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Chaotic scattering of He atoms off a Cu surface with corrugated Morse potential

Florentino Borondo, Ernest Fontich, Pau Martín

Abstract

We consider a Hamiltonian system that models the scattering of helium atoms off a copper surface. The interaction between the He and the Cu atoms is described by a corrugated Morse potential. Using corrugation coefficients values in the potential obtained by fitting to experimental values, we prove that, provided some coefficient of an auxiliary function is different from 0, there are regions of the phase space, corresponding to sufficiently large energy of the incident atom, where the scattering is chaotic. Furthermore, we prove that the system has oscillatory motions.

Chaotic scattering of He atoms off a Cu surface with corrugated Morse potential

Abstract

We consider a Hamiltonian system that models the scattering of helium atoms off a copper surface. The interaction between the He and the Cu atoms is described by a corrugated Morse potential. Using corrugation coefficients values in the potential obtained by fitting to experimental values, we prove that, provided some coefficient of an auxiliary function is different from 0, there are regions of the phase space, corresponding to sufficiently large energy of the incident atom, where the scattering is chaotic. Furthermore, we prove that the system has oscillatory motions.
Paper Structure (41 sections, 56 theorems, 457 equations, 8 figures)

This paper contains 41 sections, 56 theorems, 457 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The He-Cu scattering problem. Main statement
  3. Exponentially small splitting of invariant manifolds
  4. Structure of the paper
  5. Hamiltonian formulation of the He-Cu scattering problem and splitting of separatrices
  6. McGehee-like coordinates and fast dynamics
  7. Dynamics of $H_0$
  8. The Melnikov potential
  9. Dynamics of the full system. Splitting of the invariant manifolds
  10. Chaotic dynamics
  11. Poincaré-Cartan reduction
  12. Local coordinates around $q=p=0$ and a parabolic $\lambda$-lemma
  13. The local and the global maps
  14. Symbolic dynamics. Proof of Theorem \ref{['thm:teoremaprincipalenvariablesoriginals']}
  15. Hamilton-Jacobi equation
  16. ...and 26 more sections

Key Result

Theorem 1.1

There exists a function $f_1$ of the coefficients $r_1,r_2,\dots$ such that if $f_1 \neq 0$, then, for any $h$ large enough, there exists a section $\Sigma \subset \{H_\mathrm{CM} = h\}$ of the vector field associated to $H_\mathrm{CM}$ and a subset $\mathcal{I} \subset \Sigma$ such that the Poincar

Figures (8)

  • Figure 1: The projection of the invariant manifolds $W^0_{\nu I_0}$ onto the $(q,p)$-plane. For each $p = p_0$, $\{ (0,p_0,\theta,0)\mid \theta \in \mathbb{T} \}$ is a periodic orbit.
  • Figure 2: The domain $\mathcal{D}^+_{\kappa,\delta}$ defined in \ref{['def:DominisRaros']}.
  • Figure 3: The domain $\mathcal{D}^+_{\kappa,\delta} \cup \mathcal{D}^+_{\kappa,\mathrm{ext}}$ defined in \ref{['def:DominisRaros_extesos']}.
  • Figure 4: The domain $\widetilde{\mathcal{D}}^+_{\delta}$ defined in \ref{['def:wtDeltamesdelta']}, in gray. Compare with Figure \ref{['fig:DomRaroextes']}.
  • Figure 5: The domain $\mathcal{D}^+_{\kappa,\mathrm{in}}$ defined in \ref{['def:Dominiinner1']}, shaded in gray.
  • ...and 3 more figures

Theorems & Definitions (101)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Theorem 2.5
  • Remark 2.6
  • Corollary 2.7
  • ...and 91 more