**Principal Investigator**
Timothy Scholz **Project **g48

Department of Physics, **Machine **VP

Murdoch University

**Time-dependent Electron-Atom Scattering**

Matter is made up of atoms each of which consists of a centrally located nucleus with relatively light and small electrons orbiting around it. In many locations, such as the earth's upper atmosphere or within the sun, electrons break free from atoms and end up colliding with other atoms. The details of what happens during these collisions has a radical effect on the physical properties of the gas (e.g. the sun) in which they occur. This project seeks to study electron-atom collisions by solving the quantum mechanical equations of motion which electrons and atoms are known to obey. Unfortunately these equations are extremely complicated. The easiest way to solve them is to assume that the colliding electron is spread over an infinite distance. This is the idea behind the popular time-independent collision theory. This project does not make this dubious assumption but rather seeks to solve the more accurate time-dependent equations of motion. These equations are more difficult to solve but do lead to a more accurate description of electron-atom collisions.

**What are the basic questions addressed?**

The ultimate goal of the project is the calculated the differential cross section (DCS) for electron scattering by atomic hydrogen. This requires that the wavefunction of the electron-atom system prior to the collision be evolved in time until the collision is completed. From this final wavefunction the DCS, which tells us most of what there is to know about the collision, may be evaluated. To confidently perform this ultimate calculation we must first gain experience of the technique in one and three dimensional (3D) scattering. For example, in 3D potential scattering, we wish to test electron scattering by a Coulomb potential which has a known analytic DCS. Numerical experimentation will then reveal the sensitivity of the method on long range forces and the shape of the wavefunction for the incident electron.

**What are the results to date and future of the
work?**

The results indicate that the method used in this work is very accurate for scattering in both one and three dimensions. Where analytic solutions of the time-dependent equations exist, agreement with the calculated results is excellent. Unfortunately complete three dimensional results arise from a large number independent partial wave calculations. To date, only the results of some lower partial waves have been evaluated. However these have been found to be very accurate. In the future the higher partial waves will also be evaluated and combined with the lower partial waves to produce the full three dimensional DCSs. Initially DCSs will be calculated for two different potentials (i) the static potential of the hydrogen atom and (ii) the Coulomb potential. In the former case, accurate results from time-independent scattering results are available for comparisons and in the latter case, the analytic DCS is available. This will be an excellent test of the method. Following this we plan to perform a full six dimensional electron-hydrogen atom scattering calculation. In this case, exact results for the lowest partial wave are known and can be used for testing the method. Once satisfied with the initial results, the calculation for all partial waves will be completed yielding a final result which includes elastic, inelastic and ionisation cross sections. The latter are of most interest because they have yet to be calculated using any sound procedure.

**What computational techniques are used and why
is a supercomputer required?**

Although the time-dependent method is indisputably the most direct method for modelling electron-atom collisions processes it has largely been ignored because is computationally expensive. However methods recently developed in the study of theoretical chemical reaction dynamics has made such calculations more practical. The problem boils down to calculating the exponential of a second order differential operator. The method is in two parts (i) approximating this exponential to a Chebyshev polynomial expansion and (ii) using the fast Fourier transform (FFT) algorithm to evaluate the effect of the second order derivative. The latter requires the storage of large arrays for efficient processing so large computer memory is required. The FFT which takes most of the computational time, is also highly efficient on a vector processes, achieving about 98% vectorization on the VP2200. Without the use of a vectorizing machine with large memory the current project would not be feasible.

**Publications**

*"Time-dependent model of potential scattering"
*T.T.Scholz. Journal of Physics B: Atomic,
Molecular and Optical Physics (1996) submitted