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Equation of motion under PUSH command

Solving the equation of motion in PUSH command is much more complicated because of the possible presense of the beam field. The equation of motion is in general written in the form


The force tex2html_wrap_inline9652 includes the beam field and the external field. The tex2html_wrap_inline9654 dependence of tex2html_wrap_inline9652 comes from tex2html_wrap_inline9658 although very weak in the case of the beam field.

Given the initial variables tex2html_wrap_inline9660, a simple approximation after the time interval tex2html_wrap_inline8540 is


The error of tex2html_wrap_inline9664 by these formulas is estimated by
If this is not small enough, divide the interval tex2html_wrap_inline8540 by an integer tex2html_wrap_inline9668. Note that tex2html_wrap_inline9670 because tex2html_wrap_inline9672 is proportional to tex2html_wrap_inline8540. The total error, after multiplied by the number of intervals tex2html_wrap_inline9668, is proportional to tex2html_wrap_inline9678.

However, the above prescription is not really enough when there are extremely low energy particles (e.g., those from incoherent pair creation). It often happens that tex2html_wrap_inline9668 so determined bocomes over several hundreds. In such a case the above error estimation may not be accurate at all.

When tex2html_wrap_inline9668 is too large, CAIN tries the fourth-order Runge-Kutta integration. Starting from the whole interval tex2html_wrap_inline8540, it is divided by 2 at each step until the difference becomes small enough. This method is a little better than the simple formulas above but is still time consuming. So, the users should be aware that incoherent pair creation is expensive.

Toshiaki Tauchi
Thu Dec 3 17:27:26 JST 1998