xmpl3d95, 95th 'example' data file for CPO3D

Uses the option for users equations, replaces xmpl3d05.

This example illustrates the option for oscillatory potentials.

It is nearly the same as xmpl3d05, except that a bridging electrode has been added across the gap (which has improved the accuracy) and more importantly, the 'users equations' option has been used to define the electrodes.

The left and right electrodes have the voltage V and the shape given by

x**2 - y**2 = 1,

while the top and bottom electrodes have the voltage -V and the shape given by

y**2 - x**2 = 1,

so the potential distribution is give by

P(x,y) = V*(x**2 - y**2).

Since the symmetry planes x = 0 and y = 0 are used, and the plane x = y is defined here as a plane of voltage anti-symmetry, it is only necessary to define the shape x^2 - y^2 = 1 for the region y > 0, as given in the data file above.

The numbers of subdivisions are chosen to give approximately the same total number as in xmpl3d05, although here they are all triangles instead of rectangles.

In this example the voltage V is given the time dependence u + v*sin(wt) where u = 12.030, v = 72.218, w = (2*pi)/tau, tau = 0.001ms.

The particles are singly charged ions of energy 1eV. They start parallel to the axis, at a distance 0.001mm from the axis. The particles have a range of atomic mass near 1000 (see below).

The information below is taken from P H Dawson. (For less detailed information, see P S Farago, Free-electron physics, Penguin Books, Harmondsworth, England, 1970.)

The parameters used by Dawson are U = 2u, V = 2v, a = 2k*U, q = k*V,

where k = (2*e)/(m*w**2*r**2),

and where m and e are the ion mass and charge, and r is the distance of the centres of the electrodes from the axis (1mm in the present example).

Therefore for atomic number 1000:

k = 4.8880E-3, a = 0.23521, q = 0.70600.

These values of a and q are near those that represent the limit of stability, namely a = 0.23699, q = 0.70600, and have been chosen in the present example to give a mass resolution that should be approximately 100.

The ions are given the atomic masses 990, 991, 996, 1002 and 1003. It can be seen from the output file, or from the graphics box, that the atomic masses 991 to 1002 are stable, while the smaller and larger masses are unstable in either the x or y directions. The mean mass is therefore approximately 996 and the resolution is in the region of 80.

More accurate results would be obtained by using more segments and higher trajectory accuracy.

The periods of the 'fast' and 'slow' oscillations in the x and y directions can be deduced from the plots. The 'slow' oscillation is that of the envelope of the trajectory. It is convenient to convert the periods to frequencies and to express these in terms of the frequency w of the applied potentials. The deduced values are:

 Direction fast slow x w_xf = 0.500w w_xs = 0.0324w y w_yf = 0.996w w_ys = 0.0191w

Theoretically the values of w_xf and w_yf should be approximately w/2 and 2w respectively, for conditions near the limit of stability, as found here. Also for the present conditions the parameters beta_x and beta_y defined by Dawson have the values:

beta_x = 0.9797, beta_y = 0.0474,

which implies that:

w_xs = 0.0303w, w_ys = 0.0237w.

These values differ from those found in the present simulation, presumably because of their sensitivity to the precise conditions.

See on-line Help for information on the option for users equations.