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# CUR TOP MOD PHYSICS PHYS 101

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PHYSICAL REVIEW B VOLUME 63 155410 Experimental and theoretical analysis of the multiphonon excitation probability for Einstein like modes in atom surface scattering Mubing Li1 J R Manson1 and Andrew P Graham2 1Department of Physics and Astronomy Clemson University Clemson Smith Carolina 29634 0978 Max Planck Instilut ir Stromungsforschung BunsenstraJ e 10 D3 7073 Gottingen Germany Received 18 February 200039 revised manuscript received 2 November 2000 published 28 March 2001 The probabilities of multiple excitation multiphonon transfer of decoupled oscillating adsorbates on a surface via impact by an atom or molecule are discussed using a dynamical scattering model and comparisons are made with helium scattering results for carbon monoxide adsorbed on metal surfaces The scattering model which contains no free adjustable parameters is shown to predict correctly the angular dependence of the distribution of the multiphonon excitations in addition to the vaua tion with incident energy and adsorbate frequency Furthermore the model is used to calculate the excitation probability of other vibrational modes that have not yet been observed using heliumatom scattering DOI 101103PhysRevB63155410 I INTRODUCTION Multiple surfacephonon excitation processes are an im portant step in the accommodation of impact energy in atom or moleculesurface interactions1 These processes aid in the sticking of particles on the surface and are crucial to heat transfer between the gas and solid phases particularly at high temperature and large particle mass2 Many studies have been performed for scattering from clean wellordered sur faces particularly using heliumatom scattering HAS where the surfacephonon modes are dispersive resulting in broad energytransfer distributions3 On the other hand a substantial number of HAS studies have shown that with only a small fraction of the surface covered with adsorbed molecules or atoms the phonon characteristics of the surface become dominated by dispersionless Einsteinlike phonon modes due to vibrations of the isolated or decoupled adsorbates 7 This arises because these modes often have frequencies lower than the zoneboundary frequency of the clean surface Rayleigh and inplane phonon branches and so are easier to excite and annihilate during the scattering pro cess for a large range of momentum transfers In addition the low frequency of these modes facilitates multiple excita tion multiphonon transfer which manifests itself as over tones of the fundamental frequency in energyresolved ex periments In this paper we investigate the angular distribution of the multiphonon energytransfer processes involving adsorbed molecules for a range of incident conditions A simple model containing no freely adjustable parameters is used to analyze the experimental results for carbon monoxide adsorbed on both Cu001 and Pt111 surfaces for which all of the vi brational and adsorption parameters have been well charac terized The scattering model considers the momentum trans ferred to a diffusely scattering adsorbed molecule with a hardcore potential by assuming that the helium atom im parts a short duration impulse In this way very few assump tions need to be made about the shape of the adsorbate or its interaction potential with the surface and the helium atom Thus the lack of signi cant deviations of the measured scat 0163182920016315155410142000 631554101 PACS numbers 7920Rf 3450Dy 7320Mf tered intensities from the predictions of this model indicate that the details of the interaction potential are not important The adsorption of CO on Cu001 Refs 6 and 8711 and Pt111 Refs 5 and 12714 has been extensively studied and the vibrational frequencies and details of the scattering from isolated molecules have been measured In both cases at low coverage the CO molecule adsorbs in a vertical con guration with the carbon atom closest to the surface on top of a surface atom Of particular interest in the present case is the parallel frustrated translation mode or T mode which has a frequency of w4meV for COCu001 and u 6meV for COPt111 about a factor of 879 lower than the frequency of the next mode which in both cases is the frustrated rotation For comparison the frequencies and as signments of the various modes and other useful quantities are given in Table 1 1n the present study the multiple exci tations of the T m ode are investigated for a range of incident angle and energy combinations The angular data provide useful information on the scattering form factor or shape of TABLE 1 Comparison of the vibrational frequencies and other physical properties of carbon monoxide adsorbed on the Cu001 and Pt111 surfaces Property Designation COCu001 COPt111 Frequencies hm 00 stretch u 258 meVa 261 meVb MCO stretch Ms 428 meVa 58 meVb CO rotation 150 353 meVa 51 meVc CO translation 124T 394 meVd 6 meV Adsorption site On top On top Adsorption energy E ads 07 er 15 eVg Onset of desorption T016 150 K 400 Kg Diffusion barrier E d 32 mth 130 meVi fReference 9 gReference 13 11Reference 10 iReference 14 aReference 8 1Reference 12 cReference 13 dReference 6 Reference 5 2001 The American Physical Society MUBING LI J R MANSON AND ANDREW P GRAHAM the CO molecule with respect to the hardcore incoherent scatterer used in the calculation whereas the energy depen dence focuses on the range of available overtones which is related to the Bose and DebyeWaller factors The differing frequencies of the CO Tmode vibration on Cu001 and Pt111 also tests the applicability of this approach In this paper the details of the helium scattering apparatus are presented in Sec II followed by the experimental results in Sec III The scattering theory is detailed in Sec IV and then compared with the experimental results in Sec V Sec tion VI is a discussion of theoretical predictions for condi tions under which the higherenergy vibrational modes might be observed and the main conclusions are summarized in Sec VII II EXPERIMENT The helium scattering apparatus15 HUGO II has a nearly monoenergetic beam formed by supersonic expansion of gas from high pressure up to 500 bar through a 10um nozzle into a highvacuum chamber with a pressure of 104 10 3 Torr The central section of the expansion is ex tracted using a conical skimmer with a 05mm aperture and the angular divergence is subsequently re ned using several differentially pumped stages The incident beam energy is regulated via the nozzle temperature that can be varied in the range 20 450 K producing energies from 4 to 100 meV The helium pressure in the nozzle was optimized to give the best energy resolution typically AEE2 For timeof ight TOF experiments the beam is chopped into pulses using a mechanical chopper with pulse durations from 5 to 20 usec before striking the crystal surface which is mounted on a sixaxis manipulator in an ultrahighvacuum chamber with a base pressure of less than 5 X 10 11 mbar The helium atoms scattered from the surface at a scattering angle of 65D 95760 with respect to the incident direction are mea sured using a magneticsector mass spectrometer mounted 14 m from the crystal surface TOF spectra are measured using a homemade multichannel scaler which is cycled us ing timing pulses from the chopper The Cu001 and Pt111 crystals were oriented to better than 025 mechanically polished and then cleaned in situ with cycles of sputtering with Ar ions and annealing until a sharp He specular peak and low diffuse elastic intensities were observed No impurities could be detected using Auger spectroscopy to within the detection limit of 9 001 mea sured with respect to the clean surface atom density For most of the present CO measurements the dosage was below 023 langmuir 1 langmuir1 L 10 6 Torr sec correspond ing to a CO coverage of CO0028 as determined by the attenuation of the specularly re ected intensity17 Typical measuring times for a timeof ight spectrum were of the order of 5 30 min III RESULTS In order to investigate the intensity distribution and inci dent energy dependence of the multiphonon processes a large number of TOF spectra were measured for a range of PHYSICAL REVIEW B 63 155410 4 9000005 2 m o a a E 4 8 2 U E 0 E 6 O o 3 S 0 310 8 5 2 E o 10390088 5 o e 20 1o 0 10 Energy Transfer AE meV FIG 1 A series of heliumatom scattering inelastic timeof ight spectra converted to an energytransfer scale for a range of CO coverages on Cu001 at TS50 K The incident energy was 403 meV directed along the Cu001 100 azimuth at an incident angle of 61282O in a xed geometry in which 6f BSDi 6 with HSD9580 The peaks at multiples of AE i4 meV are attributed to the excitation of the T mode of isolated CO molecules adsorbed at on top sites and become stronger with increasing coverage 1 6 An additional peak at AE 7 52 meV is visible at higher cov erages above CO 005 which is assigned to excitations of small CO clusters incident conditions In particular the COCu001 surface was studied because this system is well understood and the T mode frequency has been measured in a number of studies6 10 16 19 To begin with it is important to determine whether the CO coverage has an in uence on the mul tiphonon intensities because effects of clustering were ob served at low coverages in a recent study10 In Fig 1 a series of TOF spectra converted to energy transfer are shown for several different coverages of CO ranging from co 0005 to CO0088 As the coverage increases another peak can be seen in addition to the T mode multiphonons at AE in X4 meV AE52 meV A previous study10 indi cated that this peak is due to interactions between CO mol ecules in close proximity ie due to small CO clusters on the surface This peak becomes signi cant at coverages above CO005 Figs 1d and 1e To investigate whether the clustering in uences the inten sity ratio between the fundamental and overtones of the T 1554102 EXPERIIVIENTAL AND THEORETICAL ANALYSIS OF 5 F 4 39 G Q 33 E 3 8 o o 4meV m2 EI EI 8meV E 2 395 C 9 E 1 O n r 000 002 004 006 008 010 Coverage FIG 2 T mode fundamental n 1 and rst overtone n 2 energy loss intensities as a function of surface coverage for an incident energy of 403 meV and an angle of 612820 The inten sities show similar monotonic increases with a constant ratio of In1In23 up to a coverage of CO006 followed by a slight intensity decrease mode the intensities of the fundamental energyloss AE 4 meV and the rstovertone AE i 8 meV peaks were extracted from the spectra in Fig l The intensities are shown in Fig 2 The intensities of the fundamental and rst overtone exhibit a nearly linear increase with coverage up to CO003 above which the rate of increase is lower with saturation occurring above COgt006 Across the whole coverage range studied the intensity ratio between the funda mental and rst overtone remains approximately constant at 3 The linear increase is consistent with a simple increase in the density of scattering centers on the surface The deviation from the linear behavior at coverages COgt 003 can be in terpreted as scattean involving more than one scattering center CO molecule due to close proximity Thus in order to avoid any coverage effects a coverage of less than 900 003 is necessary We have used a CO coverage of co 0028 for all of the following measurements Figure 3 shows several typical TOF spectra measured with an incident beam energy of 666 meV and a surface temperature of TS50K for a range of incident angles for CO0028 on Cu001 The spectra show that the total scattered intensity decreases quickly for incident angles far from the specular peak 614790 as also shown in several previous studieslg ZO 21 At the same time multiples of the fundamental T mode frequency for COCu001 of u 4 meV Refs 6 and 10 are observed to become much more dominant with the third overtone clearly visible at AE 16 meV The number of overtones excited becomes signi cantly PHYSICAL REVIEW B 63 155410 200 o COCU001 547 100 Ei666meV Ts50K 39gtquot D E O G 8 E20 8 01039 N2 0 E 839 2 439 3 E 039 839 760 4 o e 0 20 1 O O 10 20 Energy Transfer AE meV FIG 3 Several helium timeof ight spectra converted to an energytransfer scale for a CO coverage of CO0028 on Cu001 and a range of incident angles 6 with 6f BSDi 61 The surface temperature was TS 50 K and the incident energy was E 666 meV Multiple excitations of the T mode peak at AE in X4 meV become dominant features of the spectra as the incident angle is increased larger for higher incident energies as shown in Fig 4 In this case the incident angles are approximately the same but the beam energy increases from 200 to 835 meV for the same surface temperature of TS 50K and CO coverage co 0028 At 200 meV three energyloss peaks can be de tected whereas at 835 meV the seventh energyloss peak sixth overtone is clearly visible and the eighth peak is just visible above the noise Thus the number of excitations is not directly proportional to the incident beam energy How ever higher overtones may be hidden by the increased back ground level and the resulting statistical noise which is due to the enhanced CuOOl surface multiphonon processes at higher beam energies22 It is also worthwhile noting that the measured multiphonon excitations are perfectly harmonic up to the seventh overtone to within experimental error This provides some clues as to the nature of the interaction of the helium atoms with a surface as will be discussed later In addition to the measurements of COCu001 TOF spectra were obtained for a low coverage CO003 of CO on Pt111 under similar scattering conditions The CO Ptlll T mode frequency was determined in earlier measurements 14 to be w6 meV 50 higher than that for COCu001 and thus provides a useful comparison Fig 1554103 MUBING LI J R MANSON AND ANDREW P GRAHAM 100 I I 80 Ei200meV i I I 60 ei747 g 40 Ts50K i i i w I i I g 20 I I a 0 l 7 839 Ei666meV i I I g 6 95760quot I O I I l I N 4 i i i i 9 2 E o quotlt3 6 l 39 l l g Ei835meV I I g 4 ei779 l i I I i I 2 39 39 I i I l I O I 50 4O 30 20 1 O 0 10 Energy Transfer AE meV FIG 4 A series of heliumatom scattering inelastic timeof ight spectra converted to an energytransfer scale for a CO cov erage of CO0028 on Cu001 and incident energies of E 200 666 and 835 meV The surface temperature was TS 50K and the incident angle is approximately constant in the range 61747077790 At 20 meV the second overtone of the AE 4 meV T mode peak at AE 12meV is just visible whereas the seventh overtone is observed for an incident energy of 835 meV The substantially increased background at higher inci dent energies is due to multiphonon excitations of the Cu001 sub strate phonon modes ure 5 compares TOF spectra measured under nearly identical scattering and temperature conditions for CO003 CO on Cu001 and Pt111 The spectra for COPt111 exhibit fewer overtones than those for COCu001 as expected for the higher vibrational frequency However the distribution of intensity with energy transfer intensity envelope is simi lar in both cases taking into account the larger peak width of the COPt111 T mode vibration due to the shorter lifetime compared to C0Cu00110 Also shown in Fig 5 as solid curves are the theoretical calculations as discussed below in Sec V IV THEORY A Differential re ection coef cient An appropriate starting point for a general treatment of inelastic scattering between a projectile and a manybody target is the quantummechanical transition rate or general PHYSICAL REVIEW B 63 155410 10 I COCu001 8 8 Ei403meV 6 o Ts50K 6i282 4 Intensity 1O countssecmeV 0 quot 5 COPt111 4 oo Ei402meV 3 TS50KO ei279 2 1 O quotr r 71 3quot 39 I I 39 1 j I wag 20 10 o 10 20 Energy Transfer AE meV FIG 5 Comparison between the TOF spectra for helium scat tered from CO on Cu001 and Pt111 The scattering conditions are nearly identical39 the incident energies are 403 and 402 meV and the incident angles are 61282O and 61279O for CO Cu001 and COPt111 respectively The coverage was CO 003 and the surface temperature TS 50 K in both cases and the spectra have been converted from ight time to energy transfer Note that the peak width for the COPt111 T mode at AE 6 meV is larger than that for COCu001 due to the shorter vi brational lifetime The solid curves are the calculations discussed in Sec V ized Fermi golden rule for the projectile to scatter from the initial state of wave vector k to the nal state kf which is given by23 24 nfkf7k m I 27739 wkfkltE inf War 89gt 1 where is the initial manybody state of the unperturbed target 2 is a summation over all nal states of the target T is the transition operator 8f and 8 are respectively the nal and initial energy of the entire system of target plus projectile and the angular brackets signify an average over all initial target states In atomsurface scattering the measured intensities are usually differential re ection coef cients which are obtained from the transition rate wkfkl on multiplication by the density of nal scattering states as follows dR L4 m2kf d def 277W k izwa faki 2 1554104 EXPERIMENTAL AND THEORETICAL ANALYSIS OF where m is the mass of the projectile atom and k1Z is the component of the incident wave vector perpendicular to the A general expression for the transition rate for the full transition operator can be obtained in the semiclassical limit This involves making the approximation that the collision is fast in comparison to phonon periods and that it is suf cient to expand the interaction potential through terms linear in the atomic displacements These approximations have been jus ti ed in detail in connection with development of mul tiphonon exchange in atomsurface scattering and have been shown to be valid through extensive comparison with experi mental measurements 27 The result is the following form for the transition rate 26 2Wkfk l wkfgtkx 2i7fx 00 X die 1EfE t eQkfk 2 3 where QkfkK 1 is a generalized displacement correlation function and 2Wkfk Qkfk 39I027 In the semiclas sical approximation of quick collisions ie collision times that are short compared to a vibration period the time dependent correlation function Qkt becomes the displace ment correlation function Qkfgtkxl gtQkgtIltk u0k utgtgt 4 where k kfi k is the scattering vector and the argument of the DebyeWaller factor is 2WkQkI0ltkquot02gt 5 In Eq 3 the energies E f and E are respectively the nal and initial kinetic energies of the scattered projectile The factor lr 2 is the form factor that depends on the nature of the scattering center and the interaction potential The form factor is discussed in more detail in Sec IVB In the case of a vibrational displacement having only a single dispersionless Einstein mode of frequency w Eqs 3 and 4 can be evaluated together and the result is a gener alized temperaturedependent Poisson series that produces multiphonon overtones28 This expression appears as fol lows fRltkfk m2 kfl km dEfdof Zn lT Ze2Wke2WTAK AKZ xagwnIm W nwnw71 nw aZ W 5EfiElia w 6 where I az is the modi ed Bessel function of order a Tfllz is the form factor AK is the component of k parallel to the surface M A is the effective mass derived from the normal mode analysis of the CO adsorbate as shown in Sec IV C and nw is the BoseEinstein function given by PHYSICAL REVIEW B 63 155410 1 quotUHFW 7 The DebyeWaller factor arising from the contribution of the adsorbate T mode is given by 15 AK2 ZMAw l nw 2WTAK 2 8 while 2Wk is the contribution to the DebyeWaller factor arising from the substrate modes which for a Debye model of the substrate vibrations has the familiar hightemperature limit given by 3 152 k2 TS 2Wk7 Mcm 9 where M c is the mass of a substrate atom However in the calculations presented here we used the fully quantum mechanical form of Eq 9 which includes zeropoint mo tion of the substrate lattice27 The analysis of the scattering intensity measurements in this manuscript is based primarily on Eq 6 and we note that this result contains a full treat ment of the adsorbate recoil because the zeropoint motion is treated correctly throughout B The form factor In order to calculate the adsorbate multiphonon peak in tensities accurately the choice of a suitable form factor or adsorbate shape needs to be considered In the semiclassical limit the formfactor amplitude T is given by the transition matrix for scattering by the elastic part of the interaction potential extended off the energy shell27 thus permitting the form factors derived from elastic scattering experiments and calculations to be used for the present inelastic scattering calculations A complete form factor contains all the details of the scattering interaction including multiple scattering from the adsorbate and surface the Headsorbate potential and the Hemetal surface potential These details typically make a full form factor dif cult and time consuming to compute29 Fortunately in previous work on the scattering of He atoms from adsorbed CO it has been shown that both the elastic and inelastic scattering cross sections are determined largely by scattering from the hard repulsive core of the CO Furthermore this core is well approximated by a hard hemi sphere for the scattering of He from isolated CO on close packed metal surfaceszo u3930 Using these established ideas a simple model for the form factor for the present problem can be developed based on the scattering from a hard sphere In standard treatments the scattering from an isolated scattering center is described by an asymptotic wave function of the form Mas r e l lae kfur 10 For a hard sphere of radius a the form factor f 6 as a function of the scattering angle 6 is given by the following wellknown expression in the Kirchhoff limit klaaoofl 1554105 MUBING LI J R MANSON AND ANDREW P GRAHAM 6 m 39 Z39k 396 f 7 Y zexp liasmi 1cos6 a kiasm J1kla sin 0 11 where J1z is the rstorder Bessel function The use of the Kirchhoff approximation is justi ed for the present case be cause the incident wave vectors range from roughly 6 to 12 AT1 and the hardcore radius of the HeCO potential is about 23 A thus kiagt 10 for all incident energies The rst term on the righthand side of Eq 11 is called the illuminated face contribution and it produces uniform scattering intensity at all angles while the second term is the Fraunhofer contri bution If the adsorbed CO is modeled by a hemisphere on an otherwise at mirror surface then one must account for the possibility of double scattering events involving scattering from the hemisphere plus a re ection with a concomitant phase change of 7Tby the at surface This is readily effected by adding to Eq 11 a second term that accounts for the double scattering processes from the at surface For the simplest case of scattering in the sagittal plane and for a xed angle 65D between the incident beam and the detector direction the scattering form factor becomes fsf77quot 05D f6f 0139 12 where 6 and 6f are the incident and nal scattering angles with respect to the surface normal respectively The form factor corresponding to this model is given by ITflIZOCI f SIZ which consists of an envelope governed by the Fraunhofer term at small parallel momentum transfer AK and by the constant illuminated face contribution at large AK It also contains interference terms called re ection symmetry oscillations because of their similarity in origin to the symmetry oscillations observed in identical particle scattering20 An even simpler approximation to the form factor is to use only the envelope function which is given by leiIZOCIfS620C 41g2 1 1g IAKI3 13 This simple form factor combines the two essential features arising from hardcore scattering which are the Fraunhofer contribution that varies as IAKI 3 and the illuminated face contribution which is a constant It is a function of both the nal and initial energies through its dependence on the par allel momentum transfer AK Equation 13 is the form fac tor that is used in the analysis of the data in Sec V below C Normalmode analysis In order to obtain the effective mass M A of the CO mol ecules for each of the vibrational modes a normalmode analysis for the modes of CO adsorbed on a Cu001 surface was made A model adequate for the purposes of the analysis presented here is the simple ballandspring model shown in Fig 6 This model leaves the vertical and horizontal motion PHYSICAL REVIEW B 63 155410 7 X FIG 6 A schematic diagram of the ballandspring model used for the normalmode analysis for the vibrations of an isolated CO molecule For the calculations presented here kg was taken to be zero of the carbon and oxygen atoms uncoupled For vertical mo tion the spring of spring constant k2c represents the carbon metal bond and a second with spring constant kg represents the C0 bond For the horizontal motion springs with con stants kg and kg connect the carbon and oxygen atoms re spectively to the surface vertical and a third k is an angle bending spring that tends to restore the carbon and oxygen to a straightline con guration with respect to the bonding point on the surface The normalmode analysis for such a system is straight forward For the horizontally polarized modes a satisfactory match of the 4meV T mode and the 353meV R mode is TABLE II Comparison of model calculations with experiment for the isotope shifts of the T and Rmode frequencies for CO Cu001 with respect to the frequency of the 12C160 isotopomer based on the model of Fig 6 T mode shift Rmode shift Isotope Expta Present calc Expta Present calc 12C180 7458 7264 7098 7168 13C16O 7076 7028 7312 7272 13080 7509 7376 7393 7446 aReference 10 1554106 EXPERIIVIENTAL AND THEORETICAL ANALYSIS OF TABLE III Comparison of model calculations with experiment for the isotope shifts of the S mode and C0 stretch mode frequen cies of COCu001 with respect to the frequency of the 12C160 isotopomer based on our model of Fig 6 S mode shift CO stretchmode shift Isotope Expta Present calc Expta Present calc 12C180 7208 7358 7221 13C16O 7094 7161 7235 13C180 7304 7502 7463 aReference 10 obtained with k 6207 amu meVZ k2 61668 amu meVZ and kg 0 These two frequencies can also be matched with a large range of values of kggt0 with a concomitant small reduction in the values of the other two constants but the simplest choice of k 0 has been used here This model also predicts the measured isotope shifts in the frequencies well as shown in Table II The relative amplitudes of the carbon and oxygen vibra tional amplitudes are expressed in terms of the polarization 30 Intensity 103 countssecmeV u 5 o 5 0 5 10 Energy Transfer AE meV FIG 7 A series of heliumatom scattering inelastic timeof ight spectra converted to an energytransfer scale for CO0028 CO on Cu001 at TS50 K The incident energy was 200 meV The incident angles range from 61604O to 61747O with 6f 9580 7 r9 directed along the Cu001 100 azimuth The experi mental points are shown as circles while the solid lines are the results of the theory discussed in the text PHYSICAL REVIEW B 63 155410 vectors e8Kv where 8 is the Cartesian direction index K is an integer denoting the atom and v is an integer denot ing the mode number For the low frequency T mode this model gives exOTexCT 167 while for the R mode it gives exORexCR 70719 The timedependent displacement correlation function for a harmonic system such as this is given by I h m Kyye DK DV gtltnwp lleiiw nwpei 14 where M K is the mass of the Kth atom w is the frequency of the vth normal mode and N is the total number of modes Equation 14 shows that the effect of the polarization on the expression of Eq 6 for calculating the intensities can be treated through introduction of an effective mass As dis cussed above in Sec IV B the adsorbed CO molecule can be represented by a hard hemispherical pro le which derives mainly from the hard core of the oxygen atom on a hard at surface This simpli es the calculation because it can be as sumed that the only term of Eq 14 that enters into the calculation of the Einstein mode intensities of Eq 6 is that for the timedependent correlation of the oxygen atom with itself The effective mass of the oxygen atom is MA MOexOV2 where MO is the mass of an oxygen 1 0 v o Expt 3 5 Theory Intensity 103 countssecmeV i 0 5 O 5 10 Energy Transfer AE meV FIG 8 Same as Fig 7 except with the incident energy at 4029 meV and incident angles from 6282O to 23880 1554107 MUBING LI J R MANSON AND ANDREW P GRAHAM 60 40 N O Intensity 102 countssecmeV L L OED50300100100100 4 30 0 10 0 1O 20 Energy Transfer AE meV FIG 9 Same as Fig 7 except with the incident energy at 666 meV and incident angles from 61563O to 617600 atom and consequently the effective mass for the T mode can be taken to be MTMOIexOT2 16 amu08592 217 amu Similarly for the R mode an effective mass of MR MOexOR2 16 amu058424696 amu can be obtained For the vertically polarized mode a match to the 428 meV S mode and the 258meV CO stretch mode is obtained with kzc 53 394 amu meV2 and kg 437 781 amu meVZ Therefore the effective mass for the S mode is M S MOezOS2 16 amu072123002 amu The calcu lated isotope frequency shifts of the S modes and C0 stretch modes are given in Table 111 V ANALYSIS OF THE DATA Since in the present work we are mainly interested in the probabilities for inelastic scattering from the adsorbed mol ecules the broad substrate multiphonon background under the adsorbate vibrational peaks was subtracted from the ex perimental TOF spectra to leave just those peaks This was accomplished by subtracting a smooth curve tted to the intensity between the adsorbate inelastic peaks from each spectrum The resulting backgroundsubtracted data is shown in Figs 7 10 In order to test the hypothesis that the form factor for inelastic scattering from CO on metal surfaces can be ap proximated by simple scattering from a hard hemisphere the calculations using Eq 6 were rst performed for a constant PHYSICAL REVIEW B 63 155410 form factor IerIZ 1 To compare the results of the calcula tions which are a series of energy 6function peaks with the experimental measurements the calculated spectra were broadened with a xedwidth Gaussian function where the width was tted to the experimental resolution These calcu lations were tted to the backgroundsubtracted data by mul tiplying each calculation by a normalization coef cient where the intensity matching was made in most cases using the fundamental energyloss peak ie the n 1 phonon creation peak This results in a t to the experimental data which for each individual TOF spectrum is essentially a zeroparameter t This is because the differential re ection coef cient of Eq 6 with a constant form factor has no adjustable parameters aside from the DebyeWaller tempera ture appearing in the substrate DebyeWaller factor 2 Wsk which is well known to be D220K from previous work on clean Cu001 surfaces22 The resulting normalization coef cients which contain information about the form factor are shown for each of the four incident energies in Fig 11 Presented alongside the coef cient points in Fig 11 are the formfactor curves de rived using Eq 13 and a radius of a23 A solid lines This value of the radius is well established from previous investigations of the elastic scattering of He from CO on Cu001 and other metal substrates21 and consequently a cannot be considered to be a freely adjustable parameter The simple form factor given by Eq 13 matches the trend of the normalization points quite well for all four incident energies 160 120 80 4O 3O 20 1O 0 Intensity 1 O2 countssecmeV 40 20 o 2 1 o 3O 2O 1O 0 1O 20 Energy Transfer AE meV 40 FIG 10 Same as Fig 7 except with the incident energy at 834 meV and incident angles from 6563O to 27600 1554108 EXPERIIVIENTAL AND THEORETICAL ANALYSIS OF Q 834meV39 4 Form Factor 1 O countssec o i d 50 60 7o 80 Incident Angle 9i deg FIG 11 The experimentally obtained intensity of the n 1 COCu001 T mode phonon creation peak compared with the form factor of Eq 13 for the incident energies a 834 meV b 666 meV c 403 meV and d 200 meV The experimental intensity is plotted as circles and the solid line is the theoretical form factor for a hardcore radius of a 23A It is seen that with scattering at incident angles near the specular condition at 61479o the intensity rises sharply due to the IAKI 3 term at small AK in the form factor while for large incident angles the intensity saturates to a smaller constant value that can be attributed to the illumi nated face contribution from hardcore scattering Thus the observed angular intensity dependences are well represented by the form factor in Eq 13 justifyng its use together with Eq 6 to simulate the TOF spectra including the momen tumtransfer dependences of the intensities The results of the complete calculations are shown in Figs 7 10 solid lines In the process of this nal calcula tion with the new form factor it was still found to be neces sary to multiply each calculated curve by a renormalization factor N close to unity in order to obtain the best t The PHYSICAL REVIEW B 63 155410 a b 8 ei282 e e 9 0 6 0 O H 4 gt CD 2 E B 0 8 e 413 E 30 i i e 39 o C 6 39 6 8 20 L R L R m0 8 e o 10 o o I 1 a 0 E quot7 433 C 9 15 o o 39 E t g 5 E 0 2o 10 0 10 1o 0 10 Energy Transfer AE meV FIG 12 Comparison of the calculated inelastic multiphonon intensity thick solid line with experimental data circles for E 4029 meV and selected incident angles a shows the results of the calculations without including the scattering form factor of Eq 13 while b shows the same calculations including this factor At 612820 which is far from specular conditions the form factor is seen to have little effect However at 61413O and 4330 which are close to specular and where the Fraunhofer term is dominant the form factor has an important effect in bringing the calculated intensities for the inelastic overtone peaks into agreement with ex periment The two peaks marked L longitudinal resonance and R Rayleigh mode arise from the substrate vibrations need for this nal renormalization is obvious from Fig 11 because although the calculated form factor of Eq 13 fol lows the observed intensities quite well the points are scat tered about the theoretical curve The value of the nal renormalization constant is shown in each TOF plot It is seen that these nal calculations with the form factor of Eq 13 agree rather well with all of the TOF experimental data measured The reason for the residual uctuations of the inelastic intensities compared with those calculated using the simple form factor is well understood The characteristic signature of scattering from an isolated adsorbate whether elastic or inelastic is a supemumerary rainbow oscillation in the dif ferential re ection coef cient caused by multiple scattering between the adsorbate and the surface substrate18 20 21 3O These oscillations are re ection symmetry oscillations and they are most easily observed in experiments that follow the intensity of a single selected multiquantum overtone peak as a function of parallel momentum transfer AK They also 1554109 MUBING LI J R MANSON AND ANDREW P GRAHAM affect the measured results in this current set of experimental measurements However because the current observations were taken at rather widely separated values of incident angles the measurements are not suf ciently dense to exhibit these oscillations The small differences appearing in Fig 11 are clearly due to these oscillations about the envelope func tion given by Eq 13 which explicitly excludes these re ection symmetry oscillations In Figs 7710 it is apparent that often the experimentally measured zeroloss peak is substantially larger than the cal culation This is because contributions to this peak come also from other sources for example the diffuse elastic scattering from defects and other impurities on the surface The importance of the form factor is illustrated in Fig 12 which shows TOF spectra for E14029meV at three differ ent incident angles The lefthand panels show the back groundcorrected data together with calculations using a con stant form factor T 1 while the righthand panels show the same data with calculations obtained with the form factor given by Eq 13 The two peaks in the experimental data marked L and R are respectively the longitudinal resonance and the Rayleigh mode arising from the substrate vibrations At the incident angle of 61282quot which is far from the specular position and at which the parallel momentum trans fers for all observable overtone peaks are rather large and hence the form factor of Eq 13 is nearly constant there is little difference between the two calculations However for the two other incident angles that are much closer to specu lar the values of AK are much smaller and there is a large difference between the two calculations because the lAKl 3 term in the form factor has a big effect and varies strongly from peak to peak However as shown in the righthand panel of Fig 12 when the form factor of Eq 13 is included in the differential re ection coef cient the results agree well with experiment This agreement is important con rmation of the fact that Fraunhoferlike scattering from the hard re pulsive molecular core is the most important term in the d5Rkfkl mZ lkl 00 A 7 f T leZWTAKe2WRAKeZWSkze2Wk 2 IMlt g7w PHYSICAL REVIEW B 63 155410 form factor describing inelastic scattering from adsorbed CO The applicability of the present theoretical approach was further tested by comparing the calculations obtained with Eqs 6 and 13 for COCu001 with COPt111 The Tmode frequency of COPt111 is w6 meV 50 higher than that for COCu001 thus providing a useful comparison as shown in Fig 5 for two TOF spectra mea sured for similar scattering conditions As for COCu001 the present theoretical model accurately describes the inten sity ratio between the observed inelastic peaks which for these particular scattering conditions are the fundamental and rstovertone energyloss peaks and the fundamental energygain phononannihilation peak of the 6meV T mode The good quality of the t provides additional support for the present theoretical model VI PREDICTIONS FOR THE R AND S MODES In the present experimental measurements and in previ ous investigations5quot5391039113930 no evidence for excitation of the R or S modes has been reported using Heatom scattering from CO adsorbates on either Cu or Pt surfaces It is of interest to examine the question of why this is the case with the present theoretical model in order to see if experimental conditions can be predicted under which these two modes might be observed Including the R and S modes in the theoretical model of Eq 6 is relatively straightforward Since these modes have frequencies considerably larger than the T mode it is ex pected that their intensities will be small and only the rst order energyloss contribution needs to be included which is equivalent to calculating these intensities in a rstorder Born approximation However higherorder overtones and even crossterm contributions involving the simultaneous ex change of combinations of R S and T mode quanta are readily calculated within this formalism The differential re ection coef cient for the scattering including only the rst order contributions for the R and S modes is Kz Vnwnw1gt dEfde 239rr 4 klz MAw nu aZ Kz MM 712 X 5EfiElia wIl MRwRnwRnwRil W 5EfiE r wR 1 3 7125ErEz wsgt 05gt sws quotws 1 where exp72WRAK is the contribution to the Debye Waller factor arising from the R mode exp72W5AK is the contribution to the DebyeWaller factor arising from the S mode and exp72W7AK and exp72Wk are as be fore the contributions to the DebyeWaller factor arising from the T mode and substrate modes respectively Using the parameters for the existing experimental results namely a total scattering angle of 63D958 and incident energies up to 100 meV Eq 15 together with Eq 13 does not predict signi cant inelastic peak intensities for either the R or S modes in agreement with the experimental observa tions In fact from the form of Eq 15 it is clear why the highenergy modes have such small scattering intensities First the larger frequencies for wR and us make the argu 15541010 EXPERIIVIENTAL AND THEORETICAL ANALYSIS OF ments of both the Bessel function and the value of the Bose Einstein factor considerably smaller making the intensity much smaller than that for the T mode despite the coupling of the S mode to the larger perpendicular component of the momentum transfer k Second for the R mode as discussed in Sec IVC above the normalmode analysis gives a vibra tional amplitude for the oxygen atom vibration of exORexCR 072 as opposed to a value of exOTexCT 17 for the T mode For the R mode this translates into a large effective mass for the oxygen atom MRMOIexOR2 16 amu058424696 amu signi cantly reducing the calculated intensity and the contribution of the DebyeWaller factor Similarly as expected it is found that the effective mass for the S mode is close to the mass of the whole CO molecule about twice the mass of an oxygen atom MSMOIezOS2 16 amu07212 3078 amu which also tends to make the intensity of this mode smaller In particular for the R mode as noted earlier the associ ated momentum transfer is the parallel component while for the S mode it is the perpendicular component This has two important effects First the conditions in which the R mode would produce large intensities ie large parallel momen tum transfers will also be very favorable for the exchange of Tmode phonons Thus the lower frequency and smaller ef fective mass for the T mode make it dominant Further the DebyeWaller contribution due to the T mode will make both the R and Smode contributions small This effect will make it particularly dif cult to observe the R mode and probably explains why it has not yet been reported in Heatom experi ments The perpendicular momentum transfer k associated with the S mode should on the other hand help to make its in tensity large In the experiments shown in Figs 7 10 the parallel momentum transfers are small and range up to 3 4 A71 while the perpendicular components are typically larger than 10 A71 Since these momentum transfers enter as squared terms in Eq 15 this difference is substantially magni ed Thus it would at rst appear that the Smode scattering intensity should be most observable under condi tions in which the perpendicular momentum transfer is maxi mized and simultaneously the parallel momentum is mini mized so that the Tmode contribution to the DebyeWaller attenuation is small For the present experiments this would correspond to incident angles near specular and a small angle 65D between the incident beam and detector rather than the present angle of 9580 A number of calculations were consequently made using Eq 15 for realistic experimental geometries which were anticipated to show the S mode for COCu001 Figure 13 shows the predicted TOF spectra for incident heliumatom energies ranging from 50 to 100 meV and a small incident angle of 612250 with the detector placed at 6f25 making 69250 The calculations are scaled to represent a realistic incident beam ux by interpolating between the nor malization factors for the experimental data given in Figs 7 10 Thus the calculated intensities shown in Fig 13 are expected to represent realistic values realized for the present experimental apparatus with a smaller total scattering angle PHYSICAL REVIEW B 63 155410 20 v S Ei100meV 15 E 10 A g 5 T a E 0 R AA AA o g S E Ei75meV 3 4O C 3 g 20 A T gt E 0 B AA A g 300 Ei50meV 200 E 100 S M T o R A 60 40 20 O 20 40 Energy Transfer AE meV FIG 13 A series of calculated heliumatom scattering inelastic spectra using realistic incident beam uxes for isolated CO on Cu001 at TS50 K The incident angle is 6225O and the scat tering angle is Rf 250 corresponding to BSD250 The incident energies range from 50 to 100 meV In addition to the T mode multiphonons at AE in X4 meV at AE 35 meV the contri bution of the R mode appears and at AE w 43 meV the contribu tion of the S mode is observed as well Under these conditions the Smode peak at AE 743 meV exhibits a total integrated intensity relative to that of the sum of all inplane Tmode overtones ranging from 25 at Ei50 meV to 50 at E1 100 meV On the other hand the Rmode peak while clearly visible in the calculations is no larger than the smallest of the non negligible Tmode peaks As the incident energy increases from 50 to 100 meV the relative intensities of both the Tmode and the Smode peaks increase but for the R mode this increase is much less apparent Note that the widths of the S and Rmode peaks in Fig 13 were chosen to be the same as that of the Tmode overtones which is approxi mately the energy resolution of the experiment Consequently we can conclude from the calculations that the S mode should be observable under conditions in which the parallel momentum transfer is small and the perpendicu lar momentum transfer is large ie high incident energies and small incident and nal scattering angles However if the parallel momentum transfer is nonnegligible even if it is signi cantly smaller than the perpendicular momentum transfer the DebyeWaller factor due to the T mode will be so small as to severely damp out the Smode intensity The R mode on the other hand is predicted to be dif cult to ob serve under almost all experimental conditions because large parallel momentum transfers are needed Thus as a result of 155410ll MUBING LI J R MANSON AND ANDREW P GRAHAM the high frequency and higher effective mass of the R mode and the signi cant T mode multiphonon creation for large AK the DebyeWaller factors for the R mode are small resulting in lower intensities VII CONCLUSIONS In the present paper detailed experimental results for the multiple excitation of the parallel vibrational modes of CO adsorbed on Cu001 and Ptlll have been presented The angular distribution of intensity into the multiphonon peaks was described in a simple way by a comprehensive scattering theory using only the most basic aspects of the interaction between helium atoms and the adsorbed molecules on the surface From a detailed comparison between the experimen tal results and theoretical calculations the following conclu sions can be awn l The multiphonon intensity distribution is described well using a selfconsistent theory that is essentially a gen eralized temperaturedependent Poisson distribution multi plied by a form factor that describes the interaction between the probe particle and the surface It was shown that for a large range of scattering conditions the form factor can be reduced to a simple hardcore pro le and in addition that the overall features of the scattering can be reproduced by ignoring multiple scattering Furthermore because the de tails of the hardcore pro le had been determined in earlier elastic scattering measurements there were no freely adjust able parameters and the calculations could be performed in dependently of the present experimental results There are no freely adjustable parameters because the theoretical results depend only on the masses the mode frequency the hard core radius of the HeCO interaction potential and the sub strate Debye temperature all of which are well established by other independent experiments As an additional point the results depend only very slightly on the substrate Debye temperature because the substrate DebyeWaller factor has very little effect 2 The effective mass of the molecule for each vibration must be taken into account For the present cases of CO Cu001 and COPtl ll this was achieved using a balland spring model and tting the force constants to the known isotope shift measurements The good t of the scattering theory to the experimental data showed in addition that the helium atoms probe the motion of the oxygen atom which sits furthest out from the metal surface 3 The overtones of the T mode for COCuOOl ob served with HAS are equally spaced in energy and exhibit no observable increase in peak width with overtone number In particular the n 7 8 Tmode peak or seventh overtone was observed at an energy loss of AE 732meV Since other experiments have shown that the COCuOOl lateral poten tial is quite anharmonic and the diffusion barrier is only about 30 meV10 the observed multiphonon excitations do not appear to be those of single molecules This can be un derstood by considering that just as in the case of helium diffraction single helium atoms interact with an area of the surface corresponding to the coherence width of the helium wave packets which is equivalent to the coherence width of PHYSICAL REVIEW B 63 155410 the helium beam With use of a typical coherence width of 300 A and the experimental CO coverage of CO0O3 it can be seen that each helium atom interacts with about 330 CO molecules Thus even though the CO molecules are iso lated in the sense that there is no signi cant multiple scat tering between them this work provides strong evidence that the multiple overtone energy transfer clearly must be a col lective quantum effect In other words these overtones are true phonon modes ie they are collective oscillations involving a very large number of adsorbed CO molecules Such a collective and coherent interpretation of the inelastic phonon loss spectra is consistent with a coherent quantum picture of the elastic diffraction of He atoms and indeed small rstorder diffraction peaks due to the twodimensional lattice gas nature of the rare ed CO coverage are observed just as they have been observed in previous experiments on low coverages of CO on metal surfaces20 However the present interpretation of the multiphonon inelastic exchange as a coherent phenomenon involving many CO adsorbates is clearly open to objections Alterna tive explanations not invoking a coherent scattering event can be developed on assumptions that the incoming He atom scatters with only a single CO adsorbate and exchanges the energy with only that one adsorbate or the incoming He atom scatters successively with several CO adsorbates in a multicollision process and exchanges energy with several ad sorbates The main argument presented here against a localized in teraction with a single CO adsorbate is the absence of any observable anharmonicity in the measured overtone spectra even up to large overtone numbers A secondary argument is that coherence in the incoming beam over distances encom passing large numbers of CO adsorbates is clearly evidenced by the observation of small diffraction peaks With respect to the question of anharmonicity the barrier for CO diffusion on this same Cu001 surface has been measured to be 31 i 10 meV10 which is comparable to or even smaller than the n8 quanta or AEW32 meV of energy loss observed here under favorable incident conditions However the measure ments of the dynamic diffusion barrier that were carried out at temperatures of over 100 K may not represent well the actual manybody potential well of the adsorbed CO at the lower temperature of 50 K used in the present measurements Clearly it is conceptually incorrect to discuss the diffusion barrier in terms of a static potential due to the substrate especially in a case in which the barrier is small The thermal diffusion barrier is the transition point along an effective reaction coordinate in the multidimensional space of very many degrees of freedom of the CO and its surrounding Cu atoms Thus the diffusion barrier may correspond to a very unusual con guration such as large thermal excursions of surrounding substrate Cu atoms Such con gurations may be less probable during a scattering event especially one at lower temperatures Thus it is possible that mobility of the CO after direct impact is quite different from that observed in thermal diffusion an argument in favor of nearly har monic and largequantumnumber energy transfers to a single adsorbate However in contradiction to this argument there are Heatom scattering measurements of both the 15541012 EXPERIMENTAL AND THEORETICAL ANALYSIS OF T mode overtone linewidths and the small shift in mode fre quency as a function of surface temperature that show that this same COCu001 system exhibits some anharmonicity even at temperatures as low as the 50 K used here32 Multicollision processes in which the incoming He pro jectile makes successive semiclassical collisions with several CO adsorbates and exchanges only a small number of quanta at each collision are a possible explanation for the absence of observed anharmonicity in the overtone peaks However the large average distance of greater than 10 A between CO adsorbates at the low coverages used here makes such colli sions unlikely The nearly linear increase in the inelastic T mode intensities up to CO coverages of approximately 6 as exhibited in Fig 2 is consistent with the assumption of isolated scattering centers with no multiple collisions The nonlinear coverage dependence of the inelastic T mode in tensities at coverages greater than 6 implies that multiple scattering is occuring only at coverages greater than those used in this work The absence of multiple collisions is also con rmed by independent measurements and calculations of the elastic differential scattering cross sections for Heatom scattering from imilar lowcoverage COCu001 sur and by earlier work on the elastic total cross sec tions of isolated CO adsorbed on a Pt111 substrate173933 These earlier calculations of both differential and total cross sections for Heatom scattering from isolated CO adsorbates on Cu001 agreed well with the measured cross sections at low adsorbate coverages comparable to those used here and the comparison gives no evidence for multiple collisions of He with more than one CO The calculated differential cross sections provide a way of estimating the multiplecollision intensities in terms of successive single collisions and such estimates indicate that the multiple scattering contribution to the elastic cross section is signi cantly less than 1 Since the same hardcore scattering interaction is predominantly involved in both the elastic and inelastic scattering pro cesses 3930 it appears unlikely that multiple collisions play a signi cant role in the inelastic scattering 4 The present theory has shown why under the present experimental conditions the S and Rmode vibrations of adsorbed CO have not been observed to date even for experi ments using incident energies considerably higher than the energy loss associated with these modes It was found that due to the near 900 geometry of current HAS experiments the ratio of parallel to perpendicular momentum transfer fa PHYSICAL REVIEW B 63 155410 vors the creation of T mode multiphonons at the expense of the higherfrequency modes It was shown that small inci dent and total scattering angles are more favorable for S mode phonon creation whereas the R mode is expected to be weak under almost all scattering conditions and probably undetectable It is also of great interest to note that the present theoret ical formalism provides a very clear method for the determi nation of the mode characteristics of vibrations with un known polarizations provided an appropriate form factor is known For isolated molecules such as CO at low coverages the hard hemispherical potential leads to a form factor that depends on the parallel momentum transfer according to lAKl 3 for intermediate values 03A 1 SlAKlSZ A71 For low surface temperatures the DebyeWaller terms in Eq 6 tend to unity and the single T mode phonon creation can be described by the II modi ed Bessel function which is proportional to lAKlZ Thus for intermediate parallel mo mentum transfers the fundamental T mode intensity should follow a roughly lAKl 1 dependence For a total scattering angle close to 900 the perpendicular momentum transfer kz varies little with AK and the S mode intensity is conse quently dominated by the formfactor term in Eq 15 lead ing to a lAKl 3 dependence for intermediate lAKl Thus the fundamental S mode peak intensities are expected to show a strong decrease in intensity as lAKl is increased away from specular whereas the T mode dependence is an ticipated to be much weaker Finally we would like to point out that the theoretical model used to analyze the present experimental results is not limited to the scattering of helium from speci c adsorbate systems The ease of application of this approach means that it can be used to predict the interactions of other molecules with different adsorbates for a MaxwellBoltzmann distribu tion of incident energies that are typical for realistic systems ACIGVOWLEDGMENTS We would like to thank I P Toennies for support en couragement and stimulating discussions during the course of this work One of the authors IRM would like to thank the Max Planck Institut fur Stromungsforschung for hospi tality during the course of this work This work was sup ported by the National Science Foundation under Grant No DMR 9726229 and by the Department of Energy under Grant No DEFG0298ER45704 1 S C Saxena and R K Joshi ThermalAccommodation andAd sorption Coe cients of Gases CINDAS Data Series on Material Properties edited by C Y Ho Hemisphere New York 1989 2H Legge J P Toennies and J R Manson J Chem Phys 110 8767 1999 3F Hofmann J R Manson and P Toennies Surf Sci 349 L184 1996 4B F Mason and B R Williams Surf Sci 130 295 1983 5A M Lahee J P Toennies and Ch W611 Surf Sci 177 371 1986 6 Ellis J P Toennies and G Witte J Chem Phys 102 5059 1995 7A Si er B Gurnhalter J Braun A P Graham M F Bertino J P Toennies D Fuhrmann and Ch W611 Phys Rev B 59 5898 1999 8c J Hirschmugl G P Williams F M Hoffmann and Y J Chabal Phys Rev Lett 65 480 1990 9C M Truong J A Rodriguez and D W Goodman Surf Sci Lett 271 L385 1992 10A P Graham F Hofmann J P Toennies G P Williams C J 15541013 MUBING Ll J R MANSON AND ANDREW P GRAHAM Hirschmugl and J Ellis J Chem Phys 108 7825 1998 1A P Graham F Hofmann and J P Toennies J Chem Phys 104 5311 1996 12E Schweizer B N J Persson M T shaus D Hoge and A M Bradshaw Surf Sci 213 49 1989 13H Steininger S Lehwald and H Ibach Surf Sci 123 264 1982 14A P Graham and J P Toennies Europhys Lett 42 449 1998 15 J P Toennies in Surface Phonons edited by W Kress and F W de Wette Springer Series in Surface Sciences Springer 1991 Vol 21 p 111 16B Poe1sema and G Comsa in Scattering of Thermal Energy Atoms from Disordered Surfaces Springer Tracts in Modern Physics Springer Berlin 1989 Vol 115 p 1 17B Poe1sema S T de Zwart and G Comsa Phys Rev Lett 49 578 1982 51 522 1983 18M Bertino J Ellis F Hofmann J P Toennies and J R Man son Phys Rev Lett 73 605 1994 19J Braun A P Graham F Hofmann W Silvestri J P Toennies and G Witte J Chem Phys 105 3258 1996 20A M Lahee J R Manson J P Toennies and Ch W611 Phys Rev Lett 57 471 1986 J Chem Phys 86 7194 1987 PHYSICAL REVIEW B 63 155410 21A P Graham F Hofmann J P Toennies and J R Manson J Chem Phys 105 2093 1996 22F Hofmann J R Manson and J P Toennies J Chem Phys 101 10155 1994 23 L S Rodberg and R M Thaler Quantum Theory of Scattering Academic New York 1967 24J R Manson Phys Rev B 43 6924 1991 25M F Bertino J R Manson and W Silvestri J Chem Phys 108 10 239 1998 6R Brako and D M Newns Phys Rev Lett 48 1859 1982 Surf Sci 123 439 1982 27J R Manson Comput Phys Commun 80 145 1994 28J R Manson Phys Rev B 37 6750 1988 29D Lemoine J Chem Phys 101 4343 1994 3013 H Choi K T Tang and J P Toennies J Chem Phys 107 1631 1997 107 9437 1997 31P M Morse and H Feshbach Methods of Theoretical Physics McGrawHill New York 1953 p 1554 32A Graham F Hofmann and J P Toennies J Chem Phys 104 5311 1996 33H Jonsson J H Weare and A C Levi Phys Rev B 30 2241 1984 15541014

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