Ph.D. Thesis - Physics

GND
RF
r
ENDCAP
0
z 0
ENDCAP
Ring Trap Schematic
RING
ELECTRODE
Figure 4-1: A schematic of a ring Paul trap. The ion is located a distance r0 from the
nearest point on the rf electrode, and a distance z0 from the nearest point on either of the
dc endcap electrodes.
the potential is rotated (or, more accurately, “wobbled”) at the proper frequency, then a
particle can in fact be trapped.
A typical “ring” Paul trap is depicted in Fig. 4-1. It consists of one ring electrode that
is oscillating at an rf voltage Vrf, along with two “endcap” electrodes that are grounded.
These may have a dc voltage applied to them. Near the center of the trap, the electric
potential may be written as
V (x, y,z) = αx 2 + βy 2 + γz 2 . (4.1)
Since the area in the trap region is free of charge (aside from that of the trapped particle,
which is negligible), Laplace’s equation, ∇ 2 V = 0, must be satisfied. For our potential, this
means that α + β + γ = 0. For the case of the ring trap, α = β = −2γ.
To trap an ion, this potential must vary in time. Typically, an oscillating potential of
the form
V (t) = V0 cos (Ωt + φ) (4.2)
is applied. For the remainder of the thesis, we refer to V0 as the rf amplitude and Ω as the rf
frequency. In general, the solutions to this equation are solutions to the Mathieu equations,
which have the general form
üi = [ai + 2qi cos (Ωt)] Ω2
4 ui = 0, (4.3)
where the ui are the position coordinates of the ion along each direction. The parameters
qi and ai obey the following equations:
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