- Email: [email protected]
desorption
: ,.:.:;.;. :;::y> :‘:_;g:,“:,.,. .. _... .i.
Surface Science 265 North-I
(lW2) Sh-hh
surface
foiland
NO on Pt( 111): direct inelastic
Angular
momentum
cules interacting pr(~xim~tion
distributions
with a Pt(
I I I)
and the phonon
Concerning
rotational
this effect is proposed
scattering
after direct inelastic scattering
surt’ace are calculated.
The rotational
system of the solid is described
n~~gllitu~~e as well as orientational
., .
and alignment
in term& of rot~ition~~liy mediated
“prompt”
degrees of freedom oscillator.
effects of the sticking coefficient
cooling the prompt sticking-theory
_.- :
and sticking/desorption
and the so-called
as a forced
.’ ‘. .. ‘.
scienc6
turns out to be insufficient.
sticking coefficient
of NO-mole-
are treated within a quantum-sudden-apWith
reasonable
are found
parameters
the absolute
to agree with experimenta
A simplified
semiquantitative
data.
description
of
selective adsorption.
1. introduction The interaction of NO-molecules with a Ptil 11) surface has been a subject of experimentaI and theoretical research for many years. Gas particles with well-defined initial orientations or low r~)tational temperatures are analyzed after the scattering event using laser-induced fluorescence and multiphoton ionization spectroscopy. These “state to state”’ experiments serve as a fruitful field for theoretical studies. Furthermore many parameters of this system can be fixed. From thermal desorption data [1,2] one finds a tightly bound state with binding energy = 1.45 eV probably at defect sites whereas molecules on perfect Pt(l1 I)-plants have adsorption energies El;, of 1.(X% 1.I7 eV. With increasing coverage tf EC, decreases (e.g., E,, = 0.8 eV for 0 = 0.2S) and an extrinsic precursor occurs [2]. We will consider, however, surface temperatures 7; 2 450 K where the surface is clean in molecular beam experiments 131 and the well depth of the gas-sLlrface-pot~titial therefore is larger than 1 eV. According to electron impact data the NO-molecule is bound with the nitrogen-atom to the surface [4]. The sticking probability s is high and rather insensitive to variations of the surface temperature. With a singIe exception (s = 0.65 [S]) a value of s - 0.9 has been found [2,6]. Alignment effects in adsorption have been observed by Jacobs et al. [7]. The quadrupolc moment of desorbing molecules A, is considered as a function of the angular momentum j. AZ is slightly negative for particles with small angular monlenta. Higher rotational levels show positive quadr~~pole moments indicating that these desorbing molecules prefer the helicopter type of motion. Assuming a detailed balance relation between sticking and dcsorption one concludes that cartwhcelin~ particles adsorb with smaller probabilities. Kuipcrs et al. [S] observed a sticking probabiiity which depends on the initial orientation. Using an electrostatic hcxapole lens it is possible to orient incoming molecules by the Stark effect. It turns out that particles approaching with the N-end towards the surface have a larger sticking probability. Finally WC would like to mention that there exists the so-called rotational cooling, i.c., the mean rotational temperature of desorbing particles is significantly smaller than the surface temperature [I?]. A satisfying theory for the sticking probability of the system NO/Pt( 111) has to take into account the translation and rotation of the molecule as well as the phonon system of the solid. There exist two oo.3i,nr1’78/u2/$of;.oo
‘1”
IYYZ - Et\evter
Science Publishers
B.V. Ail rights rrservcd
57
C. Huug et al. / NO on Pt(lll)
classical models. Muhlhausen, Williams and Tully [9] and Tully [lo] performed a stochastic trajectory simulation. The heat bath property of the phonons is incorporated in terms of a generalized Langevin equation. They obtain a correct Arrhenius behavior of desorbing particles and for the choice of parameters in ref. [lo] the sticking probability has the right magnitude. The rotational temperatures for desorbing particles, however, are much too high. Jacobs and Zare 1111 describe the heat bath of the surface as a microcanonical ensemble consisting of four mobile surface atoms only, instead of the complicated generalized Langevin equation. They obtain in some sense a simplified version of the foregoing theory. The interaction potential is nearly the same as in ref. [9] leading to a sticking coefficient which is too low (s = 0.6). The experimental data on direct inelastic scattering and quadrupole alignment [71 are reproduced very well. The steric effect in adsorption and the rotational temperature, however, are too large. A quantum mechanical calculation describing direct scattering only was presented by Lemoine and Corey 1121. They performed a close-coupled wave-packet calculation taking into account the rotational degree of freedom of the molecule neglecting, however, the phonons of the surface. A steric effect in direct scattering, which was found experimentally by Kuipers et al. [S], is reproduced qualitatively. In the next section we derive an expression for the S matrix describing direct inelastic scattering. The rotational degree of freedom and the phonons are treated quantum mechanically. Since the diatomic molecule rotates rather slow one can apply a sudden approximation. Concerning the oscillation of the surface we assume that its essential features can be described by a forced oscillator model. Due to its large mass the gas-molecule can be treated semiclassically. 2. Theory The Hamiltonian
is given by + 4) + V(
6, 2).
2,
(1)
We consider a one-dimensional model only, denoting the corresponding translational coordinate by z and its conjugate momentum by p. The angular dependence of sticking coefficients or desorption probabilities can be calculated in our model under the assumption of normal energy scaling only. The mass of the molecule is mg = 30 amu. It is treated as a rigid rotor with angular momentum operator 1, angles 6, cp and a moment of inertia A2/21 = 2.1 x lo-” eV. Although the coupling-scheme of the II,,, ground state actually is rather complicated (Hund(a) + Hund(b) for increasing j) the main features of rotational excitation are assumed to be contained in our simplified rotor model and a proper choice of interaction parameters. The “phonons” are represented by a one-dimensional Einstein oscillator with coordinate i
and a frequency wr = 3w,/4 (w, being the bulk Debye frequency). Finite size effects an effective mass of the surface atom m, = 3Sm,, Cm,, = 195 amu). The coupling of all degrees of freedom is described by an interaction potential v( z, 6, X) =R( 6) eP2cr(z-x) _A( 8) ePu(:-a) =D(6)(eP D(6)
A( C$ = ~ 4R(6) ’
Zn(:-z,,(lY)~x)
_
2
e~n(z-z,,(;t)~.r) >>
1 2R(6) zo( 6) = --In(Y A(6)
.
are contained
in
(3)
(4)
C. Haug et al. / NO on Pt(lll)
58
V(z, 8, X) is a generalized Morse-potential with range (Y= 1.5/A. The well-depth is chosen to be 1.2 eV consistent with all experiments. The angle-dependent coefficients will be discussed in section 3. It should be mentioned, however, that corrugation is neglected (no cp-dependence) and the magnetic quantum number m is conserved in each collision. We expand for small crx which is of the order l/50 for a typical value of T, = 300 K (surface temperature) and obtain V( Z, 6, X) = D(_S)(e~24-@U + 2axD(
_ 2 e~n(ZFZll(A)))
fi)(e-2u(=-zll(fi))
= V( 2, 0) -xF(
_ e~“‘z+‘““)
2, 6).
(5)
A theory describing NO-molecules scattered at a Pt-surface has to account for the excitation of many “phonons” leading to a large sticking probability and of high angular momenta as indicated by the data of direct inelastic scattering 171. One therefore has to go beyond first-order theories for both degrees of freedom. Concerning rotations we adopt the sudden approximation [13-1.51 which can be justified if the angle of rotation does not change much during the scattering process. Since our interaction potential is very deep there is an impulsive collision at the repulsive part of the potential, where the excitation mainly takes place. In addition we note that particles which undergo excitations corresponding to high rotational states get trapped with large probabilities. The sudden approximation therefore is justified very well for direct inelastic scattering. Consequently the rotational kinetic energy can be assumed to be constant
the Hamiltonian becomes diagonal in 6 and one Schrddinger equation containing 19 as a parameter
A*
a2
has to calculate
A2 a2
2m, az2
is the classical 1 ~,~(t) = - --In ff
solution
of a particle
cash
i2 E,/m, (’
following
(7)
and in a first step we neglect recoil effects (they will of the trajectory due to the coupling -xF(z, 6). The
in the potential
1 I
for the
$(2,x,0)=0.
s
The gas particle is treated in WKB approximation be taken into account later on), i.e., perturbations coordinate z can then be replaced by ~,~(t)
z+(t)
S matrix
I
----~~+fmsw~xz+V(z,6)-xF(z,~)-(E-Ej)
i
the
l/(z,
6):
1 - 6( 8)2 at
i
- S( 19)
6(a)
+z,,(fi). !
1 6(6)
(9)
= b;1+ (-%?/D(fi))
.
The Hamiltonian then is split into one part containing particle coordinates only, and a rest describing a harmonic oscillator which is driven by an external force. Consequently the S matrix factorizes and the “phonons” can be treated as a forced oscillator [16] ,?
COS( wEt)
+
59
C. Haug et al. / NO on Pdlll)
5, is the momentum only the cosine-part
operator of the oscillator. Since the classical motion remains and one obtains by Fourier transformation
of the gas particle [17]
is symmetric,
s^‘O= exp( k41!8(wE)),
(II)
2rm,w, F,Y(WE) = Applying
cosh[ (wd/wa)(r
a
standard
- arccos a(o))]
WKB-methods,
the scattering
(14
’
sinh[ (o,/,/wa)r] phase
is given as an action
integral
(13) being
p = \/2m,E,
Integration
the
asymptotic
momentum
and
is the
z,
classical
turning
point
of
V(z, S).
can be performed A(6) /m /miA(
+cA(V
arcsin I
J
6)2 + 2m,R(
2R(G)p2
+f
+m,A(6)2
O)p2
-$ln
(14) 2P2
and
finally
one
obtains
for the S matrix
(forced
oscillator
results
are taken
from ref.
[161)
= (jf, m,, ~f,lS^(E,,)Iji, (.if, m,, nfls^(Ep,)lji,
mi, ni>a(E,-Ei)>
(15)
m,, q>
= / dfi Y,,*,,(fi, cp)exP[2i77(~)1S~~~~)E;i,,(6, cp>,
f
L”,:yQ( I a(S) I ‘) zz
$ $
5
I a( 8) I ‘)
Q’,-“‘(
JL;-“(x)
= i
The mean
number
k!
of vibrational
An( 8) = I a(S) Our
expression
exp( -
(ia(fi)*)”
..,i
-
‘U(,)‘2j,
if iZf>n,;
‘a’I”2],
if ni>n,;
(n
-k)!(m
quanta
-n
excited
+k)! An(6)
’
a(0)
:=
Fi+(%) \I’2m,Ao,
(18)
.
is given by
I 2.
for the
S matrix
(17)
I’ 1
(-x)““‘!
k=O
(iCy(7Y))nr-n’
(16)
(19) corresponding
to eq. (7) can also be derived
within
a path-integral
C‘. Hauget al. / NO on Pi(llI)
60
formalism, which has been developed for semiclassical scattering by Pechukas [20,21]. It should be noted that it contains a direct coupling between the rotation of the molecule and the phonons due to the angle-dependence of the force function FSY(:,(w,,). We apply two important improvements which take into account recoil effects: a reduced mass correction ensures the correct classical limit for the mean energy loss to phonons at cold surfaces and is performed by replacing An -+ An/(
I + /L)‘,
/.L:= mg,‘rrz,,
the energy of the gas particle energy conservation
(20)
is symmetrized
$ +;ijf(j,+ 1) + fiw,(n,+
with the energies
+) = &
$
+ gji(ji
concerning
+ 1) + Awr(ni
and final scattering
states;
+ f),
by
(21)
F
the final momentum pr is fixed for each transition p one has to calculate 2rl(i3)
of initial
separately
and if ?(I?+) is considered
as a function
=77(Of, Pi) +77(lyi), Pt): the force function
I-J, = @$,,
F,(w,)
of (22)
the particle
momentum
to be inserted
is given by (23)
+ EJ.
The S matrix then correctly describes anticorrelation effects between as for example found for the system NO/Ge [l&19]. In order to obtain mean energy transfer for non-zero surface temperature
phonon and rotational excitation the correct classical limit for the
(24) one has to modify
the expression
An( 19) --) An( ;t) -- 4
for the mean
number
P
of phonons
excited
(25)
(1 +tyY This completes our derivation of the S matrix. There remains simpIy a one-dimensional numerical integration over the angle 6 which has to be done, however, for a very large number of matrix elements. Computer time can be saved by neglecting the direct coupling between rotation and phonons and by avoiding symmetrization. The S matrix then can be calculated for rotational and phonon excitation separately. We observed that for our choice of parameters the results do not change much. After averaging with the surface temperature one finally obtains for the probability that gas molecules with tra~~slational kinetic energy E,, and rotational quantum numbers ji, mi undergo a transition to the final state jr, tlzi the expression ’ P(j,,
mf+jl,
m,:
E,,)
where I!Z,,~is determined by energy conservation. The so-called ” prompt” sticking coefficient s( ji, m i, E,,, f is simply given by the total probability that an incoming particle has a final translational energy Ep, < 0, i.e. it is inside the potential well after hitting the surface. We obtain S(ji,
r?ri, Ep,) = 1 -
C P(jf, 1,17Jt
m,+ji,
mi, E,,,).
(27)
C. Haugetal. Table 1 Interaction
potential
/ NOon
parameters
= 0.075
D=1.2eV
R,, = 1.OO
R,
(Y= 1.5/i
A,, = 2.00
A,=0.1125
3. Results
61
Pt(Ill)
R2 = 0.075 A, = 0.15
and discussion
At the beginning of this section we discuss our choice of parameters (table 1). The angle-dependent functions R(6) and A(6) in eq. (3) are expanded into Legendre polynomials with only the first two terms retained V( z, 6, X) = D{ [ R,, + R,P,(cos - [A, +A,P,(cos
8) + R,P,(cos 79) +A,P,(cos
S)]
e-2a(zpx)
S)]
e-“(Z~x)).
(28)
The value D = 1.2 eV for the well-depth is consistent with the experimental data for the clean surface as discussed in the introduction. The absolute minimum of the potential corresponds to an angle cos -i) = + 1 which we assign to the N-end-down configuration to be consistent with experiments. Consequently molecules with the O-end towards the surface are represented by - 1 < cos B < 0. For further discussion we show in fig. 1 the classical excitation function. It is simply given by the derivative of the sudden-phase shift in eq. (14), Aj = 2 dn(6)/d6. NO-molecules scattered at Pt(ll1) have a higher adsorption probability for particles approaching with the N-end towards the surface [S]. This is correctly described by our potential since molecules with angles 0 < cos 6 < 1 excite high j-values. As a result the energy transfer to the rotor is large and the sticking probability is high. The absolute magnitude of the steric effect will be evaluated below. First of all, we calculate the probability to excite final angular momenta after direct inelastic scattering and compare with experimental data [7], fig. 2. Since the initial rotational temperature was 40 K in the experiment we had to average eq. (26) with the corresponding Boltzmann distribution. The energy of the incoming particles is slightly below 0.1 eV corresponding to a beam energy of 9.2 kJ/mol and the surface 40
-2
30
._ 2
20
z
10
0
-141 0
500
1000
j*(i+l) (final) Fig. 1. Classical excitation function: the number of quanta excited Aj is plotted as a function of cos 6 which is approximately constant during the collision (sudden approximation). The initial energy of the gas particle is E, = 0.2 eV. The maxima indicate rainbow scattering.
Fig. 2. Boltzmann plot of the probability distribution p(j) for the excitation of an angular momentum j; E, = 0.095 eV, r, = 4.50 K. The experimental data (-) are taken from ref. [7].
C’. ffaug et ul / NO on Pt (11 i)
62 1.0
.15-
.10
c\I a 0, K .Y 0 z= m
.05' _ ,'_
.7
_ _
_'
Ia
,'
.'
x x
.OD'
/'-
x
*
.5
/' 1' 1'
e!
_-.
.5 500
600
I~
700
-.osl
,-_
800
900
0
15
5
T (surface)
j
Fig. 3. The sticking coefficient is plotted as function of the surface temperature. E,, = kl; CeV). J, = 0. m, = 0.
20
cinol)
Fig. 1. Quadrupole alignment A, as a function of angular momentum J for desorbing particles. The surface temperature is 7; = 553 K. The experimental data are taken from ref. [7]( X ) theory.
temperature is 1; z 450 K. It is important to note that experimentally the region of low angular momenta has a large background contribution from the incident beam (normal incidence). The incoming molecules possess an initial rotational temperature of 40 K and the measured probability for small j is much higher than the actual values for direct inelastic scattering. Having this in mind the agreement between theory and experiment is quite good over the whole range of fig. 2. It should be noted that a rotational energy corresponding to j = 19 equals the kinetic energy of the incoming particles. Since they are rotationally rather cold, higher angular momenta can be excited only if the missing energy is provided by phonons. The “prompt” sticking coefficient calculated by eq. (27) is plotted against the surface temperature in fig. 3. The kinetic energy of the incoming particles is equal to kTI,. The adsorption probability is large over the whole range and its magnitude is in agreement with experiments [2,6]. As mentioned in the introduction Kuipers et al. [8] measured a steric effect in adsorption which is defined by R := +;,;iy.;+, where
s, PJcos
(29)
’
and s_ are sticking
coefficient
for molecules
initially
oriented
according
to
0) = +( 1 * cos 6).
(30)
P, describes particles approaching with the N-end towards the surface. Within our simple rigid rotor approach we have to constr-uct linear combinations of spherical harmonics with probability distributions P,. Using the expansion [22]
VGCOS = i: (-1)1+1 I= 0
oriented molecules IPs’r for large .!)
are properly
(cos 17 I k > = 0.943Y,,,(cos
\i’T( 1 + +> (I + 1 .S)(
described
l2 - 0.25)
by truncating
8) + 0.327Y,,,(cos
(31)
r,O(cos b)3 the sum after I= 2 (the coefficients
8) - O.O604Y,,,(cos
6).
behave
as
(32)
R then can be calculated and we obtained a value of R = -0.043 which is consistent with measurements of Kuipers et al. [81, who obtained R = - 0.035 i 0.02 for a surface temperature of 573 K.
C. Haug et al. / NO on Pt(lll)
63
In the next part of this section we consider desorbing particles. provides a simple relationship between sticking coefficients s(E,, P,(E,,
The principle of detailed balance j, m> and desorption probabilities
j, m) Pd(E,,
j, m) a s(E,,
j,-
m) e-P~(Ep’“~)p.
(33)
A justification of this formula is possible if a well defined slow ing, e.g., to a single low-lying eigenvalue (i.e., well separated [23,24]. Since we are interested in the distribution of rotational has to average eq. (33) over the Boltzmann flux at T,. This can replace E, by kT,, the mean energy of a flux with temperature P,(kT,,
j, m) as(kT,,
The quadrupole
moment
j,-m)
determined
C;‘,=_,{[3m2-j(j+ A,(j)
=
component of particles exists correspondfrom all the others) in a kinetic equation states at a given surface temperature one be taken into account approximately if we T,.
epp\El. by Jacobs I)]/j(j+
Ciin= _;Pd( j, m)
(34) et al. [7] is defined l)}P,(j,
by
m) (35)
It is a measure for an alignment of molecules and is insensitive to orientation. A, has its maximum +2 for partices with I m I = j only, corresponding to the helicopter type of motion where the direction of the angular momentum classically is perpendicular to the surface. The minimum for cartwheeling I m I = 0 is A, = - 1. Our results are compared with experimental data in fig. 4. They agree within the error bars. For small momenta A, is negative. Larger values of j desorb with increasing positive A,, i.e., those molecules prefer the helicopter type of motion and higher magnetic quantum numbers m. One has to consider two effects which are contained in the sticking coefficient for a qualitative explanation. If m = j the molecules hit the surface with both atoms at approximately the same time and rotational excitation is expected to be small. The largest values of m will therefore have small sticking probabilities and A, should be negative as observed for small angular momenta. Concerning larger j this property can be overruled by a second effect. Due to conservation of the magnetic quantum number (corrugation is small and completely neglected in our calculations) a given initial m can excite j 2 m only. The distribution of final j shifts to larger angular momenta with increasing m, rotational excitation and the sticking probability will increase and A, is positive. In the last part of this section we discuss the mean rotational energy of desorbing molecules. The population of angular momenta obtained by experiments can be fitted very well by a Boltzmann distribution corresponding to a temperature T,,,. If T,,, is considered as a function of T, one observes that T,,, = i?, for T, < 350 K. If T, is increased further the rotational temperature saturates at a value of TR,, = 450 K, i.e., the population of angular momenta is cooled significantly [31. Assuming again a detailed balance relation there exists a corresponding property of the sticking coefficient [25,10], which has to be a function decreasing with initial angular momentum. The rotational temperature calculated by our theory is only about 20 K below i?, = 553 K which is much too high. The same problem occurs in classical calculations so far which do not produce enough j-dependence of s to account for the correct magnitude of rotational cooling [lo,1 11. The basic reason behind this failure to explain rotational cooling quantitatively is the large energy transfer to phonons and rotations within the first collision. In presently existing theories of sticking such a large energy transfer always leads to a large sticking coefficient close to unity in contrast to the strong decrease with angular momentum required to produce rotational cooling. We have investigated the question if rotational cooling could be due to a violation of detailed balance in the process of sticking-desorption [24]. It has turned out, that such a violation, in principle, is possible
if there are more than one slow mode in the kinetics of this process. But so far we have not been able to identify a plausible microscopic model containing such modes. Earlier attempts [27-291 to explain rotational cooling did not make use of detailed balance explicitly (although we believe that the theories used fulfill this condition). In refs. [27.29] it is assumed that in the adsorbed state rotations are strongly frustrated. Rotational cooling then is supposed to be produced by a strong transfer of high rotational to translational energy. Such a transfer, however, can only bc rclcvant, if desorption occurs more or less directly, without phonons being involved in any important way. In practice, the desorption process proceeds indirectly via many successive steps from the low-lying, perhaps frustrated, rotational states to high-lying much less frustrated states. In each step phonons will be involved and the high-lying states will have lost the memory of possible frustrations in the low-lying states. In ref. [28] no strong frustration is assumed. The initial and final state in the desorption process are treated as freely rotating. For the process of rotational de-excitation in desorption it is assumed that the statistical weights 2j + 1 of the high-lying initial states are shifted down to the low-lying final states. This increases the relative weight of low-lying rotational states and hcncc leads to rotational cooling. Such an effect would, however. imply strong m nonconservation which seems to be in contrast to the experimental results on A2 described above which indicate approximate m conservation. In this paper we propose another explanation of rotational cooling based on rotationally mediated selective adsorption resonances. Rotationally assisted selective adsorption is well known to occur for hydrogen. For a light molecule such as Hz the rotational states are widely spaced and well separated. For NO, on the other hand, this is no longer true. In fact no rotational resonances have been observed for NO. At higher energies even in a theoretical model without phonons the scattering cross sections arc smooth functions of energy. A recent careful investigation of the cross section at lower energy, however, indicated the existence of well defined resonances [26]. Since phonons were neglected in this calculation the width rii of these resonances is due to the decay of the resonance states back into the scattering continuum. If phonons are taken into account, the resonance states can also decay inelastically via phonon emission. The total width I” = ri’,,,, + I;il of the resonance will increase considerably and the resonances will overlap in general. Nevertheless the resonance states will remain well defined and orthogonal and the sticking coefficient .Ywill be given by s = Es, I
,TY’q1\‘,9i 1nel cl
t.301
Here s,. is the trapping probability into the resonance state r. Our earlier “prompt” sticking coefficient is recovered if fli can be neglected. For rotational cooling the dependence of s on the initial angular momentum is essential. Since for small incident angular momentum s is known to be of order 0.9 XY s, has to be of order one and the inelastic decay width has to be large compared to the elastic one (at least about ten times larger). The angular momentum dependence of the inelastic decay width can be expected to be small. If X,. s, also stays constant of order one the dominant j dependence of the sticking coefficient has to come from the elastic decay width. To obtain the dependence of f;Ii on the initial angular momentum we performed coupled-channel calculations which neglect any coupling to phonons. The results arc shown in fig. 5. To save computing time the calculation actually was performed for parameters corresponding to the system NO/Ag. We believe, however, that the results for NO/Pt will look similar: The Morse-potential has a strong anharmonicity and the high-lying states responsible for the resonances have spacings independent of the depth and only dependent on the width of the potential. This may be the origin of the observed independence of the effective rotational temperature on the substrate. The complicated structure of the 5’ matrix as a function of energy is due to many closely spaced resonances. Since the latter can decay only by transferring energy from the rotor to the
65
C. Haug et al. / NO on Pt(ll1)
a
-1
.o 98
100
Energy Fig. 5. The real part of the S matrix is plotted
(meV)
as a function of energy for initial angular structure is due to elastic channels only.
momenta
j = 0, 10, 20; the resonance
translational degree of freedom, the mean width of the resonances serve as an estimate for ri. One observes that rC: increases by a factor of 5 if the initial angular momentum is varied over a range 0 + 20. If r,;,, is approximately constant, eq. (36) indicates that the sticking coefficient becomes a decreasing function with j, a property which supports rotational cooling. Within a simple estimate, where all &: are assumed to be equal, we obtained T,,, = 450 K CT, = 553 K) which is close to the experimental value.
4.
Summary
We have derived a simple expression for the S matrix describing the scattering of diatomic molecules at surfaces. The translational motion of the gas-particle is treated within a WKB-approximation. The corresponding classical trajectory occurs as an input into a sudden phase shift for the rotational excitation of the molecule and into the calculation of the forced oscillator model for the phonon system. We have then applied our theory to the system NO/Pt(ll 1). The distribution of angular momenta after direct inelastic scattering is described correctly. The “prompt” sticking coefficient, which is given by the probability that an incoming gas particle has a kinetic energy Epf < 0 after the scattering event, turns out to be high for smail initial angular momenta (S = 0.9) due to a large energy transfer to phonons.
C’. Haag et al. / NO on Pt(1 I I)
66
Orientational and alignment effects contained in the adsorption probability, which have been observed in experiments, are reproduced correctly by our theory. Concerning the phenomenon of rotational cooling one has to go into a more detailed analysis of the sticking process. We considered a simple model containing rotationally mediated selective sticking resonances. The strong increase of elastic resonance widths with increasing initial angular momentum leads to a decrease of the sticking coefficient. Rotational cooling then is obtained in semiquantitative agreement with experiment.
References [I] [2] ]3] [4] [5] [6] [7] [8] [9] [IO] [ll] [12] [13] [l4] [15] [16] [l7] [18] [19] [ZO] [21] [22] [23] [24] [25] [26] 1271 [28] [29]
R.J. Gorte, L.D. Schmidt and J.L. Gland, Surf. Sci. 109 (1982) 355. CT. Campbell, G. Ertl and J. Segner, Surf. Sci. 115 (1984) 309. J. Segner. H. Robota, W. Vielhaber, G. Ertl, F. Frenkel, J. Hager, W. Krieger and H. Walter, Surf. Sci. 131 (19X3) 273. CM. Comrie, W.H. Weinberg and R.M. Lambert, Surf. Sci. 57 (1976) 619. T.H. Lin and G.A. Somorjai, Surf. Sci. 107 (1981) 573. J.A. Serri. M.J. Cardillo and G.E. Becker, J. Chem. Phys. 77 (1982) 2175. D.C. Jacobs, K.W. Kolasinski, SF. Sh,ane and R.N. Zare. J. Chem. Phys. 91 (1989) 3182. E.W. Kuipers, M.G. Tenner. A.W. Kleyn and S. Stolte, Phys. Rev. Lett. 62 (1989) 2152. C.W. Muhlhausen, L.R. Williams and J.C. Tully, J. Chem. Phys. 83 (1985) 2594. J.C. Tully, in: Kinetics of Interface Reactions, Eds. M. Grunze and H.J. Kreuzer. Vol. 8 of Springer Series in Surface Science (Springer. Berlin, 1987) p. 37. D.C. Jacobs and R.N. Zare. J. Chem. Phys. 91 (1989) 3196. D. Lemoine and G.C. Corey, J. Chem. Phys. 94 (1991) 767. T.P. Tsien and R.T. Pack, Chem. Phys. Lett. 6 (1970) 54. D. Secrest, J. Chem. Phys. 62 (1975) 710. R.B. Gerber, A.T. Yinnon, Y. Shimoni and D.J. Kouri, J. Chem. Phys. 73 (1980) 4397. P. Carruthers and M.M. Nieto, Am. J. Phys. 33 (1965) 537. L. Trilling, Surf. Sci. 21 (1970) 337. A. ModI, T. Gritsch, F. Budde, T.J. Chuang and G. Ertl, Phys. Rev. Lett. 57 (1986) 3X4. T. Brunner, R. Brako and W. Breniy. Phys. Rev. A 35 (1987) 5266. P. Pechukas, Phys. Rev. 181 (1969) 166. P. Pechukas, Phys. Rev. 181 (1969) 174. IS. Gradshteyn and I.M. Ryzhik, Table of Integrals. Series and Products (Academic Press, San Diego. 1980). W. Brenig. Z. Phys. B 48 (1982) 127. C. Haug and W. Brenig, to be published. W. Brenig, H. Kasai and H. Miiller, Surf. Sci. 161 (1985) 608. C. Haug, W. Brenig and T. Brunner, Surf. Sci. 227 (1990) 167. J.W. Gadzuk, U. Landman, E.J. Kuster, C.L. Cleveland and R.N. Barnett, Phys. Rev. Lett. 49 (1982) 426. J.M. Bowman and J.L. Gossage, Chem. Phys. Lett. 96 (1983) 481. A. Redondo, Y. Zeiri. J.J. Low and W.A. Goddard III, J. Chem. Phys. 79 (1983) 6410.
Surface Science 265 North-I
(lW2) Sh-hh
surface
foiland
NO on Pt( 111): direct inelastic
Angular
momentum
cules interacting pr(~xim~tion
distributions
with a Pt(
I I I)
and the phonon
Concerning
rotational
this effect is proposed
scattering
after direct inelastic scattering
surt’ace are calculated.
The rotational
system of the solid is described
n~~gllitu~~e as well as orientational
., .
and alignment
in term& of rot~ition~~liy mediated
“prompt”
degrees of freedom oscillator.
effects of the sticking coefficient
cooling the prompt sticking-theory
_.- :
and sticking/desorption
and the so-called
as a forced
.’ ‘. .. ‘.
scienc6
turns out to be insufficient.
sticking coefficient
of NO-mole-
are treated within a quantum-sudden-apWith
reasonable
are found
parameters
the absolute
to agree with experimenta
A simplified
semiquantitative
data.
description
of
selective adsorption.
1. introduction The interaction of NO-molecules with a Ptil 11) surface has been a subject of experimentaI and theoretical research for many years. Gas particles with well-defined initial orientations or low r~)tational temperatures are analyzed after the scattering event using laser-induced fluorescence and multiphoton ionization spectroscopy. These “state to state”’ experiments serve as a fruitful field for theoretical studies. Furthermore many parameters of this system can be fixed. From thermal desorption data [1,2] one finds a tightly bound state with binding energy = 1.45 eV probably at defect sites whereas molecules on perfect Pt(l1 I)-plants have adsorption energies El;, of 1.(X% 1.I7 eV. With increasing coverage tf EC, decreases (e.g., E,, = 0.8 eV for 0 = 0.2S) and an extrinsic precursor occurs [2]. We will consider, however, surface temperatures 7; 2 450 K where the surface is clean in molecular beam experiments 131 and the well depth of the gas-sLlrface-pot~titial therefore is larger than 1 eV. According to electron impact data the NO-molecule is bound with the nitrogen-atom to the surface [4]. The sticking probability s is high and rather insensitive to variations of the surface temperature. With a singIe exception (s = 0.65 [S]) a value of s - 0.9 has been found [2,6]. Alignment effects in adsorption have been observed by Jacobs et al. [7]. The quadrupolc moment of desorbing molecules A, is considered as a function of the angular momentum j. AZ is slightly negative for particles with small angular monlenta. Higher rotational levels show positive quadr~~pole moments indicating that these desorbing molecules prefer the helicopter type of motion. Assuming a detailed balance relation between sticking and dcsorption one concludes that cartwhcelin~ particles adsorb with smaller probabilities. Kuipcrs et al. [S] observed a sticking probabiiity which depends on the initial orientation. Using an electrostatic hcxapole lens it is possible to orient incoming molecules by the Stark effect. It turns out that particles approaching with the N-end towards the surface have a larger sticking probability. Finally WC would like to mention that there exists the so-called rotational cooling, i.c., the mean rotational temperature of desorbing particles is significantly smaller than the surface temperature [I?]. A satisfying theory for the sticking probability of the system NO/Pt( 111) has to take into account the translation and rotation of the molecule as well as the phonon system of the solid. There exist two oo.3i,nr1’78/u2/$of;.oo
‘1”
IYYZ - Et\evter
Science Publishers
B.V. Ail rights rrservcd
57
C. Huug et al. / NO on Pt(lll)
classical models. Muhlhausen, Williams and Tully [9] and Tully [lo] performed a stochastic trajectory simulation. The heat bath property of the phonons is incorporated in terms of a generalized Langevin equation. They obtain a correct Arrhenius behavior of desorbing particles and for the choice of parameters in ref. [lo] the sticking probability has the right magnitude. The rotational temperatures for desorbing particles, however, are much too high. Jacobs and Zare 1111 describe the heat bath of the surface as a microcanonical ensemble consisting of four mobile surface atoms only, instead of the complicated generalized Langevin equation. They obtain in some sense a simplified version of the foregoing theory. The interaction potential is nearly the same as in ref. [9] leading to a sticking coefficient which is too low (s = 0.6). The experimental data on direct inelastic scattering and quadrupole alignment [71 are reproduced very well. The steric effect in adsorption and the rotational temperature, however, are too large. A quantum mechanical calculation describing direct scattering only was presented by Lemoine and Corey 1121. They performed a close-coupled wave-packet calculation taking into account the rotational degree of freedom of the molecule neglecting, however, the phonons of the surface. A steric effect in direct scattering, which was found experimentally by Kuipers et al. [S], is reproduced qualitatively. In the next section we derive an expression for the S matrix describing direct inelastic scattering. The rotational degree of freedom and the phonons are treated quantum mechanically. Since the diatomic molecule rotates rather slow one can apply a sudden approximation. Concerning the oscillation of the surface we assume that its essential features can be described by a forced oscillator model. Due to its large mass the gas-molecule can be treated semiclassically. 2. Theory The Hamiltonian
is given by + 4) + V(
6, 2).
2,
(1)
We consider a one-dimensional model only, denoting the corresponding translational coordinate by z and its conjugate momentum by p. The angular dependence of sticking coefficients or desorption probabilities can be calculated in our model under the assumption of normal energy scaling only. The mass of the molecule is mg = 30 amu. It is treated as a rigid rotor with angular momentum operator 1, angles 6, cp and a moment of inertia A2/21 = 2.1 x lo-” eV. Although the coupling-scheme of the II,,, ground state actually is rather complicated (Hund(a) + Hund(b) for increasing j) the main features of rotational excitation are assumed to be contained in our simplified rotor model and a proper choice of interaction parameters. The “phonons” are represented by a one-dimensional Einstein oscillator with coordinate i
and a frequency wr = 3w,/4 (w, being the bulk Debye frequency). Finite size effects an effective mass of the surface atom m, = 3Sm,, Cm,, = 195 amu). The coupling of all degrees of freedom is described by an interaction potential v( z, 6, X) =R( 6) eP2cr(z-x) _A( 8) ePu(:-a) =D(6)(eP D(6)
A( C$ = ~ 4R(6) ’
Zn(:-z,,(lY)~x)
_
2
e~n(z-z,,(;t)~.r) >>
1 2R(6) zo( 6) = --In(Y A(6)
.
are contained
in
(3)
(4)
C. Haug et al. / NO on Pt(lll)
58
V(z, 8, X) is a generalized Morse-potential with range (Y= 1.5/A. The well-depth is chosen to be 1.2 eV consistent with all experiments. The angle-dependent coefficients will be discussed in section 3. It should be mentioned, however, that corrugation is neglected (no cp-dependence) and the magnetic quantum number m is conserved in each collision. We expand for small crx which is of the order l/50 for a typical value of T, = 300 K (surface temperature) and obtain V( Z, 6, X) = D(_S)(e~24-@U + 2axD(
_ 2 e~n(ZFZll(A)))
fi)(e-2u(=-zll(fi))
= V( 2, 0) -xF(
_ e~“‘z+‘““)
2, 6).
(5)
A theory describing NO-molecules scattered at a Pt-surface has to account for the excitation of many “phonons” leading to a large sticking probability and of high angular momenta as indicated by the data of direct inelastic scattering 171. One therefore has to go beyond first-order theories for both degrees of freedom. Concerning rotations we adopt the sudden approximation [13-1.51 which can be justified if the angle of rotation does not change much during the scattering process. Since our interaction potential is very deep there is an impulsive collision at the repulsive part of the potential, where the excitation mainly takes place. In addition we note that particles which undergo excitations corresponding to high rotational states get trapped with large probabilities. The sudden approximation therefore is justified very well for direct inelastic scattering. Consequently the rotational kinetic energy can be assumed to be constant
the Hamiltonian becomes diagonal in 6 and one Schrddinger equation containing 19 as a parameter
A*
a2
has to calculate
A2 a2
2m, az2
is the classical 1 ~,~(t) = - --In ff
solution
of a particle
cash
i2 E,/m, (’
following
(7)
and in a first step we neglect recoil effects (they will of the trajectory due to the coupling -xF(z, 6). The
in the potential
1 I
for the
$(2,x,0)=0.
s
The gas particle is treated in WKB approximation be taken into account later on), i.e., perturbations coordinate z can then be replaced by ~,~(t)
z+(t)
S matrix
I
----~~+fmsw~xz+V(z,6)-xF(z,~)-(E-Ej)
i
the
l/(z,
6):
1 - 6( 8)2 at
i
- S( 19)
6(a)
+z,,(fi). !
1 6(6)
(9)
= b;1+ (-%?/D(fi))
.
The Hamiltonian then is split into one part containing particle coordinates only, and a rest describing a harmonic oscillator which is driven by an external force. Consequently the S matrix factorizes and the “phonons” can be treated as a forced oscillator [16] ,?
COS( wEt)
+
59
C. Haug et al. / NO on Pdlll)
5, is the momentum only the cosine-part
operator of the oscillator. Since the classical motion remains and one obtains by Fourier transformation
of the gas particle [17]
is symmetric,
s^‘O= exp( k41!8(wE)),
(II)
2rm,w, F,Y(WE) = Applying
cosh[ (wd/wa)(r
a
standard
- arccos a(o))]
WKB-methods,
the scattering
(14
’
sinh[ (o,/,/wa)r] phase
is given as an action
integral
(13) being
p = \/2m,E,
Integration
the
asymptotic
momentum
and
is the
z,
classical
turning
point
of
V(z, S).
can be performed A(6) /m /miA(
+cA(V
arcsin I
J
6)2 + 2m,R(
2R(G)p2
+f
+m,A(6)2
O)p2
-$ln
(14) 2P2
and
finally
one
obtains
for the S matrix
(forced
oscillator
results
are taken
from ref.
[161)
mi, ni>a(E,-Ei)>
(15)
m,, q>
= / dfi Y,,*,,(fi, cp)exP[2i77(~)1S~~~~)E;i,,(6, cp>,
f
L”,:yQ( I a(S) I ‘) zz
$ $
5
I a( 8) I ‘)
Q’,-“‘(
JL;-“(x)
= i
The mean
number
k!
of vibrational
An( 8) = I a(S) Our
expression
exp( -
(ia(fi)*)”
..,i
-
‘U(,)‘2j,
if iZf>n,;
‘a’I”2],
if ni>n,;
(n
-k)!(m
quanta
-n
excited
+k)! An(6)
’
a(0)
:=
Fi+(%) \I’2m,Ao,
(18)
.
is given by
I 2.
for the
S matrix
(17)
I’ 1
(-x)““‘!
k=O
(iCy(7Y))nr-n’
(16)
(19) corresponding
to eq. (7) can also be derived
within
a path-integral
C‘. Hauget al. / NO on Pi(llI)
60
formalism, which has been developed for semiclassical scattering by Pechukas [20,21]. It should be noted that it contains a direct coupling between the rotation of the molecule and the phonons due to the angle-dependence of the force function FSY(:,(w,,). We apply two important improvements which take into account recoil effects: a reduced mass correction ensures the correct classical limit for the mean energy loss to phonons at cold surfaces and is performed by replacing An -+ An/(
I + /L)‘,
/.L:= mg,‘rrz,,
the energy of the gas particle energy conservation
(20)
is symmetrized
$ +;ijf(j,+ 1) + fiw,(n,+
with the energies
+) = &
$
+ gji(ji
concerning
+ 1) + Awr(ni
and final scattering
states;
+ f),
by
(21)
F
the final momentum pr is fixed for each transition p one has to calculate 2rl(i3)
of initial
separately
and if ?(I?+) is considered
as a function
=77(Of, Pi) +77(lyi), Pt): the force function
I-J, = @$,,
F,(w,)
of (22)
the particle
momentum
to be inserted
is given by (23)
+ EJ.
The S matrix then correctly describes anticorrelation effects between as for example found for the system NO/Ge [l&19]. In order to obtain mean energy transfer for non-zero surface temperature
phonon and rotational excitation the correct classical limit for the
(24) one has to modify
the expression
An( 19) --) An( ;t) -- 4
for the mean
number
P
of phonons
excited
(25)
(1 +tyY This completes our derivation of the S matrix. There remains simpIy a one-dimensional numerical integration over the angle 6 which has to be done, however, for a very large number of matrix elements. Computer time can be saved by neglecting the direct coupling between rotation and phonons and by avoiding symmetrization. The S matrix then can be calculated for rotational and phonon excitation separately. We observed that for our choice of parameters the results do not change much. After averaging with the surface temperature one finally obtains for the probability that gas molecules with tra~~slational kinetic energy E,, and rotational quantum numbers ji, mi undergo a transition to the final state jr, tlzi the expression ’ P(j,,
mf+jl,
m,:
E,,)
where I!Z,,~is determined by energy conservation. The so-called ” prompt” sticking coefficient s( ji, m i, E,,, f is simply given by the total probability that an incoming particle has a final translational energy Ep, < 0, i.e. it is inside the potential well after hitting the surface. We obtain S(ji,
r?ri, Ep,) = 1 -
C P(jf, 1,17Jt
m,+ji,
mi, E,,,).
(27)
C. Haugetal. Table 1 Interaction
potential
/ NOon
parameters
= 0.075
D=1.2eV
R,, = 1.OO
R,
(Y= 1.5/i
A,, = 2.00
A,=0.1125
3. Results
61
Pt(Ill)
R2 = 0.075 A, = 0.15
and discussion
At the beginning of this section we discuss our choice of parameters (table 1). The angle-dependent functions R(6) and A(6) in eq. (3) are expanded into Legendre polynomials with only the first two terms retained V( z, 6, X) = D{ [ R,, + R,P,(cos - [A, +A,P,(cos
8) + R,P,(cos 79) +A,P,(cos
S)]
e-2a(zpx)
S)]
e-“(Z~x)).
(28)
The value D = 1.2 eV for the well-depth is consistent with the experimental data for the clean surface as discussed in the introduction. The absolute minimum of the potential corresponds to an angle cos -i) = + 1 which we assign to the N-end-down configuration to be consistent with experiments. Consequently molecules with the O-end towards the surface are represented by - 1 < cos B < 0. For further discussion we show in fig. 1 the classical excitation function. It is simply given by the derivative of the sudden-phase shift in eq. (14), Aj = 2 dn(6)/d6. NO-molecules scattered at Pt(ll1) have a higher adsorption probability for particles approaching with the N-end towards the surface [S]. This is correctly described by our potential since molecules with angles 0 < cos 6 < 1 excite high j-values. As a result the energy transfer to the rotor is large and the sticking probability is high. The absolute magnitude of the steric effect will be evaluated below. First of all, we calculate the probability to excite final angular momenta after direct inelastic scattering and compare with experimental data [7], fig. 2. Since the initial rotational temperature was 40 K in the experiment we had to average eq. (26) with the corresponding Boltzmann distribution. The energy of the incoming particles is slightly below 0.1 eV corresponding to a beam energy of 9.2 kJ/mol and the surface 40
-2
30
._ 2
20
z
10
0
-141 0
500
1000
j*(i+l) (final) Fig. 1. Classical excitation function: the number of quanta excited Aj is plotted as a function of cos 6 which is approximately constant during the collision (sudden approximation). The initial energy of the gas particle is E, = 0.2 eV. The maxima indicate rainbow scattering.
Fig. 2. Boltzmann plot of the probability distribution p(j) for the excitation of an angular momentum j; E, = 0.095 eV, r, = 4.50 K. The experimental data (-) are taken from ref. [7].
C’. ffaug et ul / NO on Pt (11 i)
62 1.0
.15-
.10
c\I a 0, K .Y 0 z= m
.05' _ ,'_
.7
_ _
_'
Ia
,'
.'
x x
.OD'
/'-
x
*
.5
/' 1' 1'
e!
_-.
.5 500
600
I~
700
-.osl
,-_
800
900
0
15
5
T (surface)
j
Fig. 3. The sticking coefficient is plotted as function of the surface temperature. E,, = kl; CeV). J, = 0. m, = 0.
20
cinol)
Fig. 1. Quadrupole alignment A, as a function of angular momentum J for desorbing particles. The surface temperature is 7; = 553 K. The experimental data are taken from ref. [7]( X ) theory.
temperature is 1; z 450 K. It is important to note that experimentally the region of low angular momenta has a large background contribution from the incident beam (normal incidence). The incoming molecules possess an initial rotational temperature of 40 K and the measured probability for small j is much higher than the actual values for direct inelastic scattering. Having this in mind the agreement between theory and experiment is quite good over the whole range of fig. 2. It should be noted that a rotational energy corresponding to j = 19 equals the kinetic energy of the incoming particles. Since they are rotationally rather cold, higher angular momenta can be excited only if the missing energy is provided by phonons. The “prompt” sticking coefficient calculated by eq. (27) is plotted against the surface temperature in fig. 3. The kinetic energy of the incoming particles is equal to kTI,. The adsorption probability is large over the whole range and its magnitude is in agreement with experiments [2,6]. As mentioned in the introduction Kuipers et al. [8] measured a steric effect in adsorption which is defined by R := +;,;iy.;+, where
s, PJcos
(29)
’
and s_ are sticking
coefficient
for molecules
initially
oriented
according
to
0) = +( 1 * cos 6).
(30)
P, describes particles approaching with the N-end towards the surface. Within our simple rigid rotor approach we have to constr-uct linear combinations of spherical harmonics with probability distributions P,. Using the expansion [22]
VGCOS = i: (-1)1+1 I= 0
oriented molecules IPs’r for large .!)
are properly
(cos 17 I k > = 0.943Y,,,(cos
\i’T( 1 + +> (I + 1 .S)(
described
l2 - 0.25)
by truncating
8) + 0.327Y,,,(cos
(31)
r,O(cos b)3 the sum after I= 2 (the coefficients
8) - O.O604Y,,,(cos
6).
behave
as
(32)
R then can be calculated and we obtained a value of R = -0.043 which is consistent with measurements of Kuipers et al. [81, who obtained R = - 0.035 i 0.02 for a surface temperature of 573 K.
C. Haug et al. / NO on Pt(lll)
63
In the next part of this section we consider desorbing particles. provides a simple relationship between sticking coefficients s(E,, P,(E,,
The principle of detailed balance j, m> and desorption probabilities
j, m) Pd(E,,
j, m) a s(E,,
j,-
m) e-P~(Ep’“~)p.
(33)
A justification of this formula is possible if a well defined slow ing, e.g., to a single low-lying eigenvalue (i.e., well separated [23,24]. Since we are interested in the distribution of rotational has to average eq. (33) over the Boltzmann flux at T,. This can replace E, by kT,, the mean energy of a flux with temperature P,(kT,,
j, m) as(kT,,
The quadrupole
moment
j,-m)
determined
C;‘,=_,{[3m2-j(j+ A,(j)
=
component of particles exists correspondfrom all the others) in a kinetic equation states at a given surface temperature one be taken into account approximately if we T,.
epp\El. by Jacobs I)]/j(j+
Ciin= _;Pd( j, m)
(34) et al. [7] is defined l)}P,(j,
by
m) (35)
It is a measure for an alignment of molecules and is insensitive to orientation. A, has its maximum +2 for partices with I m I = j only, corresponding to the helicopter type of motion where the direction of the angular momentum classically is perpendicular to the surface. The minimum for cartwheeling I m I = 0 is A, = - 1. Our results are compared with experimental data in fig. 4. They agree within the error bars. For small momenta A, is negative. Larger values of j desorb with increasing positive A,, i.e., those molecules prefer the helicopter type of motion and higher magnetic quantum numbers m. One has to consider two effects which are contained in the sticking coefficient for a qualitative explanation. If m = j the molecules hit the surface with both atoms at approximately the same time and rotational excitation is expected to be small. The largest values of m will therefore have small sticking probabilities and A, should be negative as observed for small angular momenta. Concerning larger j this property can be overruled by a second effect. Due to conservation of the magnetic quantum number (corrugation is small and completely neglected in our calculations) a given initial m can excite j 2 m only. The distribution of final j shifts to larger angular momenta with increasing m, rotational excitation and the sticking probability will increase and A, is positive. In the last part of this section we discuss the mean rotational energy of desorbing molecules. The population of angular momenta obtained by experiments can be fitted very well by a Boltzmann distribution corresponding to a temperature T,,,. If T,,, is considered as a function of T, one observes that T,,, = i?, for T, < 350 K. If T, is increased further the rotational temperature saturates at a value of TR,, = 450 K, i.e., the population of angular momenta is cooled significantly [31. Assuming again a detailed balance relation there exists a corresponding property of the sticking coefficient [25,10], which has to be a function decreasing with initial angular momentum. The rotational temperature calculated by our theory is only about 20 K below i?, = 553 K which is much too high. The same problem occurs in classical calculations so far which do not produce enough j-dependence of s to account for the correct magnitude of rotational cooling [lo,1 11. The basic reason behind this failure to explain rotational cooling quantitatively is the large energy transfer to phonons and rotations within the first collision. In presently existing theories of sticking such a large energy transfer always leads to a large sticking coefficient close to unity in contrast to the strong decrease with angular momentum required to produce rotational cooling. We have investigated the question if rotational cooling could be due to a violation of detailed balance in the process of sticking-desorption [24]. It has turned out, that such a violation, in principle, is possible
if there are more than one slow mode in the kinetics of this process. But so far we have not been able to identify a plausible microscopic model containing such modes. Earlier attempts [27-291 to explain rotational cooling did not make use of detailed balance explicitly (although we believe that the theories used fulfill this condition). In refs. [27.29] it is assumed that in the adsorbed state rotations are strongly frustrated. Rotational cooling then is supposed to be produced by a strong transfer of high rotational to translational energy. Such a transfer, however, can only bc rclcvant, if desorption occurs more or less directly, without phonons being involved in any important way. In practice, the desorption process proceeds indirectly via many successive steps from the low-lying, perhaps frustrated, rotational states to high-lying much less frustrated states. In each step phonons will be involved and the high-lying states will have lost the memory of possible frustrations in the low-lying states. In ref. [28] no strong frustration is assumed. The initial and final state in the desorption process are treated as freely rotating. For the process of rotational de-excitation in desorption it is assumed that the statistical weights 2j + 1 of the high-lying initial states are shifted down to the low-lying final states. This increases the relative weight of low-lying rotational states and hcncc leads to rotational cooling. Such an effect would, however. imply strong m nonconservation which seems to be in contrast to the experimental results on A2 described above which indicate approximate m conservation. In this paper we propose another explanation of rotational cooling based on rotationally mediated selective adsorption resonances. Rotationally assisted selective adsorption is well known to occur for hydrogen. For a light molecule such as Hz the rotational states are widely spaced and well separated. For NO, on the other hand, this is no longer true. In fact no rotational resonances have been observed for NO. At higher energies even in a theoretical model without phonons the scattering cross sections arc smooth functions of energy. A recent careful investigation of the cross section at lower energy, however, indicated the existence of well defined resonances [26]. Since phonons were neglected in this calculation the width rii of these resonances is due to the decay of the resonance states back into the scattering continuum. If phonons are taken into account, the resonance states can also decay inelastically via phonon emission. The total width I” = ri’,,,, + I;il of the resonance will increase considerably and the resonances will overlap in general. Nevertheless the resonance states will remain well defined and orthogonal and the sticking coefficient .Ywill be given by s = Es, I
,TY’q1\‘,9i 1nel cl
t.301
Here s,. is the trapping probability into the resonance state r. Our earlier “prompt” sticking coefficient is recovered if fli can be neglected. For rotational cooling the dependence of s on the initial angular momentum is essential. Since for small incident angular momentum s is known to be of order 0.9 XY s, has to be of order one and the inelastic decay width has to be large compared to the elastic one (at least about ten times larger). The angular momentum dependence of the inelastic decay width can be expected to be small. If X,. s, also stays constant of order one the dominant j dependence of the sticking coefficient has to come from the elastic decay width. To obtain the dependence of f;Ii on the initial angular momentum we performed coupled-channel calculations which neglect any coupling to phonons. The results arc shown in fig. 5. To save computing time the calculation actually was performed for parameters corresponding to the system NO/Ag. We believe, however, that the results for NO/Pt will look similar: The Morse-potential has a strong anharmonicity and the high-lying states responsible for the resonances have spacings independent of the depth and only dependent on the width of the potential. This may be the origin of the observed independence of the effective rotational temperature on the substrate. The complicated structure of the 5’ matrix as a function of energy is due to many closely spaced resonances. Since the latter can decay only by transferring energy from the rotor to the
65
C. Haug et al. / NO on Pt(ll1)
a
-1
.o 98
100
Energy Fig. 5. The real part of the S matrix is plotted
(meV)
as a function of energy for initial angular structure is due to elastic channels only.
momenta
j = 0, 10, 20; the resonance
translational degree of freedom, the mean width of the resonances serve as an estimate for ri. One observes that rC: increases by a factor of 5 if the initial angular momentum is varied over a range 0 + 20. If r,;,, is approximately constant, eq. (36) indicates that the sticking coefficient becomes a decreasing function with j, a property which supports rotational cooling. Within a simple estimate, where all &: are assumed to be equal, we obtained T,,, = 450 K CT, = 553 K) which is close to the experimental value.
4.
Summary
We have derived a simple expression for the S matrix describing the scattering of diatomic molecules at surfaces. The translational motion of the gas-particle is treated within a WKB-approximation. The corresponding classical trajectory occurs as an input into a sudden phase shift for the rotational excitation of the molecule and into the calculation of the forced oscillator model for the phonon system. We have then applied our theory to the system NO/Pt(ll 1). The distribution of angular momenta after direct inelastic scattering is described correctly. The “prompt” sticking coefficient, which is given by the probability that an incoming gas particle has a kinetic energy Epf < 0 after the scattering event, turns out to be high for smail initial angular momenta (S = 0.9) due to a large energy transfer to phonons.
C’. Haag et al. / NO on Pt(1 I I)
66
Orientational and alignment effects contained in the adsorption probability, which have been observed in experiments, are reproduced correctly by our theory. Concerning the phenomenon of rotational cooling one has to go into a more detailed analysis of the sticking process. We considered a simple model containing rotationally mediated selective sticking resonances. The strong increase of elastic resonance widths with increasing initial angular momentum leads to a decrease of the sticking coefficient. Rotational cooling then is obtained in semiquantitative agreement with experiment.
References [I] [2] ]3] [4] [5] [6] [7] [8] [9] [IO] [ll] [12] [13] [l4] [15] [16] [l7] [18] [19] [ZO] [21] [22] [23] [24] [25] [26] 1271 [28] [29]
R.J. Gorte, L.D. Schmidt and J.L. Gland, Surf. Sci. 109 (1982) 355. CT. Campbell, G. Ertl and J. Segner, Surf. Sci. 115 (1984) 309. J. Segner. H. Robota, W. Vielhaber, G. Ertl, F. Frenkel, J. Hager, W. Krieger and H. Walter, Surf. Sci. 131 (19X3) 273. CM. Comrie, W.H. Weinberg and R.M. Lambert, Surf. Sci. 57 (1976) 619. T.H. Lin and G.A. Somorjai, Surf. Sci. 107 (1981) 573. J.A. Serri. M.J. Cardillo and G.E. Becker, J. Chem. Phys. 77 (1982) 2175. D.C. Jacobs, K.W. Kolasinski, SF. Sh,ane and R.N. Zare. J. Chem. Phys. 91 (1989) 3182. E.W. Kuipers, M.G. Tenner. A.W. Kleyn and S. Stolte, Phys. Rev. Lett. 62 (1989) 2152. C.W. Muhlhausen, L.R. Williams and J.C. Tully, J. Chem. Phys. 83 (1985) 2594. J.C. Tully, in: Kinetics of Interface Reactions, Eds. M. Grunze and H.J. Kreuzer. Vol. 8 of Springer Series in Surface Science (Springer. Berlin, 1987) p. 37. D.C. Jacobs and R.N. Zare. J. Chem. Phys. 91 (1989) 3196. D. Lemoine and G.C. Corey, J. Chem. Phys. 94 (1991) 767. T.P. Tsien and R.T. Pack, Chem. Phys. Lett. 6 (1970) 54. D. Secrest, J. Chem. Phys. 62 (1975) 710. R.B. Gerber, A.T. Yinnon, Y. Shimoni and D.J. Kouri, J. Chem. Phys. 73 (1980) 4397. P. Carruthers and M.M. Nieto, Am. J. Phys. 33 (1965) 537. L. Trilling, Surf. Sci. 21 (1970) 337. A. ModI, T. Gritsch, F. Budde, T.J. Chuang and G. Ertl, Phys. Rev. Lett. 57 (1986) 3X4. T. Brunner, R. Brako and W. Breniy. Phys. Rev. A 35 (1987) 5266. P. Pechukas, Phys. Rev. 181 (1969) 166. P. Pechukas, Phys. Rev. 181 (1969) 174. IS. Gradshteyn and I.M. Ryzhik, Table of Integrals. Series and Products (Academic Press, San Diego. 1980). W. Brenig. Z. Phys. B 48 (1982) 127. C. Haug and W. Brenig, to be published. W. Brenig, H. Kasai and H. Miiller, Surf. Sci. 161 (1985) 608. C. Haug, W. Brenig and T. Brunner, Surf. Sci. 227 (1990) 167. J.W. Gadzuk, U. Landman, E.J. Kuster, C.L. Cleveland and R.N. Barnett, Phys. Rev. Lett. 49 (1982) 426. J.M. Bowman and J.L. Gossage, Chem. Phys. Lett. 96 (1983) 481. A. Redondo, Y. Zeiri. J.J. Low and W.A. Goddard III, J. Chem. Phys. 79 (1983) 6410.