Sodium cobaltate (Na(x)CoO2) has emerged as a material of exceptional scientific interest due to the potential for thermoelectric applications, and because the strong interplay between the magnetic and superconducting properties has led to close comparisons with the physics of the superconducting copper oxides. The density x of the sodium in the intercalation layers can be altered electrochemically, directly changing the number of conduction electrons on the triangular Co layers. Recent electron diffraction measurements reveal a kaleidoscope of Na+ ion patterns as a function of concentration. Here we use single-crystal neutron diffraction supported by numerical simulations to determine the long-range three-dimensional superstructures of these ions. We show that the sodium ordering and its associated distortion field are governed by pure electrostatics, and that the organizational principle is the stabilization of charge droplets that order long range at some simple fractional fillings. Our results provide a good starting point to understand the electronic properties in terms of a Hubbard hamiltonian that takes into account the electrostatic potential from the Na superstructures. The resulting depth of potential wells in the Co layer is greater than the single-particle hopping kinetic energy and as a consequence, holes preferentially occupy the lowest potential regions. Thus we conclude that the Na+ ion patterning has a decisive role in the transport and magnetic properties.

The theory of magnetism in solid He is reviewed with emphasis on the multiple-spin-exchange theory of the magnetic interaction. Critical experiments are discussed briefly, and the most reliable present values for various quantitites are chosen. Various theories of He magnetism are considered, with the conclusion that multiple particle exchange is the most likely mechanism. The microscopic theory of exchange is presented mainly from the standpoint of tunneling ducts. The spin Hamiltonian which results from two-, three-, and four-particle ring exchange is derived and studied in the high-temperature limit, in mean-field approximation, and with spin waves. Two reasonable parameter sets are selected which fit the experimental data, with the conclusion that the exchanges are mainly threeand four-particle rings. A discussion is given of remaining problems together with some conjectures. CONTENTS D. Properties of the high-field naf phase 1. Free energy and magnetization as a function of the field H 2. Phase diagram a. Second-order transition line between the naf phase and the paramagnetic phase b. The transition "pseudoferromagnetic"~naf VIII. A Two-Parameter Model A. One-parameter models 1. Triple exchange 2. Folded four-spin exchange A+ 3. Planar four-spin exchange K& a. Low-field phase b. High-temperature series expansions B. Two-parameter model (J"lr I )