A unifying theoretical perspective has now been developed to span the range between the
weakly- and highly-magnetized regimes. This Monte-Carlo analysis (curve) is in agreement
with the experimental data (points) over 8 decades in the magnetization strength,
*R_c/b*, where *R_c* is the thermal cyclotron radius, and *b* is the thermal distance
of closest approach, as shown in Fig. 2. The analysis also gives an improved asymptotic expression
in the high magnetization limit.

We have also analyzed the recombination process that occurs when an ion is introduced into a
highly magnetized, cryogenic electron plasma. The recombination proceeds via a three-body
process in which the binding energy is carried off by a second electron. We find that the
recombination rate is reduced by an order of magnitude below the unmagnetized rate, and is
controlled by a kinetic bottleneck at a binding energy of a few *kT_e* below the ionization
energy. Within the energy range of the bottleneck, a bound electron-ion pair form a novel atom,
which we call a "guiding center atom." The calculated recombination rate is a design parameter
for proposed experiments to pass cold anti-protons through a cryogenic plasma of magnetically
confined positrons in order to produce anti-hydrogen.

Dr. Michael Glinsky received the 1993 Simon Ramo Award for his Ph.D. thesis analyzing these highly magnetized collisions and 3-body recombinations.

We have now extended this theory into the highly correlated regime; that is, we have calculated
the rate of equilibration between *Tperp* and *Tpar* for a strongly magnetized pure
electron crystal or pure ion crystal. The analysis is done in the "phonon approximation"
wherein dynamics of the crystallized plasma is described by weakly interacting phonons. There are
3 types of phonons: cyclotron, plasma and **E**x**B**. Since the frequency of the cyclotron
phonons is much larger than any plasma or **E**x**B** drift phonons, the total action in
the cyclotron phonons is an adiabatic invariant, preventing equipartition between perpendicular
and parallel motion. We find that the resonant coupling between a single cyclotron phonon and
many plasma phonons can break the invariant, leading to a rate which is exponentially small in
*1/=_c/_p*. This rate, which depends on collective "collisions," is considerably
larger than the previously calculated rate for an uncorrelated plasma, where only binary
collisions matter. This theory may have direct application to the cryogenic ion plasmas.

There are several additional features of this problem which make it rather
challenging theoretically:1) The estimated rate is found to be very small unless
is of order unity, where our theoretical analysis based on adiabatic invariance and *<<1* starts to break down.
2) The approximation that the ion dynamics are described by a linearsuperposition of phonons may
not be adequate, particularly if the system is in a liquid rather than a crystalline state, with
correlation parameter *1172*, where *e²/akT* and *a* is the
average interparticle spacing.

In order to attack these problems, we propose to construct a molecular dynamics computer
simulation of the ion crystal in order to measure the equilibration rate. Our group has expertise
in the design of codes in the strongly correlated regime. With such a code we can test the
approximations made in the theoretical analysis, and extend that analysis to the experimentally
interesting regime of *~1*, and *1*, where the equilibration rate is
largest but the analytic theory based on phonons fails.

In related work, we consider the equilibration of an anisotropic temperature distribution in a 1-dimensional chain of ions such as found in high energy storage rings or linear Paul traps. Such anisotropic distributions are common in these systems. In the limit of strong focusing, where motions transverse to the chain axis are of high frequency compared to the parallel motions, an adiabatic invariant strongly reduces the equilibration rate, just as in the case of strong magnetization.

However, the same theoretical issues exist for the 1-D chain as for the strongly magnetized
plasma: the validity of the harmonic phonon approximation must be tested, and the rate should be
extended into the experimentally relevant regimes where *~0(1)*. We plan to design a
computer simulation of the 1-D chain in order to test the predicted equilibration rate. This
simulation should be even easier to design than for the strongly magnetized plasma, since the
ion chain is one-dimensional and so only several dozen ions (certainly less than 100) should be
necessary in the simulation in order to approach the infinite chain limit. If necessary, we will
also employ periodic boundary conditions along the chain axis in order to approach this limit
more closely.

We have also studied the equipartition of various degrees of freedom in cryogenic non-neutral plasmas confined either by magnetic fields or by pondermotive forces. We have calculated the equipartition rate between the electron spin and kinetic motions, and find that the rate depends crucially on the homogeneity of the external magnetic field. This could have application to the production of polarized electron sources. Analysis of these equilibration processes formed the basis for the Ph.D. thesis of Dr. Shi-Jie Chen.