Modelling displacement correlations in computer simulations of particle-solid collisions

doi:10.1016/0010-4655(77)90010-8

Computer Physics Communications

Volume 13, Issue 3, September–October 1977, Pages 157-166

https://doi.org/10.1016/0010-4655(77)90010-8 Get rights and content

Abstract

A linear transformation method of modelling equal-time displacement correlations in computer simulations of ion-lattice interactions is presented. Test examples for the one-dimensional atomic string and the three-dimensional lattice cases are discussed with practical considerations for applications to physical systems emphasized.

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The relaxed Cu(0 0 1) surface was analyzed by high-resolution medium energy ion scattering (MEIS) using 120 keV He⁺ ions in a layer-by-layer fashion. We found the top-layer contraction of 1.6% and the 2nd-layer expansion of 0.9%, quite different from the close-packed (1 1 1) surface. Then enhanced thermal vibration amplitudes (TVAs) and the correlations between the 1st and 2nd nearest neighbor atoms in the [1 0 1]- and [0 0 1]-string were also determined simultaneously by MEIS spectrum analysis combined with Monte Carlo simulations of ion trajectories considering the surface relaxation and enhanced TVAs as well as correlations. We obtained the TVAs of the top-layer atoms strongly enhanced by 1.67 ± 0.06 and 1.55 ± 0.04 times that of the bulk (0.085 Å) in the lateral and surface normal direction, respectively and the correlation coefficients of +0.33 ± 0.04 and +0.28 ± 0.04, respectively for Cu atoms in the [1 0 1]- and [0 0 1]-axis oscillating perpendicular to each string. The correlation between the 2nd nearest neighbor atoms in the [1 0 1]-string is small probably less than +0.2. The results obtained here are compared with the other experimental reports and the Debye approximation as well as molecular dynamics simulations using the embedded atom method.
Effect of thermal vibrations and their correlations on grazing scattering of atoms from crystal surfaces
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Citation Excerpt :
The approximation of uncorrelated vibrations is e.g. used in the KALYPSO code [9]. Jackson and Barrett [10,11] have pointed out, however, that incorporation of correlations of lattice vibrations is crucial for a realistic simulation of trajectories in the channeling regime [12]. For the analysis of ion scattering experiments, correlations of thermal vibrations were shown to be important [13–17] as well as negligible [14,15,18,19].
The influence of thermal vibrations of target atoms and their correlations on grazing scattering of keV atoms from crystal surfaces is investigated. Classical 2D and 3D trajectory calculations are performed and thermal vibrations are included using the Debye model as well as its high-temperature limit and uncorrelated thermal vibrations. Ion beam triangulation, rainbow structures for scattering along low-indexed directions of the surface plane, and polar angular distributions for impact along high-indexed directions are studied. We demonstrate that ion beam triangulation can be analyzed using uncorrelated thermal vibrations, while a realistic description of angular distributions requires more detailed modeling. For the peak positions of rainbow structures thermal vibrations play a minor role and they are well described by 2D simulations based on continuum potentials.
Investigation of thermal vibration correlation of [1 1 0] silicon lattice atoms by ion scattering
2005, Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms
When a beam of fast ions hits an atom, a shadow cone is formed behind the atom. By varying the ion energy, the radius of the shadow cone can be changed and also the interaction probability between ions and lattice atoms. At strong vibration correlations, neighboring atoms are located almost inside the shadow cone and this acts to reduce the interaction probability. The Si–KL₂₃L₂₃ Auger electron emission was used to detect the reduction of the interaction between ions and lattice atoms, which is a measure for the vibration correlation. The yield of the Si–KL₂₃L₂₃ Auger electrons was measured as a function of the incidence angle around the [1 1 0] axis with ions. The correlation coefficient was determined by comparison of the minimum Auger electron yields with the results of computer simulations. It has been shown that the ion induced Auger electron emission can be used as a sensitive method for studying the thermal vibration correlation of a surface. The correlation coefficient of the normal displacement of nearest-neighbor silicon atoms along [1 1 0] at room temperature is determined to be 0.90.
Surface relaxation and near-surface atomic displacements in the N/Cu(1 0 0) self-ordered system
2001, Surface Science
The atomic displacements of Cu atoms induced by nitrogen adsorption on Cu(1 0 0) have been studied by channelling–blocking of swift $^{4}$ He ions. This study has been performed at two adsorption stages. The first one corresponds to the formation of a dense, two-dimensional, self-ordered array of square-shaped islands covered by nitrogen. The second one corresponds to uniform coverage at saturation. We have determined by nuclear reaction analysis the absolute quantity of nitrogen adsorbed at these two stages. The values obtained, when confronted to previous observations of these stages by low energy electron diffraction and by scanning tunnelling microscopy, demonstrate that nitrogen remains mostly at the sample surface and that the N concentration in bulk Cu could not exceed 1%. However, channelling measurements show that this surface adsorption generates atomic displacements of Cu atoms down to depths of a few ten (1 0 0) interplanar distances. In the mean time, blocking measurements reveal that nitrogen adsorption induces a strong surface expansion: the interplanar distance between the first two (1 0 0) planes increases of about 0.2 Å, in contrast with the weak contraction observed on bare Cu(1 0 0) surfaces. This observation supports the hypothesis that, when nitrogen is adsorbed, the surface is submitted to stress variations, from tensile to compressive stress for, respectively, bare and nitrogen-covered surface regions. The surface forces corresponding to such variations have been introduced in molecular dynamics simulations. For coverage leading to self-ordering, these simulations do indeed predict displacements of subsurface Cu atoms. The adjustment of these displacements to those measured by channelling gives the amplitude of the stress variation.
Study of atomic relaxations on clean and oxygen covered (100), (410) and (510) copper surfaces by channeling
1995, Surface Science
Atomic relaxations on Cu(100) and stepped Cu(410) and (510) surfaces were measured using a few hundred keV He⁺ ions in channeling and double alignment geometries. On stepped surfaces, we have also attempted to determine the specific relaxation of edge atoms (both the components parallel and perpendicular to the (100) terraces). For this purpose, we have analyzed blocking effects on particles backscattered near θ_lab = 90°, around directions parallel to various major crystallographic axes of the terraces, the incident beam being aligned along the [100] axis perpendicular to the terraces. The studies were performed on clean and on oxygen covered surfaces. On (100) flat surfaces, we find a small contraction of the surface plane on clean Cu (about 2% of the bulk interplanar spacing), and a marked expansion (about 10%) on oxygen covered Cu (at saturation coverage, whose value was measured by nuclear reaction analysis). On stepped surfaces, it appears difficult to discriminate between the specific relaxation of the [001] edge row perpendicular to the terraces and the relaxation of the other [001] rows inside the terraces. Concerning the component of the [001] edge row relaxation in the terrace plane, our results provide an unambiguous estimate: we find a small contraction (about 0.04 Å) on bare Cu and a larger expansion (about 0.18 Å) under oxygen coverage. In all cases, we observed that the presence of oxygen induces a definite surface disordering which can be described as random small atomic displacements of Cu surface atoms around their mean position (on the order of 0.1 Å). The results obtained on the (100) surface covered with oxygen can be interpreted in terms of a missing row model which has been proposed by some authors. However, our data can also be fitted with no missing row. Similarly, on the oxygen covered (410) stepped surface the data interpretation does not require the assumption of a missing row, but the results are not inconsistent with a description of the terrace where the fourth or possibly the third [001] row, parallel to step edge, is missing and where the edge [001] row suffers a considerable outward displacement in the terrace plane ( $≈ 0.3 A ̊$ ). The limits of high energy ion channeling for the measurement of surface atomic displacements, in particular when many sites are involved (for instance in the case of alternate relaxation) are discussed and illustrated.
Simulation analysis of ion channeling spectra: thermal vibrational amplitude in Si
1992, Nuclear Inst. and Methods in Physics Research, B
Energy spectra for 1 MeV He ions scattered by a Si single crystal over an angle of 58° were measured for incident directions along the 〈110〉 string and a number of directions 4° off the string: {100}, {110} and {111} planes as well as three non-channeling directions. The measurements were performed at a temperature of 164 K, under glancing-exit-angle conditions. The experimental spectra were reproduced by Monte Carlo simulation, and a x² analysis procedure was developed to determine the thermal vibrational amplitude in Si. Two ion-atom potentials used in the calculations (the ZBL potential and a Hartree-Fock potential) yielded essentially the same amplitude of 0.066(2) Å. The corresponding Debye temperature of 490(15) K agrees well with recent determinations by other methods.

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^∗: Operated by Union Carbide Corporation for the Energy Research and Development Administration.

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Modelling displacement correlations in computer simulations of particle-solid collisions

Abstract

Phys. Lett.

Phys. Rev.

Phil. Mag.

Acta. Cryst.

Sov. Phys. - Solid State

Sov. Phys. - Solid State