The heavy rare earth elements crystallize into hexagonally close packed (h.c.p.) structures and share a common outer electronic configuration, differing only in the number of 4f electrons they have. These chemically inert 4f electrons set up localized magnetic moments, which are coupled via an indirect exchange interaction involving the conduction electrons. This leads to the formation of a wide variety of magnetic structures, the periodicities of which are often incommensurate with the underlying crystal lattice. Such incommensurate ordering is associated with a 'webbed' topology of the momentum space surface separating the occupied and unoccupied electron states (the Fermi surface). The shape of this surface-and hence the magnetic structure-for the heavy rare earth elements is known to depend on the ratio of the interplanar spacing c and the interatomic, intraplanar spacing a of the h.c.p. lattice. A theoretical understanding of this problem is, however, far from complete. Here, using gadolinium as a prototype for all the heavy rare earth elements, we generate a unified magnetic phase diagram, which unequivocally links the magnetic structures of the heavy rare earths to their lattice parameters. In addition to verifying the importance of the c/a ratio, we find that the atomic unit cell volume plays a separate, distinct role in determining the magnetic properties: we show that the trend from ferromagnetism to incommensurate ordering as atomic number increases is connected to the concomitant decrease in unit cell volume. This volume decrease occurs because of the so-called lanthanide contraction, where the addition of electrons to the poorly shielding 4f orbitals leads to an increase in effective nuclear charge and, correspondingly, a decrease in ionic radii.

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We have used Monte Carlo simulations to observe the magnetic behaviour of Ising thin-films with cubic lattice structures as a function of temperature and thickness especially in the critical region. The fourth order Binder cumulant is used to extract critical temperatures, and an extension of finite size scaling theory for reduced geometry is derived to calculate the critical exponents. Magnetisation and magnetic susceptibility per spin in each layer are also investigated. In addition, mean-field calculations are also performed for comparison. We find that the magnetic behaviour changes from two dimensional to three dimensional character with increasing thickness of the film. The crossover of the critical temperature from a two dimensional to a bulk value is also observed with both the Monte Carlo simulations and the mean-field analysis. Nevertheless, the simulations have shown that the critical exponents only vary a little from their two dimensional values. In particular, the results for films with up to eight layers provide a strong indication of two dimensional universality.

We develop an exact gauge-invariant method for studying the two-dimensional ϮJ spin glass. It is applied to the case of an arbitrary concentration of (1Ϫ p) positive and p negative bonds and is thus a generalization of the more commonly studied pϭ50% model. The ground-state properties are examined and in particular it is shown that the spin correlation exponent remains constant over the range p c ϽpϽ50%. The value obtained is ϭ0.34Ϯ0.02. A wide range of values for is quoted in the literature. We indicate possible reasons for the discrepancies and indicate that there are potential advantages in doing calculations at concentrations markedly lower than 50%. ͓S1063-651X͑98͒07708-3͔

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