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Non-adiabatic inelastic scattering and intra-molecular vibrational excitation at metal surfaces
Surface Science 187 (1987) 289-311 North-Holland, Amsterdam
289
NON-ADIABATIC INELASTIC SCATrERING AND INTRA-MOLECULAR VIBRATIONAL EXCITATION AT METAL SURFACES N. CRISA, G. DOYEN
a n d R. H U B N E R
Institut far Physikalische Chemie, Universitiit Miinehen, Theresienstrasse 37, D-8000 Miinchen 2, Fed. Rep. of Germany Received 10 December 1986; accepted for publication 13 May 1987
Considering as energy transfer mechanism the formation of electron-hole pairs, the inelastic scattering of a diatomic molecule from a metal surface is investigated on a purely quantum mechanical basis. The coupling of an adsorbate level to the surface is treated in an Anderson-Newns-like model which neglects correlation and spin effects. The centre-of-mass (CM) motion and the stretch motion are coupled via the interaction with the metal electrons. The model does not include a mechanical translational-vibrational coupling for a rigid surface. The stretch frequency of the diatomic molecule is assumed to be independent of the molecule-surface separation. The vibrating diatomic molecule approaches the surface in an upright orientation. For the CM-potential energy curve a More-type shape plus van der Waals tail was assumed. The Born-Oppenheimer core wave functions are obtained using a variational spline interpolation (VSI). The non-adiabatic transition matrix elements are evaluated in an improved quantum sudden approximation which incorporates many-particle aspects. Sticking probabilities, vibrational excitation probabilities and angular distributions for the inelastically scattered molecule are given. The dependence of the results on the parameters of the model are discussed. Agreement with the experimentally observed trends for NO scattered from a silver surface can only be obtained for weak electronic coupling. Considering only a single adorbital and neglecting spin degeneracy vibrational excitation probabilities of the order of 10 -3 are predicted for substrate temperatures between 100 and 1000 K. This is nearly an order of magnitude smaller than found in experiment, but including spin effects and several adorbitals on the gas molecule might lead to larger probabilities.
1. Introduction The mechanism of energy transfer between a surface and an approaching m o l e c u l e is a still u n s o l v e d p r o b l e m o f c o n s i d e r a b l e i n t e r e s t . A l t h o u g h t h e p h o n o n m e c h a n i s m is o f t e n b e l i e v e d to b e o f d o m i n a n t i m p o r t a n c e , t h e existence of non-adiabatic effects has already been demonstrated experimentally [1,2]. Recently energy transfer between intra-molecular vibrations and substrate degrees of freedom have found increased attention both experimentally [3-6]
0 0 3 9 - 6 0 2 8 / 8 7 / $ 0 3 . 5 0 9 E l s e v i e r S c i e n c e P u b l i s h e r s B.V. (North-Holland Physics Publishing Division)
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N. Crisg~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
and theoretically [7-9]. The non-adiabatic electron-hole (e-h) pair mechanism is generally believed to be important, because the energy exchange is typically five to ten times the phonon Debye energy. The purely mechanical picture, i.e., exchange of energy between the translational, rotational and vibrational degrees at a rigid surface can be excluded, because vibrational excitation is found to depend strongly on the substrate temperature [3]. In this paper we present a theoretical study of a simple Hamiltonian which might be a rough model of a diatomic molecule like CO or NO interacting with a sp-band metal (e.g. Ag or Cu). The novel feature of this investigation is the purely quantum mechanical description which applies also for the core movement of the gas particle. This warrants a correct description even at very low kinetic energies. Section 2 contains the description of the model. The method of solution, which is a special version of a quantum sudden approximation, is outlined in section 3. The numerical results are presented in section 4. Finally we compare our results to experiments and to other theoretical approaches.
2. The model
The model Hamiltonian is very similar to that used in related theoretical studies [7,9,10,11]. It is here divided up into three parts: H = HCM + Hvib + Her,
(1)
HCM = - V 2 x / 2 M + Vstat( x ) .
(2)
HCM is the Hamiltonian for the center-of-mass (CM) motion, x the CM-position and M the mass of the diatomic molecule. The first term of eq. (2) is the CM-ldnetic energy. V~tat is the CM-potential energy curve which is further detailed in section 4. The intra-molecular vibration is described as usual by a harmonic oscillator of frequency %TR: Hvi b = ~asTRata,
(3)
a t and a are boson creation and annihilation operators. The coordinate for the stretch motion will in the following be denoted by y (cf. fig. 1). The electronic Hamiltonian Hel is given by the Anderson-Newns model [12] with, however, intra-adsorbate electron-electron correlation neglected (i.e., by the Wolff model [13]): Hel = E A ( z ) F / A +- s k
k -}- Vint,
(4)
V~n,= 2 [VAk(Z)CtAC k + herm. conj.]. k The fermion operator n a = CtACA describes the molecular orbital which couples most strongly to the metal surface. The one-electron states of the metal are
N, Crisgl et al. / Inelastic scattering and vibrational excitation at metal surfaces
I
y
i
L I
~ ~ ) / / , . s ' - I
z
I I >(
X
291
I L '
~'x\
I I I
I i I I
CM-position
Fig. 1. Illustration of the coordinates used in describing the motion of the gas molecule. referred to by the operators ctk, c k and n k = C~Ck. They have ionization energies Ck. Spin degeneracy is not taken into account in eq. (4). The molecule is supposed to approach the surface in an upright orientation. The molecular orbital energy E A depends on the distance z of the nearest atom from the surface. The meaning of the three coordinates x, y and z is illustrated in fig. 1. In the following the abbreviation R = (x, y) will be used as well. VAk are the quantities coupling the molecular orbital to the metal states. As E g and Vgk depend on the molecule-surface separation z, these quantities are not c-numbers but operators. This fact prevents an exact solution of the Hamiltonian.
3. M e t h o d of solution
3.1. The quantum sudden approximation
Following our pre~ious approach [14], we do not formulate the scattering problem for {he gas particle, because this leads to fundamental problems with formal scattering theory [15]. The alternative is to consider a scattering metal electron [14,16]. The gas particle is then assumed to provide a kind of moving
292
N. Cris& et al. / Inelastic scattering and vibrational excitation at metal surfaces
target or scattering center for the electrons. The fact that the molecule hits and leaves the surface will be taken into account by assuming that the (moving) scattering center is present only with a certain probability. One then requires "branch ratios" in order to derive transition probabilities for the gas particle movement. This aspect is outlined and discussed in section 3.3. After adopting tiffs philosophy one still finds that the scattering problem is not solvable exactly. The difficulty is the motion of the adparticle. We can, however, evaluate the transition matrix elements in perturbation theory by using electronic wave functions calculated for a fixed molecular position. The total many-body wave "function is then written in Born-Oppenheimer-form. The prescription reads as follows [14]: (i) Calculate electronic wave functions ~bj(R, ri) for each value of x and y, i.e., with R as parameter. (ii) Construct the Born-Oppenheimer (BO) many-particle wave function g'(x, y, z ) = q~CM(x)g2C~
det[qJj(R, ri) ] = x , ( R ) det[ffj(R, ri) ] .
is a product wave function with the first factor q,CM(x) describing the CM-motion and the other 12C~ the stretch vibration. (iii) Calculate transition matrix elements in perturbation theory using BO-wave functions. (iv) Calculate the S-matrix according to eq. (10) of ref. [14].
x,(R)
S/,, k, = 6ktS,, -- 27tiT1,. k~8(cl + % -- Ck -- % ) . Here the many-particle density of states enters. By taking the absolute square of the S-matrix elements St~' k~ one obtains the probability that a metal electron will scatter from the state l to the state k while the gas atom makes a transition from the core wave function X~ to the core wave function X,. This perturbative Born-Oppenheimer formulation has the physical meaning of a quantum sudden approximation for metal electron scattering. (It is called quantum here, because the expression "sudden approximation" is commonly used, if the semi-clasically calculated elastic S-matrix element for the scattering degrees of freedom is used for taking total scattering matrix elements [17]. Tiffs is, however, different e.g. in the BCC-IOS approximation [18].) The criterion for the validity of the quantum sudden approximation is that the energy transfer is small compared to the kinetic energy of the metal electron. Tiffs condition is always fulfilled for the present investigation. The scattering matrix element for an electron to scatter from k to l (k 4: l) while the molecule is at a position R = (x, y) is: -2~ri(llV.,.,(R)lk
+ ) 8 ( , , - ~k) = --2r
I VintGVint ] k ) 8 ( ', - "k)
= --2r = -2~riC~Ak)(R)VAt(R)6(~-
'k)
Ok),
N. Crist) et al. / Inelastic scattering and vibrational excitation at metal surfaces
293
where C(Ak)(R) is the coefficient of the adsorbate admixture in the state I k), evaluated utilizing the Lippmann-Schwinger equation (see eq. (6) of ref. [14]) and GA the Green function for an electron in the adorbital. In the sudden approximation one now integrates over R,. while weighting with the corresponding gas molecule wave functions. The result is then identical to that obtained in ref. [14]. The problem encountered is that whenever X~ and X v are unbound states, the electronic probabilities violate the unitarity condition strongly for physically reasonable values of VAk. This is an indication for the breakdown of perturbation theory. The reason for this is that in calculating the electronic metal wave functions the gas molecule is present as a scattering center during the whole time the electron moves through the solid. This results in a too strong mixing of the wave functions. Obviously, if the molecule is perturbing the metal only for a short period of time, mixing with the adorbital has to decrease. The mixing coefficient C~Ak) is then too large, because the self-energy, q g = A + i F , is not calculated correctly. It is too small and therefore GA is too large yielding a too large coefficient C~Ak). A solution of this electronic non-unitarity problem can be found as follows: If the scattered metal electron moves away from the surface, the gas particle will move away as well. This effect on GA was not considered in [14], because the molecule was bound to the surface. In order to correct for this error, it is convenient to rewrite the Hamiltonian (1) in the following form: H=Ho+
Y'~HB, #
(5)
Ho = E I kr>'k~( kr l,
(6)
k, p
I/3>ca
Ha =
(7)
(8)
k , t,
k labels the metal electron and r the gas particle core wave function describing the stretch vibration and the CM-motion. Each I/3> is a basis wave function for the situation where the electron is in the adorbital and the molecule is in some eigenfunction of HCM + H,~b. These eigenstates labelled by /3 need not concide with those labelled by r, as long as they form a complete set. The identification HCM + H~b + EA(R)nA + ~..'knk = Ho + ~ {/3)',(/3 J, k #
W= E [VAk(R)CtACk + herm. conj.] = E W~, k
(9) (10)
#
where
ffi
far x r ( R ) V A k ( R ) x B ( R )
=
VkrB
(11)
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N. Cris~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
establishes the equivalence of the two forms of the Hamiltonian. The form eqs. (5) to (8) of the Hamiltonian describes a delocalized moving electron connected with a moving gas particle in an eigenstate of the core potential. Choose now one core basis function (characterizing the molecule in the state I fl)) as extremely localized around the position R. This means that non-adiabatic interaction occurs only, if the gas particle is at position R. The matrix elements of the interaction operator W are then:
(kv I WIB) = Vk.a = (v I fl)Vkk(R#).
(12)
For the Wolff-type Hamiltonian H 0 + H# the required many-body Green function is evaluated readily: (fl l a l f l )
(13)
= ( g - ,p - q/j)-1,
with
IVk., 12
q a ( e ) -- E EZ-~k-e, k , I+
E
IVA~ 12 E - - , k _ , I(Pl/3)12.
(14)
k, ,
c k is the energy of the metal electron (near the Fermi level) and c, is the energy of the molecule. At each value E of the total energy more than one k-state contributes, because higher electronic energies c k can be compensated by lower gas molecule energies c, and vice versa. The number of k-states contributing is Y.,1 summed under the condition that there exists a state [ k ) with c, = E - c k. The electronic matrix element VAk is, however, multiplied by the overlap factor ( ~ l f l ) so that we can define an effective degeneracy by
D/j(E)= Y'.I(~,Ifl)Iz (with,k+,.--e).
(15)
At substrate temperature T~ = 0 the summation ranges from c, = 0 (deepest bound state of the CM-motion)up to c, = E - E v (elastic scattering),where E F is the Fermi energy. The self-energy q/~ can now be rewritten in the following way:
k E-c k-c,
= ~, I ( v l f l ) 1 2 q A ( E , c , ) ,
(16)
where the summation over ~, is over the specified range. Now the molecular energy ~, is in our case small (up to - 0.5 eV) whereas c~ is of the order of the Fermi energy, i.e., - 10 eV. Therefore qA(E, c,) is practically independent of c, and we can write:
q/~= Da( E)qA( E ),
(17)
where qA is now the standard electronic self-energy of the Wolff-AndersonNewns-type Hamiltonians. (fl I G[ fl) has been derived by considering H 0 + H a
3t. Cris3 et aL / Inelastic scattering and vibrational excitation at metal surfaces
295
separately for each I t ) , i.e., separately for each gas particle position R. Scattering can occur at any gas particle position R. In the case I I/~) ~ I kv) the T-matrix element for H 0 + H a is then: (I/~IW~ Ikv+ ) = ( l l ~ l W I B ) ( f l l a l B ) ( B I W I k v )
= ~A(R13)(I~Ifl)(fllGIfl)(fllv)VAk(R~).
(18)
This approximation means physically that the gas particle does not move while the metal electron scatters. The total contribution to the scattering due to all gas molecule positions R , is
(ll~lWIkv+ ) = Y'.(II~IW~ Ikv+ ). 13
(19)
Using
(20)
E = fdR x~(R)x,(R) #
and the notation
GA(R#) = ( f l l G l f l ) ,
(21)
one obtains for the total transition matrix element:
x~(R)(I~IB)~A(R)GA(R)VAk(R)(flIv)x~(R).
= f d R
(22) This result is our quantum sudden approximation, where many-particle effects have been incorporated in an improved Green function GAIf the imaginary part of the self-energy q~ is introduced
P ( R , , E) = ~rDa(E)~,lVAk 123(E - ,k) = D a ( E ) F ( R , E ) ,
(23)
k
the S-matrix elements can be written in the form
s,,,, ~ = 8,~,~,,~- 2if dR X,,(R)~(R, E)G,(R, E)x~(R).
(24)
This agrees with the expression obtained in ref. [14], except that the Green function GA and its self-energy have to be replaced by the normalized quantities indicated by the bar. Compared to the electronic Anderson-Newns-like Hamiltonian, the selfenergy qA has increased by the density factor Da. If we use this form of the Green function to evaluate the coefficient C~Ak), we go beyond first order perturbation theory. Obviously the ~coefficient becomes smaller and unitary is obeyed approximately.
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N. CrisgJ et al. / Inelastic scattering and vibrational excitation at metal surfaces
In order to get an estimate of the density factor Do(E ) one could consider scattering of a structureless particle from an infinitely hard wall: 2 f0~d Dry(E) = -~
K sin2(Kx) = K 7r
1 sin(2Kx), 2 ~r----x
(25)
with x the gas atom distance and K = (2ME) -1/2 the momentum. The second term is oscillatory and always smaller than the first for x > 0. In our numerical calculations the employed density factor accounts also for the number of bound states.
3.2. The BO-core wave functions The Schrbdinger equation for the CM-motion reads: -(1/2M)
dZq~(x)/dx 2 + V~tat(X)~(x ) = HCMq)(X ) = Eq)(x).
(26)
We want to approximate the function q~(x) in the interval 0 < x < L by a cubic spline interpolation over a set of n discrete points xj. Because q~(x) is a polynomial of third degree over each subinterval (x j, x j+l), the condition that the second derivatives are continuous at all xj leads to a system of linear equations connecting q,(xj) with its second derivatives at the end points xj of the sub-intervals. At the end points x = 0 and x = L the second derivatives are set to zero. At x = 0, which is inside the repulsive wall, this is physically reasonable. At x = L, which is in the flat part of the potential, the wave function q)(x) resembles a sine. Therefore the condition q , " ( L ) = 0 requires ~ ( L ) = 0 as well. This serves as a criterion of selecting a finite set of representative wave functions out of a continuum (see below). In matrix notation the resulting system of linear equations has the form [19]: Aq~" = Bq~,
(27)
where q~" and q~ denote the vector containing the second derivatives and the values of the function q)(x) at the points xj respectively. If the values of q~ at the end points xj of the subintervals would be known, the second derivatives (which uniquely determine the spline) could be evaluated from eq. (27). In our case the q)(x~) are unknown, but we have the SchrOdinger equation relating ~,"(xg) to the values q,(xj). In matrix notation it reads: - (1/2M)q~" + Vq) = Eq~.
(28)
Solving eq. (27) for q)" and inserting in eq. (28) yields:
[-(a/2M)(A-~)
+ V]q~ = Eq~.
(29)
This is a matrix eigenvalue problem. Because the Schrbdinger equation is equivalent to a variational principle, solving eq. (29) means choosing among all possible spline interpolations that one which has extremal properties with
N. Cris& et al. / Inelastic scattering and vibrational excitation at metal surfaces
297
respect to the energy functional (~1HcM [ if). We therefore call this method "variational spline interpolation" (VSI). The dimension of the matrices is equal to the number n of discrete points xj. One of the advantages of such a scheme is that no integrations have to be performed. The number of wave functions obtained in this way is equal to n. Obviously the continuum of unbound core states is discretized. The wave function of highest energy is that one which in the flat part of the potential completes one oscillation over three adjacent points. Hence the energy range covered depends on the length L and on the number of points in between. In our calculation L was 4.5 bohr with 200 points in between. The described method calculates the shape of the wave function accurately over the specified interval of length L. The boundary condition (second derivative equals zero at the point L ) selects out of the continuum of unbound states a finite number of wave functions. One obtains one representative gas core state within the energy interval defined by the two adjacent continuum states. Tests for exactly solvable potentials (square well, Morse) demonstrated that the levels of up to twenty bound states are given with three significant digits and the thirty lowest continuum states have the correct shape in the surface region [20]. The wave functions have to be normalized after diagonalization of eq. (29). The unbound states were normalized over a length of 2500 bohr assuming that the potential stays flat for distances larger than 3 bohr from the surface. 3.3. Transition rates and branch ratios The probability Pl~-kv for a transition from the state I k v ) to a state I/~) is obtained by taking the absolute square of the matrix element eq. (24). The transition rate R.~_ v for gas molecule scattering from core state [p) to core state 1/~) is obtained by multiplying by the number ~rk of metal electrons with excitation power arriving per unit time and summming over all initial and final metal electron states: R~ ~ = E N~e,~._ ~ .
(30)
k,l
The number of scattering metal electrons per unit time can be calculated as follows: Nk = OUkBk(~)gk"
(31)
Here Bk(T~) is the probability for a metal electron to be in the state I k ) at substrate temperaturo T~ and having the possibility of making an energy conserving transition to another empty metal state. For energy transfer to the solid this probability is unity. Foy energy transfer to the gas particle this denotes the probability for the existence of an appropriate electron-hole pair.
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N. Cris?l et al. / Inelastic scattering and vibrational excitation at metal surfaces
A Boltzmann distribution is used in the present calculation. Note that energy conservation is already built into the probabilities Pt,,- k,- Uk is the velocity of a metal electron in the state I k). gk is the number of states of kind I k ) per unit volume, o is the surface area which has to be hit by a metal electron in order to interact with the gas molecule near the surface, o was chosen to correspond to the area of a surface unit cell. Scattering probabilities for the gas particle motion are now defined by taking the ratio of the transition rate for the considered channel (e.g. total direct inelastic scattering) over the total rate, i.e., the sum over all partial rates for the separate channels. The various considered channels are detailed in section 4. As an example let us consider a separation into three different channels: total direct inelastic scattering, total sticking and elastic scattering. Recall that elastic scattering is a singular event and elastic scattering from a moving surface is a purely quantum mechanical effect. Because of the singular role played by quantum mechanical elastic scattering, we do not calculate the elastic rate from eq. (30), but connect it with the flux of the incoming gas particles:
Re, = [ K/( MbI)] Pk,~-k,.
(32)
K is the CM-momentum of the gas particle, M its mass and I the normalization length for the core wave functions. I was chosen as 2500 bohr and does not agree with the length L in the VSI. b is the width of the surface region which extends from the adsorption minimum to the classical turning point. This procedure is in the spirit of perturbation treatments and also conforms with other prescriptions used in the literature [21]. It should, however, be remembered that there exists no general proof for the validity of this procedure and that it appears questionable indeed. The gas particle scattering probabilities are: H~,_,
ErRr~
.
(33)
For the mentioned example the probability for total direct inelastic scattering is then calculated from: R inel /-Iigel = R e I .j_ R i n d + Rstick .
(34)
4. Results and comparison with experiments Numerical calculations have been performed for a heteronuclear diatomic molecule of atomic masses corresponding to 12 and 16 proton masses, respectively. These are the values for a CO-molecule, but the mass of a
N. Crisd et al. / Inelastic scattering and vibrational excitation at metal surfaces
299
NO-molecule is nearly the same (30 instead of 28). Such a difference is of no significance for a qualitative model investigation of the type presented here. The potential energy curve for the CM-motion (coordinate x) is given by
Vstat(X ) = Ul(X ) -4- U2(x),
(35)
where
U l ( x ) = Uo(e -2ax - 2 e-aX), U2(x) = - C3 e-'~/(~-~)(x - 4)-3
(36)
Ua(x) is a Morse potential describing the short range interaction with the surface. U0 is its well depth which in our case is 110 meV. /3 is the steepness parameter which, together with U0, determines the vibrational frequency ~cM of the CM-motion, and the repulsive part of the potential./3 was chosen to be 1 au, corresponding to O~cM= 10 meV. U2(x ) is the van der Waals (vdW)-like long range tail arising from fluctuating dipole-dipole interactions. The exponential factor is introduced to remove the singularity of the van der Waals potential at x = 4. C3 is a parameter describing the strength of the long range interaction (C 3 = 10.8 meV nm3). With the known polarizability of the CO-molecule (2.6 • 10-3 nm 3, [22]) this is a value which is in the range of typical values for physisorption on metal surfaces [23]. 7/ is the cut-off parameter for the vdW potential: 7/= 3.5 au. indicates the position of the singularity for the vdW potential. In our case it is = 2.67 au. Concerning the stretch motion (coordinate y) we used a harmonic potential with a characteristic frequency ~STR, independent on the CM-motion. This is not quite realistic, because o~STR is known to change upon adsorption. The consequence of our approximation is that there is no coupling between vibrational and translational motion for a rigid surface. In our model the coupling arises only via interaction with e - h pairs. Hence, vibrational excitation probabilities might be underestimated. The chosen value was Wsa-g= 260 meV. The distance dependence of the adorbital energy Eh(Z ) and the hopping parameters Va,(Z ) (cf. the electronic Hamiltonian eq. (4)) were chosen as follows: EA(Z ) = E ~ eXZ; VAk(Z) = V~ e -vz/2.
(37)
EA(Z ) is kept constant for z > 2 bohr. V~ is related to the imaginary part F of the electronic,self-energy qA which is assumed to have a semi-elliptical form:
r(,k) = (O2//,tro~) [1 --,(~k/20~)2] 1/2
(38)
The following values have been chosen: Metal bandwidth 48 = 20 eV, modelling roughly the sp-band of the noble metals. -
300
N. Cris~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
- Adorbital energy E ~ = + 4 eV with respect to the Fermi level describing in the case of CO and N O the unoccupied 2~r* orbital in interaction with the metal states. - Decay parameters for the coupling strength ('y - X) = 1.5 au. (The results depend only on the difference "y - X. (cf. ref. [14])). - Coupling strength D = 225 meV. We then have an incoming diatomic molecule (initially in the vibrational ground state v i = 0 ) impinging with a translational energy Eikin on a metal surface which has substrate temperature T~. In such a case the following outgoing channels are available: A: VibrationaUy elastic scattering (v i = 0, of = 0): The vibrational state of the molecule remains unperturbed. It may be subdivided into: A1: Elastic ( E f = Evan). i . also the translational state does not vary. A2: Direct inelastic: a certain amount of translational energy might be dissipated (energy loss; 0 < Ekinf < Ekin)i or, at T~ > 0, the adparticle might gain i translational energy from recombining e - h pairs (energy gain; E f > Ekin)A3: Sticking ( E f < 0): the energy loss is so large, that the adparticle gets trapped at the surface. B: Vibrationally inelastic scattering ( v i = 0 , o f = 1): during the interaction with the surface the molecule gains enough energy to jump into its first vibrationally excited state. The subdivision is as follows: BI: Vibrationally direct inelastic: includes all possible translational final states with E f > 0. B2: Vibration assisted sticking: the adparticle remains stuck at the surface with of = 1. In principle, further vibrational excitations (vf = 2, 3 . . . . ) as well as the rotational degrees of freedom should be taken into account. However, for numerical reasons we restrict our present investigation to the first vibrationally excited state. We shall hereafter adopt the following terminology for the branch ratios: Elastic: channel A1. Total sticking: channels A3 + B2. It consists of a part (contained in B2) directly related to the surface temperature T~ and another part that is not. Total direct inelastic: channels A2 + B1, i.e. no distinction between the different final vibrational states. Total vibrational excitation: channel B1. Figs. 2-5 display results for substrate temperature Ts = 760 K. In fig. 2 we report the probability (branch ratio) for elastic scattering, as a function of the incoming kinetic energy; in fig. 3 that for total direct inelastic scattering. The behaviour of the total vibrational excitation is shown in fig. 4. The trend with Ekm of this last quantity agrees qualitatively with experimental observations [3], i.e., there is an increase in the scattering probability with Eki n. On the other hand, as visible in fig. 5, the sticking coefficient decreases rapidly with
N. Cris~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
301
1.0 >I-1"'4
__1 < rn 0 n" 13-
0.8 0.6
o0.~ I.-{.,0 ,< ._1
~0.2 .
I
0
100
/
I
200 300 Eki n (meV)
400
Fig. 2. Elastic scattering probability versus incoming perpendicular kinetic energy (T~ = 760 K).
0 . 6 84 >-
~0.4 __J m <~ m
o0.3
0.2 < ._.1
toO.1 Z
~
0
!
260
360
Ekin (meV) Fig. 3. Total direct ~elastic probability versus incoming perpendicular kinetic energy ( Ts = 760 K).
302
N. Crisr et al. / Inelastic scattering and vibrational excitation at metal surfaces
~o2.5 X
>-.
~2.0 CI3 ,< 133
on.-l . 5 Z1.0 I--,,-I
0.5 X LLI
O.
100
0
200 3 0 Ekin(meV)
/-,00
Fig. 4. Total vibrational excitation probability (v = 0--* v = 1 ) versus incoming perpendicular kinetic energy (Ts = 760 K).
0.04 >t---
I---4
J 0.03
I,-,,.I
133 r'n 0
o.o2
Z
~0.01 Ch
O.
~
0
10
2O
30
4O
Ekin(meV) Fig. 5. Sticking probability versus incoming perpendicular kinetic energy (Ts = 760 K).
N. Crish et al. / Inelastic scattering and vibrational excitation at metal surfaces
303
0.E
.....''"'~
~0./. I-,-,4
..-" //
Ts = 1 0 0 0 K
d m
""
.
m<0.?: O
"
.~176176/ f
cY cl
c~0.2 H
.j/"
~Ts=760 K
do.1 <
Z
0.
I
0
100
I
I
200
300
/.00
Ekin (meV)
Fig. 6. Influence of the substrate temperature Ts on the total direct inelastic probability.
2.s
-'~
'~X
.-':"
>-
~.~ 2.0
~ ~
Ts-760
__..1
K
o 1.5 rr 13._ Z
o 1.0 0.5
,
Ts = 500K !
0
100
.I
|
!
200
300
1 I
400
Ekin (meV) Fig. 7. Influence of the substrate temperature Ts on the total vibrational excitation probability.
304
N. Cris~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
1000
760
I
I
600
%
500
I
/,00
I
I
-6 {...) X
1::::
-7
" ~ "
--.. . , -. - . . / Eki n = 270
-8
c"
E
meV
-~,.
-10 I
1.0
I
I
1./-,
I
I
1.8
I
~I
I'~'~
'*" TM
2.2 103/%
Fig. 8. Arrherfius plot of the total vibrational excitation probability for two different perpendicular kinetic energies.
increasing kinetic energy. We do not investigate the trapping-desorption channel, because it is obvious from the experimentally observed angular dependence that vibrational excitation occurs as a direct scattering process [3]. The branch ratios are a function of the surface temperature (eqs. (30) and (31)). This parameter was varied, resulting in figs. 6 and 7. The effect of Ts on the total vibrational excitation scattering ratio is clearly visible in fig. 8 where an Arrhenius plot of the total vibrational excitation probability versus T~-1 is displayed for different incoming translational energies. From the slope of the resulting curves the activation energy for the process is obtained. The resulting value of 260 meV corresponds to the vibrational excitation energy of the model. This result is to be expected because the excitation energy has to come from excited e - h pairs which are distributed according to a Boltzmann distribution. The same qualitative behaviour was observed experimentally [3]. The deviations from a straight line in the experimental Arrhenius plot cannot be explained within the context of our model. Because of conservation of parallel momentum in our model, the change of translational energy is conveniently displayed as angular distribution of the scattered molecules. In fig. 9 we compare the different results as a function of surface temperature for an incident angle of 45 ~ , i.e., the incident gas molecule has a parallel velocity component which is equal to the perpendicular one. Because of assumed translational invariance, the parallel component does not change during the scattering process and there is a one-to-one correspon-
N. Crisfi et a L / Inelastic scattering and vibrational excitation at metal surfaces
305
s 1000 K
M
.-~'_. c
",~(T s . , =400 K
\ \ >." t-U3. Z W k--. Z
\
\ -
\ "9...
I
'
20
I
\ ~
I
30 40 SCATTERING
50 ANGLE
6O
Fig. 9. Angular distribution of the molecules scattered back in the vibrational ground state for different substrate temperatures. The incoming perpendicular kinetic energy is Evan = 35 meV; the angle of incidence is 45 o. The elastic line has been omitted from the plot.
dence between the exit angle of the scattered molecule and its final perpendicular kinetic energy. As demonstrated in fig. 2, the elastic probability is large for our parameterization. Experimentally no elastic scattering is observed [3,4]. The elastic line has been omitted in all figures displaying angular distributions. The theoretically predicted angular width is much too small especially at higher Ekin, as shown in fig. 10, where the angular distributions are plotted for different incoming
.~.
Eki n = 180 meW-.-. ~
c ::D
I I
I
ri.
I
0
I I
~--" F--
I
I
m-t
'
Ekin = 35 meV
I
20
'
I
~
'
l
30 /+0 SCATTERING
'
I
50 ANGLE
'
60
Fig. 10. Variation of the angular distribution with incoming perpendicular kinetic energy (T~ = 760 K). The areas under the two curves are normalized to the same value. The elastic line has been omitted.
306
N. Cris& et al. / Inelastic scattering and vibrational excitation at metal surfaces
c-
..d
/
\
>-. I.--
tO Z" Iii
FZ-
~J
I
20
'
I
~
I
30 40 SCATTERING
I
50 ANGLE
'
60
Fig. 11. Angular distribution of the molecules scattered back in the vibrational ground state (full line) and in the first excited state (dashed line). (Eki ~ = 35 meV; Ts = 760 K). The areas under the two curves are normalized to the same value. The elastic line has been omitted.
translational energies. This might be due to neglected channels (phonons and rotations). Also transfer of parallel momentum and microscopic roughness will contribute to the broader experimental distribution [4]. Finally, in fig. 11 angular distributions for molecules scattered back in the vibrational groundand first excited-state, respectively, are compared. They look quite similar, in agreement with the experimental situation [3].
5. Discussion
The total vibrational excitation is found to be proportional to the Boltzmann factor exp(--~STR/kBT~). As mentioned before, this is to be expected, though it is not trivial from a theoretical point of view (see eqs. (30)-(34) for the branch ratios), since not all the channels contain this temperature factor. Due to this, only quantities arising from recombining e - h present a direct dependence on T~. The situation is obvious from figs. 6 and 7. Considering the total direct inelastic channel, we find that a considerable part of it consists of the dissipation of translational energy of the adparticle into e - h pairs, a process occurring also for T~ = 0. Consequently there is only a small variation of the probability with the substrate temperature. A different picture is obtained considering the total vibrational excitation, a process which is possible only for T~ > 0, and therefore very sensible to variations in the substrate temperature, as confirmed in fig. 8.
N. Cris~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
307
8
=3.5 eV • nd o n.13_
"~ , . . . . . . . . .
z c)
D= 0.350 eV
I---4
I,."-I
L) X UJ
........ C 0
I
50
I
'~D I
100 150 Eki n (meV)
= 0.225 eV I
200
250
Fig. 12. Dependence of the kinetic energy trend of the total vibrational excitation probability on the coupling strength D (Ts = 760 K). Our results were obtained using a coupling strength D of 225 meV. This implies vibrational lifetimes (~-) for the adsorbed molecule in the order of 2 • 10 -1~ s which disagree with both theoretical [9,14,24] and experimental [25] results. In order to match such results, one has to increase the coupling strength; reasonable values for 9 are obtained with values of D in the range 1-5 eV [14]. Larger coupling leads also, for small kinetic energies, to larger total vibrational excitation probabilities. This is demonstrated in fig. 12, which, however, reveals also that the trend with Ekin is reversed for D larger than 700 meV. In order to trace the physical origin of this behaviour, the kinetic energy dependence of the vibrational excitation transition rates (Ro= 1,--o=0) is plotted in fig. 13. Here we have normalized the rates, corresponding to the different couplings, in such a way that they are equal (1 arb. units) for Ekin = 2 meV. For small D the rates increase with Eki. because, for larger velocities, the gas molecule penetrates deeper into the surface, then experiencing a stronger coupling. However, as demonstrated by Cris& et al. [14], for very large coupling the inelasticity tends to zero again because the adsorbate level is far removed from the band (surface molecule limit). This implies that for larger values of D a deeper penetration into the surface does not increase the inelasticity. Hence, with increasing Ekm the vibrational excitation transition rate saturate~ at a certain value. For the branch ratios this has the following consequences: for small coupling the denominator of the branch ratios is dominated by the elastic rate which shows a square root behaviour with Ekin.
308
N. Cris& et al.
/ Inelastic scattering and oibrational excitation at metal surfaces
'~10.0
D o21:ev,, /
t'::3
..d
f
b 7.5 [J.J I-n-
/
D:0.350eV
5.0
z o I-
~
..,:'4/
2.5
I,,--I
...................
,.
c)
D=3.5 eV
x 1.0 w
I
0
D.-.2..22.e.V.. D : 2.25 eV ................. ..........
so
!
loo
250
Ekin (meV) Fig. 13. The same as fig. 12 for the transition rate for vibrational excitation. This variation is much weaker than that of R o = 1 ~ v=o and hence the trend of the rate is reflected in the branch ratio, except for very small E L , where the singularity of the square root reverses the trend. For large D the direct elastic rate, showing a similar trend to the vibrational one, becomes dominating in the denominator. Therefore Rv=t,__v= 0 tends to vary slightly with Ekin. At small velocities the rather complicated interplay of all rates determines the energy variation. Concerning the sticking coefficient (s) we may observe that this quantity decreases very rapidly with increasing E L . A variation in the substrate temperature leads to negligible changes in the form as well as in the values of s, which is physically reasonable, because sticking means energy transfer from the gas particle to the solid. At some substrate temperature T~ > 0 vibrationally assisted sticking becomes possible. This contribution is, however, negligible. The effect on the sticking coefficient of increasing the coupling strength D is essentially that of increasing the value of s without changing the trend with Ekin. Two papers on vibrational excitation in gas-surface collisions have been published [7,8], which seem to reproduce the experimental findings. We shall compare our results directly to those of Newns, since our physical picture is closer to his. Newns developed a theory which was also based on a BO-CM-potential energy curve and where the dominant effects occur close to the surface. The most striking difference to our treatment is that he applies a so-called
N. Crisgz et al. / Inelastic scattering and vibrational excitation at metal surfaces
309
Table 1 Parameters used in calculating vibrational excitation probabilities and derived quantities
Adpaxticle Potential depth Morse steepness parameter Adorbital energy Coupling strength Stretch vibration Decay parameters for coupling strength Electron-boson couphng factor Vibrational lifetime of trapped particle
Present work
Ref. [7]
CO U0:110 meV t : 1.0 au a) EA~ 4.0 eV D: 225 meV toS'ra: 260 meV (y - h): 1.5 au 398 meV c) -r: 2 • 10 - 10 s
NO D: 300 meV a: 0.5 au co: 3.1 eV Ao: 132 meV too: 230 meV 2a: 1 au b) h: 324 meV 2.4 • 10- lo s
a) Weakened by vdW-contribution. b) Decrease of c A a x -a. c) At classical turning point for Eki n = 0.
trajectory approximation, i.e., the motion of the adparticle core occurs on a predescribed trajectory. A disturbing aspect is that in such a picture the total energy of the system is not conserved. We used a parametrization which is quite similar to that of Newns. This holds in particular for the electronic adparticle level E ~ at the minimum of the potential, for the coupling strength D (called A 0 by Newns) at this position and for the electron-boson coupling factor, which is obtained by linearising our exponential coupling at the classical turning point for Ekm = 0. The complete comparison of parameters is given in table 1. We can say that the two different approaches describe the same physical picture. Our vibrational excitation probability is very sensible to the coupling strength. Using the trajectory approximation of Newns one finds a qualitatively similar behaviour of the kinetic energy trend as a function of D, i.e. strong increase with Eki, for small coupling, and small variations or even decrease for large coupling. This is not surprising because the same model for the electronic interaction is used. Newns was forced to use a nearly unphysically small coupling (A 0 = 132 meV) in order to reproduce the experimentally established trend. In our calculation the situation is slightly better (D being nearly twice as large as in Newns' work), but we are still far away from the values expected from both, vibrational lifetimes and chemisorption theory. The electronic Hamiltonian 'eq. (4) might be too unrealistic, because it suggests a never ending exponential increase of the hopping matrix elements (leading finally to vanishing coupling in the extreme surface molecule limit). This appears in fact to be the origin for the necessity of choosing small D-values. In reality electronic interaction will tend t~ saturate, even if the core repulsion continues to increase.
310
N. Crisd et al. / Inelastic scattering and vibrational excitation at metal surfaces
The vibrational excitation probabilities might be brought to the experimentally observed order of magnitude by taking into account several adorbitals and spin degeneracy (Newns multiplied by a factor 4 to account for degeneracy).The trend with kinetic energy might also be influenced by channels which are not included in the present work: rotations, phonons, mechanically mediated translational-rotational-vibrational coupling. Comparing the observed angular distributions with the calculated ones suggests indeed that phonons should be involved in some way, because it is well known that even the most elementary theories of gas-phonon interaction give reasonable angular distributions.
Acknowledgments The authors are grateful for the financial support by the Sonderforschungsbereich 128, Deutsche Forschungsgemeinschaft. G.D. acknowledges the support as a Heisenberg fellow of the Deutsche Forschungsgemeinschaft.
References [1] [2] [3] [4] [5] [6]
B. Kasemo, E. T6rnqvist, J.K. NSrskov and B.I. Lundqvist, Surface Sci. 89 (1979) 554. A. Amirav and M.J. Cardillo, Phys. Rev. Letters 57 (1986) 2299. C.T. Rettner, F. Fabre, J. Kimman and D.J. Auerbach, Phys. Rev. Letters 55 (1985) 1904. J. Misewich, H. Zacharias and M.M.T. Loy, Phys. Rev. Letters 55 (1985) 1919. J. Misewich and M.M.T. Loy, J. Chem. Phys. 84 (1986) 1939. J. Misewich, C.N. Plum, G. Blyholder, P.L. Houston and R.P. Merril, J. Chem. Phys. 78 (1983) 4245: A.H. Zacharias, M.M.T. Loy and P.A. Roland, Phys. Rev. Letters 49 (1982) 1790. [7] D.M. Newns, Surface Sci. 171 (1986) 600. [8] J.W. Gadzuk and S. HoUoway, Phys. Rev. B33 (1986) 4298. [9] B.N.J. Persson and M. Persson, Solid State Commun. 36 (1980) 175. [10] W. Domke and L.S. Cederbaum, Phys. Rev. B16 (1977) 1465. [11] J.W. Gadzuk, J. Chem. Phys. 81 (1984) 2828. [12] D.M. Newns, Phys. Rev. 178 (1969) 1123. [13] P.A. Wolff, Phys. Rev. 124 (1961) 1030. [14] N. Crish, G. Doyen and F. von Trentini, Surface Sci. 162 (1985) 120. [15] G. Doyen, in: Topics in Current Physics, Vol. 43 (Springer, Heidelberg, in press) ch. 8, p. 301. [16] Z. Kirson, R.B. Gerber, A. Nitzan and M.A. Rather, Surface Sci. 137 (1984) 527. [17] R. Schinke and R.B. Gerber, J. Chem. Phys. 82 (1985) 1567; R.B. Gerber, A.T. Yinnon, Y. Shimoni and D.J. Kouri, J. Chem. Phys. 73 (1980) 4397. [18] D.C. Clary, J. Chem. Phys. 78 (1983) 4916; Mol. Phys. 43 (1981) 469; J. Chem. Phys. 75 (1981) 2899. [19] J. Stoer, Einftihrung in die Numerische Mathematik I, 4th ed. (Springer, Berlin, 1983) p. 81. [20] N. Cris~ and G. Doyen, in preparation. [21] G. Doyen, Phys. Rev. B22 (1980) 497; G.P. Brivio and T.B. Grimley, Surface Sci. 161 (1985) L573.
N. Cris?, et al. / Inelastic scattering and vibrational excitation at metal surfaces
311
[22] H.H. Landolt and R. BSrnstein, Zahlenwerte und Funktionen, Vol. 1, Part 3: Molekeln II (Springer, Berlin, 1951) p. 509. [23] H. Hoinkes, Rev. Mod. Phys. 52 (1980) 933. [24] T.T. Rantala and A. Rosen, Phys. Rev. B34 (1986) 837. [25] R. Ryberg, Surface Sci. 114 (1982) 627; Phys. Rev. B32 (1985) 2671.
289
NON-ADIABATIC INELASTIC SCATrERING AND INTRA-MOLECULAR VIBRATIONAL EXCITATION AT METAL SURFACES N. CRISA, G. DOYEN
a n d R. H U B N E R
Institut far Physikalische Chemie, Universitiit Miinehen, Theresienstrasse 37, D-8000 Miinchen 2, Fed. Rep. of Germany Received 10 December 1986; accepted for publication 13 May 1987
Considering as energy transfer mechanism the formation of electron-hole pairs, the inelastic scattering of a diatomic molecule from a metal surface is investigated on a purely quantum mechanical basis. The coupling of an adsorbate level to the surface is treated in an Anderson-Newns-like model which neglects correlation and spin effects. The centre-of-mass (CM) motion and the stretch motion are coupled via the interaction with the metal electrons. The model does not include a mechanical translational-vibrational coupling for a rigid surface. The stretch frequency of the diatomic molecule is assumed to be independent of the molecule-surface separation. The vibrating diatomic molecule approaches the surface in an upright orientation. For the CM-potential energy curve a More-type shape plus van der Waals tail was assumed. The Born-Oppenheimer core wave functions are obtained using a variational spline interpolation (VSI). The non-adiabatic transition matrix elements are evaluated in an improved quantum sudden approximation which incorporates many-particle aspects. Sticking probabilities, vibrational excitation probabilities and angular distributions for the inelastically scattered molecule are given. The dependence of the results on the parameters of the model are discussed. Agreement with the experimentally observed trends for NO scattered from a silver surface can only be obtained for weak electronic coupling. Considering only a single adorbital and neglecting spin degeneracy vibrational excitation probabilities of the order of 10 -3 are predicted for substrate temperatures between 100 and 1000 K. This is nearly an order of magnitude smaller than found in experiment, but including spin effects and several adorbitals on the gas molecule might lead to larger probabilities.
1. Introduction The mechanism of energy transfer between a surface and an approaching m o l e c u l e is a still u n s o l v e d p r o b l e m o f c o n s i d e r a b l e i n t e r e s t . A l t h o u g h t h e p h o n o n m e c h a n i s m is o f t e n b e l i e v e d to b e o f d o m i n a n t i m p o r t a n c e , t h e existence of non-adiabatic effects has already been demonstrated experimentally [1,2]. Recently energy transfer between intra-molecular vibrations and substrate degrees of freedom have found increased attention both experimentally [3-6]
0 0 3 9 - 6 0 2 8 / 8 7 / $ 0 3 . 5 0 9 E l s e v i e r S c i e n c e P u b l i s h e r s B.V. (North-Holland Physics Publishing Division)
290
N. Crisg~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
and theoretically [7-9]. The non-adiabatic electron-hole (e-h) pair mechanism is generally believed to be important, because the energy exchange is typically five to ten times the phonon Debye energy. The purely mechanical picture, i.e., exchange of energy between the translational, rotational and vibrational degrees at a rigid surface can be excluded, because vibrational excitation is found to depend strongly on the substrate temperature [3]. In this paper we present a theoretical study of a simple Hamiltonian which might be a rough model of a diatomic molecule like CO or NO interacting with a sp-band metal (e.g. Ag or Cu). The novel feature of this investigation is the purely quantum mechanical description which applies also for the core movement of the gas particle. This warrants a correct description even at very low kinetic energies. Section 2 contains the description of the model. The method of solution, which is a special version of a quantum sudden approximation, is outlined in section 3. The numerical results are presented in section 4. Finally we compare our results to experiments and to other theoretical approaches.
2. The model
The model Hamiltonian is very similar to that used in related theoretical studies [7,9,10,11]. It is here divided up into three parts: H = HCM + Hvib + Her,
(1)
HCM = - V 2 x / 2 M + Vstat( x ) .
(2)
HCM is the Hamiltonian for the center-of-mass (CM) motion, x the CM-position and M the mass of the diatomic molecule. The first term of eq. (2) is the CM-ldnetic energy. V~tat is the CM-potential energy curve which is further detailed in section 4. The intra-molecular vibration is described as usual by a harmonic oscillator of frequency %TR: Hvi b = ~asTRata,
(3)
a t and a are boson creation and annihilation operators. The coordinate for the stretch motion will in the following be denoted by y (cf. fig. 1). The electronic Hamiltonian Hel is given by the Anderson-Newns model [12] with, however, intra-adsorbate electron-electron correlation neglected (i.e., by the Wolff model [13]): Hel = E A ( z ) F / A +- s k
k -}- Vint,
(4)
V~n,= 2 [VAk(Z)CtAC k + herm. conj.]. k The fermion operator n a = CtACA describes the molecular orbital which couples most strongly to the metal surface. The one-electron states of the metal are
N, Crisgl et al. / Inelastic scattering and vibrational excitation at metal surfaces
I
y
i
L I
~ ~ ) / / , . s ' - I
z
I I >(
X
291
I L '
~'x\
I I I
I i I I
CM-position
Fig. 1. Illustration of the coordinates used in describing the motion of the gas molecule. referred to by the operators ctk, c k and n k = C~Ck. They have ionization energies Ck. Spin degeneracy is not taken into account in eq. (4). The molecule is supposed to approach the surface in an upright orientation. The molecular orbital energy E A depends on the distance z of the nearest atom from the surface. The meaning of the three coordinates x, y and z is illustrated in fig. 1. In the following the abbreviation R = (x, y) will be used as well. VAk are the quantities coupling the molecular orbital to the metal states. As E g and Vgk depend on the molecule-surface separation z, these quantities are not c-numbers but operators. This fact prevents an exact solution of the Hamiltonian.
3. M e t h o d of solution
3.1. The quantum sudden approximation
Following our pre~ious approach [14], we do not formulate the scattering problem for {he gas particle, because this leads to fundamental problems with formal scattering theory [15]. The alternative is to consider a scattering metal electron [14,16]. The gas particle is then assumed to provide a kind of moving
292
N. Cris& et al. / Inelastic scattering and vibrational excitation at metal surfaces
target or scattering center for the electrons. The fact that the molecule hits and leaves the surface will be taken into account by assuming that the (moving) scattering center is present only with a certain probability. One then requires "branch ratios" in order to derive transition probabilities for the gas particle movement. This aspect is outlined and discussed in section 3.3. After adopting tiffs philosophy one still finds that the scattering problem is not solvable exactly. The difficulty is the motion of the adparticle. We can, however, evaluate the transition matrix elements in perturbation theory by using electronic wave functions calculated for a fixed molecular position. The total many-body wave "function is then written in Born-Oppenheimer-form. The prescription reads as follows [14]: (i) Calculate electronic wave functions ~bj(R, ri) for each value of x and y, i.e., with R as parameter. (ii) Construct the Born-Oppenheimer (BO) many-particle wave function g'(x, y, z ) = q~CM(x)g2C~
det[qJj(R, ri) ] = x , ( R ) det[ffj(R, ri) ] .
is a product wave function with the first factor q,CM(x) describing the CM-motion and the other 12C~ the stretch vibration. (iii) Calculate transition matrix elements in perturbation theory using BO-wave functions. (iv) Calculate the S-matrix according to eq. (10) of ref. [14].
x,(R)
S/,, k, = 6ktS,, -- 27tiT1,. k~8(cl + % -- Ck -- % ) . Here the many-particle density of states enters. By taking the absolute square of the S-matrix elements St~' k~ one obtains the probability that a metal electron will scatter from the state l to the state k while the gas atom makes a transition from the core wave function X~ to the core wave function X,. This perturbative Born-Oppenheimer formulation has the physical meaning of a quantum sudden approximation for metal electron scattering. (It is called quantum here, because the expression "sudden approximation" is commonly used, if the semi-clasically calculated elastic S-matrix element for the scattering degrees of freedom is used for taking total scattering matrix elements [17]. Tiffs is, however, different e.g. in the BCC-IOS approximation [18].) The criterion for the validity of the quantum sudden approximation is that the energy transfer is small compared to the kinetic energy of the metal electron. Tiffs condition is always fulfilled for the present investigation. The scattering matrix element for an electron to scatter from k to l (k 4: l) while the molecule is at a position R = (x, y) is: -2~ri(llV.,.,(R)lk
+ ) 8 ( , , - ~k) = --2r
I VintGVint ] k ) 8 ( ', - "k)
= --2r = -2~riC~Ak)(R)VAt(R)6(~-
'k)
Ok),
N. Crist) et al. / Inelastic scattering and vibrational excitation at metal surfaces
293
where C(Ak)(R) is the coefficient of the adsorbate admixture in the state I k), evaluated utilizing the Lippmann-Schwinger equation (see eq. (6) of ref. [14]) and GA the Green function for an electron in the adorbital. In the sudden approximation one now integrates over R,. while weighting with the corresponding gas molecule wave functions. The result is then identical to that obtained in ref. [14]. The problem encountered is that whenever X~ and X v are unbound states, the electronic probabilities violate the unitarity condition strongly for physically reasonable values of VAk. This is an indication for the breakdown of perturbation theory. The reason for this is that in calculating the electronic metal wave functions the gas molecule is present as a scattering center during the whole time the electron moves through the solid. This results in a too strong mixing of the wave functions. Obviously, if the molecule is perturbing the metal only for a short period of time, mixing with the adorbital has to decrease. The mixing coefficient C~Ak) is then too large, because the self-energy, q g = A + i F , is not calculated correctly. It is too small and therefore GA is too large yielding a too large coefficient C~Ak). A solution of this electronic non-unitarity problem can be found as follows: If the scattered metal electron moves away from the surface, the gas particle will move away as well. This effect on GA was not considered in [14], because the molecule was bound to the surface. In order to correct for this error, it is convenient to rewrite the Hamiltonian (1) in the following form: H=Ho+
Y'~HB, #
(5)
Ho = E I kr>'k~( kr l,
(6)
k, p
I/3>ca
Ha =
(7)
(8)
k , t,
k labels the metal electron and r the gas particle core wave function describing the stretch vibration and the CM-motion. Each I/3> is a basis wave function for the situation where the electron is in the adorbital and the molecule is in some eigenfunction of HCM + H,~b. These eigenstates labelled by /3 need not concide with those labelled by r, as long as they form a complete set. The identification HCM + H~b + EA(R)nA + ~..'knk = Ho + ~ {/3)',(/3 J, k #
W= E [VAk(R)CtACk + herm. conj.] = E W~, k
(9) (10)
#
where
far x r ( R ) V A k ( R ) x B ( R )
=
VkrB
(11)
294
N. Cris~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
establishes the equivalence of the two forms of the Hamiltonian. The form eqs. (5) to (8) of the Hamiltonian describes a delocalized moving electron connected with a moving gas particle in an eigenstate of the core potential. Choose now one core basis function (characterizing the molecule in the state I fl)) as extremely localized around the position R. This means that non-adiabatic interaction occurs only, if the gas particle is at position R. The matrix elements of the interaction operator W are then:
(kv I WIB) = Vk.a = (v I fl)Vkk(R#).
(12)
For the Wolff-type Hamiltonian H 0 + H# the required many-body Green function is evaluated readily: (fl l a l f l )
(13)
= ( g - ,p - q/j)-1,
with
IVk., 12
q a ( e ) -- E EZ-~k-e, k , I+
E
IVA~ 12 E - - , k _ , I(Pl/3)12.
(14)
k, ,
c k is the energy of the metal electron (near the Fermi level) and c, is the energy of the molecule. At each value E of the total energy more than one k-state contributes, because higher electronic energies c k can be compensated by lower gas molecule energies c, and vice versa. The number of k-states contributing is Y.,1 summed under the condition that there exists a state [ k ) with c, = E - c k. The electronic matrix element VAk is, however, multiplied by the overlap factor ( ~ l f l ) so that we can define an effective degeneracy by
D/j(E)= Y'.I(~,Ifl)Iz (with,k+,.--e).
(15)
At substrate temperature T~ = 0 the summation ranges from c, = 0 (deepest bound state of the CM-motion)up to c, = E - E v (elastic scattering),where E F is the Fermi energy. The self-energy q/~ can now be rewritten in the following way:
k E-c k-c,
= ~, I ( v l f l ) 1 2 q A ( E , c , ) ,
(16)
where the summation over ~, is over the specified range. Now the molecular energy ~, is in our case small (up to - 0.5 eV) whereas c~ is of the order of the Fermi energy, i.e., - 10 eV. Therefore qA(E, c,) is practically independent of c, and we can write:
q/~= Da( E)qA( E ),
(17)
where qA is now the standard electronic self-energy of the Wolff-AndersonNewns-type Hamiltonians. (fl I G[ fl) has been derived by considering H 0 + H a
3t. Cris3 et aL / Inelastic scattering and vibrational excitation at metal surfaces
295
separately for each I t ) , i.e., separately for each gas particle position R. Scattering can occur at any gas particle position R. In the case I I/~) ~ I kv) the T-matrix element for H 0 + H a is then: (I/~IW~ Ikv+ ) = ( l l ~ l W I B ) ( f l l a l B ) ( B I W I k v )
= ~A(R13)(I~Ifl)(fllGIfl)(fllv)VAk(R~).
(18)
This approximation means physically that the gas particle does not move while the metal electron scatters. The total contribution to the scattering due to all gas molecule positions R , is
(ll~lWIkv+ ) = Y'.(II~IW~ Ikv+ ). 13
(19)
Using
(20)
E = fdR x~(R)x,(R) #
and the notation
GA(R#) = ( f l l G l f l ) ,
(21)
one obtains for the total transition matrix element:
x~(R)(I~IB)~A(R)GA(R)VAk(R)(flIv)x~(R).
(22) This result is our quantum sudden approximation, where many-particle effects have been incorporated in an improved Green function GAIf the imaginary part of the self-energy q~ is introduced
P ( R , , E) = ~rDa(E)~,lVAk 123(E - ,k) = D a ( E ) F ( R , E ) ,
(23)
k
the S-matrix elements can be written in the form
s,,,, ~ = 8,~,~,,~- 2if dR X,,(R)~(R, E)G,(R, E)x~(R).
(24)
This agrees with the expression obtained in ref. [14], except that the Green function GA and its self-energy have to be replaced by the normalized quantities indicated by the bar. Compared to the electronic Anderson-Newns-like Hamiltonian, the selfenergy qA has increased by the density factor Da. If we use this form of the Green function to evaluate the coefficient C~Ak), we go beyond first order perturbation theory. Obviously the ~coefficient becomes smaller and unitary is obeyed approximately.
296
N. CrisgJ et al. / Inelastic scattering and vibrational excitation at metal surfaces
In order to get an estimate of the density factor Do(E ) one could consider scattering of a structureless particle from an infinitely hard wall: 2 f0~d Dry(E) = -~
K sin2(Kx) = K 7r
1 sin(2Kx), 2 ~r----x
(25)
with x the gas atom distance and K = (2ME) -1/2 the momentum. The second term is oscillatory and always smaller than the first for x > 0. In our numerical calculations the employed density factor accounts also for the number of bound states.
3.2. The BO-core wave functions The Schrbdinger equation for the CM-motion reads: -(1/2M)
dZq~(x)/dx 2 + V~tat(X)~(x ) = HCMq)(X ) = Eq)(x).
(26)
We want to approximate the function q~(x) in the interval 0 < x < L by a cubic spline interpolation over a set of n discrete points xj. Because q~(x) is a polynomial of third degree over each subinterval (x j, x j+l), the condition that the second derivatives are continuous at all xj leads to a system of linear equations connecting q,(xj) with its second derivatives at the end points xj of the sub-intervals. At the end points x = 0 and x = L the second derivatives are set to zero. At x = 0, which is inside the repulsive wall, this is physically reasonable. At x = L, which is in the flat part of the potential, the wave function q)(x) resembles a sine. Therefore the condition q , " ( L ) = 0 requires ~ ( L ) = 0 as well. This serves as a criterion of selecting a finite set of representative wave functions out of a continuum (see below). In matrix notation the resulting system of linear equations has the form [19]: Aq~" = Bq~,
(27)
where q~" and q~ denote the vector containing the second derivatives and the values of the function q)(x) at the points xj respectively. If the values of q~ at the end points xj of the subintervals would be known, the second derivatives (which uniquely determine the spline) could be evaluated from eq. (27). In our case the q)(x~) are unknown, but we have the SchrOdinger equation relating ~,"(xg) to the values q,(xj). In matrix notation it reads: - (1/2M)q~" + Vq) = Eq~.
(28)
Solving eq. (27) for q)" and inserting in eq. (28) yields:
[-(a/2M)(A-~)
+ V]q~ = Eq~.
(29)
This is a matrix eigenvalue problem. Because the Schrbdinger equation is equivalent to a variational principle, solving eq. (29) means choosing among all possible spline interpolations that one which has extremal properties with
N. Cris& et al. / Inelastic scattering and vibrational excitation at metal surfaces
297
respect to the energy functional (~1HcM [ if). We therefore call this method "variational spline interpolation" (VSI). The dimension of the matrices is equal to the number n of discrete points xj. One of the advantages of such a scheme is that no integrations have to be performed. The number of wave functions obtained in this way is equal to n. Obviously the continuum of unbound core states is discretized. The wave function of highest energy is that one which in the flat part of the potential completes one oscillation over three adjacent points. Hence the energy range covered depends on the length L and on the number of points in between. In our calculation L was 4.5 bohr with 200 points in between. The described method calculates the shape of the wave function accurately over the specified interval of length L. The boundary condition (second derivative equals zero at the point L ) selects out of the continuum of unbound states a finite number of wave functions. One obtains one representative gas core state within the energy interval defined by the two adjacent continuum states. Tests for exactly solvable potentials (square well, Morse) demonstrated that the levels of up to twenty bound states are given with three significant digits and the thirty lowest continuum states have the correct shape in the surface region [20]. The wave functions have to be normalized after diagonalization of eq. (29). The unbound states were normalized over a length of 2500 bohr assuming that the potential stays flat for distances larger than 3 bohr from the surface. 3.3. Transition rates and branch ratios The probability Pl~-kv for a transition from the state I k v ) to a state I/~) is obtained by taking the absolute square of the matrix element eq. (24). The transition rate R.~_ v for gas molecule scattering from core state [p) to core state 1/~) is obtained by multiplying by the number ~rk of metal electrons with excitation power arriving per unit time and summming over all initial and final metal electron states: R~ ~ = E N~e,~._ ~ .
(30)
k,l
The number of scattering metal electrons per unit time can be calculated as follows: Nk = OUkBk(~)gk"
(31)
Here Bk(T~) is the probability for a metal electron to be in the state I k ) at substrate temperaturo T~ and having the possibility of making an energy conserving transition to another empty metal state. For energy transfer to the solid this probability is unity. Foy energy transfer to the gas particle this denotes the probability for the existence of an appropriate electron-hole pair.
298
N. Cris?l et al. / Inelastic scattering and vibrational excitation at metal surfaces
A Boltzmann distribution is used in the present calculation. Note that energy conservation is already built into the probabilities Pt,,- k,- Uk is the velocity of a metal electron in the state I k). gk is the number of states of kind I k ) per unit volume, o is the surface area which has to be hit by a metal electron in order to interact with the gas molecule near the surface, o was chosen to correspond to the area of a surface unit cell. Scattering probabilities for the gas particle motion are now defined by taking the ratio of the transition rate for the considered channel (e.g. total direct inelastic scattering) over the total rate, i.e., the sum over all partial rates for the separate channels. The various considered channels are detailed in section 4. As an example let us consider a separation into three different channels: total direct inelastic scattering, total sticking and elastic scattering. Recall that elastic scattering is a singular event and elastic scattering from a moving surface is a purely quantum mechanical effect. Because of the singular role played by quantum mechanical elastic scattering, we do not calculate the elastic rate from eq. (30), but connect it with the flux of the incoming gas particles:
Re, = [ K/( MbI)] Pk,~-k,.
(32)
K is the CM-momentum of the gas particle, M its mass and I the normalization length for the core wave functions. I was chosen as 2500 bohr and does not agree with the length L in the VSI. b is the width of the surface region which extends from the adsorption minimum to the classical turning point. This procedure is in the spirit of perturbation treatments and also conforms with other prescriptions used in the literature [21]. It should, however, be remembered that there exists no general proof for the validity of this procedure and that it appears questionable indeed. The gas particle scattering probabilities are: H~,_,
ErRr~
.
(33)
For the mentioned example the probability for total direct inelastic scattering is then calculated from: R inel /-Iigel = R e I .j_ R i n d + Rstick .
(34)
4. Results and comparison with experiments Numerical calculations have been performed for a heteronuclear diatomic molecule of atomic masses corresponding to 12 and 16 proton masses, respectively. These are the values for a CO-molecule, but the mass of a
N. Crisd et al. / Inelastic scattering and vibrational excitation at metal surfaces
299
NO-molecule is nearly the same (30 instead of 28). Such a difference is of no significance for a qualitative model investigation of the type presented here. The potential energy curve for the CM-motion (coordinate x) is given by
Vstat(X ) = Ul(X ) -4- U2(x),
(35)
where
U l ( x ) = Uo(e -2ax - 2 e-aX), U2(x) = - C3 e-'~/(~-~)(x - 4)-3
(36)
Ua(x) is a Morse potential describing the short range interaction with the surface. U0 is its well depth which in our case is 110 meV. /3 is the steepness parameter which, together with U0, determines the vibrational frequency ~cM of the CM-motion, and the repulsive part of the potential./3 was chosen to be 1 au, corresponding to O~cM= 10 meV. U2(x ) is the van der Waals (vdW)-like long range tail arising from fluctuating dipole-dipole interactions. The exponential factor is introduced to remove the singularity of the van der Waals potential at x = 4. C3 is a parameter describing the strength of the long range interaction (C 3 = 10.8 meV nm3). With the known polarizability of the CO-molecule (2.6 • 10-3 nm 3, [22]) this is a value which is in the range of typical values for physisorption on metal surfaces [23]. 7/ is the cut-off parameter for the vdW potential: 7/= 3.5 au. indicates the position of the singularity for the vdW potential. In our case it is = 2.67 au. Concerning the stretch motion (coordinate y) we used a harmonic potential with a characteristic frequency ~STR, independent on the CM-motion. This is not quite realistic, because o~STR is known to change upon adsorption. The consequence of our approximation is that there is no coupling between vibrational and translational motion for a rigid surface. In our model the coupling arises only via interaction with e - h pairs. Hence, vibrational excitation probabilities might be underestimated. The chosen value was Wsa-g= 260 meV. The distance dependence of the adorbital energy Eh(Z ) and the hopping parameters Va,(Z ) (cf. the electronic Hamiltonian eq. (4)) were chosen as follows: EA(Z ) = E ~ eXZ; VAk(Z) = V~ e -vz/2.
(37)
EA(Z ) is kept constant for z > 2 bohr. V~ is related to the imaginary part F of the electronic,self-energy qA which is assumed to have a semi-elliptical form:
r(,k) = (O2//,tro~) [1 --,(~k/20~)2] 1/2
(38)
The following values have been chosen: Metal bandwidth 48 = 20 eV, modelling roughly the sp-band of the noble metals. -
300
N. Cris~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
- Adorbital energy E ~ = + 4 eV with respect to the Fermi level describing in the case of CO and N O the unoccupied 2~r* orbital in interaction with the metal states. - Decay parameters for the coupling strength ('y - X) = 1.5 au. (The results depend only on the difference "y - X. (cf. ref. [14])). - Coupling strength D = 225 meV. We then have an incoming diatomic molecule (initially in the vibrational ground state v i = 0 ) impinging with a translational energy Eikin on a metal surface which has substrate temperature T~. In such a case the following outgoing channels are available: A: VibrationaUy elastic scattering (v i = 0, of = 0): The vibrational state of the molecule remains unperturbed. It may be subdivided into: A1: Elastic ( E f = Evan). i . also the translational state does not vary. A2: Direct inelastic: a certain amount of translational energy might be dissipated (energy loss; 0 < Ekinf < Ekin)i or, at T~ > 0, the adparticle might gain i translational energy from recombining e - h pairs (energy gain; E f > Ekin)A3: Sticking ( E f < 0): the energy loss is so large, that the adparticle gets trapped at the surface. B: Vibrationally inelastic scattering ( v i = 0 , o f = 1): during the interaction with the surface the molecule gains enough energy to jump into its first vibrationally excited state. The subdivision is as follows: BI: Vibrationally direct inelastic: includes all possible translational final states with E f > 0. B2: Vibration assisted sticking: the adparticle remains stuck at the surface with of = 1. In principle, further vibrational excitations (vf = 2, 3 . . . . ) as well as the rotational degrees of freedom should be taken into account. However, for numerical reasons we restrict our present investigation to the first vibrationally excited state. We shall hereafter adopt the following terminology for the branch ratios: Elastic: channel A1. Total sticking: channels A3 + B2. It consists of a part (contained in B2) directly related to the surface temperature T~ and another part that is not. Total direct inelastic: channels A2 + B1, i.e. no distinction between the different final vibrational states. Total vibrational excitation: channel B1. Figs. 2-5 display results for substrate temperature Ts = 760 K. In fig. 2 we report the probability (branch ratio) for elastic scattering, as a function of the incoming kinetic energy; in fig. 3 that for total direct inelastic scattering. The behaviour of the total vibrational excitation is shown in fig. 4. The trend with Ekm of this last quantity agrees qualitatively with experimental observations [3], i.e., there is an increase in the scattering probability with Eki n. On the other hand, as visible in fig. 5, the sticking coefficient decreases rapidly with
N. Cris~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
301
1.0 >I-1"'4
__1 < rn 0 n" 13-
0.8 0.6
o0.~ I.-{.,0 ,< ._1
~0.2 .
I
0
100
/
I
200 300 Eki n (meV)
400
Fig. 2. Elastic scattering probability versus incoming perpendicular kinetic energy (T~ = 760 K).
0 . 6 84 >-
~0.4 __J m <~ m
o0.3
0.2 < ._.1
toO.1 Z
~
0
!
260
360
Ekin (meV) Fig. 3. Total direct ~elastic probability versus incoming perpendicular kinetic energy ( Ts = 760 K).
302
N. Crisr et al. / Inelastic scattering and vibrational excitation at metal surfaces
~o2.5 X
>-.
~2.0 CI3 ,< 133
on.-l . 5 Z1.0 I--,,-I
0.5 X LLI
O.
100
0
200 3 0 Ekin(meV)
/-,00
Fig. 4. Total vibrational excitation probability (v = 0--* v = 1 ) versus incoming perpendicular kinetic energy (Ts = 760 K).
0.04 >t---
I---4
J 0.03
I,-,,.I
133 r'n 0
o.o2
Z
~0.01 Ch
O.
~
0
10
2O
30
4O
Ekin(meV) Fig. 5. Sticking probability versus incoming perpendicular kinetic energy (Ts = 760 K).
N. Crish et al. / Inelastic scattering and vibrational excitation at metal surfaces
303
0.E
.....''"'~
~0./. I-,-,4
..-" //
Ts = 1 0 0 0 K
d m
""
.
m<0.?: O
"
.~176176/ f
cY cl
c~0.2 H
.j/"
~Ts=760 K
do.1 <
Z
0.
I
0
100
I
I
200
300
/.00
Ekin (meV)
Fig. 6. Influence of the substrate temperature Ts on the total direct inelastic probability.
2.s
-'~
'~X
.-':"
>-
~.~ 2.0
~ ~
Ts-760
__..1
K
o 1.5 rr 13._ Z
o 1.0 0.5
,
Ts = 500K !
0
100
.I
|
!
200
300
1 I
400
Ekin (meV) Fig. 7. Influence of the substrate temperature Ts on the total vibrational excitation probability.
304
N. Cris~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
1000
760
I
I
600
%
500
I
/,00
I
I
-6 {...) X
1::::
-7
" ~ "
--.. . , -. - . . / Eki n = 270
-8
c"
E
meV
-~,.
-10 I
1.0
I
I
1./-,
I
I
1.8
I
~I
I'~'~
'*" TM
2.2 103/%
Fig. 8. Arrherfius plot of the total vibrational excitation probability for two different perpendicular kinetic energies.
increasing kinetic energy. We do not investigate the trapping-desorption channel, because it is obvious from the experimentally observed angular dependence that vibrational excitation occurs as a direct scattering process [3]. The branch ratios are a function of the surface temperature (eqs. (30) and (31)). This parameter was varied, resulting in figs. 6 and 7. The effect of Ts on the total vibrational excitation scattering ratio is clearly visible in fig. 8 where an Arrhenius plot of the total vibrational excitation probability versus T~-1 is displayed for different incoming translational energies. From the slope of the resulting curves the activation energy for the process is obtained. The resulting value of 260 meV corresponds to the vibrational excitation energy of the model. This result is to be expected because the excitation energy has to come from excited e - h pairs which are distributed according to a Boltzmann distribution. The same qualitative behaviour was observed experimentally [3]. The deviations from a straight line in the experimental Arrhenius plot cannot be explained within the context of our model. Because of conservation of parallel momentum in our model, the change of translational energy is conveniently displayed as angular distribution of the scattered molecules. In fig. 9 we compare the different results as a function of surface temperature for an incident angle of 45 ~ , i.e., the incident gas molecule has a parallel velocity component which is equal to the perpendicular one. Because of assumed translational invariance, the parallel component does not change during the scattering process and there is a one-to-one correspon-
N. Crisfi et a L / Inelastic scattering and vibrational excitation at metal surfaces
305
s 1000 K
M
.-~'_. c
",~(T s . , =400 K
\ \ >." t-U3. Z W k--. Z
\
\ -
\ "9...
I
'
20
I
\ ~
I
30 40 SCATTERING
50 ANGLE
6O
Fig. 9. Angular distribution of the molecules scattered back in the vibrational ground state for different substrate temperatures. The incoming perpendicular kinetic energy is Evan = 35 meV; the angle of incidence is 45 o. The elastic line has been omitted from the plot.
dence between the exit angle of the scattered molecule and its final perpendicular kinetic energy. As demonstrated in fig. 2, the elastic probability is large for our parameterization. Experimentally no elastic scattering is observed [3,4]. The elastic line has been omitted in all figures displaying angular distributions. The theoretically predicted angular width is much too small especially at higher Ekin, as shown in fig. 10, where the angular distributions are plotted for different incoming
.~.
Eki n = 180 meW-.-. ~
c ::D
I I
I
ri.
I
0
I I
~--" F--
I
I
m-t
'
Ekin = 35 meV
I
20
'
I
~
'
l
30 /+0 SCATTERING
'
I
50 ANGLE
'
60
Fig. 10. Variation of the angular distribution with incoming perpendicular kinetic energy (T~ = 760 K). The areas under the two curves are normalized to the same value. The elastic line has been omitted.
306
N. Cris& et al. / Inelastic scattering and vibrational excitation at metal surfaces
c-
..d
/
\
>-. I.--
tO Z" Iii
FZ-
~J
I
20
'
I
~
I
30 40 SCATTERING
I
50 ANGLE
'
60
Fig. 11. Angular distribution of the molecules scattered back in the vibrational ground state (full line) and in the first excited state (dashed line). (Eki ~ = 35 meV; Ts = 760 K). The areas under the two curves are normalized to the same value. The elastic line has been omitted.
translational energies. This might be due to neglected channels (phonons and rotations). Also transfer of parallel momentum and microscopic roughness will contribute to the broader experimental distribution [4]. Finally, in fig. 11 angular distributions for molecules scattered back in the vibrational groundand first excited-state, respectively, are compared. They look quite similar, in agreement with the experimental situation [3].
5. Discussion
The total vibrational excitation is found to be proportional to the Boltzmann factor exp(--~STR/kBT~). As mentioned before, this is to be expected, though it is not trivial from a theoretical point of view (see eqs. (30)-(34) for the branch ratios), since not all the channels contain this temperature factor. Due to this, only quantities arising from recombining e - h present a direct dependence on T~. The situation is obvious from figs. 6 and 7. Considering the total direct inelastic channel, we find that a considerable part of it consists of the dissipation of translational energy of the adparticle into e - h pairs, a process occurring also for T~ = 0. Consequently there is only a small variation of the probability with the substrate temperature. A different picture is obtained considering the total vibrational excitation, a process which is possible only for T~ > 0, and therefore very sensible to variations in the substrate temperature, as confirmed in fig. 8.
N. Cris~ et al. / Inelastic scattering and vibrational excitation at metal surfaces
307
8
=3.5 eV • nd o n.13_
"~ , . . . . . . . . .
z c)
D= 0.350 eV
I---4
I,."-I
L) X UJ
........ C 0
I
50
I
'~D I
100 150 Eki n (meV)
= 0.225 eV I
200
250
Fig. 12. Dependence of the kinetic energy trend of the total vibrational excitation probability on the coupling strength D (Ts = 760 K). Our results were obtained using a coupling strength D of 225 meV. This implies vibrational lifetimes (~-) for the adsorbed molecule in the order of 2 • 10 -1~ s which disagree with both theoretical [9,14,24] and experimental [25] results. In order to match such results, one has to increase the coupling strength; reasonable values for 9 are obtained with values of D in the range 1-5 eV [14]. Larger coupling leads also, for small kinetic energies, to larger total vibrational excitation probabilities. This is demonstrated in fig. 12, which, however, reveals also that the trend with Ekin is reversed for D larger than 700 meV. In order to trace the physical origin of this behaviour, the kinetic energy dependence of the vibrational excitation transition rates (Ro= 1,--o=0) is plotted in fig. 13. Here we have normalized the rates, corresponding to the different couplings, in such a way that they are equal (1 arb. units) for Ekin = 2 meV. For small D the rates increase with Eki. because, for larger velocities, the gas molecule penetrates deeper into the surface, then experiencing a stronger coupling. However, as demonstrated by Cris& et al. [14], for very large coupling the inelasticity tends to zero again because the adsorbate level is far removed from the band (surface molecule limit). This implies that for larger values of D a deeper penetration into the surface does not increase the inelasticity. Hence, with increasing Ekm the vibrational excitation transition rate saturate~ at a certain value. For the branch ratios this has the following consequences: for small coupling the denominator of the branch ratios is dominated by the elastic rate which shows a square root behaviour with Ekin.
308
N. Cris& et al.
/ Inelastic scattering and oibrational excitation at metal surfaces
'~10.0
D o21:ev,, /
t'::3
..d
f
b 7.5 [J.J I-n-
/
D:0.350eV
5.0
z o I-
~
..,:'4/
2.5
I,,--I
...................
,.
c)
D=3.5 eV
x 1.0 w
I
0
D.-.2..22.e.V.. D : 2.25 eV ................. ..........
so
!
loo
250
Ekin (meV) Fig. 13. The same as fig. 12 for the transition rate for vibrational excitation. This variation is much weaker than that of R o = 1 ~ v=o and hence the trend of the rate is reflected in the branch ratio, except for very small E L , where the singularity of the square root reverses the trend. For large D the direct elastic rate, showing a similar trend to the vibrational one, becomes dominating in the denominator. Therefore Rv=t,__v= 0 tends to vary slightly with Ekin. At small velocities the rather complicated interplay of all rates determines the energy variation. Concerning the sticking coefficient (s) we may observe that this quantity decreases very rapidly with increasing E L . A variation in the substrate temperature leads to negligible changes in the form as well as in the values of s, which is physically reasonable, because sticking means energy transfer from the gas particle to the solid. At some substrate temperature T~ > 0 vibrationally assisted sticking becomes possible. This contribution is, however, negligible. The effect on the sticking coefficient of increasing the coupling strength D is essentially that of increasing the value of s without changing the trend with Ekin. Two papers on vibrational excitation in gas-surface collisions have been published [7,8], which seem to reproduce the experimental findings. We shall compare our results directly to those of Newns, since our physical picture is closer to his. Newns developed a theory which was also based on a BO-CM-potential energy curve and where the dominant effects occur close to the surface. The most striking difference to our treatment is that he applies a so-called
N. Crisgz et al. / Inelastic scattering and vibrational excitation at metal surfaces
309
Table 1 Parameters used in calculating vibrational excitation probabilities and derived quantities
Adpaxticle Potential depth Morse steepness parameter Adorbital energy Coupling strength Stretch vibration Decay parameters for coupling strength Electron-boson couphng factor Vibrational lifetime of trapped particle
Present work
Ref. [7]
CO U0:110 meV t : 1.0 au a) EA~ 4.0 eV D: 225 meV toS'ra: 260 meV (y - h): 1.5 au 398 meV c) -r: 2 • 10 - 10 s
NO D: 300 meV a: 0.5 au co: 3.1 eV Ao: 132 meV too: 230 meV 2a: 1 au b) h: 324 meV 2.4 • 10- lo s
a) Weakened by vdW-contribution. b) Decrease of c A a x -a. c) At classical turning point for Eki n = 0.
trajectory approximation, i.e., the motion of the adparticle core occurs on a predescribed trajectory. A disturbing aspect is that in such a picture the total energy of the system is not conserved. We used a parametrization which is quite similar to that of Newns. This holds in particular for the electronic adparticle level E ~ at the minimum of the potential, for the coupling strength D (called A 0 by Newns) at this position and for the electron-boson coupling factor, which is obtained by linearising our exponential coupling at the classical turning point for Ekm = 0. The complete comparison of parameters is given in table 1. We can say that the two different approaches describe the same physical picture. Our vibrational excitation probability is very sensible to the coupling strength. Using the trajectory approximation of Newns one finds a qualitatively similar behaviour of the kinetic energy trend as a function of D, i.e. strong increase with Eki, for small coupling, and small variations or even decrease for large coupling. This is not surprising because the same model for the electronic interaction is used. Newns was forced to use a nearly unphysically small coupling (A 0 = 132 meV) in order to reproduce the experimentally established trend. In our calculation the situation is slightly better (D being nearly twice as large as in Newns' work), but we are still far away from the values expected from both, vibrational lifetimes and chemisorption theory. The electronic Hamiltonian 'eq. (4) might be too unrealistic, because it suggests a never ending exponential increase of the hopping matrix elements (leading finally to vanishing coupling in the extreme surface molecule limit). This appears in fact to be the origin for the necessity of choosing small D-values. In reality electronic interaction will tend t~ saturate, even if the core repulsion continues to increase.
310
N. Crisd et al. / Inelastic scattering and vibrational excitation at metal surfaces
The vibrational excitation probabilities might be brought to the experimentally observed order of magnitude by taking into account several adorbitals and spin degeneracy (Newns multiplied by a factor 4 to account for degeneracy).The trend with kinetic energy might also be influenced by channels which are not included in the present work: rotations, phonons, mechanically mediated translational-rotational-vibrational coupling. Comparing the observed angular distributions with the calculated ones suggests indeed that phonons should be involved in some way, because it is well known that even the most elementary theories of gas-phonon interaction give reasonable angular distributions.
Acknowledgments The authors are grateful for the financial support by the Sonderforschungsbereich 128, Deutsche Forschungsgemeinschaft. G.D. acknowledges the support as a Heisenberg fellow of the Deutsche Forschungsgemeinschaft.
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N. Cris?, et al. / Inelastic scattering and vibrational excitation at metal surfaces
311
[22] H.H. Landolt and R. BSrnstein, Zahlenwerte und Funktionen, Vol. 1, Part 3: Molekeln II (Springer, Berlin, 1951) p. 509. [23] H. Hoinkes, Rev. Mod. Phys. 52 (1980) 933. [24] T.T. Rantala and A. Rosen, Phys. Rev. B34 (1986) 837. [25] R. Ryberg, Surface Sci. 114 (1982) 627; Phys. Rev. B32 (1985) 2671.