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Copyright (C) 1999. G. Varga. All rights reserved. No part of this material
may be translated or reproduced in any form without permission from G.
Varga.
3D classical chaotic atom scattering
G. Varga, L. Füstöss, E. Balázs,
Technical University of Budapest, Institute of Physics,
Budafoki s. 8, Budapest, Hungary, H-1111
Keywords: Classical chaos, Atom-solid interactions,
scattering, diffraction; Computer simulations; Rhodium; Single crystal
surfaces
*Tel/Fax: +36 1 242 43 16, E-mail: vargag@phy.bme.hu
Abstract
He scattering on clean Rh(311) surface has been discussed
within the frame of classical mechanics. The classical model of thermal
energy atom scattering on solid surfaces (TEAS) is based on the one particle
problem. The mass point of the He atom is scattered on an appropriately
chosen interaction potential which describes the solid surface. The motion
of the particle mass point is governed by Newton's second law. The scattering
of the atoms on the interaction potential of He-Rh(311) system is investigated
as a function of the impact parameter. Detailed computations show 2D and
3D chaotic effects in trajectories, phase diagrams, deflection angle function
and dwell time function. A crucial point of the above described model is
the numerical method to solve the system of differential equations. For
stiff problems - as the present problem is - variable order solver based
on numerical differentiation formulas is recommended.
1. Introduction
The thermal energy atomic scattering from solid surfaces
(TEAS) is very useful tool in the energy range of 10-100 meV because the
usually applied He probe particles do not penetrate into the surface but
provide information about the top layer [1]. We focused on the classical
chaotic He atom scattering from Rh(311) surface. 2D and 3D scattering are
investigated, respectively. The phenomenon of chaos arises as a chattering
region of impact parameter vs deflection angle function, a chattering region
of impact parameter vs dwell time function, trapped trajectories of particles
or semi-closed curves of real space vs momentum space diagrams. He scattering
on clean Rh(311) surface has been discussed within the frame of classical
mechanics. The classical model of TEAS is based on the one particle problem
[2]. Section 2 describes the three-dimensional classical atom surface scattering
model and the numerical method to solve the appropriate differential equation
system. Two-dimensional and three-dimensional calculations are shown in
section 3. Section 4 briefly summarises the main results.
2. Model of classical atom surface
scattering
A short description of classical model is shown as follows.
The classical model of TEAS is based on the one particle problem. It means
that the mass point of the particle is scattered on an appropriately chosen
interaction potential. This interaction potential describes the solid surface.
The motion of the particle mass point is governed by Newton's second law
in an appropriate inertial frame. Of course, we have to provide the initial
conditions before the interaction process that is outside of the interaction
region. The initial conditions are given by the position vector and the
velocity of the mass point at the initial time. Beside these it is worth
characterising the position vector as a function of so called impact parameters.
The impact parameters ensure that the mass point of the particle may scan
different parts of the surface. According to model description let us prescribe
the equations of the classical model [2]: 
,
where m is mass of the atom, r is the position vector, t
is the time and V(r) is the interaction potential. In three-dimensional
case direction x and direction y are parallel to the surface.
Direction z is perpendicular to the surface. The initial conditions
have been chosen in the following manner: 
, 
, 
, 
, 
and 
where subscript i
denotes
initial state, 
is a large
distance measured from the surface, that belongs to the asymptotic region
of the scattering. 
is the
incident angle and 
is the
azimuthal angle. 
and 
are the impact parameters in the direction x and direction y,
respectively.
The impact parameters are in the interval of [0 1]. "a" and "b"
are the lattice constants. 
, 
and 
are the initial velocities
of the direction x, direction y and direction z, respectively.
E
is the average energy of the incident He particle. Trajectories are
calculated until the condition 
is valid. The crucial point of the above described model is the numerical
method to solve the system of differential equations. For example the method
based on the explicit Runge-Kutta formula does not lead to correct results
[3]. It is only the best function to apply as a "first try" for most problems.
For stiff problems - as the present problem is - the variable order solver
based on numerical differentiation formulas (NDFs) is recommended. We should
recognise that the backward differentiation formulas (also known as Gear's
method) are usually less efficient than NDFs is [4].
3. Results and numerical method
of 2D and 3D scattering computations
He-clean Rh(311) surface scattering has been discussed
in the present paper. Above all the interaction potential was constructed
based on the results of [5]. We chose the Morse potential (eq. (3) in [5]),
but we completed it with a corrugation function which has been gained from
the inverse HCW model computations (in [5], result on p. 306.) The potential
is the following:
V(R,z)=D [ exp(-2a
(z-z (R)))-2exp(-a
(z-z (R))) ], (1)
where z (R) is the corrugation
function, D=7.74 meV, a =1.01 1/Å ,R
is parallel to the surface and z is perpendicular to the surface.
Obviously, eq. (1) does not provide the effective corrugation function
[6]. However, it is a good approach of the interaction potential in first
order.
The Fourier representation of the corrugation function
of the clean Rh(311) surface is the following [5]:
, (2)
where 
; 
; 
.
First, in classical calculation the out-of-plane scattering
has been neglected because the corrugation has strong anisotropy. We use
only the one-dimensional corrugation function. This case corresponds to
two-dimensional He-Rh(311) scattering.
The main input parameters are the following: lattice constants:
a=8.91
Å , b=2.69 Å ; incident angle: 
; 
;
He atom energy: 
. The corrugation
parameters in eq. (2) from [5]: 
Å
, 
Å , 
Å
. In figure 1A the
regular and irregular trajectories under the specified parameters can be
seen. The irregular trajectories correspond to multiple scattering (trapping),
that relates to the chaotic phenomena. According to our experience the
chaotic effect does not arise when the incident angle is small. The chaotic
scattering appears when the multiple scattering is important. The impact
parameter extends from 0.5 to 0.6 with 10 equidistant sample points in
figure
1A.Figure 1B depicts the phase diagram of direction z
vs. momentum 
of the previous
case. One can see semi-closed curves that correspond to trapping. Figure
1C is a diagram of impact parameter-deflection angle relation.
Deflection angle means the difference of the final incident angle and the
initial incident angle of the outcoming and incoming particle. 120 sample
points of impact parameter 
have been chosen in the region of [0.2 0.6]. In the regular region of impact
parameter the curve is smooth. Around »
0.54 there is a chattering region which corresponds to the chaotic phenomenon.
If one magnifies the region of »
0.54 new regular and irregular curves can be seen. Figure
1D shows a diagram of impact parameter-dwell time function.
The dwell time means the time that is spent by the particle between the
atomic source and the detector region. This curve also has a chattering
region around »
0.54 that also corresponds to the chaotic phenomenon. The classical
calculations forecast that the chaotic scattering arises
by higher probability when the incident angle is large and azimuth angle
is taken in the direction of high corrugation of the surface.
The following computations consider out-of-plane scattering
too. Since the corrugation function [5] is two-dimensional in eq. (1) the
He-Rh(311) scattering is a three-dimensional scattering problem. The corrugation
parameters in eq. (2) from [5]: 
Å
, 
Å , 
Å
,
Å , 
Å
, 
Å . Relevant
question is whether the classical chaotic scattering appears or disappears.
Similar calculations have to be performed as in 2D case, except that there
are two impact parameters:
and 
. Because of this fact
the dwell time and deflection angle functions are binary functions. It
is appropriate to demonstrate the chaotic behaviour by contours. This means
one of the impact parameters has to be held constant and the other has
to be changed.
Figure 2
provides
three-dimensional trajectories and phase diagrams. 
and 
in figure
2A and 2B. 
and 
in figure
2C and 2D. One can see regular and irregular (trapped) trajectory,
respectively in figure 2A
and 2C. Figure
2B and 2D render the phase diagrams. The semi-closed trajectory
corresponds to trapping phenomenon in figure
2D similarly to the trajectories in figure
1B.
Figure 3
shows the dwell time function as a function of impact parameters. 
and is in [0.2 0.6] interval
in figure 3A.
One can see a regular curve. Chaotic phenomenon does not arise. When 
a chattering region appears on the curve. The interval of
is identical as in figure
1D. Comparing figure
1D and figure
3B one can see similar structures of the curves. The chattering
region appears around 
again. Several values of 
have been considered in the computations and the figure
3B has been chosen. In figure
3B, 3C and 3D
,
but near the chattering region the dwell time curve is zoomed in. These
figures preserves the property that the curves contain regular and irregular
parts. The curves give self-similarity that is a convincing evidence of
3D classical chaotic scattering [7].
4. Conclusions
The above results underline the existence of two-dimensional
and three-dimensional chaotic He scattering on Rh(311) surface. When the
Rh(311) surface is characterised by one-dimensional corrugation function
chaotic scattering is received. However, this behaviour arises not from
the low order dimension of the surface structure. Namely, in the case of
the Rh(311) surface with two-dimensional corrugation function also chaotic
scattering is received. Actually, the 2D case predicts the region of impact
parameters where the effect of chaos appears. The peculiarity of chaotic
scattering is typical of both the 2D and 3D He scattering on Rh(311).
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