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Sticking and inelastic scattering at metal surfaces: The electron-hole pair mechanism
Surface Science I17 (1982) 53-59 North-Holland Publishing Company
53
STICKING AND INELASTIC SCATTERING AT METAL SURFACES: THE ELECTRON-HOLE PAIR MECHANISM K. SCHGNHAMMER
and
Received
14 September
19X1; accepted
for publication
29 October
19XI
We present calculations of the contribution of the electron-hole pair creation to the inelastic scattering probability and the sticking coefficient of atoms or molecules on metal surfaces. For light chemically reactive adsorbates this mechanism can be very important. The total inelastic scattering probability for thermal He atoms on a copper surface due to e-h pairs on the other hand is only 10-5, i.e. for rare gas atoms the phonon mechanism is the dominating one.
1. Introduction Recently there has been a large interst in the energy transfer between a surface and an atom or a molecule approaching a surface. It has been shown that inelastic scattering of rare gas atoms can give information about the elementary excitations of the surface [I]. The energy transfer processes also determine the probability that the incoming particle loses its kinetic energy and is adsorbed at the surface. In the past it has often been assumed that the dominating process for the energy transfer is the excitation of phonons [2]. For metal substrates, however, there has recently been an increasing interest in the excitation of electron-hole pairs [3-l 11. Because of the large mass of the adsorbate it is usually assumed that the adsorbate nuclear motion can be treated classically. A thermal atom approaching the metal surface then acts as a slowly vaghg localizedperturbation on the electronic system. In this paper we present a description of this electronic energy loss mechanism. A central quantity for the calculation of the sticking probability is the probability of P,,(c) that an incoming atom (molecule) with the energy e’ loses the energye during one round trip in the adsorption well. If the substrate 0039-6028/82/0000-0000/$02.75
0 1982 North-Holland
temperature S=
T, is zero the sticking
probabilitys
is
wPC.( E) dc, j c’
(1)
since the particle cannot leave the potential well if it has lost an energy 6 > E’. For T, > 0, however, there is a possibility that the adsorbate looses its initial kinetic energy during the first round trip in the adsorption well but, at a later time before it has fallen all the way to the bottom of the adsorption well, it regains enough energy from the substrate to leave the surface. To describe this effect we have followed the stochastic description of Iche and Nozieres [ 121 and assume that the different round trips in the adsorption well can be treated as statistically independent [lo]. In section 2 we derive formulas for the energy distribution function P,(C) and in section 3 these results are applied to two special cases: (1) the sticking of a light chemically reactive adsorbate with an affinity level close to the Fermi energy [8,10], and (2) the inelastic scattering of rare gas atoms [9].
2. The energy distribution’function In this section we calculate the probability P,,(C) that an incoming particle with energy C’ loses the energye during one round trip. Before the adsorbate approaches the surface the substrate is in the electronic state ) E,,,) with the probability ~(E,)=exp(-E,/kT,)/Z, where Z is the partition function. The time evolution of the electronic states is described by a time dependent Hamiltonian H, = H, + V,, where V, describes the perturbation due to the adsorbate. We introduce the state ]$,,(t)) which describes the time evolution of 1E,). Thus
with the initial condition The energy distribution J’,,(E)=
) $,,( - 00)) = (E,,,).
function
is given by
2 I(E,I~~,(~~))J*P(E,)S(~--EE,--E,))=P~~(~)+P(~), m,n
(3)
where p0 is the probability for elastic scattering. The 6’ dependence enters via V,. In the following we suppress the z’ label. For finite substrate temperatures P(E) contains loss (C > 0) as well as gain (C < 0) processes. As V, presents a slow perturbation, the deviations of the electronic state from the adiabatic state will usually be small. The lowest order expansion of 1c$,( t)) around the adiabatic state yields for noninteracting electrons [lo] (4)
55
with
(5) Here, IE, LY)’ are the adiabatic one-particle states obeying the Schrodinger equation H, It, cy>f = E[C, cu)‘. These one-particle states are labeled by the energyr and additional quantum numbers@, As the slow perturbation produces only low energy e-h pairs we can replace YeI) a1 I r;; IEl, %)’ in the integrand
in (5) by
Yfr- aI I c/cl=. %Y. where cF is the Fermi energy. difference z., - c2 only, i.e. %&,‘~2)
-)Zja,&
Now the integrations &)
Then
#a ILI2(E,, cZ) is a function
-9). in (4) can be carried
= z1 +~,&)I*[~ nlaz
out and we obtain
++>I9
(6)
with n(e) = [exp(E/kq) - l] -I being the Bose function. useful for the case of weak in~~~sti~~~~, i.e. y/j-*
of the energy
The result (6) is only
&c)dcal. --co
As shown in the appendix,
1.7 ,,Jr)
is in this limit given by
(7) where ?&Jc~ + i0) is the time derivative of the scattering Tmatrix. A simple way to improve upon the result (6) for cases of larger inelasticity is to realize that this expression for P(C) has the same form as the energy distribution function for a system of weakly linearly displaced bosom: For E>O the Bose function n(e) in the factor [I in(~)] describes the stimulated emission of the “bosons”, while for c < 0, P(E) is proportional to 11 + n(r) / = n(lc/) and this part describes the absorption of thermally excited “bosom”. Because of this correspondence we replace the result (6) by the boson result for arbitrary coupling strength [ 13, IO]
X[l
+2n(~‘)-[l
+n(~‘)]
e’r’r-n(e’)e-“”
e”‘. Ii
(8)
The step from (6) to (8) may seem like a fairly questionable approximation. We have shown previously [8,14] that (8) presents the exact result for P(E) for slow perturbations, if the eigenstates )c, a) of the unperturbed Hamiltonian can be chosen in such a way that the perturbation is diagonul in the quantum numbersa. In this case the tja,,,,(e) (see appendix A) can be expressed in terms of the Fourier transform of the time derivative s’,,,,( t) of the phase shifts at the Fermi energy [7,8] (9) For the description of the excitations created (absorbed) by a slow perturbation coherent superposition of e-h pairs first considered by Tomonaga [ 151 in this special case behave as bosons. The time derivative dC,,,( t ) describes via the Friedel sum rule the rate at which new states are created below the Fermi level. AS these new states cannot be occupied instantaneously, this is the cause of the inelastic effects.
3. Applications Nraiskov and Lundqvist [4] have proposed that the creation of e-h pairs should be very important for adsorbates which have an affinity level close to the Fermi energy of the substrate. As the adsorbate approaches the surface the level is generally shifted downwards [16] and it may move below the Fermi energy. The nonadiabatic effects turn out to be most pronounced, when the time At needed to cross the Fermi energy is much shorter than the round trip time T between the crossings. i.e. if the crossing occurs far from the surface. Assuming a time variation At/2 SC,., =
t* + (At/2)2
At/2 - ( f + T)* + (At/2)’
P(c) can be calculated P(c) =
analytically
(10) ’
for c = 0 [8]:
lAt)’ [S(c)+ T*r eCfA’]. T2
(11)
(At)‘+
The inelastic part of this expression has been discussed by Ntirskov [ 171 using semiclassical arguments. The width of P(C) is of the same order as the average energy transfer [8]. Using the definition of the sticking coefficient, we see that for At/T< 1 the sticking coefficient is close to one for kinetic energies e’ K l/At. For finite substrate temperatures T, > 0, eq. (1) is not the proper expression for the sticking coefficient. There is the possibility that the adsorbate loses its initial kinetic energy during the first round trip in the adsorption well but, at a later time before it has fallen all the way to the
bottom of the adsorption well, it regain enough leave the surface. To describe this effect we have approach by Iche and Nozibes [12]. We consider smaller than the depth of the potential well. Then for an adsorbate with energy e’ obeys the integral s(E’) =I0
P,,(e:-6’) -3c
S(E) dc,
energy from the substrate to followed the master equation the case when kT, is much the sticking probability s(E‘) equation [ 121 (12)
which has to be solved with the boundary condition S(E) - 1 when E - - 00. From the numerical solution of (12) using the T, > 0 result for P(e), which follows from (lo), we find that for a given kinetic energy of the incoming particle the sticking coefficient shows a small variation with surface temperature T, [IO]. This is due to the partial cancellation of two effects: The probability to leave the surface after the first round trip decreases due to stimulated emission of e-h pairs, but there is the possibility to leave the surface afier a few round trips, which decreases the sticking probability, However, if the elastic scattering probability is small the latter effect tends to dominate and the sticking probability decreases with q. We conclude that for the case discussed above the electronic energy loss mechanism alone can lead to a sticking coefficient of order 1. The opposite extreme is a rare gas atom on a surface, for which the adsorbate levels are far from the Fermi energy. Via the exchange repulsion a rare gas atom acts on the substrate electrons as an essentially repulsive pseudopotential, and the simplicity of this coupling makes a fairly realistic calculation possible [9]. As the total inelasticity is very small it is sufficient to work with the expression for P(E) given in eq. (6). In this limit of a small inelasticity it is also possible to treat the motion of the rare gas atoms quantum mechanically by the method of Born and Huang [ 181. Such an approach was first used in the present context by Brivio and Grimley [5]. We have compared the quantum mechanical calculation with a classical trajectory calculation and found that the classical treatment is sufficient except for very small initial velocities f9f. We have studied in detail the scattering of He from a Cu surface. For the theoretical description we have used the adiabatic potential of Zaremba and Kohn [19] and the He electron Tmatrix of Jortner et al. [ZO]. For an initial energy of the He atoms of 22.6 meV and the temperature T, = 16 K we find a total inelastic scattering probability of IF5 [9]. This is several orders of magnitude smaller than the experimental results. It follows that for this system the phonon m~hanism is the dominating one. Other adsorbate systems are probably in between these extremes. For molecules such as CO, N, and H, the interaction between the substrate and the adsorbate leads to a partial filling of the adsorbate affinity level. Model calculations [8,17] indicate, that the electronic mechanism is of importance for such light, chemically reactive adsorbates. In these cases it may be necessary to consider both the electronic and the phonon mechanism.
58
Appendix at the Fermi energy Ed The matrix elements ‘(c. (Y,11’ 1E, a?)’ evaluated determine the spectrum P(r) for slow perturbations. We will show here how these matrix elements can be related to the scattering-matrix S’ corresponding to the (static) localized potential V’. As the normalization of the one-particle eigenstates Ic, a) of the unperturbed Hamiltonian we choose (c, (Y]C’,a’)=s(e-C’)
s,,,.
(A.1)
The S matrix corresponding the standard way [21] (C, cX]SIJz’, (Y’)=s(C-C’)
to V, can be expressed
in terms of the T matrix
.s;JC)
=S(E-f’)[S,,,-2~iTaf,,(~+iO)]. AS the adiabatic
in
(A.2)
state 1c, a)’ is just the “scattering
state”
1c, a + >. (A.3)
the Tmatrix
is given by [21]
T,‘,,(c + i0) = (c, (Y(TIC, cf’)‘.
(A.4)
We now differentiate
with respect
to the label t
This matrix equation of the T matrix
can be solved formally
to express ‘(c, a ( c 1c, a’)’ in terms
(‘4.6) In the discussion of P(C) we have used this formula (a) For weak inelasticity one obtains
in two special cases:
‘(~,~J~I,,,‘)‘=~~,,(~+iO)+O(T*). (b) When the perturbation y is diagonal in the quantum is also diagonal and (A.6) simplifies to
(A.7) numbers,
1 a ‘(6, OLI l’,lf, cl’)’ = -s,,.- 2~i -at In sAa(c) = -6tdT ‘s Z~.U(t),
the T matrix
(A.8)
where we have introduced the phase shifts S,,,(t) in the usual way by writing S,‘,,(c) = S,,, exp(2i&( t)).
References [l] See, e.g., G. Brusdeylins, R.B. Doak and J.P. Toennies, Phys. Rev. Letters 45 (1980) 2040; 46 (1981) 437. [2] For a review see, e.g., F.O. Goodman and H.Y. Wachman, Dynamics of Gas-Surface Scattering (Academic Press, 1976). [3] E.G. d’Aghano, P. Kumar, W. Schaich and H. Suhl, Phys. Rev. Bl 1 (1975) 2122; A. Nourtier, J. Physique 38 (1977) 479. (41 J.K. N&skov and B.I. Lundqvist, Surface Sci. 89 (1979) 251. [S] G.P. Brivio and T.B. Grimley, Surface Sci. 89 (1979) 226; J. Phys. Cl0 (1977) 2351. [6] R. Brako and D.M. Newns, Solid State Commun. 33 (1980) 713. [7] K. Schonhammer and 0. Gunnarsson. Z. Physik B38 (1980) 127. [8] K. Schonhammer and 0. Gunnarsson, Phys. Rev. B22 (1980) 1629. [9] 0. Gunnarsson and K. Schonhammer, Phys, Rev., in press. [IO] K. Schonhammer and 0. Gunnarsson, Phys. Rev. B, in press. [l l] J.W. Gadzuk and H. Metiu, Phys. Rev. B22 (1980) 2603; H. Metiu and J.W. Gadzuk, J. Chem. Phys. 74 (1981) 2641. [ 121 G. Iche and P. Nozieres, J. Physique 37 (1976) 1313. [ 13) E. Mtiller-Hartmann, T.V. Ramakrishnan and G. Toulouse, Phys. Rev. B3 (197 1) 1102. [14] K. Schonhammer, Z. Physik B45 (1981) 23. [ 151 S. Tomonaga, Progr. Theoret. Phys. 5 (1950) 544. [ 161 N.D. Lang and A.R. Williams. Phys. Rev. Letters 37 (1976) 2 12: 0. Gunnarsson, H. Hjelmberg and J.K. Nerskov. Phys. Scripta 22 (1980) 165. [ 171 J.K. Ntirskov, J. Vacuum Sci. Technol. 18 (I 98 I) 420. [ 181 M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon. Oxford, 1954). [ 191 E. Zaremba and W. Kohn, Phys. Rev. B I5 ( 1977) 1769. [20] J. Jortner, N.R. Kestner, S.A. Rice and M.H. Cohen, J. Chem. Phys. 43 ( 1965) 2614. [21] M.K. Goldberger and K.M. Watson, Collision Theory (Wiley. New York, 1964).
53
STICKING AND INELASTIC SCATTERING AT METAL SURFACES: THE ELECTRON-HOLE PAIR MECHANISM K. SCHGNHAMMER
and
Received
14 September
19X1; accepted
for publication
29 October
19XI
We present calculations of the contribution of the electron-hole pair creation to the inelastic scattering probability and the sticking coefficient of atoms or molecules on metal surfaces. For light chemically reactive adsorbates this mechanism can be very important. The total inelastic scattering probability for thermal He atoms on a copper surface due to e-h pairs on the other hand is only 10-5, i.e. for rare gas atoms the phonon mechanism is the dominating one.
1. Introduction Recently there has been a large interst in the energy transfer between a surface and an atom or a molecule approaching a surface. It has been shown that inelastic scattering of rare gas atoms can give information about the elementary excitations of the surface [I]. The energy transfer processes also determine the probability that the incoming particle loses its kinetic energy and is adsorbed at the surface. In the past it has often been assumed that the dominating process for the energy transfer is the excitation of phonons [2]. For metal substrates, however, there has recently been an increasing interest in the excitation of electron-hole pairs [3-l 11. Because of the large mass of the adsorbate it is usually assumed that the adsorbate nuclear motion can be treated classically. A thermal atom approaching the metal surface then acts as a slowly vaghg localizedperturbation on the electronic system. In this paper we present a description of this electronic energy loss mechanism. A central quantity for the calculation of the sticking probability is the probability of P,,(c) that an incoming atom (molecule) with the energy e’ loses the energye during one round trip in the adsorption well. If the substrate 0039-6028/82/0000-0000/$02.75
0 1982 North-Holland
temperature S=
T, is zero the sticking
probabilitys
is
wPC.( E) dc, j c’
(1)
since the particle cannot leave the potential well if it has lost an energy 6 > E’. For T, > 0, however, there is a possibility that the adsorbate looses its initial kinetic energy during the first round trip in the adsorption well but, at a later time before it has fallen all the way to the bottom of the adsorption well, it regains enough energy from the substrate to leave the surface. To describe this effect we have followed the stochastic description of Iche and Nozieres [ 121 and assume that the different round trips in the adsorption well can be treated as statistically independent [lo]. In section 2 we derive formulas for the energy distribution function P,(C) and in section 3 these results are applied to two special cases: (1) the sticking of a light chemically reactive adsorbate with an affinity level close to the Fermi energy [8,10], and (2) the inelastic scattering of rare gas atoms [9].
2. The energy distribution’function In this section we calculate the probability P,,(C) that an incoming particle with energy C’ loses the energye during one round trip. Before the adsorbate approaches the surface the substrate is in the electronic state ) E,,,) with the probability ~(E,)=exp(-E,/kT,)/Z, where Z is the partition function. The time evolution of the electronic states is described by a time dependent Hamiltonian H, = H, + V,, where V, describes the perturbation due to the adsorbate. We introduce the state ]$,,(t)) which describes the time evolution of 1E,). Thus
with the initial condition The energy distribution J’,,(E)=
) $,,( - 00)) = (E,,,).
function
is given by
2 I(E,I~~,(~~))J*P(E,)S(~--EE,--E,))=P~~(~)+P(~), m,n
(3)
where p0 is the probability for elastic scattering. The 6’ dependence enters via V,. In the following we suppress the z’ label. For finite substrate temperatures P(E) contains loss (C > 0) as well as gain (C < 0) processes. As V, presents a slow perturbation, the deviations of the electronic state from the adiabatic state will usually be small. The lowest order expansion of 1c$,( t)) around the adiabatic state yields for noninteracting electrons [lo] (4)
55
with
(5) Here, IE, LY)’ are the adiabatic one-particle states obeying the Schrodinger equation H, It, cy>f = E[C, cu)‘. These one-particle states are labeled by the energyr and additional quantum numbers@, As the slow perturbation produces only low energy e-h pairs we can replace YeI) a1 I r;; IEl, %)’ in the integrand
in (5) by
Yfr- aI I c/cl=. %Y. where cF is the Fermi energy. difference z., - c2 only, i.e. %&,‘~2)
-)Zja,&
Now the integrations &)
Then
#a ILI2(E,, cZ) is a function
-9). in (4) can be carried
= z1 +~,&)I*[~ nlaz
out and we obtain
++>I9
(6)
with n(e) = [exp(E/kq) - l] -I being the Bose function. useful for the case of weak in~~~sti~~~~, i.e. y/j-*
of the energy
The result (6) is only
&c)dcal. --co
As shown in the appendix,
1.7 ,,Jr)
is in this limit given by
(7) where ?&Jc~ + i0) is the time derivative of the scattering Tmatrix. A simple way to improve upon the result (6) for cases of larger inelasticity is to realize that this expression for P(C) has the same form as the energy distribution function for a system of weakly linearly displaced bosom: For E>O the Bose function n(e) in the factor [I in(~)] describes the stimulated emission of the “bosons”, while for c < 0, P(E) is proportional to 11 + n(r) / = n(lc/) and this part describes the absorption of thermally excited “bosom”. Because of this correspondence we replace the result (6) by the boson result for arbitrary coupling strength [ 13, IO]
X[l
+2n(~‘)-[l
+n(~‘)]
e’r’r-n(e’)e-“”
e”‘. Ii
(8)
The step from (6) to (8) may seem like a fairly questionable approximation. We have shown previously [8,14] that (8) presents the exact result for P(E) for slow perturbations, if the eigenstates )c, a) of the unperturbed Hamiltonian can be chosen in such a way that the perturbation is diagonul in the quantum numbersa. In this case the tja,,,,(e) (see appendix A) can be expressed in terms of the Fourier transform of the time derivative s’,,,,( t) of the phase shifts at the Fermi energy [7,8] (9) For the description of the excitations created (absorbed) by a slow perturbation coherent superposition of e-h pairs first considered by Tomonaga [ 151 in this special case behave as bosons. The time derivative dC,,,( t ) describes via the Friedel sum rule the rate at which new states are created below the Fermi level. AS these new states cannot be occupied instantaneously, this is the cause of the inelastic effects.
3. Applications Nraiskov and Lundqvist [4] have proposed that the creation of e-h pairs should be very important for adsorbates which have an affinity level close to the Fermi energy of the substrate. As the adsorbate approaches the surface the level is generally shifted downwards [16] and it may move below the Fermi energy. The nonadiabatic effects turn out to be most pronounced, when the time At needed to cross the Fermi energy is much shorter than the round trip time T between the crossings. i.e. if the crossing occurs far from the surface. Assuming a time variation At/2 SC,., =
t* + (At/2)2
At/2 - ( f + T)* + (At/2)’
P(c) can be calculated P(c) =
analytically
(10) ’
for c = 0 [8]:
lAt)’ [S(c)+ T*r eCfA’]. T2
(11)
(At)‘+
The inelastic part of this expression has been discussed by Ntirskov [ 171 using semiclassical arguments. The width of P(C) is of the same order as the average energy transfer [8]. Using the definition of the sticking coefficient, we see that for At/T< 1 the sticking coefficient is close to one for kinetic energies e’ K l/At. For finite substrate temperatures T, > 0, eq. (1) is not the proper expression for the sticking coefficient. There is the possibility that the adsorbate loses its initial kinetic energy during the first round trip in the adsorption well but, at a later time before it has fallen all the way to the
bottom of the adsorption well, it regain enough leave the surface. To describe this effect we have approach by Iche and Nozibes [12]. We consider smaller than the depth of the potential well. Then for an adsorbate with energy e’ obeys the integral s(E’) =I0
P,,(e:-6’) -3c
S(E) dc,
energy from the substrate to followed the master equation the case when kT, is much the sticking probability s(E‘) equation [ 121 (12)
which has to be solved with the boundary condition S(E) - 1 when E - - 00. From the numerical solution of (12) using the T, > 0 result for P(e), which follows from (lo), we find that for a given kinetic energy of the incoming particle the sticking coefficient shows a small variation with surface temperature T, [IO]. This is due to the partial cancellation of two effects: The probability to leave the surface after the first round trip decreases due to stimulated emission of e-h pairs, but there is the possibility to leave the surface afier a few round trips, which decreases the sticking probability, However, if the elastic scattering probability is small the latter effect tends to dominate and the sticking probability decreases with q. We conclude that for the case discussed above the electronic energy loss mechanism alone can lead to a sticking coefficient of order 1. The opposite extreme is a rare gas atom on a surface, for which the adsorbate levels are far from the Fermi energy. Via the exchange repulsion a rare gas atom acts on the substrate electrons as an essentially repulsive pseudopotential, and the simplicity of this coupling makes a fairly realistic calculation possible [9]. As the total inelasticity is very small it is sufficient to work with the expression for P(E) given in eq. (6). In this limit of a small inelasticity it is also possible to treat the motion of the rare gas atoms quantum mechanically by the method of Born and Huang [ 181. Such an approach was first used in the present context by Brivio and Grimley [5]. We have compared the quantum mechanical calculation with a classical trajectory calculation and found that the classical treatment is sufficient except for very small initial velocities f9f. We have studied in detail the scattering of He from a Cu surface. For the theoretical description we have used the adiabatic potential of Zaremba and Kohn [19] and the He electron Tmatrix of Jortner et al. [ZO]. For an initial energy of the He atoms of 22.6 meV and the temperature T, = 16 K we find a total inelastic scattering probability of IF5 [9]. This is several orders of magnitude smaller than the experimental results. It follows that for this system the phonon m~hanism is the dominating one. Other adsorbate systems are probably in between these extremes. For molecules such as CO, N, and H, the interaction between the substrate and the adsorbate leads to a partial filling of the adsorbate affinity level. Model calculations [8,17] indicate, that the electronic mechanism is of importance for such light, chemically reactive adsorbates. In these cases it may be necessary to consider both the electronic and the phonon mechanism.
58
Appendix at the Fermi energy Ed The matrix elements ‘(c. (Y,11’ 1E, a?)’ evaluated determine the spectrum P(r) for slow perturbations. We will show here how these matrix elements can be related to the scattering-matrix S’ corresponding to the (static) localized potential V’. As the normalization of the one-particle eigenstates Ic, a) of the unperturbed Hamiltonian we choose (c, (Y]C’,a’)=s(e-C’)
s,,,.
(A.1)
The S matrix corresponding the standard way [21] (C, cX]SIJz’, (Y’)=s(C-C’)
to V, can be expressed
in terms of the T matrix
.s;JC)
=S(E-f’)[S,,,-2~iTaf,,(~+iO)]. AS the adiabatic
in
(A.2)
state 1c, a)’ is just the “scattering
state”
1c, a + >. (A.3)
the Tmatrix
is given by [21]
T,‘,,(c + i0) = (c, (Y(TIC, cf’)‘.
(A.4)
We now differentiate
with respect
to the label t
This matrix equation of the T matrix
can be solved formally
to express ‘(c, a ( c 1c, a’)’ in terms
(‘4.6) In the discussion of P(C) we have used this formula (a) For weak inelasticity one obtains
in two special cases:
‘(~,~J~I,,,‘)‘=~~,,(~+iO)+O(T*). (b) When the perturbation y is diagonal in the quantum is also diagonal and (A.6) simplifies to
(A.7) numbers,
1 a ‘(6, OLI l’,lf, cl’)’ = -s,,.- 2~i -at In sAa(c) = -6tdT ‘s Z~.U(t),
the T matrix
(A.8)
where we have introduced the phase shifts S,,,(t) in the usual way by writing S,‘,,(c) = S,,, exp(2i&( t)).
References [l] See, e.g., G. Brusdeylins, R.B. Doak and J.P. Toennies, Phys. Rev. Letters 45 (1980) 2040; 46 (1981) 437. [2] For a review see, e.g., F.O. Goodman and H.Y. Wachman, Dynamics of Gas-Surface Scattering (Academic Press, 1976). [3] E.G. d’Aghano, P. Kumar, W. Schaich and H. Suhl, Phys. Rev. Bl 1 (1975) 2122; A. Nourtier, J. Physique 38 (1977) 479. (41 J.K. N&skov and B.I. Lundqvist, Surface Sci. 89 (1979) 251. [S] G.P. Brivio and T.B. Grimley, Surface Sci. 89 (1979) 226; J. Phys. Cl0 (1977) 2351. [6] R. Brako and D.M. Newns, Solid State Commun. 33 (1980) 713. [7] K. Schonhammer and 0. Gunnarsson. Z. Physik B38 (1980) 127. [8] K. Schonhammer and 0. Gunnarsson, Phys. Rev. B22 (1980) 1629. [9] 0. Gunnarsson and K. Schonhammer, Phys, Rev., in press. [IO] K. Schonhammer and 0. Gunnarsson, Phys. Rev. B, in press. [l l] J.W. Gadzuk and H. Metiu, Phys. Rev. B22 (1980) 2603; H. Metiu and J.W. Gadzuk, J. Chem. Phys. 74 (1981) 2641. [ 121 G. Iche and P. Nozieres, J. Physique 37 (1976) 1313. [ 13) E. Mtiller-Hartmann, T.V. Ramakrishnan and G. Toulouse, Phys. Rev. B3 (197 1) 1102. [14] K. Schonhammer, Z. Physik B45 (1981) 23. [ 151 S. Tomonaga, Progr. Theoret. Phys. 5 (1950) 544. [ 161 N.D. Lang and A.R. Williams. Phys. Rev. Letters 37 (1976) 2 12: 0. Gunnarsson, H. Hjelmberg and J.K. Nerskov. Phys. Scripta 22 (1980) 165. [ 171 J.K. Ntirskov, J. Vacuum Sci. Technol. 18 (I 98 I) 420. [ 181 M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon. Oxford, 1954). [ 191 E. Zaremba and W. Kohn, Phys. Rev. B I5 ( 1977) 1769. [20] J. Jortner, N.R. Kestner, S.A. Rice and M.H. Cohen, J. Chem. Phys. 43 ( 1965) 2614. [21] M.K. Goldberger and K.M. Watson, Collision Theory (Wiley. New York, 1964).