Kapitsa's Secular Approximation

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:

$$\ddot{r}+ \frac{2e}{m(r^2_0+2z_0^2)}(V\cos \omega t)r=0$$

$$\ddot{z} - \frac{4e}{m(r^2_0+2z_0^2)}(V\cos \omega t)z=0$$ (4.1)

Define $d_o = r_0^2 + 2z_0^2$ . 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 $r_\mu$ , $z_\mu$ at the frequency of the applied potential $\omega$ . Then our equations become

$$\ddot{r}+\ddot{r_\mu} = -\frac{2e}{md^2_0}(V\cos \omega t)(r+r_\mu)$$

$$\ddot{z} + \ddot{x_\mu} = \frac{4e}{md^2_0}(V\cos \omega t)(z+z_\mu)$$ (4.2)

$$(r_\mu \ll r, \ddot{r_\mu} \gg \ddot{r}) \longrightarrow r_\mu \approx +\left(\frac{2eV}{md^2_0\omega^2}r\cos\omega t\right),$$

$$(z_\mu \ll z, \ddot{z_\mu} \gg \ddot{z}) \longrightarrow z_\mu \approx -\left(\frac{4eV}{md^2_0\omega^2}z\cos\omega t\right) \ni$$

$$\ddot{r}\approx -\left(\left(\frac{4eV}{md^2_0}\cos\omega t\right) - \left(\frac{4e^2V^2}{m^2d^4_0}\cos^2\omega t\right)\right)r,$$

$$\ddot{z}\approx \left(\left(\frac{8eV}{md^2_0}\cos\omega t\right) - \left(\frac{16e^2V^2}{m^2d^4_0}\cos^2\omega t\right)\right)z \ni$$

$$\overline{\ddot{r}} \approx \left(\frac{2e^2V^2}{m^2d_0^4\omega^2... ...ightarrow r \approx -\cos(\frac{\sqrt{2}eV}{md^2_0\omega}t) = \cos\omega_r t$$

$$\overline{\ddot{z}} \approx -\left(\frac{8e^2V^2}{m^2d_0^4\omega^... ...ightarrow z \approx \cos(\frac{2\sqrt{2}eV}{md^2_0\omega}t) = \cos\omega_z t$$

Evidently, $\omega_r = \omega_z/2$ . We can thus write,

$$r_{tot} \approx -\cos (\omega_z t/2)\left(1-\frac{2eV}{md_0^2\omega^2}\cos\omega t\right)$$

$$z_{tot} \approx \cos (\omega_z t)\left(1-\frac{4eV}{md_0^2\omega^2}\cos\omega t\right)$$

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

Figure 4.2: Secular approximation orbits