### Abstract

We present the physical principles underlying small angle multiple scattering of ions (MS) along with a renewed and comprehensive analytical approach to MS, based on probability theory, more precisely on stochastic processes. New theoretical results are derived, bearing in particular on the combination of angular and lateral spread. The scattering of ions by the screened target nuclei is governed by cross sections decreasing slowly with angle: large deflections may occur with probabilities high enough to render the basic characteristics of MS radically different from energy loss processes. These large deflections induce behaviour that may at first appear paradoxical. The width of the angular distributions presents a power law type dependence on thickness t of matter crossed, far from the familiar t^{1/2} behaviour: it varies as t^{1/ν}, where the exponent ν increases from ≈0.4 for small t, but does not exceed ≈1.8 for large t. Mathematical concepts such as Lévy flights and fractals are briefly discussed for a deeper insight into the nature of MS. The paper is intended to be self-contained, starting from first principles to present the basic elements of the physical and theoretical concepts required to describe MS processes. Projected angular distributions and the corresponding probability densities of the lateral spread of the trajectories with respect to the initial axis are calculated theoretically and numerically for a large range of thicknesses, as well as the statistical dependence between angular and lateral spread and the linear combination of their effects. The cases of both mono- and multielemental media, as well as that of thick targets are examined and the validity of the theory for projectiles heavier than the atoms of the medium and for ions with very high energies is discussed. Typical applications of MS theory are described, with particular emphasis on depth profiling of elements or isotopes in ion beam analysis. A large number of numerical data, including spatial and projected angular distribution with lateral spread calculations, are presented and are available electronically.

Original language | English |
---|---|

Pages (from-to) | 325-388 |

Number of pages | 64 |

Journal | Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms |

Volume | 201 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2003 |

### Fingerprint

### Keywords

- Angular distribution
- Ion beam analysis
- Lateral spread
- Multiple scattering
- Scattering cross section
- Screening radius
- Stochastic process

### ASJC Scopus subject areas

- Surfaces, Coatings and Films
- Instrumentation
- Surfaces and Interfaces

### Cite this

*Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms*,

*201*(2), 325-388. https://doi.org/10.1016/S0168-583X(02)01740-8

**Small angle multiple scattering of fast ions, physics, stochastic theory and numerical calculations.** / Amsel, G.; Battistig, G.; L'Hoir, A.

Research output: Contribution to journal › Article

*Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms*, vol. 201, no. 2, pp. 325-388. https://doi.org/10.1016/S0168-583X(02)01740-8

}

TY - JOUR

T1 - Small angle multiple scattering of fast ions, physics, stochastic theory and numerical calculations

AU - Amsel, G.

AU - Battistig, G.

AU - L'Hoir, A.

PY - 2003/2

Y1 - 2003/2

N2 - We present the physical principles underlying small angle multiple scattering of ions (MS) along with a renewed and comprehensive analytical approach to MS, based on probability theory, more precisely on stochastic processes. New theoretical results are derived, bearing in particular on the combination of angular and lateral spread. The scattering of ions by the screened target nuclei is governed by cross sections decreasing slowly with angle: large deflections may occur with probabilities high enough to render the basic characteristics of MS radically different from energy loss processes. These large deflections induce behaviour that may at first appear paradoxical. The width of the angular distributions presents a power law type dependence on thickness t of matter crossed, far from the familiar t1/2 behaviour: it varies as t1/ν, where the exponent ν increases from ≈0.4 for small t, but does not exceed ≈1.8 for large t. Mathematical concepts such as Lévy flights and fractals are briefly discussed for a deeper insight into the nature of MS. The paper is intended to be self-contained, starting from first principles to present the basic elements of the physical and theoretical concepts required to describe MS processes. Projected angular distributions and the corresponding probability densities of the lateral spread of the trajectories with respect to the initial axis are calculated theoretically and numerically for a large range of thicknesses, as well as the statistical dependence between angular and lateral spread and the linear combination of their effects. The cases of both mono- and multielemental media, as well as that of thick targets are examined and the validity of the theory for projectiles heavier than the atoms of the medium and for ions with very high energies is discussed. Typical applications of MS theory are described, with particular emphasis on depth profiling of elements or isotopes in ion beam analysis. A large number of numerical data, including spatial and projected angular distribution with lateral spread calculations, are presented and are available electronically.

AB - We present the physical principles underlying small angle multiple scattering of ions (MS) along with a renewed and comprehensive analytical approach to MS, based on probability theory, more precisely on stochastic processes. New theoretical results are derived, bearing in particular on the combination of angular and lateral spread. The scattering of ions by the screened target nuclei is governed by cross sections decreasing slowly with angle: large deflections may occur with probabilities high enough to render the basic characteristics of MS radically different from energy loss processes. These large deflections induce behaviour that may at first appear paradoxical. The width of the angular distributions presents a power law type dependence on thickness t of matter crossed, far from the familiar t1/2 behaviour: it varies as t1/ν, where the exponent ν increases from ≈0.4 for small t, but does not exceed ≈1.8 for large t. Mathematical concepts such as Lévy flights and fractals are briefly discussed for a deeper insight into the nature of MS. The paper is intended to be self-contained, starting from first principles to present the basic elements of the physical and theoretical concepts required to describe MS processes. Projected angular distributions and the corresponding probability densities of the lateral spread of the trajectories with respect to the initial axis are calculated theoretically and numerically for a large range of thicknesses, as well as the statistical dependence between angular and lateral spread and the linear combination of their effects. The cases of both mono- and multielemental media, as well as that of thick targets are examined and the validity of the theory for projectiles heavier than the atoms of the medium and for ions with very high energies is discussed. Typical applications of MS theory are described, with particular emphasis on depth profiling of elements or isotopes in ion beam analysis. A large number of numerical data, including spatial and projected angular distribution with lateral spread calculations, are presented and are available electronically.

KW - Angular distribution

KW - Ion beam analysis

KW - Lateral spread

KW - Multiple scattering

KW - Scattering cross section

KW - Screening radius

KW - Stochastic process

UR - http://www.scopus.com/inward/record.url?scp=0037320961&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037320961&partnerID=8YFLogxK

U2 - 10.1016/S0168-583X(02)01740-8

DO - 10.1016/S0168-583X(02)01740-8

M3 - Article

VL - 201

SP - 325

EP - 388

JO - Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms

JF - Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms

SN - 0168-583X

IS - 2

ER -