12.3 The Paul trap 265proportional to cosAτcos 2τdominate on each side, and equating their
coefficients gives− 4 B=2qxor
B=−
qx
2=−
eV 0
MΩ^2 r^20. (12.17)
This amplitude of the fast oscillation is consistent with the result for
a uniform electric field in eqn 12.6.^15 This fast oscillation is called the^15 Equation 12.10 shows that the com-
ponent of the electric field in this direc-
tion isE 0 (r)·ˆex=−V 0 x/r^20.
micromotion. To determine the angular frequencyAwe consider how the
mean displacement changes on a time-scale longer than the micromotion:
the time-average of cos^22 τ =1/2 so eqn 12.16 yields−A^2 cosAτ =
qxBcosAτ;^16 henceA=qx/
√
2 and an approximate solution is^16 The term 4ABsinAτcos 2τ time-
averages to zero.
x=x 0 cos(
qxτ
√
2+θ 0){
1+
qx
2cos 2τ}
, (12.18)
whereqxis defined in eqn 12.14.^17 We assumed thatqx1butitturns^17 An arbitrary initial phaseθ 0 has been
included to make the expression more
general but this does not affect the ar-
gument above.
out that this approximation works better than 1% forqx 0 .4(Wuerker
et al. 1959). Sinceτ=Ωt/2 the mean displacement undergoes simple
harmonic motion at an angular frequency given by
ωx=
qxΩ
2√
2
=
eV 0
√
2ΩMr^20. (12.19)
A more detailed treatment shows that ions remain trapped for
qx 0. 9 (12.20)orωx 0 .3 Ω. For a radio-frequency field oscillating at Ω = 2π×10 MHz
the ion must have a radial oscillation frequencyωx 2 π×3MHz. If we
chooseωx=2π×1 MHz (a convenient round number) then eqn 12.19
gives the numerical valuesV 0 = 500 V andr 0 =1.9 mm for trapping Mg+
ions.^18 To get this high trapping frequency the ion trap has electrodes^18 In comparison, neutral atoms in mag-
netic traps oscillate at frequencies in
the range 10–1000 Hz.
closer together than we assumed in the introduction. By symmetry,
the same considerations apply for motion in they-direction, and so we
define a radial frequencyωr≡ωx=ωy. The Paul trap has a sharp
transition from stable trapping to no trapping in the radial direction
whenqrequals the maximum value ofqxin eqn 12.20. Paul used this
feature to determine the charge-to-mass ratioe/Mof the ions and hence
perform mass spectroscopy—generally the charge state is known (e.g. it
iseor 2e, etc.) and henceMis determined.
So far we have only described confinement in thexy-plane. There
are several ways to extend trapping to all three directions. For example,
Fig. 12.3(b) shows a trap with two additional electrodes atz=±z 0 that
repel the ions. For positive ions both of these end-cap electrodes have the
same positive voltage to give a field similar to that shown in Fig. 12.1,
with a minimum in the electrostatic potential along thez-axis—in the
radial direction the static potential has a negligible effect compared to
the a.c. trapping. When the linear Paul trap has axial confinement
weaker than that in the radial direction, i.e.ωzωx=ωy, the ions
tend to lie in a string along thez-axis with only a small micromotion