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262 Ion traps

makes an impressive lecture demonstration. Such dynamic stabilisation

cannot honestly be described as ‘well known’ so it is explained carefully

here by an approximate mathematical treatment of ions in an a.c. field.

12.3.1 Equilibrium of a ball on a rotating saddle

In the mechanical analogue, the rotation of the saddle at a suitable speed

causes the ball to undergo a wobbling motion; the ball rides up and down

over the low-friction surface of the rotating saddle shape and the ball’s

(^7) Although we shall not analyse the me- mean position only changes by a small amount during each rotation. 7
chanical system in detail, it is impor-
tant to note that the wobbling motion
is not entirely up and down but has ra-
dial and tangential components. Simi-
larly, an object that floats on the sur-
face of water waves does not just bob up
and down but also oscillates back and
forth along the direction of propagation
of the waves, so its overall motion in
space is elliptical. The discussion only
applies for mechanical systems where
friction has a negligible effect so that
the ball slides smoothly over the sur-
face.
The amplitude of this wobbling increases as the ball moves further from
the centre of the saddle. For this oscillatory motion the time-averaged
potential energy is not zero and the total energy (potential plus kinetic)
increases as the object moves away from the centre. Therefore the mean
position of the object (averaged over many cycles of the rotation) moves
as if it is in an effective potential that keeps the ball near the centre of
the saddle. We will find that an ion jiggling about in an a.c. field has a
similar behaviour: a fast oscillation at a frequency close to that of the
applied field and a slower change of its mean position.

12.3.2 The effective potential in an a.c. field

To explain the operation of the Paul trap, we first look at how an ion

behaves in a.c. electric fieldE=E 0 cos(Ωt). An ion of chargeeand

massMfeels a forceF=eE 0 cos(Ωt), and so Newton’s second law gives

M

r=eE 0 cos(Ωt). (12.5)

Two successive integrations give the velocity and displacement as

r=

eE 0

MΩ

sin(Ωt),

r=r 0 −

eE 0

MΩ^2

cos(Ωt).

(12.6)

It has been assumed that the initial velocity is zero andr 0 is a constant

of integration. The field causes the ion to oscillate at angular frequency

Ω with an amplitude proportional to the electric field. From this steady-

(^8) The a.c. field does not change the state solution we see that the forced oscillation does not heat the ions. 8
ion’s average total energy because the

work done on the ion given byF·r.∝

cos(Ωt)sin(Ωt) averages to zero over
one cycle—the force and velocity have
a phase difference ofπ/2.

(These simple steps form the first part of the well-known derivation of

theplasma frequencyfor a cloud of electrons in an a.c. field, given in most

electromagnetism texts.) The following section describes an example of

this behaviour in which the amplitude of the electric field changes with

positionE 0 (r).

12.3.3 The linear Paul trap

In a linear Paul trap the ion moves in the field produced by the electrodes

shown in Fig. 12.3. The four rods lie parallel to thez-axis and at the

corners of a square in thexy-plane. Each electrode is connected to the