G. Varga*

Budapest University of Technology and Economics. Budafoki út 8, Budapest, Hungary ZIP H-1111
 

Keywords: Quantum 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

The thermal energy He scattering from solid surface (TEAS) is an important tool to investigate the processes on solid surfaces. Certain classical mechanical models of TEAS show chaotic behaviour. However, the real systems are not always governed by classical mechanics because the interaction is strong enough. In consequence of this fact the classical model may show chaotic effect but the real physical system has no chaotic effect. To resolve this contradiction a time dependent quantum mechanical model has to be applied. This model recommends not only a simulation of (chaotic) scattering but a real measurement, too. If the dwell time near the surface of scattered He atoms is stochastically fluctuating then the scattering is chaotic. The dwell time of the probe particles can be determined by the flight time measurement of He atoms. When the graph of intensity distribution vs flight time (energy) has a 'chattering' region, the scattering is quantum mechanically chaotic. The well known time of flight (TOF) measurement is suitable to explore the purely quantum chaos. The computations has been executed in the case of He atom scattering on Rh(311) surface.

1. Introduction

The thermal energy atomic scattering from solid surfaces (TEAS) is an efficient method to investigate the very top layer of the surfaces [1]. TEAS provides information about surface structure, phonon spectra and impurity. In addition to these physical properties TEAS can describe the quantum processes on the solid surface. The present paper focuses on the question of chaotic behaviour of TEAS. The interaction of neutral atoms and solid surface is strictly quantum mechanical [2]. What does it mean in the picture of quantum mechanics? Since the interaction is generally strong, Ehrenfest's theorem invalidates the picture of the trajectories. If we adhere the purely stochastic ("Heidelberg approach") discussion of quantum chaos the dwell time of atoms near the surface has to be determined as a function of appropriate physical parameters [3]. When the dwell time fluctuates the effect is quantum chaos. Therefore, flight-time measurement of the scattered atoms can support or not the chaotic scattering idea in the above mentioned situation. In the present paper the quantum chaotic scattering is looked for in the case of He scattering on Rh(311) surface. Section 2 describes the time dependent quantum mechanical model of TEAS and its numerical solution. The interaction potential of He-Rh(311) system is composed in section 3. Section 4 shows the results that try to find quantum chaotic scattering. At last one can read the conclusions in section 5.

2. Quantum mechanical model

For the description of the problem outlined in the introduction the time dependent Schrödinger equation (TDSE) will be solved. The model assumptions for solving it are: the Gaussian wave-packet to describe the incoming particle beam and the interaction potential with no restriction of periodicity and time independence. Basically, quantum mechanics is able to account for the physical processes of TEAS. In certain conditions - e.g. the probe particles are heavy atoms - semiclassical model approach is appropriate [2]. The particles of atomic beam does not interact with each other, but they have a special distribution of velocity and energy. The Gaussian wave packet characterises the atomic beam as a special quantum ensemble of independent particles. 3D Gaussian wave-packet has been chosen as an initial wave function:

(1)

Let us consider the time dependent Schrödinger equation:  , where  is the Planck constant divided by 2p and H is the Hamiltonian. A propagation scheme and an application of Hamilton operator have to be applied. Splitting Hamilton operator into two parts, the kinetic energy operator A and the potential energy operator B, in the case of time independent potential we can write the exact formal solution: . The solution of TDSE demands time propagation at every time step and requires a method to determine the effect of Hamilton operator on the wave function. We chose an efficient splitting operator method with third-order accurate formula in time [4],[5] and Fast Fourier transformation (FFT) has been applied to calculate HYat every time step [6].

3. The interaction potential of He-Rh(311) system

He-clean Rh(311) scattering has been discussed in the present paper. Above all the interaction potential was constructed based on the results of [7]. The Morse potential has been chosen, that is completed with a corrugation function resulting from the inverse hard corrugated wall (HCW) model computations [7]. The potential is the following:
 
 

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. (2) does not provide the effective corrugation function [8]. However, it is a good approach to the interaction potential in first order.

The Fourier representation of the corrugation function of the clean Rh(311) surface is [7]:

where ; ; .
 

4. Results

Quantum mechanical computations have been executed in three-dimensional space. The out-of-plane scattering has been considered. The corrugation parameters in eq. (3) are taken from [7]: Å;, Å;, Å;, Å;, Å;, Å;. In eq. (1) the main input parameters of the He beam are (in atomic units, abbreviation: a.u.): s =Ö 5, =9.8644, =12.681, =11 and k=(, , ), k=3.74. Average energy of He atom is: » 26 meV. The incident angle is: and the azimuth angle is: . 32, 96 and 64 sample points are chosen in the direction x, y and z, respectively. In figure 1A and 1B the probability density functions (PDF) are shown in the real and in the momentum space, respectively as the wave-packet approaches the classical turning point (, ‘<>‘ denotes the average). PDF is equal to . PDF is split into slices parallel to the solid surface since PDF is a tervariant function in space. Figure 1A and 1B show the slices at z=2.62 a.u. when . In that region the interaction is essential. One can see a very important fact: |k|>3.74 a.u. significantly has a probability different from zero. This corresponds to a bound state. Figure 1C and 1D show PDF after the scattering, beyond the interaction region in the real and in the momentum space, respectively. One can also see in-plane and out-of-plane scattering. The component parallel to the surface of the wave number vector k is shorter than 3.74 a.u. on the contrary when the He atom is near the top layer of the Rh(311) surface. The attractive part of the interaction potential causes longer lifetime near the surface. The determination of the escaping directions requires detailed scanning of transient probability over the solid angle. These should contain the diffraction peaks, the selective adsorption and chaotic effect. In the case of quantum chaos the probability density function should show chattering region. Unfortunately, in the case of He-Rh(311) system quantum chaotic effects could not be found.

He scattering on Rh(311) surface shows chaotic scattering within the frame of classical mechanical model [9]. The classical chaos does not imply quantum chaos by all means [9]. The quantum mechanical (QM) model washes out the "trajectories" since the trajectories do not exist by Ehrenfest's theorem when the interaction is QM. The classical chaos is thought to be caused by the strong interaction of He atom-Rh(311) system. Further investigations are needed to clarify the physical phenomena of He-Rh(311) scattering. A convincing quantum mechanical method has to determine the dwell time of He atoms near the surface as a function of energy [3] or as a function of measurement time. The measurement time means the time during the experiment. When the dwell time function provides chaotic (stochastic) behaviour, the He-Rh(311) scattering also shows quantum chaos. This is the real physical picture of the scattering. Experimentally, flight-time measurement is able to realise this idea. Namely, flight-time measurement give the function of intensity vs flight-time. When the graph is stochastic the probability of the occupied states at the interaction region fluctuates, because the intensity distribution always connects to the probability of the current state. The fluctuating occupied states mean that a small change of the scattering He beam energy or of the geometrical arrangement causes discrete steps of the He atom state.

Let us define the dwell time based on [3] as follows:

where is the time when the measurement is started, and  is the measurement time.  is the investigated volume. P(t) is a fraction of the measurement time. P(t) gives the time spent on average by the He atom of the beam in the volume . Namely, the He atom beam is a quantum mechanical ensemble of independent He particles. If the graph of P(t) behaves stohastically the wave-function of He atom beam changes stohastically in  since the region where the particle can be found, is peculiar to the state of the He beam.

P(t) is calculated by four-variant trapezoidal numerical integration. The wave function is computed by time dependent Schrödinger equation with time step short enough. Different volumes  in the physical space are chosen. Unfortunately, stochastic behaviour could not be found by this method neither. Figure 2 depicts a typical curve of dwell time P(t) when  and the time step. Small volume  has been chosen at the detector region (in atomic units): , , . The dwell time as a function of measurement time tends to an asymptotic value. The curve is regular since it is smooth enough. One can see around t=35000 a.u. that the dwell time starts to increase and around t=60000 a.u. the curve reaches an asymptotic value.

5. Conclusions

Quantum chaotic scattering of He-Rh(311) system has been looked for by time dependent Schrödinger equation in the present paper. Two methods have been applied. One of them investigated the probability density function of the He atom beam. Experimentally, it corresponds to the standard intensity distribution measurement. By the other method the dwell time of the He atom beam was analysed in the real physical space. Experimentally, this case corresponds to the time of flight measurement (TOF). None of them has shown quantum chaotic behaviour, although classical mechanics discussion provides chaotic scattering [9]. The explanation may be that quantum mechanics contains in natural mode the different phenomena of scattering (e.g. trapping, adsorption). Perhaps, TOF measurement is more evident to explore the chaotic scattering than the standard intensity measurement, because the whole solid angle has not be scanned. The energy spread of the He beam may provide the small change of the energy that is necessary to measure the chaos. Only the elastic behaviour of the scattering was investigated because of the time independent interaction potential. One can say that only the elastic scattering without diffuse scattering (frozen solid surface) does not show quantum chaos. Further investigations of more thorough quantum mechanical models have to be applied.
 

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