Poincare maps

The first applet shows how to build on the (r11/2, 2r11/2p1) plane (in the right window) Poincare section of an orbit by the r2 = 0 plane.

Click mouse in the left window to set new coordinates of the blue (right) electron.

This is the hole picture of the r2 = 0 Poincare section. Click mouse in the window to see a new (red) orbit. Initial coordinates of the first (blue) electron are shown in the status bar below.

The eZe system is conjectured to be purely hyperbolic (i.e. all orbits are unstable).

Zee subspace

In the Zee case both electrons stay on the same side of the nucleus. The inner electron oscillates between the nucleus and its outer turning point while the outer electron moves in the effective potential well formed by the attractive nuclear force and the inter-electron repulsion at short distances. This repulsion prevents the outer electron from penetrating into the region close to the nucleus.

This regular motion leads to closed torus structure on the picture below. The fixed point in the center of the structure is also stable when the electrons move in a (slightly) off-collinear arrangement. Therefore the periodic orbit is embedded in an island of stability in the hole phase space.

If the initial distance is smaller then a critical value r1 ~ 5 (while the inner electron is at the nucleus) the first electron is "kicked" out of the atom and ionizes immediately.

[1] Klaus Richter, Gregor Tanner and Dieter Wintgen
"Classical mechanics of two-electron atoms" Phys.Rev.A 48, 4182 (1993)

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updated 12 Apr 2003