0
$\begingroup$

Ideal equations for motion of a charged particle in a Penning Trap are:

$$ x(t) = \rho_{+}\cos(\omega_{+}t + \phi_{+}) + \rho_{-}\cos(\omega_{-}t + \phi_{-}) $$ $$ y(t) = (\frac{q}{\lvert q\rvert }) [\rho_{+}\sin(\omega_{+}t + \phi_{+}) + \rho_{-}\sin(\omega_{-}t + \phi_{-})] $$ $$ z(t) = \rho_{z}\cos(\omega_{z}t + \phi_{z}) $$

where,

$$ \rho_{+}^2 = \frac{2E_{+}}{m(\omega_{+}^2 - \frac{\omega_{z}^2}{2})} d^2 $$ $$ \rho_{-}^2 = \frac{2E_{-}}{m(\omega_{-}^2 - \frac{\omega_{z}^2}{2})} d^2 $$ $$ \rho_{z}^2 = \frac{E_{z}}{qU} d^2 $$

and $$ d^2 = \frac{1}{4} (2z_{0}^2 + \rho_{0}^2) $$ , $$ z_{0} , \rho_{0} \text{ are axial and radial constraints} $$

$$ E_{+},E_{-},E_{z} \text{ are kinetic Energies.} $$

$$ \text{ I assume } E_{z} = \frac{1}{2}mv_{z0}^2, v_{z0} \text{ being initial velocity in z axis. Assuming } E_{+},E_{-} \text{ are called cyclotron and magnetron kinetic Energies, }$$ How do I find the Kinetic energies assuming we know the initial velocities in x and y direction?

I have been quickly shifting though books on particle traps with the motive to understand the motion of a charged particle in a Penning Trap given U, B, q, m and initial parameters. Trying to make a simulator with Blender

$\endgroup$
  • $\begingroup$ Note that the energy of the axial mode also depends on the initial axial position! For example, consider the axial energy of a particle that is initially at rest, but with $z_0 = 1\,\text{m}$. Here, the energy is entirely potential energy. My guess is that for your Blender simulation you are looking for a transformation $(x_0, y_0, z_0, \dot{x}_0, \dot{y}_0, \dot{z}_0)$ to $(\rho_+, \rho_-, \rho_z, \phi_+, \phi_-, \phi_z)$. Such a transformation is possible, and if I find the time, I will give it a stab later. $\endgroup$ – Martin J.H. 22 hours ago

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.