Frank O. Goodman

Quantum-mechanical treatment of the one-phonon inelastic scattering of gas atoms in three dimensions by a simplified continuum model of a solid

Abstract A quantum-mechanical treatment of the one-phonon inelastic scattering of gas atoms in three dimensions by a solid surface is presented. Three major approximations made in the calculations are (i) the method of Cabrera, Celli, Goodman and Manson [Surface Science 19 (1970) 67] is used to calculate the relevant transition probabilities, (ii) atoms near the surface of the solid are treated at first as though they were in the bulk, although the fact that surface atoms differ dynamically from bulk atoms is taken into account later in a simple way, and (iii) the gas-surface interaction is represented by a potential which consists of an infinite repulsive step with zero well-depth, thus eliminating the possibility of bound states of gas atoms at the surface. The theory is a generalization to three dimensions of the one-dimensional theory reviewed by F.O. Goodman [Surface Science 24 (1971) 667]. Some of the problems associated with comparing any such theory with experiment are discussed in some detail, and, for the system HeAg, comparisons are made with experimental scattering distributions and speed measurements of the type reported by R. B. Subbarao and D. R. Miller [J. Chem. Phys. 51 (1969) 4679]. With a (bulk) Debye temperature of 226 K for Ag, satisfying qualitative agreement between theory and experiment is found.

Quantum theory of elastic and inelastic scattering of atoms by solid surfaces

The theory of scattering of atoms by solid surfaces is extended in the following three areas: (1) elastic scattering (atom diffraction) in the absence of inelastic scattering, (2) inelastic scattering (atom‐phonon scattering) in the absence of diffraction, and (3) a combination of diffraction and inelastic scattering. The elastic scattering theory is developed to the point where it is possible to interpret experimental results on up to 60 diffracted beams; the starting point is the theory of Cabrera, Celli, Goodman, and Manson [Surf. Sci. 19, 67 (1970)]. The results are applied to experimental data on the helium‐lithium fluoride system. The inelastic scattering theory is extended to a consideration of phonon scattering about the diffracted beams; the starting point is the theory cited above and that of Goodman [Surf. Sci. 30, 1 (1972)]. The results are applied to experimental data on the helium‐silver and helium‐lithium fluoride systems.

Scattering of atoms by a stationary sinusoidal hard wall: Rigorous treatment in (n+1) dimensions and comparison with the Rayleigh method

A rigorous treatment of the scattering of atoms by a stationary sinusoidal hard wall in (n+1) dimensions is presented, a previous treatment by Masel, Merrill, and Miller for n=1 being contained as a special case. Numerical comparisons are made with the GR method of Garcia, which incorporates the Rayleigh hypothesis. Advantages and disadvantages of both methods are discussed, and it is concluded that the Rayleigh GR method, if handled properly, will probably work satisfactorily in physically realistic cases.

Simple model for the velocity distribution of molecules desorbed from surfaces following recombination of atoms

Abstract A simple model for the velocity distribution of molecules desorbed from surfaces following recombination of the constituent atoms is presented in order to interpret some recent experimental measurements of Dabiri, Lee and Stickney [Surface Sci. 26 (1971) 522] on the desorption of H2 and D2 molecules from both clean and contaminated Ni and Cu surfaces. From some of these surfaces, the desorption did not follow the cosine (cosθ) law, but followed instead a law of the form cospθ where p > 1 ; further, the average energy of the desorbed molecules was greater than that expected for desorption in equilibrium with the surface. The interpretation is made from a macroscopic point of view, using macroscopic concepts such as energy accommodation coefficients.

One‐phonon scattering of atoms in three dimensions by a simplified continuum model of a surface: Thermal desorption

A model of thermal desorption of atoms from surfaces is presented. The treatment is quantum mechanical, and calculations are made using the first‐order distorted‐wave Born approximation, within the one‐phonon approximation. The atom–surface interaction potential is a Morse potential, and the modal properties of the surface are those of a bulk continuum, the work being a natural continuation of some earlier atom–surface scattering work of Goodman. Our conclusions are inconsistent with some of those of Bendow and Ying and Ying and Bendow, who reported pre‐exponential factors τ0 of order 10−5 s in the standard Frenkel formula for desorption; our values of τ0 are more in accord with the previously accepted values, of order 10−12 s. We apply the model to the astrophysical system H–C, and to interpretation of some experimental data on the systems He–Ar, He–Kr, and He–Xe; we find satisfactory agreement with experiment.

Scattering of atoms and molecules by solid surfaces

Abstract The field of scattering of atoms and molecules by solid surfaces has been reviewed more than a dozen times during this decade; among the earliest such reviews are three1–3 by the author. A good general picture of the field as it stood about 2 years ago may be obtained from References 1 to 8; the review in Reference 9 is a subset of Reference 6. The paper by Barantsev10 is particularly useful for “scattering function modeling,” as is Wehtbergs,11 for listing scattering theories and for discussing experimental aspects of lobular scatter ing. Beeby12 comments on the Debye-Waller and buk-surface separation problems. A rather specialized, but fairly complete, review of the energy accommodation coefficient (EAC) appears in Reference 13. Finally, if only for purposes of advertising, the recent monograph14 by Goodman and Wachman should be mentioned.

Determination of characteristic surface vibration temperatures by molecular beam scattering: Application to specular scattering in the HLiF (001) system

A theory of the determination of characteristic surface vibration temperatures (so-called “surface Debye temperatures”) by molecular beam scattering is outlined. The basis of the theory is the reduction in the intensities of the elastically-scattered beams which is due to the thermal motions of the surface atoms (the “Debye-Waller effect”). A combination of the theory of this effect with the diffraction theory of Cabrera, Celli, Goodman and Manson [Surface Sci. 19 (1970) 67] and of Goodman and Tan [J. Chem. Phys. 59 (1973) 1805] allows calculation of the elastic intensities as functions of the experimental parameters [in particular, as functions of the ratio of (a) the surface temperature and (b) the characteristic surface vibration temperature]. The theory is applied to some experimental data of Hoinkes, Nahr and Wilsch [Surface Sci. 33 (1972) 516] on specular scattering in the HLiF (001) system. It is concluded that a great deal of information about the experimental system, apart from the characteristic surface vibration temperature, is contained in such experimental data.

Summation of Pairwise Potentials in Gas‐Surface Interaction Calculations

A model is presented from which the over‐all gas‐surface interaction may be calculated, in simple closed form, from direct summation of microscopic, pairwise, gas atom‐solid atom interactions. A formula is derived for the Fourier components of the gas‐surface interaction potential which results from summation of the general pairwise potential over all lattice sites; for the special case of the Yukawa pairwise potential, the results are exponential functions of the distance of a gas atom from the solid surface. In this way, empirical over‐all gas‐surface interaction laws of the exponential type are related to more basic, but still empirical, pairwise interactions. The system He‐LiF is considered as an illustration of the work.

On the use of a Debye-Waller-type factor in gas-surface scattering theory

Abstract The status of the use of a Debye-Waller-type factor, in the theory of gas-surface inelastic (phonon) scattering to describe the attenuation of diffracted beams, is considered, and several conclusions are made; for example, it is concluded that the commonly-used Debye-Waller-type relation, which in the literature seems to be generally accepted as established, has no firm basis. In the context of such a relation, the role of the gas atom-solid interaction potential well is discussed; it is concluded that methods generally used in the literature for taking account of the presence of this well are arbitrary. No suggestions are made as to how the present situation may be improved.

One‐Phonon Inelastic Scattering of Gas Atoms in Three Dimensions by a Simplified Continuum Model of a Solid: Calculation of Energy Accommodation Coefficients

The quantum‐mechanical theory of the one‐phonon inelastic scattering of gas atoms in three dimensions by a solid surface, presented recently by F. O. Goodman [Surface Sci. 30, 1 (1972)] is used to calculate energy accommodation coefficients for the system He‐W. The approximations used are the same as those in the above paper, except that the gas‐surface interaction is represented by the exponential repulsive potential with zero well‐depth. The theory is compared with experimental data on both equilibrium and nonequilibrium energy accommodation coefficients for He‐W. With a gas‐surface interaction potential inverse‐length parameter corresponding to a Morse parameter for He‐W in the range 1.3–1.6 A−1, and with a (bulk) Debye temperature for W in the range 320–380°K quantitative agreement is found between theory and experiment in regimes in which the theory may be expected to be applicable. These regimes are those in which effects of bound states of gas atoms at the surface, which are neglected in the theory...