^{1,a)}and Salvador Miret-Artés

^{2}

### Abstract

A classical Wigner in-plane atom surfacescattering perturbation theory within the generalized Langevin equation formalism is proposed and discussed with applications to the Ar–Ag(111) system. The theory generalizes the well-known formula of Brako as well as the “washboard model.” Explicit expressions are derived for the joint angular and final momentum distributions, joint final energy, and angular distributions as well as average energy losses to the surface. The theory provides insight into the intertwining between the energy loss and angular dependence of the scattering. At low energies the energy loss in the horizontal direction is expected to be large, leading to a shift of the maximum of the angular distribution to subspecular angles, while at high energies the energy loss in the vertical direction dominates, leading to a superspecular maximum in the angular distribution. The same effect underlies the negative slope of the average final (relative) energy versus scattering angle at low energies which becomes positive at high energies. The theory also predicts that the full width at half maximum of the angular distribution varies as the square root of the temperature. We show how the theory provides insight into the experimental results for scattering of Ar from the Ag(111) surface.

We thank Dr. J. Moix and Dr. S. Sengupta for fruitful discussions. We gratefully acknowledge support of this work by a grant of the Israel Science Foundation. S.M.-A. would like to thank the Ministry of Science and Innovation of Spain for a project with Reference No. FIS2007-62006.

I. INTRODUCTION

II. THEORETICAL FRAMEWORK

A. The model Hamiltonian

B. The distribution of final momenta

C. Classical perturbation theory

1. The change in the final momenta

2. The angular distribution

3. Some interesting limits

4. The joint angle and energy distribution and the angle dependent final average energy

III. CLASSICAL WIGNER THEORY FOR THE SCATTERING OF Ar FROM Ag(111)

IV. DISCUSSION

### Key Topics

- Collisional energy loss
- 46.0
- Angular distribution
- 37.0
- Phonons
- 22.0
- Scattering theory
- 18.0
- Surface scattering
- 15.0

## Figures

Angular dependence of the relative final energy. The lines are for incidence energies of 210, 310, 480, 1060, 1580, and 2560 meV. The highest incidence energy is the lowest line at angles below the specular angle (40°), the other energies increase monotonically with decreasing incidence energy (at angles below specular). Note the change of slope at the incidence energy of 1060 meV.

Angular dependence of the relative final energy. The lines are for incidence energies of 210, 310, 480, 1060, 1580, and 2560 meV. The highest incidence energy is the lowest line at angles below the specular angle (40°), the other energies increase monotonically with decreasing incidence energy (at angles below specular). Note the change of slope at the incidence energy of 1060 meV.

Incidence energy dependence of the scattered angular distributions. The incidence energies are as in Fig. 1, the narrowest plot is for the highest energy, the width increases monotonically as the energy is decreased. The surface temperature is .

Incidence energy dependence of the scattered angular distributions. The incidence energies are as in Fig. 1, the narrowest plot is for the highest energy, the width increases monotonically as the energy is decreased. The surface temperature is .

Surface temperature dependence of the scattered angular distributions at an incidence energy of 1060 meV. The surface temperatures are 330, 500, 600, and 800 K. The broadest distribution is at the highest temperature and the width decreases with decreasing temperature.

Surface temperature dependence of the scattered angular distributions at an incidence energy of 1060 meV. The surface temperatures are 330, 500, 600, and 800 K. The broadest distribution is at the highest temperature and the width decreases with decreasing temperature.

Experimental square root dependence of the FWHM (in degrees) of the angular distribution on the surface temperature. The experimental widths are adapted from Ref. 12 and are plotted vs. the square root of the temperature. Note that they extrapolate nicely to 0 at while if they are plotted on a linear scale as in Ref. 12 they extrapolate to a large constant (11°) which is unphysical.

Experimental square root dependence of the FWHM (in degrees) of the angular distribution on the surface temperature. The experimental widths are adapted from Ref. 12 and are plotted vs. the square root of the temperature. Note that they extrapolate nicely to 0 at while if they are plotted on a linear scale as in Ref. 12 they extrapolate to a large constant (11°) which is unphysical.

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