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We neglect the DC potential U for now and assume the equations
of motion are of the form, for the rf Paul Ideal chamber ion trap:
![$\displaystyle \ddot{r}+ \frac{2e}{m(r^2_0+2z_0^2)}(V\cos \omega t)r=0$](img96.png) |
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![$\displaystyle \ddot{z} - \frac{4e}{m(r^2_0+2z_0^2)}(V\cos \omega t)z=0$](img97.png) |
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(4.1) |
Define
. Assume that the r and z motion can
be partitioned into large-amp slow ``secular'' motion r and z, and
small-amp high frequency micromotion
,
at the frequency
of the applied potential
. Then our equations become
![$\displaystyle \ddot{r}+\ddot{r_\mu} = -\frac{2e}{md^2_0}(V\cos \omega t)(r+r_\mu)$](img102.png) |
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![$\displaystyle \ddot{z} + \ddot{x_\mu} = \frac{4e}{md^2_0}(V\cos \omega t)(z+z_\mu)$](img103.png) |
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(4.2) |
Evidently,
. We can thus write,
The results of these approximations are graphically displayed in Figures 4.1 and 4.2
created with the following C code:
float w=53;
float wz=4;
float r,z;
float t=0.0;
while(t<1000){
r=-cos(wz*t/2)*(1-0.3*cos(w*t));
z=cos(wz*t)*(1-0.6*cos(w*t));
t+=0.01;}
Figure 4.1:
Secular approximation time series
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Figure 4.2:
Secular approximation orbits
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Next: A solution with Mathieu's
Up: Activation of the equation
Previous: Activation of the equation
tim jones
2008-07-07