0198506961.pdf

264 Ion traps

method for solving an electrostatic problem applies even though the volt-

age on the electrodes changes because at the radio-frequencies (used in

ion traps) the radiation has a wavelength much greater than the dimen-

(^10) This method would not be appropri- sions of the electrodes, e.g. a wavelength of 30 m for Ω = 2π×10 MHz. 10
ate for shorter wavelengths, e.g. micro-
wave radiation with frequencies of GHz.
The potential energyeφof an ion has a saddle point in the middle of
these electrodes that looks like the saddle shape shown in Fig. 12.2—a
potential ‘hill’ in thex-direction and a ‘valley’ in they-direction, or the
(^11) The two-dimensional quadrupole other way around. (^11) From the gradient of potential we find the electric
field between the four rods looks
superficiallylike the quadrupole field
with cylindrical symmetry in Fig. 12.1
and the analogy with a rotating saddle
applies to both. A comparison of
the potentials for the two cases in
eqn 12.9 and eqn 12.23 shows that
they are different. Note also that the
electrostatic potential oscillates ‘up
and down’ rather than rotating as in
the mechanical analogy.
field
E=E 0 (r)cos(Ωt)
=−

V 0

r^20

cos (Ωt)(xˆex−yˆey).

(12.10)

The equation of motion in thex-direction is

M

d^2 x

dt^2

=−

eV 0

r^20

cos (Ωt)x. (12.11)

A change of variable toτ=Ωt/2leadsto

d^2 x

dτ^2

=−

4 eV 0

Ω^2 Mr^20

cos (2τ)x. (12.12)

This is a simplified form of the Mathieu equation:^12

(^12) The Mathieu equation arises in a va-
riety of other physical problems, e.g.
the inverted pendulum. A pendulum is
normally considered as hanging down
from its pivot point and undergoing
simple harmonic motion with a small
amplitude. In an inverted pendulum
a rod, that is pivoted at one end, ini-
tially points vertically upwards; any
slight displacement from this unsta-
ble position causes the rod to fall and
swing about the stable equilibrium po-
sition (pointing straight down), but if
the pivot point oscillates rapidly up
and down then the rod remains up-
right whilst executing a complicated
motion—the rod can make quite large-
angle excursions from the vertical direc-
tion without falling over. The math-
ematical textbook by Acheson (1997)
gives further details of the complexities
of this fascinating system and numeri-
cal simulations can be seen on the web
site associated with that book.
d^2 x
dτ^2
+(ax− 2 qxcos 2τ)x= 0 (12.13)
withax=0.^13 It is conventional to define the parameter in front of the
(^13) This corresponds to the motion of an
ion in a trap that has no d.c. voltage. In
practice, ion traps may have some d.c.
voltage because of stray electric fields
but this can be cancelled by applying
a suitable d.c. voltage to the electrodes
(or additional electrodes near the four
rods). The solution of the Mathieu
equation withax = 0 is discussed in
the book on ion traps by Ghosh (1995).
oscillating term as 2qx(in anticipation of thisehas been used for the
ion’s charge), where
qx=
2 eV 0
Ω^2 Mr^20

. (12.14)

We look for a solution of the form

x=x 0 cosAτ{1+Bcos 2τ}. (12.15)

The arbitrary constantAgives the angular frequency of the overall mo-

tion andBis the amplitude of the fast oscillation at close to the driving

frequency. The justification for choosing this form is that we expect an

oscillating driving term to produce a periodic solution and substitution

of a function containing cosAτinto the equation leads to terms with

cosAτcos 2τ.^14 Substitution into the equation (withax=0)gives

(^14) More detailed mathematical treat-
ments of the Mathieu equation can be
found in Morse and Feshbach (1953)
and Mathews and Walker (1964).
x 0

[

− 4 BcosAτcos 2τ+4ABsinAτsin 2τ−A^2 cosAτ{1+Bcos 2τ}

]

=2qxx 0 cos 2τcosAτ{1+Bcos 2τ}.

(12.16)

We shall assume thatA 1 ,so that the function cosAτcorresponds

to a much slower oscillation than cos 2τ, and also that the amplitude

B1 (both of these assumptions are discussed below). Thus the terms