# Motion of a Particle in a Penning Trap, parameters and solutions

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

• 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. – Martin J.H. 22 hours ago