On the internal pressure of strong electrolytes

It is shown that the difference in thermal properties of dilute solutions against those of the pure solvent is due to the superposition of two effects: 1. A physical effect of the ion on the solvent which is manifested as an internal pressure varying from point to point in the solution. This pressure distribution is calculated for aqueous solutions of monovalent salts by an approximation theory in which the effect of the so-called ion atmosphere is estimated by means of the Debye-Hückel potential. In this, as in ensuing calculations, the thermal properties of the solvent are assumed to be known. The effect of this internal pressure on the thermal coefficient of expansion and on the compressibility of dilute binary solutions is calculated and the results are found to be in satisfactory agreement with experimental values. 2. Definite evidence is obtained that the second effect determining the thermal properties of solutions is due to a chemical action of the ion on the solvent. This effect increases linearly with the molal concentration. As a result of the present paper and previous work, to which reference is given, we can say that the caloric as well as the thermal behaviour of dilute solutions is duly accounted for

On the Effect of a Secondary Structure upon the Interference of X-rays

Laue's dynamic theory of x-ray interference is shown to be applicable, with only a few minor changes, to crystals having a very general type of secondary structure. It is thus applied for the purpose of obtaining a quantitative estimate of the effect of such a structure upon the nature of the x-ray interference maxima. The estimate is relative insofar as it compares the intensities of respectively the "secondarily" and the "primarily" reflected interference beams and applies only in the region where the latter have been, or can be observed. In this region the "two-dimensional lattice" type of secondary structure is found to give rise to a fine structure which, with the present insufficient resolving power, would be manifested experimentally as a weak, diffuse background. The secondary structure of this type produces no broadening of the primary lines. The existence of this type of structure, therefore, is not inconsistent with the sharpness of the interference maxima obtained from such crystals as calcite, and a possible objection to the existence of the secondary structure in such crystals is removed. The extinction effect is briefly considered, but absorption is not taken into account, except with a few qualitative remarks

Energy changes by a variation from the crystallographic group

The purpose of the paper may be briefly stated as follows: To investigate the possibility of energy minima by a systematic departure from the perfectly symmetrical crystalline lattice. A geometrically simple variation is considered. The surface (100-plane) of a heteropolar crystal of the face-centered type is contracted uniformly in patches, and the interior planes are likewise contracted in patches by amounts decreasing linearly with the distance from the surface. In this way a "mosaic" structure of a type previously suggested in the literature1 is obtained. A numerical calculation of the change in potential energy is carried out for rock-salt with certain simplifying assumptions. The change in energy is obtained as a power series in terms of the maximum displacement of the ions. Due to the slow convergence of this alternating series an approximate remainder term is introduced to take care of powers higher than the third. The series contains two parameters, namely the linear extension of the patches and the depth of the cracks. The function, in general, is found to have two minima with respect to the displacement of the ions, one at zero displacement, and a second at a finite displacement. The second minimum, depending on the parameters, may be negative with respect to the former. The function also has minima with respect to the two parameters, thus proving the static stability of a mosaic structure of definite dimensions. These considerations with only minor changes, are also applicable to a mosaic structure in the interior of the crystal

On the Stability of Certain Heteropolar Crystals

On the basis of the conventional model of heteropolar crystals it is shown that an irregular crystalline behaviour is to be expected in the region of small values of the exponent, p, of the repulsive energy. In the course of this demonstration Madelung's method of obtaining the Coulomb potential for the crystals of the cubical system is justified, and a new, very simple method of calculating the Madelung constant is described. This method is applied to the three crystals of the cubical system and other applications are suggested. The irregular behavior of crystals in the region of small values of the repulsive exponent is shown to be manifested as an instability against various variations by which the geometry of the lattice is altered. In particular, the instability is demonstrated in the case of the "calcite family" of crystals which is evolved by a continuous process from a single parameter φ. The NaCl- and CsCl-types of crystals are members of this family, and a continuous mode of transition from one to the other is thus available. These two types of crystals, in general, are unstable against a variation in which the parameter φ is changed, (φ-variation), and for small values of the exponent, p, all members of the "calcite family" are shown to have a tendency to fall apart into one-dimensional crystals of the Madelung type. Another result of these considerations is that a skew structure, such as that of calcite may be accounted for on the basis of purely central forces. Finally the regions in which respectively one-dimensional and two-dimensional crystals, and the crystals of the cubical system are most stable are calculated. The bearing of these results on the theory of the secondary structure is briefly discussed

A New Analysis of Slater's Compressibility Data

The variation of interatomic forces with interatomic distances in crystals may be determined empirically from compressibility data. The experimentally possible range of variation in interatomic distance is, however, so small that the form of the force function valid for extrapolation to much greater distances cannot be adequately determined. From an analysis of Slater's compressibility data for the alkali halides it appears that for them the interionic forces of repulsion may be derived from a common force function. Because of the relatively large variation of lattice constant between the different alkali halides it is possible to determine a form for this common force function which is valid for relatively large variations in interatomic separation. The force function so found differs but little from the inverse power or from the exponential forms for small displacements from the equilibrium position. It drops off more rapidly, however, for increasing distances and increases more rapidly for decreasing distances. The solution given has the interesting property that of all possible force functions compatible with Slater's compressibility data it is the least favorable for the formation of a secondary structure

Cohesive properties of alkali halides

We calculate cohesive properties of LiF, NaF, KF, LiCl, NaCl, and KCl with ab-initio quantum chemical methods. The coupled-cluster approach is used to correct the Hartree-Fock crystal results for correlations and to systematically improve cohesive energies, lattice constants and bulk moduli. After inclusion of correlations, we recover 95-98 % of the total cohesive energies. The lattice constants deviate from experiment by at most 1.1 %, bulk moduli by at most 8 %. We also find good agreement for spectroscopic properties of the corresponding diatomic molecules.Comment: LaTeX, 10 pages, 1 figure, accepted by Phys. Rev.

Proposal of an extended t-J Hamiltonian for high-Tc cuprates from ab initio calculations on embedded clusters

A series of accurate ab initio calculations on Cu_pO-q finite clusters, properly embedded on the Madelung potential of the infinite lattice, have been performed in order to determine the local effective interactions in the CuO_2 planes of La_{2-x}Sr_xCuO_4 compounds. The values of the first-neighbor interactions, magnetic coupling (J_{NN}=125 meV) and hopping integral (t_{NN}=-555 meV), have been confirmed. Important additional effects are evidenced, concerning essentially the second-neighbor hopping integral t_{NNN}=+110meV, the displacement of a singlet toward an adjacent colinear hole, h_{SD}^{abc}=-80 meV, a non-negligible hole-hole repulsion V_{NN}-V_{NNN}=0.8 eV and a strong anisotropic effect of the presence of an adjacent hole on the values of the first-neighbor interactions. The dependence of J_{NN} and t_{NN} on the position of neighbor hole(s) has been rationalized from the two-band model and checked from a series of additional ab initio calculations. An extended t-J model Hamiltonian has been proposed on the basis of these results. It is argued that the here-proposed three-body effects may play a role in the charge/spin separation observed in these compounds, that is, in the formation and dynamic of stripes.Comment: 24 pages, 4 figures, submitted to Phys. Rev.