A general, feasible approach is presented for the evaluation of the statistical thermodynamics of interacting lattice gases. Exact solutions are obtained for lattice systems of infinite length and increasing finite width, using the matrix method which treats all densities on an equivalent basis. Through the application of symmetry reduction and the use of an electronic computer to perform logical as well as arithmetical operations, widths of up to 24 sites for two-dimensional lattices can be handled. For examples studied, rapid convergence is obtained away from transition regions and in the vicinity of phase transitions the behavior appears to be a sufficiently regular function of the width to allow meaningful extrapolation to systems of infinite width. Two problems of two-dimensional lattice gases are solved as illustrations of the technique: the square and triangular lattice gases with infinite repulsive interactions preventing the simultaneous occupancy of adjacent lattice sites (excluded-volume effect). Both systems exhibit phase transitions which are most likely second order at densities of 74.2% (square) and 83.7% (triangular) of the close-packed density. For both lattices the compressibility is infinite at the transition point, becoming infinite linearly with the logarithm of the width of the lattice for the square lattice and perhaps slightly more strongly for the triangular lattice.

A pseudo-lattice, or homogeneous Husimi tree, is simpler statistically than a true lattice in two or three dimensions, since the pseudo-lattice contains only low-order cycles. The prototype is the ``Bethe lattice,'' or Cayley tree, containing no cycles at all. Reported here are the results of studies of the lattice statistics of the Bethe lattice of coordination number three, using the language of the hard molecule lattice gas; the size of an adsorbed molecule prevents simultaneous occupancy of the same site or any of the three nearest-neighbor sites. Using methods related to those used in enumerating graphs, a recursion relation is obtained which must be satisfied by the grand partition function. It is shown that the resulting equation of state is not the quasi-chemical equation expected because of the absence of cycles. There is no phase transition of lower than third order and in all likelihood none at all. The quasi-chemical equation of state is obtained only if ``surface'' effects are eliminated; even then the solution is valid only for activities z < 4, at which point the ``interior'' system undergoes a second-order transition to an ordered state with a finite discontinuity in compressibility.

The underlying mechanisms of several rate processes in ice are examined through cross comparisons of the processes with each other and with experimental observations. The assumption that the migration of orientational defects (Bjerrum faults) is the common origin of dielectric and elastic relaxation leads to a predicted ratio of dielectric-to-elastic relaxation time of 32, in close agreement with experiment. The conclusion that a separate process is responsible for diffusion is based on a comparison of diffusion and dielectric relaxation data. The faster diffusive motion controls the thermal equilibration of the proton spins as well as the magnetic resonance linewidths; an interstitial migration appears to be the mostlikely diffusion mechanism.

urements 11 on uranium, L vacancies were produced by irradiating samples with x rays from molybdenum. This method of L vacancy production results in an appreciable proportion of Li vacancies (15-20%), so this result is not directly comparable with the present work.It may seem surprising that the L fluorescence yield measurements in this region do not appear to fall on a smoothly rising curve (see Fig. 5). It should be pointed out, however, that Coster-Kronig transitions of the Lu~>LniMy type are energetically possible at Z=90 and above, and those of the Lu -» LmMiv type at Z=94 and above. If the Lu and L m subshell yields are different, these Coster-Kronig transitions will introduce discontinuities in a plot of coj, versus Z.No theoretical computations of L fluorescence yields for these elements were found although a program was begun by Rose and Goertzel. 12 Theoretical computations of L fluorescence yields for elements with lower Z have been carried out by Pincherle. 13 ACKNOWLEDGMENTSThe authors wish to thank Paul Day for assistance in the use of the beta spectrometer and A. H. Jaffey for advice on the statistical treatment of the data. 11 The free longitudinal relaxation of the nuclear magnetization of systems of three identical spins J at the corners of an equilateral triangle is investigated using the semiclassical form of the density operator theory. The relaxation mechanism investigated is a fluctuating dipolar interaction between the spins. Assuming only that the symmetry of the thermal motion of the molecules preserves the complete equivalence of the three spins, the general relaxation equation is obtained before adopting models for this thermal motion. The effect of correlations between different pairwise interactions is studied; this effect can be more significant in the case of anisotropic reorientations possible in solids than for isotropic motion. For systems initially describable by a spin temperature, the effect of cross correlations always is to retard the relaxation. In general the relaxation is described by a sum of four exponentials, although three suffice for isotropic motion and two for the limit of long correlation times. An "effective relaxation time" is defined, the calculation of which is far simpler than that of the complete solution.

Thermodynamic properties have been obtained using the matrix method for a two-dimensional honeycomb-lattice gas of hard molecules which prevent simultaneous occupancy of nearest-neighbor sites. Exact results have been obtained for cylinders of infinite length and finite circumference from 6 to 18 sites. Uniform trends with increasing circumference lend confidence to extrapolations used to infer the behavior of systems of infinite width. We find a great similarity with the corresponding square-lattice gas and a second-order phase transition at activity z=7.92±0.08, reduced pressure p/kT=1.12±0.05, and relative density ρ/ρ0=0.845±0.02. The compressibility appears to be infinite at the transition.

We study a multicomponent version of the ``A−B'' model of Widom and Rowlinson, generalized in a symmetric way: There is an infinite repulsive interaction between any two unlike particles. We consider both lattice and continuum versions of the model and show that the ``demixing'' transition occurs for any finite number M of components, all having the same activity. No conclusion can be drawn about this transition in the limit M→∞. It is shown, however, that another transition, in which the density is greater on one of the sublattices, appears at a finite value of M which persists for all larger M at any fixed value of the activity. In the limit M→∞, z→0, M z=ζ, const, this system apparently becomes ``equivalent'' to a one-component system with activity ζ in which there is an exclusion for occupancy of nearest neighbor sites. The latter transition then becomes the ``hard square'' transition.

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