Trapping and Cooling of Ions

Two different techniques have been developed to store ions within a small volume. In the rf quadrupole trap [13.20] the ions are confined within a hyperbolic electric radio-frequency field while in the Penning trap [13.21] a dc magnetic field with a superimposed dc electric field of hyperbolic geometry is used to trap the ions. 

The electric quadrupole field is formed by applying a voltage U between a ring electrode with hyperbolic surface as one pole and two hyperbolic caps as the other pole (Fig.13.13). The whole system has axial symmetry with respect to the z axis. In a geometrical arrangement where the inner ring radius rg is related to the cap separation 2Zg by rg = /2 Zg the electric potential 0 for points within the trap can be written as [13.20] 

The Matthieu differential equation (13,32) has stable oscillatory solutions only for certain ranges of the parameters a and b [13.22] and for certain initial conditions. Charged particles which enter the trap from outside cannot be stabilized. The ions therefore have to be produced inside the electrodes. The general stable solutions of (13.22) can be represented as a superposition of two motions a micro motion with frequency Wq of the ion around a guiding center , which itself performs slower oscillations composed of harmonic oscillations with frequency in the x and y directions and with frequency 

The frequency spectrum of this motion contains the fundamental frequency goq and its harmonics ncog with sidebands at na)Q w.  

The trapped ions can be detected either by the rf voltage of frequency they induce in an rf circuit with the cap electrodes serving as capacitance [13.23] or by laser-induced fluorescence [13.24]. The last method is very sensitive. If the laser transition is chosen properly to ensure that all ions pumped out of the initial level E. return into this level by spontaneous emission, each ion can be recycled n l/x times. This means that at a spontaneous 

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