Rectangular Cube Research Articles

Approximate and exact virial coefficients for hard rectangular parallelepipeds and cubes on a simple cubic lattice

The first five virial coefficients for lattice gases of hard rectangular parallelepipeds of ld2 sites (l=2–6, d=2) and of hard cubes of l3 sites (l=2–6) were calculated exactly and in the Percus–Yevick, hypernetted chain, and complete star approximations. The corresponding terms in the direct correlation function for hard cubes were calculated in the Percus–Yevick approximation and it was concluded that simple analytical solutions are improbable for this class of model. The relevance of these results to recent model making for defect complexes in UO2+x was indicated.

The effect of quasispherical solutes on the nematic to isotropic transition in liquid crystals

When nonmesomorphic solutes are added to nematic liquid crystals, the nematic–isotropic transition temperature is depressed and a two-phase region is observed. The slopes, βn and βi, of the nematic and isotropic phase boundary lines in the reduced nematic–isotropic transition temperature (T*) vs solute mole fraction (x) diagrams are reported for binary mixtures of quasispherical solutes with nematic liquid crystals. The systems studied are Me4Sn, Et4C, Et4Sn, Pr4Sn, Bu4Pb, and Bu4Sn (Me = methyl, Et = ethyl, Pr = propyl, and Bu = butyl), mixed with p-methoxybenzylidene-p′-n-butylaniline (MBBA), and CCl4 mixed with p-p′-di-n-hexyloxyazoxybenzene (DHAB). Also reported are infinite dilution solute activity coefficients for several mixtures involving Et4C and CCl4. The experimental results are compared with predicted values from statistical–mechanical theories based on (a) a lattice model of hard rectangular parallelepipeds and cubes, (b) a virial expansion treatment of hard spherocylinders and spheres, and (c) a proposed modification of a mean field treatment (Maier–Saupe type) of axially symmetric and spherical molecules. All three models correctly predict the observed trend of increasing βn and βi with increasing solute size and yield predicted slopes which are at least within a factor of 2 of the experimental slopes. The modified mean field approach is found to give the best agreement with the activity coefficient data. A simple experimental method is proposed for estimating the effect of potential impurities (x?0.01) on the nematic–isotropic transition, and possible directions for future research in this area are suggested.

Moments of Inertia of Solid Rectangular Parallelopipeds, Cubes, and Twin Cubes, and Two Other Regular Polyhedra

The derivation and verification of the values of the moments of inertia of simple solids such as rectangular blocks and cubes and combinations of two cubes and also of the regular polyhedra afford good exercise in theory and practice. The formulas lead readily from one to the next. The verification of the values obtained have been made with a torsion wire and frame plus a good stopwatch, and has afforded many hours of quiet pleasure.

A method for recording spatial vectorcardiograms

Three perpendicular bipolar axes which constitute a spatial reference system have been described. The electrodes comprising the reference system are remote or equidistant. The angle subtended to the force measured in each instance was as great as the body would accommodate and not violate the distance necessary for remoteness. This reference system was compared in one hundred instances with the rectangular cube method used by Grishman and others. The chief difference in the two reference systems was found in the measurement of the anterior-posterior electromotive force. In each instance, using our modification, the anterior-posterior component was greater. We believe this change is due to the difference in the angle subtended by electrodes comprising this bipolar lead in the two systems. It is recognized that the electrode placement of this system, as with other spatial reference systems, may not always be considered remote. However, this factor is not inherently greater in the suggested reference system than other currently employed methods. Therefore, we suggest that the method described here may more closely reflect the true potential manifested in the heart.