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Properties of atom scattering from percolation lattices: He scattering from substitutionally disordered Xe+Kr monolayers
Abstract
He scattering from substitutionally disordered Xe+Kr monolayers is studied theoretically. The angular distribution of the scattered He, and lifetimes of scattering resonances at the surface, are calculated for different Xe:kr mixing ratios. In addition to Bragg diffraction, new intensity maxima are found. These are interpreted as rainbow effects (disorder rainbows), and may be very useful in characterizing disordered surfaces. Also, trapping resonances are found, which are sharply sensitive to the percolation thresholds for the mixtures. The results show He scattering can be a powerful tool in studying 2D percolation behavior.
Atomic beam techniques are presently being used in many branches of surface physics such as studies of the particle-surface physisorption potential, surface structure, surface phonons, nucleation and growth on metal and insulator surfaces, surface diffusion and accommodation and sticking of molecules. This review concentrates on diffractive phenomena from surfaces, which up to now were investigated mainly with helium. The theoretical background for diffraction calculations is outlined and representative examples of different applications are given. The main subjects covered are: structural determinations of chemisorbed and physisorbed systems, investigations of disordered surfaces, selective adsorption resonances, diffusion and nucleation studies and investigations of growth and phase transitions on surfaces. Diffraction results obtained with Ne, Ar, and are also summarized.
The diffraction of thermal He atoms from mixed Xe+Kr monolayers on Pt(111) was measured, and the results were compared with theoretical studies of these systems. The results shed light on the structural properties of these disordered systems, and on their relation to the He diffraction intensities. Experimentally, the specular (0,0), the (1,0), and the (2,0) Bragg peak intensities were measured for monolayers of different Kr:Xe concentration ratios. The theoretical calculations included Monte Carlo simulations of the mixed disordered monolayers, and quantum calculations in the Sudden approximation of the scattering intensities from the simulated disordered structures. The following main results were obtained: (1) Both experiment and the Monte Carlo simulations suggest that the mixed Xe+Kr monolayers are periodic for all Xe:Kr concentration ratios, the lattice constant varies linearly with the Xe:Kr ratio. The domain size of the 2D crystals, from experiment and theory, is found to be larger than 100 Å. (2) The Monte Carlo simulations suggest that the Xe+Kr monolayers form an almost ideal substitutionally disordered lattice. (3) Using a semiempirical Debye–Waller factor, reasonable agreement is found between the theoretical and the measured diffraction intensities, thus supporting the calculated structural model for the disordered surface. (4) The theoretical scattering calculations show that in addition to the diffraction peaks, there are also intensity maxima at non‐Bragg positions. These are entirely due to the lattice disorder, and are identified as a recently found new type of Rainbow effect that can furnish important information on disordered surfaces. The results demonstrate the power of He scattering as a tool for exploring substitutionally disordered surfaces.
Molecular beam scattering experiments have emerged in recent years as an important tool for investigating structural disorder on surfaces. This paper gives a brief overview of the developing theory of scattering from disordered surfaces, which is essential for the interpretation of such experiments, and presents new results on this topic. A brief discussion is given of the methods developed for calculating atom scattering disordered systems (e.g. time-dependent quantum wavepackets; distorted-wave approximation; sudden approximation) and of simulations of disordered surface structures (by Monte Carlo or molecular dynamics). The paper focuses, however, on features of the scattering intensities and their relation to disordered surface structures. The main topics include: (1) The concept of cross section for He scattering by an isolated defect on a surface, and its application to the study of interactions between defects. Comsa and his coworkers developed this into a most powerful tool for studying surface defects and defect interactions. We examine the assumptions involved, and the information contained in the measured cross section on defect shape and geometry, and on He/defect potentials. (2) Angular intensity distribution for He scattering from isolated defects; defect rainbow and Fraunhofer effects. (3) He scattering from periodic, substitutionally disordered surfaces. New results are presented on He scattering from mixed Xe + Kr monolayers on Pt(111), comparing theory with the experiments of Comsa et al. The results indicate that these monolayers are indeed periodic and (almost) perfectly substitutionally disordered for all Xe: Kr ratios. Rainbow intensity features due to the substitutional disorder are calculated and analyzed. (4) He scattering from amorphous mixed monolayers, with application to Xe + Ar mixtures on Pt(111). (5) Atom scattering from liquid surfaces, e.g. structure and dynamics of liquid Hg probed by Ar scattering.The paper concludes by emphasizing some of the outstanding open problems in the field.
Successive adsorption of submonolayer amounts of Kr and Ar onto a Pt(111) surface leads to the formation of separate 2D islands of almost pure Kr and Ar content. Upon annealing, the phase separated Kr and Ar start to mix resulting eventually in a complete stochastic mixture of the two components at about Tmix = 30–36 K, depending on the Ar:Kr mixing ratio. This continuous mixing transition proceeds primarily via the dissolution of Kr atoms into the Ar 2D islands and not vice versa, as might be expected for kinetic reasons. Instead, the intermixing behaviour can be explained by the fact that the binding between an Ar and a Kr atom is stronger than between two Ar atoms but weaker than between two Kr atoms. As a consequence the incorporation of a Kr atom from the edge of an Ar island into its center is energetically favored while the dissolution of Ar into Kr is hindered.
Krypton and xenon co-adsorbed on the Pt(111) surface at 40 K are found to form homogenous stochastic mixtures for all composition ratios and in the entire coverage regime below one monolayer. These quasi two-dimensional Kr1−χ−Xeχalloys (0⩽χ⩽ 1) are characterized by narrow diffraction peaks indicating a high degree of positional as well as orientational order. The position of the diffraction peaks of the mixture varies linearly as a function of the molar fraction χ between those of the pure Kr and Xe components (Vegard's law). The half-widths of the diffraction peaks are larger than for the pure Kr system but even at χ = 0.5 quasi crystalline order still extends over a length scale of about 100 Å. The diffraction line-shapes are well described within a simple model of lattice disorder.
We review the projected continuum approach to quantum scattering from disordered surfaces, placing it in the context of previous time-independent and time-dependent approaches. The method provides a one K-vector expansion of the scattering with a coherent field potential average, which leads to a unitary set of equations describing specular and non-specular scattering. Of particular significance is the capability to describe sticking due to surface disorder, which increases at low collision energies. Application to a model partial monolayer of molecular adsorbate is made, illustrating the efficiency and usefulness of the approach for study of finite coverage, disordered adsorbate systems.
We study multiple-spin coherence transfers in linear Ising spin chains with
nearest neighbor couplings. These constitute a model for efficient information
transfers in future quantum computing devices and for many multi-dimensional
experiments for the assignment of complex spectra in nuclear magnetic resonance
spectroscopy. We complement prior analytic techniques for multiple-spin
coherence transfers with a systematic numerical study where we obtain strong
evidence that a certain analytically-motivated family of restricted controls is
sufficient for time-optimality. In the case of a linear three-spin system,
additional evidence suggests that prior analytic pulse sequences using this
family of restricted controls are time-optimal even for arbitrary local
controls. In addition, we compare the pulse sequences for linear Ising spin
chains to pulse sequences for more realistic spin systems with additional
long-range couplings between non-adjacent spins. We experimentally implement
the derived pulse sequences in three and four spin systems and demonstrate that
they are applicable in realistic settings under relaxation and experimental
imperfections-in particular-by deriving broadband pulse sequences which are
robust with respect to frequency offsets.
The Sudden Approximation (SA) for scattering of atoms from surfaces is
generalized to allow for double collision events and scattering from
time-dependent quantum liquid surfaces. The resulting new schemes retain the
simplicity of the original SA, while requiring little extra computational
effort. The results suggest that inert atom (and in particular He) scattering
can be used profitably to study hitherto unexplored forms of complex surface
disorder.
Large-angle-diffraction oscillations in the intensity of helium atoms
elastically scattered from randomly stepped Pt(111) are reported. A
formalism based on the concept of reflection-symmetry interference
explains the observed features and the data analysis yields a value of
4.4-4.6 Å for the hard-core repulsive cross section of a single
step. The observed oscillations are sensitive to the presence of
adsorbates at step sites.
Large-angle-diffraction oscillations in the intensity of helium atoms elastically scattered from randomly stepped Pt(111) are reported. A formalism based on the concept of reflection-symmetry interference explains the observed features and the data analysis yields a value of 4.4-4.6 Å for the hard-core repulsive cross section of a single step. The observed oscillations are sensitive to the presence of adsorbates at step sites.
Exact quantum-mechanical calculations are present for He scattering from one-dimensional models of disordered, mixed Xe + Ar overlayers. A time-dependent wavepacket approach is used with a recent technique for solving the Schrödinger equation. Results are given for several overlayers of different Xe : Ar concentration ratios. The dependence of scattering intensities on the disordered structures is discussed. The results provide a reference for testing approximations for scattering from disordered surfaces.
This paper is the first in a series covering our experiments examining the physical properties of ordered overlayers of Ar, Kr, and Xe physisorbed on Ag(111). In this paper, we examine the desorption kinetics of the Xe monolayer/Ag(111) system. Desorption is zeroth order until approx.90% of the monolayer has desorbed, then becomes first order. We also measured the inelastic scattering of an 18 meV He beam from unconstrained monolayers of (111) oriented Ar, Kr, and Xe. The transitions are mapped across the entire surface Brillouin zone from Gamma-bar to M-bar. The data are dispersionless, indicating that for the measured mode the adatoms are behaving as independent Einstein oscillators. Parametrized physisorption potentials for RG--Ag(111) (RG = Ar, Kr, Xe) are constructed using these results. Inelastic scattering probabilities and linewidths are also presented. The inelastic scattering probabilities vary by at most a factor of 3 across the entire surface Brillouin zone, and are reported as a function of incident angle, final wave vector, and surface temperature. Variations in the inelastic scattering probabilities are indicative of dynamical adatom--substrate coupling. Experimental techniques which turn these dynamical couplings on or off for the same phonon energy are discussed. Limited results for clean Ag(111) are also presented. It is hoped that these measurements, on such ideal systems as ordered rare gas monolayers, will provide further impetus for developing improved theoretical treatments of inelastic single phonon scattering.
The van der Waals interaction between an atom and an adsorbate consists of the usual two-body direct interaction as well as three-body interactions from the image force and an interference effect. The three-body interactions are caused by the presence of the surface and render the resulting potential anisotropic even when the adsorbate is spherically symmetrical. Expressions for the leading coefficients of the van der Waals forces and the corresponding reference-plane positions are given in terms of the polarizabilities of the atom, the adsorbate and the substrate. A theory of atomic scattering from a single adsorbate is given and numerical results for the total cross-sections of the He–Xe/Ag system are presented.
The recent application of the scattering of thermal He beams to the study of disordered adsorbates is featured by both an extreme sensitivity (coverages down to θ≈ 0.001 are accessible) and non-destructiveness. The sensitivity is a consequence of the very large cross-section (ca. 100 Å) for diffuse scattering exhibited by adsorbates. The cross-section overlap is substantial even at low coverages and the degree of overlap depends on the nature of the adsorbate distribution along the surface, i.e. on the mutual interaction of the adsorbates. Thus information on the latter can be obtained from the former; this is the overlap approach. A recent development has shown that this is also true if ‘monovacancy’ is substituted for ‘adsorbate’.
The scattering of an atomic beam from a randomly stepped surface has been calculated using the hard corrugated wall model. Under the basic assumption that scattering from the step edges may be neglected the scattering equation can be solved without any further approximation. The solution displays the usual diffraction peaks each of them is broaden by a term characterizing the step configuration. If the step distribution is ergodic and stationnary in space this term is the characteristic function of the difference of level between two point of the surface considered as a random variable. Statistical models for the step repartition at the surface are proposed and the scattering intensity is derived in a closed form. The main result is that the broadening of the peaks varies from no broadening to a maximum according as the interferences from waves reemitted by the various terraces are constructive or destructive. A comparison is made with previous experimental data from which an estimation of the average step separation can be drawn. The sensitivity of the atomic beam scattering to steps is found to be more than one step every one hundred crystal atoms.
The scattering cross section of an adsorbed CO molecule is about 1 order of magnitude larger than expected from the van der Waals radius. The dependences on the He and H2 incident velocities reveal glory structures similar to those in corresponding gas-phase experiments. Accordingly the long-standing puzzle of the extreme sensitivity of molecular beams to adsorbates is explained by inhomogeneities of the attractive potential.
The capability of a He-scattering experiment to detect the migration of adsorbed molecules on monocrystal surfaces is demonstrated for the first time. CO migration and island formation can be investigated at very low coverages because of the very large cross section for He scattering of CO adsorbed on Pt(111). The activation energy for CO migration on Pt(111) is found to be 7 kcal/mole. No island formation at low CO coverages on a defect-free Pt(111) surface was detected in the temperature range 120-400 K.
A theory for light-atom scattering by low-coverage adsorbates on metal surfaces is presented. A model including attractive adatom potentials successfully explains data for CO and Xe adsorption. The radius of the repulsive potential for CO is found to increase by 10-20% upon adsorption. It is shown theoretically that random adsorption and island formation can be distinguished, all data for CO being consistent with the former. Xe data show both mechanisms, depending on coverage.
Excellent agreement is found between measured and calculated H-atom scattering by (strongly corrugated) Xe overlayers on graphite. Elastic scattering is calculated exactly using a surface potential generated by summing gas-phase H-Xe potentials. The diffraction peaks are carefully simulated by convolution over the velocity distribution, correcting for inelasticity and geometric effects, and including an inelastic background generated by a new method. Thresholds are found to affect strongly the resonance data, and the behavior of resonances at some crossings is different from that seen in weakly corrugated systems.
The three-dimensional simple-cubic spin-½ axial next-nearest-neighbor Ising model is studied by means of Kikuchi's cluster-variation method employing a new technique described previously [J. S. Desjardins and O. Steinsvoll, Phys. Scr. 28 565 (1983) for the solution of the general equations of equilibrium. The particular solution employed in this paper is equivalent to Bethe's first approximation and yields a surprisingly rich phase diagram with modulated structures appearing up to a repeat distance of 15 planes (the highest studied). The phase diagram obtained by our technique resembles closely the mean-field, spin-½ phase diagram of von Boehm and Bak with some significant differences: The second-order boundary of the paramagnetic region is at a significantly lower temperature with a minimum at |κ|∼0.4, where κ is the ratio of the antiferromagnetic to the ferromagnetic coupling constant. In addition, we are able to follow the temperature dependence of the shape of the modulated solutions from the squared-off, low-temperature behavior of Selke and Fisher to the sinusoidal behavior of the high-temperature, mean-field results. The position of our Lifshitz point is in good agreement with previous results, as is the conclusion that transitions between phases are of first order. By contrast, in two dimensions the same approximation completely fails to reproduce reported features of the phase diagram.
The scattering of He atoms from a CO molecule adsorbed on a Pt surface is studied theoretically by methods that include: (1) Numerically exact solutions of the time‐dependent Schrödinger equation for the scattered wavepacket; (2) The sudden approximation; (3) Classical trajectories. The methods are used to obtain detailed insight into the collision dynamics, and to predict and understand interesting features in the angular intensity distribution of the scattered atoms. The analysis and interpretation of the exact quantum results is facilitated by calculations of the probability current density of the scattered particles. Some of the main results are: (i) The angular intensity distribution exhibits nonspecular maxima of two types: Several of the peaks are rainbow effects induced by the adsorbate, while others (at angles nearer to the specular) are Fraunhofer diffraction interferences. Both types of peaks contain useful, largely complementary, information on adsorbate geometry and on the He/adsorbate interaction. (ii) The angular intensity distribution is quantitatively sensitive to the adsorbate distance from the surface, suggesting possible determination of that distance from experimental data. (iii) The corrugation due to the adsorbate leads to scattering resonances associated with temporary trapping of the scattered atom at the defect site. This is a new effect of potential importance for experimental studies of atom/defect interactions. The results obtained here suggest that He scattering from isolated adsorbates exhibits distinct, substantial effects, measurement of which should yield very useful data on the adsorbates and on their interactions with gas‐phase atoms.
The angular intensity distribution for atom scattering from surfaces with isolated defects is calculated. For a two‐dimensional model time‐dependent wave packet calculations are carried out for potentials with different steepness and variable defect size and height. The influence of the potential softness is studied by comparison with results from a corresponding hard wall model. Within the hard wall model the Eikonal approximation (EA) and an improved Eikonal approximation (IEA) which accounts for double collision effects are tested against numerically exact results. While, in general, the EA works well only in the vicinity of the specular peak the IEA is quite accurate over the entire angular range. Finally, wave packet calculations are carried out for a model of a flat surface with isolated steps, fitted to represent a stepped Pt(111) surface. The results are compared to recent measurements of the incoherent elastic diffraction for this system. Basic measured features like a rainbow maximum and large angle interference oscillations are well reproduced by the calculations.
It was recently proposed that the sudden approximation should be a powerful tool for the calculation of the angular intensity distribution in high‐energy atom scattering from disordered surfaces. In the present study the sudden approximation is applied to scattering from one‐ and two‐dimensional models of: (1) Isolated adsorbed impurities on crystalline surfaces (Ar on Cu); (2) Mixed overlayers on an underlying surface (Xe+Ar mixtures on a smooth surface). The results are tested against numerically exact quantum‐mechanical wave packet calculations. Except for very low collision energies, the sudden approximation gives results of excellent quantitative accuracy for both types of noncrystalline surfaces. At low energies, several features of the intensity distribution are not produced correctly by the sudden: These are found to be due mainly to double collision effects. The accuracy and validity range of the method are discussed in the light of the results obtained in the test calculations.
This is the second of three papers discussing our studies of overlayers of Ar, Kr, and Xe physisorbed on Ag(111). All of these rare gases form ordered structures, which are azimuthally aligned but translationally incommensurate with the Ag substrate. In this paper, we discuss experiments in which we utilized the angle resolved time of flight of inelastically scattered 18 MeV He atoms to examine the surface dynamics of multilayer rare gas films along the Γ– direction. This was done on a layer‐by‐layer basis for 2, 3, and ≥20 layers for each of the rare gases. Unlike the monolayers, the vibrational modes observed for the multilayers show dispersion across the surface Brillouin zone, the amount of dispersion increasing with the number of adsorbed layers. These results reveal in detail how the surface dynamical properties of a thin film evolve towards those of a thick crystal as a function of increasing dimension. Lattice dynamics calculations, which utilize realistic gas phase pair potentials, reproduce the experimentally observed phonon dispersion relations quite well. We also examine the inelastic scattering probabilities and linewidths of the transitions. One of the more notable results is that the inelastic scattering probabilities vary by at most a factor of 2–3 across the entire surface Brillouin zone. Isothermal desorption measurements for the Xe overlayers are also discussed. Like the monolayer, the bilayer and trilayer exhibit nearly zeroth order desorption until ∼90% of the top layer has desorbed, where the desorption kinetics become first order.
The angular intensity distribution of atomic beams scattered from partially disordered solid surfaces is investigated. Quantitative relations are established between correlation functions that describe the structural disorder, and several features of the scattered intensity distribution. To simplify the collision dynamics, the study is carried out in the framework of the sudden approximation, which assumes that momentum changes perpendicular to the surface are large compared with momentum transfer due to surface corrugation. Simple relations are obtained between the structural correlation functions and the scattering at near‐specular angles, at extreme off‐specular angles, and for directions close to that at which the surface scattering rainbow appears for the corresponding ordered lattice system. Detailed expressions for the case of step disorder are presented. The results suggest that molecular beam scattering should be a powerful probe of structural properties of disordered surfaces.
A recent method for time-dependent wavepacket scattering calculations is applied to He scattering from a Cu surface with isolated Ar impurities. Several effects are found: (i) broad tails superimposed on each diffraction spike; (ii) shallow impurity rainbow maxima; (iii) impurity—surface interference peaks. A sudden approximation is applied to the interpretation of the newly found features.
Two versions of the sudden approximation are introduced to decouple and solve the equations that describe atom- surface scattering with many open diffraction channels. Both approximations require a high incident beam wave number compared with the magnitude of the reciprocal space vector of the lattice. In this framework, simple explicit expressions are obtained for the observable diffraction intensifies, making calculations feasible even for systems with hundreds of open diffraction channels. Further considerable simplifications ensue when the approximations are specialized to the case of a Lennard-Jones-Devonshire potential, or to that of a weakly corrugated surface. The approximations were applied to the systems He/LiF(001); Ne/LiF(001) and Ne/W(110) and the results are compared with other calculations or with experiment. The sudden approximation is found to be of good accuracy in these cases.
Differential cross sections have been measured for the scattering of helium atoms from isolated CO molecules on a Pt(111) surface. The cross sections reveal an oscillatory structure as a function of scattering angle extending to large momentum transfer on both sides of the specular peak. Using a hard hemisphere model to approximate the interaction potential, the data can be well reproduced by a hard core of radius about 2.5 Å. This result is compared to the charge density profile of the adsorbed molecule and also to the gas phase interaction potential, and in both cases good agreement with the experimentally predicted classical turning points is found.
The use of thermal energy atom scattering (TEAS) as a surface investigation tool is reviewed. Specific properties of thermal He scattering, which are particularly advantageous in surface analysis, are discussed; the main causes which hampered so far the broad application of this technique to the investigation of ordered surfaces are analysed. The recent development of TEAS based on the exploitation of the large cross section for He diffuse scattering of adsorbates and vacancies is reviewed. This development extends the capabilities of TEAS to a straightforward investigation of various aspects of disordered surfaces so far hardly accessible to other methods. Examples concerning the lateral distribution of adsorbates and defects, the mutual attraction and repulsion, 2D-phase transitions, mobility and surface diffusion are given.
A time dependent wavepacket approach is applied to diffractive scattering from surfaces with imperfections. This semiclassical method is based on Gaussian wavefunctions whose average positions and momenta are those of classical trajectories. The approach can be applied to arbitrary potentials. Positions, shapes and magnitudes of the diffraction peaks are obtained in a single calculation. In a first example the wavepacket method is applied to a stepped surface. The diffraction pattern is calculated for different incident scattering angles and for various regular and random distributions of steps with two different sizes. Second, the approach is used to study the scattering for an interaction potential modeling a corrugated surface with adsorbed atoms. The diffraction intensities are examined for several final scattering directions and again for regular and random arrangements of the adsorbed atoms.
A pronounced triangular background, observed near the specular beam with He diffraction from Ni(100)-c(2\ifmmode\times\else\texttimes\fi{}2)O, is explained by hard-wall calculations as originating from a random distribution of {}15% empty oxygen sites in good agreement with Auger calibration. Lateral adatom relaxations of {}0.25 A\r{} away from the empty sites fit both the background shape and intensity. This is the first application of He scattering for defect characterization at high coverages.