0198506961.pdf

12.3 The Paul trap 261

Fig. 12.1The electric field lines be-

tween two equal positive charges. Mid-

way between the charges at the point P

theelectricfieldsfromthetwocharges

cancel. At this position the ion expe-

riences no force but it is not in stable

equilibrium. At all other positions the

resultant electric field accelerates the

ion.



Fig. 12.2A ball on a saddle-shaped

surface has a gravitational potential en-

ergy that resembles the electrostatic

potential energy of an ion in a Paul

trap. Rotation of the surface about a

vertical axis, at a suitable speed, pre-

vents the ball rolling off the sides of the

saddle and gives stable confinement.

12.3 The Paul trap

The analogy with a ball moving on the saddle-shaped surface shown in
Fig. 12.2 provides a good way of understanding the method for confining
ions invented by Wolfgang Paul. The gravitational potential energy of
the ball on the surface has the same form as the potential energy of an
ion close to a saddle point of the electrostatic potential. We assume
here a symmetric saddle whose curvature has the same magnitude, but
opposite sign, along the principal axes:

z=

κ

2

[

(x′)

2

−(y′)

2

]

, (12.4)

wherex′=rcos Ωtandy′=rsin Ωtare coordinates in a frame rotating
with respect to the laboratory frame of reference. The time average of
this potential is zero. Rotation of the saddle shape around the vertical
axis turns the unstable situation into stable mechanical equilibrium, and