First consider the axial position of a single particle in an axially-confining quadrupole potential:
The resultant frequency of axial oscillation is:
Now consider the radial position of a single particle subjected to an axially-directed magnetic field. The magnetic field gives rise to a radial cyclotron orbit whose frequency, , is given by equating the magnetic force to the centripetal force:
In the equation above, only the radial component of velocity is of interest, since the axial component contributes neither to the magnetic force nor to the radial centripetal force.
For the single ion subjected to both the axially-focusing quadrupole potential and the axially-directed, homogeneous magnetic field, the radial motion is influenced by the defocusing force of the electrostatic potential as well as by the magnetic field:
Rewriting the above equation in terms of the cyclotron and axial frequencies and substituting, as does Milonni, and leads to:
where is the perturbed (by the exponentially defocusing component of the quadrupole field) cyclotron frequency and describes the slower magnetron motion:[26,36]
In order that radial motion is confined, and must be purely real. Hence, single-ion stability in the Penning trap requires that:
Returning to the previous definitions of and , Eq. is equivalent to the stability constraint:
In general, Penning trap parameters are chosen such that the radius of magnetron motion exceeds that of the cyclotron motion by a factor on the order of 100 and:[26,37]
In this case, , since and .
The desirability of large frequency discrepancies follows from the fact that the potential energy associated with magnetron orbits decreases for larger values of r. Consequently, there is a tendency for the metastable magnetron motion to defocus the radial position as energy is radiated from the circulating ion. In order to enable long-term (measured in years) Penning trap confinement, a slow magnetron motion--- which corresponds to long-lived metastable magnetron orbits--- is necessary. Hence, the large frequency discrepancies expressed in Eq. are required for quasi-permanent Penning trap confinement.
Although it is experimentally desirable for axial, cyclotron, and magnetron frequencies to differ by several orders of magnitude, a wide range of frequencies complicates computer simulation. In order that a molecular dynamics simulation is stable and accurate, the time step must be chosen as a fraction of the period of the fastest oscillation--- in this case, a fraction of the cyclotron period. However, understanding a system's long-term behavior requires integration over several periods of the slowest oscillation.
These requirements of a small time step and a relatively long integration time render Penning trap simulations computationally expensive and slow. One strategy for mitigating the large time requirement is to select parameters such that:
such that the frequencies of cyclotron, axial, and magnetron motions are comparable. This method is satisfactory for modeling undamped particles, but when a large number of viscous-damped particles is investigated under the condition of Eq. , the tendency of the magnetron radius to increase with loss of energy precludes observation of theoretically predicted cylindrical ring structures.
A more sophisticated method of dealing with the computational intensity of Penning trap simulations is employed by Dubin, et al., who use guiding-center equations with time-averaged cyclotron dynamics to model Penning-trapped particles in the strong B-field limit.