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Clouds of two to about fifty simultaneously stored, laser-cooled Mg + ions in a Paul trap were observed in two phases, which are clearly distinguishable by their excitation spectra. Transitions between these phases can be induced either by a variation of the power of the laser radiation used to cool the ions or by a change of the size of the radio-frequency voltage applied to the trap. Transitions between a "crystalline" phase and a "gaseous" phase can be repeatedly observed by variation of the appropriate parameters. The two phases and the transitions between them have also been recorded by a photon-counting image system.There is growing interest in the use of particle traps for high-resolution spectroscopy. The ultimate resolution limits which can be achieved in principle are beyond all other known experimental possibilities. It seems quite probable that particle traps will be the basis of future frequency standards. 1,2 The traps most extensively employed to date for ions are the static Penning trap and especially the dynamic Paul trap. Storage in the Penning trap is achieved by use of a strong magnetic field; the Paul trap uses an oscillating electric quadrupole field. 2 ' 3In the Paul trap a single ion can, in principle, be kept at rest at a point of vanishing field amplitude in the center of the trap. By cooling of single Mg + , Ba + , and Hg + ions 4 " 7 in the millikelvin range, previous experiments have demonstrated that this ideal situation can be realized in very good approximation. In these experiments the kinetic energy of the ions («1 eV after the loading cycle) was reduced by radiation-pressure cooling. This was done by the excitation of a strong dipole transition with laser light whose frequency was detuned from resonance towards longer wavelengths. In this way a single Ba + ion may be localized in a region with dimensions small compared with the optical wavelength of the exciting light. 6 In this so-called Lamb-Dicke regime, the first-order Doppler effect is suppressed and, in addition, the second-order Doppler effect is negligible, so that single laser-cooled ions have been proposed as the basis for frequency standards. 8 The Lamb-Dicke regime can easily be obtained for microwave transitions. Because of the longer wavelength, even clouds of 10 5 ions can be confined to a space-charge-limited spatial region which is inside the Lamb-Dicke regime. 2 In this connection the question arises of how a cloud of ions behaves when cooled to low temperatures. Crystallization is expected when the Coulomb coupling constant T, which corresponds to the ratio of the Coulomb energy to the kinetic energy of a particle, is much bigger than l. 9 Experiments with laser-cooled Be + ions have shown that the observation of such a phase transition in a Penning trap should be possible. I0 The cooling of an ion crystal is predicted by theory to be as efficient as the cooling of independent particles. 11

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A single H g 'ion that is confined in an rf (Paul) trap can be laser cooled so that the amplitude of its motion is much less than a wavelength (the Dicke limit) for optical transitions.cooling to reach the zero point of motion. This realizes for the first time the fundamental limit of laser cooling for a bound atom and the ideal of an isolated atomic particle at rest in space to within the quantum mechanical limits imposed by the surrounding apparatus. all orders, the interrogation time is long and the fundamental shot noise detection limit of a single atom is readily attained.