Coulomb Coupling Constant for Spatially Confined Single-Species Plasmas

where 2d is the average interparticle distance, T is the plasma temperature, and kB is the Boltzmann constant. This parameter has been employed even for the characterization of a nonneutral system consisting of a single species of particles. Such nonneutral plasmas are spatially localized by artificial electromagnetic forces that overcome internal Coulomb repulsion. In a Paul trap, for instance, a radiofrequency (rf) quadrupole field is utilized to confine a large number of particular ions. Charged-particle beams traveling in accelerators can also be regarded as a sort of nonneutral plasma focused by discrete electromagnetic elements; observing a beam from the rest frame, we actually see that it is almost equivalent to a single-species plasma in a Paul trap. The purpose of this note is to show that the conventional definition of the coupling constant is not relevant to these plasmas confined by time-dependent forces. As demonstrated below, the application of a non-static confinement field causes the plasma to ‘‘breathe’’, which can considerably reduce the magnitude of . In what follows, we consider, as an example, a single-species plasma column confined in a typical Paul trap. For the sake of simplicity, we only look at the transverse degrees of freedom, assuming that the plasma is longitudinally uniform. The transverse kinetic energy of a plasma is calculated from kBT 1⁄4 ðhpxi þ hpyiÞ=2m where ðpx; pyÞ are the horizontal and vertical kinetic momenta, m is the rest mass of the particle, and hAi stands for the average of the quantity ‘‘A’’ over the whole phase space. Since the plasma temperature is usually high according to this definition, the coupling constant is much less than unity, which means that the plasma is in a gaseous phase. In order to enhance the Coulomb coupling, we have to ‘‘cool’’ the plasma through dissipative interactions. It is said that is typically on the order of 1 to 10 in a liquid phase and, if it goes beyond 170, the plasma will be Coulomb crystallized. This argument relies on the plausible expectation that the average kinetic energy can be made arbitrarily low, at least, in theory. We, however, readily understand that the transverse momenta of off-axis particles never vanish as long as the external restoring force depends on time. Even after a perfect crystalline state is reached, kBT is always finite unless all particles are aligned exactly along the trap axis. The concept of temperature T must, therefore, be somewhat generalized in the present case so that the simple definition in eq. (1) remains appropriate. A possible way to avoid the ambiguity of T is the use of the emittance, i.e., the -space volume occupied by the plasma. This concept is quite popular in the particle accelerator community and has been used as a measure of beam quality. The root-mean-squared (rms) emittance projected on the horizontal phase space can be introduced as