[Orbits of the two electrons in the classical helium atom.]
[Orbits of the two electrons in the classical helium atom.]
We continue our study of dynamical systems by studying systems with additional degrees of freedom.
The Classical Helium Model is a relatively simple example of a three-body problem and is similar to the gravitational three-body problem of a heavy sun and two light planets. The important difference is that the
helium atom's two electrons repel one another, unlike the planetary case where the intraplanetary interaction is attractive. If we ignore the small motion of the heavy nucleus, the equations of motion for the two electrons can be written as
where the r1 and r2 vectors are measured from the fixed nucleus at the origin, and r12 is the scalar distance between the two electrons. Units are chosen such that the mass and charge of the electron are both unity. The charge of the helium nucleus is two in these units.
Because the electrons are sometimes very close to the nucleus, their acceleration can become very large, and a very small time step Δt is required. It is not efficient to use the same small time step throughout the simulation and instead a variable time step or an adaptive step size algorithm is suggested. An adaptive step size algorithm can be used with any standard numerical algorithm for solving differential equations. The RK45 algorithm described in is adaptive and is a good all-around choice for these types of problems.
See: An Introduction to Computer Simulation Methods (3rd ed.) Section 5.12 and Project 5.19.
Most conditions result in unstable orbits in which one electron eventually leaves the atom (autoionization). Test the initial
condition r1=(1.4,0), r2=(-1,0), v1=(0,0.86) and v2=(0,-1) and observer the braided
orbit. Make small changes to this condition and observer autoionization.
The following differential equation models will be discussed in class.
Additional models may be be posted for self-study.
The Classical Helium Model was adapted by Wolfgang Christian from An Introduction to Computer Simulation Methods (3rd ed.) by H. Gould, J. Tobochnik and W. Christian. You can examine and modify a compiled EJS model if you run the model (double click on the model's jar file), right-click within a plot, and select "Open Ejs Model" from the pop-up menu. You must, of course, have the the Easy Java Simulations (EJS) version 4.1 authoring and modeling tool installed on your computer.
Information about EJS is available at <http://www.um.es/fem/Ejs/> and in the OSP comPADRE collection <http://www.compadre.org/OSP/>.