we used the normalization in which the energy of the perpendicular-to-the-surface motion is a good quantum number, namely: (filfi’) = S(E,
- E,,).
(4)
The passage from one type of normalization to the other is achieved by making use of the equality En = h2ki/2m. A typical behaviour of r and rb for normal He atom incidence within the normalization (4) is sketched in fig. 1. This r corresponds to the decay
E (meV) 10
0
20
I
I
I
/ 2.0
------____ I
0.5
I
____ I
EC 10-3Kyd J
1.0
Fig. 1. Decay rates r and Tb induced by scattering mechanism II (see section 1) for a He atom impinging normally with energy E onto a surface of a metal characterized by rS= 2. Results obtained within the normalization given by eq. (4). E, and E, denote the energies of two lowest bound states of V(z).
B. Gumhalter, .k. Crljen / Effect of electronic surface response
235
processes caused by the overlap induced pseudopotential 0 (cf. II) which induces nonadiabatic excitation of e-h pairs in an rs = 2 metal. A qualitatively similar picture has also been obtained for the polarization induced scattering potential discussed in I. Fig. 1 reveals several interesting features of r and fb at low incoming energies. Firstly, in the limit E = E,, -+ 0 we find that Tb gives a dominant contribution to r. This is in fact hardly surprising since at low enough atom incident energies the available phase space for inelastic scattering would consist mainly of bound states. Here in accord with the Heisenberg time-energy uncertainty relations, the major contribution to r comes from the two lowest bound states. This is so because the transition into them may involve the largest energy uncertainty. As the energy is increased, continuum to continuum transitions set in and Tb appears to be only a small fraction of r. At 20 meV it already becomes negligible (see also next section). This property can be understood in terms of a decreasing overlap between the itinerant and localized wavefunctions belonging to the spectrum of V{ z), and the interaction potential. Secondly, by using the numerical exact wavefunctions corresponding to V(Z) we have found that lim T,,,(E)#O,
E-+0
K=O,
(5)
even at energies several orders of magnitude lower than the potential well depth of v(z). This seems to be in accord with findings of Goodman [lo] who treated a similar problem of He sticking induced by phonons. In our work, we have not been able to reach the results of Brenig and co-workers [ll] who claim different behaviour, namely r(O) = 0 even for long range potentials. The only noticeable difference between F’s due to overlap and polarization induced dissipation within the same normalization appears to be in the overall order of magnitude of the decay rates for each particular scattering channel. In the case of polarization induced dissipative interaction, r and rb are about two orders of magnitude smaller than the corresponding overlap induced decay rates.
3. Sticking probabilities The sticking probability within the present perturbational approach may be formulated as the transition probability per unit time normalized to the unit current in the direction normal to the surface: 0)
= WME).
(6)
This definition of s rests on the assumption of an infinitely extended surface, i.e., all the particles for whichj, # 0 will eventually hit the surface with which
236
B. Gumhalter,
2. Crljen / Effect of electronic surface response
they may interact in either elastic or inelastic fashion [12]. The correctness of such a procedure was also checked by a semi-classical treatment of the particle-surface collision event and this revealed a one-to-one correspondence between the two approaches in the semiclassical limit [13]. The magnitude and the dimension of the current density depends also on the normalization used in the calculation of r and thereby P. Within the normalization utilized in I, one has dim[ P’] = dim[time]-‘. The current is then obtained from the equation:
and dim[ j,‘] = dim[ time] - ’ since dim[ $,,( z)] * = dim[length]-‘. Hence s defined by eq. (6) is a dimensionless quantity as it should be. On the other hand, within the normalization utilized in II we get dim[ P”] = dim[tz-‘1 and the current density is given by [14]: j,”
=
1/27rA,
(8)
which again yields a dimensionless quantity for s. Eqs. (6)-(g) enable us now to make use of the results of I and II to calculate the corresponding sticking probabiIities. With respect to experiments, of particular interest are the sticking probabilities for the atoms in beams of energy around 20 meV which is a typical nozzle beam energy for He [l] and at energies below 1 meV (- 10 K) utilized recently [2]. The results of calculations for s(E) induced by mechanisms I and II are summarized in table 1. They show that, as regards the nonadiabatic surface electronic response as a dissipative mechanism, the overlap induced potential gives rise to a much more efficient inelastic interaction then the one brought about by the polarization forces. (In a recent work, Sols et al. [15] have derived a similar result pertaining to mechanism I.)
Table 1 Sticking probabilities for He atoms approaching a metal surface at normal incidence for two typical experimental energies; I and II refer to the models of atom surface dissipative interactions of refs. [4] and [7], respectively; surface electronic response properties corresponding to an rs = 2 metal Dissipative
mechanism
s(E) E=l
I
II
meV
0.85~10-~ 1.9x1o-4
E=20meV 0.38x10-’ 1.2x1o-5
B. Gumhalter,
.k. Crljen / Effectof electronicsurfaceresponse
237
4. Conclusion Even though the overlap induced potential as a dominating interaction gives rise to a rapid increase of s by more than one order or magnitude as the energy of the incoming beam is lowered from 20 meV to below 1 meV, the sticking probability originating from this mechanism saturates at the value of the order 10e4. As this value was derived from a parameter free, strictly quantum model, it cannot be improved upon introducing any adjustable quantities into our calculations. Thus our conclusion is that neither polarization induced nor overlap induced dissipation of the atomic kinetic energy to the surface electronic density fluctuations is sufficiently effective to explain the sticking probability of the order of unity observed by Sinvani et al. [2]. Our results, although obtained for a perfectly flat surface, should not depend much on the corrugation of the tails of the electronic density profile extending out onto the vacuum. In any case, the corrugation of the low index metal surfaces is weak in the region of the classical turning points, for energies considered in table 1. A natural question which arises in connection with our results is whether the phonon induced dissipation could be responsible for the sticking probabilities observed. Goodman [lo] treated phonon induced dissipation for He/W system and obtained values for s which would be too low to explain recent experiments [2]. A possibility which offers itself in the direction of overcoming this controversy is to study the non-linear electron-hole or phonon excitations or to include the effects of the surface roughness and contamination into the consideration of atom scattering and sticking at metallic surfaces.
Acknowledgements One of the authors (B.G.) wishes to thank the Natural Sciences and Engineering Research Council of Canada for the award of a grant under its International Scientific Exchange Programme.
References [l] For more recent works on the scattering on solid surfaces, see e.g.: G. Boato, P. Cantini and R. Tatarek, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, Eds. R. Dobrozemsky et al. (Vienna, 1977) p. 1337; D.R. Miller and J.M. Horne, ibid., p. 1385; W. Cole and D.R. Frank], Surface Sci. 70 (1978) 585; H.U. Finzel, H. Frank, H. Hoinkes, H. Luschka, H. Nahr, H. Wilsch and U. Wonka, Surface Sci. 49 (1975) 577; G. Brusdeylins, R.B. Doak and J.P. Toennies, Phys. Rev. Letters 44 (1980) 1417; K.H. Rieder and T. Engel, Phys. Rev. Letters 47 (1980) 824;
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