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## Material Information- Title:
- Scattering of atoms by molecules adsorbed at solid surfaces
- Creator:
- Parra, Zaida, 1947-
- Publication Date:
- 1988
- Language:
- English
- Physical Description:
- vii, 166 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Atomic interactions ( jstor )
Atoms ( jstor ) Energy transfer ( jstor ) Ions ( jstor ) Molecular interactions ( jstor ) Molecules ( jstor ) Projectiles ( jstor ) Symmetry ( jstor ) Trajectories ( jstor ) Vibration mode ( jstor ) Adsorption ( lcsh ) Atoms -- Scattering ( lcsh ) Chemisorption ( lcsh ) Chemistry thesis Ph. D Collisions (Physics) ( lcsh ) Dissertations, Academic -- Chemistry -- UF Energy transfer ( lcsh ) Vibrational spectra ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1988.
- Bibliography:
- Includes bibliographical references.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Zaida Parra.
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SCATTERING OF ATOMS BY MOLECULES ADSORBED AT SOLID SURFACES By ZAIDA PARRA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 r'P!MsrY OF FLORIDA LIBRAM TO GILBERTO CARLOS AND CAMILO ACKNOWLEDGEMENTS I would like to thank my advisor, Professor David A. Micha, for his invaluable ideas, guidance and support during the development of this work. Also, I wish to extend my appreciation to the professors of the Quantum Theory Project who have contributed to my education and made the Quantum Theory Project such a stimulating and enriching environment. I would particularly like to express my gratitude toward Professor Per-Olov L6wdin who made possible my participation in the 1983 Summer School in Quantum Theory in Uppsala, Sweden. My deepest appreciation is to my husband Roy Little, for his support and patience during these years of studies. Finally, my thanks go to Robin Bastanzi for her help in the typing of this dissertation. TABLE OF CONTENTS ACKNOWLDGEMENTS ............................................... ABSTRACT ï¿½..iii . .................................................... ...vi CHAPTERS 1. INTRODUCTION .............................................. 1 2. COLLISIONAL TIME-CORRELATION FUNCTION APPROACH TO ENERGY TRANSFER .............................. 7 2.1 Many-Body Approach .................................. 8 2.2 The Impulse Approximation .......................... 12 2.3 Application to Scattering by Adsorbates ............ 16 3. VIBRATIONAL FREQUENCIES AND NORMAL MODES OF MOLECULES ADSORBED ON SURFACES .... 3.1 Cluster Models and Force Fields ....... 3.2 Normal Modes Analyses ................. 3.3 Numerical Results for CO on Ni(001)... ............. 19 ............. 21 ............. 28 ............. 33 4. ENERGY TRANSFER INTO MOLECULAR ADSORBATES ............... 46 4.1 The Vibrational Correlation Function ............... 47 4.2 Short Time Expansion ............................... 50 4.3 Statistical Model for Single and Double Collisions .................................. 51 5. GAS-ADSORBATE INTERACTION POTENTIALS: EFFECTIVE DIFFERENTIAL CROSS-SECTIONS ...... ............. 57 5.1 Atom (Ion)-Adsorbate Interaction Potential ......... 58 5.2 Effective Classical Differential Cross-Sections .... 80 6. VIBRATIONAL ENERGY TRANSFER IN HYPERTHERMAL COLLISIONS OF He AND Li+ WITH CO ADSORBED ON Ni(O01) ............... 94 6.1 Double Differential Cross-Sections for the System He/OC-Ni(O01) ....................... 94 6.2 Double Differential Cross-Sections for the System Li + /OC-Ni(O01) ...................... 97 7. SUMMARY AND CONCLUSIONS ................................ 146 APPENDICES A. COEFFICIENTS FOR TRANSFORMATION s=AX ................... 155 B. SYMMETRY ADAPTED COORDINATES ........................... 158 C. COEFFICIENTS FOR TRANSFORMATION Q=CX ................... 159 REFERENCES ........................................................ 162 BIOGRAPHICAL SKETCH ................................................ 166 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SCATTERING OF ATOMS BY MOLECULES ADSORBED AT SOLID SURFACES By ZAIDA PARRA August, 1988 Chairman: David A. Micha Major Department: Chemistry The formalism of collisional time-correlation functions, appropriate for scattering by many-body targets, is implemented to study energy transfer in the scattering of atoms and ions from molecules adsorbed on metal surfaces. Double differential cross-sections for the energy and angular distributions of atoms and ions scattered by a molecule adsorbed on a metal surface are derived in the limit of impulsive collisions and within a statistical model that accounts for single and double collisions. They are found to be given by the product of an effective cross-section that accounts for the probability of deflection into a solid angle times a probability per unit energy transfer. A cluster model is introduced for the vibrations of an adsorbed molecule which includes the molecular atoms, the surface atoms binding the molecule, and their nearest neighbors. The vibrational modes of CO adsorbed on a Ni(001) metal surface are obtained using two different cluster models to represent the ontop and bridge-bonding situations. A short-time expansion is introduced for the slow modes of the target, which leads to Gaussian distributions of the energy transferred into the slow modes. A He/OC-Ni(001) potential is constructed from a strongly repulsive potential of He interacting with the oxygen atom in the CO molecule and a van der Waals attraction accounting for the He interaction with the free Ni(O01) surface. A potential is also presented for the Li+/OC-Ni(O01) where a coulombic term is introduced to account for the image force. Trajectory studies are performed in three dimensions to obtain effective classical cross-sections for the He/OC-Ni(001) and Li+ /OCNi(001) systems. These cross-sections are then used to calculate the double differential cross-sections per unit solid angle and energy transfer for the same systems. Results for the double differential cross-sections are presented as functions of scattering angles, energy transfer and collisional energy. Temperature dependence results are also analyzed. Extensions of the approach and inclusion of effects such as anharmonicity, collisions at lower energies, and applications of the approach to higher coverages are discussed. CHAPTER 1 INTRODUCTION In the past few years, there has been an increasing interest in the study of metal surfaces by a wide variety of experimental techniques. An important aspect of metal surfaces is their ability to effectively adsorb atoms and molecules, leading to chemisorption, surface reactions, or catalysis. The experimental study of the vibrational modes of atoms and molecules adsorbed at surfaces has proved to be a powerful means of deriving information about changes in molecular bonding which occur during chemisorption processes and also on the preferred adsorption sites (Rocca et al., 1986). Vibrational spectra of adsorbates have been observed with infrared reflection spectroscopy (IRS) and with low-energy electron scattering (EELS). These techniques are in practice sensitive only to the dipole-active high frequency modes of vibration, which correspond to either intrinsic vibrations of the free molecule, or to the high frequency vibration of the molecule perpendicular to the surface. A further class of low frequency modes, the hindered rotations and translations parallel to the surface, also exists but their observation is prevented by the instrumental resolution in EELS and the limited spectral range and sensitivity of IRS. A technique which is in principle an ideal method for studying these vibrations is inelastic atom scattering spectroscopy. It also has the capability to observe the dynamics of surface adsorbate layers and is sensitive to very small amounts of adatoms on smooth metal substrates and can, therefore, give information about isolated adatoms (Gadzuk, 1987). Several articles have appeared on experimental aspects of scattering by adsorbates. Systems which have been studied by neutral atom scattering include, among others, He with either 0 or CO on Ni(O01) (Ibafiez et al., 1983) and He with CO on Pt(lll). In the latter system cross-sections have been measured as a function of the velocity of the incident He atoms and the angle of incidence (Poelsema et al., 1983) and as a function of scattering angle and momentum transfer (Lahee et al., 1987). All of the experiments with neutral atoms have been done with energies in the thermal regime. Ions have also been used as probes in experiments of scattering by adsorbates. In one of the pioneer works in the area (Hulpke, 1975), energy and angular distributions of Li+ ions scattered by clean W(110) surfaces and by adsorbates, such as 02 and CO, were measured for beam energies between 2 and 20 eV. Scattering intensities in the systems Li+-O/Ni(llO) and He+-O/Ni(llO) have been measured as a function of the scattering angles and relative energies of the ions (Englert et al., 1983). Recently, measurements on angle and energy distributions of K+ ions scattered from W(110) in the energy range of 12 to 100 eV have been published (Tenner et al., 1986). These measurements have been done at normal incidence. In general, most of the theoretical approaches to the problem of scattering by adsorbates have concentrated on elastic scattering. Furthermore, since a detailed theoretical treatment is virtually impossible to accomplish because of the computing time and memory required, approximate schemes are a necessity. Several approaches to the scattering of atoms with thermal energies by adsorbates have been based on extensions of methods which have been useful in the description of scattering by clean surfaces. The general theory of atom scattering in the eikonal approximation has been extended to scattering by overlayers (Levi, 1982). In this theory the motion of the adsorbate is assumed to be separated from the motion of the substrate and does not contain information on the site occupied by the adsorbate. A theory for scattering by adatoms at low coverage has been presented (Jonsson et al., 1984). Here the surface is considered to be a hard wall and the cross-section for the isolated adatom is simply given by an effective scattering amplitude obtained by subtracting the scattering amplitude of a hard wall from the gasphase scattering amplitude for the adatom. This theory has been applied to the system He-OC/Pt using the distorted wave Born approximation (Liu and Gunhalter, 1987), and the cross-sections have been calculated from the formalism of close coupled equations. However, the possibility of inelastic scattering increases the computational complexity of the problem. Quantal approaches have been based on wavepackets (Gerber et al., 1984) and on multiplecollision expansions (Singh et al., 1986). With a few exceptions (Vilallonga and Rabitz, 1986) theoreticians have considered only elastic Bragg scattering which gives information on the structure of adsorbate-covered surfaces or the elastic component of diffuse scattering from adsorbates. However, the inelastic scattering is of great interest because it involves the excitation of surface phonons and the vibrational modes of the adsorbed molecule. When an atom collides with a molecule adsorbed on a metal surface, kinetic energy can be transferred into any of the infinite number of vibrational modes of the target. Consequently, one is not interested in the state-to-state cross-sections familiar in scattering by isolated molecules, but instead one wants double differential cross-sections, per unit of solid angle and of energy transfer. Furthermore, the quantities of experimental interest are averages of these cross-sections with respect to statistical distributions of target states for the given surface temperature. The purpose of this dissertation is to present a model of energy transfer for the scattering of atoms and ions from diatomic molecules adsorbed on a metal surface. Recent experiments have shown (Berndt et al., 1987) that information on the type of interaction between atoms and adsorbed molecules as well as on the vibrational modes of adsorbed molecules can be obtained from isolated adsorbates. Therefore, we are interested in a model which would be applicable to low coverages, and in a simple theory which could provide insights into the type of interactions and also be able to predict trends in the energy and angular distributions of the scattered particles. Standard treatments of molecular collisions, based on target state expansions, are not convenient because excited target states are usually unknown. Instead, one can use the formalism of collisional time-correlation functions (Micha, 1986) which is appropriate for scattering by many-body targets and includes statistical averages. For hyperthermal collisions, the interaction is dominated by strong short range repulsive forces and lasts a short time compared with the periods of internal motions of the target, so that an impulsive model is appropriate. Furthermore, as long as the projectile is light compared with target atoms, only a single collision occurs with the adsorbate before the projectile flies away. These two features lead to a simple expression for the timecorrelation, which can then be obtained analytically from the normal vibrational modes of the target. The vibrations of an adsorbed molecule can be modelled by a cluster which includes the molecular atoms, the surface atoms binding the molecule, and their nearest neighbors. The normal modes of the target can be grouped into two sets, one with high frequencies (fast modes) similar to those of the isolated molecule and the other with low frequencies (slow modes) characteristic of the surface vibrations. When the target involves slow modes, one can approximate the time-correlation functions by short time expansions which lead to Gaussian distributions of the energy transferred into the slow modes. The present work is presented here as follows. In Chapter 2 the formalism of collision time-correlation functions is presented together with expressions for the differential cross-sections within the impulsive model, and its application to scattering by adsorbates. Two models are developed in Chapter 3 to represent on-top and bridge adsorption sites of a CO molecule adsorbed on Ni(001). Depending on the angle of scattering, the interaction between the projectile and the target may involve only the single collision with the adsorbate, or that followed by a collision with the solid surface. In Chapter 4 we develop a statistical model to calculate contributions of double collisions to the cross-sections. In order to calculate the double differential cross-sections, one first needs to obtain effective classical cross-sections. We use the facts that the potentials of the separate adsorbates do not overlap and that the translational wavelength of the projectile is much shorter than the range of the intermolecular potential. Thus, translation of the projectile is obtained by a classical trajectory. In Chapter 5 the interaction potential for the systems Li+-OC/Ni(001) and He-OC/Ni(O01 are constructed and trajectory studies are performed to obtain the effective classical cross-sections. In Chapter 6 results of double differential cross-sections for the systems Li+-OC/Ni(001) and HeOC/Ni(O01) are given as a function of energy transfer and scattering angles. Finally, in Chapter 7 the advantages and disadvantages of the approach are presented together with some suggestions on possible improvements of the approach. CHAPTER 2 COLLISIONAL TIME-CORRELATION FUNCTION APPROACH TO ENERGY TRANSFER When an atom is scattered by a target, energy can be transferred from the projectile into the target or out of the target to the projectile. The amount of energy transferred is determined to a large extent by the intramolecular dynamics of the target, therefore it is expected that atomic motions within the target will place certain constraints on the process. The theoretical description of the energy transfer between atoms and extended targets can be done using classical, semiclassical or quantum mechanical methods. The quantum mechanical methods are usually based on expansions in target states but are computationally impractical and costly due to the large number of energetically accessible levels of the target. An alternative approach which takes into account the quantal nature of the target motion, but avoids expansions in target states has been developed (Micha, 1979a) and used in the study of atom-polyatomic collisions (Micha, 1979b). In such an approach the target is described as a many-atom system and atom-pair correlation functions of the target play an important role in the description of the pocess. In this chapter we review the main features of this many body theory in its exact form and then discuss a simplification of the approach, namely the impulse approximation, which makes the approach more tractable for its application to the scattering of atoms by adsorbates. 2.1 Many-Body Approach The type of process we are interested can in general be described by A(Ei) + X(v) -> A(Ef) + X(\'), where A denotes an atom with an initial kinetic energy E1 and X represents a many atom system, e.g., an adsorbate on a metal surface, in an initial internal state v. During the collision an amount of energy s is transferred between them, promoting X to a final internal state \' and leaving A with a final kinetic energy Ef. In the laboratory frame, with A denoting the position of the projectile the Hamiltonian of the projectile is simply its kinetic energy HA -2m A (2.1) and the internal Hamiltonian HX for the target may be written as M2 4X HX = E -2 V + V , (2.2) a a a where ra is the position of atom a and r = [ra) the collection of these vectors. In the coordinate system fixed to the center of mass of the pair (A,X), the system Hamiltonian is given by HH 0 + V ,(2.3a) where H = K + H4X (2.3b) r Here K is the operator for the kinetic energy of the relative motion given by S2 K -f- 4 Y(2.3c) where is the position of the projectile A with respect to that of the target and M is the reduced mass of the system, M = MAMx/(MA + Mx), with MA and M the masses of the projectile and the target, A XX respectively. The Hamiltonian of the isolated target is H4X, with r r denoting the position of the atoms in the target. The operator V represents the interaction potential which depends on and on r Indicating with E and Iv > the internal energy and the internal state of X, respectively, we can write for the unperturbed motion HOl i > = 2 + EJI i > , (2.4a) < I Xl > = (2R)-312 e'" < r4XI > , (2.4b) where hK is the relative momentum. Given an interaction potential V between the projectile A and the target X, the collision probability amplitudes may be obtained from the transition operator, T, which satisfies the Lippman-Schwinger equation T = V + V G0 T , (2.5) where G0 = (E + iO - H0)- is the propagator for free relative motion of A and X, but involves all the internal motions of X. Considering scattering from an initial state kv >, where p = is the relative momentum and v is the internal state of the target, into a final state IP'v'>, then the scattering rate is given by R(kV 4 k'V') = (21/14) I< '\' ITI v >21 (E-E' ) , (2.6) where the 8 function ensures conservation of the total energy of the system. However, since experimental measurements usually correspond to thermal averages over initial distributions, wV , and they only specify 9 and i', we must consider the total rate for scattering between initial and final momenta, that is R(k -k') = (2n/1) Ew I ' ' ITI k' >12 6(E - E') (2.7) The total initial and final energies E and E' must be equal, as it is insured with the 8 function. The sum over final states in Eq. (2.7) usually contains a very large number of terms, specially at hyperthermal collision energies. However, by using the completeness of the states and the integral representation of the 8 function 8(E - E') = j dt/(2nH) exp[-i(E - E')t/] , (2.8) the sum over final states may be formally eliminated (Micha, 1986), this gives the result (2.9a) x exp(iH xt/h) T exp(-iH xt/h)I ' >, where Hx is the target Hamiltonian, and c is the energy transfer defined by, 4= K2k - (k) 2]/2M (2.9b) The operator T depends on all variables, so its matrix elements T _, = < i' ITI > , (2.10) vary only with the internal variables. Substituting Eq. (2.10) into (2.9a) and the latter into Eq. (2.7) one arrives at the following expression for the total scattering rate, R( k-1') = M-2 dt e- it/h Fj,j(t) (2.11a) Fi,k(t) = << T ,k(O) T , (t) >> (2.1lb) Tj,j(t) = exp(iHUt/K) T--,j exp(-iHxt/h) (2.11c) where the double brackets indicate the quantum mechanical and thermal averages over initial states. The integral in Eq. (2.11a) contains the time correlation function (TCF) of the matrix elements T which are transition operators on internal variables. The time dependence of the transition is given by the internal Hamiltonian of the target. This general equation for the total scattering rate is particularly useful when compared with other approaches which would fail when many internal states are involved. Once the transition operators are expressed in the variables of a given system, the time dependence of the latter may be followed much as one would in a classical treatment. The double differential cross-section for scattering into a unit solid angle 9 and with a transfer of energy of c may be obtained from the ratio of the transition rate to the incident flux (Micha, 1979a), to give a) d 2a/(dc dQ) = (2T/K)4 M 2(k'/k) dt exp(-ict/K) x (2.12a) x << T , (0) T ,k( t) >>/(2nh), where the relation between the final momentum and the energy transfer is given by 2 2 1/2 k' = (k -2Mg/h2) (2.12b) The cross-section is thus expressed as the Fourier transform of the time-correlation function of the transition operator. This correlation function approach allows the development of dynamical approximation to the transition operator in a systematic way, as it is shown in the following section. 2.2 The Impulse Approximation In the regime of hyperthermal collisions, the projectile probes the internal regions of the interaction potential V which is of a multicenter nature. In these cases the potential may be conveniently represented by an expansion about the atomic centers of the target (Micha, 1979a) r = v ,) , (2.13) a The potential v represents the interaction between the (A,a) a atom pair and depends on the electron distribution of A and on that of the valence state of a in X. In the formalism of multiple scattering (Rodberg and Thaer, 1967) the many-body transition operator presented in Eq. (2.5) can be expressed in terms of the two body potentials v a' leading to the final channel decomposition T E T(a), (2.14a) a T(a) = Ta + Ta G 0 T(b), (2.14b) b a Ta = va + va G0 Ta t (2.14c) where T(a) corresponds to a final interaction between A and atom a mediated by va, and Ta is the transition operator when the only allowed interaction is v a, but in which all internal potentials of the target are included in GO. Introducing the assumption that the majority of collisions involve a single encounter between the projectile atom A and the target atom a, allows us to write T(a) T a. However, in order to calculate Ta, a N + 1 body problem must still be solved because GO involves the motions of all the atoms in the target. This may be substantially simplified whenever the energy transfer occurs in the course of an impulsive collision of the projectile A with atom a. The basic assumption of the impulse approximation is that a large force F acts on an atom a for a period of time At so short, that all a the other forces may be neglected during that time. The position of atom a does not change but its kinetic energy < Ka>v in the internal state v jumps to a new value < Ka > Energy and momentum are transferred to the target through the interaction between the pair (A,a), while the remaining N-1 atoms in the target provide the restoring forces on a which determine its momentum distribution within the target. To incorporate these ideas into formulas one can introduce the (A,a) pair-Hamiltonian h0,a = KR + Ka + E0 (2.15a) KR = _K2V2 /(2M) , (2.15b) RR K = -h2V2 /(2m (2.15c) a r a a E =E -< K> -< K > , (2.15d) and the propagator -l go,a (E) = (E + iv - hOa) 1 (2.16) so that the propagator for free relative motion Go can be written as G0(E) = go,a (E) + go,a(E) (HX - E0 - K a) Go(E) (2.17) Substituting Eq. (1.17) in the equation for Ta, one finds Ta = ta + ta(G0 - go0a) T , (2.18) where t a(E) = va + va go'a(E) ta(E) (2.19) Provided G and go,a are close in regions where ta is different from zero, one can drop the second term in Eq. (2.18) and write Ta t (E) which constitutes the impulse approximation. Replacing T = ta in the equation for the total rate, one finds a (a) R(k 4 k') - . Rba , (2.20a) a,b R (ba) =2n k',0't I ko >* H VV b V N (2.20b) x < k'v Ita I kv > S(E - E') which shows the contribution of each (ab) atom pair in the target. Double differential cross-sections may now be obtained (Micha, 1979b) from the Eqs. (2.20) and one finds dM r2 rg ,R(ba) M dE Rb (2.21a) a,b R(ba) 2n b ( A* 4 a) () (2.21b) (ba)dt -it/h -iKrb(0) i ra(t) sb(KS) = w T J e (2.21c) 4 Here K is the momentum transfer defined by MK = M(U_'). The factor a ( ',A) involves only the relative motion of the pair (A,a). The second factor, S(ab)(K,C), is the Fourier transform of the atom-pair time-correlation function (Van Hove, 1954) and depends only on the internal dynamics of the isolated target. The correlation function for a pair of different atoms (aeb) contains phases that depend on the initial values of the dynamical variables of the target. Experimental conditions usually correspond to initial averages over random phase so that the terms with a=b average to zero and the cross-sections are then given by d 2 44 2 (aa) 4F de S S (K) . (2.22) a Each term in the sum above is a product of a deflection probability and a target absorption probability. The deflection probability can be written as an effective atomic differential cross-section a(k',k) k CI T -(,' ) , (2.23) in terms of which Eq. (2.22) becomes de d - a S(aa) ( (2.24) a The impulse approximation presented here will hold provided collisions occur in times short compared with internal motions, and provided they distort only a local region of the target around a. On the other hand, multiple collision effects may be neglected only for certain conditions on the kinematics and potentials. For example, multiple collision terms would be small for mA/mX << 1 and kRab >> 1, that is for light projectiles and wavelengths short compared with interatomic distances in the target. Geometric considerations will also play a role when considering multiple collision effects. In general the validity of these assumptions must be reconsidered in each particular case. 2.3 Application to Scattering by Adsorbates Several changes in the structure of a crystal surface can occur when adsorbates are introduced. One of the most important changes is the loss of periodicity which is associated with bare crystal surfaces. In general three cases can be distinguished. In the first case the adsorbates form a regular lattice on the substrate. In such a case the periodicity parallel to the surface is maintained but a new unit cell, usually larger than the original unit cell of the substrate, is needed in order to describe the system. In the second case the periodicity parallel to the surface is destroyed. Here we may think of a single adsorbate on the surface, or a non-periodic arrangement of adsorbates. The final case is that in which several layers of atoms are deposited on a metal substrate. In these cases the periodicity of the system as a whole, parallel to the crystal surface, is lost, usually because the adsorbate spacing differs from the atom spacing in the substrate. The theoretical description of collisions between atoms and adsorbates requires a variety of approaches, depending on the magnitudes of collision energies and on the extent and type of surface coverage. The many-body approach has been previously used to study energy transfer in hyperthermal collisions of ions with a solid surface (Micha, 1981). In such cases the scattering intensity is concentrated around and within surface rainbow angles, er, with single-atom collisions contributing over all scattering angles e while double collisions may contribute at certain angles. For scattering by light projectiles with energies in the hyperthermal region, where the projectile probes the repulsive region of the adsorbate and of the atoms in the uppermost layer of the substrate, the double differential cross-section is given in a similar way as in the case of surface scattering d 2a (11) dc d2Q E a (k,S) S (K,s) , (2.25) 1 where 1 denotes atoms in the target, which can now be adsorbate atoms or surface atoms. Here a1(k,Q) is the elastic differential crosssection for the projectile-atom 1 in the target. The surface atoms lie on the plane (x,y,z=O) and the adsorbate is placed at a distance Z a, given by the equilibrium position of the adsorbate above the surface plane. The projectile moves in the direction of decreasing Z, with a certain incident angle 6. measured from the Z axis normal to the surface. The potential at center I derives from the valence electronic state of atom 1 in the target and can be obtained from semiempirical potentials whose parameters have been fitted to reproduce experimental data. The elastic differential cross-section a1(k,Q) may then be obtained from a bundle of trajectories with impact parameters b within a bare surface unit cell. The factor S(11) has the meaning of a probability per unit energy for adsorption of momentum hK and energy transfer c. This self-correlation function, S (11) be obtained from a , , must b bandfo model Hamiltonian for the system. A procedure which can be used, appropriate in cases of low coverage, is to describe the target by a finite cluster of N atoms, including the adsorbate and a few layers of the substrate (Parra and Micha, 1986). This model would provide the vibrational modes which are localized in the vicinity of the surface and which are the ones that can be detected by experimental tools. Since scattering from adsorbates resembles scattering from clean surfaces, it is expected that double collisions may contribute at certain scattering angles. The application of Eq. (2.25) is only valid in regions of single collisions. However a statistical approach, which uses the geometry of the problem and allows for the description of double collisions, will be presented in a later chapter. CHAPTER 3 VIBRATIONAL FREQUENCIES AND NORMAL MODES OF MOLECULES ADSORBED ON SURFACES The vibrational spectra of molecules adsorbed on metal surfaces can be studied with a variety of experimental techniques involving scattering of electrons (Ibach and Mills, 1982), photons, neutrons and neutral or ionized atoms (Willis, 1980). The spacings and shapes of spectral lines provide information about the conformation of the adsorbate and its substrate, and about the force field that determines the vibration dynamics. When the probing particles are atoms, translational energy is transferred into both adsorbate and substrate, possibly resulting in the shift and reshaping of spectral lines. The theoretical interpretation of the spectra can be simplified when the internal dynamics of the adsorbate can be identified and separated from the collision dynamics. The normal modes of adsorbates on solid surfaces can be obtained from generalizations of theories developed for surface lattice dynamics. Approaches can be based on Green functions (Rahman, Black and Mills, 1982), slab models with a small number of layers (Strong et al., 1982), and cluster models. Green functions require-ektensive work while slab calculations, although less involved, may not al-ay properly describe the influence of site symmetry (Rahman et al., 1983). Furthermore, they have been used together with simple force fields of the central-field type. However, one expects that more detailed force fields will be needed, containing in particular 19 bending forces, when one wishes to describe the collisional excitation of adsorbate vibrations. Cluster models have been found adequate to describe adatoms (Black et al., 1982), and in some cases have been as effective as models based on periodic small clusters (Black, 1982). Applied to molecular adsorbates, those models can be parametrized to incorporate what is known about the vibrational modes of the isolated molecule and clean surface. They also allow for detailed valence-force fields and for the incorporation of the point group symmetry of the adsorption site. Experimental studies (Andersson, 1977) of CO on Ni(O01) have provided evidence that CO molecules can bind on a top site above a Ni atom, on a bridge site between two of them or in a mixture of both (Bertolini and Tardy, 1981). For a dilute CO lattice gas on Ni(O01), two peaks are observed, at 239 and 256 meV, in electron energy loss spectra (EELS). However, at higher coverages only a single peak at 256 meV is observed. These findings have been interpreted as due to the presence of both top and bridge species in the case of low coverages, but of only the on-top species in the higher coverages (Andersson and Pendry, 1979). Model calculations on small clusters have been performed by several workers to obtain vibrational frequencies of a CO molecule adsorbed on a nickel surface. The frequencies of a Ni5CO cluster for %5 the on-top site and a Ni6CO cluster for the bridge site have been analyzed (Richardson and Bradshaw, 1979), in terms of semi-empirical force fields, with only the C and 0 atoms moving. The frequencies of a linear NiCO molecule have been calculated using ab initio methods (Allison and Goddard, 1982) based on general equations for XYZ type molecules. In this chapter, we develop a simple cluster model of diatomic adsorption and apply it to CO adsorbed on Ni(O01). It is based on a more detailed, valence-force field than previously found in the literature, and it incorporates site symmetry. It also allows for moving surface atoms so that a more detailed picture of the adsorbate dynamics can be derived. Furthermore, the substrate cluster contains two surface modes whose frequencies can be fitted to known values at the r-point of the surface Brillouin zone. We analyze the frequencies of a CO molecule adsorbed on Ni(O01) using two different clusters, intended to model the on-top and bridge species. Section 3.1 describes the conformation and force fields for the clusters. Section 3.2 presents the normal mode analysis based on the FG method of molecular spectroscopy (Wilson, Decius and Cross, 1955). The numerical results and figures showing the normal modes are given in section 3.3. 3.1 Cluster Models and Force Fields To model adsorption at the on-top site, we use an eleven-atom cluster, shown in Fig. 3.1, containing C, 0 and nine Ni atoms, with C, 0 and one Ni atom allowed to move. This cluster contains all metal atoms that are nearest neighbors to the bonding site. four in the first layer and four in the second layer, with the CO bonded to a Ni atom in the face centered cubic structure. In the bridge-bonded situation, we have chosen a 16-atom cluster, shown in Fig. 3.2, that contains fourteen Ni atoms, of which the two bonded to C are allowed to move. This cluster also contains all Figure 3.1: 0 C On-Top Cluster. Nickel atoms with numbers 1 to 5 correspond to surface atoms, and with numbers 6 to 9 to nearest neighbors in the second layer; C,O and nickel atom 1 are the only ones allowed to move. 0 C Figure 3.2: Bridge Cluster. Nickel atoms with numbers 1 to 8 correspond to surface atoms, and with numbers 9 to 14 to nearest neighbors in the second layer; C,O and nickel atoms 1 and 2 are the only ones allowed to move. The height of the C atom above the surface (dc) is determined by the angle Ni-C-Ni. metal atoms that are nearest neighbors to the two metal atoms representing the bonding site, six in the first layer and six in the second layer. The carbon atom of the CO molecule is bonded to two nickel atoms, one in a position corresponding to a face center atom and the other one corner in the face centered cubic structure. A C4v point group can be associated to the top bonding situation whereas a C2v point group is more appropriate for the bridge model. Since on a real surface all translation and rotations are restricted, we expect all 3N degrees of freedom to be vibrational modes. Therefore, we should find 3 x 3 = 9 vibrational modes for the top case. These can be classified as three hindered translations, two hindered rotations and four modes which are equivalent to the vibrational modes of an XYZ isolated linear molecule in the gas phase. The latter could also be considered as arising from the vibration of the isolated CO molecule interacting with two phonon modes of the substrate represented by a single nickel atom. For the bridge bonded situation two of the metal atoms are allowed to move, given a total of 3 x 4 = 12 vibrational modes; six of these correspond to the molecular modes of a X2CO isolated molecule in the gas phase. Again this could also be considered as the vibration of the CO molecule interacting with four phonon modes of the substrate represented by two nickel atoms. In order to carry out calculations of the frequencies, to obtain the actual form of the normal modes and to classify the different vibrational modes according to the symmetry of the clusters, we have chosen a valence force field to represent the potential energy of the clusters. Usually, the potential energy is expressed in terms of the changes in the internuclear distances and in a number of angles. For the on-top position we used a total of 12 internal coordinates, eight of these coordinates to represent the changes in the Ni-Ni internuclear distances, one each for the changes in CO and NiC bond distances, one for the change in the angle between the CO and C-Ni bond and one for the change in the angle between C-Ni bond and the adjacent Ni-Ni bond. Denoting by s.. the changes in the Ni-Ni internuclear distances, by sco and sCNi the changes in CO and CNi bond distances and by s a and sb the changes in the angles described above, the potential energy VT for the on-top cluster is given by 5 9 2VT(S) = f(1) E 2 + f(2) E s2 +Cf s2 Ni -l lk CNi CNi j=2 k=6 + f 2 + 2 + f 2 CO CO a a b b (3.1) In the above expression the subscript 1 had been used to denote the metal atom bonded to the CO molecule, f(l) and f(2) denote the force Ni Ni constants for the interaction between neighboring atoms in the first layer and that between first and second layer atoms. A total of 22 coordinates were used for the bridge bonded model; 15 to represent changes in the Ni-Ni internuclear distances, 3 to represent changes in the bond distances of the adsorbed molecule and four to represent changes in the angles. Denoting by sd the change in the angle between the CNi bonds, by s the change in the angle between the CO bond and the plane Ni-C-Ni, and by sa and sb the same variables as for the on-top position, the potential energy VB can be written as 5 8 12 14 2VB(s) = fl) E Sl2+ E s2j + fSl) E ik2 + S2k j=2 j=6 k=9 k=9,12 (3.2) + 2 + f s2 2 + 2 +fS2 +fs2 CO CO CNi CNi a a d d g g b b Phonon dispersion curves have been obtained (Black et al., 1981) for the Ni(O01) surface at different points of the substrate Brillouin -1 zone. Two surface modes with frequencies of 104.7 and 125.5 cm and polarizations parallel to the surface and normal to the surface, respectively, were found at the r zone boundary point. These values were used to fit the two force constants f(l) and f (2) Ni NiinE.(2) The interatomic distances and the other force constants were taken from data for Ni(CO)4, (Jones, 1960). For the bridging species the force constants fd and f were fitted to reproduce the frequencies for a molecule of the type X2CO (Herzberg, 1945), which is the limiting case of the bridge cluster after the neighboring atoms of the Ni bonded to the CO molecule have been taken away. The chosen force constants and geometric parameters are presented in Table 3.1 and 3.2 respectively, where dNi is the distance between nearest neighbors in the bulk crystal structure, and dC is the distance from the C atom to the surface in Fig. 3.2. As a preliminary check of the model and the chosen field, we calculated the frequencies of an isolated XYZ molecule to represent the limiting case for the oh- top position cluster. The results are within 1 cm , as can be seen in Table 3.3, from those obtained by an ab initio calculation on the same molecule (Allison and Goddard, 1982). For the cluster representing the bridge bonded situation, the limiting case was H2CO, obtained by replacing in our cluster the Table 3.1 Force Constants for Internal Coordinate Sets Force Constants Top Bridge fC0 (mdyne A-) 17.0 14.7 fCNi (mdyne A-) 2.1 1.9 f (1) -1 0.15 0.15 ~Ni(2 fNl(2) (mdyne A-') 0.27 0.27 f (mdyne A rad- 2) 0.38 0.38 a fb (mdyne A rad-2) 0.23 0.23 fd (mdyne A rad-2) 0.23 f (mdyne A rad-2 ) 0.38 g Table 3.2 Geometric Parameters of Clusters Geometric Parameters Top Bridge dCO (A) dCNi (A) dNi (A) dC (A) Ni-C-Ni angle (deg) 1.15 1.84 2.49 1.15 1.60 2.49 0.84 112.00 -1 Table 3.3 Comparison of Vibrational Frequencies (in cm) for a Linear NiCO Molecule Normal Mode Our Results (Allison and Goddard, 1982) CO Stretch 2130.5 2129 NiC Stretch 400.6 401 NiCO Bend 327.8 327 masses and force constants of the two nickel atoms bonded to CO by hydrogen masses and force constants, and setting to zero all other unnecessary force constants. The calculated frequencies were found to be within less than 100 cm1 when compared to experimental results for formaldehyde (Clouthier and Ramsay, 1983). 3.2 Normal Mode Analyses Expressions (3.1) and (3.2) in Section 3.1, for the potential energy of the clusters, contain far more coordinates than the degrees of freedom of the clusters. In order to reduce the number of coordinates to equal the number of degrees of freedom of the clusters we recall that only a reduced number of atoms are allowed to move; this relates the internal coordinates {s} to the atomic cartesian displacements (x,y,z). After expanding around the equilibrium position and setting to zero HIP displacements of the atoms other than Ni, C and 0, we obtain the transformation s = A X . (3.3) Here X is a column vector with the cartesian displacements of the three moving atoms. Using the point group symmetry of the clusters, a set of symmetry coordinates (S) was constructed from the atomic displacements X. The transformation S = U X , (3.4) allowed us to determine a new force constant matrix which is blockdiagonal according to the symmetry of each cluster. The successive transformation are represented by t A t U 2V(s) = S f s A X k X St F S (3.5) The relations between the different coordinates are listed in the Appendices A and B. The final F matrices are presented in Table 3.4 for the on-top cluster and Table 3.5 for the bridge. Frequencies were calculated using F and G matrix method (Wilson, Decius and Cross, 1955). The G matrix is defined by the kinetic energy T. Starting with 2T = (X)t M X , and, using S = U X, one can then write (3.6) 2T = ( -)t G- 1 (3.7) where G- = (U-l)t M U- and M is the diagonal matrix of masses. From these F and G matrices, one obtains the frequencies corresponding to the normal modes by solving (E) Ak = Xk Ak P (3.8) where, introducing the speed of light c, frequencies v k are given by (3.9) Xk = (2mcvk)2 , and the corresponding normal modes by Q=A-I S or Q = (A-3 U) X (3.10) Table 3.4 F Matrix for On-top Cluster F 2 f (2)+f F = Ni CNi 12 f CNi F22 -fCNi + f - fc0 F 44 2f(I) f(2) + (f + f )/d 2 F 44 Ni Ni a b CNi 33 f cO F 46 f( +d)( 2 -f/d2 F = 46 fa(dCNi dc)/ COCNi b CNi 55 44 F48 -fa/(d cod cN) F57 -F46 F66 - fa[(dCNi + dco)/(dCNidCO)2 + fb/(dCNi)2 F59 - 48 F68 =-fa (dcNi + dcO)/(dcNid0) = F a C~i co Ci CO77 F66 2 F88 f aC/(dCO) F99 F 88 Table 3.5 F Matrix for Bridge Cluster A1 Symmetry F + )2 + 4b 2 11 = 2(a3 a4) fa 2(dC/dcNi CNi + 1 g F12 - 2a4(a3 - a4) f CO F 3= -(2/-2) a3(a - a f - (2/2)(d /d c 2 f - (4T-2) b 2f 13 aa3 a4, a ' C CNi' CNi 1 g F14 = -(2/2) bldCfCNi - (2/4f2) a2 (a3 - a4) fa + (4/4-f) b2a2fg F 2 2a2 f + f 22 4 a CO F 23= (2/4-2) a 3a 4fa Table 3.5 (continued) F2= -(2/ W) a a fa 2 2+2)2 F33 = a3fa +(dc/dcNi) fCNi + 2(dc/dcNidNi)2 fb +2b~fg + (40/21) f(2) F34 - - 2 d /Nd F3 2a3fa (dc/dNi)(I/dcNi)2 fb - (cdNi/2dNi) fCNi - 2bla2fg - (4/7)(I/ W) f1(2) 44a2 ++ 2a f +(I2 )b F4 a a +(dNi/dc) FCNi 2g (/dNi) b + (17/5)f1) + 8/ ) where, a2 = dc/dco a4 = -4 dd/(dNidc0) = 2d + )/CNi dNidcNi) bI = _ dNi/ (2 dCNi) B2 Symmetry 11-2 FI 3 fa + (dc/dc~ F12 = (2/4-f') a3(a2 - a )2 fC + (40/21) f(2) + 2dcdici2f 2 ll fa - (i/4[2)(dNidc/dcNi) fCNi =(2/4-2) al3fa 2 2 F22 = (I/T-2)(dNi/dc)2 fCNi + 2(a2 - a1)2 fa F2 - 2a1(a2 - a1) fa F3 =-2a f aI = I/d0 F13 Table 3.5 (continued) F24 ..- (2/4-2) a2 (a2 - a1) fa (1/2f-2) (dNi /dCNi)2 fCNi F = - (2/4-2) a1a2fa F - a2f + /2d )2f + f/(2dcNi) fb + (7/5) f li + (8/7) f) A2 Symmetry F11= (8/5) f )+ (20/21) f(2) + (2/dCNi) fb B1 Symmetry F11 = (4/dNi) fb + [(dc + dc0)/(dc dc0)2 fd F -- 2 F12 - fd (dc + dc0)/(dc dc0) F13 4f=- dCNi) fd(dc dc0)/(42 dcd 2 13 ~ dC C CO C~ F22 -fd/d 2 F23 = f /(42 d d 22 dCO 23 d C CO ff) (2) 2 2dN F 3 (8/5) + (20/21) + f /(/d 33 Ni Ni d dC) b CNi The matrices GF- constructed in this way reflect the symmetries of the clusters, thus simplifying the calculation and the classification of the modes according to the symmetry of the clusters. In the on-top case the resulting GF matrix is block-diagonalized into one 3 x 3 matrix belonging to the A1 irreducible representation, and one 6 x 6 matrix containing two coupled modes belonging to the E irreducible representation. The GF matrix in the bridge case is 12 x 12 but block-diagonalizes into two 4 x 4 matrices, one belonging to the A1 and the other to the B2 irreducible representations, one 3 x 3 matrix belonging to the B1 and one 1 x i matrix belonging to the A2 irreducible representation. 3.3 Numerical Results for CO on Ni(OO1) The vibrational modes for the on-top and bridge clusters are shown in Figures 3.3 and 3.4, respectively. The resulting coefficients for the transformation from cartesian to normal coordinates are presented for the on-top and bridge clusters, in Appendix C. We separately discuss the on-top and bridge cases. 3.3.1 The On-Top Cluster The calculated frequencies along with previous experimental and theoretical results are presented in Table 3.6 for the on-top cluster. In this case we find one frequency v with value comparable -l to the CO stretching for the free molecule, 2143.2 cm , and two 2 and V with values similar to the experimental -1 -1 frequencies (of 381 cm-I for CNi stretch, and of 461 cm- for NiCO bend) found in carbonyl compounds, Ni(CO)4 (Crawford and Horwitz, 1948). We also find three frequencies v3' V5 and v6 with values lower than the largest frequency for the bare substrate phonon band, -l 295 cm (Black et al., 1981). The mode with the highest frequency, v1, is related to the free molecule CO stretch, v2 relates to the Ni-C stretching mode, and the V3 mode can be considered as a frustrated translation along the direction normal to the surface. 10 ï¿½ z x Ni I IM I 2 3 4 5 6 Figure 3.3: Normal Vibrational Displacements for the On-Top Cluster. The v-symbol label different irreducible representations for each symmetry group. 7IAi 1I 1/Al 2 N 1 4/A 3l Figure 3.4: Normal Vibrational Displacements for the Bridge Cluster. BB Z' Z6 5E 2/ Figure 3.4--continued Figure 3.4--continued 1 - ,/B2 8 Z/B2 9 ZiB2 10 /B2 -I Table 3.6. Comparison of Results, On-Top Cluster (Frequencies in cm ) A A A E E E 1 2 3 4 5 6 Experiment (Andersson 1977) Our Results NiCO (Richardson and Bradshaw, 1979) Ni5CO (Black et al., 1982) Ni2CO Periodic Cluster (Black, 1982) Spectral Density (Black et al. 1982) 2069 2088 2088 2088.3 2087.5 2088.3 480 431 437 431.7 430.6 440.3 102 98.6 84.7 148 389 411 399.2 399.6 400.6 44 1 71.6 67.3 37 Three modes which belong to the E representation, v4' V5 and 6' can be considered as two frustrated rotations (v4 and v5), about the x and y axes, and a frustrated translation (v6) along a direction parallel to the surface. These assignments are shown in Table 3.7. The values obtained for the C-O stretch and Ni-C stretch are in fairly good agreement with the values assigned to the on-top bonded CO (Andersson, 1977). For the three higher frequencies we obtain results almost identical to the ones for a (Ni)5CO cluster and also to the ones for a periodic lattice of small clusters, (Ni)2CO (Black, 1982); the latter used only nearest neighbor interactions and a central-force field. -i We obtain two frequencies at 389 and 44 cm , with values higher and lower than the largest bare substrate phonon frequency that are Table 3.7 Designation, Character, Species and Frequencies of the Normal Modes for On-Top Cluster Description A1 E Frequency (cm -) C-O Stretching V 2088 C-Ni Stretching V2 431 Ni-C-O Frustrated V 3 102 Translation (Perpendicular to Surface) 0-C-Ni Bending V4 389 Ni-C-O Frustrated V5 131 Rotation Ni-C-O Frustrated V6 44 Translation (Parallel to Surface) in very good agreement with two 8-function like modes at 37 and 400.6 -1 cm obtained in calculations of the spectral density function of a periodic (Ni)2CO cluster using green function methods (Black, 1982). We also obtained a value of 102 cm- for the mode which involves A1 motion perpendicular to the surface (v33 ); this value, although higher than the one obtained by Black for the overlayer and for the isolated (Ni)5CO cluster, could be associated to the fairly broad -i peak centered at 148 cm obtained in the spectral density calculations for the full crystal. 3.3.2 The Bridge Cluster For the bridge model we found a total of 12 modes, four with Asymmetry, three with B1 symmetry, four with B2 symmetry and one A1 A1 A1 B2 B2 with A2 symmetry. The frequencies v 'V 2 '23 '5 ' 8 and B2 V9 . have values similar to molecular mode frequencies, and the A B1 B1 B2 B2 A2 remaining '34 '36 '3 7 Y V10, %3ll and v 12 have values lower than the largest bare substrate phonon frequency (295 cm- ). These calculated frequencies are presented in Table 3.8 along with results from other calculations and previous experimental results. The A1 mode with the largest frequency, vi' can be easily recognized as a CO stretch, '32 is a NiC symmetric stretch and the other two, '33 and v 49 can be considered as frustrated translations in a plane normal to the surface, with the last two modes involving CNi-Ni bending. Three bending modes belonging to the B1 representation involve motions out of the plane formed by the two nickel atoms and the carbon; one of these, v5Y can be considered as a frustrated translation and the other two, '36 and 7' as OC(Ni)2 bendings. A summary of these assignments is shown in Table 3.9. Table 3.8 Comparison of ResulIs, Bridge Cluster, (Frequencies in cm ) Frequency Experiment Our (Richardson & (Andersson, 1977) Results Bradshaw, 1979) 1932 1981 530 2099 359 657 143 Table 3.9 Designation, Character, Species and Frequencies of the Normal Modes for Bridge Cluster Description A1 B1 B2 A2 Frequenjies (cm-) C-0 Stretching V1 1981 C-Ni Symmetric V2 530 Stretching C-Ni Assymmetric V3 366 Stretching Frustrated Trans- V 4 53 lation (z axes) OC(Ni)2 Bending V5 569 Frustrated Rotation V6 125 (y axes) Frustrated Trans- V7 71 lation (x axes) Antisymmetric NiC V8 764 Stretching Frustrated Rotation V9 397 (x axes) Frustrated Trans- V10 68 lation (y axes) Combination of 8 V12 and 9 Frustrated Rotation V12 143 (z axes) The present procedure includes several metal atoms to represent the metal substrate and at the same time restricts the number of degree of freedom of the cluster to describe the vibrational modes of a molecule attached to metal atoms at a surface. The calculations yield vibrational frequencies which are in good agreement with experimental results available on the system. Some of the frequencies presented here describe new modes for the bridge cluster, which have not been presented in previous works. A fairly clear distinction, see energy diagram shown in Figure 3.5, can be made between modes with relative high frequencies (molecular modes) and those which correspond to soft modes coming from the interaction of the adsorbate with the modes of the bare substrate. This separation of modes is more evident for the bridge case than for the top case. These cluster models and their frequencies and normal modes can be used as a first step in the study of atoms scattered by adsorbates. Figure 3.5 Comparison on Energies for the isolated CO Molecule, the Bare Nickel Substrate and the Results Obtained for the On-Top and Bridge Cluster. BRIDGE substrate substrate+ mde( FREE MOLECULE kAd ON-TOP substrote+moecule ZIAl 2080 1980 800 2 9__ Al E If1A, A, 0 "- __ A i OL I A, '4E substrate 700 600 E U UIl 20 I0 1/,E E3 6 CHAPTER 4 ENERGY TRANSFER INTO MOLECULAR ADSORBATES In scattering by adsorbates leading to energy transfer, the translational degrees of freedom can frequently be described classically while the remaining ones, only vibrational degrees of freedom in our case, must be treated as quantized. It was pointed out in Chapter 2, that by relating inelastic scattering cross-sections to time-correlation functions (TCF's) of transition amplitudes, it is possible to study energy transfer in scattering by extended targets without having to expand wavefunctions in the many excited states of the target. Inelastic scattering cross-sections for collisions in which the relative momentum changes from li. to M f were shown to be given by the Fourier transform of the TCFs of the operators For experimental conditions of high relative energy and light projectiles, the interaction is dominated by strong short range repulsive forces, and lasts a short time compared with the periods of internal motions of the target, so that an impulsive model is appropriate. Under these conditions, as was shown in Chapter 2, the double differential cross-section is obtained from TCFs involving the position operators of the target atoms. We show in Sec. 4.1 how to obtain the vibrational TCFs analytically from the normal vibrational modes of the target. Furthermore, since the normal modes of the target can be grouped in two sets, slow and fast modes, we can approximate the TCFs by short-time expansions which lead to Gaussian distributions of the energy transferred into the slow modes, as shown in Sec. 4.2. Finally, in Sec. 4.3, we introduce a statistical model which allows us to obtain a simple description of double collisions. 4.1 The Vibrational Correlation Function The TCF of position of the target atoms that we wish to calculate is given by F(aa) 4 -4 Fa(K,t)= << exp[-i .ra (0)] exp[iK.r a(t)] , (4.1a) with << ... >>= E wN < vi...I > , (4.1b) V which includes the quantum-statistical average over the distribution w of initial vibrational states 1v> of the target. Indicating operators with carets, ra(t) corresponds to the instantaneous position of atom a, which evolves in time in accordance with the Hamiltonian HX of the isolated target as follows: ra (t) = exp(iHxt/M) ra exp(-iHxt/h) (4.2) For each atom a in the target we can introduce its equilibrium position ca and displacement from equilibrium u ay so that r =d + U (4.3) a a a In a body-fixed reference frame we have N = 3N free vibrational V coordinates, where N is the number of atoms in the target. Using Eq. (4.3) in Eq. (4.1a) we obtain F(aa) -* F aa)(K,t) = << exp{-iK.[d a(0) + u a(O)] (4.4) x exp(i.[da(t) + Ua(t)]) "v The target Hamiltonian is nothing but Hv and from [H v,d = 0, we find that F 4(t) = << exp[-iK.u (0)] exp[iK'u (t)] >>v (4.5) vK a a v Assuming harmonic vibrations as a zeroth order approximation, which is acceptable except for highly excited vibrational levels near the dissociation energy, and using internal coordinates s. which are linearly related to the displacements by n v 4 = ua = s. , (4.6) i=l the vibrational TCF is most conveniently evaluated in terms of the normal displacements Qj(t) = Qj(0) cos.jt, which are related to the internal displacements by a linear transformation s(t) = UQ(t) . (4.7) The standard normal-mode analysis (Goldstein, 1950) allows one to determine the normal frequencies wj, as well as the vectors e aj= E ai Ui. of the transformation from normal displacements to i the u a displacements. Since the normal modes are independent, the vibrational TCF factors into a product of correlation functions for each mode as shown below: n V Fv-(t) = << exp[iK. exp[i.aQ(t)] >>v. (4.7) VK FT..() x eP i aj v j=l The correlation of each mode is readily evaluated by means of the algebra of creation and annihilation operators at and a. (Messiah, J J 1961), through = 2/(2 j)1 (a. + a 3 (4.8) Introducing the notation 1/2 1aj = E (K /K) 1 Ca ,j (4.9) where & denotes the cartesian component (x,y, or z) and using Eq. (4.8) into Eq. (4.7), we can write N V Fv4(t) = T[ F(J)(t) (4.10a) VVK j=l V F (t); << exp[-iKl a(a. + a.)o aj j 0 VK x exp[iKlaj (a. + a >v (4.10b) By following a previous procedure (Micha, 1979b), one then arrives at F(0)(t) = E exp(-injW.t) p ' (4.11a) VK " J n . n. J Pn. = exp(-njcaj - j cosh j In (X ) , (4.11b) Xj = 1(K C. cos ) /(2w j sinh aj) , (4.11c) aj= Mwoj/2kBT , (4.l1d) where n. is the number of quanta transferred into a mode of energy J ho)J, In (X) is the modified Bessel function; j is the angle between K and the vector C., and kB is Boltzmann's constant. The JB total vibrational correlation function FVK in Eq. (4.10) can be rearranged as a sum of products of exponentials FvK(t) = E exp(-inwvt) P , (4.12a) n Pn(K) = p n. (4.12b) j J Where wv and n are column matrices with elements w. and integers n., respectively. It is immediately seen that the Fourier transform of the vibrational TCF would consist of a group of 8-function peaks located at s=-nt. --V 4.2 Short Time Expansion The calculation of the vibrational TCF given in Eq. (4.10) can be simplified whenever the vibrational modes of the target satisfy Hj < kBT. Therefore, one can approximate FM by a short time expansion around t = 0 (Micha, 1981), of the form ln F(t) = ln F(O) + t F(O)/F(O) +I t2 [F(0)/F(0) - F(0) 2/F(0) 2 (4.13) where a dot signifies a time derivative. Performing the derivatives from Eq. (4.10), the expression for each of the low frequency modes j is given by M 2 2t)2 ln FvJ(t) = it. - t2 r l/2 , (4.14a) v j J w= . X. sinh . , J J J J (4. 14b) r 2 Y,2 W2 X. cosh a.)/2 (4.14c) J J J J where X. and a. are the same quantities as given by Eq. (4.11). J 1 Collecting all the low frequency mode contributions one obtains the final expression 2 22 in F v(t) = it A &j/M - t2 E r l /K (4.15) J J The Fourier transform with respect to time of Eq. (4.15) gives a Gaussian distribution peaking at the energy transfer Z . with a width r = r.. Both quantities would increase J J with momentum transfer. 4.3 Statistical Model for Single and Double Collisions When an atom collides with a molecular adsorbate it causes transitions among the target's vibrational states. The most likely event is excitation of both adsorbate and substrate vibrations, although deexcitation is also possible from the thermally populated excited states of the target. The mechanism of energy transfer depends on the scattering angle, E, defined as the angle between the incoming and outgoing momentum vectors. For an atom approaching along the perpendicular to the surface with a small impact parameter, b, collisions with the 0 atom of the adsorbate will lead to deflections by an angle larger than n/2, in which case only one encounter takes place. This can be seen in Figure 4.1, which has been drawn using covalent atomic radii plus a van der Waals radius for He atoms. If, however, the impact parameter is large, the first deflection is smaller than n/2 and the I 7r/21 I I II I b,,.- Ni Figure 4.1: Trajectories showing regions of single and double collisions. Model drawn using covalent atomic radii plus a van der Waals radius for He atoms. atom next collides with the surface before flying away at an angle larger than n/2. Hence, the deflection function of such a collision would show two branches for E > n/2. It is then obvious that an atom which had been deflected by either mechanism could be detected at the same final scattering angle. Therefore, to obtain the double differential cross-section one needs to add all contributions involving the same amount of energy transfer and having the same scattering angles regardless of the mechanism by which this occurred. In Chapter 2 an expression for the double differential cross-section, for single collisions and under the impulse approximation, was shown to be given in terms of the TCF. Using the vibrational TCF obtained in section 4.1, one can write 21 dd d= drt1 -it exp(-inv t) Pn (it) .(4.16) n Here one can recognize two factors, the first, (da/dQ), is an effective classical cross-section and the second has the meaning of a probability per unit energy. Then equation (4.16) can be rewritten as d2 ( [da dcdg dj d (4.17a) where dP f dt -iit/nw- t d -, e E e -v P () E dP /dF (4.17b) n n In particular, for a scattering characterized by the energy transfer s = -nKw, we find the partial differential cross-section to be given by d2a d d J/d dP/de (4.17c) This way, the total differential cross-section is given by dd 2 ada/d] dP /de (4.18) n A detailed cross-section should account for the quantal phase interference of branches I and II. However, in our case we must average the detailed cross-sections over the target thermal distribution, which would lead to cancellation of the interference term except possibly near e = n/2. Denoting the two branches corresponding to single and double collisions by I and II, respectively, we can then write the cross-section d2a (I) )(II) (II) dcdQ = (da/dQ) (dP/d ) + (da/d2) (dP/dc) (4.19) where the first term is obtained from branch I at angle e, and the second term from branch II at the same angle. The probability per unit energy transfer corresponding to single collisions, (dP/dc)(I), follows directly from the Fourier transform of the vibrational TCF given by Eq. (4.16). The second term in Eq. (4.19) is more difficult to obtain because it is a combined probability since it accounts for the transfer of two amounts of energy occurring in a double collision. They are the energy c transferred during the first collision into the cluster plus the amount 6m transferred during the second collision into the metal away from the adsorbate. This term can be derived using statistical arguments as follows. We first separate the collection n of normal mode quantum numbers of the whole target into the subsets n for a cluster and nm for the metal surface away from the adsorbate. The latter contains the parallel and perpendicular vibrational modes of the clean surface. For the double collisions we then write iE t/h (d/d)(I) "dt -i Et/h 1 (II = 2- dn e E e P n(C) , (4.20a) n 1 Pn( ) = I dy P c(y) P m [(l-y)E] (4.20b) ~ n n where E = -E n.M. and P is the probability of transferring n nj j n~ quanta when the total energy transfer is z c + em. The probability P has been written as the convolution of the probabilities P and P for excitation of the cluster and the clean c m n n surface, corresponding to a sequence of two independent events. The integration variable y in Eq. (4.20) is defined by y = c/C. Using En +E we then find ~ n n 1 (d~dg)ll) J"dt -ictl/h FcFT -d-Jd e dy F (t,y) Fm (t,l-y) , (4.21a) 0 F (t,y) = E exp(iE ct/1) Pc (YE) , (4.21b) n n c n Fm(t,l-y) = exp(iE mt/M) P m[(l-y)] . (4.21c) n n m 12 n Here, FC(t,y) and F m(t,l-y) are the vibrational TCFs for the cluster and clean metal respectively. Each of these two TCFs could be calculated from the expressions in Eq. (4.11). However, it is usually found, as was the case in Chapter 3 for a CO molecule adsorbed on a Nickel surface, that frequencies can be separated into high and low ones. Thus, for each low frequency mode, 1, one can use the short time expansion given in Eq. (4.14) which, when replaced in Eq. (4.21), leads to 2 (II) d2 n (d-d d d' (4.22a) fl d2 an(II) 1 de dQ dy pn (y) f ap0 f (4.22b) 1/2 x -"2 xp-c _6 2 /2 where nf is the number of quanta gained by the fast mode of energy MWf, and es = c + n fMf is the energy gained by the slow modes. The total shift 6 and width r2 are sums over all slow modes 1 and are functions of l-y. Equation (4.22) may then be used to calculate double differential cross-sections (DDCS) versus the scattering angle 0, and versus the amount of energy s, transferred into the slow modes, for each fast mode nf. CHAPTER 5 GAS-ADSORBATE INTERACTIONS POTENTIALS: EFFECTIVE DIFFERENTIAL CROSS-SECTIONS In previous chapters we have shown that double differential cross-sections for atom-adsorbate collisions can be obtained from a product of deflection probability and a target absorption probability, where the former can be given by an effective differential cross-section. In order to obtain this effective differential cross-section one must first specify an interaction potential between the projectile and the target. In this chapter we first review general qualitative features of the atom surface interaction potential and the usual procedure one can follow to obtain analytic representations which are desirable for scattering calculations. Since not much is known quantitatively about the behavior of the potential, one is forced to introduce model potentials. A model potential is then presented for the interaction between helium atoms and lithium ions with a carbon monoxide molecule adsorbed on a metal surface. These model potentials are then used in trajectory studies of the two systems, He - adsorbed CO molecule and Li+ - adsorbed CO molecule, to obtain the desired effective differential crosssections. 5.1 Atom (Ion)-Adsorbate Interaction Potential The general features of gas-adsorbate potentials are expected to be similar to the qualitative picture of the gas-surface potentials. In the case of sufficiently small distances from the surface the potential has a strong repulsive part in front of the solid. The reason for this repulsion is the overlapping of the wave functions of the electrons of the gas atom and of the electrons at the surface of the solid. This gives rise to a periodic modulation of the repulsive part of the potential parallel to the surface. At longer distances the gas-surface potential is dominated by an attractive van der Waals interaction given by the distance-dependent polarization energy between a gas atom and the solid. This attractive part of the potential is caused by the interaction of the gas atom with a relatively large number of crystal atoms, so that the attractive part can be assumed to be practically constant parallel to the surface. There have been only a relatively small number of attempts to calculate the physical gas-surface interaction potential from first principles. Calculations of He on metals have been done (Zaremba and Kohn, 1977) to calculate the attractive van der Waals interaction of large distances and the repulsive interaction at small distances and to construct from these two parts a complete potential as a function of the distance Z of the gas atom from the surface. In this procedure the position of the reference plane Z = Z0, from which the distance of the gas atom should be measured, appears as an essential parameter. More recently, methods for generating surface potentials from surface electron densities have been presented. Some authors (Esbjerg and Norskov, 1980) have suggested a theory which gives a relation of proportionality between the helium potential energy and the surface-electron charge density. However, when experimental results are compared with theoretical predictions, they strongly disagree. In particular, for the Ni(ll0) surface the corrugation parameter across the close-packed rows is found to be three times the experimental value (Annett and Haydock, 1984). Similarly, several ab initio calculations of potentials for weakly corrugated surfaces have not been very successful, predicting approximately two times too large a corrugation (Batra et al., 1985). However, semi-empirical methods, allowing the adjustment of some parameter, have led to good agreement with the He scattering data (Harris and Liebsch, 1982b). An approximation frequently used that allows calculating physisorption potential energies of gas-surface systems is based on the additivity assumption. The thermally averaged solid surface is assumed to have perfect periodicity and thus may be expressed in a Fourier expansion as follows: v(r= v (Z) exp(i&-5) , (5.1) where r= (x,y,z) = ( sZ), with the Z-axis extending in the direction normal to the surface. The two dimensional vectors are the reciprocal lattice vectors of the surface. The first term in the expansion, vd(Z), is considered the over-all average potential and is the large potential in perturbation theory which causes specular diffraction. The vM(Z) for e # 0 are treated as perturbations which cause non-specular diffraction. Diffraction experiments give information on both the dimensions of the unit cell via the angular location of Bragg peaks and the distribution of the scattering centers within the unit cell via the intensities of diffracted beams. However, to obtain a solution to the diffraction problem in closed form, the potentials ve(Z), including e = 6, must be simple. A convenient empirical representation of the ve(Z) is the Morse/exponential-repulsion representation (MERR) (Goodman, 1987) which represents v6(Z) by a Morse potential and ve(Z) by an exponential repulsion for e 0. v6(Z) = Dfexp(-2aZ) - 2exp(-aZ)} , (5.2a) ve(Z)/D = Ke exp(-2cZ) e # 0 , (5.2b) where Ke are the so-called diffraction strengths. In the 4G model where only the four smallest nonzero reciprocal lattice vectors are used, the sum over U reduces to four terms and the Ke reduces to a single value. The qualitative picture of the gas-surface interaction we have described has to be modified in the case of ion-surface interactions. In these cases and at short distances, the important contribution to the surface interaction comes from the overlap of the electronic charge densities, which in the case of closed shell ions, is reasonably considered as purely repulsive. At large distances from the surface the interaction is attractive and is well described by the macroscopic image potential, Vim = -e 2/4Z. The image picture, however, loses validity when the particle approaches to within a few angstroms of the surface. Here again, as in the case of the van der Waals interaction, a parameter may be introduced to define the position of the image plane which improves the validity of the image potential. Recently, a way to construct the interaction potential between an ion and a surface of a monatomic solid has been presented (Mann et al., 1987). The K+ W(IO0) potential constructed this way was found to agree with an empirical potential used to fit scattering data for the same system (Hulpke and Mann, 1985). The interaction of an atom with an adsorbate will be comprised of direct potentials (the attractive van der Waals and the repulsive short range), as occur in the gas phase, and of indirect surfacemediate potential which arise due to the proximity of a third polarizable body, whose presence modifies the simple, two-body gas phase interaction. The surface-mediated potential is expected to be of particular importance in the case of long range interactions and only for the case of low incoming atom energies. The induced polarization interaction of closed shell atoms with anisotropic adsorbed molecules has been investigated and a theory of the long range van der Waals potentials acting between a He atom and a CO molecule adsorbed on a metal surface has been presented (Liu and Gumhalter, 1987), and the full potential has been given as the sum of the long range interaction plus the short range repulsion for the atom-gas phase CO molecule interaction. A simple approach to the full description of an atom-adsorbate interaction consists in superimposing an ab initio atom-gas phase molecule interaction potential on an empirical atom-surface potential. This procedure has been used in the system He/OC-Pt(lll) (Lahee et al., 1987) and has been found to give good agreement with the experimentally predicted classical turning points. 5.1.1 He/OC-Ni(001) Interaction Potential Since our final goal is to obtain an interaction potential simple enough to be used in trajectory studies, we follow the assumption that the atom-adsorbate potential can be given by superimposing an atom-molecule potential on an atom-surface potential. For the atom-surface potential we have followed the procedure given by Harris (Harris and Liebsch, 1982a). Basically, their theory, a generalization of that used for jellium surfaces (Zaremba and Kohn, 1977), relates the interaction to shifts of the band energies due to the presence of the helium. In this theory, the Hartree-Fock interaction energy to lowest order in the overlap is given by gF VR(r) = { de p(c,r) g(E) , (5.3) where cF is the Fermi energy, g(e) is a smooth function of energy and P(C,rN) is the local density of states at the helium nucleus. In the case of face centered cubic (fcc) transition metals VR(x,y,z) = V0 e- z 1 + LO exp [O(Z-Zo) h(x~y) , (5.4a) h(x,y) = cos(gxX) + cos(g yy) , (5.4b) 1/2 = 2[2(w+A)] , (5.4c) 1/2 = [2(w+A) 2 - 2 + " (5.4d) Where w is the work function of the metal, A is an energy shift that depends primarily on the band width, and gx gy are the magnitudes of the smallest reciprocal vectors in the x and y direction, respectively. The attractive part of the helium interaction is taken to be -3 given by the van der Waals expression - Cvw (Z-Zvw) with the origin of z lying one-half a layer spacing outside the topmost plane of nuclei. The full He-adsorbate potential can then be written as the sum of the three contributions: repulsive He - surface, attractive He surface and He - gas phase molecule, that is: V(x,y,z) = V a(R a) + V rep(R) + V vw(z) , (5.5a) V a(R a) = Aa exp(-Ba R a) , (5.5b) V rep(R) = V s(Z) + V corr(x,y,z) , (5.5c) Vs(z) = As exp(-Bsz) , (5.5d) Vcorr(X,y,z) = Vs (z) h(x,y,z) , (5.5e) h(x,y,z) = Bs D exp[-O(z-z0)]h(xy) , (5.5f) Svw(Z) = -C vw/(Z-Zvw)3 (5.5g) where R represents the distance between the He atom and the oxygen a atom of the adsorbed CO molecule and it can be written as R =[R2+R2 2R R cos]1/2 a m m and Rm is the distance between the surface and the oxygen atom in the CO molecule. In the case of CO on Ni(O01), R has been determined m experimentally (Andersson and Pendry, 1980). The geometry of the problem is given in Fig. 5.1, where the surface defines the (x,y) plane and the z coordinate is perpendicular to the surface and passes through the center of the adsorbed molecule. From electronic calculations of CO on a cluster of surface atoms there is overwhelming evidence that there is a chemical bond formation between the substrate valence band orbitals and the adsorbed CO valence molecular orbitals. The CO 5a molecular orbital may acquire partial metal character and thereby donate some of its electronic charge into the metal states. Simultaneously, some of the * metal charge may be back-donated into the formerly unoccupied CO 2rt I/ 0 0 / 0 Figure 5.1: System of coordinates and geometry used for the interaction potential. a) On-top model; b) Bridge model. I I / RI molecular orbital which may become a bonding one with respect to the metal-CO interaction (Bauschlicher, 1986). This back donation into the 2n orbital is then responsible for the weakening of the molecular bond and an increase of the CO bond length. The C-0 bond length in a CO molecule is 1.128 A, whereas in a CO2 molecule, it is 1.159 A which is closer to the value found in carbonyl compounds and in a CO molecule adsorbed on a Nickel surface. In view of the above arguments, we have chosen to use the parameters given by the fitting of a SCF interaction potential energy for He - CO2 (Clary, 1982), for the He - adsorbate repulsive part of the potential. Diffraction data have not been reported on the HeNi(O01) system, so we have used the Harris and Liebsich parameters for the He - surface interaction. A list of all the parameters we have used in this work is given in Table 5.1 for the case of a CO molecule adsorbed on top. The only parameter which changes when one considers the bridge bonding situation is the height of the adsorbed molecule. This has been chosen according to the geometric parameters of the clusters presented in Chapter 3. It is known from the study of atomic and molecular interactions that the van der Waals interactions must be damped at short distances, because of electronic overlap of wave functions. In order to accomplish this damping, we have used a function of the type f(z) = exp[O(z-z0)]/{l + exp[O(z-z0)j) which for z The interaction potentials for the on-top and bridge case are shown in figures 5.2 and 5.3, respectively. It can be seen that in both cases the interaction is steeply repulsive in the region of hyperthermal energies, i.e., > 1 eV, in which we are interested. The Table 5.1 Interaction Potential Parameters for He/OC-Ni(OO) He/CO He/Ni A a(eV) 2157.21 B (A-1) 4.7 A (eV) 12.0 S B (A-) 2.63 D(A) 0.0127 O(A-I) 0.8692 Cvw(eVA3) 0.22537 z0(A) 3.01644 zvw(A) 0.24396 g(A-1) 2.489 Figure 5.2: Attractive (---) and full (-) potential for the on top He/OC-Ni(O01) system over the surface point x=y=O. The inset shows the region around the minimum. 0 1 2 3 4 5 6 Z/Angstroms 7 8 9 10 1.50 1.23 0.97 0.70 0.43 0.17 -0.10 Figure 5.3: Attractive (---) and full (-) potential for the bridge He/OC-Ni(OO1) system over the surface point x=y=O. Inset shows region around the minimum. 1.50 1.23 0.97 a 0.70 0.43 2 3 4 5 a 7 a 9 o Z/knstroms 0.17 -0.10 / I I I I I 0 1 2 3 4 5 6 7 8 9 10 Z/Angstroms interaction for the on-top case gives a larger turning point than the case of the bridge, as was expected due to the shorter distance of the CO molecule to the surface in the latter case. It also can be seen in the inset of both figures that the bridge potential presents a deeper well than the top potential. This is because the attractive part of the interaction between the He atom and the surface is stronger when the He atom is over the space between two surface atoms, rather than directly on top of a surface atom, and in the bridge model, the adsorbate is positioned in between two surface atoms. In figures 5.4 and 5.5, the dependence of the potential in the (y,z) plane is shown for the two models. Again the bridge gives larger turning points than the top model but now the effect of the adsorbate becomes smaller as one moves away from it and finally, when y>a, the two models give identical turning points. This is due to the fact that for y>a one is only seeing the He - surface interaction. 5.1.2 Li +/OC-Ni(O01) Interaction Potential The ion-adsorbate interaction potential can in general be written as the sum of three contributions: the ion-molecule repulsive interaction, the ion-surface interaction, and the image interaction. As was mentioned in Sect. 5.1, a recent approach to construct the interaction potential between an ion and the surface of a monatomic solid (Mann et al., 1987) is found to agree with an empirical potential used to fit scattering data for the system Li+/W(OO) (Hulpke and Mann, 1985). We have chosen to represent the Li+-Ni(O01) potential by a modified MEER potential which has been used to fit scattering data for the system Li+/Ni(001) (Gerlach and Hulpke, 1.50 1.23 0.97 0.70 0.43 0.17 -0.10 0.0 Figure 5.4: 0.8 1.5 2.3 3.0 3.8 Z/Angstroms 4.5 5.3 6.0 Potential for on-top He/OC-Ni(O01) system over several points on the (y-z) plane. Starting from first line on the right, the points correspond to y=O, a/4, a/2, 3a/4, a; where a is the lattice spacing. 1.50 1.23 0.97 V S0.70 0.43 0.17 -0.10 0.0 Figure 5.5: 0.8 1.5 2.3 3.0 3.8 4.5 5.3 6.0 Z/Angstroms Potential for bridge He/OC-Ni(O01) system over several points on the (y-z) plane. Starting from first line on the right, the points correspond to y=O, a/4, a/2, 3a/4, a; where a is the lattice spacing 1977). Here again we can write a similar equation to Eq. (5.5) V(x,y,z) = Va (Ra) + V s(R) + V imag(z) . (5.6) The term V a(R a) is identical to the one for the helium potential but now the parameters are obtained from Li + /C02 data (Vilallonga and Micha, 1983a). The term V s(R) contains a repulsive and an attractive term given by V s(R) = V rep(Z) + V attr(X,y,z) (5.7a) V rep(z) = As exp(-B sZ) (5.7b) Vattr(x,y,z) = D(exp[-20(z-z)1 - 2 exp[-O(z-zl)I) (5.7c) + 2coD[-20(z-zl)] h(x,y) (5.7c) and V imag(z) = -e 2/4(z-z0) (5.7d) The parameter D represents the well depth and 0 its width. The meaning of each of these terms has already being explained in Section 5.1. However, one should mention that in this model the Coulomb term, at sufficiently small distances z from the surface, is cancelled by the repulsive part included in V attr. The potential parameters used for the Li+/OC-Ni(0010) system are presented in Table 5.2. The shape of the potential calculated from this construction is again found to be steeply repulsive around the energies of interest which in this case are 10 to 50 eV. Figures 5.6 and 5.7 show the potentials for the top and bridge model respectively. One can observe that the trends for turning points and well depths, when compared between models, are the same as the ones found for the He potential. However, an important difference in the behaviour of the potential in the y-z plane can be found when one compares the two Table 5.2 Interaction Potential Parameters for Li + /OC-Ni(O01) Ti +Ic 1777.55 5.27 A (eV) B a(A-1) A s(eV) B S(A- ) D(eV) z0(A) z1(A) 2400.0 3.0 1.0 1.1 -0.1 -0.75 0.50 Li +/Ni(001) Figure 5.6: Attractive (- - -), Repulsive (---) and full (-) potential for the on-top Li +/OC-Ni(001) system over the surface point x=y=O. Inset shows region around minimum. 02 3 4 5 6 Z/Angstroms 25.0 20.5 16.0 11.5 7.0 2.5 -2.0 7 8 9 10 Figure 5.7: Attractive (- - -), Repulsive (---) and full (-) potential for the bridge Li + /OC-Ni(001) system over the surface point x=y=O. Inset shows region around minimum. 25.0 20.5 16.0 11.5 7.0 -ao 4 5 2.5 X -2.0 I , I I , 0 1 2 3 4 5 6 Z/Angstroms 7 8 9 10 different systems. In figures 5.8 and 5.9 for the Li+/OC-Ni(OO1) system, one can observe that the turning points, as one moves away from the adsorbate, do not change as much as they do in the He/CO-Ni(001) system. This is because the Li+-Ni(O01) interaction is more attractive on top of the surface atoms than in between atoms; this is the opposite of what occurs for the He-Ni(001) interaction. 5.2 Effective Classical Differential Cross-Sections The interaction potential in the system atom(ion)-adsorbatesurface is not only dependent on the separation between the atom and the adsorbate but also depends on the separation between the atom and the surface. This represents a two center interaction problem, which means that the problem is not of a central force type and the classical cross-sections can not be evaluated from the formulation for isotropic potentials. For anisotropic potentials, the differential classical crosssections can be evaluated from a classical trajectory (t) determined by M = - (5.8) at2 3 To obtain the differential cross-sections one needs to go from an initial element of area determined by the impact parameter, b, and the initial azfmuthal angle, 40 to a final element of area determined by the final polar, 9, and azimuthal, , angles. Therefore, one needs the transformation from the set (b, 0) to the set (0,J). This transformation is obtained by the Jacobian and the differential cross-section (Vilallonga and Micha, 1987) is then given by 0 1 2 3 4 5 6 Z/Angstroms 7 8 9 10 Figure 5.8: Potential for on-top Li+/OC-Ni(O01) system over several points on the (y-z) plane. Starting from solid line on the right the points correspond to y=O, a/4, a/2, 3a/4, -a; where a is the lattice spacing. 25.0 20.5 16.0 11.5 7.0 2.5 -2.0 0 1 2 3 4 5 6 Z/Angstroms 7 8 9 10 Figure 5.9: Potential for points on the the right the a; where a is bridge Li+/OC-Ni(001) system over several (y-z) plane. StaLting from solid line on points correspond to y=0, a/4, a/2, 3a/4, the lattice spacing. 25.0 20.5 16.0 11.5 7.0 2.5 -2.0 d c d (b/sinO) a(E, ) (5.9) The above procedure involves a significant amount of computing time. In order to reduce the computational efforts we have used a simplified procedure based on histogram methods which have been widely used in the computation of elastic and inelastic and inclastic molecular differential cross-sections (Gentry, 1979). In physical applications of scattering by adsorbates we are concerned not with the deflection of a single particle but with the scattering of a beam of identical particles, with a specified kinetic energy, El, striking a sample of the material under study. The atoms in the beam have different impact parameters and are scattered through different azimuthal and polar scattering angles. Using a bundle of trajectories, the density of the incident beam can be represented by the ratio of the number of impinging trajectories, dNi, over an area dxdy of the sample. In conditions of normal incidence, that is, for an initial angle e. = 0 deg, measured 1 from the normal to the surface under study, the area, dxdy, can be represented by a grid of (xi,Yi) points. These points are related to the initial conditions, (b,0), of the trajectories by x = b cos40 and y = bsin'0. Denoting by dNf the number of trajectories with final scattering angles within the intervals (Oido) and ($ï¿½d'), the differential cross-sections may then be obtained from da/d2 = (dNf/sinO dO df)/(dN./dxdy) (5.10) The quantity, dNf, can be extracted from studies of trajectories by analyses of the final polar and azimuthal angles of the individual trajectories. Trajectories were obtained by numerically solving Hamilton's equations in the coordinate system shown in Fig. 5.1. The set of six first order differential equations was integrated by a method which uses a modified form of the Adams Pece formulas and local extrapolation (Shampine and Gordon, 1975). The initial conditions (b, O, Rmax ) were chosen by requiring that the interaction potential be small with respect to the relative translational energy, e.g., by a ratio of 10- 3, and the same criterion was used to terminate each individual trajectory. The area of the target sampled by the trajectories is related to the maximum impact parameter, bmax, defined as the largest impact parameter for which the trajectories show the influence of the adsorbate. The area was chosen to be given by Axy, where Ax = xmax - x, and xmax = bmax cosP0 (b max). Here 0(b ma) is the initial azimuthal angle for which b is found. This method will assure max that the bundle of trajectories will sample the region where the interaction between the projectile and the adsorbate is of importance. In general, the trajectories behave differently according to three separate regions. The first region, for small impact parameters is where the trajectories are deflected mainly by the strong repulsive interaction with the adsorbed molecule. These trajectories can be said to be single encounter collisions. The second region is where the trajectories are deflected twice. The first deflection is caused by the interaction with the adsorbate and the second by the interaction with the surface. These trajectories are double encounter collisions. Finally, the third region is where the interaction is mainly due to the effect of the surface. These trajectories have impact parameters b>bmax and are detected at very large polar scattering angles and do not stay in the same plane from where they originated. Energy conservation was checked at every step in the calculation and was kept to about 2%. Several of the trajectories were back integrated and always reproduced the initial values with an accuracy of at least two significant figures. Calculations for lithium were done both with and without the effect of the image. The image effect appears to be of no significance for large deflections. Only trajectories with polar angles around 900 seem to be significantly affected. This can be seen in figure 5.10 where trajectories, in the (x-z) plane, for the Li+ with 10 eV of energy are shown. The trajectories show the shape and size of the adsorbate and, as was expected, they reflect the different sizes of the adsorbate in the two models, see figure 5.11. In the case of helium atoms, the trajectories end at larger polar angles than the ones with similar impact parameters for lithium atoms, and trapping does not occur. This is due to the absence of the large attractive potential in the case of helium ions as compared to the one for lithium atoms. The turning points are much larger for helium atoms than for the lithium ions. Compare Figures 5.10 and 5.12. This is to be expected, considering the difference in the kinetic energy of the two. The dependence of the polar scattering angle on the impact parameter shows a minimum in all the cases; that is, for both projectiles and both models. This minimum corresponds to the impact parameter at which the separation between th, regions of single and 8 7 6 0 In4 o 3 2 1 00 0 1 2 3 4 5 6 7 8 9 10 X/Angstroms 8 7 6 0 $4 N 3 2 1 1 2 3 4 5 6 7 8 9 10 X/Angstroms Figure 5.10: Li+ trajectories in (x-z) plane, on-top model; for impact parameters b = 0.2, 0.4, ..., 1.6, 2.0, 2.6, 3.0 A; a) including image, b) without image. 0 1 2 3 4 5 6 X/Angstroms Figure 5.11: 7 8 9 10 Li+ trajectories in (x-z) plane, bridge model; for impact parameters b = 0.2, 0.4, ..., 1.4, 1.8, 2.2, 3.0 A. 8 7 6 O2 5 0 4- -II m4 to N3 2 0 0 1 2 3 4 5 6 7 8 9 10 X/Angstroms 8 7 6 5 4 N 3 2 0 J 0 1 2 3 4 5 6 7 8 9 10 X/Angst. Figure 5.12: He trajectories in (x-z) plane for impact parameters b = 0.2, 0.4, ..., 1.6, 2.0, 2.6, 3.0 A; a) On-top model, b) Bridge model. double encounters occurs. At large impact parameters, the deflection function reaches a maximum which indicates the beginning of the third region where the interaction is mainly between the projectile and the surface. These features are shown for the Li+ ions in Figure 5.13 and for the He atoms in Figure 5.14. The corrugation appears to be larger in the case of the interaction with He. The differential cross-sections for the on-top model are presented in Figure 5.15 and Figure 5.16 for Li+ and He atoms, respectively, for two different azimuthal angles. The main feature of these results is that the scattering is predominantly backscattering. In the case of Li+ ions, double collisions represent the main contribution for all scattering polar angles. The ratio of single to double contributions depends on the azimuthal angle. For lithium ions, the single collision contribution is the same for different directions on the x-y plane, but the double contribution is not symmetric with respect to the z axis. In the case of helium atoms, both contributions are angle dependent. A possible explanation for this is the larger corrugation present in the helium case. Also, one can notice that at certain azimuthal angles, e.g. 1=450, the importance of single collisions with respect to double ones is reversed for certain ranges of polar angles. This factor arises from the symmetry of the model. That is, the nearest neighbor distance along the y axis, 1=90', is approximately 2.489 A, whereas, in the direction with 4=450, it is 3.52 A, thus giving a larger double collision contribution for 4=450. 0.5 1.0 1.5 2.0 2.5 b/Angst. 3.0 3.5 4.0 Figure 5.13: Li + deflection function, on-top model. = 0'(-) 45O(---). The inset shows an enlargement for large impact parameters. 185 165 145 125 105 85 0.0 180 ,. -.- -.. _ 170 160 150 140 ï¿½ 130 120 110 100 90 1 n 1 1 I 0 1 2 3 4 5 6 b/Angstroms Figure 5.14: He deflection function, on-top model. 0(-), 45()---). The inset shows an enlargemenY for large impact parameters. 100 110 120 130 140 @/deg 150 160 170 180 25 20 15 10 90 100 110 120 130 140 O/deg 150 160 170 180 Figure 5.15: Li+ differential cross-section, da/dS2, on-top model, for single (-) and double (---) collisions; a) 4 = 00 b) t = 450. .I I I I i I 10[ 0 90 I I I I I I i m I i i i I i 100 110 120 130 140 @/deg I I 100 110 120 130 140 150 160 170 @/deg 150 160 170 180 180 Figure 5.16: He differential cross-section, do/dQ, on-top model, for single (-) and double (---) collisions; a) $ = 0' b) = 450. 30 F .- 25 ~20 to cn 15 I I I I 0 90 70 60 50 40 30 10 0 90 I I I I I I I I I I I |

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PAGE 1 SCATTERING OF ATOMS B Y MOLECULES A DSORBED AT SOLID SURFACES B y ZAIDA PARRA A DISSERT A TION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY U NIVERSITY OF FLORIDA 1988 OF flORIDA PAGE 2 TO GILBERTO CARLOS AND CAMILO PAGE 3 ACKNOIJLEDGEMENT S I would like to thank my ad v isor, Professor D a vid A. Micha, for his invaluable ideas, guidance and support during the development of this work. Also, I wish to extend my appreciation to the professors of the Quantum Theory Project who have contributed to m y education and made the Quantum Theory P roject such a stimulating and enrichin g environment. I would particularl y like to e xpress my gratitude toward Professor Per-Olov L6wdin who made possible m y participation in the 1983 Summer School in Quantum Theory in Uppsala, Sweden. My deepest appreciation is to m y husb and Roy Little, for his support and patience during these years of studies. Finally, my thanks go to Robin Bastanz i fo r her help in the typing of this dissertation. iii PAGE 4 TABLE OF CONTENTS ACKNOWLDGEMENTS ................. . . ................................ iii ABSTRACT ............................ ........... .......... ....... v i CHA PTERS 1 . INTRODUC TION .................. ...... ........ ............. 1 2 . COLLISIONAL TIME-CORRELATION FUNCTION APPROACH TO ENERG Y T R A N SFER ......... .................. ... 7 2.1 Many -Bod y Approa c h . . ..... ...................... . . ... 8 2 . 2 The I mpulse Appro x i mation ....................... . . . 1 2 2 . 3 Application to S cattering b y Adsorba tes ...... ... ... 1 6 3 . VIBRATIONAL FREQUENCIES AND NORMAL MODES OF MOLECULES ADSORBED ON S U R FACES ..... ............ 1 9 3.1 Cluster Mode l s and Force Fields . . ..... ...... ....... 2 1 3 . 2 Nor mal Modes Analyses .. .................... . ....... 2 8 3 . 3 Numerical Results fo r CO on Ni(OOl) ..... . . . ... ..... 33 4. ENERGY TRANSFER INTO MOLECULA R ADSORBATES ............... 4 6 4.1 The Vibrational C orrelation Function . .............. 4 7 4 . 2 Short Time E x p a n sion .... .................... ..... . . 50 4 . 3 S tatistica l Mode l for S i n g l e and D oub l e Collisions ...... . ..... ... ... . ............... 5 1 5 . G A S -ADSORBATE I NTERACTION P O TENTIALS: EFFECTIVE DIFFEREN TI A L CROSS-SECT I ONS ................ . . . 57 5.1 Atom (Ion)-Ads o r bate Interacti o n Pote n tial ...... . . . 58 5 . 2 Effective C lassical Diffe r e n t i a l Cr o ss-Sections .... 8 0 6. VIBRATIONAL E N ERGY TRANSFER I N HYPERTHERMAL C O LLISIONS OF H e AND Li + W I T H CO A D SORBED ON Ni(OOl) . . . ............ 94 6 . 1 Double Differential Cross-Sections for the S yste m H e /OC-Ni( O O l ) . . ....... . ............ . 9 4 6.2 Double Differential Cross-Sections for the System Li+ /OC-Ni(OOl) ....... . . .... ...... ... 97 i v PAGE 5 7. SUMMARY AND CONCLUSIONS ................................ 146 APPENDICES A. COEFFICIENTS FOR TRANSFORMATION s =AX ................... 155 B. SYMMETRY ADAPTED COORDINATES . ... :.:: ................... 158 C. COEFFICIENTS FOR TRANSFORMATION ................... 159 REFERENCES ........................................................ 162 BIOGRAPHICAL SKETCH ............................................... 166 v PAGE 6 Abstrac t of Dissertation Present ed to the Graduate S chool of the University of Florida i n Partial Fulfillme n t of the Requirements for the Degree o f Doctor of Philosophy SCATTERING OF ATOMS B Y MOLECULES A D SORBED AT SOLID SURFACES Chairman: David A. Micha Major Department: Chemistry B y ZAID A PARRA August, 1 988 The formalism of collisional time-correlation fun ctions , appropriate for scattering b y m a ny-body targets , i s implemented to study en ergy transfer in the scattering of atom s and i ons from molecules adsorbed on metal surfaces . Double differential cross-section s for the energy and a ngular distributions of atom s and i ons scattered b y a molecule adsorbed on a metal surface are d erive d in the limit of impulsive collisions and within a statistical model that accounts fo r single and double collisions. The y are found to b e given b y the product of an effectiv e cross-section that accounts for the probability of deflection into a solid angle t imes a probability per unit energy transfer. A cluster m o del is introduced for o f an adsorbed molecul e whic h includes the mole cular atom s , the surface atom s binding the molecule , and their nearest n eighbors . v i PAGE 7 The vibrational modes of CO adsorbed on a Ni(OOl) metal surface are obtained using two different cluster models to represent the ontop and bridge-bonding situations. A short-time e xpansion i s introduced for the slow modes of the target, which leads to Gaussian distributions of the energy transferred into the slow modes . A He/OC-Ni(OOl) potential is con structed from a strongly repulsive potential of He interacting with the oxygen atom in the CO molecule and a van der Waal s attraction accounting for the H e interaction with the free Ni(OOl) surface. A potential is also presented for the Li+/OC-Ni(OOl) w here a coulombic term i s introduced to account for the image force. Trajectory studies are performed in three dimensions to obtain effective classical cross-sections for the He/OCNi(OOl) and Li+/OC Ni(OOl) systems. These cross-sections are then used to calculate the double differential cross-section s per unit solid angle and energy transfer for the same systems. Results for the double differential cross-sections are presented a s functions of scattering angles, energy transfer and collisional energy. Temperature dep ende nce results are also analyzed. Extensions of the approach and inclusion of effects such as anharmonicity, collisions at lower energies, and applications of the approach to higher coverages are discussed. vii PAGE 8 CHAPTER 1 INTRODUCTION In the past few years, there ha s been an increasing interes t in the study of metal surfaces by a wide variety of experimental techniques. An important aspect of metal surfaces is their ability to effectively adsorb atoms and molecules, leading to chemisorption, surface reactions, or catalysis. The experimental study of the vibrational modes of atoms and molecules adsorbed at surfaces has prove d to be a powerful mean s of deriving information about changes in molecular bonding which occur during chemisorption processes and also on the preferred adsorption sites (Rocca et al., 1986). Vibrational spectra of adsorbates have been observe d with infrared reflection spectroscopy (IRS) and withlow-energy electron scattering (EELS). These techniques are in practice sensitive only to the dipole-active high frequency modes of vibration, which correspond to either intrinsic vibrations of the free molecule, or to the high frequency vibration of the molecule perpendicular to the surface. A further class of low frequency modes, the hindered translations parallel to the surface, also exists but their observation is prevente d by the instrumental resolution in EELS and the limited spectral range and sensitivity of IRS. A technique which is in principle an ideal method for s tud ying these vibrations is inelastic atom scattering spectroscopy . It also ha s the capability to observe the dynamics of surface adsorbate laye r s and i s 1 PAGE 9 2 sensitive to very small amounts of ada toms on smooth metal substrates and can, therefore, give information about isolated ada toms (Gadzuk, 1987). Several articles have appeared on experimental aspects of scattering by adsorbates. Systems which have been studied by neutral atom scattering include, among others, H e with either 0 or CO on Ni(OOl) (Ibanez et al., 198 3 ) and H e with CO on Pt(lll). In the latter system cross-sections ha ve been measured as a function of th e velocity of the incident H e atoms and the angle of incidence (Poelsema et al., 1983) and as a fun ction of scattering angle and momentum transfer (Lahee et al., 1 987). All of the experiments with neutral atom s ha ve been done with energies in the thermal regime. Ions have also been used as probes in experiments of scattering by adsorbates. In one of the pioneer works in the area (Hulpke, 1975), energy and angular distributions of Li+ ions scattered by clean W(110) surfaces and b y adsorbates, such as O 2 and CO, were measured for beam energies between 2 and 20 eV. Scattering intensities in the systems Li+-0/Ni(110) and He+-0/Ni(110) ha ve been measured as a function of the scatterin g angles and relative energies of the ions (Englert et al., 198 3). Recently, measurements on angle and energy distributions of K+ ions scattered from W(110) in the energy range of 12 to 100 eV ha ve been published (Tenner et al., 1986). These measurements have been done at normal incidence. In general, mos t of the theoretical approaches to the problem of scattering b y adsorbates ha ve concentrated on elastic scattering. Furthermore, since a detailed theoretical treatment i s virtually impossible to accomplish because of the computing tim e and m emory PAGE 10 3 required, approximate schemes are a necessity . Several approaches to the scattering of atom s with therm a l energies b y adsorbates have been based on e xtensions of methods which h a v e been useful in the description of scattering b y clean s urfaces. The general theory of atom scattering in the eikonal approximation has been extended to scattering b y overlayers (Levi, 1 982). In this theory the motion of the adsorbate is assumed to be separated from the motion of the substrate and does not contain information on the site occupied b y the adsorbate. A theory for scattering b y adatom s at low coverage has been presented (Jonsson et al., 1984). Here the surface i s considered to be a hard wall and the cross-section for the isolate d adatom is simply give n b y an effective scatterin g amplitude obtained by subtracting the scattering ampl i tud e of a hard wall from the gasphase scattering a mplitude for the adatom . This theory has been applied to the system He-OC/Pt using the distorted wave Bor n approximation (Liu and Gunhalter, 1987), and t h e cross-sections h a v e been calculated from the formalism of clos e coupled equation s . However, the possibility of inelastic scattering increases the computational complexity of the problem. Quantal approaches have be en based on wavepackets (Gerber et al., 19 84 ) and on multiplecollision e xpansions (Sing h et al. , 1986). Vith a few exceptions (Vilallonga and Rabitz, 1 986 ) theoretic ians ha v e considered only elastic Bragg scattering whic h give s i n formation o n the structure of adsorbatecovered surfaces or the elastic component of diffuse scattering from adsorbates. However, t h e inelasti c scattering i s of great interes t because it invol v e s the e x citation of surface phonon s and the vibrational modes of the adsorbed molecule . PAGE 11 Vhen an atom collides with a mol ecule ad sorbe d on a metal surface, kinetic energy can be transferred into an y of the infinite number of vibrational modes of the target. Consequently , one i s n o t interested in the state-to-state cross-sections familiar in scattering by isolated molecules, but instead one wants double differential cross-sections, per unit of solid angle and of energy transfer. Furthermore, the quantities of e xperimental interest are averages of these cross-sections with respect to statistical distributions of target states for the give n surfac e temperature. 4 The purpose of this dissertation is to p r e sent a model of energy transfer for the scattering of atom s and ions from diatomic molecules adsorbed on a metal surface. Recent experiments ha v e shown (Berndt et al., 1987) that information on the type of interaction between atoms and adsorbed molecules as well as on the vibrational modes of adsorbed molecules can be obtained from isolated ad sorbates. Therefore, we are interested in a model whi c h would b e applicable to low co verages , and in a simple theory which could provide insights into the t y pe of interactions and also be able to predic t tre n ds in the energy and angular distributions of the s cattered partic les. Standard treatments of mole cular collis i o n s , bas ed on targe t state expansions, are not c on venient because excited targe t states are usually unkno wn. Instead, on e c an use the form alism of collisional time-correlation functions (Mic h a , 1986) which is appropriate for scattering b y many-body target s and includes statistical averages. For hyper thermal collisions , the interaction i s dominated b y strong short range repuls i ve forces and last s a short time compar e d PAGE 12 5 vith the periods of internal motions of the target, so that an impulsive model is appropriate. Furthermore, as long as the projectile is light co m pared vith target atoms, only a single collision occurs vith the adsorbate before the projectile flies a vay . These tvo features lead to a simple expression for the timecorrelation, vhich can then b e obtained analytically from the normal vibrational modes of the target. The vibrations of an adsorb ed molecule can be modelled b y a cluster vhich includes the molecular atoms, the surface atoms binding the molecule, and their neares t neighbors . The normal modes of the target can be grouped into t v o sets, one vith high frequencies (fast modes) similar to those of the i solated molecule and the other vith lov frequencies (slov modes) characteristic of the surface vibrations. Vhen the target involves slov modes , one can approximate the time-correlation functions b y short tim e expansion s vhich lead to Gaussian distributions of the energy transferred into the slov modes . The present vork is presented here a s follovs. In Chapter 2 the formalism of collision time-corre lation functions i s presented together vith expressions for the differential cross-sections vithin the impulsive model, and its application to scattering b y adsorbates. Tvo models are develop ed in Chapter 3 to represent on-top and bridge adsorption sites of a CO molecule adsorbed on Ni(OOl). D ependin$ on the angle of scattering, the interaction betveen th e projectile and the target may involve only the single collision with the adsorbate, or that fo llove d b y a collision vith the solid surface. In Chapter 4 ve develop a statistical model to calculate co ntributions of double collisions to the cross-sections . In order to calculate the double PAGE 13 6 differential cross-sections , on e firs t needs to obtain effective classical cross-sections . W e use the facts that the potentials of the separate ad sorbates d o not overlap and that the translational wavelength of the projectile i s much shorter than the range of the intermolecular potential. Thus, translation of the projectile i s obtained by a classical trajectory . In Chapter 5 the interaction potential for the systems Li+-OC/ Ni(OOl) and H e-OC/Ni(OO l are constructed and trajectory studies are performe d to obtain the effectiv e classical cross-sections . In Chapter 6 results of double differential cross-sections for the systems Li+ -OC/Ni(OOl) and He OC/Ni(OOl) are given a s a function of e nergy trans fer and scattering angles. Finally, in Chapter 7 the ad vantages and disadvantages of the approach are presented together with some suggestions on possible improvements of the approach. PAGE 14 CHAPTER 2 COLLISIONAL TIME-CORRELATION FUNCTION APPROACH TO ENERGY TRANSFER Vhen an atom is scattered b y a target, energy can b e transferred from the projectile into the target or out of the target to the projectile. The amount of energy transferred i s determined to a large extent by the intramolecular d y namic s of the target, therefore it is expected that atomic motions within the target will place certain constraints on the process. The theoretical description of the energy transfer between atom s and extended targets can be done using classical, semiclassical or quantum mechanical method s . The qua ntum mechanical methods are usually based on e xpansions in target states but are computationally impractical and costly due to the large number of energetically accessible levels of the target. An alternative approach whic h takes into account the quantal nature of the target motion, but avoids expansions in target states has be en developed (Micha, 1979a) and used in the study of atom-polyatomic collisions (Micha, 1 979b). In such an approach the target i s describe d as a many-atom system and atom -pair correlation functions of the target play an important role in the description of thi In this chapter we rev iew the main features of this many bod y theory in its exact form and then discuss a simplification of the approach, namel y the impulse approximation, which makes the approach more tractable for its application to the scattering of atoms b y adsorbates. 7 PAGE 15 8 2.1 ManyBody Approa ch The type of process we are interested can in general be described b y where A denotes an atom with an initial kinetic energy E1 and X represents a many atom system, e.g., an adsorbate on a metal surface, in an initial internal state v. During the collision an amount of energy is transferred between them, promoting X to a final internal state v, and leaving A with a final kinetic energy E f . In the laboratory frame, with RA denoting the position of the projectile the Hamiltonian of the projectile is simply its kinetic energy , A (2.1) and the internal Hamiltonian HX for the target may be written as HX = L: 112 2m a where -7 is the r a these vectors. -7X r + VX(r ) a a position of atom a and -7X r (2.2) {i } the collection of a In the coordinate system fixe d to the center of mass of the pair ( A,X), the system Hamiltonian is give n b y where K + H-7X r (2 .3a) (2.3b) Here K is the operator for the kinetic energy of th e relative motion given by PAGE 16 K (2.3c ) where R is the position of the projectile A with respect to that of the target and M is the reduced mass of the system, M = MAMX/(MA + M X)' with MA and MX the masses of the projectile and the target, 9 respectively. -7X The Hamiltonian of the isolated targe t i s H-7X, with r r denoting the position of the atom s in the target. The operator V ::7 -7X represents the interaction potential which depends on K and on r . Indicating with E and Iv > the internal energy and the internal v state of X, respectively, we can write for the unperturbed motion ( +Ev ]1 > , (2.4a) (2.4b) where is the relative momentum. Given an interaction potential V between the projectile A and the target X, the collision probability amplitudes m ay b e obtained from the transition operator, T, which satisfies the Lippman-Sch winger equation T = V + V GO T , ( 2.5) -1 where GO = (E + in HO) i s the propagator for free relative motion of A and X, but involves all the internal motions of X . -7 -7-7 Considering scattering from an initial state Ikv >, w here p = Mk is the relative momentum and v is the internal state of the target, into a final state Ik'v'>, then the scattering rate i s given by PAGE 17 10 (2.6) where the function ensures conservation of the total energy of the system. However, since e xperimental meas urem ents u sually correspond to thermal averages over initial distributions , w ,and they only v :-') :-') specify k and k', we must consider the total rate for scattering between initial and final momenta, that i s ( 2 Ttl Y\ ) L w v 1 < it I V 'iT 1 it v > 12 0 (E E ') . (2.7) v, v I The total initial and final energies E and E' must be equal, as it is insured with the function. The sum over final states in Eq. ( 2.7) usually contains a very large number of terms, specially at hyper thermal collision energies. How e ver, b y using the completeness of the states and the integral representation of the function -E') = J dt/(2nYl) e x p[ -i(E E')t/Y\j , (2.8) the sum over final states may be formally eliminated (Micha, 1986), this gives the result CD v' IT J dt/(2nl1) e xp(-istlY\) < v'I _CD (2.9a) where HX is the target Hamiltonian, and s i s the e n ergy trans fer defined b y , s = (2.9b) The operator T depends on all variables, so its matrix elements (2.10) PAGE 18 11 vary only with the internal variables. Sub stituting Eq. ( 2. 1 0 ) into (2.9a) and the latter into Eq. ( 2.7) on e arrives at the following e xpression for the total scatterin g rate, ( 2.11a) ( 2.11b) (2.11c) where the double bracket s indicate the qu antum mechanical and thermal averages over initial states. The integral in Eq. ( 2.11a) contains the time correlation function (TeF) of the matrix elements Tkfk, which are transition operators on internal variables. The time d ependence of the transition is given b y the internal Hamiltonian of the target. This general equation for the tota l scatterin g rate i s particularly useful when compared with other approaches which would fail when many internal states are involved. Once the tran sition o perators are e xpressed in the vari ables of a given system, the time dependence of the latter may be followed muc h a s one would in a classical treatment. The double differential cross-section for scattering into a unit solid an gle Q and with a transfer of energy of may b e obtaine d from the ratio of the transition rate to the incident flux (Micha, 1 979a), to giv e OJ 2 d 0-1 (d d Q) (2n/ M)4 M 2(kf/k) J dt exp(-it/M) x ( 2.12a) ",;;. PAGE 19 12 where the relation between the final momentu m and the energy transfer is given by k' = (k2 _ 2ME/h2)1/2 (2.12b) The cross-section is thus expressed as the Fourier transform of the time-correlation function of the transition operator. This correlation function approach allows the development of dynamical approximation to the transition operator in a systematic way, as it is shown in the following section. 2.2 The Impulse Appro ximation In the regime of hyper thermal collisions, the projectile probes the internal regions of the interaction potential V which is of a multicenter nature. In these cases the potential may be con veniently represented by an expansion about the atomic centers of the target (Micha, 1979a) (2.l3) a The potential v represents the interaction between the (A,a) a atom pair and depends on the electron distribution of A and on that of the valence state of a in X . In the formalism of multiple scattering ( Rodberg and 1967) the many-body transiti o n operator presented In Eq. ( 2.5) can b e e xpressed in terms of the two bod y potentials va' leading to the final channel decompo sition T ( 2.14a) a PAGE 20 13 (2.14b) (2 . 14c) where T(a) corresponds to a final interaction between A and atom a mediated by v , and T is the transition operator when the only a a allowed interaction is va' but in which all internal potentials of the target are included in GO' Introducing the assumption that the majority of collisions involve a single encounter between the projectile atom A and the target atom a, allows us to write T(a) T. However, in order to a calculate Ta' a N + 1 body problem must still be solved because GO involves the motions of all the atoms in the target. This may be substantially simplified whenever the energy transfer occurs in the course of an impulsive collision of the projectile A with atom a. The basic assumption of the impulse approximation is that a large force F acts on an atom a for a period of time 6t so short, that all a the other forces may be neglected during that time. The position of atom a does not change but its kinetic energy < K > in the internal a \l state \l jumps to a new value < K > I' Energy and momentum are a \l transferred to the target through the interaction betwee n the pair (A,a), while the remaining N-1 atoms in the target provide the restoring forces on a which determine its momentum distribution within the target. To incorporate these ideas into formulas on e can introduce the (A,a) pair-Hamiltonian ( 2.15a ) ( 2.15b) PAGE 21 K a and the propagator E", -< K > , , v a v 14 (2.15c ) (2.15d) (2.16 ) so that the propagator fo r free relative motion GO can be a s GO(E) = gO,a(E) + gO,a(E) (HX EO -Ka) GO(E) , Substituting Eq. (1.17) in the equation for T , one finds a T = t + t (GO go ) T a a a ,a a ta(E) = v + v go (E) t (E) a a,a a (2.17) (2.18) (2.19) Provided GO and gO,a are close in regions ta is different from zero, one can drop the second term in Eq. ( 2.18) and T t (E) a a constitutes the impulse approximation. Replacing T R(k k') R(ba) 2IT l1 x < t in the equation for the total rate, one finds a a R(ba) ( 2 .20a) a,b < k'v' Itbl * kv > v V v' ( 2 . 20b) k'v'lt I kv> o(E -E') a , shows the contribution of each ( ab ) atom p air in the target. Double differential cross-sections may n o w b e obtained (Mic h a , 1 979b ) from the Eqs. ( 2 .20) and one finds ( 2 . 21a ) PAGE 22 15 ( 2 . 21b) J -ik.; ( 0 ) ik.; (t) \' dt istlh I b a l.J w \) 2 JIYI e < \) eel \) > . \) (2.21c ) Here k is the momentum transfer d efined b y Ylk = The factor L (k' ,k) involves only the relative motion of the pair ( A,a). The a second factor, S(ab)(k,S), i s the Fourier trans form of the atom-pair time-correlation function (Va n Hove, 1954) and d e pend s only on the internal dynamics of the isolat ed target. The correlation function for a pair of different a toms (atb) contains phases that depend on the initial values of the d y namical variables of the target. Experimental conditions u sually correspond to initial averages over random phase so that the terms with a=b a verage to zero and the cross-sections are then given b y (2.22) Each term in the sum abo ve is a product of a deflection probability and a target absorption probability . The deflection probability can be written as an effective atomic differential cross-section 2 L ( k', k ) I , a ( 2 . 23) in terms of which Eq. ( 2 .22) becomes L: (j a (k' ,k) s (aa) (k, s ) . ( 2 .24) a The impulse approximation present ed here will hold provided collision s occur in t imes s hort compared with internal motions, and PAGE 23 1 6 provided they distort only a loca l region o f the targe t around a. O n the othe r hand, multiple collis i o n effect s m ay b e n eg lect e d only for certain conditions on the kine m atics and p o t entia l s . For example, multiple collision terms vould b e small for m A/mX 1 and kRa b 1, that i s for light p rojectiles an d vavel e ngth s short c o mpare d vith interatomic distances in the target. Geometric co nsideration s viII also play a role v hen con sidering multiple collision effects. In general the validity of these assu mptions mus t b e reconsidered in each particular cas e . 2.3 Application to Scatt ering b y Adsorbates Se veral changes in the structure of a cryst a l s u r face c an occur vhen adsorbates are introduced . One o f the most important changes is t h e loss of periodicity vhic h is associate d with bare crystal surfaces. In general three cases can b e distinguis h e d . In the first case the ad sorbates for m a r egular lattice o n t h e substrate. In such a case the periodicity parall e l t o the surface i s m aintai ne d but a nev unit cell, u s u ally larger tha n the o r iginal unit cell of the s ubstrate, is need e d in o r der to descr i b e the syst e m . In t h e second case the periodicity paralle l to the s u r face is destroyed . Here ve may think o f a sing l e a dsorb a t e o n t h e s urface, or a n on -periodic arrangement of adsorbates. The final case is tha t in vhic h several layers of atom s are deposited on a met a l substrate. In these cases the periodicity of the system as a w hole, parallel t o the crystal surface, is lost, u sually because the adsorbate s p a cing differs from the atom spaci n g in the substrate. The t heoretical description of collisions between atoms and ad s orbates requires a variet y of approaches, d e pending on the PAGE 24 magnitudes of collision energies and on the extent and t ype of surface coverage. The many-bod y approach has be en previously used to study energy transfer in hyper thermal collisions of ions with a solid surface ( Micha, 1981). In such cases the scattering intensity is concentrated around and within surface rainbow angles, 8 , with r single-atom collisions contributing over all scattering angles 8 while dou ble collisions may co ntribute at certain angles. For scattering b y light projectiles with energies in the 17 hyperthermal region, where the projectile probes the repulsive region of the adsorbate and of the atom s in the uppermost layer of the substrate, the double differential cross-section is given in a similar way as in the case of surface scattering l: (2.25) 1 where 1 denotes atoms in the target, which can now be adsorbate atoms o r surface atoms . Here crl(k,Q) is the elastic differential cross-section for the projectileatom 1 in the target. The surface atom s lie on the plane ( x,y,z=O) and the adsorbate i s placed at a distance Z , given b y the equilibrium position of the ad sorbate above the a surface plane. The projectile moves in the direction of decreasing Z, with a certain incident angle 8. measured from the Z a xis normal 1 to the surface. The potential a t center 1 derives from the va lence e l ectronic state of atom 1 in the target and can be obtaine d from semiempirical potentials whose parameters hav e been fitted to reproduce e xperimental data. PAGE 25 1 8 The elastic differential cross-section 01(k,Q) may the n b e obtained from a bundle of trajectories with impact parameters b within a bare surface unit cell. The factor Sell) has the meaning of a probability per unit energy for ad sorption of momentum and energy transfer E. This self-correlatio n fun ction, Sell), mus t b e obtained fro m a model Hamiltonian for the syst em. A procedure which can b e used, appropriate in cases of low coverage, i s t o describe the target by a finite cluster of N atoms, including the adsorbate and a few layers of the substrate (Parra and Micha, 1 986). This model would provide the vibrational modes which are localized in the vicinity of the surface and which are the ones that can be detected by e xperimental tools. Since scattering from ad sorbates r esembles scattering from clean surfaces, it is e xpected that double collisions m ay contribute at certain scattering angles. The applicati on of Eq. ( 2.25 ) i s on l y valid in regions of single collisions. H o weve r a statistical approach, which uses the geometry of the proble m a n d allows for the description of double collisi o n s , will be presente d in a later chapter. PAGE 26 CHAPTER 3 VIBRATIONAL FREQUENCIES AND NORMAL M ODES OF MOLECULES ADSORBED ON SURFACES The vibrational spectr a of mole cules ad sorbe d on m etal surfac e s can be studied with a variety of e x p erimenta l techniques involving scattering of electrons (Ibach and Mills , 198 2), photons , neutrons and neutral or ionized atom s (Willis , 1980). The spacings and s h a pes of spectral lines provide information about the conformation of the adsorbate and its substrate, and about the force field that determines the vibration d y namic s . When the probing particles are atom s , tra n s l ational en ergy i s transferred into both ad s o rbate and substrate, possibly resulting in the shift and reshaping of spectral lines. The theoretical interpretation of the spectra can b e simplifie d w hen the interna l dynamics of the adsorbate c a n be identifie d a n d separated from the collision dynamics. The normal modes of ad sorbates o n solid surface s can b e ob tained from generalizations of theories developed for surface lattic e dynamics. Approaches can b e based o n G reen functions (Ra h man, Black and Mills , 1982), slab mode l s w i t h a small n umber of layers (Stron g et a1., 1982), and clus ter mode l s . Green f u nction s e ' x tensive work while slab calculatio n s , alt h o u g h less involved, may n o t a J ' J ays properly describe the influence of sit e symmetry (Ra hman e t al .. 1983). Furthermore , they have been used together w i t h s i mple force fields of the c entral-field typ e . H owevpr , o n e ex pect s tha t more detailed force fields w ill b e need ed , co ntainin g in pctrticular 1 9 PAGE 27 bending forces, when one wishes to describe the collisional excitation of adsorbate vibrations. 20 Cluster models have been found adequate to describe adatom s (Black et al., 1982), and in some cases ha ve been as effective as models based on periodic small clusters (Black, 1 982). Applied to molecular adsorbates, those models can b e parametrized to incorporate what is known about the vibrational modes of the isolated molecule and clean surface. The y also allow for detailed valence-force fields and for the incorporation of the point group s y mmetry of the adsorption site. Experimental studies (Andersson, 1977) of CO on Ni(O Ol) ha ve provided evidence that CO molecules can bind on a top site abo ve a Ni atom, on a bridge site between two of them or in a mixture of both (Bertolini and Tardy, 1981). For a dilute CO lattice gas on Ni(OOl), two peaks are observed, at 239 and 256 meV, in electron energy loss spectra (EELS). However, at higher coverages only a single peak at 2S6 meV is observed. These findings ha ve been interpreted as due to the presence of both top and bridge species in the case of low coverages, but of only the on-top species in the higher co verages (Andersson and Pendry, 1979). Model calculations on small clusters ha ve be e n performed b y several workers to obtain vibrational frequencies of a CO molecule adsorbed on a nickel surface. The of a NiSCO cluster for the on-top site and a Ni6C O cluster for the bridge site ha ve been analyzed (Richardson and Bradshaw, 1979), in term s of semi-empirical force fields, with only the C and 0 atoms moving. The frequencies of a linear NiCO molecule ha ve bee n calculated using ab initi o method s PAGE 28 (Allison and Goddard, 1982) based on general equations for XYZ type molecules. 21 In this chapter, we develop a simple c l u ster model of diatomic adsorption and apply it t o CO ad sorbe d on Ni( 001). It i s based on a more detailed, valence-force field than previously found in the literature, and it incorporates site symmetry . It also allows for moving surface atom s so that a more detail e d picture of the adsorbate dynamics can be derived. Furthermore, the substrate cluster contains two surface modes whose frequencies can be fitted to kno w n values a t the f -point of the surface Brillouin zone. We analyze the frequencies of a CO molecul e adsorbed on Ni(001) using two different clusters, intended to mode l the on-top and bridge species. Section 3.1 describes the conformation and force fields for the clusters. Section 3.2 presents the normal mode an a lysis based on the FG met hod of molecular spectroscop y (Wilson, Decius and Cross, 1955). The numerical results and figures showing the normal modes are giv en in section 3.3. 3.1 Cluster Mode l s and Force Fields To model adsorption at the on-top site, we use an eleve n-atom cluster, shown in Fig. 3.1, containing C, 0 and nine Ni atoms, with C, 0 and one Ni atom allowed to move . This cluster contains all metal atom s that are neighbors t o the bonding f ,)llr in the firs t layer and fou r in the secon d layer, with t h e CO bon ded to a Ni atom in the face centered cubic structure . In the bridge-bond ed situation, we have c hosen a 16-atom cluster, show n in Fig. 3 . 2, that contains fourteen Ni atoms, of whic h the two bonded to C are allowed to move. This cluster also contains all PAGE 29 o c Figure 3.1, On-Top CLuster. Nickel atoms with numbers I to 5 correspond to Surface atoms, and With numbers 6 to 9 to nearest neighbors in the second laYer ; C,O and nickel atom I are the only ones allowed to move . 22 PAGE 30 o Pigure 3.2: Brid ge Cluster. Nickel atom s with numbers 1 to B correspon d to Surfac e atoms, and with numbers 9 to 14 to nearest neighbors in th e s e c ond layer; C,O and nickel a toms 1 and 2 are the only ones allowed to mow e . The height of th e C atom abowe th e sUrface ( dc ) is d e t ermi ned b y the angle NiC -Ni. 23 PAGE 31 24 metal atom s that are nearest n eighhors to the two m etal atom s representing the bonding site, six in the first laye r and six in the second layer. The carbon atom of the CO mole cule i s bond ed to two nickel atoms, one in a position corresponding to a face center atom and the other one corner in the face centered cubic structure. A C 4v point group can be associated to the top bonding situation whereas a C 2 v point group is more appropriate for the bridge model. Since on a real surface all translation and rotations are restricted, we e xpect all 3N degrees of freedom to b e vibration a l modes. Therefore, we should find 3 x 3 = 9 vibrationa l modes for the top case. These can b e classified as three hindered translations , two hindered rotations and four modes which are equivalent to the vibrational modes of an XYZ isolated linear molecule in the gas phase. The latter could also be con sidered as arising from the vibration of the isolated CO molecule interactin g with t w o phonon modes of the substrate represented b y a single nickel atom. For the bridge bond ed situation t w o of the metal atoms are allowed to move, given a tota l of 3 x 4 = 1 2 vibrational modes ; s i x of these correspond to the molecular m o des ofaX2CO isolated molecule in the gas phase. Again this could a l s o b e con sidered as the vibration of the CO molecule interacting with four phonon modes of the substrate r epresented b y two nickel atom s . In order to carry out calculations of the frequencies, to obtain the actual form of the normal modes a nd t o classify the different vibrational modes according to the symmetry of the clusters, we have chosen a valence force field to represent the potential energy of the clusters. PAGE 32 25 Usually, the potential energy is expressed in terms of the changes in the internuclear distances and in a number of angles. For the on-top position we used a total of 12 internal coordinates , eight of these coordinates to represent the changes in the Ni-Ni internuclear distances, one each for the changes in CO and NiC bond distances, one for the chang e in the angle b e t w e e n the CO and C -Ni bond and one for the change in the angle between C-Ni bond and the adjacent Ni-Ni bond. Denoting by s .. the changes in the Ni-Ni internuclear distances , 1J b y sco and sCNi the changes in CO and CNi bond distances and by sa and sb the changes in the angles described abo ve, the potential T energy V for the on-top cluster is giv en by 5 9 2VT(s) 2 2 + fCNi 2 Slj + Slk sCNi j=2 k=6 + fCO 2 f 2 2 ( 3.1) sco + s + fb s b a a In the above e xpression the subscript 1 had been us e d to denote the metal atom bonded to the CO molecule , and denote the force constants for the interaction between neighboring atoms in the firs t layer and that between first and sec ond layer atoms. A total of 22 coordinates were used for the bridge bonded model; 15 to represent changes in the Ni-Ni internuc l ear distances , 3 t o represent changes in the bond dis t a nces of the ad sorbe d molecule and four to represent changes in the an g les. Denoting by sd the change in the angle b e tween the CNi bond s , b y s the change in the angle b e t wee n the CO b ond and t h e plane NiC -Ni, g and by s and sb the same variables a s fo r the on-top position, the a potential energy VB can b e written a s PAGE 33 26 [ 5 8 1 ' [ 1 2 14 2VB(s) l: 2 l: 2 l: 2 , E 1 Slj + S 2j S 1 k j=2 j =6 k=9 k=9,12 (3.2) + fCO 2 fCNi 2 + f 2 2 + f 2 + fb 2 sco + sCNi sa + fd sd s sb a g g Phonon dispersion curves ha ve been obtained (Black et al., 1981) for the Ni(OOl) surface at different points of the substrate Brillouin zone. -1 Two surface modes with frequencies of 104.7 and 125.5 cm and polarizations parallel to the surface and no r mal to the surface, respectively, were found at the r zone boundary point. These values were used to fit the t w o force constants and in Eq. (3.2). The interatomic distances and the other force constants were taken from data for Ni(CO)4' (Jones , 196 0). For the b ridging species the force constants fd and fg were fitte d to r eproduce the frequencies for a molecule of the type X 2CO (Herzberg, 1 945), which is the limiting case of the bridge cluster after the neighboring atom s of the Ni bonded to the CO molecule h a ve been taken away. The chosen force constants and geometric parame ters are presented in Table 3.1 and 3.2 respective l y , w here dNi i s the distance b e t wee n neares t neighbors in the bulk crystal structure, and d C is the distance from the C atom to the surface in Fig. 3 . 2 . As a preliminary check of the model and the chosen field, we cal culated the freque ncies of isolated XYZ mole cule to represen t the limiting case for the on-top position cluster. The results are within 1 em-I, a s can b e seen in Table 3.3, from those obtained b y an ab initio calculation on the same molecule ( Allison and Goddard, 19 82). For the cluster representing the bridge bond e d situation, the limiting case was H 2 C O , obtain e d b y replacing in our c luster the PAGE 34 Table 3.1 Force Constants for Internal Coordinate Sets Force Constants Top -1 fCO (mdyne A ) 17.0 .-1 fCNi (mdyne A ) 2.1 f (1) -1 0.15 Nl (mdyne A ) fNl (2) -1 (mdyne A ) 0 .27 f (mdyne A -2 0.38 a rad ) fb (mdyne A -2 rad ) 0.23 fd (mdyne A -2 rad ) f (mdyne A -2 g rad ) Table 3.2 Geometric Parameters of Clusters Geometric Parameters dCO (A) d CNi (A) dNi (A) dc (A) Ni-C-Ni angle (deg) Top 1.15 1.84 2 .49 Bridge 14 . 7 1.9 0.15 0 .27 0.38 0.23 0.23 0.38 Bridge 1.15 1. 60 2.49 0.84 112.00 27 PAGE 35 Table 3.3 Normal Mode CO Stretch NiC Stretch NiCO Bend -1 Comparison of Vibrational Freque n cies (in cm ) for a Linear NiCO Molecule Our Results ( Allison and Goddard, 1 982) 2130.5 2129 400.6 4 0 1 327.8 327 masses and force constants of the t w o nickel atoms bond ed to CO b y hydrogen masse s and force con stants , and setting to zero all other unnecessary force constants. The calculate d frequencies were found 28 1 to be within less than 1 00 cm w h en compared to experimental results for formaldehyde (Clouthier and Ramsay, 1 983). 3.2 Normal Mode Analyses E xpressions ( 3.1) and ( 3 . 2 ) in Section 3.1, for the potential energy of the clusters, contain far more coordinates than the degrees of freedom of the clusters. In order to reduce the number of coordinates to equal the number of degrees of freedom of the clusters we recall that only a reduced number of atom s are allowed to move; this r elates the internal coordinates to the atomic cartesian displ acements ( x , y,z). Afte r e xpanding around the position and setting to zero thp displacements of the atoms other than Ni, C and 0 , we obtain the transformation s = A X ( 3 . 3 ) --Here X i s a co lumn vector w i t h the c artesian displacements of the three moving atom s . U sing the point grou p symmetry of tIl e clu s t e r s , PAGE 36 29 a set of symmetry [S1 was constructed from the atomic displacements X. The transformation S = , (3.4) allowed us to determine a new force constant matrix which i s block-diagonal according to the symmetry of each clus t er. The successive transformation are represented b y AUt 2V(s) = f xt k X S F S . ( 3.5) The relations between the different coordinates are listed in the Appendices A and B. The final F matrices are presented in Table 3.4 for the on -top cluster and Table 3.5 for the bridge. Frequencies were calculate d lIsing F and G matrix method (Vilson, Decius and Cross, 1955) . The G matrix i s d efined b y the kinetic energy T. Starting with . t . 2T = , ( 3 . 6 ) and, using S = on e can then writ e ( 3.7) 1 1 t 1 where = ) M U and M i s the diagonal m atrix of masses. From these F and G matrices, one obtains the frequencies corresponding to the normal modes b y sol ving (3.8) where, introducing the speed of light c , fre qu e n cies v k are give n b y Ak 2 = (2ncvk ) , ( 3 .9) and the corresponding norm a l modes b y Q A-I S or Q X (3.10) PAGE 37 30 Table 3.4 F Matrix fo r Ontop Clu ster Fll 2 + fCNi FI2 fCNi F22 fCNi + fCO F23 fCO F44 + + (f 2 fcO + fb)/ dCNi F33 a F46 2 fa(dCNi + dCO ) /(dCOdCNi) 2 fb/dCNi FSS F44 F48 f/(dCOdCNi) FS7 F46 F66 2 fa[(dCNi + dCO)/(dCNidCO)] + fb / (dCNi) 2 FS9 F48 F68 -f (dCNi 2 a + dCO)/ (dCNidCO) Fn F66 F88 fa/(dCO) 2 F99 F88 Table 3.S F Matrix fo r Bridge Clu s ter Al S ymmetry PAGE 38 Table 3.5 ( continue d) F33 + (dC/dCNi)2 fCNi + 2(dC/dCNidNi) 2 fb + 2bifg + (40/21) 2 a 2 a 3 f a -(dC/dNi)(l/dCNi) fb -2b a f (4/7)(1/{2) f N ( 21,) 1 2 g 222 2 a 2 fa + (dNi/dC ) FCNi + 2a 2 f g + (1 /2dCNi) fb + (17/5) + (8 1 7) where, B 2 S y mmetry 31 PAGE 39 Table 3.5 (continued) F44 + (dNi/2dCNi) 2 fCNi + fb + (7/5) + (817) A2 Symmetry B1 S ymmetry Fll 2 (4/dCNi) fb + [( d C + dCO)/(dC dc O ) 1 2 fd F12 fd (d c + dcO)/(dc 2 dc O ) F 13 -4fbl (IT -fd(dc + dc O ) /(IT 32 The matrices GF-constructed in this way reflect the symmetries of the clusters, thus simplifying the calculation and th e classification of the modes according to the symmetry of the clus ters. In the on-top case the resulting GF matrix is block-diagonalize d into one 3 x 3 matrix b elonging to the A1 irreducible representation, and one 6 x 6 matrix containing t w o coupled modes b elonging to the E irreducible PAGE 40 33 representation. T he GF matrix in the bridge case i s 1 2 x 12 but block-diagonalizes into t v o 4 x 4 m atrices, one b elonging to the Al and the other to the B2 irreducible r epresentations, one 3 x 3 matrix belonging to the Bl and o ne 1 x 1 matrix belonging to the A2 irreducible representation. 3.3 Numerical Results fo r CO on Ni( 001 ) The vibrational modes fo r the on-top and bridge clusters are shovn in Figures 3.3 and 3.4, respectively. The resulting coefficients for the transformation from cartesian to normal coordinates are presented for the ont op and b ridge clusters, in Appendix C. Ue separately discuss the on-top and b ridge cases. 3.3.1 The OnTop Cluster The calculated frequencies along vith previous experimental and theoretical results are presented in Table 3 . 6 for the on-top cluster. In this case ve find on e frequency with value comparable 1 to the CO stretching for the free molecule, 2143 . 2 cm , and t v o frequencies a n d with values similar to the experimental 1 and of 461 cml for Nl'CO frequencies (of 381 cm for CNi stretch , bend) found in carbonyl compound s , Ni(CO)4 (Cravford and Horvitz, 1948). Ue also find three frequencies v3 ' v5 and v6 vith values lover than the largest freque nc y fo r the bare substrate phonon band, -1 295 cm (Black et al., 1981). The mode vith the highest freque n cy , vI' i s related to the free molecul e CO stretch, v 2 relates to the NiC stretching mode, and the v3 mode can b e considered as a frl1 strated trRnslrttion along the direction normal to the surface . PAGE 41 34 0 c x Ni I 1 t I 1 ! â€¢ 1 I VA l UAI lLAI I 2 3 Figure 3.3: Normal Vibrational Displacements for the On-Top Cluster. The v-symbol label different irreducible representations for each symmetry group. PAGE 42 35 o c X Ni Ni 1 1 1 t â€¢ â€¢ / VAl UAI I 2 1 1 1 ! I I Figure 3 . 4: Normal Vibrational Displacements for the Bridge Cluster. PAGE 43 36 /" â€¢ â€¢ Figure 3.4--continued PAGE 44 37 ( . 1 1 ( â€¢ ! 1 VB2 : VB2 10 II Figure 3.4--continued PAGE 45 Table 3 .6. Comparison of Results, On-Top Cluster (Frequencies in cm-1 ) Experiment (Andersson 1977) Our Results NiCO (Richardson and Bradshavr, 1979) t,' NisCO (Black et al., 1982) Ni2CO Periodic Cluster (Black, 1 9 82) Spectral Density (Black et al. 1 9 8 2) A "'II 2069 2088 2088 2088.3 2087 . 5 2088 . 3 A "'21 480 431 437 431. 7 430 . 6 440 . 3 A "'31 102 9 8 . 6 84 . 7 148 E "'4 389 411 399 . 2 399 . 6 400.6 E "'5 131 82 182 . 1 208 190 E "'6 44 71. 6 67.3 37 UJ 00 PAGE 46 39 Three modes belong to the E representation, and can be considered a s two frustrate d rotations and about the x and y axes, and a frustrated translation (v6 ) along a direction parallel to the surface. These ass ignm ents are in Table 3 .7. The values obtained for the C O stretch and Ni-C stretch are in fairly good agreement the values assigned to the on -top bonded co (Andersson, 1977). For the three higher frequencies obtain results almost identical to the ones for a (Ni)SCO cluster and also to the ones for a periodic lattice of small clusters , (Ni)2CO (Black, 1982); the latter used only nearest neighbor interactions and a central-force field. 1 We obtain frequencie s at 389 and 44 em values higher and than the largest bare substrate phonon frequency that are Table 3 . 7 Designation, Character, Species and Frequencies of the Normal Modes for On-Top Clu s t e r Description Al E Frequency -1 (cm ) C-O Stretching vI 2 0 88 C-Ni Stretching 431 Ni-C-O Frustrated v 3 102 Translation (Perpendicular to Surface) O-C-Ni Bending 3 89 Ni-C-O Frustrated 131 Rotation Ni-C-O Frustrated 44 Translation (Parallel to Surface) PAGE 47 40 in very good agreement with two S fun ction like modes at 37 and 4 00.6 cm1 obtained in calculations of the spectral density function of a periodic (Ni)2CO cluster using green fun ction met hod s (Black, 1982). Ve also obtained a value of 102 1 cm for the mode which involves motion perpendicular to the surface A1 (v3 ); this value, although higher than the one obtaine d b y Black for the overlayer and for the isolate d ( Ni)SCO cluster, could b e associated to the fairly broad 1 peak centered at 148 cm obtained in the spectral density calculations for the full crystal. 3 . 3 . 2 The Bridge Cluster For the bridge model we found a total of 1 2 modes, four with A 1 s ymmetry, three with B1 symmetry , four with B2 symmetry and one A1 A1 A1 B2 B2 with A2 symmetry. The frequencies v 1 ' v 2 ' v3 ' Vs ,va ,and B2 v9 ' have values similar to molecular mode frequencies, and the A1 B1 B1 B2 B2 A2 remaining v 4 ' v6 ' v 7 ' vlO' v1l' and v1 2 ' have values lower than 1 the largest bare sub strate phonon frequency ( 29S cm ). These calculated fre quencies are presented in Table 3 . 8 a l ong with results from other calculation s an d previous ex p e rim ental results. The A1 mode with the largest frequency, v1 ' can b e easily recognize d as a CO stretch, v 2 i s a NiC symmetric stretch and the other two, v3 and v 4 ' can be considered a s frustrated translations in a plane normal to the surface, with the last two modes involving C -Ni-Ni bending. Three bendin g modes belonging to the B 1 representation invol ve motions out of the formed b y the two nickel atoms and the carbon; one of these, vs ' can be considered as a frustrated translation and the other two, v 6 and v7 ' a s OC(Ni)2 bendings. A summary of these ass i gnments i s show n in Table 3 . 9 . PAGE 48 Table 3.8 Comparison of Resu!!s, Bridge Cluster, (Frequencies in cm ) Frequency Experiment Our (Richardson & (Andersson, 1977) Results Brad shaw, 1979) A \)11 1932 1981 2099 A \)21 530 A \)31 359 366 457 A \)41 53 B \)51 569 572 B \)61 125 B \)71 71 76 B \)82 657 764 649 B \)92 397 184 B2 68 \)10 B2 2 \)11 A2 143 \)12 41 PAGE 49 Table 3.9 Designation, Character, Species and Frequ encies of the Normal Modes for B ridge Clu ster Description Al B1 B 2 A2 (cm ) C-O Stretching "I 1981 C-Ni Symmetric " 2 530 Stretching C-Ni Assymmetric " 3 366 Stretching Frustrated Trans"4 53 lation ( 2 axes) OC(Ni)2 Bending "5 569 Frustrated Rotation "6 125 (y axes) Frustrated Trans"7 71 lation ( x axes) Antisymmetric NiC "8 764 Stretching Frustrated Rotation "9 397 ( x axes) Frustrated Trans"10 68 lation (y axes) Combination of 8 "11 2 and 9 Frustrated Rotation "12 143 (2 axes) The present proce dure includes several metal atoms to represent the metal substrate and at the same time restric t s the number of 42 degree of freedom of the cluster to describe the vibrational modes of a molecule attached to metal atom s at a s urface. The calculations yield vibrational frequencies which are in good agreement with e xperimental results a vailable on the syst em. Some of the PAGE 50 43 frequencies presented here describe modes for the bridge cluster, have not b een presented in previous works . A fairly clear distinction, see en ergy diagram in Figure 3.5, can be made modes relative high freque n cies (molecular modes) and those corres pond to soft modes coming from the interaction of the adsorbate the modes of the bare substrate. This separation of modes is more evident for the bridge case than for the top case . These cluster models and their frequencies and normal modes can be used as a first step in the study of atom s scattered b y adsorbates. PAGE 51 Figure 3.5 Comparison on Energies for the isolated CO Molecule, the Bare Nickel Substrate and the Results Obtained for the On-Top and Bridge Cluster. , , ' PAGE 52 BRIDGE FREE MOLECULE ON-TOP su bstrate substrate + molecule substrote+molecule substrate 2080r v,A, I --v,A, I 800 UB2 8 700 I I' 600 I-UBI 5 uA , 2 -UzA, 'E UB2 ..g u.E w 9 uA , 4 3 ,Z/' UA2 , 200 r :' ' ''-UB2 ,-12 E ,,' 2 ,/ _ ,,-liB, uE /-Uz 100 r _ -J_lI.A, , ' B, \-lI. A , ,,"-11 .. I E 3 \-v.B2 I ..,.. .. 10 zt V1 Ol I \-uA , 4 PAGE 53 CHAPTER 4 ENERGY TRANSFER INTO MOLECULAR ADSORBATES In scattering b y adsorbates leadin g to e nergy transfer, the translational degrees of freedom can frequently b e described classically while the remaining one s , only vibrational degrees of freedom in our case, must be treated a s quantized . It was pointed out in Chapter 2, that b y r elating inelastic scattering cross-sections to time-correlation functions (TCF's) of transition amplitudes, it is possible to study energy transfer in scatterin g b y exten ded targets without ha ving to expand wavefunctions in the many e xcited states of the target. Inelastic scattering cross-sections for collisions in whic h the relative momentum changes f lb' b IF' f f rom n i to n f were Slown to e glve n y tle ourler trans orm 0 the TCFs of the operators PAGE 54 47 target can be grouped in two sets , slow and fast modes , we can approximate the TCFs by short-time e x pan sions which lead to Gaussian distributions of the energy transferred into the slow modes, as show n in Sec. 4.2. Finally , in Sec. 4.3, we introduce a statistical mode l which allows us to obtain a simple description of double collisions . 4 . 1 The Vibrational Correlation Fun ction The TeF of position of the target atoms that w e wish to calculate is given by (0)] (t)] , a a (4.1a) with .. . I: w < vi I v > , v (4.1b) v which includes the quantum -statistical average over the distribution w of initial vibrational states Iv > of the target. Indicating v operators with carets, r (t) corresponds t o the instantaneous a position of atom a, which e volves in time in acco r dan c e with the Hamiltonian HX of the isolated targe t a s follows: r (t) a For each atom a in the target we can introduce its equilibrium position and displacement from equilibrium , so that a a r a a + u a (4.2) ( 4 . 3 ) In a body-fixe d referenc e frame we have N = 3N free vibrational v coordinates , where N is the number of atoms in the target. U sing Eq. (4.3) in Eq. (4.1a) we obtain PAGE 55 (4 . 4) x exp(ik.[d (t) + U (t)]} a a v The target Hamiltonian is nothing but H and from [H ,d ] v v a 0, we find that (0)] exp[iK U (t)] a a v (4.5) Assuming harmonic vibrations as a zeroth order approximation, which is acceptable except for highly excited vibrational levels near the dissociation energy, and using internal coordinates s. which are 1 linearly related to the displacements b y n v I: A u ai s. a 1 (4.6) i=1 the vibrational TeF is most conveniently evaluated in terms of the normal displacements Q.(t) = Q.(O) cosw.t, whic h are related to the J J J internal displacements by a linear transformation s ( t) = UQ ( t) . (4.7) The standard normal-mode analysis (Goldstein, 195 0 ) allows one to determine the normal frequencies w., as well as the vectors J c. I: A . U .. of the transformation from normal displacements to aJ a1 1J i the u displacements . a Since the normal modes are independent, the vibrational TeF factors into a product of correlation functions for each mode as shown below: 48 PAGE 56 n v 49 n .0.(0)] x .O.(t)] . (4.7) aJ J aJ J v The correlation of each mode is readily e valuated b y means of the -+ algebra o f creation and annihilation operators a. and a. (Messiah, J J 1961), through Q. = [M/(2oo.)]1/2 (a. + . J J J J Introducing the notation 1 . aJ (M_] 1/2 200. J Cr , a<.,,] where E, denotes the cartesian component ( x,y, or z) and using Eq. ( 4 .8) in to Eq. (4.7), we can wri te N v = e xp[-iKl .(a. + aJT)ol vK aJ J -r x e x p [ i Kl . (a. + a.) t ] . aJ J J v (4.8) (4.9) (4.10a) (4.10b ) By following a previous procedure (Micha, 1979b), one then arrives at F(j)(t) 1.= e xp(-in.oo.t) Pn. vK J J n . J (4.11a) J Pn. = exp(-n.ex.. -x. cosh ex.. ) I ( X . ) , J J J J n. J J J (4.11b) x . M(K C. 2 sinh ex..) cos f3.) /(200. , J J J J J (4.11c) ex.. Moo/2k B T J (4.11d) PAGE 57 where n. is the number of quanta transferre d into a mode of ene r gy J 1100., I (X.) is the modified Bess e l fun ction; i s the angle J n. J J between K and the ve ctor Cj , and kB i s Boltz m a n n ' S con s t ant. The total vibrational correlation function F in Eq. (4.10) can be v K rearranged as a sum of products of e xponentials 5 0 L: exp(-inoo t) F VK( t) P (K) --v n (4.12a) n n P (K) Pn. n (4.12b) j J 00 and n are column matrices with elements oo. and integers n., v J J respectively. It is immediately s e e n that the Fou rier trans form of the vibrational TCF would con sist of a group of & fun ction peaks located at 8=-l1ntoo . v 4.2 Short Time E xpansion The calculation of the vibrational TCF give n in Eq. (4.10) can b e simplified whene ver the vibrational modes of the t arge t satisfy l100j kBT. Therefore, one can approximate by a short time expansion around t = 0 (Micha, 1981), of the form In F( t) In F( O ) + t F(O ) /F(O ) + t t2 [F(0) /F(0 ) -F(0)2/ F ( 0)21 (4.13 ) where a dot signifies a time d erivative . P erfo r ming the d erivatives from Eq. (4.10), the e xpression fo r e a c h of the low frequ e n cy modes j is given by (4.14a) 6. J l>\oo. X. sinh J J J (4.14b) PAGE 58 51 J (4.14c ) where X. and a . are the same quantities a s given b y Eq. (4.11). J 1 Collecting all the low frequency mode contributions one obtains the final expression In F (t) v j (4.15) j The Fourier transform with respect to time of Eq. (4 . 15) gives a Gaussian distribution peaking at the energy transfer = I: 6. with a width r J j with momentum transfer. I: r .. Both quantities would increase J j 4 . 3 Statistical Model for Single and Double Collisions an atom collides with a molecular ad sorbate it causes transitions among the target's vibrationa l states. The mos t like l y event is excitation of both adsorbate and substrate vibrations, although deexcitation i s also possible from the thermally populated excited states of the target. The mechanism of energy tra n s fer d e pends on the scattering angle , 8, defined as the angle between the incoming and outgoing momentum vectors. For an atom approaching along the perpendicular to the surface with a small impact param e ter, b, collision s with n. atom of the adsorbate will lead to deflections b y an angle larger than n/2, in which case only on e encounter takes place. Thi s can b e seen in Figure 4 .1, which has been draw n u sing co vale n t atomic radii plus a van der radius fo r H e atom s . If, however, the i mpact parameter is large, the first d eflection i s smaller than n /2 and the PAGE 59 I , f-b1Tl2"j i I I I I I: I I I I I I I I I I I I I I I -.......... / '" \ 52 Figure 4.1: Trajectories showing regions of sing l e and double collisions. Model draw n u sing c o valent atomic radii plus a van der Waals radius for H e atom s . PAGE 60 53 atom ne x t collides with the surfac e before flying away a t a n angle larger than n/2. Hence, the deflection fun ction of such a collision would show two branches for e > n/2. It is then obvious that an atom which had been deflected by either mechanis m could be detected at the same final scattering angle. Therefore, to obtain the double differential cross-section one need s to add all contributions involving the same amount of energy trans f e r and having the sam e scattering angles regardless of the mechanism by \Jhic h this oc curred. In Chapter 2 an e xpression for the double differential cross-section, for single collisions and under the impulse approximation, was s hown to be giv en in terms of the TCF. Using the vibrational TCF obtaine d in section 4.1, one can write (dcr] \' exp ( inw t) P ( k ) dQ 2nY1 f...., --v n (4.16) n Here one can recogniz e t w o fac tors, the first, (dcr/dQ), i s a n effectiv e classical cross-section an d the second has the meaning of a probability per unit energy. Then equation (4.16 ) can be rewritten as where dP Jdt -itIYi 2nY1 e \' inw t f...., e --v P ( K ) n n (4.17a ) L: d P Id n (4.17 b) n In particular, for a scattering c haracterize d b y the energy transfer = w e find the p artial cross-section to be g i ve n by PAGE 61 54 (4.17c) This way, the total differential cross section i s given by (4.18) A detailed cross-section should account for the quantal pha s e interference of branches I and II. However, in our case w e mus t average the detailed cross-sections o ve r the target t h e rmal distribution, which would lead t o c ancellatio n of the interf e r e nce term e xcept possibly near e = n/2 . D enoting the t w o branc he s corresponding to single and double collisions b y I and II, respectively, we can then write the cross-section 2 = (do/dQ) (I) (dP/d8) (I) + (dcr/dQ) (II) (dP/d8) (II) (4.19) where the first term i s obtained from branc h I a t a n gle e, and the second term from branc h II at the same a n g l e . The probability per unit energy t r ansfer c orresponding to single collisions, (dP/d8)(I), follows directly from the F o u r i e r tra n s form of the vibrational TeF g i v e n b y Eq. ( 4.16 ) . The s e cond term in Eq. (4.19) is more difficult to obtain because i t i s a c ombined probability since it accounts for the tra n s f e r of t w o amount s of energy occurring in a double collision. c They are the energy 8 transferred during the first collision into the cluster plus the amount 8 m transferred during the second collision into the m etal awa y from the adsorbate. This term can b e d erived u sing statistical arguments as follows. W e first separate t h e collection n of nor mal mode quantum number s of the whole targe t into the subset s nC for a PAGE 62 55 cluster and nm for the metal surface away from th e ad sorbate. The latter contains the parallel and perpendicular vibrational modes of the clean surface. For the double collisions we then write (dP/d) (II) 1 P () n J o Jdt -it/l1\, 2 nYi e L., e n iE t/Yi n dy P c (y) P m [(l-y)] , n n p () n (4.20a) (4.20b ) where E = L: n .l1w. and P i s the probabili t y of transferring n n J J 12 quanta when the total energy transfer i s = c + m. The probability P has b een written as the convolution of the n probabilities P and P for excitation of the cluster and the clean c m n n surface, corresponding to a sequence of two inde p endent events. The integration variable y in Eq. (4.20) i s defined b y y = c/. Using E n E + E we then find c m n n (dP/d) (II) c F (t,y) m F (t,1-y) 1 Jdt -itll1 J c m 2nl1 e d y F (t,y ) F (t,1-y) , o L: exp(iE t/l1) P c c ( y ) , c n n n L: e xp(iE tll1) P [(l-y) 1 m m m n n n (4.21a) (4.21b ) (4.21c ) c m Here, F (t,y) and F (t,1-y) are the vibrationa l TCFs for the cluster and clean metal respectively. Each of these two TCFs could b e calculated f rom the expressions in Eq. (4.11). PAGE 63 56 However , it is u sually found , as was the case in Chapter 3 for a co molecule adsorb ed on a Nickel surface, that freque n c ies can be separated into high and low ones. Thus, for each low frequency mode, 1, one can use the short time expansion given in Eq. (4.1 4 ) which, when replaced in Eq. (4.21), leads to ds dQ d y Pn ( y) f [ n ]1/2 2 2 x r 2 e xp [ (ss-6) / ( 2 f )] , ( 4 .22a) ( 4 . 22b) where n f is the number of quanta gained b y the fast mode of energy s + nfMoof is the energy gained b y the slow modes. The total shift 6 and width r2 are sums o ve r all slow modes I and are functions of 1-y. Equation (4.22) may the n be used to calculate double differentia l cross-section s (DOCS) versus the scatterin g angle e, and versus the amount of energy s , tra nsferred into the slow modes, for each fast s mode nf" PAGE 64 CHAPTER 5 GAS-ADSORBATE INTERACTIONS POTENTIALS: EFFECTIVE DIFFERENTIAL CROSS-SECTIONS In previous chapters we ha ve shown that double differential cross-sections for atom-adsorbate collisions can b e obtained from a product of deflection probability and a target absorption probability, where the former can be give n by an effective differential cross-section. In order to obtain this effective differential cross-section one must first specify an interaction potential between the projectile and the target. In this chapter we first review general qualitative features of the atom surface interaction potential and the usual procedure one can follow to obtain analytic representations which are de sirable for scattering calculations. Since not muc h i s kno w n quantitatively about the behavior of the potential, one i s forced to introduce model potentials. A model potential is then presented for the interaction between helium atoms and lithium ions with a carbon monoxide molecule adsorbed on a metal surface. These model potentials are then used in trajectory studies of the + "< two systems, He -adsorbed CO molecule and Li -adsorbed CO molecule, to obtain the de sired effective differential cross-sections. 57 PAGE 65 58 5.1 Atom (Ion)-Adsorbate Interaction Potential The general features of gas-ad sorbate potentiaJs are e xpected to be similar to the qu alitative picture of th e gas-surface potentials . In the case of sufficiently small distances from the surface the potential has a strong r epu lsive part in front of the solid. The reason for this repulsion is the overlapping of the wave functions of the electrons of the gas atom and of the electrons at the surface of the solid. This gives rise to a pe riodic modulation of the repulsiv e part of the potential parallel to the surface. At longer distances the gas-surface potential i s dominated b y an attractive van der Vaals interaction given b y the distance-dependent p o larization energy between a gas atom and the solid. This attractiv e part of the potential is caused b y the interaction of the gas atom with a relatively large numbe r of crystal atoms, so that the attractiv e part can be assumed to be practicall y constant parallel to the surface. There have been only a relatively small number of attempts t o calculate the physical gas-surface interaction potential from first principles. Calculations of H e on metals ha ve been don e (Zaremba and Kohn, 1977) to calculate the attractive van der Vaals interaction of large distances and the repulsive interaction at small distances and to construct from these t w o parts a compl e t e potential a s a fun ction of the distance Z of the gas atom from the surface. In this procedure the position of the plane Z = ZO' from which the distance of the gas atom should be measured, appears as an essential param eter. More recently, methods for generating surface potentials from surface electron densities h a v e been presented. Some authors (Esbjerg and Norskov, 198 0 ) have suggested a theory whic h gives a PAGE 66 59 relation of proportionality b etween the helium potential energy and the surface-electron charge d e n sity. However, when experimental results are compared with theoretical predictions, they strongly disagree. In particular, for the Ni(110) surface the corrugation parameter across the close-packed rows i s found to b e three times the e xperimental value ( A nnett and Haydock, 1 984). Similarly, several ab initio calculations of potentials for weakly corrugate d surfaces have not been very successful, predicting approximately t w o times too large a corrugation (Batra et al., 1985). H owever , semi-empirical methods, allowing the adjustment of some parameter, have led to good agreement with the H e scatterin g data (Harr i s and Li e bsch, 198 2b). An approximation frequently used that allows calculating physisorption potential energies of gas-surface system s is based on the additivity assu mption. The thermally averaged solid surface i s assumed to hav e perfect periodicity and thus may be expressed in a Fourier e x pan sion as follows: v Cr) exp(iC.R ) , G s where r = ( x , y , z ) (RS'Z), with the Z-axi s extending in the (5.1) direction normal to the surface. The t\JO dim e n sional vectors Care the reciprocal lattice vectors of the surface. The first t erm i n the e x pansion, vO(Z), i s con sidered the over-all average potential and is the large in perturbation theory w hich causes specular diffraction. The v C ( Z ) for C t 0 are treated a s p erturbations whic h cause non-specular diffraction. Diffraction ex perime n t s give informatio n on both the dimensions of the unit cell via the angular location of Bragg peaks and the distribution of the scatteri n g centers within the unit cell v i a the inte n s i t ies of diffrac t e d b eams . PAGE 67 60 However, to obtain a solution to th e d iffraction problem in closed form, the potentials including = mus t b e simple. A con venient empirical representation of the vC(Z) is the Mors e/expon ential-repulsion represelltation (MERR) ( Goodman, 1987) which represents b y a Morse p o t ential and b y an e xponential repulsion for i O . = D(e xp(-2oZ) 2exp(-oZ)} , = e xp(-2oZ) i 0 , (5.2a ) (5.2b) where are the so-calle d diffraction strengths. In the 4G model where only the four smallest nonzero reeiprocal l attice vectors are used, the sum o ver reduces to four t erms and the reduces to a single value. The qualitative picture of the gassurface interaction we have described has to be modified in the case of ion-surface interactions . In these cases and at short dis t a nces, t he important contribution to the surface interaction comes from the o verlap of the e lectronic charge densities, which in the case of closed shell ions, is reasonably co nsidered as purely repulsive. At large distances from the surface the interaction i s attracti ve and i s ylell described b y the macroscopic image potential, V . 1m 2 = -e /4Z . The image picture, however, loses validity w hen the particle ap proaches to within a few angstroms of the surface. H e r e again, as in th e case of the va n der Waals interaction, a parameter may b e introduced to define the position of the i mage plane whic h i mproves th e validity of the image potential. Recently, a way to construct the interaction potential between an ion and a surfac e of a monatomi c solid has been presented (Mann et al., 1987). The K + V(1 00 ) potential co n structed this way was found PAGE 68 61 to agree with an empirical potential used to fit scatterin g data for the same system (Hulpke and Mann, 1 985). The interaction of an atom with an adsorbate will b e comprised of direct potentials (the attractive van der Waals and the repulsive short range), as occur in the gas phase, and of indirect surfacemediate potential which arise du e to the proximity of a third polarizable body, whose presence modifies the simple, t wo-body ga s phase interaction. The surface-mediated potential is e x pected to be of particular importance in the case of long range interactions and only for the case of low incoming atom energies. The induced polarization interaction of closed shell atoms with anisotropic adsorbed molecules ha s been investigated and a theory of the long range van der Waals potentials acting between a H e atom and a CO molecule adsorbed on a metal surface has been presented (Liu and Gumhalter, 1987), and the full potential has been given as the sum of the long range interaction plus the short range repulsion for the atom-gas phase CO molecule interaction. A simple approach to the full description of an atom-adsorbate interaction consists in superimposing an ab initio atom-gas phase molecule interaction potential on an empirical at?m-surface potential. This procedure ha s been used in the system H e/OC-Pt(111) (Lahee et al., 1987) and has been found to give good agreement with the e xperimentally predicte d classical turning points. 5.1.1 He/OC-Ni(O Ol) Interaction Potential Since our final goal i s to obtain an interaction potential simple enough to be used in trajectory studies, we follow the assumption that the atom ad sorbate potential can be given b y superimposing an atom-molecule potential on an atom-surface potential. PAGE 69 62 For the atom-surface potential we hav e follow ed the procedure given by Harris (Harris and Liebsch, 1982a). Basically, their theory, a generalization of that u s ed for jellium surfaces (Zaremba and Kohn, 1977), relates the interaction to s h ifts of the b and energies due to the presence of the helium. In this theory, the Hartree-Fock interaction energy to lowest order in the o verlap is given by 7 d p(,r) g() , (5.3) _co where F is the Fermi energy , g() is a smooth function of energy and 7 p(,rN ) is the local density of states at the helium nu cleus. In the case of face centered cubic (fcc) transition metals h(x , y ) = cos(g x ) + co s(g y ) , x y 2]112 + g y 1 2 ex â€¢ (S.4a) (S.4b) (S.4c ) (S.4d) w is the work function of the metal, 6 i s an energy shift tha t depends primarily on the band width, and g x ' g y are the magnitudes of the smallest reciprocal vectors in the x and y direction, respectively. The attractive part of the h elium interaction is tak en to be -3 given by the van der Vaals e xpression C ( z z ) with the origin VYl v w of z lying one-half a laye r spacing outside the topmo s t plane of nuclei. PAGE 70 63 The full He-adsorbate potential can then b e written a s the sum of the three contributions: repulsive He -surface, at.tractive He -surface and He gas phase molecule, that is: V(x,y,z) = V (R ) + V (R) + V (z), a a rep vw Vcorr(x,y,z) = Vs(z) h(x,y,z) , h(x,y,z) = Bs D exp[-S(z-zO)]h(x,y) , 3 Vvw(z) = -Cvw/(z-zvw) , (5.5a) (5.5b) (5.5c) (S.Sd) (5.5e ) (5.5f) (5.5g) where R represents the distance between the He atom and the oxygen a atom of the adsorbed CO molecule and it can be written as R a 2 2 1/2 [R +R -2R R cose] , m m and R is the distance between the surface and the oxygen atom in the m CO molecule. In the case of CO on Ni(OOl), R has been determined m experimentally (Andersson and Pendry, 1980). The geometry of the problem is given in Fig. 5.1, wher e the surface defines the (x,y) plane and the z coordinate is perpendicular to the surface and passe s through the center of the ad sorbe d molecule . From electronic calculations of CO on a cluster of surface atom s there is overwhelming evidence that there is a chemical bond formation between the substrate valence band orbitals and the adsorbed CO valence molecular orbitals. The CO S a molecular orbital may acquire partial metal character and thereby donate some of its electronic charge into the metal states . Simulta neou sly, some of the * metal charge may be ba c k -donate d into the form erly unoccupied CO 2n PAGE 71 .. ) .... /, i / ..... I', R , . ' I a , "," I ." I " .,,' I ,,-' I ,.,'" I I I I I / 1 / / / / ,f / 0 /1 R I I I I I I o I 1 1 1 1 Z / / 1 r----/ / / / / 1 / / 0 / / / / / o / / / / / / / / ---fo-------fr--------------ft--/ / / / / / / / / 0 / 0 / / / / / / / / / / F igure 5.1: System o f co ordina tes an d geometry used for t h e interaction potentia l . a) Ont op mode l ; b) B r i d ge model . 64 y PAGE 72 65 molecular orbital which may become a bonding one with respect to the metal-CO interaction (Bauschlicher, 198 6). This back donation into * the 2n orbital is then responsible for the weak ening of the molecular bond and an increase of the CO bond l ength. The C-O bond length in a CO molecule is 1.128 A, whereas in a CO2 molecule, it is 1.159 A which is closer to the value found in carbonyl compounds and in a CO molecule adsorbed on a Nickel surface. In view of the above arguments, we hav e chosen to use the parameters given b y the fitting of a SCF interaction potential energy for He CO2 (Clary, 1982), for the He ad sorbate repulsive part of the potential. Diffraction data ha v e not been reported on the HeNi(OOI) system, so we have used the Harris and Li e b s i c h parameters for the He -surface interaction. A list of all the parameters we have used in this work is giv en in Table 5.1 for the case of a CO molecule adsorbed on top. The only parameter which changes when one considers the bridge bonding situation is the height of the adsorbed molecule. This has been chosen according to the geometric parameters of the clusters presented in Chapter 3. It is known from the study of atomic and molecular interactions that the van der Waals interactions mus t b e damped at short distances, because of electronic overlap of wave fun ctions . In orde r to accomplish this damping, we ha ve used a function of the type fe z ) = + which for z PAGE 73 66 Table 5.1 Interaction P o t e n t ial Parameters for H e/OC-Ni (OOl) H e/CO He/Ni A (eV) 2 157. 2 1 a B (A-I) 4 . 7 a A (eV) 1 2 . 0 s B (A-I) 2 .63 s D(A ) 0 . 0127 f3(A 1 ) 0.8692 C (eVA3 ) 0.22537 v w zO(A ) 3 . 01644 z (A) 0 . 24396 vw g(A-1 ) 2.489 PAGE 74 " Figure 5.2: Attractive (---) and full (---) potential for the on top He/OCNi(OOl) system over the surface point x=y=O. The inset shows the region around the minimum. PAGE 75 68 0 I I I I I 2 t-ao Q') .. .... a CO . I 0 !l \ C) â€¢ ----, â€¢ \ "\ "," 1:"\ , â€¢ t-'"' CIl CO S 0 N H ... '" ., N I N I I -+oJ ,,,,aI/ ( Z 'wlI)A LOCIl OD I I I N I C'J I I I C\2 \ t\ "-'" I I I I I 0 0 C'J t--0 C'J t--0 LO C\2 Q') t--0 0 0 0 0 I Aa/(Z'BH)A PAGE 76 Figure 5.3: Attractive (---) and full (---) potential for the bridge He/OC-Ni(OOl) system over the surface point x=y=O. Inset shows region around the minimum. PAGE 77 7 0 0 C> CO ., r-. E .,n ... \ "-.oN ---\ U) \ COS \ 0 h n ....,J Lr) U) QD N ., on M N i N r:: I I PAGE 78 71 interaction for the on -top case giv e s a larger turning point than the case of the bridge, a s was e x p ecte d du e to the s h orte r dis t a nce of the CO molecule to the sur face in the l atte r case. It also can b e seen in the inset of both figures that the bridge potential presents a deeper well than the top potential. This i s because the attractive part of the interaction between the H e atom and the surface i s stronger when the He atom is o ver the space between two surface atoms, rather than directly on top of a surface atom, and in the bridge model, the adsorbate is positione d in b e t ween two surfac e atoms. In figures 5.4 and 5.5, the depend e n c e of the potential in the (y,z) plane is show n for the t w o mode l s . A gain the bridge gives larger turning points than the top mode l but no w the effec t of the adsorbate becomes smaller as one moves away f rom it and finally , w h e n the two models give identical turning points . This i s du e to the fact that for one is only seeing the H e -surface interaction. 5.1.2 Li+/OC-Ni(OOl) Interaction Potential The ion-adsorbate interaction potential can in general be writte n as the sum of three contributions : the ionm o lecule repuls i v e interaction, the ion-surfac e interaction, and the image interaction. As was mentioned in S ect. 5.1, a recent approach to con struc t the interaction potential b e tween an ion and the s u r face of a monatomic solid (Mann et al., 1987) is found to agree with an empirical potential used to fit scattering data for the syst e m Li+/V(OOl) (Hulpke and Mann, 1985). V e ha ve chosen to r epresent the Li+ -Ni(OOl) potential by a modified MEER potential which ha s been used to fit scattering data for the system Li+/Ni(001) (Gerlac h and Hulpke, PAGE 79 > Q) "'" -.. N aj 0:: -> 1.23 0.97 0 .70 0.43 0.17 -0.10 0.0 Figure 5.4: 0.8 1.5 2.3 3.0 3.8 Z/ Angstroms I I I I , I , I I , , , I , I I I \ \ \ 4.5 72 5.3 Potential for on-top He/OC-Ni(OOl) system over several points on the (y-z) plane. Starting from first line on the right, the points correspond to y=O, a/4, a/2, 3a/4, aj where a is the lattice spacing. 6.0 PAGE 80 1.50 1.23 0.97 :> Q) "'--. N 0.70 a:l 0:: '-" :> 0.43 0.17 -0. 10 0.0 Figure 5 .5: 7 3 \ \ \ , \ , \ , , \ , \ , , \ , \ , , \ , \ , , \ , " \ " " " " \ " \ "'-"-......... 0 . 8 1.5 2.3 3.0 3.8 Z/ Angstroms 4.5 5 . 3 6 . 0 '" Potential for bridge H e/OC-Ni ( OOl) system over several points on the (y-z) plane . Starting from first line on the righ t, the pain ts correspond t o y=O, a/4, a/2, 3a/4, a; w here a i s the lattice spacing PAGE 81 74 1977). Here again we can write a sjmilar equation to Eq. (5.5) V(x,y,z) = V ( R ) + V (R) + Vimag(z) (5. 6) a a s The term V (R ) is identical to the one for the helium potential but a a noVi the parameters are obtained from Li+/C0 2 data (Vilallonga and Micha, 1983a). The term V (R) contains a repulsive and an attractive s term given b y V (R) = V ( z ) + V tt (x,y,z) s rep a r (s.7a) V (z) = A exp( B z) rep s s (s.7b) (s.7c) + h(x,y) (5 .7c) and 2 V . (z) = -e /4( z-zO) (s. 7d) Imag The parameter 0 represents the Vlell depth and its Vlidth. The meaning of each of these term s has already b eing explained in Section 5 . 1. HOViever , one should mention that in this model the Coulomb term, at sufficiently small distances z from the surface, is cancelled by the repulsive part included in Vattr . The potential parameters used for the Li +/OC-Ni(OOlO) system are presented in Table 5 .2. The shape of the potential calculated from thi s construction is again found to be steeply repulsive around the energies of interest <. Vlhich in this case are 10 to 50 eV. Figures 5.6 and 5.7 shoVi the potentials for the top and bridge model respectiv e l y . One can observe that the trends fo r turning points and Vlell d epths, Vlh e n compared betVleen models, are the same as the on e s found for the He potential. HOViever, an important differen ce in the behaviour of the potential in the y-z plane can b e f ou n d "/h e n one the PAGE 82 75 Table 5.2 Interaction Potential Parameters for Li+/OC-Ni(OOl) Lj + ICO Li+/Ni(OOl) A (eV) 1777.55 a B ( A -1) 5.27 a A ( eV) 2400 . 0 s B (A-I) 3 . 0 s D(eV) 1.0 1.1 0: -0.1 zO(A) -0.75 zl(A) 0 . 50 PAGE 83 Figure 5.6: Attractive (---), Repulsive (---) and full (---) potential for the on-top Li+/OC-Ni(OOl) system over the surface point x=y=O. Inset shows region around minimum. PAGE 84 77 0 0 LO LD 0 C\1 C\l \ \ \ ., '" \ . \ , o l"e M e 0 CO o e M e I LD l"e I Aa /(Z'BH)A o i 0 . r-co coCf.l S 0 h ......, LDCf.l / 0.0 ./ \ \ N ' \ \ C") \ 0 LO 0 C\l C\1 I PAGE 85 Figul:"e 5.7: Attl:"active (---), Repulsive (---) and full (---) potential fOl:" the bl:"idge Li+/OC-Ni(OOl) system ovel:" the surface point x =y=O. Inset shows region around minimum. ,.' PAGE 86 0 LD LD 0 C\l C\l ro 0 CO ...-4 ., o o o ., o I LD ...-4 ...-4 , r-0 I "< Aa /(Z'13H)A , , 79 co cofll S 0 J...4 +J LDfIl I:lD â€¢ < I N ., C! i \ \ C'J \ 0 0 LD 0 . l:'-C\l C\l I PAGE 87 80 different systems. In figures 5.8 and 5 . 9 fo r the Li+/OCNi(OOI) system, one can observe that the turning points , as one moves away from the adsorbate, do not change a s much as th ey do in the He/CONi(OOI ) system. Thi s is because the Li+-Ni(OOI) interaction is more attractive on top of the surface atom s than in betwe e n atoms; this is the opposite of what occurs fo r the He-Ni(OOI) interaction. 5 . 2 Effective Classical Differential Cross-Sections The interaction potential i n the system atom(ion)-adsorbate-surface is not only dep endent on the separation between the atom and t he adsorbate but also depends on the separation between the atom and the surface. This represents a two center interaction problem , which means that the problem i s not of a central force type and the classical cross-sections can not b e evaluated from t h e formulation for isotropic potentials. For anisotropic potentials, the differential classical crosssections can be evaluated from a classical trajectory determined b y a v (5.8) To obtain the differential cross-sections one need s to go from an initial element of area d e t ermine d by the impact parameter, b, and the initial azfmuthal angle, to a final e lement of area determined by the fina l polar, G, and a zimuthal, an g les. Therefore, o n e need s the transformation from the set to the set This transformation i s obtaine d b y the Jacobian and the differential cross-section (Vilallonga and Mic ha, 1987) i s the n given by PAGE 88 81 25.0 I I I I I I I I I 20. 5 I 16.0 l-I I :>-I Q) I "---. N 11.5 --:>-7.0 I I I , , 2.5 I-:'\ " " " -2.0 I I I I I I I I I 0 1 2 3 4 5 6 7 8 9 Z/ Angstroms Figure 5 . 8 : Potential for on -top Li+/OC-Ni(OOl) system over several points on the (y-z) plane. Starting from solid line on t he r ight the points correspond to y=O, a/4, a/2, 3a/4, .a; where a is the lattice spacing. ----10 PAGE 89 25.0 20.5 r 16.0 f-:> "11.5 f----p:: --:> 7.0 r 2.5 f--2.0 0 Figure 5.9: 82 I I I I I I I I I I I I I I I , I I I I I I I I I , I , , \ \ \ \ \ \ .... " I I I I I I I I I 1 2 3 4 5 6 Z/ Angstroms 7 8 9 Potential for bridge Li+/OC-Ni (OOl) system o ver several points on the (y-z) plane. Starti ng from solid line on the righ t the points correspond to y=O, a/4, a/2, 3a/4, aj vhere a is the lattice spacing. --10 PAGE 90 da c dQ (b/sin8) 8 3 (5.9 ) The abo v e procedure involves a significant amount of computing time. In order to reduce the co mputationa l efforts we have used a simplified procedure based on histogram m e thod s whic h ha ve been widely used in the computation of elasti c and inelastic an d inelastic molecular differentia l cross-section s (Gentry, 1 9 7 9). In physical applications of scatteLing b y ad sorba tes we are concerned not with the d eflection of a sing l e partic l e but with the scattering of a beam of identica l particles, w ith a s pecified k inetic energy, E., striking a sample of the materia l unde r study . The atoms 1 in the beam hav e different impa c t p arameters a n d are s catte r ed through different a zimutha l and polar scattering an gles . Using a bundle of trajectories, the den sity of the incident beam can be represented b y the ratio of the num ber of impinging trajectories , dN., o ve r an area d x d y o f the sample. In conditio n s of 1 normal incidence, tha t i s , fo r an i nitial a n g l e 8 . = 0 d eg, measured 1 from the normal to the surface und e r stu d y , t h e a rea, dxdy , ca n b e represented by a grid of ( x . ,y.) p oint s . These points are r elated to 1 1 the initial conditions , of the tra jectories b y x = b and y = D enoting b y d N f the number of trajectori e s with final scattering angles within the inte r va l s ( 8 d8) and the differential cross-sections may the n b e obtained from daldQ = (dNf / sin8 dA ( 5.10 ) The quantity, dNf , can b e extrac t e d from s t u d ies of trajectories b y analyses of the fina l p o lar an d a zimuthal a n g les oE t h e indiv idual t rajectories . PAGE 91 84 Trajectories were obtained by numerically solving Hamilton's equations in the coordinate system shown in Fig. 5.1. The set of six first order differential equations was integrated b y a m e thod which uses a modified form of the Adams Pece fo r mulas and local extrapolation (Shampine and Gordon, 1975). The initial conditions R ) were chosen b y requiring that the interaction potential max be small with respect to the relative translational energy, e.g., b y 3 a ratio of 10 ,and the same criterion was used to terminate each individual trajectory. The area of the target sampled by the trajectories is related to the maximum impact parameter, b ,defined a s the largest impact max parameter for which the trajectories show the influe nce of the adsorbate. The area was chosen to b e given b y where 6 x = x max Her e ) i s the initial m ax azimuthal angle for which b i s found. This m e thod will assure m ax that the bundle of trajectories will sample the region where the interaction between the projectile and the adsorbat e i s of importance. In general, the trajectories behave differently according to three separate regions. The first region, for small impact parameters is where the trajectories are d eflected mainly b y the strong repulsive interaction with the adsorbed m o lecule. These can b e said to be single encounter colli sions . The second region is where the trajectories are d eflect e d t w ice. The first deflection i s caused b y the interaction with the adsorbate and the second b y the interaction with the surface. These trajectories are double encounter collisions. Finally, the third region i s where the interaction is mainly due to the effect of the surface. These PAGE 92 85 trajectories have impact parameters b>b and are detected at ve ry max large polar scattering angles and do not stay in the same plane from where they originated. Energy conservation was checked at every step in the calculation and was kept to about 2 % . Se veral of the trajectories were back integrated and always reproduced the initial values with an accuracy of at least two significant figures. Calculations for lithium were don e both with and without the effect of the image. The image effect appears to be of no significance for large deflections . Only trajectories with polar angles around 90 seem to be significantly affected. This can be seen in figure 5.10 where trajectories, in the (x-z) plane, for the Li + with 10 eV of energy are shown. The trajectories show the shape and size of the adsorbate and, as was expected, they reflect the different sizes of the adsorbate in the two models, see figure 5.11. In the case of helium atoms, th e trajectories end at larger polar angles than the ones with similar impact parameters for lithium atoms, and trapping does not occur. This i s du e to the absence of the large attractive potential in the case of helium ions as compar e d to the one for lithium atoms. The turning points are much larger for helium a toms than for the lithium ions. Compar e Figures 5.10 and 5 .12. This i s to be expected, considering the difference in the kinetic energy of the two. The dependence of the polar scattering angle on the impact parameter shows a minimum in all the cases; that is, for both projectiles and both models . Thi s minimum corresponds to the impact parameter at which the separation b e tween tl1 P regions oE single and PAGE 93 86 B 7 6 In 5 e 0 I-< ..... 4 In bO .......... 3 N 2 1 0 0 1 2 3 4 5 6 7 B 9 10 XI Angstroms B 7 6 In 5 e 0 I-< ..... 4 In bO .......... 3 N 2 1 0 0 1 2 3 456 XI Angstroms 7 B 9 10 Figure 5.10: Li+ trajectories in (x-z) plane, on-top model; for impact parameters b = 0.2, 0.4, ... , 1. 6, 2.0, 2.6, 3 . 0 A; a) including image, b) without image . PAGE 94 8 7 6 O'l 5 E 0 M +' 4 O'l bO "" 3 N 2 1 0 0 1 2 3 4 5 6 XI Angstroms 7 8 9 10 Figure 5 .11: Li+ trajectories in (x-z) plane , bridge model; for impact parameters b = 0 . 2 , 0.4, ... , 1.4, 1.8, 2 . 2, 3 . 0 A . 87 PAGE 95 88 7 6 5 2 1 o L__ L__ L__ L__ o 1 7 6 5 3 2 1 2 3 456 X / Angstroms 7 B 9 10 o L__ L__ L__ L__ 10 o 1 Figure 5 . 12: 2 3 456 X/Angst. 7 B 9 He trajectories in (x-z) plane for impact parameters b = 0.2,0.4, ... , 1.6, 2 . 0 , 2.6, 3 . 0 A; a) On-top model, b) Bridge model. PAGE 96 89 double encounters occurs. At large impa c t parameters, th e d eflection function reaches a maximum which indicates th e b eginning of the third region where the interaction i s main l y between the projectile and th e surface. These features are shown for the Li + ions in Figure 5 . 1 3 and for the He atoms in Figure 5.14. The corrugation appears to b e larger in the case of the interaction with H e . The differential cross-sections for the on-top model are presented in Figure 5.15 and Figure 5.16 for Li+ and H e atoms, respectively, for t w o different azimuthal angles. The main feature of these results is that the scattering is predominantly backscattering. In the case of Li + ions , double collisions represent the main contribution for all scattering polar angles. The rati o of single to double contributions d e pends on the a zimuthal angle. For lithium ions, the single collision contributio n is the same for different directions on the x-y plane , but the double contribution is no t symmetric with respect to the z axis. In the case of helium atoms, both contributions are angle dep endent. A possible explanation for this is the larger corrugation present in the helium case. Also, one can notice that at certain a zimuthal angles, e . g . 1 =45, the importance of single collisions with respect to double ones is reversed for certain ranges of polar angles. This factor arises from the symmetry of the model . That is, the nearest n eighbor distance along the y a x i s , 1 =90, i s approximately 2.489 A, w hereas, in the direction with 1 =45, it i s 3 . 52 A, thus giving a larger double collision contribution for 1 = 45 . PAGE 97 90 185 165 145 160.0 tlD Q) '"0 "'IIG ., ." 179 . 5 125 ....... 179 . 0 105 176.5 1.5 2 . 0 2 . 5 3 . 0 b/Angsl. 3 . 5 85 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3 . 5 4.0 bl Angst. Figure 5.13: Li+ deflection function, on -top model. PAGE 98 180 170 160 150 bD 140 v '0 ""-.. (!) 130 120 110 100 90 0 Figure 5.14: 1 2 3 4 5 6 b / Angstroms He deflection function, on-top model. 4>0 = 0(-), 45(__ ). The inset shows an enlargement fo r large impact parameters. 91 PAGE 99 ,-... +-' Pl ':-+-' Pl bO 0-m ......... "'" rn u ,-... 80 50 40 30 20 10 r ---' __ L_ __ J o r---F--I 90 25 5 100 110 120 130 140 8/deg 150 ,. ---' , , L ___ , 180 .-, __ -oJ __ 170 180 o __ __ __ __ __ __ ____ L_ __ 90 100 110 120 130 140 8/deg 150 180 170 180 92 Figure 5.15: Li+ differential cross-section, dcr/dQ, ontop model, for single (-) and double (---) collisions; a) = 0 b) = 45. PAGE 100 93 Figure 5.16: H e differential cross-section, da/dQ , on-top model, for single (-) and double (---) collisions; a) = 00 b) = 450 â€¢ PAGE 101 CHAPTER 6 VIBRATIONAL ENERGY TRANSFER IN HYPERTHERMAL COLLISIONS OF He AND Li + CO ADSORBED ON Ni(OOl) In this chapter we apply the theoretical mode l presented in Chapter 4 to the scattering of H e and Li+ from a CO molecule ad sorbed on a Ni(OOl) surface. calculate double differential cross-sections for the energy and angular distributions of the H e atom s an d the Li+ ions. Results are obtaine d for ex p e rim ental conditions o f high relative energy, 1 eV for H e atom s and 10 eV for Li+, for norm a l incidence and for the low co verages where the theoretical model i s applicable. 6.1 Double Differential Cross-S ection fo r the System H e / O C Ni(OOl The calculations are base d on the e xpressions obtaine d in Chapter 4 using the short time e x pan sion of the vibrational TCFs for the low frequency modes of the targe t which, in our case, are all the surface and cluster modes e xcept the CO stre t ch. According to Eq. (4.22), the pa rtial d ouble differential cross-section for a given numbe r of quanta, n f , g aine d b y the C O stretc h will be given by a Gaussian distribution of en ergy transfer, with a shift IJ and width r, b y the slow modes . Thi s would be s o for both mechanisms we are con sidering h e r e ; that is, for single and double collisions. have done the calculations con sidering single and double collisions. find that single collisions are the preriominant mechanism for helium scattering. In figures 6.1 and 6.2, the partia l 94 PAGE 102 95 differential cross-sections for excitation of th e modes ( -n=O), are presented for th e top and bridge model, respectively. In both cases the single collision contribution is large r than the double. The same is found to be true energy is also transferred into the fast mode. This is in Figures 6.3 and 6.4 for n = l and -n=2, respectively, it also can be seen that single collisions are even more predominant a large amount of energy i s transferred into the CO stretch. Double differential cross-sections (DOCS) at scattering angles of 0=140 and are for the top and bridge models in Figures 6.5 and 6.6, respectively. In both cases the DOCS increase increasing , means increasing s' since f i s fixed b y n. As is further increased, the DOCS goes through a maximum and then decreases to zero. In the case of no e xcitation of the CO stretch, that is, for -n=O, there is no apparent difference the models. for e xcitation of CO b y n =l, 2, and 3 quanta, the bridge model gives larger quantities for the DOCS and Gaussians centered at slightly smaller values of transferred energy. This i s found to be in accordance the fac t that the frequency of the CO stretch in the bridge model is slightly smaller than in the top case and equation (4.22 ) tells us the that the center of the Gaussian is given b y ( +6). nf We have studied the angle dependence of the DOCS. Figures 6.7, and 6.8, and 6.9 present DOCS for the top and bridge models as a function of energy transferred for three differen t scattering angles , 0=100, 140, and 170, respectively. In both models the DOC S increases the polar scattering angle increases. This trend is easily understood one recalls that the CO stretch is a mode PAGE 103 Al symmetry and motions p erpendicular to the surface. Therefore, collisions at small impact p aramet e r s , l eading to backward deflections, would b e the mos t effic i ent to e xcite the CO stretch. It is also seen that the same trend i s followed w h e n n =O. In this 9 6 case the overall process involve s the simultaneous e xcitation of the slow modes, some of which are E type (motions paralle l to the surface) and less likely to b e excite d at s m all impact p arameters , and some of which are A type and more like l y to b e e xcited. The two trends compound to give the resulting angle d e p endence which indicates that the A type modes are the domin ant one s . All the results presented and discussed up to now were obtained for a temperature of 3000 K . The effec t of the temperatur e on the probabilities of e xcitation for each mode is different and is determined by its magnitude in relation to the qu antity KBT. Modes whose frequencies satisfy will h a v e excitation probabilit ies given b y a Poisson distributio n of n., inde p endent of temperature for J In our case, we find thi'lt the t e m pe r a ture d e p ende nce i s due to the slow modes, whose freque n c ies satis f y kBT, and is reflected in the width of the Gaussians . From equation (4.22) on e obtains that the width at halfh e i ght i s propo r tional to r whic h i s giv en by equations (4 . 14 and 4 .11). These equ ations indic a t e smaller widths for smaller temperatures. In Figures 6.10 , 6.11, and 6.12 , we present DOCS for the top and bridge m ode l s a t a !emp erat'lfE' d for scattering angles 1 =0, and 8 = 1 00 , 14 0 , and 1700 , res pecti ve l y . As expected, the peaks a r e narrower tha n the ones presented in Figures 6.7, and 6.8, and 6.9 , whic h are obtained for a temp erature of 3000 K. PAGE 104 97 In the calculation for 1000K and at an an g l e of 8 =170, a very small peak appears in the region of very small energy transfer. In Figure 6.13 this peak is i solated and found to be centered at a point where the energy transferred, is about 17 m eV, and presents a Poisson, rather than a Gaussian, distribution. This seems to indicate that it corresponds to a m ode with a frequency satisfying the minimum condition of Hw. > 8 meV. In our analyses of normal J modes we have modes corresponding to frustrated rotations with frequencies of 15.5 meV and 16 . 2 MeV in the bridge and top cases, respectively. This peak may the n b e associated with these modes, especially considering that it appears in both models. Finally , the dependence of the DOCS with the azimuthal angle t for both top and bridge models is presented in Figures 6.14 and 6.15. The trends followed b y the two model s are essentially the same, with a smaller DOCS in the (11 0 ) direction (t=O, 90) than in the (100) direction (8=45). Thi s result indicates that the projectile doe s not seem to notice the corrugation of the surface. 6.2 Double Differential Cross-Sections for the System Li+/OC-Ni(OOl) In the calculations fo r this sys tem, we have chosen energies in the range of 1 0 to 20 eV, whic h i s co n sidered an intermediate regime between low energy (thermal) and hig h energy, or Rutherford scattering, where the dynamics are rather simple. The calculations are based on the same formulation used for the H e/OC-Ni(OOl). As before, we consider the separation of the modes into fast ones, that is the CO stretch, and slow modes, whic h are all the rest. PAGE 105 In this system, since the energy of the projectile is larger, more energy is available to be transferred into the target. As a consequence large values for the number of quanta transferred into the fast mode mus t be considered. 98 We first consider the partial double differential cross-sections as a function of energy transferred for a pair of scattering angles, o This is shown in the case of the top model and for 0=100 and in Figure 6.16 . In this case, contrary to w hat was obtained for He atoms, t w o peaks, well separated in energy, are found. The peak at high energies is found to originate from the double collision mechanism and the one at lower energies from single collisions. Double collision contributions are dominant for values up to n=10 . As more energy is transferred into the fast mode, that is, for increasing values of n, the single collision starts to b ecome more important. At this value of 0 , the excitation of the cO stretch b y n = 5 quanta is the one with the highest probability. That this is not always the case can b e seen from Figures 6.17, 6.18, and 6.19, in which similar results are represented for 0 =110,150, and 170, respectively. As one goe s from small to large 0, the probability of e xciting the CO stretch b y a do ub l e collision mechanism becomes smaller at the same time that the probability of excitin g the slow modes, represented by n=O, b ecomes larger by the same mechanism. This trend could be e xplaine d if one tries to visualize the trajectory of a double collision with a large scattering angle. Such a trajectory would b e on e in whic h the projectile h ad suffered a very small deflection b y the adsorbate so that it would strike the metal surface with a very small angle of incidence. Consequently, considering an almost flat surface, the projectile would be deflected PAGE 106 99 by the surface into a large scattering an g l e . A ccording to this argument, the double collision peak would b e m ainly due to the slow modes of the metal surface . The fact that the single collision p eak almost disappears in going from 8 = 1 00 to 8 = 170 seems to suggest that the modes with E symmetry in the cluster have higher probabilities of e xcitation than the modes with A symmetry, giving lower DOCS for larger scattering angles. The total DOCS obtained b y summing over all n is shown in Figure 6.20 a s a function of and 8. The same trend observed in the individual contributions is seen here. Results obtained for the bridge model are not muc h different than the ones obtained for the top, although some differences can b e pointed out. In the bridge model at 8 = 10 0, the e xcitation of CO stretch by n=5 and n=6 ha ve almost the same probabilities . Additionally, the peaks in the bridge mode l appear to be wider than in the case of the top model. This can b e a consequence of the larger number of modes considered a s slow modes in the bridge model. In Figure 6.22 a comparison of the DOCS summed over all n for the two models is presented for 0=100. It is found that the bridge model gives smaller DOCS than the top model . The effect of the temperature is show n in Figure 6.23 where the DOCS summed over all n are presented for a temperature of 1 00 K . A s in the case of scattering b y He atoms, the width of th e peaks decreases w h en decreasing the temperature. The effect seem s to b e more noticeable for the single : collision peak than for the double collision one . PAGE 107 Figure 6.1: Single ( ---) and double (---) collision contribution to the partial double differential cross-sections at scattering angles 0=100, for n=O. C O molecule adsorbed on top. " PAGE 108 101 0 m 0 CO 0 t-r , 0 II CD 0 0"'-kl 0 C"'J 0 "-C\2 "-0 \ 0 0 C\2 0 0 (P.rlS A8 -bS)/SJGG PAGE 109 I:' Figure 6 . 2: Single (---) and double (---) collision contributions to the partial double differential cross-sections at scattering angles 8=100 for n=O. CO molecule adsorbed on bridge site. PAGE 110 1 03 a ---/ ...-/ / \ '-"'-"'-....... ....... --/ ----/ --/ I "-"-..... "'-..... " m a co a 'f'a '-C\l \ a a PAGE 111 Figure 6.3: Single (---) and double (---) collision contributions to the partial double differential cross-sections at angles 8=100, for excitation of CO stretch by -n=l. PAGE 112 o C\2 o LO o ( P.ItS 11.8 -bS)/S:)QQ "-"-" 1 05 0 OJ -0 CO 0 l.'-0 CO 0 Co.) \ 0 cr:> 0 C\2 0 0 0 00 0 PAGE 113 Figure 6.4: Single (---) and double (---) collision contributions to the partial double differential cross-sections at angles 0=100, for excitation of CO stretch by -n=2. 1'< PAGE 114 1 0 7 0 ...-i .----// m -/ 0 / I CD --0 /' \ \ \ l'--"-0 \ 0-.. CD 0 0"-'V 0 C'J 0 C\1 0 ...-i 0 0 LO 0 LO 0 L() 0 lO 0 10 0 0 C'J C'J C\1 C\1 ...-i ...-i 0 0 0 0 0 0 0 0 0 0 0 0 (P.I1 S A2J 'bS)/SJGG PAGE 115 Figure 6.5: ' xcitation of the CO stretch by n quanta, top model. Scattering angles <1>=0, 0=140, T=300 k, E=l eV. PAGE 116 109 N o PAGE 117 Figure 6.6: E xcitation of the CO stretch b y n quanta, bridge model. Scattering angles 0=140, T =300 k , E=l eV. PAGE 118 111 (.ItS x 8:)(1<1 PAGE 119 Figure 6.7: He DOCS for CO stretch excitation b y -n=0,1,2 and 3 quanta. Top (---) and bridge (---) models. 8=100, T=300 1(. 1.\ PAGE 120 11 3 0 .....-l ill 0 CO 0 l'-0 CO 0 0'-...,. c...l 0 C'J 0 C\1 0 .....-l '" 0 0 00 PAGE 121 Figure 6 . 8 : He DOCS for CO stretch excitation by -n=O,1,2 and 3 quanta. Top (---) and bridge ( --) models . 8=140, T=300 K . " PAGE 122 11 5 0 m 0 CO 0 l:'-0 CO 0 0"-.. G0 0 C'J 0 C\2 0 0 0 LD 0 0 (P.ItS Ad "bS)/SJQQ PAGE 123 " Figure 6.9: He DOCS for CO stretch excitation by -n=0,1,2 and 3 quanta. Top (---) and bridge (-) models . 8=170, T=300 K . PAGE 124 0 0 0 0 0 LD C'J C\2 ........l (P.I'tS A8 /SUIO.I'tS!1UV 'bS)/SJGG 117 o m o co o CD 0 ci""'-c.v 0 C'J 0 C\2 0 ........l 0 0 00 PAGE 125 Figure 6.10: He DOCS for CO stretch excitation by -n=0,1,2 and 3 quanta. Top (-) and bridge (---) models . 0=100, T=100 K. PAGE 126 11 9 0 en 0 CD 0 t"-O CO 0 1 \ ./ G.j 0 C'J 0 C\l 0 0 0 00 (P.ItS A8 'bS)/S:JGG PAGE 127 (!) bO -0 .r-< I-> ..0 -0 c: ct! .-.... I ......., 0.. a E-' ct! '-' c: ct! :l cr M -0 c: ct! N .-i 0 II c: I ..0 c: a .r-< '-'0 ct!0 '-' .-i .r-< II <.JE-' X (!) 0 ..r::0 <.J- PAGE 128 1 2 1 0 m 0 CO / 0 \ / r--0 CO 0 cu ...... '\ 0 C'J 0 C\l 0 0 0 co C\l 00 PAGE 129 Figure 6 . 1 2 : He DOCS for CO stretch excitation by -n=0,1,2 and 3 quanta. Top (---) and bridge (---) models . 8 =170, T = 100 K . PAGE 130 123 o o CD o LD 0 0 0 C") C\2 (P.ItS A8 'bS)/SJGG / 0 ......-4 ( m I . o co r-0 CD 0 0"'-GV 0 C") 0 C\2 0 ......-4 0 0 0 0 PAGE 131 Figure 6.13: He DOCS in the region of very small energy transfers for T=100 K and 9=170. PAGE 132 125 to () '/ / '/ '/ / '/ '/ / '/ / / / / / / / / / / / / () () () C\2 () () M () () () () C'J C\2 M 00 0 0 () () 0 o (P.I't S A8 /SUlO.I'tSEJUV obS)/SJQQ PAGE 133 ..--i Cl) " o E 0. o E-< o o o .-t c8 .... Cl) > o " Cl) E E ;:s UJ U'J U CI CI Cl) ::c PAGE 134 127 o ...... PAGE 135 o o o .--i II CD I-> o "'-' I-> Q) :: o "0 Q) E E :l UJ C/J (.) Q Q Q) :r:: PAGE 136 129 co co o PAGE 137 Figure 6.16: Li+ DOCS for excitation of the CO stretch by n quanta, Top model E .=10 eV, T = 300 K, 0=100, 1 PAGE 138 131 PAGE 139 > Q) 0 rl II ..... W r-i Q) '" 0 E 0.. 0 E-< ro ..., c ro ::l t::r C >, .0 ..c: c.J ..., Q) I-< ..., UJ 0 U Q) ..c: ..., 0 C 0 ..... ..., roo ..., II ..... >eo c.J XO Q)rl rl I-< II 0 PAGE 140 133 PAGE 141 Figure 6.18: Li+ DOCS for excitation of the CO stretch by n quanta. Top model, E=lO eV, T=3000 K, 8=150, PAGE 142 135 PAGE 143 FiguLe 6.19: Li+ DOCS fOL excitation of the CO stLetch by n quanta. Top model, E=10 eV, T =300 K, 8=170, PAGE 144 137 PAGE 145 Q) '"0 o E 0. o ....., o o o C"1 " E-< o o I I 0& > Q) o ..... " til Q) > o '"0 Q) E E UJ '-" tI) U Cl Cl + .,..; ....J PAGE 146 139 co Q) :>-Q) '" . 0 X C\1(J;) ..-4 PAGE 147 6.21: Li+ DOCS for excitation of the CO stretch by n quanta. E=10 eV, 0=100, T=300 K. model. PAGE 148 1 4 1 = PAGE 149 o o o C"l II E--< >OJ o ...-i II ....-i OJ "0 o E ,-... I OJ bD "0 rl I-< .D "0 c:: rn ,-... I '-" 0. 0 E--< ,-... c:: I-< OJ > 0 "0 OJ E EO ::l0 UJ...-i tf) U 00 0 II >& + rl ...J N N \0 OJ I-< ::l bD rl I:.t... PAGE 150 143 0 .,......; ---------------------------co \ / / / >/ I t-Q) "" I c.v I , \ CO ____ ____ ______ ____ ______ ____ -L ____ o C':J o C\l o .,......; L.O o o o PAGE 151 o o o ri II E--< :> OJ o ri II W OJ "'0 o E OJ be "'0 'M s... ..0 "'0 C C1l -'"' I '-" 0.. 0 E--< -'"' c s... OJ > 0 "'0 OJ E EO ='0 (/)ri '-" II PAGE 152 1 4 5 ____ ______ ____ ____ ______ ____ ______ 0 I I I I I I ()) I-:>..., r-- PAGE 153 CHAPTER 7 SUMMARY AND CONCLUSIONS In this work we have presente d a theoretical model suitable for the study of scattering of atom s or ions with h y pertherma l energies by molecules adsorbed on m e tal surfaces at low coverages. A simple model to obtain the vibrational modes of m o lecules adsorbed on surfaces has been presented and frequencies have been calculated for a CO molecule adsorbe d on a N i (OOl) surface. The procedure allows for the modelling of different adsorption sites and makes the classification of the modes acco rding to symmetry an easy task. An interaction potential ha s been constructed and a trajectory study of the elastic scattering of H e atom s and Li+ i o n s by a CO molecule adsorbed on a metal surface ha s been carried out. I t i s found that at the h yper therm a l energies of 1 eV for H e atoms and of 10 eV for Li + atoms, the effectiv e classical cross-sections for the adsorbed molecu le-surface comple x d o not s ho w the corrugation of th e surface to a great extent and, in the case of Li+ ions , the imag e potential effect i s almost n egligible e xcept around p o lar scattering angles of n/2. We hav e obtained double differential cross-sections as fun ctions of energy transfer, scattering angles, incident energy and metal temperature for the cases of a CO molecule adsorbed either at the bridge or on -top configuration. 146 PAGE 154 147 The approach we ha ve followed allows u s to predic t whi c h excitation process is the most likely to occur at a given s cattering angle and for a certain energy transfer. For example , we are able to predict that in the scattering o f H e atoms, the mos t like l y energy loss would be found at 0.4 eV and would b e du e to a collective e xcitation of modes with lower freque n c ies and not to the excitation of the CO stretch mode. U e could also say that features f ound at large transfer of energy would most likely correspond to multiple excitations of the CO stretch of the ad sorbe d molecule. From the temperature effect, we can say tha t more information on the excitation of individual modes can b e obtained b y pe rforming experiments at lower temperatures where features of the low frequency modes such as hindered rotations and translations would b e enhanced. Related to this point, we would like to m ention that in an e xperimental study of H e scattered b y CO molecules (Lahee, et al.) at low CO co verages on a P t (OOl) surface, a peak at 6 . 0 m e V was attributed to CO vibrations. Another very weak stru cture was seen at 16.5 meV and was assigned to the hindered rotation. Thi s assignm ent was based on the results of some calculations (Richardson and Bradshaw ) that ha ve give n a value of 22.8 meV for this mode . However , in our calculations of the vibrationa l modes w e obtaine d values for the frustrated rotations for the bridge and top mode ls, of 15.5 and 16.2 meV, respecti ve l y . These values are very c lose to the one measured in the experiments. Furthermore, a s was discussed in Chapter 6, in ou r cal culatio n s fo r DOCS , we obtain a very weak peak centered at about 17 meV fo r both models. U e b elieve this peak should corres pond to the frustrate d rotations on eithe r model. PAGE 155 148 The results obtained for the scattering oE Li + ions suggest that a more complicated process occurs in compar i s on with the H e case. For on e thing, double collisions seem to play a major role in the process. Although e xcitations of th e C O stretch involve large numbers of quanta that are add e d up, they give results where predictions can be made. For example , on e can predic t that mos t of the energy loss peaks should b e found at large energy transfers. On can also say that, depending on the scattering angle , on e can expect to find either a single peak at large values of transferred energy o r two peaks, one broad and at lower values of transferred energy and another narrow peak centered at larger values of transferred energy. In order to obtain more information about the individual modes , a more detailed calculation, where more modes are specified by the quantum numbers n., is need ed . Thi s can b e readily obtained from our J approach when the short time e x pan sion approximation is only used for a few modes. For e xample, the bridge site, wher e the distinction between high and low frequency modes i s not so sharp, could b e treated in this way. A more rigorous treatment of double and multiple collisions can be based on the multiple scattering expansion of the transition operator and can be develope d within the time-correlation formalism. One ob vious limitation in our model i s that it i gnores the large number of low -frequency bulk modes of the s ubstrat e . These modes could in principle b e obtaine d with the standard methods of lattice dynamics, and have actually been used in the problem of adsorbate vibrations (Rahman et al., 1 982). However, these method s can be applied to molecular ad sorbates with great difficulty and PAGE 156 149 since they rely on lattice periodicity their application to disordered systems is even more complex. A possible method which could b e used to deal with this limitation, but would still have the convenience of a cluster model, consists of including periodic boundary conditions for the clusters . Periodicity in the directions paralle l to the crystal surface is accomplished by adding ne w bond-stret ching internal coordinates, in additional to the ones within the cluster, betwee n edge atoms located on opposite sides of a layer in the crystal and between edge atoms on two adjacent layers of the crystal. This method woul d p r o vide modes with well defined values of the momentum parallel to the surface, allowing thus to examine the dis persion of the m o d e frequency and the energy transfer into these modes a s a function of the parallel momentum. This method ha ve been us e d to generate the spectra of the O/Ni (OOl) system ( Lloyd and Hemminger, 1985) and found to reproduce almost e xactly the results obtaine d with the Green's fun ction lattice dynamics technique. However in our treatment we co n sider hyper thermal energies which are dominated b y s hort-time dynamics and these modes are not like l y to have an effect in the process . Another aspect of the model that needs attention is the assumption of harmonicity. One of the most interesting aspects associated with the vibrational states of chemisorbed mole cu les i s the change in vibrational frequencies and lifetime of a molecule upon adsorption. One of the mechanism s which give rise to the shift and broadening of the vibrational state of a c hemisorbed molecule on a metal surface is vibrational energy relaxation . This mechanism i s associated with the coupling between the localized vibrational modes of the adsorbated and the phonon modes of the metal. The PAGE 157 150 effectiveness of this m e chanism rle p ends of the anharmonicity of the potential and also on the frequency of the localized v ibrational mode in relation to the highest pho non frequency of the metal. By analogy with the study of multiphonon processes in the bulk (Mills et al., 1980) one may infer that fo r modes with frequencies greater than twice the highest phonon freque nc y of the m e tal the damping provide d by anharmonic processes woul d b e very weak. On the other h and, for low-frequency modes, suc h a s the C-Ni stretching vibration for CO adsorbed on a Ni(OOI) surface, de cay via two-ph o non emission is energetically possible and probably the relevant energy relaxation process. Again be cause of the short-time d y namics in hyper thermal collisions this effect can be ignored. If necessary , at low collision energies, anharmonicity can be introduced in the formalism of the time-correlation functions for adsorbates on metal surfaces along the same lines as it is don e in the case of polyatomic s (Vilallonga and Micha, 1 983b). In a general way one can write the vibrational potential in terms of a harmonic part, V h , and an anh armon i c on e Va' s u c h that in terms of the normal coordinates that diagonalize the h armon i c part of the vibrational Hamiltonian, the total vibrational H amiltonian is given by where P is the column matrix of momenta P. = -iha/ a Q.. One then 1 1 finds that the TCFs are give n in term of displaceme nt-displacement --T correlation functions (DOCS), Q( O ) Q (t>, whicll can be evaluated from the corresponding double-time Green functions . The procedure PAGE 158 1 5 1 involves the introduction of a set of time-inde p endent parameters, which relates to the harmoni c frequ e n cies , and the new effective anharmonic ones w b y T 2 -2 the equation 12 = w the anharmonic frequencies w can b e obtained from their relation with the DOCS, that is I1w cotha: where a and f i s the column matrix o f the anharmonic forces tha t a c t on the Q.. The vibrational correlation fun ction can 1 then be give n by the e xpression F (k, t) v where and X. J c = N v e x p{-X. co s ha:. + J J + X.[ex p( -a:. iw.t) + e xp (a:. + iw.t)]f2 } , J J J J J -2 h(kc.) f(2w. sinha.) --J J J C D This e xpression i s similar to the one obtaine d from ha rmonic motions but the vibrational probabilities now depend on the anharmonic forces. In order to study the scatterin g of atoms b y adsorbates at thermal energies, a semi-classical approach could b e used . a semi -classical model, the intermo lecular potential V(t)=V(R(t), rX) depends on the motion of the projectile relative to the center of mass of the target. Furth ermo re, Vet) is an op erator tha t a cts on the internal states of the target. During a collision, an internal PAGE 159 15 2 state of the target, In >, evolves in time in accordance vith the semiclassical Hamiltonian H(t), so that the probability for a transition betveen an initial state, In >, and a final one, In'>, i s given by W = lim n'ex. 2 I PAGE 160 1 53 CD dP/d8 J F(t) where and is given by so that the differential probability of energ y trans f e r i s give n b y the Fourier transform of the TeF of the semiclassical e volution operator. Then the procedure to follow will involve: firs t d efining the Hamiltonian of the non -interacting parts , and n ex t obtaining U(t,t.). This requires the integration of the differential equation 1 of the evolution operator, that is [il1Cl/Clt -H'(t,ti)] U(t,ti) 0 with U(t,t) 1 and H'(t,ti) 6K(t) + e xp[i(Hi(t-ti)/l1] V(t) e x p[ -iHi(tt i ) /l1] where 6K( t) being the initial relativ e mome ntum . This approac h woul d prov i de the means to tackle the problem of scattering of atom s at therm a l energies by adsorbates . PAGE 161 154 As one can see collisional TCFs are of general validity, and can be used in conjunction with approximations proved to be valid in special cases. Several aspects of the approach that we have presented can be developed in future work. In particular,in view of the increasing amount of e xperimental work currently in progress on the scattering of atoms at thermal energies, it would b e fruitful to look into the semiclassical method. The approach proposed in this work in its present form could be applied without major changes to different systems. Calculations could be done for example for a different crystal face of the same metal or any other metal, or for a different type of adsorbed molecule and even for a different type of projectile. Also, calculations could be done for incident angles different from zero. The case of higher con verages could also be studied with the present approach, including the interaction between the adsorbates. PAGE 162 APPENDIX A COEFFICIENTS FOR THE TRANSFORMATION s AX a) On-Top Cluster s13 s14 S15 s16 s17 s18 s a b) -YNi -s12 -s13 -(XNi + YNi)/2 + IT -(XNi YNi)/2 + IT (XNi + YNi)/2 + Bridge Cluster _y(l) Ni IT ZNi/2 ZN/2 ) ZN/2 ) 15 5 PAGE 163 Appendix A (continued) s15 = (-2/ .[5 ) -(1/.[5) y(1) Ni s12 y(1) _ y(2) Ni Ni s26 = -(2/.[5) + (1/.[5) s27 = s28 ( 21.[5) + (1 1.[5) s19 = (11 D) x(l) Ni + Z(1 ) Ni sl,10 (1/.[7) -(21.[7) + sl,11 (1 I .[7) ) (2/D) + r;:;(2) (2) s2,9 = (1li3) .XNi + ('l2/3) ZNi s2,12 -(liD) X(2) Ni + z(2) Ni s2,13 -(1/.[7) + (21.[7) + s2,14 (liD) + (2/,[7) + Z(2) Ni sCNi(l) Y C ) + (d C/dcNi)(ZC sCNi(2) = (dNi/2dCNi)(Y C + (d C/dCNi)(ZC s = a 156 PAGE 164 s g Appendix A (continued ) 2 (1) (2) 2 (1) ( 2 ) (dC/dCNi)(YNi YNi ) + (dNi/dCNi)(ZNi )(ZNi ) + + [(dc + dCO ) /(dc dcO) Xc -(l/dcO) Xo 157 PAGE 165 158 APPENDIX B SYMMETRY ADAPTED COORDINATES a) On-Top Clu s ter b) Bridge Cluster A1 Sl A1 Sl Zc A1 52 Zc A1 52 Zo A1 S3 = Zo A 1 S 3 + )/IT 5 E ( X N i + YNi )/IT A1 4 S4 SE (XC + Yc)/IT B1 Xc 6 S5 SE (XO + YO)/IT B1 Xo 8 S6 B1 S7 + )/IT B2 S8 Y C B2 S9 YO B2 S10 + B 2 Sl1 / IT A? ( x ( 2 ) / IT S 1 2 ' Ni PAGE 166 APPENDIX C COEFFICIENTS FOR TRANSFORMATION Q =CX a) On-Top Cluster XNi YNi ZNi Xc Y C Zc QAl 1 0 0 0.025 0 0 -0.778 QA1 2 0 0 0.602 0 0 -0.49 0 oA1 2 0 0 0.798 0 0 0 . 393 oE 4 0.135 0 . 135 0 -0.622 -0.622 0 QE 5 0.692 0.692 0 0 .093 0.093 0 QE 6 0 . 061 0 . 061 0 0.323 0.323 0 Xo YO 0 0 0 0 0 0 0 . 308 0.308 -0.115 -0.115 0.626 0.626 Zo 0.628 -0.631 0 .456 0 0 0 f-" U1 \0 PAGE 167 Appendix C (continued) b) Bridge Cluster XNi (1) YNi (1) ZNi (1) XNi(2) YNi(2) ZNi(2) Xc OAI 1 0 0 . 00 7 -0.001 0 -0.007 -0.00 1 0 OAI 2 0 -0.559 -0. 296 0 -0. 559 -0. 296 0 OAI 3 0 0 . 428 0 . 301 0 -0.428 0 .301 0 A 0 4 1 0 0 . 065 -0.568 0 0.065 -0.568 0 OBI 5 0 . 146 0 0 0 .146 0 0 0 . 935 B Q6 1 0 . 672 0 0 -0.672 0 0 -0.122 OBI 7 0.165 0 0 0.165 0 0 0 . 332 Y C 2C Xo 0 0.758 0 0 0.258 0 0 -0.444 0 0 -0. 385 0 0 0 -0.287 0 0 0 .286 0 0 0 .914 YO 0 0 0 0 0 0 0 20 -0.653 0 . 344 -0.50 7 -0. 446 0 0 0 t--' (J\ o PAGE 168 Appendi x C (continued) X N i (1) Y N i (1) ZNi (1) XNi(2) YNi( 2 ) Z Ni(2 ) OB2 8 0 0 . 0 1 3 0.405 0 0 . 0 13 0 . 4 0 5 O B 2 9 0 -0. 443 0.299 0 -0. 443 0 . 299 B2 0 1 0 0 0 . 549 0.260 0 0.549 -0. 260 B 2 0 0 . 037 0.424 0 -0.037 -0. 424 A2 1 2 -0.707 0 0 0 . 707 0 0 X c Y C Zc Xo 0 0 . 239 0 0 0 0 . 4 84 0 0 0 0.333 0 0 0 0 . 773 0 0 0 0 0 0 YO -0. 7 8 4 0.441 0 . 3 8 8 0 . 200 0 Zo 0 0 0 0 0 ...... a...... PAGE 169 REFERENCES Allison, J.N., and W.A. (1982), Surface Sci., 115, 553. Andersson, S., (1977), in: 3rd Int. Conf. on Solid Surface, Vienn a . Andersson, S., and J.B. Pendry (1979), Phys . Rev . Letters, 43, 363. Andersson, S., and J.B. Pendr y (1980), J. Phys., C13, 3547. Annett, J.F., and R. Haydock (1984), Phys. Rev . Lett., 53, 838. Batra I.P., P.S. Bagus, and J.A. Barker (1985), Phys. Rev., B31, 1737. Bauschlicher, C.W., P.S. Bagus, C.J. Nelin, and B.O. Ross (1986), J. Chern. Phys. 85, 354. Berndt R., J.P. Toennies and Ch. Wall (1987), Abstract to the 5th International Congress on Vibrations at Surfaces, Grainau-Erbsee, Germany. Bertolini, J.C., and B. Tard y (1981), Surface Sci., 102, 131. Black, J.E., (1982), Surface Sci., 116, 240. Black, J.E., P. Bopp, K. Lutzenkirchen, and M. Uolfberg (1982), J. Chern., Phys., 76, 6431. Black, J.E., D.A. Campbell, and R.F. Uallis (1981), Surface Sci., 105, 629. Clary, D.C. (1982), Chern. Phys., 65, 247. Clouthier, D.J., and D.A. Ramsay (1983), Ann. Rev . Phys. Chern., 34, 31. Crawford, B.L., and A . Horwitz (1948), J. Chern. Phys., 16, 147. Englert, W., E. Taglauer, U . Heiland, and D.P. Jac kson (1983), Physica Scripta, T6. 38 Esbjerg N., and J.K. Norsko v (1980), Phys. Rev . L ett., 45, 807 . Gadzuk, J.U. (1987), in: Spectroscopy of Mole cules on Surfaces, eds., J.T. Yate s and T.E. PLellum Press, New York . 162 PAGE 170 Gentry, (1979), in: Atom-Molecule Collision Theory, ed. R.B. Berstein, Plenum Press, New York. Gerber, R.B., A.T. Yinon, and R. Kosloff (1984), Chem. Phys. Lett. 105, 523. Goldberger, M.L., and K.M. Watson (1964), Collision Theory, New York. 163 Goldstein, H. (1950), Classical Mechanics, Addison-Wesley, Cambridg e . Goodman, F.A. (1987), J. Chem. Phys . 58, 5530. Harris, J., and A. Liebsch (1982a), J. Phys. CIS, 2275. Harris, J., and A. Liebsch (1982b), Phys. Rev . Lett., 49, 341. Herzberg, G. (1945), Spectra of Diatomic Molecules, D. van Nostrand Co., New York. Hulpke E. (1975), Surface Sci 52, 615. Hulpke, E., and K. Mann (1985), Surface Sci., 157, 245 . Ibach, H., and D.L. Mills (198 2), Electron Energy Loss Spectroscopy and Surface Vibrations, Academic, New York . Ibanez, J., N. Garcia, and J.M. Rojo (1 983), Phys . Rev., B28, 3164. Jones, L.H. (1960), J. Mol. Spectrosc., 133. Jonsson, H., J.H. and A.C. L ev i (1984), Surface Sci. , 148, 126. Lahee, A.M., J.R. Manson, and J.P. Toennies, Ch. (1987), J. Chem. Phys., 86. 7194. Lehwald, S., and H. Ibach (1 982), in: Vibrations at Surfaces, ed. b y R. Caudano, J.M. Gilles, A . A . Lucan, Plenum, New York. Levi, A.C. (1982), in: Dynamics of Gas-Surface Interaction, eds. G. Benedek and V. Balbusa, Springer, Berlin. Liu, and B. Gumhalter (1 987), Surface Sci. , 180, 169. Llo yd, K.G., and J.C. H emminger (1985), J. C hem. Phys., 82, 3858 . Mann, K . , V. Celli, and J.P. Toennies (1 987), Surface Sci. , 1 85 . 269. Messiah, A . (1961), Q uantum Mechanics, Vol . 1, North-Holland, Amsterdam. Micha, D.A . (1979a), J. C hem. Phys., 70, 565 . Micha, D.A. (1979b) , J. Chem. Phys., 70, 3 165. PAGE 171 Micha, D.A. (1981), J. Chern. Phys., 74, 2054 . Micha, D .A. (1986), Intern. J. Quantum Chern. 20, 773. Mills, D.L., C.J. Duthler, and M. Sparks in: Disordered Solids, Optical Properties, Dynamical Properties of Solids, vol. 4, eds. G.K. Horton, and A . A . Haradudin, North-Holland, Amsterdam . Parra, Z., and D.A. Hicha (198 6), Chern. Phys. Letters, 132, 488. Poelsema, B., R.L. Pallmer, S . T . De Zwart and G . Comsa (1 983), Surface Sci., 126, 641 . Rahman, T.S., J.E. Black, and D.L. Hills (1982), Phys. Rev . , B25, 883. Rahman, T.S., D.L. Hills, and J .E. Black (1983), Phys. Rev. , B27, 4059. Richardson, N.V., and A.M. (1 979), Surface Sci., 88, 255 . Rocca, H., H. Ibach, S. Lehwald, and T . S . Rachman (1986), in: 164 Structure and Dynamic s of Surfaces I, eds. V. Schommers and P. von Blanckenhagen, Springer-Verlag, Berlin. Rodberg , L.S., and R.H. Thaler (1967), in: Introduction to the Quantum Theory of Scattering, Academic Press, New York. Shampine, L.F., and H . K . Gordon (1975), Computer Solution of Ordinary Differential Equations: The Initial Value Problem, Freeman, San Francisco. Singh, H., O.K. Dacol, and H. Rabitz (1986), J. Chern. Phys. , 84, 1373. Steele, V.A. (1974), The Interaction of Gases with Solid Surfaces, Pergamon Press, O xford. Strong, R.L., B. Firey, G . V . de Vette, and J.L. Erskine (1982), Phys. Rev., B26, 3483 . Tenner A.D., K.T. Gillen, T .C.M. Horn, J. Los, and A .V. Kle y n (1986), Surface Sci., 172, 90 . Van Hove, L . (1954), Phys. Rev., 95, 249 . Vilallonga, E., and D.A. Micha (1983a), J. C hern. Phys., 78, 3942 . Vilallonga E., and D.A. Mic ha (1983b), J . Chern. Phys., 79, 3794 . Vilallonga, E., and D.A. Hicha (1987), J. Chern., Phys., 87, 750 . Vilallonga, E., and H. Rabitz (1986), J. C h em. Phys .. 85, 2300 . Villis, R.F. (19 80 ) , in: Vibrational Spectroscop y of Adsorbates, Springer, Berlin. PAGE 172 Vi1son, E.B., J.C. Decius, and P.C. Cross (1955), Molecular Vibrations, McGraw-Hill, New York. Zaremba, E., and V. Kohn (1977), Phy s . Rev., B15, 1769. 165 PAGE 173 BIOGRAPHICAL SKETCH Zaida Parra was born in Maracay, Venezu ela, on August 3, 1947. She attended school in Venezuela and received her Licenciado in Chemistry from the University of the Andes, Merida, Venezuela, in July of 1976 . From September of 1976 until February of 1980 she was employed as Instructor of Chemistry at the Uni versity of the Andes. She was promoted to her present position as A ssistant Professor in February 1980. In February of 1981 she was a warded a fellowship to study for the Ph.D. degree at the Uni versity of Florida's Quantum Theory Project, where she has been until the present. In August of 1983, she attended the Summer Institute in Quantum Theory in Uppsala, Sweden. 166 PAGE 174 I certify that I have read this study and that in my opInIon it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David A. :7 Professor of Chemistry and Physics I certify that I have read this study and that in my opInIon it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. R. Eyler ssor of Chemi I certify that I have read this study and that in my opInIon it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 'Villis B. Person Professor of Chemistry I certify that I have read this study and that in my opInIon it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality , as a dissertation for the degree of Doctor of Philosophy. Henk Monkhorst Profe !,Eor of Physics and Chemis try PAGE 175 I certify that I have read this study and that in my opInIon it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Samuel B. Tric ey Professor of P ysics and Chemistry This dissertation was submitted to the Graduate Faculty of the Department of Chemistry in the College of Liberal Arts and Sciences and to the Graduate School, and was accepted a s partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1988 Dean, Graduate School |